LMFDB - Embedded newform 4830.2.a.i.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} -14.0000 q^{61} +8.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +12.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} -1.00000 q^{72} +10.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} -2.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -1.00000 q^{84} +6.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -10.0000 q^{89} +1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} -8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−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
$26$	−2.00000	−0.392232
$27$	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
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	4.00000	0.696311
$34$	6.00000	1.02899
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	1.00000	0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	2.00000	0.277350
$53$	−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
$57$	−4.00000	−0.529813
$58$	−2.00000	−0.262613
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−1.00000	−0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000	1.01600
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−4.00000	−0.492366
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−6.00000	−0.727607
$69$	1.00000	0.120386
$70$	−1.00000	−0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	−2.00000	−0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−1.00000	−0.109109
$85$	6.00000	0.650791
$86$	4.00000	0.431331
$87$	2.00000	0.214423
$88$	−4.00000	−0.426401
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	1.00000	0.105409
$91$	−2.00000	−0.209657
$92$	1.00000	0.104257
$93$	−8.00000	−0.829561
$94$	0	0
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	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
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	6.00000	0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−2.00000	−0.196116
$105$	1.00000	0.0975900
$106$	6.00000	0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	4.00000	0.381385
$111$	6.00000	0.569495
$112$	−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	$0.728811\pi$
$114$	4.00000	0.374634
$115$	−1.00000	−0.0932505
$116$	2.00000	0.185695
$117$	2.00000	0.184900
$118$	8.00000	0.736460
$119$	6.00000	0.550019
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.00000	0.348155
$133$	4.00000	0.346844
$134$	−12.0000	−1.03664
$135$	−1.00000	−0.0860663
$136$	6.00000	0.514496
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−1.00000	−0.0851257
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	−10.0000	−0.827606
$147$	1.00000	0.0824786
$148$	6.00000	0.493197
$149$	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
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	4.00000	0.322329
$155$	8.00000	0.642575
$156$	2.00000	0.160128
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−4.00000	−0.318223
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	−1.00000	−0.0788110
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	1.00000	0.0771517
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−2.00000	−0.151620
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	−8.00000	−0.601317
$178$	10.0000	0.749532
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	2.00000	0.148250
$183$	−14.0000	−1.03491
$184$	−1.00000	−0.0737210
$185$	−6.00000	−0.441129
$186$	8.00000	0.586588
$187$	−24.0000	−1.75505
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	−4.00000	−0.290191
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	1.00000	0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−2.00000	−0.143592
$195$	−2.00000	−0.143223
$196$	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
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	12.0000	0.846415
$202$	14.0000	0.985037
$203$	−2.00000	−0.140372
$204$	−6.00000	−0.420084
$205$	6.00000	0.419058
$206$	−8.00000	−0.557386
$207$	1.00000	0.0695048
$208$	2.00000	0.138675
$209$	−16.0000	−1.10674
$210$	−1.00000	−0.0690066
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	8.00000	0.543075
$218$	−10.0000	−0.677285
$219$	10.0000	0.675737
$220$	−4.00000	−0.269680
$221$	−12.0000	−0.807207
$222$	−6.00000	−0.402694
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−4.00000	−0.264906
$229$	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
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−8.00000	−0.520756
$237$	4.00000	0.259828
$238$	−6.00000	−0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−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
$244$	−14.0000	−0.896258
$245$	−1.00000	−0.0638877
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	0	0
$250$	1.00000	0.0632456
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	−4.00000	−0.250982
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	4.00000	0.249029
$259$	−6.00000	−0.372822
$260$	−2.00000	−0.124035
$261$	2.00000	0.123797
$262$	0	0
$263$	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
$266$	−4.00000	−0.245256
$267$	−10.0000	−0.611990
$268$	12.0000	0.733017
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	1.00000	0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−6.00000	−0.363803
$273$	−2.00000	−0.121046
$274$	−18.0000	−1.08742
$275$	4.00000	0.241209
$276$	1.00000	0.0601929
$277$	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
$279$	−8.00000	−0.478947
$280$	−1.00000	−0.0597614
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	−8.00000	−0.473050
$287$	6.00000	0.354169
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	2.00000	0.117444
$291$	2.00000	0.117242
$292$	10.0000	0.585206
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	−1.00000	−0.0583212
$295$	8.00000	0.465778
$296$	−6.00000	−0.348743
$297$	4.00000	0.232104
$298$	−10.0000	−0.579284
$299$	2.00000	0.115663
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	16.0000	0.920697
$303$	−14.0000	−0.804279
$304$	−4.00000	−0.229416
$305$	14.0000	0.801638
$306$	6.00000	0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−4.00000	−0.227921
$309$	8.00000	0.455104
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−2.00000	−0.113228
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−14.0000	−0.790066
$315$	1.00000	0.0563436
$316$	4.00000	0.225018
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	6.00000	0.336463
$319$	8.00000	0.447914
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	1.00000	0.0557278
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−4.00000	−0.221540
$327$	10.0000	0.553001
$328$	6.00000	0.331295
$329$	0	0
$330$	4.00000	0.220193
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	24.0000	1.31322
$335$	−12.0000	−0.655630
$336$	−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	$0.482652\pi$
$338$	9.00000	0.489535
$339$	−14.0000	−0.760376
$340$	6.00000	0.325396
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	−1.00000	−0.0538382
$346$	14.0000	0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	1.00000	0.0534522
$351$	2.00000	0.106752
$352$	−4.00000	−0.213201
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	8.00000	0.425195
$355$	0	0
$356$	−10.0000	−0.529999
$357$	6.00000	0.317554
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	14.0000	0.735824
$363$	5.00000	0.262432
$364$	−2.00000	−0.104828
$365$	−10.0000	−0.523424
$366$	14.0000	0.731792
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	1.00000	0.0521286
$369$	−6.00000	−0.312348
$370$	6.00000	0.311925
$371$	6.00000	0.311504
$372$	−8.00000	−0.414781
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	24.0000	1.24101
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	1.00000	0.0514344
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	4.00000	0.205196
$381$	4.00000	0.204926
$382$	16.0000	0.818631
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	4.00000	0.203859
$386$	6.00000	0.305392
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	2.00000	0.101274
$391$	−6.00000	−0.303433
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−4.00000	−0.201262
$396$	4.00000	0.201008
$397$	−6.00000	−0.301131	−0.150566	−	0.988600i	$-0.548110\pi$
$397$	−6.00000	−0.301131	−0.150566	+	0.988600i	$0.548110\pi$
$398$	4.00000	0.200502
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	−12.0000	−0.598506
$403$	−16.0000	−0.797017
$404$	−14.0000	−0.696526
$405$	−1.00000	−0.0496904
$406$	2.00000	0.0992583
$407$	24.0000	1.18964
$408$	6.00000	0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−6.00000	−0.296319
$411$	18.0000	0.887875
$412$	8.00000	0.394132
$413$	8.00000	0.393654
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−12.0000	−0.587643
$418$	16.0000	0.782586
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	1.00000	0.0487950
$421$	−30.0000	−1.46211	−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000	−1.46211	−0.731055	+	0.682318i	$0.760972\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	0	0
$427$	14.0000	0.677507
$428$	0	0
$429$	8.00000	0.386244
$430$	−4.00000	−0.192897
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−8.00000	−0.384012
$435$	−2.00000	−0.0958927
$436$	10.0000	0.478913
$437$	−4.00000	−0.191346
$438$	−10.0000	−0.477818
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	4.00000	0.190693
$441$	1.00000	0.0476190
$442$	12.0000	0.570782
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	6.00000	0.284747
$445$	10.0000	0.474045
$446$	4.00000	0.189405
$447$	10.0000	0.472984
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−1.00000	−0.0471405
$451$	−24.0000	−1.13012
$452$	−14.0000	−0.658505
$453$	−16.0000	−0.751746
$454$	24.0000	1.12638
$455$	2.00000	0.0937614
$456$	4.00000	0.187317
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	−10.0000	−0.467269
$459$	−6.00000	−0.280056
$460$	−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
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	2.00000	0.0928477
$465$	8.00000	0.370991
$466$	10.0000	0.463241
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	2.00000	0.0924500
$469$	−12.0000	−0.554109
$470$	0	0
$471$	14.0000	0.645086
$472$	8.00000	0.368230
$473$	−16.0000	−0.735681
$474$	−4.00000	−0.183726
$475$	−4.00000	−0.183533
$476$	6.00000	0.275010
$477$	−6.00000	−0.274721
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	1.00000	0.0456435
$481$	12.0000	0.547153
$482$	−10.0000	−0.455488
$483$	−1.00000	−0.0455016
$484$	5.00000	0.227273
$485$	−2.00000	−0.0908153
$486$	−1.00000	−0.0453609
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	14.0000	0.633750
$489$	4.00000	0.180886
$490$	1.00000	0.0451754
$491$	32.0000	1.44414	0.722070	−	0.691820i	$-0.243191\pi$
$491$	32.0000	1.44414	0.722070	+	0.691820i	$0.243191\pi$
$492$	−6.00000	−0.270501
$493$	−12.0000	−0.540453
$494$	8.00000	0.359937
$495$	−4.00000	−0.179787
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	−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
$501$	−24.0000	−1.07224
$502$	4.00000	0.178529
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	1.00000	0.0445435
$505$	14.0000	0.622992
$506$	−4.00000	−0.177822
$507$	−9.00000	−0.399704
$508$	4.00000	0.177471
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	−6.00000	−0.265684
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	2.00000	0.0882162
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	0	0
$518$	6.00000	0.263625
$519$	−14.0000	−0.614532
$520$	2.00000	0.0877058
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	−2.00000	−0.0875376
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	−16.0000	−0.697633
$527$	48.0000	2.09091
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	−8.00000	−0.347170
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	10.0000	0.432742
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	−26.0000	−1.12094
$539$	4.00000	0.172292
$540$	−1.00000	−0.0430331
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−8.00000	−0.343629
$543$	−14.0000	−0.600798
$544$	6.00000	0.257248
$545$	−10.0000	−0.428353
$546$	2.00000	0.0855921
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	18.0000	0.768922
$549$	−14.0000	−0.597505
$550$	−4.00000	−0.170561
$551$	−8.00000	−0.340811
$552$	−1.00000	−0.0425628
$553$	−4.00000	−0.170097
$554$	−10.0000	−0.424859
$555$	−6.00000	−0.254686
$556$	−12.0000	−0.508913
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	8.00000	0.338667
$559$	−8.00000	−0.338364
$560$	1.00000	0.0422577
$561$	−24.0000	−1.01328
$562$	2.00000	0.0843649
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	−20.0000	−0.840663
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	−4.00000	−0.167542
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	8.00000	0.334497
$573$	−16.0000	−0.668410
$574$	−6.00000	−0.250435
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−19.0000	−0.790296
$579$	−6.00000	−0.249351
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	−2.00000	−0.0829027
$583$	−24.0000	−0.993978
$584$	−10.0000	−0.413803
$585$	−2.00000	−0.0826898
$586$	30.0000	1.23929
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	1.00000	0.0412393
$589$	32.0000	1.31854
$590$	−8.00000	−0.329355
$591$	−6.00000	−0.246807
$592$	6.00000	0.246598
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−4.00000	−0.164122
$595$	−6.00000	−0.245976
$596$	10.0000	0.409616
$597$	−4.00000	−0.163709
$598$	−2.00000	−0.0817861
$599$	−48.0000	−1.96123	−0.980613	−	0.195952i	$-0.937220\pi$
$599$	−48.0000	−1.96123	−0.980613	+	0.195952i	$0.937220\pi$
$600$	−1.00000	−0.0408248
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−4.00000	−0.163028
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	−5.00000	−0.203279
$606$	14.0000	0.568711
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	4.00000	0.162221
$609$	−2.00000	−0.0810441
$610$	−14.0000	−0.566843
$611$	0	0
$612$	−6.00000	−0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	20.0000	0.807134
$615$	6.00000	0.241943
$616$	4.00000	0.161165
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	−8.00000	−0.321807
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	8.00000	0.321288
$621$	1.00000	0.0401286
$622$	0	0
$623$	10.0000	0.400642
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	−16.0000	−0.638978
$628$	14.0000	0.558661
$629$	−36.0000	−1.43541
$630$	−1.00000	−0.0398410
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−4.00000	−0.159111
$633$	−20.0000	−0.794929
$634$	30.0000	1.19145
$635$	−4.00000	−0.158735
$636$	−6.00000	−0.237915
$637$	2.00000	0.0792429
$638$	−8.00000	−0.316723
$639$	0	0
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−1.00000	−0.0394055
$645$	4.00000	0.157500
$646$	−24.0000	−0.944267
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	−1.00000	−0.0392837
$649$	−32.0000	−1.25611
$650$	−2.00000	−0.0784465
$651$	8.00000	0.313545
$652$	4.00000	0.156652
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	−6.00000	−0.234261
$657$	10.0000	0.390137
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−4.00000	−0.155700
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	20.0000	0.777322
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−4.00000	−0.155113
$666$	−6.00000	−0.232495
$667$	2.00000	0.0774403
$668$	−24.0000	−0.928588
$669$	−4.00000	−0.154649
$670$	12.0000	0.463600
$671$	−56.0000	−2.16186
$672$	1.00000	0.0385758
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−2.00000	−0.0770371
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	14.0000	0.537667
$679$	−2.00000	−0.0767530
$680$	−6.00000	−0.230089
$681$	−24.0000	−0.919682
$682$	32.0000	1.22534
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−4.00000	−0.152944
$685$	−18.0000	−0.687745
$686$	1.00000	0.0381802
$687$	10.0000	0.381524
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	1.00000	0.0380693
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−14.0000	−0.532200
$693$	−4.00000	−0.151947
$694$	12.0000	0.455514
$695$	12.0000	0.455186
$696$	−2.00000	−0.0758098
$697$	36.0000	1.36360
$698$	10.0000	0.378506
$699$	−10.0000	−0.378235
$700$	−1.00000	−0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	−2.00000	−0.0754851
$703$	−24.0000	−0.905177
$704$	4.00000	0.150756
$705$	0	0
$706$	−14.0000	−0.526897
$707$	14.0000	0.526524
$708$	−8.00000	−0.300658
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	10.0000	0.374766
$713$	−8.00000	−0.299602
$714$	−6.00000	−0.224544
$715$	−8.00000	−0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−1.00000	−0.0372678
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	10.0000	0.371904
$724$	−14.0000	−0.520306
$725$	2.00000	0.0742781
$726$	−5.00000	−0.185567
$727$	48.0000	1.78022	0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000	1.78022	0.890111	+	0.455744i	$0.150627\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	24.0000	0.887672
$732$	−14.0000	−0.517455
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−24.0000	−0.885856
$735$	−1.00000	−0.0368856
$736$	−1.00000	−0.0368605
$737$	48.0000	1.76810
$738$	6.00000	0.220863
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	−6.00000	−0.220564
$741$	−8.00000	−0.293887
$742$	−6.00000	−0.220267
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	8.00000	0.293294
$745$	−10.0000	−0.366372
$746$	−14.0000	−0.512576
$747$	0	0
$748$	−24.0000	−0.877527
$749$	0	0
$750$	1.00000	0.0365148
$751$	52.0000	1.89751	0.948753	−	0.316017i	$-0.102346\pi$
$751$	52.0000	1.89751	0.948753	+	0.316017i	$0.102346\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	−4.00000	−0.145671
$755$	16.0000	0.582300
$756$	−1.00000	−0.0363696
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	4.00000	0.145287
$759$	4.00000	0.145191
$760$	−4.00000	−0.145095
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−4.00000	−0.144905
$763$	−10.0000	−0.362024
$764$	−16.0000	−0.578860
$765$	6.00000	0.216930
$766$	−16.0000	−0.578103
$767$	−16.0000	−0.577727
$768$	1.00000	0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	−4.00000	−0.144150
$771$	−2.00000	−0.0720282
$772$	−6.00000	−0.215945
$773$	50.0000	1.79838	0.899188	−	0.437564i	$-0.144158\pi$
$773$	50.0000	1.79838	0.899188	+	0.437564i	$0.144158\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	−2.00000	−0.0717958
$777$	−6.00000	−0.215249
$778$	−10.0000	−0.358517
$779$	24.0000	0.859889
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	6.00000	0.214560
$783$	2.00000	0.0714742
$784$	1.00000	0.0357143
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	−6.00000	−0.213741
$789$	16.0000	0.569615
$790$	4.00000	0.142314
$791$	14.0000	0.497783
$792$	−4.00000	−0.142134
$793$	−28.0000	−0.994309
$794$	6.00000	0.212932
$795$	6.00000	0.212798
$796$	−4.00000	−0.141776
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−10.0000	−0.353333
$802$	26.0000	0.918092
$803$	40.0000	1.41157
$804$	12.0000	0.423207
$805$	1.00000	0.0352454
$806$	16.0000	0.563576
$807$	26.0000	0.915243
$808$	14.0000	0.492518
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	1.00000	0.0351364
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−2.00000	−0.0701862
$813$	8.00000	0.280572
$814$	−24.0000	−0.841200
$815$	−4.00000	−0.140114
$816$	−6.00000	−0.210042
$817$	16.0000	0.559769
$818$	−10.0000	−0.349642
$819$	−2.00000	−0.0698857
$820$	6.00000	0.209529
$821$	26.0000	0.907406	0.453703	−	0.891153i	$-0.350103\pi$
$821$	26.0000	0.907406	0.453703	+	0.891153i	$0.350103\pi$
$822$	−18.0000	−0.627822
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	−8.00000	−0.278693
$825$	4.00000	0.139262
$826$	−8.00000	−0.278356
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	1.00000	0.0347524
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	2.00000	0.0693375
$833$	−6.00000	−0.207888
$834$	12.0000	0.415526
$835$	24.0000	0.830554
$836$	−16.0000	−0.553372
$837$	−8.00000	−0.276520
$838$	−12.0000	−0.414533
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−1.00000	−0.0345033
$841$	−25.0000	−0.862069
$842$	30.0000	1.03387
$843$	−2.00000	−0.0688837
$844$	−20.0000	−0.688428
$845$	9.00000	0.309609
$846$	0	0
$847$	−5.00000	−0.171802
$848$	−6.00000	−0.206041
$849$	20.0000	0.686398
$850$	6.00000	0.205798
$851$	6.00000	0.205677
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	−14.0000	−0.479070
$855$	4.00000	0.136797
$856$	0	0
$857$	46.0000	1.57133	0.785665	−	0.618652i	$-0.212321\pi$
$857$	46.0000	1.57133	0.785665	+	0.618652i	$0.212321\pi$
$858$	−8.00000	−0.273115
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	4.00000	0.136399
$861$	6.00000	0.204479
$862$	16.0000	0.544962
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	−1.00000	−0.0340207
$865$	14.0000	0.476014
$866$	−2.00000	−0.0679628
$867$	19.0000	0.645274
$868$	8.00000	0.271538
$869$	16.0000	0.542763
$870$	2.00000	0.0678064
$871$	24.0000	0.813209
$872$	−10.0000	−0.338643
$873$	2.00000	0.0676897
$874$	4.00000	0.135302
$875$	1.00000	0.0338062
$876$	10.0000	0.337869
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−24.0000	−0.809961
$879$	−30.0000	−1.01187
$880$	−4.00000	−0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−1.00000	−0.0336718
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−12.0000	−0.403604
$885$	8.00000	0.268917
$886$	36.0000	1.20944
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−6.00000	−0.201347
$889$	−4.00000	−0.134156
$890$	−10.0000	−0.335201
$891$	4.00000	0.134005
$892$	−4.00000	−0.133930
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	1.00000	0.0334077
$897$	2.00000	0.0667781
$898$	6.00000	0.200223
$899$	−16.0000	−0.533630
$900$	1.00000	0.0333333
$901$	36.0000	1.19933
$902$	24.0000	0.799113
$903$	4.00000	0.133112
$904$	14.0000	0.465633
$905$	14.0000	0.465376
$906$	16.0000	0.531564
$907$	52.0000	1.72663	0.863316	−	0.504664i	$-0.168384\pi$
$907$	52.0000	1.72663	0.863316	+	0.504664i	$0.168384\pi$
$908$	−24.0000	−0.796468
$909$	−14.0000	−0.464351
$910$	−2.00000	−0.0662994
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	6.00000	0.198462
$915$	14.0000	0.462826
$916$	10.0000	0.330409
$917$	0	0
$918$	6.00000	0.198030
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	1.00000	0.0329690
$921$	−20.0000	−0.659022
$922$	−2.00000	−0.0658665
$923$	0	0
$924$	−4.00000	−0.131590
$925$	6.00000	0.197279
$926$	−4.00000	−0.131448
$927$	8.00000	0.262754
$928$	−2.00000	−0.0656532
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−8.00000	−0.262330
$931$	−4.00000	−0.131095
$932$	−10.0000	−0.327561
$933$	0	0
$934$	24.0000	0.785304
$935$	24.0000	0.784884
$936$	−2.00000	−0.0653720
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	12.0000	0.391814
$939$	−6.00000	−0.195803
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	−14.0000	−0.456145
$943$	−6.00000	−0.195387
$944$	−8.00000	−0.260378
$945$	1.00000	0.0325300
$946$	16.0000	0.520205
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	4.00000	0.129914
$949$	20.0000	0.649227
$950$	4.00000	0.129777
$951$	−30.0000	−0.972817
$952$	−6.00000	−0.194461
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	6.00000	0.194257
$955$	16.0000	0.517748
$956$	0	0
$957$	8.00000	0.258603
$958$	−24.0000	−0.775405
$959$	−18.0000	−0.581250
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	−12.0000	−0.386896
$963$	0	0
$964$	10.0000	0.322078
$965$	6.00000	0.193147
$966$	1.00000	0.0321745
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	−5.00000	−0.160706
$969$	24.0000	0.770991
$970$	2.00000	0.0642161
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	1.00000	0.0320750
$973$	12.0000	0.384702
$974$	−12.0000	−0.384505
$975$	2.00000	0.0640513
$976$	−14.0000	−0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−4.00000	−0.127906
$979$	−40.0000	−1.27841
$980$	−1.00000	−0.0319438
$981$	10.0000	0.319275
$982$	−32.0000	−1.02116
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	12.0000	0.382158
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−4.00000	−0.127193
$990$	4.00000	0.127128
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000	0.254000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	4.00000	0.126618
$999$	6.00000	0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.i.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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