LMFDB - Embedded newform 75.9.f.d.43.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(7,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.5533957546$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 192 x^{9} + 27713 x^{8} - 24384 x^{7} + 18432 x^{6} - 2072064 x^{5} + 128589064 x^{4} + \cdots + 846810000 $ x^12 - 192x^9 + 27713x^8 - 24384x^7 + 18432x^6 - 2072064x^5 + 128589064x^4 - 223046400x^3 + 184320000x^2 + 558720000*x + 846810000
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{18}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.4
Root			$5.92676 - 5.92676i$ of defining polynomial
Character	$\chi$	$=$	75.43
Dual form			75.9.f.d.7.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.10647 - 3.10647i) q^{2} +(33.0681 + 33.0681i) q^{3} +236.700i q^{4} +205.450 q^{6} +(-974.502 + 974.502i) q^{7} +(1530.56 + 1530.56i) q^{8} +2187.00i q^{9} +O(q^{10})$	\(q+(3.10647 - 3.10647i) q^{2} +(33.0681 + 33.0681i) q^{3} +236.700i q^{4} +205.450 q^{6} +(-974.502 + 974.502i) q^{7} +(1530.56 + 1530.56i) q^{8} +2187.00i q^{9} -4996.30 q^{11} +(-7827.21 + 7827.21i) q^{12} +(14944.9 + 14944.9i) q^{13} +6054.51i q^{14} -51085.9 q^{16} +(-11795.2 + 11795.2i) q^{17} +(6793.84 + 6793.84i) q^{18} +52539.8i q^{19} -64449.9 q^{21} +(-15520.8 + 15520.8i) q^{22} +(-245594. - 245594. i) q^{23} +101225. i q^{24} +92851.4 q^{26} +(-72320.0 + 72320.0i) q^{27} +(-230664. - 230664. i) q^{28} -1.02151e6i q^{29} -165534. q^{31} +(-550519. + 550519. i) q^{32} +(-165218. - 165218. i) q^{33} +73282.9i q^{34} -517662. q^{36} +(-2.47866e6 + 2.47866e6i) q^{37} +(163213. + 163213. i) q^{38} +988397. i q^{39} +3.10533e6 q^{41} +(-200211. + 200211. i) q^{42} +(-1.63711e6 - 1.63711e6i) q^{43} -1.18262e6i q^{44} -1.52586e6 q^{46} +(-5.87036e6 + 5.87036e6i) q^{47} +(-1.68931e6 - 1.68931e6i) q^{48} +3.86549e6i q^{49} -780091. q^{51} +(-3.53744e6 + 3.53744e6i) q^{52} +(-8.41070e6 - 8.41070e6i) q^{53} +449319. i q^{54} -2.98306e6 q^{56} +(-1.73739e6 + 1.73739e6i) q^{57} +(-3.17328e6 - 3.17328e6i) q^{58} +9.42589e6i q^{59} +3.74107e6 q^{61} +(-514226. + 514226. i) q^{62} +(-2.13123e6 - 2.13123e6i) q^{63} -9.65765e6i q^{64} -1.02649e6 q^{66} +(-1.23693e7 + 1.23693e7i) q^{67} +(-2.79193e6 - 2.79193e6i) q^{68} -1.62427e7i q^{69} +1.38777e7 q^{71} +(-3.34732e6 + 3.34732e6i) q^{72} +(2.14552e7 + 2.14552e7i) q^{73} +1.53998e7i q^{74} -1.24361e7 q^{76} +(4.86890e6 - 4.86890e6i) q^{77} +(3.07042e6 + 3.07042e6i) q^{78} -4.04271e7i q^{79} -4.78297e6 q^{81} +(9.64661e6 - 9.64661e6i) q^{82} +(1.85669e7 + 1.85669e7i) q^{83} -1.52553e7i q^{84} -1.01712e7 q^{86} +(3.37794e7 - 3.37794e7i) q^{87} +(-7.64712e6 - 7.64712e6i) q^{88} +5.33621e7i q^{89} -2.91276e7 q^{91} +(5.81321e7 - 5.81321e7i) q^{92} +(-5.47390e6 - 5.47390e6i) q^{93} +3.64722e7i q^{94} -3.64092e7 q^{96} +(8.55969e7 - 8.55969e7i) q^{97} +(1.20080e7 + 1.20080e7i) q^{98} -1.09269e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2268 q^{6}+O(q^{10}) $ 12 * q + 2268 * q^6	$ 12 q + 2268 q^{6} - 88920 q^{11} - 485796 q^{16} + 79704 q^{21} - 4055976 q^{26} + 5658696 q^{31} + 5065092 q^{36} + 1798056 q^{41} - 10882464 q^{46} + 4329936 q^{51} + 7268040 q^{56} + 33649848 q^{61} - 141361848 q^{66} + 335506464 q^{71} - 395386536 q^{76} - 57395628 q^{81} + 489958560 q^{86} - 216875664 q^{91} - 710311356 q^{96}+O(q^{100}) $ 12 * q + 2268 * q^6 - 88920 * q^11 - 485796 * q^16 + 79704 * q^21 - 4055976 * q^26 + 5658696 * q^31 + 5065092 * q^36 + 1798056 * q^41 - 10882464 * q^46 + 4329936 * q^51 + 7268040 * q^56 + 33649848 * q^61 - 141361848 * q^66 + 335506464 * q^71 - 395386536 * q^76 - 57395628 * q^81 + 489958560 * q^86 - 216875664 * q^91 - 710311356 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	3.10647	−	3.10647i	0.194154	−	0.194154i	−0.603334	−	0.797488i	$-0.706162\pi$
$2$	3.10647	−	3.10647i	0.194154	−	0.194154i	0.797488	+	0.603334i	$0.206162\pi$
$3$	33.0681	+	33.0681i	0.408248	+	0.408248i
$3$	33.0681	+	33.0681i	0.408248	+	0.408248i
$4$			236.700i			0.924608i
$5$		0			0
$5$		0			0
$6$	205.450			0.158526
$7$	−974.502	+	974.502i	−0.405873	+	0.405873i	−0.880297	−	0.474424i	$-0.842657\pi$
$7$	−974.502	+	974.502i	−0.405873	+	0.405873i	0.474424	+	0.880297i	$0.342657\pi$
$8$	1530.56	+	1530.56i	0.373671	+	0.373671i
$9$			2187.00i			0.333333i
$10$		0			0
$11$	−4996.30			−0.341254			−0.170627	−	0.985336i	$-0.554579\pi$
$11$	−4996.30			−0.341254			−0.170627	+	0.985336i	$0.554579\pi$
$12$	−7827.21	+	7827.21i	−0.377470	+	0.377470i
$13$	14944.9	+	14944.9i	0.523261	+	0.523261i	0.918555	−	0.395294i	$-0.129357\pi$
$13$	14944.9	+	14944.9i	0.523261	+	0.523261i	−0.395294	+	0.918555i	$0.629357\pi$
$14$			6054.51i			0.157604i
$15$		0			0
$16$	−51085.9			−0.779509
$17$	−11795.2	+	11795.2i	−0.141225	+	0.141225i	−0.774185	−	0.632960i	$-0.781840\pi$
$17$	−11795.2	+	11795.2i	−0.141225	+	0.141225i	0.632960	+	0.774185i	$0.281840\pi$
$18$	6793.84	+	6793.84i	0.0647181	+	0.0647181i
$19$			52539.8i			0.403157i	0.979472	+	0.201578i	$0.0646070\pi$
$19$			52539.8i			0.403157i	−0.979472	+	0.201578i	$0.935393\pi$
$20$		0			0
$21$	−64449.9			−0.331394
$22$	−15520.8	+	15520.8i	−0.0662559	+	0.0662559i
$23$	−245594.	−	245594.i	−0.877620	−	0.877620i	0.115668	−	0.993288i	$-0.463099\pi$
$23$	−245594.	−	245594.i	−0.877620	−	0.877620i	−0.993288	+	0.115668i	$0.963099\pi$
$24$			101225.i			0.305101i
$25$		0			0
$26$	92851.4			0.203187
$27$	−72320.0	+	72320.0i	−0.136083	+	0.136083i
$28$	−230664.	−	230664.i	−0.375274	−	0.375274i
$29$		−	1.02151e6i		−	1.44428i	−0.691749	−	0.722138i	$-0.743159\pi$
$29$		−	1.02151e6i		−	1.44428i	0.691749	−	0.722138i	$-0.256841\pi$
$30$		0			0
$31$	−165534.			−0.179242			−0.0896211	−	0.995976i	$-0.528566\pi$
$31$	−165534.			−0.179242			−0.0896211	+	0.995976i	$0.528566\pi$
$32$	−550519.	+	550519.i	−0.525016	+	0.525016i
$33$	−165218.	−	165218.i	−0.139316	−	0.139316i
$34$			73282.9i			0.0548387i
$35$		0			0
$36$	−517662.			−0.308203
$37$	−2.47866e6	+	2.47866e6i	−1.32255	+	1.32255i	−0.410837	+	0.911709i	$0.634764\pi$
$37$	−2.47866e6	+	2.47866e6i	−1.32255	+	1.32255i	−0.911709	+	0.410837i	$0.865236\pi$
$38$	163213.	+	163213.i	0.0782745	+	0.0782745i
$39$			988397.i			0.427241i
$40$		0			0
$41$	3.10533e6			1.09894			0.549468	−	0.835515i	$-0.314830\pi$
$41$	3.10533e6			1.09894			0.549468	+	0.835515i	$0.314830\pi$
$42$	−200211.	+	200211.i	−0.0643415	+	0.0643415i
$43$	−1.63711e6	−	1.63711e6i	−0.478854	−	0.478854i	0.425911	−	0.904765i	$-0.359954\pi$
$43$	−1.63711e6	−	1.63711e6i	−0.478854	−	0.478854i	−0.904765	+	0.425911i	$0.859954\pi$
$44$		−	1.18262e6i		−	0.315526i
$45$		0			0
$46$	−1.52586e6			−0.340787
$47$	−5.87036e6	+	5.87036e6i	−1.20302	+	1.20302i	−0.229779	+	0.973243i	$0.573800\pi$
$47$	−5.87036e6	+	5.87036e6i	−1.20302	+	1.20302i	−0.973243	+	0.229779i	$0.926200\pi$
$48$	−1.68931e6	−	1.68931e6i	−0.318233	−	0.318233i
$49$			3.86549e6i			0.670534i
$50$		0			0
$51$	−780091.			−0.115309
$52$	−3.53744e6	+	3.53744e6i	−0.483812	+	0.483812i
$53$	−8.41070e6	−	8.41070e6i	−1.06593	−	1.06593i	−0.997667	−	0.0682627i	$-0.978254\pi$
$53$	−8.41070e6	−	8.41070e6i	−1.06593	−	1.06593i	−0.0682627	−	0.997667i	$-0.521746\pi$
$54$			449319.i			0.0528421i
$55$		0			0
$56$	−2.98306e6			−0.303326
$57$	−1.73739e6	+	1.73739e6i	−0.164588	+	0.164588i
$58$	−3.17328e6	−	3.17328e6i	−0.280412	−	0.280412i
$59$			9.42589e6i			0.777883i	0.921262	+	0.388942i	$0.127159\pi$
$59$			9.42589e6i			0.777883i	−0.921262	+	0.388942i	$0.872841\pi$
$60$		0			0
$61$	3.74107e6			0.270194			0.135097	−	0.990832i	$-0.456865\pi$
$61$	3.74107e6			0.270194			0.135097	+	0.990832i	$0.456865\pi$
$62$	−514226.	+	514226.i	−0.0348006	+	0.0348006i
$63$	−2.13123e6	−	2.13123e6i	−0.135291	−	0.135291i
$64$		−	9.65765e6i		−	0.575641i
$65$		0			0
$66$	−1.02649e6			−0.0540977
$67$	−1.23693e7	+	1.23693e7i	−0.613827	+	0.613827i	−0.943941	−	0.330114i	$-0.892913\pi$
$67$	−1.23693e7	+	1.23693e7i	−0.613827	+	0.613827i	0.330114	+	0.943941i	$0.392913\pi$
$68$	−2.79193e6	−	2.79193e6i	−0.130577	−	0.130577i
$69$		−	1.62427e7i		−	0.716574i
$70$		0			0
$71$	1.38777e7			0.546116			0.273058	−	0.961998i	$-0.411965\pi$
$71$	1.38777e7			0.546116			0.273058	+	0.961998i	$0.411965\pi$
$72$	−3.34732e6	+	3.34732e6i	−0.124557	+	0.124557i
$73$	2.14552e7	+	2.14552e7i	0.755511	+	0.755511i	0.975502	−	0.219991i	$-0.0706027\pi$
$73$	2.14552e7	+	2.14552e7i	0.755511	+	0.755511i	−0.219991	+	0.975502i	$0.570603\pi$
$74$			1.53998e7i			0.513556i
$75$		0			0
$76$	−1.24361e7			−0.372762
$77$	4.86890e6	−	4.86890e6i	0.138506	−	0.138506i
$78$	3.07042e6	+	3.07042e6i	0.0829506	+	0.0829506i
$79$		−	4.04271e7i		−	1.03792i	−0.854798	−	0.518961i	$-0.826319\pi$
$79$		−	4.04271e7i		−	1.03792i	0.854798	−	0.518961i	$-0.173681\pi$
$80$		0			0
$81$	−4.78297e6			−0.111111
$82$	9.64661e6	−	9.64661e6i	0.213363	−	0.213363i
$83$	1.85669e7	+	1.85669e7i	0.391225	+	0.391225i	0.875124	−	0.483899i	$-0.160780\pi$
$83$	1.85669e7	+	1.85669e7i	0.391225	+	0.391225i	−0.483899	+	0.875124i	$0.660780\pi$
$84$		−	1.52553e7i		−	0.306410i
$85$		0			0
$86$	−1.01712e7			−0.185943
$87$	3.37794e7	−	3.37794e7i	0.589623	−	0.589623i
$88$	−7.64712e6	−	7.64712e6i	−0.127517	−	0.127517i
$89$			5.33621e7i			0.850497i	0.905077	+	0.425248i	$0.139813\pi$
$89$			5.33621e7i			0.850497i	−0.905077	+	0.425248i	$0.860187\pi$
$90$		0			0
$91$	−2.91276e7			−0.424755
$92$	5.81321e7	−	5.81321e7i	0.811455	−	0.811455i
$93$	−5.47390e6	−	5.47390e6i	−0.0731753	−	0.0731753i
$94$			3.64722e7i			0.467143i
$95$		0			0
$96$	−3.64092e7			−0.428673
$97$	8.55969e7	−	8.55969e7i	0.966877	−	0.966877i	−0.0325916	−	0.999469i	$-0.510376\pi$
$97$	8.55969e7	−	8.55969e7i	0.966877	−	0.966877i	0.999469	+	0.0325916i	$0.0103761\pi$
$98$	1.20080e7	+	1.20080e7i	0.130187	+	0.130187i
$99$		−	1.09269e7i		−	0.113751i
$100$		0			0
$101$	1.76856e8			1.69955			0.849774	−	0.527147i	$-0.176738\pi$
$101$	1.76856e8			1.69955			0.849774	+	0.527147i	$0.176738\pi$
$102$	−2.42333e6	+	2.42333e6i	−0.0223878	+	0.0223878i
$103$	7.41794e7	+	7.41794e7i	0.659074	+	0.659074i	0.955161	−	0.296087i	$-0.0956818\pi$
$103$	7.41794e7	+	7.41794e7i	0.659074	+	0.659074i	−0.296087	+	0.955161i	$0.595682\pi$
$104$			4.57479e7i			0.391055i
$105$		0			0
$106$	−5.22551e7			−0.413910
$107$	4.71593e7	−	4.71593e7i	0.359776	−	0.359776i	−0.503955	−	0.863730i	$-0.668122\pi$
$107$	4.71593e7	−	4.71593e7i	0.359776	−	0.359776i	0.863730	+	0.503955i	$0.168122\pi$
$108$	−1.71181e7	−	1.71181e7i	−0.125823	−	0.125823i
$109$		−	1.75445e7i		−	0.124290i	−0.998067	−	0.0621449i	$-0.980206\pi$
$109$		−	1.75445e7i		−	0.124290i	0.998067	−	0.0621449i	$-0.0197941\pi$
$110$		0			0
$111$	−1.63929e8			−1.07985
$112$	4.97833e7	−	4.97833e7i	0.316382	−	0.316382i
$113$	1.15312e8	+	1.15312e8i	0.707229	+	0.707229i	0.965952	−	0.258722i	$-0.0833015\pi$
$113$	1.15312e8	+	1.15312e8i	0.707229	+	0.707229i	−0.258722	+	0.965952i	$0.583301\pi$
$114$			1.07943e7i			0.0639109i
$115$		0			0
$116$	2.41791e8			1.33539
$117$	−3.26844e7	+	3.26844e7i	−0.174420	+	0.174420i
$118$	2.92812e7	+	2.92812e7i	0.151029	+	0.151029i
$119$		−	2.29889e7i		−	0.114639i
$120$		0			0
$121$	−1.89396e8			−0.883546
$122$	1.16215e7	−	1.16215e7i	0.0524594	−	0.0524594i
$123$	1.02687e8	+	1.02687e8i	0.448639	+	0.448639i
$124$		−	3.91819e7i		−	0.165729i
$125$		0			0
$126$	−1.32412e7			−0.0525347
$127$	−1.05771e8	+	1.05771e8i	−0.406585	+	0.406585i	−0.880546	−	0.473961i	$-0.842824\pi$
$127$	−1.05771e8	+	1.05771e8i	−0.406585	+	0.406585i	0.473961	+	0.880546i	$0.342824\pi$
$128$	−1.70934e8	−	1.70934e8i	−0.636779	−	0.636779i
$129$		−	1.08272e8i		−	0.390983i
$130$		0			0
$131$	4.98885e8			1.69401			0.847004	−	0.531587i	$-0.178404\pi$
$131$	4.98885e8			1.69401			0.847004	+	0.531587i	$0.178404\pi$
$132$	3.91071e7	−	3.91071e7i	0.128813	−	0.128813i
$133$	−5.12001e7	−	5.12001e7i	−0.163630	−	0.163630i
$134$			7.68496e7i			0.238354i
$135$		0			0
$136$	−3.61065e7			−0.105543
$137$	−1.96007e8	+	1.96007e8i	−0.556403	+	0.556403i	−0.928281	−	0.371879i	$-0.878714\pi$
$137$	−1.96007e8	+	1.96007e8i	−0.556403	+	0.556403i	0.371879	+	0.928281i	$0.378714\pi$
$138$	−5.04573e7	−	5.04573e7i	−0.139126	−	0.139126i
$139$		−	3.18232e8i		−	0.852480i	−0.904610	−	0.426240i	$-0.859838\pi$
$139$		−	3.18232e8i		−	0.852480i	0.904610	−	0.426240i	$-0.140162\pi$
$140$		0			0
$141$	−3.88244e8			−0.982263
$142$	4.31107e7	−	4.31107e7i	0.106031	−	0.106031i
$143$	−7.46690e7	−	7.46690e7i	−0.178565	−	0.178565i
$144$		−	1.11725e8i		−	0.259836i
$145$		0			0
$146$	1.33300e8			0.293371
$147$	−1.27825e8	+	1.27825e8i	−0.273744	+	0.273744i
$148$	−5.86699e8	−	5.86699e8i	−1.22284	−	1.22284i
$149$			7.86775e8i			1.59627i	0.602481	+	0.798134i	$0.294179\pi$
$149$			7.86775e8i			1.59627i	−0.602481	+	0.798134i	$0.705821\pi$
$150$		0			0
$151$	−7.06754e8			−1.35944			−0.679721	−	0.733471i	$-0.737899\pi$
$151$	−7.06754e8			−1.35944			−0.679721	+	0.733471i	$0.737899\pi$
$152$	−8.04150e7	+	8.04150e7i	−0.150648	+	0.150648i
$153$	−2.57961e7	−	2.57961e7i	−0.0470749	−	0.0470749i
$154$		−	3.02502e7i		−	0.0537830i
$155$		0			0
$156$	−2.33953e8			−0.395031
$157$	−7.32328e8	+	7.32328e8i	−1.20533	+	1.20533i	−0.232811	+	0.972522i	$0.574792\pi$
$157$	−7.32328e8	+	7.32328e8i	−1.20533	+	1.20533i	−0.972522	+	0.232811i	$0.925208\pi$
$158$	−1.25586e8	−	1.25586e8i	−0.201517	−	0.201517i
$159$		−	5.56252e8i		−	0.870328i
$160$		0			0
$161$	4.78664e8			0.712405
$162$	−1.48581e7	+	1.48581e7i	−0.0215727	+	0.0215727i
$163$	−3.22135e7	−	3.22135e7i	−0.0456339	−	0.0456339i	0.683922	−	0.729555i	$-0.260273\pi$
$163$	−3.22135e7	−	3.22135e7i	−0.0456339	−	0.0456339i	−0.729555	+	0.683922i	$0.760273\pi$
$164$			7.35031e8i			1.01609i
$165$		0			0
$166$	1.15355e8			0.151916
$167$	−7.92458e8	+	7.92458e8i	−1.01885	+	1.01885i	−0.0190311	+	0.999819i	$0.506058\pi$
$167$	−7.92458e8	+	7.92458e8i	−1.01885	+	1.01885i	−0.999819	+	0.0190311i	$0.993942\pi$
$168$	−9.86441e7	−	9.86441e7i	−0.123832	−	0.123832i
$169$		−	3.69033e8i		−	0.452395i
$170$		0			0
$171$	−1.14904e8			−0.134386
$172$	3.87503e8	−	3.87503e8i	0.442752	−	0.442752i
$173$	6.76294e8	+	6.76294e8i	0.755008	+	0.755008i	0.975409	−	0.220402i	$-0.0707368\pi$
$173$	6.76294e8	+	6.76294e8i	0.755008	+	0.755008i	−0.220402	+	0.975409i	$0.570737\pi$
$174$		−	2.09869e8i		−	0.228956i
$175$		0			0
$176$	2.55241e8			0.266011
$177$	−3.11697e8	+	3.11697e8i	−0.317570	+	0.317570i
$178$	1.65767e8	+	1.65767e8i	0.165127	+	0.165127i
$179$			8.49985e8i			0.827940i	0.910290	+	0.413970i	$0.135858\pi$
$179$			8.49985e8i			0.827940i	−0.910290	+	0.413970i	$0.864142\pi$
$180$		0			0
$181$	1.37723e8			0.128319			0.0641596	−	0.997940i	$-0.479563\pi$
$181$	1.37723e8			0.128319			0.0641596	+	0.997940i	$0.479563\pi$
$182$	−9.04839e7	+	9.04839e7i	−0.0824680	+	0.0824680i
$183$	1.23710e8	+	1.23710e8i	0.110306	+	0.110306i
$184$		−	7.51791e8i		−	0.655882i
$185$		0			0
$186$	−3.40090e7			−0.0284146
$187$	5.89325e7	−	5.89325e7i	0.0481935	−	0.0481935i
$188$	−1.38951e9	−	1.38951e9i	−1.11232	−	1.11232i
$189$		−	1.40952e8i		−	0.110465i
$190$		0			0
$191$	1.83102e9			1.37581			0.687905	−	0.725800i	$-0.258530\pi$
$191$	1.83102e9			1.37581			0.687905	+	0.725800i	$0.258530\pi$
$192$	3.19360e8	−	3.19360e8i	0.235004	−	0.235004i
$193$	4.47959e8	+	4.47959e8i	0.322856	+	0.322856i	0.849862	−	0.527006i	$-0.176685\pi$
$193$	4.47959e8	+	4.47959e8i	0.322856	+	0.322856i	−0.527006	+	0.849862i	$0.676685\pi$
$194$		−	5.31808e8i		−	0.375446i
$195$		0			0
$196$	−9.14961e8			−0.619981
$197$	1.01717e9	−	1.01717e9i	0.675351	−	0.675351i	−0.283594	−	0.958945i	$-0.591527\pi$
$197$	1.01717e9	−	1.01717e9i	0.675351	−	0.675351i	0.958945	+	0.283594i	$0.0915267\pi$
$198$	−3.39441e7	−	3.39441e7i	−0.0220853	−	0.0220853i
$199$			4.81408e8i			0.306973i	0.988151	+	0.153487i	$0.0490502\pi$
$199$			4.81408e8i			0.306973i	−0.988151	+	0.153487i	$0.950950\pi$
$200$		0			0
$201$	−8.18059e8			−0.501187
$202$	5.49396e8	−	5.49396e8i	0.329974	−	0.329974i
$203$	9.95463e8	+	9.95463e8i	0.586193	+	0.586193i
$204$		−	1.84647e8i		−	0.106616i
$205$		0			0
$206$	4.60871e8			0.255924
$207$	5.37114e8	−	5.37114e8i	0.292540	−	0.292540i
$208$	−7.63472e8	−	7.63472e8i	−0.407887	−	0.407887i
$209$		−	2.62505e8i		−	0.137579i
$210$		0			0
$211$	5.57618e7			0.0281324			0.0140662	−	0.999901i	$-0.495522\pi$
$211$	5.57618e7			0.0281324			0.0140662	+	0.999901i	$0.495522\pi$
$212$	1.99081e9	−	1.99081e9i	0.985568	−	0.985568i
$213$	4.58910e8	+	4.58910e8i	0.222951	+	0.222951i
$214$		−	2.92997e8i		−	0.139704i
$215$		0			0
$216$	−2.21379e8			−0.101700
$217$	1.61313e8	−	1.61313e8i	0.0727496	−	0.0727496i
$218$	−5.45015e7	−	5.45015e7i	−0.0241314	−	0.0241314i
$219$			1.41897e9i			0.616872i
$220$		0			0
$221$	−3.52556e8			−0.147795
$222$	−5.09241e8	+	5.09241e8i	−0.209658	+	0.209658i
$223$	3.04995e9	+	3.04995e9i	1.23331	+	1.23331i	0.962684	+	0.270629i	$0.0872316\pi$
$223$	3.04995e9	+	3.04995e9i	1.23331	+	1.23331i	0.270629	+	0.962684i	$0.412768\pi$
$224$		−	1.07296e9i		−	0.426180i
$225$		0			0
$226$	7.16425e8			0.274623
$227$	6.19777e8	−	6.19777e8i	0.233417	−	0.233417i	−0.580701	−	0.814117i	$-0.697221\pi$
$227$	6.19777e8	−	6.19777e8i	0.233417	−	0.233417i	0.814117	+	0.580701i	$0.197221\pi$
$228$	−4.11240e8	−	4.11240e8i	−0.152179	−	0.152179i
$229$			3.01662e9i			1.09693i	0.836173	+	0.548465i	$0.184788\pi$
$229$			3.01662e9i			1.09693i	−0.836173	+	0.548465i	$0.815212\pi$
$230$		0			0
$231$	3.22011e8			0.113090
$232$	1.56348e9	−	1.56348e9i	0.539684	−	0.539684i
$233$	2.53460e9	+	2.53460e9i	0.859974	+	0.859974i	0.991335	−	0.131361i	$-0.0419346\pi$
$233$	2.53460e9	+	2.53460e9i	0.859974	+	0.859974i	−0.131361	+	0.991335i	$0.541935\pi$
$234$			2.03066e8i			0.0677289i
$235$		0			0
$236$	−2.23111e9			−0.719238
$237$	1.33685e9	−	1.33685e9i	0.423730	−	0.423730i
$238$	−7.14143e7	−	7.14143e7i	−0.0222576	−	0.0222576i
$239$		−	2.93731e9i		−	0.900240i	−0.892968	−	0.450120i	$-0.851381\pi$
$239$		−	2.93731e9i		−	0.900240i	0.892968	−	0.450120i	$-0.148619\pi$
$240$		0			0
$241$	−2.48664e9			−0.737132			−0.368566	−	0.929602i	$-0.620151\pi$
$241$	−2.48664e9			−0.737132			−0.368566	+	0.929602i	$0.620151\pi$
$242$	−5.88352e8	+	5.88352e8i	−0.171544	+	0.171544i
$243$	−1.58164e8	−	1.58164e8i	−0.0453609	−	0.0453609i
$244$			8.85510e8i			0.249824i
$245$		0			0
$246$	6.37990e8			0.174210
$247$	−7.85200e8	+	7.85200e8i	−0.210956	+	0.210956i
$248$	−2.53359e8	−	2.53359e8i	−0.0669776	−	0.0669776i
$249$			1.22794e9i			0.319434i
$250$		0			0
$251$	−2.67826e9			−0.674772			−0.337386	−	0.941366i	$-0.609543\pi$
$251$	−2.67826e9			−0.674772			−0.337386	+	0.941366i	$0.609543\pi$
$252$	5.04463e8	−	5.04463e8i	0.125091	−	0.125091i
$253$	1.22706e9	+	1.22706e9i	0.299492	+	0.299492i
$254$			6.57147e8i			0.157880i
$255$		0			0
$256$	1.41036e9			0.328374
$257$	−1.40231e9	+	1.40231e9i	−0.321449	+	0.321449i	−0.849323	−	0.527874i	$-0.822989\pi$
$257$	−1.40231e9	+	1.40231e9i	−0.321449	+	0.321449i	0.527874	+	0.849323i	$0.322989\pi$
$258$	−3.36344e8	−	3.36344e8i	−0.0759109	−	0.0759109i
$259$		−	4.83092e9i		−	1.07357i
$260$		0			0
$261$	2.23404e9			0.481426
$262$	1.54977e9	−	1.54977e9i	0.328899	−	0.328899i
$263$	−3.19839e9	−	3.19839e9i	−0.668510	−	0.668510i	0.288861	−	0.957371i	$-0.406723\pi$
$263$	−3.19839e9	−	3.19839e9i	−0.668510	−	0.668510i	−0.957371	+	0.288861i	$0.906723\pi$
$264$		−	5.05751e8i		−	0.104117i
$265$		0			0
$266$	−3.18103e8			−0.0635391
$267$	−1.76458e9	+	1.76458e9i	−0.347214	+	0.347214i
$268$	−2.92781e9	−	2.92781e9i	−0.567549	−	0.567549i
$269$		−	2.77450e9i		−	0.529877i	−0.964265	−	0.264939i	$-0.914648\pi$
$269$		−	2.77450e9i		−	0.529877i	0.964265	−	0.264939i	$-0.0853517\pi$
$270$		0			0
$271$	9.95720e9			1.84612			0.923060	−	0.384656i	$-0.125680\pi$
$271$	9.95720e9			1.84612			0.923060	+	0.384656i	$0.125680\pi$
$272$	6.02569e8	−	6.02569e8i	0.110086	−	0.110086i
$273$	−9.63194e8	−	9.63194e8i	−0.173406	−	0.173406i
$274$			1.21778e9i			0.216056i
$275$		0			0
$276$	3.84464e9			0.662550
$277$	−2.87012e9	+	2.87012e9i	−0.487508	+	0.487508i	−0.907519	−	0.420011i	$-0.862026\pi$
$277$	−2.87012e9	+	2.87012e9i	−0.487508	+	0.487508i	0.420011	+	0.907519i	$0.362026\pi$
$278$	−9.88577e8	−	9.88577e8i	−0.165513	−	0.165513i
$279$		−	3.62023e8i		−	0.0597474i
$280$		0			0
$281$	−1.02942e10			−1.65108			−0.825540	−	0.564344i	$-0.809129\pi$
$281$	−1.02942e10			−1.65108			−0.825540	+	0.564344i	$0.809129\pi$
$282$	−1.20607e9	+	1.20607e9i	−0.190711	+	0.190711i
$283$	−3.13540e9	−	3.13540e9i	−0.488818	−	0.488818i	0.419115	−	0.907933i	$-0.362340\pi$
$283$	−3.13540e9	−	3.13540e9i	−0.488818	−	0.488818i	−0.907933	+	0.419115i	$0.862340\pi$
$284$			3.28486e9i			0.504944i
$285$		0			0
$286$	−4.63914e8			−0.0693383
$287$	−3.02615e9	+	3.02615e9i	−0.446029	+	0.446029i
$288$	−1.20398e9	−	1.20398e9i	−0.175005	−	0.175005i
$289$			6.69750e9i			0.960111i
$290$		0			0
$291$	5.66106e9			0.789452
$292$	−5.07844e9	+	5.07844e9i	−0.698552	+	0.698552i
$293$	−2.31877e9	−	2.31877e9i	−0.314620	−	0.314620i	0.532076	−	0.846697i	$-0.321412\pi$
$293$	−2.31877e9	−	2.31877e9i	−0.314620	−	0.314620i	−0.846697	+	0.532076i	$0.821412\pi$
$294$			7.94166e8i			0.106297i
$295$		0			0
$296$	−7.58747e9			−0.988393
$297$	3.61332e8	−	3.61332e8i	0.0464388	−	0.0464388i
$298$	2.44409e9	+	2.44409e9i	0.309922	+	0.309922i
$299$		−	7.34074e9i		−	0.918449i
$300$		0			0
$301$	3.19073e9			0.388708
$302$	−2.19551e9	+	2.19551e9i	−0.263941	+	0.263941i
$303$	5.84828e9	+	5.84828e9i	0.693838	+	0.693838i
$304$		−	2.68404e9i		−	0.314264i
$305$		0			0
$306$	−1.60270e8			−0.0182796
$307$	−7.65906e9	+	7.65906e9i	−0.862228	+	0.862228i	−0.991597	−	0.129368i	$-0.958705\pi$
$307$	−7.65906e9	+	7.65906e9i	−0.862228	+	0.862228i	0.129368	+	0.991597i	$0.458705\pi$
$308$	1.15247e9	+	1.15247e9i	0.128064	+	0.128064i
$309$			4.90594e9i			0.538132i
$310$		0			0
$311$	−9.30214e9			−0.994355			−0.497178	−	0.867649i	$-0.665630\pi$
$311$	−9.30214e9			−0.994355			−0.497178	+	0.867649i	$0.665630\pi$
$312$	−1.51280e9	+	1.51280e9i	−0.159647	+	0.159647i
$313$	−6.96927e9	−	6.96927e9i	−0.726122	−	0.726122i	0.243723	−	0.969845i	$-0.421631\pi$
$313$	−6.96927e9	−	6.96927e9i	−0.726122	−	0.726122i	−0.969845	+	0.243723i	$0.921631\pi$
$314$			4.54990e9i			0.468041i
$315$		0			0
$316$	9.56909e9			0.959671
$317$	−5.65467e9	+	5.65467e9i	−0.559977	+	0.559977i	−0.929301	−	0.369324i	$-0.879589\pi$
$317$	−5.65467e9	+	5.65467e9i	−0.559977	+	0.559977i	0.369324	+	0.929301i	$0.379589\pi$
$318$	−1.72798e9	−	1.72798e9i	−0.168978	−	0.168978i
$319$			5.10377e9i			0.492865i
$320$		0			0
$321$	3.11894e9			0.293756
$322$	1.48695e9	−	1.48695e9i	0.138316	−	0.138316i
$323$	−6.19718e8	−	6.19718e8i	−0.0569356	−	0.0569356i
$324$		−	1.13213e9i		−	0.102734i
$325$		0			0
$326$	−2.00140e8			−0.0177200
$327$	5.80164e8	−	5.80164e8i	0.0507411	−	0.0507411i
$328$	4.75288e9	+	4.75288e9i	0.410640	+	0.410640i
$329$		−	1.14414e10i		−	0.976549i
$330$		0			0
$331$	−1.43122e10			−1.19232			−0.596160	−	0.802865i	$-0.703308\pi$
$331$	−1.43122e10			−1.19232			−0.596160	+	0.802865i	$0.703308\pi$
$332$	−4.39477e9	+	4.39477e9i	−0.361730	+	0.361730i
$333$	−5.42084e9	−	5.42084e9i	−0.440849	−	0.440849i
$334$			4.92349e9i			0.395628i
$335$		0			0
$336$	3.29248e9			0.258325
$337$	1.43343e10	−	1.43343e10i	1.11136	−	1.11136i	0.118399	−	0.992966i	$-0.462224\pi$
$337$	1.43343e10	−	1.43343e10i	1.11136	−	1.11136i	0.992966	−	0.118399i	$-0.0377761\pi$
$338$	−1.14639e9	−	1.14639e9i	−0.0878345	−	0.0878345i
$339$			7.62629e9i			0.577450i
$340$		0			0
$341$	8.27058e8			0.0611672
$342$	−3.56947e8	+	3.56947e8i	−0.0260915	+	0.0260915i
$343$	−9.38474e9	−	9.38474e9i	−0.678025	−	0.678025i
$344$		−	5.01137e9i		−	0.357868i
$345$		0			0
$346$	4.20177e9			0.293176
$347$	1.41952e10	−	1.41952e10i	0.979092	−	0.979092i	−0.0206935	−	0.999786i	$-0.506587\pi$
$347$	1.41952e10	−	1.41952e10i	0.979092	−	0.979092i	0.999786	+	0.0206935i	$0.00658742\pi$
$348$	7.99557e9	+	7.99557e9i	0.545171	+	0.545171i
$349$		−	1.42593e9i		−	0.0961159i	−0.998845	−	0.0480579i	$-0.984697\pi$
$349$		−	1.42593e9i		−	0.0961159i	0.998845	−	0.0480579i	$-0.0153032\pi$
$350$		0			0
$351$	−2.16162e9			−0.142414
$352$	2.75056e9	−	2.75056e9i	0.179164	−	0.179164i
$353$	6.11946e9	+	6.11946e9i	0.394107	+	0.394107i	0.876149	−	0.482041i	$-0.160104\pi$
$353$	6.11946e9	+	6.11946e9i	0.394107	+	0.394107i	−0.482041	+	0.876149i	$0.660104\pi$
$354$			1.93655e9i			0.123315i
$355$		0			0
$356$	−1.26308e10			−0.786376
$357$	7.60200e8	−	7.60200e8i	0.0468010	−	0.0468010i
$358$	2.64045e9	+	2.64045e9i	0.160748	+	0.160748i
$359$		−	2.72763e10i		−	1.64213i	−0.570833	−	0.821066i	$-0.693380\pi$
$359$		−	2.72763e10i		−	1.64213i	0.570833	−	0.821066i	$-0.306620\pi$
$360$		0			0
$361$	1.42231e10			0.837465
$362$	4.27831e8	−	4.27831e8i	0.0249137	−	0.0249137i
$363$	−6.26296e9	−	6.26296e9i	−0.360706	−	0.360706i
$364$		−	6.89449e9i		−	0.392732i
$365$		0			0
$366$	7.68603e8			0.0428329
$367$	4.35880e9	−	4.35880e9i	0.240271	−	0.240271i	−0.576691	−	0.816962i	$-0.695656\pi$
$367$	4.35880e9	−	4.35880e9i	0.240271	−	0.240271i	0.816962	+	0.576691i	$0.195656\pi$
$368$	1.25464e10	+	1.25464e10i	0.684113	+	0.684113i
$369$			6.79136e9i			0.366312i
$370$		0			0
$371$	1.63925e10			0.865265
$372$	1.29567e9	−	1.29567e9i	0.0676585	−	0.0676585i
$373$	8.61815e9	+	8.61815e9i	0.445224	+	0.445224i	0.893763	−	0.448539i	$-0.148055\pi$
$373$	8.61815e9	+	8.61815e9i	0.445224	+	0.445224i	−0.448539	+	0.893763i	$0.648055\pi$
$374$		−	3.66144e8i		−	0.0187139i
$375$		0			0
$376$	−1.79698e10			−0.899068
$377$	1.52663e10	−	1.52663e10i	0.755734	−	0.755734i
$378$	−4.37862e8	−	4.37862e8i	−0.0214472	−	0.0214472i
$379$			1.62972e10i			0.789873i	0.918708	+	0.394936i	$0.129233\pi$
$379$			1.62972e10i			0.789873i	−0.918708	+	0.394936i	$0.870767\pi$
$380$		0			0
$381$	−6.99528e9			−0.331975
$382$	5.68799e9	−	5.68799e9i	0.267119	−	0.267119i
$383$	−9.04711e8	−	9.04711e8i	−0.0420450	−	0.0420450i	0.685772	−	0.727817i	$-0.259465\pi$
$383$	−9.04711e8	−	9.04711e8i	−0.0420450	−	0.0420450i	−0.727817	+	0.685772i	$0.759465\pi$
$384$		−	1.13049e10i		−	0.519928i
$385$		0			0
$386$	2.78314e9			0.125368
$387$	3.58035e9	−	3.58035e9i	0.159618	−	0.159618i
$388$	2.02608e10	+	2.02608e10i	0.893983	+	0.893983i
$389$			1.38210e10i			0.603591i	0.953373	+	0.301795i	$0.0975860\pi$
$389$			1.38210e10i			0.603591i	−0.953373	+	0.301795i	$0.902414\pi$
$390$		0			0
$391$	5.79367e9			0.247883
$392$	−5.91635e9	+	5.91635e9i	−0.250559	+	0.250559i
$393$	1.64972e10	+	1.64972e10i	0.691576	+	0.691576i
$394$		−	6.31962e9i		−	0.262244i
$395$		0			0
$396$	2.58640e9			0.105175
$397$	1.18087e10	−	1.18087e10i	0.475378	−	0.475378i	−0.428272	−	0.903650i	$-0.640878\pi$
$397$	1.18087e10	−	1.18087e10i	0.475378	−	0.475378i	0.903650	+	0.428272i	$0.140878\pi$
$398$	1.49548e9	+	1.49548e9i	0.0596002	+	0.0596002i
$399$		−	3.38618e9i		−	0.133604i
$400$		0			0
$401$	2.44172e10			0.944319			0.472160	−	0.881513i	$-0.343475\pi$
$401$	2.44172e10			0.944319			0.472160	+	0.881513i	$0.343475\pi$
$402$	−2.54127e9	+	2.54127e9i	−0.0973076	+	0.0973076i
$403$	−2.47388e9	−	2.47388e9i	−0.0937905	−	0.0937905i
$404$			4.18617e10i			1.57142i
$405$		0			0
$406$	6.18474e9			0.227624
$407$	1.23842e10	−	1.23842e10i	0.451324	−	0.451324i
$408$	−1.19397e9	−	1.19397e9i	−0.0430877	−	0.0430877i
$409$			1.20856e9i			0.0431894i	0.999767	+	0.0215947i	$0.00687433\pi$
$409$			1.20856e9i			0.0431894i	−0.999767	+	0.0215947i	$0.993126\pi$
$410$		0			0
$411$	−1.29632e10			−0.454301
$412$	−1.75582e10	+	1.75582e10i	−0.609385	+	0.609385i
$413$	−9.18555e9	−	9.18555e9i	−0.315722	−	0.315722i
$414$		−	3.33706e9i		−	0.113596i
$415$		0			0
$416$	−1.64549e10			−0.549441
$417$	1.05233e10	−	1.05233e10i	0.348024	−	0.348024i
$418$	−8.15462e8	−	8.15462e8i	−0.0267115	−	0.0267115i
$419$			2.99061e10i			0.970296i	0.874432	+	0.485148i	$0.161234\pi$
$419$			2.99061e10i			0.970296i	−0.874432	+	0.485148i	$0.838766\pi$
$420$		0			0
$421$	−3.13382e10			−0.997574			−0.498787	−	0.866725i	$-0.666221\pi$
$421$	−3.13382e10			−0.997574			−0.498787	+	0.866725i	$0.666221\pi$
$422$	1.73222e8	−	1.73222e8i	0.00546203	−	0.00546203i
$423$	−1.28385e10	−	1.28385e10i	−0.401007	−	0.401007i
$424$		−	2.57461e10i		−	0.796614i
$425$		0			0
$426$	2.85118e9			0.0865737
$427$	−3.64568e9	+	3.64568e9i	−0.109665	+	0.109665i
$428$	1.11626e10	+	1.11626e10i	0.332652	+	0.332652i
$429$		−	4.93833e9i		−	0.145798i
$430$		0			0
$431$	1.97968e10			0.573701			0.286850	−	0.957975i	$-0.407392\pi$
$431$	1.97968e10			0.573701			0.286850	+	0.957975i	$0.407392\pi$
$432$	3.69453e9	−	3.69453e9i	0.106078	−	0.106078i
$433$	−2.04786e10	−	2.04786e10i	−0.582570	−	0.582570i	0.353039	−	0.935609i	$-0.385148\pi$
$433$	−2.04786e10	−	2.04786e10i	−0.582570	−	0.582570i	−0.935609	+	0.353039i	$0.885148\pi$
$434$		−	1.00223e9i		−	0.0282493i
$435$		0			0
$436$	4.15278e9			0.114919
$437$	1.29035e10	−	1.29035e10i	0.353818	−	0.353818i
$438$	4.40797e9	+	4.40797e9i	0.119768	+	0.119768i
$439$		−	1.52813e10i		−	0.411435i	−0.978611	−	0.205717i	$-0.934047\pi$
$439$		−	1.52813e10i		−	0.411435i	0.978611	−	0.205717i	$-0.0659528\pi$
$440$		0			0
$441$	−8.45384e9			−0.223511
$442$	−1.09520e9	+	1.09520e9i	−0.0286950	+	0.0286950i
$443$	−4.19068e10	−	4.19068e10i	−1.08810	−	1.08810i	−0.995724	−	0.0923778i	$-0.970553\pi$
$443$	−4.19068e10	−	4.19068e10i	−1.08810	−	1.08810i	−0.0923778	−	0.995724i	$-0.529447\pi$
$444$		−	3.88021e10i		−	0.998442i
$445$		0			0
$446$	1.89491e10			0.478906
$447$	−2.60172e10	+	2.60172e10i	−0.651673	+	0.651673i
$448$	9.41140e9	+	9.41140e9i	0.233637	+	0.233637i
$449$			3.76958e9i			0.0927487i	0.998924	+	0.0463743i	$0.0147667\pi$
$449$			3.76958e9i			0.0927487i	−0.998924	+	0.0463743i	$0.985233\pi$
$450$		0			0
$451$	−1.55152e10			−0.375017
$452$	−2.72943e10	+	2.72943e10i	−0.653910	+	0.653910i
$453$	−2.33710e10	−	2.33710e10i	−0.554990	−	0.554990i
$454$		−	3.85063e9i		−	0.0906377i
$455$		0			0
$456$	−5.31835e9			−0.123003
$457$	−1.58058e10	+	1.58058e10i	−0.362370	+	0.362370i	−0.864685	−	0.502315i	$-0.832482\pi$
$457$	−1.58058e10	+	1.58058e10i	−0.362370	+	0.362370i	0.502315	+	0.864685i	$0.332482\pi$
$458$	9.37104e9	+	9.37104e9i	0.212974	+	0.212974i
$459$		−	1.70606e9i		−	0.0384365i
$460$		0			0
$461$	5.93819e10			1.31477			0.657386	−	0.753554i	$-0.271662\pi$
$461$	5.93819e10			1.31477			0.657386	+	0.753554i	$0.271662\pi$
$462$	1.00032e9	−	1.00032e9i	0.0219568	−	0.0219568i
$463$	−3.75293e9	−	3.75293e9i	−0.0816670	−	0.0816670i	0.665093	−	0.746760i	$-0.268392\pi$
$463$	−3.75293e9	−	3.75293e9i	−0.0816670	−	0.0816670i	−0.746760	+	0.665093i	$0.768392\pi$
$464$			5.21847e10i			1.12583i
$465$		0			0
$466$	1.57473e10			0.333935
$467$	−6.17134e10	+	6.17134e10i	−1.29751	+	1.29751i	−0.367484	+	0.930030i	$0.619781\pi$
$467$	−6.17134e10	+	6.17134e10i	−1.29751	+	1.29751i	−0.930030	+	0.367484i	$0.880219\pi$
$468$	−7.73639e9	−	7.73639e9i	−0.161271	−	0.161271i
$469$		−	2.41078e10i		−	0.498272i
$470$		0			0
$471$	−4.84334e10			−0.984150
$472$	−1.44269e10	+	1.44269e10i	−0.290672	+	0.290672i
$473$	8.17948e9	+	8.17948e9i	0.163411	+	0.163411i
$474$		−	8.30575e9i		−	0.164538i
$475$		0			0
$476$	5.44147e9			0.105996
$477$	1.83942e10	−	1.83942e10i	0.355310	−	0.355310i
$478$	−9.12466e9	−	9.12466e9i	−0.174785	−	0.174785i
$479$		−	2.65972e9i		−	0.0505235i	−0.999681	−	0.0252617i	$-0.991958\pi$
$479$		−	2.65972e9i		−	0.0505235i	0.999681	−	0.0252617i	$-0.00804191\pi$
$480$		0			0
$481$	−7.40866e10			−1.38407
$482$	−7.72467e9	+	7.72467e9i	−0.143117	+	0.143117i
$483$	1.58285e10	+	1.58285e10i	0.290838	+	0.290838i
$484$		−	4.48299e10i		−	0.816934i
$485$		0			0
$486$	−9.82661e8			−0.0176140
$487$	−4.38176e10	+	4.38176e10i	−0.778991	+	0.778991i	−0.979659	−	0.200668i	$-0.935689\pi$
$487$	−4.38176e10	+	4.38176e10i	−0.778991	+	0.778991i	0.200668	+	0.979659i	$0.435689\pi$
$488$	5.72591e9	+	5.72591e9i	0.100964	+	0.100964i
$489$		−	2.13048e9i		−	0.0372599i
$490$		0			0
$491$	6.09667e10			1.04898			0.524490	−	0.851417i	$-0.324256\pi$
$491$	6.09667e10			1.04898			0.524490	+	0.851417i	$0.324256\pi$
$492$	−2.43061e10	+	2.43061e10i	−0.414815	+	0.414815i
$493$	1.20489e10	+	1.20489e10i	0.203967	+	0.203967i
$494$			4.87839e9i			0.0819161i
$495$		0			0
$496$	8.45645e9			0.139721
$497$	−1.35239e10	+	1.35239e10i	−0.221654	+	0.221654i
$498$	3.81456e9	+	3.81456e9i	0.0620194	+	0.0620194i
$499$			6.14633e10i			0.991319i	0.868517	+	0.495660i	$0.165074\pi$
$499$			6.14633e10i			0.991319i	−0.868517	+	0.495660i	$0.834926\pi$
$500$		0			0
$501$	−5.24102e10			−0.831888
$502$	−8.31992e9	+	8.31992e9i	−0.131010	+	0.131010i
$503$	−4.56188e10	−	4.56188e10i	−0.712642	−	0.712642i	0.254445	−	0.967087i	$-0.418107\pi$
$503$	−4.56188e10	−	4.56188e10i	−0.712642	−	0.712642i	−0.967087	+	0.254445i	$0.918107\pi$
$504$		−	6.52395e9i		−	0.101109i
$505$		0			0
$506$	7.62366e9			0.116295
$507$	1.22032e10	−	1.22032e10i	0.184690	−	0.184690i
$508$	−2.50359e10	−	2.50359e10i	−0.375931	−	0.375931i
$509$		−	9.24908e10i		−	1.37793i	−0.724794	−	0.688966i	$-0.758065\pi$
$509$		−	9.24908e10i		−	1.37793i	0.724794	−	0.688966i	$-0.241935\pi$
$510$		0			0
$511$	−4.18162e10			−0.613284
$512$	4.81403e10	−	4.81403e10i	0.700534	−	0.700534i
$513$	−3.79967e9	−	3.79967e9i	−0.0548627	−	0.0548627i
$514$			8.71248e9i			0.124821i
$515$		0			0
$516$	2.56280e10			0.361506
$517$	2.93301e10	−	2.93301e10i	0.410536	−	0.410536i
$518$	−1.50071e10	−	1.50071e10i	−0.208438	−	0.208438i
$519$			4.47276e10i			0.616461i
$520$		0			0
$521$	8.59534e10			1.16657			0.583287	−	0.812266i	$-0.301766\pi$
$521$	8.59534e10			1.16657			0.583287	+	0.812266i	$0.301766\pi$
$522$	6.93997e9	−	6.93997e9i	0.0934708	−	0.0934708i
$523$	−1.53299e10	−	1.53299e10i	−0.204896	−	0.204896i	0.597198	−	0.802094i	$-0.296281\pi$
$523$	−1.53299e10	−	1.53299e10i	−0.204896	−	0.204896i	−0.802094	+	0.597198i	$0.796281\pi$
$524$			1.18086e11i			1.56629i
$525$		0			0
$526$	−1.98714e10			−0.259588
$527$	1.95251e9	−	1.95251e9i	0.0253134	−	0.0253134i
$528$	8.44032e9	+	8.44032e9i	0.108598	+	0.108598i
$529$			4.23220e10i			0.540435i
$530$		0			0
$531$	−2.06144e10			−0.259294
$532$	1.21190e10	−	1.21190e10i	0.151294	−	0.151294i
$533$	4.64088e10	+	4.64088e10i	0.575031	+	0.575031i
$534$			1.09632e10i			0.134826i
$535$		0			0
$536$	−3.78638e10			−0.458738
$537$	−2.81074e10	+	2.81074e10i	−0.338005	+	0.338005i
$538$	−8.61889e9	−	8.61889e9i	−0.102878	−	0.102878i
$539$		−	1.93132e10i		−	0.228822i
$540$		0			0
$541$	−1.92085e9			−0.0224236			−0.0112118	−	0.999937i	$-0.503569\pi$
$541$	−1.92085e9			−0.0224236			−0.0112118	+	0.999937i	$0.503569\pi$
$542$	3.09317e10	−	3.09317e10i	0.358432	−	0.358432i
$543$	4.55423e9	+	4.55423e9i	0.0523861	+	0.0523861i
$544$		−	1.29870e10i		−	0.148290i
$545$		0			0
$546$	−5.98426e9			−0.0673349
$547$	4.24260e10	−	4.24260e10i	0.473896	−	0.473896i	−0.429277	−	0.903173i	$-0.641232\pi$
$547$	4.24260e10	−	4.24260e10i	0.473896	−	0.473896i	0.903173	+	0.429277i	$0.141232\pi$
$548$	−4.63948e10	−	4.63948e10i	−0.514454	−	0.514454i
$549$			8.18172e9i			0.0900648i
$550$		0			0
$551$	5.36699e10			0.582270
$552$	2.48603e10	−	2.48603e10i	0.267763	−	0.267763i
$553$	3.93963e10	+	3.93963e10i	0.421265	+	0.421265i
$554$			1.78319e10i			0.189303i
$555$		0			0
$556$	7.53254e10			0.788210
$557$	7.68174e10	−	7.68174e10i	0.798066	−	0.798066i	−0.184725	−	0.982790i	$-0.559139\pi$
$557$	7.68174e10	−	7.68174e10i	0.798066	−	0.798066i	0.982790	+	0.184725i	$0.0591393\pi$
$558$	−1.12461e9	−	1.12461e9i	−0.0116002	−	0.0116002i
$559$		−	4.89327e10i		−	0.501132i
$560$		0			0
$561$	3.89757e9			0.0393498
$562$	−3.19786e10	+	3.19786e10i	−0.320564	+	0.320564i
$563$	1.20218e11	+	1.20218e11i	1.19657	+	1.19657i	0.975188	+	0.221377i	$0.0710551\pi$
$563$	1.20218e11	+	1.20218e11i	1.19657	+	1.19657i	0.221377	+	0.975188i	$0.428945\pi$
$564$		−	9.18972e10i		−	0.908209i
$565$		0			0
$566$	−1.94800e10			−0.189812
$567$	4.66101e9	−	4.66101e9i	0.0450970	−	0.0450970i
$568$	2.12406e10	+	2.12406e10i	0.204068	+	0.204068i
$569$		−	1.12597e11i		−	1.07418i	−0.843525	−	0.537090i	$-0.819523\pi$
$569$		−	1.12597e11i		−	1.07418i	0.843525	−	0.537090i	$-0.180477\pi$
$570$		0			0
$571$	−7.88582e9			−0.0741827			−0.0370913	−	0.999312i	$-0.511809\pi$
$571$	−7.88582e9			−0.0741827			−0.0370913	+	0.999312i	$0.511809\pi$
$572$	1.76741e10	−	1.76741e10i	0.165103	−	0.165103i
$573$	6.05483e10	+	6.05483e10i	0.561672	+	0.561672i
$574$			1.88013e10i			0.173197i
$575$		0			0
$576$	2.11213e10			0.191880
$577$	1.35022e11	−	1.35022e11i	1.21815	−	1.21815i	0.249872	−	0.968279i	$-0.419611\pi$
$577$	1.35022e11	−	1.35022e11i	1.21815	−	1.21815i	0.968279	−	0.249872i	$-0.0803885\pi$
$578$	2.08056e10	+	2.08056e10i	0.186410	+	0.186410i
$579$			2.96263e10i			0.263611i
$580$		0			0
$581$	−3.61869e10			−0.317575
$582$	1.75859e10	−	1.75859e10i	0.153275	−	0.153275i
$583$	4.20224e10	+	4.20224e10i	0.363753	+	0.363753i
$584$			6.56767e10i			0.564625i
$585$		0			0
$586$	−1.44064e10			−0.122170
$587$	−1.24270e11	+	1.24270e11i	−1.04668	+	1.04668i	−0.0478229	+	0.998856i	$0.515228\pi$
$587$	−1.24270e11	+	1.24270e11i	−1.04668	+	1.04668i	−0.998856	+	0.0478229i	$0.984772\pi$
$588$	−3.02560e10	−	3.02560e10i	−0.253106	−	0.253106i
$589$		−	8.69712e9i		−	0.0722627i
$590$		0			0
$591$	6.72719e10			0.551422
$592$	1.26625e11	−	1.26625e11i	1.03094	−	1.03094i
$593$	6.87856e10	+	6.87856e10i	0.556261	+	0.556261i	0.928241	−	0.371980i	$-0.121321\pi$
$593$	6.87856e10	+	6.87856e10i	0.556261	+	0.556261i	−0.371980	+	0.928241i	$0.621321\pi$
$594$		−	2.24493e9i		−	0.0180326i
$595$		0			0
$596$	−1.86229e11			−1.47592
$597$	−1.59192e10	+	1.59192e10i	−0.125321	+	0.125321i
$598$	−2.28038e10	−	2.28038e10i	−0.178321	−	0.178321i
$599$		−	4.22834e10i		−	0.328445i	−0.986423	−	0.164222i	$-0.947489\pi$
$599$		−	4.22834e10i		−	0.328445i	0.986423	−	0.164222i	$-0.0525115\pi$
$600$		0			0
$601$	1.67538e11			1.28415			0.642075	−	0.766642i	$-0.278074\pi$
$601$	1.67538e11			1.28415			0.642075	+	0.766642i	$0.278074\pi$
$602$	9.91189e9	−	9.91189e9i	0.0754693	−	0.0754693i
$603$	−2.70517e10	−	2.70517e10i	−0.204609	−	0.204609i
$604$		−	1.67288e11i		−	1.25695i
$605$		0			0
$606$	3.63350e10			0.269423
$607$	−7.21743e10	+	7.21743e10i	−0.531653	+	0.531653i	−0.921064	−	0.389411i	$-0.872678\pi$
$607$	−7.21743e10	+	7.21743e10i	−0.531653	+	0.531653i	0.389411	+	0.921064i	$0.372678\pi$
$608$	−2.89241e10	−	2.89241e10i	−0.211664	−	0.211664i
$609$			6.58361e10i			0.478625i
$610$		0			0
$611$	−1.75464e11			−1.25899
$612$	6.10594e9	−	6.10594e9i	0.0435258	−	0.0435258i
$613$	1.16637e11	+	1.16637e11i	0.826024	+	0.826024i	0.986964	−	0.160940i	$-0.0514525\pi$
$613$	1.16637e11	+	1.16637e11i	0.826024	+	0.826024i	−0.160940	+	0.986964i	$0.551452\pi$
$614$			4.75853e10i			0.334810i
$615$		0			0
$616$	1.49043e10			0.103511
$617$	1.32769e11	−	1.32769e11i	0.916126	−	0.916126i	−0.0806194	−	0.996745i	$-0.525690\pi$
$617$	1.32769e11	−	1.32769e11i	0.916126	−	0.916126i	0.996745	+	0.0806194i	$0.0256898\pi$
$618$	1.52401e10	+	1.52401e10i	0.104480	+	0.104480i
$619$			2.20817e11i			1.50408i	0.659118	+	0.752040i	$0.270930\pi$
$619$			2.20817e11i			1.50408i	−0.659118	+	0.752040i	$0.729070\pi$
$620$		0			0
$621$	3.55227e10			0.238858
$622$	−2.88968e10	+	2.88968e10i	−0.193058	+	0.193058i
$623$	−5.20014e10	−	5.20014e10i	−0.345194	−	0.345194i
$624$		−	5.04931e10i		−	0.333038i
$625$		0			0
$626$	−4.32996e10			−0.281959
$627$	8.68053e9	−	8.68053e9i	0.0561663	−	0.0561663i
$628$	−1.73342e11	−	1.73342e11i	−1.11446	−	1.11446i
$629$		−	5.84728e10i		−	0.373552i
$630$		0			0
$631$	−5.85541e10			−0.369352			−0.184676	−	0.982800i	$-0.559123\pi$
$631$	−5.85541e10			−0.369352			−0.184676	+	0.982800i	$0.559123\pi$
$632$	6.18760e10	−	6.18760e10i	0.387841	−	0.387841i
$633$	1.84394e9	+	1.84394e9i	0.0114850	+	0.0114850i
$634$			3.51321e10i			0.217444i
$635$		0			0
$636$	1.31665e11			0.804713
$637$	−5.77693e10	+	5.77693e10i	−0.350864	+	0.350864i
$638$	1.58547e10	+	1.58547e10i	0.0956919	+	0.0956919i
$639$			3.03506e10i			0.182039i
$640$		0			0
$641$	−9.71973e10			−0.575734			−0.287867	−	0.957670i	$-0.592946\pi$
$641$	−9.71973e10			−0.575734			−0.287867	+	0.957670i	$0.592946\pi$
$642$	9.68887e9	−	9.68887e9i	0.0570339	−	0.0570339i
$643$	−1.13687e11	−	1.13687e11i	−0.665071	−	0.665071i	0.291500	−	0.956571i	$-0.405846\pi$
$643$	−1.13687e11	−	1.13687e11i	−0.665071	−	0.665071i	−0.956571	+	0.291500i	$0.905846\pi$
$644$			1.13300e11i			0.658696i
$645$		0			0
$646$	−3.85027e9			−0.0221086
$647$	−1.17592e11	+	1.17592e11i	−0.671061	+	0.671061i	−0.957961	−	0.286899i	$-0.907375\pi$
$647$	−1.17592e11	+	1.17592e11i	−0.671061	+	0.671061i	0.286899	+	0.957961i	$0.407375\pi$
$648$	−7.32060e9	−	7.32060e9i	−0.0415190	−	0.0415190i
$649$		−	4.70946e10i		−	0.265456i
$650$		0			0
$651$	1.06686e10			0.0593998
$652$	7.62492e9	−	7.62492e9i	0.0421935	−	0.0421935i
$653$	−2.53561e10	−	2.53561e10i	−0.139454	−	0.139454i	0.633934	−	0.773387i	$-0.281439\pi$
$653$	−2.53561e10	−	2.53561e10i	−0.139454	−	0.139454i	−0.773387	+	0.633934i	$0.781439\pi$
$654$		−	3.60452e9i		−	0.0197032i
$655$		0			0
$656$	−1.58639e11			−0.856631
$657$	−4.69225e10	+	4.69225e10i	−0.251837	+	0.251837i
$658$	−3.55422e10	−	3.55422e10i	−0.189601	−	0.189601i
$659$		−	2.75271e11i		−	1.45955i	−0.683689	−	0.729774i	$-0.739625\pi$
$659$		−	2.75271e11i		−	1.45955i	0.683689	−	0.729774i	$-0.260375\pi$
$660$		0			0
$661$	6.27595e10			0.328756			0.164378	−	0.986397i	$-0.447438\pi$
$661$	6.27595e10			0.328756			0.164378	+	0.986397i	$0.447438\pi$
$662$	−4.44602e10	+	4.44602e10i	−0.231494	+	0.231494i
$663$	−1.16584e10	−	1.16584e10i	−0.0603369	−	0.0603369i
$664$			5.68353e10i			0.292379i
$665$		0			0
$666$	−3.36793e10			−0.171185
$667$	−2.50877e11	+	2.50877e11i	−1.26753	+	1.26753i
$668$	−1.87575e11	−	1.87575e11i	−0.942037	−	0.942037i
$669$			2.01712e11i			1.00700i
$670$		0			0
$671$	−1.86915e10			−0.0922050
$672$	3.54809e10	−	3.54809e10i	0.173987	−	0.173987i
$673$	2.01482e11	+	2.01482e11i	0.982147	+	0.982147i	0.999843	−	0.0176962i	$-0.00563317\pi$
$673$	2.01482e11	+	2.01482e11i	0.982147	+	0.982147i	−0.0176962	+	0.999843i	$0.505633\pi$
$674$		−	8.90580e10i		−	0.431552i
$675$		0			0
$676$	8.73500e10			0.418289
$677$	−4.34668e10	+	4.34668e10i	−0.206920	+	0.206920i	−0.802957	−	0.596037i	$-0.796741\pi$
$677$	−4.34668e10	+	4.34668e10i	−0.206920	+	0.206920i	0.596037	+	0.802957i	$0.296741\pi$
$678$	2.36908e10	+	2.36908e10i	0.112114	+	0.112114i
$679$			1.66829e11i			0.784859i
$680$		0			0
$681$	4.09897e10			0.190584
$682$	2.56923e9	−	2.56923e9i	0.0118759	−	0.0118759i
$683$	−1.33784e10	−	1.33784e10i	−0.0614780	−	0.0614780i	0.675699	−	0.737177i	$-0.263842\pi$
$683$	−1.33784e10	−	1.33784e10i	−0.0614780	−	0.0614780i	−0.737177	+	0.675699i	$0.763842\pi$
$684$		−	2.71979e10i		−	0.124254i
$685$		0			0
$686$	−5.83068e10			−0.263283
$687$	−9.97540e10	+	9.97540e10i	−0.447820	+	0.447820i
$688$	8.36331e10	+	8.36331e10i	0.373271	+	0.373271i
$689$		−	2.51394e11i		−	1.11552i
$690$		0			0
$691$	2.54479e11			1.11619			0.558097	−	0.829775i	$-0.311532\pi$
$691$	2.54479e11			1.11619			0.558097	+	0.829775i	$0.311532\pi$
$692$	−1.60079e11	+	1.60079e11i	−0.698086	+	0.698086i
$693$	1.06483e10	+	1.06483e10i	0.0461686	+	0.0461686i
$694$		−	8.81938e10i		−	0.380190i
$695$		0			0
$696$	1.03402e11			0.440650
$697$	−3.66281e10	+	3.66281e10i	−0.155197	+	0.155197i
$698$	−4.42959e9	−	4.42959e9i	−0.0186613	−	0.0186613i
$699$			1.67629e11i			0.702166i
$700$		0			0
$701$	1.18657e11			0.491384			0.245692	−	0.969348i	$-0.420985\pi$
$701$	1.18657e11			0.491384			0.245692	+	0.969348i	$0.420985\pi$
$702$	−6.71501e9	+	6.71501e9i	−0.0276502	+	0.0276502i
$703$	−1.30228e11	−	1.30228e11i	−0.533193	−	0.533193i
$704$			4.82525e10i			0.196440i
$705$		0			0
$706$	3.80198e10			0.153035
$707$	−1.72346e11	+	1.72346e11i	−0.689801	+	0.689801i
$708$	−7.37785e10	−	7.37785e10i	−0.293627	−	0.293627i
$709$			3.20598e11i			1.26875i	0.773025	+	0.634375i	$0.218743\pi$
$709$			3.20598e11i			1.26875i	−0.773025	+	0.634375i	$0.781257\pi$
$710$		0			0
$711$	8.84141e10			0.345974
$712$	−8.16736e10	+	8.16736e10i	−0.317806	+	0.317806i
$713$	4.06542e10	+	4.06542e10i	0.157307	+	0.157307i
$714$		−	4.72307e9i		−	0.0181732i
$715$		0			0
$716$	−2.01191e11			−0.765520
$717$	9.71313e10	−	9.71313e10i	0.367521	−	0.367521i
$718$	−8.47330e10	−	8.47330e10i	−0.318827	−	0.318827i
$719$		−	4.23512e11i		−	1.58471i	−0.610059	−	0.792356i	$-0.708854\pi$
$719$		−	4.23512e11i		−	1.58471i	0.610059	−	0.792356i	$-0.291146\pi$
$720$		0			0
$721$	−1.44576e11			−0.535001
$722$	4.41837e10	−	4.41837e10i	0.162597	−	0.162597i
$723$	−8.22286e10	−	8.22286e10i	−0.300933	−	0.300933i
$724$			3.25990e10i			0.118645i
$725$		0			0
$726$	−3.89114e10			−0.140065
$727$	−9.00351e10	+	9.00351e10i	−0.322310	+	0.322310i	−0.849653	−	0.527342i	$-0.823189\pi$
$727$	−9.00351e10	+	9.00351e10i	−0.322310	+	0.322310i	0.527342	+	0.849653i	$0.323189\pi$
$728$	−4.45814e10	−	4.45814e10i	−0.158719	−	0.158719i
$729$		−	1.04604e10i		−	0.0370370i
$730$		0			0
$731$	3.86201e10			0.135252
$732$	−2.92821e10	+	2.92821e10i	−0.101990	+	0.101990i
$733$	1.38110e10	+	1.38110e10i	0.0478421	+	0.0478421i	0.730623	−	0.682781i	$-0.239230\pi$
$733$	1.38110e10	+	1.38110e10i	0.0478421	+	0.0478421i	−0.682781	+	0.730623i	$0.739230\pi$
$734$		−	2.70809e10i		−	0.0932994i
$735$		0			0
$736$	2.70408e11			0.921529
$737$	6.18007e10	−	6.18007e10i	0.209471	−	0.209471i
$738$	2.10971e10	+	2.10971e10i	0.0711210	+	0.0711210i
$739$		−	3.81206e11i		−	1.27815i	−0.769144	−	0.639075i	$-0.779317\pi$
$739$		−	3.81206e11i		−	1.27815i	0.769144	−	0.639075i	$-0.220683\pi$
$740$		0			0
$741$	−5.19301e10			−0.172245
$742$	5.09227e10	−	5.09227e10i	0.167995	−	0.167995i
$743$	−1.18224e11	−	1.18224e11i	−0.387927	−	0.387927i	0.486020	−	0.873947i	$-0.338448\pi$
$743$	−1.18224e11	−	1.18224e11i	−0.387927	−	0.387927i	−0.873947	+	0.486020i	$0.838448\pi$
$744$		−	1.67562e10i		−	0.0546870i
$745$		0			0
$746$	5.35440e10			0.172884
$747$	−4.06058e10	+	4.06058e10i	−0.130408	+	0.130408i
$748$	1.39493e10	+	1.39493e10i	0.0445601	+	0.0445601i
$749$			9.19135e10i			0.292047i
$750$		0			0
$751$	−4.45619e11			−1.40089			−0.700445	−	0.713707i	$-0.747015\pi$
$751$	−4.45619e11			−1.40089			−0.700445	+	0.713707i	$0.747015\pi$
$752$	2.99893e11	−	2.99893e11i	0.937766	−	0.937766i
$753$	−8.85649e10	−	8.85649e10i	−0.275475	−	0.275475i
$754$		−	9.48486e10i		−	0.293458i
$755$		0			0
$756$	3.33633e10			0.102137
$757$	4.26149e11	−	4.26149e11i	1.29771	−	1.29771i	0.367808	−	0.929902i	$-0.380108\pi$
$757$	4.26149e11	−	4.26149e11i	1.29771	−	1.29771i	0.929902	−	0.367808i	$-0.119892\pi$
$758$	5.06268e10	+	5.06268e10i	0.153357	+	0.153357i
$759$			8.11533e10i			0.244534i
$760$		0			0
$761$	−2.56636e11			−0.765207			−0.382603	−	0.923913i	$-0.624972\pi$
$761$	−2.56636e11			−0.765207			−0.382603	+	0.923913i	$0.624972\pi$
$762$	−2.17306e10	+	2.17306e10i	−0.0644543	+	0.0644543i
$763$	1.70972e10	+	1.70972e10i	0.0504459	+	0.0504459i
$764$			4.33401e11i			1.27209i
$765$		0			0
$766$	−5.62091e9			−0.0163264
$767$	−1.40869e11	+	1.40869e11i	−0.407036	+	0.407036i
$768$	4.66379e10	+	4.66379e10i	0.134058	+	0.134058i
$769$			5.23066e11i			1.49572i	0.663855	+	0.747861i	$0.268919\pi$
$769$			5.23066e11i			1.49572i	−0.663855	+	0.747861i	$0.731081\pi$
$770$		0			0
$771$	−9.27437e10			−0.262462
$772$	−1.06032e11	+	1.06032e11i	−0.298516	+	0.298516i
$773$	2.31264e11	+	2.31264e11i	0.647723	+	0.647723i	0.952442	−	0.304719i	$-0.0985627\pi$
$773$	2.31264e11	+	2.31264e11i	0.647723	+	0.647723i	−0.304719	+	0.952442i	$0.598563\pi$
$774$		−	2.22445e10i		−	0.0619810i
$775$		0			0
$776$	2.62022e11			0.722587
$777$	1.59750e11	−	1.59750e11i	0.438284	−	0.438284i
$778$	4.29346e10	+	4.29346e10i	0.117190	+	0.117190i
$779$			1.63153e11i			0.443044i
$780$		0			0
$781$	−6.93373e10			−0.186364
$782$	1.79979e10	−	1.79979e10i	0.0481275	−	0.0481275i
$783$	7.38755e10	+	7.38755e10i	0.196541	+	0.196541i
$784$		−	1.97472e11i		−	0.522687i
$785$		0			0
$786$	1.02496e11			0.268545
$787$	−9.83798e9	+	9.83798e9i	−0.0256453	+	0.0256453i	−0.719813	−	0.694168i	$-0.755773\pi$
$787$	−9.83798e9	+	9.83798e9i	−0.0256453	+	0.0256453i	0.694168	+	0.719813i	$0.255773\pi$
$788$	2.40764e11	+	2.40764e11i	0.624435	+	0.624435i
$789$		−	2.11529e11i		−	0.545836i
$790$		0			0
$791$	−2.24743e11			−0.574091
$792$	1.67242e10	−	1.67242e10i	0.0425056	−	0.0425056i
$793$	5.59098e10	+	5.59098e10i	0.141382	+	0.141382i
$794$		−	7.33665e10i		−	0.184593i
$795$		0			0
$796$	−1.13949e11			−0.283830
$797$	5.12588e11	−	5.12588e11i	1.27038	−	1.27038i	0.324497	−	0.945887i	$-0.394805\pi$
$797$	5.12588e11	−	5.12588e11i	1.27038	−	1.27038i	0.945887	−	0.324497i	$-0.105195\pi$
$798$	−1.05191e10	−	1.05191e10i	−0.0259397	−	0.0259397i
$799$		−	1.38484e11i		−	0.339793i
$800$		0			0
$801$	−1.16703e11			−0.283499
$802$	7.58513e10	−	7.58513e10i	0.183344	−	0.183344i
$803$	−1.07197e11	−	1.07197e11i	−0.257821	−	0.257821i
$804$		−	1.93634e11i		−	0.463402i
$805$		0			0
$806$	−1.53701e10			−0.0364196
$807$	9.17474e10	−	9.17474e10i	0.216322	−	0.216322i
$808$	2.70687e11	+	2.70687e11i	0.635071	+	0.635071i
$809$			2.69111e11i			0.628258i	0.949380	+	0.314129i	$0.101712\pi$
$809$			2.69111e11i			0.628258i	−0.949380	+	0.314129i	$0.898288\pi$
$810$		0			0
$811$	5.11333e11			1.18201			0.591004	−	0.806669i	$-0.298732\pi$
$811$	5.11333e11			1.18201			0.591004	+	0.806669i	$0.298732\pi$
$812$	−2.35626e11	+	2.35626e11i	−0.541999	+	0.541999i
$813$	3.29266e11	+	3.29266e11i	0.753675	+	0.753675i
$814$		−	7.69419e10i		−	0.175253i
$815$		0			0
$816$	3.98517e10			0.0898847
$817$	8.60132e10	−	8.60132e10i	0.193053	−	0.193053i
$818$	3.75437e9	+	3.75437e9i	0.00838539	+	0.00838539i
$819$		−	6.37020e10i		−	0.141585i
$820$		0			0
$821$	1.54755e11			0.340623			0.170311	−	0.985390i	$-0.445523\pi$
$821$	1.54755e11			0.340623			0.170311	+	0.985390i	$0.445523\pi$
$822$	−4.02696e10	+	4.02696e10i	−0.0882044	+	0.0882044i
$823$	9.01212e10	+	9.01212e10i	0.196439	+	0.196439i	0.798471	−	0.602033i	$-0.205642\pi$
$823$	9.01212e10	+	9.01212e10i	0.196439	+	0.196439i	−0.602033	+	0.798471i	$0.705642\pi$
$824$			2.27071e11i			0.492553i
$825$		0			0
$826$	−5.70692e10			−0.122598
$827$	2.24761e11	−	2.24761e11i	0.480506	−	0.480506i	−0.424787	−	0.905293i	$-0.639651\pi$
$827$	2.24761e11	−	2.24761e11i	0.480506	−	0.480506i	0.905293	+	0.424787i	$0.139651\pi$
$828$	1.27135e11	+	1.27135e11i	0.270485	+	0.270485i
$829$		−	7.91622e11i		−	1.67610i	−0.545594	−	0.838050i	$-0.683696\pi$
$829$		−	7.91622e11i		−	1.67610i	0.545594	−	0.838050i	$-0.316304\pi$
$830$		0			0
$831$	−1.89819e11			−0.398048
$832$	1.44332e11	−	1.44332e11i	0.301211	−	0.301211i
$833$	−4.55944e10	−	4.55944e10i	−0.0946959	−	0.0946959i
$834$		−	6.53807e10i		−	0.135140i
$835$		0			0
$836$	6.21348e10			0.127207
$837$	1.19714e10	−	1.19714e10i	0.0243918	−	0.0243918i
$838$	9.29024e10	+	9.29024e10i	0.188387	+	0.188387i
$839$			3.46532e11i			0.699352i	0.936871	+	0.349676i	$0.113708\pi$
$839$			3.46532e11i			0.699352i	−0.936871	+	0.349676i	$0.886292\pi$
$840$		0			0
$841$	−5.43235e11			−1.08593
$842$	−9.73510e10	+	9.73510e10i	−0.193683	+	0.193683i
$843$	−3.40410e11	−	3.40410e11i	−0.674051	−	0.674051i
$844$			1.31988e10i			0.0260115i
$845$		0			0
$846$	−7.97647e10			−0.155714
$847$	1.84567e11	−	1.84567e11i	0.358608	−	0.358608i
$848$	4.29668e11	+	4.29668e11i	0.830902	+	0.830902i
$849$		−	2.07363e11i		−	0.399118i
$850$		0			0
$851$	1.21749e12			2.32139
$852$	−1.08624e11	+	1.08624e11i	−0.206142	+	0.206142i
$853$	−2.32474e11	−	2.32474e11i	−0.439114	−	0.439114i	0.452600	−	0.891714i	$-0.350497\pi$
$853$	−2.32474e11	−	2.32474e11i	−0.439114	−	0.439114i	−0.891714	+	0.452600i	$0.850497\pi$
$854$			2.26504e10i			0.0425837i
$855$		0			0
$856$	1.44360e11			0.268875
$857$	3.14575e11	−	3.14575e11i	0.583177	−	0.583177i	−0.352598	−	0.935775i	$-0.614702\pi$
$857$	3.14575e11	−	3.14575e11i	0.583177	−	0.583177i	0.935775	+	0.352598i	$0.114702\pi$
$858$	−1.53408e10	−	1.53408e10i	−0.0283072	−	0.0283072i
$859$			4.37243e11i			0.803063i	0.915845	+	0.401532i	$0.131522\pi$
$859$			4.37243e11i			0.803063i	−0.915845	+	0.401532i	$0.868478\pi$
$860$		0			0
$861$	−2.00138e11			−0.364181
$862$	6.14980e10	−	6.14980e10i	0.111386	−	0.111386i
$863$	−2.88466e11	−	2.88466e11i	−0.520057	−	0.520057i	0.397532	−	0.917588i	$-0.369867\pi$
$863$	−2.88466e11	−	2.88466e11i	−0.520057	−	0.520057i	−0.917588	+	0.397532i	$0.869867\pi$
$864$		−	7.96270e10i		−	0.142891i
$865$		0			0
$866$	−1.27232e11			−0.226217
$867$	−2.21474e11	+	2.21474e11i	−0.391964	+	0.391964i
$868$	3.81828e10	+	3.81828e10i	0.0672649	+	0.0672649i
$869$			2.01986e11i			0.354195i
$870$		0			0
$871$	−3.69715e11			−0.642383
$872$	2.68529e10	−	2.68529e10i	0.0464435	−	0.0464435i
$873$	1.87201e11	+	1.87201e11i	0.322292	+	0.322292i
$874$		−	8.01683e10i		−	0.137391i
$875$		0			0
$876$	−3.35869e11			−0.570365
$877$	6.00399e11	−	6.00399e11i	1.01494	−	1.01494i	0.0150556	−	0.999887i	$-0.495207\pi$
$877$	6.00399e11	−	6.00399e11i	1.01494	−	1.01494i	0.999887	−	0.0150556i	$-0.00479254\pi$
$878$	−4.74707e10	−	4.74707e10i	−0.0798818	−	0.0798818i
$879$		−	1.53355e11i		−	0.256887i
$880$		0			0
$881$	1.08213e12			1.79629			0.898147	−	0.439694i	$-0.144913\pi$
$881$	1.08213e12			1.79629			0.898147	+	0.439694i	$0.144913\pi$
$882$	−2.62616e10	+	2.62616e10i	−0.0433956	+	0.0433956i
$883$	−7.01877e11	−	7.01877e11i	−1.15456	−	1.15456i	−0.985627	−	0.168938i	$-0.945966\pi$
$883$	−7.01877e11	−	7.01877e11i	−1.15456	−	1.15456i	−0.168938	−	0.985627i	$-0.554034\pi$
$884$		−	8.34499e10i		−	0.136652i
$885$		0			0
$886$	−2.60364e11			−0.422519
$887$	−4.78658e11	+	4.78658e11i	−0.773270	+	0.773270i	−0.978677	−	0.205407i	$-0.934148\pi$
$887$	−4.78658e11	+	4.78658e11i	−0.773270	+	0.773270i	0.205407	+	0.978677i	$0.434148\pi$
$888$	−2.50903e11	−	2.50903e11i	−0.403510	−	0.403510i
$889$		−	2.06148e11i		−	0.330044i
$890$		0			0
$891$	2.38972e10			0.0379171
$892$	−7.21922e11	+	7.21922e11i	−1.14033	+	1.14033i
$893$	−3.08428e11	−	3.08428e11i	−0.485006	−	0.485006i
$894$			1.61643e11i			0.253050i
$895$		0			0
$896$	3.33151e11			0.516903
$897$	2.42744e11	−	2.42744e11i	0.374955	−	0.374955i
$898$	1.17101e10	+	1.17101e10i	0.0180075	+	0.0180075i
$899$			1.69095e11i			0.258875i
$900$		0			0
$901$	1.98412e11			0.301071
$902$	−4.81974e10	+	4.81974e10i	−0.0728110	+	0.0728110i
$903$	1.05511e11	+	1.05511e11i	0.158689	+	0.158689i
$904$			3.52982e11i			0.528542i
$905$		0			0
$906$	−1.45203e11			−0.215507
$907$	−5.94503e11	+	5.94503e11i	−0.878465	+	0.878465i	−0.993376	−	0.114911i	$-0.963342\pi$
$907$	−5.94503e11	+	5.94503e11i	−0.878465	+	0.878465i	0.114911	+	0.993376i	$0.463342\pi$
$908$	1.46701e11	+	1.46701e11i	0.215819	+	0.215819i
$909$			3.86783e11i			0.566516i
$910$		0			0
$911$	1.94927e10			0.0283008			0.0141504	−	0.999900i	$-0.495496\pi$
$911$	1.94927e10			0.0283008			0.0141504	+	0.999900i	$0.495496\pi$
$912$	8.87562e10	−	8.87562e10i	0.128298	−	0.128298i
$913$	−9.27657e10	−	9.27657e10i	−0.133507	−	0.133507i
$914$			9.82006e10i			0.140711i
$915$		0			0
$916$	−7.14034e11			−1.01423
$917$	−4.86164e11	+	4.86164e11i	−0.687552	+	0.687552i
$918$	−5.29982e9	−	5.29982e9i	−0.00746260	−	0.00746260i
$919$		−	5.36385e11i		−	0.751995i	−0.926621	−	0.375997i	$-0.877300\pi$
$919$		−	5.36385e11i		−	0.751995i	0.926621	−	0.375997i	$-0.122700\pi$
$920$		0			0
$921$	−5.06542e11			−0.704006
$922$	1.84468e11	−	1.84468e11i	0.255268	−	0.255268i
$923$	2.07401e11	+	2.07401e11i	0.285761	+	0.285761i
$924$			7.62199e10i			0.104564i
$925$		0			0
$926$	−2.33167e10			−0.0317120
$927$	−1.62230e11	+	1.62230e11i	−0.219691	+	0.219691i
$928$	5.62360e11	+	5.62360e11i	0.758268	+	0.758268i
$929$			2.91516e11i			0.391381i	0.980666	+	0.195691i	$0.0626948\pi$
$929$			2.91516e11i			0.391381i	−0.980666	+	0.195691i	$0.937305\pi$
$930$		0			0
$931$	−2.03092e11			−0.270330
$932$	−5.99939e11	+	5.99939e11i	−0.795139	+	0.795139i
$933$	−3.07604e11	−	3.07604e11i	−0.405944	−	0.405944i
$934$			3.83421e11i			0.503835i
$935$		0			0
$936$	−1.00051e11			−0.130352
$937$	−3.52799e11	+	3.52799e11i	−0.457688	+	0.457688i	−0.897896	−	0.440208i	$-0.854905\pi$
$937$	−3.52799e11	+	3.52799e11i	−0.457688	+	0.457688i	0.440208	+	0.897896i	$0.354905\pi$
$938$	−7.48901e10	−	7.48901e10i	−0.0967415	−	0.0967415i
$939$		−	4.60921e11i		−	0.592876i
$940$		0			0
$941$	−1.96719e9			−0.00250893			−0.00125446	−	0.999999i	$-0.500399\pi$
$941$	−1.96719e9			−0.00250893			−0.00125446	+	0.999999i	$0.500399\pi$
$942$	−1.50457e11	+	1.50457e11i	−0.191077	+	0.191077i
$943$	−7.62651e11	−	7.62651e11i	−0.964449	−	0.964449i
$944$		−	4.81530e11i		−	0.606367i
$945$		0			0
$946$	5.08186e10			0.0634538
$947$	−2.52474e11	+	2.52474e11i	−0.313918	+	0.313918i	−0.846426	−	0.532507i	$-0.821250\pi$
$947$	−2.52474e11	+	2.52474e11i	−0.313918	+	0.313918i	0.532507	+	0.846426i	$0.321250\pi$
$948$	3.16432e11	+	3.16432e11i	0.391784	+	0.391784i
$949$			6.41290e11i			0.790659i
$950$		0			0
$951$	−3.73978e11			−0.457219
$952$	3.51858e10	−	3.51858e10i	0.0428371	−	0.0428371i
$953$	−1.39641e11	−	1.39641e11i	−0.169294	−	0.169294i	0.617375	−	0.786669i	$-0.288196\pi$
$953$	−1.39641e11	−	1.39641e11i	−0.169294	−	0.169294i	−0.786669	+	0.617375i	$0.788196\pi$
$954$		−	1.14282e11i		−	0.137970i
$955$		0			0
$956$	6.95260e11			0.832369
$957$	−1.68772e11	+	1.68772e11i	−0.201211	+	0.201211i
$958$	−8.26232e9	−	8.26232e9i	−0.00980934	−	0.00980934i
$959$		−	3.82018e11i		−	0.451658i
$960$		0			0
$961$	−8.25490e11			−0.967872
$962$	−2.30148e11	+	2.30148e11i	−0.268724	+	0.268724i
$963$	1.03137e11	+	1.03137e11i	0.119925	+	0.119925i
$964$		−	5.88587e11i		−	0.681558i
$965$		0			0
$966$	9.83415e10			0.112935
$967$	−1.22373e12	+	1.22373e12i	−1.39953	+	1.39953i	−0.598124	+	0.801403i	$0.704087\pi$
$967$	−1.22373e12	+	1.22373e12i	−1.39953	+	1.39953i	−0.801403	+	0.598124i	$0.795913\pi$
$968$	−2.89881e11	−	2.89881e11i	−0.330155	−	0.330155i
$969$		−	4.09858e10i		−	0.0464877i
$970$		0			0
$971$	1.85260e11			0.208403			0.104202	−	0.994556i	$-0.466771\pi$
$971$	1.85260e11			0.208403			0.104202	+	0.994556i	$0.466771\pi$
$972$	3.74373e10	−	3.74373e10i	0.0419411	−	0.0419411i
$973$	3.10117e11	+	3.10117e11i	0.345999	+	0.345999i
$974$			2.72236e11i			0.302489i
$975$		0			0
$976$	−1.91116e11			−0.210619
$977$	8.96530e11	−	8.96530e11i	0.983981	−	0.983981i	−0.0158927	−	0.999874i	$-0.505059\pi$
$977$	8.96530e11	−	8.96530e11i	0.983981	−	0.983981i	0.999874	+	0.0158927i	$0.00505901\pi$
$978$	−6.61826e9	−	6.61826e9i	−0.00723417	−	0.00723417i
$979$		−	2.66613e11i		−	0.290235i
$980$		0			0
$981$	3.83699e10			0.0414299
$982$	1.89391e11	−	1.89391e11i	0.203664	−	0.203664i
$983$	−7.43264e11	−	7.43264e11i	−0.796029	−	0.796029i	0.186438	−	0.982467i	$-0.440306\pi$
$983$	−7.43264e11	−	7.43264e11i	−0.796029	−	0.796029i	−0.982467	+	0.186438i	$0.940306\pi$
$984$			3.14338e11i			0.335286i
$985$		0			0
$986$	7.48592e10			0.0792022
$987$	3.78344e11	−	3.78344e11i	0.398674	−	0.398674i
$988$	−1.85857e11	−	1.85857e11i	−0.195052	−	0.195052i
$989$			8.04128e11i			0.840504i
$990$		0			0
$991$	1.69152e12			1.75381			0.876904	−	0.480665i	$-0.159605\pi$
$991$	1.69152e12			1.75381			0.876904	+	0.480665i	$0.159605\pi$
$992$	9.11296e10	−	9.11296e10i	0.0941050	−	0.0941050i
$993$	−4.73276e11	−	4.73276e11i	−0.486763	−	0.486763i
$994$			8.40229e10i			0.0860701i
$995$		0			0
$996$	−2.90654e11			−0.295351
$997$	6.16013e11	−	6.16013e11i	0.623461	−	0.623461i	−0.322953	−	0.946415i	$-0.604676\pi$
$997$	6.16013e11	−	6.16013e11i	0.623461	−	0.623461i	0.946415	+	0.322953i	$0.104676\pi$
$998$	1.90934e11	+	1.90934e11i	0.192469	+	0.192469i
$999$		−	3.58514e11i		−	0.359951i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.f.d.43.4	yes	12
5.2	odd	4	inner	75.9.f.d.7.4	yes	12
5.3	odd	4	inner	75.9.f.d.7.3	✓	12
5.4	even	2	inner	75.9.f.d.43.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.f.d.7.3	✓	12	5.3	odd	4	inner
75.9.f.d.7.4	yes	12	5.2	odd	4	inner
75.9.f.d.43.3	yes	12	5.4	even	2	inner
75.9.f.d.43.4	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.10647 - 3.10647i) q^{2} +(33.0681 + 33.0681i) q^{3} +236.700i q^{4} +205.450 q^{6} +(-974.502 + 974.502i) q^{7} +(1530.56 + 1530.56i) q^{8} +2187.00i q^{9} +O(q^{10})\)	\(q+(3.10647 - 3.10647i) q^{2} +(33.0681 + 33.0681i) q^{3} +236.700i q^{4} +205.450 q^{6} +(-974.502 + 974.502i) q^{7} +(1530.56 + 1530.56i) q^{8} +2187.00i q^{9} -4996.30 q^{11} +(-7827.21 + 7827.21i) q^{12} +(14944.9 + 14944.9i) q^{13} +6054.51i q^{14} -51085.9 q^{16} +(-11795.2 + 11795.2i) q^{17} +(6793.84 + 6793.84i) q^{18} +52539.8i q^{19} -64449.9 q^{21} +(-15520.8 + 15520.8i) q^{22} +(-245594. - 245594. i) q^{23} +101225. i q^{24} +92851.4 q^{26} +(-72320.0 + 72320.0i) q^{27} +(-230664. - 230664. i) q^{28} -1.02151e6i q^{29} -165534. q^{31} +(-550519. + 550519. i) q^{32} +(-165218. - 165218. i) q^{33} +73282.9i q^{34} -517662. q^{36} +(-2.47866e6 + 2.47866e6i) q^{37} +(163213. + 163213. i) q^{38} +988397. i q^{39} +3.10533e6 q^{41} +(-200211. + 200211. i) q^{42} +(-1.63711e6 - 1.63711e6i) q^{43} -1.18262e6i q^{44} -1.52586e6 q^{46} +(-5.87036e6 + 5.87036e6i) q^{47} +(-1.68931e6 - 1.68931e6i) q^{48} +3.86549e6i q^{49} -780091. q^{51} +(-3.53744e6 + 3.53744e6i) q^{52} +(-8.41070e6 - 8.41070e6i) q^{53} +449319. i q^{54} -2.98306e6 q^{56} +(-1.73739e6 + 1.73739e6i) q^{57} +(-3.17328e6 - 3.17328e6i) q^{58} +9.42589e6i q^{59} +3.74107e6 q^{61} +(-514226. + 514226. i) q^{62} +(-2.13123e6 - 2.13123e6i) q^{63} -9.65765e6i q^{64} -1.02649e6 q^{66} +(-1.23693e7 + 1.23693e7i) q^{67} +(-2.79193e6 - 2.79193e6i) q^{68} -1.62427e7i q^{69} +1.38777e7 q^{71} +(-3.34732e6 + 3.34732e6i) q^{72} +(2.14552e7 + 2.14552e7i) q^{73} +1.53998e7i q^{74} -1.24361e7 q^{76} +(4.86890e6 - 4.86890e6i) q^{77} +(3.07042e6 + 3.07042e6i) q^{78} -4.04271e7i q^{79} -4.78297e6 q^{81} +(9.64661e6 - 9.64661e6i) q^{82} +(1.85669e7 + 1.85669e7i) q^{83} -1.52553e7i q^{84} -1.01712e7 q^{86} +(3.37794e7 - 3.37794e7i) q^{87} +(-7.64712e6 - 7.64712e6i) q^{88} +5.33621e7i q^{89} -2.91276e7 q^{91} +(5.81321e7 - 5.81321e7i) q^{92} +(-5.47390e6 - 5.47390e6i) q^{93} +3.64722e7i q^{94} -3.64092e7 q^{96} +(8.55969e7 - 8.55969e7i) q^{97} +(1.20080e7 + 1.20080e7i) q^{98} -1.09269e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2268 q^{6}+O(q^{10}) \) 12 * q + 2268 * q^6	\( 12 q + 2268 q^{6} - 88920 q^{11} - 485796 q^{16} + 79704 q^{21} - 4055976 q^{26} + 5658696 q^{31} + 5065092 q^{36} + 1798056 q^{41} - 10882464 q^{46} + 4329936 q^{51} + 7268040 q^{56} + 33649848 q^{61} - 141361848 q^{66} + 335506464 q^{71} - 395386536 q^{76} - 57395628 q^{81} + 489958560 q^{86} - 216875664 q^{91} - 710311356 q^{96}+O(q^{100}) \) 12 * q + 2268 * q^6 - 88920 * q^11 - 485796 * q^16 + 79704 * q^21 - 4055976 * q^26 + 5658696 * q^31 + 5065092 * q^36 + 1798056 * q^41 - 10882464 * q^46 + 4329936 * q^51 + 7268040 * q^56 + 33649848 * q^61 - 141361848 * q^66 + 335506464 * q^71 - 395386536 * q^76 - 57395628 * q^81 + 489958560 * q^86 - 216875664 * q^91 - 710311356 * q^96

Introduction

L-functions

Modular forms

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