LMFDB - Embedded newform 4830.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.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:	$38.5677441763$
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	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} -6.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} -1.00000 q^{42} +8.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +10.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} +4.00000 q^{66} -16.0000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +4.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} -4.00000 q^{77} +6.00000 q^{78} -16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -2.00000 q^{85} -8.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} -6.00000 q^{91} +1.00000 q^{92} +12.0000 q^{94} +1.00000 q^{96} +18.0000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	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	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	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
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	1.00000	0.218218
$22$	−4.00000	−0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−2.00000	−0.342997
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	1.00000	0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−1.00000	−0.154303
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	4.00000	0.603023
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	−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
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	6.00000	0.832050
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	−4.00000	−0.539360
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	4.00000	0.492366
$67$	−16.0000	−1.95471	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−16.0000	−1.95471	−0.977356	+	0.211604i	$0.932131\pi$
$68$	2.00000	0.242536
$69$	−1.00000	−0.120386
$70$	−1.00000	−0.119523
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−10.0000	−1.16248
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	6.00000	0.679366
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	1.00000	0.109109
$85$	−2.00000	−0.216930
$86$	−8.00000	−0.862662
$87$	−2.00000	−0.214423
$88$	−4.00000	−0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	−6.00000	−0.628971
$92$	1.00000	0.104257
$93$	0	0
$94$	12.0000	1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	−1.00000	−0.101015
$99$	4.00000	0.402015
$100$	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	$0.596490\pi$
$102$	2.00000	0.198030
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−6.00000	−0.588348
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	4.00000	0.381385
$111$	−10.0000	−0.949158
$112$	−1.00000	−0.0944911
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	2.00000	0.185695
$117$	6.00000	0.554700
$118$	−12.0000	−1.10469
$119$	−2.00000	−0.183340
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	6.00000	0.526235
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	16.0000	1.38219
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	1.00000	0.0851257
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	1.00000	0.0845154
$141$	12.0000	1.01058
$142$	−4.00000	−0.335673
$143$	24.0000	2.00698
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	6.00000	0.496564
$147$	−1.00000	−0.0824786
$148$	10.0000	0.821995
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	4.00000	0.322329
$155$	0	0
$156$	−6.00000	−0.480384
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	16.0000	1.27289
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	−1.00000	−0.0788110
$162$	−1.00000	−0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	6.00000	0.468521
$165$	4.00000	0.311400
$166$	4.00000	0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	23.0000	1.76923
$170$	2.00000	0.153393
$171$	0	0
$172$	8.00000	0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	2.00000	0.151620
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	−12.0000	−0.901975
$178$	2.00000	0.149906
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.00000	−0.0745356
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	6.00000	0.444750
$183$	−10.0000	−0.739221
$184$	−1.00000	−0.0737210
$185$	−10.0000	−0.735215
$186$	0	0
$187$	8.00000	0.585018
$188$	−12.0000	−0.875190
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−18.0000	−1.29232
$195$	6.00000	0.429669
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	16.0000	1.12855
$202$	6.00000	0.422159
$203$	−2.00000	−0.140372
$204$	−2.00000	−0.140028
$205$	−6.00000	−0.419058
$206$	−16.0000	−1.11477
$207$	1.00000	0.0695048
$208$	6.00000	0.416025
$209$	0	0
$210$	1.00000	0.0690066
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	−6.00000	−0.412082
$213$	−4.00000	−0.274075
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	6.00000	0.405442
$220$	−4.00000	−0.269680
$221$	12.0000	0.807207
$222$	10.0000	0.671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	1.00000	0.0659380
$231$	4.00000	0.263181
$232$	−2.00000	−0.131306
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	−6.00000	−0.392232
$235$	12.0000	0.782794
$236$	12.0000	0.781133
$237$	16.0000	1.03931
$238$	2.00000	0.129641
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	1.00000	0.0645497
$241$	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
$243$	−1.00000	−0.0641500
$244$	10.0000	0.640184
$245$	−1.00000	−0.0638877
$246$	6.00000	0.382546
$247$	0	0
$248$	0	0
$249$	4.00000	0.253490
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	0	0
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	8.00000	0.498058
$259$	−10.0000	−0.621370
$260$	−6.00000	−0.372104
$261$	2.00000	0.123797
$262$	−12.0000	−0.741362
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	4.00000	0.246183
$265$	6.00000	0.368577
$266$	0	0
$267$	2.00000	0.122398
$268$	−16.0000	−0.977356
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	2.00000	0.121268
$273$	6.00000	0.363137
$274$	2.00000	0.120824
$275$	4.00000	0.241209
$276$	−1.00000	−0.0601929
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	−12.0000	−0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−24.0000	−1.41915
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	−18.0000	−1.05518
$292$	−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	$0.481394\pi$
$294$	1.00000	0.0583212
$295$	−12.0000	−0.698667
$296$	−10.0000	−0.581238
$297$	−4.00000	−0.232104
$298$	−10.0000	−0.579284
$299$	6.00000	0.346989
$300$	−1.00000	−0.0577350
$301$	−8.00000	−0.461112
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−10.0000	−0.572598
$306$	−2.00000	−0.114332
$307$	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
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	6.00000	0.339683
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	10.0000	0.564333
$315$	1.00000	0.0563436
$316$	−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	$0.482113\pi$
$318$	−6.00000	−0.336463
$319$	8.00000	0.447914
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	1.00000	0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	6.00000	0.332820
$326$	−8.00000	−0.443079
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	12.0000	0.661581
$330$	−4.00000	−0.220193
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−4.00000	−0.219529
$333$	10.0000	0.547997
$334$	12.0000	0.656611
$335$	16.0000	0.874173
$336$	1.00000	0.0545545
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−23.0000	−1.25104
$339$	18.0000	0.977626
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−8.00000	−0.431331
$345$	1.00000	0.0538382
$346$	6.00000	0.322562
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	−2.00000	−0.107211
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	1.00000	0.0534522
$351$	−6.00000	−0.320256
$352$	−4.00000	−0.213201
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	12.0000	0.637793
$355$	−4.00000	−0.212298
$356$	−2.00000	−0.106000
$357$	2.00000	0.105851
$358$	−4.00000	−0.211407
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	−18.0000	−0.946059
$363$	−5.00000	−0.262432
$364$	−6.00000	−0.314485
$365$	6.00000	0.314054
$366$	10.0000	0.522708
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	1.00000	0.0521286
$369$	6.00000	0.312348
$370$	10.0000	0.519875
$371$	6.00000	0.311504
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−8.00000	−0.413670
$375$	1.00000	0.0516398
$376$	12.0000	0.618853
$377$	12.0000	0.618031
$378$	−1.00000	−0.0514344
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	12.0000	0.613973
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	4.00000	0.203859
$386$	−10.0000	−0.508987
$387$	8.00000	0.406663
$388$	18.0000	0.913812
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	−6.00000	−0.303822
$391$	2.00000	0.101144
$392$	−1.00000	−0.0505076
$393$	−12.0000	−0.605320
$394$	6.00000	0.302276
$395$	16.0000	0.805047
$396$	4.00000	0.201008
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−16.0000	−0.798007
$403$	0	0
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	2.00000	0.0992583
$407$	40.0000	1.98273
$408$	2.00000	0.0990148
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	6.00000	0.296319
$411$	2.00000	0.0986527
$412$	16.0000	0.788263
$413$	−12.0000	−0.590481
$414$	−1.00000	−0.0491473
$415$	4.00000	0.196352
$416$	−6.00000	−0.294174
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	−1.00000	−0.0487950
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	−28.0000	−1.36302
$423$	−12.0000	−0.583460
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	4.00000	0.193801
$427$	−10.0000	−0.483934
$428$	−12.0000	−0.580042
$429$	−24.0000	−1.15873
$430$	8.00000	0.385794
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	−1.00000	−0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	−10.0000	−0.478913
$437$	0	0
$438$	−6.00000	−0.286691
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	4.00000	0.190693
$441$	1.00000	0.0476190
$442$	−12.0000	−0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−10.0000	−0.474579
$445$	2.00000	0.0948091
$446$	16.0000	0.757622
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	−18.0000	−0.846649
$453$	0	0
$454$	12.0000	0.563188
$455$	6.00000	0.281284
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−10.0000	−0.467269
$459$	−2.00000	−0.0933520
$460$	−1.00000	−0.0466252
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	−4.00000	−0.186097
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	6.00000	0.277350
$469$	16.0000	0.738811
$470$	−12.0000	−0.553519
$471$	10.0000	0.460776
$472$	−12.0000	−0.552345
$473$	32.0000	1.47136
$474$	−16.0000	−0.734904
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−6.00000	−0.274721
$478$	−4.00000	−0.182956
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	−1.00000	−0.0456435
$481$	60.0000	2.73576
$482$	−10.0000	−0.455488
$483$	1.00000	0.0455016
$484$	5.00000	0.227273
$485$	−18.0000	−0.817338
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−10.0000	−0.452679
$489$	−8.00000	−0.361773
$490$	1.00000	0.0451754
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	4.00000	0.180151
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	−4.00000	−0.179425
$498$	−4.00000	−0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	−12.0000	−0.535586
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	1.00000	0.0445435
$505$	6.00000	0.266996
$506$	−4.00000	−0.177822
$507$	−23.0000	−1.02147
$508$	0	0
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	−2.00000	−0.0885615
$511$	6.00000	0.265424
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	22.0000	0.970378
$515$	−16.0000	−0.705044
$516$	−8.00000	−0.352180
$517$	−48.0000	−2.11104
$518$	10.0000	0.439375
$519$	6.00000	0.263371
$520$	6.00000	0.263117
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	−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	$0.472127\pi$
$524$	12.0000	0.524222
$525$	1.00000	0.0436436
$526$	−16.0000	−0.697633
$527$	0	0
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	12.0000	0.520756
$532$	0	0
$533$	36.0000	1.55933
$534$	−2.00000	−0.0865485
$535$	12.0000	0.518805
$536$	16.0000	0.691095
$537$	−4.00000	−0.172613
$538$	−18.0000	−0.776035
$539$	4.00000	0.172292
$540$	1.00000	0.0430331
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	16.0000	0.687259
$543$	−18.0000	−0.772454
$544$	−2.00000	−0.0857493
$545$	10.0000	0.428353
$546$	−6.00000	−0.256776
$547$	24.0000	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$548$	−2.00000	−0.0854358
$549$	10.0000	0.426790
$550$	−4.00000	−0.170561
$551$	0	0
$552$	1.00000	0.0425628
$553$	16.0000	0.680389
$554$	−2.00000	−0.0849719
$555$	10.0000	0.424476
$556$	8.00000	0.339276
$557$	−46.0000	−1.94908	−0.974541	−	0.224208i	$-0.928020\pi$
$557$	−46.0000	−1.94908	−0.974541	+	0.224208i	$0.928020\pi$
$558$	0	0
$559$	48.0000	2.03018
$560$	1.00000	0.0422577
$561$	−8.00000	−0.337760
$562$	−26.0000	−1.09674
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	12.0000	0.505291
$565$	18.0000	0.757266
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	−4.00000	−0.167836
$569$	−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
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	24.0000	1.00349
$573$	12.0000	0.501307
$574$	6.00000	0.250435
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	13.0000	0.540729
$579$	−10.0000	−0.415586
$580$	−2.00000	−0.0830455
$581$	4.00000	0.165948
$582$	18.0000	0.746124
$583$	−24.0000	−0.993978
$584$	6.00000	0.248282
$585$	−6.00000	−0.248069
$586$	−2.00000	−0.0826192
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	−1.00000	−0.0412393
$589$	0	0
$590$	12.0000	0.494032
$591$	6.00000	0.246807
$592$	10.0000	0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	4.00000	0.164122
$595$	2.00000	0.0819920
$596$	10.0000	0.409616
$597$	−16.0000	−0.654836
$598$	−6.00000	−0.245358
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	1.00000	0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	8.00000	0.326056
$603$	−16.0000	−0.651570
$604$	0	0
$605$	−5.00000	−0.203279
$606$	−6.00000	−0.243733
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	10.0000	0.404888
$611$	−72.0000	−2.91281
$612$	2.00000	0.0808452
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−4.00000	−0.161427
$615$	6.00000	0.241943
$616$	4.00000	0.161165
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	16.0000	0.643614
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	32.0000	1.28308
$623$	2.00000	0.0801283
$624$	−6.00000	−0.240192
$625$	1.00000	0.0400000
$626$	22.0000	0.879297
$627$	0	0
$628$	−10.0000	−0.399043
$629$	20.0000	0.797452
$630$	−1.00000	−0.0398410
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	16.0000	0.636446
$633$	−28.0000	−1.11290
$634$	−2.00000	−0.0794301
$635$	0	0
$636$	6.00000	0.237915
$637$	6.00000	0.237729
$638$	−8.00000	−0.316723
$639$	4.00000	0.158238
$640$	1.00000	0.0395285
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−12.0000	−0.473602
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	−1.00000	−0.0394055
$645$	8.00000	0.315000
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	48.0000	1.88416
$650$	−6.00000	−0.235339
$651$	0	0
$652$	8.00000	0.313304
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	−10.0000	−0.391031
$655$	−12.0000	−0.468879
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	−12.0000	−0.467809
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	4.00000	0.155700
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−20.0000	−0.777322
$663$	−12.0000	−0.466041
$664$	4.00000	0.155230
$665$	0	0
$666$	−10.0000	−0.387492
$667$	2.00000	0.0774403
$668$	−12.0000	−0.464294
$669$	16.0000	0.618596
$670$	−16.0000	−0.618134
$671$	40.0000	1.54418
$672$	−1.00000	−0.0385758
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	−2.00000	−0.0770371
$675$	−1.00000	−0.0384900
$676$	23.0000	0.884615
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−18.0000	−0.691286
$679$	−18.0000	−0.690777
$680$	2.00000	0.0766965
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	1.00000	0.0381802
$687$	−10.0000	−0.381524
$688$	8.00000	0.304997
$689$	−36.0000	−1.37149
$690$	−1.00000	−0.0380693
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−6.00000	−0.228086
$693$	−4.00000	−0.151947
$694$	−4.00000	−0.151838
$695$	−8.00000	−0.303457
$696$	2.00000	0.0758098
$697$	12.0000	0.454532
$698$	−18.0000	−0.681310
$699$	2.00000	0.0756469
$700$	−1.00000	−0.0377964
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	6.00000	0.226455
$703$	0	0
$704$	4.00000	0.150756
$705$	−12.0000	−0.451946
$706$	−26.0000	−0.978523
$707$	6.00000	0.225653
$708$	−12.0000	−0.450988
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	4.00000	0.150117
$711$	−16.0000	−0.600047
$712$	2.00000	0.0749532
$713$	0	0
$714$	−2.00000	−0.0748481
$715$	−24.0000	−0.897549
$716$	4.00000	0.149487
$717$	−4.00000	−0.149383
$718$	12.0000	0.447836
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	−1.00000	−0.0372678
$721$	−16.0000	−0.595871
$722$	19.0000	0.707107
$723$	−10.0000	−0.371904
$724$	18.0000	0.668965
$725$	2.00000	0.0742781
$726$	5.00000	0.185567
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	6.00000	0.222375
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	16.0000	0.591781
$732$	−10.0000	−0.369611
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	−16.0000	−0.590571
$735$	1.00000	0.0368856
$736$	−1.00000	−0.0368605
$737$	−64.0000	−2.35747
$738$	−6.00000	−0.220863
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	−10.0000	−0.367607
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−26.0000	−0.951928
$747$	−4.00000	−0.146352
$748$	8.00000	0.292509
$749$	12.0000	0.438470
$750$	−1.00000	−0.0365148
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	−12.0000	−0.437595
$753$	−12.0000	−0.437304
$754$	−12.0000	−0.437014
$755$	0	0
$756$	1.00000	0.0363696
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	−4.00000	−0.145287
$759$	−4.00000	−0.145191
$760$	0	0
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	−12.0000	−0.434145
$765$	−2.00000	−0.0723102
$766$	−12.0000	−0.433578
$767$	72.0000	2.59977
$768$	−1.00000	−0.0360844
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	−4.00000	−0.144150
$771$	22.0000	0.792311
$772$	10.0000	0.359908
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	−8.00000	−0.287554
$775$	0	0
$776$	−18.0000	−0.646162
$777$	10.0000	0.358748
$778$	−10.0000	−0.358517
$779$	0	0
$780$	6.00000	0.214834
$781$	16.0000	0.572525
$782$	−2.00000	−0.0715199
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	10.0000	0.356915
$786$	12.0000	0.428026
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−6.00000	−0.213741
$789$	−16.0000	−0.569615
$790$	−16.0000	−0.569254
$791$	18.0000	0.640006
$792$	−4.00000	−0.142134
$793$	60.0000	2.13066
$794$	−38.0000	−1.34857
$795$	−6.00000	−0.212798
$796$	16.0000	0.567105
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	30.0000	1.05934
$803$	−24.0000	−0.846942
$804$	16.0000	0.564276
$805$	1.00000	0.0352454
$806$	0	0
$807$	−18.0000	−0.633630
$808$	6.00000	0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	1.00000	0.0351364
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	−2.00000	−0.0701862
$813$	16.0000	0.561144
$814$	−40.0000	−1.40200
$815$	−8.00000	−0.280228
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	−26.0000	−0.909069
$819$	−6.00000	−0.209657
$820$	−6.00000	−0.209529
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	−2.00000	−0.0697580
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−16.0000	−0.557386
$825$	−4.00000	−0.139262
$826$	12.0000	0.417533
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	1.00000	0.0347524
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	−4.00000	−0.138842
$831$	−2.00000	−0.0693792
$832$	6.00000	0.208013
$833$	2.00000	0.0692959
$834$	8.00000	0.277017
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	20.0000	0.690889
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	1.00000	0.0345033
$841$	−25.0000	−0.862069
$842$	18.0000	0.620321
$843$	−26.0000	−0.895488
$844$	28.0000	0.963800
$845$	−23.0000	−0.791224
$846$	12.0000	0.412568
$847$	−5.00000	−0.171802
$848$	−6.00000	−0.206041
$849$	4.00000	0.137280
$850$	−2.00000	−0.0685994
$851$	10.0000	0.342796
$852$	−4.00000	−0.137038
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	10.0000	0.342193
$855$	0	0
$856$	12.0000	0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	24.0000	0.819346
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−8.00000	−0.272798
$861$	6.00000	0.204479
$862$	12.0000	0.408722
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	6.00000	0.204006
$866$	−18.0000	−0.611665
$867$	13.0000	0.441503
$868$	0	0
$869$	−64.0000	−2.17105
$870$	−2.00000	−0.0678064
$871$	−96.0000	−3.25284
$872$	10.0000	0.338643
$873$	18.0000	0.609208
$874$	0	0
$875$	1.00000	0.0338062
$876$	6.00000	0.202721
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	40.0000	1.34993
$879$	−2.00000	−0.0674583
$880$	−4.00000	−0.134840
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	−1.00000	−0.0336718
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	12.0000	0.403604
$885$	12.0000	0.403376
$886$	−4.00000	−0.134383
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	10.0000	0.335578
$889$	0	0
$890$	−2.00000	−0.0670402
$891$	4.00000	0.134005
$892$	−16.0000	−0.535720
$893$	0	0
$894$	10.0000	0.334450
$895$	−4.00000	−0.133705
$896$	1.00000	0.0334077
$897$	−6.00000	−0.200334
$898$	22.0000	0.734150
$899$	0	0
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	−24.0000	−0.799113
$903$	8.00000	0.266223
$904$	18.0000	0.598671
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−12.0000	−0.398234
$909$	−6.00000	−0.199007
$910$	−6.00000	−0.198898
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	−18.0000	−0.595387
$915$	10.0000	0.330590
$916$	10.0000	0.330409
$917$	−12.0000	−0.396275
$918$	2.00000	0.0660098
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	1.00000	0.0329690
$921$	−4.00000	−0.131804
$922$	−2.00000	−0.0658665
$923$	24.0000	0.789970
$924$	4.00000	0.131590
$925$	10.0000	0.328798
$926$	24.0000	0.788689
$927$	16.0000	0.525509
$928$	−2.00000	−0.0656532
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	32.0000	1.04763
$934$	−12.0000	−0.392652
$935$	−8.00000	−0.261628
$936$	−6.00000	−0.196116
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	−16.0000	−0.522419
$939$	22.0000	0.717943
$940$	12.0000	0.391397
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	−10.0000	−0.325818
$943$	6.00000	0.195387
$944$	12.0000	0.390567
$945$	−1.00000	−0.0325300
$946$	−32.0000	−1.04041
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	16.0000	0.519656
$949$	−36.0000	−1.16861
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	2.00000	0.0648204
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	6.00000	0.194257
$955$	12.0000	0.388311
$956$	4.00000	0.129369
$957$	−8.00000	−0.258603
$958$	−8.00000	−0.258468
$959$	2.00000	0.0645834
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	−60.0000	−1.93448
$963$	−12.0000	−0.386695
$964$	10.0000	0.322078
$965$	−10.0000	−0.321911
$966$	−1.00000	−0.0321745
$967$	−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
$970$	18.0000	0.577945
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	−1.00000	−0.0320750
$973$	−8.00000	−0.256468
$974$	−24.0000	−0.769010
$975$	−6.00000	−0.192154
$976$	10.0000	0.320092
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	8.00000	0.255812
$979$	−8.00000	−0.255681
$980$	−1.00000	−0.0319438
$981$	−10.0000	−0.319275
$982$	−12.0000	−0.382935
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	−4.00000	−0.127386
$987$	−12.0000	−0.381964
$988$	0	0
$989$	8.00000	0.254385
$990$	4.00000	0.127128
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	4.00000	0.126872
$995$	−16.0000	−0.507234
$996$	4.00000	0.126745
$997$	−50.0000	−1.58352	−0.791758	−	0.610835i	$-0.790834\pi$
$997$	−50.0000	−1.58352	−0.791758	+	0.610835i	$0.790834\pi$
$998$	4.00000	0.126618
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.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:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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