LMFDB - Embedded newform 2916.3.c.b.1457.3

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			1457.3
Character	$\chi$	$=$	2916.1457
Dual form			2916.3.c.b.1457.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.78294i q^{5} +3.61169 q^{7} +O(q^{10})$	$q-7.78294i q^{5} +3.61169 q^{7} -4.04124i q^{11} +13.3122 q^{13} +23.9807i q^{17} -27.1966 q^{19} +11.6113i q^{23} -35.5742 q^{25} +23.2183i q^{29} +4.05768 q^{31} -28.1096i q^{35} -70.7471 q^{37} +43.7499i q^{41} +46.1859 q^{43} +82.9064i q^{47} -35.9557 q^{49} +28.9765i q^{53} -31.4528 q^{55} +52.6257i q^{59} -4.35030 q^{61} -103.608i q^{65} -130.999 q^{67} -69.7998i q^{71} -130.917 q^{73} -14.5957i q^{77} -24.2397 q^{79} -41.6347i q^{83} +186.641 q^{85} -97.2245i q^{89} +48.0794 q^{91} +211.670i q^{95} +48.6620 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 7.78294i	− 1.55659i	−0.627900	−	0.778294i	$-0.716085\pi$
$5$	− 7.78294i	− 1.55659i	0.627900	−	0.778294i	$-0.283915\pi$
$6$	0	0
$7$	3.61169	0.515956	0.257978	−	0.966151i	$-0.416944\pi$
$7$	3.61169	0.515956	0.257978	+	0.966151i	$0.416944\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.04124i	− 0.367386i	−0.982984	−	0.183693i	$-0.941195\pi$
$11$	− 4.04124i	− 0.367386i	0.982984	−	0.183693i	$-0.0588052\pi$
$12$	0	0
$13$	13.3122	1.02401	0.512007	−	0.858982i	$-0.328902\pi$
$13$	13.3122	1.02401	0.512007	+	0.858982i	$0.328902\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	23.9807i	1.41063i	0.708893	+	0.705316i	$0.249195\pi$
$17$	23.9807i	1.41063i	−0.708893	+	0.705316i	$0.750805\pi$
$18$	0	0
$19$	−27.1966	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$19$	−27.1966	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	11.6113i	0.504837i	0.967618	+	0.252419i	$0.0812260\pi$
$23$	11.6113i	0.504837i	−0.967618	+	0.252419i	$0.918774\pi$
$24$	0	0
$25$	−35.5742	−1.42297
$26$	0	0
$27$	0	0
$28$	0	0
$29$	23.2183i	0.800631i	0.916377	+	0.400315i	$0.131099\pi$
$29$	23.2183i	0.800631i	−0.916377	+	0.400315i	$0.868901\pi$
$30$	0	0
$31$	4.05768	0.130893	0.0654465	−	0.997856i	$-0.479153\pi$
$31$	4.05768	0.130893	0.0654465	+	0.997856i	$0.479153\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 28.1096i	− 0.803131i
$36$	0	0
$37$	−70.7471	−1.91208	−0.956041	−	0.293232i	$-0.905269\pi$
$37$	−70.7471	−1.91208	−0.956041	+	0.293232i	$0.905269\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	43.7499i	1.06707i	0.845778	+	0.533535i	$0.179137\pi$
$41$	43.7499i	1.06707i	−0.845778	+	0.533535i	$0.820863\pi$
$42$	0	0
$43$	46.1859	1.07409	0.537045	−	0.843554i	$-0.319541\pi$
$43$	46.1859	1.07409	0.537045	+	0.843554i	$0.319541\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	82.9064i	1.76397i	0.471281	+	0.881983i	$0.343792\pi$
$47$	82.9064i	1.76397i	−0.471281	+	0.881983i	$0.656208\pi$
$48$	0	0
$49$	−35.9557	−0.733790
$50$	0	0
$51$	0	0
$52$	0	0
$53$	28.9765i	0.546726i	0.961911	+	0.273363i	$0.0881361\pi$
$53$	28.9765i	0.546726i	−0.961911	+	0.273363i	$0.911864\pi$
$54$	0	0
$55$	−31.4528	−0.571868
$56$	0	0
$57$	0	0
$58$	0	0
$59$	52.6257i	0.891961i	0.895043	+	0.445981i	$0.147145\pi$
$59$	52.6257i	0.891961i	−0.895043	+	0.445981i	$0.852855\pi$
$60$	0	0
$61$	−4.35030	−0.0713164	−0.0356582	−	0.999364i	$-0.511353\pi$
$61$	−4.35030	−0.0713164	−0.0356582	+	0.999364i	$0.511353\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 103.608i	− 1.59397i
$66$	0	0
$67$	−130.999	−1.95521	−0.977604	−	0.210454i	$-0.932506\pi$
$67$	−130.999	−1.95521	−0.977604	+	0.210454i	$0.932506\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 69.7998i	− 0.983096i	−0.870850	−	0.491548i	$-0.836431\pi$
$71$	− 69.7998i	− 0.983096i	0.870850	−	0.491548i	$-0.163569\pi$
$72$	0	0
$73$	−130.917	−1.79338	−0.896690	−	0.442660i	$-0.854035\pi$
$73$	−130.917	−1.79338	−0.896690	+	0.442660i	$0.854035\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 14.5957i	− 0.189555i
$78$	0	0
$79$	−24.2397	−0.306831	−0.153416	−	0.988162i	$-0.549027\pi$
$79$	−24.2397	−0.306831	−0.153416	+	0.988162i	$0.549027\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 41.6347i	− 0.501622i	−0.968036	−	0.250811i	$-0.919303\pi$
$83$	− 41.6347i	− 0.501622i	0.968036	−	0.250811i	$-0.0806973\pi$
$84$	0	0
$85$	186.641	2.19577
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 97.2245i	− 1.09241i	−0.837652	−	0.546205i	$-0.816072\pi$
$89$	− 97.2245i	− 1.09241i	0.837652	−	0.546205i	$-0.183928\pi$
$90$	0	0
$91$	48.0794	0.528346
$92$	0	0
$93$	0	0
$94$	0	0
$95$	211.670i	2.22810i
$96$	0	0
$97$	48.6620	0.501670	0.250835	−	0.968030i	$-0.419295\pi$
$97$	48.6620	0.501670	0.250835	+	0.968030i	$0.419295\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 37.9006i	− 0.375253i	−0.982240	−	0.187627i	$-0.939920\pi$
$101$	− 37.9006i	− 0.375253i	0.982240	−	0.187627i	$-0.0600795\pi$
$102$	0	0
$103$	15.3680	0.149204	0.0746020	−	0.997213i	$-0.476231\pi$
$103$	15.3680	0.149204	0.0746020	+	0.997213i	$0.476231\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	116.068i	1.08475i	0.840138	+	0.542373i	$0.182474\pi$
$107$	116.068i	1.08475i	−0.840138	+	0.542373i	$0.817526\pi$
$108$	0	0
$109$	108.821	0.998362	0.499181	−	0.866498i	$-0.333634\pi$
$109$	108.821	0.998362	0.499181	+	0.866498i	$0.333634\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 49.4349i	− 0.437477i	−0.975783	−	0.218739i	$-0.929806\pi$
$113$	− 49.4349i	− 0.437477i	0.975783	−	0.218739i	$-0.0701942\pi$
$114$	0	0
$115$	90.3697	0.785824
$116$	0	0
$117$	0	0
$118$	0	0
$119$	86.6110i	0.727824i
$120$	0	0
$121$	104.668	0.865028
$122$	0	0
$123$	0	0
$124$	0	0
$125$	82.2981i	0.658385i
$126$	0	0
$127$	−65.2718	−0.513951	−0.256975	−	0.966418i	$-0.582726\pi$
$127$	−65.2718	−0.513951	−0.256975	+	0.966418i	$0.582726\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.2197i	− 0.154349i	−0.997018	−	0.0771746i	$-0.975410\pi$
$131$	− 20.2197i	− 0.154349i	0.997018	−	0.0771746i	$-0.0245899\pi$
$132$	0	0
$133$	−98.2259	−0.738540
$134$	0	0
$135$	0	0
$136$	0	0
$137$	42.2966i	0.308734i	0.988014	+	0.154367i	$0.0493338\pi$
$137$	42.2966i	0.308734i	−0.988014	+	0.154367i	$0.950666\pi$
$138$	0	0
$139$	−88.0616	−0.633537	−0.316768	−	0.948503i	$-0.602598\pi$
$139$	−88.0616	−0.633537	−0.316768	+	0.948503i	$0.602598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 53.7977i	− 0.376208i
$144$	0	0
$145$	180.707	1.24625
$146$	0	0
$147$	0	0
$148$	0	0
$149$	211.934i	1.42238i	0.703002	+	0.711188i	$0.251842\pi$
$149$	211.934i	1.42238i	−0.703002	+	0.711188i	$0.748158\pi$
$150$	0	0
$151$	−176.464	−1.16864	−0.584319	−	0.811524i	$-0.698638\pi$
$151$	−176.464	−1.16864	−0.584319	+	0.811524i	$0.698638\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 31.5807i	− 0.203746i
$156$	0	0
$157$	−5.60779	−0.0357184	−0.0178592	−	0.999841i	$-0.505685\pi$
$157$	−5.60779	−0.0357184	−0.0178592	+	0.999841i	$0.505685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	41.9363i	0.260474i
$162$	0	0
$163$	176.753	1.08437	0.542186	−	0.840258i	$-0.317597\pi$
$163$	176.753	1.08437	0.542186	+	0.840258i	$0.317597\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	297.175i	1.77949i	0.456459	+	0.889744i	$0.349118\pi$
$167$	297.175i	1.77949i	−0.456459	+	0.889744i	$0.650882\pi$
$168$	0	0
$169$	8.21388	0.0486028
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 141.296i	− 0.816739i	−0.912817	−	0.408370i	$-0.866097\pi$
$173$	− 141.296i	− 0.816739i	0.912817	−	0.408370i	$-0.133903\pi$
$174$	0	0
$175$	−128.483	−0.734188
$176$	0	0
$177$	0	0
$178$	0	0
$179$	175.887i	0.982608i	0.870988	+	0.491304i	$0.163480\pi$
$179$	175.887i	0.982608i	−0.870988	+	0.491304i	$0.836520\pi$
$180$	0	0
$181$	241.107	1.33208	0.666042	−	0.745914i	$-0.267987\pi$
$181$	241.107	1.33208	0.666042	+	0.745914i	$0.267987\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	550.620i	2.97632i
$186$	0	0
$187$	96.9120	0.518246
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 162.090i	− 0.848637i	−0.905513	−	0.424319i	$-0.860514\pi$
$191$	− 162.090i	− 0.848637i	0.905513	−	0.424319i	$-0.139486\pi$
$192$	0	0
$193$	14.4069	0.0746469	0.0373235	−	0.999303i	$-0.488117\pi$
$193$	14.4069	0.0746469	0.0373235	+	0.999303i	$0.488117\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	217.658i	1.10486i	0.833559	+	0.552431i	$0.186300\pi$
$197$	217.658i	1.10486i	−0.833559	+	0.552431i	$0.813700\pi$
$198$	0	0
$199$	−41.0305	−0.206183	−0.103092	−	0.994672i	$-0.532874\pi$
$199$	−41.0305	−0.206183	−0.103092	+	0.994672i	$0.532874\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	83.8573i	0.413090i
$204$	0	0
$205$	340.503	1.66099
$206$	0	0
$207$	0	0
$208$	0	0
$209$	109.908i	0.525877i
$210$	0	0
$211$	−123.752	−0.586504	−0.293252	−	0.956035i	$-0.594738\pi$
$211$	−123.752	−0.586504	−0.293252	+	0.956035i	$0.594738\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 359.462i	− 1.67192i
$216$	0	0
$217$	14.6551	0.0675350
$218$	0	0
$219$	0	0
$220$	0	0
$221$	319.236i	1.44451i
$222$	0	0
$223$	−261.574	−1.17298	−0.586488	−	0.809958i	$-0.699490\pi$
$223$	−261.574	−1.17298	−0.586488	+	0.809958i	$0.699490\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 82.6522i	− 0.364107i	−0.983289	−	0.182053i	$-0.941726\pi$
$227$	− 82.6522i	− 0.364107i	0.983289	−	0.182053i	$-0.0582743\pi$
$228$	0	0
$229$	132.478	0.578507	0.289254	−	0.957252i	$-0.406593\pi$
$229$	132.478	0.578507	0.289254	+	0.957252i	$0.406593\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 410.338i	− 1.76111i	−0.473947	−	0.880553i	$-0.657171\pi$
$233$	− 410.338i	− 1.76111i	0.473947	−	0.880553i	$-0.342829\pi$
$234$	0	0
$235$	645.256	2.74577
$236$	0	0
$237$	0	0
$238$	0	0
$239$	113.300i	0.474058i	0.971503	+	0.237029i	$0.0761736\pi$
$239$	113.300i	0.474058i	−0.971503	+	0.237029i	$0.923826\pi$
$240$	0	0
$241$	−105.955	−0.439647	−0.219823	−	0.975540i	$-0.570548\pi$
$241$	−105.955	−0.439647	−0.219823	+	0.975540i	$0.570548\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	279.841i	1.14221i
$246$	0	0
$247$	−362.046	−1.46577
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.7745i	0.0628466i	0.999506	+	0.0314233i	$0.0100040\pi$
$251$	15.7745i	0.0628466i	−0.999506	+	0.0314233i	$0.989996\pi$
$252$	0	0
$253$	46.9239	0.185470
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 295.092i	− 1.14822i	−0.818779	−	0.574109i	$-0.805349\pi$
$257$	− 295.092i	− 1.14822i	0.818779	−	0.574109i	$-0.194651\pi$
$258$	0	0
$259$	−255.516	−0.986550
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.2302i	0.0731186i	0.999331	+	0.0365593i	$0.0116398\pi$
$263$	19.2302i	0.0731186i	−0.999331	+	0.0365593i	$0.988360\pi$
$264$	0	0
$265$	225.522	0.851028
$266$	0	0
$267$	0	0
$268$	0	0
$269$	190.266i	0.707309i	0.935376	+	0.353654i	$0.115061\pi$
$269$	190.266i	0.707309i	−0.935376	+	0.353654i	$0.884939\pi$
$270$	0	0
$271$	−121.179	−0.447154	−0.223577	−	0.974686i	$-0.571773\pi$
$271$	−121.179	−0.447154	−0.223577	+	0.974686i	$0.571773\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	143.764i	0.522778i
$276$	0	0
$277$	−62.4401	−0.225415	−0.112708	−	0.993628i	$-0.535952\pi$
$277$	−62.4401	−0.225415	−0.112708	+	0.993628i	$0.535952\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	254.617i	0.906111i	0.891482	+	0.453056i	$0.149666\pi$
$281$	254.617i	0.906111i	−0.891482	+	0.453056i	$0.850334\pi$
$282$	0	0
$283$	218.026	0.770412	0.385206	−	0.922831i	$-0.374130\pi$
$283$	218.026	0.770412	0.385206	+	0.922831i	$0.374130\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	158.011i	0.550561i
$288$	0	0
$289$	−286.076	−0.989881
$290$	0	0
$291$	0	0
$292$	0	0
$293$	255.990i	0.873686i	0.899538	+	0.436843i	$0.143904\pi$
$293$	255.990i	0.873686i	−0.899538	+	0.436843i	$0.856096\pi$
$294$	0	0
$295$	409.583	1.38842
$296$	0	0
$297$	0	0
$298$	0	0
$299$	154.571i	0.516960i
$300$	0	0
$301$	166.809	0.554183
$302$	0	0
$303$	0	0
$304$	0	0
$305$	33.8581i	0.111010i
$306$	0	0
$307$	−353.345	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$307$	−353.345	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	48.5633i	0.156152i	0.996947	+	0.0780761i	$0.0248777\pi$
$311$	48.5633i	0.156152i	−0.996947	+	0.0780761i	$0.975122\pi$
$312$	0	0
$313$	−65.7457	−0.210050	−0.105025	−	0.994470i	$-0.533492\pi$
$313$	−65.7457	−0.210050	−0.105025	+	0.994470i	$0.533492\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 509.208i	− 1.60633i	−0.595754	−	0.803167i	$-0.703147\pi$
$317$	− 509.208i	− 1.60633i	0.595754	−	0.803167i	$-0.296853\pi$
$318$	0	0
$319$	93.8308	0.294140
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 652.196i	− 2.01918i
$324$	0	0
$325$	−473.569	−1.45714
$326$	0	0
$327$	0	0
$328$	0	0
$329$	299.432i	0.910129i
$330$	0	0
$331$	444.598	1.34320	0.671598	−	0.740916i	$-0.265608\pi$
$331$	444.598	1.34320	0.671598	+	0.740916i	$0.265608\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1019.56i	3.04345i
$336$	0	0
$337$	354.719	1.05258	0.526289	−	0.850306i	$-0.323583\pi$
$337$	354.719	1.05258	0.526289	+	0.850306i	$0.323583\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 16.3981i	− 0.0480882i
$342$	0	0
$343$	−306.834	−0.894559
$344$	0	0
$345$	0	0
$346$	0	0
$347$	458.748i	1.32204i	0.750369	+	0.661020i	$0.229876\pi$
$347$	458.748i	1.32204i	−0.750369	+	0.661020i	$0.770124\pi$
$348$	0	0
$349$	418.275	1.19849	0.599247	−	0.800564i	$-0.295467\pi$
$349$	418.275	1.19849	0.599247	+	0.800564i	$0.295467\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 635.186i	− 1.79939i	−0.436514	−	0.899697i	$-0.643787\pi$
$353$	− 635.186i	− 1.79939i	0.436514	−	0.899697i	$-0.356213\pi$
$354$	0	0
$355$	−543.248	−1.53028
$356$	0	0
$357$	0	0
$358$	0	0
$359$	447.522i	1.24658i	0.781991	+	0.623290i	$0.214204\pi$
$359$	447.522i	1.24658i	−0.781991	+	0.623290i	$0.785796\pi$
$360$	0	0
$361$	378.657	1.04891
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1018.92i	2.79155i
$366$	0	0
$367$	−570.876	−1.55552	−0.777760	−	0.628561i	$-0.783644\pi$
$367$	−570.876	−1.55552	−0.777760	+	0.628561i	$0.783644\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	104.654i	0.282087i
$372$	0	0
$373$	221.762	0.594535	0.297268	−	0.954794i	$-0.403925\pi$
$373$	221.762	0.594535	0.297268	+	0.954794i	$0.403925\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	309.086i	0.819856i
$378$	0	0
$379$	−523.969	−1.38250	−0.691252	−	0.722614i	$-0.742941\pi$
$379$	−523.969	−1.38250	−0.691252	+	0.722614i	$0.742941\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	408.849i	1.06749i	0.845645	+	0.533746i	$0.179216\pi$
$383$	408.849i	1.06749i	−0.845645	+	0.533746i	$0.820784\pi$
$384$	0	0
$385$	−113.598	−0.295059
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 398.851i	− 1.02532i	−0.858590	−	0.512662i	$-0.828659\pi$
$389$	− 398.851i	− 1.02532i	0.858590	−	0.512662i	$-0.171341\pi$
$390$	0	0
$391$	−278.447	−0.712139
$392$	0	0
$393$	0	0
$394$	0	0
$395$	188.656i	0.477610i
$396$	0	0
$397$	−277.477	−0.698934	−0.349467	−	0.936949i	$-0.613637\pi$
$397$	−277.477	−0.698934	−0.349467	+	0.936949i	$0.613637\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	145.880i	0.363790i	0.983318	+	0.181895i	$0.0582231\pi$
$401$	145.880i	0.363790i	−0.983318	+	0.181895i	$0.941777\pi$
$402$	0	0
$403$	54.0165	0.134036
$404$	0	0
$405$	0	0
$406$	0	0
$407$	285.906i	0.702472i
$408$	0	0
$409$	113.984	0.278689	0.139344	−	0.990244i	$-0.455501\pi$
$409$	113.984	0.278689	0.139344	+	0.990244i	$0.455501\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	190.068i	0.460213i
$414$	0	0
$415$	−324.040	−0.780819
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 522.170i	− 1.24623i	−0.782130	−	0.623115i	$-0.785867\pi$
$419$	− 522.170i	− 1.24623i	0.782130	−	0.623115i	$-0.214133\pi$
$420$	0	0
$421$	−395.418	−0.939235	−0.469617	−	0.882870i	$-0.655608\pi$
$421$	−395.418	−0.939235	−0.469617	+	0.882870i	$0.655608\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 853.095i	− 2.00728i
$426$	0	0
$427$	−15.7119	−0.0367961
$428$	0	0
$429$	0	0
$430$	0	0
$431$	555.362i	1.28854i	0.764797	+	0.644272i	$0.222839\pi$
$431$	555.362i	1.28854i	−0.764797	+	0.644272i	$0.777161\pi$
$432$	0	0
$433$	13.3027	0.0307222	0.0153611	−	0.999882i	$-0.495110\pi$
$433$	13.3027	0.0307222	0.0153611	+	0.999882i	$0.495110\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 315.787i	− 0.722625i
$438$	0	0
$439$	−297.557	−0.677805	−0.338903	−	0.940821i	$-0.610056\pi$
$439$	−297.557	−0.677805	−0.338903	+	0.940821i	$0.610056\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 99.6456i	− 0.224934i	−0.993655	−	0.112467i	$-0.964125\pi$
$443$	− 99.6456i	− 0.224934i	0.993655	−	0.112467i	$-0.0358752\pi$
$444$	0	0
$445$	−756.692	−1.70043
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 490.945i	− 1.09342i	−0.837322	−	0.546710i	$-0.815880\pi$
$449$	− 490.945i	− 1.09342i	0.837322	−	0.546710i	$-0.184120\pi$
$450$	0	0
$451$	176.804	0.392026
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 374.199i	− 0.822416i
$456$	0	0
$457$	−115.530	−0.252801	−0.126400	−	0.991979i	$-0.540342\pi$
$457$	−115.530	−0.252801	−0.126400	+	0.991979i	$0.540342\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 115.145i	− 0.249772i	−0.992171	−	0.124886i	$-0.960144\pi$
$461$	− 115.145i	− 0.249772i	0.992171	−	0.124886i	$-0.0398565\pi$
$462$	0	0
$463$	45.6831	0.0986675	0.0493338	−	0.998782i	$-0.484290\pi$
$463$	45.6831	0.0986675	0.0493338	+	0.998782i	$0.484290\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 397.192i	− 0.850519i	−0.905072	−	0.425259i	$-0.860183\pi$
$467$	− 397.192i	− 0.850519i	0.905072	−	0.425259i	$-0.139817\pi$
$468$	0	0
$469$	−473.128	−1.00880
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 186.648i	− 0.394605i
$474$	0	0
$475$	967.498	2.03684
$476$	0	0
$477$	0	0
$478$	0	0
$479$	782.566i	1.63375i	0.576815	+	0.816875i	$0.304295\pi$
$479$	782.566i	1.63375i	−0.576815	+	0.816875i	$0.695705\pi$
$480$	0	0
$481$	−941.797	−1.95800
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 378.733i	− 0.780893i
$486$	0	0
$487$	−505.750	−1.03850	−0.519250	−	0.854622i	$-0.673789\pi$
$487$	−505.750	−1.03850	−0.519250	+	0.854622i	$0.673789\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	921.951i	1.87770i	0.344325	+	0.938851i	$0.388108\pi$
$491$	921.951i	1.87770i	−0.344325	+	0.938851i	$0.611892\pi$
$492$	0	0
$493$	−556.792	−1.12940
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 252.095i	− 0.507234i
$498$	0	0
$499$	619.830	1.24215	0.621073	−	0.783753i	$-0.286697\pi$
$499$	619.830	1.24215	0.621073	+	0.783753i	$0.286697\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.97061i	0.00391771i	0.999998	+	0.00195885i	$0.000623523\pi$
$503$	1.97061i	0.00391771i	−0.999998	+	0.00195885i	$0.999376\pi$
$504$	0	0
$505$	−294.978	−0.584115
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 12.6050i	− 0.0247642i	−0.999923	−	0.0123821i	$-0.996059\pi$
$509$	− 12.6050i	− 0.0247642i	0.999923	−	0.0123821i	$-0.00394144\pi$
$510$	0	0
$511$	−472.831	−0.925305
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 119.608i	− 0.232249i
$516$	0	0
$517$	335.045	0.648056
$518$	0	0
$519$	0	0
$520$	0	0
$521$	262.877i	0.504562i	0.967654	+	0.252281i	$0.0811808\pi$
$521$	262.877i	0.504562i	−0.967654	+	0.252281i	$0.918819\pi$
$522$	0	0
$523$	−269.666	−0.515613	−0.257807	−	0.966197i	$-0.583000\pi$
$523$	−269.666	−0.515613	−0.257807	+	0.966197i	$0.583000\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	97.3062i	0.184642i
$528$	0	0
$529$	394.179	0.745139
$530$	0	0
$531$	0	0
$532$	0	0
$533$	582.405i	1.09269i
$534$	0	0
$535$	903.348	1.68850
$536$	0	0
$537$	0	0
$538$	0	0
$539$	145.306i	0.269584i
$540$	0	0
$541$	986.273	1.82306	0.911528	−	0.411238i	$-0.134903\pi$
$541$	986.273	1.82306	0.911528	+	0.411238i	$0.134903\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 846.951i	− 1.55404i
$546$	0	0
$547$	673.602	1.23145	0.615724	−	0.787962i	$-0.288864\pi$
$547$	673.602	1.23145	0.615724	+	0.787962i	$0.288864\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 631.460i	− 1.14602i
$552$	0	0
$553$	−87.5462	−0.158311
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 221.715i	− 0.398052i	−0.979994	−	0.199026i	$-0.936222\pi$
$557$	− 221.715i	− 0.398052i	0.979994	−	0.199026i	$-0.0637779\pi$
$558$	0	0
$559$	614.834	1.09988
$560$	0	0
$561$	0	0
$562$	0	0
$563$	99.7845i	0.177237i	0.996066	+	0.0886185i	$0.0282452\pi$
$563$	99.7845i	0.177237i	−0.996066	+	0.0886185i	$0.971755\pi$
$564$	0	0
$565$	−384.749	−0.680972
$566$	0	0
$567$	0	0
$568$	0	0
$569$	610.761i	1.07339i	0.843775	+	0.536697i	$0.180328\pi$
$569$	610.761i	1.07339i	−0.843775	+	0.536697i	$0.819672\pi$
$570$	0	0
$571$	593.332	1.03911	0.519555	−	0.854437i	$-0.326098\pi$
$571$	593.332	1.03911	0.519555	+	0.854437i	$0.326098\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 413.061i	− 0.718367i
$576$	0	0
$577$	401.938	0.696600	0.348300	−	0.937383i	$-0.386759\pi$
$577$	401.938	0.696600	0.348300	+	0.937383i	$0.386759\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 150.372i	− 0.258815i
$582$	0	0
$583$	117.101	0.200859
$584$	0	0
$585$	0	0
$586$	0	0
$587$	404.628i	0.689314i	0.938729	+	0.344657i	$0.112005\pi$
$587$	404.628i	0.689314i	−0.938729	+	0.344657i	$0.887995\pi$
$588$	0	0
$589$	−110.355	−0.187360
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 397.833i	− 0.670882i	−0.942061	−	0.335441i	$-0.891115\pi$
$593$	− 397.833i	− 0.670882i	0.942061	−	0.335441i	$-0.108885\pi$
$594$	0	0
$595$	674.088	1.13292
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 978.414i	− 1.63341i	−0.577054	−	0.816706i	$-0.695798\pi$
$599$	− 978.414i	− 1.63341i	0.577054	−	0.816706i	$-0.304202\pi$
$600$	0	0
$601$	308.080	0.512612	0.256306	−	0.966596i	$-0.417495\pi$
$601$	308.080	0.512612	0.256306	+	0.966596i	$0.417495\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 814.628i	− 1.34649i
$606$	0	0
$607$	136.949	0.225616	0.112808	−	0.993617i	$-0.464015\pi$
$607$	136.949	0.225616	0.112808	+	0.993617i	$0.464015\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1103.66i	1.80632i
$612$	0	0
$613$	−331.181	−0.540263	−0.270132	−	0.962823i	$-0.587067\pi$
$613$	−331.181	−0.540263	−0.270132	+	0.962823i	$0.587067\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 490.266i	− 0.794596i	−0.917690	−	0.397298i	$-0.869948\pi$
$617$	− 490.266i	− 0.794596i	0.917690	−	0.397298i	$-0.130052\pi$
$618$	0	0
$619$	−256.807	−0.414874	−0.207437	−	0.978248i	$-0.566512\pi$
$619$	−256.807	−0.414874	−0.207437	+	0.978248i	$0.566512\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 351.145i	− 0.563635i
$624$	0	0
$625$	−248.833	−0.398133
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1696.57i	− 2.69724i
$630$	0	0
$631$	641.311	1.01634	0.508171	−	0.861256i	$-0.330322\pi$
$631$	641.311	1.01634	0.508171	+	0.861256i	$0.330322\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	508.006i	0.800010i
$636$	0	0
$637$	−478.648	−0.751410
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 867.460i	− 1.35329i	−0.736309	−	0.676646i	$-0.763433\pi$
$641$	− 867.460i	− 1.35329i	0.736309	−	0.676646i	$-0.236567\pi$
$642$	0	0
$643$	−16.7349	−0.0260263	−0.0130132	−	0.999915i	$-0.504142\pi$
$643$	−16.7349	−0.0260263	−0.0130132	+	0.999915i	$0.504142\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 96.9735i	− 0.149882i	−0.997188	−	0.0749409i	$-0.976123\pi$
$647$	− 96.9735i	− 0.149882i	0.997188	−	0.0749409i	$-0.0238768\pi$
$648$	0	0
$649$	212.673	0.327694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 364.001i	− 0.557429i	−0.960374	−	0.278715i	$-0.910092\pi$
$653$	− 364.001i	− 0.557429i	0.960374	−	0.278715i	$-0.0899083\pi$
$654$	0	0
$655$	−157.369	−0.240258
$656$	0	0
$657$	0	0
$658$	0	0
$659$	384.053i	0.582781i	0.956604	+	0.291390i	$0.0941179\pi$
$659$	384.053i	0.582781i	−0.956604	+	0.291390i	$0.905882\pi$
$660$	0	0
$661$	266.206	0.402733	0.201366	−	0.979516i	$-0.435462\pi$
$661$	266.206	0.402733	0.201366	+	0.979516i	$0.435462\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	764.486i	1.14960i
$666$	0	0
$667$	−269.594	−0.404188
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17.5806i	0.0262006i
$672$	0	0
$673$	−88.6987	−0.131796	−0.0658980	−	0.997826i	$-0.520991\pi$
$673$	−88.6987	−0.131796	−0.0658980	+	0.997826i	$0.520991\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 447.341i	− 0.660769i	−0.943846	−	0.330385i	$-0.892822\pi$
$677$	− 447.341i	− 0.660769i	0.943846	−	0.330385i	$-0.107178\pi$
$678$	0	0
$679$	175.752	0.258839
$680$	0	0
$681$	0	0
$682$	0	0
$683$	334.656i	0.489980i	0.969526	+	0.244990i	$0.0787846\pi$
$683$	334.656i	0.489980i	−0.969526	+	0.244990i	$0.921215\pi$
$684$	0	0
$685$	329.192	0.480572
$686$	0	0
$687$	0	0
$688$	0	0
$689$	385.740i	0.559855i
$690$	0	0
$691$	417.226	0.603800	0.301900	−	0.953340i	$-0.402379\pi$
$691$	417.226	0.603800	0.301900	+	0.953340i	$0.402379\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	685.379i	0.986156i
$696$	0	0
$697$	−1049.15	−1.50524
$698$	0	0
$699$	0	0
$700$	0	0
$701$	613.675i	0.875428i	0.899114	+	0.437714i	$0.144212\pi$
$701$	613.675i	0.875428i	−0.899114	+	0.437714i	$0.855788\pi$
$702$	0	0
$703$	1924.08	2.73696
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 136.885i	− 0.193614i
$708$	0	0
$709$	839.395	1.18391	0.591957	−	0.805969i	$-0.298355\pi$
$709$	839.395	1.18391	0.591957	+	0.805969i	$0.298355\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	47.1148i	0.0660796i
$714$	0	0
$715$	−418.704	−0.585601
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1001.00i	− 1.39222i	−0.717937	−	0.696108i	$-0.754913\pi$
$719$	− 1001.00i	− 1.39222i	0.717937	−	0.696108i	$-0.245087\pi$
$720$	0	0
$721$	55.5045	0.0769827
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 825.971i	− 1.13927i
$726$	0	0
$727$	−1060.32	−1.45849	−0.729246	−	0.684252i	$-0.760129\pi$
$727$	−1060.32	−1.45849	−0.729246	+	0.684252i	$0.760129\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1107.57i	1.51515i
$732$	0	0
$733$	−351.925	−0.480116	−0.240058	−	0.970759i	$-0.577166\pi$
$733$	−351.925	−0.480116	−0.240058	+	0.970759i	$0.577166\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	529.398i	0.718315i
$738$	0	0
$739$	834.559	1.12931	0.564654	−	0.825328i	$-0.309010\pi$
$739$	834.559	1.12931	0.564654	+	0.825328i	$0.309010\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	134.270i	0.180714i	0.995909	+	0.0903568i	$0.0288007\pi$
$743$	134.270i	0.180714i	−0.995909	+	0.0903568i	$0.971199\pi$
$744$	0	0
$745$	1649.47	2.21405
$746$	0	0
$747$	0	0
$748$	0	0
$749$	419.201i	0.559681i
$750$	0	0
$751$	−306.416	−0.408011	−0.204005	−	0.978970i	$-0.565396\pi$
$751$	−306.416	−0.408011	−0.204005	+	0.978970i	$0.565396\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1373.41i	1.81909i
$756$	0	0
$757$	248.812	0.328682	0.164341	−	0.986404i	$-0.447450\pi$
$757$	248.812	0.328682	0.164341	+	0.986404i	$0.447450\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 125.274i	− 0.164618i	−0.996607	−	0.0823091i	$-0.973771\pi$
$761$	− 125.274i	− 0.164618i	0.996607	−	0.0823091i	$-0.0262295\pi$
$762$	0	0
$763$	393.029	0.515111
$764$	0	0
$765$	0	0
$766$	0	0
$767$	700.562i	0.913380i
$768$	0	0
$769$	−926.914	−1.20535	−0.602675	−	0.797987i	$-0.705898\pi$
$769$	−926.914	−1.20535	−0.602675	+	0.797987i	$0.705898\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	234.036i	0.302763i	0.988475	+	0.151381i	$0.0483722\pi$
$773$	234.036i	0.302763i	−0.988475	+	0.151381i	$0.951628\pi$
$774$	0	0
$775$	−144.349	−0.186256
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1189.85i	− 1.52741i
$780$	0	0
$781$	−282.078	−0.361175
$782$	0	0
$783$	0	0
$784$	0	0
$785$	43.6451i	0.0555989i
$786$	0	0
$787$	1209.80	1.53723	0.768617	−	0.639710i	$-0.220945\pi$
$787$	1209.80	1.53723	0.768617	+	0.639710i	$0.220945\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 178.544i	− 0.225719i
$792$	0	0
$793$	−57.9119	−0.0730289
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 824.725i	− 1.03479i	−0.855748	−	0.517393i	$-0.826902\pi$
$797$	− 824.725i	− 1.03479i	0.855748	−	0.517393i	$-0.173098\pi$
$798$	0	0
$799$	−1988.16	−2.48831
$800$	0	0
$801$	0	0
$802$	0	0
$803$	529.066i	0.658862i
$804$	0	0
$805$	326.388	0.405450
$806$	0	0
$807$	0	0
$808$	0	0
$809$	786.815i	0.972577i	0.873798	+	0.486289i	$0.161650\pi$
$809$	786.815i	0.972577i	−0.873798	+	0.486289i	$0.838350\pi$
$810$	0	0
$811$	−935.082	−1.15300	−0.576500	−	0.817097i	$-0.695582\pi$
$811$	−935.082	−1.15300	−0.576500	+	0.817097i	$0.695582\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1375.66i	− 1.68792i
$816$	0	0
$817$	−1256.10	−1.53745
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 702.400i	− 0.855542i	−0.903887	−	0.427771i	$-0.859299\pi$
$821$	− 702.400i	− 0.855542i	0.903887	−	0.427771i	$-0.140701\pi$
$822$	0	0
$823$	−588.119	−0.714604	−0.357302	−	0.933989i	$-0.616303\pi$
$823$	−588.119	−0.714604	−0.357302	+	0.933989i	$0.616303\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 494.834i	− 0.598348i	−0.954199	−	0.299174i	$-0.903289\pi$
$827$	− 494.834i	− 0.598348i	0.954199	−	0.299174i	$-0.0967111\pi$
$828$	0	0
$829$	762.667	0.919984	0.459992	−	0.887923i	$-0.347852\pi$
$829$	762.667	0.919984	0.459992	+	0.887923i	$0.347852\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 862.244i	− 1.03511i
$834$	0	0
$835$	2312.89	2.76993
$836$	0	0
$837$	0	0
$838$	0	0
$839$	637.193i	0.759467i	0.925096	+	0.379734i	$0.123984\pi$
$839$	637.193i	0.759467i	−0.925096	+	0.379734i	$0.876016\pi$
$840$	0	0
$841$	301.911	0.358990
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 63.9282i	− 0.0756546i
$846$	0	0
$847$	378.030	0.446316
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 821.462i	− 0.965291i
$852$	0	0
$853$	366.768	0.429974	0.214987	−	0.976617i	$-0.431029\pi$
$853$	366.768	0.429974	0.214987	+	0.976617i	$0.431029\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	166.918i	0.194770i	0.995247	+	0.0973852i	$0.0310479\pi$
$857$	166.918i	0.194770i	−0.995247	+	0.0973852i	$0.968952\pi$
$858$	0	0
$859$	1088.81	1.26753	0.633766	−	0.773525i	$-0.281508\pi$
$859$	1088.81	1.26753	0.633766	+	0.773525i	$0.281508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1151.23i	− 1.33399i	−0.745062	−	0.666995i	$-0.767580\pi$
$863$	− 1151.23i	− 1.33399i	0.745062	−	0.666995i	$-0.232420\pi$
$864$	0	0
$865$	−1099.70	−1.27133
$866$	0	0
$867$	0	0
$868$	0	0
$869$	97.9584i	0.112725i
$870$	0	0
$871$	−1743.88	−2.00216
$872$	0	0
$873$	0	0
$874$	0	0
$875$	297.235i	0.339698i
$876$	0	0
$877$	−1062.06	−1.21102	−0.605510	−	0.795838i	$-0.707031\pi$
$877$	−1062.06	−1.21102	−0.605510	+	0.795838i	$0.707031\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 527.852i	− 0.599150i	−0.954073	−	0.299575i	$-0.903155\pi$
$881$	− 527.852i	− 0.599150i	0.954073	−	0.299575i	$-0.0968450\pi$
$882$	0	0
$883$	−996.532	−1.12858	−0.564288	−	0.825578i	$-0.690849\pi$
$883$	−996.532	−1.12858	−0.564288	+	0.825578i	$0.690849\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1624.61i	− 1.83158i	−0.401659	−	0.915789i	$-0.631566\pi$
$887$	− 1624.61i	− 1.83158i	0.401659	−	0.915789i	$-0.368434\pi$
$888$	0	0
$889$	−235.741	−0.265176
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2254.78i	− 2.52495i
$894$	0	0
$895$	1368.92	1.52952
$896$	0	0
$897$	0	0
$898$	0	0
$899$	94.2124i	0.104797i
$900$	0	0
$901$	−694.878	−0.771229
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1876.52i	− 2.07351i
$906$	0	0
$907$	220.982	0.243641	0.121820	−	0.992552i	$-0.461127\pi$
$907$	220.982	0.243641	0.121820	+	0.992552i	$0.461127\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 796.569i	− 0.874389i	−0.899367	−	0.437195i	$-0.855972\pi$
$911$	− 796.569i	− 0.874389i	0.899367	−	0.437195i	$-0.144028\pi$
$912$	0	0
$913$	−168.256	−0.184289
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 73.0274i	− 0.0796373i
$918$	0	0
$919$	−445.659	−0.484939	−0.242469	−	0.970159i	$-0.577957\pi$
$919$	−445.659	−0.484939	−0.242469	+	0.970159i	$0.577957\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 929.187i	− 1.00670i
$924$	0	0
$925$	2516.77	2.72083
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 647.150i	− 0.696610i	−0.937381	−	0.348305i	$-0.886757\pi$
$929$	− 647.150i	− 0.696610i	0.937381	−	0.348305i	$-0.113243\pi$
$930$	0	0
$931$	977.874	1.05035
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 754.260i	− 0.806695i
$936$	0	0
$937$	664.431	0.709105	0.354552	−	0.935036i	$-0.384633\pi$
$937$	664.431	0.709105	0.354552	+	0.935036i	$0.384633\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	304.298i	0.323378i	0.986842	+	0.161689i	$0.0516941\pi$
$941$	304.298i	0.323378i	−0.986842	+	0.161689i	$0.948306\pi$
$942$	0	0
$943$	−507.991	−0.538697
$944$	0	0
$945$	0	0
$946$	0	0
$947$	366.957i	0.387494i	0.981052	+	0.193747i	$0.0620641\pi$
$947$	366.957i	0.387494i	−0.981052	+	0.193747i	$0.937936\pi$
$948$	0	0
$949$	−1742.79	−1.83644
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1527.63i	1.60297i	0.598016	+	0.801484i	$0.295956\pi$
$953$	1527.63i	1.60297i	−0.598016	+	0.801484i	$0.704044\pi$
$954$	0	0
$955$	−1261.53	−1.32098
$956$	0	0
$957$	0	0
$958$	0	0
$959$	152.762i	0.159293i
$960$	0	0
$961$	−944.535	−0.982867
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 112.128i	− 0.116195i
$966$	0	0
$967$	−438.428	−0.453390	−0.226695	−	0.973966i	$-0.572792\pi$
$967$	−438.428	−0.453390	−0.226695	+	0.973966i	$0.572792\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 447.249i	− 0.460607i	−0.973119	−	0.230303i	$-0.926028\pi$
$971$	− 447.249i	− 0.460607i	0.973119	−	0.230303i	$-0.0739718\pi$
$972$	0	0
$973$	−318.051	−0.326877
$974$	0	0
$975$	0	0
$976$	0	0
$977$	249.685i	0.255563i	0.991802	+	0.127782i	$0.0407857\pi$
$977$	249.685i	0.255563i	−0.991802	+	0.127782i	$0.959214\pi$
$978$	0	0
$979$	−392.908	−0.401336
$980$	0	0
$981$	0	0
$982$	0	0
$983$	865.364i	0.880330i	0.897917	+	0.440165i	$0.145080\pi$
$983$	865.364i	0.880330i	−0.897917	+	0.440165i	$0.854920\pi$
$984$	0	0
$985$	1694.02	1.71981
$986$	0	0
$987$	0	0
$988$	0	0
$989$	536.276i	0.542241i
$990$	0	0
$991$	−1372.22	−1.38468	−0.692342	−	0.721569i	$-0.743421\pi$
$991$	−1372.22	−1.38468	−0.692342	+	0.721569i	$0.743421\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	319.338i	0.320943i
$996$	0	0
$997$	−378.291	−0.379430	−0.189715	−	0.981839i	$-0.560756\pi$
$997$	−378.291	−0.379430	−0.189715	+	0.981839i	$0.560756\pi$
$998$	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.1457.3		36
3.2	odd	2	inner	2916.3.c.b.1457.34		36
27.5	odd	18		324.3.k.a.89.6		36
27.11	odd	18		108.3.k.a.41.5	yes	36
27.16	even	9		324.3.k.a.233.6		36
27.22	even	9		108.3.k.a.29.5	✓	36
108.11	even	18		432.3.bc.b.257.2		36
108.103	odd	18		432.3.bc.b.353.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.29.5	✓	36	27.22	even	9
108.3.k.a.41.5	yes	36	27.11	odd	18
324.3.k.a.89.6		36	27.5	odd	18
324.3.k.a.233.6		36	27.16	even	9
432.3.bc.b.257.2		36	108.11	even	18
432.3.bc.b.353.2		36	108.103	odd	18
2916.3.c.b.1457.3		36	1.1	even	1	trivial
2916.3.c.b.1457.34		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)

\(f(q)\)	\(=\)	\(q-7.78294i q^{5} +3.61169 q^{7} +O(q^{10})\)	\(q-7.78294i q^{5} +3.61169 q^{7} -4.04124i q^{11} +13.3122 q^{13} +23.9807i q^{17} -27.1966 q^{19} +11.6113i q^{23} -35.5742 q^{25} +23.2183i q^{29} +4.05768 q^{31} -28.1096i q^{35} -70.7471 q^{37} +43.7499i q^{41} +46.1859 q^{43} +82.9064i q^{47} -35.9557 q^{49} +28.9765i q^{53} -31.4528 q^{55} +52.6257i q^{59} -4.35030 q^{61} -103.608i q^{65} -130.999 q^{67} -69.7998i q^{71} -130.917 q^{73} -14.5957i q^{77} -24.2397 q^{79} -41.6347i q^{83} +186.641 q^{85} -97.2245i q^{89} +48.0794 q^{91} +211.670i q^{95} +48.6620 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 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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