LMFDB - Embedded newform 585.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, 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("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	585.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:	$4.67124851824$
Analytic rank:	$0$
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	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} +4.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} +4.00000 q^{34} +2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} -8.00000 q^{43} -4.00000 q^{44} +8.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} +4.00000 q^{53} +4.00000 q^{55} -6.00000 q^{56} +4.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} -1.00000 q^{65} -10.0000 q^{67} -4.00000 q^{68} +2.00000 q^{70} -6.00000 q^{73} -2.00000 q^{74} -6.00000 q^{76} +8.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} +4.00000 q^{85} -8.00000 q^{86} -12.0000 q^{88} -14.0000 q^{89} -2.00000 q^{91} +8.00000 q^{94} +6.00000 q^{95} -14.0000 q^{97} -3.00000 q^{98} +O(q^{100})$

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	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	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	$0.338794\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−2.00000	−0.377964
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	4.00000	0.685994
$35$	2.00000	0.338062
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−3.00000	−0.474342
$41$	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
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	−6.00000	−0.801784
$57$	0	0
$58$	4.00000	0.525226
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−6.00000	−0.688247
$77$	8.00000	0.911685
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.00000	0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−12.0000	−1.27920
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	8.00000	0.825137
$95$	6.00000	0.615587
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	4.00000	0.388514
$107$	−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
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	0	0
$124$	10.0000	0.898027
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	−10.0000	−0.863868
$135$	0	0
$136$	−12.0000	−1.02899
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	4.00000	0.332182
$146$	−6.00000	−0.496564
$147$	0	0
$148$	2.00000	0.164399
$149$	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
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−18.0000	−1.45999
$153$	0	0
$154$	8.00000	0.644658
$155$	−10.0000	−0.803219
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	5.00000	0.395285
$161$	0	0
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.00000	0.306786
$171$	0	0
$172$	8.00000	0.609994
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−14.0000	−1.04934
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	16.0000	1.17004
$188$	−8.00000	−0.583460
$189$	0	0
$190$	6.00000	0.435286
$191$	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
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	3.00000	0.214286
$197$	−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
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−12.0000	−0.844317
$203$	8.00000	0.561490
$204$	0	0
$205$	6.00000	0.419058
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000	0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	−4.00000	−0.273434
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−20.0000	−1.35769
$218$	−2.00000	−0.135457
$219$	0	0
$220$	−4.00000	−0.269680
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	−16.0000	−1.06430
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−12.0000	−0.787839
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	12.0000	0.781133
$237$	0	0
$238$	8.00000	0.518563
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−6.00000	−0.381771
$248$	30.0000	1.90500
$249$	0	0
$250$	1.00000	0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	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
$259$	−4.00000	−0.248548
$260$	1.00000	0.0620174
$261$	0	0
$262$	16.0000	0.988483
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	12.0000	0.735767
$267$	0	0
$268$	10.0000	0.610847
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	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
$274$	−10.0000	−0.604122
$275$	4.00000	0.241209
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	−6.00000	−0.358569
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	−4.00000	−0.236525
$287$	12.0000	0.708338
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	4.00000	0.234888
$291$	0	0
$292$	6.00000	0.351123
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	6.00000	0.348743
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	10.0000	0.575435
$303$	0	0
$304$	−6.00000	−0.344124
$305$	2.00000	0.114520
$306$	0	0
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	−10.0000	−0.567962
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	7.00000	0.391312
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	6.00000	0.332309
$327$	0	0
$328$	−18.0000	−0.993884
$329$	16.0000	0.882109
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	24.0000	1.31322
$335$	−10.0000	−0.546358
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−4.00000	−0.216930
$341$	−40.0000	−2.16612
$342$	0	0
$343$	−20.0000	−1.07990
$344$	24.0000	1.29399
$345$	0	0
$346$	−24.0000	−1.29025
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	20.0000	1.06600
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	0	0
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	0	0
$364$	2.00000	0.104828
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	0	0
$370$	−2.00000	−0.103975
$371$	8.00000	0.415339
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−24.0000	−1.23771
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	12.0000	0.613973
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	−26.0000	−1.32337
$387$	0	0
$388$	14.0000	0.710742
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	0	0
$394$	−18.0000	−0.906827
$395$	12.0000	0.603786
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	12.0000	0.597022
$405$	0	0
$406$	8.00000	0.397033
$407$	−8.00000	−0.396545
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	4.00000	0.197066
$413$	−24.0000	−1.18096
$414$	0	0
$415$	4.00000	0.196352
$416$	−5.00000	−0.245145
$417$	0	0
$418$	24.0000	1.17388
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	−12.0000	−0.582772
$425$	4.00000	0.194029
$426$	0	0
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	0	0
$430$	−8.00000	−0.385794
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	−36.0000	−1.71819	−0.859093	−	0.511819i	$-0.828972\pi$
$439$	−36.0000	−1.71819	−0.859093	+	0.511819i	$0.828972\pi$
$440$	−12.0000	−0.572078
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	−14.0000	−0.662919
$447$	0	0
$448$	14.0000	0.661438
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	16.0000	0.752577
$453$	0	0
$454$	20.0000	0.938647
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	8.00000	0.370593
$467$	−8.00000	−0.370196	−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000	−0.370196	−0.185098	+	0.982720i	$0.559260\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	8.00000	0.369012
$471$	0	0
$472$	36.0000	1.65703
$473$	−32.0000	−1.47136
$474$	0	0
$475$	6.00000	0.275299
$476$	−8.00000	−0.366679
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	22.0000	1.00207
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−22.0000	−0.996915	−0.498458	−	0.866914i	$-0.666100\pi$
$487$	−22.0000	−0.996915	−0.498458	+	0.866914i	$0.666100\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	−3.00000	−0.135526
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	16.0000	0.720604
$494$	−6.00000	−0.269953
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	8.00000	0.354943
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−8.00000	−0.352865
$515$	−4.00000	−0.176261
$516$	0	0
$517$	32.0000	1.40736
$518$	−4.00000	−0.175750
$519$	0	0
$520$	3.00000	0.131559
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	4.00000	0.174408
$527$	−40.0000	−1.74243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	4.00000	0.173749
$531$	0	0
$532$	−12.0000	−0.520266
$533$	−6.00000	−0.259889
$534$	0	0
$535$	−4.00000	−0.172935
$536$	30.0000	1.29580
$537$	0	0
$538$	−12.0000	−0.517357
$539$	−12.0000	−0.516877
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	20.0000	0.857493
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	10.0000	0.427179
$549$	0	0
$550$	4.00000	0.170561
$551$	24.0000	1.02243
$552$	0	0
$553$	24.0000	1.02058
$554$	10.0000	0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	30.0000	1.26547
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	−4.00000	−0.168133
$567$	0	0
$568$	0	0
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	−4.00000	−0.166091
$581$	8.00000	0.331896
$582$	0	0
$583$	16.0000	0.662652
$584$	18.0000	0.744845
$585$	0	0
$586$	2.00000	0.0826192
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	−12.0000	−0.494032
$591$	0	0
$592$	2.00000	0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	−18.0000	−0.737309
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	−16.0000	−0.652111
$603$	0	0
$604$	−10.0000	−0.406894
$605$	5.00000	0.203279
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	30.0000	1.21666
$609$	0	0
$610$	2.00000	0.0809776
$611$	−8.00000	−0.323645
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−24.0000	−0.966988
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	22.0000	0.884255	0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000	0.884255	0.442127	+	0.896952i	$0.354224\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	4.00000	0.160385
$623$	−28.0000	−1.12180
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.00000	0.0799361
$627$	0	0
$628$	−18.0000	−0.718278
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	−36.0000	−1.43200
$633$	0	0
$634$	−18.0000	−0.714871
$635$	−8.00000	−0.317470
$636$	0	0
$637$	3.00000	0.118864
$638$	16.0000	0.633446
$639$	0	0
$640$	−3.00000	−0.118585
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	0	0
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−6.00000	−0.234978
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	−6.00000	−0.234261
$657$	0	0
$658$	16.0000	0.623745
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	−12.0000	−0.465690
$665$	12.0000	0.465340
$666$	0	0
$667$	0	0
$668$	−24.0000	−0.928588
$669$	0	0
$670$	−10.0000	−0.386334
$671$	8.00000	0.308837
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	−28.0000	−1.07454
$680$	−12.0000	−0.460179
$681$	0	0
$682$	−40.0000	−1.53168
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−10.0000	−0.382080
$686$	−20.0000	−0.763604
$687$	0	0
$688$	8.00000	0.304997
$689$	−4.00000	−0.152388
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	24.0000	0.912343
$693$	0	0
$694$	−32.0000	−1.21470
$695$	12.0000	0.455186
$696$	0	0
$697$	24.0000	0.909065
$698$	−6.00000	−0.227103
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	28.0000	1.05529
$705$	0	0
$706$	6.00000	0.225813
$707$	−24.0000	−0.902613
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	0	0
$712$	42.0000	1.57402
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	0	0
$718$	16.0000	0.597115
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	17.0000	0.632674
$723$	0	0
$724$	10.0000	0.371647
$725$	4.00000	0.148556
$726$	0	0
$727$	−24.0000	−0.890111	−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000	−0.890111	−0.445055	+	0.895503i	$0.646816\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	−6.00000	−0.222070
$731$	−32.0000	−1.18356
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	8.00000	0.293689
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	22.0000	0.805477
$747$	0	0
$748$	−16.0000	−0.585018
$749$	−8.00000	−0.292314
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	−4.00000	−0.145671
$755$	10.0000	0.363937
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−2.00000	−0.0726433
$759$	0	0
$760$	−18.0000	−0.652929
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	−12.0000	−0.434145
$765$	0	0
$766$	24.0000	0.867155
$767$	12.0000	0.433295
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	26.0000	0.935760
$773$	50.0000	1.79838	0.899188	−	0.437564i	$-0.144158\pi$
$773$	50.0000	1.79838	0.899188	+	0.437564i	$0.144158\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	42.0000	1.50771
$777$	0	0
$778$	−8.00000	−0.286814
$779$	36.0000	1.28983
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	18.0000	0.642448
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	12.0000	0.426941
$791$	−32.0000	−1.13779
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−8.00000	−0.283552
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	5.00000	0.176777
$801$	0	0
$802$	−22.0000	−0.776847
$803$	−24.0000	−0.846942
$804$	0	0
$805$	0	0
$806$	10.0000	0.352235
$807$	0	0
$808$	36.0000	1.26648
$809$	−28.0000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$809$	−28.0000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$810$	0	0
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	−8.00000	−0.280400
$815$	6.00000	0.210171
$816$	0	0
$817$	−48.0000	−1.67931
$818$	30.0000	1.04893
$819$	0	0
$820$	−6.00000	−0.209529
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	−24.0000	−0.835067
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	4.00000	0.138842
$831$	0	0
$832$	−7.00000	−0.242681
$833$	−12.0000	−0.415775
$834$	0	0
$835$	24.0000	0.830554
$836$	−24.0000	−0.830057
$837$	0	0
$838$	16.0000	0.552711
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	22.0000	0.758170
$843$	0	0
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	10.0000	0.343604
$848$	−4.00000	−0.137361
$849$	0	0
$850$	4.00000	0.137199
$851$	0	0
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	12.0000	0.410152
$857$	48.0000	1.63965	0.819824	−	0.572615i	$-0.194071\pi$
$857$	48.0000	1.63965	0.819824	+	0.572615i	$0.194071\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	40.0000	1.36241
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	20.0000	0.678844
$869$	48.0000	1.62829
$870$	0	0
$871$	10.0000	0.338837
$872$	6.00000	0.203186
$873$	0	0
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−36.0000	−1.21494
$879$	0	0
$880$	−4.00000	−0.134840
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−20.0000	−0.671913
$887$	56.0000	1.88030	0.940148	−	0.340766i	$-0.110687\pi$
$887$	56.0000	1.88030	0.940148	+	0.340766i	$0.110687\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−14.0000	−0.469281
$891$	0	0
$892$	14.0000	0.468755
$893$	48.0000	1.60626
$894$	0	0
$895$	0	0
$896$	−6.00000	−0.200446
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−40.0000	−1.33407
$900$	0	0
$901$	16.0000	0.533037
$902$	24.0000	0.799113
$903$	0	0
$904$	48.0000	1.59646
$905$	−10.0000	−0.332411
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	−2.00000	−0.0662994
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	32.0000	1.05673
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	0	0
$921$	0	0
$922$	14.0000	0.461065
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	18.0000	0.591517
$927$	0	0
$928$	20.0000	0.656532
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−8.00000	−0.262049
$933$	0	0
$934$	−8.00000	−0.261768
$935$	16.0000	0.523256
$936$	0	0
$937$	54.0000	1.76410	0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000	1.76410	0.882052	+	0.471153i	$0.156162\pi$
$938$	−20.0000	−0.653023
$939$	0	0
$940$	−8.00000	−0.260931
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−32.0000	−1.04041
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	6.00000	0.194666
$951$	0	0
$952$	−24.0000	−0.777844
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	0	0
$958$	24.0000	0.775405
$959$	−20.0000	−0.645834
$960$	0	0
$961$	69.0000	2.22581
$962$	2.00000	0.0644826
$963$	0	0
$964$	−22.0000	−0.708572
$965$	−26.0000	−0.836970
$966$	0	0
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	−15.0000	−0.482118
$969$	0	0
$970$	−14.0000	−0.449513
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	24.0000	0.769405
$974$	−22.0000	−0.704925
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	0	0
$979$	−56.0000	−1.78977
$980$	3.00000	0.0958315
$981$	0	0
$982$	−36.0000	−1.14881
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	16.0000	0.509544
$987$	0	0
$988$	6.00000	0.190885
$989$	0	0
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	−50.0000	−1.58750
$993$	0	0
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−38.0000	−1.20287
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.2.a.i.1.1	yes	1
3.2	odd	2		585.2.a.d.1.1	✓	1
4.3	odd	2		9360.2.a.be.1.1		1
5.2	odd	4		2925.2.c.k.2224.2		2
5.3	odd	4		2925.2.c.k.2224.1		2
5.4	even	2		2925.2.a.c.1.1		1
12.11	even	2		9360.2.a.i.1.1		1
13.12	even	2		7605.2.a.c.1.1		1
15.2	even	4		2925.2.c.g.2224.1		2
15.8	even	4		2925.2.c.g.2224.2		2
15.14	odd	2		2925.2.a.m.1.1		1
39.38	odd	2		7605.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.d.1.1	✓	1	3.2	odd	2
585.2.a.i.1.1	yes	1	1.1	even	1	trivial
2925.2.a.c.1.1		1	5.4	even	2
2925.2.a.m.1.1		1	15.14	odd	2
2925.2.c.g.2224.1		2	15.2	even	4
2925.2.c.g.2224.2		2	15.8	even	4
2925.2.c.k.2224.1		2	5.3	odd	4
2925.2.c.k.2224.2		2	5.2	odd	4
7605.2.a.c.1.1		1	13.12	even	2
7605.2.a.q.1.1		1	39.38	odd	2
9360.2.a.i.1.1		1	12.11	even	2
9360.2.a.be.1.1		1	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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