LMFDB - Embedded newform 75.4.e.d.32.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(32,75)]

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([2, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("75.32");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.e (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:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 36x^{14} + 562x^{12} - 3672x^{10} + 16413x^{8} - 6588x^{6} + 43024x^{4} + 499896x^{2} + 532900 $ x^16 - 36x^14 + 562x^12 - 3672x^10 + 16413x^8 - 6588x^6 + 43024x^4 + 499896*x^2 + 532900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{8}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			32.1
Root			$1.55042 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.32
Dual form			75.4.e.d.68.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.39887 + 3.39887i) q^{2} +(0.624963 + 5.15843i) q^{3} -15.1047i q^{4} +(-19.6570 - 15.4087i) q^{6} +(-19.7241 - 19.7241i) q^{7} +(24.1479 + 24.1479i) q^{8} +(-26.2188 + 6.44766i) q^{9} +O(q^{10})$	\(q+(-3.39887 + 3.39887i) q^{2} +(0.624963 + 5.15843i) q^{3} -15.1047i q^{4} +(-19.6570 - 15.4087i) q^{6} +(-19.7241 - 19.7241i) q^{7} +(24.1479 + 24.1479i) q^{8} +(-26.2188 + 6.44766i) q^{9} -9.19707i q^{11} +(77.9165 - 9.43987i) q^{12} +(22.4300 - 22.4300i) q^{13} +134.080 q^{14} -43.3141 q^{16} +(-50.6273 + 50.6273i) q^{17} +(67.1998 - 111.029i) q^{18} -16.5813i q^{19} +(89.4187 - 114.072i) q^{21} +(31.2597 + 31.2597i) q^{22} +(-48.2959 - 48.2959i) q^{23} +(-109.474 + 139.657i) q^{24} +152.474i q^{26} +(-49.6456 - 131.219i) q^{27} +(-297.927 + 297.927i) q^{28} -203.298 q^{29} -27.4766 q^{31} +(-45.9644 + 45.9644i) q^{32} +(47.4425 - 5.74783i) q^{33} -344.152i q^{34} +(97.3898 + 396.027i) q^{36} +(-130.592 - 130.592i) q^{37} +(56.3576 + 56.3576i) q^{38} +(129.722 + 101.686i) q^{39} +9.19707i q^{41} +(83.7948 + 691.641i) q^{42} +(-63.3021 + 63.3021i) q^{43} -138.919 q^{44} +328.303 q^{46} +(383.520 - 383.520i) q^{47} +(-27.0697 - 223.433i) q^{48} +435.083i q^{49} +(-292.798 - 229.517i) q^{51} +(-338.799 - 338.799i) q^{52} +(-441.142 - 441.142i) q^{53} +(614.735 + 277.256i) q^{54} -952.594i q^{56} +(85.5333 - 10.3627i) q^{57} +(690.986 - 690.986i) q^{58} -314.626 q^{59} -431.664 q^{61} +(93.3894 - 93.3894i) q^{62} +(644.318 + 389.969i) q^{63} -658.967i q^{64} +(-141.715 + 180.787i) q^{66} +(649.685 + 649.685i) q^{67} +(764.709 + 764.709i) q^{68} +(218.948 - 279.314i) q^{69} +722.186i q^{71} +(-788.828 - 477.433i) q^{72} +(-662.432 + 662.432i) q^{73} +887.733 q^{74} -250.455 q^{76} +(-181.404 + 181.404i) q^{77} +(-786.526 + 95.2905i) q^{78} -206.816i q^{79} +(645.855 - 338.100i) q^{81} +(-31.2597 - 31.2597i) q^{82} +(544.335 + 544.335i) q^{83} +(-1723.03 - 1350.64i) q^{84} -430.312i q^{86} +(-127.054 - 1048.70i) q^{87} +(222.090 - 222.090i) q^{88} -563.910 q^{89} -884.827 q^{91} +(-729.494 + 729.494i) q^{92} +(-17.1718 - 141.736i) q^{93} +2607.07i q^{94} +(-265.830 - 208.378i) q^{96} +(66.0080 + 66.0080i) q^{97} +(-1478.79 - 1478.79i) q^{98} +(59.2996 + 241.137i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 84 q^{6}+O(q^{10}) $ 16 * q - 84 * q^6	$ 16 q - 84 q^{6} - 232 q^{16} + 816 q^{21} - 1208 q^{31} + 252 q^{36} + 1872 q^{46} + 156 q^{51} - 1528 q^{61} - 3420 q^{66} + 1064 q^{76} + 6876 q^{81} - 10008 q^{91} - 8172 q^{96}+O(q^{100}) $ 16 * q - 84 * q^6 - 232 * q^16 + 816 * q^21 - 1208 * q^31 + 252 * q^36 + 1872 * q^46 + 156 * q^51 - 1528 * q^61 - 3420 * q^66 + 1064 * q^76 + 6876 * q^81 - 10008 * q^91 - 8172 * q^96

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	−3.39887	+	3.39887i	−1.20168	+	1.20168i	−0.228029	+	0.973654i	$0.573228\pi$
$2$	−3.39887	+	3.39887i	−1.20168	+	1.20168i	−0.973654	+	0.228029i	$0.926772\pi$
$3$	0.624963	+	5.15843i	0.120274	+	0.992741i
$3$	0.624963	+	5.15843i	0.120274	+	0.992741i
$4$		−	15.1047i		−	1.88809i
$5$		0			0
$5$		0			0
$6$	−19.6570	−	15.4087i	−1.33749	−	1.04843i
$7$	−19.7241	−	19.7241i	−1.06500	−	1.06500i	−0.997735	−	0.0672681i	$-0.978572\pi$
$7$	−19.7241	−	19.7241i	−1.06500	−	1.06500i	−0.0672681	−	0.997735i	$-0.521428\pi$
$8$	24.1479	+	24.1479i	1.06720	+	1.06720i
$9$	−26.2188	+	6.44766i	−0.971068	+	0.238802i
$10$		0			0
$11$		−	9.19707i		−	0.252093i	−0.992024	−	0.126046i	$-0.959771\pi$
$11$		−	9.19707i		−	0.252093i	0.992024	−	0.126046i	$-0.0402288\pi$
$12$	77.9165	−	9.43987i	1.87438	−	0.227088i
$13$	22.4300	−	22.4300i	0.478537	−	0.478537i	−0.426127	−	0.904663i	$-0.640122\pi$
$13$	22.4300	−	22.4300i	0.478537	−	0.478537i	0.904663	+	0.426127i	$0.140122\pi$
$14$	134.080			2.55959
$15$		0			0
$16$	−43.3141			−0.676782
$17$	−50.6273	+	50.6273i	−0.722290	+	0.722290i	−0.969071	−	0.246782i	$-0.920627\pi$
$17$	−50.6273	+	50.6273i	−0.722290	+	0.722290i	0.246782	+	0.969071i	$0.420627\pi$
$18$	67.1998	−	111.029i	0.879952	−	1.45388i
$19$		−	16.5813i		−	0.200211i	−0.994977	−	0.100105i	$-0.968082\pi$
$19$		−	16.5813i		−	0.200211i	0.994977	−	0.100105i	$-0.0319180\pi$
$20$		0			0
$21$	89.4187	−	114.072i	0.929180	−	1.18536i
$22$	31.2597	+	31.2597i	0.302936	+	0.302936i
$23$	−48.2959	−	48.2959i	−0.437843	−	0.437843i	0.453443	−	0.891285i	$-0.350196\pi$
$23$	−48.2959	−	48.2959i	−0.437843	−	0.437843i	−0.891285	+	0.453443i	$0.850196\pi$
$24$	−109.474	+	139.657i	−0.931094	+	1.18781i
$25$		0			0
$26$			152.474i			1.15010i
$27$	−49.6456	−	131.219i	−0.353863	−	0.935297i
$28$	−297.927	+	297.927i	−2.01082	+	2.01082i
$29$	−203.298			−1.30178			−0.650889	−	0.759173i	$-0.725604\pi$
$29$	−203.298			−1.30178			−0.650889	+	0.759173i	$0.725604\pi$
$30$		0			0
$31$	−27.4766			−0.159192			−0.0795958	−	0.996827i	$-0.525363\pi$
$31$	−27.4766			−0.159192			−0.0795958	+	0.996827i	$0.525363\pi$
$32$	−45.9644	+	45.9644i	−0.253920	+	0.253920i
$33$	47.4425	−	5.74783i	0.250263	−	0.0303203i
$34$		−	344.152i		−	1.73593i
$35$		0			0
$36$	97.3898	+	396.027i	0.450879	+	1.83346i
$37$	−130.592	−	130.592i	−0.580250	−	0.580250i	0.354722	−	0.934972i	$-0.384575\pi$
$37$	−130.592	−	130.592i	−0.580250	−	0.580250i	−0.934972	+	0.354722i	$0.884575\pi$
$38$	56.3576	+	56.3576i	0.240590	+	0.240590i
$39$	129.722	+	101.686i	0.532618	+	0.417507i
$40$		0			0
$41$			9.19707i			0.0350327i	0.999847	+	0.0175164i	$0.00557592\pi$
$41$			9.19707i			0.0350327i	−0.999847	+	0.0175164i	$0.994424\pi$
$42$	83.7948	+	691.641i	0.307853	+	2.54101i
$43$	−63.3021	+	63.3021i	−0.224499	+	0.224499i	−0.810390	−	0.585891i	$-0.800745\pi$
$43$	−63.3021	+	63.3021i	−0.224499	+	0.224499i	0.585891	+	0.810390i	$0.300745\pi$
$44$	−138.919			−0.475973
$45$		0			0
$46$	328.303			1.05230
$47$	383.520	−	383.520i	1.19026	−	1.19026i	0.213265	−	0.976994i	$-0.431590\pi$
$47$	383.520	−	383.520i	1.19026	−	1.19026i	0.976994	−	0.213265i	$-0.0684098\pi$
$48$	−27.0697	−	223.433i	−0.0813994	−	0.671869i
$49$			435.083i			1.26846i
$50$		0			0
$51$	−292.798	−	229.517i	−0.803919	−	0.630173i
$52$	−338.799	−	338.799i	−0.903518	−	0.903518i
$53$	−441.142	−	441.142i	−1.14331	−	1.14331i	−0.987840	−	0.155471i	$-0.950310\pi$
$53$	−441.142	−	441.142i	−1.14331	−	1.14331i	−0.155471	−	0.987840i	$-0.549690\pi$
$54$	614.735	+	277.256i	1.54916	+	0.698700i
$55$		0			0
$56$		−	952.594i		−	2.27314i
$57$	85.5333	−	10.3627i	0.198757	−	0.0240802i
$58$	690.986	−	690.986i	1.56433	−	1.56433i
$59$	−314.626			−0.694251			−0.347126	−	0.937819i	$-0.612842\pi$
$59$	−314.626			−0.694251			−0.347126	+	0.937819i	$0.612842\pi$
$60$		0			0
$61$	−431.664			−0.906048			−0.453024	−	0.891498i	$-0.649655\pi$
$61$	−431.664			−0.906048			−0.453024	+	0.891498i	$0.649655\pi$
$62$	93.3894	−	93.3894i	0.191298	−	0.191298i
$63$	644.318	+	389.969i	1.28852	+	0.779866i
$64$		−	658.967i		−	1.28705i
$65$		0			0
$66$	−141.715	+	180.787i	−0.264301	+	0.337172i
$67$	649.685	+	649.685i	1.18465	+	1.18465i	0.978525	+	0.206126i	$0.0660858\pi$
$67$	649.685	+	649.685i	1.18465	+	1.18465i	0.206126	+	0.978525i	$0.433914\pi$
$68$	764.709	+	764.709i	1.36374	+	1.36374i
$69$	218.948	−	279.314i	0.382003	−	0.487326i
$70$		0			0
$71$			722.186i			1.20715i	0.797306	+	0.603575i	$0.206258\pi$
$71$			722.186i			1.20715i	−0.797306	+	0.603575i	$0.793742\pi$
$72$	−788.828	−	477.433i	−1.29117	−	0.781473i
$73$	−662.432	+	662.432i	−1.06208	+	1.06208i	−0.0641383	+	0.997941i	$0.520430\pi$
$73$	−662.432	+	662.432i	−1.06208	+	1.06208i	−0.997941	+	0.0641383i	$0.979570\pi$
$74$	887.733			1.39455
$75$		0			0
$76$	−250.455			−0.378015
$77$	−181.404	+	181.404i	−0.268480	+	0.268480i
$78$	−786.526	+	95.2905i	−1.14175	+	0.138327i
$79$		−	206.816i		−	0.294539i	−0.989096	−	0.147269i	$-0.952952\pi$
$79$		−	206.816i		−	0.294539i	0.989096	−	0.147269i	$-0.0470484\pi$
$80$		0			0
$81$	645.855	−	338.100i	0.885947	−	0.463786i
$82$	−31.2597	−	31.2597i	−0.0420982	−	0.0420982i
$83$	544.335	+	544.335i	0.719862	+	0.719862i	0.968577	−	0.248715i	$-0.0800083\pi$
$83$	544.335	+	544.335i	0.719862	+	0.719862i	−0.248715	+	0.968577i	$0.580008\pi$
$84$	−1723.03	−	1350.64i	−2.23807	−	1.75437i
$85$		0			0
$86$		−	430.312i		−	0.539555i
$87$	−127.054	−	1048.70i	−0.156570	−	1.29233i
$88$	222.090	−	222.090i	0.269033	−	0.269033i
$89$	−563.910			−0.671622			−0.335811	−	0.941929i	$-0.609010\pi$
$89$	−563.910			−0.671622			−0.335811	+	0.941929i	$0.609010\pi$
$90$		0			0
$91$	−884.827			−1.01929
$92$	−729.494	+	729.494i	−0.826685	+	0.826685i
$93$	−17.1718	−	141.736i	−0.0191466	−	0.158036i
$94$			2607.07i			2.86063i
$95$		0			0
$96$	−265.830	−	208.378i	−0.282617	−	0.221537i
$97$	66.0080	+	66.0080i	0.0690938	+	0.0690938i	0.740809	−	0.671715i	$-0.234442\pi$
$97$	66.0080	+	66.0080i	0.0690938	+	0.0690938i	−0.671715	+	0.740809i	$0.734442\pi$
$98$	−1478.79	−	1478.79i	−1.52429	−	1.52429i
$99$	59.2996	+	241.137i	0.0602003	+	0.244799i
$100$		0			0
$101$			1108.46i			1.09204i	0.837772	+	0.546021i	$0.183858\pi$
$101$			1108.46i			1.09204i	−0.837772	+	0.546021i	$0.816142\pi$
$102$	1775.28	−	215.082i	1.72332	−	0.208787i
$103$	823.002	−	823.002i	0.787308	−	0.787308i	−0.193744	−	0.981052i	$-0.562063\pi$
$103$	823.002	−	823.002i	0.787308	−	0.787308i	0.981052	+	0.193744i	$0.0620630\pi$
$104$	1083.28			1.02139
$105$		0			0
$106$	2998.77			2.74780
$107$	−918.923	+	918.923i	−0.830239	+	0.830239i	−0.987549	−	0.157310i	$-0.949718\pi$
$107$	−918.923	+	918.923i	−0.830239	+	0.830239i	0.157310	+	0.987549i	$0.449718\pi$
$108$	−1982.02	+	749.881i	−1.76592	+	0.668124i
$109$			4.75472i			0.00417816i	0.999998	+	0.00208908i	$0.000664976\pi$
$109$			4.75472i			0.00417816i	−0.999998	+	0.00208908i	$0.999335\pi$
$110$		0			0
$111$	592.036	−	755.267i	0.506248	−	0.645826i
$112$	854.332	+	854.332i	0.720775	+	0.720775i
$113$	−791.114	−	791.114i	−0.658600	−	0.658600i	0.296449	−	0.955049i	$-0.404198\pi$
$113$	−791.114	−	791.114i	−0.658600	−	0.658600i	−0.955049	+	0.296449i	$0.904198\pi$
$114$	−255.495	+	325.938i	−0.209906	+	0.267780i
$115$		0			0
$116$			3070.76i			2.45787i
$117$	−443.469	+	732.711i	−0.350416	+	0.578967i
$118$	1069.37	−	1069.37i	0.834270	−	0.834270i
$119$	1997.16			1.53848
$120$		0			0
$121$	1246.41			0.936449
$122$	1467.17	−	1467.17i	1.08878	−	1.08878i
$123$	−47.4425	+	5.74783i	−0.0347784	+	0.00421353i
$124$			415.025i			0.300567i
$125$		0			0
$126$	−3515.41	+	864.500i	−2.48554	+	0.611236i
$127$	9.96644	+	9.96644i	0.00696361	+	0.00696361i	0.710580	−	0.703616i	$-0.248433\pi$
$127$	9.96644	+	9.96644i	0.00696361	+	0.00696361i	−0.703616	+	0.710580i	$0.748433\pi$
$128$	1872.03	+	1872.03i	1.29270	+	1.29270i
$129$	−366.101	−	286.978i	−0.249871	−	0.195868i
$130$		0			0
$131$		−	2663.20i		−	1.77622i	−0.459631	−	0.888110i	$-0.652018\pi$
$131$		−	2663.20i		−	1.77622i	0.459631	−	0.888110i	$-0.347982\pi$
$132$	−86.8192	−	716.604i	−0.0572473	−	0.472518i
$133$	−327.051	+	327.051i	−0.213225	+	0.213225i
$134$	−4416.39			−2.84715
$135$		0			0
$136$	−2445.09			−1.54165
$137$	−351.396	+	351.396i	−0.219137	+	0.219137i	−0.808135	−	0.588998i	$-0.799523\pi$
$137$	−351.396	+	351.396i	−0.219137	+	0.219137i	0.588998	+	0.808135i	$0.299523\pi$
$138$	205.177	+	1693.53i	0.126564	+	1.04466i
$139$		−	1738.04i		−	1.06057i	−0.847820	−	0.530283i	$-0.822086\pi$
$139$		−	1738.04i		−	1.06057i	0.847820	−	0.530283i	$-0.177914\pi$
$140$		0			0
$141$	2218.05	+	1738.68i	1.32478	+	1.03846i
$142$	−2454.62	−	2454.62i	−1.45061	−	1.45061i
$143$	−206.291	−	206.291i	−0.120636	−	0.120636i
$144$	1135.64	−	279.274i	0.657202	−	0.161617i
$145$		0			0
$146$		−	4503.04i		−	2.55257i
$147$	−2244.34	+	271.911i	−1.25925	+	0.152563i
$148$	−1972.55	+	1972.55i	−1.09556	+	1.09556i
$149$	183.941			0.101135			0.0505674	−	0.998721i	$-0.483897\pi$
$149$	183.941			0.101135			0.0505674	+	0.998721i	$0.483897\pi$
$150$		0			0
$151$	−805.273			−0.433988			−0.216994	−	0.976173i	$-0.569625\pi$
$151$	−805.273			−0.433988			−0.216994	+	0.976173i	$0.569625\pi$
$152$	400.403	−	400.403i	0.213664	−	0.213664i
$153$	1000.96	−	1653.82i	0.528908	−	0.873877i
$154$		−	1233.14i		−	0.645255i
$155$		0			0
$156$	1535.93	−	1959.41i	0.788289	−	1.00563i
$157$	1626.56	+	1626.56i	0.826840	+	0.826840i	0.987078	−	0.160239i	$-0.0512264\pi$
$157$	1626.56	+	1626.56i	0.826840	+	0.826840i	−0.160239	+	0.987078i	$0.551226\pi$
$158$	702.940	+	702.940i	0.353942	+	0.353942i
$159$	1999.90	−	2551.30i	0.997501	−	1.27252i
$160$		0			0
$161$			1905.19i			0.932608i
$162$	−1046.02	+	3344.34i	−0.507304	+	1.62195i
$163$	395.910	−	395.910i	0.190246	−	0.190246i	−0.605556	−	0.795802i	$-0.707049\pi$
$163$	395.910	−	395.910i	0.190246	−	0.190246i	0.795802	+	0.605556i	$0.207049\pi$
$164$	138.919			0.0661448
$165$		0			0
$166$	−3700.25			−1.73009
$167$	875.168	−	875.168i	0.405524	−	0.405524i	−0.474650	−	0.880174i	$-0.657426\pi$
$167$	875.168	−	875.168i	0.405524	−	0.405524i	0.880174	+	0.474650i	$0.157426\pi$
$168$	4913.89	−	595.336i	2.25664	−	0.273400i
$169$			1190.79i			0.542005i
$170$		0			0
$171$	106.910	+	434.741i	0.0478107	+	0.194418i
$172$	956.158	+	956.158i	0.423874	+	0.423874i
$173$	1048.43	+	1048.43i	0.460753	+	0.460753i	0.898902	−	0.438149i	$-0.144366\pi$
$173$	1048.43	+	1048.43i	0.460753	+	0.460753i	−0.438149	+	0.898902i	$0.644366\pi$
$174$	3996.24	+	3132.56i	1.74112	+	1.36482i
$175$		0			0
$176$			398.363i			0.170612i
$177$	−196.630	−	1622.98i	−0.0835005	−	0.689212i
$178$	1916.66	−	1916.66i	0.807077	−	0.807077i
$179$	210.570			0.0879259			0.0439629	−	0.999033i	$-0.486002\pi$
$179$	210.570			0.0879259			0.0439629	+	0.999033i	$0.486002\pi$
$180$		0			0
$181$	−128.633			−0.0528244			−0.0264122	−	0.999651i	$-0.508408\pi$
$181$	−128.633			−0.0528244			−0.0264122	+	0.999651i	$0.508408\pi$
$182$	3007.41	−	3007.41i	1.22486	−	1.22486i
$183$	−269.774	−	2226.71i	−0.108974	−	0.899471i
$184$		−	2332.49i		−	0.934530i
$185$		0			0
$186$	540.108	+	423.378i	0.212917	+	0.166901i
$187$	465.623	+	465.623i	0.182084	+	0.182084i
$188$	−5792.96	−	5792.96i	−2.24731	−	2.24731i
$189$	−1608.96	+	3567.39i	−0.619229	+	1.37296i
$190$		0			0
$191$		−	4291.51i		−	1.62577i	−0.582421	−	0.812887i	$-0.697894\pi$
$191$		−	4291.51i		−	1.62577i	0.582421	−	0.812887i	$-0.302106\pi$
$192$	3399.24	−	411.830i	1.27770	−	0.154798i
$193$	−3567.00	+	3567.00i	−1.33035	+	1.33035i	−0.425303	+	0.905051i	$0.639832\pi$
$193$	−3567.00	+	3567.00i	−1.33035	+	1.33035i	−0.905051	+	0.425303i	$0.860168\pi$
$194$	−448.706			−0.166058
$195$		0			0
$196$	6571.79			2.39497
$197$	421.067	−	421.067i	0.152283	−	0.152283i	−0.626854	−	0.779137i	$-0.715658\pi$
$197$	421.067	−	421.067i	0.152283	−	0.152283i	0.779137	+	0.626854i	$0.215658\pi$
$198$	−1021.14	−	618.041i	−0.366513	−	0.221830i
$199$			3024.07i			1.07724i	0.842549	+	0.538619i	$0.181054\pi$
$199$			3024.07i			1.07724i	−0.842549	+	0.538619i	$0.818946\pi$
$200$		0			0
$201$	−2945.33	+	3757.39i	−1.03357	+	1.31853i
$202$	−3767.53	−	3767.53i	−1.31229	−	1.31229i
$203$	4009.88	+	4009.88i	1.38640	+	1.38640i
$204$	−3466.79	+	4422.62i	−1.18982	+	1.51787i
$205$		0			0
$206$			5594.56i			1.89219i
$207$	1577.66	+	954.867i	0.529733	+	0.320617i
$208$	−971.537	+	971.537i	−0.323865	+	0.323865i
$209$	−152.499			−0.0504717
$210$		0			0
$211$	−2881.37			−0.940104			−0.470052	−	0.882639i	$-0.655765\pi$
$211$	−2881.37			−0.940104			−0.470052	+	0.882639i	$0.655765\pi$
$212$	−6663.31	+	6663.31i	−2.15867	+	2.15867i
$213$	−3725.35	+	451.339i	−1.19839	+	0.145189i
$214$		−	6246.60i		−	1.99537i
$215$		0			0
$216$	1969.82	−	4367.50i	0.620505	−	1.37579i
$217$	541.951	+	541.951i	0.169539	+	0.169539i
$218$	−16.1607	−	16.1607i	−0.00502083	−	0.00502083i
$219$	−3831.11	−	3003.11i	−1.18211	−	0.926629i
$220$		0			0
$221$			2271.15i			0.691284i
$222$	554.800	+	4579.31i	0.167729	+	1.38443i
$223$	1334.90	−	1334.90i	0.400858	−	0.400858i	−0.477677	−	0.878535i	$-0.658521\pi$
$223$	1334.90	−	1334.90i	0.400858	−	0.400858i	0.878535	+	0.477677i	$0.158521\pi$
$224$	1813.22			0.540851
$225$		0			0
$226$	5377.79			1.58286
$227$	−337.339	+	337.339i	−0.0986342	+	0.0986342i	−0.754702	−	0.656068i	$-0.772219\pi$
$227$	−337.339	+	337.339i	−0.0986342	+	0.0986342i	0.656068	+	0.754702i	$0.272219\pi$
$228$	−156.525	−	1291.95i	−0.0454654	−	0.375271i
$229$		−	4575.71i		−	1.32040i	−0.751090	−	0.660200i	$-0.770471\pi$
$229$		−	4575.71i		−	1.32040i	0.751090	−	0.660200i	$-0.229529\pi$
$230$		0			0
$231$	−1049.13	−	822.391i	−0.298822	−	0.234240i
$232$	−4909.24	−	4909.24i	−1.38926	−	1.38926i
$233$	−529.682	−	529.682i	−0.148930	−	0.148930i	0.628710	−	0.777640i	$-0.283583\pi$
$233$	−529.682	−	529.682i	−0.148930	−	0.148930i	−0.777640	+	0.628710i	$0.783583\pi$
$234$	−983.099	−	3997.69i	−0.274646	−	1.11682i
$235$		0			0
$236$			4752.33i			1.31081i
$237$	1066.84	−	129.252i	0.292401	−	0.0354254i
$238$	−6788.09	+	6788.09i	−1.84877	+	1.84877i
$239$	871.265			0.235805			0.117903	−	0.993025i	$-0.462383\pi$
$239$	871.265			0.235805			0.117903	+	0.993025i	$0.462383\pi$
$240$		0			0
$241$	42.9221			0.0114724			0.00573622	−	0.999984i	$-0.498174\pi$
$241$	42.9221			0.0114724			0.00573622	+	0.999984i	$0.498174\pi$
$242$	−4236.40	+	4236.40i	−1.12532	+	1.12532i
$243$	2147.70	+	3120.30i	0.566976	+	0.823734i
$244$			6520.15i			1.71070i
$245$		0			0
$246$	141.715	−	180.787i	0.0367293	−	0.0468560i
$247$	−371.918	−	371.918i	−0.0958081	−	0.0958081i
$248$	−663.502	−	663.502i	−0.169889	−	0.169889i
$249$	−2467.73	+	3148.10i	−0.628055	+	0.801217i
$250$		0			0
$251$			50.7994i			0.0127746i	0.999980	+	0.00638731i	$0.00203316\pi$
$251$			50.7994i			0.0127746i	−0.999980	+	0.00638731i	$0.997967\pi$
$252$	5890.37	−	9732.23i	1.47245	−	2.43283i
$253$	−444.181	+	444.181i	−0.110377	+	0.110377i
$254$	−67.7493			−0.0167361
$255$		0			0
$256$	−7453.85			−1.81979
$257$	4271.29	−	4271.29i	1.03671	−	1.03671i	0.0374152	−	0.999300i	$-0.488088\pi$
$257$	4271.29	−	4271.29i	1.03671	−	1.03671i	0.999300	−	0.0374152i	$-0.0119124\pi$
$258$	2219.73	−	268.929i	0.535638	−	0.0648945i
$259$			5151.64i			1.23593i
$260$		0			0
$261$	5330.25	−	1310.80i	1.26412	−	0.310867i
$262$	9051.88	+	9051.88i	2.13445	+	2.13445i
$263$	−1151.74	−	1151.74i	−0.270036	−	0.270036i	0.559079	−	0.829114i	$-0.311155\pi$
$263$	−1151.74	−	1151.74i	−0.270036	−	0.270036i	−0.829114	+	0.559079i	$0.811155\pi$
$264$	1284.44	+	1006.84i	0.299438	+	0.234722i
$265$		0			0
$266$		−	2223.21i		−	0.512458i
$267$	−352.423	−	2908.89i	−0.0807787	−	0.666746i
$268$	9813.29	−	9813.29i	2.23672	−	2.23672i
$269$	1684.46			0.381796			0.190898	−	0.981610i	$-0.438860\pi$
$269$	1684.46			0.381796			0.190898	+	0.981610i	$0.438860\pi$
$270$		0			0
$271$	−3308.36			−0.741581			−0.370791	−	0.928717i	$-0.620913\pi$
$271$	−3308.36			−0.741581			−0.370791	+	0.928717i	$0.620913\pi$
$272$	2192.87	−	2192.87i	0.488833	−	0.488833i
$273$	−552.984	−	4564.32i	−0.122594	−	1.01189i
$274$		−	2388.70i		−	0.526666i
$275$		0			0
$276$	−4218.95	−	3307.14i	−0.920112	−	0.721255i
$277$	−2178.55	−	2178.55i	−0.472550	−	0.472550i	0.430189	−	0.902739i	$-0.358447\pi$
$277$	−2178.55	−	2178.55i	−0.472550	−	0.472550i	−0.902739	+	0.430189i	$0.858447\pi$
$278$	5907.39	+	5907.39i	1.27447	+	1.27447i
$279$	720.404	−	177.159i	0.154586	−	0.0380153i
$280$		0			0
$281$		−	4972.10i		−	1.05555i	−0.849383	−	0.527776i	$-0.823026\pi$
$281$		−	4972.10i		−	1.05555i	0.849383	−	0.527776i	$-0.176974\pi$
$282$	−13448.4	+	1629.32i	−2.83986	+	0.344060i
$283$	3325.61	−	3325.61i	0.698542	−	0.698542i	−0.265554	−	0.964096i	$-0.585555\pi$
$283$	3325.61	−	3325.61i	0.698542	−	0.698542i	0.964096	+	0.265554i	$0.0855550\pi$
$284$	10908.4			2.27920
$285$		0			0
$286$	1402.31			0.289932
$287$	181.404	−	181.404i	0.0373100	−	0.0373100i
$288$	908.771	−	1501.50i	0.185937	−	0.307210i
$289$		−	213.245i		−	0.0434043i
$290$		0			0
$291$	−299.245	+	381.750i	−0.0602820	+	0.0769024i
$292$	10005.8	+	10005.8i	2.00530	+	2.00530i
$293$	−3473.24	−	3473.24i	−0.692521	−	0.692521i	0.270265	−	0.962786i	$-0.412889\pi$
$293$	−3473.24	−	3473.24i	−0.692521	−	0.692521i	−0.962786	+	0.270265i	$0.912889\pi$
$294$	6704.06	−	8552.43i	1.32989	−	1.69656i
$295$		0			0
$296$		−	6307.07i		−	1.23848i
$297$	−1206.83	+	456.594i	−0.235782	+	0.0892063i
$298$	−625.194	+	625.194i	−0.121532	+	0.121532i
$299$	−2166.56			−0.419048
$300$		0			0
$301$	2497.16			0.478185
$302$	2737.02	−	2737.02i	0.521516	−	0.521516i
$303$	−5717.93	+	692.748i	−1.08411	+	0.131344i
$304$			718.201i			0.135499i
$305$		0			0
$306$	2218.97	+	9023.26i	0.414543	+	1.68570i
$307$	−4105.90	−	4105.90i	−0.763310	−	0.763310i	0.213609	−	0.976919i	$-0.431478\pi$
$307$	−4105.90	−	4105.90i	−0.763310	−	0.763310i	−0.976919	+	0.213609i	$0.931478\pi$
$308$	2740.05	+	2740.05i	0.506913	+	0.506913i
$309$	4759.74	+	3731.05i	0.876286	+	0.686900i
$310$		0			0
$311$		−	3937.21i		−	0.717874i	−0.933362	−	0.358937i	$-0.883139\pi$
$311$		−	3937.21i		−	0.717874i	0.933362	−	0.358937i	$-0.116861\pi$
$312$	677.009	+	5588.02i	0.122846	+	1.01397i
$313$	160.559	−	160.559i	0.0289946	−	0.0289946i	−0.692461	−	0.721455i	$-0.743473\pi$
$313$	160.559	−	160.559i	0.0289946	−	0.0289946i	0.721455	+	0.692461i	$0.243473\pi$
$314$	−11057.0			−1.98720
$315$		0			0
$316$	−3123.88			−0.556115
$317$	−6408.72	+	6408.72i	−1.13549	+	1.13549i	−0.146239	+	0.989249i	$0.546717\pi$
$317$	−6408.72	+	6408.72i	−1.13549	+	1.13549i	−0.989249	+	0.146239i	$0.953283\pi$
$318$	1874.12	+	15469.0i	0.330489	+	2.72785i
$319$			1869.75i			0.328169i
$320$		0			0
$321$	−5314.49	−	4165.91i	−0.924069	−	0.724356i
$322$	−6475.49	−	6475.49i	−1.12070	−	1.12070i
$323$	839.464	+	839.464i	0.144610	+	0.144610i
$324$	−5106.90	−	9755.44i	−0.875668	−	1.67274i
$325$		0			0
$326$			2691.30i			0.457231i
$327$	−24.5269	+	2.97153i	−0.00414783	+	0.000502525i
$328$	−222.090	+	222.090i	−0.0373868	+	0.0373868i
$329$	−15129.2			−2.53526
$330$		0			0
$331$	8806.84			1.46244			0.731220	−	0.682142i	$-0.238951\pi$
$331$	8806.84			1.46244			0.731220	+	0.682142i	$0.238951\pi$
$332$	8222.01	−	8222.01i	1.35916	−	1.35916i
$333$	4265.99	+	2581.96i	0.702027	+	0.424897i
$334$			5949.17i			0.974623i
$335$		0			0
$336$	−3873.09	+	4940.94i	−0.628852	+	0.802233i
$337$	−5690.38	−	5690.38i	−0.919806	−	0.919806i	0.0772088	−	0.997015i	$-0.475399\pi$
$337$	−5690.38	−	5690.38i	−0.919806	−	0.919806i	−0.997015	+	0.0772088i	$0.975399\pi$
$338$	−4047.33	−	4047.33i	−0.651319	−	0.651319i
$339$	3586.49	−	4575.33i	0.574606	−	0.733031i
$340$		0			0
$341$			252.704i			0.0401311i
$342$	−1841.01	−	1114.26i	−0.291082	−	0.176176i
$343$	1816.25	−	1816.25i	0.285914	−	0.285914i
$344$	−3057.23			−0.479171
$345$		0			0
$346$	−7126.93			−1.10736
$347$	4974.95	−	4974.95i	0.769652	−	0.769652i	−0.208393	−	0.978045i	$-0.566823\pi$
$347$	4974.95	−	4974.95i	0.769652	−	0.769652i	0.978045	+	0.208393i	$0.0668234\pi$
$348$	−15840.3	+	1919.11i	−2.44003	+	0.295618i
$349$		−	6686.71i		−	1.02559i	−0.858511	−	0.512795i	$-0.828610\pi$
$349$		−	6686.71i		−	1.02559i	0.858511	−	0.512795i	$-0.171390\pi$
$350$		0			0
$351$	−4056.79	−	1829.69i	−0.616910	−	0.278238i
$352$	422.738	+	422.738i	0.0640114	+	0.0640114i
$353$	−2025.97	−	2025.97i	−0.305472	−	0.305472i	0.537678	−	0.843150i	$-0.319302\pi$
$353$	−2025.97	−	2025.97i	−0.305472	−	0.305472i	−0.843150	+	0.537678i	$0.819302\pi$
$354$	6184.61	+	4847.98i	0.928555	+	0.727873i
$355$		0			0
$356$			8517.68i			1.26808i
$357$	1248.15	+	10302.2i	0.185040	+	1.52731i
$358$	−715.700	+	715.700i	−0.105659	+	0.105659i
$359$	7178.33			1.05531			0.527657	−	0.849458i	$-0.323071\pi$
$359$	7178.33			1.05531			0.527657	+	0.849458i	$0.323071\pi$
$360$		0			0
$361$	6584.06			0.959916
$362$	437.207	−	437.207i	0.0634782	−	0.0634782i
$363$	778.962	+	6429.54i	0.112631	+	0.929651i
$364$			13365.0i			1.92450i
$365$		0			0
$366$	8485.23	+	6651.38i	1.21183	+	0.949926i
$367$	5129.64	+	5129.64i	0.729605	+	0.729605i	0.970541	−	0.240936i	$-0.0774545\pi$
$367$	5129.64	+	5129.64i	0.729605	+	0.729605i	−0.240936	+	0.970541i	$0.577454\pi$
$368$	2091.89	+	2091.89i	0.296324	+	0.296324i
$369$	−59.2996	−	241.137i	−0.00836589	−	0.0340192i
$370$		0			0
$371$			17402.3i			2.43526i
$372$	−2140.88	+	259.375i	−0.298385	+	0.0361505i
$373$	1332.77	−	1332.77i	0.185009	−	0.185009i	−0.608525	−	0.793534i	$-0.708239\pi$
$373$	1332.77	−	1332.77i	0.185009	−	0.185009i	0.793534	+	0.608525i	$0.208239\pi$
$374$	−3165.19			−0.437615
$375$		0			0
$376$	18522.4			2.54049
$377$	−4559.99	+	4559.99i	−0.622949	+	0.622949i
$378$	−6656.47	−	17593.7i	−0.905745	−	2.39398i
$379$		−	5792.28i		−	0.785039i	−0.919744	−	0.392519i	$-0.871604\pi$
$379$		−	5792.28i		−	0.785039i	0.919744	−	0.392519i	$-0.128396\pi$
$380$		0			0
$381$	−45.1825	+	57.6399i	−0.00607552	+	0.00775060i
$382$	14586.3	+	14586.3i	1.95367	+	1.95367i
$383$	−5023.31	−	5023.31i	−0.670181	−	0.670181i	0.287577	−	0.957758i	$-0.407150\pi$
$383$	−5023.31	−	5023.31i	−0.670181	−	0.670181i	−0.957758	+	0.287577i	$0.907150\pi$
$384$	−8486.79	+	10826.7i	−1.12784	+	1.43880i
$385$		0			0
$386$		−	24247.6i		−	3.19733i
$387$	1251.56	−	2067.86i	0.164393	−	0.271615i
$388$	997.030	−	997.030i	0.130455	−	0.130455i
$389$	10198.8			1.32931			0.664654	−	0.747151i	$-0.268579\pi$
$389$	10198.8			1.32931			0.664654	+	0.747151i	$0.268579\pi$
$390$		0			0
$391$	4890.18			0.632498
$392$	−10506.3	+	10506.3i	−1.35370	+	1.35370i
$393$	13737.9	−	1664.40i	1.76333	−	0.213633i
$394$			2862.31i			0.365993i
$395$		0			0
$396$	3642.29	−	895.701i	0.462202	−	0.113663i
$397$	90.2345	+	90.2345i	0.0114074	+	0.0114074i	0.712787	−	0.701380i	$-0.247432\pi$
$397$	90.2345	+	90.2345i	0.0114074	+	0.0114074i	−0.701380	+	0.712787i	$0.747432\pi$
$398$	−10278.4	−	10278.4i	−1.29450	−	1.29450i
$399$	−1891.46	−	1482.67i	−0.237322	−	0.186032i
$400$		0			0
$401$		−	597.810i		−	0.0744469i	−0.999307	−	0.0372234i	$-0.988149\pi$
$401$		−	597.810i		−	0.0744469i	0.999307	−	0.0372234i	$-0.0118513\pi$
$402$	−2760.08	−	22781.7i	−0.342439	−	2.82648i
$403$	−616.301	+	616.301i	−0.0761790	+	0.0761790i
$404$	16743.0			2.06187
$405$		0			0
$406$	−27258.2			−3.33202
$407$	−1201.07	+	1201.07i	−0.146277	+	0.146277i
$408$	−1528.09	−	12612.8i	−0.185421	−	1.53046i
$409$			10819.0i			1.30799i	0.756500	+	0.653994i	$0.226908\pi$
$409$			10819.0i			1.30799i	−0.756500	+	0.653994i	$0.773092\pi$
$410$		0			0
$411$	−2032.26	−	1593.04i	−0.243903	−	0.191190i
$412$	−12431.2	−	12431.2i	−1.48651	−	1.48651i
$413$	6205.73	+	6205.73i	0.739380	+	0.739380i
$414$	−8607.73	+	2116.79i	−1.02185	+	0.251291i
$415$		0			0
$416$			2061.97i			0.243020i
$417$	8965.57	−	1086.21i	1.05287	−	0.127559i
$418$	518.325	−	518.325i	0.0606510	−	0.0606510i
$419$	−10849.8			−1.26503			−0.632514	−	0.774549i	$-0.717977\pi$
$419$	−10849.8			−1.26503			−0.632514	+	0.774549i	$0.717977\pi$
$420$		0			0
$421$	−12341.0			−1.42865			−0.714324	−	0.699815i	$-0.753266\pi$
$421$	−12341.0			−1.42865			−0.714324	+	0.699815i	$0.753266\pi$
$422$	9793.43	−	9793.43i	1.12971	−	1.12971i
$423$	−7582.65	+	12528.3i	−0.871587	+	1.44006i
$424$		−	21305.3i		−	2.44028i
$425$		0			0
$426$	11127.9	−	14196.0i	1.26561	−	1.61455i
$427$	8514.20	+	8514.20i	0.964944	+	0.964944i
$428$	13880.0	+	13880.0i	1.56756	+	1.56756i
$429$	935.213	−	1193.06i	0.105251	−	0.134269i
$430$		0			0
$431$			10648.0i			1.19001i	0.803721	+	0.595006i	$0.202850\pi$
$431$			10648.0i			1.19001i	−0.803721	+	0.595006i	$0.797150\pi$
$432$	2150.35	+	5683.61i	0.239488	+	0.632993i
$433$	−10825.2	+	10825.2i	−1.20144	+	1.20144i	−0.227716	+	0.973728i	$0.573126\pi$
$433$	−10825.2	+	10825.2i	−1.20144	+	1.20144i	−0.973728	+	0.227716i	$0.926874\pi$
$434$	−3684.05			−0.407465
$435$		0			0
$436$	71.8186			0.00788873
$437$	−800.806	+	800.806i	−0.0876607	+	0.0876607i
$438$	23228.7	−	2814.24i	2.53404	−	0.307008i
$439$			1560.21i			0.169623i	0.996397	+	0.0848117i	$0.0270289\pi$
$439$			1560.21i			0.169623i	−0.996397	+	0.0848117i	$0.972971\pi$
$440$		0			0
$441$	−2805.26	−	11407.4i	−0.302912	−	1.23176i
$442$	−7719.34	−	7719.34i	−0.830705	−	0.830705i
$443$	−2588.92	−	2588.92i	−0.277660	−	0.277660i	0.554514	−	0.832174i	$-0.312904\pi$
$443$	−2588.92	−	2588.92i	−0.277660	−	0.277660i	−0.832174	+	0.554514i	$0.812904\pi$
$444$	−11408.1	−	8942.52i	−1.21938	−	0.955840i
$445$		0			0
$446$			9074.30i			0.963409i
$447$	114.957	+	948.850i	0.0121639	+	0.100401i
$448$	−12997.6	+	12997.6i	−1.37071	+	1.37071i
$449$	14291.3			1.50211			0.751057	−	0.660238i	$-0.229544\pi$
$449$	14291.3			1.50211			0.751057	+	0.660238i	$0.229544\pi$
$450$		0			0
$451$	84.5862			0.00883150
$452$	−11949.5	+	11949.5i	−1.24349	+	1.24349i
$453$	−503.266	−	4153.95i	−0.0521976	−	0.430838i
$454$		−	2293.14i		−	0.237054i
$455$		0			0
$456$	2315.69	+	1815.21i	0.237812	+	0.186415i
$457$	−7928.59	−	7928.59i	−0.811562	−	0.811562i	0.173306	−	0.984868i	$-0.444555\pi$
$457$	−7928.59	−	7928.59i	−0.811562	−	0.811562i	−0.984868	+	0.173306i	$0.944555\pi$
$458$	15552.3	+	15552.3i	1.58670	+	1.58670i
$459$	9156.66	+	4129.82i	0.931147	+	0.419964i
$460$		0			0
$461$			10685.6i			1.07957i	0.841804	+	0.539783i	$0.181494\pi$
$461$			10685.6i			1.07957i	−0.841804	+	0.539783i	$0.818506\pi$
$462$	6361.07	−	770.667i	0.640571	−	0.0776075i
$463$	9059.56	−	9059.56i	0.909360	−	0.909360i	−0.0868606	−	0.996220i	$-0.527683\pi$
$463$	9059.56	−	9059.56i	0.909360	−	0.909360i	0.996220	+	0.0868606i	$0.0276835\pi$
$464$	8805.68			0.881020
$465$		0			0
$466$	3600.65			0.357933
$467$	−9692.99	+	9692.99i	−0.960467	+	0.960467i	−0.999248	−	0.0387806i	$-0.987653\pi$
$467$	−9692.99	+	9692.99i	−0.960467	+	0.960467i	0.0387806	+	0.999248i	$0.487653\pi$
$468$	11067.4	+	6698.46i	1.09314	+	0.661616i
$469$		−	25628.9i		−	2.52332i
$470$		0			0
$471$	−7373.97	+	9407.05i	−0.721390	+	0.920285i
$472$	−7597.57	−	7597.57i	−0.740904	−	0.740904i
$473$	582.194	+	582.194i	0.0565947	+	0.0565947i
$474$	−3186.76	+	4065.38i	−0.308803	+	0.393943i
$475$		0			0
$476$		−	30166.5i		−	2.90478i
$477$	14410.6	+	8721.90i	1.38326	+	0.837208i
$478$	−2961.32	+	2961.32i	−0.283363	+	0.283363i
$479$	−5193.79			−0.495428			−0.247714	−	0.968833i	$-0.579679\pi$
$479$	−5193.79			−0.495428			−0.247714	+	0.968833i	$0.579679\pi$
$480$		0			0
$481$	−5858.38			−0.555341
$482$	−145.887	+	145.887i	−0.0137862	+	0.0137862i
$483$	−9827.78	+	1190.67i	−0.925838	+	0.112169i
$484$		−	18826.7i		−	1.76810i
$485$		0			0
$486$	−17905.3	−	3305.74i	−1.67119	−	0.308542i
$487$	1020.47	+	1020.47i	0.0949523	+	0.0949523i	0.752987	−	0.658035i	$-0.228612\pi$
$487$	1020.47	+	1020.47i	0.0949523	+	0.0949523i	−0.658035	+	0.752987i	$0.728612\pi$
$488$	−10423.8	−	10423.8i	−0.966932	−	0.966932i
$489$	2289.70	+	1794.85i	0.211746	+	0.165983i
$490$		0			0
$491$		−	7350.62i		−	0.675619i	−0.941215	−	0.337809i	$-0.890314\pi$
$491$		−	7350.62i		−	0.675619i	0.941215	−	0.337809i	$-0.109686\pi$
$492$	86.8192	+	716.604i	0.00795551	+	0.0656646i
$493$	10292.4	−	10292.4i	0.940261	−	0.940261i
$494$	2528.21			0.230262
$495$		0			0
$496$	1190.12			0.107738
$497$	14244.5	−	14244.5i	1.28562	−	1.28562i
$498$	−2312.52	−	19087.5i	−0.208085	−	1.71753i
$499$			15309.0i			1.37339i	0.726944	+	0.686697i	$0.240940\pi$
$499$			15309.0i			1.37339i	−0.726944	+	0.686697i	$0.759060\pi$
$500$		0			0
$501$	5061.44	+	3967.55i	0.451354	+	0.353806i
$502$	−172.661	−	172.661i	−0.0153511	−	0.0153511i
$503$	738.534	+	738.534i	0.0654664	+	0.0654664i	0.739082	−	0.673616i	$-0.235260\pi$
$503$	738.534	+	738.534i	0.0654664	+	0.0654664i	−0.673616	+	0.739082i	$0.735260\pi$
$504$	6142.00	+	24975.9i	0.542830	+	2.20737i
$505$		0			0
$506$		−	3019.43i		−	0.265277i
$507$	−6142.59	+	744.197i	−0.538071	+	0.0651892i
$508$	150.540	−	150.540i	0.0131479	−	0.0131479i
$509$	4477.48			0.389903			0.194952	−	0.980813i	$-0.437545\pi$
$509$	4477.48			0.389903			0.194952	+	0.980813i	$0.437545\pi$
$510$		0			0
$511$	26131.8			2.26224
$512$	10358.5	−	10358.5i	0.894109	−	0.894109i
$513$	−2175.77	+	823.186i	−0.187256	+	0.0708471i
$514$			29035.2i			2.49161i
$515$		0			0
$516$	−4334.71	+	5529.84i	−0.369816	+	0.471778i
$517$	−3527.27	−	3527.27i	−0.300056	−	0.300056i
$518$	−17509.8	−	17509.8i	−1.48520	−	1.48520i
$519$	−4753.00	+	6063.46i	−0.401992	+	0.512825i
$520$		0			0
$521$		−	14580.8i		−	1.22610i	−0.790045	−	0.613049i	$-0.789943\pi$
$521$		−	14580.8i		−	1.22610i	0.790045	−	0.613049i	$-0.210057\pi$
$522$	−13661.6	+	22572.1i	−1.14550	+	1.89263i
$523$	2457.46	−	2457.46i	0.205463	−	0.205463i	−0.596873	−	0.802336i	$-0.703590\pi$
$523$	2457.46	−	2457.46i	0.205463	−	0.205463i	0.802336	+	0.596873i	$0.203590\pi$
$524$	−40226.8			−3.35366
$525$		0			0
$526$	7829.24			0.648994
$527$	1391.06	−	1391.06i	0.114982	−	0.114982i
$528$	−2054.93	+	248.962i	−0.169373	+	0.0205202i
$529$		−	7502.02i		−	0.616587i
$530$		0			0
$531$	8249.13	−	2028.60i	0.674165	−	0.165789i
$532$	4940.00	+	4940.00i	0.402587	+	0.402587i
$533$	206.291	+	206.291i	0.0167644	+	0.0167644i
$534$	11084.8	+	8689.11i	0.898288	+	0.704147i
$535$		0			0
$536$			31377.1i			2.52852i
$537$	131.598	+	1086.21i	0.0105752	+	0.0872876i
$538$	−5725.26	+	5725.26i	−0.458798	+	0.458798i
$539$	4001.49			0.319770
$540$		0			0
$541$	375.804			0.0298652			0.0149326	−	0.999889i	$-0.495247\pi$
$541$	375.804			0.0298652			0.0149326	+	0.999889i	$0.495247\pi$
$542$	11244.7	−	11244.7i	0.891146	−	0.891146i
$543$	−80.3909	−	663.545i	−0.00635341	−	0.0524409i
$544$		−	4654.11i		−	0.366808i
$545$		0			0
$546$	17393.1	+	13634.0i	1.36329	+	1.06865i
$547$	3656.92	+	3656.92i	0.285848	+	0.285848i	0.835436	−	0.549588i	$-0.185215\pi$
$547$	3656.92	+	3656.92i	0.285848	+	0.285848i	−0.549588	+	0.835436i	$0.685215\pi$
$548$	5307.72	+	5307.72i	0.413749	+	0.413749i
$549$	11317.7	−	2783.22i	0.879834	−	0.216366i
$550$		0			0
$551$			3370.94i			0.260630i
$552$	12032.0	−	1457.72i	0.927746	−	0.112400i
$553$	−4079.26	+	4079.26i	−0.313685	+	0.313685i
$554$	14809.2			1.13571
$555$		0			0
$556$	−26252.6			−2.00244
$557$	−4512.51	+	4512.51i	−0.343270	+	0.343270i	−0.857595	−	0.514325i	$-0.828042\pi$
$557$	−4512.51	+	4512.51i	−0.343270	+	0.343270i	0.514325	+	0.857595i	$0.328042\pi$
$558$	−1846.42	+	3050.70i	−0.140081	+	0.231446i
$559$			2839.74i			0.214862i
$560$		0			0
$561$	−2110.89	+	2692.88i	−0.158862	+	0.202662i
$562$	16899.5	+	16899.5i	1.26844	+	1.26844i
$563$	−15527.9	−	15527.9i	−1.16238	−	1.16238i	−0.983953	−	0.178430i	$-0.942898\pi$
$563$	−15527.9	−	15527.9i	−1.16238	−	1.16238i	−0.178430	−	0.983953i	$-0.557102\pi$
$564$	26262.2	−	33502.9i	1.96070	−	2.50129i
$565$		0			0
$566$			22606.7i			1.67885i
$567$	−19407.7	−	6070.20i	−1.43747	−	0.449603i
$568$	−17439.3	+	17439.3i	−1.28827	+	1.28827i
$569$	−12965.1			−0.955230			−0.477615	−	0.878569i	$-0.658499\pi$
$569$	−12965.1			−0.955230			−0.477615	+	0.878569i	$0.658499\pi$
$570$		0			0
$571$	−6452.24			−0.472886			−0.236443	−	0.971645i	$-0.575982\pi$
$571$	−6452.24			−0.472886			−0.236443	+	0.971645i	$0.575982\pi$
$572$	−3115.96	+	3115.96i	−0.227771	+	0.227771i
$573$	22137.5	−	2682.04i	1.61397	−	0.195539i
$574$			1233.14i			0.0896695i
$575$		0			0
$576$	4248.79	+	17277.4i	0.307349	+	1.24981i
$577$	−11704.9	−	11704.9i	−0.844508	−	0.844508i	0.144934	−	0.989441i	$-0.453703\pi$
$577$	−11704.9	−	11704.9i	−0.844508	−	0.844508i	−0.989441	+	0.144934i	$0.953703\pi$
$578$	724.794	+	724.794i	0.0521582	+	0.0521582i
$579$	−20629.4	−	16170.9i	−1.48070	−	1.16069i
$580$		0			0
$581$		−	21473.1i		−	1.53331i
$582$	−280.424	−	2314.62i	−0.0199725	−	0.164852i
$583$	−4057.22	+	4057.22i	−0.288221	+	0.288221i
$584$	−31992.7			−2.26690
$585$		0			0
$586$	23610.2			1.66438
$587$	−6441.65	+	6441.65i	−0.452939	+	0.452939i	−0.896329	−	0.443390i	$-0.853776\pi$
$587$	−6441.65	+	6441.65i	−0.452939	+	0.452939i	0.443390	+	0.896329i	$0.353776\pi$
$588$	4107.12	+	33900.1i	0.288053	+	2.37758i
$589$			455.596i			0.0318718i
$590$		0			0
$591$	2435.20	+	1908.90i	0.169494	+	0.132862i
$592$	5656.48	+	5656.48i	0.392703	+	0.392703i
$593$	12555.8	+	12555.8i	0.869487	+	0.869487i	0.992416	−	0.122928i	$-0.0392285\pi$
$593$	12555.8	+	12555.8i	0.869487	+	0.869487i	−0.122928	+	0.992416i	$0.539228\pi$
$594$	2549.95	−	5653.76i	0.176137	−	0.390533i
$595$		0			0
$596$		−	2778.38i		−	0.190951i
$597$	−15599.4	+	1889.93i	−1.06942	+	0.129564i
$598$	7363.85	−	7363.85i	0.503563	−	0.503563i
$599$	−24700.6			−1.68487			−0.842437	−	0.538795i	$-0.818880\pi$
$599$	−24700.6			−1.68487			−0.842437	+	0.538795i	$0.818880\pi$
$600$		0			0
$601$	−22487.6			−1.52627			−0.763135	−	0.646239i	$-0.776341\pi$
$601$	−22487.6			−1.52627			−0.763135	+	0.646239i	$0.776341\pi$
$602$	−8487.52	+	8487.52i	−0.574627	+	0.574627i
$603$	−21222.9	−	12845.0i	−1.43327	−	0.867480i
$604$			12163.4i			0.819407i
$605$		0			0
$606$	17080.0	−	21789.1i	1.14493	−	1.46060i
$607$	−13817.0	−	13817.0i	−0.923912	−	0.923912i	0.0733910	−	0.997303i	$-0.476618\pi$
$607$	−13817.0	−	13817.0i	−0.923912	−	0.923912i	−0.997303	+	0.0733910i	$0.976618\pi$
$608$	762.148	+	762.148i	0.0508375	+	0.0508375i
$609$	−18178.7	+	23190.7i	−1.20959	+	1.54308i
$610$		0			0
$611$		−	17204.8i		−	1.13917i
$612$	−24980.4	−	15119.2i	−1.64995	−	0.998624i
$613$	−7861.86	+	7861.86i	−0.518005	+	0.518005i	−0.916967	−	0.398962i	$-0.869371\pi$
$613$	−7861.86	+	7861.86i	−0.518005	+	0.518005i	0.398962	+	0.916967i	$0.369371\pi$
$614$	27910.9			1.83451
$615$		0			0
$616$	−8761.08			−0.573042
$617$	17057.5	−	17057.5i	1.11298	−	1.11298i	0.120233	−	0.992746i	$-0.461636\pi$
$617$	17057.5	−	17057.5i	1.11298	−	1.11298i	0.992746	−	0.120233i	$-0.0383641\pi$
$618$	−28859.1	+	3496.39i	−1.87845	+	0.227582i
$619$			24671.5i			1.60199i	0.598671	+	0.800995i	$0.295696\pi$
$619$			24671.5i			1.60199i	−0.598671	+	0.800995i	$0.704304\pi$
$620$		0			0
$621$	−3939.64	+	8734.99i	−0.254577	+	0.564450i
$622$	13382.1	+	13382.1i	0.862657	+	0.862657i
$623$	11122.6	+	11122.6i	0.715279	+	0.715279i
$624$	−5618.78	−	4404.43i	−0.360467	−	0.282561i
$625$		0			0
$626$			1091.44i			0.0696847i
$627$	−95.3062	−	786.656i	−0.00607044	−	0.0501053i
$628$	24568.7	−	24568.7i	1.56114	−	1.56114i
$629$	13223.1			0.838216
$630$		0			0
$631$	13039.4			0.822648			0.411324	−	0.911489i	$-0.365066\pi$
$631$	13039.4			0.822648			0.411324	+	0.911489i	$0.365066\pi$
$632$	4994.17	−	4994.17i	0.314331	−	0.314331i
$633$	−1800.75	−	14863.4i	−0.113070	−	0.933280i
$634$		−	43564.9i		−	2.72900i
$635$		0			0
$636$	−38536.6	−	30207.9i	−2.40263	−	1.88337i
$637$	9758.93	+	9758.93i	0.607006	+	0.607006i
$638$	−6355.05	−	6355.05i	−0.394355	−	0.394355i
$639$	−4656.41	−	18934.9i	−0.288270	−	1.17223i
$640$		0			0
$641$		−	7413.70i		−	0.456823i	−0.973565	−	0.228412i	$-0.926647\pi$
$641$		−	7413.70i		−	0.456823i	0.973565	−	0.228412i	$-0.0733532\pi$
$642$	32222.7	−	3903.90i	1.98088	−	0.239991i
$643$	−3856.84	+	3856.84i	−0.236546	+	0.236546i	−0.815418	−	0.578872i	$-0.803493\pi$
$643$	−3856.84	+	3856.84i	−0.236546	+	0.236546i	0.578872	+	0.815418i	$0.303493\pi$
$644$	28777.3			1.76084
$645$		0			0
$646$	−5706.46			−0.347551
$647$	−762.540	+	762.540i	−0.0463347	+	0.0463347i	−0.729894	−	0.683560i	$-0.760431\pi$
$647$	−762.540	+	762.540i	−0.0463347	+	0.0463347i	0.683560	+	0.729894i	$0.260431\pi$
$648$	23760.5	+	7431.65i	1.44043	+	0.450529i
$649$			2893.64i			0.175016i
$650$		0			0
$651$	−2456.92	+	3134.32i	−0.147918	+	0.188700i
$652$	−5980.10	−	5980.10i	−0.359200	−	0.359200i
$653$	18549.1	+	18549.1i	1.11161	+	1.11161i	0.992933	+	0.118679i	$0.0378661\pi$
$653$	18549.1	+	18549.1i	1.11161	+	1.11161i	0.118679	+	0.992933i	$0.462134\pi$
$654$	73.2641	−	93.4637i	0.00438051	−	0.00558826i
$655$		0			0
$656$		−	398.363i		−	0.0237095i
$657$	13097.1	−	21639.3i	0.777725	−	1.28498i
$658$	51422.3	−	51422.3i	3.04658	−	3.04658i
$659$	27210.3			1.60844			0.804222	−	0.594329i	$-0.202582\pi$
$659$	27210.3			1.60844			0.804222	+	0.594329i	$0.202582\pi$
$660$		0			0
$661$	17912.8			1.05405			0.527026	−	0.849849i	$-0.323307\pi$
$661$	17912.8			1.05405			0.527026	+	0.849849i	$0.323307\pi$
$662$	−29933.3	+	29933.3i	−1.75739	+	1.75739i
$663$	−11715.5	+	1419.38i	−0.686266	+	0.0831436i
$664$			26289.1i			1.53647i
$665$		0			0
$666$	−23275.3	+	5723.80i	−1.35421	+	0.333022i
$667$	9818.47	+	9818.47i	0.569974	+	0.569974i
$668$	−13219.1	−	13219.1i	−0.765664	−	0.765664i
$669$	7720.24	+	6051.72i	0.446161	+	0.349735i
$670$		0			0
$671$			3970.05i			0.228408i
$672$	1133.19	+	9353.36i	0.0650504	+	0.536925i
$673$	13447.5	−	13447.5i	0.770225	−	0.770225i	−0.207921	−	0.978146i	$-0.566670\pi$
$673$	13447.5	−	13447.5i	0.770225	−	0.770225i	0.978146	+	0.207921i	$0.0666696\pi$
$674$	38681.8			2.21063
$675$		0			0
$676$	17986.4			1.02335
$677$	−14818.2	+	14818.2i	−0.841228	+	0.841228i	−0.989019	−	0.147791i	$-0.952784\pi$
$677$	−14818.2	+	14818.2i	−0.841228	+	0.841228i	0.147791	+	0.989019i	$0.452784\pi$
$678$	3360.92	+	27741.0i	0.190377	+	1.57137i
$679$		−	2603.90i		−	0.147170i
$680$		0			0
$681$	−1950.96	−	1529.32i	−0.109781	−	0.0860550i
$682$	−858.909	−	858.909i	−0.0482248	−	0.0482248i
$683$	−10932.8	−	10932.8i	−0.612493	−	0.612493i	0.331102	−	0.943595i	$-0.392579\pi$
$683$	−10932.8	−	10932.8i	−0.612493	−	0.612493i	−0.943595	+	0.331102i	$0.892579\pi$
$684$	6566.63	−	1614.85i	0.367078	−	0.0902707i
$685$		0			0
$686$			12346.4i			0.687156i
$687$	23603.5	−	2859.65i	1.31082	−	0.158810i
$688$	2741.87	−	2741.87i	0.151937	−	0.151937i
$689$	−19789.7			−1.09423
$690$		0			0
$691$	−6859.09			−0.377615			−0.188807	−	0.982014i	$-0.560462\pi$
$691$	−6859.09			−0.377615			−0.188807	+	0.982014i	$0.560462\pi$
$692$	15836.1	−	15836.1i	0.869942	−	0.869942i
$693$	3586.58	−	5925.84i	0.196599	−	0.324826i
$694$			33818.4i			1.84976i
$695$		0			0
$696$	22255.9	−	28392.1i	1.21208	−	1.54626i
$697$	−465.623	−	465.623i	−0.0253038	−	0.0253038i
$698$	22727.3	+	22727.3i	1.23244	+	1.23244i
$699$	2401.30	−	3063.36i	0.129936	−	0.165761i
$700$		0			0
$701$			26912.7i			1.45004i	0.688727	+	0.725021i	$0.258170\pi$
$701$			26912.7i			1.45004i	−0.688727	+	0.725021i	$0.741830\pi$
$702$	20007.4	−	7569.65i	1.07568	−	0.406977i
$703$	−2165.38	+	2165.38i	−0.116172	+	0.116172i
$704$	−6060.57			−0.324455
$705$		0			0
$706$	13772.1			0.734162
$707$	21863.5	−	21863.5i	1.16303	−	1.16303i
$708$	−24514.6	+	2970.03i	−1.30129	+	0.157656i
$709$		−	12724.4i		−	0.674011i	−0.941503	−	0.337005i	$-0.890586\pi$
$709$		−	12724.4i		−	0.674011i	0.941503	−	0.337005i	$-0.109414\pi$
$710$		0			0
$711$	1333.48	+	5422.46i	0.0703365	+	0.286017i
$712$	−13617.3	−	13617.3i	−0.716753	−	0.716753i
$713$	1327.00	+	1327.00i	0.0697009	+	0.0697009i
$714$	−39258.2	−	30773.6i	−2.05771	−	1.61299i
$715$		0			0
$716$		−	3180.59i		−	0.166012i
$717$	544.508	+	4494.36i	0.0283613	+	0.234093i
$718$	−24398.2	+	24398.2i	−1.26815	+	1.26815i
$719$	1776.43			0.0921413			0.0460707	−	0.998938i	$-0.485330\pi$
$719$	1776.43			0.0921413			0.0460707	+	0.998938i	$0.485330\pi$
$720$		0			0
$721$	−32466.0			−1.67697
$722$	−22378.4	+	22378.4i	−1.15351	+	1.15351i
$723$	26.8247	+	221.411i	0.00137984	+	0.0113892i
$724$			1942.96i			0.0997370i
$725$		0			0
$726$	−24500.8	−	19205.6i	−1.25249	−	0.981800i
$727$	11790.0	+	11790.0i	0.601465	+	0.601465i	0.940701	−	0.339236i	$-0.110169\pi$
$727$	11790.0	+	11790.0i	0.601465	+	0.601465i	−0.339236	+	0.940701i	$0.610169\pi$
$728$	−21366.7	−	21366.7i	−1.08778	−	1.08778i
$729$	−14753.6	+	13028.9i	−0.749562	+	0.661934i
$730$		0			0
$731$		−	6409.63i		−	0.324307i
$732$	−33633.7	+	4074.85i	−1.69828	+	0.205753i
$733$	−22491.0	+	22491.0i	−1.13332	+	1.13332i	−0.143696	+	0.989622i	$0.545899\pi$
$733$	−22491.0	+	22491.0i	−1.13332	+	1.13332i	−0.989622	+	0.143696i	$0.954101\pi$
$734$	−34870.0			−1.75351
$735$		0			0
$736$	4439.78			0.222354
$737$	5975.20	−	5975.20i	0.298642	−	0.298642i
$738$	1021.14	+	618.041i	0.0509334	+	0.0308271i
$739$			9723.29i			0.484002i	0.970276	+	0.242001i	$0.0778037\pi$
$739$			9723.29i			0.484002i	−0.970276	+	0.242001i	$0.922196\pi$
$740$		0			0
$741$	1686.08	−	2150.95i	0.0835894	−	0.106636i
$742$	−59148.2	−	59148.2i	−2.92641	−	2.92641i
$743$	3640.61	+	3640.61i	0.179759	+	0.179759i	0.791251	−	0.611492i	$-0.209430\pi$
$743$	3640.61	+	3640.61i	0.179759	+	0.179759i	−0.611492	+	0.791251i	$0.709430\pi$
$744$	3007.97	−	3837.30i	0.148222	−	0.189089i
$745$		0			0
$746$			9059.85i			0.444645i
$747$	−17781.5	−	10762.1i	−0.870939	−	0.527130i
$748$	7033.09	−	7033.09i	0.343790	−	0.343790i
$749$	36249.9			1.76841
$750$		0			0
$751$	−39901.6			−1.93879			−0.969395	−	0.245508i	$-0.921045\pi$
$751$	−39901.6			−1.93879			−0.969395	+	0.245508i	$0.921045\pi$
$752$	−16611.8	+	16611.8i	−0.805546	+	0.805546i
$753$	−262.045	+	31.7477i	−0.0126819	+	0.00153646i
$754$		−	30997.7i		−	1.49717i
$755$		0			0
$756$	53884.3	+	24302.8i	2.59227	+	1.16916i
$757$	19152.6	+	19152.6i	0.919568	+	0.919568i	0.996998	−	0.0774296i	$-0.0246713\pi$
$757$	19152.6	+	19152.6i	0.919568	+	0.919568i	−0.0774296	+	0.996998i	$0.524671\pi$
$758$	19687.2	+	19687.2i	0.943368	+	0.943368i
$759$	−2568.87	−	2013.68i	−0.122851	−	0.0963003i
$760$		0			0
$761$		−	28330.4i		−	1.34951i	−0.738043	−	0.674753i	$-0.764250\pi$
$761$		−	28330.4i		−	1.34951i	0.738043	−	0.674753i	$-0.235750\pi$
$762$	−42.3408	−	349.480i	−0.00201292	−	0.0166146i
$763$	93.7828	−	93.7828i	0.00444976	−	0.00444976i
$764$	−64821.9			−3.06960
$765$		0			0
$766$	34147.2			1.61069
$767$	−7057.08	+	7057.08i	−0.332225	+	0.332225i
$768$	−4658.38	−	38450.2i	−0.218874	−	1.80658i
$769$		−	12352.6i		−	0.579256i	−0.957139	−	0.289628i	$-0.906468\pi$
$769$		−	12352.6i		−	0.579256i	0.957139	−	0.289628i	$-0.0935316\pi$
$770$		0			0
$771$	24702.6	+	19363.8i	1.15388	+	0.904499i
$772$	53878.4	+	53878.4i	2.51182	+	2.51182i
$773$	−12713.0	−	12713.0i	−0.591532	−	0.591532i	0.346513	−	0.938045i	$-0.387366\pi$
$773$	−12713.0	−	12713.0i	−0.591532	−	0.591532i	−0.938045	+	0.346513i	$0.887366\pi$
$774$	2774.50	+	11282.3i	0.128847	+	0.523944i
$775$		0			0
$776$			3187.91i			0.147474i
$777$	−26574.4	+	3219.58i	−1.22696	+	0.148651i
$778$	−34664.5	+	34664.5i	−1.59741	+	1.59741i
$779$	152.499			0.00701392
$780$		0			0
$781$	6642.00			0.304314
$782$	−16621.1	+	16621.1i	−0.760063	+	0.760063i
$783$	10092.9	+	26676.5i	0.460651	+	1.21755i
$784$		−	18845.2i		−	0.858473i
$785$		0			0
$786$	−41036.4	+	52350.6i	−1.86224	+	2.37568i
$787$	11888.6	+	11888.6i	0.538481	+	0.538481i	0.923083	−	0.384602i	$-0.125661\pi$
$787$	11888.6	+	11888.6i	0.538481	+	0.538481i	−0.384602	+	0.923083i	$0.625661\pi$
$788$	−6360.09	−	6360.09i	−0.287524	−	0.287524i
$789$	5221.38	−	6660.97i	0.235597	−	0.300554i
$790$		0			0
$791$			31208.1i			1.40282i
$792$	−4390.99	+	7254.91i	−0.197004	+	0.325495i
$793$	−9682.25	+	9682.25i	−0.433577	+	0.433577i
$794$	−613.391			−0.0274162
$795$		0			0
$796$	45677.6			2.03392
$797$	8060.57	−	8060.57i	0.358243	−	0.358243i	−0.504922	−	0.863165i	$-0.668479\pi$
$797$	8060.57	−	8060.57i	0.358243	−	0.358243i	0.863165	+	0.504922i	$0.168479\pi$
$798$	11468.3	−	1389.42i	0.508737	−	0.0616354i
$799$			38833.2i			1.71942i
$800$		0			0
$801$	14785.1	−	3635.90i	0.652191	−	0.160385i
$802$	2031.88	+	2031.88i	0.0894616	+	0.0894616i
$803$	6092.44	+	6092.44i	0.267743	+	0.267743i
$804$	56754.1	+	44488.2i	2.48951	+	1.95147i
$805$		0			0
$806$		−	4189.46i		−	0.183086i
$807$	1052.72	+	8689.16i	0.0459202	+	0.379025i
$808$	−26767.1	+	26767.1i	−1.16542	+	1.16542i
$809$	−13824.2			−0.600782			−0.300391	−	0.953816i	$-0.597117\pi$
$809$	−13824.2			−0.600782			−0.300391	+	0.953816i	$0.597117\pi$
$810$		0			0
$811$	−20584.0			−0.891249			−0.445625	−	0.895220i	$-0.647018\pi$
$811$	−20584.0			−0.891249			−0.445625	+	0.895220i	$0.647018\pi$
$812$	60568.1	−	60568.1i	2.61764	−	2.61764i
$813$	−2067.60	−	17065.9i	−0.0891931	−	0.736198i
$814$		−	8164.55i		−	0.351557i
$815$		0			0
$816$	12682.3	+	9941.32i	0.544078	+	0.426490i
$817$	1049.63	+	1049.63i	0.0449472	+	0.0449472i
$818$	−36772.6	−	36772.6i	−1.57179	−	1.57179i
$819$	23199.1	−	5705.06i	0.989796	−	0.243408i
$820$		0			0
$821$		−	7269.14i		−	0.309007i	−0.987992	−	0.154503i	$-0.950622\pi$
$821$		−	7269.14i		−	0.309007i	0.987992	−	0.154503i	$-0.0493778\pi$
$822$	12321.9	−	1492.85i	0.522843	−	0.0633443i
$823$	22295.9	−	22295.9i	0.944334	−	0.944334i	−0.0541961	−	0.998530i	$-0.517260\pi$
$823$	22295.9	−	22295.9i	0.944334	−	0.944334i	0.998530	+	0.0541961i	$0.0172596\pi$
$824$	39747.6			1.68043
$825$		0			0
$826$	−42185.0			−1.77700
$827$	4878.42	−	4878.42i	0.205126	−	0.205126i	−0.597066	−	0.802192i	$-0.703667\pi$
$827$	4878.42	−	4878.42i	0.205126	−	0.205126i	0.802192	+	0.597066i	$0.203667\pi$
$828$	14423.0	−	23830.0i	0.605353	−	1.00018i
$829$		−	28818.4i		−	1.20736i	−0.797225	−	0.603682i	$-0.793700\pi$
$829$		−	28818.4i		−	1.20736i	0.797225	−	0.603682i	$-0.206300\pi$
$830$		0			0
$831$	9876.38	−	12599.4i	0.412284	−	0.525955i
$832$	−14780.7	−	14780.7i	−0.615898	−	0.615898i
$833$	−22027.1	−	22027.1i	−0.916197	−	0.916197i
$834$	−26780.9	+	34164.7i	−1.11193	+	1.41850i
$835$		0			0
$836$			2303.45i			0.0952948i
$837$	1364.09	+	3605.44i	0.0563320	+	0.148891i
$838$	36877.1	−	36877.1i	1.52016	−	1.52016i
$839$	−19129.2			−0.787144			−0.393572	−	0.919294i	$-0.628761\pi$
$839$	−19129.2			−0.787144			−0.393572	+	0.919294i	$0.628761\pi$
$840$		0			0
$841$	16941.2			0.694627
$842$	41945.3	−	41945.3i	1.71678	−	1.71678i
$843$	25648.2	−	3107.38i	1.04789	−	0.126956i
$844$			43522.3i			1.77500i
$845$		0			0
$846$	−16809.5	−	68354.5i	−0.683125	−	2.77787i
$847$	−24584.4	−	24584.4i	−0.997321	−	0.997321i
$848$	19107.6	+	19107.6i	0.773773	+	0.773773i
$849$	19233.3	+	15076.6i	0.777487	+	0.609454i
$850$		0			0
$851$			12614.1i			0.508116i
$852$	6817.34	+	56270.2i	0.274129	+	2.26266i
$853$	−415.608	+	415.608i	−0.0166825	+	0.0166825i	−0.715399	−	0.698716i	$-0.753755\pi$
$853$	−415.608	+	415.608i	−0.0166825	+	0.0166825i	0.698716	+	0.715399i	$0.253755\pi$
$854$	−57877.4			−2.31911
$855$		0			0
$856$	−44380.2			−1.77206
$857$	−21659.9	+	21659.9i	−0.863347	+	0.863347i	−0.991725	−	0.128378i	$-0.959023\pi$
$857$	−21659.9	+	21659.9i	−0.863347	+	0.863347i	0.128378	+	0.991725i	$0.459023\pi$
$858$	876.393	+	7233.74i	0.0348713	+	0.287827i
$859$		−	20261.0i		−	0.804769i	−0.915471	−	0.402384i	$-0.868182\pi$
$859$		−	20261.0i		−	0.804769i	0.915471	−	0.402384i	$-0.131818\pi$
$860$		0			0
$861$	1049.13	+	822.391i	0.0415265	+	0.0325517i
$862$	−36191.2	−	36191.2i	−1.43002	−	1.43002i
$863$	23444.8	+	23444.8i	0.924762	+	0.924762i	0.997361	−	0.0725989i	$-0.0231293\pi$
$863$	23444.8	+	23444.8i	0.924762	+	0.924762i	−0.0725989	+	0.997361i	$0.523129\pi$
$864$	8313.32	+	3749.46i	0.327344	+	0.147638i
$865$		0			0
$866$		−	73586.9i		−	2.88751i
$867$	1100.01	−	133.270i	0.0430892	−	0.00522041i
$868$	8186.01	−	8186.01i	0.320105	−	0.320105i
$869$	−1902.10			−0.0742512
$870$		0			0
$871$	29144.9			1.13380
$872$	−114.817	+	114.817i	−0.00445893	+	0.00445893i
$873$	−2156.25	−	1305.06i	−0.0835945	−	0.0505950i
$874$		−	5443.68i		−	0.210681i
$875$		0			0
$876$	−45361.1	+	57867.6i	−1.74955	+	2.23193i
$877$	11791.4	+	11791.4i	0.454012	+	0.454012i	0.896684	−	0.442672i	$-0.145969\pi$
$877$	11791.4	+	11791.4i	0.454012	+	0.454012i	−0.442672	+	0.896684i	$0.645969\pi$
$878$	−5302.95	−	5302.95i	−0.203834	−	0.203834i
$879$	15745.8	−	20087.1i	0.604201	−	0.770786i
$880$		0			0
$881$			30662.9i			1.17260i	0.810095	+	0.586299i	$0.199416\pi$
$881$			30662.9i			1.17260i	−0.810095	+	0.586299i	$0.800584\pi$
$882$	48306.9	+	29237.5i	1.84419	+	1.11619i
$883$	−1717.16	+	1717.16i	−0.0654439	+	0.0654439i	−0.739071	−	0.673627i	$-0.764735\pi$
$883$	−1717.16	+	1717.16i	−0.0654439	+	0.0654439i	0.673627	+	0.739071i	$0.264735\pi$
$884$	34304.9			1.30520
$885$		0			0
$886$	17598.8			0.667318
$887$	−13438.0	+	13438.0i	−0.508687	+	0.508687i	−0.914123	−	0.405436i	$-0.867120\pi$
$887$	−13438.0	+	13438.0i	−0.508687	+	0.508687i	0.405436	+	0.914123i	$0.367120\pi$
$888$	32534.6	−	3941.68i	1.22949	−	0.148957i
$889$		−	393.159i		−	0.0148325i
$890$		0			0
$891$	−3109.53	−	5939.98i	−0.116917	−	0.223341i
$892$	−20163.2	−	20163.2i	−0.756854	−	0.756854i
$893$	−6359.25	−	6359.25i	−0.238303	−	0.238303i
$894$	−3615.74	−	2834.30i	−0.135267	−	0.106033i
$895$		0			0
$896$		−	73848.4i		−	2.75346i
$897$	−1354.02	−	11176.0i	−0.0504006	−	0.416006i
$898$	−48574.4	+	48574.4i	−1.80506	+	1.80506i
$899$	5585.94			0.207232
$900$		0			0
$901$	44667.6			1.65160
$902$	−287.498	+	287.498i	−0.0106127	+	0.0106127i
$903$	1560.63	+	12881.4i	0.0575133	+	0.474714i
$904$		−	38207.5i		−	1.40571i
$905$		0			0
$906$	15829.3	+	12408.2i	0.580455	+	0.455006i
$907$	14586.4	+	14586.4i	0.533994	+	0.533994i	0.921759	−	0.387764i	$-0.126753\pi$
$907$	14586.4	+	14586.4i	0.533994	+	0.533994i	−0.387764	+	0.921759i	$0.626753\pi$
$908$	5095.40	+	5095.40i	0.186230	+	0.186230i
$909$	−7146.99	−	29062.6i	−0.260782	−	1.06045i
$910$		0			0
$911$		−	27679.6i		−	1.00666i	−0.864095	−	0.503329i	$-0.832108\pi$
$911$		−	27679.6i		−	1.00666i	0.864095	−	0.503329i	$-0.167892\pi$
$912$	−3704.79	+	448.849i	−0.134515	+	0.0162970i
$913$	5006.29	−	5006.29i	0.181472	−	0.181472i
$914$	53896.6			1.95048
$915$		0			0
$916$	−69114.7			−2.49303
$917$	−52529.3	+	52529.3i	−1.89168	+	1.89168i
$918$	−45159.1	+	17085.6i	−1.62361	+	0.614280i
$919$		−	44894.9i		−	1.61147i	−0.592273	−	0.805737i	$-0.701769\pi$
$919$		−	44894.9i		−	1.61147i	0.592273	−	0.805737i	$-0.298231\pi$
$920$		0			0
$921$	18614.0	−	23746.1i	0.665963	−	0.849576i
$922$	−36319.1	−	36319.1i	−1.29730	−	1.29730i
$923$	16198.7	+	16198.7i	0.577666	+	0.577666i
$924$	−12422.0	+	15846.8i	−0.442264	+	0.564201i
$925$		0			0
$926$			61584.6i			2.18553i
$927$	−16271.7	+	26884.6i	−0.576519	+	0.952541i
$928$	9344.50	−	9344.50i	0.330548	−	0.330548i
$929$	−29033.0			−1.02534			−0.512670	−	0.858586i	$-0.671344\pi$
$929$	−29033.0			−1.02534			−0.512670	+	0.858586i	$0.671344\pi$
$930$		0			0
$931$	7214.22			0.253960
$932$	−8000.68	+	8000.68i	−0.281192	+	0.281192i
$933$	20309.8	−	2460.61i	0.712662	−	0.0863416i
$934$		−	65890.5i		−	2.30835i
$935$		0			0
$936$	−28402.3	+	6984.61i	−0.991836	+	0.243909i
$937$	−35681.0	−	35681.0i	−1.24402	−	1.24402i	−0.958318	−	0.285703i	$-0.907773\pi$
$937$	−35681.0	−	35681.0i	−1.24402	−	1.24402i	−0.285703	−	0.958318i	$-0.592227\pi$
$938$	87109.6	+	87109.6i	3.03223	+	3.03223i
$939$	928.574	+	727.888i	0.0322714	+	0.0252968i
$940$		0			0
$941$			28986.4i			1.00418i	0.864816	+	0.502089i	$0.167435\pi$
$941$			28986.4i			1.00418i	−0.864816	+	0.502089i	$0.832565\pi$
$942$	−6910.19	−	57036.6i	−0.239009	−	1.97277i
$943$	444.181	−	444.181i	0.0153388	−	0.0153388i
$944$	13627.7			0.469857
$945$		0			0
$946$	−3957.61			−0.136018
$947$	10086.1	−	10086.1i	0.346099	−	0.346099i	−0.512555	−	0.858654i	$-0.671301\pi$
$947$	10086.1	−	10086.1i	0.346099	−	0.346099i	0.858654	+	0.512555i	$0.171301\pi$
$948$	−1952.31	−	16114.3i	−0.0668862	−	0.552078i
$949$			29716.8i			1.01649i
$950$		0			0
$951$	−37064.2	−	29053.8i	−1.26382	−	0.990676i
$952$	48227.3	+	48227.3i	1.64186	+	1.64186i
$953$	−35251.1	−	35251.1i	−1.19821	−	1.19821i	−0.974702	−	0.223509i	$-0.928249\pi$
$953$	−35251.1	−	35251.1i	−1.19821	−	1.19821i	−0.223509	−	0.974702i	$-0.571751\pi$
$954$	−78624.3	+	19335.1i	−2.66830	+	0.656180i
$955$		0			0
$956$		−	13160.2i		−	0.445220i
$957$	−9644.98	+	1168.52i	−0.325787	+	0.0394703i
$958$	17653.0	−	17653.0i	0.595348	−	0.595348i
$959$	13861.9			0.466763
$960$		0			0
$961$	−29036.0			−0.974658
$962$	19911.9	−	19911.9i	0.667344	−	0.667344i
$963$	18168.2	−	30018.0i	0.607956	−	1.00448i
$964$		−	648.325i		−	0.0216610i
$965$		0			0
$966$	29356.5	−	37450.3i	0.977773	−	1.24735i
$967$	29437.7	+	29437.7i	0.978958	+	0.978958i	0.999783	−	0.0208251i	$-0.00662933\pi$
$967$	29437.7	+	29437.7i	0.978958	+	0.978958i	−0.0208251	+	0.999783i	$0.506629\pi$
$968$	30098.3	+	30098.3i	0.999377	+	0.999377i
$969$	−3805.68	+	4854.95i	−0.126167	+	0.160953i
$970$		0			0
$971$			14117.9i			0.466595i	0.972405	+	0.233298i	$0.0749516\pi$
$971$			14117.9i			0.466595i	−0.972405	+	0.233298i	$0.925048\pi$
$972$	47131.2	−	32440.4i	1.55528	−	1.07050i
$973$	−34281.4	+	34281.4i	−1.12951	+	1.12951i
$974$	−6936.87			−0.228205
$975$		0			0
$976$	18697.1			0.613197
$977$	18396.9	−	18396.9i	0.602423	−	0.602423i	−0.338532	−	0.940955i	$-0.609930\pi$
$977$	18396.9	−	18396.9i	0.602423	−	0.602423i	0.940955	+	0.338532i	$0.109930\pi$
$978$	−13882.9	+	1681.96i	−0.453911	+	0.0549930i
$979$			5186.32i			0.169311i
$980$		0			0
$981$	−30.6568	−	124.663i	−0.000997754	−	0.00405728i
$982$	24983.8	+	24983.8i	0.811880	+	0.811880i
$983$	−37328.5	−	37328.5i	−1.21119	−	1.21119i	−0.970637	−	0.240549i	$-0.922673\pi$
$983$	−37328.5	−	37328.5i	−1.21119	−	1.21119i	−0.240549	−	0.970637i	$-0.577327\pi$
$984$	−1284.44	−	1006.84i	−0.0416121	−	0.0326188i
$985$		0			0
$986$			69965.5i			2.25979i
$987$	−9455.20	−	78043.0i	−0.304926	−	2.51686i
$988$	−5617.71	+	5617.71i	−0.180894	+	0.180894i
$989$	6114.46			0.196591
$990$		0			0
$991$	−26565.6			−0.851547			−0.425773	−	0.904830i	$-0.639998\pi$
$991$	−26565.6			−0.851547			−0.425773	+	0.904830i	$0.639998\pi$
$992$	1262.95	−	1262.95i	0.0404219	−	0.0404219i
$993$	5503.95	+	45429.5i	0.175894	+	1.45182i
$994$			96830.4i			3.08981i
$995$		0			0
$996$	47551.1	+	37274.2i	1.51277	+	1.18582i
$997$	3172.34	+	3172.34i	0.100771	+	0.100771i	0.755695	−	0.654924i	$-0.227299\pi$
$997$	3172.34	+	3172.34i	0.100771	+	0.100771i	−0.654924	+	0.755695i	$0.727299\pi$
$998$	−52033.3	−	52033.3i	−1.65039	−	1.65039i
$999$	−10652.8	+	23619.5i	−0.337377	+	0.748035i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.e.d.32.1	✓	16
3.2	odd	2	inner	75.4.e.d.32.7	yes	16
5.2	odd	4	inner	75.4.e.d.68.2	yes	16
5.3	odd	4	inner	75.4.e.d.68.7	yes	16
5.4	even	2	inner	75.4.e.d.32.8	yes	16
15.2	even	4	inner	75.4.e.d.68.8	yes	16
15.8	even	4	inner	75.4.e.d.68.1	yes	16
15.14	odd	2	inner	75.4.e.d.32.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.e.d.32.1	✓	16	1.1	even	1	trivial
75.4.e.d.32.2	yes	16	15.14	odd	2	inner
75.4.e.d.32.7	yes	16	3.2	odd	2	inner
75.4.e.d.32.8	yes	16	5.4	even	2	inner
75.4.e.d.68.1	yes	16	15.8	even	4	inner
75.4.e.d.68.2	yes	16	5.2	odd	4	inner
75.4.e.d.68.7	yes	16	5.3	odd	4	inner
75.4.e.d.68.8	yes	16	15.2	even	4	inner

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-3.39887 + 3.39887i) q^{2} +(0.624963 + 5.15843i) q^{3} -15.1047i q^{4} +(-19.6570 - 15.4087i) q^{6} +(-19.7241 - 19.7241i) q^{7} +(24.1479 + 24.1479i) q^{8} +(-26.2188 + 6.44766i) q^{9} +O(q^{10})\)	\(q+(-3.39887 + 3.39887i) q^{2} +(0.624963 + 5.15843i) q^{3} -15.1047i q^{4} +(-19.6570 - 15.4087i) q^{6} +(-19.7241 - 19.7241i) q^{7} +(24.1479 + 24.1479i) q^{8} +(-26.2188 + 6.44766i) q^{9} -9.19707i q^{11} +(77.9165 - 9.43987i) q^{12} +(22.4300 - 22.4300i) q^{13} +134.080 q^{14} -43.3141 q^{16} +(-50.6273 + 50.6273i) q^{17} +(67.1998 - 111.029i) q^{18} -16.5813i q^{19} +(89.4187 - 114.072i) q^{21} +(31.2597 + 31.2597i) q^{22} +(-48.2959 - 48.2959i) q^{23} +(-109.474 + 139.657i) q^{24} +152.474i q^{26} +(-49.6456 - 131.219i) q^{27} +(-297.927 + 297.927i) q^{28} -203.298 q^{29} -27.4766 q^{31} +(-45.9644 + 45.9644i) q^{32} +(47.4425 - 5.74783i) q^{33} -344.152i q^{34} +(97.3898 + 396.027i) q^{36} +(-130.592 - 130.592i) q^{37} +(56.3576 + 56.3576i) q^{38} +(129.722 + 101.686i) q^{39} +9.19707i q^{41} +(83.7948 + 691.641i) q^{42} +(-63.3021 + 63.3021i) q^{43} -138.919 q^{44} +328.303 q^{46} +(383.520 - 383.520i) q^{47} +(-27.0697 - 223.433i) q^{48} +435.083i q^{49} +(-292.798 - 229.517i) q^{51} +(-338.799 - 338.799i) q^{52} +(-441.142 - 441.142i) q^{53} +(614.735 + 277.256i) q^{54} -952.594i q^{56} +(85.5333 - 10.3627i) q^{57} +(690.986 - 690.986i) q^{58} -314.626 q^{59} -431.664 q^{61} +(93.3894 - 93.3894i) q^{62} +(644.318 + 389.969i) q^{63} -658.967i q^{64} +(-141.715 + 180.787i) q^{66} +(649.685 + 649.685i) q^{67} +(764.709 + 764.709i) q^{68} +(218.948 - 279.314i) q^{69} +722.186i q^{71} +(-788.828 - 477.433i) q^{72} +(-662.432 + 662.432i) q^{73} +887.733 q^{74} -250.455 q^{76} +(-181.404 + 181.404i) q^{77} +(-786.526 + 95.2905i) q^{78} -206.816i q^{79} +(645.855 - 338.100i) q^{81} +(-31.2597 - 31.2597i) q^{82} +(544.335 + 544.335i) q^{83} +(-1723.03 - 1350.64i) q^{84} -430.312i q^{86} +(-127.054 - 1048.70i) q^{87} +(222.090 - 222.090i) q^{88} -563.910 q^{89} -884.827 q^{91} +(-729.494 + 729.494i) q^{92} +(-17.1718 - 141.736i) q^{93} +2607.07i q^{94} +(-265.830 - 208.378i) q^{96} +(66.0080 + 66.0080i) q^{97} +(-1478.79 - 1478.79i) q^{98} +(59.2996 + 241.137i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 84 q^{6}+O(q^{10}) \) 16 * q - 84 * q^6	\( 16 q - 84 q^{6} - 232 q^{16} + 816 q^{21} - 1208 q^{31} + 252 q^{36} + 1872 q^{46} + 156 q^{51} - 1528 q^{61} - 3420 q^{66} + 1064 q^{76} + 6876 q^{81} - 10008 q^{91} - 8172 q^{96}+O(q^{100}) \) 16 * q - 84 * q^6 - 232 * q^16 + 816 * q^21 - 1208 * q^31 + 252 * q^36 + 1872 * q^46 + 156 * q^51 - 1528 * q^61 - 3420 * q^66 + 1064 * q^76 + 6876 * q^81 - 10008 * q^91 - 8172 * q^96

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