LMFDB - Embedded newform 5025.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5025,2,Mod(1,5025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5025 = 3 \cdot 5^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5025.a (trivial)

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:	yes
Analytic conductor:	$40.1248270157$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1005)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -8.00000 q^{23} +3.00000 q^{24} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{31} -5.00000 q^{32} -4.00000 q^{34} -1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} +8.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} +4.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +2.00000 q^{58} -6.00000 q^{59} +14.0000 q^{61} -2.00000 q^{62} +7.00000 q^{64} +1.00000 q^{67} -4.00000 q^{68} -8.00000 q^{69} +14.0000 q^{71} +3.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +16.0000 q^{83} -2.00000 q^{87} -18.0000 q^{89} +8.00000 q^{92} +2.00000 q^{93} +12.0000 q^{94} -5.00000 q^{96} +8.00000 q^{97} +7.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−4.00000	−0.648886
$39$	−4.00000	−0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	8.00000	1.17954
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	4.00000	0.560112
$52$	4.00000	0.554700
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	4.00000	0.529813
$58$	2.00000	0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	−4.00000	−0.485071
$69$	−8.00000	−0.963087
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	3.00000	0.353553
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	4.00000	0.452911
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	8.00000	0.834058
$93$	2.00000	0.207390
$94$	12.0000	1.23771
$95$	0	0
$96$	−5.00000	−0.510310
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−4.00000	−0.396059
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	−1.00000	−0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	−4.00000	−0.369800
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	6.00000	0.541002
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	−2.00000	−0.174741	−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000	−0.174741	−0.0873704	+	0.996176i	$0.527846\pi$
$132$	0	0
$133$	0	0
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	12.0000	1.02899
$137$	22.0000	1.87959	0.939793	−	0.341743i	$-0.111017\pi$
$137$	22.0000	1.87959	0.939793	+	0.341743i	$0.111017\pi$
$138$	8.00000	0.681005
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	−14.0000	−1.17485
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	−7.00000	−0.577350
$148$	6.00000	0.493197
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	12.0000	0.973329
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	10.0000	0.795557
$159$	2.00000	0.158610
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−16.0000	−1.24184
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	−6.00000	−0.450988
$178$	18.0000	1.34916
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	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
$183$	14.0000	1.03491
$184$	−24.0000	−1.76930
$185$	0	0
$186$	−2.00000	−0.146647
$187$	0	0
$188$	12.0000	0.875190
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	7.00000	0.505181
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	7.00000	0.500000
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	6.00000	0.422159
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	16.0000	1.11477
$207$	−8.00000	−0.556038
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−2.00000	−0.137361
$213$	14.0000	0.959264
$214$	16.0000	1.09374
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−16.0000	−1.07628
$222$	6.00000	0.402694
$223$	−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
$226$	6.00000	0.399114
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−4.00000	−0.264906
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	6.00000	0.390567
$237$	−10.0000	−0.649570
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−16.0000	−1.01806
$248$	6.00000	0.381000
$249$	16.0000	1.01396
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	2.00000	0.123560
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−18.0000	−1.10158
$268$	−1.00000	−0.0610847
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−22.0000	−1.32907
$275$	0	0
$276$	8.00000	0.481543
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−14.0000	−0.839664
$279$	2.00000	0.119737
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	12.0000	0.714590
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−5.00000	−0.294628
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	2.00000	0.117041
$293$	−32.0000	−1.86946	−0.934730	−	0.355359i	$-0.884359\pi$
$293$	−32.0000	−1.86946	−0.934730	+	0.355359i	$0.884359\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−18.0000	−1.04623
$297$	0	0
$298$	14.0000	0.810998
$299$	32.0000	1.85061
$300$	0	0
$301$	0	0
$302$	12.0000	0.690522
$303$	−6.00000	−0.344691
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	−12.0000	−0.679366
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	10.0000	0.562544
$317$	8.00000	0.449325	0.224662	−	0.974437i	$-0.427872\pi$
$317$	8.00000	0.449325	0.224662	+	0.974437i	$0.427872\pi$
$318$	−2.00000	−0.112154
$319$	0	0
$320$	0	0
$321$	−16.0000	−0.893033
$322$	0	0
$323$	16.0000	0.890264
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	10.0000	0.553001
$328$	18.0000	0.993884
$329$	0	0
$330$	0	0
$331$	−22.0000	−1.20923	−0.604615	−	0.796518i	$-0.706673\pi$
$331$	−22.0000	−1.20923	−0.604615	+	0.796518i	$0.706673\pi$
$332$	−16.0000	−0.878114
$333$	−6.00000	−0.328798
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	−3.00000	−0.163178
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	−4.00000	−0.216295
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−8.00000	−0.430083
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	2.00000	0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	6.00000	0.318896
$355$	0	0
$356$	18.0000	0.953998
$357$	0	0
$358$	16.0000	0.845626
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	−11.0000	−0.577350
$364$	0	0
$365$	0	0
$366$	−14.0000	−0.731792
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	8.00000	0.417029
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	−2.00000	−0.103695
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	0	0
$375$	0	0
$376$	−36.0000	−1.85656
$377$	8.00000	0.412021
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	0	0
$382$	−24.0000	−1.22795
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	6.00000	0.305392
$387$	0	0
$388$	−8.00000	−0.406138
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	−21.0000	−1.06066
$393$	−2.00000	−0.100887
$394$	2.00000	0.100759
$395$	0	0
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	−1.00000	−0.0498755
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	12.0000	0.594089
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	16.0000	0.788263
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	20.0000	0.980581
$417$	14.0000	0.685583
$418$	0	0
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	8.00000	0.389434
$423$	−12.0000	−0.583460
$424$	6.00000	0.291386
$425$	0	0
$426$	−14.0000	−0.678302
$427$	0	0
$428$	16.0000	0.773389
$429$	0	0
$430$	0	0
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	−1.00000	−0.0481125
$433$	−8.00000	−0.384455	−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000	−0.384455	−0.192228	+	0.981350i	$0.561571\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−32.0000	−1.53077
$438$	2.00000	0.0955637
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	16.0000	0.761042
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	24.0000	1.13643
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	6.00000	0.282216
$453$	−12.0000	−0.563809
$454$	−4.00000	−0.187729
$455$	0	0
$456$	12.0000	0.561951
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	−6.00000	−0.280362
$459$	4.00000	0.186704
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	−18.0000	−0.828517
$473$	0	0
$474$	10.0000	0.459315
$475$	0	0
$476$	0	0
$477$	2.00000	0.0915737
$478$	−4.00000	−0.182956
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	0	0
$481$	24.0000	1.09431
$482$	−14.0000	−0.637683
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	42.0000	1.90125
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−22.0000	−0.992846	−0.496423	−	0.868081i	$-0.665354\pi$
$491$	−22.0000	−0.992846	−0.496423	+	0.868081i	$0.665354\pi$
$492$	−6.00000	−0.270501
$493$	−8.00000	−0.360302
$494$	16.0000	0.719874
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	−16.0000	−0.716977
$499$	30.0000	1.34298	0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	20.0000	0.892644
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	4.00000	0.176604
$514$	8.00000	0.352865
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	8.00000	0.351161
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	2.00000	0.0875376
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	2.00000	0.0873704
$525$	0	0
$526$	4.00000	0.174408
$527$	8.00000	0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	−24.0000	−1.03956
$534$	18.0000	0.778936
$535$	0	0
$536$	3.00000	0.129580
$537$	−16.0000	−0.690451
$538$	30.0000	1.29339
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−14.0000	−0.601351
$543$	−22.0000	−0.944110
$544$	−20.0000	−0.857493
$545$	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	$0.527253\pi$
$548$	−22.0000	−0.939793
$549$	14.0000	0.597505
$550$	0	0
$551$	−8.00000	−0.340811
$552$	−24.0000	−1.02151
$553$	0	0
$554$	22.0000	0.934690
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−16.0000	−0.677942	−0.338971	−	0.940797i	$-0.610079\pi$
$557$	−16.0000	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$558$	−2.00000	−0.0846668
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	42.0000	1.76228
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	1.00000	0.0415945
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	−8.00000	−0.331611
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	32.0000	1.32191
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	7.00000	0.288675
$589$	8.00000	0.329634
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	6.00000	0.246598
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	14.0000	0.573462
$597$	−8.00000	−0.327418
$598$	−32.0000	−1.30858
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	12.0000	0.488273
$605$	0	0
$606$	6.00000	0.243733
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	0	0
$611$	48.0000	1.94187
$612$	−4.00000	−0.161690
$613$	42.0000	1.69636	0.848182	−	0.529705i	$-0.177697\pi$
$613$	42.0000	1.69636	0.848182	+	0.529705i	$0.177697\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	16.0000	0.643614
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	4.00000	0.160385
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	12.0000	0.479616
$627$	0	0
$628$	2.00000	0.0798087
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	−30.0000	−1.19334
$633$	−8.00000	−0.317971
$634$	−8.00000	−0.317721
$635$	0	0
$636$	−2.00000	−0.0793052
$637$	28.0000	1.10940
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	16.0000	0.631470
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	3.00000	0.117851
$649$	0	0
$650$	0	0
$651$	0	0
$652$	20.0000	0.783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−42.0000	−1.63361	−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000	−1.63361	−0.816805	+	0.576913i	$0.804257\pi$
$662$	22.0000	0.855054
$663$	−16.0000	−0.621389
$664$	48.0000	1.86276
$665$	0	0
$666$	6.00000	0.232495
$667$	16.0000	0.619522
$668$	−12.0000	−0.464294
$669$	−24.0000	−0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	−3.00000	−0.115385
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−8.00000	−0.304114
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	24.0000	0.909065
$698$	−2.00000	−0.0757011
$699$	22.0000	0.832116
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	4.00000	0.150970
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	6.00000	0.225494
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	−54.0000	−2.02374
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	16.0000	0.597948
$717$	4.00000	0.149383
$718$	−26.0000	−0.970311
$719$	26.0000	0.969636	0.484818	−	0.874615i	$-0.338886\pi$
$719$	26.0000	0.969636	0.484818	+	0.874615i	$0.338886\pi$
$720$	0	0
$721$	0	0
$722$	3.00000	0.111648
$723$	14.0000	0.520666
$724$	22.0000	0.817624
$725$	0	0
$726$	11.0000	0.408248
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−14.0000	−0.517455
$733$	−28.0000	−1.03420	−0.517102	−	0.855924i	$-0.672989\pi$
$733$	−28.0000	−1.03420	−0.517102	+	0.855924i	$0.672989\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	40.0000	1.47442
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	0	0
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	6.00000	0.219971
$745$	0	0
$746$	32.0000	1.17160
$747$	16.0000	0.585409
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	12.0000	0.437595
$753$	−20.0000	−0.728841
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	52.0000	1.88997	0.944986	−	0.327111i	$-0.106075\pi$
$757$	52.0000	1.88997	0.944986	+	0.327111i	$0.106075\pi$
$758$	22.0000	0.799076
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	0	0
$764$	−24.0000	−0.868290
$765$	0	0
$766$	0	0
$767$	24.0000	0.866590
$768$	−17.0000	−0.613435
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	−8.00000	−0.288113
$772$	6.00000	0.215945
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	24.0000	0.861550
$777$	0	0
$778$	−2.00000	−0.0717035
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	32.0000	1.14432
$783$	−2.00000	−0.0714742
$784$	7.00000	0.250000
$785$	0	0
$786$	2.00000	0.0713376
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	2.00000	0.0712470
$789$	−4.00000	−0.142404
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−56.0000	−1.98862
$794$	2.00000	0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	−16.0000	−0.566749	−0.283375	−	0.959009i	$-0.591454\pi$
$797$	−16.0000	−0.566749	−0.283375	+	0.959009i	$0.591454\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	−18.0000	−0.635999
$802$	18.0000	0.635602
$803$	0	0
$804$	−1.00000	−0.0352673
$805$	0	0
$806$	8.00000	0.281788
$807$	−30.0000	−1.05605
$808$	−18.0000	−0.633238
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	0	0
$815$	0	0
$816$	−4.00000	−0.140028
$817$	0	0
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−22.0000	−0.767338
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−48.0000	−1.67216
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	8.00000	0.278019
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	−28.0000	−0.970725
$833$	−28.0000	−0.970143
$834$	−14.0000	−0.484780
$835$	0	0
$836$	0	0
$837$	2.00000	0.0691301
$838$	−14.0000	−0.483622
$839$	−46.0000	−1.58810	−0.794048	−	0.607855i	$-0.792030\pi$
$839$	−46.0000	−1.58810	−0.794048	+	0.607855i	$0.792030\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	18.0000	0.619953
$844$	8.00000	0.275371
$845$	0	0
$846$	12.0000	0.412568
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	4.00000	0.137280
$850$	0	0
$851$	48.0000	1.64542
$852$	−14.0000	−0.479632
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	0	0
$856$	−48.0000	−1.64061
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	10.0000	0.340601
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	8.00000	0.271851
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	30.0000	1.01593
$873$	8.00000	0.270759
$874$	32.0000	1.08242
$875$	0	0
$876$	2.00000	0.0675737
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	8.00000	0.269987
$879$	−32.0000	−1.07933
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	7.00000	0.235702
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	4.00000	0.134383
$887$	4.00000	0.134307	0.0671534	−	0.997743i	$-0.478608\pi$
$887$	4.00000	0.134307	0.0671534	+	0.997743i	$0.478608\pi$
$888$	−18.0000	−0.604040
$889$	0	0
$890$	0	0
$891$	0	0
$892$	24.0000	0.803579
$893$	−48.0000	−1.60626
$894$	14.0000	0.468230
$895$	0	0
$896$	0	0
$897$	32.0000	1.06845
$898$	6.00000	0.200223
$899$	−4.00000	−0.133407
$900$	0	0
$901$	8.00000	0.266519
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	12.0000	0.398673
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−4.00000	−0.132745
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−14.0000	−0.463079
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	−4.00000	−0.132020
$919$	−6.00000	−0.197922	−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000	−0.197922	−0.0989609	+	0.995091i	$0.531552\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	−18.0000	−0.592798
$923$	−56.0000	−1.84326
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	−16.0000	−0.525509
$928$	10.0000	0.328266
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	−22.0000	−0.720634
$933$	−4.00000	−0.130954
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−12.0000	−0.392232
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	2.00000	0.0651635
$943$	−48.0000	−1.56310
$944$	6.00000	0.195283
$945$	0	0
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	10.0000	0.324785
$949$	8.00000	0.259691
$950$	0	0
$951$	8.00000	0.259418
$952$	0	0
$953$	52.0000	1.68445	0.842223	−	0.539130i	$-0.181247\pi$
$953$	52.0000	1.68445	0.842223	+	0.539130i	$0.181247\pi$
$954$	−2.00000	−0.0647524
$955$	0	0
$956$	−4.00000	−0.129369
$957$	0	0
$958$	26.0000	0.840022
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−24.0000	−0.773791
$963$	−16.0000	−0.515593
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	−33.0000	−1.06066
$969$	16.0000	0.513994
$970$	0	0
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	40.0000	1.28168
$975$	0	0
$976$	−14.0000	−0.448129
$977$	28.0000	0.895799	0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000	0.895799	0.447900	+	0.894084i	$0.352172\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	22.0000	0.702048
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	18.0000	0.573819
$985$	0	0
$986$	8.00000	0.254772
$987$	0	0
$988$	16.0000	0.509028
$989$	0	0
$990$	0	0
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	−10.0000	−0.317500
$993$	−22.0000	−0.698149
$994$	0	0
$995$	0	0
$996$	−16.0000	−0.506979
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	−30.0000	−0.949633
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5025.2.a.b.1.1		1
5.4	even	2		1005.2.a.b.1.1	✓	1
15.14	odd	2		3015.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1005.2.a.b.1.1	✓	1	5.4	even	2
3015.2.a.a.1.1		1	15.14	odd	2
5025.2.a.b.1.1		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5025 = 3 \cdot 5^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5025.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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