LMFDB - Embedded newform 4608.2.f.i.2303.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(2303,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4608.2303");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.f (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:	$36.7950652514$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2303.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4608.2303
Dual form			4608.2.f.i.2303.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.24264i q^{7} +O(q^{10})$	$q-4.24264i q^{7} -4.00000i q^{11} +6.00000i q^{13} +4.24264i q^{17} -2.82843 q^{19} -6.00000 q^{23} -5.00000 q^{25} +8.48528 q^{29} -4.24264i q^{31} -6.00000i q^{37} +1.41421i q^{41} +2.82843 q^{43} -6.00000 q^{47} -11.0000 q^{49} -8.48528 q^{53} +4.00000i q^{59} -6.00000i q^{61} -11.3137 q^{67} +6.00000 q^{71} -6.00000 q^{73} -16.9706 q^{77} -4.24264i q^{79} +16.0000i q^{83} +12.7279i q^{89} +25.4558 q^{91} -12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{23} - 20 q^{25} - 24 q^{47} - 44 q^{49} + 24 q^{71} - 24 q^{73} - 48 q^{97}+O(q^{100}) $ 4 * q - 24 * q^23 - 20 * q^25 - 24 * q^47 - 44 * q^49 + 24 * q^71 - 24 * q^73 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times$.

$n$	$2053$	$3583$	$4097$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	− 4.24264i	− 1.60357i	−0.597614	−	0.801784i	$-0.703885\pi$
$7$	− 4.24264i	− 1.60357i	0.597614	−	0.801784i	$-0.296115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264i	1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$	4.24264i	1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	0	0
$31$	− 4.24264i	− 0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$	− 4.24264i	− 0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.41421i	0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$	1.41421i	0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.48528	−1.16554	−0.582772	−	0.812636i	$-0.698032\pi$
$53$	−8.48528	−1.16554	−0.582772	+	0.812636i	$0.698032\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	− 6.00000i	− 0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$	− 6.00000i	− 0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16.9706	−1.93398
$78$	0	0
$79$	− 4.24264i	− 0.477334i	−0.971101	−	0.238667i	$-0.923290\pi$
$79$	− 4.24264i	− 0.477334i	0.971101	−	0.238667i	$-0.0767105\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.0000i	1.75623i	0.478451	+	0.878114i	$0.341198\pi$
$83$	16.0000i	1.75623i	−0.478451	+	0.878114i	$0.658802\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.7279i	1.34916i	0.738203	+	0.674579i	$0.235675\pi$
$89$	12.7279i	1.34916i	−0.738203	+	0.674579i	$0.764325\pi$
$90$	0	0
$91$	25.4558	2.66850
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	0	0
$103$	12.7279i	1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$	12.7279i	1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0000i	− 1.54678i	−0.633932	−	0.773389i	$-0.718560\pi$
$107$	− 16.0000i	− 1.54678i	0.633932	−	0.773389i	$-0.281440\pi$
$108$	0	0
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.41421i	0.133038i	0.997785	+	0.0665190i	$0.0211893\pi$
$113$	1.41421i	0.133038i	−0.997785	+	0.0665190i	$0.978811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	18.0000	1.65006
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.7279i	1.12942i	0.825289	+	0.564710i	$0.191012\pi$
$127$	12.7279i	1.12942i	−0.825289	+	0.564710i	$0.808988\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000i	0.698963i	0.936943	+	0.349482i	$0.113642\pi$
$131$	8.00000i	0.698963i	−0.936943	+	0.349482i	$0.886358\pi$
$132$	0	0
$133$	12.0000i	1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.5563i	1.32907i	0.747258	+	0.664534i	$0.231370\pi$
$137$	15.5563i	1.32907i	−0.747258	+	0.664534i	$0.768630\pi$
$138$	0	0
$139$	−5.65685	−0.479808	−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685	−0.479808	−0.239904	+	0.970797i	$0.577116\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	24.0000	2.00698
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.9706	1.39028	0.695141	−	0.718873i	$-0.255342\pi$
$149$	16.9706	1.39028	0.695141	+	0.718873i	$0.255342\pi$
$150$	0	0
$151$	4.24264i	0.345261i	0.984987	+	0.172631i	$0.0552267\pi$
$151$	4.24264i	0.345261i	−0.984987	+	0.172631i	$0.944773\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	25.4558i	2.00620i
$162$	0	0
$163$	−14.1421	−1.10770	−0.553849	−	0.832617i	$-0.686841\pi$
$163$	−14.1421	−1.10770	−0.553849	+	0.832617i	$0.686841\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.9706	1.29025	0.645124	−	0.764078i	$-0.276806\pi$
$173$	16.9706	1.29025	0.645124	+	0.764078i	$0.276806\pi$
$174$	0	0
$175$	21.2132i	1.60357i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	18.0000i	1.33793i	0.743294	+	0.668965i	$0.233262\pi$
$181$	18.0000i	1.33793i	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.9706	1.24101
$188$	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
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	0	0
$199$	− 4.24264i	− 0.300753i	−0.988629	−	0.150376i	$-0.951951\pi$
$199$	− 4.24264i	− 0.300753i	0.988629	−	0.150376i	$-0.0480486\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 36.0000i	− 2.52670i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.3137i	0.782586i
$210$	0	0
$211$	−22.6274	−1.55774	−0.778868	−	0.627188i	$-0.784206\pi$
$211$	−22.6274	−1.55774	−0.778868	+	0.627188i	$0.784206\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−25.4558	−1.71235
$222$	0	0
$223$	4.24264i	0.284108i	0.989859	+	0.142054i	$0.0453707\pi$
$223$	4.24264i	0.284108i	−0.989859	+	0.142054i	$0.954629\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	6.00000i	0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$	6.00000i	0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.24264i	− 0.277945i	−0.990296	−	0.138972i	$-0.955620\pi$
$233$	− 4.24264i	− 0.277945i	0.990296	−	0.138972i	$-0.0443799\pi$
$234$	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
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 16.9706i	− 1.07981i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	28.0000i	1.76734i	0.468106	+	0.883672i	$0.344936\pi$
$251$	28.0000i	1.76734i	−0.468106	+	0.883672i	$0.655064\pi$
$252$	0	0
$253$	24.0000i	1.50887i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.41421i	0.0882162i	0.999027	+	0.0441081i	$0.0140446\pi$
$257$	1.41421i	0.0882162i	−0.999027	+	0.0441081i	$0.985955\pi$
$258$	0	0
$259$	−25.4558	−1.58175
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.4558	1.55207	0.776035	−	0.630690i	$-0.217228\pi$
$269$	25.4558	1.55207	0.776035	+	0.630690i	$0.217228\pi$
$270$	0	0
$271$	− 29.6985i	− 1.80405i	−0.431679	−	0.902027i	$-0.642079\pi$
$271$	− 29.6985i	− 1.80405i	0.431679	−	0.902027i	$-0.357921\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.0000i	1.20605i
$276$	0	0
$277$	− 30.0000i	− 1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	− 30.0000i	− 1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.7279i	− 0.759284i	−0.925133	−	0.379642i	$-0.876047\pi$
$281$	− 12.7279i	− 0.759284i	0.925133	−	0.379642i	$-0.123953\pi$
$282$	0	0
$283$	22.6274	1.34506	0.672530	−	0.740070i	$-0.265208\pi$
$283$	22.6274	1.34506	0.672530	+	0.740070i	$0.265208\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 36.0000i	− 2.08193i
$300$	0	0
$301$	− 12.0000i	− 0.691669i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.65685	0.322854	0.161427	−	0.986885i	$-0.448390\pi$
$307$	5.65685	0.322854	0.161427	+	0.986885i	$0.448390\pi$
$308$	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
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.48528	−0.476581	−0.238290	−	0.971194i	$-0.576587\pi$
$317$	−8.48528	−0.476581	−0.238290	+	0.971194i	$0.576587\pi$
$318$	0	0
$319$	− 33.9411i	− 1.90034i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	0	0
$325$	− 30.0000i	− 1.66410i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	25.4558i	1.40343i
$330$	0	0
$331$	5.65685	0.310929	0.155464	−	0.987841i	$-0.450313\pi$
$331$	5.65685	0.310929	0.155464	+	0.987841i	$0.450313\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.9706	−0.919007
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0000i	0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$	16.0000i	0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$	0	0
$349$	18.0000i	0.963518i	0.876304	+	0.481759i	$0.160002\pi$
$349$	18.0000i	0.963518i	−0.876304	+	0.481759i	$0.839998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.89949i	− 0.526897i	−0.964673	−	0.263448i	$-0.915140\pi$
$353$	− 9.89949i	− 0.526897i	0.964673	−	0.263448i	$-0.0848599\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.24264i	− 0.221464i	−0.993850	−	0.110732i	$-0.964680\pi$
$367$	− 4.24264i	− 0.221464i	0.993850	−	0.110732i	$-0.0353195\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	36.0000i	1.86903i
$372$	0	0
$373$	− 30.0000i	− 1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$	− 30.0000i	− 1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	50.9117i	2.62209i
$378$	0	0
$379$	19.7990	1.01701	0.508503	−	0.861060i	$-0.330199\pi$
$379$	19.7990	1.01701	0.508503	+	0.861060i	$0.330199\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.9706	−0.860442	−0.430221	−	0.902724i	$-0.641564\pi$
$389$	−16.9706	−0.860442	−0.430221	+	0.902724i	$0.641564\pi$
$390$	0	0
$391$	− 25.4558i	− 1.28736i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.2132i	1.05934i	0.848205	+	0.529668i	$0.177684\pi$
$401$	21.2132i	1.05934i	−0.848205	+	0.529668i	$0.822316\pi$
$402$	0	0
$403$	25.4558	1.26805
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.9706	0.835067
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.00000i	0.390826i	0.980721	+	0.195413i	$0.0626047\pi$
$419$	8.00000i	0.390826i	−0.980721	+	0.195413i	$0.937395\pi$
$420$	0	0
$421$	18.0000i	0.877266i	0.898666	+	0.438633i	$0.144537\pi$
$421$	18.0000i	0.877266i	−0.898666	+	0.438633i	$0.855463\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 21.2132i	− 1.02899i
$426$	0	0
$427$	−25.4558	−1.23189
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.9706	0.811812
$438$	0	0
$439$	4.24264i	0.202490i	0.994862	+	0.101245i	$0.0322826\pi$
$439$	4.24264i	0.202490i	−0.994862	+	0.101245i	$0.967717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.00000i	0.380091i	0.981775	+	0.190046i	$0.0608636\pi$
$443$	8.00000i	0.380091i	−0.981775	+	0.190046i	$0.939136\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4.24264i	− 0.200223i	−0.994976	−	0.100111i	$-0.968080\pi$
$449$	− 4.24264i	− 0.200223i	0.994976	−	0.100111i	$-0.0319199\pi$
$450$	0	0
$451$	5.65685	0.266371
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	21.2132i	0.985861i	0.870069	+	0.492931i	$0.164074\pi$
$463$	21.2132i	0.985861i	−0.870069	+	0.492931i	$0.835926\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.0000i	− 0.740392i	−0.928954	−	0.370196i	$-0.879291\pi$
$467$	− 16.0000i	− 0.740392i	0.928954	−	0.370196i	$-0.120709\pi$
$468$	0	0
$469$	48.0000i	2.21643i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 11.3137i	− 0.520205i
$474$	0	0
$475$	14.1421	0.648886
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.7279i	− 0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$487$	− 12.7279i	− 0.576757i	0.957516	−	0.288379i	$-0.0931162\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 20.0000i	− 0.902587i	−0.892375	−	0.451294i	$-0.850963\pi$
$491$	− 20.0000i	− 0.902587i	0.892375	−	0.451294i	$-0.149037\pi$
$492$	0	0
$493$	36.0000i	1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 25.4558i	− 1.14185i
$498$	0	0
$499$	−5.65685	−0.253236	−0.126618	−	0.991952i	$-0.540412\pi$
$499$	−5.65685	−0.253236	−0.126618	+	0.991952i	$0.540412\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.48528	0.376103	0.188052	−	0.982159i	$-0.439783\pi$
$509$	8.48528	0.376103	0.188052	+	0.982159i	$0.439783\pi$
$510$	0	0
$511$	25.4558i	1.12610i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24.0000i	1.05552i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.07107i	0.309789i	0.987931	+	0.154895i	$0.0495038\pi$
$521$	7.07107i	0.309789i	−0.987931	+	0.154895i	$0.950496\pi$
$522$	0	0
$523$	−36.7696	−1.60782	−0.803910	−	0.594751i	$-0.797251\pi$
$523$	−36.7696	−1.60782	−0.803910	+	0.594751i	$0.797251\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.0000	0.784092
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.48528	−0.367538
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	44.0000i	1.89521i
$540$	0	0
$541$	− 18.0000i	− 0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$	− 18.0000i	− 0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−36.7696	−1.57215	−0.786076	−	0.618130i	$-0.787891\pi$
$547$	−36.7696	−1.57215	−0.786076	+	0.618130i	$0.787891\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.9411	−1.43813	−0.719066	−	0.694942i	$-0.755430\pi$
$557$	−33.9411	−1.43813	−0.719066	+	0.694942i	$0.755430\pi$
$558$	0	0
$559$	16.9706i	0.717778i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 35.3553i	− 1.48217i	−0.671410	−	0.741086i	$-0.734311\pi$
$569$	− 35.3553i	− 1.48217i	0.671410	−	0.741086i	$-0.265689\pi$
$570$	0	0
$571$	39.5980	1.65712	0.828562	−	0.559897i	$-0.189159\pi$
$571$	39.5980	1.65712	0.828562	+	0.559897i	$0.189159\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	30.0000	1.25109
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	67.8823	2.81623
$582$	0	0
$583$	33.9411i	1.40570i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	12.0000i	0.494451i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	32.5269i	1.33572i	0.744287	+	0.667860i	$0.232790\pi$
$593$	32.5269i	1.33572i	−0.744287	+	0.667860i	$0.767210\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.7279i	0.516610i	0.966063	+	0.258305i	$0.0831640\pi$
$607$	12.7279i	0.516610i	−0.966063	+	0.258305i	$0.916836\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 36.0000i	− 1.45640i
$612$	0	0
$613$	− 30.0000i	− 1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$	− 30.0000i	− 1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 46.6690i	− 1.87883i	−0.342789	−	0.939413i	$-0.611371\pi$
$617$	− 46.6690i	− 1.87883i	0.342789	−	0.939413i	$-0.388629\pi$
$618$	0	0
$619$	−28.2843	−1.13684	−0.568420	−	0.822738i	$-0.692445\pi$
$619$	−28.2843	−1.13684	−0.568420	+	0.822738i	$0.692445\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	54.0000	2.16346
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	25.4558	1.01499
$630$	0	0
$631$	− 12.7279i	− 0.506691i	−0.967376	−	0.253345i	$-0.918469\pi$
$631$	− 12.7279i	− 0.506691i	0.967376	−	0.253345i	$-0.0815309\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 66.0000i	− 2.61502i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 21.2132i	− 0.837871i	−0.908016	−	0.418936i	$-0.862403\pi$
$641$	− 21.2132i	− 0.837871i	0.908016	−	0.418936i	$-0.137597\pi$
$642$	0	0
$643$	31.1127	1.22697	0.613483	−	0.789708i	$-0.289768\pi$
$643$	31.1127	1.22697	0.613483	+	0.789708i	$0.289768\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 20.0000i	− 0.779089i	−0.921008	−	0.389545i	$-0.872632\pi$
$659$	− 20.0000i	− 0.779089i	0.921008	−	0.389545i	$-0.127368\pi$
$660$	0	0
$661$	30.0000i	1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$	30.0000i	1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−50.9117	−1.97131
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.9706	−0.652232	−0.326116	−	0.945330i	$-0.605740\pi$
$677$	−16.9706	−0.652232	−0.326116	+	0.945330i	$0.605740\pi$
$678$	0	0
$679$	50.9117i	1.95381i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 50.9117i	− 1.93958i
$690$	0	0
$691$	19.7990	0.753189	0.376595	−	0.926378i	$-0.377095\pi$
$691$	19.7990	0.753189	0.376595	+	0.926378i	$0.377095\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.48528	0.320485	0.160242	−	0.987078i	$-0.448772\pi$
$701$	8.48528	0.320485	0.160242	+	0.987078i	$0.448772\pi$
$702$	0	0
$703$	16.9706i	0.640057i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 36.0000i	− 1.35392i
$708$	0	0
$709$	6.00000i	0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$	6.00000i	0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	25.4558i	0.953329i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.0000	−1.56634	−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000	−1.56634	−0.783168	+	0.621810i	$0.786397\pi$
$720$	0	0
$721$	54.0000	2.01107
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−42.4264	−1.57568
$726$	0	0
$727$	− 21.2132i	− 0.786754i	−0.919377	−	0.393377i	$-0.871307\pi$
$727$	− 21.2132i	− 0.786754i	0.919377	−	0.393377i	$-0.128693\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	12.0000i	0.443836i
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	45.2548i	1.66698i
$738$	0	0
$739$	28.2843	1.04045	0.520227	−	0.854028i	$-0.325847\pi$
$739$	28.2843	1.04045	0.520227	+	0.854028i	$0.325847\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−67.8823	−2.48036
$750$	0	0
$751$	− 46.6690i	− 1.70298i	−0.524373	−	0.851489i	$-0.675700\pi$
$751$	− 46.6690i	− 1.70298i	0.524373	−	0.851489i	$-0.324300\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.00000i	0.218074i	0.994038	+	0.109037i	$0.0347767\pi$
$757$	6.00000i	0.218074i	−0.994038	+	0.109037i	$0.965223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 24.0416i	− 0.871508i	−0.900066	−	0.435754i	$-0.856482\pi$
$761$	− 24.0416i	− 0.871508i	0.900066	−	0.435754i	$-0.143518\pi$
$762$	0	0
$763$	−25.4558	−0.921563
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.48528	−0.305194	−0.152597	−	0.988288i	$-0.548764\pi$
$773$	−8.48528	−0.305194	−0.152597	+	0.988288i	$0.548764\pi$
$774$	0	0
$775$	21.2132i	0.762001i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4.00000i	− 0.143315i
$780$	0	0
$781$	− 24.0000i	− 0.858788i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	53.7401	1.91563	0.957814	−	0.287388i	$-0.0927871\pi$
$787$	53.7401	1.91563	0.957814	+	0.287388i	$0.0927871\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.9411	1.20226	0.601128	−	0.799153i	$-0.294718\pi$
$797$	33.9411	1.20226	0.601128	+	0.799153i	$0.294718\pi$
$798$	0	0
$799$	− 25.4558i	− 0.900563i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.3848i	0.646374i	0.946335	+	0.323187i	$0.104754\pi$
$809$	18.3848i	0.646374i	−0.946335	+	0.323187i	$0.895246\pi$
$810$	0	0
$811$	−2.82843	−0.0993195	−0.0496598	−	0.998766i	$-0.515814\pi$
$811$	−2.82843	−0.0993195	−0.0496598	+	0.998766i	$0.515814\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.4558	−0.888415	−0.444208	−	0.895924i	$-0.646515\pi$
$821$	−25.4558	−0.888415	−0.444208	+	0.895924i	$0.646515\pi$
$822$	0	0
$823$	46.6690i	1.62678i	0.581718	+	0.813390i	$0.302381\pi$
$823$	46.6690i	1.62678i	−0.581718	+	0.813390i	$0.697619\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	0	0
$829$	18.0000i	0.625166i	0.949890	+	0.312583i	$0.101194\pi$
$829$	18.0000i	0.625166i	−0.949890	+	0.312583i	$0.898806\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 46.6690i	− 1.61699i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.0000	−1.86429	−0.932144	−	0.362089i	$-0.882064\pi$
$839$	−54.0000	−1.86429	−0.932144	+	0.362089i	$0.882064\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	21.2132i	0.728894i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	36.0000i	1.23406i
$852$	0	0
$853$	− 42.0000i	− 1.43805i	−0.694983	−	0.719026i	$-0.744588\pi$
$853$	− 42.0000i	− 1.43805i	0.694983	−	0.719026i	$-0.255412\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.89949i	− 0.338160i	−0.985602	−	0.169080i	$-0.945920\pi$
$857$	− 9.89949i	− 0.338160i	0.985602	−	0.169080i	$-0.0540797\pi$
$858$	0	0
$859$	−14.1421	−0.482523	−0.241262	−	0.970460i	$-0.577561\pi$
$859$	−14.1421	−0.482523	−0.241262	+	0.970460i	$0.577561\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−16.9706	−0.575687
$870$	0	0
$871$	− 67.8823i	− 2.30010i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 32.5269i	− 1.09586i	−0.836524	−	0.547930i	$-0.815416\pi$
$881$	− 32.5269i	− 1.09586i	0.836524	−	0.547930i	$-0.184584\pi$
$882$	0	0
$883$	−19.7990	−0.666289	−0.333145	−	0.942876i	$-0.608110\pi$
$883$	−19.7990	−0.666289	−0.333145	+	0.942876i	$0.608110\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.9706	0.567898
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 36.0000i	− 1.20067i
$900$	0	0
$901$	− 36.0000i	− 1.19933i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.1421	−0.469582	−0.234791	−	0.972046i	$-0.575441\pi$
$907$	−14.1421	−0.469582	−0.234791	+	0.972046i	$0.575441\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	64.0000	2.11809
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.9411	1.12083
$918$	0	0
$919$	− 21.2132i	− 0.699759i	−0.936795	−	0.349880i	$-0.886223\pi$
$919$	− 21.2132i	− 0.699759i	0.936795	−	0.349880i	$-0.113777\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.0000i	1.18495i
$924$	0	0
$925$	30.0000i	0.986394i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 12.7279i	− 0.417590i	−0.977959	−	0.208795i	$-0.933046\pi$
$929$	− 12.7279i	− 0.417590i	0.977959	−	0.208795i	$-0.0669541\pi$
$930$	0	0
$931$	31.1127	1.01968
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.0000	0.588034	0.294017	−	0.955800i	$-0.405008\pi$
$937$	18.0000	0.588034	0.294017	+	0.955800i	$0.405008\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50.9117	1.65967	0.829837	−	0.558006i	$-0.188433\pi$
$941$	50.9117	1.65967	0.829837	+	0.558006i	$0.188433\pi$
$942$	0	0
$943$	− 8.48528i	− 0.276319i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	− 36.0000i	− 1.16861i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7.07107i	0.229054i	0.993420	+	0.114527i	$0.0365353\pi$
$953$	7.07107i	0.229054i	−0.993420	+	0.114527i	$0.963465\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	66.0000	2.13125
$960$	0	0
$961$	13.0000	0.419355
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 46.6690i	− 1.50078i	−0.660998	−	0.750388i	$-0.729867\pi$
$967$	− 46.6690i	− 1.50078i	0.660998	−	0.750388i	$-0.270133\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.0000i	1.28366i	0.766846	+	0.641831i	$0.221825\pi$
$971$	40.0000i	1.28366i	−0.766846	+	0.641831i	$0.778175\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.2132i	− 0.678671i	−0.940666	−	0.339335i	$-0.889798\pi$
$977$	− 21.2132i	− 0.678671i	0.940666	−	0.339335i	$-0.110202\pi$
$978$	0	0
$979$	50.9117	1.62714
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.9706	−0.539633
$990$	0	0
$991$	− 46.6690i	− 1.48249i	−0.671234	−	0.741246i	$-0.734235\pi$
$991$	− 46.6690i	− 1.48249i	0.671234	−	0.741246i	$-0.265765\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30.0000i	0.950110i	0.879956	+	0.475055i	$0.157572\pi$
$997$	30.0000i	0.950110i	−0.879956	+	0.475055i	$0.842428\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.f.i.2303.1		4
3.2	odd	2		4608.2.f.k.2303.2		4
4.3	odd	2		4608.2.f.k.2303.4		4
8.3	odd	2		4608.2.f.k.2303.3		4
8.5	even	2	inner	4608.2.f.i.2303.2		4
12.11	even	2	inner	4608.2.f.i.2303.3		4
16.3	odd	4		4608.2.c.g.4607.1	yes	2
16.5	even	4		4608.2.c.h.4607.2	yes	2
16.11	odd	4		4608.2.c.b.4607.1	yes	2
16.13	even	4		4608.2.c.a.4607.2	yes	2
24.5	odd	2		4608.2.f.k.2303.1		4
24.11	even	2	inner	4608.2.f.i.2303.4		4
48.5	odd	4		4608.2.c.b.4607.2	yes	2
48.11	even	4		4608.2.c.h.4607.1	yes	2
48.29	odd	4		4608.2.c.g.4607.2	yes	2
48.35	even	4		4608.2.c.a.4607.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.c.a.4607.1	✓	2	48.35	even	4
4608.2.c.a.4607.2	yes	2	16.13	even	4
4608.2.c.b.4607.1	yes	2	16.11	odd	4
4608.2.c.b.4607.2	yes	2	48.5	odd	4
4608.2.c.g.4607.1	yes	2	16.3	odd	4
4608.2.c.g.4607.2	yes	2	48.29	odd	4
4608.2.c.h.4607.1	yes	2	48.11	even	4
4608.2.c.h.4607.2	yes	2	16.5	even	4
4608.2.f.i.2303.1		4	1.1	even	1	trivial
4608.2.f.i.2303.2		4	8.5	even	2	inner
4608.2.f.i.2303.3		4	12.11	even	2	inner
4608.2.f.i.2303.4		4	24.11	even	2	inner
4608.2.f.k.2303.1		4	24.5	odd	2
4608.2.f.k.2303.2		4	3.2	odd	2
4608.2.f.k.2303.3		4	8.3	odd	2
4608.2.f.k.2303.4		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.24264i q^{7} +O(q^{10})\)	\(q-4.24264i q^{7} -4.00000i q^{11} +6.00000i q^{13} +4.24264i q^{17} -2.82843 q^{19} -6.00000 q^{23} -5.00000 q^{25} +8.48528 q^{29} -4.24264i q^{31} -6.00000i q^{37} +1.41421i q^{41} +2.82843 q^{43} -6.00000 q^{47} -11.0000 q^{49} -8.48528 q^{53} +4.00000i q^{59} -6.00000i q^{61} -11.3137 q^{67} +6.00000 q^{71} -6.00000 q^{73} -16.9706 q^{77} -4.24264i q^{79} +16.0000i q^{83} +12.7279i q^{89} +25.4558 q^{91} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{23} - 20 q^{25} - 24 q^{47} - 44 q^{49} + 24 q^{71} - 24 q^{73} - 48 q^{97}+O(q^{100}) \) 4 * q - 24 * q^23 - 20 * q^25 - 24 * q^47 - 44 * q^49 + 24 * q^71 - 24 * q^73 - 48 * q^97

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