LMFDB - Embedded newform 69.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,2,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, 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("69.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	69.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:	$0.550967773947$
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	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{21} +4.00000 q^{22} -1.00000 q^{23} -3.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} +2.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} -6.00000 q^{39} +2.00000 q^{41} -2.00000 q^{42} +10.0000 q^{43} -4.00000 q^{44} -1.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -5.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} +6.00000 q^{56} +2.00000 q^{57} +2.00000 q^{58} -12.0000 q^{59} -6.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} -10.0000 q^{67} -4.00000 q^{68} -1.00000 q^{69} +8.00000 q^{71} -3.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} -5.00000 q^{75} -2.00000 q^{76} -8.00000 q^{77} -6.00000 q^{78} +10.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +10.0000 q^{86} +2.00000 q^{87} -12.0000 q^{88} -16.0000 q^{89} +12.0000 q^{91} +1.00000 q^{92} +4.00000 q^{93} +5.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	1.00000	0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	4.00000	0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−3.00000	−0.612372
$25$	−5.00000	−1.00000
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	4.00000	0.685994
$35$	0	0
$36$	−1.00000	−0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	−6.00000	−0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−5.00000	−0.707107
$51$	4.00000	0.560112
$52$	6.00000	0.832050
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	6.00000	0.801784
$57$	2.00000	0.264906
$58$	2.00000	0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	4.00000	0.508001
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−4.00000	−0.485071
$69$	−1.00000	−0.120386
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−3.00000	−0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	2.00000	0.232495
$75$	−5.00000	−0.577350
$76$	−2.00000	−0.229416
$77$	−8.00000	−0.911685
$78$	−6.00000	−0.679366
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	10.0000	1.07833
$87$	2.00000	0.214423
$88$	−12.0000	−1.27920
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	1.00000	0.104257
$93$	4.00000	0.414781
$94$	0	0
$95$	0	0
$96$	5.00000	0.510310
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−3.00000	−0.303046
$99$	4.00000	0.402015
$100$	5.00000	0.500000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	4.00000	0.396059
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	18.0000	1.76505
$105$	0	0
$106$	−12.0000	−1.16554
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000	0.188982
$113$	16.0000	1.50515	0.752577	−	0.658505i	$-0.228811\pi$
$113$	16.0000	1.50515	0.752577	+	0.658505i	$0.228811\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−6.00000	−0.554700
$118$	−12.0000	−1.10469
$119$	−8.00000	−0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	2.00000	0.180334
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−3.00000	−0.265165
$129$	10.0000	0.880451
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−4.00000	−0.348155
$133$	−4.00000	−0.346844
$134$	−10.0000	−0.863868
$135$	0	0
$136$	−12.0000	−1.02899
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−1.00000	−0.0851257
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	−24.0000	−2.00698
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	−3.00000	−0.247436
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−5.00000	−0.408248
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−6.00000	−0.486664
$153$	4.00000	0.323381
$154$	−8.00000	−0.644658
$155$	0	0
$156$	6.00000	0.480384
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	10.0000	0.795557
$159$	−12.0000	−0.951662
$160$	0	0
$161$	2.00000	0.157622
$162$	1.00000	0.0785674
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	6.00000	0.462910
$169$	23.0000	1.76923
$170$	0	0
$171$	2.00000	0.152944
$172$	−10.0000	−0.762493
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	2.00000	0.151620
$175$	10.0000	0.755929
$176$	−4.00000	−0.301511
$177$	−12.0000	−0.901975
$178$	−16.0000	−1.19925
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	12.0000	0.889499
$183$	−6.00000	−0.443533
$184$	3.00000	0.221163
$185$	0	0
$186$	4.00000	0.293294
$187$	16.0000	1.17004
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	7.00000	0.505181
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	3.00000	0.214286
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	4.00000	0.284268
$199$	−6.00000	−0.425329	−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000	−0.425329	−0.212664	+	0.977125i	$0.568214\pi$
$200$	15.0000	1.06066
$201$	−10.0000	−0.705346
$202$	14.0000	0.985037
$203$	−4.00000	−0.280745
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−6.00000	−0.418040
$207$	−1.00000	−0.0695048
$208$	6.00000	0.416025
$209$	8.00000	0.553372
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	12.0000	0.824163
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	0	0
$216$	−3.00000	−0.204124
$217$	−8.00000	−0.543075
$218$	14.0000	0.948200
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−24.0000	−1.61441
$222$	2.00000	0.134231
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−10.0000	−0.668153
$225$	−5.00000	−0.333333
$226$	16.0000	1.06430
$227$	−24.0000	−1.59294	−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000	−1.59294	−0.796468	+	0.604681i	$0.793301\pi$
$228$	−2.00000	−0.132453
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	−6.00000	−0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	10.0000	0.649570
$238$	−8.00000	−0.518563
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	−12.0000	−0.763542
$248$	−12.0000	−0.762001
$249$	12.0000	0.760469
$250$	0	0
$251$	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
$253$	−4.00000	−0.251478
$254$	−12.0000	−0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	10.0000	0.622573
$259$	−4.00000	−0.248548
$260$	0	0
$261$	2.00000	0.123797
$262$	−4.00000	−0.247121
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	−4.00000	−0.245256
$267$	−16.0000	−0.979184
$268$	10.0000	0.610847
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−4.00000	−0.242536
$273$	12.0000	0.726273
$274$	8.00000	0.483298
$275$	−20.0000	−1.20605
$276$	1.00000	0.0601929
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	4.00000	0.239474
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	−4.00000	−0.236113
$288$	5.00000	0.294628
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−10.0000	−0.586210
$292$	14.0000	0.819288
$293$	20.0000	1.16841	0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000	1.16841	0.584206	+	0.811605i	$0.301406\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−6.00000	−0.348743
$297$	4.00000	0.232104
$298$	0	0
$299$	6.00000	0.346989
$300$	5.00000	0.288675
$301$	−20.0000	−1.15278
$302$	8.00000	0.460348
$303$	14.0000	0.804279
$304$	−2.00000	−0.114708
$305$	0	0
$306$	4.00000	0.228665
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	8.00000	0.455842
$309$	−6.00000	−0.341328
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	18.0000	1.01905
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	−12.0000	−0.672927
$319$	8.00000	0.447914
$320$	0	0
$321$	12.0000	0.669775
$322$	2.00000	0.111456
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	30.0000	1.66410
$326$	8.00000	0.443079
$327$	14.0000	0.774202
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−12.0000	−0.658586
$333$	2.00000	0.109599
$334$	−16.0000	−0.875481
$335$	0	0
$336$	2.00000	0.109109
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	23.0000	1.25104
$339$	16.0000	0.869001
$340$	0	0
$341$	16.0000	0.866449
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	−30.0000	−1.61749
$345$	0	0
$346$	2.00000	0.107521
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−2.00000	−0.107211
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	10.0000	0.534522
$351$	−6.00000	−0.320256
$352$	20.0000	1.06600
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	16.0000	0.847998
$357$	−8.00000	−0.423405
$358$	12.0000	0.634220
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−2.00000	−0.105118
$363$	5.00000	0.262432
$364$	−12.0000	−0.628971
$365$	0	0
$366$	−6.00000	−0.313625
$367$	−38.0000	−1.98358	−0.991792	−	0.127862i	$-0.959188\pi$
$367$	−38.0000	−1.98358	−0.991792	+	0.127862i	$0.959188\pi$
$368$	1.00000	0.0521286
$369$	2.00000	0.104116
$370$	0	0
$371$	24.0000	1.24602
$372$	−4.00000	−0.207390
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	−2.00000	−0.102869
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	−12.0000	−0.613973
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−18.0000	−0.916176
$387$	10.0000	0.508329
$388$	10.0000	0.507673
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	9.00000	0.454569
$393$	−4.00000	−0.201773
$394$	18.0000	0.906827
$395$	0	0
$396$	−4.00000	−0.201008
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−6.00000	−0.300753
$399$	−4.00000	−0.200250
$400$	5.00000	0.250000
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	−10.0000	−0.498755
$403$	−24.0000	−1.19553
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−4.00000	−0.198517
$407$	8.00000	0.396545
$408$	−12.0000	−0.594089
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	6.00000	0.295599
$413$	24.0000	1.18096
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−30.0000	−1.47087
$417$	4.00000	0.195881
$418$	8.00000	0.391293
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	36.0000	1.74831
$425$	−20.0000	−0.970143
$426$	8.00000	0.387601
$427$	12.0000	0.580721
$428$	−12.0000	−0.580042
$429$	−24.0000	−1.15873
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−2.00000	−0.0956730
$438$	−14.0000	−0.668946
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−24.0000	−1.14156
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−14.0000	−0.661438
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	−5.00000	−0.235702
$451$	8.00000	0.376705
$452$	−16.0000	−0.752577
$453$	8.00000	0.375873
$454$	−24.0000	−1.12638
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	6.00000	0.280362
$459$	4.00000	0.186704
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	−8.00000	−0.372194
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	6.00000	0.277350
$469$	20.0000	0.923514
$470$	0	0
$471$	10.0000	0.460776
$472$	36.0000	1.65703
$473$	40.0000	1.83920
$474$	10.0000	0.459315
$475$	−10.0000	−0.458831
$476$	8.00000	0.366679
$477$	−12.0000	−0.549442
$478$	8.00000	0.365911
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	−2.00000	−0.0910975
$483$	2.00000	0.0910032
$484$	−5.00000	−0.227273
$485$	0	0
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	18.0000	0.814822
$489$	8.00000	0.361773
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−2.00000	−0.0901670
$493$	8.00000	0.360302
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−16.0000	−0.717698
$498$	12.0000	0.537733
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	24.0000	1.07117
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	6.00000	0.267261
$505$	0	0
$506$	−4.00000	−0.177822
$507$	23.0000	1.02147
$508$	12.0000	0.532414
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	0	0
$511$	28.0000	1.23865
$512$	−11.0000	−0.486136
$513$	2.00000	0.0883022
$514$	18.0000	0.793946
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	−4.00000	−0.175750
$519$	2.00000	0.0877903
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	2.00000	0.0875376
$523$	38.0000	1.66162	0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000	1.66162	0.830812	+	0.556553i	$0.187876\pi$
$524$	4.00000	0.174741
$525$	10.0000	0.436436
$526$	−12.0000	−0.523225
$527$	16.0000	0.696971
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	0	0
$531$	−12.0000	−0.520756
$532$	4.00000	0.173422
$533$	−12.0000	−0.519778
$534$	−16.0000	−0.692388
$535$	0	0
$536$	30.0000	1.29580
$537$	12.0000	0.517838
$538$	10.0000	0.431131
$539$	−12.0000	−0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−20.0000	−0.859074
$543$	−2.00000	−0.0858282
$544$	20.0000	0.857493
$545$	0	0
$546$	12.0000	0.513553
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−8.00000	−0.341743
$549$	−6.00000	−0.256074
$550$	−20.0000	−0.852803
$551$	4.00000	0.170406
$552$	3.00000	0.127688
$553$	−20.0000	−0.850487
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	4.00000	0.169334
$559$	−60.0000	−2.53773
$560$	0	0
$561$	16.0000	0.675521
$562$	−24.0000	−1.01238
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	0	0
$566$	6.00000	0.252199
$567$	−2.00000	−0.0839921
$568$	−24.0000	−1.00702
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	24.0000	1.00349
$573$	−12.0000	−0.501307
$574$	−4.00000	−0.166957
$575$	5.00000	0.208514
$576$	7.00000	0.291667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−1.00000	−0.0415945
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−24.0000	−0.995688
$582$	−10.0000	−0.414513
$583$	−48.0000	−1.98796
$584$	42.0000	1.73797
$585$	0	0
$586$	20.0000	0.826192
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	3.00000	0.123718
$589$	8.00000	0.329634
$590$	0	0
$591$	18.0000	0.740421
$592$	−2.00000	−0.0821995
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	0	0
$597$	−6.00000	−0.245564
$598$	6.00000	0.245358
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	15.0000	0.612372
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	−20.0000	−0.815139
$603$	−10.0000	−0.407231
$604$	−8.00000	−0.325515
$605$	0	0
$606$	14.0000	0.568711
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	10.0000	0.405554
$609$	−4.00000	−0.162088
$610$	0	0
$611$	0	0
$612$	−4.00000	−0.161690
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	24.0000	0.966988
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	−6.00000	−0.241355
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	32.0000	1.28205
$624$	6.00000	0.240192
$625$	25.0000	1.00000
$626$	2.00000	0.0799361
$627$	8.00000	0.319489
$628$	−10.0000	−0.399043
$629$	8.00000	0.318981
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	−30.0000	−1.19334
$633$	8.00000	0.317971
$634$	−30.0000	−1.19145
$635$	0	0
$636$	12.0000	0.475831
$637$	18.0000	0.713186
$638$	8.00000	0.316723
$639$	8.00000	0.316475
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	12.0000	0.473602
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	8.00000	0.314756
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	−3.00000	−0.117851
$649$	−48.0000	−1.88416
$650$	30.0000	1.17670
$651$	−8.00000	−0.313545
$652$	−8.00000	−0.313304
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−14.0000	−0.546192
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	−36.0000	−1.39707
$665$	0	0
$666$	2.00000	0.0774984
$667$	−2.00000	−0.0774403
$668$	16.0000	0.619059
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	−10.0000	−0.385758
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−2.00000	−0.0770371
$675$	−5.00000	−0.192450
$676$	−23.0000	−0.884615
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	16.0000	0.614476
$679$	20.0000	0.767530
$680$	0	0
$681$	−24.0000	−0.919682
$682$	16.0000	0.612672
$683$	−52.0000	−1.98972	−0.994862	−	0.101237i	$-0.967720\pi$
$683$	−52.0000	−1.98972	−0.994862	+	0.101237i	$0.967720\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	20.0000	0.763604
$687$	6.00000	0.228914
$688$	−10.0000	−0.381246
$689$	72.0000	2.74298
$690$	0	0
$691$	16.0000	0.608669	0.304334	−	0.952565i	$-0.401566\pi$
$691$	16.0000	0.608669	0.304334	+	0.952565i	$0.401566\pi$
$692$	−2.00000	−0.0760286
$693$	−8.00000	−0.303895
$694$	12.0000	0.455514
$695$	0	0
$696$	−6.00000	−0.227429
$697$	8.00000	0.303022
$698$	2.00000	0.0757011
$699$	10.0000	0.378235
$700$	−10.0000	−0.377964
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	−6.00000	−0.226455
$703$	4.00000	0.150863
$704$	28.0000	1.05529
$705$	0	0
$706$	6.00000	0.225813
$707$	−28.0000	−1.05305
$708$	12.0000	0.450988
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	48.0000	1.79888
$713$	−4.00000	−0.149801
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	8.00000	0.298765
$718$	20.0000	0.746393
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	−15.0000	−0.558242
$723$	−2.00000	−0.0743808
$724$	2.00000	0.0743294
$725$	−10.0000	−0.371391
$726$	5.00000	0.185567
$727$	6.00000	0.222528	0.111264	−	0.993791i	$-0.464510\pi$
$727$	6.00000	0.222528	0.111264	+	0.993791i	$0.464510\pi$
$728$	−36.0000	−1.33425
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	6.00000	0.221766
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−38.0000	−1.40261
$735$	0	0
$736$	−5.00000	−0.184302
$737$	−40.0000	−1.47342
$738$	2.00000	0.0736210
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	24.0000	0.881068
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	34.0000	1.24483
$747$	12.0000	0.439057
$748$	−16.0000	−0.585018
$749$	−24.0000	−0.876941
$750$	0	0
$751$	30.0000	1.09472	0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000	1.09472	0.547358	+	0.836899i	$0.315634\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	−12.0000	−0.437014
$755$	0	0
$756$	2.00000	0.0727393
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	10.0000	0.363216
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−12.0000	−0.434714
$763$	−28.0000	−1.01367
$764$	12.0000	0.434145
$765$	0	0
$766$	−28.0000	−1.01168
$767$	72.0000	2.59977
$768$	−17.0000	−0.613435
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	18.0000	0.647834
$773$	−32.0000	−1.15096	−0.575480	−	0.817816i	$-0.695185\pi$
$773$	−32.0000	−1.15096	−0.575480	+	0.817816i	$0.695185\pi$
$774$	10.0000	0.359443
$775$	−20.0000	−0.718421
$776$	30.0000	1.07694
$777$	−4.00000	−0.143499
$778$	−20.0000	−0.717035
$779$	4.00000	0.143315
$780$	0	0
$781$	32.0000	1.14505
$782$	−4.00000	−0.143040
$783$	2.00000	0.0714742
$784$	3.00000	0.107143
$785$	0	0
$786$	−4.00000	−0.142675
$787$	2.00000	0.0712923	0.0356462	−	0.999364i	$-0.488651\pi$
$787$	2.00000	0.0712923	0.0356462	+	0.999364i	$0.488651\pi$
$788$	−18.0000	−0.641223
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−32.0000	−1.13779
$792$	−12.0000	−0.426401
$793$	36.0000	1.27840
$794$	−18.0000	−0.638796
$795$	0	0
$796$	6.00000	0.212664
$797$	44.0000	1.55856	0.779280	−	0.626676i	$-0.215585\pi$
$797$	44.0000	1.55856	0.779280	+	0.626676i	$0.215585\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	−25.0000	−0.883883
$801$	−16.0000	−0.565332
$802$	−4.00000	−0.141245
$803$	−56.0000	−1.97620
$804$	10.0000	0.352673
$805$	0	0
$806$	−24.0000	−0.845364
$807$	10.0000	0.352017
$808$	−42.0000	−1.47755
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	4.00000	0.140372
$813$	−20.0000	−0.701431
$814$	8.00000	0.280400
$815$	0	0
$816$	−4.00000	−0.140028
$817$	20.0000	0.699711
$818$	−26.0000	−0.909069
$819$	12.0000	0.419314
$820$	0	0
$821$	−26.0000	−0.907406	−0.453703	−	0.891153i	$-0.649897\pi$
$821$	−26.0000	−0.907406	−0.453703	+	0.891153i	$0.649897\pi$
$822$	8.00000	0.279032
$823$	52.0000	1.81261	0.906303	−	0.422628i	$-0.138892\pi$
$823$	52.0000	1.81261	0.906303	+	0.422628i	$0.138892\pi$
$824$	18.0000	0.627060
$825$	−20.0000	−0.696311
$826$	24.0000	0.835067
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	1.00000	0.0347524
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	−42.0000	−1.45609
$833$	−12.0000	−0.415775
$834$	4.00000	0.138509
$835$	0	0
$836$	−8.00000	−0.276686
$837$	4.00000	0.138260
$838$	−4.00000	−0.138178
$839$	−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
$841$	−25.0000	−0.862069
$842$	38.0000	1.30957
$843$	−24.0000	−0.826604
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	−10.0000	−0.343604
$848$	12.0000	0.412082
$849$	6.00000	0.205919
$850$	−20.0000	−0.685994
$851$	−2.00000	−0.0685591
$852$	−8.00000	−0.274075
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	−36.0000	−1.23045
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	−24.0000	−0.819346
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	2.00000	0.0679628
$867$	−1.00000	−0.0339618
$868$	8.00000	0.271538
$869$	40.0000	1.35691
$870$	0	0
$871$	60.0000	2.03302
$872$	−42.0000	−1.42230
$873$	−10.0000	−0.338449
$874$	−2.00000	−0.0676510
$875$	0	0
$876$	14.0000	0.473016
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	8.00000	0.269987
$879$	20.0000	0.674583
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−3.00000	−0.101015
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	−12.0000	−0.403148
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	−6.00000	−0.201347
$889$	24.0000	0.804934
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	6.00000	0.200446
$897$	6.00000	0.200334
$898$	10.0000	0.333704
$899$	8.00000	0.266815
$900$	5.00000	0.166667
$901$	−48.0000	−1.59911
$902$	8.00000	0.266371
$903$	−20.0000	−0.665558
$904$	−48.0000	−1.59646
$905$	0	0
$906$	8.00000	0.265782
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	24.0000	0.796468
$909$	14.0000	0.464351
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	−2.00000	−0.0662266
$913$	48.0000	1.58857
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	8.00000	0.264183
$918$	4.00000	0.132020
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	2.00000	0.0658665
$923$	−48.0000	−1.57994
$924$	8.00000	0.263181
$925$	−10.0000	−0.328798
$926$	−8.00000	−0.262896
$927$	−6.00000	−0.197066
$928$	10.0000	0.328266
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−10.0000	−0.327561
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	18.0000	0.588348
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	20.0000	0.653023
$939$	2.00000	0.0652675
$940$	0	0
$941$	40.0000	1.30396	0.651981	−	0.758235i	$-0.273938\pi$
$941$	40.0000	1.30396	0.651981	+	0.758235i	$0.273938\pi$
$942$	10.0000	0.325818
$943$	−2.00000	−0.0651290
$944$	12.0000	0.390567
$945$	0	0
$946$	40.0000	1.30051
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	−10.0000	−0.324785
$949$	84.0000	2.72676
$950$	−10.0000	−0.324443
$951$	−30.0000	−0.972817
$952$	24.0000	0.777844
$953$	12.0000	0.388718	0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000	0.388718	0.194359	+	0.980930i	$0.437737\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	−8.00000	−0.258738
$957$	8.00000	0.258603
$958$	16.0000	0.516937
$959$	−16.0000	−0.516667
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−12.0000	−0.386896
$963$	12.0000	0.386695
$964$	2.00000	0.0644157
$965$	0	0
$966$	2.00000	0.0643489
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	−15.0000	−0.482118
$969$	8.00000	0.256997
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	−1.00000	−0.0320750
$973$	−8.00000	−0.256468
$974$	24.0000	0.769010
$975$	30.0000	0.960769
$976$	6.00000	0.192055
$977$	16.0000	0.511885	0.255943	−	0.966692i	$-0.417614\pi$
$977$	16.0000	0.511885	0.255943	+	0.966692i	$0.417614\pi$
$978$	8.00000	0.255812
$979$	−64.0000	−2.04545
$980$	0	0
$981$	14.0000	0.446986
$982$	−20.0000	−0.638226
$983$	−16.0000	−0.510321	−0.255160	−	0.966899i	$-0.582128\pi$
$983$	−16.0000	−0.510321	−0.255160	+	0.966899i	$0.582128\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	8.00000	0.254772
$987$	0	0
$988$	12.0000	0.381771
$989$	−10.0000	−0.317982
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	20.0000	0.635001
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	−12.0000	−0.380235
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	−8.00000	−0.253236
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.a.a.1.1	✓	1
3.2	odd	2		207.2.a.a.1.1		1
4.3	odd	2		1104.2.a.c.1.1		1
5.2	odd	4		1725.2.b.g.1174.2		2
5.3	odd	4		1725.2.b.g.1174.1		2
5.4	even	2		1725.2.a.e.1.1		1
7.6	odd	2		3381.2.a.k.1.1		1
8.3	odd	2		4416.2.a.x.1.1		1
8.5	even	2		4416.2.a.f.1.1		1
11.10	odd	2		8349.2.a.a.1.1		1
12.11	even	2		3312.2.a.k.1.1		1
15.14	odd	2		5175.2.a.v.1.1		1
23.22	odd	2		1587.2.a.e.1.1		1
69.68	even	2		4761.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.a.1.1	✓	1	1.1	even	1	trivial
207.2.a.a.1.1		1	3.2	odd	2
1104.2.a.c.1.1		1	4.3	odd	2
1587.2.a.e.1.1		1	23.22	odd	2
1725.2.a.e.1.1		1	5.4	even	2
1725.2.b.g.1174.1		2	5.3	odd	4
1725.2.b.g.1174.2		2	5.2	odd	4
3312.2.a.k.1.1		1	12.11	even	2
3381.2.a.k.1.1		1	7.6	odd	2
4416.2.a.f.1.1		1	8.5	even	2
4416.2.a.x.1.1		1	8.3	odd	2
4761.2.a.b.1.1		1	69.68	even	2
5175.2.a.v.1.1		1	15.14	odd	2
8349.2.a.a.1.1		1	11.10	odd	2

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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