LMFDB - Embedded newform 2916.3.c.b.1457.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2916,3,Mod(1457,2916)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2916, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2916.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2916 = 2^{2} \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2916.c (of order $2$, degree $1$, 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:	$79.4552450875$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.18
Character	$\chi$	$=$	2916.1457
Dual form			2916.3.c.b.1457.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.200781i q^{5} -8.51501 q^{7} +O(q^{10})$	$q-0.200781i q^{5} -8.51501 q^{7} +7.55263i q^{11} -16.3949 q^{13} -6.02618i q^{17} +0.379891 q^{19} +28.1090i q^{23} +24.9597 q^{25} +41.4339i q^{29} +13.5683 q^{31} +1.70965i q^{35} +4.52381 q^{37} +76.7567i q^{41} -1.74489 q^{43} -76.5424i q^{47} +23.5054 q^{49} -85.8739i q^{53} +1.51642 q^{55} -18.2273i q^{59} -37.5279 q^{61} +3.29178i q^{65} -70.3881 q^{67} +44.9385i q^{71} -102.499 q^{73} -64.3107i q^{77} +84.4674 q^{79} -68.7695i q^{83} -1.20994 q^{85} -137.692i q^{89} +139.603 q^{91} -0.0762748i q^{95} -120.162 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q+O(q^{10}) $ 36 * q	$ 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) $ 36 * q - 180 * q^25 + 252 * q^49 + 18 * q^61 - 90 * q^67 + 126 * q^73 - 198 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times$.

$n$	$1459$	$2189$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 0.200781i	− 0.0401562i	−0.999798	−	0.0200781i	$-0.993609\pi$
$5$	− 0.200781i	− 0.0401562i	0.999798	−	0.0200781i	$-0.00639148\pi$
$6$	0	0
$7$	−8.51501	−1.21643	−0.608215	−	0.793772i	$-0.708114\pi$
$7$	−8.51501	−1.21643	−0.608215	+	0.793772i	$0.708114\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.55263i	0.686603i	0.939225	+	0.343301i	$0.111545\pi$
$11$	7.55263i	0.686603i	−0.939225	+	0.343301i	$0.888455\pi$
$12$	0	0
$13$	−16.3949	−1.26115	−0.630573	−	0.776130i	$-0.717180\pi$
$13$	−16.3949	−1.26115	−0.630573	+	0.776130i	$0.717180\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.02618i	− 0.354481i	−0.984168	−	0.177241i	$-0.943283\pi$
$17$	− 6.02618i	− 0.354481i	0.984168	−	0.177241i	$-0.0567171\pi$
$18$	0	0
$19$	0.379891	0.0199943	0.00999714	−	0.999950i	$-0.496818\pi$
$19$	0.379891	0.0199943	0.00999714	+	0.999950i	$0.496818\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	28.1090i	1.22213i	0.791581	+	0.611064i	$0.209258\pi$
$23$	28.1090i	1.22213i	−0.791581	+	0.611064i	$0.790742\pi$
$24$	0	0
$25$	24.9597	0.998387
$26$	0	0
$27$	0	0
$28$	0	0
$29$	41.4339i	1.42876i	0.699760	+	0.714378i	$0.253290\pi$
$29$	41.4339i	1.42876i	−0.699760	+	0.714378i	$0.746710\pi$
$30$	0	0
$31$	13.5683	0.437689	0.218844	−	0.975760i	$-0.429771\pi$
$31$	13.5683	0.437689	0.218844	+	0.975760i	$0.429771\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.70965i	0.0488472i
$36$	0	0
$37$	4.52381	0.122265	0.0611325	−	0.998130i	$-0.480529\pi$
$37$	4.52381	0.122265	0.0611325	+	0.998130i	$0.480529\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	76.7567i	1.87212i	0.351847	+	0.936058i	$0.385554\pi$
$41$	76.7567i	1.87212i	−0.351847	+	0.936058i	$0.614446\pi$
$42$	0	0
$43$	−1.74489	−0.0405789	−0.0202895	−	0.999794i	$-0.506459\pi$
$43$	−1.74489	−0.0405789	−0.0202895	+	0.999794i	$0.506459\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 76.5424i	− 1.62856i	−0.580472	−	0.814280i	$-0.697132\pi$
$47$	− 76.5424i	− 1.62856i	0.580472	−	0.814280i	$-0.302868\pi$
$48$	0	0
$49$	23.5054	0.479703
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 85.8739i	− 1.62026i	−0.586249	−	0.810131i	$-0.699396\pi$
$53$	− 85.8739i	− 1.62026i	0.586249	−	0.810131i	$-0.300604\pi$
$54$	0	0
$55$	1.51642	0.0275713
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 18.2273i	− 0.308937i	−0.987998	−	0.154469i	$-0.950633\pi$
$59$	− 18.2273i	− 0.308937i	0.987998	−	0.154469i	$-0.0493665\pi$
$60$	0	0
$61$	−37.5279	−0.615212	−0.307606	−	0.951514i	$-0.599528\pi$
$61$	−37.5279	−0.615212	−0.307606	+	0.951514i	$0.599528\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.29178i	0.0506428i
$66$	0	0
$67$	−70.3881	−1.05057	−0.525285	−	0.850927i	$-0.676041\pi$
$67$	−70.3881	−1.05057	−0.525285	+	0.850927i	$0.676041\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	44.9385i	0.632936i	0.948603	+	0.316468i	$0.102497\pi$
$71$	44.9385i	0.632936i	−0.948603	+	0.316468i	$0.897503\pi$
$72$	0	0
$73$	−102.499	−1.40409	−0.702047	−	0.712130i	$-0.747731\pi$
$73$	−102.499	−1.40409	−0.702047	+	0.712130i	$0.747731\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 64.3107i	− 0.835204i
$78$	0	0
$79$	84.4674	1.06921	0.534604	−	0.845103i	$-0.320461\pi$
$79$	84.4674	1.06921	0.534604	+	0.845103i	$0.320461\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 68.7695i	− 0.828548i	−0.910152	−	0.414274i	$-0.864036\pi$
$83$	− 68.7695i	− 0.828548i	0.910152	−	0.414274i	$-0.135964\pi$
$84$	0	0
$85$	−1.20994	−0.0142346
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 137.692i	− 1.54710i	−0.633734	−	0.773551i	$-0.718479\pi$
$89$	− 137.692i	− 1.54710i	0.633734	−	0.773551i	$-0.281521\pi$
$90$	0	0
$91$	139.603	1.53410
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 0.0762748i	0 0.000802893i
$96$	0	0
$97$	−120.162	−1.23879	−0.619394	−	0.785081i	$-0.712621\pi$
$97$	−120.162	−1.23879	−0.619394	+	0.785081i	$0.712621\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 13.3213i	− 0.131894i	−0.997823	−	0.0659469i	$-0.978993\pi$
$101$	− 13.3213i	− 0.131894i	0.997823	−	0.0659469i	$-0.0210068\pi$
$102$	0	0
$103$	194.916	1.89239	0.946194	−	0.323600i	$-0.104893\pi$
$103$	194.916	1.89239	0.946194	+	0.323600i	$0.104893\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 155.702i	− 1.45516i	−0.686024	−	0.727579i	$-0.740645\pi$
$107$	− 155.702i	− 1.45516i	0.686024	−	0.727579i	$-0.259355\pi$
$108$	0	0
$109$	70.1664	0.643729	0.321864	−	0.946786i	$-0.395691\pi$
$109$	70.1664	0.643729	0.321864	+	0.946786i	$0.395691\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 104.845i	− 0.927828i	−0.885880	−	0.463914i	$-0.846445\pi$
$113$	− 104.845i	− 0.927828i	0.885880	−	0.463914i	$-0.153555\pi$
$114$	0	0
$115$	5.64374	0.0490760
$116$	0	0
$117$	0	0
$118$	0	0
$119$	51.3130i	0.431202i
$120$	0	0
$121$	63.9578	0.528577
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 10.0309i	− 0.0802476i
$126$	0	0
$127$	−155.690	−1.22591	−0.612954	−	0.790119i	$-0.710019\pi$
$127$	−155.690	−1.22591	−0.612954	+	0.790119i	$0.710019\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 142.811i	− 1.09016i	−0.838384	−	0.545081i	$-0.816499\pi$
$131$	− 142.811i	− 1.09016i	0.838384	−	0.545081i	$-0.183501\pi$
$132$	0	0
$133$	−3.23478	−0.0243216
$134$	0	0
$135$	0	0
$136$	0	0
$137$	192.044i	1.40178i	0.713269	+	0.700891i	$0.247214\pi$
$137$	192.044i	1.40178i	−0.713269	+	0.700891i	$0.752786\pi$
$138$	0	0
$139$	−71.4337	−0.513912	−0.256956	−	0.966423i	$-0.582719\pi$
$139$	−71.4337	−0.513912	−0.256956	+	0.966423i	$0.582719\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 123.825i	− 0.865906i
$144$	0	0
$145$	8.31914	0.0573734
$146$	0	0
$147$	0	0
$148$	0	0
$149$	99.8115i	0.669876i	0.942240	+	0.334938i	$0.108715\pi$
$149$	99.8115i	0.669876i	−0.942240	+	0.334938i	$0.891285\pi$
$150$	0	0
$151$	96.8640	0.641484	0.320742	−	0.947167i	$-0.396068\pi$
$151$	96.8640	0.641484	0.320742	+	0.947167i	$0.396068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.72426i	− 0.0175759i
$156$	0	0
$157$	−240.351	−1.53090	−0.765449	−	0.643496i	$-0.777483\pi$
$157$	−240.351	−1.53090	−0.765449	+	0.643496i	$0.777483\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 239.348i	− 1.48663i
$162$	0	0
$163$	−58.5417	−0.359152	−0.179576	−	0.983744i	$-0.557473\pi$
$163$	−58.5417	−0.359152	−0.179576	+	0.983744i	$0.557473\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 65.5012i	− 0.392223i	−0.980582	−	0.196111i	$-0.937169\pi$
$167$	− 65.5012i	− 0.392223i	0.980582	−	0.196111i	$-0.0628314\pi$
$168$	0	0
$169$	99.7926	0.590489
$170$	0	0
$171$	0	0
$172$	0	0
$173$	75.6348i	0.437195i	0.975815	+	0.218598i	$0.0701482\pi$
$173$	75.6348i	0.437195i	−0.975815	+	0.218598i	$0.929852\pi$
$174$	0	0
$175$	−212.532	−1.21447
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 299.776i	− 1.67473i	−0.546648	−	0.837363i	$-0.684096\pi$
$179$	− 299.776i	− 1.67473i	0.546648	−	0.837363i	$-0.315904\pi$
$180$	0	0
$181$	297.120	1.64155	0.820774	−	0.571253i	$-0.193543\pi$
$181$	297.120	1.64155	0.820774	+	0.571253i	$0.193543\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 0.908294i	− 0.00490970i
$186$	0	0
$187$	45.5135	0.243388
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 81.8527i	− 0.428548i	−0.976774	−	0.214274i	$-0.931261\pi$
$191$	− 81.8527i	− 0.428548i	0.976774	−	0.214274i	$-0.0687386\pi$
$192$	0	0
$193$	117.721	0.609955	0.304977	−	0.952360i	$-0.401351\pi$
$193$	117.721	0.609955	0.304977	+	0.952360i	$0.401351\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 80.6677i	− 0.409481i	−0.978816	−	0.204740i	$-0.934365\pi$
$197$	− 80.6677i	− 0.409481i	0.978816	−	0.204740i	$-0.0656350\pi$
$198$	0	0
$199$	−175.912	−0.883981	−0.441991	−	0.897020i	$-0.645728\pi$
$199$	−175.912	−0.883981	−0.441991	+	0.897020i	$0.645728\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 352.810i	− 1.73798i
$204$	0	0
$205$	15.4113	0.0751769
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.86918i	0.0137281i
$210$	0	0
$211$	36.1935	0.171533	0.0857665	−	0.996315i	$-0.472666\pi$
$211$	36.1935	0.171533	0.0857665	+	0.996315i	$0.472666\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.350341i	0.00162949i
$216$	0	0
$217$	−115.535	−0.532418
$218$	0	0
$219$	0	0
$220$	0	0
$221$	98.7986i	0.447052i
$222$	0	0
$223$	−403.667	−1.81016	−0.905082	−	0.425237i	$-0.860191\pi$
$223$	−403.667	−1.81016	−0.905082	+	0.425237i	$0.860191\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 175.416i	− 0.772758i	−0.922340	−	0.386379i	$-0.873726\pi$
$227$	− 175.416i	− 0.772758i	0.922340	−	0.386379i	$-0.126274\pi$
$228$	0	0
$229$	134.205	0.586049	0.293024	−	0.956105i	$-0.405338\pi$
$229$	134.205	0.586049	0.293024	+	0.956105i	$0.405338\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 395.568i	− 1.69772i	−0.528620	−	0.848858i	$-0.677290\pi$
$233$	− 395.568i	− 1.69772i	0.528620	−	0.848858i	$-0.322710\pi$
$234$	0	0
$235$	−15.3682	−0.0653967
$236$	0	0
$237$	0	0
$238$	0	0
$239$	449.383i	1.88026i	0.340810	+	0.940132i	$0.389299\pi$
$239$	449.383i	1.88026i	−0.340810	+	0.940132i	$0.610701\pi$
$240$	0	0
$241$	−227.933	−0.945782	−0.472891	−	0.881121i	$-0.656790\pi$
$241$	−227.933	−0.945782	−0.472891	+	0.881121i	$0.656790\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.71944i	− 0.0192630i
$246$	0	0
$247$	−6.22828	−0.0252157
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 355.096i	− 1.41472i	−0.706852	−	0.707362i	$-0.749885\pi$
$251$	− 355.096i	− 1.41472i	0.706852	−	0.707362i	$-0.250115\pi$
$252$	0	0
$253$	−212.297	−0.839117
$254$	0	0
$255$	0	0
$256$	0	0
$257$	191.068i	0.743454i	0.928342	+	0.371727i	$0.121234\pi$
$257$	191.068i	0.743454i	−0.928342	+	0.371727i	$0.878766\pi$
$258$	0	0
$259$	−38.5203	−0.148727
$260$	0	0
$261$	0	0
$262$	0	0
$263$	60.7565i	0.231013i	0.993307	+	0.115507i	$0.0368491\pi$
$263$	60.7565i	0.231013i	−0.993307	+	0.115507i	$0.963151\pi$
$264$	0	0
$265$	−17.2418	−0.0650635
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 1.02338i	− 0.00380438i	−0.999998	−	0.00190219i	$-0.999395\pi$
$269$	− 1.02338i	− 0.00380438i	0.999998	−	0.00190219i	$-0.000605486\pi$
$270$	0	0
$271$	264.552	0.976205	0.488102	−	0.872786i	$-0.337689\pi$
$271$	264.552	0.976205	0.488102	+	0.872786i	$0.337689\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	188.511i	0.685496i
$276$	0	0
$277$	363.358	1.31176	0.655882	−	0.754864i	$-0.272297\pi$
$277$	363.358	1.31176	0.655882	+	0.754864i	$0.272297\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 189.525i	− 0.674465i	−0.941421	−	0.337233i	$-0.890509\pi$
$281$	− 189.525i	− 0.674465i	0.941421	−	0.337233i	$-0.109491\pi$
$282$	0	0
$283$	286.812	1.01347	0.506736	−	0.862102i	$-0.330852\pi$
$283$	286.812	1.01347	0.506736	+	0.862102i	$0.330852\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 653.584i	− 2.27730i
$288$	0	0
$289$	252.685	0.874343
$290$	0	0
$291$	0	0
$292$	0	0
$293$	200.349i	0.683785i	0.939739	+	0.341893i	$0.111068\pi$
$293$	200.349i	0.683785i	−0.939739	+	0.341893i	$0.888932\pi$
$294$	0	0
$295$	−3.65969	−0.0124057
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 460.844i	− 1.54128i
$300$	0	0
$301$	14.8578	0.0493614
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.53488i	0.0247045i
$306$	0	0
$307$	175.560	0.571856	0.285928	−	0.958251i	$-0.407698\pi$
$307$	175.560	0.571856	0.285928	+	0.958251i	$0.407698\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 143.494i	− 0.461396i	−0.973025	−	0.230698i	$-0.925899\pi$
$311$	− 143.494i	− 0.461396i	0.973025	−	0.230698i	$-0.0741009\pi$
$312$	0	0
$313$	500.059	1.59763	0.798816	−	0.601575i	$-0.205460\pi$
$313$	500.059	1.59763	0.798816	+	0.601575i	$0.205460\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	161.670i	0.509999i	0.966941	+	0.254999i	$0.0820753\pi$
$317$	161.670i	0.509999i	−0.966941	+	0.254999i	$0.917925\pi$
$318$	0	0
$319$	−312.935	−0.980988
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.28929i	− 0.00708759i
$324$	0	0
$325$	−409.211	−1.25911
$326$	0	0
$327$	0	0
$328$	0	0
$329$	651.759i	1.98103i
$330$	0	0
$331$	460.137	1.39014	0.695071	−	0.718941i	$-0.255373\pi$
$331$	460.137	1.39014	0.695071	+	0.718941i	$0.255373\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.1326i	0.0421868i
$336$	0	0
$337$	343.132	1.01820	0.509098	−	0.860708i	$-0.329979\pi$
$337$	343.132	1.01820	0.509098	+	0.860708i	$0.329979\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	102.477i	0.300518i
$342$	0	0
$343$	217.086	0.632905
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 274.713i	− 0.791681i	−0.918319	−	0.395840i	$-0.870453\pi$
$347$	− 274.713i	− 0.791681i	0.918319	−	0.395840i	$-0.129547\pi$
$348$	0	0
$349$	−492.168	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$349$	−492.168	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	110.831i	0.313969i	0.987601	+	0.156984i	$0.0501772\pi$
$353$	110.831i	0.313969i	−0.987601	+	0.156984i	$0.949823\pi$
$354$	0	0
$355$	9.02278	0.0254163
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 555.325i	− 1.54687i	−0.633878	−	0.773433i	$-0.718538\pi$
$359$	− 555.325i	− 1.54687i	0.633878	−	0.773433i	$-0.281462\pi$
$360$	0	0
$361$	−360.856	−0.999600
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.5798i	0.0563830i
$366$	0	0
$367$	−398.764	−1.08655	−0.543275	−	0.839555i	$-0.682816\pi$
$367$	−398.764	−1.08655	−0.543275	+	0.839555i	$0.682816\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	731.217i	1.97094i
$372$	0	0
$373$	−201.572	−0.540408	−0.270204	−	0.962803i	$-0.587091\pi$
$373$	−201.572	−0.540408	−0.270204	+	0.962803i	$0.587091\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 679.305i	− 1.80187i
$378$	0	0
$379$	135.501	0.357522	0.178761	−	0.983892i	$-0.442791\pi$
$379$	135.501	0.357522	0.178761	+	0.983892i	$0.442791\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 240.674i	− 0.628392i	−0.949358	−	0.314196i	$-0.898265\pi$
$383$	− 240.674i	− 0.628392i	0.949358	−	0.314196i	$-0.101735\pi$
$384$	0	0
$385$	−12.9124	−0.0335386
$386$	0	0
$387$	0	0
$388$	0	0
$389$	100.238i	0.257680i	0.991665	+	0.128840i	$0.0411254\pi$
$389$	100.238i	0.257680i	−0.991665	+	0.128840i	$0.958875\pi$
$390$	0	0
$391$	169.390	0.433222
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 16.9594i	− 0.0429352i
$396$	0	0
$397$	63.1233	0.159001	0.0795003	−	0.996835i	$-0.474668\pi$
$397$	63.1233	0.159001	0.0795003	+	0.996835i	$0.474668\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 388.121i	− 0.967883i	−0.875100	−	0.483941i	$-0.839205\pi$
$401$	− 388.121i	− 0.967883i	0.875100	−	0.483941i	$-0.160795\pi$
$402$	0	0
$403$	−222.452	−0.551989
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.1666i	0.0839475i
$408$	0	0
$409$	−522.802	−1.27824	−0.639122	−	0.769106i	$-0.720702\pi$
$409$	−522.802	−1.27824	−0.639122	+	0.769106i	$0.720702\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	155.206i	0.375801i
$414$	0	0
$415$	−13.8076	−0.0332713
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 505.586i	− 1.20665i	−0.797496	−	0.603325i	$-0.793842\pi$
$419$	− 505.586i	− 1.20665i	0.797496	−	0.603325i	$-0.206158\pi$
$420$	0	0
$421$	65.3960	0.155335	0.0776675	−	0.996979i	$-0.475253\pi$
$421$	65.3960	0.155335	0.0776675	+	0.996979i	$0.475253\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 150.412i	− 0.353909i
$426$	0	0
$427$	319.551	0.748362
$428$	0	0
$429$	0	0
$430$	0	0
$431$	449.973i	1.04402i	0.852939	+	0.522010i	$0.174818\pi$
$431$	449.973i	1.04402i	−0.852939	+	0.522010i	$0.825182\pi$
$432$	0	0
$433$	57.6415	0.133121	0.0665606	−	0.997782i	$-0.478797\pi$
$433$	57.6415	0.133121	0.0665606	+	0.997782i	$0.478797\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.6783i	0.0244356i
$438$	0	0
$439$	−230.876	−0.525914	−0.262957	−	0.964807i	$-0.584698\pi$
$439$	−230.876	−0.525914	−0.262957	+	0.964807i	$0.584698\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	284.788i	0.642862i	0.946933	+	0.321431i	$0.104164\pi$
$443$	284.788i	0.642862i	−0.946933	+	0.321431i	$0.895836\pi$
$444$	0	0
$445$	−27.6459	−0.0621257
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 288.628i	− 0.642823i	−0.946940	−	0.321412i	$-0.895843\pi$
$449$	− 288.628i	− 0.642823i	0.946940	−	0.321412i	$-0.104157\pi$
$450$	0	0
$451$	−579.715	−1.28540
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 28.0295i	− 0.0616034i
$456$	0	0
$457$	74.4253	0.162856	0.0814281	−	0.996679i	$-0.474052\pi$
$457$	74.4253	0.162856	0.0814281	+	0.996679i	$0.474052\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 126.355i	− 0.274090i	−0.990565	−	0.137045i	$-0.956240\pi$
$461$	− 126.355i	− 0.274090i	0.990565	−	0.137045i	$-0.0437605\pi$
$462$	0	0
$463$	45.2142	0.0976549	0.0488275	−	0.998807i	$-0.484452\pi$
$463$	45.2142	0.0976549	0.0488275	+	0.998807i	$0.484452\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 456.678i	− 0.977898i	−0.872312	−	0.488949i	$-0.837380\pi$
$467$	− 456.678i	− 0.977898i	0.872312	−	0.488949i	$-0.162620\pi$
$468$	0	0
$469$	599.356	1.27794
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 13.1785i	− 0.0278616i
$474$	0	0
$475$	9.48196	0.0199620
$476$	0	0
$477$	0	0
$478$	0	0
$479$	375.927i	0.784816i	0.919791	+	0.392408i	$0.128358\pi$
$479$	375.927i	0.784816i	−0.919791	+	0.392408i	$0.871642\pi$
$480$	0	0
$481$	−74.1674	−0.154194
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.1263i	0.0497449i
$486$	0	0
$487$	51.3988	0.105542	0.0527708	−	0.998607i	$-0.483195\pi$
$487$	51.3988	0.105542	0.0527708	+	0.998607i	$0.483195\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 744.279i	− 1.51584i	−0.652346	−	0.757922i	$-0.726215\pi$
$491$	− 744.279i	− 1.51584i	0.652346	−	0.757922i	$-0.273785\pi$
$492$	0	0
$493$	249.688	0.506467
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 382.652i	− 0.769923i
$498$	0	0
$499$	−381.627	−0.764783	−0.382392	−	0.924000i	$-0.624899\pi$
$499$	−381.627	−0.764783	−0.382392	+	0.924000i	$0.624899\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 195.642i	− 0.388951i	−0.980907	−	0.194475i	$-0.937700\pi$
$503$	− 195.642i	− 0.388951i	0.980907	−	0.194475i	$-0.0623004\pi$
$504$	0	0
$505$	−2.67466	−0.00529635
$506$	0	0
$507$	0	0
$508$	0	0
$509$	436.567i	0.857696i	0.903377	+	0.428848i	$0.141080\pi$
$509$	436.567i	0.857696i	−0.903377	+	0.428848i	$0.858920\pi$
$510$	0	0
$511$	872.780	1.70798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 39.1354i	− 0.0759910i
$516$	0	0
$517$	578.096	1.11817
$518$	0	0
$519$	0	0
$520$	0	0
$521$	293.037i	0.562451i	0.959642	+	0.281226i	$0.0907409\pi$
$521$	293.037i	0.562451i	−0.959642	+	0.281226i	$0.909259\pi$
$522$	0	0
$523$	−333.512	−0.637691	−0.318845	−	0.947807i	$-0.603295\pi$
$523$	−333.512	−0.637691	−0.318845	+	0.947807i	$0.603295\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 81.7653i	− 0.155152i
$528$	0	0
$529$	−261.114	−0.493599
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1258.42i	− 2.36101i
$534$	0	0
$535$	−31.2620	−0.0584336
$536$	0	0
$537$	0	0
$538$	0	0
$539$	177.528i	0.329365i
$540$	0	0
$541$	−51.1409	−0.0945304	−0.0472652	−	0.998882i	$-0.515051\pi$
$541$	−51.1409	−0.0945304	−0.0472652	+	0.998882i	$0.515051\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 14.0881i	− 0.0258497i
$546$	0	0
$547$	−450.669	−0.823893	−0.411947	−	0.911208i	$-0.635151\pi$
$547$	−450.669	−0.823893	−0.411947	+	0.911208i	$0.635151\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.7404i	0.0285669i
$552$	0	0
$553$	−719.241	−1.30062
$554$	0	0
$555$	0	0
$556$	0	0
$557$	294.799i	0.529262i	0.964350	+	0.264631i	$0.0852501\pi$
$557$	294.799i	0.529262i	−0.964350	+	0.264631i	$0.914750\pi$
$558$	0	0
$559$	28.6073	0.0511759
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 579.409i	− 1.02915i	−0.857447	−	0.514573i	$-0.827950\pi$
$563$	− 579.409i	− 1.02915i	0.857447	−	0.514573i	$-0.172050\pi$
$564$	0	0
$565$	−21.0508	−0.0372580
$566$	0	0
$567$	0	0
$568$	0	0
$569$	208.878i	0.367096i	0.983011	+	0.183548i	$0.0587583\pi$
$569$	208.878i	0.367096i	−0.983011	+	0.183548i	$0.941242\pi$
$570$	0	0
$571$	−478.309	−0.837669	−0.418835	−	0.908062i	$-0.637561\pi$
$571$	−478.309	−0.837669	−0.418835	+	0.908062i	$0.637561\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	701.591i	1.22016i
$576$	0	0
$577$	−188.740	−0.327106	−0.163553	−	0.986535i	$-0.552296\pi$
$577$	−188.740	−0.327106	−0.163553	+	0.986535i	$0.552296\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	585.573i	1.00787i
$582$	0	0
$583$	648.574	1.11248
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 572.121i	− 0.974653i	−0.873220	−	0.487326i	$-0.837972\pi$
$587$	− 572.121i	− 0.974653i	0.873220	−	0.487326i	$-0.162028\pi$
$588$	0	0
$589$	5.15450	0.00875127
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1075.40i	− 1.81349i	−0.421682	−	0.906744i	$-0.638560\pi$
$593$	− 1075.40i	− 1.81349i	0.421682	−	0.906744i	$-0.361440\pi$
$594$	0	0
$595$	10.3027	0.0173154
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 425.765i	− 0.710793i	−0.934716	−	0.355397i	$-0.884346\pi$
$599$	− 425.765i	− 0.710793i	0.934716	−	0.355397i	$-0.115654\pi$
$600$	0	0
$601$	−739.547	−1.23053	−0.615263	−	0.788322i	$-0.710950\pi$
$601$	−739.547	−1.23053	−0.615263	+	0.788322i	$0.710950\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 12.8415i	− 0.0212256i
$606$	0	0
$607$	−495.047	−0.815564	−0.407782	−	0.913079i	$-0.633698\pi$
$607$	−495.047	−0.815564	−0.407782	+	0.913079i	$0.633698\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1254.90i	2.05385i
$612$	0	0
$613$	1043.04	1.70153	0.850766	−	0.525545i	$-0.176139\pi$
$613$	1043.04	1.70153	0.850766	+	0.525545i	$0.176139\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	157.228i	0.254827i	0.991850	+	0.127413i	$0.0406675\pi$
$617$	157.228i	0.254827i	−0.991850	+	0.127413i	$0.959333\pi$
$618$	0	0
$619$	270.724	0.437357	0.218679	−	0.975797i	$-0.429825\pi$
$619$	270.724	0.437357	0.218679	+	0.975797i	$0.429825\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1172.45i	1.88194i
$624$	0	0
$625$	621.978	0.995165
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 27.2613i	− 0.0433407i
$630$	0	0
$631$	452.185	0.716616	0.358308	−	0.933603i	$-0.383354\pi$
$631$	452.185	0.716616	0.358308	+	0.933603i	$0.383354\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	31.2596i	0.0492277i
$636$	0	0
$637$	−385.369	−0.604975
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 692.695i	− 1.08065i	−0.841457	−	0.540324i	$-0.818302\pi$
$641$	− 692.695i	− 1.08065i	0.841457	−	0.540324i	$-0.181698\pi$
$642$	0	0
$643$	786.380	1.22299	0.611493	−	0.791250i	$-0.290569\pi$
$643$	786.380	1.22299	0.611493	+	0.791250i	$0.290569\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 62.1427i	− 0.0960475i	−0.998846	−	0.0480237i	$-0.984708\pi$
$647$	− 62.1427i	− 0.0960475i	0.998846	−	0.0480237i	$-0.0152923\pi$
$648$	0	0
$649$	137.664	0.212117
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 296.250i	− 0.453675i	−0.973933	−	0.226838i	$-0.927161\pi$
$653$	− 296.250i	− 0.453675i	0.973933	−	0.226838i	$-0.0728387\pi$
$654$	0	0
$655$	−28.6737	−0.0437767
$656$	0	0
$657$	0	0
$658$	0	0
$659$	276.904i	0.420189i	0.977681	+	0.210094i	$0.0673771\pi$
$659$	276.904i	0.420189i	−0.977681	+	0.210094i	$0.932623\pi$
$660$	0	0
$661$	638.938	0.966624	0.483312	−	0.875448i	$-0.339434\pi$
$661$	638.938	0.966624	0.483312	+	0.875448i	$0.339434\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.649481i	0 0.000976663i
$666$	0	0
$667$	−1164.66	−1.74612
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 283.434i	− 0.422406i
$672$	0	0
$673$	472.178	0.701602	0.350801	−	0.936450i	$-0.385909\pi$
$673$	472.178	0.701602	0.350801	+	0.936450i	$0.385909\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 309.579i	− 0.457281i	−0.973511	−	0.228641i	$-0.926572\pi$
$677$	− 309.579i	− 0.457281i	0.973511	−	0.228641i	$-0.0734281\pi$
$678$	0	0
$679$	1023.18	1.50690
$680$	0	0
$681$	0	0
$682$	0	0
$683$	755.907i	1.10675i	0.832934	+	0.553373i	$0.186660\pi$
$683$	755.907i	1.10675i	−0.832934	+	0.553373i	$0.813340\pi$
$684$	0	0
$685$	38.5588	0.0562901
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1407.89i	2.04339i
$690$	0	0
$691$	−757.557	−1.09632	−0.548160	−	0.836374i	$-0.684671\pi$
$691$	−757.557	−1.09632	−0.548160	+	0.836374i	$0.684671\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.3425i	0.0206367i
$696$	0	0
$697$	462.550	0.663629
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1368.47i	1.95217i	0.217397	+	0.976083i	$0.430243\pi$
$701$	1368.47i	1.95217i	−0.217397	+	0.976083i	$0.569757\pi$
$702$	0	0
$703$	1.71855	0.00244460
$704$	0	0
$705$	0	0
$706$	0	0
$707$	113.431i	0.160440i
$708$	0	0
$709$	407.731	0.575079	0.287540	−	0.957769i	$-0.407163\pi$
$709$	407.731	0.575079	0.287540	+	0.957769i	$0.407163\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	381.392i	0.534912i
$714$	0	0
$715$	−24.8616	−0.0347715
$716$	0	0
$717$	0	0
$718$	0	0
$719$	248.224i	0.345235i	0.984989	+	0.172617i	$0.0552224\pi$
$719$	248.224i	0.345235i	−0.984989	+	0.172617i	$0.944778\pi$
$720$	0	0
$721$	−1659.71	−2.30196
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1034.18i	1.42645i
$726$	0	0
$727$	299.973	0.412617	0.206308	−	0.978487i	$-0.433855\pi$
$727$	299.973	0.412617	0.206308	+	0.978487i	$0.433855\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10.5150i	0.0143845i
$732$	0	0
$733$	−763.952	−1.04223	−0.521113	−	0.853487i	$-0.674483\pi$
$733$	−763.952	−1.04223	−0.521113	+	0.853487i	$0.674483\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 531.616i	− 0.721324i
$738$	0	0
$739$	875.825	1.18515	0.592575	−	0.805515i	$-0.298111\pi$
$739$	875.825	1.18515	0.592575	+	0.805515i	$0.298111\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 90.1112i	− 0.121280i	−0.998160	−	0.0606401i	$-0.980686\pi$
$743$	− 90.1112i	− 0.121280i	0.998160	−	0.0606401i	$-0.0193142\pi$
$744$	0	0
$745$	20.0402	0.0268996
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1325.80i	1.77010i
$750$	0	0
$751$	450.573	0.599964	0.299982	−	0.953945i	$-0.403019\pi$
$751$	450.573	0.599964	0.299982	+	0.953945i	$0.403019\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 19.4484i	− 0.0257595i
$756$	0	0
$757$	67.0196	0.0885332	0.0442666	−	0.999020i	$-0.485905\pi$
$757$	67.0196	0.0885332	0.0442666	+	0.999020i	$0.485905\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	905.846i	1.19034i	0.803601	+	0.595168i	$0.202914\pi$
$761$	905.846i	1.19034i	−0.803601	+	0.595168i	$0.797086\pi$
$762$	0	0
$763$	−597.468	−0.783051
$764$	0	0
$765$	0	0
$766$	0	0
$767$	298.835i	0.389615i
$768$	0	0
$769$	−922.570	−1.19970	−0.599850	−	0.800112i	$-0.704773\pi$
$769$	−922.570	−1.19970	−0.599850	+	0.800112i	$0.704773\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1521.11i	1.96780i	0.178722	+	0.983900i	$0.442804\pi$
$773$	1521.11i	1.96780i	−0.178722	+	0.983900i	$0.557196\pi$
$774$	0	0
$775$	338.662	0.436983
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.1592i	0.0374316i
$780$	0	0
$781$	−339.404	−0.434576
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.2579i	0.0614750i
$786$	0	0
$787$	−786.172	−0.998948	−0.499474	−	0.866329i	$-0.666473\pi$
$787$	−786.172	−0.998948	−0.499474	+	0.866329i	$0.666473\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	892.752i	1.12864i
$792$	0	0
$793$	615.266	0.775872
$794$	0	0
$795$	0	0
$796$	0	0
$797$	688.537i	0.863910i	0.901895	+	0.431955i	$0.142176\pi$
$797$	688.537i	0.863910i	−0.901895	+	0.431955i	$0.857824\pi$
$798$	0	0
$799$	−461.258	−0.577294
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 774.136i	− 0.964055i
$804$	0	0
$805$	−48.0565	−0.0596975
$806$	0	0
$807$	0	0
$808$	0	0
$809$	267.972i	0.331238i	0.986190	+	0.165619i	$0.0529623\pi$
$809$	267.972i	0.331238i	−0.986190	+	0.165619i	$0.947038\pi$
$810$	0	0
$811$	457.328	0.563906	0.281953	−	0.959428i	$-0.409018\pi$
$811$	457.328	0.563906	0.281953	+	0.959428i	$0.409018\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.7540i	0.0144221i
$816$	0	0
$817$	−0.662869	−0.000811345 0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	207.949i	0.253287i	0.991948	+	0.126643i	$0.0404204\pi$
$821$	207.949i	0.253287i	−0.991948	+	0.126643i	$0.959580\pi$
$822$	0	0
$823$	−312.245	−0.379399	−0.189699	−	0.981842i	$-0.560751\pi$
$823$	−312.245	−0.379399	−0.189699	+	0.981842i	$0.560751\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1045.51i	1.26422i	0.774878	+	0.632111i	$0.217811\pi$
$827$	1045.51i	1.26422i	−0.774878	+	0.632111i	$0.782189\pi$
$828$	0	0
$829$	−741.310	−0.894222	−0.447111	−	0.894478i	$-0.647547\pi$
$829$	−741.310	−0.894222	−0.447111	+	0.894478i	$0.647547\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 141.648i	− 0.170046i
$834$	0	0
$835$	−13.1514	−0.0157502
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 84.2278i	− 0.100391i	−0.998739	−	0.0501953i	$-0.984016\pi$
$839$	− 84.2278i	− 0.100391i	0.998739	−	0.0501953i	$-0.0159844\pi$
$840$	0	0
$841$	−875.771	−1.04134
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 20.0364i	− 0.0237118i
$846$	0	0
$847$	−544.601	−0.642977
$848$	0	0
$849$	0	0
$850$	0	0
$851$	127.160i	0.149424i
$852$	0	0
$853$	17.4786	0.0204907	0.0102454	−	0.999948i	$-0.496739\pi$
$853$	17.4786	0.0204907	0.0102454	+	0.999948i	$0.496739\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	761.512i	0.888579i	0.895883	+	0.444289i	$0.146544\pi$
$857$	761.512i	0.888579i	−0.895883	+	0.444289i	$0.853456\pi$
$858$	0	0
$859$	−946.183	−1.10149	−0.550747	−	0.834672i	$-0.685657\pi$
$859$	−946.183	−1.10149	−0.550747	+	0.834672i	$0.685657\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 357.864i	− 0.414675i	−0.978270	−	0.207337i	$-0.933520\pi$
$863$	− 357.864i	− 0.414675i	0.978270	−	0.207337i	$-0.0664798\pi$
$864$	0	0
$865$	15.1860	0.0175561
$866$	0	0
$867$	0	0
$868$	0	0
$869$	637.951i	0.734120i
$870$	0	0
$871$	1154.01	1.32492
$872$	0	0
$873$	0	0
$874$	0	0
$875$	85.4136i	0.0976156i
$876$	0	0
$877$	1088.04	1.24064	0.620320	−	0.784349i	$-0.287003\pi$
$877$	1088.04	1.24064	0.620320	+	0.784349i	$0.287003\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	771.560i	0.875778i	0.899029	+	0.437889i	$0.144274\pi$
$881$	771.560i	0.875778i	−0.899029	+	0.437889i	$0.855726\pi$
$882$	0	0
$883$	1210.00	1.37033	0.685166	−	0.728387i	$-0.259730\pi$
$883$	1210.00	1.37033	0.685166	+	0.728387i	$0.259730\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1354.60i	− 1.52717i	−0.645708	−	0.763585i	$-0.723438\pi$
$887$	− 1354.60i	− 1.52717i	0.645708	−	0.763585i	$-0.276562\pi$
$888$	0	0
$889$	1325.70	1.49123
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 29.0778i	− 0.0325619i
$894$	0	0
$895$	−60.1892	−0.0672505
$896$	0	0
$897$	0	0
$898$	0	0
$899$	562.190i	0.625350i
$900$	0	0
$901$	−517.491	−0.574352
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 59.6560i	− 0.0659182i
$906$	0	0
$907$	1539.24	1.69706	0.848531	−	0.529145i	$-0.177487\pi$
$907$	1539.24	1.69706	0.848531	+	0.529145i	$0.177487\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 314.668i	− 0.345410i	−0.984974	−	0.172705i	$-0.944749\pi$
$911$	− 314.668i	− 0.345410i	0.984974	−	0.172705i	$-0.0552507\pi$
$912$	0	0
$913$	519.390	0.568883
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1216.04i	1.32611i
$918$	0	0
$919$	−956.851	−1.04119	−0.520594	−	0.853805i	$-0.674289\pi$
$919$	−956.851	−1.04119	−0.520594	+	0.853805i	$0.674289\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 736.762i	− 0.798225i
$924$	0	0
$925$	112.913	0.122068
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1090.95i	− 1.17433i	−0.809466	−	0.587166i	$-0.800244\pi$
$929$	− 1090.95i	− 1.17433i	0.809466	−	0.587166i	$-0.199756\pi$
$930$	0	0
$931$	8.92951	0.00959131
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 9.13823i	− 0.00977351i
$936$	0	0
$937$	447.050	0.477107	0.238554	−	0.971129i	$-0.423327\pi$
$937$	447.050	0.477107	0.238554	+	0.971129i	$0.423327\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.47921i	− 0.00157196i	−1.00000		0.000785980i	$-0.999750\pi$
$941$	− 1.47921i	− 0.00157196i	1.00000		0.000785980i	$-0.000250185\pi$
$942$	0	0
$943$	−2157.55	−2.28797
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1552.07i	− 1.63893i	−0.573128	−	0.819466i	$-0.694270\pi$
$947$	− 1552.07i	− 1.63893i	0.573128	−	0.819466i	$-0.305730\pi$
$948$	0	0
$949$	1680.46	1.77077
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1693.86i	1.77740i	0.458493	+	0.888698i	$0.348389\pi$
$953$	1693.86i	1.77740i	−0.458493	+	0.888698i	$0.651611\pi$
$954$	0	0
$955$	−16.4345	−0.0172089
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1635.26i	− 1.70517i
$960$	0	0
$961$	−776.900	−0.808429
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 23.6362i	− 0.0244934i
$966$	0	0
$967$	−593.289	−0.613535	−0.306768	−	0.951784i	$-0.599247\pi$
$967$	−593.289	−0.613535	−0.306768	+	0.951784i	$0.599247\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	597.827i	0.615682i	0.951438	+	0.307841i	$0.0996065\pi$
$971$	597.827i	0.615682i	−0.951438	+	0.307841i	$0.900394\pi$
$972$	0	0
$973$	608.259	0.625138
$974$	0	0
$975$	0	0
$976$	0	0
$977$	129.700i	0.132754i	0.997795	+	0.0663768i	$0.0211439\pi$
$977$	129.700i	0.132754i	−0.997795	+	0.0663768i	$0.978856\pi$
$978$	0	0
$979$	1039.94	1.06224
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 426.075i	− 0.433444i	−0.976233	−	0.216722i	$-0.930464\pi$
$983$	− 426.075i	− 0.433444i	0.976233	−	0.216722i	$-0.0695364\pi$
$984$	0	0
$985$	−16.1965	−0.0164432
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 49.0471i	− 0.0495926i
$990$	0	0
$991$	389.526	0.393064	0.196532	−	0.980497i	$-0.437032\pi$
$991$	389.526	0.393064	0.196532	+	0.980497i	$0.437032\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	35.3198i	0.0354973i
$996$	0	0
$997$	−659.426	−0.661410	−0.330705	−	0.943734i	$-0.607286\pi$
$997$	−659.426	−0.661410	−0.330705	+	0.943734i	$0.607286\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2916.3.c.b.1457.18		36
3.2	odd	2	inner	2916.3.c.b.1457.19		36
27.4	even	9		324.3.k.a.197.4		36
27.7	even	9		108.3.k.a.5.3	✓	36
27.20	odd	18		324.3.k.a.125.4		36
27.23	odd	18		108.3.k.a.65.3	yes	36
108.7	odd	18		432.3.bc.b.113.4		36
108.23	even	18		432.3.bc.b.65.4		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.5.3	✓	36	27.7	even	9
108.3.k.a.65.3	yes	36	27.23	odd	18
324.3.k.a.125.4		36	27.20	odd	18
324.3.k.a.197.4		36	27.4	even	9
432.3.bc.b.65.4		36	108.23	even	18
432.3.bc.b.113.4		36	108.7	odd	18
2916.3.c.b.1457.18		36	1.1	even	1	trivial
2916.3.c.b.1457.19		36	3.2	odd	2	inner

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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:	\( N \)	\(=\)	\( 2916 = 2^{2} \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2916.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.200781i q^{5} -8.51501 q^{7} +O(q^{10})\)	\(q-0.200781i q^{5} -8.51501 q^{7} +7.55263i q^{11} -16.3949 q^{13} -6.02618i q^{17} +0.379891 q^{19} +28.1090i q^{23} +24.9597 q^{25} +41.4339i q^{29} +13.5683 q^{31} +1.70965i q^{35} +4.52381 q^{37} +76.7567i q^{41} -1.74489 q^{43} -76.5424i q^{47} +23.5054 q^{49} -85.8739i q^{53} +1.51642 q^{55} -18.2273i q^{59} -37.5279 q^{61} +3.29178i q^{65} -70.3881 q^{67} +44.9385i q^{71} -102.499 q^{73} -64.3107i q^{77} +84.4674 q^{79} -68.7695i q^{83} -1.20994 q^{85} -137.692i q^{89} +139.603 q^{91} -0.0762748i q^{95} -120.162 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 36 * q	\( 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) \) 36 * q - 180 * q^25 + 252 * q^49 + 18 * q^61 - 90 * q^67 + 126 * q^73 - 198 * q^91

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