LMFDB - Embedded newform 9225.2.a.be.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9225,2,Mod(1,9225)] 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("9225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9225 = 3^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,2,0,0,1,0,0,0,-6,0,5,2,0,-4,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.6619958646$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{7} -6.00000 q^{11} +5.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +6.00000 q^{17} -3.00000 q^{19} -12.0000 q^{22} -8.00000 q^{23} +10.0000 q^{26} +2.00000 q^{28} -3.00000 q^{31} -8.00000 q^{32} +12.0000 q^{34} +6.00000 q^{37} -6.00000 q^{38} -1.00000 q^{41} -1.00000 q^{43} -12.0000 q^{44} -16.0000 q^{46} +2.00000 q^{47} -6.00000 q^{49} +10.0000 q^{52} -6.00000 q^{53} -2.00000 q^{59} -13.0000 q^{61} -6.00000 q^{62} -8.00000 q^{64} +5.00000 q^{67} +12.0000 q^{68} -12.0000 q^{71} +6.00000 q^{73} +12.0000 q^{74} -6.00000 q^{76} -6.00000 q^{77} -4.00000 q^{79} -2.00000 q^{82} +10.0000 q^{83} -2.00000 q^{86} -12.0000 q^{89} +5.00000 q^{91} -16.0000 q^{92} +4.00000 q^{94} -3.00000 q^{97} -12.0000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	−12.0000	−2.55841
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	0	0
$26$	10.0000	1.96116
$27$	0	0
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	12.0000	2.05798
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−12.0000	−1.80907
$45$	0	0
$46$	−16.0000	−2.35907
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	10.0000	1.38675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	12.0000	1.45521
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	12.0000	1.39497
$75$	0	0
$76$	−6.00000	−0.688247
$77$	−6.00000	−0.683763
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	−2.00000	−0.220863
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	−16.0000	−1.66812
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	−12.0000	−1.21218
$99$	0	0
$100$	0	0
$101$	−16.0000	−1.59206	−0.796030	−	0.605257i	$-0.793070\pi$
$101$	−16.0000	−1.59206	−0.796030	+	0.605257i	$0.793070\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−4.00000	−0.368230
$119$	6.00000	0.550019
$120$	0	0
$121$	25.0000	2.27273
$122$	−26.0000	−2.35393
$123$	0	0
$124$	−6.00000	−0.538816
$125$	0	0
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	10.0000	0.863868
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	−24.0000	−2.01404
$143$	−30.0000	−2.50873
$144$	0	0
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	12.0000	0.986394
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	1.00000	0.0813788	0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000	0.0813788	0.0406894	+	0.999172i	$0.487045\pi$
$152$	0	0
$153$	0	0
$154$	−12.0000	−0.966988
$155$	0	0
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	20.0000	1.55230
$167$	−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
$170$	0	0
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	0	0
$176$	24.0000	1.80907
$177$	0	0
$178$	−24.0000	−1.79888
$179$	−26.0000	−1.94333	−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000	−1.94333	−0.971666	+	0.236360i	$0.924046\pi$
$180$	0	0
$181$	−25.0000	−1.85824	−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000	−1.85824	−0.929118	+	0.369784i	$0.879432\pi$
$182$	10.0000	0.741249
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−36.0000	−2.63258
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	−12.0000	−0.857143
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−23.0000	−1.63043	−0.815213	−	0.579161i	$-0.803380\pi$
$199$	−23.0000	−1.63043	−0.815213	+	0.579161i	$0.803380\pi$
$200$	0	0
$201$	0	0
$202$	−32.0000	−2.25151
$203$	0	0
$204$	0	0
$205$	0	0
$206$	32.0000	2.22955
$207$	0	0
$208$	−20.0000	−1.38675
$209$	18.0000	1.24509
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	−8.00000	−0.546869
$215$	0	0
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−2.00000	−0.135457
$219$	0	0
$220$	0	0
$221$	30.0000	2.01802
$222$	0	0
$223$	7.00000	0.468755	0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000	0.468755	0.234377	+	0.972146i	$0.424695\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	−4.00000	−0.266076
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	23.0000	1.51988	0.759941	−	0.649992i	$-0.225228\pi$
$229$	23.0000	1.51988	0.759941	+	0.649992i	$0.225228\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	12.0000	0.777844
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−3.00000	−0.193247	−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000	−0.193247	−0.0966235	+	0.995321i	$0.530804\pi$
$242$	50.0000	3.21412
$243$	0	0
$244$	−26.0000	−1.66448
$245$	0	0
$246$	0	0
$247$	−15.0000	−0.954427
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	48.0000	3.01773
$254$	−8.00000	−0.501965
$255$	0	0
$256$	16.0000	1.00000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	4.00000	0.247121
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	10.0000	0.610847
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−24.0000	−1.45521
$273$	0	0
$274$	−24.0000	−1.44989
$275$	0	0
$276$	0	0
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	−24.0000	−1.43942
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	−24.0000	−1.42414
$285$	0	0
$286$	−60.0000	−3.54787
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	12.0000	0.702247
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	36.0000	2.08542
$299$	−40.0000	−2.31326
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	2.00000	0.115087
$303$	0	0
$304$	12.0000	0.688247
$305$	0	0
$306$	0	0
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	−12.0000	−0.683763
$309$	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
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−16.0000	−0.891645
$323$	−18.0000	−1.00155
$324$	0	0
$325$	0	0
$326$	26.0000	1.44001
$327$	0	0
$328$	0	0
$329$	2.00000	0.110264
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	20.0000	1.09764
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	35.0000	1.90657	0.953286	−	0.302070i	$-0.0976776\pi$
$337$	35.0000	1.90657	0.953286	+	0.302070i	$0.0976776\pi$
$338$	24.0000	1.30543
$339$	0	0
$340$	0	0
$341$	18.0000	0.974755
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	0	0
$346$	−32.0000	−1.72033
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	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
$351$	0	0
$352$	48.0000	2.55841
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	0	0
$356$	−24.0000	−1.27200
$357$	0	0
$358$	−52.0000	−2.74829
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−50.0000	−2.62794
$363$	0	0
$364$	10.0000	0.524142
$365$	0	0
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	32.0000	1.66812
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	25.0000	1.29445	0.647225	−	0.762299i	$-0.275929\pi$
$373$	25.0000	1.29445	0.647225	+	0.762299i	$0.275929\pi$
$374$	−72.0000	−3.72303
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	21.0000	1.07870	0.539349	−	0.842082i	$-0.318670\pi$
$379$	21.0000	1.07870	0.539349	+	0.842082i	$0.318670\pi$
$380$	0	0
$381$	0	0
$382$	12.0000	0.613973
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−6.00000	−0.304604
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−48.0000	−2.42746
$392$	0	0
$393$	0	0
$394$	−36.0000	−1.81365
$395$	0	0
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	−46.0000	−2.30577
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	−32.0000	−1.59206
$405$	0	0
$406$	0	0
$407$	−36.0000	−1.78445
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	32.0000	1.57653
$413$	−2.00000	−0.0984136
$414$	0	0
$415$	0	0
$416$	−40.0000	−1.96116
$417$	0	0
$418$	36.0000	1.76082
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−13.0000	−0.629114
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−27.0000	−1.29754	−0.648769	−	0.760986i	$-0.724716\pi$
$433$	−27.0000	−1.29754	−0.648769	+	0.760986i	$0.724716\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	24.0000	1.14808
$438$	0	0
$439$	3.00000	0.143182	0.0715911	−	0.997434i	$-0.477192\pi$
$439$	3.00000	0.143182	0.0715911	+	0.997434i	$0.477192\pi$
$440$	0	0
$441$	0	0
$442$	60.0000	2.85391
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	0	0
$445$	0	0
$446$	14.0000	0.662919
$447$	0	0
$448$	−8.00000	−0.377964
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−4.00000	−0.188144
$453$	0	0
$454$	36.0000	1.68956
$455$	0	0
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	46.0000	2.14944
$459$	0	0
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	0	0
$466$	32.0000	1.48237
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	12.0000	0.548867
$479$	2.00000	0.0913823	0.0456912	−	0.998956i	$-0.485451\pi$
$479$	2.00000	0.0913823	0.0456912	+	0.998956i	$0.485451\pi$
$480$	0	0
$481$	30.0000	1.36788
$482$	−6.00000	−0.273293
$483$	0	0
$484$	50.0000	2.27273
$485$	0	0
$486$	0	0
$487$	−43.0000	−1.94852	−0.974258	−	0.225436i	$-0.927619\pi$
$487$	−43.0000	−1.94852	−0.974258	+	0.225436i	$0.927619\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	0	0
$494$	−30.0000	−1.34976
$495$	0	0
$496$	12.0000	0.538816
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−39.0000	−1.74588	−0.872940	−	0.487828i	$-0.837789\pi$
$499$	−39.0000	−1.74588	−0.872940	+	0.487828i	$0.837789\pi$
$500$	0	0
$501$	0	0
$502$	24.0000	1.07117
$503$	−14.0000	−0.624229	−0.312115	−	0.950044i	$-0.601037\pi$
$503$	−14.0000	−0.624229	−0.312115	+	0.950044i	$0.601037\pi$
$504$	0	0
$505$	0	0
$506$	96.0000	4.26772
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	32.0000	1.41421
$513$	0	0
$514$	−36.0000	−1.58789
$515$	0	0
$516$	0	0
$517$	−12.0000	−0.527759
$518$	12.0000	0.527250
$519$	0	0
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	41.0000	1.79280	0.896402	−	0.443241i	$-0.146171\pi$
$523$	41.0000	1.79280	0.896402	+	0.443241i	$0.146171\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	24.0000	1.04645
$527$	−18.0000	−0.784092
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	0	0
$532$	−6.00000	−0.260133
$533$	−5.00000	−0.216574
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	−56.0000	−2.40541
$543$	0	0
$544$	−48.0000	−2.05798
$545$	0	0
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−24.0000	−1.02523
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.00000	−0.170097
$554$	−34.0000	−1.44452
$555$	0	0
$556$	−24.0000	−1.01783
$557$	32.0000	1.35588	0.677942	−	0.735116i	$-0.262872\pi$
$557$	32.0000	1.35588	0.677942	+	0.735116i	$0.262872\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	0	0
$565$	0	0
$566$	2.00000	0.0840663
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	−60.0000	−2.50873
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	38.0000	1.58059
$579$	0	0
$580$	0	0
$581$	10.0000	0.414870
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	60.0000	2.47858
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	0	0
$591$	0	0
$592$	−24.0000	−0.986394
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	36.0000	1.47462
$597$	0	0
$598$	−80.0000	−3.27144
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	13.0000	0.530281	0.265141	−	0.964210i	$-0.414582\pi$
$601$	13.0000	0.530281	0.265141	+	0.964210i	$0.414582\pi$
$602$	−2.00000	−0.0815139
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	24.0000	0.973329
$609$	0	0
$610$	0	0
$611$	10.0000	0.404557
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−38.0000	−1.53356
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	15.0000	0.602901	0.301450	−	0.953482i	$-0.402529\pi$
$619$	15.0000	0.602901	0.301450	+	0.953482i	$0.402529\pi$
$620$	0	0
$621$	0	0
$622$	−32.0000	−1.28308
$623$	−12.0000	−0.480770
$624$	0	0
$625$	0	0
$626$	22.0000	0.879297
$627$	0	0
$628$	14.0000	0.558661
$629$	36.0000	1.43541
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	0	0
$633$	0	0
$634$	24.0000	0.953162
$635$	0	0
$636$	0	0
$637$	−30.0000	−1.18864
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−36.0000	−1.41640
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	26.0000	1.01824
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	0	0
$655$	0	0
$656$	4.00000	0.156174
$657$	0	0
$658$	4.00000	0.155936
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	56.0000	2.17650
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−24.0000	−0.928588
$669$	0	0
$670$	0	0
$671$	78.0000	3.01116
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	70.0000	2.69630
$675$	0	0
$676$	24.0000	0.923077
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	0	0
$679$	−3.00000	−0.115129
$680$	0	0
$681$	0	0
$682$	36.0000	1.37851
$683$	10.0000	0.382639	0.191320	−	0.981528i	$-0.438723\pi$
$683$	10.0000	0.382639	0.191320	+	0.981528i	$0.438723\pi$
$684$	0	0
$685$	0	0
$686$	−26.0000	−0.992685
$687$	0	0
$688$	4.00000	0.152499
$689$	−30.0000	−1.14291
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−32.0000	−1.21646
$693$	0	0
$694$	−36.0000	−1.36654
$695$	0	0
$696$	0	0
$697$	−6.00000	−0.227266
$698$	4.00000	0.151402
$699$	0	0
$700$	0	0
$701$	−38.0000	−1.43524	−0.717620	−	0.696435i	$-0.754769\pi$
$701$	−38.0000	−1.43524	−0.717620	+	0.696435i	$0.754769\pi$
$702$	0	0
$703$	−18.0000	−0.678883
$704$	48.0000	1.80907
$705$	0	0
$706$	−20.0000	−0.752710
$707$	−16.0000	−0.601742
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	−52.0000	−1.94333
$717$	0	0
$718$	60.0000	2.23918
$719$	−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
$721$	16.0000	0.595871
$722$	−20.0000	−0.744323
$723$	0	0
$724$	−50.0000	−1.85824
$725$	0	0
$726$	0	0
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.00000	−0.221918
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	34.0000	1.25496
$735$	0	0
$736$	64.0000	2.35907
$737$	−30.0000	−1.10506
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	0	0
$746$	50.0000	1.83063
$747$	0	0
$748$	−72.0000	−2.63258
$749$	−4.00000	−0.146157
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33.0000	1.19941	0.599703	−	0.800223i	$-0.295286\pi$
$757$	33.0000	1.19941	0.599703	+	0.800223i	$0.295286\pi$
$758$	42.0000	1.52551
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	−1.00000	−0.0362024
$764$	12.0000	0.434145
$765$	0	0
$766$	−24.0000	−0.867155
$767$	−10.0000	−0.361079
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	16.0000	0.573628
$779$	3.00000	0.107486
$780$	0	0
$781$	72.0000	2.57636
$782$	−96.0000	−3.43295
$783$	0	0
$784$	24.0000	0.857143
$785$	0	0
$786$	0	0
$787$	−3.00000	−0.106938	−0.0534692	−	0.998569i	$-0.517028\pi$
$787$	−3.00000	−0.106938	−0.0534692	+	0.998569i	$0.517028\pi$
$788$	−36.0000	−1.28245
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−65.0000	−2.30822
$794$	−50.0000	−1.77443
$795$	0	0
$796$	−46.0000	−1.63043
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	0	0
$801$	0	0
$802$	36.0000	1.27120
$803$	−36.0000	−1.27041
$804$	0	0
$805$	0	0
$806$	−30.0000	−1.05670
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	35.0000	1.22902	0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000	1.22902	0.614508	+	0.788911i	$0.289355\pi$
$812$	0	0
$813$	0	0
$814$	−72.0000	−2.52360
$815$	0	0
$816$	0	0
$817$	3.00000	0.104957
$818$	50.0000	1.74821
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−35.0000	−1.22002	−0.610012	−	0.792392i	$-0.708835\pi$
$823$	−35.0000	−1.22002	−0.610012	+	0.792392i	$0.708835\pi$
$824$	0	0
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	−40.0000	−1.38675
$833$	−36.0000	−1.24733
$834$	0	0
$835$	0	0
$836$	36.0000	1.24509
$837$	0	0
$838$	−28.0000	−0.967244
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	20.0000	0.689246
$843$	0	0
$844$	−22.0000	−0.757271
$845$	0	0
$846$	0	0
$847$	25.0000	0.859010
$848$	24.0000	0.824163
$849$	0	0
$850$	0	0
$851$	−48.0000	−1.64542
$852$	0	0
$853$	−3.00000	−0.102718	−0.0513590	−	0.998680i	$-0.516355\pi$
$853$	−3.00000	−0.102718	−0.0513590	+	0.998680i	$0.516355\pi$
$854$	−26.0000	−0.889702
$855$	0	0
$856$	0	0
$857$	52.0000	1.77629	0.888143	−	0.459567i	$-0.151995\pi$
$857$	52.0000	1.77629	0.888143	+	0.459567i	$0.151995\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	−36.0000	−1.22616
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	−54.0000	−1.83499
$867$	0	0
$868$	−6.00000	−0.203653
$869$	24.0000	0.814144
$870$	0	0
$871$	25.0000	0.847093
$872$	0	0
$873$	0	0
$874$	48.0000	1.62362
$875$	0	0
$876$	0	0
$877$	45.0000	1.51954	0.759771	−	0.650191i	$-0.225311\pi$
$877$	45.0000	1.51954	0.759771	+	0.650191i	$0.225311\pi$
$878$	6.00000	0.202490
$879$	0	0
$880$	0	0
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	0	0
$883$	17.0000	0.572096	0.286048	−	0.958215i	$-0.407658\pi$
$883$	17.0000	0.572096	0.286048	+	0.958215i	$0.407658\pi$
$884$	60.0000	2.01802
$885$	0	0
$886$	−36.0000	−1.20944
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	14.0000	0.468755
$893$	−6.00000	−0.200782
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−4.00000	−0.133482
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	12.0000	0.399556
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	36.0000	1.19470
$909$	0	0
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	−28.0000	−0.926158
$915$	0	0
$916$	46.0000	1.51988
$917$	2.00000	0.0660458
$918$	0	0
$919$	−51.0000	−1.68233	−0.841167	−	0.540775i	$-0.818131\pi$
$919$	−51.0000	−1.68233	−0.841167	+	0.540775i	$0.818131\pi$
$920$	0	0
$921$	0	0
$922$	−56.0000	−1.84426
$923$	−60.0000	−1.97492
$924$	0	0
$925$	0	0
$926$	−80.0000	−2.62896
$927$	0	0
$928$	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
$931$	18.0000	0.589926
$932$	32.0000	1.04819
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	0	0
$937$	−17.0000	−0.555366	−0.277683	−	0.960673i	$-0.589566\pi$
$937$	−17.0000	−0.555366	−0.277683	+	0.960673i	$0.589566\pi$
$938$	10.0000	0.326512
$939$	0	0
$940$	0	0
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	8.00000	0.260378
$945$	0	0
$946$	12.0000	0.390154
$947$	−38.0000	−1.23483	−0.617417	−	0.786636i	$-0.711821\pi$
$947$	−38.0000	−1.23483	−0.617417	+	0.786636i	$0.711821\pi$
$948$	0	0
$949$	30.0000	0.973841
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	4.00000	0.129234
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	60.0000	1.93448
$963$	0	0
$964$	−6.00000	−0.193247
$965$	0	0
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	−86.0000	−2.75562
$975$	0	0
$976$	52.0000	1.66448
$977$	−56.0000	−1.79160	−0.895799	−	0.444459i	$-0.853396\pi$
$977$	−56.0000	−1.79160	−0.895799	+	0.444459i	$0.853396\pi$
$978$	0	0
$979$	72.0000	2.30113
$980$	0	0
$981$	0	0
$982$	72.0000	2.29761
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−30.0000	−0.954427
$989$	8.00000	0.254385
$990$	0	0
$991$	55.0000	1.74713	0.873566	−	0.486705i	$-0.161801\pi$
$991$	55.0000	1.74713	0.873566	+	0.486705i	$0.161801\pi$
$992$	24.0000	0.762001
$993$	0	0
$994$	−24.0000	−0.761234
$995$	0	0
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	−78.0000	−2.46905
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9225.2.a.be.1.1	yes	1
3.2	odd	2		9225.2.a.b.1.1	yes	1
5.4	even	2		9225.2.a.a.1.1	✓	1
15.14	odd	2		9225.2.a.bd.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9225.2.a.a.1.1	✓	1	5.4	even	2
9225.2.a.b.1.1	yes	1	3.2	odd	2
9225.2.a.bd.1.1	yes	1	15.14	odd	2
9225.2.a.be.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9225 = 3^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more