LMFDB - Embedded newform 9386.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9386,2,Mod(1,9386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9386.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9386 = 2 \cdot 13 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9386.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:	$74.9475873372$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} +6.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} -3.00000 q^{20} +1.00000 q^{21} +6.00000 q^{22} -1.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} +3.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} +3.00000 q^{35} -2.00000 q^{36} +7.00000 q^{37} +1.00000 q^{39} -3.00000 q^{40} +1.00000 q^{42} -1.00000 q^{43} +6.00000 q^{44} +6.00000 q^{45} +3.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} +4.00000 q^{50} +3.00000 q^{51} -1.00000 q^{52} +5.00000 q^{54} -18.0000 q^{55} -1.00000 q^{56} -6.00000 q^{58} +6.00000 q^{59} +3.00000 q^{60} +8.00000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{65} -6.00000 q^{66} -14.0000 q^{67} -3.00000 q^{68} +3.00000 q^{70} +3.00000 q^{71} -2.00000 q^{72} +2.00000 q^{73} +7.00000 q^{74} -4.00000 q^{75} -6.00000 q^{77} +1.00000 q^{78} -8.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +12.0000 q^{83} +1.00000 q^{84} +9.00000 q^{85} -1.00000 q^{86} +6.00000 q^{87} +6.00000 q^{88} +6.00000 q^{89} +6.00000 q^{90} +1.00000 q^{91} -4.00000 q^{93} +3.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} -6.00000 q^{98} -12.0000 q^{99} +O(q^{100})\)

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−3.00000	−0.948683
$11$	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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.00000	−0.267261
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	−2.00000	−0.471405
$19$	0	0
$19$	0	0
$20$	−3.00000	−0.670820
$21$	1.00000	0.218218
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	−1.00000	−0.196116
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	3.00000	0.547723
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	−3.00000	−0.514496
$35$	3.00000	0.507093
$36$	−2.00000	−0.333333
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	1.00000	0.154303
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	6.00000	0.904534
$45$	6.00000	0.894427
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	−6.00000	−0.857143
$50$	4.00000	0.565685
$51$	3.00000	0.420084
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	5.00000	0.680414
$55$	−18.0000	−2.42712
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.00000	−0.787839
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	3.00000	0.387298
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000	0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	−6.00000	−0.738549
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	3.00000	0.358569
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	−2.00000	−0.235702
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	7.00000	0.813733
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−6.00000	−0.683763
$78$	1.00000	0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−3.00000	−0.335410
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	1.00000	0.109109
$85$	9.00000	0.976187
$86$	−1.00000	−0.107833
$87$	6.00000	0.643268
$88$	6.00000	0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	6.00000	0.632456
$91$	1.00000	0.104828
$92$	0	0
$93$	−4.00000	−0.414781
$94$	3.00000	0.309426
$95$	0	0
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−6.00000	−0.606092
$99$	−12.0000	−1.20605
$100$	4.00000	0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	3.00000	0.297044
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−1.00000	−0.0980581
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	5.00000	0.481125
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−18.0000	−1.71623
$111$	−7.00000	−0.664411
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	2.00000	0.184900
$118$	6.00000	0.552345
$119$	3.00000	0.275010
$120$	3.00000	0.273861
$121$	25.0000	2.27273
$122$	8.00000	0.724286
$123$	0	0
$124$	4.00000	0.359211
$125$	3.00000	0.268328
$126$	2.00000	0.178174
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	1.00000	0.0880451
$130$	3.00000	0.263117
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−14.0000	−1.20942
$135$	−15.0000	−1.29099
$136$	−3.00000	−0.257248
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	3.00000	0.253546
$141$	−3.00000	−0.252646
$142$	3.00000	0.251754
$143$	−6.00000	−0.501745
$144$	−2.00000	−0.166667
$145$	18.0000	1.49482
$146$	2.00000	0.165521
$147$	6.00000	0.494872
$148$	7.00000	0.575396
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−4.00000	−0.326599
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	−6.00000	−0.483494
$155$	−12.0000	−0.963863
$156$	1.00000	0.0800641
$157$	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
$160$	−3.00000	−0.237171
$161$	0	0
$162$	1.00000	0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	18.0000	1.40130
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	1.00000	0.0771517
$169$	1.00000	0.0769231
$170$	9.00000	0.690268
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	6.00000	0.454859
$175$	−4.00000	−0.302372
$176$	6.00000	0.452267
$177$	−6.00000	−0.450988
$178$	6.00000	0.449719
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	6.00000	0.447214
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	1.00000	0.0741249
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−21.0000	−1.54395
$186$	−4.00000	−0.293294
$187$	−18.0000	−1.31629
$188$	3.00000	0.218797
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−1.00000	−0.0721688
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	10.0000	0.717958
$195$	−3.00000	−0.214834
$196$	−6.00000	−0.428571
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	−12.0000	−0.852803
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	4.00000	0.282843
$201$	14.0000	0.987484
$202$	−12.0000	−0.844317
$203$	6.00000	0.421117
$204$	3.00000	0.210042
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	0	0
$210$	−3.00000	−0.207020
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	−3.00000	−0.205557
$214$	−12.0000	−0.820303
$215$	3.00000	0.204598
$216$	5.00000	0.340207
$217$	−4.00000	−0.271538
$218$	7.00000	0.474100
$219$	−2.00000	−0.135147
$220$	−18.0000	−1.21356
$221$	3.00000	0.201802
$222$	−7.00000	−0.469809
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	−1.00000	−0.0668153
$225$	−8.00000	−0.533333
$226$	6.00000	0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	−6.00000	−0.393919
$233$	−27.0000	−1.76883	−0.884414	−	0.466702i	$-0.845442\pi$
$233$	−27.0000	−1.76883	−0.884414	+	0.466702i	$0.845442\pi$
$234$	2.00000	0.130744
$235$	−9.00000	−0.587095
$236$	6.00000	0.390567
$237$	8.00000	0.519656
$238$	3.00000	0.194461
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	3.00000	0.193649
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	25.0000	1.60706
$243$	−16.0000	−1.02640
$244$	8.00000	0.512148
$245$	18.0000	1.14998
$246$	0	0
$247$	0	0
$248$	4.00000	0.254000
$249$	−12.0000	−0.760469
$250$	3.00000	0.189737
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−20.0000	−1.25491
$255$	−9.00000	−0.563602
$256$	1.00000	0.0625000
$257$	−9.00000	−0.561405	−0.280702	−	0.959795i	$-0.590567\pi$
$257$	−9.00000	−0.561405	−0.280702	+	0.959795i	$0.590567\pi$
$258$	1.00000	0.0622573
$259$	−7.00000	−0.434959
$260$	3.00000	0.186052
$261$	12.0000	0.742781
$262$	−21.0000	−1.29738
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−6.00000	−0.369274
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−14.0000	−0.855186
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	−15.0000	−0.912871
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−3.00000	−0.181902
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	24.0000	1.44725
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−13.0000	−0.779688
$279$	−8.00000	−0.478947
$280$	3.00000	0.179284
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	−3.00000	−0.178647
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	18.0000	1.05700
$291$	−10.0000	−0.586210
$292$	2.00000	0.117041
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	6.00000	0.349927
$295$	−18.0000	−1.04800
$296$	7.00000	0.406867
$297$	30.0000	1.74078
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−4.00000	−0.230940
$301$	1.00000	0.0576390
$302$	−17.0000	−0.978240
$303$	12.0000	0.689382
$304$	0	0
$305$	−24.0000	−1.37424
$306$	6.00000	0.342997
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−6.00000	−0.341882
$309$	−4.00000	−0.227552
$310$	−12.0000	−0.681554
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	1.00000	0.0566139
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	14.0000	0.790066
$315$	−6.00000	−0.338062
$316$	−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
$319$	−36.0000	−2.01561
$320$	−3.00000	−0.167705
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−16.0000	−0.886158
$327$	−7.00000	−0.387101
$328$	0	0
$329$	−3.00000	−0.165395
$330$	18.0000	0.990867
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	12.0000	0.658586
$333$	−14.0000	−0.767195
$334$	0	0
$335$	42.0000	2.29471
$336$	1.00000	0.0545545
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	1.00000	0.0543928
$339$	−6.00000	−0.325875
$340$	9.00000	0.488094
$341$	24.0000	1.29967
$342$	0	0
$343$	13.0000	0.701934
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	0	0
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	6.00000	0.321634
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	−4.00000	−0.213809
$351$	−5.00000	−0.266880
$352$	6.00000	0.319801
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−6.00000	−0.318896
$355$	−9.00000	−0.477670
$356$	6.00000	0.317999
$357$	−3.00000	−0.158777
$358$	−3.00000	−0.158555
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	6.00000	0.316228
$361$	0	0
$362$	−20.0000	−1.05118
$363$	−25.0000	−1.31216
$364$	1.00000	0.0524142
$365$	−6.00000	−0.314054
$366$	−8.00000	−0.418167
$367$	26.0000	1.35719	0.678594	−	0.734513i	$-0.262589\pi$
$367$	26.0000	1.35719	0.678594	+	0.734513i	$0.262589\pi$
$368$	0	0
$369$	0	0
$370$	−21.0000	−1.09174
$371$	0	0
$372$	−4.00000	−0.207390
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	−18.0000	−0.930758
$375$	−3.00000	−0.154919
$376$	3.00000	0.154713
$377$	6.00000	0.309016
$378$	−5.00000	−0.257172
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	−18.0000	−0.920960
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	−1.00000	−0.0510310
$385$	18.0000	0.917365
$386$	4.00000	0.203595
$387$	2.00000	0.101666
$388$	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	$0.548606\pi$
$390$	−3.00000	−0.151911
$391$	0	0
$392$	−6.00000	−0.303046
$393$	21.0000	1.05931
$394$	3.00000	0.151138
$395$	24.0000	1.20757
$396$	−12.0000	−0.603023
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	2.00000	0.100251
$399$	0	0
$400$	4.00000	0.200000
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	14.0000	0.698257
$403$	−4.00000	−0.199254
$404$	−12.0000	−0.597022
$405$	−3.00000	−0.149071
$406$	6.00000	0.297775
$407$	42.0000	2.08186
$408$	3.00000	0.148522
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	4.00000	0.197066
$413$	−6.00000	−0.295241
$414$	0	0
$415$	−36.0000	−1.76717
$416$	−1.00000	−0.0490290
$417$	13.0000	0.636613
$418$	0	0
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	−3.00000	−0.146385
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	13.0000	0.632830
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−12.0000	−0.582086
$426$	−3.00000	−0.145350
$427$	−8.00000	−0.387147
$428$	−12.0000	−0.580042
$429$	6.00000	0.289683
$430$	3.00000	0.144673
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	5.00000	0.240563
$433$	25.0000	1.20142	0.600712	−	0.799466i	$-0.294884\pi$
$433$	25.0000	1.20142	0.600712	+	0.799466i	$0.294884\pi$
$434$	−4.00000	−0.192006
$435$	−18.0000	−0.863034
$436$	7.00000	0.335239
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	−18.0000	−0.858116
$441$	12.0000	0.571429
$442$	3.00000	0.142695
$443$	21.0000	0.997740	0.498870	−	0.866677i	$-0.333748\pi$
$443$	21.0000	0.997740	0.498870	+	0.866677i	$0.333748\pi$
$444$	−7.00000	−0.332205
$445$	−18.0000	−0.853282
$446$	19.0000	0.899676
$447$	6.00000	0.283790
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−8.00000	−0.377124
$451$	0	0
$452$	6.00000	0.282216
$453$	17.0000	0.798730
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−13.0000	−0.607450
$459$	−15.0000	−0.700140
$460$	0	0
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	6.00000	0.279145
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−6.00000	−0.278543
$465$	12.0000	0.556487
$466$	−27.0000	−1.25075
$467$	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
$469$	14.0000	0.646460
$470$	−9.00000	−0.415139
$471$	−14.0000	−0.645086
$472$	6.00000	0.276172
$473$	−6.00000	−0.275880
$474$	8.00000	0.367452
$475$	0	0
$476$	3.00000	0.137505
$477$	0	0
$478$	15.0000	0.686084
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	3.00000	0.136931
$481$	−7.00000	−0.319173
$482$	10.0000	0.455488
$483$	0	0
$484$	25.0000	1.13636
$485$	−30.0000	−1.36223
$486$	−16.0000	−0.725775
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	8.00000	0.362143
$489$	16.0000	0.723545
$490$	18.0000	0.813157
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	36.0000	1.61808
$496$	4.00000	0.179605
$497$	−3.00000	−0.134568
$498$	−12.0000	−0.537733
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	3.00000	0.134164
$501$	0	0
$502$	24.0000	1.07117
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	2.00000	0.0890871
$505$	36.0000	1.60198
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	−20.0000	−0.887357
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	−9.00000	−0.398527
$511$	−2.00000	−0.0884748
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.00000	−0.396973
$515$	−12.0000	−0.528783
$516$	1.00000	0.0440225
$517$	18.0000	0.791639
$518$	−7.00000	−0.307562
$519$	0	0
$520$	3.00000	0.131559
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	12.0000	0.525226
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−21.0000	−0.917389
$525$	4.00000	0.174574
$526$	−12.0000	−0.523225
$527$	−12.0000	−0.522728
$528$	−6.00000	−0.261116
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	36.0000	1.55642
$536$	−14.0000	−0.604708
$537$	3.00000	0.129460
$538$	−24.0000	−1.03471
$539$	−36.0000	−1.55063
$540$	−15.0000	−0.645497
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	11.0000	0.472490
$543$	20.0000	0.858282
$544$	−3.00000	−0.128624
$545$	−21.0000	−0.899541
$546$	−1.00000	−0.0427960
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	0	0
$549$	−16.0000	−0.682863
$550$	24.0000	1.02336
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	−28.0000	−1.18961
$555$	21.0000	0.891400
$556$	−13.0000	−0.551323
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	−8.00000	−0.338667
$559$	1.00000	0.0422955
$560$	3.00000	0.126773
$561$	18.0000	0.759961
$562$	6.00000	0.253095
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	−3.00000	−0.126323
$565$	−18.0000	−0.757266
$566$	−4.00000	−0.168133
$567$	−1.00000	−0.0419961
$568$	3.00000	0.125877
$569$	−15.0000	−0.628833	−0.314416	−	0.949285i	$-0.601809\pi$
$569$	−15.0000	−0.628833	−0.314416	+	0.949285i	$0.601809\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	−6.00000	−0.250873
$573$	18.0000	0.751961
$574$	0	0
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	−4.00000	−0.166234
$580$	18.0000	0.747409
$581$	−12.0000	−0.497844
$582$	−10.0000	−0.414513
$583$	0	0
$584$	2.00000	0.0827606
$585$	−6.00000	−0.248069
$586$	−21.0000	−0.867502
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	6.00000	0.247436
$589$	0	0
$590$	−18.0000	−0.741048
$591$	−3.00000	−0.123404
$592$	7.00000	0.287698
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	30.0000	1.23091
$595$	−9.00000	−0.368964
$596$	−6.00000	−0.245770
$597$	−2.00000	−0.0818546
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	−4.00000	−0.163299
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	1.00000	0.0407570
$603$	28.0000	1.14025
$604$	−17.0000	−0.691720
$605$	−75.0000	−3.04918
$606$	12.0000	0.487467
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	−6.00000	−0.243132
$610$	−24.0000	−0.971732
$611$	−3.00000	−0.121367
$612$	6.00000	0.242536
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−6.00000	−0.241747
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	−4.00000	−0.160904
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	−12.0000	−0.481932
$621$	0	0
$622$	−30.0000	−1.20289
$623$	−6.00000	−0.240385
$624$	1.00000	0.0400320
$625$	−29.0000	−1.16000
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	14.0000	0.558661
$629$	−21.0000	−0.837325
$630$	−6.00000	−0.239046
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	−8.00000	−0.318223
$633$	−13.0000	−0.516704
$634$	6.00000	0.238290
$635$	60.0000	2.38103
$636$	0	0
$637$	6.00000	0.237729
$638$	−36.0000	−1.42525
$639$	−6.00000	−0.237356
$640$	−3.00000	−0.118585
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	12.0000	0.473602
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	−3.00000	−0.118125
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	1.00000	0.0392837
$649$	36.0000	1.41312
$650$	−4.00000	−0.156893
$651$	4.00000	0.156772
$652$	−16.0000	−0.626608
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−7.00000	−0.273722
$655$	63.0000	2.46161
$656$	0	0
$657$	−4.00000	−0.156055
$658$	−3.00000	−0.116952
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	18.0000	0.700649
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−8.00000	−0.310929
$663$	−3.00000	−0.116510
$664$	12.0000	0.465690
$665$	0	0
$666$	−14.0000	−0.542489
$667$	0	0
$668$	0	0
$669$	−19.0000	−0.734582
$670$	42.0000	1.62260
$671$	48.0000	1.85302
$672$	1.00000	0.0385758
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−23.0000	−0.885927
$675$	20.0000	0.769800
$676$	1.00000	0.0384615
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	−6.00000	−0.230429
$679$	−10.0000	−0.383765
$680$	9.00000	0.345134
$681$	0	0
$682$	24.0000	0.919007
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	13.0000	0.495981
$688$	−1.00000	−0.0381246
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	3.00000	0.113878
$695$	39.0000	1.47935
$696$	6.00000	0.227429
$697$	0	0
$698$	−19.0000	−0.719161
$699$	27.0000	1.02123
$700$	−4.00000	−0.151186
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−5.00000	−0.188713
$703$	0	0
$704$	6.00000	0.226134
$705$	9.00000	0.338960
$706$	24.0000	0.903252
$707$	12.0000	0.451306
$708$	−6.00000	−0.225494
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−9.00000	−0.337764
$711$	16.0000	0.600047
$712$	6.00000	0.224860
$713$	0	0
$714$	−3.00000	−0.112272
$715$	18.0000	0.673162
$716$	−3.00000	−0.112115
$717$	−15.0000	−0.560185
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	6.00000	0.223607
$721$	−4.00000	−0.148968
$722$	0	0
$723$	−10.0000	−0.371904
$724$	−20.0000	−0.743294
$725$	−24.0000	−0.891338
$726$	−25.0000	−0.927837
$727$	−10.0000	−0.370879	−0.185440	−	0.982656i	$-0.559371\pi$
$727$	−10.0000	−0.370879	−0.185440	+	0.982656i	$0.559371\pi$
$728$	1.00000	0.0370625
$729$	13.0000	0.481481
$730$	−6.00000	−0.222070
$731$	3.00000	0.110959
$732$	−8.00000	−0.295689
$733$	23.0000	0.849524	0.424762	−	0.905305i	$-0.360358\pi$
$733$	23.0000	0.849524	0.424762	+	0.905305i	$0.360358\pi$
$734$	26.0000	0.959678
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−84.0000	−3.09418
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	−4.00000	−0.146647
$745$	18.0000	0.659469
$746$	4.00000	0.146450
$747$	−24.0000	−0.878114
$748$	−18.0000	−0.658145
$749$	12.0000	0.438470
$750$	−3.00000	−0.109545
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	3.00000	0.109399
$753$	−24.0000	−0.874609
$754$	6.00000	0.218507
$755$	51.0000	1.85608
$756$	−5.00000	−0.181848
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	20.0000	0.724524
$763$	−7.00000	−0.253417
$764$	−18.0000	−0.651217
$765$	−18.0000	−0.650791
$766$	−21.0000	−0.758761
$767$	−6.00000	−0.216647
$768$	−1.00000	−0.0360844
$769$	32.0000	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$769$	32.0000	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$770$	18.0000	0.648675
$771$	9.00000	0.324127
$772$	4.00000	0.143963
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	2.00000	0.0718885
$775$	16.0000	0.574737
$776$	10.0000	0.358979
$777$	7.00000	0.251124
$778$	−6.00000	−0.215110
$779$	0	0
$780$	−3.00000	−0.107417
$781$	18.0000	0.644091
$782$	0	0
$783$	−30.0000	−1.07211
$784$	−6.00000	−0.214286
$785$	−42.0000	−1.49904
$786$	21.0000	0.749045
$787$	40.0000	1.42585	0.712923	−	0.701242i	$-0.247371\pi$
$787$	40.0000	1.42585	0.712923	+	0.701242i	$0.247371\pi$
$788$	3.00000	0.106871
$789$	12.0000	0.427211
$790$	24.0000	0.853882
$791$	−6.00000	−0.213335
$792$	−12.0000	−0.426401
$793$	−8.00000	−0.284088
$794$	−34.0000	−1.20661
$795$	0	0
$796$	2.00000	0.0708881
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	4.00000	0.141421
$801$	−12.0000	−0.423999
$802$	−36.0000	−1.27120
$803$	12.0000	0.423471
$804$	14.0000	0.493742
$805$	0	0
$806$	−4.00000	−0.140894
$807$	24.0000	0.844840
$808$	−12.0000	−0.422159
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	−3.00000	−0.105409
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	6.00000	0.210559
$813$	−11.0000	−0.385787
$814$	42.0000	1.47210
$815$	48.0000	1.68137
$816$	3.00000	0.105021
$817$	0	0
$818$	−32.0000	−1.11885
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	4.00000	0.139347
$825$	−24.0000	−0.835573
$826$	−6.00000	−0.208767
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	−36.0000	−1.24958
$831$	28.0000	0.971309
$832$	−1.00000	−0.0346688
$833$	18.0000	0.623663
$834$	13.0000	0.450153
$835$	0	0
$836$	0	0
$837$	20.0000	0.691301
$838$	9.00000	0.310900
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−3.00000	−0.103510
$841$	7.00000	0.241379
$842$	−17.0000	−0.585859
$843$	−6.00000	−0.206651
$844$	13.0000	0.447478
$845$	−3.00000	−0.103203
$846$	−6.00000	−0.206284
$847$	−25.0000	−0.859010
$848$	0	0
$849$	4.00000	0.137280
$850$	−12.0000	−0.411597
$851$	0	0
$852$	−3.00000	−0.102778
$853$	−37.0000	−1.26686	−0.633428	−	0.773802i	$-0.718353\pi$
$853$	−37.0000	−1.26686	−0.633428	+	0.773802i	$0.718353\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	6.00000	0.204837
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	3.00000	0.102299
$861$	0	0
$862$	33.0000	1.12398
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	25.0000	0.849535
$867$	8.00000	0.271694
$868$	−4.00000	−0.135769
$869$	−48.0000	−1.62829
$870$	−18.0000	−0.610257
$871$	14.0000	0.474372
$872$	7.00000	0.237050
$873$	−20.0000	−0.676897
$874$	0	0
$875$	−3.00000	−0.101419
$876$	−2.00000	−0.0675737
$877$	13.0000	0.438979	0.219489	−	0.975615i	$-0.429561\pi$
$877$	13.0000	0.438979	0.219489	+	0.975615i	$0.429561\pi$
$878$	−26.0000	−0.877457
$879$	21.0000	0.708312
$880$	−18.0000	−0.606780
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	12.0000	0.404061
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	3.00000	0.100901
$885$	18.0000	0.605063
$886$	21.0000	0.705509
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	−7.00000	−0.234905
$889$	20.0000	0.670778
$890$	−18.0000	−0.603361
$891$	6.00000	0.201008
$892$	19.0000	0.636167
$893$	0	0
$894$	6.00000	0.200670
$895$	9.00000	0.300837
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−24.0000	−0.800445
$900$	−8.00000	−0.266667
$901$	0	0
$902$	0	0
$903$	−1.00000	−0.0332779
$904$	6.00000	0.199557
$905$	60.0000	1.99447
$906$	17.0000	0.564787
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	0	0
$909$	24.0000	0.796030
$910$	−3.00000	−0.0994490
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	−10.0000	−0.330771
$915$	24.0000	0.793416
$916$	−13.0000	−0.429532
$917$	21.0000	0.693481
$918$	−15.0000	−0.495074
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	9.00000	0.296399
$923$	−3.00000	−0.0987462
$924$	6.00000	0.197386
$925$	28.0000	0.920634
$926$	−40.0000	−1.31448
$927$	−8.00000	−0.262754
$928$	−6.00000	−0.196960
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	12.0000	0.393496
$931$	0	0
$932$	−27.0000	−0.884414
$933$	30.0000	0.982156
$934$	36.0000	1.17796
$935$	54.0000	1.76599
$936$	2.00000	0.0653720
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	14.0000	0.457116
$939$	1.00000	0.0326338
$940$	−9.00000	−0.293548
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	−14.0000	−0.456145
$943$	0	0
$944$	6.00000	0.195283
$945$	15.0000	0.487950
$946$	−6.00000	−0.195077
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	8.00000	0.259828
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	−6.00000	−0.194563
$952$	3.00000	0.0972306
$953$	−15.0000	−0.485898	−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000	−0.485898	−0.242949	+	0.970039i	$0.578115\pi$
$954$	0	0
$955$	54.0000	1.74740
$956$	15.0000	0.485135
$957$	36.0000	1.16371
$958$	−21.0000	−0.678479
$959$	0	0
$960$	3.00000	0.0968246
$961$	−15.0000	−0.483871
$962$	−7.00000	−0.225689
$963$	24.0000	0.773389
$964$	10.0000	0.322078
$965$	−12.0000	−0.386294
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−30.0000	−0.963242
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	−16.0000	−0.513200
$973$	13.0000	0.416761
$974$	16.0000	0.512673
$975$	4.00000	0.128103
$976$	8.00000	0.256074
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	16.0000	0.511624
$979$	36.0000	1.15056
$980$	18.0000	0.574989
$981$	−14.0000	−0.446986
$982$	−9.00000	−0.287202
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	−9.00000	−0.286764
$986$	18.0000	0.573237
$987$	3.00000	0.0954911
$988$	0	0
$989$	0	0
$990$	36.0000	1.14416
$991$	−2.00000	−0.0635321	−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000	−0.0635321	−0.0317660	+	0.999495i	$0.510113\pi$
$992$	4.00000	0.127000
$993$	8.00000	0.253872
$994$	−3.00000	−0.0951542
$995$	−6.00000	−0.190213
$996$	−12.0000	−0.380235
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−40.0000	−1.26618
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9386.2.a.j.1.1		1
19.18	odd	2		26.2.a.a.1.1	✓	1
57.56	even	2		234.2.a.e.1.1		1
76.75	even	2		208.2.a.a.1.1		1
95.18	even	4		650.2.b.d.599.2		2
95.37	even	4		650.2.b.d.599.1		2
95.94	odd	2		650.2.a.j.1.1		1
133.18	odd	6		1274.2.f.p.79.1		2
133.37	odd	6		1274.2.f.p.1145.1		2
133.75	even	6		1274.2.f.r.1145.1		2
133.94	even	6		1274.2.f.r.79.1		2
133.132	even	2		1274.2.a.d.1.1		1
152.37	odd	2		832.2.a.d.1.1		1
152.75	even	2		832.2.a.i.1.1		1
171.56	even	6		2106.2.e.b.1405.1		2
171.94	odd	6		2106.2.e.ba.703.1		2
171.113	even	6		2106.2.e.b.703.1		2
171.151	odd	6		2106.2.e.ba.1405.1		2
209.208	even	2		3146.2.a.n.1.1		1
228.227	odd	2		1872.2.a.q.1.1		1
247.18	even	4		338.2.b.c.337.2		2
247.37	even	12		338.2.e.a.147.2		4
247.56	odd	6		338.2.c.a.315.1		2
247.75	odd	6		338.2.c.a.191.1		2
247.94	odd	6		338.2.c.d.191.1		2
247.113	odd	6		338.2.c.d.315.1		2
247.132	even	12		338.2.e.a.147.1		4
247.151	even	4		338.2.b.c.337.1		2
247.189	even	12		338.2.e.a.23.1		4
247.227	even	12		338.2.e.a.23.2		4
247.246	odd	2		338.2.a.f.1.1		1
285.113	odd	4		5850.2.e.a.5149.1		2
285.227	odd	4		5850.2.e.a.5149.2		2
285.284	even	2		5850.2.a.p.1.1		1
304.37	odd	4		3328.2.b.m.1665.1		2
304.75	even	4		3328.2.b.j.1665.2		2
304.189	odd	4		3328.2.b.m.1665.2		2
304.227	even	4		3328.2.b.j.1665.1		2
323.322	odd	2		7514.2.a.c.1.1		1
380.379	even	2		5200.2.a.x.1.1		1
456.227	odd	2		7488.2.a.h.1.1		1
456.341	even	2		7488.2.a.g.1.1		1
741.398	odd	4		3042.2.b.a.1351.2		2
741.512	odd	4		3042.2.b.a.1351.1		2
741.740	even	2		3042.2.a.a.1.1		1
988.151	odd	4		2704.2.f.d.337.1		2
988.759	odd	4		2704.2.f.d.337.2		2
988.987	even	2		2704.2.a.f.1.1		1
1235.1234	odd	2		8450.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	19.18	odd	2
208.2.a.a.1.1		1	76.75	even	2
234.2.a.e.1.1		1	57.56	even	2
338.2.a.f.1.1		1	247.246	odd	2
338.2.b.c.337.1		2	247.151	even	4
338.2.b.c.337.2		2	247.18	even	4
338.2.c.a.191.1		2	247.75	odd	6
338.2.c.a.315.1		2	247.56	odd	6
338.2.c.d.191.1		2	247.94	odd	6
338.2.c.d.315.1		2	247.113	odd	6
338.2.e.a.23.1		4	247.189	even	12
338.2.e.a.23.2		4	247.227	even	12
338.2.e.a.147.1		4	247.132	even	12
338.2.e.a.147.2		4	247.37	even	12
650.2.a.j.1.1		1	95.94	odd	2
650.2.b.d.599.1		2	95.37	even	4
650.2.b.d.599.2		2	95.18	even	4
832.2.a.d.1.1		1	152.37	odd	2
832.2.a.i.1.1		1	152.75	even	2
1274.2.a.d.1.1		1	133.132	even	2
1274.2.f.p.79.1		2	133.18	odd	6
1274.2.f.p.1145.1		2	133.37	odd	6
1274.2.f.r.79.1		2	133.94	even	6
1274.2.f.r.1145.1		2	133.75	even	6
1872.2.a.q.1.1		1	228.227	odd	2
2106.2.e.b.703.1		2	171.113	even	6
2106.2.e.b.1405.1		2	171.56	even	6
2106.2.e.ba.703.1		2	171.94	odd	6
2106.2.e.ba.1405.1		2	171.151	odd	6
2704.2.a.f.1.1		1	988.987	even	2
2704.2.f.d.337.1		2	988.151	odd	4
2704.2.f.d.337.2		2	988.759	odd	4
3042.2.a.a.1.1		1	741.740	even	2
3042.2.b.a.1351.1		2	741.512	odd	4
3042.2.b.a.1351.2		2	741.398	odd	4
3146.2.a.n.1.1		1	209.208	even	2
3328.2.b.j.1665.1		2	304.227	even	4
3328.2.b.j.1665.2		2	304.75	even	4
3328.2.b.m.1665.1		2	304.37	odd	4
3328.2.b.m.1665.2		2	304.189	odd	4
5200.2.a.x.1.1		1	380.379	even	2
5850.2.a.p.1.1		1	285.284	even	2
5850.2.e.a.5149.1		2	285.113	odd	4
5850.2.e.a.5149.2		2	285.227	odd	4
7488.2.a.g.1.1		1	456.341	even	2
7488.2.a.h.1.1		1	456.227	odd	2
7514.2.a.c.1.1		1	323.322	odd	2
8450.2.a.c.1.1		1	1235.1234	odd	2
9386.2.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:	\( N \)	\(=\)	\( 9386 = 2 \cdot 13 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9386.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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