LMFDB - Embedded newform 3330.2.a.j.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:	$1$
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)

$f(q)$

$=$

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

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} -8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -2.00000 q^{29} -1.00000 q^{32} +6.00000 q^{34} -1.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{44} +8.00000 q^{46} -6.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} +2.00000 q^{55} +2.00000 q^{58} -4.00000 q^{61} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +6.00000 q^{73} +1.00000 q^{74} +6.00000 q^{76} +8.00000 q^{79} +1.00000 q^{80} +2.00000 q^{82} -12.0000 q^{83} -6.00000 q^{85} -2.00000 q^{88} -6.00000 q^{89} -8.00000 q^{92} +6.00000 q^{94} +6.00000 q^{95} +8.00000 q^{97} +7.00000 q^{98} +O(q^{100})$

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
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	−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
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	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
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	6.00000	0.688247
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	2.00000	0.220863
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−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
$91$	0	0
$92$	−8.00000	−0.834058
$93$	0	0
$94$	6.00000	0.618853
$95$	6.00000	0.615587
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	7.00000	0.707107
$99$	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
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−2.00000	−0.185695
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	4.00000	0.362143
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.00000	0.175412
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	16.0000	1.36697	0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000	1.36697	0.683486	+	0.729964i	$0.260463\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	−2.00000	−0.166091
$146$	−6.00000	−0.496564
$147$	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
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	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
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	0	0
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	8.00000	0.589768
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−6.00000	−0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	−7.00000	−0.500000
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	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
$203$	0	0
$204$	0	0
$205$	−2.00000	−0.139686
$206$	−10.0000	−0.696733
$207$	0	0
$208$	−2.00000	−0.138675
$209$	12.0000	0.830057
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	0	0
$218$	16.0000	1.08366
$219$	0	0
$220$	2.00000	0.134840
$221$	12.0000	0.807207
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−30.0000	−1.98246	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000	−1.98246	−0.991228	+	0.132164i	$0.957808\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	2.00000	0.131306
$233$	−12.0000	−0.786146	−0.393073	−	0.919507i	$-0.628588\pi$
$233$	−12.0000	−0.786146	−0.393073	+	0.919507i	$0.628588\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−4.00000	−0.256074
$245$	−7.00000	−0.447214
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−16.0000	−1.00591
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	−2.00000	−0.124035
$261$	0	0
$262$	−8.00000	−0.494242
$263$	−2.00000	−0.123325	−0.0616626	−	0.998097i	$-0.519640\pi$
$263$	−2.00000	−0.123325	−0.0616626	+	0.998097i	$0.519640\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−8.00000	−0.487769	−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000	−0.487769	−0.243884	+	0.969804i	$0.578422\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−16.0000	−0.966595
$275$	2.00000	0.120605
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	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	$-0.576418\pi$
$283$	−8.00000	−0.475551	−0.237775	+	0.971320i	$0.576418\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	2.00000	0.117444
$291$	0	0
$292$	6.00000	0.351123
$293$	2.00000	0.116841	0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000	0.116841	0.0584206	+	0.998292i	$0.481394\pi$
$294$	0	0
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	−12.0000	−0.695141
$299$	16.0000	0.925304
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	0	0
$304$	6.00000	0.344124
$305$	−4.00000	−0.229039
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.0000	−1.81455	−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000	−1.81455	−0.907277	+	0.420534i	$0.861843\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	8.00000	0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	−36.0000	−2.00309
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−8.00000	−0.443079
$327$	0	0
$328$	2.00000	0.110432
$329$	0	0
$330$	0	0
$331$	10.0000	0.549650	0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000	0.549650	0.274825	+	0.961494i	$0.411380\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−6.00000	−0.325396
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	10.0000	0.537603
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	2.00000	0.105118
$363$	0	0
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	1.00000	0.0519875
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	6.00000	0.309426
$377$	4.00000	0.206010
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	6.00000	0.307794
$381$	0	0
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	0	0
$388$	8.00000	0.406138
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	48.0000	2.42746
$392$	7.00000	0.353553
$393$	0	0
$394$	−2.00000	−0.100759
$395$	8.00000	0.402524
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	10.0000	0.492665
$413$	0	0
$414$	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
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	2.00000	0.0971286
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	−48.0000	−2.29615
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−12.0000	−0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	6.00000	0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	30.0000	1.40181
$459$	0	0
$460$	−8.00000	−0.373002
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	12.0000	0.555889
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	0	0
$470$	6.00000	0.276759
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	0	0
$478$	8.00000	0.365911
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	−26.0000	−1.18427
$483$	0	0
$484$	−7.00000	−0.318182
$485$	8.00000	0.363261
$486$	0	0
$487$	−42.0000	−1.90320	−0.951601	−	0.307337i	$-0.900562\pi$
$487$	−42.0000	−1.90320	−0.951601	+	0.307337i	$0.900562\pi$
$488$	4.00000	0.181071
$489$	0	0
$490$	7.00000	0.316228
$491$	14.0000	0.631811	0.315906	−	0.948791i	$-0.397692\pi$
$491$	14.0000	0.631811	0.315906	+	0.948791i	$0.397692\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	12.0000	0.539906
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−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
$506$	16.0000	0.711287
$507$	0	0
$508$	−16.0000	−0.709885
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	10.0000	0.440653
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	0	0
$520$	2.00000	0.0877058
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	2.00000	0.0872041
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	2.00000	0.0868744
$531$	0	0
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	−8.00000	−0.345870
$536$	4.00000	0.172774
$537$	0	0
$538$	8.00000	0.344904
$539$	−14.0000	−0.603023
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	6.00000	0.257248
$545$	−16.0000	−0.685365
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	16.0000	0.683486
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−22.0000	−0.928014
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	0	0
$575$	−8.00000	−0.333623
$576$	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
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	0	0
$583$	−4.00000	−0.165663
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−8.00000	−0.328521	−0.164260	−	0.986417i	$-0.552524\pi$
$593$	−8.00000	−0.328521	−0.164260	+	0.986417i	$0.552524\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	0	0
$598$	−16.0000	−0.654289
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	−8.00000	−0.325515
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−46.0000	−1.86708	−0.933541	−	0.358470i	$-0.883298\pi$
$607$	−46.0000	−1.86708	−0.933541	+	0.358470i	$0.883298\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	4.00000	0.161955
$611$	12.0000	0.485468
$612$	0	0
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	0	0
$617$	−40.0000	−1.61034	−0.805170	−	0.593045i	$-0.797926\pi$
$617$	−40.0000	−1.61034	−0.805170	+	0.593045i	$0.797926\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	0	0
$622$	32.0000	1.28308
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	24.0000	0.959233
$627$	0	0
$628$	−22.0000	−0.877896
$629$	6.00000	0.239236
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	18.0000	0.714871
$635$	−16.0000	−0.634941
$636$	0	0
$637$	14.0000	0.554700
$638$	4.00000	0.158362
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	0	0
$646$	36.0000	1.41640
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	8.00000	0.313304
$653$	50.0000	1.95665	0.978326	−	0.207072i	$-0.0663936\pi$
$653$	50.0000	1.95665	0.978326	+	0.207072i	$0.0663936\pi$
$654$	0	0
$655$	8.00000	0.312586
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	0	0
$659$	−22.0000	−0.856998	−0.428499	−	0.903542i	$-0.640958\pi$
$659$	−22.0000	−0.856998	−0.428499	+	0.903542i	$0.640958\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	12.0000	0.464294
$669$	0	0
$670$	4.00000	0.154533
$671$	−8.00000	−0.308837
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	−9.00000	−0.346154
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	0	0
$679$	0	0
$680$	6.00000	0.230089
$681$	0	0
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	16.0000	0.611329
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	24.0000	0.913003	0.456502	−	0.889723i	$-0.349102\pi$
$691$	24.0000	0.913003	0.456502	+	0.889723i	$0.349102\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	28.0000	1.06287
$695$	−4.00000	−0.151729
$696$	0	0
$697$	12.0000	0.454532
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	34.0000	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$701$	34.0000	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	2.00000	0.0753778
$705$	0	0
$706$	−22.0000	−0.827981
$707$	0	0
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	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
$721$	0	0
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	0	0
$730$	−6.00000	−0.222070
$731$	0	0
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	8.00000	0.294884
$737$	−8.00000	−0.294684
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	0	0
$743$	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
$746$	10.0000	0.366126
$747$	0	0
$748$	−12.0000	−0.438763
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−4.00000	−0.145671
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−30.0000	−1.09037	−0.545184	−	0.838316i	$-0.683540\pi$
$757$	−30.0000	−1.09037	−0.545184	+	0.838316i	$0.683540\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−6.00000	−0.217643
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	4.00000	0.144526
$767$	0	0
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	24.0000	0.863779
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	−18.0000	−0.645331
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	−48.0000	−1.71648
$783$	0	0
$784$	−7.00000	−0.250000
$785$	−22.0000	−0.785214
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	2.00000	0.0712470
$789$	0	0
$790$	−8.00000	−0.284627
$791$	0	0
$792$	0	0
$793$	8.00000	0.284088
$794$	−30.0000	−1.06466
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−22.0000	−0.779280	−0.389640	−	0.920967i	$-0.627401\pi$
$797$	−22.0000	−0.779280	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−38.0000	−1.34183
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	8.00000	0.280228
$816$	0	0
$817$	0	0
$818$	−6.00000	−0.209785
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	0	0
$827$	52.0000	1.80822	0.904109	−	0.427303i	$-0.140536\pi$
$827$	52.0000	1.80822	0.904109	+	0.427303i	$0.140536\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	42.0000	1.45521
$834$	0	0
$835$	12.0000	0.415277
$836$	12.0000	0.415029
$837$	0	0
$838$	6.00000	0.207267
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	28.0000	0.964944
$843$	0	0
$844$	−8.00000	−0.275371
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	6.00000	0.205798
$851$	8.00000	0.274236
$852$	0	0
$853$	34.0000	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$853$	34.0000	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	2.00000	0.0679628
$867$	0	0
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	8.00000	0.271070
$872$	16.0000	0.541828
$873$	0	0
$874$	48.0000	1.62362
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	−4.00000	−0.134993
$879$	0	0
$880$	2.00000	0.0674200
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	0	0
$889$	0	0
$890$	6.00000	0.201120
$891$	0	0
$892$	16.0000	0.535720
$893$	−36.0000	−1.20469
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	30.0000	1.00111
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	4.00000	0.133185
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	8.00000	0.264616
$915$	0	0
$916$	−30.0000	−0.991228
$917$	0	0
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	8.00000	0.263752
$921$	0	0
$922$	−42.0000	−1.38320
$923$	0	0
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−34.0000	−1.11731
$927$	0	0
$928$	2.00000	0.0656532
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−42.0000	−1.37649
$932$	−12.0000	−0.393073
$933$	0	0
$934$	36.0000	1.17796
$935$	−12.0000	−0.392442
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	0	0
$940$	−6.00000	−0.195698
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.0000	0.649913	0.324956	−	0.945729i	$-0.394650\pi$
$947$	20.0000	0.649913	0.324956	+	0.945729i	$0.394650\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	−6.00000	−0.194666
$951$	0	0
$952$	0	0
$953$	16.0000	0.518291	0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000	0.518291	0.259145	+	0.965838i	$0.416559\pi$
$954$	0	0
$955$	0	0
$956$	−8.00000	−0.258738
$957$	0	0
$958$	8.00000	0.258468
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−2.00000	−0.0644826
$963$	0	0
$964$	26.0000	0.837404
$965$	24.0000	0.772587
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−8.00000	−0.256865
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	0	0
$973$	0	0
$974$	42.0000	1.34577
$975$	0	0
$976$	−4.00000	−0.128037
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	−7.00000	−0.223607
$981$	0	0
$982$	−14.0000	−0.446758
$983$	14.0000	0.446531	0.223265	−	0.974758i	$-0.428328\pi$
$983$	14.0000	0.446531	0.223265	+	0.974758i	$0.428328\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−12.0000	−0.381771
$989$	0	0
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	6.00000	0.189927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.j.1.1		1
3.2	odd	2		1110.2.a.i.1.1	✓	1
12.11	even	2		8880.2.a.v.1.1		1
15.14	odd	2		5550.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.i.1.1	✓	1	3.2	odd	2
3330.2.a.j.1.1		1	1.1	even	1	trivial
5550.2.a.q.1.1		1	15.14	odd	2
8880.2.a.v.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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