LMFDB - Embedded newform 4410.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +6.00000 q^{11} -4.00000 q^{13} +1.00000 q^{16} +2.00000 q^{19} +1.00000 q^{20} +6.00000 q^{22} +3.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +3.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -9.00000 q^{41} -7.00000 q^{43} +6.00000 q^{44} +3.00000 q^{46} +1.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} +6.00000 q^{55} +3.00000 q^{58} +6.00000 q^{59} +5.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +5.00000 q^{67} +6.00000 q^{71} -16.0000 q^{73} -4.00000 q^{74} +2.00000 q^{76} +2.00000 q^{79} +1.00000 q^{80} -9.00000 q^{82} -3.00000 q^{83} -7.00000 q^{86} +6.00000 q^{88} +15.0000 q^{89} +3.00000 q^{92} +2.00000 q^{95} +14.0000 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	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
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	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
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	3.00000	0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	−4.00000	−0.554700
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	3.00000	0.393919
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−9.00000	−0.993884
$83$	−3.00000	−0.329293	−0.164646	−	0.986353i	$-0.552648\pi$
$83$	−3.00000	−0.329293	−0.164646	+	0.986353i	$0.552648\pi$
$84$	0	0
$85$	0	0
$86$	−7.00000	−0.754829
$87$	0	0
$88$	6.00000	0.639602
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	6.00000	0.572078
$111$	0	0
$112$	0	0
$113$	−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
$115$	3.00000	0.279751
$116$	3.00000	0.278543
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	5.00000	0.452679
$123$	0	0
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	5.00000	0.431934
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	6.00000	0.503509
$143$	−24.0000	−2.00698
$144$	0	0
$145$	3.00000	0.249136
$146$	−16.0000	−1.32417
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−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
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−3.00000	−0.232845
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−7.00000	−0.533745
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	15.0000	1.12430
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	0	0
$184$	3.00000	0.221163
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	2.00000	0.145095
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−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
$202$	−15.0000	−1.05540
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	−4.00000	−0.277350
$209$	12.0000	0.830057
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	15.0000	1.02538
$215$	−7.00000	−0.477396
$216$	0	0
$217$	0	0
$218$	11.0000	0.745014
$219$	0	0
$220$	6.00000	0.404520
$221$	0	0
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	0	0
$225$	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
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	3.00000	0.197814
$231$	0	0
$232$	3.00000	0.196960
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	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
$244$	5.00000	0.320092
$245$	0	0
$246$	0	0
$247$	−8.00000	−0.509028
$248$	8.00000	0.508001
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	−4.00000	−0.248069
$261$	0	0
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	5.00000	0.305424
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	0	0
$274$	−12.0000	−0.724947
$275$	6.00000	0.361814
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	3.00000	0.176166
$291$	0	0
$292$	−16.0000	−0.936329
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−15.0000	−0.868927
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	−4.00000	−0.230174
$303$	0	0
$304$	2.00000	0.114708
$305$	5.00000	0.286299
$306$	0	0
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	2.00000	0.112509
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−4.00000	−0.221540
$327$	0	0
$328$	−9.00000	−0.496942
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−3.00000	−0.164646
$333$	0	0
$334$	3.00000	0.164153
$335$	5.00000	0.273179
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	0	0
$341$	48.0000	2.59935
$342$	0	0
$343$	0	0
$344$	−7.00000	−0.377415
$345$	0	0
$346$	0	0
$347$	−9.00000	−0.483145	−0.241573	−	0.970383i	$-0.577663\pi$
$347$	−9.00000	−0.483145	−0.241573	+	0.970383i	$0.577663\pi$
$348$	0	0
$349$	17.0000	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$350$	0	0
$351$	0	0
$352$	6.00000	0.319801
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	15.0000	0.794998
$357$	0	0
$358$	24.0000	1.26844
$359$	−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
$361$	−15.0000	−0.789474
$362$	11.0000	0.578147
$363$	0	0
$364$	0	0
$365$	−16.0000	−0.837478
$366$	0	0
$367$	35.0000	1.82699	0.913493	−	0.406855i	$-0.133375\pi$
$367$	35.0000	1.82699	0.913493	+	0.406855i	$0.133375\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	−4.00000	−0.207950
$371$	0	0
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	6.00000	0.306987
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	14.0000	0.710742
$389$	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
$392$	0	0
$393$	0	0
$394$	6.00000	0.302276
$395$	2.00000	0.100631
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	−15.0000	−0.749064	−0.374532	−	0.927214i	$-0.622197\pi$
$401$	−15.0000	−0.749064	−0.374532	+	0.927214i	$0.622197\pi$
$402$	0	0
$403$	−32.0000	−1.59403
$404$	−15.0000	−0.746278
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	−9.00000	−0.444478
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	0	0
$414$	0	0
$415$	−3.00000	−0.147264
$416$	−4.00000	−0.196116
$417$	0	0
$418$	12.0000	0.586939
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	−10.0000	−0.486792
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	15.0000	0.725052
$429$	0	0
$430$	−7.00000	−0.337570
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	−22.0000	−1.05725	−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000	−1.05725	−0.528626	+	0.848855i	$0.677293\pi$
$434$	0	0
$435$	0	0
$436$	11.0000	0.526804
$437$	6.00000	0.287019
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	0	0
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	−28.0000	−1.32584
$447$	0	0
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−54.0000	−2.54276
$452$	−6.00000	−0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	3.00000	0.139876
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	12.0000	0.555889
$467$	15.0000	0.694117	0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000	0.694117	0.347059	+	0.937843i	$0.387180\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	−42.0000	−1.93116
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	2.00000	0.0910975
$483$	0	0
$484$	25.0000	1.13636
$485$	14.0000	0.635707
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	5.00000	0.226339
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	−8.00000	−0.359937
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.0000	0.535586
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	−15.0000	−0.667491
$506$	18.0000	0.800198
$507$	0	0
$508$	8.00000	0.354943
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	0	0
$515$	−1.00000	−0.0440653
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−4.00000	−0.175412
$521$	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
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	0	0
$525$	0	0
$526$	21.0000	0.915644
$527$	0	0
$528$	0	0
$529$	−14.0000	−0.608696
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	15.0000	0.648507
$536$	5.00000	0.215967
$537$	0	0
$538$	−15.0000	−0.646696
$539$	0	0
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	0	0
$545$	11.0000	0.471188
$546$	0	0
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	6.00000	0.255841
$551$	6.00000	0.255609
$552$	0	0
$553$	0	0
$554$	8.00000	0.339887
$555$	0	0
$556$	−10.0000	−0.424094
$557$	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
$559$	28.0000	1.18427
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	0	0
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−24.0000	−1.00349
$573$	0	0
$574$	0	0
$575$	3.00000	0.125109
$576$	0	0
$577$	26.0000	1.08239	0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000	1.08239	0.541197	+	0.840896i	$0.317971\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	3.00000	0.124568
$581$	0	0
$582$	0	0
$583$	36.0000	1.49097
$584$	−16.0000	−0.662085
$585$	0	0
$586$	−12.0000	−0.495715
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	6.00000	0.247016
$591$	0	0
$592$	−4.00000	−0.164399
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	−15.0000	−0.614424
$597$	0	0
$598$	−12.0000	−0.490716
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	0	0
$604$	−4.00000	−0.162758
$605$	25.0000	1.01639
$606$	0	0
$607$	23.0000	0.933541	0.466771	−	0.884378i	$-0.345417\pi$
$607$	23.0000	0.933541	0.466771	+	0.884378i	$0.345417\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	5.00000	0.202444
$611$	0	0
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	5.00000	0.201784
$615$	0	0
$616$	0	0
$617$	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
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	18.0000	0.721734
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	0	0
$628$	−22.0000	−0.877896
$629$	0	0
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	2.00000	0.0795557
$633$	0	0
$634$	−12.0000	−0.476581
$635$	8.00000	0.317470
$636$	0	0
$637$	0	0
$638$	18.0000	0.712627
$639$	0	0
$640$	1.00000	0.0395285
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−4.00000	−0.156652
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	0	0
$655$	0	0
$656$	−9.00000	−0.351391
$657$	0	0
$658$	0	0
$659$	−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
$661$	41.0000	1.59472	0.797358	−	0.603507i	$-0.206231\pi$
$661$	41.0000	1.59472	0.797358	+	0.603507i	$0.206231\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−3.00000	−0.116423
$665$	0	0
$666$	0	0
$667$	9.00000	0.348481
$668$	3.00000	0.116073
$669$	0	0
$670$	5.00000	0.193167
$671$	30.0000	1.15814
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	3.00000	0.115385
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	48.0000	1.83801
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	0	0
$688$	−7.00000	−0.266872
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	0	0
$693$	0	0
$694$	−9.00000	−0.341635
$695$	−10.0000	−0.379322
$696$	0	0
$697$	0	0
$698$	17.0000	0.643459
$699$	0	0
$700$	0	0
$701$	3.00000	0.113308	0.0566542	−	0.998394i	$-0.481957\pi$
$701$	3.00000	0.113308	0.0566542	+	0.998394i	$0.481957\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	6.00000	0.226134
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	15.0000	0.562149
$713$	24.0000	0.898807
$714$	0	0
$715$	−24.0000	−0.897549
$716$	24.0000	0.896922
$717$	0	0
$718$	−24.0000	−0.895672
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	0	0
$724$	11.0000	0.408812
$725$	3.00000	0.111417
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	0	0
$730$	−16.0000	−0.592187
$731$	0	0
$732$	0	0
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	35.0000	1.29187
$735$	0	0
$736$	3.00000	0.110581
$737$	30.0000	1.10506
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	0	0
$743$	−39.0000	−1.43077	−0.715386	−	0.698730i	$-0.753749\pi$
$743$	−39.0000	−1.43077	−0.715386	+	0.698730i	$0.753749\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	−4.00000	−0.146450
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−34.0000	−1.23494
$759$	0	0
$760$	2.00000	0.0725476
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	0	0
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	15.0000	0.541972
$767$	−24.0000	−0.866590
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	2.00000	0.0719816
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	14.0000	0.502571
$777$	0	0
$778$	30.0000	1.07555
$779$	−18.0000	−0.644917
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.0000	−0.785214
$786$	0	0
$787$	−43.0000	−1.53278	−0.766392	−	0.642373i	$-0.777950\pi$
$787$	−43.0000	−1.53278	−0.766392	+	0.642373i	$0.777950\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	2.00000	0.0711568
$791$	0	0
$792$	0	0
$793$	−20.0000	−0.710221
$794$	14.0000	0.496841
$795$	0	0
$796$	−4.00000	−0.141776
$797$	48.0000	1.70025	0.850124	−	0.526583i	$-0.176527\pi$
$797$	48.0000	1.70025	0.850124	+	0.526583i	$0.176527\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	−15.0000	−0.529668
$803$	−96.0000	−3.38777
$804$	0	0
$805$	0	0
$806$	−32.0000	−1.12715
$807$	0	0
$808$	−15.0000	−0.527698
$809$	−21.0000	−0.738321	−0.369160	−	0.929366i	$-0.620355\pi$
$809$	−21.0000	−0.738321	−0.369160	+	0.929366i	$0.620355\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	0	0
$814$	−24.0000	−0.841200
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−14.0000	−0.489798
$818$	−13.0000	−0.454534
$819$	0	0
$820$	−9.00000	−0.314294
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	−1.00000	−0.0348367
$825$	0	0
$826$	0	0
$827$	−15.0000	−0.521601	−0.260801	−	0.965393i	$-0.583986\pi$
$827$	−15.0000	−0.521601	−0.260801	+	0.965393i	$0.583986\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−3.00000	−0.104132
$831$	0	0
$832$	−4.00000	−0.138675
$833$	0	0
$834$	0	0
$835$	3.00000	0.103819
$836$	12.0000	0.415029
$837$	0	0
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	17.0000	0.585859
$843$	0	0
$844$	−10.0000	−0.344214
$845$	3.00000	0.103203
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	0	0
$856$	15.0000	0.512689
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	−7.00000	−0.238698
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−27.0000	−0.919091	−0.459545	−	0.888154i	$-0.651988\pi$
$863$	−27.0000	−0.919091	−0.459545	+	0.888154i	$0.651988\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−20.0000	−0.677674
$872$	11.0000	0.372507
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	6.00000	0.202260
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	−21.0000	−0.705509
$887$	21.0000	0.705111	0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000	0.705111	0.352555	+	0.935791i	$0.385313\pi$
$888$	0	0
$889$	0	0
$890$	15.0000	0.502801
$891$	0	0
$892$	−28.0000	−0.937509
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	9.00000	0.300334
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	−54.0000	−1.79800
$903$	0	0
$904$	−6.00000	−0.199557
$905$	11.0000	0.365652
$906$	0	0
$907$	−25.0000	−0.830111	−0.415056	−	0.909796i	$-0.636238\pi$
$907$	−25.0000	−0.830111	−0.415056	+	0.909796i	$0.636238\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−18.0000	−0.595713
$914$	32.0000	1.05847
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	0	0
$919$	14.0000	0.461817	0.230909	−	0.972975i	$-0.425830\pi$
$919$	14.0000	0.461817	0.230909	+	0.972975i	$0.425830\pi$
$920$	3.00000	0.0989071
$921$	0	0
$922$	−18.0000	−0.592798
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−13.0000	−0.427207
$927$	0	0
$928$	3.00000	0.0984798
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	0	0
$932$	12.0000	0.393073
$933$	0	0
$934$	15.0000	0.490815
$935$	0	0
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	6.00000	0.195283
$945$	0	0
$946$	−42.0000	−1.36554
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	0	0
$949$	64.0000	2.07753
$950$	2.00000	0.0648886
$951$	0	0
$952$	0	0
$953$	−60.0000	−1.94359	−0.971795	−	0.235826i	$-0.924220\pi$
$953$	−60.0000	−1.94359	−0.971795	+	0.235826i	$0.924220\pi$
$954$	0	0
$955$	6.00000	0.194155
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−12.0000	−0.387702
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	16.0000	0.515861
$963$	0	0
$964$	2.00000	0.0644157
$965$	2.00000	0.0643823
$966$	0	0
$967$	35.0000	1.12552	0.562762	−	0.826619i	$-0.309739\pi$
$967$	35.0000	1.12552	0.562762	+	0.826619i	$0.309739\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	14.0000	0.449513
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	5.00000	0.160046
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	90.0000	2.87641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	−8.00000	−0.254514
$989$	−21.0000	−0.667761
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	8.00000	0.254000
$993$	0	0
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bm.1.1		1
3.2	odd	2		490.2.a.c.1.1		1
7.2	even	3		630.2.k.b.361.1		2
7.4	even	3		630.2.k.b.541.1		2
7.6	odd	2		4410.2.a.bd.1.1		1
12.11	even	2		3920.2.a.p.1.1		1
15.2	even	4		2450.2.c.g.99.1		2
15.8	even	4		2450.2.c.g.99.2		2
15.14	odd	2		2450.2.a.w.1.1		1
21.2	odd	6		70.2.e.c.11.1	✓	2
21.5	even	6		490.2.e.h.361.1		2
21.11	odd	6		70.2.e.c.51.1	yes	2
21.17	even	6		490.2.e.h.471.1		2
21.20	even	2		490.2.a.b.1.1		1
84.11	even	6		560.2.q.g.401.1		2
84.23	even	6		560.2.q.g.81.1		2
84.83	odd	2		3920.2.a.bc.1.1		1
105.2	even	12		350.2.j.b.249.1		4
105.23	even	12		350.2.j.b.249.2		4
105.32	even	12		350.2.j.b.149.2		4
105.44	odd	6		350.2.e.e.151.1		2
105.53	even	12		350.2.j.b.149.1		4
105.62	odd	4		2450.2.c.l.99.1		2
105.74	odd	6		350.2.e.e.51.1		2
105.83	odd	4		2450.2.c.l.99.2		2
105.104	even	2		2450.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.c.11.1	✓	2	21.2	odd	6
70.2.e.c.51.1	yes	2	21.11	odd	6
350.2.e.e.51.1		2	105.74	odd	6
350.2.e.e.151.1		2	105.44	odd	6
350.2.j.b.149.1		4	105.53	even	12
350.2.j.b.149.2		4	105.32	even	12
350.2.j.b.249.1		4	105.2	even	12
350.2.j.b.249.2		4	105.23	even	12
490.2.a.b.1.1		1	21.20	even	2
490.2.a.c.1.1		1	3.2	odd	2
490.2.e.h.361.1		2	21.5	even	6
490.2.e.h.471.1		2	21.17	even	6
560.2.q.g.81.1		2	84.23	even	6
560.2.q.g.401.1		2	84.11	even	6
630.2.k.b.361.1		2	7.2	even	3
630.2.k.b.541.1		2	7.4	even	3
2450.2.a.w.1.1		1	15.14	odd	2
2450.2.a.bc.1.1		1	105.104	even	2
2450.2.c.g.99.1		2	15.2	even	4
2450.2.c.g.99.2		2	15.8	even	4
2450.2.c.l.99.1		2	105.62	odd	4
2450.2.c.l.99.2		2	105.83	odd	4
3920.2.a.p.1.1		1	12.11	even	2
3920.2.a.bc.1.1		1	84.83	odd	2
4410.2.a.bd.1.1		1	7.6	odd	2
4410.2.a.bm.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

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