LMFDB - Embedded newform 5850.2.e.a.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 26)
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.a.5149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000 q^{19} +6.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +7.00000i q^{37} +2.00000i q^{38} -1.00000i q^{43} +6.00000 q^{44} +3.00000i q^{47} +6.00000 q^{49} -1.00000i q^{52} -1.00000 q^{56} -6.00000i q^{58} -6.00000 q^{59} +8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -14.0000i q^{67} +3.00000i q^{68} +3.00000 q^{71} +2.00000i q^{73} +7.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -8.00000 q^{79} -12.0000i q^{83} -1.00000 q^{86} -6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} +3.00000 q^{94} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 12 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 12 q^{29} - 8 q^{31} - 6 q^{34} + 12 q^{44} + 12 q^{49} - 2 q^{56} - 12 q^{59} + 16 q^{61} - 2 q^{64} + 6 q^{71} + 14 q^{74} + 4 q^{76} - 16 q^{79} - 2 q^{86} - 12 q^{89} - 2 q^{91} + 6 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 12 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 12 * q^29 - 8 * q^31 - 6 * q^34 + 12 * q^44 + 12 * q^49 - 2 * q^56 - 12 * q^59 + 16 * q^61 - 2 * q^64 + 6 * q^71 + 14 * q^74 + 4 * q^76 - 16 * q^79 - 2 * q^86 - 12 * q^89 - 2 * q^91 + 6 * q^94

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
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−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
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	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
$22$	6.00000i	1.27920i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	2.00000i	0.324443i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 1.00000i	− 0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$	− 1.00000i	− 0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	0	0
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	3.00000i	0.363803i
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	7.00000	0.813733
$75$	0	0
$76$	2.00000	0.229416
$77$	− 6.00000i	− 0.683763i
$78$	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
$82$	0	0
$83$	− 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
$86$	−1.00000	−0.107833
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	−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
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	3.00000	0.309426
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 6.00000i	− 0.606092i
$99$	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
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0	0
$107$	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
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	6.00000i	0.552345i
$119$	3.00000	0.275010
$120$	0	0
$121$	25.0000	2.27273
$122$	− 8.00000i	− 0.724286i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 20.0000i	− 1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	− 20.0000i	− 1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	0	0
$133$	− 2.00000i	− 0.173422i
$134$	−14.0000	−1.20942
$135$	0	0
$136$	3.00000	0.257248
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	− 3.00000i	− 0.251754i
$143$	− 6.00000i	− 0.501745i
$144$	0	0
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	− 7.00000i	− 0.575396i
$149$	−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
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	− 2.00000i	− 0.162221i
$153$	0	0
$154$	−6.00000	−0.483494
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	0	0
$165$	0	0
$166$	−12.0000	−0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	1.00000i	0.0762493i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	6.00000i	0.449719i
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	1.00000i	0.0741249i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.0000i	1.31629i
$188$	− 3.00000i	− 0.218797i
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−6.00000	−0.428571
$197$	3.00000i	0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$	3.00000i	0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	0	0
$202$	− 12.0000i	− 0.844317i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	12.0000	0.830057
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	0	0
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	− 7.00000i	− 0.474100i
$219$	0	0
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	27.0000i	1.76883i	0.466702	+	0.884414i	$0.345442\pi$
$233$	27.0000i	1.76883i	−0.466702	+	0.884414i	$0.654558\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	− 3.00000i	− 0.194461i
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	− 25.0000i	− 1.60706i
$243$	0	0
$244$	−8.00000	−0.512148
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	0	0
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000i	0.561405i	0.959795	+	0.280702i	$0.0905674\pi$
$257$	9.00000i	0.561405i	−0.959795	+	0.280702i	$0.909433\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	0	0
$262$	− 21.0000i	− 1.29738i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	−2.00000	−0.122628
$267$	0	0
$268$	14.0000i	0.855186i
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	28.0000i	1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$	28.0000i	1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$	− 13.0000i	− 0.779688i
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 2.00000i	− 0.117041i
$293$	− 21.0000i	− 1.22683i	−0.789760	−	0.613417i	$-0.789795\pi$
$293$	− 21.0000i	− 1.22683i	0.789760	−	0.613417i	$-0.210205\pi$
$294$	0	0
$295$	0	0
$296$	−7.00000	−0.406867
$297$	0	0
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	− 17.0000i	− 0.978240i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	6.00000i	0.341882i
$309$	0	0
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	− 1.00000i	− 0.0565233i	−0.999601	−	0.0282617i	$-0.991003\pi$
$313$	− 1.00000i	− 0.0565233i	0.999601	−	0.0282617i	$-0.00899717\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	8.00000	0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.00000i	0.333849i
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	12.0000i	0.658586i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 23.0000i	− 1.25289i	−0.779466	−	0.626445i	$-0.784509\pi$
$337$	− 23.0000i	− 1.25289i	0.779466	−	0.626445i	$-0.215491\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	13.0000i	0.701934i
$344$	1.00000	0.0539164
$345$	0	0
$346$	0	0
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	0	0
$352$	6.00000i	0.319801i
$353$	− 24.0000i	− 1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$	− 24.0000i	− 1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 3.00000i	− 0.158555i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 20.0000i	− 1.05118i
$363$	0	0
$364$	1.00000	0.0524142
$365$	0	0
$366$	0	0
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	−3.00000	−0.154713
$377$	6.00000i	0.309016i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	− 21.0000i	− 1.07305i	−0.843884	−	0.536525i	$-0.819737\pi$
$383$	− 21.0000i	− 1.07305i	0.843884	−	0.536525i	$-0.180263\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	6.00000i	0.303046i
$393$	0	0
$394$	3.00000	0.151138
$395$	0	0
$396$	0	0
$397$	34.0000i	1.70641i	0.521575	+	0.853206i	$0.325345\pi$
$397$	34.0000i	1.70641i	−0.521575	+	0.853206i	$0.674655\pi$
$398$	2.00000i	0.100251i
$399$	0	0
$400$	0	0
$401$	−36.0000	−1.79775	−0.898877	−	0.438201i	$-0.855616\pi$
$401$	−36.0000	−1.79775	−0.898877	+	0.438201i	$0.855616\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	6.00000	0.297775
$407$	− 42.0000i	− 2.08186i
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	− 12.0000i	− 0.586939i
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	13.0000i	0.632830i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	33.0000	1.58955	0.794777	−	0.606902i	$-0.207588\pi$
$431$	33.0000	1.58955	0.794777	+	0.606902i	$0.207588\pi$
$432$	0	0
$433$	− 25.0000i	− 1.20142i	−0.799466	−	0.600712i	$-0.794884\pi$
$433$	− 25.0000i	− 1.20142i	0.799466	−	0.600712i	$-0.205116\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−7.00000	−0.335239
$437$	0	0
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	− 3.00000i	− 0.142695i
$443$	− 21.0000i	− 0.997740i	−0.866677	−	0.498870i	$-0.833748\pi$
$443$	− 21.0000i	− 0.997740i	0.866677	−	0.498870i	$-0.166252\pi$
$444$	0	0
$445$	0	0
$446$	−19.0000	−0.899676
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	− 6.00000i	− 0.282216i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 13.0000i	− 0.607450i
$459$	0	0
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	27.0000	1.25075
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	0	0
$472$	− 6.00000i	− 0.276172i
$473$	6.00000i	0.275880i
$474$	0	0
$475$	0	0
$476$	−3.00000	−0.137505
$477$	0	0
$478$	− 15.0000i	− 0.686084i
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	8.00000i	0.362143i
$489$	0	0
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	3.00000i	0.134568i
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	0	0
$502$	24.0000i	1.07117i
$503$	30.0000i	1.33763i	0.743427	+	0.668817i	$0.233199\pi$
$503$	30.0000i	1.33763i	−0.743427	+	0.668817i	$0.766801\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	20.0000i	0.887357i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	9.00000	0.396973
$515$	0	0
$516$	0	0
$517$	− 18.0000i	− 0.791639i
$518$	7.00000i	0.307562i
$519$	0	0
$520$	0	0
$521$	9.00000	0.394297	0.197149	−	0.980374i	$-0.436832\pi$
$521$	9.00000	0.394297	0.197149	+	0.980374i	$0.436832\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−21.0000	−0.917389
$525$	0	0
$526$	12.0000	0.523225
$527$	12.0000i	0.522728i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	2.00000i	0.0867110i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	14.0000	0.604708
$537$	0	0
$538$	− 24.0000i	− 1.03471i
$539$	−36.0000	−1.55063
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	− 11.0000i	− 0.472490i
$543$	0	0
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	− 17.0000i	− 0.726868i	−0.931620	−	0.363434i	$-0.881604\pi$
$547$	− 17.0000i	− 0.726868i	0.931620	−	0.363434i	$-0.118396\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	28.0000	1.18961
$555$	0	0
$556$	−13.0000	−0.551323
$557$	3.00000i	0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$	3.00000i	0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	− 6.00000i	− 0.253095i
$563$	− 39.0000i	− 1.64365i	−0.569737	−	0.821827i	$-0.692955\pi$
$563$	− 39.0000i	− 1.64365i	0.569737	−	0.821827i	$-0.307045\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	3.00000i	0.125877i
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	5.00000	0.209243	0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000	0.209243	0.104622	+	0.994512i	$0.466637\pi$
$572$	6.00000i	0.250873i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−21.0000	−0.867502
$587$	24.0000i	0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$	24.0000i	0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	0	0
$592$	7.00000i	0.287698i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	− 1.00000i	− 0.0407570i
$603$	0	0
$604$	−17.0000	−0.691720
$605$	0	0
$606$	0	0
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	2.00000i	0.0811107i
$609$	0	0
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	6.00000	0.241747
$617$	− 24.0000i	− 0.966204i	−0.875564	−	0.483102i	$-0.839510\pi$
$617$	− 24.0000i	− 0.966204i	0.875564	−	0.483102i	$-0.160490\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	− 30.0000i	− 1.20289i
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	14.0000i	0.558661i
$629$	21.0000	0.837325
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−6.00000	−0.238290
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	36.0000i	1.42525i
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	− 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
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	3.00000i	0.116952i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	− 8.00000i	− 0.310929i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−48.0000	−1.85302
$672$	0	0
$673$	− 19.0000i	− 0.732396i	−0.930537	−	0.366198i	$-0.880659\pi$
$673$	− 19.0000i	− 0.732396i	0.930537	−	0.366198i	$-0.119341\pi$
$674$	−23.0000	−0.885927
$675$	0	0
$676$	1.00000	0.0384615
$677$	48.0000i	1.84479i	0.386248	+	0.922395i	$0.373771\pi$
$677$	48.0000i	1.84479i	−0.386248	+	0.922395i	$0.626229\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	− 24.0000i	− 0.919007i
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	− 1.00000i	− 0.0381246i
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	3.00000	0.113878
$695$	0	0
$696$	0	0
$697$	0	0
$698$	− 19.0000i	− 0.719161i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 14.0000i	− 0.528020i
$704$	6.00000	0.226134
$705$	0	0
$706$	−24.0000	−0.903252
$707$	12.0000i	0.451306i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	0	0
$712$	− 6.00000i	− 0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	0	0
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	15.0000i	0.558242i
$723$	0	0
$724$	−20.0000	−0.743294
$725$	0	0
$726$	0	0
$727$	10.0000i	0.370879i	0.982656	+	0.185440i	$0.0593710\pi$
$727$	10.0000i	0.370879i	−0.982656	+	0.185440i	$0.940629\pi$
$728$	− 1.00000i	− 0.0370625i
$729$	0	0
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	23.0000i	0.849524i	0.905305	+	0.424762i	$0.139642\pi$
$733$	23.0000i	0.849524i	−0.905305	+	0.424762i	$0.860358\pi$
$734$	−26.0000	−0.959678
$735$	0	0
$736$	0	0
$737$	84.0000i	3.09418i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.00000i	0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$	9.00000i	0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$	0	0
$745$	0	0
$746$	−4.00000	−0.146450
$747$	0	0
$748$	− 18.0000i	− 0.658145i
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	3.00000i	0.109399i
$753$	0	0
$754$	6.00000	0.218507
$755$	0	0
$756$	0	0
$757$	16.0000i	0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$	16.0000i	0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	7.00000i	0.253417i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−21.0000	−0.758761
$767$	− 6.00000i	− 0.216647i
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	4.00000i	0.143963i
$773$	39.0000i	1.40273i	0.712801	+	0.701366i	$0.247426\pi$
$773$	39.0000i	1.40273i	−0.712801	+	0.701366i	$0.752574\pi$
$774$	0	0
$775$	0	0
$776$	−10.0000	−0.358979
$777$	0	0
$778$	6.00000i	0.215110i
$779$	0	0
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	− 3.00000i	− 0.106871i
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	8.00000i	0.284088i
$794$	34.0000	1.20661
$795$	0	0
$796$	2.00000	0.0708881
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	0	0
$802$	36.0000i	1.27120i
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	12.0000i	0.422159i
$809$	−33.0000	−1.16022	−0.580109	−	0.814539i	$-0.696990\pi$
$809$	−33.0000	−1.16022	−0.580109	+	0.814539i	$0.696990\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	−42.0000	−1.47210
$815$	0	0
$816$	0	0
$817$	2.00000i	0.0699711i
$818$	32.0000i	1.11885i
$819$	0	0
$820$	0	0
$821$	3.00000	0.104701	0.0523504	−	0.998629i	$-0.483329\pi$
$821$	3.00000	0.104701	0.0523504	+	0.998629i	$0.483329\pi$
$822$	0	0
$823$	14.0000i	0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$	14.0000i	0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−6.00000	−0.208767
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	− 9.00000i	− 0.310900i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 17.0000i	− 0.585859i
$843$	0	0
$844$	13.0000	0.447478
$845$	0	0
$846$	0	0
$847$	25.0000i	0.859010i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 37.0000i	− 1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$853$	− 37.0000i	− 1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 42.0000i	− 1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$	− 42.0000i	− 1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	− 33.0000i	− 1.12398i
$863$	45.0000i	1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$863$	45.0000i	1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$864$	0	0
$865$	0	0
$866$	−25.0000	−0.849535
$867$	0	0
$868$	4.00000i	0.135769i
$869$	48.0000	1.62829
$870$	0	0
$871$	14.0000	0.474372
$872$	7.00000i	0.237050i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.0000i	0.438979i	0.975615	+	0.219489i	$0.0704391\pi$
$877$	13.0000i	0.438979i	−0.975615	+	0.219489i	$0.929561\pi$
$878$	26.0000i	0.877457i
$879$	0	0
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	29.0000i	0.975928i	0.872864	+	0.487964i	$0.162260\pi$
$883$	29.0000i	0.975928i	−0.872864	+	0.487964i	$0.837740\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−21.0000	−0.705509
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	20.0000	0.670778
$890$	0	0
$891$	0	0
$892$	19.0000i	0.636167i
$893$	− 6.00000i	− 0.200782i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	− 6.00000i	− 0.200223i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	37.0000i	1.22856i	0.789086	+	0.614282i	$0.210554\pi$
$907$	37.0000i	1.22856i	−0.789086	+	0.614282i	$0.789446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	72.0000i	2.38285i
$914$	10.0000	0.330771
$915$	0	0
$916$	−13.0000	−0.429532
$917$	21.0000i	0.693481i
$918$	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
$922$	9.00000i	0.296399i
$923$	3.00000i	0.0987462i
$924$	0	0
$925$	0	0
$926$	−40.0000	−1.31448
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	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
$931$	−12.0000	−0.393284
$932$	− 27.0000i	− 0.884414i
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	34.0000i	1.11073i	0.831606	+	0.555366i	$0.187422\pi$
$937$	34.0000i	1.11073i	−0.831606	+	0.555366i	$0.812578\pi$
$938$	− 14.0000i	− 0.457116i
$939$	0	0
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	0	0
$944$	−6.00000	−0.195283
$945$	0	0
$946$	6.00000	0.195077
$947$	6.00000i	0.194974i	0.995237	+	0.0974869i	$0.0310804\pi$
$947$	6.00000i	0.194974i	−0.995237	+	0.0974869i	$0.968920\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	0	0
$952$	3.00000i	0.0972306i
$953$	− 15.0000i	− 0.485898i	−0.970039	−	0.242949i	$-0.921885\pi$
$953$	− 15.0000i	− 0.485898i	0.970039	−	0.242949i	$-0.0781147\pi$
$954$	0	0
$955$	0	0
$956$	−15.0000	−0.485135
$957$	0	0
$958$	21.0000i	0.678479i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	7.00000i	0.225689i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	31.0000i	0.996893i	0.866921	+	0.498446i	$0.166096\pi$
$967$	31.0000i	0.996893i	−0.866921	+	0.498446i	$0.833904\pi$
$968$	25.0000i	0.803530i
$969$	0	0
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	0	0
$973$	13.0000i	0.416761i
$974$	16.0000	0.512673
$975$	0	0
$976$	8.00000	0.256074
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	− 9.00000i	− 0.287202i
$983$	− 39.0000i	− 1.24391i	−0.783054	−	0.621953i	$-0.786339\pi$
$983$	− 39.0000i	− 1.24391i	0.783054	−	0.621953i	$-0.213661\pi$
$984$	0	0
$985$	0	0
$986$	−18.0000	−0.573237
$987$	0	0
$988$	2.00000i	0.0636285i
$989$	0	0
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	3.00000	0.0951542
$995$	0	0
$996$	0	0
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	− 40.0000i	− 1.26618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.a.5149.1		2
3.2	odd	2		650.2.b.d.599.2		2
5.2	odd	4		234.2.a.e.1.1		1
5.3	odd	4		5850.2.a.p.1.1		1
5.4	even	2	inner	5850.2.e.a.5149.2		2
15.2	even	4		26.2.a.a.1.1	✓	1
15.8	even	4		650.2.a.j.1.1		1
15.14	odd	2		650.2.b.d.599.1		2
20.7	even	4		1872.2.a.q.1.1		1
40.27	even	4		7488.2.a.h.1.1		1
40.37	odd	4		7488.2.a.g.1.1		1
45.2	even	12		2106.2.e.ba.1405.1		2
45.7	odd	12		2106.2.e.b.1405.1		2
45.22	odd	12		2106.2.e.b.703.1		2
45.32	even	12		2106.2.e.ba.703.1		2
60.23	odd	4		5200.2.a.x.1.1		1
60.47	odd	4		208.2.a.a.1.1		1
65.12	odd	4		3042.2.a.a.1.1		1
65.47	even	4		3042.2.b.a.1351.2		2
65.57	even	4		3042.2.b.a.1351.1		2
105.2	even	12		1274.2.f.p.1145.1		2
105.17	odd	12		1274.2.f.r.79.1		2
105.32	even	12		1274.2.f.p.79.1		2
105.47	odd	12		1274.2.f.r.1145.1		2
105.62	odd	4		1274.2.a.d.1.1		1
120.77	even	4		832.2.a.d.1.1		1
120.107	odd	4		832.2.a.i.1.1		1
165.32	odd	4		3146.2.a.n.1.1		1
195.2	odd	12		338.2.e.a.147.1		4
195.17	even	12		338.2.c.a.315.1		2
195.32	odd	12		338.2.e.a.23.2		4
195.38	even	4		8450.2.a.c.1.1		1
195.47	odd	4		338.2.b.c.337.1		2
195.62	even	12		338.2.c.a.191.1		2
195.77	even	4		338.2.a.f.1.1		1
195.107	even	12		338.2.c.d.191.1		2
195.122	odd	4		338.2.b.c.337.2		2
195.137	odd	12		338.2.e.a.23.1		4
195.152	even	12		338.2.c.d.315.1		2
195.167	odd	12		338.2.e.a.147.2		4
240.77	even	4		3328.2.b.m.1665.2		2
240.107	odd	4		3328.2.b.j.1665.2		2
240.197	even	4		3328.2.b.m.1665.1		2
240.227	odd	4		3328.2.b.j.1665.1		2
255.152	even	4		7514.2.a.c.1.1		1
285.227	odd	4		9386.2.a.j.1.1		1
780.47	even	4		2704.2.f.d.337.1		2
780.467	odd	4		2704.2.a.f.1.1		1
780.707	even	4		2704.2.f.d.337.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.a.1.1	✓	1	15.2	even	4
208.2.a.a.1.1		1	60.47	odd	4
234.2.a.e.1.1		1	5.2	odd	4
338.2.a.f.1.1		1	195.77	even	4
338.2.b.c.337.1		2	195.47	odd	4
338.2.b.c.337.2		2	195.122	odd	4
338.2.c.a.191.1		2	195.62	even	12
338.2.c.a.315.1		2	195.17	even	12
338.2.c.d.191.1		2	195.107	even	12
338.2.c.d.315.1		2	195.152	even	12
338.2.e.a.23.1		4	195.137	odd	12
338.2.e.a.23.2		4	195.32	odd	12
338.2.e.a.147.1		4	195.2	odd	12
338.2.e.a.147.2		4	195.167	odd	12
650.2.a.j.1.1		1	15.8	even	4
650.2.b.d.599.1		2	15.14	odd	2
650.2.b.d.599.2		2	3.2	odd	2
832.2.a.d.1.1		1	120.77	even	4
832.2.a.i.1.1		1	120.107	odd	4
1274.2.a.d.1.1		1	105.62	odd	4
1274.2.f.p.79.1		2	105.32	even	12
1274.2.f.p.1145.1		2	105.2	even	12
1274.2.f.r.79.1		2	105.17	odd	12
1274.2.f.r.1145.1		2	105.47	odd	12
1872.2.a.q.1.1		1	20.7	even	4
2106.2.e.b.703.1		2	45.22	odd	12
2106.2.e.b.1405.1		2	45.7	odd	12
2106.2.e.ba.703.1		2	45.32	even	12
2106.2.e.ba.1405.1		2	45.2	even	12
2704.2.a.f.1.1		1	780.467	odd	4
2704.2.f.d.337.1		2	780.47	even	4
2704.2.f.d.337.2		2	780.707	even	4
3042.2.a.a.1.1		1	65.12	odd	4
3042.2.b.a.1351.1		2	65.57	even	4
3042.2.b.a.1351.2		2	65.47	even	4
3146.2.a.n.1.1		1	165.32	odd	4
3328.2.b.j.1665.1		2	240.227	odd	4
3328.2.b.j.1665.2		2	240.107	odd	4
3328.2.b.m.1665.1		2	240.197	even	4
3328.2.b.m.1665.2		2	240.77	even	4
5200.2.a.x.1.1		1	60.23	odd	4
5850.2.a.p.1.1		1	5.3	odd	4
5850.2.e.a.5149.1		2	1.1	even	1	trivial
5850.2.e.a.5149.2		2	5.4	even	2	inner
7488.2.a.g.1.1		1	40.37	odd	4
7488.2.a.h.1.1		1	40.27	even	4
7514.2.a.c.1.1		1	255.152	even	4
8450.2.a.c.1.1		1	195.38	even	4
9386.2.a.j.1.1		1	285.227	odd	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} -2.00000 q^{19} +6.00000i q^{22} +1.00000 q^{26} -1.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +7.00000i q^{37} +2.00000i q^{38} -1.00000i q^{43} +6.00000 q^{44} +3.00000i q^{47} +6.00000 q^{49} -1.00000i q^{52} -1.00000 q^{56} -6.00000i q^{58} -6.00000 q^{59} +8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -14.0000i q^{67} +3.00000i q^{68} +3.00000 q^{71} +2.00000i q^{73} +7.00000 q^{74} +2.00000 q^{76} -6.00000i q^{77} -8.00000 q^{79} -12.0000i q^{83} -1.00000 q^{86} -6.00000i q^{88} -6.00000 q^{89} -1.00000 q^{91} +3.00000 q^{94} +10.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 12 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{19} + 2 q^{26} + 12 q^{29} - 8 q^{31} - 6 q^{34} + 12 q^{44} + 12 q^{49} - 2 q^{56} - 12 q^{59} + 16 q^{61} - 2 q^{64} + 6 q^{71} + 14 q^{74} + 4 q^{76} - 16 q^{79} - 2 q^{86} - 12 q^{89} - 2 q^{91} + 6 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 12 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^19 + 2 * q^26 + 12 * q^29 - 8 * q^31 - 6 * q^34 + 12 * q^44 + 12 * q^49 - 2 * q^56 - 12 * q^59 + 16 * q^61 - 2 * q^64 + 6 * q^71 + 14 * q^74 + 4 * q^76 - 16 * q^79 - 2 * q^86 - 12 * q^89 - 2 * q^91 + 6 * q^94

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