LMFDB - Embedded newform 4410.2.a.bf.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 90)
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)

$f(q)$

$=$

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

$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -6.00000 q^{11} +4.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +4.00000 q^{19} +1.00000 q^{20} -6.00000 q^{22} +1.00000 q^{25} +4.00000 q^{26} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +8.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} +8.00000 q^{43} -6.00000 q^{44} +1.00000 q^{50} +4.00000 q^{52} +6.00000 q^{53} -6.00000 q^{55} +6.00000 q^{58} +6.00000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +12.0000 q^{71} +10.0000 q^{73} +8.00000 q^{74} +4.00000 q^{76} -4.00000 q^{79} +1.00000 q^{80} +12.0000 q^{83} -6.00000 q^{85} +8.00000 q^{86} -6.00000 q^{88} +12.0000 q^{89} +4.00000 q^{95} -2.00000 q^{97} +O(q^{100})$

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	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	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
$26$	4.00000	0.784465
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	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
$47$	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
$58$	6.00000	0.787839
$59$	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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	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
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	0	0
$83$	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
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−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
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−2.00000	−0.181071
$123$	0	0
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.00000	−0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	−24.0000	−2.00698
$144$	0	0
$145$	6.00000	0.498273
$146$	10.0000	0.827606
$147$	0	0
$148$	8.00000	0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−6.00000	−0.460179
$171$	0	0
$172$	8.00000	0.609994
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	12.0000	0.899438
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	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
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	36.0000	2.63258
$188$	0	0
$189$	0	0
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−2.00000	−0.139347
$207$	0	0
$208$	4.00000	0.277350
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−12.0000	−0.820303
$215$	8.00000	0.545595
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	−6.00000	−0.404520
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	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	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	4.00000	0.254000
$249$	0	0
$250$	1.00000	0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	0	0
$262$	−6.00000	−0.370681
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−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	$0.422742\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−24.0000	−1.41915
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	0	0
$292$	10.0000	0.585206
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	8.00000	0.464991
$297$	0	0
$298$	−6.00000	−0.347571
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−16.0000	−0.920697
$303$	0	0
$304$	4.00000	0.229416
$305$	−2.00000	−0.114520
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	4.00000	0.227185
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−4.00000	−0.225018
$317$	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
$323$	−24.0000	−1.33540
$324$	0	0
$325$	4.00000	0.221880
$326$	20.0000	1.10770
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−24.0000	−1.31322
$335$	−4.00000	−0.218543
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	−6.00000	−0.325396
$341$	−24.0000	−1.29967
$342$	0	0
$343$	0	0
$344$	8.00000	0.431331
$345$	0	0
$346$	18.0000	0.967686
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	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
$355$	12.0000	0.636894
$356$	12.0000	0.635999
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−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
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−26.0000	−1.35719	−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000	−1.35719	−0.678594	+	0.734513i	$0.737411\pi$
$368$	0	0
$369$	0	0
$370$	8.00000	0.415900
$371$	0	0
$372$	0	0
$373$	−28.0000	−1.44979	−0.724893	−	0.688862i	$-0.758111\pi$
$373$	−28.0000	−1.44979	−0.724893	+	0.688862i	$0.758111\pi$
$374$	36.0000	1.86152
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	4.00000	0.205196
$381$	0	0
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	−2.00000	−0.101535
$389$	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
$391$	0	0
$392$	0	0
$393$	0	0
$394$	18.0000	0.906827
$395$	−4.00000	−0.201262
$396$	0	0
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	−48.0000	−2.37927
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	−2.00000	−0.0985329
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	4.00000	0.196116
$417$	0	0
$418$	−24.0000	−1.17388
$419$	−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
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	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	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	−24.0000	−1.14156
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	−26.0000	−1.23114
$447$	0	0
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	0	0
$452$	−6.00000	−0.282216
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−14.0000	−0.654177
$459$	0	0
$460$	0	0
$461$	−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
$463$	−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
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	6.00000	0.276172
$473$	−48.0000	−2.20704
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	12.0000	0.548867
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	10.0000	0.455488
$483$	0	0
$484$	25.0000	1.13636
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	0	0
$491$	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
$494$	16.0000	0.719874
$495$	0	0
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	18.0000	0.803379
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	0	0
$508$	2.00000	0.0887357
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	4.00000	0.175412
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−12.0000	−0.518805
$536$	−4.00000	−0.172774
$537$	0	0
$538$	18.0000	0.776035
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	−6.00000	−0.257248
$545$	2.00000	0.0856706
$546$	0	0
$547$	−40.0000	−1.71028	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000	−1.71028	−0.855138	+	0.518400i	$0.826528\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	−6.00000	−0.255841
$551$	24.0000	1.02243
$552$	0	0
$553$	0	0
$554$	8.00000	0.339887
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	4.00000	0.168133
$567$	0	0
$568$	12.0000	0.503509
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	−24.0000	−1.00349
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	6.00000	0.249136
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	10.0000	0.413803
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−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
$590$	6.00000	0.247016
$591$	0	0
$592$	8.00000	0.328798
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	0	0
$604$	−16.0000	−0.651031
$605$	25.0000	1.01639
$606$	0	0
$607$	−38.0000	−1.54237	−0.771186	−	0.636610i	$-0.780336\pi$
$607$	−38.0000	−1.54237	−0.771186	+	0.636610i	$0.780336\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	−12.0000	−0.481156
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	0	0
$628$	4.00000	0.159617
$629$	−48.0000	−1.91389
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	18.0000	0.714871
$635$	2.00000	0.0793676
$636$	0	0
$637$	0	0
$638$	−36.0000	−1.42525
$639$	0	0
$640$	1.00000	0.0395285
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−48.0000	−1.88707	−0.943537	−	0.331266i	$-0.892524\pi$
$647$	−48.0000	−1.88707	−0.943537	+	0.331266i	$0.892524\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	4.00000	0.156893
$651$	0	0
$652$	20.0000	0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−24.0000	−0.928588
$669$	0	0
$670$	−4.00000	−0.154533
$671$	12.0000	0.463255
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	3.00000	0.115385
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	−6.00000	−0.230089
$681$	0	0
$682$	−24.0000	−0.919007
$683$	−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
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	8.00000	0.304997
$689$	24.0000	0.914327
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	0	0
$698$	22.0000	0.832712
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	0	0
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	12.0000	0.450352
$711$	0	0
$712$	12.0000	0.449719
$713$	0	0
$714$	0	0
$715$	−24.0000	−0.897549
$716$	−6.00000	−0.224231
$717$	0	0
$718$	−24.0000	−0.895672
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	0	0
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	6.00000	0.222834
$726$	0	0
$727$	10.0000	0.370879	0.185440	−	0.982656i	$-0.440629\pi$
$727$	10.0000	0.370879	0.185440	+	0.982656i	$0.440629\pi$
$728$	0	0
$729$	0	0
$730$	10.0000	0.370117
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	−26.0000	−0.959678
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	8.00000	0.294086
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−28.0000	−1.02515
$747$	0	0
$748$	36.0000	1.31629
$749$	0	0
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	0	0
$754$	24.0000	0.874028
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	4.00000	0.145095
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	18.0000	0.645331
$779$	0	0
$780$	0	0
$781$	−72.0000	−2.57636
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−4.00000	−0.142314
$791$	0	0
$792$	0	0
$793$	−8.00000	−0.284088
$794$	16.0000	0.567819
$795$	0	0
$796$	16.0000	0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	36.0000	1.27120
$803$	−60.0000	−2.11735
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	−6.00000	−0.211079
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	0	0
$814$	−48.0000	−1.68240
$815$	20.0000	0.700569
$816$	0	0
$817$	32.0000	1.11954
$818$	22.0000	0.769212
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−2.00000	−0.0696733
$825$	0	0
$826$	0	0
$827$	−24.0000	−0.834562	−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000	−0.834562	−0.417281	+	0.908778i	$0.637017\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	4.00000	0.138675
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	−24.0000	−0.830057
$837$	0	0
$838$	−6.00000	−0.207267
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000	0.482472
$843$	0	0
$844$	−4.00000	−0.137686
$845$	3.00000	0.103203
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−6.00000	−0.205798
$851$	0	0
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	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
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	−38.0000	−1.29129
$867$	0	0
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−16.0000	−0.542139
$872$	2.00000	0.0677285
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−36.0000	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	24.0000	0.806296
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	0	0
$890$	12.0000	0.402241
$891$	0	0
$892$	−26.0000	−0.870544
$893$	0	0
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	0	0
$898$	12.0000	0.400445
$899$	24.0000	0.800445
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	26.0000	0.860004
$915$	0	0
$916$	−14.0000	−0.462573
$917$	0	0
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	0	0
$921$	0	0
$922$	−18.0000	−0.592798
$923$	48.0000	1.57994
$924$	0	0
$925$	8.00000	0.263038
$926$	−34.0000	−1.11731
$927$	0	0
$928$	6.00000	0.196960
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	−12.0000	−0.392652
$935$	36.0000	1.17733
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	0	0
$944$	6.00000	0.195283
$945$	0	0
$946$	−48.0000	−1.56061
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	40.0000	1.29845
$950$	4.00000	0.129777
$951$	0	0
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	12.0000	0.388108
$957$	0	0
$958$	36.0000	1.16311
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	32.0000	1.03172
$963$	0	0
$964$	10.0000	0.322078
$965$	−10.0000	−0.321911
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	38.0000	1.21760
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	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
$979$	−72.0000	−2.30113
$980$	0	0
$981$	0	0
$982$	6.00000	0.191468
$983$	−48.0000	−1.53096	−0.765481	−	0.643458i	$-0.777499\pi$
$983$	−48.0000	−1.53096	−0.765481	+	0.643458i	$0.777499\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	−36.0000	−1.14647
$987$	0	0
$988$	16.0000	0.509028
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	−44.0000	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bf.1.1		1
3.2	odd	2		4410.2.a.k.1.1		1
7.6	odd	2		90.2.a.b.1.1	yes	1
21.20	even	2		90.2.a.a.1.1	✓	1
28.27	even	2		720.2.a.b.1.1		1
35.13	even	4		450.2.c.a.199.1		2
35.27	even	4		450.2.c.a.199.2		2
35.34	odd	2		450.2.a.a.1.1		1
56.13	odd	2		2880.2.a.bf.1.1		1
56.27	even	2		2880.2.a.u.1.1		1
63.13	odd	6		810.2.e.e.541.1		2
63.20	even	6		810.2.e.h.271.1		2
63.34	odd	6		810.2.e.e.271.1		2
63.41	even	6		810.2.e.h.541.1		2
84.83	odd	2		720.2.a.g.1.1		1
105.62	odd	4		450.2.c.d.199.1		2
105.83	odd	4		450.2.c.d.199.2		2
105.104	even	2		450.2.a.e.1.1		1
140.27	odd	4		3600.2.f.u.2449.1		2
140.83	odd	4		3600.2.f.u.2449.2		2
140.139	even	2		3600.2.a.bj.1.1		1
168.83	odd	2		2880.2.a.h.1.1		1
168.125	even	2		2880.2.a.k.1.1		1
420.83	even	4		3600.2.f.a.2449.2		2
420.167	even	4		3600.2.f.a.2449.1		2
420.419	odd	2		3600.2.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.a.a.1.1	✓	1	21.20	even	2
90.2.a.b.1.1	yes	1	7.6	odd	2
450.2.a.a.1.1		1	35.34	odd	2
450.2.a.e.1.1		1	105.104	even	2
450.2.c.a.199.1		2	35.13	even	4
450.2.c.a.199.2		2	35.27	even	4
450.2.c.d.199.1		2	105.62	odd	4
450.2.c.d.199.2		2	105.83	odd	4
720.2.a.b.1.1		1	28.27	even	2
720.2.a.g.1.1		1	84.83	odd	2
810.2.e.e.271.1		2	63.34	odd	6
810.2.e.e.541.1		2	63.13	odd	6
810.2.e.h.271.1		2	63.20	even	6
810.2.e.h.541.1		2	63.41	even	6
2880.2.a.h.1.1		1	168.83	odd	2
2880.2.a.k.1.1		1	168.125	even	2
2880.2.a.u.1.1		1	56.27	even	2
2880.2.a.bf.1.1		1	56.13	odd	2
3600.2.a.ba.1.1		1	420.419	odd	2
3600.2.a.bj.1.1		1	140.139	even	2
3600.2.f.a.2449.1		2	420.167	even	4
3600.2.f.a.2449.2		2	420.83	even	4
3600.2.f.u.2449.1		2	140.27	odd	4
3600.2.f.u.2449.2		2	140.83	odd	4
4410.2.a.k.1.1		1	3.2	odd	2
4410.2.a.bf.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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more