LMFDB - Embedded newform 9522.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,-1,0,0,-1,0,1,0,0,-5,0,0,1,2,0,-4,-1,0,0,0,0,-4,5,0, 0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3174)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{8} +1.00000 q^{10} -5.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{25} +5.00000 q^{26} +5.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +10.0000 q^{37} +4.00000 q^{38} +1.00000 q^{40} +5.00000 q^{41} -8.00000 q^{43} +4.00000 q^{47} -7.00000 q^{49} +4.00000 q^{50} -5.00000 q^{52} +11.0000 q^{53} -5.00000 q^{58} +4.00000 q^{59} +1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +5.00000 q^{65} +4.00000 q^{67} +2.00000 q^{68} +12.0000 q^{71} +3.00000 q^{73} -10.0000 q^{74} -4.00000 q^{76} +4.00000 q^{79} -1.00000 q^{80} -5.00000 q^{82} -16.0000 q^{83} -2.00000 q^{85} +8.00000 q^{86} -9.00000 q^{89} -4.00000 q^{94} +4.00000 q^{95} +5.00000 q^{97} +7.00000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−4.00000	−0.800000
$26$	5.00000	0.980581
$27$	0	0
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	4.00000	0.565685
$51$	0	0
$52$	−5.00000	−0.693375
$53$	11.0000	1.51097	0.755483	−	0.655168i	$-0.227402\pi$
$53$	11.0000	1.51097	0.755483	+	0.655168i	$0.227402\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−5.00000	−0.656532
$59$	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
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	5.00000	0.620174
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.00000	0.242536
$69$	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
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−5.00000	−0.552158
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−4.00000	−0.412568
$95$	4.00000	0.410391
$96$	0	0
$97$	5.00000	0.507673	0.253837	−	0.967247i	$-0.418307\pi$
$97$	5.00000	0.507673	0.253837	+	0.967247i	$0.418307\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−11.0000	−1.06841
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	13.0000	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−21.0000	−1.97551	−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−21.0000	−1.97551	−0.987757	+	0.156001i	$0.950140\pi$
$114$	0	0
$115$	0	0
$116$	5.00000	0.464238
$117$	0	0
$118$	−4.00000	−0.368230
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	4.00000	0.359211
$125$	9.00000	0.804984
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.00000	−0.438529
$131$	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
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−2.00000	−0.171499
$137$	19.0000	1.62328	0.811640	−	0.584158i	$-0.198575\pi$
$137$	19.0000	1.62328	0.811640	+	0.584158i	$0.198575\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	−12.0000	−1.00702
$143$	0	0
$144$	0	0
$145$	−5.00000	−0.415227
$146$	−3.00000	−0.248282
$147$	0	0
$148$	10.0000	0.821995
$149$	11.0000	0.901155	0.450578	−	0.892737i	$-0.351218\pi$
$149$	11.0000	0.901155	0.450578	+	0.892737i	$0.351218\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	16.0000	1.24184
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	2.00000	0.153393
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	9.00000	0.674579
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	0	0
$188$	4.00000	0.291730
$189$	0	0
$190$	−4.00000	−0.290191
$191$	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
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−5.00000	−0.358979
$195$	0	0
$196$	−7.00000	−0.500000
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	15.0000	1.05540
$203$	0	0
$204$	0	0
$205$	−5.00000	−0.349215
$206$	8.00000	0.557386
$207$	0	0
$208$	−5.00000	−0.346688
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	11.0000	0.755483
$213$	0	0
$214$	−8.00000	−0.546869
$215$	8.00000	0.545595
$216$	0	0
$217$	0	0
$218$	−13.0000	−0.880471
$219$	0	0
$220$	0	0
$221$	−10.0000	−0.672673
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	0	0
$226$	21.0000	1.39690
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	−5.00000	−0.328266
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	13.0000	0.837404	0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000	0.837404	0.418702	+	0.908124i	$0.362485\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	1.00000	0.0640184
$245$	7.00000	0.447214
$246$	0	0
$247$	20.0000	1.27257
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−9.00000	−0.569210
$251$	−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
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	0	0
$260$	5.00000	0.310087
$261$	0	0
$262$	−12.0000	−0.741362
$263$	32.0000	1.97320	0.986602	−	0.163144i	$-0.0521635\pi$
$263$	32.0000	1.97320	0.986602	+	0.163144i	$0.0521635\pi$
$264$	0	0
$265$	−11.0000	−0.675725
$266$	0	0
$267$	0	0
$268$	4.00000	0.244339
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−19.0000	−1.14783
$275$	0	0
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	5.00000	0.293610
$291$	0	0
$292$	3.00000	0.175562
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−11.0000	−0.637213
$299$	0	0
$300$	0	0
$301$	0	0
$302$	20.0000	1.15087
$303$	0	0
$304$	−4.00000	−0.229416
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	−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
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	−1.00000	−0.0564333
$315$	0	0
$316$	4.00000	0.225018
$317$	−23.0000	−1.29181	−0.645904	−	0.763418i	$-0.723520\pi$
$317$	−23.0000	−1.29181	−0.645904	+	0.763418i	$0.723520\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	20.0000	1.10940
$326$	16.0000	0.886158
$327$	0	0
$328$	−5.00000	−0.276079
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	−16.0000	−0.878114
$333$	0	0
$334$	12.0000	0.656611
$335$	−4.00000	−0.218543
$336$	0	0
$337$	33.0000	1.79762	0.898812	−	0.438334i	$-0.144431\pi$
$337$	33.0000	1.79762	0.898812	+	0.438334i	$0.144431\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	11.0000	0.591364
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.0000	−1.01127	−0.505634	−	0.862748i	$-0.668741\pi$
$353$	−19.0000	−1.01127	−0.505634	+	0.862748i	$0.668741\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	−9.00000	−0.476999
$357$	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
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	0	0
$364$	0	0
$365$	−3.00000	−0.157027
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	0	0
$369$	0	0
$370$	10.0000	0.519875
$371$	0	0
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	−4.00000	−0.206284
$377$	−25.0000	−1.28757
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−12.0000	−0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	5.00000	0.254493
$387$	0	0
$388$	5.00000	0.253837
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	0	0
$392$	7.00000	0.353553
$393$	0	0
$394$	7.00000	0.352655
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−9.00000	−0.451697	−0.225849	−	0.974162i	$-0.572515\pi$
$397$	−9.00000	−0.451697	−0.225849	+	0.974162i	$0.572515\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−17.0000	−0.848939	−0.424470	−	0.905442i	$-0.639539\pi$
$401$	−17.0000	−0.848939	−0.424470	+	0.905442i	$0.639539\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	−15.0000	−0.746278
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	5.00000	0.246932
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	0	0
$415$	16.0000	0.785409
$416$	5.00000	0.245145
$417$	0	0
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−11.0000	−0.534207
$425$	−8.00000	−0.388057
$426$	0	0
$427$	0	0
$428$	8.00000	0.386695
$429$	0	0
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−35.0000	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$434$	0	0
$435$	0	0
$436$	13.0000	0.622587
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	10.0000	0.475651
$443$	−28.0000	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$443$	−28.0000	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$444$	0	0
$445$	9.00000	0.426641
$446$	24.0000	1.13643
$447$	0	0
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	0	0
$452$	−21.0000	−0.987757
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	−15.0000	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	0	0
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	4.00000	0.184506
$471$	0	0
$472$	−4.00000	−0.184115
$473$	0	0
$474$	0	0
$475$	16.0000	0.734130
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−50.0000	−2.27980
$482$	−13.0000	−0.592134
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−5.00000	−0.227038
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	−7.00000	−0.316228
$491$	44.0000	1.98569	0.992846	−	0.119401i	$-0.0380974\pi$
$491$	44.0000	1.98569	0.992846	+	0.119401i	$0.0380974\pi$
$492$	0	0
$493$	10.0000	0.450377
$494$	−20.0000	−0.899843
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	28.0000	1.24970
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	0	0
$505$	15.0000	0.667491
$506$	0	0
$507$	0	0
$508$	−20.0000	−0.887357
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.00000	0.132324
$515$	8.00000	0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−5.00000	−0.219265
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−32.0000	−1.39527
$527$	8.00000	0.348485
$528$	0	0
$529$	0	0
$530$	11.0000	0.477809
$531$	0	0
$532$	0	0
$533$	−25.0000	−1.08287
$534$	0	0
$535$	−8.00000	−0.345870
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−2.00000	−0.0862261
$539$	0	0
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−13.0000	−0.556859
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	19.0000	0.811640
$549$	0	0
$550$	0	0
$551$	−20.0000	−0.852029
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	4.00000	0.169638
$557$	11.0000	0.466085	0.233042	−	0.972467i	$-0.425132\pi$
$557$	11.0000	0.466085	0.233042	+	0.972467i	$0.425132\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	0	0
$561$	0	0
$562$	22.0000	0.928014
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	21.0000	0.883477
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−12.0000	−0.503509
$569$	19.0000	0.796521	0.398261	−	0.917272i	$-0.369614\pi$
$569$	19.0000	0.796521	0.398261	+	0.917272i	$0.369614\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−5.00000	−0.207614
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−3.00000	−0.124141
$585$	0	0
$586$	−3.00000	−0.123929
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	4.00000	0.164677
$591$	0	0
$592$	10.0000	0.410997
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	0	0
$596$	11.0000	0.450578
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	3.00000	0.122373	0.0611863	−	0.998126i	$-0.480512\pi$
$601$	3.00000	0.122373	0.0611863	+	0.998126i	$0.480512\pi$
$602$	0	0
$603$	0	0
$604$	−20.0000	−0.813788
$605$	11.0000	0.447214
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	1.00000	0.0404888
$611$	−20.0000	−0.809113
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	16.0000	0.641542
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	1.00000	0.0399043
$629$	20.0000	0.797452
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	23.0000	0.913447
$635$	20.0000	0.793676
$636$	0	0
$637$	35.0000	1.38675
$638$	0	0
$639$	0	0
$640$	1.00000	0.0395285
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	0	0
$646$	8.00000	0.314756
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	0	0
$649$	0	0
$650$	−20.0000	−0.784465
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−35.0000	−1.36966	−0.684828	−	0.728705i	$-0.740123\pi$
$653$	−35.0000	−1.36966	−0.684828	+	0.728705i	$0.740123\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	5.00000	0.195217
$657$	0	0
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	−3.00000	−0.116686	−0.0583432	−	0.998297i	$-0.518582\pi$
$661$	−3.00000	−0.116686	−0.0583432	+	0.998297i	$0.518582\pi$
$662$	16.0000	0.621858
$663$	0	0
$664$	16.0000	0.620920
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	4.00000	0.154533
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	−33.0000	−1.27111
$675$	0	0
$676$	12.0000	0.461538
$677$	−13.0000	−0.499631	−0.249815	−	0.968294i	$-0.580370\pi$
$677$	−13.0000	−0.499631	−0.249815	+	0.968294i	$0.580370\pi$
$678$	0	0
$679$	0	0
$680$	2.00000	0.0766965
$681$	0	0
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	−19.0000	−0.725953
$686$	0	0
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−55.0000	−2.09533
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−11.0000	−0.418157
$693$	0	0
$694$	24.0000	0.911028
$695$	−4.00000	−0.151729
$696$	0	0
$697$	10.0000	0.378777
$698$	−30.0000	−1.13552
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−40.0000	−1.50863
$704$	0	0
$705$	0	0
$706$	19.0000	0.715074
$707$	0	0
$708$	0	0
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	9.00000	0.337289
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−12.0000	−0.447836
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	0	0
$724$	−22.0000	−0.817624
$725$	−20.0000	−0.742781
$726$	0	0
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	0	0
$729$	0	0
$730$	3.00000	0.111035
$731$	−16.0000	−0.591781
$732$	0	0
$733$	33.0000	1.21888	0.609441	−	0.792831i	$-0.291394\pi$
$733$	33.0000	1.21888	0.609441	+	0.792831i	$0.291394\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	−10.0000	−0.367607
$741$	0	0
$742$	0	0
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0	0
$745$	−11.0000	−0.403009
$746$	6.00000	0.219676
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	25.0000	0.910446
$755$	20.0000	0.727875
$756$	0	0
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	−4.00000	−0.145095
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	0	0
$764$	12.0000	0.434145
$765$	0	0
$766$	−4.00000	−0.144526
$767$	−20.0000	−0.722158
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	−5.00000	−0.179954
$773$	−26.0000	−0.935155	−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000	−0.935155	−0.467578	+	0.883952i	$0.654873\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	−5.00000	−0.179490
$777$	0	0
$778$	10.0000	0.358517
$779$	−20.0000	−0.716574
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−7.00000	−0.250000
$785$	−1.00000	−0.0356915
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−7.00000	−0.249365
$789$	0	0
$790$	4.00000	0.142314
$791$	0	0
$792$	0	0
$793$	−5.00000	−0.177555
$794$	9.00000	0.319398
$795$	0	0
$796$	8.00000	0.283552
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	4.00000	0.141421
$801$	0	0
$802$	17.0000	0.600291
$803$	0	0
$804$	0	0
$805$	0	0
$806$	20.0000	0.704470
$807$	0	0
$808$	15.0000	0.527698
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.0000	0.560456
$816$	0	0
$817$	32.0000	1.11954
$818$	22.0000	0.769212
$819$	0	0
$820$	−5.00000	−0.174608
$821$	−31.0000	−1.08191	−0.540954	−	0.841052i	$-0.681937\pi$
$821$	−31.0000	−1.08191	−0.540954	+	0.841052i	$0.681937\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	−16.0000	−0.555368
$831$	0	0
$832$	−5.00000	−0.173344
$833$	−14.0000	−0.485071
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	−36.0000	−1.24360
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	6.00000	0.206774
$843$	0	0
$844$	−12.0000	−0.413057
$845$	−12.0000	−0.412813
$846$	0	0
$847$	0	0
$848$	11.0000	0.377742
$849$	0	0
$850$	8.00000	0.274398
$851$	0	0
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	0	0
$856$	−8.00000	−0.273434
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	11.0000	0.374011
$866$	35.0000	1.18935
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−20.0000	−0.677674
$872$	−13.0000	−0.440236
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−16.0000	−0.539974
$879$	0	0
$880$	0	0
$881$	59.0000	1.98776	0.993880	−	0.110463i	$-0.0352333\pi$
$881$	59.0000	1.98776	0.993880	+	0.110463i	$0.0352333\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	−10.0000	−0.336336
$885$	0	0
$886$	28.0000	0.940678
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	0	0
$890$	−9.00000	−0.301681
$891$	0	0
$892$	−24.0000	−0.803579
$893$	−16.0000	−0.535420
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	15.0000	0.500556
$899$	20.0000	0.667037
$900$	0	0
$901$	22.0000	0.732926
$902$	0	0
$903$	0	0
$904$	21.0000	0.698450
$905$	22.0000	0.731305
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	0	0
$914$	15.0000	0.496156
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	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
$921$	0	0
$922$	15.0000	0.493999
$923$	−60.0000	−1.97492
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	0	0
$928$	−5.00000	−0.164133
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	1.00000	0.0327561
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	0	0
$940$	−4.00000	−0.130466
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	−15.0000	−0.486921
$950$	−16.0000	−0.519109
$951$	0	0
$952$	0	0
$953$	−13.0000	−0.421111	−0.210556	−	0.977582i	$-0.567527\pi$
$953$	−13.0000	−0.421111	−0.210556	+	0.977582i	$0.567527\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	12.0000	0.388108
$957$	0	0
$958$	24.0000	0.775405
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	50.0000	1.61206
$963$	0	0
$964$	13.0000	0.418702
$965$	5.00000	0.160956
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	5.00000	0.160540
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	0	0
$973$	0	0
$974$	−28.0000	−0.897178
$975$	0	0
$976$	1.00000	0.0320092
$977$	−17.0000	−0.543878	−0.271939	−	0.962314i	$-0.587665\pi$
$977$	−17.0000	−0.543878	−0.271939	+	0.962314i	$0.587665\pi$
$978$	0	0
$979$	0	0
$980$	7.00000	0.223607
$981$	0	0
$982$	−44.0000	−1.40410
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	7.00000	0.223039
$986$	−10.0000	−0.318465
$987$	0	0
$988$	20.0000	0.636285
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	8.00000	0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.b.1.1		1
3.2	odd	2		3174.2.a.g.1.1	yes	1
23.22	odd	2		9522.2.a.c.1.1		1
69.68	even	2		3174.2.a.f.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3174.2.a.f.1.1	✓	1	69.68	even	2
3174.2.a.g.1.1	yes	1	3.2	odd	2
9522.2.a.b.1.1		1	1.1	even	1	trivial
9522.2.a.c.1.1		1	23.22	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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