LMFDB - Embedded newform 4830.2.a.m.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -10.0000 q^{29} -1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +4.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} +4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} +1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} +4.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{56} +2.00000 q^{57} +10.0000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} -4.00000 q^{68} -1.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +4.00000 q^{77} -4.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +6.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +8.00000 q^{86} -10.0000 q^{87} +4.00000 q^{88} +4.00000 q^{89} -1.00000 q^{90} -4.00000 q^{91} -1.00000 q^{92} +2.00000 q^{93} -6.00000 q^{94} +2.00000 q^{95} -1.00000 q^{96} -4.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	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
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	−1.00000	−0.182574
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	4.00000	0.685994
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−2.00000	−0.324443
$39$	4.00000	0.640513
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	1.00000	0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−4.00000	−0.603023
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−4.00000	−0.560112
$52$	4.00000	0.554700
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	−4.00000	−0.539360
$56$	1.00000	0.133631
$57$	2.00000	0.264906
$58$	10.0000	1.31306
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	1.00000	0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−2.00000	−0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	4.00000	0.492366
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−4.00000	−0.485071
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	2.00000	0.229416
$77$	4.00000	0.455842
$78$	−4.00000	−0.452911
$79$	−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
$82$	2.00000	0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−1.00000	−0.109109
$85$	−4.00000	−0.433861
$86$	8.00000	0.862662
$87$	−10.0000	−1.07211
$88$	4.00000	0.426401
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	−1.00000	−0.105409
$91$	−4.00000	−0.419314
$92$	−1.00000	−0.104257
$93$	2.00000	0.207390
$94$	−6.00000	−0.618853
$95$	2.00000	0.205196
$96$	−1.00000	−0.102062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−1.00000	−0.101015
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	4.00000	0.396059
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−4.00000	−0.392232
$105$	−1.00000	−0.0975900
$106$	10.0000	0.971286
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	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
$111$	2.00000	0.189832
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−2.00000	−0.187317
$115$	−1.00000	−0.0932505
$116$	−10.0000	−0.928477
$117$	4.00000	0.369800
$118$	4.00000	0.368230
$119$	4.00000	0.366679
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	−2.00000	−0.180334
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	−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
$129$	−8.00000	−0.704361
$130$	−4.00000	−0.350823
$131$	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
$133$	−2.00000	−0.173422
$134$	0	0
$135$	1.00000	0.0860663
$136$	4.00000	0.342997
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	1.00000	0.0851257
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−1.00000	−0.0845154
$141$	6.00000	0.505291
$142$	−8.00000	−0.671345
$143$	−16.0000	−1.33799
$144$	1.00000	0.0833333
$145$	−10.0000	−0.830455
$146$	2.00000	0.165521
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−1.00000	−0.0816497
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−2.00000	−0.162221
$153$	−4.00000	−0.323381
$154$	−4.00000	−0.322329
$155$	2.00000	0.160644
$156$	4.00000	0.320256
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	16.0000	1.27289
$159$	−10.0000	−0.793052
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	−2.00000	−0.156174
$165$	−4.00000	−0.311400
$166$	−6.00000	−0.465690
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	1.00000	0.0771517
$169$	3.00000	0.230769
$170$	4.00000	0.306786
$171$	2.00000	0.152944
$172$	−8.00000	−0.609994
$173$	−20.0000	−1.52057	−0.760286	−	0.649589i	$-0.774941\pi$
$173$	−20.0000	−1.52057	−0.760286	+	0.649589i	$0.774941\pi$
$174$	10.0000	0.758098
$175$	−1.00000	−0.0755929
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	−4.00000	−0.299813
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	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	$0.674182\pi$
$182$	4.00000	0.296500
$183$	2.00000	0.147844
$184$	1.00000	0.0737210
$185$	2.00000	0.147043
$186$	−2.00000	−0.146647
$187$	16.0000	1.17004
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	−2.00000	−0.145095
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	1.00000	0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	4.00000	0.287183
$195$	4.00000	0.286446
$196$	1.00000	0.0714286
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	10.0000	0.701862
$204$	−4.00000	−0.280056
$205$	−2.00000	−0.139686
$206$	4.00000	0.278693
$207$	−1.00000	−0.0695048
$208$	4.00000	0.277350
$209$	−8.00000	−0.553372
$210$	1.00000	0.0690066
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−10.0000	−0.686803
$213$	8.00000	0.548151
$214$	8.00000	0.546869
$215$	−8.00000	−0.545595
$216$	−1.00000	−0.0680414
$217$	−2.00000	−0.135769
$218$	10.0000	0.677285
$219$	−2.00000	−0.135147
$220$	−4.00000	−0.269680
$221$	−16.0000	−1.07628
$222$	−2.00000	−0.134231
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	2.00000	0.132453
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	1.00000	0.0659380
$231$	4.00000	0.263181
$232$	10.0000	0.656532
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−4.00000	−0.261488
$235$	6.00000	0.391397
$236$	−4.00000	−0.260378
$237$	−16.0000	−1.03931
$238$	−4.00000	−0.259281
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	1.00000	0.0645497
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	1.00000	0.0638877
$246$	2.00000	0.127515
$247$	8.00000	0.509028
$248$	−2.00000	−0.127000
$249$	6.00000	0.380235
$250$	−1.00000	−0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	−1.00000	−0.0629941
$253$	4.00000	0.251478
$254$	8.00000	0.501965
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	8.00000	0.498058
$259$	−2.00000	−0.124274
$260$	4.00000	0.248069
$261$	−10.0000	−0.618984
$262$	−12.0000	−0.741362
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	4.00000	0.246183
$265$	−10.0000	−0.614295
$266$	2.00000	0.122628
$267$	4.00000	0.244796
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	−1.00000	−0.0608581
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	−4.00000	−0.242536
$273$	−4.00000	−0.242091
$274$	18.0000	1.08742
$275$	−4.00000	−0.241209
$276$	−1.00000	−0.0601929
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	1.00000	0.0597614
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	−6.00000	−0.357295
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	8.00000	0.474713
$285$	2.00000	0.118470
$286$	16.0000	0.946100
$287$	2.00000	0.118056
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	10.0000	0.587220
$291$	−4.00000	−0.234484
$292$	−2.00000	−0.117041
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	−1.00000	−0.0583212
$295$	−4.00000	−0.232889
$296$	−2.00000	−0.116248
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	−4.00000	−0.231326
$300$	1.00000	0.0577350
$301$	8.00000	0.461112
$302$	4.00000	0.230174
$303$	0	0
$304$	2.00000	0.114708
$305$	2.00000	0.114520
$306$	4.00000	0.228665
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	4.00000	0.227921
$309$	−4.00000	−0.227552
$310$	−2.00000	−0.113592
$311$	10.0000	0.567048	0.283524	−	0.958965i	$-0.408496\pi$
$311$	10.0000	0.567048	0.283524	+	0.958965i	$0.408496\pi$
$312$	−4.00000	−0.226455
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−22.0000	−1.24153
$315$	−1.00000	−0.0563436
$316$	−16.0000	−0.900070
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	10.0000	0.560772
$319$	40.0000	2.23957
$320$	1.00000	0.0559017
$321$	−8.00000	−0.446516
$322$	−1.00000	−0.0557278
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−20.0000	−1.10770
$327$	−10.0000	−0.553001
$328$	2.00000	0.110432
$329$	−6.00000	−0.330791
$330$	4.00000	0.220193
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	6.00000	0.329293
$333$	2.00000	0.109599
$334$	6.00000	0.328305
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−3.00000	−0.163178
$339$	−6.00000	−0.325875
$340$	−4.00000	−0.216930
$341$	−8.00000	−0.433224
$342$	−2.00000	−0.108148
$343$	−1.00000	−0.0539949
$344$	8.00000	0.431331
$345$	−1.00000	−0.0538382
$346$	20.0000	1.07521
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−10.0000	−0.536056
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	1.00000	0.0534522
$351$	4.00000	0.213504
$352$	4.00000	0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	4.00000	0.212598
$355$	8.00000	0.424596
$356$	4.00000	0.212000
$357$	4.00000	0.211702
$358$	12.0000	0.634220
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	14.0000	0.735824
$363$	5.00000	0.262432
$364$	−4.00000	−0.209657
$365$	−2.00000	−0.104685
$366$	−2.00000	−0.104542
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	−2.00000	−0.103975
$371$	10.0000	0.519174
$372$	2.00000	0.103695
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−16.0000	−0.827340
$375$	1.00000	0.0516398
$376$	−6.00000	−0.309426
$377$	−40.0000	−2.06010
$378$	1.00000	0.0514344
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	2.00000	0.102598
$381$	−8.00000	−0.409852
$382$	−8.00000	−0.409316
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−1.00000	−0.0510310
$385$	4.00000	0.203859
$386$	18.0000	0.916176
$387$	−8.00000	−0.406663
$388$	−4.00000	−0.203069
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−4.00000	−0.202548
$391$	4.00000	0.202289
$392$	−1.00000	−0.0505076
$393$	12.0000	0.605320
$394$	18.0000	0.906827
$395$	−16.0000	−0.805047
$396$	−4.00000	−0.201008
$397$	−12.0000	−0.602263	−0.301131	−	0.953583i	$-0.597364\pi$
$397$	−12.0000	−0.602263	−0.301131	+	0.953583i	$0.597364\pi$
$398$	4.00000	0.200502
$399$	−2.00000	−0.100125
$400$	1.00000	0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	1.00000	0.0496904
$406$	−10.0000	−0.496292
$407$	−8.00000	−0.396545
$408$	4.00000	0.198030
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	2.00000	0.0987730
$411$	−18.0000	−0.887875
$412$	−4.00000	−0.197066
$413$	4.00000	0.196827
$414$	1.00000	0.0491473
$415$	6.00000	0.294528
$416$	−4.00000	−0.196116
$417$	0	0
$418$	8.00000	0.391293
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	−1.00000	−0.0487950
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−20.0000	−0.973585
$423$	6.00000	0.291730
$424$	10.0000	0.485643
$425$	−4.00000	−0.194029
$426$	−8.00000	−0.387601
$427$	−2.00000	−0.0967868
$428$	−8.00000	−0.386695
$429$	−16.0000	−0.772487
$430$	8.00000	0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	2.00000	0.0960031
$435$	−10.0000	−0.479463
$436$	−10.0000	−0.478913
$437$	−2.00000	−0.0956730
$438$	2.00000	0.0955637
$439$	6.00000	0.286364	0.143182	−	0.989696i	$-0.454267\pi$
$439$	6.00000	0.286364	0.143182	+	0.989696i	$0.454267\pi$
$440$	4.00000	0.190693
$441$	1.00000	0.0476190
$442$	16.0000	0.761042
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	2.00000	0.0949158
$445$	4.00000	0.189618
$446$	22.0000	1.04173
$447$	−6.00000	−0.283790
$448$	−1.00000	−0.0472456
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	−1.00000	−0.0471405
$451$	8.00000	0.376705
$452$	−6.00000	−0.282216
$453$	−4.00000	−0.187936
$454$	−18.0000	−0.844782
$455$	−4.00000	−0.187523
$456$	−2.00000	−0.0936586
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−2.00000	−0.0934539
$459$	−4.00000	−0.186704
$460$	−1.00000	−0.0466252
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	−4.00000	−0.186097
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	−10.0000	−0.464238
$465$	2.00000	0.0927478
$466$	10.0000	0.463241
$467$	−14.0000	−0.647843	−0.323921	−	0.946084i	$-0.605001\pi$
$467$	−14.0000	−0.647843	−0.323921	+	0.946084i	$0.605001\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	−6.00000	−0.276759
$471$	22.0000	1.01371
$472$	4.00000	0.184115
$473$	32.0000	1.47136
$474$	16.0000	0.734904
$475$	2.00000	0.0917663
$476$	4.00000	0.183340
$477$	−10.0000	−0.457869
$478$	−8.00000	−0.365911
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	−1.00000	−0.0456435
$481$	8.00000	0.364769
$482$	−4.00000	−0.182195
$483$	1.00000	0.0455016
$484$	5.00000	0.227273
$485$	−4.00000	−0.181631
$486$	−1.00000	−0.0453609
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	−2.00000	−0.0905357
$489$	20.0000	0.904431
$490$	−1.00000	−0.0451754
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−2.00000	−0.0901670
$493$	40.0000	1.80151
$494$	−8.00000	−0.359937
$495$	−4.00000	−0.179787
$496$	2.00000	0.0898027
$497$	−8.00000	−0.358849
$498$	−6.00000	−0.268866
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	1.00000	0.0447214
$501$	−6.00000	−0.268060
$502$	−18.0000	−0.803379
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	−4.00000	−0.177822
$507$	3.00000	0.133235
$508$	−8.00000	−0.354943
$509$	16.0000	0.709188	0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000	0.709188	0.354594	+	0.935020i	$0.384619\pi$
$510$	4.00000	0.177123
$511$	2.00000	0.0884748
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	−10.0000	−0.441081
$515$	−4.00000	−0.176261
$516$	−8.00000	−0.352180
$517$	−24.0000	−1.05552
$518$	2.00000	0.0878750
$519$	−20.0000	−0.877903
$520$	−4.00000	−0.175412
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	10.0000	0.437688
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	12.0000	0.524222
$525$	−1.00000	−0.0436436
$526$	8.00000	0.348817
$527$	−8.00000	−0.348485
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	10.0000	0.434372
$531$	−4.00000	−0.173585
$532$	−2.00000	−0.0867110
$533$	−8.00000	−0.346518
$534$	−4.00000	−0.173097
$535$	−8.00000	−0.345870
$536$	0	0
$537$	−12.0000	−0.517838
$538$	−16.0000	−0.689809
$539$	−4.00000	−0.172292
$540$	1.00000	0.0430331
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	−10.0000	−0.429537
$543$	−14.0000	−0.600798
$544$	4.00000	0.171499
$545$	−10.0000	−0.428353
$546$	4.00000	0.171184
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−18.0000	−0.768922
$549$	2.00000	0.0853579
$550$	4.00000	0.170561
$551$	−20.0000	−0.852029
$552$	1.00000	0.0425628
$553$	16.0000	0.680389
$554$	−18.0000	−0.764747
$555$	2.00000	0.0848953
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−2.00000	−0.0846668
$559$	−32.0000	−1.35346
$560$	−1.00000	−0.0422577
$561$	16.0000	0.675521
$562$	−2.00000	−0.0843649
$563$	30.0000	1.26435	0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000	1.26435	0.632175	+	0.774826i	$0.282163\pi$
$564$	6.00000	0.252646
$565$	−6.00000	−0.252422
$566$	2.00000	0.0840663
$567$	−1.00000	−0.0419961
$568$	−8.00000	−0.335673
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	−2.00000	−0.0837708
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	−16.0000	−0.668994
$573$	8.00000	0.334205
$574$	−2.00000	−0.0834784
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−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
$579$	−18.0000	−0.748054
$580$	−10.0000	−0.415227
$581$	−6.00000	−0.248922
$582$	4.00000	0.165805
$583$	40.0000	1.65663
$584$	2.00000	0.0827606
$585$	4.00000	0.165380
$586$	18.0000	0.743573
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	1.00000	0.0412393
$589$	4.00000	0.164817
$590$	4.00000	0.164677
$591$	−18.0000	−0.740421
$592$	2.00000	0.0821995
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	4.00000	0.164122
$595$	4.00000	0.163984
$596$	−6.00000	−0.245770
$597$	−4.00000	−0.163709
$598$	4.00000	0.163572
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	−1.00000	−0.0408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	−4.00000	−0.162758
$605$	5.00000	0.203279
$606$	0	0
$607$	−18.0000	−0.730597	−0.365299	−	0.930890i	$-0.619033\pi$
$607$	−18.0000	−0.730597	−0.365299	+	0.930890i	$0.619033\pi$
$608$	−2.00000	−0.0811107
$609$	10.0000	0.405220
$610$	−2.00000	−0.0809776
$611$	24.0000	0.970936
$612$	−4.00000	−0.161690
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	8.00000	0.322854
$615$	−2.00000	−0.0806478
$616$	−4.00000	−0.161165
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	4.00000	0.160904
$619$	−2.00000	−0.0803868	−0.0401934	−	0.999192i	$-0.512797\pi$
$619$	−2.00000	−0.0803868	−0.0401934	+	0.999192i	$0.512797\pi$
$620$	2.00000	0.0803219
$621$	−1.00000	−0.0401286
$622$	−10.0000	−0.400963
$623$	−4.00000	−0.160257
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	−8.00000	−0.319744
$627$	−8.00000	−0.319489
$628$	22.0000	0.877896
$629$	−8.00000	−0.318981
$630$	1.00000	0.0398410
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	16.0000	0.636446
$633$	20.0000	0.794929
$634$	18.0000	0.714871
$635$	−8.00000	−0.317470
$636$	−10.0000	−0.396526
$637$	4.00000	0.158486
$638$	−40.0000	−1.58362
$639$	8.00000	0.316475
$640$	−1.00000	−0.0395285
$641$	−14.0000	−0.552967	−0.276483	−	0.961019i	$-0.589169\pi$
$641$	−14.0000	−0.552967	−0.276483	+	0.961019i	$0.589169\pi$
$642$	8.00000	0.315735
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	1.00000	0.0394055
$645$	−8.00000	−0.315000
$646$	8.00000	0.314756
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	−1.00000	−0.0392837
$649$	16.0000	0.628055
$650$	−4.00000	−0.156893
$651$	−2.00000	−0.0783862
$652$	20.0000	0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	10.0000	0.391031
$655$	12.0000	0.468879
$656$	−2.00000	−0.0780869
$657$	−2.00000	−0.0780274
$658$	6.00000	0.233904
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	−4.00000	−0.155700
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−4.00000	−0.155464
$663$	−16.0000	−0.621389
$664$	−6.00000	−0.232845
$665$	−2.00000	−0.0775567
$666$	−2.00000	−0.0774984
$667$	10.0000	0.387202
$668$	−6.00000	−0.232147
$669$	−22.0000	−0.850569
$670$	0	0
$671$	−8.00000	−0.308837
$672$	1.00000	0.0385758
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	22.0000	0.847408
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	6.00000	0.230429
$679$	4.00000	0.153506
$680$	4.00000	0.153393
$681$	18.0000	0.689761
$682$	8.00000	0.306336
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	2.00000	0.0764719
$685$	−18.0000	−0.687745
$686$	1.00000	0.0381802
$687$	2.00000	0.0763048
$688$	−8.00000	−0.304997
$689$	−40.0000	−1.52388
$690$	1.00000	0.0380693
$691$	40.0000	1.52167	0.760836	−	0.648944i	$-0.224789\pi$
$691$	40.0000	1.52167	0.760836	+	0.648944i	$0.224789\pi$
$692$	−20.0000	−0.760286
$693$	4.00000	0.151947
$694$	4.00000	0.151838
$695$	0	0
$696$	10.0000	0.379049
$697$	8.00000	0.303022
$698$	−20.0000	−0.757011
$699$	−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	$0.691719\pi$
$702$	−4.00000	−0.150970
$703$	4.00000	0.150863
$704$	−4.00000	−0.150756
$705$	6.00000	0.225973
$706$	−6.00000	−0.225813
$707$	0	0
$708$	−4.00000	−0.150329
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−8.00000	−0.300235
$711$	−16.0000	−0.600047
$712$	−4.00000	−0.149906
$713$	−2.00000	−0.0749006
$714$	−4.00000	−0.149696
$715$	−16.0000	−0.598366
$716$	−12.0000	−0.448461
$717$	8.00000	0.298765
$718$	24.0000	0.895672
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	1.00000	0.0372678
$721$	4.00000	0.148968
$722$	15.0000	0.558242
$723$	4.00000	0.148762
$724$	−14.0000	−0.520306
$725$	−10.0000	−0.371391
$726$	−5.00000	−0.185567
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	4.00000	0.148250
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	32.0000	1.18356
$732$	2.00000	0.0739221
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	−28.0000	−1.03350
$735$	1.00000	0.0368856
$736$	1.00000	0.0368605
$737$	0	0
$738$	2.00000	0.0736210
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	2.00000	0.0735215
$741$	8.00000	0.293887
$742$	−10.0000	−0.367112
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	−2.00000	−0.0733236
$745$	−6.00000	−0.219823
$746$	6.00000	0.219676
$747$	6.00000	0.219529
$748$	16.0000	0.585018
$749$	8.00000	0.292314
$750$	−1.00000	−0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	6.00000	0.218797
$753$	18.0000	0.655956
$754$	40.0000	1.45671
$755$	−4.00000	−0.145575
$756$	−1.00000	−0.0363696
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	8.00000	0.290573
$759$	4.00000	0.145191
$760$	−2.00000	−0.0725476
$761$	26.0000	0.942499	0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	8.00000	0.289809
$763$	10.0000	0.362024
$764$	8.00000	0.289430
$765$	−4.00000	−0.144620
$766$	16.0000	0.578103
$767$	−16.0000	−0.577727
$768$	1.00000	0.0360844
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	−4.00000	−0.144150
$771$	10.0000	0.360141
$772$	−18.0000	−0.647834
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	8.00000	0.287554
$775$	2.00000	0.0718421
$776$	4.00000	0.143592
$777$	−2.00000	−0.0717496
$778$	26.0000	0.932145
$779$	−4.00000	−0.143315
$780$	4.00000	0.143223
$781$	−32.0000	−1.14505
$782$	−4.00000	−0.143040
$783$	−10.0000	−0.357371
$784$	1.00000	0.0357143
$785$	22.0000	0.785214
$786$	−12.0000	−0.428026
$787$	42.0000	1.49714	0.748569	−	0.663057i	$-0.230741\pi$
$787$	42.0000	1.49714	0.748569	+	0.663057i	$0.230741\pi$
$788$	−18.0000	−0.641223
$789$	−8.00000	−0.284808
$790$	16.0000	0.569254
$791$	6.00000	0.213335
$792$	4.00000	0.142134
$793$	8.00000	0.284088
$794$	12.0000	0.425864
$795$	−10.0000	−0.354663
$796$	−4.00000	−0.141776
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	2.00000	0.0707992
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	4.00000	0.141333
$802$	−18.0000	−0.635602
$803$	8.00000	0.282314
$804$	0	0
$805$	1.00000	0.0352454
$806$	−8.00000	−0.281788
$807$	16.0000	0.563227
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	−1.00000	−0.0351364
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	10.0000	0.350931
$813$	10.0000	0.350715
$814$	8.00000	0.280400
$815$	20.0000	0.700569
$816$	−4.00000	−0.140028
$817$	−16.0000	−0.559769
$818$	−34.0000	−1.18878
$819$	−4.00000	−0.139771
$820$	−2.00000	−0.0698430
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	18.0000	0.627822
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	4.00000	0.139347
$825$	−4.00000	−0.139262
$826$	−4.00000	−0.139178
$827$	−32.0000	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$828$	−1.00000	−0.0347524
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	−6.00000	−0.208263
$831$	18.0000	0.624413
$832$	4.00000	0.138675
$833$	−4.00000	−0.138592
$834$	0	0
$835$	−6.00000	−0.207639
$836$	−8.00000	−0.276686
$837$	2.00000	0.0691301
$838$	6.00000	0.207267
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	1.00000	0.0345033
$841$	71.0000	2.44828
$842$	26.0000	0.896019
$843$	2.00000	0.0688837
$844$	20.0000	0.688428
$845$	3.00000	0.103203
$846$	−6.00000	−0.206284
$847$	−5.00000	−0.171802
$848$	−10.0000	−0.343401
$849$	−2.00000	−0.0686398
$850$	4.00000	0.137199
$851$	−2.00000	−0.0685591
$852$	8.00000	0.274075
$853$	20.0000	0.684787	0.342393	−	0.939557i	$-0.388762\pi$
$853$	20.0000	0.684787	0.342393	+	0.939557i	$0.388762\pi$
$854$	2.00000	0.0684386
$855$	2.00000	0.0683986
$856$	8.00000	0.273434
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	16.0000	0.546231
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	−8.00000	−0.272798
$861$	2.00000	0.0681598
$862$	0	0
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	−1.00000	−0.0340207
$865$	−20.0000	−0.680020
$866$	28.0000	0.951479
$867$	−1.00000	−0.0339618
$868$	−2.00000	−0.0678844
$869$	64.0000	2.17105
$870$	10.0000	0.339032
$871$	0	0
$872$	10.0000	0.338643
$873$	−4.00000	−0.135379
$874$	2.00000	0.0676510
$875$	−1.00000	−0.0338062
$876$	−2.00000	−0.0675737
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−6.00000	−0.202490
$879$	−18.0000	−0.607125
$880$	−4.00000	−0.134840
$881$	56.0000	1.88669	0.943344	−	0.331816i	$-0.107661\pi$
$881$	56.0000	1.88669	0.943344	+	0.331816i	$0.107661\pi$
$882$	−1.00000	−0.0336718
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−16.0000	−0.538138
$885$	−4.00000	−0.134459
$886$	−12.0000	−0.403148
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	−2.00000	−0.0671156
$889$	8.00000	0.268311
$890$	−4.00000	−0.134080
$891$	−4.00000	−0.134005
$892$	−22.0000	−0.736614
$893$	12.0000	0.401565
$894$	6.00000	0.200670
$895$	−12.0000	−0.401116
$896$	1.00000	0.0334077
$897$	−4.00000	−0.133556
$898$	2.00000	0.0667409
$899$	−20.0000	−0.667037
$900$	1.00000	0.0333333
$901$	40.0000	1.33259
$902$	−8.00000	−0.266371
$903$	8.00000	0.266223
$904$	6.00000	0.199557
$905$	−14.0000	−0.465376
$906$	4.00000	0.132891
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	4.00000	0.132599
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	2.00000	0.0662266
$913$	−24.0000	−0.794284
$914$	10.0000	0.330771
$915$	2.00000	0.0661180
$916$	2.00000	0.0660819
$917$	−12.0000	−0.396275
$918$	4.00000	0.132020
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	1.00000	0.0329690
$921$	−8.00000	−0.263609
$922$	0	0
$923$	32.0000	1.05329
$924$	4.00000	0.131590
$925$	2.00000	0.0657596
$926$	−20.0000	−0.657241
$927$	−4.00000	−0.131377
$928$	10.0000	0.328266
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	−2.00000	−0.0655826
$931$	2.00000	0.0655474
$932$	−10.0000	−0.327561
$933$	10.0000	0.327385
$934$	14.0000	0.458094
$935$	16.0000	0.523256
$936$	−4.00000	−0.130744
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	6.00000	0.195698
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−22.0000	−0.716799
$943$	2.00000	0.0651290
$944$	−4.00000	−0.130189
$945$	−1.00000	−0.0325300
$946$	−32.0000	−1.04041
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−16.0000	−0.519656
$949$	−8.00000	−0.259691
$950$	−2.00000	−0.0648886
$951$	−18.0000	−0.583690
$952$	−4.00000	−0.129641
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	10.0000	0.323762
$955$	8.00000	0.258874
$956$	8.00000	0.258738
$957$	40.0000	1.29302
$958$	−20.0000	−0.646171
$959$	18.0000	0.581250
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	−8.00000	−0.257930
$963$	−8.00000	−0.257796
$964$	4.00000	0.128831
$965$	−18.0000	−0.579441
$966$	−1.00000	−0.0321745
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	−5.00000	−0.160706
$969$	−8.00000	−0.256997
$970$	4.00000	0.128432
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	12.0000	0.384505
$975$	4.00000	0.128103
$976$	2.00000	0.0640184
$977$	−58.0000	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$977$	−58.0000	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$978$	−20.0000	−0.639529
$979$	−16.0000	−0.511362
$980$	1.00000	0.0319438
$981$	−10.0000	−0.319275
$982$	36.0000	1.14881
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	2.00000	0.0637577
$985$	−18.0000	−0.573528
$986$	−40.0000	−1.27386
$987$	−6.00000	−0.190982
$988$	8.00000	0.254514
$989$	8.00000	0.254385
$990$	4.00000	0.127128
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	−2.00000	−0.0635001
$993$	4.00000	0.126936
$994$	8.00000	0.253745
$995$	−4.00000	−0.126809
$996$	6.00000	0.190117
$997$	44.0000	1.39349	0.696747	−	0.717317i	$-0.254630\pi$
$997$	44.0000	1.39349	0.696747	+	0.717317i	$0.254630\pi$
$998$	−20.0000	−0.633089
$999$	2.00000	0.0632772

Twists

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

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