LMFDB - Embedded newform 3600.2.f.s.2449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(2449,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3600.2449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.f (of order $2$, degree $1$, not 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:	$28.7461447277$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1800)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3600.2449
Dual form			3600.2.f.s.2449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} +4.00000 q^{11} +1.00000i q^{13} -4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} +4.00000 q^{29} +5.00000 q^{31} -6.00000i q^{37} -12.0000 q^{41} +5.00000i q^{43} +8.00000i q^{47} +6.00000 q^{49} -12.0000i q^{53} -8.00000 q^{59} +7.00000 q^{61} -13.0000i q^{67} +12.0000 q^{71} +6.00000i q^{73} -4.00000i q^{77} +12.0000 q^{79} +8.00000i q^{83} +1.00000 q^{91} -13.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{11} + 2 q^{19} + 8 q^{29} + 10 q^{31} - 24 q^{41} + 12 q^{49} - 16 q^{59} + 14 q^{61} + 24 q^{71} + 24 q^{79} + 2 q^{91}+O(q^{100}) $ 2 * q + 8 * q^11 + 2 * q^19 + 8 * q^29 + 10 * q^31 - 24 * q^41 + 12 * q^49 - 16 * q^59 + 14 * q^61 + 24 * q^71 + 24 * q^79 + 2 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.0000i	− 1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	− 13.0000i	− 1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.0000i	− 1.31995i	−0.751288	−	0.659975i	$-0.770567\pi$
$97$	− 13.0000i	− 1.31995i	0.751288	−	0.659975i	$-0.229433\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	− 1.00000i	− 0.0867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.00000i	− 0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	− 8.00000i	− 0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 19.0000i	− 1.51637i	−0.652042	−	0.758183i	$-0.726088\pi$
$157$	− 19.0000i	− 1.51637i	0.652042	−	0.758183i	$-0.273912\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	23.0000i	1.80150i	0.434339	+	0.900750i	$0.356982\pi$
$163$	23.0000i	1.80150i	−0.434339	+	0.900750i	$0.643018\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.00000i	0.304114i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	4.00000i	0.304114i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 16.0000i	− 1.17004i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	7.00000i	0.503871i	0.967744	+	0.251936i	$0.0810671\pi$
$193$	7.00000i	0.503871i	−0.967744	+	0.251936i	$0.918933\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.00000i	− 0.339422i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 23.0000i	− 1.54019i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$	− 23.0000i	− 1.54019i	0.637927	−	0.770097i	$-0.279792\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 12.0000i	− 0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$	− 12.0000i	− 0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.00000i	0.0636285i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	16.0000i	1.00591i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.0000i	1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$	24.0000i	1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−28.0000	−1.70719	−0.853595	−	0.520937i	$-0.825583\pi$
$269$	−28.0000	−1.70719	−0.853595	+	0.520937i	$0.825583\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.00000i	0.420589i	0.977638	+	0.210295i	$0.0674423\pi$
$277$	7.00000i	0.420589i	−0.977638	+	0.210295i	$0.932558\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	31.0000i	1.84276i	0.388664	+	0.921379i	$0.372937\pi$
$283$	31.0000i	1.84276i	−0.388664	+	0.921379i	$0.627063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	32.0000i	1.86946i	0.355359	+	0.934730i	$0.384359\pi$
$293$	32.0000i	1.86946i	−0.355359	+	0.934730i	$0.615641\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	5.00000	0.288195
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 11.0000i	− 0.627803i	−0.949456	−	0.313902i	$-0.898364\pi$
$307$	− 11.0000i	− 0.627803i	0.949456	−	0.313902i	$-0.101636\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	− 13.0000i	− 0.734803i	−0.930062	−	0.367402i	$-0.880247\pi$
$313$	− 13.0000i	− 0.734803i	0.930062	−	0.367402i	$-0.119753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0000i	1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$	24.0000i	1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.00000i	− 0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$	− 8.00000i	− 0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.00000i	0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$	4.00000i	0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.0000i	0.887393i	0.896177	+	0.443696i	$0.146333\pi$
$367$	17.0000i	0.887393i	−0.896177	+	0.443696i	$0.853667\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	− 5.00000i	− 0.258890i	−0.991587	−	0.129445i	$-0.958680\pi$
$373$	− 5.00000i	− 0.258890i	0.991587	−	0.129445i	$-0.0413196\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000i	0.206010i
$378$	0	0
$379$	−29.0000	−1.48963	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−29.0000	−1.48963	−0.744815	+	0.667271i	$0.767462\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7.00000i	− 0.351320i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$	− 7.00000i	− 0.351320i	0.984451	−	0.175660i	$-0.0562059\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	5.00000i	0.249068i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 24.0000i	− 1.18964i
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 7.00000i	− 0.338754i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	11.0000i	0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$	11.0000i	0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	19.0000	0.906821	0.453410	−	0.891302i	$-0.350207\pi$
$439$	19.0000	0.906821	0.453410	+	0.891302i	$0.350207\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	−48.0000	−2.26023
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	0	0
$469$	−13.0000	−0.600284
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.0000i	0.919601i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	13.0000i	0.589086i	0.955638	+	0.294543i	$0.0951675\pi$
$487$	13.0000i	0.589086i	−0.955638	+	0.294543i	$0.904833\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	− 16.0000i	− 0.720604i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	−19.0000	−0.850557	−0.425278	−	0.905063i	$-0.639824\pi$
$499$	−19.0000	−0.850557	−0.425278	+	0.905063i	$0.639824\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.0000	1.77297	0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000	1.77297	0.886484	+	0.462758i	$0.153140\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.0000i	1.40736i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	35.0000i	1.53044i	0.643767	+	0.765222i	$0.277371\pi$
$523$	35.0000i	1.53044i	−0.643767	+	0.765222i	$0.722629\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.0000i	− 0.871214i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	24.0000	1.03375
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	− 12.0000i	− 0.510292i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16.0000i	− 0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$	− 16.0000i	− 0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	−25.0000	−1.04622	−0.523109	−	0.852266i	$-0.675228\pi$
$571$	−25.0000	−1.04622	−0.523109	+	0.852266i	$0.675228\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.0000i	− 0.457936i	−0.973434	−	0.228968i	$-0.926465\pi$
$577$	− 11.0000i	− 0.457936i	0.973434	−	0.228968i	$-0.0735351\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	− 48.0000i	− 1.98796i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	5.00000	0.206021
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.0000i	0.492781i	0.969171	+	0.246390i	$0.0792446\pi$
$593$	12.0000i	0.492781i	−0.969171	+	0.246390i	$0.920755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.0000i	1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$	36.0000i	1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.0000i	1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$	36.0000i	1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$	0	0
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	17.0000	0.676759	0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000	0.676759	0.338380	+	0.941010i	$0.390121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	24.0000i	0.946468i	0.880937	+	0.473234i	$0.156913\pi$
$643$	24.0000i	0.946468i	−0.880937	+	0.473234i	$0.843087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	0	0
$649$	−32.0000	−1.25611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 32.0000i	− 1.25226i	−0.779720	−	0.626128i	$-0.784639\pi$
$653$	− 32.0000i	− 1.25226i	0.779720	−	0.626128i	$-0.215361\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	28.0000	1.08093
$672$	0	0
$673$	− 30.0000i	− 1.15642i	−0.815890	−	0.578208i	$-0.803752\pi$
$673$	− 30.0000i	− 1.15642i	0.815890	−	0.578208i	$-0.196248\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.0000i	1.07613i	0.842904	+	0.538064i	$0.180844\pi$
$677$	28.0000i	1.07613i	−0.842904	+	0.538064i	$0.819156\pi$
$678$	0	0
$679$	−13.0000	−0.498894
$680$	0	0
$681$	0	0
$682$	0	0
$683$	48.0000i	1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$	48.0000i	1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	48.0000i	1.81813i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	− 6.00000i	− 0.226294i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	−29.0000	−1.08912	−0.544559	−	0.838723i	$-0.683303\pi$
$709$	−29.0000	−1.08912	−0.544559	+	0.838723i	$0.683303\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.0000i	0.749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	47.0000i	1.74313i	0.490277	+	0.871567i	$0.336896\pi$
$727$	47.0000i	1.74313i	−0.490277	+	0.871567i	$0.663104\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20.0000	0.739727
$732$	0	0
$733$	18.0000i	0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$	18.0000i	0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 52.0000i	− 1.91544i
$738$	0	0
$739$	−48.0000	−1.76571	−0.882854	−	0.469647i	$-0.844381\pi$
$739$	−48.0000	−1.76571	−0.882854	+	0.469647i	$0.844381\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.0000i	1.56286i	0.623992	+	0.781431i	$0.285510\pi$
$757$	43.0000i	1.56286i	−0.623992	+	0.781431i	$0.714490\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	− 7.00000i	− 0.253417i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.00000i	− 0.288863i
$768$	0	0
$769$	17.0000	0.613036	0.306518	−	0.951865i	$-0.400836\pi$
$769$	17.0000	0.613036	0.306518	+	0.951865i	$0.400836\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	48.0000	1.71758
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 7.00000i	− 0.249523i	−0.992187	−	0.124762i	$-0.960183\pi$
$787$	− 7.00000i	− 0.249523i	0.992187	−	0.124762i	$-0.0398166\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.00000i	0.248577i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.0000i	− 0.850124i	−0.905164	−	0.425062i	$-0.860252\pi$
$797$	− 24.0000i	− 0.850124i	0.905164	−	0.425062i	$-0.139748\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36.0000	−1.26569	−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000	−1.26569	−0.632846	+	0.774277i	$0.718114\pi$
$810$	0	0
$811$	−53.0000	−1.86108	−0.930541	−	0.366188i	$-0.880663\pi$
$811$	−53.0000	−1.86108	−0.930541	+	0.366188i	$0.880663\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.00000i	0.174928i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	0	0
$823$	− 31.0000i	− 1.08059i	−0.841475	−	0.540296i	$-0.818312\pi$
$823$	− 31.0000i	− 1.08059i	0.841475	−	0.540296i	$-0.181688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 24.0000i	− 0.831551i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	− 49.0000i	− 1.67773i	−0.544341	−	0.838864i	$-0.683220\pi$
$853$	− 49.0000i	− 1.67773i	0.544341	−	0.838864i	$-0.316780\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	13.0000	0.440488
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19.0000i	− 0.641584i	−0.947150	−	0.320792i	$-0.896051\pi$
$877$	− 19.0000i	− 0.641584i	0.947150	−	0.320792i	$-0.103949\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	25.0000i	0.841317i	0.907219	+	0.420658i	$0.138201\pi$
$883$	25.0000i	0.841317i	−0.907219	+	0.420658i	$0.861799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.00000i	0.267710i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.0000	0.667037
$900$	0	0
$901$	−48.0000	−1.59911
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	24.0000i	0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$	24.0000i	0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	32.0000i	1.05905i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	5.00000	0.164935	0.0824674	−	0.996594i	$-0.473720\pi$
$919$	5.00000	0.164935	0.0824674	+	0.996594i	$0.473720\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	52.0000	1.70606	0.853032	−	0.521858i	$-0.174761\pi$
$929$	52.0000	1.70606	0.853032	+	0.521858i	$0.174761\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.0000i	0.947389i	0.880689	+	0.473694i	$0.157080\pi$
$937$	29.0000i	0.947389i	−0.880689	+	0.473694i	$0.842920\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−16.0000	−0.521585	−0.260793	−	0.965395i	$-0.583984\pi$
$941$	−16.0000	−0.521585	−0.260793	+	0.965395i	$0.583984\pi$
$942$	0	0
$943$	− 48.0000i	− 1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 48.0000i	− 1.55487i	−0.628962	−	0.777436i	$-0.716520\pi$
$953$	− 48.0000i	− 1.55487i	0.628962	−	0.777436i	$-0.283480\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	36.0000i	1.15768i	0.815440	+	0.578841i	$0.196495\pi$
$967$	36.0000i	1.15768i	−0.815440	+	0.578841i	$0.803505\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.00000i	0.255943i	0.991778	+	0.127971i	$0.0408466\pi$
$977$	8.00000i	0.255943i	−0.991778	+	0.127971i	$0.959153\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48.0000i	1.53096i	0.643458	+	0.765481i	$0.277499\pi$
$983$	48.0000i	1.53096i	−0.643458	+	0.765481i	$0.722501\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.0000	−0.635963
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	42.0000i	1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$	42.0000i	1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.f.s.2449.1		2
3.2	odd	2		3600.2.f.d.2449.1		2
4.3	odd	2		1800.2.f.b.649.2		2
5.2	odd	4		3600.2.a.y.1.1		1
5.3	odd	4		3600.2.a.r.1.1		1
5.4	even	2	inner	3600.2.f.s.2449.2		2
12.11	even	2		1800.2.f.i.649.2		2
15.2	even	4		3600.2.a.w.1.1		1
15.8	even	4		3600.2.a.p.1.1		1
15.14	odd	2		3600.2.f.d.2449.2		2
20.3	even	4		1800.2.a.o.1.1	yes	1
20.7	even	4		1800.2.a.k.1.1	✓	1
20.19	odd	2		1800.2.f.b.649.1		2
60.23	odd	4		1800.2.a.p.1.1	yes	1
60.47	odd	4		1800.2.a.l.1.1	yes	1
60.59	even	2		1800.2.f.i.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.2.a.k.1.1	✓	1	20.7	even	4
1800.2.a.l.1.1	yes	1	60.47	odd	4
1800.2.a.o.1.1	yes	1	20.3	even	4
1800.2.a.p.1.1	yes	1	60.23	odd	4
1800.2.f.b.649.1		2	20.19	odd	2
1800.2.f.b.649.2		2	4.3	odd	2
1800.2.f.i.649.1		2	60.59	even	2
1800.2.f.i.649.2		2	12.11	even	2
3600.2.a.p.1.1		1	15.8	even	4
3600.2.a.r.1.1		1	5.3	odd	4
3600.2.a.w.1.1		1	15.2	even	4
3600.2.a.y.1.1		1	5.2	odd	4
3600.2.f.d.2449.1		2	3.2	odd	2
3600.2.f.d.2449.2		2	15.14	odd	2
3600.2.f.s.2449.1		2	1.1	even	1	trivial
3600.2.f.s.2449.2		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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +4.00000 q^{11} +1.00000i q^{13} -4.00000i q^{17} +1.00000 q^{19} +4.00000i q^{23} +4.00000 q^{29} +5.00000 q^{31} -6.00000i q^{37} -12.0000 q^{41} +5.00000i q^{43} +8.00000i q^{47} +6.00000 q^{49} -12.0000i q^{53} -8.00000 q^{59} +7.00000 q^{61} -13.0000i q^{67} +12.0000 q^{71} +6.00000i q^{73} -4.00000i q^{77} +12.0000 q^{79} +8.00000i q^{83} +1.00000 q^{91} -13.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{11} + 2 q^{19} + 8 q^{29} + 10 q^{31} - 24 q^{41} + 12 q^{49} - 16 q^{59} + 14 q^{61} + 24 q^{71} + 24 q^{79} + 2 q^{91}+O(q^{100}) \) 2 * q + 8 * q^11 + 2 * q^19 + 8 * q^29 + 10 * q^31 - 24 * q^41 + 12 * q^49 - 16 * q^59 + 14 * q^61 + 24 * q^71 + 24 * q^79 + 2 * q^91

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