LMFDB - Embedded newform 4830.2.a.p.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -1.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +12.0000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +1.00000 q^{45} +1.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} +12.0000 q^{59} +1.00000 q^{60} +14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -4.00000 q^{67} -1.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} -4.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +1.00000 q^{84} +4.00000 q^{86} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -1.00000 q^{92} -4.00000 q^{93} -12.0000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	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
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−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
$22$	0	0
$23$	−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
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	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	$0.447432\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	−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	$0.598656\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−4.00000	−0.529813
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	−1.00000	−0.119523
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	−1.00000	−0.104257
$93$	−4.00000	−0.414781
$94$	−12.0000	−1.23771
$95$	−4.00000	−0.410391
$96$	−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
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.00000	−0.196116
$105$	1.00000	0.0975900
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	1.00000	0.0962250
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	1.00000	0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	4.00000	0.374634
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	−12.0000	−1.10469
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	−14.0000	−1.26750
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	1.00000	0.0894427
$126$	−1.00000	−0.0890871
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	0	0
$137$	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
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	1.00000	0.0845154
$141$	12.0000	1.01058
$142$	12.0000	1.00702
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	1.00000	0.0824786
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−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
$153$	0	0
$154$	0	0
$155$	−4.00000	−0.321288
$156$	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	$0.311315\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−1.00000	−0.0788110
$162$	−1.00000	−0.0785674
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	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
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	12.0000	0.901975
$178$	−6.00000	−0.449719
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	1.00000	0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−2.00000	−0.148250
$183$	14.0000	1.03491
$184$	1.00000	0.0737210
$185$	2.00000	0.147043
$186$	4.00000	0.293294
$187$	0	0
$188$	12.0000	0.875190
$189$	1.00000	0.0727393
$190$	4.00000	0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	10.0000	0.717958
$195$	2.00000	0.143223
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	12.0000	0.838116
$206$	4.00000	0.278693
$207$	−1.00000	−0.0695048
$208$	2.00000	0.138675
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	−6.00000	−0.410152
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	−4.00000	−0.271538
$218$	−8.00000	−0.541828
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	−1.00000	−0.0668153
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	0	0
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−2.00000	−0.130744
$235$	12.0000	0.782794
$236$	12.0000	0.781133
$237$	8.00000	0.519656
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	1.00000	0.0645497
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	1.00000	0.0638877
$246$	−12.0000	−0.765092
$247$	−8.00000	−0.509028
$248$	4.00000	0.254000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	1.00000	0.0629941
$253$	0	0
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	4.00000	0.249029
$259$	2.00000	0.124274
$260$	2.00000	0.124035
$261$	0	0
$262$	12.0000	0.741362
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	−1.00000	−0.0608581
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	4.00000	0.239904
$279$	−4.00000	−0.239474
$280$	−1.00000	−0.0597614
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−12.0000	−0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	−4.00000	−0.236940
$286$	0	0
$287$	12.0000	0.708338
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−4.00000	−0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−1.00000	−0.0583212
$295$	12.0000	0.698667
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−2.00000	−0.115663
$300$	1.00000	0.0577350
$301$	−4.00000	−0.230556
$302$	16.0000	0.920697
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	14.0000	0.801638
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	4.00000	0.227185
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	−2.00000	−0.113228
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−14.0000	−0.790066
$315$	1.00000	0.0563436
$316$	8.00000	0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	1.00000	0.0559017
$321$	6.00000	0.334887
$322$	1.00000	0.0557278
$323$	0	0
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	−14.0000	−0.775388
$327$	8.00000	0.442401
$328$	−12.0000	−0.662589
$329$	12.0000	0.661581
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	24.0000	1.31322
$335$	−4.00000	−0.218543
$336$	1.00000	0.0545545
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	9.00000	0.489535
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	1.00000	0.0539949
$344$	4.00000	0.215666
$345$	−1.00000	−0.0538382
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−1.00000	−0.0534522
$351$	2.00000	0.106752
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−12.0000	−0.637793
$355$	−12.0000	−0.636894
$356$	6.00000	0.317999
$357$	0	0
$358$	6.00000	0.317110
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	22.0000	1.15629
$363$	−11.0000	−0.577350
$364$	2.00000	0.104828
$365$	−4.00000	−0.209370
$366$	−14.0000	−0.731792
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−1.00000	−0.0521286
$369$	12.0000	0.624695
$370$	−2.00000	−0.103975
$371$	0	0
$372$	−4.00000	−0.207390
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	−12.0000	−0.618853
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−4.00000	−0.205196
$381$	20.0000	1.02463
$382$	0	0
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−4.00000	−0.203331
$388$	−10.0000	−0.507673
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	−12.0000	−0.605320
$394$	6.00000	0.302276
$395$	8.00000	0.402524
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−14.0000	−0.701757
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	4.00000	0.199502
$403$	−8.00000	−0.398508
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−12.0000	−0.592638
$411$	18.0000	0.887875
$412$	−4.00000	−0.197066
$413$	12.0000	0.590481
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	−4.00000	−0.195881
$418$	0	0
$419$	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
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	16.0000	0.778868
$423$	12.0000	0.583460
$424$	0	0
$425$	0	0
$426$	12.0000	0.581402
$427$	14.0000	0.677507
$428$	6.00000	0.290021
$429$	0	0
$430$	4.00000	0.192897
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	8.00000	0.383131
$437$	4.00000	0.191346
$438$	4.00000	0.191127
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	2.00000	0.0949158
$445$	6.00000	0.284427
$446$	−26.0000	−1.23114
$447$	6.00000	0.283790
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	18.0000	0.846649
$453$	−16.0000	−0.751746
$454$	−12.0000	−0.563188
$455$	2.00000	0.0937614
$456$	4.00000	0.187317
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	30.0000	1.38972
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	2.00000	0.0924500
$469$	−4.00000	−0.184703
$470$	−12.0000	−0.553519
$471$	14.0000	0.645086
$472$	−12.0000	−0.552345
$473$	0	0
$474$	−8.00000	−0.367452
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−1.00000	−0.0456435
$481$	4.00000	0.182384
$482$	−20.0000	−0.910975
$483$	−1.00000	−0.0455016
$484$	−11.0000	−0.500000
$485$	−10.0000	−0.454077
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−14.0000	−0.633750
$489$	14.0000	0.633102
$490$	−1.00000	−0.0451754
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	1.00000	0.0447214
$501$	−24.0000	−1.07224
$502$	−12.0000	−0.535586
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	−1.00000	−0.0445435
$505$	6.00000	0.266996
$506$	0	0
$507$	−9.00000	−0.399704
$508$	20.0000	0.887357
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	6.00000	0.264649
$515$	−4.00000	−0.176261
$516$	−4.00000	−0.176090
$517$	0	0
$518$	−2.00000	−0.0878750
$519$	6.00000	0.263371
$520$	−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	$0.370988\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−12.0000	−0.524222
$525$	1.00000	0.0436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	12.0000	0.520756
$532$	−4.00000	−0.173422
$533$	24.0000	1.03956
$534$	−6.00000	−0.259645
$535$	6.00000	0.259403
$536$	4.00000	0.172774
$537$	−6.00000	−0.258919
$538$	−18.0000	−0.776035
$539$	0	0
$540$	1.00000	0.0430331
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	4.00000	0.171815
$543$	−22.0000	−0.944110
$544$	0	0
$545$	8.00000	0.342682
$546$	−2.00000	−0.0855921
$547$	26.0000	1.11168	0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	18.0000	0.768922
$549$	14.0000	0.597505
$550$	0	0
$551$	0	0
$552$	1.00000	0.0425628
$553$	8.00000	0.340195
$554$	28.0000	1.18961
$555$	2.00000	0.0848953
$556$	−4.00000	−0.169638
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	4.00000	0.169334
$559$	−8.00000	−0.338364
$560$	1.00000	0.0422577
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	12.0000	0.505291
$565$	18.0000	0.757266
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	12.0000	0.503509
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	4.00000	0.167542
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	0	0
$573$	0	0
$574$	−12.0000	−0.500870
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	17.0000	0.707107
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	10.0000	0.414513
$583$	0	0
$584$	4.00000	0.165521
$585$	2.00000	0.0826898
$586$	−6.00000	−0.247858
$587$	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
$589$	16.0000	0.659269
$590$	−12.0000	−0.494032
$591$	−6.00000	−0.246807
$592$	2.00000	0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	14.0000	0.572982
$598$	2.00000	0.0817861
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	−1.00000	−0.0408248
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	4.00000	0.163028
$603$	−4.00000	−0.162893
$604$	−16.0000	−0.651031
$605$	−11.0000	−0.447214
$606$	−6.00000	−0.243733
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−14.0000	−0.566843
$611$	24.0000	0.970936
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	16.0000	0.645707
$615$	12.0000	0.483887
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	4.00000	0.160904
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−4.00000	−0.160644
$621$	−1.00000	−0.0401286
$622$	6.00000	0.240578
$623$	6.00000	0.240385
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	22.0000	0.879297
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	−1.00000	−0.0398410
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−8.00000	−0.318223
$633$	−16.0000	−0.635943
$634$	−6.00000	−0.238290
$635$	20.0000	0.793676
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−12.0000	−0.474713
$640$	−1.00000	−0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−6.00000	−0.236801
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−1.00000	−0.0394055
$645$	−4.00000	−0.157500
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	−4.00000	−0.156772
$652$	14.0000	0.548282
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−8.00000	−0.312825
$655$	−12.0000	−0.468879
$656$	12.0000	0.468521
$657$	−4.00000	−0.156055
$658$	−12.0000	−0.467809
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	0	0
$665$	−4.00000	−0.155113
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−24.0000	−0.928588
$669$	26.0000	1.00522
$670$	4.00000	0.154533
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	−14.0000	−0.539260
$675$	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	$0.201038\pi$
$678$	−18.0000	−0.691286
$679$	−10.0000	−0.383765
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−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
$687$	−22.0000	−0.839352
$688$	−4.00000	−0.152499
$689$	0	0
$690$	1.00000	0.0380693
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−4.00000	−0.151729
$696$	0	0
$697$	0	0
$698$	−2.00000	−0.0757011
$699$	−30.0000	−1.13470
$700$	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
$703$	−8.00000	−0.301726
$704$	0	0
$705$	12.0000	0.451946
$706$	−18.0000	−0.677439
$707$	6.00000	0.225653
$708$	12.0000	0.450988
$709$	32.0000	1.20179	0.600893	−	0.799330i	$-0.294812\pi$
$709$	32.0000	1.20179	0.600893	+	0.799330i	$0.294812\pi$
$710$	12.0000	0.450352
$711$	8.00000	0.300023
$712$	−6.00000	−0.224860
$713$	4.00000	0.149801
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	12.0000	0.448148
$718$	−24.0000	−0.895672
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	1.00000	0.0372678
$721$	−4.00000	−0.148968
$722$	3.00000	0.111648
$723$	20.0000	0.743808
$724$	−22.0000	−0.817624
$725$	0	0
$726$	11.0000	0.408248
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	0	0
$732$	14.0000	0.517455
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−8.00000	−0.295285
$735$	1.00000	0.0368856
$736$	1.00000	0.0368605
$737$	0	0
$738$	−12.0000	−0.441726
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	2.00000	0.0735215
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	4.00000	0.146647
$745$	6.00000	0.219823
$746$	−38.0000	−1.39128
$747$	0	0
$748$	0	0
$749$	6.00000	0.219235
$750$	−1.00000	−0.0365148
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	12.0000	0.437595
$753$	12.0000	0.437304
$754$	0	0
$755$	−16.0000	−0.582300
$756$	1.00000	0.0363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	34.0000	1.23494
$759$	0	0
$760$	4.00000	0.145095
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−20.0000	−0.724524
$763$	8.00000	0.289619
$764$	0	0
$765$	0	0
$766$	18.0000	0.650366
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	14.0000	0.503871
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	−4.00000	−0.143684
$776$	10.0000	0.358979
$777$	2.00000	0.0717496
$778$	6.00000	0.215110
$779$	−48.0000	−1.71978
$780$	2.00000	0.0716115
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	14.0000	0.499681
$786$	12.0000	0.428026
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−8.00000	−0.284627
$791$	18.0000	0.640006
$792$	0	0
$793$	28.0000	0.994309
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	18.0000	0.635602
$803$	0	0
$804$	−4.00000	−0.141069
$805$	−1.00000	−0.0352454
$806$	8.00000	0.281788
$807$	18.0000	0.633630
$808$	−6.00000	−0.211079
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−1.00000	−0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	0	0
$815$	14.0000	0.490399
$816$	0	0
$817$	16.0000	0.559769
$818$	−26.0000	−0.909069
$819$	2.00000	0.0698857
$820$	12.0000	0.419058
$821$	48.0000	1.67521	0.837606	−	0.546275i	$-0.183955\pi$
$821$	48.0000	1.67521	0.837606	+	0.546275i	$0.183955\pi$
$822$	−18.0000	−0.627822
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−12.0000	−0.417533
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	−1.00000	−0.0347524
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	2.00000	0.0693375
$833$	0	0
$834$	4.00000	0.138509
$835$	−24.0000	−0.830554
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−12.0000	−0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−1.00000	−0.0345033
$841$	−29.0000	−1.00000
$842$	28.0000	0.964944
$843$	18.0000	0.619953
$844$	−16.0000	−0.550743
$845$	−9.00000	−0.309609
$846$	−12.0000	−0.412568
$847$	−11.0000	−0.377964
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	−12.0000	−0.411113
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	−14.0000	−0.479070
$855$	−4.00000	−0.136797
$856$	−6.00000	−0.205076
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−4.00000	−0.136399
$861$	12.0000	0.408959
$862$	24.0000	0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	6.00000	0.204006
$866$	34.0000	1.15537
$867$	−17.0000	−0.577350
$868$	−4.00000	−0.135769
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−8.00000	−0.270914
$873$	−10.0000	−0.338449
$874$	−4.00000	−0.135302
$875$	1.00000	0.0338062
$876$	−4.00000	−0.135147
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	−20.0000	−0.674967
$879$	6.00000	0.202375
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−1.00000	−0.0336718
$883$	14.0000	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$883$	14.0000	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	−36.0000	−1.20944
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	−2.00000	−0.0671156
$889$	20.0000	0.670778
$890$	−6.00000	−0.201120
$891$	0	0
$892$	26.0000	0.870544
$893$	−48.0000	−1.60626
$894$	−6.00000	−0.200670
$895$	−6.00000	−0.200558
$896$	−1.00000	−0.0334077
$897$	−2.00000	−0.0667781
$898$	−6.00000	−0.200223
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	0	0
$903$	−4.00000	−0.133112
$904$	−18.0000	−0.598671
$905$	−22.0000	−0.731305
$906$	16.0000	0.531564
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	12.0000	0.398234
$909$	6.00000	0.199007
$910$	−2.00000	−0.0662994
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	−38.0000	−1.25693
$915$	14.0000	0.462826
$916$	−22.0000	−0.726900
$917$	−12.0000	−0.396275
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	1.00000	0.0329690
$921$	−16.0000	−0.527218
$922$	30.0000	0.987997
$923$	−24.0000	−0.789970
$924$	0	0
$925$	2.00000	0.0657596
$926$	4.00000	0.131448
$927$	−4.00000	−0.131377
$928$	0	0
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	4.00000	0.131165
$931$	−4.00000	−0.131095
$932$	−30.0000	−0.982683
$933$	−6.00000	−0.196431
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	4.00000	0.130605
$939$	−22.0000	−0.717943
$940$	12.0000	0.391397
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	−14.0000	−0.456145
$943$	−12.0000	−0.390774
$944$	12.0000	0.390567
$945$	1.00000	0.0325300
$946$	0	0
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	8.00000	0.259828
$949$	−8.00000	−0.259691
$950$	4.00000	0.129777
$951$	6.00000	0.194563
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	24.0000	0.775405
$959$	18.0000	0.581250
$960$	1.00000	0.0322749
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	6.00000	0.193347
$964$	20.0000	0.644157
$965$	14.0000	0.450676
$966$	1.00000	0.0321745
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	10.0000	0.321081
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000	0.0320750
$973$	−4.00000	−0.128234
$974$	16.0000	0.512673
$975$	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	$0.407031\pi$
$978$	−14.0000	−0.447671
$979$	0	0
$980$	1.00000	0.0319438
$981$	8.00000	0.255420
$982$	−6.00000	−0.191468
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	−12.0000	−0.382546
$985$	−6.00000	−0.191176
$986$	0	0
$987$	12.0000	0.381964
$988$	−8.00000	−0.254514
$989$	4.00000	0.127193
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	4.00000	0.127000
$993$	−28.0000	−0.888553
$994$	12.0000	0.380617
$995$	14.0000	0.443830
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	16.0000	0.506471
$999$	2.00000	0.0632772

Twists

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

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