LMFDB - Embedded newform 9310.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9310,2,Mod(1,9310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9310, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9310.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9310 = 2 \cdot 5 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9310.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:	$74.3407242818$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9310.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +5.00000 q^{27} -3.00000 q^{29} +1.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -2.00000 q^{36} -10.0000 q^{37} -1.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{43} +2.00000 q^{45} +3.00000 q^{46} -1.00000 q^{48} +1.00000 q^{50} -3.00000 q^{51} +1.00000 q^{52} +3.00000 q^{53} +5.00000 q^{54} +1.00000 q^{57} -3.00000 q^{58} -3.00000 q^{59} +1.00000 q^{60} -8.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -7.00000 q^{67} +3.00000 q^{68} -3.00000 q^{69} +12.0000 q^{71} -2.00000 q^{72} +13.0000 q^{73} -10.0000 q^{74} -1.00000 q^{75} -1.00000 q^{76} -1.00000 q^{78} +14.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -3.00000 q^{85} +2.00000 q^{86} +3.00000 q^{87} -6.00000 q^{89} +2.00000 q^{90} +3.00000 q^{92} +2.00000 q^{93} +1.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.00000	−0.666667
$10$	−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
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.00000	−0.471405
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	5.00000	0.962250
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	1.00000	0.182574
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−1.00000	−0.162221
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	−3.00000	−0.420084
$52$	1.00000	0.138675
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	−3.00000	−0.393919
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	1.00000	0.129099
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	3.00000	0.363803
$69$	−3.00000	−0.361158
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−2.00000	−0.235702
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	−10.0000	−1.16248
$75$	−1.00000	−0.115470
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−1.00000	−0.113228
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	−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	$0.606810\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	2.00000	0.215666
$87$	3.00000	0.321634
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	3.00000	0.312772
$93$	2.00000	0.207390
$94$	0	0
$95$	1.00000	0.102598
$96$	−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	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−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
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	3.00000	0.291386
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	5.00000	0.481125
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	1.00000	0.0936586
$115$	−3.00000	−0.279751
$116$	−3.00000	−0.278543
$117$	−2.00000	−0.184900
$118$	−3.00000	−0.276172
$119$	0	0
$120$	1.00000	0.0912871
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	6.00000	0.541002
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	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
$129$	−2.00000	−0.176090
$130$	−1.00000	−0.0877058
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	−7.00000	−0.604708
$135$	−5.00000	−0.430331
$136$	3.00000	0.257248
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	−3.00000	−0.255377
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	0	0
$144$	−2.00000	−0.166667
$145$	3.00000	0.249136
$146$	13.0000	1.07589
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	−1.00000	−0.0816497
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	−1.00000	−0.0811107
$153$	−6.00000	−0.485071
$154$	0	0
$155$	2.00000	0.160644
$156$	−1.00000	−0.0800641
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	14.0000	1.11378
$159$	−3.00000	−0.237915
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−3.00000	−0.230089
$171$	2.00000	0.152944
$172$	2.00000	0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	0	0
$177$	3.00000	0.225494
$178$	−6.00000	−0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.00000	0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	3.00000	0.221163
$185$	10.0000	0.735215
$186$	2.00000	0.146647
$187$	0	0
$188$	0	0
$189$	0	0
$190$	1.00000	0.0725476
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	−1.00000	−0.0721688
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	10.0000	0.717958
$195$	1.00000	0.0716115
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	1.00000	0.0707107
$201$	7.00000	0.493742
$202$	−12.0000	−0.844317
$203$	0	0
$204$	−3.00000	−0.210042
$205$	6.00000	0.419058
$206$	−8.00000	−0.557386
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	−25.0000	−1.72107	−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000	−1.72107	−0.860535	+	0.509390i	$0.829871\pi$
$212$	3.00000	0.206041
$213$	−12.0000	−0.822226
$214$	15.0000	1.02538
$215$	−2.00000	−0.136399
$216$	5.00000	0.340207
$217$	0	0
$218$	11.0000	0.745014
$219$	−13.0000	−0.878459
$220$	0	0
$221$	3.00000	0.201802
$222$	10.0000	0.671156
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	−12.0000	−0.798228
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	1.00000	0.0662266
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	−3.00000	−0.196960
$233$	−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
$235$	0	0
$236$	−3.00000	−0.195283
$237$	−14.0000	−0.909398
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	1.00000	0.0645497
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	−11.0000	−0.707107
$243$	−16.0000	−1.02640
$244$	−8.00000	−0.512148
$245$	0	0
$246$	6.00000	0.382546
$247$	−1.00000	−0.0636285
$248$	−2.00000	−0.127000
$249$	6.00000	0.380235
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	2.00000	0.125491
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	−1.00000	−0.0620174
$261$	6.00000	0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	6.00000	0.367194
$268$	−7.00000	−0.427593
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−5.00000	−0.304290
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	9.00000	0.543710
$275$	0	0
$276$	−3.00000	−0.180579
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−8.00000	−0.479808
$279$	4.00000	0.239474
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	12.0000	0.712069
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	0	0
$288$	−2.00000	−0.117851
$289$	−8.00000	−0.470588
$290$	3.00000	0.176166
$291$	−10.0000	−0.586210
$292$	13.0000	0.760767
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−12.0000	−0.695141
$299$	3.00000	0.173494
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	−10.0000	−0.575435
$303$	12.0000	0.689382
$304$	−1.00000	−0.0573539
$305$	8.00000	0.458079
$306$	−6.00000	−0.342997
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	2.00000	0.113592
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	−1.00000	−0.0566139
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	14.0000	0.787562
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	−3.00000	−0.168232
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−15.0000	−0.837218
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	−22.0000	−1.21847
$327$	−11.0000	−0.608301
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	5.00000	0.274825	0.137412	−	0.990514i	$-0.456121\pi$
$331$	5.00000	0.274825	0.137412	+	0.990514i	$0.456121\pi$
$332$	−6.00000	−0.329293
$333$	20.0000	1.09599
$334$	−18.0000	−0.984916
$335$	7.00000	0.382451
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−12.0000	−0.652714
$339$	12.0000	0.651751
$340$	−3.00000	−0.162698
$341$	0	0
$342$	2.00000	0.108148
$343$	0	0
$344$	2.00000	0.107833
$345$	3.00000	0.161515
$346$	6.00000	0.322562
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	3.00000	0.160817
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	5.00000	0.266880
$352$	0	0
$353$	33.0000	1.75641	0.878206	−	0.478282i	$-0.158740\pi$
$353$	33.0000	1.75641	0.878206	+	0.478282i	$0.158740\pi$
$354$	3.00000	0.159448
$355$	−12.0000	−0.636894
$356$	−6.00000	−0.317999
$357$	0	0
$358$	0	0
$359$	−33.0000	−1.74167	−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000	−1.74167	−0.870837	+	0.491572i	$0.836422\pi$
$360$	2.00000	0.105409
$361$	1.00000	0.0526316
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	0	0
$365$	−13.0000	−0.680451
$366$	8.00000	0.418167
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	3.00000	0.156386
$369$	12.0000	0.624695
$370$	10.0000	0.519875
$371$	0	0
$372$	2.00000	0.103695
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.00000	−0.154508
$378$	0	0
$379$	35.0000	1.79783	0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000	1.79783	0.898915	+	0.438124i	$0.144357\pi$
$380$	1.00000	0.0512989
$381$	−2.00000	−0.102463
$382$	−27.0000	−1.38144
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	−4.00000	−0.203331
$388$	10.0000	0.507673
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	1.00000	0.0506370
$391$	9.00000	0.455150
$392$	0	0
$393$	0	0
$394$	6.00000	0.302276
$395$	−14.0000	−0.704416
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	19.0000	0.952384
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	7.00000	0.349128
$403$	−2.00000	−0.0996271
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	−3.00000	−0.148522
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	6.00000	0.296319
$411$	−9.00000	−0.443937
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−6.00000	−0.294884
$415$	6.00000	0.294528
$416$	1.00000	0.0490290
$417$	8.00000	0.391762
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−25.0000	−1.21698
$423$	0	0
$424$	3.00000	0.145693
$425$	3.00000	0.145521
$426$	−12.0000	−0.581402
$427$	0	0
$428$	15.0000	0.725052
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	5.00000	0.240563
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	−3.00000	−0.143839
$436$	11.0000	0.526804
$437$	−3.00000	−0.143509
$438$	−13.0000	−0.621164
$439$	−38.0000	−1.81364	−0.906821	−	0.421517i	$-0.861498\pi$
$439$	−38.0000	−1.81364	−0.906821	+	0.421517i	$0.861498\pi$
$440$	0	0
$441$	0	0
$442$	3.00000	0.142695
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	10.0000	0.474579
$445$	6.00000	0.284427
$446$	−14.0000	−0.662919
$447$	12.0000	0.567581
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−2.00000	−0.0942809
$451$	0	0
$452$	−12.0000	−0.564433
$453$	10.0000	0.469841
$454$	15.0000	0.703985
$455$	0	0
$456$	1.00000	0.0468293
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	−26.0000	−1.21490
$459$	15.0000	0.700140
$460$	−3.00000	−0.139876
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−3.00000	−0.139272
$465$	−2.00000	−0.0927478
$466$	−6.00000	−0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	−3.00000	−0.138086
$473$	0	0
$474$	−14.0000	−0.643041
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−15.0000	−0.686084
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	1.00000	0.0456435
$481$	−10.0000	−0.455961
$482$	4.00000	0.182195
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−10.0000	−0.454077
$486$	−16.0000	−0.725775
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−8.00000	−0.362143
$489$	22.0000	0.994874
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	6.00000	0.270501
$493$	−9.00000	−0.405340
$494$	−1.00000	−0.0449921
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	6.00000	0.268866
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	−1.00000	−0.0447214
$501$	18.0000	0.804181
$502$	0	0
$503$	−39.0000	−1.73892	−0.869462	−	0.494000i	$-0.835534\pi$
$503$	−39.0000	−1.73892	−0.869462	+	0.494000i	$0.835534\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	12.0000	0.532939
$508$	2.00000	0.0887357
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	3.00000	0.132842
$511$	0	0
$512$	1.00000	0.0441942
$513$	−5.00000	−0.220755
$514$	−18.0000	−0.793946
$515$	8.00000	0.352522
$516$	−2.00000	−0.0880451
$517$	0	0
$518$	0	0
$519$	−6.00000	−0.263371
$520$	−1.00000	−0.0438529
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	6.00000	0.262613
$523$	−11.0000	−0.480996	−0.240498	−	0.970650i	$-0.577311\pi$
$523$	−11.0000	−0.480996	−0.240498	+	0.970650i	$0.577311\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−3.00000	−0.130312
$531$	6.00000	0.260378
$532$	0	0
$533$	−6.00000	−0.259889
$534$	6.00000	0.259645
$535$	−15.0000	−0.648507
$536$	−7.00000	−0.302354
$537$	0	0
$538$	−18.0000	−0.776035
$539$	0	0
$540$	−5.00000	−0.215166
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	7.00000	0.300676
$543$	2.00000	0.0858282
$544$	3.00000	0.128624
$545$	−11.0000	−0.471188
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	9.00000	0.384461
$549$	16.0000	0.682863
$550$	0	0
$551$	3.00000	0.127804
$552$	−3.00000	−0.127688
$553$	0	0
$554$	8.00000	0.339887
$555$	−10.0000	−0.424476
$556$	−8.00000	−0.339276
$557$	−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
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	18.0000	0.759284
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−14.0000	−0.588464
$567$	0	0
$568$	12.0000	0.503509
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	−1.00000	−0.0418854
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	27.0000	1.12794
$574$	0	0
$575$	3.00000	0.125109
$576$	−2.00000	−0.0833333
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	−8.00000	−0.332756
$579$	22.0000	0.914289
$580$	3.00000	0.124568
$581$	0	0
$582$	−10.0000	−0.414513
$583$	0	0
$584$	13.0000	0.537944
$585$	2.00000	0.0826898
$586$	−21.0000	−0.867502
$587$	−30.0000	−1.23823	−0.619116	−	0.785299i	$-0.712509\pi$
$587$	−30.0000	−1.23823	−0.619116	+	0.785299i	$0.712509\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	3.00000	0.123508
$591$	−6.00000	−0.246807
$592$	−10.0000	−0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	−19.0000	−0.777618
$598$	3.00000	0.122679
$599$	−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
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	14.0000	0.570124
$604$	−10.0000	−0.406894
$605$	11.0000	0.447214
$606$	12.0000	0.487467
$607$	10.0000	0.405887	0.202944	−	0.979190i	$-0.434949\pi$
$607$	10.0000	0.405887	0.202944	+	0.979190i	$0.434949\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	8.00000	0.323911
$611$	0	0
$612$	−6.00000	−0.242536
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−20.0000	−0.807134
$615$	−6.00000	−0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	8.00000	0.321807
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	2.00000	0.0803219
$621$	15.0000	0.601929
$622$	3.00000	0.120289
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	1.00000	0.0399680
$627$	0	0
$628$	10.0000	0.399043
$629$	−30.0000	−1.19618
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	14.0000	0.556890
$633$	25.0000	0.993661
$634$	9.00000	0.357436
$635$	−2.00000	−0.0793676
$636$	−3.00000	−0.118958
$637$	0	0
$638$	0	0
$639$	−24.0000	−0.949425
$640$	−1.00000	−0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	−15.0000	−0.592003
$643$	−50.0000	−1.97181	−0.985904	−	0.167313i	$-0.946491\pi$
$643$	−50.0000	−1.97181	−0.985904	+	0.167313i	$0.946491\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	−3.00000	−0.118033
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	−22.0000	−0.861586
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	−11.0000	−0.430134
$655$	0	0
$656$	−6.00000	−0.234261
$657$	−26.0000	−1.01436
$658$	0	0
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	49.0000	1.90588	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	49.0000	1.90588	0.952940	+	0.303160i	$0.0980418\pi$
$662$	5.00000	0.194331
$663$	−3.00000	−0.116510
$664$	−6.00000	−0.232845
$665$	0	0
$666$	20.0000	0.774984
$667$	−9.00000	−0.348481
$668$	−18.0000	−0.696441
$669$	14.0000	0.541271
$670$	7.00000	0.270434
$671$	0	0
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	2.00000	0.0770371
$675$	5.00000	0.192450
$676$	−12.0000	−0.461538
$677$	15.0000	0.576497	0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000	0.576497	0.288248	+	0.957556i	$0.406927\pi$
$678$	12.0000	0.460857
$679$	0	0
$680$	−3.00000	−0.115045
$681$	−15.0000	−0.574801
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	2.00000	0.0764719
$685$	−9.00000	−0.343872
$686$	0	0
$687$	26.0000	0.991962
$688$	2.00000	0.0762493
$689$	3.00000	0.114291
$690$	3.00000	0.114208
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−30.0000	−1.13878
$695$	8.00000	0.303457
$696$	3.00000	0.113715
$697$	−18.0000	−0.681799
$698$	−14.0000	−0.529908
$699$	6.00000	0.226941
$700$	0	0
$701$	48.0000	1.81293	0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000	1.81293	0.906467	+	0.422276i	$0.138769\pi$
$702$	5.00000	0.188713
$703$	10.0000	0.377157
$704$	0	0
$705$	0	0
$706$	33.0000	1.24197
$707$	0	0
$708$	3.00000	0.112747
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	−12.0000	−0.450352
$711$	−28.0000	−1.05008
$712$	−6.00000	−0.224860
$713$	−6.00000	−0.224702
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.0000	0.560185
$718$	−33.0000	−1.23155
$719$	−33.0000	−1.23069	−0.615346	−	0.788257i	$-0.710984\pi$
$719$	−33.0000	−1.23069	−0.615346	+	0.788257i	$0.710984\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	1.00000	0.0372161
$723$	−4.00000	−0.148762
$724$	−2.00000	−0.0743294
$725$	−3.00000	−0.111417
$726$	11.0000	0.408248
$727$	13.0000	0.482143	0.241072	−	0.970507i	$-0.422501\pi$
$727$	13.0000	0.482143	0.241072	+	0.970507i	$0.422501\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−13.0000	−0.481152
$731$	6.00000	0.221918
$732$	8.00000	0.295689
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	3.00000	0.110581
$737$	0	0
$738$	12.0000	0.441726
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	10.0000	0.367607
$741$	1.00000	0.0367359
$742$	0	0
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	2.00000	0.0733236
$745$	12.0000	0.439646
$746$	−31.0000	−1.13499
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	1.00000	0.0365148
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	0	0
$754$	−3.00000	−0.109254
$755$	10.0000	0.363937
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	35.0000	1.27126
$759$	0	0
$760$	1.00000	0.0362738
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	−2.00000	−0.0724524
$763$	0	0
$764$	−27.0000	−0.976826
$765$	6.00000	0.216930
$766$	24.0000	0.867155
$767$	−3.00000	−0.108324
$768$	−1.00000	−0.0360844
$769$	55.0000	1.98335	0.991675	−	0.128763i	$-0.0411007\pi$
$769$	55.0000	1.98335	0.991675	+	0.128763i	$0.0411007\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	−22.0000	−0.791797
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	−4.00000	−0.143777
$775$	−2.00000	−0.0718421
$776$	10.0000	0.358979
$777$	0	0
$778$	−24.0000	−0.860442
$779$	6.00000	0.214972
$780$	1.00000	0.0358057
$781$	0	0
$782$	9.00000	0.321839
$783$	−15.0000	−0.536056
$784$	0	0
$785$	−10.0000	−0.356915
$786$	0	0
$787$	−5.00000	−0.178231	−0.0891154	−	0.996021i	$-0.528404\pi$
$787$	−5.00000	−0.178231	−0.0891154	+	0.996021i	$0.528404\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−14.0000	−0.498098
$791$	0	0
$792$	0	0
$793$	−8.00000	−0.284088
$794$	−8.00000	−0.283909
$795$	3.00000	0.106399
$796$	19.0000	0.673437
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	12.0000	0.423999
$802$	6.00000	0.211867
$803$	0	0
$804$	7.00000	0.246871
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	18.0000	0.633630
$808$	−12.0000	−0.422159
$809$	33.0000	1.16022	0.580109	−	0.814539i	$-0.303010\pi$
$809$	33.0000	1.16022	0.580109	+	0.814539i	$0.303010\pi$
$810$	−1.00000	−0.0351364
$811$	7.00000	0.245803	0.122902	−	0.992419i	$-0.460780\pi$
$811$	7.00000	0.245803	0.122902	+	0.992419i	$0.460780\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	0	0
$815$	22.0000	0.770626
$816$	−3.00000	−0.105021
$817$	−2.00000	−0.0699711
$818$	22.0000	0.769212
$819$	0	0
$820$	6.00000	0.209529
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	−9.00000	−0.313911
$823$	−31.0000	−1.08059	−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000	−1.08059	−0.540296	+	0.841475i	$0.681688\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	0	0
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	−6.00000	−0.208514
$829$	13.0000	0.451509	0.225754	−	0.974184i	$-0.427515\pi$
$829$	13.0000	0.451509	0.225754	+	0.974184i	$0.427515\pi$
$830$	6.00000	0.208263
$831$	−8.00000	−0.277517
$832$	1.00000	0.0346688
$833$	0	0
$834$	8.00000	0.277017
$835$	18.0000	0.622916
$836$	0	0
$837$	−10.0000	−0.345651
$838$	30.0000	1.03633
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	17.0000	0.585859
$843$	−18.0000	−0.619953
$844$	−25.0000	−0.860535
$845$	12.0000	0.412813
$846$	0	0
$847$	0	0
$848$	3.00000	0.103020
$849$	14.0000	0.480479
$850$	3.00000	0.102899
$851$	−30.0000	−1.02839
$852$	−12.0000	−0.411113
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	15.0000	0.512689
$857$	−48.0000	−1.63965	−0.819824	−	0.572615i	$-0.805929\pi$
$857$	−48.0000	−1.63965	−0.819824	+	0.572615i	$0.805929\pi$
$858$	0	0
$859$	16.0000	0.545913	0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000	0.545913	0.272956	+	0.962026i	$0.411998\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	24.0000	0.817443
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	5.00000	0.170103
$865$	−6.00000	−0.204006
$866$	4.00000	0.135926
$867$	8.00000	0.271694
$868$	0	0
$869$	0	0
$870$	−3.00000	−0.101710
$871$	−7.00000	−0.237186
$872$	11.0000	0.372507
$873$	−20.0000	−0.676897
$874$	−3.00000	−0.101477
$875$	0	0
$876$	−13.0000	−0.439229
$877$	−31.0000	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$877$	−31.0000	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$878$	−38.0000	−1.28244
$879$	21.0000	0.708312
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	3.00000	0.100901
$885$	−3.00000	−0.100844
$886$	−24.0000	−0.806296
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	10.0000	0.335578
$889$	0	0
$890$	6.00000	0.201120
$891$	0	0
$892$	−14.0000	−0.468755
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	0	0
$897$	−3.00000	−0.100167
$898$	−30.0000	−1.00111
$899$	6.00000	0.200111
$900$	−2.00000	−0.0666667
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	−12.0000	−0.399114
$905$	2.00000	0.0664822
$906$	10.0000	0.332228
$907$	35.0000	1.16216	0.581078	−	0.813848i	$-0.302631\pi$
$907$	35.0000	1.16216	0.581078	+	0.813848i	$0.302631\pi$
$908$	15.0000	0.497792
$909$	24.0000	0.796030
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	1.00000	0.0331133
$913$	0	0
$914$	35.0000	1.15770
$915$	−8.00000	−0.264472
$916$	−26.0000	−0.859064
$917$	0	0
$918$	15.0000	0.495074
$919$	−1.00000	−0.0329870	−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	−1.00000	−0.0329870	−0.0164935	+	0.999864i	$0.505250\pi$
$920$	−3.00000	−0.0989071
$921$	20.0000	0.659022
$922$	−6.00000	−0.197599
$923$	12.0000	0.394985
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−16.0000	−0.525793
$927$	16.0000	0.525509
$928$	−3.00000	−0.0984798
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	−2.00000	−0.0655826
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−3.00000	−0.0982156
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	0	0
$939$	−1.00000	−0.0326338
$940$	0	0
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	−10.0000	−0.325818
$943$	−18.0000	−0.586161
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−14.0000	−0.454699
$949$	13.0000	0.421998
$950$	−1.00000	−0.0324443
$951$	−9.00000	−0.291845
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	−6.00000	−0.194257
$955$	27.0000	0.873699
$956$	−15.0000	−0.485135
$957$	0	0
$958$	24.0000	0.775405
$959$	0	0
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	−10.0000	−0.322413
$963$	−30.0000	−0.966736
$964$	4.00000	0.128831
$965$	22.0000	0.708205
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	−11.0000	−0.353553
$969$	3.00000	0.0963739
$970$	−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	$0.561675\pi$
$972$	−16.0000	−0.513200
$973$	0	0
$974$	2.00000	0.0640841
$975$	−1.00000	−0.0320256
$976$	−8.00000	−0.256074
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	22.0000	0.703482
$979$	0	0
$980$	0	0
$981$	−22.0000	−0.702406
$982$	30.0000	0.957338
$983$	−54.0000	−1.72233	−0.861166	−	0.508323i	$-0.830265\pi$
$983$	−54.0000	−1.72233	−0.861166	+	0.508323i	$0.830265\pi$
$984$	6.00000	0.191273
$985$	−6.00000	−0.191176
$986$	−9.00000	−0.286618
$987$	0	0
$988$	−1.00000	−0.0318142
$989$	6.00000	0.190789
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	−2.00000	−0.0635001
$993$	−5.00000	−0.158670
$994$	0	0
$995$	−19.0000	−0.602340
$996$	6.00000	0.190117
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	−10.0000	−0.316544
$999$	−50.0000	−1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9310.2.a.o.1.1		1
7.6	odd	2		190.2.a.c.1.1	✓	1
21.20	even	2		1710.2.a.d.1.1		1
28.27	even	2		1520.2.a.d.1.1		1
35.13	even	4		950.2.b.e.799.1		2
35.27	even	4		950.2.b.e.799.2		2
35.34	odd	2		950.2.a.a.1.1		1
56.13	odd	2		6080.2.a.h.1.1		1
56.27	even	2		6080.2.a.p.1.1		1
105.104	even	2		8550.2.a.bd.1.1		1
133.132	even	2		3610.2.a.b.1.1		1
140.139	even	2		7600.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.c.1.1	✓	1	7.6	odd	2
950.2.a.a.1.1		1	35.34	odd	2
950.2.b.e.799.1		2	35.13	even	4
950.2.b.e.799.2		2	35.27	even	4
1520.2.a.d.1.1		1	28.27	even	2
1710.2.a.d.1.1		1	21.20	even	2
3610.2.a.b.1.1		1	133.132	even	2
6080.2.a.h.1.1		1	56.13	odd	2
6080.2.a.p.1.1		1	56.27	even	2
7600.2.a.m.1.1		1	140.139	even	2
8550.2.a.bd.1.1		1	105.104	even	2
9310.2.a.o.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:	\( N \)	\(=\)	\( 9310 = 2 \cdot 5 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9310.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more