LMFDB - Embedded newform 5850.2.e.r.5149.1

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -6.00000i q^{23} -1.00000 q^{26} -2.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} +2.00000i q^{37} +2.00000i q^{38} +6.00000 q^{41} +4.00000i q^{43} -6.00000 q^{46} +3.00000 q^{49} +1.00000i q^{52} -6.00000i q^{53} -2.00000 q^{56} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} -8.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} -8.00000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} +6.00000i q^{92} +8.00000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{31} + 12 q^{41} - 12 q^{46} + 6 q^{49} - 4 q^{56} - 20 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 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 - 8 * q^31 + 12 * q^41 - 12 * q^46 + 6 * q^49 - 4 * q^56 - 20 * q^61 - 2 * q^64 + 4 * q^74 + 4 * q^76 - 16 * 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
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 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	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	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
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	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
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	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	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	6.00000i	0.625543i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	− 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
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	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
$122$	10.0000i	0.905357i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	8.00000	0.691095
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	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
$146$	−8.00000	−0.662085
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	− 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	$0.974569\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−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
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	− 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
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	− 20.0000i	− 1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$	− 20.0000i	− 1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	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
$206$	−8.00000	−0.557386
$207$	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
$217$	− 8.00000i	− 0.543075i
$218$	− 16.0000i	− 1.08366i
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	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
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	2.00000i	0.127257i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 12.0000i	− 0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$	− 12.0000i	− 0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	0	0
$262$	− 18.0000i	− 1.11204i
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	− 8.00000i	− 0.488678i
$269$	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
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	8.00000i	0.479808i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	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
$292$	8.00000i	0.468165i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	− 20.0000i	− 1.15087i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 14.0000i	− 0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$	− 14.0000i	− 0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	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
$326$	−20.0000	−1.10770
$327$	0	0
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	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
$346$	−6.00000	−0.322562
$347$	− 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
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 18.0000i	− 0.951330i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	22.0000i	1.15629i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	− 6.00000i	− 0.312772i
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	0	0
$382$	24.0000i	1.22795i
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	0	0
$386$	−20.0000	−1.01797
$387$	0	0
$388$	− 8.00000i	− 0.406138i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	0	0
$392$	3.00000i	0.151523i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−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
$412$	8.00000i	0.394132i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	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
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	12.0000i	0.574038i
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	12.0000i	0.564433i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	32.0000i	1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$	32.0000i	1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$	− 16.0000i	− 0.747631i
$459$	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
$463$	− 14.0000i	− 0.650635i	−0.945605	−	0.325318i	$-0.894529\pi$
$463$	− 14.0000i	− 0.650635i	0.945605	−	0.325318i	$-0.105471\pi$
$464$	0	0
$465$	0	0
$466$	24.0000	1.11178
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	24.0000i	1.09773i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 2.00000i	− 0.0910975i
$483$	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	$0.985571\pi$
$488$	− 10.0000i	− 0.452679i
$489$	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
$496$	−4.00000	−0.179605
$497$	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
$502$	18.0000i	0.803379i
$503$	6.00000i	0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$	6.00000i	0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	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
$511$	16.0000	0.707798
$512$	− 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
$521$	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
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−18.0000	−0.786334
$525$	0	0
$526$	6.00000	0.261612
$527$	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
$536$	−8.00000	−0.345547
$537$	0	0
$538$	− 12.0000i	− 0.517357i
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	4.00000i	0.171815i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 4.00000i	− 0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$	− 4.00000i	− 0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$	18.0000i	0.768922i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	8.00000	0.339276
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	12.0000	0.500870
$575$	0	0
$576$	0	0
$577$	− 4.00000i	− 0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$	− 4.00000i	− 0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$	− 17.0000i	− 0.707107i
$579$	0	0
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	8.00000	0.331042
$585$	0	0
$586$	6.00000	0.247858
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	6.00000i	0.245358i
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	8.00000i	0.326056i
$603$	0	0
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000i	0.480770i
$624$	0	0
$625$	0	0
$626$	−14.0000	−0.559553
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	0	0
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	30.0000	1.19145
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	0	0
$647$	18.0000i	0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$	18.0000i	0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	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
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	− 26.0000i	− 1.01052i
$663$	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
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	−34.0000	−1.30963
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	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
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	−24.0000	−0.911028
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 16.0000i	− 0.605609i
$699$	0	0
$700$	0	0
$701$	36.0000	1.35970	0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000	1.35970	0.679851	+	0.733351i	$0.262045\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	24.0000i	0.902613i
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	−18.0000	−0.672692
$717$	0	0
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	15.0000i	0.558242i
$723$	0	0
$724$	22.0000	0.817624
$725$	0	0
$726$	0	0
$727$	− 28.0000i	− 1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$	− 28.0000i	− 1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$	2.00000i	0.0741249i
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 26.0000i	− 0.960332i	−0.877178	−	0.480166i	$-0.840576\pi$
$733$	− 26.0000i	− 0.960332i	0.877178	−	0.480166i	$-0.159424\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−6.00000	−0.221163
$737$	0	0
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	0	0
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	14.0000i	0.508839i	0.967094	+	0.254419i	$0.0818843\pi$
$757$	14.0000i	0.508839i	−0.967094	+	0.254419i	$0.918116\pi$
$758$	− 10.0000i	− 0.363216i
$759$	0	0
$760$	0	0
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	0	0
$763$	32.0000i	1.15848i
$764$	24.0000	0.868290
$765$	0	0
$766$	−36.0000	−1.30073
$767$	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
$772$	20.0000i	0.719816i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	0	0
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	− 12.0000i	− 0.430221i
$779$	−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
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	10.0000i	0.355110i
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 6.00000i	− 0.211867i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	12.0000i	0.422159i
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	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
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	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
$838$	18.0000i	0.621800i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	4.00000i	0.137849i
$843$	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	$-0.146836\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	12.0000	0.410152
$857$	− 12.0000i	− 0.409912i	−0.978771	−	0.204956i	$-0.934295\pi$
$857$	− 12.0000i	− 0.409912i	0.978771	−	0.204956i	$-0.0657052\pi$
$858$	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
$863$	36.0000i	1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$	36.0000i	1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$	0	0
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	8.00000i	0.271538i
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	16.0000i	0.541828i
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	− 34.0000i	− 1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	− 34.0000i	− 1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	14.0000i	0.468755i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	18.0000i	0.600668i
$899$	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
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	0	0
$914$	32.0000	1.05847
$915$	0	0
$916$	−16.0000	−0.528655
$917$	36.0000i	1.18882i
$918$	0	0
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	0	0
$921$	0	0
$922$	18.0000i	0.592798i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−14.0000	−0.460069
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 24.0000i	− 0.786146i
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	26.0000i	0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$	26.0000i	0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	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
$949$	−8.00000	−0.259691
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.0000i	1.55487i	0.628962	+	0.777436i	$0.283480\pi$
$953$	48.0000i	1.55487i	−0.628962	+	0.777436i	$0.716520\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	24.0000i	0.775405i
$959$	36.0000	1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 2.00000i	− 0.0644826i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	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
$971$	−18.0000	−0.577647	−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000	−0.577647	−0.288824	+	0.957382i	$0.593264\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	2.00000	0.0640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	6.00000i	0.191468i
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 2.00000i	− 0.0636285i
$989$	24.0000	0.763156
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000i	0.127000i
$993$	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.r.5149.1		2
3.2	odd	2		1950.2.e.k.1249.2		2
5.2	odd	4		5850.2.a.bk.1.1		1
5.3	odd	4		1170.2.a.g.1.1		1
5.4	even	2	inner	5850.2.e.r.5149.2		2
15.2	even	4		1950.2.a.b.1.1		1
15.8	even	4		390.2.a.g.1.1	✓	1
15.14	odd	2		1950.2.e.k.1249.1		2
20.3	even	4		9360.2.a.bg.1.1		1
60.23	odd	4		3120.2.a.b.1.1		1
195.8	odd	4		5070.2.b.n.1351.2		2
195.38	even	4		5070.2.a.k.1.1		1
195.83	odd	4		5070.2.b.n.1351.1		2

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.00000i q^{8} -1.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{19} -6.00000i q^{23} -1.00000 q^{26} -2.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} +2.00000i q^{37} +2.00000i q^{38} +6.00000 q^{41} +4.00000i q^{43} -6.00000 q^{46} +3.00000 q^{49} +1.00000i q^{52} -6.00000i q^{53} -2.00000 q^{56} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +8.00000i q^{67} -8.00000i q^{73} +2.00000 q^{74} +2.00000 q^{76} -8.00000 q^{79} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +6.00000 q^{89} +2.00000 q^{91} +6.00000i q^{92} +8.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 4 q^{14} + 2 q^{16} - 4 q^{19} - 2 q^{26} - 8 q^{31} + 12 q^{41} - 12 q^{46} + 6 q^{49} - 4 q^{56} - 20 q^{61} - 2 q^{64} + 4 q^{74} + 4 q^{76} - 16 q^{79} + 8 q^{86} + 12 q^{89} + 4 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^14 + 2 * q^16 - 4 * q^19 - 2 * q^26 - 8 * q^31 + 12 * q^41 - 12 * q^46 + 6 * q^49 - 4 * q^56 - 20 * q^61 - 2 * q^64 + 4 * q^74 + 4 * q^76 - 16 * q^79 + 8 * q^86 + 12 * q^89 + 4 * q^91

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