LMFDB - Embedded newform 75.9.d.d.74.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,9,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	75.d (of order $2$, degree $1$, not 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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 3278 x^{18} + 4245491 x^{16} + 2854629536 x^{14} + 1117319469691 x^{12} + 266849787342054 x^{10} + \cdots + 89\!\cdots\!00 $ x^20 + 3278x^18 + 4245491x^16 + 2854629536x^14 + 1117319469691x^12 + 266849787342054x^10 + 38935163332201981x^8 + 3308855301815965532x^6 + 143709036159536320436x^4 + 2262672788809149782000*x^2 + 8995759125454185640000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{22}\cdot 5^{26} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			74.8
Root			$14.7835i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.9.d.d.74.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.2639 q^{2} +(22.8278 + 77.7167i) q^{3} +8.51375 q^{4} +(-371.269 - 1263.98i) q^{6} +569.895i q^{7} +4025.09 q^{8} +(-5518.78 + 3548.20i) q^{9} +O(q^{10})$	\(q-16.2639 q^{2} +(22.8278 + 77.7167i) q^{3} +8.51375 q^{4} +(-371.269 - 1263.98i) q^{6} +569.895i q^{7} +4025.09 q^{8} +(-5518.78 + 3548.20i) q^{9} +6582.70i q^{11} +(194.350 + 661.661i) q^{12} -28214.2i q^{13} -9268.70i q^{14} -67643.0 q^{16} -151139. q^{17} +(89756.8 - 57707.6i) q^{18} -41572.9 q^{19} +(-44290.4 + 13009.4i) q^{21} -107060. i q^{22} +173241. q^{23} +(91883.9 + 312817. i) q^{24} +458873. i q^{26} +(-401737. - 347904. i) q^{27} +4851.94i q^{28} +783419. i q^{29} -215271. q^{31} +69716.1 q^{32} +(-511586. + 150269. i) q^{33} +2.45811e6 q^{34} +(-46985.6 + 30208.5i) q^{36} -2.71385e6i q^{37} +676137. q^{38} +(2.19272e6 - 644068. i) q^{39} -676092. i q^{41} +(720333. - 211584. i) q^{42} -4.21531e6i q^{43} +56043.5i q^{44} -2.81758e6 q^{46} +8.87092e6 q^{47} +(-1.54414e6 - 5.25700e6i) q^{48} +5.44002e6 q^{49} +(-3.45018e6 - 1.17461e7i) q^{51} -240209. i q^{52} +6.53406e6 q^{53} +(6.53379e6 + 5.65827e6i) q^{54} +2.29387e6i q^{56} +(-949019. - 3.23091e6i) q^{57} -1.27414e7i q^{58} -1.54958e7i q^{59} -8.68344e6 q^{61} +3.50114e6 q^{62} +(-2.02210e6 - 3.14512e6i) q^{63} +1.61828e7 q^{64} +(8.32037e6 - 2.44395e6i) q^{66} -2.87523e7i q^{67} -1.28676e6 q^{68} +(3.95472e6 + 1.34638e7i) q^{69} +3.73955e7i q^{71} +(-2.22136e7 + 1.42818e7i) q^{72} +3.75089e7i q^{73} +4.41378e7i q^{74} -353942. q^{76} -3.75145e6 q^{77} +(-3.56621e7 + 1.04751e7i) q^{78} -4.22725e7 q^{79} +(1.78672e7 - 3.91635e7i) q^{81} +1.09959e7i q^{82} +7.28474e7 q^{83} +(-377077. + 110759. i) q^{84} +6.85572e7i q^{86} +(-6.08847e7 + 1.78837e7i) q^{87} +2.64959e7i q^{88} -5.99236e7i q^{89} +1.60791e7 q^{91} +1.47494e6 q^{92} +(-4.91416e6 - 1.67302e7i) q^{93} -1.44276e8 q^{94} +(1.59146e6 + 5.41811e6i) q^{96} +1.15177e7i q^{97} -8.84758e7 q^{98} +(-2.33568e7 - 3.63285e7i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) $ 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9	$ 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) $ 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9 + 89268 * q^16 + 287868 * q^19 + 1346856 * q^21 + 2033718 * q^24 - 6028120 * q^31 - 9954292 * q^34 + 9001054 * q^36 + 15026564 * q^39 - 27877272 * q^46 - 17907092 * q^49 + 12418574 * q^51 + 17772544 * q^54 + 3040440 * q^61 + 9073996 * q^64 + 20930590 * q^66 - 22789956 * q^69 - 59058092 * q^76 - 17099792 * q^79 + 45225890 * q^81 + 273766584 * q^84 - 214448528 * q^91 - 712237192 * q^94 + 1050848002 * q^96 + 764842670 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−16.2639			−1.01649			−0.508246	−	0.861212i	$-0.669706\pi$
$2$	−16.2639			−1.01649			−0.508246	+	0.861212i	$0.669706\pi$
$3$	22.8278	+	77.7167i	0.281825	+	0.959466i
$3$	22.8278	+	77.7167i	0.281825	+	0.959466i
$4$	8.51375			0.0332569
$5$		0			0
$5$		0			0
$6$	−371.269	−	1263.98i	−0.286473	−	0.975290i
$7$			569.895i			0.237357i	0.992933	+	0.118679i	$0.0378658\pi$
$7$			569.895i			0.237357i	−0.992933	+	0.118679i	$0.962134\pi$
$8$	4025.09			0.982687
$9$	−5518.78	+	3548.20i	−0.841150	+	0.540802i
$10$		0			0
$11$			6582.70i			0.449607i	0.974404	+	0.224804i	$0.0721741\pi$
$11$			6582.70i			0.449607i	−0.974404	+	0.224804i	$0.927826\pi$
$12$	194.350	+	661.661i	0.00937260	+	0.0319088i
$13$		−	28214.2i		−	0.987858i	−0.869502	−	0.493929i	$-0.835560\pi$
$13$		−	28214.2i		−	0.987858i	0.869502	−	0.493929i	$-0.164440\pi$
$14$		−	9268.70i		−	0.241272i
$15$		0			0
$16$	−67643.0			−1.03215
$17$	−151139.			−1.80960			−0.904798	−	0.425840i	$-0.859979\pi$
$17$	−151139.			−1.80960			−0.904798	+	0.425840i	$0.859979\pi$
$18$	89756.8	−	57707.6i	0.855022	−	0.549721i
$19$	−41572.9			−0.319004			−0.159502	−	0.987198i	$-0.550989\pi$
$19$	−41572.9			−0.319004			−0.159502	+	0.987198i	$0.550989\pi$
$20$		0			0
$21$	−44290.4	+	13009.4i	−0.227736	+	0.0668931i
$22$		−	107060.i		−	0.457022i
$23$	173241.			0.619071			0.309536	−	0.950888i	$-0.399826\pi$
$23$	173241.			0.619071			0.309536	+	0.950888i	$0.399826\pi$
$24$	91883.9	+	312817.i	0.276945	+	0.942855i
$25$		0			0
$26$			458873.i			1.00415i
$27$	−401737.	−	347904.i	−0.755938	−	0.654643i
$28$			4851.94i			0.00789375i
$29$			783419.i			1.10765i	0.832634	+	0.553824i	$0.186832\pi$
$29$			783419.i			1.10765i	−0.832634	+	0.553824i	$0.813168\pi$
$30$		0			0
$31$	−215271.			−0.233098			−0.116549	−	0.993185i	$-0.537183\pi$
$31$	−215271.			−0.233098			−0.116549	+	0.993185i	$0.537183\pi$
$32$	69716.1			0.0664864
$33$	−511586.	+	150269.i	−0.431383	+	0.126710i
$34$	2.45811e6			1.83944
$35$		0			0
$36$	−46985.6	+	30208.5i	−0.0279740	+	0.0179854i
$37$		−	2.71385e6i		−	1.44804i	−0.689781	−	0.724018i	$-0.742293\pi$
$37$		−	2.71385e6i		−	1.44804i	0.689781	−	0.724018i	$-0.257707\pi$
$38$	676137.			0.324265
$39$	2.19272e6	−	644068.i	0.947816	−	0.278403i
$40$		0			0
$41$		−	676092.i		−	0.239260i	−0.992819	−	0.119630i	$-0.961829\pi$
$41$		−	676092.i		−	0.239260i	0.992819	−	0.119630i	$-0.0381709\pi$
$42$	720333.	−	211584.i	0.231492	−	0.0679963i
$43$		−	4.21531e6i		−	1.23298i	−0.787364	−	0.616489i	$-0.788555\pi$
$43$		−	4.21531e6i		−	1.23298i	0.787364	−	0.616489i	$-0.211445\pi$
$44$			56043.5i			0.0149525i
$45$		0			0
$46$	−2.81758e6			−0.629281
$47$	8.87092e6			1.81793			0.908965	−	0.416872i	$-0.136874\pi$
$47$	8.87092e6			1.81793			0.908965	+	0.416872i	$0.136874\pi$
$48$	−1.54414e6	−	5.25700e6i	−0.290886	−	0.990314i
$49$	5.44002e6			0.943662
$50$		0			0
$51$	−3.45018e6	−	1.17461e7i	−0.509989	−	1.73625i
$52$		−	240209.i		−	0.0328531i
$53$	6.53406e6			0.828094			0.414047	−	0.910255i	$-0.364115\pi$
$53$	6.53406e6			0.828094			0.414047	+	0.910255i	$0.364115\pi$
$54$	6.53379e6	+	5.65827e6i	0.768405	+	0.665440i
$55$		0			0
$56$			2.29387e6i			0.233248i
$57$	−949019.	−	3.23091e6i	−0.0899032	−	0.306074i
$58$		−	1.27414e7i		−	1.12592i
$59$		−	1.54958e7i		−	1.27881i	−0.768871	−	0.639403i	$-0.779181\pi$
$59$		−	1.54958e7i		−	1.27881i	0.768871	−	0.639403i	$-0.220819\pi$
$60$		0			0
$61$	−8.68344e6			−0.627152			−0.313576	−	0.949563i	$-0.601527\pi$
$61$	−8.68344e6			−0.627152			−0.313576	+	0.949563i	$0.601527\pi$
$62$	3.50114e6			0.236942
$63$	−2.02210e6	−	3.14512e6i	−0.128363	−	0.199653i
$64$	1.61828e7			0.964568
$65$		0			0
$66$	8.32037e6	−	2.44395e6i	0.438497	−	0.128800i
$67$		−	2.87523e7i		−	1.42684i	−0.700739	−	0.713418i	$-0.747146\pi$
$67$		−	2.87523e7i		−	1.42684i	0.700739	−	0.713418i	$-0.252854\pi$
$68$	−1.28676e6			−0.0601815
$69$	3.95472e6	+	1.34638e7i	0.174470	+	0.593978i
$70$		0			0
$71$			3.73955e7i			1.47159i	0.677207	+	0.735793i	$0.263190\pi$
$71$			3.73955e7i			1.47159i	−0.677207	+	0.735793i	$0.736810\pi$
$72$	−2.22136e7	+	1.42818e7i	−0.826587	+	0.531439i
$73$			3.75089e7i			1.32082i	0.750907	+	0.660408i	$0.229617\pi$
$73$			3.75089e7i			1.32082i	−0.750907	+	0.660408i	$0.770383\pi$
$74$			4.41378e7i			1.47192i
$75$		0			0
$76$	−353942.			−0.0106091
$77$	−3.75145e6			−0.106718
$78$	−3.56621e7	+	1.04751e7i	−0.963448	+	0.282994i
$79$	−4.22725e7			−1.08530			−0.542649	−	0.839959i	$-0.682579\pi$
$79$	−4.22725e7			−1.08530			−0.542649	+	0.839959i	$0.682579\pi$
$80$		0			0
$81$	1.78672e7	−	3.91635e7i	0.415066	−	0.909791i
$82$			1.09959e7i			0.243206i
$83$	7.28474e7			1.53498			0.767488	−	0.641064i	$-0.221507\pi$
$83$	7.28474e7			1.53498			0.767488	+	0.641064i	$0.221507\pi$
$84$	−377077.	+	110759.i	−0.00757379	+	0.00222465i
$85$		0			0
$86$			6.85572e7i			1.25331i
$87$	−6.08847e7	+	1.78837e7i	−1.06275	+	0.312163i
$88$			2.64959e7i			0.441823i
$89$		−	5.99236e7i		−	0.955077i	−0.878611	−	0.477538i	$-0.841529\pi$
$89$		−	5.99236e7i		−	0.955077i	0.878611	−	0.477538i	$-0.158471\pi$
$90$		0			0
$91$	1.60791e7			0.234475
$92$	1.47494e6			0.0205884
$93$	−4.91416e6	−	1.67302e7i	−0.0656928	−	0.223650i
$94$	−1.44276e8			−1.84791
$95$		0			0
$96$	1.59146e6	+	5.41811e6i	0.0187375	+	0.0637915i
$97$			1.15177e7i			0.130100i	0.997882	+	0.0650500i	$0.0207207\pi$
$97$			1.15177e7i			0.130100i	−0.997882	+	0.0650500i	$0.979279\pi$
$98$	−8.84758e7			−0.959225
$99$	−2.33568e7	−	3.63285e7i	−0.243149	−	0.378187i
$100$		0			0
$101$		−	9.81566e7i		−	0.943266i	−0.881795	−	0.471633i	$-0.843665\pi$
$101$		−	9.81566e7i		−	0.943266i	0.881795	−	0.471633i	$-0.156335\pi$
$102$	5.61133e7	+	1.91036e8i	0.518400	+	1.76488i
$103$			1.66920e7i			0.148306i	0.997247	+	0.0741529i	$0.0236253\pi$
$103$			1.66920e7i			0.148306i	−0.997247	+	0.0741529i	$0.976375\pi$
$104$		−	1.13565e8i		−	0.970755i
$105$		0			0
$106$	−1.06269e8			−0.841751
$107$	4.28172e7			0.326651			0.163325	−	0.986572i	$-0.447778\pi$
$107$	4.28172e7			0.326651			0.163325	+	0.986572i	$0.447778\pi$
$108$	−3.42029e6	−	2.96197e6i	−0.0251401	−	0.0217714i
$109$	−1.49703e8			−1.06053			−0.530266	−	0.847831i	$-0.677908\pi$
$109$	−1.49703e8			−1.06053			−0.530266	+	0.847831i	$0.677908\pi$
$110$		0			0
$111$	2.10912e8	−	6.19513e7i	1.38934	−	0.408092i
$112$		−	3.85494e7i		−	0.244988i
$113$	1.27819e7			0.0783937			0.0391969	−	0.999232i	$-0.487520\pi$
$113$	1.27819e7			0.0783937			0.0391969	+	0.999232i	$0.487520\pi$
$114$	1.54347e7	+	5.25472e7i	0.0913860	+	0.311122i
$115$		0			0
$116$			6.66983e6i			0.0368369i
$117$	1.00110e8	+	1.55708e8i	0.534236	+	0.830937i
$118$			2.52021e8i			1.29990i
$119$		−	8.61335e7i		−	0.429521i
$120$		0			0
$121$	1.71027e8			0.797853
$122$	1.41226e8			0.637495
$123$	5.25437e7	−	1.54337e7i	0.229562	−	0.0674295i
$124$	−1.83276e6			−0.00775211
$125$		0			0
$126$	3.28872e7	+	5.11519e7i	0.130480	+	0.202946i
$127$		−	2.68115e8i		−	1.03064i	−0.856998	−	0.515320i	$-0.827673\pi$
$127$		−	2.68115e8i		−	1.03064i	0.856998	−	0.515320i	$-0.172327\pi$
$128$	−2.81042e8			−1.04696
$129$	3.27600e8	−	9.62261e7i	1.18300	−	0.347484i
$130$		0			0
$131$			5.50651e6i			0.0186978i	0.999956	+	0.00934892i	$0.00297590\pi$
$131$			5.50651e6i			0.0186978i	−0.999956	+	0.00934892i	$0.997024\pi$
$132$	−4.35552e6	+	1.27935e6i	−0.0143464	+	0.00421399i
$133$		−	2.36922e7i		−	0.0757179i
$134$			4.67624e8i			1.45037i
$135$		0			0
$136$	−6.08349e8			−1.77827
$137$	−5.57045e8			−1.58128			−0.790639	−	0.612283i	$-0.790251\pi$
$137$	−5.57045e8			−1.58128			−0.790639	+	0.612283i	$0.790251\pi$
$138$	−6.43191e7	−	2.18973e8i	−0.177347	−	0.603774i
$139$	3.34413e7			0.0895827			0.0447914	−	0.998996i	$-0.485738\pi$
$139$	3.34413e7			0.0895827			0.0447914	+	0.998996i	$0.485738\pi$
$140$		0			0
$141$	2.02504e8	+	6.89419e8i	0.512338	+	1.74424i
$142$		−	6.08195e8i		−	1.49586i
$143$	1.85726e8			0.444148
$144$	3.73307e8	−	2.40011e8i	0.868193	−	0.558190i
$145$		0			0
$146$		−	6.10040e8i		−	1.34260i
$147$	1.24184e8	+	4.22781e8i	0.265947	+	0.905411i
$148$		−	2.31051e7i		−	0.0481571i
$149$			4.46572e8i			0.906037i	0.891501	+	0.453019i	$0.149653\pi$
$149$			4.46572e8i			0.906037i	−0.891501	+	0.453019i	$0.850347\pi$
$150$		0			0
$151$	−2.41495e8			−0.464515			−0.232257	−	0.972654i	$-0.574611\pi$
$151$	−2.41495e8			−0.464515			−0.232257	+	0.972654i	$0.574611\pi$
$152$	−1.67335e8			−0.313481
$153$	8.34105e8	−	5.36273e8i	1.52214	−	0.978634i
$154$	6.10130e7			0.108478
$155$		0			0
$156$	1.86683e7	−	5.48344e6i	0.0315214	−	0.00925880i
$157$		−	1.12488e9i		−	1.85143i	−0.378226	−	0.925713i	$-0.623466\pi$
$157$		−	1.12488e9i		−	1.85143i	0.378226	−	0.925713i	$-0.376534\pi$
$158$	6.87514e8			1.10320
$159$	1.49158e8	+	5.07806e8i	0.233377	+	0.794528i
$160$		0			0
$161$			9.87294e7i			0.146941i
$162$	−2.90590e8	+	6.36951e8i	−0.421911	+	0.924796i
$163$		−	1.05083e9i		−	1.48861i	−0.667838	−	0.744307i	$-0.732780\pi$
$163$		−	1.05083e9i		−	1.48861i	0.667838	−	0.744307i	$-0.267220\pi$
$164$		−	5.75609e6i		−	0.00795705i
$165$		0			0
$166$	−1.18478e9			−1.56029
$167$	1.52565e8			0.196150			0.0980751	−	0.995179i	$-0.468731\pi$
$167$	1.52565e8			0.196150			0.0980751	+	0.995179i	$0.468731\pi$
$168$	−1.78272e8	+	5.23641e7i	−0.223793	+	0.0657350i
$169$	1.96888e7			0.0241364
$170$		0			0
$171$	2.29432e8	−	1.47509e8i	0.268330	−	0.172518i
$172$		−	3.58881e7i		−	0.0410050i
$173$	−1.48867e9			−1.66193			−0.830967	−	0.556321i	$-0.812212\pi$
$173$	−1.48867e9			−1.66193			−0.830967	+	0.556321i	$0.812212\pi$
$174$	9.90222e8	−	2.90859e8i	1.08028	−	0.317311i
$175$		0			0
$176$		−	4.45274e8i		−	0.464063i
$177$	1.20428e9	−	3.53734e8i	1.22697	−	0.360399i
$178$			9.74591e8i			0.970828i
$179$		−	6.73543e8i		−	0.656074i	−0.944665	−	0.328037i	$-0.893613\pi$
$179$		−	6.73543e8i		−	0.656074i	0.944665	−	0.328037i	$-0.106387\pi$
$180$		0			0
$181$	9.31938e8			0.868306			0.434153	−	0.900839i	$-0.357048\pi$
$181$	9.31938e8			0.868306			0.434153	+	0.900839i	$0.357048\pi$
$182$	−2.61509e8			−0.238342
$183$	−1.98224e8	−	6.74849e8i	−0.176747	−	0.601731i
$184$	6.97312e8			0.608353
$185$		0			0
$186$	7.99233e7	+	2.72097e8i	0.0667762	+	0.227338i
$187$		−	9.94905e8i		−	0.813608i
$188$	7.55248e7			0.0604586
$189$	1.98269e8	−	2.28947e8i	0.155384	−	0.179427i
$190$		0			0
$191$		−	6.44104e7i		−	0.0483974i	−0.999707	−	0.0241987i	$-0.992297\pi$
$191$		−	6.44104e7i		−	0.0483974i	0.999707	−	0.0241987i	$-0.00770344\pi$
$192$	3.69417e8	+	1.25767e9i	0.271839	+	0.925470i
$193$			9.15124e8i			0.659554i	0.944059	+	0.329777i	$0.106974\pi$
$193$			9.15124e8i			0.659554i	−0.944059	+	0.329777i	$0.893026\pi$
$194$		−	1.87322e8i		−	0.132246i
$195$		0			0
$196$	4.63150e7			0.0313832
$197$	1.99042e9			1.32154			0.660769	−	0.750589i	$-0.270230\pi$
$197$	1.99042e9			1.32154			0.660769	+	0.750589i	$0.270230\pi$
$198$	3.79872e8	+	5.90842e8i	0.247159	+	0.384424i
$199$	−2.33793e9			−1.49080			−0.745400	−	0.666618i	$-0.767741\pi$
$199$	−2.33793e9			−1.49080			−0.745400	+	0.666618i	$0.767741\pi$
$200$		0			0
$201$	2.23454e9	−	6.56352e8i	1.36900	−	0.402117i
$202$			1.59641e9i			0.958823i
$203$	−4.46466e8			−0.262908
$204$	−2.93740e7	−	1.00003e8i	−0.0169606	−	0.0577421i
$205$		0			0
$206$		−	2.71476e8i		−	0.150752i
$207$	−9.56082e8	+	6.14696e8i	−0.520732	+	0.334795i
$208$			1.90850e9i			1.01962i
$209$		−	2.73662e8i		−	0.143427i
$210$		0			0
$211$	−3.72705e7			−0.0188034			−0.00940168	−	0.999956i	$-0.502993\pi$
$211$	−3.72705e7			−0.0188034			−0.00940168	+	0.999956i	$0.502993\pi$
$212$	5.56294e7			0.0275398
$213$	−2.90625e9	+	8.53656e8i	−1.41194	+	0.414729i
$214$	−6.96374e8			−0.332038
$215$		0			0
$216$	−1.61702e9	−	1.40034e9i	−0.742851	−	0.643309i
$217$		−	1.22682e8i		−	0.0553275i
$218$	2.43475e9			1.07802
$219$	−2.91507e9	+	8.56245e8i	−1.26728	+	0.372239i
$220$		0			0
$221$			4.26428e9i			1.78762i
$222$	−3.43024e9	+	1.00757e9i	−1.41225	+	0.414823i
$223$		−	2.51046e9i		−	1.01516i	−0.861606	−	0.507578i	$-0.830541\pi$
$223$		−	2.51046e9i		−	1.01516i	0.861606	−	0.507578i	$-0.169459\pi$
$224$			3.97308e7i			0.0157810i
$225$		0			0
$226$	−2.07883e8			−0.0796866
$227$	−1.11767e9			−0.420930			−0.210465	−	0.977601i	$-0.567498\pi$
$227$	−1.11767e9			−0.420930			−0.210465	+	0.977601i	$0.567498\pi$
$228$	−8.07971e6	−	2.75072e7i	−0.00298990	−	0.0101790i
$229$	−3.29155e9			−1.19690			−0.598450	−	0.801160i	$-0.704217\pi$
$229$	−3.29155e9			−1.19690			−0.598450	+	0.801160i	$0.704217\pi$
$230$		0			0
$231$	−8.56372e7	−	2.91550e8i	−0.0300756	−	0.102392i
$232$			3.15333e9i			1.08847i
$233$	1.74654e9			0.592591			0.296296	−	0.955096i	$-0.404249\pi$
$233$	1.74654e9			0.592591			0.296296	+	0.955096i	$0.404249\pi$
$234$	−1.62817e9	−	2.53242e9i	−0.543047	−	0.844641i
$235$		0			0
$236$		−	1.31927e8i		−	0.0425291i
$237$	−9.64988e8	−	3.28528e9i	−0.305864	−	1.04131i
$238$			1.40086e9i			0.436605i
$239$		−	3.20404e9i		−	0.981988i	−0.871163	−	0.490994i	$-0.836634\pi$
$239$		−	3.20404e9i		−	0.981988i	0.871163	−	0.490994i	$-0.163366\pi$
$240$		0			0
$241$	2.32540e9			0.689334			0.344667	−	0.938725i	$-0.387992\pi$
$241$	2.32540e9			0.689334			0.344667	+	0.938725i	$0.387992\pi$
$242$	−2.78156e9			−0.811012
$243$	3.45153e9	+	4.94565e8i	0.989890	+	0.141840i
$244$	−7.39287e7			−0.0208571
$245$		0			0
$246$	−8.54564e8	+	2.51012e8i	−0.233348	+	0.0685415i
$247$			1.17295e9i			0.315131i
$248$	−8.66484e8			−0.229062
$249$	1.66294e9	+	5.66146e9i	0.432594	+	1.47276i
$250$		0			0
$251$			3.38503e9i			0.852840i	0.904525	+	0.426420i	$0.140225\pi$
$251$			3.38503e9i			0.852840i	−0.904525	+	0.426420i	$0.859775\pi$
$252$	−1.72157e7	−	2.67768e7i	−0.00426896	−	0.00663983i
$253$			1.14040e9i			0.278339i
$254$			4.36059e9i			1.04764i
$255$		0			0
$256$	4.28043e8			0.0996614
$257$	−9.71066e8			−0.222595			−0.111298	−	0.993787i	$-0.535501\pi$
$257$	−9.71066e8			−0.222595			−0.111298	+	0.993787i	$0.535501\pi$
$258$	−5.32804e9	+	1.56501e9i	−1.20251	+	0.353214i
$259$	1.54661e9			0.343702
$260$		0			0
$261$	−2.77973e9	−	4.32352e9i	−0.599019	−	0.931698i
$262$		−	8.95573e7i		−	0.0190062i
$263$	−2.06605e8			−0.0431835			−0.0215918	−	0.999767i	$-0.506873\pi$
$263$	−2.06605e8			−0.0431835			−0.0215918	+	0.999767i	$0.506873\pi$
$264$	−2.05918e9	+	6.04844e8i	−0.423914	+	0.124517i
$265$		0			0
$266$			3.85327e8i			0.0769667i
$267$	4.65707e9	−	1.36792e9i	0.916363	−	0.269164i
$268$		−	2.44790e8i		−	0.0474520i
$269$		−	3.12793e9i		−	0.597376i	−0.954351	−	0.298688i	$-0.903451\pi$
$269$		−	3.12793e9i		−	0.597376i	0.954351	−	0.298688i	$-0.0965489\pi$
$270$		0			0
$271$	4.61051e9			0.854814			0.427407	−	0.904059i	$-0.359427\pi$
$271$	4.61051e9			0.854814			0.427407	+	0.904059i	$0.359427\pi$
$272$	1.02235e10			1.86778
$273$	3.67051e8	+	1.24962e9i	0.0660809	+	0.224971i
$274$	9.05972e9			1.60736
$275$		0			0
$276$	3.36695e7	+	1.14627e8i	0.00580231	+	0.0197538i
$277$			1.77445e9i			0.301401i	0.988579	+	0.150700i	$0.0481529\pi$
$277$			1.77445e9i			0.301401i	−0.988579	+	0.150700i	$0.951847\pi$
$278$	−5.43886e8			−0.0910602
$279$	1.18803e9	−	7.63825e8i	0.196070	−	0.126060i
$280$		0			0
$281$			7.80582e7i			0.0125197i	0.999980	+	0.00625984i	$0.00199258\pi$
$281$			7.80582e7i			0.0125197i	−0.999980	+	0.00625984i	$0.998007\pi$
$282$	−3.29349e9	−	1.12126e10i	−0.520787	−	1.77301i
$283$		−	8.12145e9i		−	1.26616i	−0.774088	−	0.633079i	$-0.781791\pi$
$283$		−	8.12145e9i		−	1.26616i	0.774088	−	0.633079i	$-0.218209\pi$
$284$			3.18376e8i			0.0489403i
$285$		0			0
$286$	−3.02062e9			−0.451473
$287$	3.85301e8			0.0567902
$288$	−3.84748e8	+	2.47367e8i	−0.0559250	+	0.0359560i
$289$	1.58673e10			2.27464
$290$		0			0
$291$	−8.95115e8	+	2.62923e8i	−0.124826	+	0.0366654i
$292$			3.19341e8i			0.0439262i
$293$	−2.52625e9			−0.342773			−0.171386	−	0.985204i	$-0.554825\pi$
$293$	−2.52625e9			−0.342773			−0.171386	+	0.985204i	$0.554825\pi$
$294$	−2.01971e9	−	6.87605e9i	−0.270333	−	0.920344i
$295$		0			0
$296$		−	1.09235e10i		−	1.42297i
$297$	2.29015e9	−	2.64451e9i	0.294332	−	0.339875i
$298$		−	7.26299e9i		−	0.920980i
$299$		−	4.88787e9i		−	0.611554i
$300$		0			0
$301$	2.40228e9			0.292656
$302$	3.92764e9			0.472176
$303$	7.62841e9	−	2.24070e9i	0.905032	−	0.265836i
$304$	2.81212e9			0.329260
$305$		0			0
$306$	−1.35658e10	+	8.72188e9i	−1.54725	+	0.994774i
$307$			1.03150e10i			1.16123i	0.814180	+	0.580613i	$0.197187\pi$
$307$			1.03150e10i			1.16123i	−0.814180	+	0.580613i	$0.802813\pi$
$308$	−3.19389e7			−0.00354909
$309$	−1.29724e9	+	3.81040e8i	−0.142294	+	0.0417962i
$310$		0			0
$311$		−	1.72354e10i		−	1.84239i	−0.389106	−	0.921193i	$-0.627216\pi$
$311$		−	1.72354e10i		−	1.84239i	0.389106	−	0.921193i	$-0.372784\pi$
$312$	8.82587e9	−	2.59243e9i	0.931407	−	0.273583i
$313$		−	5.42800e9i		−	0.565539i	−0.959188	−	0.282770i	$-0.908747\pi$
$313$		−	5.42800e9i		−	0.565539i	0.959188	−	0.282770i	$-0.0912532\pi$
$314$			1.82949e10i			1.88196i
$315$		0			0
$316$	−3.59897e8			−0.0360936
$317$	−1.16981e10			−1.15845			−0.579226	−	0.815167i	$-0.696645\pi$
$317$	−1.16981e10			−1.15845			−0.579226	+	0.815167i	$0.696645\pi$
$318$	−2.42589e9	−	8.25889e9i	−0.237226	−	0.807632i
$319$	−5.15701e9			−0.498007
$320$		0			0
$321$	9.77423e8	+	3.32761e9i	0.0920582	+	0.313410i
$322$		−	1.60572e9i		−	0.149364i
$323$	6.28331e9			0.577269
$324$	1.52117e8	−	3.33429e8i	0.0138038	−	0.0302568i
$325$		0			0
$326$			1.70906e10i			1.51316i
$327$	−3.41738e9	−	1.16344e10i	−0.298884	−	1.01754i
$328$		−	2.72133e9i		−	0.235118i
$329$			5.05549e9i			0.431499i
$330$		0			0
$331$	−6.03632e9			−0.502875			−0.251438	−	0.967873i	$-0.580903\pi$
$331$	−6.03632e9			−0.502875			−0.251438	+	0.967873i	$0.580903\pi$
$332$	6.20205e8			0.0510485
$333$	9.62930e9	+	1.49772e10i	0.783101	+	1.21801i
$334$	−2.48130e9			−0.199385
$335$		0			0
$336$	2.99593e9	−	8.79998e8i	0.235058	−	0.0690438i
$337$			1.35475e9i			0.105037i	0.998620	+	0.0525183i	$0.0167248\pi$
$337$			1.35475e9i			0.105037i	−0.998620	+	0.0525183i	$0.983275\pi$
$338$	−3.20216e8			−0.0245345
$339$	2.91782e8	+	9.93367e8i	0.0220933	+	0.0752161i
$340$		0			0
$341$		−	1.41706e9i		−	0.104803i
$342$	−3.73146e9	+	2.39907e9i	−0.272756	+	0.175363i
$343$			6.38557e9i			0.461342i
$344$		−	1.69670e10i		−	1.21163i
$345$		0			0
$346$	2.42115e10			1.68934
$347$	−2.47345e10			−1.70602			−0.853012	−	0.521891i	$-0.825227\pi$
$347$	−2.47345e10			−1.70602			−0.853012	+	0.521891i	$0.825227\pi$
$348$	−5.18358e8	+	1.52258e8i	−0.0353437	+	0.0103815i
$349$	9.08306e9			0.612253			0.306126	−	0.951991i	$-0.400967\pi$
$349$	9.08306e9			0.612253			0.306126	+	0.951991i	$0.400967\pi$
$350$		0			0
$351$	−9.81584e9	+	1.13347e10i	−0.646694	+	0.746760i
$352$			4.58920e8i			0.0298928i
$353$	−1.66294e10			−1.07097			−0.535485	−	0.844545i	$-0.679871\pi$
$353$	−1.66294e10			−1.07097			−0.535485	+	0.844545i	$0.679871\pi$
$354$	−1.95863e10	+	5.75309e9i	−1.24721	+	0.366343i
$355$		0			0
$356$		−	5.10175e8i		−	0.0317628i
$357$	6.69401e9	−	1.96624e9i	0.412111	−	0.121050i
$358$			1.09544e10i			0.666894i
$359$			6.51832e9i			0.392426i	0.980561	+	0.196213i	$0.0628644\pi$
$359$			6.51832e9i			0.392426i	−0.980561	+	0.196213i	$0.937136\pi$
$360$		0			0
$361$	−1.52553e10			−0.898236
$362$	−1.51569e10			−0.882626
$363$	3.90417e9	+	1.32917e10i	0.224855	+	0.765513i
$364$	1.36894e8			0.00779791
$365$		0			0
$366$	3.22389e9	+	1.09757e10i	0.179662	+	0.611655i
$367$			1.72068e10i			0.948498i	0.880391	+	0.474249i	$0.157280\pi$
$367$			1.72068e10i			0.948498i	−0.880391	+	0.474249i	$0.842720\pi$
$368$	−1.17186e10			−0.638975
$369$	2.39891e9	+	3.73121e9i	0.129393	+	0.201254i
$370$		0			0
$371$			3.72373e9i			0.196554i
$372$	−4.18380e7	−	1.42436e8i	−0.00218474	−	0.00743789i
$373$			1.47580e10i			0.762418i	0.924489	+	0.381209i	$0.124492\pi$
$373$			1.47580e10i			0.762418i	−0.924489	+	0.381209i	$0.875508\pi$
$374$			1.61810e10i			0.827026i
$375$		0			0
$376$	3.57062e10			1.78646
$377$	2.21035e10			1.09420
$378$	−3.22462e9	+	3.72357e9i	−0.157947	+	0.182387i
$379$	−3.28781e10			−1.59349			−0.796746	−	0.604315i	$-0.793447\pi$
$379$	−3.28781e10			−1.59349			−0.796746	+	0.604315i	$0.793447\pi$
$380$		0			0
$381$	2.08370e10	−	6.12048e9i	0.988863	−	0.290460i
$382$			1.04756e9i			0.0491956i
$383$	−1.11696e10			−0.519089			−0.259544	−	0.965731i	$-0.583572\pi$
$383$	−1.11696e10			−0.519089			−0.259544	+	0.965731i	$0.583572\pi$
$384$	−6.41557e9	−	2.18417e10i	−0.295060	−	1.00452i
$385$		0			0
$386$		−	1.48835e10i		−	0.670432i
$387$	1.49568e10	+	2.32634e10i	0.666797	+	1.03712i
$388$			9.80585e7i			0.00432672i
$389$		−	3.14195e10i		−	1.37215i	−0.727533	−	0.686073i	$-0.759333\pi$
$389$		−	3.14195e10i		−	1.37215i	0.727533	−	0.686073i	$-0.240667\pi$
$390$		0			0
$391$	−2.61836e10			−1.12027
$392$	2.18966e10			0.927324
$393$	−4.27948e8	+	1.25702e8i	−0.0179399	+	0.00526951i
$394$	−3.23719e10			−1.34333
$395$		0			0
$396$	−1.98854e8	−	3.09292e8i	−0.00808636	−	0.0125773i
$397$			8.73430e9i			0.351614i	0.984425	+	0.175807i	$0.0562534\pi$
$397$			8.73430e9i			0.351614i	−0.984425	+	0.175807i	$0.943747\pi$
$398$	3.80238e10			1.51539
$399$	1.84128e9	−	5.40841e8i	0.0726488	−	0.0213392i
$400$		0			0
$401$		−	1.27867e9i		−	0.0494516i	−0.999694	−	0.0247258i	$-0.992129\pi$
$401$		−	1.27867e9i		−	0.0494516i	0.999694	−	0.0247258i	$-0.00787127\pi$
$402$	−3.63422e10	+	1.06748e10i	−1.39158	+	0.408749i
$403$			6.07370e9i			0.230268i
$404$		−	8.35682e8i		−	0.0313701i
$405$		0			0
$406$	7.26127e9			0.267244
$407$	1.78645e10			0.651047
$408$	−1.38873e10	−	4.72789e10i	−0.501160	−	1.70619i
$409$	−1.02810e10			−0.367401			−0.183701	−	0.982982i	$-0.558808\pi$
$409$	−1.02810e10			−0.367401			−0.183701	+	0.982982i	$0.558808\pi$
$410$		0			0
$411$	−1.27161e10	−	4.32917e10i	−0.445643	−	1.51718i
$412$			1.42111e8i			0.00493219i
$413$	8.83095e9			0.303534
$414$	1.55496e10	−	9.99734e9i	0.529320	−	0.340317i
$415$		0			0
$416$		−	1.96698e9i		−	0.0656792i
$417$	7.63392e8	+	2.59895e9i	0.0252466	+	0.0859516i
$418$			4.45081e9i			0.145792i
$419$		−	3.60565e10i		−	1.16984i	−0.811090	−	0.584922i	$-0.801125\pi$
$419$		−	3.60565e10i		−	1.16984i	0.811090	−	0.584922i	$-0.198875\pi$
$420$		0			0
$421$	−3.63986e10			−1.15866			−0.579331	−	0.815092i	$-0.696686\pi$
$421$	−3.63986e10			−1.15866			−0.579331	+	0.815092i	$0.696686\pi$
$422$	6.06163e8			0.0191135
$423$	−4.89567e10	+	3.14758e10i	−1.52915	+	0.983141i
$424$	2.63002e10			0.813757
$425$		0			0
$426$	4.72670e10	−	1.38838e10i	1.43522	−	0.421569i
$427$		−	4.94865e9i		−	0.148859i
$428$	3.64535e8			0.0108634
$429$	4.23971e9	+	1.44340e10i	0.125172	+	0.426145i
$430$		0			0
$431$			3.03555e10i			0.879686i	0.898075	+	0.439843i	$0.144966\pi$
$431$			3.03555e10i			0.879686i	−0.898075	+	0.439843i	$0.855034\pi$
$432$	2.71747e10	+	2.35333e10i	0.780242	+	0.675690i
$433$		−	3.45811e10i		−	0.983757i	−0.870664	−	0.491878i	$-0.836310\pi$
$433$		−	3.45811e10i		−	0.983757i	0.870664	−	0.491878i	$-0.163690\pi$
$434$			1.99528e9i			0.0562400i
$435$		0			0
$436$	−1.27453e9			−0.0352700
$437$	−7.20216e9			−0.197486
$438$	4.74103e10	−	1.39259e10i	1.28818	−	0.378378i
$439$	1.76997e9			0.0476549			0.0238275	−	0.999716i	$-0.492415\pi$
$439$	1.76997e9			0.0476549			0.0238275	+	0.999716i	$0.492415\pi$
$440$		0			0
$441$	−3.00223e10	+	1.93023e10i	−0.793761	+	0.510334i
$442$		−	6.93537e10i		−	1.81711i
$443$	−3.19676e10			−0.830033			−0.415017	−	0.909814i	$-0.636224\pi$
$443$	−3.19676e10			−0.830033			−0.415017	+	0.909814i	$0.636224\pi$
$444$	1.79565e9	−	5.27438e8i	0.0462051	−	0.0135719i
$445$		0			0
$446$			4.08297e10i			1.03190i
$447$	−3.47061e10	+	1.01942e10i	−0.869312	+	0.255344i
$448$			9.22247e9i			0.228947i
$449$			8.00746e10i			1.97020i	0.171994	+	0.985098i	$0.444979\pi$
$449$			8.00746e10i			1.97020i	−0.171994	+	0.985098i	$0.555021\pi$
$450$		0			0
$451$	4.45051e9			0.107573
$452$	1.08822e8			0.00260713
$453$	−5.51279e9	−	1.87682e10i	−0.130912	−	0.445686i
$454$	1.81777e10			0.427873
$455$		0			0
$456$	−3.81988e9	−	1.30047e10i	−0.0883468	−	0.300775i
$457$		−	5.45219e10i		−	1.24999i	−0.780629	−	0.624994i	$-0.785101\pi$
$457$		−	5.45219e10i		−	1.24999i	0.780629	−	0.624994i	$-0.214899\pi$
$458$	5.35333e10			1.21664
$459$	6.07182e10	+	5.25820e10i	1.36794	+	1.18464i
$460$		0			0
$461$		−	3.66539e8i		−	0.00811552i	−0.999992	−	0.00405776i	$-0.998708\pi$
$461$		−	3.66539e8i		−	0.00811552i	0.999992	−	0.00405776i	$-0.00129163\pi$
$462$	1.39279e9	+	4.74174e9i	0.0305716	+	0.104081i
$463$			3.22822e10i			0.702488i	0.936284	+	0.351244i	$0.114241\pi$
$463$			3.22822e10i			0.702488i	−0.936284	+	0.351244i	$0.885759\pi$
$464$		−	5.29928e10i		−	1.14326i
$465$		0			0
$466$	−2.84055e10			−0.602365
$467$	2.11777e10			0.445258			0.222629	−	0.974903i	$-0.428536\pi$
$467$	2.11777e10			0.445258			0.222629	+	0.974903i	$0.428536\pi$
$468$	8.52310e8	+	1.32566e9i	0.0177670	+	0.0276343i
$469$	1.63858e10			0.338670
$470$		0			0
$471$	8.74218e10	−	2.56785e10i	1.77638	−	0.521778i
$472$		−	6.23718e10i		−	1.25667i
$473$	2.77481e10			0.554356
$474$	1.56944e10	+	5.34314e10i	0.310908	+	1.05848i
$475$		0			0
$476$		−	7.33319e8i		−	0.0142845i
$477$	−3.60601e10	+	2.31842e10i	−0.696551	+	0.447835i
$478$			5.21101e10i			0.998184i
$479$			4.69016e10i			0.890934i	0.895298	+	0.445467i	$0.146962\pi$
$479$			4.69016e10i			0.890934i	−0.895298	+	0.445467i	$0.853038\pi$
$480$		0			0
$481$	−7.65692e10			−1.43045
$482$	−3.78201e10			−0.700703
$483$	−7.67293e9	+	2.25377e9i	−0.140985	+	0.0414116i
$484$	1.45608e9			0.0265341
$485$		0			0
$486$	−5.61353e10	−	8.04354e9i	−1.00622	−	0.144179i
$487$		−	1.34767e10i		−	0.239590i	−0.992799	−	0.119795i	$-0.961776\pi$
$487$		−	1.34767e10i		−	0.239590i	0.992799	−	0.119795i	$-0.0382237\pi$
$488$	−3.49516e10			−0.616294
$489$	8.16671e10	−	2.39881e10i	1.42827	−	0.419528i
$490$		0			0
$491$			8.27160e10i			1.42319i	0.702589	+	0.711596i	$0.252027\pi$
$491$			8.27160e10i			1.42319i	−0.702589	+	0.711596i	$0.747973\pi$
$492$	4.47344e8	−	1.31399e8i	0.00763451	−	0.00224249i
$493$		−	1.18405e11i		−	2.00440i
$494$		−	1.90767e10i		−	0.320328i
$495$		0			0
$496$	1.45616e10			0.240592
$497$	−2.13115e10			−0.349291
$498$	−2.70459e10	−	9.20773e10i	−0.439728	−	1.49705i
$499$	−1.96950e10			−0.317653			−0.158827	−	0.987307i	$-0.550771\pi$
$499$	−1.96950e10			−0.317653			−0.158827	+	0.987307i	$0.550771\pi$
$500$		0			0
$501$	3.48272e9	+	1.18569e10i	0.0552800	+	0.188199i
$502$		−	5.50537e10i		−	0.866906i
$503$	5.14640e10			0.803955			0.401978	−	0.915649i	$-0.368323\pi$
$503$	5.14640e10			0.803955			0.401978	+	0.915649i	$0.368323\pi$
$504$	−8.13914e9	−	1.26594e10i	−0.126141	−	0.196196i
$505$		0			0
$506$		−	1.85473e10i		−	0.282929i
$507$	4.49452e8	+	1.53015e9i	0.00680223	+	0.0231580i
$508$		−	2.28267e9i		−	0.0342758i
$509$		−	7.84617e8i		−	0.0116893i	−0.999983	−	0.00584463i	$-0.998140\pi$
$509$		−	7.84617e8i		−	0.0116893i	0.999983	−	0.00584463i	$-0.00186041\pi$
$510$		0			0
$511$	−2.13761e10			−0.313505
$512$	6.49851e10			0.945657
$513$	1.67014e10	+	1.44634e10i	0.241147	+	0.208834i
$514$	1.57933e10			0.226267
$515$		0			0
$516$	2.78910e9	−	8.19246e8i	0.0393429	−	0.0115562i
$517$			5.83946e10i			0.817355i
$518$	−2.51539e10			−0.349370
$519$	−3.39830e10	−	1.15695e11i	−0.468374	−	1.59457i
$520$		0			0
$521$		−	1.39001e11i		−	1.88654i	−0.332024	−	0.943271i	$-0.607731\pi$
$521$		−	1.39001e11i		−	1.88654i	0.332024	−	0.943271i	$-0.392269\pi$
$522$	4.52092e10	+	7.03172e10i	0.608898	+	0.947064i
$523$			9.91022e9i			0.132457i	0.997804	+	0.0662287i	$0.0210967\pi$
$523$			9.91022e9i			0.132457i	−0.997804	+	0.0662287i	$0.978903\pi$
$524$			4.68811e7i			0.000621831i
$525$		0			0
$526$	3.36020e9			0.0438957
$527$	3.25359e10			0.421814
$528$	3.46052e10	−	1.01646e10i	0.445252	−	0.130784i
$529$	−4.82984e10			−0.616751
$530$		0			0
$531$	5.49821e10	+	8.55178e10i	0.691582	+	1.07567i
$532$		−	2.01710e8i		−	0.00251814i
$533$	−1.90754e10			−0.236355
$534$	−7.57420e10	+	2.22478e10i	−0.931477	+	0.273603i
$535$		0			0
$536$		−	1.15731e11i		−	1.40213i
$537$	5.23455e10	−	1.53755e10i	0.629481	−	0.184898i
$538$			5.08722e10i			0.607228i
$539$			3.58100e10i			0.424277i
$540$		0			0
$541$	−2.74155e10			−0.320042			−0.160021	−	0.987114i	$-0.551156\pi$
$541$	−2.74155e10			−0.320042			−0.160021	+	0.987114i	$0.551156\pi$
$542$	−7.49848e10			−0.868912
$543$	2.12741e10	+	7.24272e10i	0.244710	+	0.833110i
$544$	−1.05368e10			−0.120314
$545$		0			0
$546$	−5.96967e9	−	2.03236e10i	−0.0671707	−	0.228681i
$547$			8.62303e10i			0.963187i	0.876395	+	0.481594i	$0.159942\pi$
$547$			8.62303e10i			0.963187i	−0.876395	+	0.481594i	$0.840058\pi$
$548$	−4.74255e9			−0.0525883
$549$	4.79220e10	−	3.08106e10i	0.527528	−	0.339165i
$550$		0			0
$551$		−	3.25690e10i		−	0.353344i
$552$	1.59181e10	+	5.41928e10i	0.171449	+	0.583694i
$553$		−	2.40909e10i		−	0.257603i
$554$		−	2.88594e10i		−	0.306372i
$555$		0			0
$556$	2.84711e8			0.00297924
$557$	6.87189e10			0.713929			0.356965	−	0.934118i	$-0.383812\pi$
$557$	6.87189e10			0.713929			0.356965	+	0.934118i	$0.383812\pi$
$558$	−1.93220e10	+	1.24228e10i	−0.199304	+	0.128139i
$559$	−1.18932e11			−1.21801
$560$		0			0
$561$	7.73208e10	−	2.27115e10i	0.780629	−	0.229295i
$562$		−	1.26953e9i		−	0.0127262i
$563$	−9.04024e10			−0.899801			−0.449901	−	0.893079i	$-0.648541\pi$
$563$	−9.04024e10			−0.899801			−0.449901	+	0.893079i	$0.648541\pi$
$564$	1.72407e9	+	5.86954e9i	0.0170387	+	0.0580080i
$565$		0			0
$566$			1.32086e11i			1.28704i
$567$	2.23191e10	+	1.01824e10i	0.215946	+	0.0985188i
$568$			1.50520e11i			1.44611i
$569$		−	1.77922e11i		−	1.69739i	−0.528886	−	0.848693i	$-0.677390\pi$
$569$		−	1.77922e11i		−	1.69739i	0.528886	−	0.848693i	$-0.322610\pi$
$570$		0			0
$571$	1.96988e11			1.85309			0.926543	−	0.376189i	$-0.122766\pi$
$571$	1.96988e11			1.85309			0.926543	+	0.376189i	$0.122766\pi$
$572$	1.58122e9			0.0147710
$573$	5.00576e9	−	1.47035e9i	0.0464357	−	0.0136396i
$574$	−6.26650e9			−0.0577268
$575$		0			0
$576$	−8.93092e10	+	5.74197e10i	−0.811346	+	0.521641i
$577$		−	2.39252e10i		−	0.215850i	−0.994159	−	0.107925i	$-0.965579\pi$
$577$		−	2.39252e10i		−	0.215850i	0.994159	−	0.107925i	$-0.0344206\pi$
$578$	−2.58064e11			−2.31215
$579$	−7.11204e10	+	2.08903e10i	−0.632820	+	0.185879i
$580$		0			0
$581$			4.15153e10i			0.364337i
$582$	1.45580e10	−	4.27614e9i	0.126885	−	0.0372701i
$583$			4.30118e10i			0.372317i
$584$			1.50976e11i			1.29795i
$585$		0			0
$586$	4.10866e10			0.348426
$587$	−2.02670e11			−1.70701			−0.853505	−	0.521085i	$-0.825528\pi$
$587$	−2.02670e11			−1.70701			−0.853505	+	0.521085i	$0.825528\pi$
$588$	1.05727e9	+	3.59945e9i	0.00884456	+	0.0301111i
$589$	8.94945e9			0.0743593
$590$		0			0
$591$	4.54369e10	+	1.54689e11i	0.372442	+	1.26797i
$592$			1.83573e11i			1.49459i
$593$	1.89947e11			1.53608			0.768039	−	0.640403i	$-0.221233\pi$
$593$	1.89947e11			1.53608			0.768039	+	0.640403i	$0.221233\pi$
$594$	−3.72467e10	+	4.30100e10i	−0.299187	+	0.345481i
$595$		0			0
$596$			3.80200e9i			0.0301320i
$597$	−5.33698e10	−	1.81696e11i	−0.420144	−	1.43037i
$598$			7.94958e10i			0.621640i
$599$			3.15053e10i			0.244724i	0.992486	+	0.122362i	$0.0390469\pi$
$599$			3.15053e10i			0.244724i	−0.992486	+	0.122362i	$0.960953\pi$
$600$		0			0
$601$	2.47794e10			0.189930			0.0949649	−	0.995481i	$-0.469726\pi$
$601$	2.47794e10			0.189930			0.0949649	+	0.995481i	$0.469726\pi$
$602$	−3.90704e10			−0.297483
$603$	1.02019e11	+	1.58678e11i	0.771636	+	1.20018i
$604$	−2.05603e9			−0.0154483
$605$		0			0
$606$	−1.24068e11	+	3.64425e10i	−0.919958	+	0.270220i
$607$		−	6.54251e10i		−	0.481936i	−0.970533	−	0.240968i	$-0.922535\pi$
$607$		−	6.54251e10i		−	0.481936i	0.970533	−	0.240968i	$-0.0774649\pi$
$608$	−2.89830e9			−0.0212094
$609$	−1.01918e10	−	3.46979e10i	−0.0740940	−	0.252252i
$610$		0			0
$611$		−	2.50286e11i		−	1.79586i
$612$	7.10137e9	−	4.56570e9i	0.0506216	−	0.0325463i
$613$		−	1.70377e11i		−	1.20661i	−0.797509	−	0.603307i	$-0.793849\pi$
$613$		−	1.70377e11i		−	1.20661i	0.797509	−	0.603307i	$-0.206151\pi$
$614$		−	1.67762e11i		−	1.18038i
$615$		0			0
$616$	−1.50999e10			−0.104870
$617$	−8.96664e10			−0.618713			−0.309356	−	0.950946i	$-0.600114\pi$
$617$	−8.96664e10			−0.618713			−0.309356	+	0.950946i	$0.600114\pi$
$618$	2.10982e10	−	6.19720e9i	0.144641	−	0.0424856i
$619$	−1.34052e11			−0.913085			−0.456542	−	0.889702i	$-0.650912\pi$
$619$	−1.34052e11			−0.913085			−0.456542	+	0.889702i	$0.650912\pi$
$620$		0			0
$621$	−6.95974e10	−	6.02714e10i	−0.467979	−	0.405271i
$622$			2.80315e11i			1.87277i
$623$	3.41502e10			0.226694
$624$	−1.48322e11	+	4.35667e10i	−0.978289	+	0.287354i
$625$		0			0
$626$			8.82804e10i			0.574867i
$627$	2.12681e10	−	6.24711e9i	0.137613	−	0.0404212i
$628$		−	9.57693e9i		−	0.0615726i
$629$			4.10170e11i			2.62036i
$630$		0			0
$631$	−3.60073e9			−0.0227130			−0.0113565	−	0.999936i	$-0.503615\pi$
$631$	−3.60073e9			−0.0227130			−0.0113565	+	0.999936i	$0.503615\pi$
$632$	−1.70150e11			−1.06651
$633$	−8.50804e8	−	2.89654e9i	−0.00529925	−	0.0180412i
$634$	1.90256e11			1.17756
$635$		0			0
$636$	1.26990e9	+	4.32333e9i	0.00776140	+	0.0264235i
$637$		−	1.53486e11i		−	0.932204i
$638$	8.38730e10			0.506220
$639$	−1.32687e11	−	2.06377e11i	−0.795837	−	1.23782i
$640$		0			0
$641$			1.77443e11i			1.05106i	0.850775	+	0.525530i	$0.176133\pi$
$641$			1.77443e11i			1.05106i	−0.850775	+	0.525530i	$0.823867\pi$
$642$	−1.58967e10	−	5.41199e10i	−0.0935764	−	0.318579i
$643$		−	1.47027e11i		−	0.860110i	−0.902803	−	0.430055i	$-0.858494\pi$
$643$		−	1.47027e11i		−	0.860110i	0.902803	−	0.430055i	$-0.141506\pi$
$644$			8.40558e8i			0.00488680i
$645$		0			0
$646$	−1.02191e11			−0.586789
$647$	2.73529e11			1.56094			0.780471	−	0.625192i	$-0.214979\pi$
$647$	2.73529e11			1.56094			0.780471	+	0.625192i	$0.214979\pi$
$648$	7.19171e10	−	1.57637e11i	0.407880	−	0.894040i
$649$	1.02004e11			0.574961
$650$		0			0
$651$	9.53443e9	−	2.80055e9i	0.0530849	−	0.0155927i
$652$		−	8.94651e9i		−	0.0495066i
$653$	1.23992e11			0.681933			0.340967	−	0.940075i	$-0.389246\pi$
$653$	1.23992e11			0.681933			0.340967	+	0.940075i	$0.389246\pi$
$654$	5.55799e10	+	1.89221e11i	0.303813	+	1.03433i
$655$		0			0
$656$			4.57329e10i			0.246953i
$657$	−1.33089e11	−	2.07003e11i	−0.714301	−	1.11100i
$658$		−	8.22219e10i		−	0.438615i
$659$		−	2.37796e11i		−	1.26085i	−0.776251	−	0.630424i	$-0.782881\pi$
$659$		−	2.37796e11i		−	1.26085i	0.776251	−	0.630424i	$-0.217119\pi$
$660$		0			0
$661$	8.33114e10			0.436414			0.218207	−	0.975903i	$-0.429979\pi$
$661$	8.33114e10			0.436414			0.218207	+	0.975903i	$0.429979\pi$
$662$	9.81740e10			0.511169
$663$	−3.31406e11	+	9.73441e10i	−1.71517	+	0.503797i
$664$	2.93217e11			1.50840
$665$		0			0
$666$	−1.56610e11	−	2.43587e11i	−0.796016	−	1.23810i
$667$			1.35721e11i			0.685713i
$668$	1.29890e9			0.00652334
$669$	1.95104e11	−	5.73082e10i	0.974008	−	0.286096i
$670$		0			0
$671$		−	5.71605e10i		−	0.281972i
$672$	−3.08775e9	+	9.06967e8i	−0.0151414	+	0.00444748i
$673$		−	3.96318e11i		−	1.93189i	−0.258742	−	0.965947i	$-0.583308\pi$
$673$		−	3.96318e11i		−	1.93189i	0.258742	−	0.965947i	$-0.416692\pi$
$674$		−	2.20335e10i		−	0.106769i
$675$		0			0
$676$	1.67626e8			0.000802700
$677$	3.22020e11			1.53295			0.766474	−	0.642275i	$-0.222009\pi$
$677$	3.22020e11			1.53295			0.766474	+	0.642275i	$0.222009\pi$
$678$	−4.74551e9	−	1.61560e10i	−0.0224577	−	0.0764566i
$679$	−6.56385e9			−0.0308802
$680$		0			0
$681$	−2.55140e10	−	8.68617e10i	−0.118629	−	0.403868i
$682$			2.30470e10i			0.106531i
$683$	3.92810e10			0.180510			0.0902548	−	0.995919i	$-0.471232\pi$
$683$	3.92810e10			0.180510			0.0902548	+	0.995919i	$0.471232\pi$
$684$	1.95333e9	−	1.25586e9i	0.00892382	−	0.00573741i
$685$		0			0
$686$		−	1.03854e11i		−	0.468951i
$687$	−7.51387e10	−	2.55808e11i	−0.337316	−	1.14839i
$688$			2.85136e11i			1.27262i
$689$		−	1.84353e11i		−	0.818039i
$690$		0			0
$691$	3.54272e10			0.155391			0.0776954	−	0.996977i	$-0.475244\pi$
$691$	3.54272e10			0.155391			0.0776954	+	0.996977i	$0.475244\pi$
$692$	−1.26742e10			−0.0552707
$693$	2.07034e10	−	1.33109e10i	0.0897654	−	0.0577131i
$694$	4.02279e11			1.73416
$695$		0			0
$696$	−2.45066e11	+	7.19835e10i	−1.04435	+	0.306758i
$697$			1.02184e11i			0.432965i
$698$	−1.47726e11			−0.622350
$699$	3.98697e10	+	1.35736e11i	0.167007	+	0.568571i
$700$		0			0
$701$		−	3.64764e9i		−	0.0151057i	−0.999971	−	0.00755284i	$-0.997596\pi$
$701$		−	3.64764e9i		−	0.0151057i	0.999971	−	0.00755284i	$-0.00240417\pi$
$702$	1.59644e11	−	1.84346e11i	0.657360	−	0.759075i
$703$			1.12823e11i			0.461929i
$704$			1.06526e11i			0.433677i
$705$		0			0
$706$	2.70458e11			1.08863
$707$	5.59389e10			0.223891
$708$	1.02529e10	−	3.01161e9i	0.0408052	−	0.0119857i
$709$	2.79311e11			1.10536			0.552678	−	0.833395i	$-0.313606\pi$
$709$	2.79311e11			1.10536			0.552678	+	0.833395i	$0.313606\pi$
$710$		0			0
$711$	2.33293e11	−	1.49991e11i	0.912899	−	0.586932i
$712$		−	2.41198e11i		−	0.938541i
$713$	−3.72939e10			−0.144304
$714$	−1.08871e11	+	3.19787e10i	−0.418907	+	0.123046i
$715$		0			0
$716$		−	5.73438e9i		−	0.0218190i
$717$	2.49008e11	−	7.31412e10i	0.942184	−	0.276749i
$718$		−	1.06013e11i		−	0.398898i
$719$			1.75259e11i			0.655789i	0.944714	+	0.327895i	$0.106339\pi$
$719$			1.75259e11i			0.655789i	−0.944714	+	0.327895i	$0.893661\pi$
$720$		0			0
$721$	−9.51265e9			−0.0352015
$722$	2.48110e11			0.913050
$723$	5.30838e10	+	1.80723e11i	0.194271	+	0.661393i
$724$	7.93429e9			0.0288771
$725$		0			0
$726$	−6.34969e10	−	2.16174e11i	−0.228563	−	0.778138i
$727$			3.76305e11i			1.34711i	0.739137	+	0.673555i	$0.235233\pi$
$727$			3.76305e11i			1.34711i	−0.739137	+	0.673555i	$0.764767\pi$
$728$	6.47199e10			0.230416
$729$	4.03549e10	+	2.79532e11i	0.142885	+	0.989739i
$730$		0			0
$731$			6.37098e11i			2.23119i
$732$	−1.68763e9	−	5.74550e9i	−0.00587804	−	0.0200117i
$733$		−	2.94004e11i		−	1.01845i	−0.860635	−	0.509223i	$-0.829933\pi$
$733$		−	2.94004e11i		−	1.01845i	0.860635	−	0.509223i	$-0.170067\pi$
$734$		−	2.79850e11i		−	0.964141i
$735$		0			0
$736$	1.20777e10			0.0411598
$737$	1.89268e11			0.641515
$738$	−3.90156e10	−	6.06839e10i	−0.131527	−	0.204573i
$739$	−2.57966e11			−0.864939			−0.432470	−	0.901649i	$-0.642358\pi$
$739$	−2.57966e11			−0.864939			−0.432470	+	0.901649i	$0.642358\pi$
$740$		0			0
$741$	−9.11577e10	+	2.67758e10i	−0.302357	+	0.0888117i
$742$		−	6.05622e10i		−	0.199796i
$743$	−8.70436e10			−0.285615			−0.142808	−	0.989750i	$-0.545613\pi$
$743$	−8.70436e10			−0.285615			−0.142808	+	0.989750i	$0.545613\pi$
$744$	−1.97799e10	−	6.73403e10i	−0.0645555	−	0.219778i
$745$		0			0
$746$		−	2.40023e11i		−	0.774992i
$747$	−4.02029e11	+	2.58477e11i	−1.29114	+	0.830118i
$748$		−	8.47038e9i		−	0.0270580i
$749$			2.44013e10i			0.0775328i
$750$		0			0
$751$	−3.07477e11			−0.966613			−0.483306	−	0.875451i	$-0.660564\pi$
$751$	−3.07477e11			−0.966613			−0.483306	+	0.875451i	$0.660564\pi$
$752$	−6.00056e11			−1.87638
$753$	−2.63074e11	+	7.72728e10i	−0.818271	+	0.240351i
$754$	−3.59489e11			−1.11225
$755$		0			0
$756$	1.68801e9	−	1.94920e9i	0.00516759	−	0.00596719i
$757$		−	3.68112e11i		−	1.12098i	−0.828163	−	0.560488i	$-0.810614\pi$
$757$		−	3.68112e11i		−	1.12098i	0.828163	−	0.560488i	$-0.189386\pi$
$758$	5.34725e11			1.61977
$759$	−8.86279e10	+	2.60327e10i	−0.267057	+	0.0784428i
$760$		0			0
$761$			1.79849e11i			0.536252i	0.963384	+	0.268126i	$0.0864043\pi$
$761$			1.79849e11i			0.536252i	−0.963384	+	0.268126i	$0.913596\pi$
$762$	−3.38891e11	+	9.95427e10i	−1.00517	+	0.295250i
$763$		−	8.53148e10i		−	0.251725i
$764$		−	5.48374e8i		−	0.00160955i
$765$		0			0
$766$	1.81661e11			0.527650
$767$	−4.37201e11			−1.26328
$768$	9.77127e9	+	3.32661e10i	0.0280871	+	0.0956218i
$769$	4.66289e11			1.33337			0.666684	−	0.745341i	$-0.267713\pi$
$769$	4.66289e11			1.33337			0.666684	+	0.745341i	$0.267713\pi$
$770$		0			0
$771$	−2.21673e10	−	7.54681e10i	−0.0627329	−	0.213573i
$772$			7.79114e9i			0.0219347i
$773$	3.85685e10			0.108023			0.0540114	−	0.998540i	$-0.482799\pi$
$773$	3.85685e10			0.108023			0.0540114	+	0.998540i	$0.482799\pi$
$774$	−2.43255e11	−	3.78352e11i	−0.677794	−	1.05422i
$775$		0			0
$776$			4.63596e10i			0.127848i
$777$	3.53057e10	+	1.20197e11i	0.0968636	+	0.329770i
$778$			5.11002e11i			1.39478i
$779$			2.81072e10i			0.0763250i
$780$		0			0
$781$	−2.46163e11			−0.661636
$782$	4.25847e11			1.13875
$783$	2.72555e11	−	3.14728e11i	0.725114	−	0.837314i
$784$	−3.67980e11			−0.974001
$785$		0			0
$786$	6.96010e9	−	2.04439e9i	0.0182358	−	0.00535642i
$787$			5.13745e11i			1.33921i	0.742717	+	0.669605i	$0.233537\pi$
$787$			5.13745e11i			1.33921i	−0.742717	+	0.669605i	$0.766463\pi$
$788$	1.69459e10			0.0439502
$789$	−4.71634e9	−	1.60567e10i	−0.0121702	−	0.0414331i
$790$		0			0
$791$			7.28433e9i			0.0186073i
$792$	−9.40130e10	−	1.46225e11i	−0.238939	−	0.371640i
$793$			2.44996e11i			0.619537i
$794$		−	1.42054e11i		−	0.357413i
$795$		0			0
$796$	−1.99046e10			−0.0495793
$797$	−5.20760e11			−1.29064			−0.645318	−	0.763914i	$-0.723275\pi$
$797$	−5.20760e11			−1.29064			−0.645318	+	0.763914i	$0.723275\pi$
$798$	−2.99464e10	+	8.79617e9i	−0.0738469	+	0.0216911i
$799$	−1.34074e12			−3.28972
$800$		0			0
$801$	2.12621e11	+	3.30706e11i	0.516508	+	0.803362i
$802$			2.07961e10i			0.0502672i
$803$	−2.46910e11			−0.593849
$804$	1.90243e10	−	5.58802e9i	0.0455286	−	0.0133732i
$805$		0			0
$806$		−	9.87820e10i		−	0.234066i
$807$	2.43092e11	−	7.14037e10i	0.573162	−	0.168355i
$808$		−	3.95089e11i		−	0.926935i
$809$			7.88009e11i			1.83966i	0.392320	+	0.919829i	$0.371673\pi$
$809$			7.88009e11i			1.83966i	−0.392320	+	0.919829i	$0.628327\pi$
$810$		0			0
$811$	5.15883e11			1.19253			0.596263	−	0.802789i	$-0.296651\pi$
$811$	5.15883e11			1.19253			0.596263	+	0.802789i	$0.296651\pi$
$812$	−3.80110e9			−0.00874350
$813$	1.05248e11	+	3.58314e11i	0.240908	+	0.820165i
$814$	−2.90546e11			−0.661785
$815$		0			0
$816$	2.33381e11	+	7.94539e11i	0.526386	+	1.79207i
$817$			1.75243e11i			0.393325i
$818$	1.67208e11			0.373461
$819$	−8.87372e10	+	5.70520e10i	−0.197229	+	0.126805i
$820$		0			0
$821$		−	5.32514e11i		−	1.17208i	−0.810281	−	0.586041i	$-0.800686\pi$
$821$		−	5.32514e11i		−	1.17208i	0.810281	−	0.586041i	$-0.199314\pi$
$822$	2.06813e11	+	7.04092e11i	0.452993	+	1.54220i
$823$			6.35863e11i			1.38600i	0.720936	+	0.693002i	$0.243712\pi$
$823$			6.35863e11i			1.38600i	−0.720936	+	0.693002i	$0.756288\pi$
$824$			6.71865e10i			0.145738i
$825$		0			0
$826$	−1.43626e11			−0.308540
$827$	−1.67343e10			−0.0357756			−0.0178878	−	0.999840i	$-0.505694\pi$
$827$	−1.67343e10			−0.0357756			−0.0178878	+	0.999840i	$0.505694\pi$
$828$	−8.13985e9	+	5.23337e9i	−0.0173179	+	0.0111342i
$829$	5.88172e11			1.24534			0.622668	−	0.782486i	$-0.286049\pi$
$829$	5.88172e11			1.24534			0.622668	+	0.782486i	$0.286049\pi$
$830$		0			0
$831$	−1.37904e11	+	4.05068e10i	−0.289184	+	0.0849422i
$832$		−	4.56584e11i		−	0.952856i
$833$	−8.22201e11			−1.70765
$834$	−1.24157e10	−	4.22690e10i	−0.0256630	−	0.0873691i
$835$		0			0
$836$		−	2.32989e9i		−	0.00476992i
$837$	8.64822e10	+	7.48937e10i	0.176208	+	0.152596i
$838$			5.86419e11i			1.18914i
$839$		−	9.03790e11i		−	1.82398i	−0.410215	−	0.911989i	$-0.634546\pi$
$839$		−	9.03790e11i		−	1.82398i	0.410215	−	0.911989i	$-0.365454\pi$
$840$		0			0
$841$	−1.13498e11			−0.226885
$842$	5.91983e11			1.17777
$843$	−6.06643e9	+	1.78190e9i	−0.0120122	+	0.00352836i
$844$	−3.17312e8			−0.000625341
$845$		0			0
$846$	7.96225e11	−	5.11919e11i	1.55437	−	0.999355i
$847$			9.74673e10i			0.189376i
$848$	−4.41984e11			−0.854718
$849$	6.31172e11	−	1.85395e11i	1.21483	−	0.356834i
$850$		0			0
$851$		−	4.70152e11i		−	0.896437i
$852$	−2.47431e10	+	7.26782e9i	−0.0469566	+	0.0137926i
$853$			2.17472e11i			0.410779i	0.978680	+	0.205389i	$0.0658461\pi$
$853$			2.17472e11i			0.410779i	−0.978680	+	0.205389i	$0.934154\pi$
$854$			8.04842e10i			0.151314i
$855$		0			0
$856$	1.72343e11			0.320995
$857$	−5.72821e10			−0.106193			−0.0530964	−	0.998589i	$-0.516909\pi$
$857$	−5.72821e10			−0.106193			−0.0530964	+	0.998589i	$0.516909\pi$
$858$	−6.89541e10	−	2.34753e11i	−0.127236	−	0.433173i
$859$	−2.59635e11			−0.476860			−0.238430	−	0.971160i	$-0.576633\pi$
$859$	−2.59635e11			−0.476860			−0.238430	+	0.971160i	$0.576633\pi$
$860$		0			0
$861$	8.79558e9	+	2.99444e10i	0.0160049	+	0.0544882i
$862$		−	4.93697e11i		−	0.894194i
$863$	9.75428e10			0.175854			0.0879270	−	0.996127i	$-0.471976\pi$
$863$	9.75428e10			0.175854			0.0879270	+	0.996127i	$0.471976\pi$
$864$	−2.80075e10	−	2.42545e10i	−0.0502596	−	0.0435249i
$865$		0			0
$866$			5.62424e11i			0.999982i
$867$	3.62216e11	+	1.23316e12i	0.641050	+	2.18244i
$868$		−	1.04448e9i		−	0.00184002i
$869$		−	2.78267e11i		−	0.487958i
$870$		0			0
$871$	−8.11224e11			−1.40951
$872$	−6.02566e11			−1.04217
$873$	−4.08670e10	−	6.35634e10i	−0.0703584	−	0.109434i
$874$	1.17135e11			0.200743
$875$		0			0
$876$	−2.48182e10	+	7.28986e9i	−0.0421457	+	0.0123795i
$877$		−	9.16421e11i		−	1.54916i	−0.632476	−	0.774580i	$-0.717961\pi$
$877$		−	9.16421e11i		−	1.54916i	0.632476	−	0.774580i	$-0.282039\pi$
$878$	−2.87866e10			−0.0484409
$879$	−5.76688e10	−	1.96332e11i	−0.0966018	−	0.328879i
$880$		0			0
$881$		−	5.22036e11i		−	0.866557i	−0.901260	−	0.433279i	$-0.857357\pi$
$881$		−	5.22036e11i		−	0.866557i	0.901260	−	0.433279i	$-0.142643\pi$
$882$	4.88279e11	−	3.13930e11i	0.806852	−	0.518751i
$883$		−	2.66487e11i		−	0.438363i	−0.975684	−	0.219181i	$-0.929661\pi$
$883$		−	2.66487e11i		−	0.438363i	0.975684	−	0.219181i	$-0.0703386\pi$
$884$			3.63050e10i			0.0594508i
$885$		0			0
$886$	5.19917e11			0.843722
$887$	4.32506e11			0.698710			0.349355	−	0.936990i	$-0.386401\pi$
$887$	4.32506e11			0.698710			0.349355	+	0.936990i	$0.386401\pi$
$888$	8.48938e11	−	2.49359e11i	1.36529	−	0.401027i
$889$	1.52797e11			0.244630
$890$		0			0
$891$	2.57802e11	+	1.17615e11i	0.409049	+	0.186617i
$892$		−	2.13734e10i		−	0.0337609i
$893$	−3.68790e11			−0.579927
$894$	5.64456e11	−	1.65798e11i	0.883649	−	0.259555i
$895$		0			0
$896$		−	1.60164e11i		−	0.248504i
$897$	3.79870e11	−	1.11579e11i	0.586766	−	0.172351i
$898$		−	1.30232e12i		−	2.00269i
$899$		−	1.68647e11i		−	0.258191i
$900$		0			0
$901$	−9.87554e11			−1.49852
$902$	−7.23826e10			−0.109347
$903$	5.48388e10	+	1.86697e11i	0.0824777	+	0.280794i
$904$	5.14482e10			0.0770365
$905$		0			0
$906$	8.96593e10	+	3.05243e11i	0.133071	+	0.453037i
$907$			1.46416e10i			0.0216352i	0.999941	+	0.0108176i	$0.00344341\pi$
$907$			1.46416e10i			0.0216352i	−0.999941	+	0.0108176i	$0.996557\pi$
$908$	−9.51557e9			−0.0139988
$909$	3.48280e11	+	5.41705e11i	0.510120	+	0.793428i
$910$		0			0
$911$		−	4.82789e11i		−	0.700944i	−0.936573	−	0.350472i	$-0.886021\pi$
$911$		−	4.82789e11i		−	0.700944i	0.936573	−	0.350472i	$-0.113979\pi$
$912$	6.41945e10	+	2.18549e11i	0.0927937	+	0.315914i
$913$			4.79532e11i			0.690136i
$914$			8.86737e11i			1.27060i
$915$		0			0
$916$	−2.80234e10			−0.0398051
$917$	−3.13813e9			−0.00443807
$918$	−9.87513e11	−	8.55187e11i	−1.39050	−	1.20418i
$919$	−2.79672e11			−0.392090			−0.196045	−	0.980595i	$-0.562810\pi$
$919$	−2.79672e11			−0.392090			−0.196045	+	0.980595i	$0.562810\pi$
$920$		0			0
$921$	−8.01650e11	+	2.35469e11i	−1.11416	+	0.327262i
$922$			5.96134e9i			0.00824936i
$923$	1.05508e12			1.45372
$924$	−7.29094e8	−	2.48219e9i	−0.00100022	−	0.00340523i
$925$		0			0
$926$		−	5.25034e11i		−	0.714074i
$927$	−5.92264e10	−	9.21193e10i	−0.0802041	−	0.124747i
$928$			5.46169e10i			0.0736436i
$929$			9.95357e10i			0.133634i	0.997765	+	0.0668169i	$0.0212843\pi$
$929$			9.95357e10i			0.133634i	−0.997765	+	0.0668169i	$0.978716\pi$
$930$		0			0
$931$	−2.26158e11			−0.301032
$932$	1.48696e10			0.0197077
$933$	1.33948e12	−	3.93447e11i	1.76771	−	0.519230i
$934$	−3.44432e11			−0.452601
$935$		0			0
$936$	4.02951e11	+	6.26739e11i	0.524987	+	0.816551i
$937$		−	8.34167e11i		−	1.08217i	−0.840968	−	0.541084i	$-0.818014\pi$
$937$		−	8.34167e11i		−	1.08217i	0.840968	−	0.541084i	$-0.181986\pi$
$938$	−2.66497e11			−0.344255
$939$	4.21847e11	−	1.23909e11i	0.542616	−	0.159383i
$940$		0			0
$941$			2.45126e11i			0.312630i	0.987707	+	0.156315i	$0.0499615\pi$
$941$			2.45126e11i			0.312630i	−0.987707	+	0.156315i	$0.950039\pi$
$942$	−1.42182e12	+	4.17632e11i	−1.80568	+	0.530383i
$943$		−	1.17127e11i		−	0.148119i
$944$			1.04818e12i			1.31992i
$945$		0			0
$946$	−4.51292e11			−0.563498
$947$	−1.34423e12			−1.67137			−0.835685	−	0.549209i	$-0.814929\pi$
$947$	−1.34423e12			−1.67137			−0.835685	+	0.549209i	$0.814929\pi$
$948$	−8.21567e9	−	2.79701e10i	−0.0101721	−	0.0346306i
$949$	1.05828e12			1.30478
$950$		0			0
$951$	−2.67042e11	−	9.09137e11i	−0.326480	−	1.11149i
$952$		−	3.46695e11i		−	0.422085i
$953$	1.02998e12			1.24870			0.624351	−	0.781144i	$-0.285363\pi$
$953$	1.02998e12			1.24870			0.624351	+	0.781144i	$0.285363\pi$
$954$	5.86477e11	−	3.77065e11i	0.708039	−	0.455221i
$955$		0			0
$956$		−	2.72784e10i		−	0.0326578i
$957$	−1.17723e11	−	4.00786e11i	−0.140351	−	0.477820i
$958$		−	7.62802e11i		−	0.905628i
$959$		−	3.17457e11i		−	0.375328i
$960$		0			0
$961$	−8.06549e11			−0.945665
$962$	1.24531e12			1.45405
$963$	−2.36299e11	+	1.51924e11i	−0.274762	+	0.176653i
$964$	1.97979e10			0.0229251
$965$		0			0
$966$	1.24792e11	−	3.66551e10i	0.143310	−	0.0420946i
$967$		−	8.83788e11i		−	1.01075i	−0.862901	−	0.505373i	$-0.831355\pi$
$967$		−	8.83788e11i		−	1.01075i	0.862901	−	0.505373i	$-0.168645\pi$
$968$	6.88398e11			0.784040
$969$	1.43434e11	+	4.88318e11i	0.162689	+	0.553870i
$970$		0			0
$971$			1.36295e12i			1.53321i	0.642117	+	0.766607i	$0.278057\pi$
$971$			1.36295e12i			1.53321i	−0.642117	+	0.766607i	$0.721943\pi$
$972$	2.93855e10	+	4.21060e9i	0.0329206	+	0.00471714i
$973$			1.90580e10i			0.0212631i
$974$			2.19184e11i			0.243541i
$975$		0			0
$976$	5.87374e11			0.647315
$977$	8.27099e10			0.0907777			0.0453888	−	0.998969i	$-0.485547\pi$
$977$	8.27099e10			0.0907777			0.0453888	+	0.998969i	$0.485547\pi$
$978$	−1.32822e12	+	3.90140e11i	−1.45183	+	0.426447i
$979$	3.94459e11			0.429409
$980$		0			0
$981$	8.26177e11	−	5.31176e11i	0.892066	−	0.573538i
$982$		−	1.34528e12i		−	1.44666i
$983$	−1.05894e12			−1.13412			−0.567058	−	0.823678i	$-0.691918\pi$
$983$	−1.05894e12			−1.13412			−0.567058	+	0.823678i	$0.691918\pi$
$984$	2.11493e11	−	6.21220e10i	0.225588	−	0.0662621i
$985$		0			0
$986$			1.92573e12i			2.03745i
$987$	−3.92896e11	+	1.15406e11i	−0.414008	+	0.121607i
$988$			9.98619e9i			0.0104803i
$989$		−	7.30266e11i		−	0.763301i
$990$		0			0
$991$	1.36731e11			0.141766			0.0708828	−	0.997485i	$-0.477418\pi$
$991$	1.36731e11			0.141766			0.0708828	+	0.997485i	$0.477418\pi$
$992$	−1.50078e10			−0.0154979
$993$	−1.37796e11	−	4.69123e11i	−0.141723	−	0.482492i
$994$	3.46607e11			0.355052
$995$		0			0
$996$	1.41579e10	+	4.82003e10i	0.0143867	+	0.0489793i
$997$			1.12309e12i			1.13667i	0.822798	+	0.568334i	$0.192412\pi$
$997$			1.12309e12i			1.13667i	−0.822798	+	0.568334i	$0.807588\pi$
$998$	3.20317e11			0.322892
$999$	−9.44160e11	+	1.09025e12i	−0.947947	+	1.09463i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.9.d.d.74.8		20
3.2	odd	2	inner	75.9.d.d.74.14		20
5.2	odd	4		75.9.c.f.26.4	yes	10
5.3	odd	4		75.9.c.e.26.7	yes	10
5.4	even	2	inner	75.9.d.d.74.13		20
15.2	even	4		75.9.c.f.26.7	yes	10
15.8	even	4		75.9.c.e.26.4	✓	10
15.14	odd	2	inner	75.9.d.d.74.7		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.9.c.e.26.4	✓	10	15.8	even	4
75.9.c.e.26.7	yes	10	5.3	odd	4
75.9.c.f.26.4	yes	10	5.2	odd	4
75.9.c.f.26.7	yes	10	15.2	even	4
75.9.d.d.74.7		20	15.14	odd	2	inner
75.9.d.d.74.8		20	1.1	even	1	trivial
75.9.d.d.74.13		20	5.4	even	2	inner
75.9.d.d.74.14		20	3.2	odd	2	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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-16.2639 q^{2} +(22.8278 + 77.7167i) q^{3} +8.51375 q^{4} +(-371.269 - 1263.98i) q^{6} +569.895i q^{7} +4025.09 q^{8} +(-5518.78 + 3548.20i) q^{9} +O(q^{10})\)	\(q-16.2639 q^{2} +(22.8278 + 77.7167i) q^{3} +8.51375 q^{4} +(-371.269 - 1263.98i) q^{6} +569.895i q^{7} +4025.09 q^{8} +(-5518.78 + 3548.20i) q^{9} +6582.70i q^{11} +(194.350 + 661.661i) q^{12} -28214.2i q^{13} -9268.70i q^{14} -67643.0 q^{16} -151139. q^{17} +(89756.8 - 57707.6i) q^{18} -41572.9 q^{19} +(-44290.4 + 13009.4i) q^{21} -107060. i q^{22} +173241. q^{23} +(91883.9 + 312817. i) q^{24} +458873. i q^{26} +(-401737. - 347904. i) q^{27} +4851.94i q^{28} +783419. i q^{29} -215271. q^{31} +69716.1 q^{32} +(-511586. + 150269. i) q^{33} +2.45811e6 q^{34} +(-46985.6 + 30208.5i) q^{36} -2.71385e6i q^{37} +676137. q^{38} +(2.19272e6 - 644068. i) q^{39} -676092. i q^{41} +(720333. - 211584. i) q^{42} -4.21531e6i q^{43} +56043.5i q^{44} -2.81758e6 q^{46} +8.87092e6 q^{47} +(-1.54414e6 - 5.25700e6i) q^{48} +5.44002e6 q^{49} +(-3.45018e6 - 1.17461e7i) q^{51} -240209. i q^{52} +6.53406e6 q^{53} +(6.53379e6 + 5.65827e6i) q^{54} +2.29387e6i q^{56} +(-949019. - 3.23091e6i) q^{57} -1.27414e7i q^{58} -1.54958e7i q^{59} -8.68344e6 q^{61} +3.50114e6 q^{62} +(-2.02210e6 - 3.14512e6i) q^{63} +1.61828e7 q^{64} +(8.32037e6 - 2.44395e6i) q^{66} -2.87523e7i q^{67} -1.28676e6 q^{68} +(3.95472e6 + 1.34638e7i) q^{69} +3.73955e7i q^{71} +(-2.22136e7 + 1.42818e7i) q^{72} +3.75089e7i q^{73} +4.41378e7i q^{74} -353942. q^{76} -3.75145e6 q^{77} +(-3.56621e7 + 1.04751e7i) q^{78} -4.22725e7 q^{79} +(1.78672e7 - 3.91635e7i) q^{81} +1.09959e7i q^{82} +7.28474e7 q^{83} +(-377077. + 110759. i) q^{84} +6.85572e7i q^{86} +(-6.08847e7 + 1.78837e7i) q^{87} +2.64959e7i q^{88} -5.99236e7i q^{89} +1.60791e7 q^{91} +1.47494e6 q^{92} +(-4.91416e6 - 1.67302e7i) q^{93} -1.44276e8 q^{94} +(1.59146e6 + 5.41811e6i) q^{96} +1.15177e7i q^{97} -8.84758e7 q^{98} +(-2.33568e7 - 3.63285e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9}+O(q^{10}) \) 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9	\( 20 q + 3108 q^{4} + 4514 q^{6} + 22414 q^{9} + 89268 q^{16} + 287868 q^{19} + 1346856 q^{21} + 2033718 q^{24} - 6028120 q^{31} - 9954292 q^{34} + 9001054 q^{36} + 15026564 q^{39} - 27877272 q^{46} - 17907092 q^{49} + 12418574 q^{51} + 17772544 q^{54} + 3040440 q^{61} + 9073996 q^{64} + 20930590 q^{66} - 22789956 q^{69} - 59058092 q^{76} - 17099792 q^{79} + 45225890 q^{81} + 273766584 q^{84} - 214448528 q^{91} - 712237192 q^{94} + 1050848002 q^{96} + 764842670 q^{99}+O(q^{100}) \) 20 * q + 3108 * q^4 + 4514 * q^6 + 22414 * q^9 + 89268 * q^16 + 287868 * q^19 + 1346856 * q^21 + 2033718 * q^24 - 6028120 * q^31 - 9954292 * q^34 + 9001054 * q^36 + 15026564 * q^39 - 27877272 * q^46 - 17907092 * q^49 + 12418574 * q^51 + 17772544 * q^54 + 3040440 * q^61 + 9073996 * q^64 + 20930590 * q^66 - 22789956 * q^69 - 59058092 * q^76 - 17099792 * q^79 + 45225890 * q^81 + 273766584 * q^84 - 214448528 * q^91 - 712237192 * q^94 + 1050848002 * q^96 + 764842670 * q^99

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