LMFDB - Embedded newform 9300.2.g.a.3349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,-8,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.a.3349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +4.00000i q^{13} -2.00000 q^{21} +1.00000i q^{27} +6.00000 q^{29} +1.00000 q^{31} +4.00000i q^{33} -8.00000i q^{37} +4.00000 q^{39} -10.0000 q^{41} +4.00000i q^{43} +10.0000i q^{47} +3.00000 q^{49} +4.00000 q^{59} -6.00000 q^{61} +2.00000i q^{63} -2.00000i q^{67} +8.00000 q^{71} -4.00000i q^{73} +8.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -6.00000i q^{87} +18.0000 q^{89} +8.00000 q^{91} -1.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 8 q^{11} - 4 q^{21} + 12 q^{29} + 2 q^{31} + 8 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{59} - 12 q^{61} + 16 q^{71} - 32 q^{79} + 2 q^{81} + 36 q^{89} + 16 q^{91} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 8 * q^11 - 4 * q^21 + 12 * q^29 + 2 * q^31 + 8 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^59 - 12 * q^61 + 16 * q^71 - 32 * q^79 + 2 * q^81 + 36 * q^89 + 16 * q^91 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	4.00000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0000i	1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$	10.0000i	1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.00000i	0.911685i
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	− 6.00000i	− 0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$	− 6.00000i	− 0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	10.0000i	0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$	10.0000i	0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	10.0000	0.842152
$142$	0	0
$143$	− 16.0000i	− 1.33799i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 18.0000i	− 1.40987i	−0.709273	−	0.704934i	$-0.750976\pi$
$163$	− 18.0000i	− 1.40987i	0.709273	−	0.704934i	$-0.249024\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.00000i	− 0.300658i
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	6.00000i	0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	− 8.00000i	− 0.548151i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.00000i	− 0.135769i
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.00000i	− 0.132745i	−0.997795	−	0.0663723i	$-0.978857\pi$
$227$	− 2.00000i	− 0.132745i	0.997795	−	0.0663723i	$-0.0211425\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	20.0000i	1.23325i	0.787256	+	0.616626i	$0.211501\pi$
$263$	20.0000i	1.23325i	−0.787256	+	0.616626i	$0.788499\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.0000i	− 1.10158i
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	− 8.00000i	− 0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 26.0000i	− 1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$	− 26.0000i	− 1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.0000i	1.18056i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.00000i	− 0.232104i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	− 16.0000i	− 0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$	− 16.0000i	− 0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.00000i	− 0.110600i
$328$	0	0
$329$	20.0000	1.10264
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	8.00000i	0.438397i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8.00000i	− 0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$	− 8.00000i	− 0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4.00000i	− 0.203331i
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 4.00000i	− 0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32.0000i	1.58618i
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	− 10.0000i	− 0.486217i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.0000i	0.580721i
$428$	0	0
$429$	−16.0000	−0.772487
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	8.00000i	0.384455i	0.981350	+	0.192228i	$0.0615712\pi$
$433$	8.00000i	0.384455i	−0.981350	+	0.192228i	$0.938429\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	− 6.00000i	− 0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$	− 6.00000i	− 0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.0000i	− 0.662177i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.0000i	0.935561i	0.883845	+	0.467780i	$0.154946\pi$
$457$	20.0000i	0.935561i	−0.883845	+	0.467780i	$0.845054\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 14.0000i	− 0.647843i	−0.946084	−	0.323921i	$-0.894999\pi$
$467$	− 14.0000i	− 0.647843i	0.946084	−	0.323921i	$-0.105001\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	28.0000i	1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$	28.0000i	1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	0	0
$489$	−18.0000	−0.813988
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.0000i	− 0.717698i
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	2.00000i	0.0891756i	0.999005	+	0.0445878i	$0.0141974\pi$
$503$	2.00000i	0.0891756i	−0.999005	+	0.0445878i	$0.985803\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 40.0000i	− 1.75920i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 20.0000i	− 0.863064i
$538$	0	0
$539$	−12.0000	−0.516877
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	2.00000i	0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.00000i	0.256541i	0.991739	+	0.128271i	$0.0409426\pi$
$547$	6.00000i	0.256541i	−0.991739	+	0.128271i	$0.959057\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000i	1.36078i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	42.0000i	1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$	42.0000i	1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 16.0000i	− 0.654836i
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10.0000i	− 0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$	− 10.0000i	− 0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−40.0000	−1.61823
$612$	0	0
$613$	36.0000i	1.45403i	0.686624	+	0.727013i	$0.259092\pi$
$613$	36.0000i	1.45403i	−0.686624	+	0.727013i	$0.740908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 36.0000i	− 1.44231i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000i	0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$	12.0000i	0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.00000i	0.156055i
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−20.0000	−0.773245
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	4.00000i	0.154189i	0.997024	+	0.0770943i	$0.0245643\pi$
$673$	4.00000i	0.154189i	−0.997024	+	0.0770943i	$0.975436\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	0	0
$681$	−2.00000	−0.0766402
$682$	0	0
$683$	26.0000i	0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$	26.0000i	0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.0000i	0.381524i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−16.0000	−0.608669	−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000	−0.608669	−0.304334	+	0.952565i	$0.598434\pi$
$692$	0	0
$693$	− 8.00000i	− 0.303895i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−14.0000	−0.529529
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8.00000i	− 0.298765i
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	− 14.0000i	− 0.520666i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.00000i	0.222528i	0.993791	+	0.111264i	$0.0354899\pi$
$727$	6.00000i	0.222528i	−0.993791	+	0.111264i	$0.964510\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000i	0.294684i
$738$	0	0
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.00000i	0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$	4.00000i	0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	28.0000i	1.02038i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000i	0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$	12.0000i	0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.0000i	0.577727i
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	0	0
$773$	− 4.00000i	− 0.143870i	−0.997409	−	0.0719350i	$-0.977083\pi$
$773$	− 4.00000i	− 0.143870i	0.997409	−	0.0719350i	$-0.0229174\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.0000i	0.573997i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	6.00000i	0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	0	0
$789$	20.0000	0.712019
$790$	0	0
$791$	20.0000	0.711118
$792$	0	0
$793$	− 24.0000i	− 0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.0000i	1.13350i	0.823890	+	0.566749i	$0.191799\pi$
$797$	32.0000i	1.13350i	−0.823890	+	0.566749i	$0.808201\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	16.0000i	0.564628i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000i	0.211210i
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	− 24.0000i	− 0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$	− 24.0000i	− 0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 44.0000i	− 1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	− 44.0000i	− 1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 10.0000i	− 0.343604i
$848$	0	0
$849$	−26.0000	−0.892318
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	0	0
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 17.0000i	− 0.577350i
$868$	0	0
$869$	64.0000	2.17105
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	− 14.0000i	− 0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	48.0000i	1.61533i	0.589643	+	0.807664i	$0.299269\pi$
$883$	48.0000i	1.61533i	−0.589643	+	0.807664i	$0.700731\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	−4.00000	−0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.00000	0.200111
$900$	0	0
$901$	0	0
$902$	0	0
$903$	− 8.00000i	− 0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 38.0000i	− 1.26177i	−0.775877	−	0.630885i	$-0.782692\pi$
$907$	− 38.0000i	− 1.26177i	0.775877	−	0.630885i	$-0.217308\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 8.00000i	− 0.264183i
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−22.0000	−0.724925
$922$	0	0
$923$	32.0000i	1.05329i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.00000i	0.197066i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.0000i	0.392862i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.0000i	− 1.37208i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$	− 42.0000i	− 1.37208i	0.727564	−	0.686040i	$-0.240653\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	24.0000i	0.775810i
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 52.0000i	− 1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$	− 52.0000i	− 1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	− 24.0000i	− 0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	44.0000i	1.40338i	0.712481	+	0.701691i	$0.247571\pi$
$983$	44.0000i	1.40338i	−0.712481	+	0.701691i	$0.752429\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 20.0000i	− 0.636607i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 54.0000i	− 1.71020i	−0.518465	−	0.855099i	$-0.673497\pi$
$997$	− 54.0000i	− 1.71020i	0.518465	−	0.855099i	$-0.326503\pi$
$998$	0	0
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.a.3349.1		2
5.2	odd	4		9300.2.a.e.1.1		1
5.3	odd	4		1860.2.a.b.1.1	✓	1
5.4	even	2	inner	9300.2.g.a.3349.2		2
15.8	even	4		5580.2.a.b.1.1		1
20.3	even	4		7440.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.b.1.1	✓	1	5.3	odd	4
5580.2.a.b.1.1		1	15.8	even	4
7440.2.a.l.1.1		1	20.3	even	4
9300.2.a.e.1.1		1	5.2	odd	4
9300.2.g.a.3349.1		2	1.1	even	1	trivial
9300.2.g.a.3349.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +4.00000i q^{13} -2.00000 q^{21} +1.00000i q^{27} +6.00000 q^{29} +1.00000 q^{31} +4.00000i q^{33} -8.00000i q^{37} +4.00000 q^{39} -10.0000 q^{41} +4.00000i q^{43} +10.0000i q^{47} +3.00000 q^{49} +4.00000 q^{59} -6.00000 q^{61} +2.00000i q^{63} -2.00000i q^{67} +8.00000 q^{71} -4.00000i q^{73} +8.00000i q^{77} -16.0000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -6.00000i q^{87} +18.0000 q^{89} +8.00000 q^{91} -1.00000i q^{93} +14.0000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 8 q^{11} - 4 q^{21} + 12 q^{29} + 2 q^{31} + 8 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{59} - 12 q^{61} + 16 q^{71} - 32 q^{79} + 2 q^{81} + 36 q^{89} + 16 q^{91} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 8 * q^11 - 4 * q^21 + 12 * q^29 + 2 * q^31 + 8 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^59 - 12 * q^61 + 16 * q^71 - 32 * q^79 + 2 * q^81 + 36 * q^89 + 16 * q^91 + 8 * q^99

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