LMFDB - Embedded newform 3330.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -6.00000 q^{11} +2.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} -1.00000 q^{20} -6.00000 q^{22} +1.00000 q^{25} +2.00000 q^{26} -4.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} -6.00000 q^{44} -6.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +6.00000 q^{55} -4.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +8.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} +14.0000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +24.0000 q^{77} -16.0000 q^{79} -1.00000 q^{80} +6.00000 q^{82} +12.0000 q^{83} -6.00000 q^{85} +8.00000 q^{86} -6.00000 q^{88} -6.00000 q^{89} -8.00000 q^{91} -6.00000 q^{94} -2.00000 q^{95} +8.00000 q^{97} +9.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−6.00000	−1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	−4.00000	−0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	4.00000	0.676123
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	2.00000	0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	−4.00000	−0.534522
$57$	0	0
$58$	−6.00000	−0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	24.0000	2.73505
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	8.00000	0.862662
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−2.00000	−0.205196
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	12.0000	1.10469
$119$	−24.0000	−2.20008
$120$	0	0
$121$	25.0000	2.27273
$122$	8.00000	0.724286
$123$	0	0
$124$	8.00000	0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	6.00000	0.498273
$146$	14.0000	1.15865
$147$	0	0
$148$	1.00000	0.0821995
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	24.0000	1.93398
$155$	−8.00000	−0.642575
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	−16.0000	−1.27289
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	0	0
$172$	8.00000	0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	−6.00000	−0.452267
$177$	0	0
$178$	−6.00000	−0.449719
$179$	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
$181$	−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
$183$	0	0
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−36.0000	−2.63258
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−2.00000	−0.145095
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	9.00000	0.642857
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	−6.00000	−0.419058
$206$	14.0000	0.975426
$207$	0	0
$208$	2.00000	0.138675
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−32.0000	−2.17230
$218$	20.0000	1.35457
$219$	0	0
$220$	6.00000	0.404520
$221$	12.0000	0.807207
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	12.0000	0.781133
$237$	0	0
$238$	−24.0000	−1.55569
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	8.00000	0.512148
$245$	−9.00000	−0.574989
$246$	0	0
$247$	4.00000	0.254514
$248$	8.00000	0.508001
$249$	0	0
$250$	−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
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	−2.00000	−0.124035
$261$	0	0
$262$	12.0000	0.741362
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−8.00000	−0.490511
$267$	0	0
$268$	−4.00000	−0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	12.0000	0.724947
$275$	−6.00000	−0.361814
$276$	0	0
$277$	−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
$279$	0	0
$280$	4.00000	0.239046
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	−12.0000	−0.709575
$287$	−24.0000	−1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	0	0
$292$	14.0000	0.819288
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	1.00000	0.0581238
$297$	0	0
$298$	12.0000	0.695141
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	−16.0000	−0.920697
$303$	0	0
$304$	2.00000	0.114708
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	24.0000	1.36753
$309$	0	0
$310$	−8.00000	−0.454369
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	2.00000	0.110940
$326$	−16.0000	−0.886158
$327$	0	0
$328$	6.00000	0.331295
$329$	24.0000	1.32316
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−12.0000	−0.656611
$335$	4.00000	0.218543
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−48.0000	−2.59935
$342$	0	0
$343$	−8.00000	−0.431959
$344$	8.00000	0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−10.0000	−0.525588
$363$	0	0
$364$	−8.00000	−0.419314
$365$	−14.0000	−0.732793
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	0	0
$370$	−1.00000	−0.0519875
$371$	24.0000	1.24602
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−12.0000	−0.618031
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	−16.0000	−0.814379
$387$	0	0
$388$	8.00000	0.406138
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	0	0
$394$	−18.0000	−0.906827
$395$	16.0000	0.805047
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	0	0
$405$	0	0
$406$	24.0000	1.19110
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	14.0000	0.689730
$413$	−48.0000	−2.36193
$414$	0	0
$415$	−12.0000	−0.589057
$416$	2.00000	0.0980581
$417$	0	0
$418$	−12.0000	−0.586939
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	−16.0000	−0.778868
$423$	0	0
$424$	−6.00000	−0.291386
$425$	6.00000	0.291043
$426$	0	0
$427$	−32.0000	−1.54859
$428$	0	0
$429$	0	0
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	20.0000	0.957826
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	12.0000	0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−4.00000	−0.188982
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	−6.00000	−0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	8.00000	0.375046
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	26.0000	1.21490
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	−34.0000	−1.58011	−0.790057	−	0.613033i	$-0.789949\pi$
$463$	−34.0000	−1.58011	−0.790057	+	0.613033i	$0.789949\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	24.0000	1.11178
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	6.00000	0.276759
$471$	0	0
$472$	12.0000	0.552345
$473$	−48.0000	−2.20704
$474$	0	0
$475$	2.00000	0.0917663
$476$	−24.0000	−1.10004
$477$	0	0
$478$	−24.0000	−1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	2.00000	0.0910975
$483$	0	0
$484$	25.0000	1.13636
$485$	−8.00000	−0.363261
$486$	0	0
$487$	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
$489$	0	0
$490$	−9.00000	−0.406579
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	4.00000	0.179969
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−4.00000	−0.177471
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.0000	−1.32324
$515$	−14.0000	−0.616914
$516$	0	0
$517$	36.0000	1.58328
$518$	−4.00000	−0.175750
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	6.00000	0.261612
$527$	48.0000	2.09091
$528$	0	0
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	0	0
$532$	−8.00000	−0.346844
$533$	12.0000	0.519778
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	24.0000	1.03471
$539$	−54.0000	−2.32594
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	6.00000	0.257248
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	12.0000	0.512615
$549$	0	0
$550$	−6.00000	−0.255841
$551$	−12.0000	−0.511217
$552$	0	0
$553$	64.0000	2.72156
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	4.00000	0.169031
$561$	0	0
$562$	30.0000	1.26547
$563$	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
$565$	6.00000	0.252422
$566$	−16.0000	−0.672530
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	0	0
$577$	20.0000	0.832611	0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000	0.832611	0.416305	+	0.909225i	$0.363325\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	6.00000	0.249136
$581$	−48.0000	−1.99138
$582$	0	0
$583$	36.0000	1.49097
$584$	14.0000	0.579324
$585$	0	0
$586$	30.0000	1.23929
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−12.0000	−0.494032
$591$	0	0
$592$	1.00000	0.0410997
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	12.0000	0.491539
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−32.0000	−1.30422
$603$	0	0
$604$	−16.0000	−0.651031
$605$	−25.0000	−1.01639
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−8.00000	−0.323911
$611$	−12.0000	−0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	24.0000	0.966988
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	14.0000	0.558661
$629$	6.00000	0.239236
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	−16.0000	−0.636446
$633$	0	0
$634$	18.0000	0.714871
$635$	4.00000	0.158735
$636$	0	0
$637$	18.0000	0.713186
$638$	36.0000	1.42525
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−72.0000	−2.82625
$650$	2.00000	0.0784465
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	6.00000	0.234261
$657$	0	0
$658$	24.0000	0.935617
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	12.0000	0.465690
$665$	8.00000	0.310227
$666$	0	0
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	4.00000	0.154533
$671$	−48.0000	−1.85302
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−32.0000	−1.22805
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−48.0000	−1.83801
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	−8.00000	−0.305441
$687$	0	0
$688$	8.00000	0.304997
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	12.0000	0.455514
$695$	4.00000	0.151729
$696$	0	0
$697$	36.0000	1.36360
$698$	26.0000	0.984115
$699$	0	0
$700$	−4.00000	−0.151186
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	12.0000	0.448775
$716$	12.0000	0.448461
$717$	0	0
$718$	24.0000	0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	−56.0000	−2.08555
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−10.0000	−0.371647
$725$	−6.00000	−0.222834
$726$	0	0
$727$	38.0000	1.40934	0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000	1.40934	0.704671	+	0.709534i	$0.251095\pi$
$728$	−8.00000	−0.296500
$729$	0	0
$730$	−14.0000	−0.518163
$731$	48.0000	1.77534
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	24.0000	0.881068
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	−22.0000	−0.805477
$747$	0	0
$748$	−36.0000	−1.31629
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−12.0000	−0.437014
$755$	16.0000	0.582300
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	−2.00000	−0.0725476
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−80.0000	−2.89619
$764$	0	0
$765$	0	0
$766$	12.0000	0.433578
$767$	24.0000	0.866590
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	−24.0000	−0.864900
$771$	0	0
$772$	−16.0000	−0.575853
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	8.00000	0.287183
$777$	0	0
$778$	30.0000	1.07555
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	9.00000	0.321429
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	16.0000	0.569254
$791$	24.0000	0.853342
$792$	0	0
$793$	16.0000	0.568177
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	1.00000	0.0353553
$801$	0	0
$802$	−18.0000	−0.635602
$803$	−84.0000	−2.96430
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	−6.00000	−0.210300
$815$	16.0000	0.560456
$816$	0	0
$817$	16.0000	0.559769
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	−48.0000	−1.67013
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	−12.0000	−0.416526
$831$	0	0
$832$	2.00000	0.0693375
$833$	54.0000	1.87099
$834$	0	0
$835$	12.0000	0.415277
$836$	−12.0000	−0.415029
$837$	0	0
$838$	−6.00000	−0.207267
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	8.00000	0.275698
$843$	0	0
$844$	−16.0000	−0.550743
$845$	9.00000	0.309609
$846$	0	0
$847$	−100.000	−3.43604
$848$	−6.00000	−0.206041
$849$	0	0
$850$	6.00000	0.205798
$851$	0	0
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−32.0000	−1.09502
$855$	0	0
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	38.0000	1.29129
$867$	0	0
$868$	−32.0000	−1.08615
$869$	96.0000	3.25658
$870$	0	0
$871$	−8.00000	−0.271070
$872$	20.0000	0.677285
$873$	0	0
$874$	0	0
$875$	4.00000	0.135225
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	20.0000	0.674967
$879$	0	0
$880$	6.00000	0.202260
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	12.0000	0.403148
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	6.00000	0.201120
$891$	0	0
$892$	−4.00000	−0.133930
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−12.0000	−0.401116
$896$	−4.00000	−0.133631
$897$	0	0
$898$	18.0000	0.600668
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−36.0000	−1.19933
$902$	−36.0000	−1.19867
$903$	0	0
$904$	−6.00000	−0.199557
$905$	10.0000	0.332411
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	8.00000	0.265197
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	8.00000	0.264616
$915$	0	0
$916$	26.0000	0.859064
$917$	−48.0000	−1.58510
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	0	0
$922$	6.00000	0.197599
$923$	0	0
$924$	0	0
$925$	1.00000	0.0328798
$926$	−34.0000	−1.11731
$927$	0	0
$928$	−6.00000	−0.196960
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	24.0000	0.786146
$933$	0	0
$934$	12.0000	0.392652
$935$	36.0000	1.17733
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	6.00000	0.195698
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	2.00000	0.0648886
$951$	0	0
$952$	−24.0000	−0.777844
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	24.0000	0.775405
$959$	−48.0000	−1.55000
$960$	0	0
$961$	33.0000	1.06452
$962$	2.00000	0.0644826
$963$	0	0
$964$	2.00000	0.0644157
$965$	16.0000	0.515058
$966$	0	0
$967$	−46.0000	−1.47926	−0.739630	−	0.673014i	$-0.765000\pi$
$967$	−46.0000	−1.47926	−0.739630	+	0.673014i	$0.765000\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−8.00000	−0.256865
$971$	54.0000	1.73294	0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000	1.73294	0.866471	+	0.499227i	$0.166383\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	2.00000	0.0640841
$975$	0	0
$976$	8.00000	0.256074
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	−9.00000	−0.287494
$981$	0	0
$982$	6.00000	0.191468
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−36.0000	−1.14647
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	14.0000	0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.m.1.1		1
3.2	odd	2		1110.2.a.h.1.1	✓	1
12.11	even	2		8880.2.a.n.1.1		1
15.14	odd	2		5550.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.h.1.1	✓	1	3.2	odd	2
3330.2.a.m.1.1		1	1.1	even	1	trivial
5550.2.a.bd.1.1		1	15.14	odd	2
8880.2.a.n.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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