LMFDB - Embedded newform 2916.3.c.b.1457.9

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			1457.9
Character	$\chi$	$=$	2916.1457
Dual form			2916.3.c.b.1457.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.39539i q^{5} -3.57384 q^{7} +O(q^{10})$	$q-4.39539i q^{5} -3.57384 q^{7} +14.6813i q^{11} +3.91381 q^{13} +6.88924i q^{17} -28.2823 q^{19} -1.29527i q^{23} +5.68054 q^{25} -37.9974i q^{29} +54.4584 q^{31} +15.7084i q^{35} -37.0667 q^{37} +41.1578i q^{41} -3.54682 q^{43} -42.8169i q^{47} -36.2276 q^{49} +47.8007i q^{53} +64.5299 q^{55} -62.0518i q^{59} -11.1077 q^{61} -17.2027i q^{65} +111.902 q^{67} -105.064i q^{71} +97.0235 q^{73} -52.4686i q^{77} +110.992 q^{79} -131.274i q^{83} +30.2809 q^{85} +101.456i q^{89} -13.9873 q^{91} +124.312i q^{95} -165.580 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 4.39539i	− 0.879078i	−0.898223	−	0.439539i	$-0.855142\pi$
$5$	− 4.39539i	− 0.879078i	0.898223	−	0.439539i	$-0.144858\pi$
$6$	0	0
$7$	−3.57384	−0.510549	−0.255275	−	0.966869i	$-0.582166\pi$
$7$	−3.57384	−0.510549	−0.255275	+	0.966869i	$0.582166\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.6813i	1.33466i	0.744762	+	0.667331i	$0.232563\pi$
$11$	14.6813i	1.33466i	−0.744762	+	0.667331i	$0.767437\pi$
$12$	0	0
$13$	3.91381	0.301062	0.150531	−	0.988605i	$-0.451902\pi$
$13$	3.91381	0.301062	0.150531	+	0.988605i	$0.451902\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.88924i	0.405250i	0.979256	+	0.202625i	$0.0649472\pi$
$17$	6.88924i	0.405250i	−0.979256	+	0.202625i	$0.935053\pi$
$18$	0	0
$19$	−28.2823	−1.48854	−0.744272	−	0.667877i	$-0.767203\pi$
$19$	−28.2823	−1.48854	−0.744272	+	0.667877i	$0.767203\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.29527i	− 0.0563161i	−0.999603	−	0.0281580i	$-0.991036\pi$
$23$	− 1.29527i	− 0.0563161i	0.999603	−	0.0281580i	$-0.00896416\pi$
$24$	0	0
$25$	5.68054	0.227222
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 37.9974i	− 1.31026i	−0.755518	−	0.655128i	$-0.772615\pi$
$29$	− 37.9974i	− 1.31026i	0.755518	−	0.655128i	$-0.227385\pi$
$30$	0	0
$31$	54.4584	1.75672	0.878361	−	0.477999i	$-0.158638\pi$
$31$	54.4584	1.75672	0.878361	+	0.477999i	$0.158638\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.7084i	0.448813i
$36$	0	0
$37$	−37.0667	−1.00180	−0.500902	−	0.865504i	$-0.666998\pi$
$37$	−37.0667	−1.00180	−0.500902	+	0.865504i	$0.666998\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	41.1578i	1.00385i	0.864912	+	0.501924i	$0.167374\pi$
$41$	41.1578i	1.00385i	−0.864912	+	0.501924i	$0.832626\pi$
$42$	0	0
$43$	−3.54682	−0.0824842	−0.0412421	−	0.999149i	$-0.513131\pi$
$43$	−3.54682	−0.0824842	−0.0412421	+	0.999149i	$0.513131\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 42.8169i	− 0.910998i	−0.890236	−	0.455499i	$-0.849461\pi$
$47$	− 42.8169i	− 0.910998i	0.890236	−	0.455499i	$-0.150539\pi$
$48$	0	0
$49$	−36.2276	−0.739340
$50$	0	0
$51$	0	0
$52$	0	0
$53$	47.8007i	0.901900i	0.892549	+	0.450950i	$0.148915\pi$
$53$	47.8007i	0.901900i	−0.892549	+	0.450950i	$0.851085\pi$
$54$	0	0
$55$	64.5299	1.17327
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 62.0518i	− 1.05173i	−0.850569	−	0.525863i	$-0.823742\pi$
$59$	− 62.0518i	− 1.05173i	0.850569	−	0.525863i	$-0.176258\pi$
$60$	0	0
$61$	−11.1077	−0.182093	−0.0910464	−	0.995847i	$-0.529021\pi$
$61$	−11.1077	−0.182093	−0.0910464	+	0.995847i	$0.529021\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 17.2027i	− 0.264657i
$66$	0	0
$67$	111.902	1.67018	0.835089	−	0.550115i	$-0.185416\pi$
$67$	111.902	1.67018	0.835089	+	0.550115i	$0.185416\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 105.064i	− 1.47978i	−0.672727	−	0.739890i	$-0.734877\pi$
$71$	− 105.064i	− 1.47978i	0.672727	−	0.739890i	$-0.265123\pi$
$72$	0	0
$73$	97.0235	1.32909	0.664545	−	0.747249i	$-0.268626\pi$
$73$	97.0235	1.32909	0.664545	+	0.747249i	$0.268626\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 52.4686i	− 0.681410i
$78$	0	0
$79$	110.992	1.40497	0.702483	−	0.711700i	$-0.252075\pi$
$79$	110.992	1.40497	0.702483	+	0.711700i	$0.252075\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 131.274i	− 1.58162i	−0.612063	−	0.790809i	$-0.709660\pi$
$83$	− 131.274i	− 1.58162i	0.612063	−	0.790809i	$-0.290340\pi$
$84$	0	0
$85$	30.2809	0.356246
$86$	0	0
$87$	0	0
$88$	0	0
$89$	101.456i	1.13996i	0.821659	+	0.569979i	$0.193049\pi$
$89$	101.456i	1.13996i	−0.821659	+	0.569979i	$0.806951\pi$
$90$	0	0
$91$	−13.9873	−0.153707
$92$	0	0
$93$	0	0
$94$	0	0
$95$	124.312i	1.30855i
$96$	0	0
$97$	−165.580	−1.70701	−0.853506	−	0.521082i	$-0.825528\pi$
$97$	−165.580	−1.70701	−0.853506	+	0.521082i	$0.825528\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	22.1030i	0.218842i	0.993995	+	0.109421i	$0.0348997\pi$
$101$	22.1030i	0.218842i	−0.993995	+	0.109421i	$0.965100\pi$
$102$	0	0
$103$	−28.2929	−0.274688	−0.137344	−	0.990523i	$-0.543857\pi$
$103$	−28.2929	−0.274688	−0.137344	+	0.990523i	$0.543857\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 67.0830i	− 0.626944i	−0.949597	−	0.313472i	$-0.898508\pi$
$107$	− 67.0830i	− 0.626944i	0.949597	−	0.313472i	$-0.101492\pi$
$108$	0	0
$109$	82.0479	0.752733	0.376367	−	0.926471i	$-0.377173\pi$
$109$	82.0479	0.752733	0.376367	+	0.926471i	$0.377173\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	141.726i	1.25421i	0.778935	+	0.627105i	$0.215760\pi$
$113$	141.726i	1.25421i	−0.778935	+	0.627105i	$0.784240\pi$
$114$	0	0
$115$	−5.69322	−0.0495062
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 24.6211i	− 0.206900i
$120$	0	0
$121$	−94.5399	−0.781321
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 134.853i	− 1.07882i
$126$	0	0
$127$	−72.2517	−0.568911	−0.284455	−	0.958689i	$-0.591813\pi$
$127$	−72.2517	−0.568911	−0.284455	+	0.958689i	$0.591813\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 125.219i	− 0.955871i	−0.878395	−	0.477935i	$-0.841385\pi$
$131$	− 125.219i	− 0.955871i	0.878395	−	0.477935i	$-0.158615\pi$
$132$	0	0
$133$	101.077	0.759975
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 130.074i	− 0.949444i	−0.880136	−	0.474722i	$-0.842549\pi$
$137$	− 130.074i	− 0.949444i	0.880136	−	0.474722i	$-0.157451\pi$
$138$	0	0
$139$	−84.4849	−0.607805	−0.303902	−	0.952703i	$-0.598290\pi$
$139$	−84.4849	−0.607805	−0.303902	+	0.952703i	$0.598290\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	57.4597i	0.401816i
$144$	0	0
$145$	−167.014	−1.15182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 142.771i	− 0.958194i	−0.877762	−	0.479097i	$-0.840964\pi$
$149$	− 142.771i	− 0.958194i	0.877762	−	0.479097i	$-0.159036\pi$
$150$	0	0
$151$	42.6929	0.282734	0.141367	−	0.989957i	$-0.454850\pi$
$151$	42.6929	0.282734	0.141367	+	0.989957i	$0.454850\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 239.366i	− 1.54430i
$156$	0	0
$157$	−67.9774	−0.432977	−0.216489	−	0.976285i	$-0.569460\pi$
$157$	−67.9774	−0.432977	−0.216489	+	0.976285i	$0.569460\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.62909i	0.0287521i
$162$	0	0
$163$	−52.2555	−0.320586	−0.160293	−	0.987069i	$-0.551244\pi$
$163$	−52.2555	−0.320586	−0.160293	+	0.987069i	$0.551244\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 208.936i	− 1.25111i	−0.780180	−	0.625556i	$-0.784872\pi$
$167$	− 208.936i	− 1.25111i	0.780180	−	0.625556i	$-0.215128\pi$
$168$	0	0
$169$	−153.682	−0.909362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 289.919i	− 1.67583i	−0.545799	−	0.837916i	$-0.683774\pi$
$173$	− 289.919i	− 1.67583i	0.545799	−	0.837916i	$-0.316226\pi$
$174$	0	0
$175$	−20.3014	−0.116008
$176$	0	0
$177$	0	0
$178$	0	0
$179$	97.4469i	0.544396i	0.962241	+	0.272198i	$0.0877506\pi$
$179$	97.4469i	0.544396i	−0.962241	+	0.272198i	$0.912249\pi$
$180$	0	0
$181$	−37.3205	−0.206191	−0.103095	−	0.994671i	$-0.532875\pi$
$181$	−37.3205	−0.206191	−0.103095	+	0.994671i	$0.532875\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	162.923i	0.880664i
$186$	0	0
$187$	−101.143	−0.540871
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 350.318i	− 1.83412i	−0.398745	−	0.917062i	$-0.630554\pi$
$191$	− 350.318i	− 1.83412i	0.398745	−	0.917062i	$-0.369446\pi$
$192$	0	0
$193$	11.0531	0.0572698	0.0286349	−	0.999590i	$-0.490884\pi$
$193$	11.0531	0.0572698	0.0286349	+	0.999590i	$0.490884\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 318.101i	− 1.61472i	−0.590057	−	0.807362i	$-0.700895\pi$
$197$	− 318.101i	− 1.61472i	0.590057	−	0.807362i	$-0.299105\pi$
$198$	0	0
$199$	176.697	0.887925	0.443963	−	0.896045i	$-0.353572\pi$
$199$	176.697	0.887925	0.443963	+	0.896045i	$0.353572\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	135.797i	0.668950i
$204$	0	0
$205$	180.905	0.882461
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 415.221i	− 1.98670i
$210$	0	0
$211$	−207.349	−0.982697	−0.491349	−	0.870963i	$-0.663496\pi$
$211$	−207.349	−0.982697	−0.491349	+	0.870963i	$0.663496\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	15.5897i	0.0725100i
$216$	0	0
$217$	−194.626	−0.896893
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.9632i	0.122005i
$222$	0	0
$223$	413.610	1.85475	0.927376	−	0.374129i	$-0.122058\pi$
$223$	413.610	1.85475	0.927376	+	0.374129i	$0.122058\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 96.3769i	− 0.424568i	−0.977208	−	0.212284i	$-0.931910\pi$
$227$	− 96.3769i	− 0.424568i	0.977208	−	0.212284i	$-0.0680901\pi$
$228$	0	0
$229$	−201.271	−0.878911	−0.439456	−	0.898264i	$-0.644829\pi$
$229$	−201.271	−0.878911	−0.439456	+	0.898264i	$0.644829\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	350.203i	1.50302i	0.659724	+	0.751508i	$0.270673\pi$
$233$	350.203i	1.50302i	−0.659724	+	0.751508i	$0.729327\pi$
$234$	0	0
$235$	−188.197	−0.800839
$236$	0	0
$237$	0	0
$238$	0	0
$239$	156.604i	0.655245i	0.944809	+	0.327622i	$0.106247\pi$
$239$	156.604i	0.655245i	−0.944809	+	0.327622i	$0.893753\pi$
$240$	0	0
$241$	−345.240	−1.43253	−0.716265	−	0.697829i	$-0.754150\pi$
$241$	−345.240	−1.43253	−0.716265	+	0.697829i	$0.754150\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	159.235i	0.649937i
$246$	0	0
$247$	−110.692	−0.448144
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.4217i	0.105266i	0.998614	+	0.0526329i	$0.0167613\pi$
$251$	26.4217i	0.105266i	−0.998614	+	0.0526329i	$0.983239\pi$
$252$	0	0
$253$	19.0162	0.0751629
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 118.859i	− 0.462485i	−0.972896	−	0.231242i	$-0.925721\pi$
$257$	− 118.859i	− 0.462485i	0.972896	−	0.231242i	$-0.0742790\pi$
$258$	0	0
$259$	132.471	0.511470
$260$	0	0
$261$	0	0
$262$	0	0
$263$	171.364i	0.651573i	0.945443	+	0.325787i	$0.105629\pi$
$263$	171.364i	0.651573i	−0.945443	+	0.325787i	$0.894371\pi$
$264$	0	0
$265$	210.103	0.792841
$266$	0	0
$267$	0	0
$268$	0	0
$269$	147.131i	0.546957i	0.961878	+	0.273478i	$0.0881742\pi$
$269$	147.131i	0.546957i	−0.961878	+	0.273478i	$0.911826\pi$
$270$	0	0
$271$	42.2256	0.155814	0.0779071	−	0.996961i	$-0.475176\pi$
$271$	42.2256	0.155814	0.0779071	+	0.996961i	$0.475176\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	83.3976i	0.303264i
$276$	0	0
$277$	−315.896	−1.14042	−0.570209	−	0.821500i	$-0.693138\pi$
$277$	−315.896	−1.14042	−0.570209	+	0.821500i	$0.693138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 431.857i	− 1.53686i	−0.639935	−	0.768429i	$-0.721039\pi$
$281$	− 431.857i	− 1.53686i	0.639935	−	0.768429i	$-0.278961\pi$
$282$	0	0
$283$	54.7963	0.193627	0.0968133	−	0.995303i	$-0.469135\pi$
$283$	54.7963	0.193627	0.0968133	+	0.995303i	$0.469135\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 147.091i	− 0.512514i
$288$	0	0
$289$	241.538	0.835773
$290$	0	0
$291$	0	0
$292$	0	0
$293$	140.561i	0.479730i	0.970806	+	0.239865i	$0.0771031\pi$
$293$	140.561i	0.479730i	−0.970806	+	0.239865i	$0.922897\pi$
$294$	0	0
$295$	−272.742	−0.924549
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 5.06943i	− 0.0169546i
$300$	0	0
$301$	12.6758	0.0421122
$302$	0	0
$303$	0	0
$304$	0	0
$305$	48.8225i	0.160074i
$306$	0	0
$307$	427.882	1.39375	0.696876	−	0.717192i	$-0.254573\pi$
$307$	427.882	1.39375	0.696876	+	0.717192i	$0.254573\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 416.442i	− 1.33904i	−0.742793	−	0.669521i	$-0.766499\pi$
$311$	− 416.442i	− 1.33904i	0.742793	−	0.669521i	$-0.233501\pi$
$312$	0	0
$313$	144.643	0.462120	0.231060	−	0.972940i	$-0.425781\pi$
$313$	144.643	0.462120	0.231060	+	0.972940i	$0.425781\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 542.853i	− 1.71247i	−0.516587	−	0.856234i	$-0.672798\pi$
$317$	− 542.853i	− 1.71247i	0.516587	−	0.856234i	$-0.327202\pi$
$318$	0	0
$319$	557.851	1.74875
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 194.844i	− 0.603231i
$324$	0	0
$325$	22.2325	0.0684078
$326$	0	0
$327$	0	0
$328$	0	0
$329$	153.021i	0.465109i
$330$	0	0
$331$	221.587	0.669446	0.334723	−	0.942317i	$-0.391357\pi$
$331$	221.587	0.669446	0.334723	+	0.942317i	$0.391357\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 491.853i	− 1.46822i
$336$	0	0
$337$	−38.6214	−0.114603	−0.0573017	−	0.998357i	$-0.518250\pi$
$337$	−38.6214	−0.114603	−0.0573017	+	0.998357i	$0.518250\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	799.518i	2.34463i
$342$	0	0
$343$	304.590	0.888018
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 209.244i	− 0.603009i	−0.953465	−	0.301504i	$-0.902511\pi$
$347$	− 209.244i	− 0.603009i	0.953465	−	0.301504i	$-0.0974888\pi$
$348$	0	0
$349$	279.721	0.801492	0.400746	−	0.916189i	$-0.368751\pi$
$349$	279.721	0.801492	0.400746	+	0.916189i	$0.368751\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 84.0045i	− 0.237973i	−0.992896	−	0.118987i	$-0.962035\pi$
$353$	− 84.0045i	− 0.237973i	0.992896	−	0.118987i	$-0.0379645\pi$
$354$	0	0
$355$	−461.799	−1.30084
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 255.492i	− 0.711677i	−0.934548	−	0.355838i	$-0.884195\pi$
$359$	− 255.492i	− 0.711677i	0.934548	−	0.355838i	$-0.115805\pi$
$360$	0	0
$361$	438.890	1.21576
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 426.456i	− 1.16837i
$366$	0	0
$367$	−257.545	−0.701756	−0.350878	−	0.936421i	$-0.614117\pi$
$367$	−257.545	−0.701756	−0.350878	+	0.936421i	$0.614117\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 170.832i	− 0.460464i
$372$	0	0
$373$	481.344	1.29047	0.645234	−	0.763985i	$-0.276760\pi$
$373$	481.344	1.29047	0.645234	+	0.763985i	$0.276760\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 148.715i	− 0.394469i
$378$	0	0
$379$	−370.571	−0.977759	−0.488880	−	0.872351i	$-0.662594\pi$
$379$	−370.571	−0.977759	−0.488880	+	0.872351i	$0.662594\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	489.240i	1.27739i	0.769461	+	0.638694i	$0.220525\pi$
$383$	489.240i	1.27739i	−0.769461	+	0.638694i	$0.779475\pi$
$384$	0	0
$385$	−230.620	−0.599013
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 314.462i	− 0.808385i	−0.914674	−	0.404193i	$-0.867553\pi$
$389$	− 314.462i	− 0.808385i	0.914674	−	0.404193i	$-0.132447\pi$
$390$	0	0
$391$	8.92342	0.0228221
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 487.855i	− 1.23508i
$396$	0	0
$397$	89.6043	0.225704	0.112852	−	0.993612i	$-0.464001\pi$
$397$	89.6043	0.225704	0.112852	+	0.993612i	$0.464001\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	85.9469i	0.214331i	0.994241	+	0.107166i	$0.0341775\pi$
$401$	85.9469i	0.214331i	−0.994241	+	0.107166i	$0.965822\pi$
$402$	0	0
$403$	213.140	0.528882
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 544.187i	− 1.33707i
$408$	0	0
$409$	626.579	1.53198	0.765989	−	0.642854i	$-0.222250\pi$
$409$	626.579	1.53198	0.765989	+	0.642854i	$0.222250\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	221.764i	0.536958i
$414$	0	0
$415$	−577.002	−1.39037
$416$	0	0
$417$	0	0
$418$	0	0
$419$	352.293i	0.840795i	0.907340	+	0.420397i	$0.138109\pi$
$419$	352.293i	0.840795i	−0.907340	+	0.420397i	$0.861891\pi$
$420$	0	0
$421$	154.607	0.367236	0.183618	−	0.982998i	$-0.441219\pi$
$421$	154.607	0.367236	0.183618	+	0.982998i	$0.441219\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	39.1346i	0.0920815i
$426$	0	0
$427$	39.6970	0.0929673
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 226.253i	− 0.524949i	−0.964939	−	0.262474i	$-0.915461\pi$
$431$	− 226.253i	− 0.524949i	0.964939	−	0.262474i	$-0.0845386\pi$
$432$	0	0
$433$	−648.539	−1.49778	−0.748890	−	0.662695i	$-0.769413\pi$
$433$	−648.539	−1.49778	−0.748890	+	0.662695i	$0.769413\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	36.6332i	0.0838289i
$438$	0	0
$439$	301.958	0.687831	0.343915	−	0.939001i	$-0.388247\pi$
$439$	301.958	0.687831	0.343915	+	0.939001i	$0.388247\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 465.948i	− 1.05180i	−0.850546	−	0.525900i	$-0.823729\pi$
$443$	− 465.948i	− 1.05180i	0.850546	−	0.525900i	$-0.176271\pi$
$444$	0	0
$445$	445.940	1.00211
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 41.4426i	− 0.0922997i	−0.998935	−	0.0461498i	$-0.985305\pi$
$449$	− 41.4426i	− 0.0922997i	0.998935	−	0.0461498i	$-0.0146952\pi$
$450$	0	0
$451$	−604.249	−1.33980
$452$	0	0
$453$	0	0
$454$	0	0
$455$	61.4798i	0.135120i
$456$	0	0
$457$	−432.020	−0.945340	−0.472670	−	0.881240i	$-0.656710\pi$
$457$	−432.020	−0.945340	−0.472670	+	0.881240i	$0.656710\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	657.839i	1.42698i	0.700664	+	0.713492i	$0.252887\pi$
$461$	657.839i	1.42698i	−0.700664	+	0.713492i	$0.747113\pi$
$462$	0	0
$463$	−542.462	−1.17162	−0.585812	−	0.810447i	$-0.699224\pi$
$463$	−542.462	−1.17162	−0.585812	+	0.810447i	$0.699224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 82.4792i	− 0.176615i	−0.996093	−	0.0883075i	$-0.971854\pi$
$467$	− 82.4792i	− 0.176615i	0.996093	−	0.0883075i	$-0.0281458\pi$
$468$	0	0
$469$	−399.920	−0.852708
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 52.0718i	− 0.110088i
$474$	0	0
$475$	−160.659	−0.338229
$476$	0	0
$477$	0	0
$478$	0	0
$479$	84.2621i	0.175912i	0.996124	+	0.0879562i	$0.0280336\pi$
$479$	84.2621i	0.175912i	−0.996124	+	0.0879562i	$0.971966\pi$
$480$	0	0
$481$	−145.072	−0.301605
$482$	0	0
$483$	0	0
$484$	0	0
$485$	727.790i	1.50060i
$486$	0	0
$487$	566.361	1.16296	0.581479	−	0.813561i	$-0.302474\pi$
$487$	566.361	1.16296	0.581479	+	0.813561i	$0.302474\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 766.535i	− 1.56117i	−0.625049	−	0.780586i	$-0.714921\pi$
$491$	− 766.535i	− 1.56117i	0.625049	−	0.780586i	$-0.285079\pi$
$492$	0	0
$493$	261.774	0.530981
$494$	0	0
$495$	0	0
$496$	0	0
$497$	375.484i	0.755501i
$498$	0	0
$499$	220.697	0.442278	0.221139	−	0.975242i	$-0.429023\pi$
$499$	220.697	0.442278	0.221139	+	0.975242i	$0.429023\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 904.591i	− 1.79839i	−0.437546	−	0.899196i	$-0.644152\pi$
$503$	− 904.591i	− 1.79839i	0.437546	−	0.899196i	$-0.355848\pi$
$504$	0	0
$505$	97.1515	0.192379
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 342.589i	− 0.673063i	−0.941672	−	0.336532i	$-0.890746\pi$
$509$	− 342.589i	− 0.673063i	0.941672	−	0.336532i	$-0.109254\pi$
$510$	0	0
$511$	−346.747	−0.678565
$512$	0	0
$513$	0	0
$514$	0	0
$515$	124.358i	0.241472i
$516$	0	0
$517$	628.607	1.21587
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 139.220i	− 0.267216i	−0.991034	−	0.133608i	$-0.957344\pi$
$521$	− 139.220i	− 0.267216i	0.991034	−	0.133608i	$-0.0426564\pi$
$522$	0	0
$523$	24.7642	0.0473502	0.0236751	−	0.999720i	$-0.492463\pi$
$523$	24.7642	0.0473502	0.0236751	+	0.999720i	$0.492463\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	375.177i	0.711910i
$528$	0	0
$529$	527.322	0.996829
$530$	0	0
$531$	0	0
$532$	0	0
$533$	161.084i	0.302221i
$534$	0	0
$535$	−294.856	−0.551133
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 531.868i	− 0.986768i
$540$	0	0
$541$	476.086	0.880011	0.440006	−	0.897995i	$-0.354976\pi$
$541$	476.086	0.880011	0.440006	+	0.897995i	$0.354976\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 360.633i	− 0.661711i
$546$	0	0
$547$	−294.789	−0.538920	−0.269460	−	0.963012i	$-0.586845\pi$
$547$	−294.789	−0.538920	−0.269460	+	0.963012i	$0.586845\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1074.66i	1.95037i
$552$	0	0
$553$	−396.669	−0.717304
$554$	0	0
$555$	0	0
$556$	0	0
$557$	873.721i	1.56862i	0.620370	+	0.784309i	$0.286982\pi$
$557$	873.721i	1.56862i	−0.620370	+	0.784309i	$0.713018\pi$
$558$	0	0
$559$	−13.8816	−0.0248329
$560$	0	0
$561$	0	0
$562$	0	0
$563$	509.959i	0.905788i	0.891564	+	0.452894i	$0.149608\pi$
$563$	509.959i	0.905788i	−0.891564	+	0.452894i	$0.850392\pi$
$564$	0	0
$565$	622.940	1.10255
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 887.184i	− 1.55920i	−0.626279	−	0.779599i	$-0.715423\pi$
$569$	− 887.184i	− 1.55920i	0.626279	−	0.779599i	$-0.284577\pi$
$570$	0	0
$571$	526.290	0.921699	0.460850	−	0.887478i	$-0.347545\pi$
$571$	526.290	0.921699	0.460850	+	0.887478i	$0.347545\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 7.35783i	− 0.0127962i
$576$	0	0
$577$	−137.156	−0.237705	−0.118853	−	0.992912i	$-0.537922\pi$
$577$	−137.156	−0.237705	−0.118853	+	0.992912i	$0.537922\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	469.154i	0.807494i
$582$	0	0
$583$	−701.776	−1.20373
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 178.975i	− 0.304898i	−0.988311	−	0.152449i	$-0.951284\pi$
$587$	− 178.975i	− 0.304898i	0.988311	−	0.152449i	$-0.0487160\pi$
$588$	0	0
$589$	−1540.21	−2.61496
$590$	0	0
$591$	0	0
$592$	0	0
$593$	52.0285i	0.0877377i	0.999037	+	0.0438689i	$0.0139684\pi$
$593$	52.0285i	0.0877377i	−0.999037	+	0.0438689i	$0.986032\pi$
$594$	0	0
$595$	−108.219	−0.181881
$596$	0	0
$597$	0	0
$598$	0	0
$599$	116.260i	0.194091i	0.995280	+	0.0970454i	$0.0309392\pi$
$599$	116.260i	0.194091i	−0.995280	+	0.0970454i	$0.969061\pi$
$600$	0	0
$601$	−525.657	−0.874638	−0.437319	−	0.899307i	$-0.644072\pi$
$601$	−525.657	−0.874638	−0.437319	+	0.899307i	$0.644072\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	415.540i	0.686842i
$606$	0	0
$607$	−85.6418	−0.141090	−0.0705451	−	0.997509i	$-0.522474\pi$
$607$	−85.6418	−0.141090	−0.0705451	+	0.997509i	$0.522474\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 167.577i	− 0.274267i
$612$	0	0
$613$	−163.293	−0.266383	−0.133191	−	0.991090i	$-0.542523\pi$
$613$	−163.293	−0.266383	−0.133191	+	0.991090i	$0.542523\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 278.558i	− 0.451472i	−0.974189	−	0.225736i	$-0.927521\pi$
$617$	− 278.558i	− 0.451472i	0.974189	−	0.225736i	$-0.0724786\pi$
$618$	0	0
$619$	712.713	1.15139	0.575697	−	0.817663i	$-0.304731\pi$
$619$	712.713	1.15139	0.575697	+	0.817663i	$0.304731\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 362.589i	− 0.582005i
$624$	0	0
$625$	−450.718	−0.721149
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 255.362i	− 0.405981i
$630$	0	0
$631$	622.058	0.985829	0.492914	−	0.870078i	$-0.335932\pi$
$631$	622.058	0.985829	0.492914	+	0.870078i	$0.335932\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	317.574i	0.500117i
$636$	0	0
$637$	−141.788	−0.222587
$638$	0	0
$639$	0	0
$640$	0	0
$641$	254.212i	0.396587i	0.980143	+	0.198294i	$0.0635400\pi$
$641$	254.212i	0.396587i	−0.980143	+	0.198294i	$0.936460\pi$
$642$	0	0
$643$	212.140	0.329923	0.164961	−	0.986300i	$-0.447250\pi$
$643$	212.140	0.329923	0.164961	+	0.986300i	$0.447250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	330.267i	0.510459i	0.966880	+	0.255230i	$0.0821511\pi$
$647$	330.267i	0.510459i	−0.966880	+	0.255230i	$0.917849\pi$
$648$	0	0
$649$	911.000	1.40370
$650$	0	0
$651$	0	0
$652$	0	0
$653$	334.307i	0.511956i	0.966683	+	0.255978i	$0.0823974\pi$
$653$	334.307i	0.511956i	−0.966683	+	0.255978i	$0.917603\pi$
$654$	0	0
$655$	−550.387	−0.840285
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1036.95i	1.57352i	0.617259	+	0.786760i	$0.288243\pi$
$659$	1036.95i	1.57352i	−0.617259	+	0.786760i	$0.711757\pi$
$660$	0	0
$661$	−422.540	−0.639243	−0.319622	−	0.947545i	$-0.603556\pi$
$661$	−422.540	−0.639243	−0.319622	+	0.947545i	$0.603556\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 444.271i	− 0.668077i
$666$	0	0
$667$	−49.2169	−0.0737885
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 163.075i	− 0.243032i
$672$	0	0
$673$	15.2879	0.0227160	0.0113580	−	0.999935i	$-0.496385\pi$
$673$	15.2879	0.0227160	0.0113580	+	0.999935i	$0.496385\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	303.061i	0.447652i	0.974629	+	0.223826i	$0.0718548\pi$
$677$	303.061i	0.447652i	−0.974629	+	0.223826i	$0.928145\pi$
$678$	0	0
$679$	591.758	0.871514
$680$	0	0
$681$	0	0
$682$	0	0
$683$	795.226i	1.16431i	0.813077	+	0.582157i	$0.197791\pi$
$683$	795.226i	1.16431i	−0.813077	+	0.582157i	$0.802209\pi$
$684$	0	0
$685$	−571.725	−0.834636
$686$	0	0
$687$	0	0
$688$	0	0
$689$	187.083i	0.271528i
$690$	0	0
$691$	47.8610	0.0692634	0.0346317	−	0.999400i	$-0.488974\pi$
$691$	47.8610	0.0692634	0.0346317	+	0.999400i	$0.488974\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	371.344i	0.534308i
$696$	0	0
$697$	−283.546	−0.406809
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 32.9301i	− 0.0469759i	−0.999724	−	0.0234880i	$-0.992523\pi$
$701$	− 32.9301i	− 0.0469759i	0.999724	−	0.0234880i	$-0.00747714\pi$
$702$	0	0
$703$	1048.33	1.49123
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 78.9928i	− 0.111730i
$708$	0	0
$709$	921.258	1.29938	0.649688	−	0.760201i	$-0.274899\pi$
$709$	921.258	1.29938	0.649688	+	0.760201i	$0.274899\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 70.5383i	− 0.0989316i
$714$	0	0
$715$	252.558	0.353228
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 469.195i	− 0.652566i	−0.945272	−	0.326283i	$-0.894204\pi$
$719$	− 469.195i	− 0.652566i	0.945272	−	0.326283i	$-0.105796\pi$
$720$	0	0
$721$	101.114	0.140242
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 215.846i	− 0.297719i
$726$	0	0
$727$	−56.7264	−0.0780280	−0.0390140	−	0.999239i	$-0.512422\pi$
$727$	−56.7264	−0.0780280	−0.0390140	+	0.999239i	$0.512422\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 24.4349i	− 0.0334267i
$732$	0	0
$733$	−1010.35	−1.37837	−0.689185	−	0.724585i	$-0.742031\pi$
$733$	−1010.35	−1.37837	−0.689185	+	0.724585i	$0.742031\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1642.86i	2.22912i
$738$	0	0
$739$	−737.364	−0.997786	−0.498893	−	0.866664i	$-0.666260\pi$
$739$	−737.364	−0.997786	−0.498893	+	0.866664i	$0.666260\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	92.0738i	0.123922i	0.998079	+	0.0619609i	$0.0197354\pi$
$743$	92.0738i	0.123922i	−0.998079	+	0.0619609i	$0.980265\pi$
$744$	0	0
$745$	−627.534	−0.842328
$746$	0	0
$747$	0	0
$748$	0	0
$749$	239.744i	0.320086i
$750$	0	0
$751$	−376.779	−0.501703	−0.250852	−	0.968026i	$-0.580711\pi$
$751$	−376.779	−0.501703	−0.250852	+	0.968026i	$0.580711\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 187.652i	− 0.248545i
$756$	0	0
$757$	98.9786	0.130751	0.0653755	−	0.997861i	$-0.479175\pi$
$757$	98.9786	0.130751	0.0653755	+	0.997861i	$0.479175\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1051.29i	− 1.38146i	−0.723113	−	0.690730i	$-0.757289\pi$
$761$	− 1051.29i	− 1.38146i	0.723113	−	0.690730i	$-0.242711\pi$
$762$	0	0
$763$	−293.226	−0.384307
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 242.859i	− 0.316635i
$768$	0	0
$769$	−14.9634	−0.0194582	−0.00972912	−	0.999953i	$-0.503097\pi$
$769$	−14.9634	−0.0194582	−0.00972912	+	0.999953i	$0.503097\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 523.668i	− 0.677448i	−0.940886	−	0.338724i	$-0.890005\pi$
$773$	− 523.668i	− 0.677448i	0.940886	−	0.338724i	$-0.109995\pi$
$774$	0	0
$775$	309.353	0.399165
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1164.04i	− 1.49427i
$780$	0	0
$781$	1542.48	1.97501
$782$	0	0
$783$	0	0
$784$	0	0
$785$	298.787i	0.380621i
$786$	0	0
$787$	157.602	0.200257	0.100128	−	0.994975i	$-0.468075\pi$
$787$	157.602	0.200257	0.100128	+	0.994975i	$0.468075\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 506.506i	− 0.640336i
$792$	0	0
$793$	−43.4732	−0.0548212
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.3481i	− 0.0205120i	−0.999947	−	0.0102560i	$-0.996735\pi$
$797$	− 16.3481i	− 0.0205120i	0.999947	−	0.0102560i	$-0.00326465\pi$
$798$	0	0
$799$	294.976	0.369182
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1424.43i	1.77388i
$804$	0	0
$805$	20.3467	0.0252754
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1248.37i	1.54310i	0.636168	+	0.771551i	$0.280519\pi$
$809$	1248.37i	1.54310i	−0.636168	+	0.771551i	$0.719481\pi$
$810$	0	0
$811$	−796.242	−0.981803	−0.490901	−	0.871215i	$-0.663332\pi$
$811$	−796.242	−0.981803	−0.490901	+	0.871215i	$0.663332\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	229.683i	0.281820i
$816$	0	0
$817$	100.312	0.122781
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1390.54i	1.69371i	0.531823	+	0.846856i	$0.321507\pi$
$821$	1390.54i	1.69371i	−0.531823	+	0.846856i	$0.678493\pi$
$822$	0	0
$823$	22.4296	0.0272535	0.0136267	−	0.999907i	$-0.495662\pi$
$823$	22.4296	0.0272535	0.0136267	+	0.999907i	$0.495662\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 91.7876i	− 0.110989i	−0.998459	−	0.0554943i	$-0.982327\pi$
$827$	− 91.7876i	− 0.110989i	0.998459	−	0.0554943i	$-0.0176735\pi$
$828$	0	0
$829$	797.611	0.962136	0.481068	−	0.876683i	$-0.340249\pi$
$829$	797.611	0.962136	0.481068	+	0.876683i	$0.340249\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 249.581i	− 0.299617i
$834$	0	0
$835$	−918.354	−1.09982
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1656.32i	1.97416i	0.160219	+	0.987081i	$0.448780\pi$
$839$	1656.32i	1.97416i	−0.160219	+	0.987081i	$0.551220\pi$
$840$	0	0
$841$	−602.805	−0.716772
$842$	0	0
$843$	0	0
$844$	0	0
$845$	675.493i	0.799400i
$846$	0	0
$847$	337.871	0.398903
$848$	0	0
$849$	0	0
$850$	0	0
$851$	48.0114i	0.0564177i
$852$	0	0
$853$	−1241.57	−1.45554	−0.727769	−	0.685822i	$-0.759443\pi$
$853$	−1241.57	−1.45554	−0.727769	+	0.685822i	$0.759443\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1477.10i	1.72357i	0.507270	+	0.861787i	$0.330655\pi$
$857$	1477.10i	1.72357i	−0.507270	+	0.861787i	$0.669345\pi$
$858$	0	0
$859$	−996.484	−1.16005	−0.580025	−	0.814598i	$-0.696957\pi$
$859$	−996.484	−1.16005	−0.580025	+	0.814598i	$0.696957\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 671.209i	− 0.777763i	−0.921288	−	0.388881i	$-0.872862\pi$
$863$	− 671.209i	− 0.777763i	0.921288	−	0.388881i	$-0.127138\pi$
$864$	0	0
$865$	−1274.31	−1.47319
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1629.51i	1.87515i
$870$	0	0
$871$	437.963	0.502827
$872$	0	0
$873$	0	0
$874$	0	0
$875$	481.943i	0.550793i
$876$	0	0
$877$	649.802	0.740937	0.370468	−	0.928845i	$-0.379197\pi$
$877$	649.802	0.740937	0.370468	+	0.928845i	$0.379197\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	226.836i	0.257476i	0.991679	+	0.128738i	$0.0410926\pi$
$881$	226.836i	0.257476i	−0.991679	+	0.128738i	$0.958907\pi$
$882$	0	0
$883$	−68.9989	−0.0781415	−0.0390707	−	0.999236i	$-0.512440\pi$
$883$	−68.9989	−0.0781415	−0.0390707	+	0.999236i	$0.512440\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	904.525i	1.01976i	0.860246	+	0.509879i	$0.170310\pi$
$887$	904.525i	1.01976i	−0.860246	+	0.509879i	$0.829690\pi$
$888$	0	0
$889$	258.216	0.290457
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1210.96i	1.35606i
$894$	0	0
$895$	428.317	0.478567
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2069.28i	− 2.30176i
$900$	0	0
$901$	−329.311	−0.365495
$902$	0	0
$903$	0	0
$904$	0	0
$905$	164.038i	0.181258i
$906$	0	0
$907$	−161.386	−0.177933	−0.0889667	−	0.996035i	$-0.528356\pi$
$907$	−161.386	−0.177933	−0.0889667	+	0.996035i	$0.528356\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	80.8154i	0.0887107i	0.999016	+	0.0443553i	$0.0141234\pi$
$911$	80.8154i	0.0887107i	−0.999016	+	0.0443553i	$0.985877\pi$
$912$	0	0
$913$	1927.27	2.11092
$914$	0	0
$915$	0	0
$916$	0	0
$917$	447.513i	0.488019i
$918$	0	0
$919$	−656.985	−0.714891	−0.357445	−	0.933934i	$-0.616352\pi$
$919$	−656.985	−0.714891	−0.357445	+	0.933934i	$0.616352\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 411.202i	− 0.445506i
$924$	0	0
$925$	−210.559	−0.227632
$926$	0	0
$927$	0	0
$928$	0	0
$929$	248.318i	0.267296i	0.991029	+	0.133648i	$0.0426692\pi$
$929$	248.318i	0.267296i	−0.991029	+	0.133648i	$0.957331\pi$
$930$	0	0
$931$	1024.60	1.10054
$932$	0	0
$933$	0	0
$934$	0	0
$935$	444.562i	0.475468i
$936$	0	0
$937$	−936.945	−0.999941	−0.499971	−	0.866042i	$-0.666656\pi$
$937$	−936.945	−0.999941	−0.499971	+	0.866042i	$0.666656\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1399.26i	1.48700i	0.668738	+	0.743498i	$0.266835\pi$
$941$	1399.26i	1.48700i	−0.668738	+	0.743498i	$0.733165\pi$
$942$	0	0
$943$	53.3104	0.0565328
$944$	0	0
$945$	0	0
$946$	0	0
$947$	729.884i	0.770732i	0.922764	+	0.385366i	$0.125925\pi$
$947$	729.884i	0.770732i	−0.922764	+	0.385366i	$0.874075\pi$
$948$	0	0
$949$	379.731	0.400138
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1348.06i	1.41455i	0.706940	+	0.707274i	$0.250075\pi$
$953$	1348.06i	1.41455i	−0.706940	+	0.707274i	$0.749925\pi$
$954$	0	0
$955$	−1539.78	−1.61234
$956$	0	0
$957$	0	0
$958$	0	0
$959$	464.864i	0.484738i
$960$	0	0
$961$	2004.71	2.08607
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 48.5825i	− 0.0503446i
$966$	0	0
$967$	715.280	0.739690	0.369845	−	0.929093i	$-0.379411\pi$
$967$	715.280	0.739690	0.369845	+	0.929093i	$0.379411\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1288.41i	− 1.32689i	−0.748226	−	0.663444i	$-0.769094\pi$
$971$	− 1288.41i	− 1.32689i	0.748226	−	0.663444i	$-0.230906\pi$
$972$	0	0
$973$	301.936	0.310314
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 842.925i	− 0.862769i	−0.902168	−	0.431384i	$-0.858025\pi$
$977$	− 842.925i	− 0.862769i	0.902168	−	0.431384i	$-0.141975\pi$
$978$	0	0
$979$	−1489.51	−1.52146
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 510.355i	− 0.519181i	−0.965719	−	0.259590i	$-0.916412\pi$
$983$	− 510.355i	− 0.519181i	0.965719	−	0.259590i	$-0.0835876\pi$
$984$	0	0
$985$	−1398.18	−1.41947
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.59409i	0.00464518i
$990$	0	0
$991$	373.597	0.376990	0.188495	−	0.982074i	$-0.439639\pi$
$991$	373.597	0.376990	0.188495	+	0.982074i	$0.439639\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 776.653i	− 0.780556i
$996$	0	0
$997$	−1876.48	−1.88212	−0.941062	−	0.338235i	$-0.890170\pi$
$997$	−1876.48	−1.88212	−0.941062	+	0.338235i	$0.890170\pi$
$998$	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.1457.9		36
3.2	odd	2	inner	2916.3.c.b.1457.28		36
27.2	odd	18		324.3.k.a.17.4		36
27.13	even	9		324.3.k.a.305.4		36
27.14	odd	18		108.3.k.a.101.2	yes	36
27.25	even	9		108.3.k.a.77.2	✓	36
108.79	odd	18		432.3.bc.b.401.5		36
108.95	even	18		432.3.bc.b.209.5		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.77.2	✓	36	27.25	even	9
108.3.k.a.101.2	yes	36	27.14	odd	18
324.3.k.a.17.4		36	27.2	odd	18
324.3.k.a.305.4		36	27.13	even	9
432.3.bc.b.209.5		36	108.95	even	18
432.3.bc.b.401.5		36	108.79	odd	18
2916.3.c.b.1457.9		36	1.1	even	1	trivial
2916.3.c.b.1457.28		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)

\(f(q)\)	\(=\)	\(q-4.39539i q^{5} -3.57384 q^{7} +O(q^{10})\)	\(q-4.39539i q^{5} -3.57384 q^{7} +14.6813i q^{11} +3.91381 q^{13} +6.88924i q^{17} -28.2823 q^{19} -1.29527i q^{23} +5.68054 q^{25} -37.9974i q^{29} +54.4584 q^{31} +15.7084i q^{35} -37.0667 q^{37} +41.1578i q^{41} -3.54682 q^{43} -42.8169i q^{47} -36.2276 q^{49} +47.8007i q^{53} +64.5299 q^{55} -62.0518i q^{59} -11.1077 q^{61} -17.2027i q^{65} +111.902 q^{67} -105.064i q^{71} +97.0235 q^{73} -52.4686i q^{77} +110.992 q^{79} -131.274i q^{83} +30.2809 q^{85} +101.456i q^{89} -13.9873 q^{91} +124.312i q^{95} -165.580 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 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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