LMFDB - Embedded newform 4830.2.a.j.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} -6.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -4.00000 q^{29} +1.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +6.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -6.00000 q^{55} +1.00000 q^{56} +6.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -6.00000 q^{66} -2.00000 q^{67} -1.00000 q^{69} -1.00000 q^{70} +10.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +6.00000 q^{76} -6.00000 q^{77} +2.00000 q^{78} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -1.00000 q^{84} -6.00000 q^{86} -4.00000 q^{87} -6.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} +2.00000 q^{93} -2.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	−1.00000	−0.223607
$21$	−1.00000	−0.218218
$22$	−6.00000	−1.27920
$23$	−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
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	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
$33$	6.00000	1.04447
$34$	0	0
$35$	1.00000	0.169031
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−6.00000	−0.973329
$39$	−2.00000	−0.320256
$40$	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
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	6.00000	0.904534
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	−6.00000	−0.809040
$56$	1.00000	0.133631
$57$	6.00000	0.794719
$58$	4.00000	0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−2.00000	−0.254000
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	−6.00000	−0.738549
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	−1.00000	−0.119523
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	−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	$0.614200\pi$
$74$	6.00000	0.697486
$75$	1.00000	0.115470
$76$	6.00000	0.688247
$77$	−6.00000	−0.683763
$78$	2.00000	0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−6.00000	−0.646997
$87$	−4.00000	−0.428845
$88$	−6.00000	−0.639602
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	−1.00000	−0.104257
$93$	2.00000	0.207390
$94$	−2.00000	−0.206284
$95$	−6.00000	−0.615587
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−1.00000	−0.101015
$99$	6.00000	0.603023
$100$	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	$0.745272\pi$
$102$	0	0
$103$	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
$106$	2.00000	0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	6.00000	0.572078
$111$	−6.00000	−0.569495
$112$	−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	$0.591070\pi$
$114$	−6.00000	−0.561951
$115$	1.00000	0.0932505
$116$	−4.00000	−0.371391
$117$	−2.00000	−0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	1.00000	0.0912871
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	6.00000	0.541002
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	1.00000	0.0890871
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−1.00000	−0.0883883
$129$	6.00000	0.528271
$130$	−2.00000	−0.175412
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	6.00000	0.522233
$133$	−6.00000	−0.520266
$134$	2.00000	0.172774
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	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
$141$	2.00000	0.168430
$142$	−10.0000	−0.839181
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	6.00000	0.496564
$147$	1.00000	0.0824786
$148$	−6.00000	−0.493197
$149$	−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
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	6.00000	0.483494
$155$	−2.00000	−0.160644
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	1.00000	0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	6.00000	0.468521
$165$	−6.00000	−0.467099
$166$	6.00000	0.465690
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	1.00000	0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	6.00000	0.457496
$173$	24.0000	1.82469	0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000	1.82469	0.912343	+	0.409426i	$0.134271\pi$
$174$	4.00000	0.303239
$175$	−1.00000	−0.0755929
$176$	6.00000	0.452267
$177$	4.00000	0.300658
$178$	−10.0000	−0.749532
$179$	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
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	−2.00000	−0.148250
$183$	−2.00000	−0.147844
$184$	1.00000	0.0737210
$185$	6.00000	0.441129
$186$	−2.00000	−0.146647
$187$	0	0
$188$	2.00000	0.145865
$189$	−1.00000	−0.0727393
$190$	6.00000	0.435286
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	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	$0.569283\pi$
$194$	10.0000	0.717958
$195$	2.00000	0.143223
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−6.00000	−0.426401
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	−1.00000	−0.0707107
$201$	−2.00000	−0.141069
$202$	14.0000	0.985037
$203$	4.00000	0.280745
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−8.00000	−0.557386
$207$	−1.00000	−0.0695048
$208$	−2.00000	−0.138675
$209$	36.0000	2.49017
$210$	−1.00000	−0.0690066
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−2.00000	−0.137361
$213$	10.0000	0.685189
$214$	−12.0000	−0.820303
$215$	−6.00000	−0.409197
$216$	−1.00000	−0.0680414
$217$	−2.00000	−0.135769
$218$	6.00000	0.406371
$219$	−6.00000	−0.405442
$220$	−6.00000	−0.404520
$221$	0	0
$222$	6.00000	0.402694
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	6.00000	0.397360
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−1.00000	−0.0659380
$231$	−6.00000	−0.394771
$232$	4.00000	0.262613
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	2.00000	0.130744
$235$	−2.00000	−0.130466
$236$	4.00000	0.260378
$237$	0	0
$238$	0	0
$239$	10.0000	0.646846	0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000	0.646846	0.323423	+	0.946254i	$0.395166\pi$
$240$	−1.00000	−0.0645497
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	−25.0000	−1.60706
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	−12.0000	−0.763542
$248$	−2.00000	−0.127000
$249$	−6.00000	−0.380235
$250$	1.00000	0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	−1.00000	−0.0629941
$253$	−6.00000	−0.377217
$254$	−2.00000	−0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−6.00000	−0.373544
$259$	6.00000	0.372822
$260$	2.00000	0.124035
$261$	−4.00000	−0.247594
$262$	−20.0000	−1.23560
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	−6.00000	−0.369274
$265$	2.00000	0.122859
$266$	6.00000	0.367884
$267$	10.0000	0.611990
$268$	−2.00000	−0.122169
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	1.00000	0.0608581
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	−6.00000	−0.362473
$275$	6.00000	0.361814
$276$	−1.00000	−0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	−1.00000	−0.0597614
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	−2.00000	−0.119098
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	10.0000	0.593391
$285$	−6.00000	−0.355409
$286$	12.0000	0.709575
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−4.00000	−0.234888
$291$	−10.0000	−0.586210
$292$	−6.00000	−0.351123
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	−1.00000	−0.0583212
$295$	−4.00000	−0.232889
$296$	6.00000	0.348743
$297$	6.00000	0.348155
$298$	6.00000	0.347571
$299$	2.00000	0.115663
$300$	1.00000	0.0577350
$301$	−6.00000	−0.345834
$302$	8.00000	0.460348
$303$	−14.0000	−0.804279
$304$	6.00000	0.344124
$305$	2.00000	0.114520
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−6.00000	−0.341882
$309$	8.00000	0.455104
$310$	2.00000	0.113592
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	2.00000	0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	10.0000	0.564333
$315$	1.00000	0.0563436
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	2.00000	0.112154
$319$	−24.0000	−1.34374
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−12.0000	−0.664619
$327$	−6.00000	−0.331801
$328$	−6.00000	−0.331295
$329$	−2.00000	−0.110264
$330$	6.00000	0.330289
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−6.00000	−0.329293
$333$	−6.00000	−0.328798
$334$	−18.0000	−0.984916
$335$	2.00000	0.109272
$336$	−1.00000	−0.0545545
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	0	0
$341$	12.0000	0.649836
$342$	−6.00000	−0.324443
$343$	−1.00000	−0.0539949
$344$	−6.00000	−0.323498
$345$	1.00000	0.0538382
$346$	−24.0000	−1.29025
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−4.00000	−0.214423
$349$	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
$351$	−2.00000	−0.106752
$352$	−6.00000	−0.319801
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	−4.00000	−0.212598
$355$	−10.0000	−0.530745
$356$	10.0000	0.529999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	1.00000	0.0527046
$361$	17.0000	0.894737
$362$	−6.00000	−0.315353
$363$	25.0000	1.31216
$364$	2.00000	0.104828
$365$	6.00000	0.314054
$366$	2.00000	0.104542
$367$	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
$370$	−6.00000	−0.311925
$371$	2.00000	0.103835
$372$	2.00000	0.103695
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−2.00000	−0.103142
$377$	8.00000	0.412021
$378$	1.00000	0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−6.00000	−0.307794
$381$	2.00000	0.102463
$382$	−16.0000	−0.818631
$383$	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
$385$	6.00000	0.305788
$386$	6.00000	0.305392
$387$	6.00000	0.304997
$388$	−10.0000	−0.507673
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	20.0000	1.00887
$394$	−6.00000	−0.302276
$395$	0	0
$396$	6.00000	0.301511
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−4.00000	−0.200502
$399$	−6.00000	−0.300376
$400$	1.00000	0.0500000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	2.00000	0.0997509
$403$	−4.00000	−0.199254
$404$	−14.0000	−0.696526
$405$	−1.00000	−0.0496904
$406$	−4.00000	−0.198517
$407$	−36.0000	−1.78445
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	6.00000	0.296319
$411$	6.00000	0.295958
$412$	8.00000	0.394132
$413$	−4.00000	−0.196827
$414$	1.00000	0.0491473
$415$	6.00000	0.294528
$416$	2.00000	0.0980581
$417$	0	0
$418$	−36.0000	−1.76082
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	1.00000	0.0487950
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000	0.194717
$423$	2.00000	0.0972433
$424$	2.00000	0.0971286
$425$	0	0
$426$	−10.0000	−0.484502
$427$	2.00000	0.0967868
$428$	12.0000	0.580042
$429$	−12.0000	−0.579365
$430$	6.00000	0.289346
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	2.00000	0.0960031
$435$	4.00000	0.191785
$436$	−6.00000	−0.287348
$437$	−6.00000	−0.287019
$438$	6.00000	0.286691
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	6.00000	0.286039
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−6.00000	−0.284747
$445$	−10.0000	−0.474045
$446$	−16.0000	−0.757622
$447$	−6.00000	−0.283790
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	36.0000	1.69517
$452$	−6.00000	−0.282216
$453$	−8.00000	−0.375873
$454$	−10.0000	−0.469323
$455$	−2.00000	−0.0937614
$456$	−6.00000	−0.280976
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	1.00000	0.0466252
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	6.00000	0.279145
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	−4.00000	−0.185695
$465$	−2.00000	−0.0927478
$466$	6.00000	0.277945
$467$	−34.0000	−1.57333	−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000	−1.57333	−0.786666	+	0.617379i	$0.788195\pi$
$468$	−2.00000	−0.0924500
$469$	2.00000	0.0923514
$470$	2.00000	0.0922531
$471$	−10.0000	−0.460776
$472$	−4.00000	−0.184115
$473$	36.0000	1.65528
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−10.0000	−0.457389
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	1.00000	0.0456435
$481$	12.0000	0.547153
$482$	−12.0000	−0.546585
$483$	1.00000	0.0455016
$484$	25.0000	1.13636
$485$	10.0000	0.454077
$486$	−1.00000	−0.0453609
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	2.00000	0.0905357
$489$	12.0000	0.542659
$490$	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
$493$	0	0
$494$	12.0000	0.539906
$495$	−6.00000	−0.269680
$496$	2.00000	0.0898027
$497$	−10.0000	−0.448561
$498$	6.00000	0.268866
$499$	−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
$501$	18.0000	0.804181
$502$	−20.0000	−0.892644
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	1.00000	0.0445435
$505$	14.0000	0.622992
$506$	6.00000	0.266733
$507$	−9.00000	−0.399704
$508$	2.00000	0.0887357
$509$	26.0000	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$509$	26.0000	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	−1.00000	−0.0441942
$513$	6.00000	0.264906
$514$	−2.00000	−0.0882162
$515$	−8.00000	−0.352522
$516$	6.00000	0.264135
$517$	12.0000	0.527759
$518$	−6.00000	−0.263625
$519$	24.0000	1.05348
$520$	−2.00000	−0.0877058
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	4.00000	0.175075
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	20.0000	0.873704
$525$	−1.00000	−0.0436436
$526$	−16.0000	−0.697633
$527$	0	0
$528$	6.00000	0.261116
$529$	1.00000	0.0434783
$530$	−2.00000	−0.0868744
$531$	4.00000	0.173585
$532$	−6.00000	−0.260133
$533$	−12.0000	−0.519778
$534$	−10.0000	−0.432742
$535$	−12.0000	−0.518805
$536$	2.00000	0.0863868
$537$	4.00000	0.172613
$538$	−6.00000	−0.258678
$539$	6.00000	0.258438
$540$	−1.00000	−0.0430331
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	6.00000	0.257722
$543$	6.00000	0.257485
$544$	0	0
$545$	6.00000	0.257012
$546$	−2.00000	−0.0855921
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	6.00000	0.256307
$549$	−2.00000	−0.0853579
$550$	−6.00000	−0.255841
$551$	−24.0000	−1.02243
$552$	1.00000	0.0425628
$553$	0	0
$554$	−8.00000	−0.339887
$555$	6.00000	0.254686
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−2.00000	−0.0846668
$559$	−12.0000	−0.507546
$560$	1.00000	0.0422577
$561$	0	0
$562$	8.00000	0.337460
$563$	−30.0000	−1.26435	−0.632175	−	0.774826i	$-0.717837\pi$
$563$	−30.0000	−1.26435	−0.632175	+	0.774826i	$0.717837\pi$
$564$	2.00000	0.0842152
$565$	6.00000	0.252422
$566$	−28.0000	−1.17693
$567$	−1.00000	−0.0419961
$568$	−10.0000	−0.419591
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	6.00000	0.251312
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−12.0000	−0.501745
$573$	16.0000	0.668410
$574$	6.00000	0.250435
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	17.0000	0.707107
$579$	−6.00000	−0.249351
$580$	4.00000	0.166091
$581$	6.00000	0.248922
$582$	10.0000	0.414513
$583$	−12.0000	−0.496989
$584$	6.00000	0.248282
$585$	2.00000	0.0826898
$586$	−26.0000	−1.07405
$587$	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
$589$	12.0000	0.494451
$590$	4.00000	0.164677
$591$	6.00000	0.246807
$592$	−6.00000	−0.246598
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	−6.00000	−0.245770
$597$	4.00000	0.163709
$598$	−2.00000	−0.0817861
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−1.00000	−0.0408248
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	6.00000	0.244542
$603$	−2.00000	−0.0814463
$604$	−8.00000	−0.325515
$605$	−25.0000	−1.01639
$606$	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	$0.578306\pi$
$608$	−6.00000	−0.243332
$609$	4.00000	0.162088
$610$	−2.00000	−0.0809776
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−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
$616$	6.00000	0.241747
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−8.00000	−0.321807
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	−2.00000	−0.0803219
$621$	−1.00000	−0.0401286
$622$	−4.00000	−0.160385
$623$	−10.0000	−0.400642
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−14.0000	−0.559553
$627$	36.0000	1.43770
$628$	−10.0000	−0.399043
$629$	0	0
$630$	−1.00000	−0.0398410
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	−2.00000	−0.0794301
$635$	−2.00000	−0.0793676
$636$	−2.00000	−0.0793052
$637$	−2.00000	−0.0792429
$638$	24.0000	0.950169
$639$	10.0000	0.395594
$640$	1.00000	0.0395285
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	−12.0000	−0.473602
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	1.00000	0.0394055
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	−1.00000	−0.0392837
$649$	24.0000	0.942082
$650$	2.00000	0.0784465
$651$	−2.00000	−0.0783862
$652$	12.0000	0.469956
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	6.00000	0.234619
$655$	−20.0000	−0.781465
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	2.00000	0.0779681
$659$	−14.0000	−0.545363	−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000	−0.545363	−0.272681	+	0.962104i	$0.587910\pi$
$660$	−6.00000	−0.233550
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	6.00000	0.232845
$665$	6.00000	0.232670
$666$	6.00000	0.232495
$667$	4.00000	0.154881
$668$	18.0000	0.696441
$669$	16.0000	0.618596
$670$	−2.00000	−0.0772667
$671$	−12.0000	−0.463255
$672$	1.00000	0.0385758
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−8.00000	−0.308148
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	6.00000	0.230429
$679$	10.0000	0.383765
$680$	0	0
$681$	10.0000	0.383201
$682$	−12.0000	−0.459504
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	6.00000	0.229416
$685$	−6.00000	−0.229248
$686$	1.00000	0.0381802
$687$	−26.0000	−0.991962
$688$	6.00000	0.228748
$689$	4.00000	0.152388
$690$	−1.00000	−0.0380693
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	24.0000	0.912343
$693$	−6.00000	−0.227921
$694$	−12.0000	−0.455514
$695$	0	0
$696$	4.00000	0.151620
$697$	0	0
$698$	−20.0000	−0.757011
$699$	−6.00000	−0.226941
$700$	−1.00000	−0.0377964
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	2.00000	0.0754851
$703$	−36.0000	−1.35777
$704$	6.00000	0.226134
$705$	−2.00000	−0.0753244
$706$	26.0000	0.978523
$707$	14.0000	0.526524
$708$	4.00000	0.150329
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	10.0000	0.375293
$711$	0	0
$712$	−10.0000	−0.374766
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	12.0000	0.448775
$716$	4.00000	0.149487
$717$	10.0000	0.373457
$718$	32.0000	1.19423
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	−1.00000	−0.0372678
$721$	−8.00000	−0.297936
$722$	−17.0000	−0.632674
$723$	12.0000	0.446285
$724$	6.00000	0.222988
$725$	−4.00000	−0.148556
$726$	−25.0000	−0.927837
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−16.0000	−0.590571
$735$	−1.00000	−0.0368856
$736$	1.00000	0.0368605
$737$	−12.0000	−0.442026
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	6.00000	0.220564
$741$	−12.0000	−0.440831
$742$	−2.00000	−0.0734223
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−2.00000	−0.0733236
$745$	6.00000	0.219823
$746$	−10.0000	−0.366126
$747$	−6.00000	−0.219529
$748$	0	0
$749$	−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	$0.546628\pi$
$752$	2.00000	0.0729325
$753$	20.0000	0.728841
$754$	−8.00000	−0.291343
$755$	8.00000	0.291150
$756$	−1.00000	−0.0363696
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	20.0000	0.726433
$759$	−6.00000	−0.217786
$760$	6.00000	0.217643
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	−2.00000	−0.0724524
$763$	6.00000	0.217215
$764$	16.0000	0.578860
$765$	0	0
$766$	−12.0000	−0.433578
$767$	−8.00000	−0.288863
$768$	1.00000	0.0360844
$769$	24.0000	0.865462	0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000	0.865462	0.432731	+	0.901523i	$0.357550\pi$
$770$	−6.00000	−0.216225
$771$	2.00000	0.0720282
$772$	−6.00000	−0.215945
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−6.00000	−0.215666
$775$	2.00000	0.0718421
$776$	10.0000	0.358979
$777$	6.00000	0.215249
$778$	18.0000	0.645331
$779$	36.0000	1.28983
$780$	2.00000	0.0716115
$781$	60.0000	2.14697
$782$	0	0
$783$	−4.00000	−0.142948
$784$	1.00000	0.0357143
$785$	10.0000	0.356915
$786$	−20.0000	−0.713376
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	6.00000	0.213741
$789$	16.0000	0.569615
$790$	0	0
$791$	6.00000	0.213335
$792$	−6.00000	−0.213201
$793$	4.00000	0.142044
$794$	18.0000	0.638796
$795$	2.00000	0.0709327
$796$	4.00000	0.141776
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	6.00000	0.212398
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	10.0000	0.353333
$802$	−36.0000	−1.27120
$803$	−36.0000	−1.27041
$804$	−2.00000	−0.0705346
$805$	−1.00000	−0.0352454
$806$	4.00000	0.140894
$807$	6.00000	0.211210
$808$	14.0000	0.492518
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	4.00000	0.140372
$813$	−6.00000	−0.210429
$814$	36.0000	1.26180
$815$	−12.0000	−0.420342
$816$	0	0
$817$	36.0000	1.25948
$818$	6.00000	0.209785
$819$	2.00000	0.0698857
$820$	−6.00000	−0.209529
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	−6.00000	−0.209274
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	−8.00000	−0.278693
$825$	6.00000	0.208893
$826$	4.00000	0.139178
$827$	−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
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	−6.00000	−0.208263
$831$	8.00000	0.277517
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	−18.0000	−0.622916
$836$	36.0000	1.24509
$837$	2.00000	0.0691301
$838$	−24.0000	−0.829066
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	−1.00000	−0.0345033
$841$	−13.0000	−0.448276
$842$	10.0000	0.344623
$843$	−8.00000	−0.275535
$844$	−4.00000	−0.137686
$845$	9.00000	0.309609
$846$	−2.00000	−0.0687614
$847$	−25.0000	−0.859010
$848$	−2.00000	−0.0686803
$849$	28.0000	0.960958
$850$	0	0
$851$	6.00000	0.205677
$852$	10.0000	0.342594
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	−2.00000	−0.0684386
$855$	−6.00000	−0.205196
$856$	−12.0000	−0.410152
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	12.0000	0.409673
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	−6.00000	−0.204598
$861$	−6.00000	−0.204479
$862$	4.00000	0.136241
$863$	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
$865$	−24.0000	−0.816024
$866$	26.0000	0.883516
$867$	−17.0000	−0.577350
$868$	−2.00000	−0.0678844
$869$	0	0
$870$	−4.00000	−0.135613
$871$	4.00000	0.135535
$872$	6.00000	0.203186
$873$	−10.0000	−0.338449
$874$	6.00000	0.202953
$875$	1.00000	0.0338062
$876$	−6.00000	−0.202721
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	−10.0000	−0.337484
$879$	26.0000	0.876958
$880$	−6.00000	−0.202260
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	−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	$0.609252\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	−4.00000	−0.134383
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	6.00000	0.201347
$889$	−2.00000	−0.0670778
$890$	10.0000	0.335201
$891$	6.00000	0.201008
$892$	16.0000	0.535720
$893$	12.0000	0.401565
$894$	6.00000	0.200670
$895$	−4.00000	−0.133705
$896$	1.00000	0.0334077
$897$	2.00000	0.0667781
$898$	18.0000	0.600668
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	0	0
$902$	−36.0000	−1.19867
$903$	−6.00000	−0.199667
$904$	6.00000	0.199557
$905$	−6.00000	−0.199447
$906$	8.00000	0.265782
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	10.0000	0.331862
$909$	−14.0000	−0.464351
$910$	2.00000	0.0662994
$911$	−52.0000	−1.72284	−0.861418	−	0.507896i	$-0.830423\pi$
$911$	−52.0000	−1.72284	−0.861418	+	0.507896i	$0.830423\pi$
$912$	6.00000	0.198680
$913$	−36.0000	−1.19143
$914$	32.0000	1.05847
$915$	2.00000	0.0661180
$916$	−26.0000	−0.859064
$917$	−20.0000	−0.660458
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	−1.00000	−0.0329690
$921$	−20.0000	−0.659022
$922$	−10.0000	−0.329332
$923$	−20.0000	−0.658308
$924$	−6.00000	−0.197386
$925$	−6.00000	−0.197279
$926$	6.00000	0.197172
$927$	8.00000	0.262754
$928$	4.00000	0.131306
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	2.00000	0.0655826
$931$	6.00000	0.196642
$932$	−6.00000	−0.196537
$933$	4.00000	0.130954
$934$	34.0000	1.11251
$935$	0	0
$936$	2.00000	0.0653720
$937$	−46.0000	−1.50275	−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000	−1.50275	−0.751377	+	0.659873i	$0.770610\pi$
$938$	−2.00000	−0.0653023
$939$	14.0000	0.456873
$940$	−2.00000	−0.0652328
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	10.0000	0.325818
$943$	−6.00000	−0.195387
$944$	4.00000	0.130189
$945$	1.00000	0.0325300
$946$	−36.0000	−1.17046
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	−6.00000	−0.194666
$951$	2.00000	0.0648544
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	2.00000	0.0647524
$955$	−16.0000	−0.517748
$956$	10.0000	0.323423
$957$	−24.0000	−0.775810
$958$	32.0000	1.03387
$959$	−6.00000	−0.193750
$960$	−1.00000	−0.0322749
$961$	−27.0000	−0.870968
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	12.0000	0.386494
$965$	6.00000	0.193147
$966$	−1.00000	−0.0321745
$967$	−62.0000	−1.99379	−0.996893	−	0.0787703i	$-0.974901\pi$
$967$	−62.0000	−1.99379	−0.996893	+	0.0787703i	$0.974901\pi$
$968$	−25.0000	−0.803530
$969$	0	0
$970$	−10.0000	−0.321081
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−34.0000	−1.08943
$975$	−2.00000	−0.0640513
$976$	−2.00000	−0.0640184
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−12.0000	−0.383718
$979$	60.0000	1.91761
$980$	−1.00000	−0.0319438
$981$	−6.00000	−0.191565
$982$	−12.0000	−0.382935
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−6.00000	−0.191273
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−2.00000	−0.0636607
$988$	−12.0000	−0.381771
$989$	−6.00000	−0.190789
$990$	6.00000	0.190693
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−2.00000	−0.0635001
$993$	−20.0000	−0.634681
$994$	10.0000	0.317181
$995$	−4.00000	−0.126809
$996$	−6.00000	−0.190117
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	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.j.1.1	✓	1

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