LMFDB - Embedded newform 1122.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1122,2,Mod(1,1122)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1122, 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("1122.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1122 = 2 \cdot 3 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1122.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:	$8.95921510679$
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	$\chi$	$=$	1122.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} +2.00000 q^{20} -2.00000 q^{21} -1.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +2.00000 q^{29} +2.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{38} +4.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} -2.00000 q^{42} +2.00000 q^{43} -1.00000 q^{44} +2.00000 q^{45} +2.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +4.00000 q^{52} -12.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -2.00000 q^{56} +2.00000 q^{57} +2.00000 q^{58} -14.0000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +8.00000 q^{65} -1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +2.00000 q^{69} -4.00000 q^{70} -2.00000 q^{71} +1.00000 q^{72} -8.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} +2.00000 q^{77} +4.00000 q^{78} -2.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -12.0000 q^{83} -2.00000 q^{84} +2.00000 q^{85} +2.00000 q^{86} +2.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +2.00000 q^{90} -8.00000 q^{91} +2.00000 q^{92} -4.00000 q^{93} +4.00000 q^{95} +1.00000 q^{96} -6.00000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−2.00000	−0.534522
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	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	$0.426318\pi$
$20$	2.00000	0.447214
$21$	−2.00000	−0.436436
$22$	−1.00000	−0.213201
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	4.00000	0.784465
$27$	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	$0.440546\pi$
$30$	2.00000	0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	2.00000	0.324443
$39$	4.00000	0.640513
$40$	2.00000	0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−1.00000	−0.150756
$45$	2.00000	0.298142
$46$	2.00000	0.294884
$47$	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
$50$	−1.00000	−0.141421
$51$	1.00000	0.140028
$52$	4.00000	0.554700
$53$	−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
$55$	−2.00000	−0.269680
$56$	−2.00000	−0.267261
$57$	2.00000	0.264906
$58$	2.00000	0.262613
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	2.00000	0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	−1.00000	−0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.00000	0.121268
$69$	2.00000	0.240772
$70$	−4.00000	−0.478091
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	1.00000	0.117851
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	6.00000	0.697486
$75$	−1.00000	−0.115470
$76$	2.00000	0.229416
$77$	2.00000	0.227921
$78$	4.00000	0.452911
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−2.00000	−0.218218
$85$	2.00000	0.216930
$86$	2.00000	0.215666
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	2.00000	0.210819
$91$	−8.00000	−0.838628
$92$	2.00000	0.208514
$93$	−4.00000	−0.414781
$94$	0	0
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−3.00000	−0.303046
$99$	−1.00000	−0.100504
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	1.00000	0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	4.00000	0.392232
$105$	−4.00000	−0.390360
$106$	−12.0000	−1.16554
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−2.00000	−0.190693
$111$	6.00000	0.569495
$112$	−2.00000	−0.188982
$113$	12.0000	1.12887	0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000	1.12887	0.564433	+	0.825479i	$0.309095\pi$
$114$	2.00000	0.187317
$115$	4.00000	0.373002
$116$	2.00000	0.185695
$117$	4.00000	0.369800
$118$	−14.0000	−1.28880
$119$	−2.00000	−0.183340
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	6.00000	0.541002
$124$	−4.00000	−0.359211
$125$	−12.0000	−1.07331
$126$	−2.00000	−0.178174
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	8.00000	0.701646
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	−1.00000	−0.0870388
$133$	−4.00000	−0.346844
$134$	−4.00000	−0.345547
$135$	2.00000	0.172133
$136$	1.00000	0.0857493
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	2.00000	0.170251
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	−2.00000	−0.167836
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−8.00000	−0.662085
$147$	−3.00000	−0.247436
$148$	6.00000	0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000	0.162221
$153$	1.00000	0.0808452
$154$	2.00000	0.161165
$155$	−8.00000	−0.642575
$156$	4.00000	0.320256
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−2.00000	−0.159111
$159$	−12.0000	−0.951662
$160$	2.00000	0.158114
$161$	−4.00000	−0.315244
$162$	1.00000	0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	−12.0000	−0.931381
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	2.00000	0.153393
$171$	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	$0.426753\pi$
$174$	2.00000	0.151620
$175$	2.00000	0.151186
$176$	−1.00000	−0.0753778
$177$	−14.0000	−1.05230
$178$	6.00000	0.449719
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	2.00000	0.149071
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	−8.00000	−0.592999
$183$	6.00000	0.443533
$184$	2.00000	0.147442
$185$	12.0000	0.882258
$186$	−4.00000	−0.293294
$187$	−1.00000	−0.0731272
$188$	0	0
$189$	−2.00000	−0.145479
$190$	4.00000	0.290191
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−6.00000	−0.430775
$195$	8.00000	0.572892
$196$	−3.00000	−0.214286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	2.00000	0.140720
$203$	−4.00000	−0.280745
$204$	1.00000	0.0700140
$205$	12.0000	0.838116
$206$	−8.00000	−0.557386
$207$	2.00000	0.139010
$208$	4.00000	0.277350
$209$	−2.00000	−0.138343
$210$	−4.00000	−0.276026
$211$	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
$213$	−2.00000	−0.137038
$214$	4.00000	0.273434
$215$	4.00000	0.272798
$216$	1.00000	0.0680414
$217$	8.00000	0.543075
$218$	−6.00000	−0.406371
$219$	−8.00000	−0.540590
$220$	−2.00000	−0.134840
$221$	4.00000	0.269069
$222$	6.00000	0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−2.00000	−0.133631
$225$	−1.00000	−0.0666667
$226$	12.0000	0.798228
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	2.00000	0.132453
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	4.00000	0.263752
$231$	2.00000	0.131590
$232$	2.00000	0.131306
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−14.0000	−0.911322
$237$	−2.00000	−0.129914
$238$	−2.00000	−0.129641
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	2.00000	0.129099
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	−6.00000	−0.383326
$246$	6.00000	0.382546
$247$	8.00000	0.509028
$248$	−4.00000	−0.254000
$249$	−12.0000	−0.760469
$250$	−12.0000	−0.758947
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−2.00000	−0.125988
$253$	−2.00000	−0.125739
$254$	−8.00000	−0.501965
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	2.00000	0.124515
$259$	−12.0000	−0.745644
$260$	8.00000	0.496139
$261$	2.00000	0.123797
$262$	20.0000	1.23560
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−1.00000	−0.0615457
$265$	−24.0000	−1.47431
$266$	−4.00000	−0.245256
$267$	6.00000	0.367194
$268$	−4.00000	−0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	2.00000	0.121716
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	1.00000	0.0606339
$273$	−8.00000	−0.484182
$274$	18.0000	1.08742
$275$	1.00000	0.0603023
$276$	2.00000	0.120386
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	−4.00000	−0.239046
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	−2.00000	−0.118678
$285$	4.00000	0.236940
$286$	−4.00000	−0.236525
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	4.00000	0.234888
$291$	−6.00000	−0.351726
$292$	−8.00000	−0.468165
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	−28.0000	−1.63022
$296$	6.00000	0.348743
$297$	−1.00000	−0.0580259
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	−4.00000	−0.230174
$303$	2.00000	0.114897
$304$	2.00000	0.114708
$305$	12.0000	0.687118
$306$	1.00000	0.0571662
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	2.00000	0.113961
$309$	−8.00000	−0.455104
$310$	−8.00000	−0.454369
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	4.00000	0.226455
$313$	34.0000	1.92179	0.960897	−	0.276907i	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	−14.0000	−0.790066
$315$	−4.00000	−0.225374
$316$	−2.00000	−0.112509
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	−12.0000	−0.672927
$319$	−2.00000	−0.111979
$320$	2.00000	0.111803
$321$	4.00000	0.223258
$322$	−4.00000	−0.222911
$323$	2.00000	0.111283
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−12.0000	−0.664619
$327$	−6.00000	−0.331801
$328$	6.00000	0.331295
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−12.0000	−0.658586
$333$	6.00000	0.328798
$334$	12.0000	0.656611
$335$	−8.00000	−0.437087
$336$	−2.00000	−0.109109
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	3.00000	0.163178
$339$	12.0000	0.651751
$340$	2.00000	0.108465
$341$	4.00000	0.216612
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	2.00000	0.107833
$345$	4.00000	0.215353
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	2.00000	0.106904
$351$	4.00000	0.213504
$352$	−1.00000	−0.0533002
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−14.0000	−0.744092
$355$	−4.00000	−0.212298
$356$	6.00000	0.317999
$357$	−2.00000	−0.105851
$358$	−10.0000	−0.528516
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	2.00000	0.105409
$361$	−15.0000	−0.789474
$362$	−10.0000	−0.525588
$363$	1.00000	0.0524864
$364$	−8.00000	−0.419314
$365$	−16.0000	−0.837478
$366$	6.00000	0.313625
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	2.00000	0.104257
$369$	6.00000	0.312348
$370$	12.0000	0.623850
$371$	24.0000	1.24602
$372$	−4.00000	−0.207390
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−1.00000	−0.0517088
$375$	−12.0000	−0.619677
$376$	0	0
$377$	8.00000	0.412021
$378$	−2.00000	−0.102869
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	4.00000	0.205196
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	1.00000	0.0510310
$385$	4.00000	0.203859
$386$	−4.00000	−0.203595
$387$	2.00000	0.101666
$388$	−6.00000	−0.304604
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	8.00000	0.405096
$391$	2.00000	0.101144
$392$	−3.00000	−0.151523
$393$	20.0000	1.00887
$394$	−6.00000	−0.302276
$395$	−4.00000	−0.201262
$396$	−1.00000	−0.0502519
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−20.0000	−1.00251
$399$	−4.00000	−0.200250
$400$	−1.00000	−0.0500000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	−4.00000	−0.199502
$403$	−16.0000	−0.797017
$404$	2.00000	0.0995037
$405$	2.00000	0.0993808
$406$	−4.00000	−0.198517
$407$	−6.00000	−0.297409
$408$	1.00000	0.0495074
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	12.0000	0.592638
$411$	18.0000	0.887875
$412$	−8.00000	−0.394132
$413$	28.0000	1.37779
$414$	2.00000	0.0982946
$415$	−24.0000	−1.17811
$416$	4.00000	0.196116
$417$	0	0
$418$	−2.00000	−0.0978232
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	−4.00000	−0.195180
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	−12.0000	−0.582772
$425$	−1.00000	−0.0485071
$426$	−2.00000	−0.0969003
$427$	−12.0000	−0.580721
$428$	4.00000	0.193347
$429$	−4.00000	−0.193122
$430$	4.00000	0.192897
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	1.00000	0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	8.00000	0.384012
$435$	4.00000	0.191785
$436$	−6.00000	−0.287348
$437$	4.00000	0.191346
$438$	−8.00000	−0.382255
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	−2.00000	−0.0953463
$441$	−3.00000	−0.142857
$442$	4.00000	0.190261
$443$	34.0000	1.61539	0.807694	−	0.589601i	$-0.200715\pi$
$443$	34.0000	1.61539	0.807694	+	0.589601i	$0.200715\pi$
$444$	6.00000	0.284747
$445$	12.0000	0.568855
$446$	0	0
$447$	6.00000	0.283790
$448$	−2.00000	−0.0944911
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	−1.00000	−0.0471405
$451$	−6.00000	−0.282529
$452$	12.0000	0.564433
$453$	−4.00000	−0.187936
$454$	−12.0000	−0.563188
$455$	−16.0000	−0.750092
$456$	2.00000	0.0936586
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−22.0000	−1.02799
$459$	1.00000	0.0466760
$460$	4.00000	0.186501
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	2.00000	0.0930484
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	2.00000	0.0928477
$465$	−8.00000	−0.370991
$466$	−18.0000	−0.833834
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	4.00000	0.184900
$469$	8.00000	0.369406
$470$	0	0
$471$	−14.0000	−0.645086
$472$	−14.0000	−0.644402
$473$	−2.00000	−0.0919601
$474$	−2.00000	−0.0918630
$475$	−2.00000	−0.0917663
$476$	−2.00000	−0.0916698
$477$	−12.0000	−0.549442
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	2.00000	0.0912871
$481$	24.0000	1.09431
$482$	−4.00000	−0.182195
$483$	−4.00000	−0.182006
$484$	1.00000	0.0454545
$485$	−12.0000	−0.544892
$486$	1.00000	0.0453609
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	6.00000	0.271607
$489$	−12.0000	−0.542659
$490$	−6.00000	−0.271052
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	6.00000	0.270501
$493$	2.00000	0.0900755
$494$	8.00000	0.359937
$495$	−2.00000	−0.0898933
$496$	−4.00000	−0.179605
$497$	4.00000	0.179425
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−12.0000	−0.536656
$501$	12.0000	0.536120
$502$	−2.00000	−0.0892644
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−2.00000	−0.0890871
$505$	4.00000	0.177998
$506$	−2.00000	−0.0889108
$507$	3.00000	0.133235
$508$	−8.00000	−0.354943
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	2.00000	0.0885615
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−22.0000	−0.970378
$515$	−16.0000	−0.705044
$516$	2.00000	0.0880451
$517$	0	0
$518$	−12.0000	−0.527250
$519$	6.00000	0.263371
$520$	8.00000	0.350823
$521$	−28.0000	−1.22670	−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000	−1.22670	−0.613351	+	0.789810i	$0.710179\pi$
$522$	2.00000	0.0875376
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	20.0000	0.873704
$525$	2.00000	0.0872872
$526$	8.00000	0.348817
$527$	−4.00000	−0.174243
$528$	−1.00000	−0.0435194
$529$	−19.0000	−0.826087
$530$	−24.0000	−1.04249
$531$	−14.0000	−0.607548
$532$	−4.00000	−0.173422
$533$	24.0000	1.03956
$534$	6.00000	0.259645
$535$	8.00000	0.345870
$536$	−4.00000	−0.172774
$537$	−10.0000	−0.431532
$538$	10.0000	0.431131
$539$	3.00000	0.129219
$540$	2.00000	0.0860663
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.00000	−0.343629
$543$	−10.0000	−0.429141
$544$	1.00000	0.0428746
$545$	−12.0000	−0.514024
$546$	−8.00000	−0.342368
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	18.0000	0.768922
$549$	6.00000	0.256074
$550$	1.00000	0.0426401
$551$	4.00000	0.170406
$552$	2.00000	0.0851257
$553$	4.00000	0.170097
$554$	−14.0000	−0.594803
$555$	12.0000	0.509372
$556$	0	0
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	−4.00000	−0.169334
$559$	8.00000	0.338364
$560$	−4.00000	−0.169031
$561$	−1.00000	−0.0422200
$562$	22.0000	0.928014
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	24.0000	1.00969
$566$	8.00000	0.336265
$567$	−2.00000	−0.0839921
$568$	−2.00000	−0.0839181
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	4.00000	0.167542
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	−4.00000	−0.167248
$573$	12.0000	0.501307
$574$	−12.0000	−0.500870
$575$	−2.00000	−0.0834058
$576$	1.00000	0.0416667
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	1.00000	0.0415945
$579$	−4.00000	−0.166234
$580$	4.00000	0.166091
$581$	24.0000	0.995688
$582$	−6.00000	−0.248708
$583$	12.0000	0.496989
$584$	−8.00000	−0.331042
$585$	8.00000	0.330759
$586$	6.00000	0.247858
$587$	2.00000	0.0825488	0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000	0.0825488	0.0412744	+	0.999148i	$0.486858\pi$
$588$	−3.00000	−0.123718
$589$	−8.00000	−0.329634
$590$	−28.0000	−1.15274
$591$	−6.00000	−0.246807
$592$	6.00000	0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−1.00000	−0.0410305
$595$	−4.00000	−0.163984
$596$	6.00000	0.245770
$597$	−20.0000	−0.818546
$598$	8.00000	0.327144
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	−1.00000	−0.0408248
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	−4.00000	−0.163028
$603$	−4.00000	−0.162893
$604$	−4.00000	−0.162758
$605$	2.00000	0.0813116
$606$	2.00000	0.0812444
$607$	−42.0000	−1.70473	−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000	−1.70473	−0.852364	+	0.522949i	$0.824832\pi$
$608$	2.00000	0.0811107
$609$	−4.00000	−0.162088
$610$	12.0000	0.485866
$611$	0	0
$612$	1.00000	0.0404226
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	22.0000	0.887848
$615$	12.0000	0.483887
$616$	2.00000	0.0805823
$617$	16.0000	0.644136	0.322068	−	0.946717i	$-0.395622\pi$
$617$	16.0000	0.644136	0.322068	+	0.946717i	$0.395622\pi$
$618$	−8.00000	−0.321807
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	−8.00000	−0.321288
$621$	2.00000	0.0802572
$622$	30.0000	1.20289
$623$	−12.0000	−0.480770
$624$	4.00000	0.160128
$625$	−19.0000	−0.760000
$626$	34.0000	1.35891
$627$	−2.00000	−0.0798723
$628$	−14.0000	−0.558661
$629$	6.00000	0.239236
$630$	−4.00000	−0.159364
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	−2.00000	−0.0795557
$633$	8.00000	0.317971
$634$	30.0000	1.19145
$635$	−16.0000	−0.634941
$636$	−12.0000	−0.475831
$637$	−12.0000	−0.475457
$638$	−2.00000	−0.0791808
$639$	−2.00000	−0.0791188
$640$	2.00000	0.0790569
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	4.00000	0.157867
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	−4.00000	−0.157622
$645$	4.00000	0.157500
$646$	2.00000	0.0786889
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	1.00000	0.0392837
$649$	14.0000	0.549548
$650$	−4.00000	−0.156893
$651$	8.00000	0.313545
$652$	−12.0000	−0.469956
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−6.00000	−0.234619
$655$	40.0000	1.56293
$656$	6.00000	0.234261
$657$	−8.00000	−0.312110
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−2.00000	−0.0778499
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−20.0000	−0.777322
$663$	4.00000	0.155347
$664$	−12.0000	−0.465690
$665$	−8.00000	−0.310227
$666$	6.00000	0.232495
$667$	4.00000	0.154881
$668$	12.0000	0.464294
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−6.00000	−0.231627
$672$	−2.00000	−0.0771517
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	8.00000	0.308148
$675$	−1.00000	−0.0384900
$676$	3.00000	0.115385
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	12.0000	0.460857
$679$	12.0000	0.460518
$680$	2.00000	0.0766965
$681$	−12.0000	−0.459841
$682$	4.00000	0.153168
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	2.00000	0.0764719
$685$	36.0000	1.37549
$686$	20.0000	0.763604
$687$	−22.0000	−0.839352
$688$	2.00000	0.0762493
$689$	−48.0000	−1.82865
$690$	4.00000	0.152277
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	6.00000	0.228086
$693$	2.00000	0.0759737
$694$	−12.0000	−0.455514
$695$	0	0
$696$	2.00000	0.0758098
$697$	6.00000	0.227266
$698$	−4.00000	−0.151402
$699$	−18.0000	−0.680823
$700$	2.00000	0.0755929
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	4.00000	0.150970
$703$	12.0000	0.452589
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−4.00000	−0.150435
$708$	−14.0000	−0.526152
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	−4.00000	−0.150117
$711$	−2.00000	−0.0750059
$712$	6.00000	0.224860
$713$	−8.00000	−0.299602
$714$	−2.00000	−0.0748481
$715$	−8.00000	−0.299183
$716$	−10.0000	−0.373718
$717$	0	0
$718$	−32.0000	−1.19423
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	2.00000	0.0745356
$721$	16.0000	0.595871
$722$	−15.0000	−0.558242
$723$	−4.00000	−0.148762
$724$	−10.0000	−0.371647
$725$	−2.00000	−0.0742781
$726$	1.00000	0.0371135
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	−16.0000	−0.592187
$731$	2.00000	0.0739727
$732$	6.00000	0.221766
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	4.00000	0.147643
$735$	−6.00000	−0.221313
$736$	2.00000	0.0737210
$737$	4.00000	0.147342
$738$	6.00000	0.220863
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	12.0000	0.441129
$741$	8.00000	0.293887
$742$	24.0000	0.881068
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	−4.00000	−0.146647
$745$	12.0000	0.439646
$746$	20.0000	0.732252
$747$	−12.0000	−0.439057
$748$	−1.00000	−0.0365636
$749$	−8.00000	−0.292314
$750$	−12.0000	−0.438178
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	−2.00000	−0.0728841
$754$	8.00000	0.291343
$755$	−8.00000	−0.291150
$756$	−2.00000	−0.0727393
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	4.00000	0.145287
$759$	−2.00000	−0.0725954
$760$	4.00000	0.145095
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	−8.00000	−0.289809
$763$	12.0000	0.434429
$764$	12.0000	0.434145
$765$	2.00000	0.0723102
$766$	4.00000	0.144526
$767$	−56.0000	−2.02204
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	4.00000	0.144150
$771$	−22.0000	−0.792311
$772$	−4.00000	−0.143963
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	2.00000	0.0718885
$775$	4.00000	0.143684
$776$	−6.00000	−0.215387
$777$	−12.0000	−0.430498
$778$	24.0000	0.860442
$779$	12.0000	0.429945
$780$	8.00000	0.286446
$781$	2.00000	0.0715656
$782$	2.00000	0.0715199
$783$	2.00000	0.0714742
$784$	−3.00000	−0.107143
$785$	−28.0000	−0.999363
$786$	20.0000	0.713376
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−6.00000	−0.213741
$789$	8.00000	0.284808
$790$	−4.00000	−0.142314
$791$	−24.0000	−0.853342
$792$	−1.00000	−0.0355335
$793$	24.0000	0.852265
$794$	−14.0000	−0.496841
$795$	−24.0000	−0.851192
$796$	−20.0000	−0.708881
$797$	−20.0000	−0.708436	−0.354218	−	0.935163i	$-0.615253\pi$
$797$	−20.0000	−0.708436	−0.354218	+	0.935163i	$0.615253\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	28.0000	0.988714
$803$	8.00000	0.282314
$804$	−4.00000	−0.141069
$805$	−8.00000	−0.281963
$806$	−16.0000	−0.563576
$807$	10.0000	0.352017
$808$	2.00000	0.0703598
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	2.00000	0.0702728
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	−4.00000	−0.140372
$813$	−8.00000	−0.280572
$814$	−6.00000	−0.210300
$815$	−24.0000	−0.840683
$816$	1.00000	0.0350070
$817$	4.00000	0.139942
$818$	30.0000	1.04893
$819$	−8.00000	−0.279543
$820$	12.0000	0.419058
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	18.0000	0.627822
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	−8.00000	−0.278693
$825$	1.00000	0.0348155
$826$	28.0000	0.974245
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	2.00000	0.0695048
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	−24.0000	−0.833052
$831$	−14.0000	−0.485655
$832$	4.00000	0.138675
$833$	−3.00000	−0.103944
$834$	0	0
$835$	24.0000	0.830554
$836$	−2.00000	−0.0691714
$837$	−4.00000	−0.138260
$838$	−20.0000	−0.690889
$839$	−2.00000	−0.0690477	−0.0345238	−	0.999404i	$-0.510991\pi$
$839$	−2.00000	−0.0690477	−0.0345238	+	0.999404i	$0.510991\pi$
$840$	−4.00000	−0.138013
$841$	−25.0000	−0.862069
$842$	−34.0000	−1.17172
$843$	22.0000	0.757720
$844$	8.00000	0.275371
$845$	6.00000	0.206406
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	−12.0000	−0.412082
$849$	8.00000	0.274559
$850$	−1.00000	−0.0342997
$851$	12.0000	0.411355
$852$	−2.00000	−0.0685189
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−12.0000	−0.410632
$855$	4.00000	0.136797
$856$	4.00000	0.136717
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−4.00000	−0.136558
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	4.00000	0.136399
$861$	−12.0000	−0.408959
$862$	−16.0000	−0.544962
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	1.00000	0.0340207
$865$	12.0000	0.408012
$866$	14.0000	0.475739
$867$	1.00000	0.0339618
$868$	8.00000	0.271538
$869$	2.00000	0.0678454
$870$	4.00000	0.135613
$871$	−16.0000	−0.542139
$872$	−6.00000	−0.203186
$873$	−6.00000	−0.203069
$874$	4.00000	0.135302
$875$	24.0000	0.811348
$876$	−8.00000	−0.270295
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−10.0000	−0.337484
$879$	6.00000	0.202375
$880$	−2.00000	−0.0674200
$881$	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	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	4.00000	0.134535
$885$	−28.0000	−0.941210
$886$	34.0000	1.14225
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	6.00000	0.201347
$889$	16.0000	0.536623
$890$	12.0000	0.402241
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	6.00000	0.200670
$895$	−20.0000	−0.668526
$896$	−2.00000	−0.0668153
$897$	8.00000	0.267112
$898$	32.0000	1.06785
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	−12.0000	−0.399778
$902$	−6.00000	−0.199778
$903$	−4.00000	−0.133112
$904$	12.0000	0.399114
$905$	−20.0000	−0.664822
$906$	−4.00000	−0.132891
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	−12.0000	−0.398234
$909$	2.00000	0.0663358
$910$	−16.0000	−0.530395
$911$	46.0000	1.52405	0.762024	−	0.647549i	$-0.224206\pi$
$911$	46.0000	1.52405	0.762024	+	0.647549i	$0.224206\pi$
$912$	2.00000	0.0662266
$913$	12.0000	0.397142
$914$	22.0000	0.727695
$915$	12.0000	0.396708
$916$	−22.0000	−0.726900
$917$	−40.0000	−1.32092
$918$	1.00000	0.0330049
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	4.00000	0.131876
$921$	22.0000	0.724925
$922$	−18.0000	−0.592798
$923$	−8.00000	−0.263323
$924$	2.00000	0.0657952
$925$	−6.00000	−0.197279
$926$	−16.0000	−0.525793
$927$	−8.00000	−0.262754
$928$	2.00000	0.0656532
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	−8.00000	−0.262330
$931$	−6.00000	−0.196642
$932$	−18.0000	−0.589610
$933$	30.0000	0.982156
$934$	−6.00000	−0.196326
$935$	−2.00000	−0.0654070
$936$	4.00000	0.130744
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	8.00000	0.261209
$939$	34.0000	1.10955
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	−14.0000	−0.456145
$943$	12.0000	0.390774
$944$	−14.0000	−0.455661
$945$	−4.00000	−0.130120
$946$	−2.00000	−0.0650256
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−2.00000	−0.0649570
$949$	−32.0000	−1.03876
$950$	−2.00000	−0.0648886
$951$	30.0000	0.972817
$952$	−2.00000	−0.0648204
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	−12.0000	−0.388514
$955$	24.0000	0.776622
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	12.0000	0.387702
$959$	−36.0000	−1.16250
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	24.0000	0.773791
$963$	4.00000	0.128898
$964$	−4.00000	−0.128831
$965$	−8.00000	−0.257529
$966$	−4.00000	−0.128698
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	1.00000	0.0321412
$969$	2.00000	0.0642493
$970$	−12.0000	−0.385297
$971$	10.0000	0.320915	0.160458	−	0.987043i	$-0.448703\pi$
$971$	10.0000	0.320915	0.160458	+	0.987043i	$0.448703\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	28.0000	0.897178
$975$	−4.00000	−0.128103
$976$	6.00000	0.192055
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−12.0000	−0.383718
$979$	−6.00000	−0.191761
$980$	−6.00000	−0.191663
$981$	−6.00000	−0.191565
$982$	−8.00000	−0.255290
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	6.00000	0.191273
$985$	−12.0000	−0.382352
$986$	2.00000	0.0636930
$987$	0	0
$988$	8.00000	0.254514
$989$	4.00000	0.127193
$990$	−2.00000	−0.0635642
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−4.00000	−0.127000
$993$	−20.0000	−0.634681
$994$	4.00000	0.126872
$995$	−40.0000	−1.26809
$996$	−12.0000	−0.380235
$997$	46.0000	1.45683	0.728417	−	0.685134i	$-0.240256\pi$
$997$	46.0000	1.45683	0.728417	+	0.685134i	$0.240256\pi$
$998$	4.00000	0.126618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1122.2.a.l.1.1	✓	1
3.2	odd	2		3366.2.a.c.1.1		1
4.3	odd	2		8976.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1122.2.a.l.1.1	✓	1	1.1	even	1	trivial
3366.2.a.c.1.1		1	3.2	odd	2
8976.2.a.r.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1122.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more