LMFDB - Embedded newform 7200.2.h.l.1151.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1151,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7200.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.h (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:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	no (minimal twist has level 1440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.4
Root			$1.41127 - 0.0912546i$ of defining polynomial
Character	$\chi$	$=$	7200.1151
Dual form			7200.2.h.l.1151.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64002i q^{7} +O(q^{10})$	$q-2.64002i q^{7} +3.00504 q^{11} +0.640023 q^{13} -0.685698i q^{17} -5.28005i q^{19} -2.27653 q^{23} +8.15281i q^{29} +2.96972i q^{31} +1.60975 q^{37} -7.42430i q^{41} -11.2195i q^{43} +4.19982 q^{47} +0.0302761 q^{49} -9.60984i q^{53} -7.20487 q^{59} +10.4995 q^{61} +8.49954i q^{67} +13.1240 q^{71} -14.2498 q^{73} -7.93338i q^{77} -11.4693i q^{79} +13.1240 q^{83} +10.2527i q^{89} -1.68968i q^{91} -8.31032 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 24 q^{13} - 16 q^{37} + 4 q^{49} - 8 q^{61} - 104 q^{73} - 40 q^{97}+O(q^{100}) $ 12 * q - 24 * q^13 - 16 * q^37 + 4 * q^49 - 8 * q^61 - 104 * q^73 - 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$6401$	$6751$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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	0
$5$	0	0
$6$	0	0
$7$	− 2.64002i	− 0.997835i	−0.866649	−	0.498918i	$-0.833731\pi$
$7$	− 2.64002i	− 0.997835i	0.866649	−	0.498918i	$-0.166269\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00504	0.906054	0.453027	−	0.891497i	$-0.350344\pi$
$11$	3.00504	0.906054	0.453027	+	0.891497i	$0.350344\pi$
$12$	0	0
$13$	0.640023	0.177511	0.0887553	−	0.996053i	$-0.471711\pi$
$13$	0.640023	0.177511	0.0887553	+	0.996053i	$0.471711\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.685698i	− 0.166306i	−0.996537	−	0.0831531i	$-0.973501\pi$
$17$	− 0.685698i	− 0.166306i	0.996537	−	0.0831531i	$-0.0264991\pi$
$18$	0	0
$19$	− 5.28005i	− 1.21133i	−0.795721	−	0.605663i	$-0.792908\pi$
$19$	− 5.28005i	− 1.21133i	0.795721	−	0.605663i	$-0.207092\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.27653	−0.474689	−0.237344	−	0.971426i	$-0.576277\pi$
$23$	−2.27653	−0.474689	−0.237344	+	0.971426i	$0.576277\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.15281i	1.51394i	0.653450	+	0.756970i	$0.273321\pi$
$29$	8.15281i	1.51394i	−0.653450	+	0.756970i	$0.726679\pi$
$30$	0	0
$31$	2.96972i	0.533378i	0.963783	+	0.266689i	$0.0859297\pi$
$31$	2.96972i	0.533378i	−0.963783	+	0.266689i	$0.914070\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.60975	0.264641	0.132320	−	0.991207i	$-0.457757\pi$
$37$	1.60975	0.264641	0.132320	+	0.991207i	$0.457757\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.42430i	− 1.15948i	−0.814801	−	0.579740i	$-0.803154\pi$
$41$	− 7.42430i	− 1.15948i	0.814801	−	0.579740i	$-0.196846\pi$
$42$	0	0
$43$	− 11.2195i	− 1.71096i	−0.517838	−	0.855478i	$-0.673263\pi$
$43$	− 11.2195i	− 1.71096i	0.517838	−	0.855478i	$-0.326737\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.19982	0.612607	0.306304	−	0.951934i	$-0.400908\pi$
$47$	4.19982	0.612607	0.306304	+	0.951934i	$0.400908\pi$
$48$	0	0
$49$	0.0302761	0.00432516
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.60984i	− 1.32001i	−0.751260	−	0.660007i	$-0.770553\pi$
$53$	− 9.60984i	− 1.32001i	0.751260	−	0.660007i	$-0.229447\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.20487	−0.937994	−0.468997	−	0.883200i	$-0.655384\pi$
$59$	−7.20487	−0.937994	−0.468997	+	0.883200i	$0.655384\pi$
$60$	0	0
$61$	10.4995	1.34433	0.672164	−	0.740402i	$-0.265365\pi$
$61$	10.4995	1.34433	0.672164	+	0.740402i	$0.265365\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.49954i	1.03838i	0.854658	+	0.519192i	$0.173767\pi$
$67$	8.49954i	1.03838i	−0.854658	+	0.519192i	$0.826233\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.1240	1.55753	0.778764	−	0.627317i	$-0.215847\pi$
$71$	13.1240	1.55753	0.778764	+	0.627317i	$0.215847\pi$
$72$	0	0
$73$	−14.2498	−1.66781	−0.833905	−	0.551908i	$-0.813900\pi$
$73$	−14.2498	−1.66781	−0.833905	+	0.551908i	$0.813900\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.93338i	− 0.904093i
$78$	0	0
$79$	− 11.4693i	− 1.29039i	−0.764017	−	0.645197i	$-0.776775\pi$
$79$	− 11.4693i	− 1.29039i	0.764017	−	0.645197i	$-0.223225\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.1240	1.44054	0.720271	−	0.693692i	$-0.244017\pi$
$83$	13.1240	1.44054	0.720271	+	0.693692i	$0.244017\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.2527i	1.08679i	0.839478	+	0.543393i	$0.182861\pi$
$89$	10.2527i	1.08679i	−0.839478	+	0.543393i	$0.817139\pi$
$90$	0	0
$91$	− 1.68968i	− 0.177126i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.31032	−0.843785	−0.421893	−	0.906646i	$-0.638634\pi$
$97$	−8.31032	−0.843785	−0.421893	+	0.906646i	$0.638634\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 3.51413i	− 0.349669i	−0.984598	−	0.174834i	$-0.944061\pi$
$101$	− 3.51413i	− 0.349669i	0.984598	−	0.174834i	$-0.0559390\pi$
$102$	0	0
$103$	3.13957i	0.309351i	0.987965	+	0.154675i	$0.0494331\pi$
$103$	3.13957i	0.309351i	−0.987965	+	0.154675i	$0.950567\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.46711	−0.721873	−0.360937	−	0.932590i	$-0.617543\pi$
$107$	−7.46711	−0.721873	−0.360937	+	0.932590i	$0.617543\pi$
$108$	0	0
$109$	−2.96972	−0.284448	−0.142224	−	0.989835i	$-0.545425\pi$
$109$	−2.96972	−0.284448	−0.142224	+	0.989835i	$0.545425\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.67761i	0.534105i	0.963682	+	0.267053i	$0.0860497\pi$
$113$	5.67761i	0.534105i	−0.963682	+	0.267053i	$0.913950\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.81026	−0.165946
$120$	0	0
$121$	−1.96972	−0.179066
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 19.1396i	− 1.69836i	−0.528102	−	0.849181i	$-0.677096\pi$
$127$	− 19.1396i	− 1.69836i	0.528102	−	0.849181i	$-0.322904\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.5760	−1.01140	−0.505698	−	0.862711i	$-0.668765\pi$
$131$	−11.5760	−1.01140	−0.505698	+	0.862711i	$0.668765\pi$
$132$	0	0
$133$	−13.9394	−1.20870
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.49596i	0.213244i	0.994300	+	0.106622i	$0.0340035\pi$
$137$	2.49596i	0.213244i	−0.994300	+	0.106622i	$0.965997\pi$
$138$	0	0
$139$	7.46927i	0.633535i	0.948503	+	0.316767i	$0.102597\pi$
$139$	7.46927i	0.633535i	−0.948503	+	0.316767i	$0.897403\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.92330	0.160834
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 9.60984i	− 0.787269i	−0.919267	−	0.393634i	$-0.871218\pi$
$149$	− 9.60984i	− 0.787269i	0.919267	−	0.393634i	$-0.128782\pi$
$150$	0	0
$151$	19.4693i	1.58439i	0.610270	+	0.792193i	$0.291061\pi$
$151$	19.4693i	1.58439i	−0.610270	+	0.792193i	$0.708939\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.8292	−1.02388	−0.511942	−	0.859020i	$-0.671074\pi$
$157$	−12.8292	−1.02388	−0.511942	+	0.859020i	$0.671074\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.01008i	0.473661i
$162$	0	0
$163$	− 21.2800i	− 1.66678i	−0.552684	−	0.833391i	$-0.686396\pi$
$163$	− 21.2800i	− 1.66678i	0.552684	−	0.833391i	$-0.313604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.8676	1.76955	0.884774	−	0.466020i	$-0.154312\pi$
$167$	22.8676	1.76955	0.884774	+	0.466020i	$0.154312\pi$
$168$	0	0
$169$	−12.5904	−0.968490
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 5.98932i	− 0.455360i	−0.973736	−	0.227680i	$-0.926886\pi$
$173$	− 5.98932i	− 0.455360i	0.973736	−	0.227680i	$-0.0731140\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.65181	−0.198206	−0.0991029	−	0.995077i	$-0.531597\pi$
$179$	−2.65181	−0.198206	−0.0991029	+	0.995077i	$0.531597\pi$
$180$	0	0
$181$	−11.4693	−0.852504	−0.426252	−	0.904605i	$-0.640166\pi$
$181$	−11.4693	−0.852504	−0.426252	+	0.904605i	$0.640166\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.06055i	− 0.150682i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.30362	0.383757	0.191878	−	0.981419i	$-0.438542\pi$
$191$	5.30362	0.383757	0.191878	+	0.981419i	$0.438542\pi$
$192$	0	0
$193$	−10.0606	−0.724174	−0.362087	−	0.932144i	$-0.617936\pi$
$193$	−10.0606	−0.724174	−0.362087	+	0.932144i	$0.617936\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.79065i	0.412567i	0.978492	+	0.206283i	$0.0661369\pi$
$197$	5.79065i	0.412567i	−0.978492	+	0.206283i	$0.933863\pi$
$198$	0	0
$199$	24.0899i	1.70769i	0.520529	+	0.853844i	$0.325735\pi$
$199$	24.0899i	1.70769i	−0.520529	+	0.853844i	$0.674265\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	21.5236	1.51066
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 15.8668i	− 1.09753i
$210$	0	0
$211$	− 23.4693i	− 1.61569i	−0.589394	−	0.807845i	$-0.700634\pi$
$211$	− 23.4693i	− 1.61569i	0.589394	−	0.807845i	$-0.299366\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.84014	0.532223
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.438863i	− 0.0295211i
$222$	0	0
$223$	− 13.3600i	− 0.894650i	−0.894371	−	0.447325i	$-0.852377\pi$
$223$	− 13.3600i	− 0.894650i	0.894371	−	0.447325i	$-0.147623\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.57092	0.568872	0.284436	−	0.958695i	$-0.408194\pi$
$227$	8.57092	0.568872	0.284436	+	0.958695i	$0.408194\pi$
$228$	0	0
$229$	8.90917	0.588735	0.294367	−	0.955692i	$-0.404891\pi$
$229$	8.90917	0.588735	0.294367	+	0.955692i	$0.404891\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 24.0196i	− 1.57357i	−0.617224	−	0.786787i	$-0.711743\pi$
$233$	− 24.0196i	− 1.57357i	0.617224	−	0.786787i	$-0.288257\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.4097	0.932088	0.466044	−	0.884762i	$-0.345679\pi$
$239$	14.4097	0.932088	0.466044	+	0.884762i	$0.345679\pi$
$240$	0	0
$241$	3.87890	0.249862	0.124931	−	0.992165i	$-0.460129\pi$
$241$	3.87890	0.249862	0.124931	+	0.992165i	$0.460129\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.37935i	− 0.215023i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.1754	−1.52594	−0.762970	−	0.646434i	$-0.776259\pi$
$251$	−24.1754	−1.52594	−0.762970	+	0.646434i	$0.776259\pi$
$252$	0	0
$253$	−6.84106	−0.430094
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.4383i	− 0.775878i	−0.921685	−	0.387939i	$-0.873187\pi$
$257$	− 12.4383i	− 0.775878i	0.921685	−	0.387939i	$-0.126813\pi$
$258$	0	0
$259$	− 4.24977i	− 0.264068i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.2471	1.18683	0.593413	−	0.804898i	$-0.297780\pi$
$263$	19.2471	1.18683	0.593413	+	0.804898i	$0.297780\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 15.8875i	− 0.968679i	−0.874880	−	0.484340i	$-0.839060\pi$
$269$	− 15.8875i	− 0.968679i	0.874880	−	0.484340i	$-0.160940\pi$
$270$	0	0
$271$	− 6.02936i	− 0.366258i	−0.983089	−	0.183129i	$-0.941377\pi$
$271$	− 6.02936i	− 0.366258i	0.983089	−	0.183129i	$-0.0586225\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.3591	0.742584	0.371292	−	0.928516i	$-0.378915\pi$
$277$	12.3591	0.742584	0.371292	+	0.928516i	$0.378915\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.7198i	1.05708i	0.848909	+	0.528538i	$0.177260\pi$
$281$	17.7198i	1.05708i	−0.848909	+	0.528538i	$0.822740\pi$
$282$	0	0
$283$	− 20.6206i	− 1.22577i	−0.790172	−	0.612885i	$-0.790009\pi$
$283$	− 20.6206i	− 1.22577i	0.790172	−	0.612885i	$-0.209991\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−19.6003	−1.15697
$288$	0	0
$289$	16.5298	0.972342
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 21.6574i	− 1.26524i	−0.774463	−	0.632620i	$-0.781980\pi$
$293$	− 21.6574i	− 1.26524i	0.774463	−	0.632620i	$-0.218020\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.45703	−0.0842622
$300$	0	0
$301$	−29.6197	−1.70725
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.87890i	− 0.221380i	−0.993855	−	0.110690i	$-0.964694\pi$
$307$	− 3.87890i	− 0.221380i	0.993855	−	0.110690i	$-0.0353061\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.71654	−0.380860	−0.190430	−	0.981701i	$-0.560988\pi$
$311$	−6.71654	−0.380860	−0.190430	+	0.981701i	$0.560988\pi$
$312$	0	0
$313$	−21.9394	−1.24009	−0.620045	−	0.784566i	$-0.712886\pi$
$313$	−21.9394	−1.24009	−0.620045	+	0.784566i	$0.712886\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 14.3616i	− 0.806626i	−0.915062	−	0.403313i	$-0.867859\pi$
$317$	− 14.3616i	− 0.806626i	0.915062	−	0.403313i	$-0.132141\pi$
$318$	0	0
$319$	24.4995i	1.37171i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.62052	−0.201451
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 11.0876i	− 0.611281i
$330$	0	0
$331$	− 4.78051i	− 0.262760i	−0.991332	−	0.131380i	$-0.958059\pi$
$331$	− 4.78051i	− 0.262760i	0.991332	−	0.131380i	$-0.0419409\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.1883	−1.80788	−0.903941	−	0.427657i	$-0.859339\pi$
$337$	−33.1883	−1.80788	−0.903941	+	0.427657i	$0.859339\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.92414i	0.483270i
$342$	0	0
$343$	− 18.5601i	− 1.00215i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.8304	−0.742456	−0.371228	−	0.928542i	$-0.621063\pi$
$347$	−13.8304	−0.742456	−0.371228	+	0.928542i	$0.621063\pi$
$348$	0	0
$349$	−20.4390	−1.09407	−0.547037	−	0.837108i	$-0.684244\pi$
$349$	−20.4390	−1.09407	−0.547037	+	0.837108i	$0.684244\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 20.8379i	− 1.10909i	−0.832154	−	0.554545i	$-0.812892\pi$
$353$	− 20.8379i	− 1.10909i	0.832154	−	0.554545i	$-0.187108\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.0481	1.16365	0.581827	−	0.813312i	$-0.302338\pi$
$359$	22.0481	1.16365	0.581827	+	0.813312i	$0.302338\pi$
$360$	0	0
$361$	−8.87890	−0.467310
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.70058i	− 0.245368i	−0.992446	−	0.122684i	$-0.960850\pi$
$367$	− 4.70058i	− 0.245368i	0.992446	−	0.122684i	$-0.0391502\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−25.3702	−1.31716
$372$	0	0
$373$	−35.5104	−1.83866	−0.919330	−	0.393486i	$-0.871269\pi$
$373$	−35.5104	−1.83866	−0.919330	+	0.393486i	$0.871269\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.21799i	0.268740i
$378$	0	0
$379$	1.52982i	0.0785815i	0.999228	+	0.0392907i	$0.0125098\pi$
$379$	1.52982i	0.0785815i	−0.999228	+	0.0392907i	$0.987490\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.16349	−0.110549	−0.0552746	−	0.998471i	$-0.517603\pi$
$383$	−2.16349	−0.110549	−0.0552746	+	0.998471i	$0.517603\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.0207603i	0.00105259i	1.00000		0.000526294i	$0.000167524\pi$
$389$	0.0207603i	0.00105259i	−1.00000		0.000526294i	$0.999832\pi$
$390$	0	0
$391$	1.56101i	0.0789437i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−36.6694	−1.84038	−0.920192	−	0.391468i	$-0.871967\pi$
$397$	−36.6694	−1.84038	−0.920192	+	0.391468i	$0.871967\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 7.59556i	− 0.379304i	−0.981851	−	0.189652i	$-0.939264\pi$
$401$	− 7.59556i	− 0.379304i	0.981851	−	0.189652i	$-0.0607360\pi$
$402$	0	0
$403$	1.90069i	0.0946803i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.83736	0.239779
$408$	0	0
$409$	26.5289	1.31177	0.655885	−	0.754861i	$-0.272296\pi$
$409$	26.5289	1.31177	0.655885	+	0.754861i	$0.272296\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.0210i	0.935963i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.841553	−0.0411125	−0.0205563	−	0.999789i	$-0.506544\pi$
$419$	−0.841553	−0.0411125	−0.0205563	+	0.999789i	$0.506544\pi$
$420$	0	0
$421$	26.1505	1.27450	0.637248	−	0.770659i	$-0.280073\pi$
$421$	26.1505	1.27450	0.637248	+	0.770659i	$0.280073\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 27.7190i	− 1.34142i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	38.8474	1.87121	0.935607	−	0.353044i	$-0.114853\pi$
$431$	38.8474	1.87121	0.935607	+	0.353044i	$0.114853\pi$
$432$	0	0
$433$	30.7787	1.47913	0.739564	−	0.673086i	$-0.235032\pi$
$433$	30.7787	1.47913	0.739564	+	0.673086i	$0.235032\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0202i	0.575003i
$438$	0	0
$439$	− 26.1505i	− 1.24809i	−0.781387	−	0.624047i	$-0.785487\pi$
$439$	− 26.1505i	− 1.24809i	0.781387	−	0.624047i	$-0.214513\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.8304	−0.657103	−0.328552	−	0.944486i	$-0.606561\pi$
$443$	−13.8304	−0.657103	−0.328552	+	0.944486i	$0.606561\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.07915i	− 0.0981212i	−0.998796	−	0.0490606i	$-0.984377\pi$
$449$	− 2.07915i	− 0.0981212i	0.998796	−	0.0490606i	$-0.0156228\pi$
$450$	0	0
$451$	− 22.3103i	− 1.05055i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.6282	0.871391	0.435695	−	0.900094i	$-0.356502\pi$
$457$	18.6282	0.871391	0.435695	+	0.900094i	$0.356502\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.0014i	1.07128i	0.844446	+	0.535641i	$0.179930\pi$
$461$	23.0014i	1.07128i	−0.844446	+	0.535641i	$0.820070\pi$
$462$	0	0
$463$	− 25.9806i	− 1.20742i	−0.797203	−	0.603711i	$-0.793688\pi$
$463$	− 25.9806i	− 1.20742i	0.797203	−	0.603711i	$-0.206312\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.3177	1.54176	0.770880	−	0.636981i	$-0.219817\pi$
$467$	33.3177	1.54176	0.770880	+	0.636981i	$0.219817\pi$
$468$	0	0
$469$	22.4390	1.03614
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 33.7151i	− 1.55022i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.0380	0.732796	0.366398	−	0.930458i	$-0.380591\pi$
$479$	16.0380	0.732796	0.366398	+	0.930458i	$0.380591\pi$
$480$	0	0
$481$	1.03028	0.0469765
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	19.1396i	0.867296i	0.901082	+	0.433648i	$0.142774\pi$
$487$	19.1396i	0.867296i	−0.901082	+	0.433648i	$0.857226\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−36.3669	−1.64121	−0.820607	−	0.571493i	$-0.806364\pi$
$491$	−36.3669	−1.64121	−0.820607	+	0.571493i	$0.806364\pi$
$492$	0	0
$493$	5.59037	0.251778
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 34.6476i	− 1.55416i
$498$	0	0
$499$	− 8.65940i	− 0.387648i	−0.981036	−	0.193824i	$-0.937911\pi$
$499$	− 8.65940i	− 0.387648i	0.981036	−	0.193824i	$-0.0620891\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.1542	1.38910	0.694549	−	0.719445i	$-0.255604\pi$
$503$	31.1542	1.38910	0.694549	+	0.719445i	$0.255604\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.3907i	1.25839i	0.777246	+	0.629197i	$0.216616\pi$
$509$	28.3907i	1.25839i	−0.777246	+	0.629197i	$0.783384\pi$
$510$	0	0
$511$	37.6197i	1.66420i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.6206	0.555055
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.4341i	0.719990i	0.932954	+	0.359995i	$0.117222\pi$
$521$	16.4341i	0.719990i	−0.932954	+	0.359995i	$0.882778\pi$
$522$	0	0
$523$	14.2791i	0.624383i	0.950019	+	0.312191i	$0.101063\pi$
$523$	14.2791i	0.624383i	−0.950019	+	0.312191i	$0.898937\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.03633	0.0887041
$528$	0	0
$529$	−17.8174	−0.774671
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.75172i	− 0.205820i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.0909809	0.00391883
$540$	0	0
$541$	1.40871	0.0605653	0.0302827	−	0.999541i	$-0.490359\pi$
$541$	1.40871	0.0605653	0.0302827	+	0.999541i	$0.490359\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.3406i	− 0.655917i	−0.944692	−	0.327958i	$-0.893639\pi$
$547$	− 15.3406i	− 0.655917i	0.944692	−	0.327958i	$-0.106361\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	43.0472	1.83387
$552$	0	0
$553$	−30.2791	−1.28760
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.53199i	0.319141i	0.987187	+	0.159570i	$0.0510109\pi$
$557$	7.53199i	0.319141i	−0.987187	+	0.159570i	$0.948989\pi$
$558$	0	0
$559$	− 7.18074i	− 0.303713i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40.0784	1.68910	0.844551	−	0.535475i	$-0.179868\pi$
$563$	40.0784	1.68910	0.844551	+	0.535475i	$0.179868\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.6803i	1.20234i	0.799121	+	0.601171i	$0.205299\pi$
$569$	28.6803i	1.20234i	−0.799121	+	0.601171i	$0.794701\pi$
$570$	0	0
$571$	− 33.9007i	− 1.41870i	−0.704857	−	0.709350i	$-0.748989\pi$
$571$	− 33.9007i	− 1.41870i	0.704857	−	0.709350i	$-0.251011\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.4002	0.766012	0.383006	−	0.923746i	$-0.374889\pi$
$577$	18.4002	0.766012	0.383006	+	0.923746i	$0.374889\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 34.6476i	− 1.43742i
$582$	0	0
$583$	− 28.8780i	− 1.19600i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.8869	1.15102	0.575508	−	0.817796i	$-0.304804\pi$
$587$	27.8869	1.15102	0.575508	+	0.817796i	$0.304804\pi$
$588$	0	0
$589$	15.6803	0.646095
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.2295i	1.40564i	0.711370	+	0.702818i	$0.248075\pi$
$593$	34.2295i	1.40564i	−0.711370	+	0.702818i	$0.751925\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.7546	−0.929727	−0.464863	−	0.885382i	$-0.653897\pi$
$599$	−22.7546	−0.929727	−0.464863	+	0.885382i	$0.653897\pi$
$600$	0	0
$601$	11.5298	0.470311	0.235156	−	0.971958i	$-0.424440\pi$
$601$	11.5298	0.470311	0.235156	+	0.971958i	$0.424440\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.0790i	0.693216i	0.938010	+	0.346608i	$0.112667\pi$
$607$	17.0790i	0.693216i	−0.938010	+	0.346608i	$0.887333\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.68799	0.108744
$612$	0	0
$613$	−43.1084	−1.74113	−0.870565	−	0.492053i	$-0.836247\pi$
$613$	−43.1084	−1.74113	−0.870565	+	0.492053i	$0.836247\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.7935i	0.957890i	0.877845	+	0.478945i	$0.158981\pi$
$617$	23.7935i	0.957890i	−0.877845	+	0.478945i	$0.841019\pi$
$618$	0	0
$619$	− 32.4683i	− 1.30501i	−0.757783	−	0.652507i	$-0.773717\pi$
$619$	− 32.4683i	− 1.30501i	0.757783	−	0.652507i	$-0.226283\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	27.0674	1.08443
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.10380i	− 0.0440114i
$630$	0	0
$631$	13.0303i	0.518727i	0.965780	+	0.259364i	$0.0835128\pi$
$631$	13.0303i	0.518727i	−0.965780	+	0.259364i	$0.916487\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.0193774	0.000767761 0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 38.1397i	− 1.50643i	−0.657777	−	0.753213i	$-0.728503\pi$
$641$	− 38.1397i	− 1.50643i	0.657777	−	0.753213i	$-0.271497\pi$
$642$	0	0
$643$	17.1589i	0.676683i	0.941023	+	0.338341i	$0.109866\pi$
$643$	17.1589i	0.676683i	−0.941023	+	0.338341i	$0.890134\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.4478	−1.19702	−0.598512	−	0.801113i	$-0.704241\pi$
$647$	−30.4478	−1.19702	−0.598512	+	0.801113i	$0.704241\pi$
$648$	0	0
$649$	−21.6509	−0.849873
$650$	0	0
$651$	0	0
$652$	0	0
$653$	11.7151i	0.458447i	0.973374	+	0.229224i	$0.0736187\pi$
$653$	11.7151i	0.458447i	−0.973374	+	0.229224i	$0.926381\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.1189	−0.394177	−0.197089	−	0.980386i	$-0.563149\pi$
$659$	−10.1189	−0.394177	−0.197089	+	0.980386i	$0.563149\pi$
$660$	0	0
$661$	11.1202	0.432525	0.216263	−	0.976335i	$-0.430613\pi$
$661$	11.1202	0.432525	0.216263	+	0.976335i	$0.430613\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 18.5601i	− 0.718650i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.5516	1.21803
$672$	0	0
$673$	−15.0984	−0.582000	−0.291000	−	0.956723i	$-0.593988\pi$
$673$	−15.0984	−0.582000	−0.291000	+	0.956723i	$0.593988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.82699i	− 0.300816i	−0.988624	−	0.150408i	$-0.951941\pi$
$677$	− 7.82699i	− 0.300816i	0.988624	−	0.150408i	$-0.0480587\pi$
$678$	0	0
$679$	21.9394i	0.841959i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−34.4763	−1.31920	−0.659600	−	0.751617i	$-0.729274\pi$
$683$	−34.4763	−1.31920	−0.659600	+	0.751617i	$0.729274\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 6.15052i	− 0.234316i
$690$	0	0
$691$	36.7181i	1.39682i	0.715696	+	0.698412i	$0.246109\pi$
$691$	36.7181i	1.39682i	−0.715696	+	0.698412i	$0.753891\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−5.09083	−0.192829
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.47779i	− 0.0558154i	−0.999611	−	0.0279077i	$-0.991116\pi$
$701$	− 1.47779i	− 0.0558154i	0.999611	−	0.0279077i	$-0.00888445\pi$
$702$	0	0
$703$	− 8.49954i	− 0.320566i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.27737	−0.348912
$708$	0	0
$709$	−6.02936	−0.226437	−0.113219	−	0.993570i	$-0.536116\pi$
$709$	−6.02936	−0.226437	−0.113219	+	0.993570i	$0.536116\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 6.76066i	− 0.253189i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−13.3059	−0.496227	−0.248114	−	0.968731i	$-0.579811\pi$
$719$	−13.3059	−0.496227	−0.248114	+	0.968731i	$0.579811\pi$
$720$	0	0
$721$	8.28853	0.308681
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	17.7384i	0.657881i	0.944351	+	0.328941i	$0.106692\pi$
$727$	17.7384i	0.657881i	−0.944351	+	0.328941i	$0.893308\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.69319	−0.284543
$732$	0	0
$733$	29.6391	1.09475	0.547373	−	0.836889i	$-0.315628\pi$
$733$	29.6391	1.09475	0.547373	+	0.836889i	$0.315628\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.5415i	0.940832i
$738$	0	0
$739$	− 19.2195i	− 0.707001i	−0.935434	−	0.353500i	$-0.884991\pi$
$739$	− 19.2195i	− 0.707001i	0.935434	−	0.353500i	$-0.115009\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.8768	−0.802584	−0.401292	−	0.915950i	$-0.631439\pi$
$743$	−21.8768	−0.802584	−0.401292	+	0.915950i	$0.631439\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19.7134i	0.720310i
$750$	0	0
$751$	14.5913i	0.532444i	0.963912	+	0.266222i	$0.0857754\pi$
$751$	14.5913i	0.532444i	−0.963912	+	0.266222i	$0.914225\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−5.26067	−0.191202	−0.0956011	−	0.995420i	$-0.530477\pi$
$757$	−5.26067	−0.191202	−0.0956011	+	0.995420i	$0.530477\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 22.6261i	− 0.820196i	−0.912041	−	0.410098i	$-0.865494\pi$
$761$	− 22.6261i	− 0.820196i	0.912041	−	0.410098i	$-0.134506\pi$
$762$	0	0
$763$	7.84014i	0.283832i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.61128	−0.166504
$768$	0	0
$769$	−0.150464	−0.00542586	−0.00271293	−	0.999996i	$-0.500864\pi$
$769$	−0.150464	−0.00542586	−0.00271293	+	0.999996i	$0.500864\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.23760i	0.0445133i	0.999752	+	0.0222567i	$0.00708510\pi$
$773$	1.23760i	0.0445133i	−0.999752	+	0.0222567i	$0.992915\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−39.2006	−1.40451
$780$	0	0
$781$	39.4381	1.41121
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.84106i	0.243857i	0.992539	+	0.121929i	$0.0389079\pi$
$787$	6.84106i	0.243857i	−0.992539	+	0.121929i	$0.961092\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.9890	0.532949
$792$	0	0
$793$	6.71995	0.238633
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 32.9437i	− 1.16693i	−0.812140	−	0.583463i	$-0.801697\pi$
$797$	− 32.9437i	− 1.16693i	0.812140	−	0.583463i	$-0.198303\pi$
$798$	0	0
$799$	− 2.87981i	− 0.101880i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−42.8212	−1.51113
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	53.6532i	1.88635i	0.332303	+	0.943173i	$0.392174\pi$
$809$	53.6532i	1.88635i	−0.332303	+	0.943173i	$0.607826\pi$
$810$	0	0
$811$	− 19.5904i	− 0.687911i	−0.938986	−	0.343955i	$-0.888233\pi$
$811$	− 19.5904i	− 0.687911i	0.938986	−	0.343955i	$-0.111767\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−59.2395	−2.07253
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.49596i	− 0.0871095i	−0.999051	−	0.0435548i	$-0.986132\pi$
$821$	− 2.49596i	− 0.0871095i	0.999051	−	0.0435548i	$-0.0138683\pi$
$822$	0	0
$823$	4.04117i	0.140866i	0.997516	+	0.0704332i	$0.0224382\pi$
$823$	4.04117i	0.140866i	−0.997516	+	0.0704332i	$0.977562\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.76066	−0.235091	−0.117546	−	0.993067i	$-0.537503\pi$
$827$	−6.76066	−0.235091	−0.117546	+	0.993067i	$0.537503\pi$
$828$	0	0
$829$	−29.5298	−1.02561	−0.512806	−	0.858504i	$-0.671394\pi$
$829$	−29.5298	−1.02561	−0.512806	+	0.858504i	$0.671394\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 0.0207603i	0 0.000719301i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10.7344	0.370593	0.185296	−	0.982683i	$-0.440675\pi$
$839$	10.7344	0.370593	0.185296	+	0.982683i	$0.440675\pi$
$840$	0	0
$841$	−37.4683	−1.29201
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.20012i	0.178678i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.66463	−0.125622
$852$	0	0
$853$	19.0109	0.650921	0.325460	−	0.945556i	$-0.394481\pi$
$853$	19.0109	0.650921	0.325460	+	0.945556i	$0.394481\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.4584i	− 0.835484i	−0.908566	−	0.417742i	$-0.862822\pi$
$857$	− 24.4584i	− 0.835484i	0.908566	−	0.417742i	$-0.137178\pi$
$858$	0	0
$859$	− 23.4693i	− 0.800761i	−0.916349	−	0.400381i	$-0.868878\pi$
$859$	− 23.4693i	− 0.800761i	0.916349	−	0.400381i	$-0.131122\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.5539	0.393299	0.196650	−	0.980474i	$-0.436994\pi$
$863$	11.5539	0.393299	0.196650	+	0.980474i	$0.436994\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 34.4656i	− 1.16917i
$870$	0	0
$871$	5.43991i	0.184324i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.38934	0.249520	0.124760	−	0.992187i	$-0.460184\pi$
$877$	7.38934	0.249520	0.124760	+	0.992187i	$0.460184\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.05321i	0.102865i	0.998676	+	0.0514326i	$0.0163787\pi$
$881$	3.05321i	0.102865i	−0.998676	+	0.0514326i	$0.983621\pi$
$882$	0	0
$883$	21.2800i	0.716131i	0.933697	+	0.358065i	$0.116563\pi$
$883$	21.2800i	0.716131i	−0.933697	+	0.358065i	$0.883437\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.9737	1.07357	0.536786	−	0.843718i	$-0.319638\pi$
$887$	31.9737	1.07357	0.536786	+	0.843718i	$0.319638\pi$
$888$	0	0
$889$	−50.5289	−1.69468
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 22.1753i	− 0.742067i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−24.2116	−0.807502
$900$	0	0
$901$	−6.58945	−0.219527
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.6594i	0.553166i	0.960990	+	0.276583i	$0.0892021\pi$
$907$	16.6594i	0.553166i	−0.960990	+	0.276583i	$0.910798\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	23.3339	0.773086	0.386543	−	0.922271i	$-0.373669\pi$
$911$	23.3339	0.773086	0.386543	+	0.922271i	$0.373669\pi$
$912$	0	0
$913$	39.4381	1.30521
$914$	0	0
$915$	0	0
$916$	0	0
$917$	30.5608i	1.00921i
$918$	0	0
$919$	21.0303i	0.693725i	0.937916	+	0.346862i	$0.112753\pi$
$919$	21.0303i	0.693725i	−0.937916	+	0.346862i	$0.887247\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.39965	0.276478
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 39.8536i	− 1.30755i	−0.756687	−	0.653777i	$-0.773184\pi$
$929$	− 39.8536i	− 1.30755i	0.756687	−	0.653777i	$-0.226816\pi$
$930$	0	0
$931$	− 0.159859i	− 0.00523917i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.5601	1.39038	0.695189	−	0.718827i	$-0.255321\pi$
$937$	42.5601	1.39038	0.695189	+	0.718827i	$0.255321\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.05710i	0.0670594i	0.999438	+	0.0335297i	$0.0106748\pi$
$941$	2.05710i	0.0670594i	−0.999438	+	0.0335297i	$0.989325\pi$
$942$	0	0
$943$	16.9016i	0.550392i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.4740	0.860290	0.430145	−	0.902760i	$-0.358462\pi$
$947$	26.4740	0.860290	0.430145	+	0.902760i	$0.358462\pi$
$948$	0	0
$949$	−9.12019	−0.296054
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.9717i	1.39199i	0.718047	+	0.695994i	$0.245036\pi$
$953$	42.9717i	1.39199i	−0.718047	+	0.695994i	$0.754964\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.58939	0.212782
$960$	0	0
$961$	22.1807	0.715508
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 11.7990i	− 0.379429i	−0.981839	−	0.189715i	$-0.939244\pi$
$967$	− 11.7990i	− 0.379429i	0.981839	−	0.189715i	$-0.0607563\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.23112	−0.103691	−0.0518457	−	0.998655i	$-0.516510\pi$
$971$	−3.23112	−0.103691	−0.0518457	+	0.998655i	$0.516510\pi$
$972$	0	0
$973$	19.7190	0.632163
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 47.6210i	− 1.52353i	−0.647852	−	0.761766i	$-0.724333\pi$
$977$	− 47.6210i	− 1.52353i	0.647852	−	0.761766i	$-0.275667\pi$
$978$	0	0
$979$	30.8099i	0.984688i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13.4772	−0.429856	−0.214928	−	0.976630i	$-0.568952\pi$
$983$	−13.4772	−0.429856	−0.214928	+	0.976630i	$0.568952\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.5415i	0.812172i
$990$	0	0
$991$	47.0284i	1.49391i	0.664876	+	0.746954i	$0.268484\pi$
$991$	47.0284i	1.49391i	−0.664876	+	0.746954i	$0.731516\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.2295	0.989047	0.494524	−	0.869164i	$-0.335343\pi$
$997$	31.2295	0.989047	0.494524	+	0.869164i	$0.335343\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7200.2.h.l.1151.4		12
3.2	odd	2	inner	7200.2.h.l.1151.3		12
4.3	odd	2	inner	7200.2.h.l.1151.9		12
5.2	odd	4		1440.2.o.a.1439.11	yes	12
5.3	odd	4		1440.2.o.b.1439.1	yes	12
5.4	even	2		7200.2.h.m.1151.10		12
12.11	even	2	inner	7200.2.h.l.1151.10		12
15.2	even	4		1440.2.o.a.1439.2	yes	12
15.8	even	4		1440.2.o.b.1439.12	yes	12
15.14	odd	2		7200.2.h.m.1151.9		12
20.3	even	4		1440.2.o.a.1439.1	✓	12
20.7	even	4		1440.2.o.b.1439.11	yes	12
20.19	odd	2		7200.2.h.m.1151.3		12
40.3	even	4		2880.2.o.f.2879.12		12
40.13	odd	4		2880.2.o.e.2879.12		12
40.27	even	4		2880.2.o.e.2879.2		12
40.37	odd	4		2880.2.o.f.2879.2		12
60.23	odd	4		1440.2.o.a.1439.12	yes	12
60.47	odd	4		1440.2.o.b.1439.2	yes	12
60.59	even	2		7200.2.h.m.1151.4		12
120.53	even	4		2880.2.o.e.2879.1		12
120.77	even	4		2880.2.o.f.2879.11		12
120.83	odd	4		2880.2.o.f.2879.1		12
120.107	odd	4		2880.2.o.e.2879.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.o.a.1439.1	✓	12	20.3	even	4
1440.2.o.a.1439.2	yes	12	15.2	even	4
1440.2.o.a.1439.11	yes	12	5.2	odd	4
1440.2.o.a.1439.12	yes	12	60.23	odd	4
1440.2.o.b.1439.1	yes	12	5.3	odd	4
1440.2.o.b.1439.2	yes	12	60.47	odd	4
1440.2.o.b.1439.11	yes	12	20.7	even	4
1440.2.o.b.1439.12	yes	12	15.8	even	4
2880.2.o.e.2879.1		12	120.53	even	4
2880.2.o.e.2879.2		12	40.27	even	4
2880.2.o.e.2879.11		12	120.107	odd	4
2880.2.o.e.2879.12		12	40.13	odd	4
2880.2.o.f.2879.1		12	120.83	odd	4
2880.2.o.f.2879.2		12	40.37	odd	4
2880.2.o.f.2879.11		12	120.77	even	4
2880.2.o.f.2879.12		12	40.3	even	4
7200.2.h.l.1151.3		12	3.2	odd	2	inner
7200.2.h.l.1151.4		12	1.1	even	1	trivial
7200.2.h.l.1151.9		12	4.3	odd	2	inner
7200.2.h.l.1151.10		12	12.11	even	2	inner
7200.2.h.m.1151.3		12	20.19	odd	2
7200.2.h.m.1151.4		12	60.59	even	2
7200.2.h.m.1151.9		12	15.14	odd	2
7200.2.h.m.1151.10		12	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64002i q^{7} +O(q^{10})\)	\(q-2.64002i q^{7} +3.00504 q^{11} +0.640023 q^{13} -0.685698i q^{17} -5.28005i q^{19} -2.27653 q^{23} +8.15281i q^{29} +2.96972i q^{31} +1.60975 q^{37} -7.42430i q^{41} -11.2195i q^{43} +4.19982 q^{47} +0.0302761 q^{49} -9.60984i q^{53} -7.20487 q^{59} +10.4995 q^{61} +8.49954i q^{67} +13.1240 q^{71} -14.2498 q^{73} -7.93338i q^{77} -11.4693i q^{79} +13.1240 q^{83} +10.2527i q^{89} -1.68968i q^{91} -8.31032 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 24 q^{13} - 16 q^{37} + 4 q^{49} - 8 q^{61} - 104 q^{73} - 40 q^{97}+O(q^{100}) \) 12 * q - 24 * q^13 - 16 * q^37 + 4 * q^49 - 8 * q^61 - 104 * q^73 - 40 * q^97

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