LMFDB - Embedded newform 630.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,2,Mod(1,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, 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("630.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	630.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:	$5.03057532734$
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	$\chi$	$=$	630.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.00000 q^{11} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -1.00000 q^{35} +8.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} +10.0000 q^{41} -2.00000 q^{43} -4.00000 q^{44} +10.0000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{52} +14.0000 q^{53} -4.00000 q^{55} +1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +6.00000 q^{67} +4.00000 q^{68} +1.00000 q^{70} -2.00000 q^{71} -10.0000 q^{73} -8.00000 q^{74} +6.00000 q^{76} +4.00000 q^{77} +16.0000 q^{79} +1.00000 q^{80} -10.0000 q^{82} -8.00000 q^{83} +4.00000 q^{85} +2.00000 q^{86} +4.00000 q^{88} +2.00000 q^{89} -6.00000 q^{91} -10.0000 q^{94} +6.00000 q^{95} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	1.00000	0.267261
$15$	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
$26$	−6.00000	−1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	1.00000	0.133631
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	1.00000	0.119523
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	6.00000	0.688247
$77$	4.00000	0.455842
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−10.0000	−1.10432
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	2.00000	0.215666
$87$	0	0
$88$	4.00000	0.426401
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	0	0
$94$	−10.0000	−1.03142
$95$	6.00000	0.615587
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−14.0000	−1.35980
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	4.00000	0.368230
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	8.00000	0.724286
$123$	0	0
$124$	−4.00000	−0.359211
$125$	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	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−6.00000	−0.526235
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−4.00000	−0.342997
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	2.00000	0.167836
$143$	−24.0000	−2.00698
$144$	0	0
$145$	−6.00000	−0.498273
$146$	10.0000	0.827606
$147$	0	0
$148$	8.00000	0.657596
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	−4.00000	−0.322329
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	8.00000	0.620920
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	−16.0000	−1.17004
$188$	10.0000	0.729325
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	6.00000	0.421117
$204$	0	0
$205$	10.0000	0.698430
$206$	8.00000	0.557386
$207$	0	0
$208$	6.00000	0.416025
$209$	−24.0000	−1.66011
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	14.0000	0.961524
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−2.00000	−0.136399
$216$	0	0
$217$	4.00000	0.271538
$218$	18.0000	1.21911
$219$	0	0
$220$	−4.00000	−0.269680
$221$	24.0000	1.61441
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	10.0000	0.652328
$236$	−4.00000	−0.260378
$237$	0	0
$238$	4.00000	0.259281
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−8.00000	−0.512148
$245$	1.00000	0.0638877
$246$	0	0
$247$	36.0000	2.29063
$248$	4.00000	0.254000
$249$	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
$256$	1.00000	0.0625000
$257$	16.0000	0.998053	0.499026	−	0.866587i	$-0.333691\pi$
$257$	16.0000	0.998053	0.499026	+	0.866587i	$0.333691\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	6.00000	0.372104
$261$	0	0
$262$	−4.00000	−0.247121
$263$	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
$265$	14.0000	0.860013
$266$	6.00000	0.367884
$267$	0	0
$268$	6.00000	0.366508
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	1.00000	0.0597614
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	24.0000	1.41915
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	6.00000	0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	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
$295$	−4.00000	−0.232889
$296$	−8.00000	−0.464991
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	16.0000	0.920697
$303$	0	0
$304$	6.00000	0.344124
$305$	−8.00000	−0.458079
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	4.00000	0.227185
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	16.0000	0.900070
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	24.0000	1.34374
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	0	0
$325$	6.00000	0.332820
$326$	10.0000	0.553849
$327$	0	0
$328$	−10.0000	−0.552158
$329$	−10.0000	−0.551318
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	−6.00000	−0.328305
$335$	6.00000	0.327815
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	4.00000	0.216930
$341$	16.0000	0.866449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.00000	0.107833
$345$	0	0
$346$	2.00000	0.107521
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	4.00000	0.213201
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	2.00000	0.106000
$357$	0	0
$358$	24.0000	1.26844
$359$	34.0000	1.79445	0.897226	−	0.441572i	$-0.145579\pi$
$359$	34.0000	1.79445	0.897226	+	0.441572i	$0.145579\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	8.00000	0.420471
$363$	0	0
$364$	−6.00000	−0.314485
$365$	−10.0000	−0.523424
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	0	0
$370$	−8.00000	−0.415900
$371$	−14.0000	−0.726844
$372$	0	0
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−10.0000	−0.515711
$377$	−36.0000	−1.85409
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	18.0000	0.920960
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	10.0000	0.508987
$387$	0	0
$388$	2.00000	0.101535
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−22.0000	−1.10834
$395$	16.0000	0.805047
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	−24.0000	−1.19553
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−32.0000	−1.58618
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	−10.0000	−0.493865
$411$	0	0
$412$	−8.00000	−0.394132
$413$	4.00000	0.196827
$414$	0	0
$415$	−8.00000	−0.392705
$416$	−6.00000	−0.294174
$417$	0	0
$418$	24.0000	1.17388
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	−14.0000	−0.679900
$425$	4.00000	0.194029
$426$	0	0
$427$	8.00000	0.387147
$428$	12.0000	0.580042
$429$	0	0
$430$	2.00000	0.0964486
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	4.00000	0.190693
$441$	0	0
$442$	−24.0000	−1.14156
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	−16.0000	−0.757622
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	−14.0000	−0.658505
$453$	0	0
$454$	20.0000	0.938647
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−14.0000	−0.654892	−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000	−0.654892	−0.327446	+	0.944870i	$0.606188\pi$
$458$	−4.00000	−0.186908
$459$	0	0
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	−10.0000	−0.461266
$471$	0	0
$472$	4.00000	0.184115
$473$	8.00000	0.367840
$474$	0	0
$475$	6.00000	0.275299
$476$	−4.00000	−0.183340
$477$	0	0
$478$	22.0000	1.00626
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	48.0000	2.18861
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	2.00000	0.0908153
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	−24.0000	−1.08091
$494$	−36.0000	−1.61972
$495$	0	0
$496$	−4.00000	−0.179605
$497$	2.00000	0.0897123
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−16.0000	−0.705730
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−40.0000	−1.75920
$518$	8.00000	0.351500
$519$	0	0
$520$	−6.00000	−0.263117
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	−16.0000	−0.696971
$528$	0	0
$529$	−23.0000	−1.00000
$530$	−14.0000	−0.608121
$531$	0	0
$532$	−6.00000	−0.260133
$533$	60.0000	2.59889
$534$	0	0
$535$	12.0000	0.518805
$536$	−6.00000	−0.259161
$537$	0	0
$538$	−18.0000	−0.776035
$539$	−4.00000	−0.172292
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	−4.00000	−0.171499
$545$	−18.0000	−0.771035
$546$	0	0
$547$	30.0000	1.28271	0.641354	−	0.767245i	$-0.278373\pi$
$547$	30.0000	1.28271	0.641354	+	0.767245i	$0.278373\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	4.00000	0.170561
$551$	−36.0000	−1.53365
$552$	0	0
$553$	−16.0000	−0.680389
$554$	12.0000	0.509831
$555$	0	0
$556$	10.0000	0.424094
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	8.00000	0.337460
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	4.00000	0.168133
$567$	0	0
$568$	2.00000	0.0839181
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	−24.0000	−1.00349
$573$	0	0
$574$	10.0000	0.417392
$575$	0	0
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−6.00000	−0.249136
$581$	8.00000	0.331896
$582$	0	0
$583$	−56.0000	−2.31928
$584$	10.0000	0.413803
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	4.00000	0.164677
$591$	0	0
$592$	8.00000	0.328798
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	−2.00000	−0.0815139
$603$	0	0
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	8.00000	0.323911
$611$	60.0000	2.42734
$612$	0	0
$613$	48.0000	1.93870	0.969351	−	0.245680i	$-0.0790114\pi$
$613$	48.0000	1.93870	0.969351	+	0.245680i	$0.0790114\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−4.00000	−0.161165
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	12.0000	0.481156
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	0	0
$628$	−14.0000	−0.558661
$629$	32.0000	1.27592
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	−8.00000	−0.317470
$636$	0	0
$637$	6.00000	0.237729
$638$	−24.0000	−0.950169
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	2.00000	0.0786281	0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000	0.0786281	0.0393141	+	0.999227i	$0.487483\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−10.0000	−0.391630
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	10.0000	0.390434
$657$	0	0
$658$	10.0000	0.389841
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	−16.0000	−0.621858
$663$	0	0
$664$	8.00000	0.310460
$665$	−6.00000	−0.232670
$666$	0	0
$667$	0	0
$668$	6.00000	0.232147
$669$	0	0
$670$	−6.00000	−0.231800
$671$	32.0000	1.23535
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	23.0000	0.884615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	−4.00000	−0.153393
$681$	0	0
$682$	−16.0000	−0.612672
$683$	52.0000	1.98972	0.994862	−	0.101237i	$-0.0322800\pi$
$683$	52.0000	1.98972	0.994862	+	0.101237i	$0.0322800\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	1.00000	0.0381802
$687$	0	0
$688$	−2.00000	−0.0762493
$689$	84.0000	3.20015
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−20.0000	−0.759190
$695$	10.0000	0.379322
$696$	0	0
$697$	40.0000	1.51511
$698$	−28.0000	−1.05982
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	48.0000	1.81035
$704$	−4.00000	−0.150756
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	2.00000	0.0750587
$711$	0	0
$712$	−2.00000	−0.0749532
$713$	0	0
$714$	0	0
$715$	−24.0000	−0.897549
$716$	−24.0000	−0.896922
$717$	0	0
$718$	−34.0000	−1.26887
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−8.00000	−0.297318
$725$	−6.00000	−0.222834
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	10.0000	0.370117
$731$	−8.00000	−0.295891
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	8.00000	0.294086
$741$	0	0
$742$	14.0000	0.513956
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−32.0000	−1.17160
$747$	0	0
$748$	−16.0000	−0.585018
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	10.0000	0.364662
$753$	0	0
$754$	36.0000	1.31104
$755$	−16.0000	−0.582300
$756$	0	0
$757$	40.0000	1.45382	0.726912	−	0.686730i	$-0.240955\pi$
$757$	40.0000	1.45382	0.726912	+	0.686730i	$0.240955\pi$
$758$	0	0
$759$	0	0
$760$	−6.00000	−0.217643
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	18.0000	0.651644
$764$	−18.0000	−0.651217
$765$	0	0
$766$	30.0000	1.08394
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	−4.00000	−0.144150
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	−22.0000	−0.788738
$779$	60.0000	2.14972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	−14.0000	−0.499681
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	−16.0000	−0.569254
$791$	14.0000	0.497783
$792$	0	0
$793$	−48.0000	−1.70453
$794$	26.0000	0.922705
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	16.0000	0.564980
$803$	40.0000	1.41157
$804$	0	0
$805$	0	0
$806$	24.0000	0.845364
$807$	0	0
$808$	6.00000	0.211079
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	32.0000	1.12160
$815$	−10.0000	−0.350285
$816$	0	0
$817$	−12.0000	−0.419827
$818$	−18.0000	−0.629355
$819$	0	0
$820$	10.0000	0.349215
$821$	−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
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	6.00000	0.208013
$833$	4.00000	0.138592
$834$	0	0
$835$	6.00000	0.207639
$836$	−24.0000	−0.830057
$837$	0	0
$838$	36.0000	1.24360
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−14.0000	−0.482472
$843$	0	0
$844$	16.0000	0.550743
$845$	23.0000	0.791224
$846$	0	0
$847$	−5.00000	−0.171802
$848$	14.0000	0.480762
$849$	0	0
$850$	−4.00000	−0.137199
$851$	0	0
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	0	0
$859$	30.0000	1.02359	0.511793	−	0.859109i	$-0.328981\pi$
$859$	30.0000	1.02359	0.511793	+	0.859109i	$0.328981\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	−30.0000	−1.02180
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	22.0000	0.747590
$867$	0	0
$868$	4.00000	0.135769
$869$	−64.0000	−2.17105
$870$	0	0
$871$	36.0000	1.21981
$872$	18.0000	0.609557
$873$	0	0
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	−4.00000	−0.134840
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	4.00000	0.134383
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−2.00000	−0.0670402
$891$	0	0
$892$	16.0000	0.535720
$893$	60.0000	2.00782
$894$	0	0
$895$	−24.0000	−0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	12.0000	0.400445
$899$	24.0000	0.800445
$900$	0	0
$901$	56.0000	1.86563
$902$	40.0000	1.33185
$903$	0	0
$904$	14.0000	0.465633
$905$	−8.00000	−0.265929
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	6.00000	0.198898
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	32.0000	1.05905
$914$	14.0000	0.463079
$915$	0	0
$916$	4.00000	0.132164
$917$	−4.00000	−0.132092
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	−2.00000	−0.0658665
$923$	−12.0000	−0.394985
$924$	0	0
$925$	8.00000	0.263038
$926$	4.00000	0.131448
$927$	0	0
$928$	6.00000	0.196960
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−6.00000	−0.196537
$933$	0	0
$934$	32.0000	1.04707
$935$	−16.0000	−0.523256
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	6.00000	0.195907
$939$	0	0
$940$	10.0000	0.326164
$941$	58.0000	1.89075	0.945373	−	0.325991i	$-0.105698\pi$
$941$	58.0000	1.89075	0.945373	+	0.325991i	$0.105698\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−8.00000	−0.260102
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	−6.00000	−0.194666
$951$	0	0
$952$	4.00000	0.129641
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	−22.0000	−0.711531
$957$	0	0
$958$	−36.0000	−1.16311
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−48.0000	−1.54758
$963$	0	0
$964$	10.0000	0.322078
$965$	−10.0000	−0.321911
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−8.00000	−0.256074
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	1.00000	0.0319438
$981$	0	0
$982$	−24.0000	−0.765871
$983$	−38.0000	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$983$	−38.0000	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$984$	0	0
$985$	22.0000	0.700978
$986$	24.0000	0.764316
$987$	0	0
$988$	36.0000	1.14531
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	−2.00000	−0.0634361
$995$	−16.0000	−0.507234
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	24.0000	0.759707
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.2.a.e.1.1	✓	1
3.2	odd	2		630.2.a.g.1.1	yes	1
4.3	odd	2		5040.2.a.bp.1.1		1
5.2	odd	4		3150.2.g.b.2899.1		2
5.3	odd	4		3150.2.g.b.2899.2		2
5.4	even	2		3150.2.a.bh.1.1		1
7.6	odd	2		4410.2.a.a.1.1		1
12.11	even	2		5040.2.a.n.1.1		1
15.2	even	4		3150.2.g.s.2899.2		2
15.8	even	4		3150.2.g.s.2899.1		2
15.14	odd	2		3150.2.a.s.1.1		1
21.20	even	2		4410.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.a.e.1.1	✓	1	1.1	even	1	trivial
630.2.a.g.1.1	yes	1	3.2	odd	2
3150.2.a.s.1.1		1	15.14	odd	2
3150.2.a.bh.1.1		1	5.4	even	2
3150.2.g.b.2899.1		2	5.2	odd	4
3150.2.g.b.2899.2		2	5.3	odd	4
3150.2.g.s.2899.1		2	15.8	even	4
3150.2.g.s.2899.2		2	15.2	even	4
4410.2.a.a.1.1		1	7.6	odd	2
4410.2.a.bl.1.1		1	21.20	even	2
5040.2.a.n.1.1		1	12.11	even	2
5040.2.a.bp.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:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	630.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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Database

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