LMFDB - Embedded newform 5850.2.e.o.5149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(5149,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5850.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.o.5149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} -1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -6.00000i q^{23} +1.00000 q^{26} -2.00000i q^{28} +8.00000 q^{31} +1.00000i q^{32} +2.00000i q^{37} -2.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} +6.00000 q^{46} +3.00000 q^{49} +1.00000i q^{52} -6.00000i q^{53} +2.00000 q^{56} +14.0000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +4.00000i q^{73} -2.00000 q^{74} +2.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -6.00000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -4.00000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 4 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 16 q^{31} - 12 q^{41} + 12 q^{46} + 6 q^{49} + 4 q^{56} + 28 q^{61} - 2 q^{64} - 4 q^{74} + 4 q^{76} + 32 q^{79} - 8 q^{86} - 12 q^{89} + 4 q^{91}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 16 * q^31 - 12 * q^41 + 12 * q^46 + 6 * q^49 + 4 * q^56 + 28 * q^61 - 2 * q^64 - 4 * q^74 + 4 * q^76 + 32 * q^79 - 8 * q^86 - 12 * q^89 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	− 2.00000i	− 0.324443i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	2.00000	0.267261
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	− 6.00000i	− 0.662589i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	−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
$91$	2.00000	0.209657
$92$	6.00000i	0.625543i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	14.0000i	1.26750i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−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
$133$	− 4.00000i	− 0.346844i
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	2.00000i	0.162221i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	16.0000i	1.27289i
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 4.00000i	− 0.304997i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	−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	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 8.00000i	− 0.575853i	−0.957653	−	0.287926i	$-0.907034\pi$
$193$	− 8.00000i	− 0.575853i	0.957653	−	0.287926i	$-0.0929658\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	12.0000i	0.844317i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	4.00000i	0.270914i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−12.0000	−0.798228
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	− 11.0000i	− 0.707107i
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	0	0
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	0	0
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000i	0.748539i	0.927320	+	0.374270i	$0.122107\pi$
$257$	12.0000i	0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	30.0000i	1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$	30.0000i	1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	4.00000i	0.244339i
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	− 8.00000i	− 0.479808i
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	18.0000i	1.04271i
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	8.00000i	0.460348i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	12.0000i	0.668734i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000i	1.07990i
$344$	4.00000	0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	−20.0000	−1.07058	−0.535288	−	0.844670i	$-0.679797\pi$
$349$	−20.0000	−1.07058	−0.535288	+	0.844670i	$0.679797\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 6.00000i	− 0.317110i
$359$	−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
$362$	2.00000i	0.105118i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	8.00000	0.407189
$387$	0	0
$388$	4.00000i	0.203069i
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	16.0000i	0.802008i
$399$	0	0
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	28.0000i	1.35501i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	− 14.0000i	− 0.672797i	−0.941720	−	0.336399i	$-0.890791\pi$
$433$	− 14.0000i	− 0.672797i	0.941720	−	0.336399i	$-0.109209\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	−4.00000	−0.191565
$437$	12.0000i	0.574038i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	−10.0000	−0.473514
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	−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
$451$	0	0
$452$	− 12.0000i	− 0.564433i
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	− 28.0000i	− 1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$	− 28.0000i	− 1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$	4.00000i	0.186908i
$459$	0	0
$460$	0	0
$461$	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
$463$	10.0000i	0.464739i	0.972628	+	0.232370i	$0.0746479\pi$
$463$	10.0000i	0.464739i	−0.972628	+	0.232370i	$0.925352\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	26.0000i	1.18427i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	− 14.0000i	− 0.633750i
$489$	0	0
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	0	0
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	0	0
$502$	30.0000i	1.33897i
$503$	− 18.0000i	− 0.802580i	−0.915951	−	0.401290i	$-0.868562\pi$
$503$	− 18.0000i	− 0.802580i	0.915951	−	0.401290i	$-0.131438\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	16.0000i	0.709885i
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 4.00000i	− 0.175750i
$519$	0	0
$520$	0	0
$521$	42.0000	1.84005	0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000	1.84005	0.920027	+	0.391856i	$0.128167\pi$
$522$	0	0
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−30.0000	−1.30806
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	6.00000i	0.259889i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	12.0000i	0.517357i
$539$	0	0
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	8.00000i	0.343629i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	32.0000i	1.36078i
$554$	−26.0000	−1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	30.0000i	1.26547i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	−28.0000	−1.17693
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	− 16.0000i	− 0.666089i	−0.942911	−	0.333044i	$-0.891924\pi$
$577$	− 16.0000i	− 0.666089i	0.942911	−	0.333044i	$-0.108076\pi$
$578$	17.0000i	0.707107i
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	4.00000	0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	− 6.00000i	− 0.245358i
$599$	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	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	− 8.00000i	− 0.326056i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	− 2.00000i	− 0.0811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.0000i	0.888572i	0.895885	+	0.444286i	$0.146543\pi$
$613$	22.0000i	0.888572i	−0.895885	+	0.444286i	$0.853457\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	− 12.0000i	− 0.480770i
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	0	0
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	− 16.0000i	− 0.636446i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 16.0000i	− 0.626608i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	2.00000i	0.0777322i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	12.0000i	0.464294i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	1.00000	0.0384615
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	4.00000i	0.152499i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	26.0000	0.989087	0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000	0.989087	0.494543	+	0.869153i	$0.335335\pi$
$692$	− 18.0000i	− 0.684257i
$693$	0	0
$694$	24.0000	0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 20.0000i	− 0.757011i
$699$	0	0
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	−6.00000	−0.225813
$707$	24.0000i	0.902613i
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	− 48.0000i	− 1.79761i
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	− 15.0000i	− 0.558242i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	− 4.00000i	− 0.148352i	−0.997245	−	0.0741759i	$-0.976367\pi$
$727$	− 4.00000i	− 0.148352i	0.997245	−	0.0741759i	$-0.0236326\pi$
$728$	− 2.00000i	− 0.0741249i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 50.0000i	− 1.84679i	−0.383849	−	0.923396i	$-0.625402\pi$
$733$	− 50.0000i	− 1.84679i	0.383849	−	0.923396i	$-0.374598\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	0	0
$739$	−38.0000	−1.39785	−0.698926	−	0.715194i	$-0.746338\pi$
$739$	−38.0000	−1.39785	−0.698926	+	0.715194i	$0.746338\pi$
$740$	0	0
$741$	0	0
$742$	12.0000i	0.440534i
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	0	0
$746$	26.0000	0.951928
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	34.0000i	1.23494i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	0	0
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	8.00000i	0.287926i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	−4.00000	−0.143592
$777$	0	0
$778$	36.0000i	1.29066i
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	3.00000	0.107143
$785$	0	0
$786$	0	0
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	− 14.0000i	− 0.497155i
$794$	−26.0000	−0.922705
$795$	0	0
$796$	−16.0000	−0.567105
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	18.0000i	0.635602i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	− 12.0000i	− 0.422159i
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.00000i	− 0.279885i
$818$	− 14.0000i	− 0.489499i
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	− 32.0000i	− 1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$	− 32.0000i	− 1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	6.00000i	0.207267i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	32.0000i	1.10279i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 22.0000i	− 0.755929i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	12.0000	0.410152
$857$	12.0000i	0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$	12.0000i	0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	0	0
$865$	0	0
$866$	14.0000	0.475739
$867$	0	0
$868$	− 16.0000i	− 0.543075i
$869$	0	0
$870$	0	0
$871$	−4.00000	−0.135535
$872$	− 4.00000i	− 0.135457i
$873$	0	0
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	− 58.0000i	− 1.95852i	−0.202606	−	0.979260i	$-0.564941\pi$
$877$	− 58.0000i	− 1.95852i	0.202606	−	0.979260i	$-0.435059\pi$
$878$	− 32.0000i	− 1.07995i
$879$	0	0
$880$	0	0
$881$	−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
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.0000i	1.00730i	0.863907	+	0.503651i	$0.168010\pi$
$887$	30.0000i	1.00730i	−0.863907	+	0.503651i	$0.831990\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	0	0
$892$	− 10.0000i	− 0.334825i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	− 30.0000i	− 1.00111i
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	12.0000	0.399114
$905$	0	0
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	28.0000	0.926158
$915$	0	0
$916$	−4.00000	−0.132164
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	42.0000i	1.38320i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−10.0000	−0.328620
$927$	0	0
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	2.00000i	0.0653372i	0.999466	+	0.0326686i	$0.0104006\pi$
$937$	2.00000i	0.0653372i	−0.999466	+	0.0326686i	$0.989599\pi$
$938$	8.00000i	0.261209i
$939$	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
$943$	36.0000i	1.17232i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	2.00000i	0.0644826i
$963$	0	0
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	50.0000i	1.60789i	0.594703	+	0.803946i	$0.297270\pi$
$967$	50.0000i	1.60789i	−0.594703	+	0.803946i	$0.702730\pi$
$968$	11.0000i	0.353553i
$969$	0	0
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	14.0000	0.448129
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 6.00000i	− 0.191468i
$983$	36.0000i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$	36.0000i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	24.0000	0.763156
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 46.0000i	− 1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$	− 46.0000i	− 1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$	− 14.0000i	− 0.443162i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.o.5149.2		2
3.2	odd	2		1950.2.e.d.1249.1		2
5.2	odd	4		5850.2.a.g.1.1		1
5.3	odd	4		1170.2.a.k.1.1		1
5.4	even	2	inner	5850.2.e.o.5149.1		2
15.2	even	4		1950.2.a.o.1.1		1
15.8	even	4		390.2.a.d.1.1	✓	1
15.14	odd	2		1950.2.e.d.1249.2		2
20.3	even	4		9360.2.a.g.1.1		1
60.23	odd	4		3120.2.a.j.1.1		1
195.8	odd	4		5070.2.b.m.1351.1		2
195.38	even	4		5070.2.a.t.1.1		1
195.83	odd	4		5070.2.b.m.1351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.d.1.1	✓	1	15.8	even	4
1170.2.a.k.1.1		1	5.3	odd	4
1950.2.a.o.1.1		1	15.2	even	4
1950.2.e.d.1249.1		2	3.2	odd	2
1950.2.e.d.1249.2		2	15.14	odd	2
3120.2.a.j.1.1		1	60.23	odd	4
5070.2.a.t.1.1		1	195.38	even	4
5070.2.b.m.1351.1		2	195.8	odd	4
5070.2.b.m.1351.2		2	195.83	odd	4
5850.2.a.g.1.1		1	5.2	odd	4
5850.2.e.o.5149.1		2	5.4	even	2	inner
5850.2.e.o.5149.2		2	1.1	even	1	trivial
9360.2.a.g.1.1		1	20.3	even	4

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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.00000i q^{8} -1.00000i q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -6.00000i q^{23} +1.00000 q^{26} -2.00000i q^{28} +8.00000 q^{31} +1.00000i q^{32} +2.00000i q^{37} -2.00000i q^{38} -6.00000 q^{41} +4.00000i q^{43} +6.00000 q^{46} +3.00000 q^{49} +1.00000i q^{52} -6.00000i q^{53} +2.00000 q^{56} +14.0000 q^{61} +8.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +4.00000i q^{73} -2.00000 q^{74} +2.00000 q^{76} +16.0000 q^{79} -6.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -6.00000 q^{89} +2.00000 q^{91} +6.00000i q^{92} -4.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 4 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 16 q^{31} - 12 q^{41} + 12 q^{46} + 6 q^{49} + 4 q^{56} + 28 q^{61} - 2 q^{64} - 4 q^{74} + 4 q^{76} + 32 q^{79} - 8 q^{86} - 12 q^{89} + 4 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 16 * q^31 - 12 * q^41 + 12 * q^46 + 6 * q^49 + 4 * q^56 + 28 * q^61 - 2 * q^64 - 4 * q^74 + 4 * q^76 + 32 * q^79 - 8 * q^86 - 12 * q^89 + 4 * q^91

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