LMFDB - Embedded newform 4608.2.c.n.4607.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(4607,4608)]

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, 0, 1]))

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

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

chi := DirichletCharacter("4608.4607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.c (of order $2$, degree $1$, 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:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4607.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	4608.4607
Dual form			4608.2.c.n.4607.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{5} +1.41421i q^{7} +O(q^{10})$	$q-2.00000i q^{5} +1.41421i q^{7} -2.82843 q^{11} -4.24264i q^{17} +6.00000 q^{23} +1.00000 q^{25} -2.00000i q^{29} -1.41421i q^{31} +2.82843 q^{35} -8.48528 q^{37} -4.24264i q^{41} +12.0000i q^{43} -6.00000 q^{47} +5.00000 q^{49} -2.00000i q^{53} +5.65685i q^{55} -8.48528 q^{61} -12.0000i q^{67} -6.00000 q^{71} +6.00000 q^{73} -4.00000i q^{77} -7.07107i q^{79} +2.82843 q^{83} -8.48528 q^{85} -4.24264i q^{89} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 24 q^{23} + 4 q^{25} - 24 q^{47} + 20 q^{49} - 24 q^{71} + 24 q^{73} - 32 q^{97}+O(q^{100}) $ 4 * q + 24 * q^23 + 4 * q^25 - 24 * q^47 + 20 * q^49 - 24 * q^71 + 24 * q^73 - 32 * 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
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	1.41421i	0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$	1.41421i	0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.24264i	− 1.02899i	−0.857493	−	0.514496i	$-0.827979\pi$
$17$	− 4.24264i	− 1.02899i	0.857493	−	0.514496i	$-0.172021\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	− 1.41421i	− 0.254000i	−0.991903	−	0.127000i	$-0.959465\pi$
$31$	− 1.41421i	− 0.254000i	0.991903	−	0.127000i	$-0.0405349\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.24264i	− 0.662589i	−0.943527	−	0.331295i	$-0.892515\pi$
$41$	− 4.24264i	− 0.662589i	0.943527	−	0.331295i	$-0.107485\pi$
$42$	0	0
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	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
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	5.65685i	0.762770i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	− 7.07107i	− 0.795557i	−0.917481	−	0.397779i	$-0.869781\pi$
$79$	− 7.07107i	− 0.795557i	0.917481	−	0.397779i	$-0.130219\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	−8.48528	−0.920358
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.24264i	− 0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$	− 4.24264i	− 0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	− 15.5563i	− 1.53281i	−0.642356	−	0.766406i	$-0.722043\pi$
$103$	− 15.5563i	− 1.53281i	0.642356	−	0.766406i	$-0.277957\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.9706	−1.64061	−0.820303	−	0.571929i	$-0.806195\pi$
$107$	−16.9706	−1.64061	−0.820303	+	0.571929i	$0.806195\pi$
$108$	0	0
$109$	−16.9706	−1.62549	−0.812743	−	0.582623i	$-0.802026\pi$
$109$	−16.9706	−1.62549	−0.812743	+	0.582623i	$0.802026\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.24264i	0.399114i	0.979886	+	0.199557i	$0.0639503\pi$
$113$	4.24264i	0.399114i	−0.979886	+	0.199557i	$0.936050\pi$
$114$	0	0
$115$	− 12.0000i	− 1.11901i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	9.89949i	0.878438i	0.898380	+	0.439219i	$0.144745\pi$
$127$	9.89949i	0.878438i	−0.898380	+	0.439219i	$0.855255\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.24264i	0.362473i	0.983440	+	0.181237i	$0.0580100\pi$
$137$	4.24264i	0.362473i	−0.983440	+	0.181237i	$0.941990\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.00000	−0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 14.0000i	− 1.14692i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	− 14.0000i	− 1.14692i	0.819232	−	0.573462i	$-0.194400\pi$
$150$	0	0
$151$	15.5563i	1.26596i	0.774169	+	0.632979i	$0.218168\pi$
$151$	15.5563i	1.26596i	−0.774169	+	0.632979i	$0.781832\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.82843	−0.227185
$156$	0	0
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.48528i	0.668734i
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	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
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	0	0
$175$	1.41421i	0.106904i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.9706	1.26844	0.634220	−	0.773153i	$-0.281321\pi$
$179$	16.9706	1.26844	0.634220	+	0.773153i	$0.281321\pi$
$180$	0	0
$181$	−16.9706	−1.26141	−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706	−1.26141	−0.630706	+	0.776022i	$0.717235\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	16.9706i	1.24770i
$186$	0	0
$187$	12.0000i	0.877527i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000i	0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$	10.0000i	0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$	0	0
$199$	− 15.5563i	− 1.10276i	−0.834254	−	0.551380i	$-0.814101\pi$
$199$	− 15.5563i	− 1.10276i	0.834254	−	0.551380i	$-0.185899\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.82843	0.198517
$204$	0	0
$205$	−8.48528	−0.592638
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	24.0000	1.63679
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.07107i	0.473514i	0.971569	+	0.236757i	$0.0760845\pi$
$223$	7.07107i	0.473514i	−0.971569	+	0.236757i	$0.923916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	0	0
$229$	−16.9706	−1.12145	−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706	−1.12145	−0.560723	+	0.828003i	$0.689477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.7279i	0.833834i	0.908945	+	0.416917i	$0.136889\pi$
$233$	12.7279i	0.833834i	−0.908945	+	0.416917i	$0.863111\pi$
$234$	0	0
$235$	12.0000i	0.782794i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 10.0000i	− 0.638877i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	−16.9706	−1.06693
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 12.7279i	− 0.793946i	−0.917830	−	0.396973i	$-0.870061\pi$
$257$	− 12.7279i	− 0.793946i	0.917830	−	0.396973i	$-0.129939\pi$
$258$	0	0
$259$	− 12.0000i	− 0.745644i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.0000i	0.853595i	0.904347	+	0.426798i	$0.140358\pi$
$269$	14.0000i	0.853595i	−0.904347	+	0.426798i	$0.859642\pi$
$270$	0	0
$271$	7.07107i	0.429537i	0.976665	+	0.214768i	$0.0688997\pi$
$271$	7.07107i	0.429537i	−0.976665	+	0.214768i	$0.931100\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	−16.9706	−1.01966	−0.509831	−	0.860274i	$-0.670292\pi$
$277$	−16.9706	−1.01966	−0.509831	+	0.860274i	$0.670292\pi$
$278$	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
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	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
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−16.9706	−0.978167
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.9706i	0.971732i
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.0000i	− 0.561656i	−0.959758	−	0.280828i	$-0.909391\pi$
$317$	− 10.0000i	− 0.561656i	0.959758	−	0.280828i	$-0.0906090\pi$
$318$	0	0
$319$	5.65685i	0.316723i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.48528i	− 0.467809i
$330$	0	0
$331$	− 12.0000i	− 0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$	− 12.0000i	− 0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.0000	−1.31126
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.00000i	0.216612i
$342$	0	0
$343$	16.9706i	0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.1421	−0.759190	−0.379595	−	0.925153i	$-0.623937\pi$
$347$	−14.1421	−0.759190	−0.379595	+	0.925153i	$0.623937\pi$
$348$	0	0
$349$	8.48528	0.454207	0.227103	−	0.973871i	$-0.427074\pi$
$349$	8.48528	0.454207	0.227103	+	0.973871i	$0.427074\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.24264i	0.225813i	0.993606	+	0.112906i	$0.0360161\pi$
$353$	4.24264i	0.225813i	−0.993606	+	0.112906i	$0.963984\pi$
$354$	0	0
$355$	12.0000i	0.636894i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	− 35.3553i	− 1.84553i	−0.385359	−	0.922767i	$-0.625922\pi$
$367$	− 35.3553i	− 1.84553i	0.385359	−	0.922767i	$-0.374078\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.82843	0.146845
$372$	0	0
$373$	8.48528	0.439351	0.219676	−	0.975573i	$-0.429500\pi$
$373$	8.48528	0.439351	0.219676	+	0.975573i	$0.429500\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	24.0000i	1.23280i	0.787434	+	0.616399i	$0.211409\pi$
$379$	24.0000i	1.23280i	−0.787434	+	0.616399i	$0.788591\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 10.0000i	− 0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$	− 10.0000i	− 0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$	0	0
$391$	− 25.4558i	− 1.28736i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.1421	−0.711568
$396$	0	0
$397$	25.4558	1.27759	0.638796	−	0.769376i	$-0.279433\pi$
$397$	25.4558	1.27759	0.638796	+	0.769376i	$0.279433\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.24264i	− 0.211867i	−0.994373	−	0.105934i	$-0.966217\pi$
$401$	− 4.24264i	− 0.211867i	0.994373	−	0.105934i	$-0.0337831\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 5.65685i	− 0.277684i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.1421	0.690889	0.345444	−	0.938439i	$-0.387728\pi$
$419$	14.1421	0.690889	0.345444	+	0.938439i	$0.387728\pi$
$420$	0	0
$421$	−33.9411	−1.65419	−0.827095	−	0.562063i	$-0.810008\pi$
$421$	−33.9411	−1.65419	−0.827095	+	0.562063i	$0.810008\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 4.24264i	− 0.205798i
$426$	0	0
$427$	− 12.0000i	− 0.580721i
$428$	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
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	26.8701i	1.28244i	0.767358	+	0.641219i	$0.221571\pi$
$439$	26.8701i	1.28244i	−0.767358	+	0.641219i	$0.778429\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.82843	0.134383	0.0671913	−	0.997740i	$-0.478596\pi$
$443$	2.82843	0.134383	0.0671913	+	0.997740i	$0.478596\pi$
$444$	0	0
$445$	−8.48528	−0.402241
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 29.6985i	− 1.40156i	−0.713378	−	0.700779i	$-0.752836\pi$
$449$	− 29.6985i	− 1.40156i	0.713378	−	0.700779i	$-0.247164\pi$
$450$	0	0
$451$	12.0000i	0.565058i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 10.0000i	− 0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$	− 10.0000i	− 0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$	0	0
$463$	1.41421i	0.0657241i	0.999460	+	0.0328620i	$0.0104622\pi$
$463$	1.41421i	0.0657241i	−0.999460	+	0.0328620i	$0.989538\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.1127	1.43972	0.719862	−	0.694117i	$-0.244205\pi$
$467$	31.1127	1.43972	0.719862	+	0.694117i	$0.244205\pi$
$468$	0	0
$469$	16.9706	0.783628
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 33.9411i	− 1.56061i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000i	0.726523i
$486$	0	0
$487$	− 24.0416i	− 1.08943i	−0.838621	−	0.544715i	$-0.816638\pi$
$487$	− 24.0416i	− 1.08943i	0.838621	−	0.544715i	$-0.183362\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	33.9411	1.53174	0.765871	−	0.642995i	$-0.222308\pi$
$491$	33.9411	1.53174	0.765871	+	0.642995i	$0.222308\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.48528i	− 0.380617i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.0000	0.802580	0.401290	−	0.915951i	$-0.368562\pi$
$503$	18.0000	0.802580	0.401290	+	0.915951i	$0.368562\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.00000i	− 0.0886484i	−0.999017	−	0.0443242i	$-0.985887\pi$
$509$	− 2.00000i	− 0.0886484i	0.999017	−	0.0443242i	$-0.0141135\pi$
$510$	0	0
$511$	8.48528i	0.375367i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−31.1127	−1.37099
$516$	0	0
$517$	16.9706	0.746364
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 21.2132i	− 0.929367i	−0.885477	−	0.464684i	$-0.846168\pi$
$521$	− 21.2132i	− 0.929367i	0.885477	−	0.464684i	$-0.153832\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.00000	−0.261364
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	33.9411i	1.46740i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−14.1421	−0.609145
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	33.9411i	1.45388i
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.0000i	1.44063i	0.693649	+	0.720313i	$0.256002\pi$
$557$	34.0000i	1.44063i	−0.693649	+	0.720313i	$0.743998\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.7696	−1.54965	−0.774826	−	0.632175i	$-0.782163\pi$
$563$	−36.7696	−1.54965	−0.774826	+	0.632175i	$0.782163\pi$
$564$	0	0
$565$	8.48528	0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1838i	1.60075i	0.599502	+	0.800373i	$0.295365\pi$
$569$	38.1838i	1.60075i	−0.599502	+	0.800373i	$0.704635\pi$
$570$	0	0
$571$	− 24.0000i	− 1.00437i	−0.864761	−	0.502184i	$-0.832530\pi$
$571$	− 24.0000i	− 1.00437i	0.864761	−	0.502184i	$-0.167470\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	36.0000	1.49870	0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000	1.49870	0.749350	+	0.662174i	$0.230366\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.00000i	0.165948i
$582$	0	0
$583$	5.65685i	0.234283i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.9706	0.700450	0.350225	−	0.936666i	$-0.386105\pi$
$587$	16.9706	0.700450	0.350225	+	0.936666i	$0.386105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 38.1838i	− 1.56802i	−0.620749	−	0.784010i	$-0.713171\pi$
$593$	− 38.1838i	− 1.56802i	0.620749	−	0.784010i	$-0.286829\pi$
$594$	0	0
$595$	− 12.0000i	− 0.491952i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	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
$605$	6.00000i	0.243935i
$606$	0	0
$607$	9.89949i	0.401808i	0.979611	+	0.200904i	$0.0643879\pi$
$607$	9.89949i	0.401808i	−0.979611	+	0.200904i	$0.935612\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	25.4558	1.02815	0.514076	−	0.857745i	$-0.328135\pi$
$613$	25.4558	1.02815	0.514076	+	0.857745i	$0.328135\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.2132i	0.854011i	0.904249	+	0.427006i	$0.140432\pi$
$617$	21.2132i	0.854011i	−0.904249	+	0.427006i	$0.859568\pi$
$618$	0	0
$619$	− 24.0000i	− 0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$	− 24.0000i	− 0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	36.0000i	1.43541i
$630$	0	0
$631$	9.89949i	0.394093i	0.980394	+	0.197046i	$0.0631349\pi$
$631$	9.89949i	0.394093i	−0.980394	+	0.197046i	$0.936865\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.7990	0.785699
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 12.7279i	− 0.502723i	−0.967893	−	0.251361i	$-0.919122\pi$
$641$	− 12.7279i	− 0.502723i	0.967893	−	0.251361i	$-0.0808782\pi$
$642$	0	0
$643$	12.0000i	0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$	12.0000i	0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.00000i	0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$	2.00000i	0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$	0	0
$655$	− 33.9411i	− 1.32619i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.9706	−0.661079	−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706	−0.661079	−0.330540	+	0.943792i	$0.607231\pi$
$660$	0	0
$661$	−8.48528	−0.330039	−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528	−0.330039	−0.165020	+	0.986290i	$0.552769\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.00000i	− 0.0768662i	−0.999261	−	0.0384331i	$-0.987763\pi$
$677$	− 2.00000i	− 0.0768662i	0.999261	−	0.0384331i	$-0.0122367\pi$
$678$	0	0
$679$	− 11.3137i	− 0.434180i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.7990	−0.757587	−0.378794	−	0.925481i	$-0.623661\pi$
$683$	−19.7990	−0.757587	−0.378794	+	0.925481i	$0.623661\pi$
$684$	0	0
$685$	8.48528	0.324206
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	36.0000i	1.36950i	0.728776	+	0.684752i	$0.240090\pi$
$691$	36.0000i	1.36950i	−0.728776	+	0.684752i	$0.759910\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.0000	0.910372
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.0000i	0.982006i	0.871158	+	0.491003i	$0.163370\pi$
$701$	26.0000i	0.982006i	−0.871158	+	0.491003i	$0.836630\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.1421	0.531870
$708$	0	0
$709$	33.9411	1.27469	0.637343	−	0.770580i	$-0.280034\pi$
$709$	33.9411	1.27469	0.637343	+	0.770580i	$0.280034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 8.48528i	− 0.317776i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	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
$721$	22.0000	0.819323
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	0	0
$727$	− 26.8701i	− 0.996555i	−0.867018	−	0.498278i	$-0.833966\pi$
$727$	− 26.8701i	− 0.996555i	0.867018	−	0.498278i	$-0.166034\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	50.9117	1.88304
$732$	0	0
$733$	33.9411	1.25364	0.626822	−	0.779162i	$-0.284355\pi$
$733$	33.9411	1.25364	0.626822	+	0.779162i	$0.284355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.9411i	1.25024i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−28.0000	−1.02584
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 24.0000i	− 0.876941i
$750$	0	0
$751$	52.3259i	1.90940i	0.297571	+	0.954700i	$0.403823\pi$
$751$	52.3259i	1.90940i	−0.297571	+	0.954700i	$0.596177\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	31.1127	1.13231
$756$	0	0
$757$	16.9706	0.616806	0.308403	−	0.951256i	$-0.400206\pi$
$757$	16.9706	0.616806	0.308403	+	0.951256i	$0.400206\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.2132i	0.768978i	0.923130	+	0.384489i	$0.125622\pi$
$761$	21.2132i	0.768978i	−0.923130	+	0.384489i	$0.874378\pi$
$762$	0	0
$763$	− 24.0000i	− 0.868858i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	50.0000i	1.79838i	0.437564	+	0.899188i	$0.355842\pi$
$773$	50.0000i	1.79838i	−0.437564	+	0.899188i	$0.644158\pi$
$774$	0	0
$775$	− 1.41421i	− 0.0508001i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	16.9706	0.607254
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16.9706i	0.605705i
$786$	0	0
$787$	24.0000i	0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$	24.0000i	0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	25.4558i	0.900563i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.9706	−0.598878
$804$	0	0
$805$	16.9706	0.598134
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.24264i	− 0.149163i	−0.997215	−	0.0745817i	$-0.976238\pi$
$809$	− 4.24264i	− 0.149163i	0.997215	−	0.0745817i	$-0.0237621\pi$
$810$	0	0
$811$	− 24.0000i	− 0.842754i	−0.906886	−	0.421377i	$-0.861547\pi$
$811$	− 24.0000i	− 0.842754i	0.906886	−	0.421377i	$-0.138453\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−24.0000	−0.840683
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 34.0000i	− 1.18661i	−0.804978	−	0.593304i	$-0.797823\pi$
$821$	− 34.0000i	− 1.18661i	0.804978	−	0.593304i	$-0.202177\pi$
$822$	0	0
$823$	1.41421i	0.0492964i	0.999696	+	0.0246482i	$0.00784656\pi$
$823$	1.41421i	0.0492964i	−0.999696	+	0.0246482i	$0.992153\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	50.9117	1.76824	0.884118	−	0.467264i	$-0.154760\pi$
$829$	50.9117	1.76824	0.884118	+	0.467264i	$0.154760\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 21.2132i	− 0.734994i
$834$	0	0
$835$	48.0000i	1.66111i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	26.0000i	0.894427i
$846$	0	0
$847$	− 4.24264i	− 0.145779i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−50.9117	−1.74523
$852$	0	0
$853$	25.4558	0.871592	0.435796	−	0.900046i	$-0.356467\pi$
$853$	25.4558	0.871592	0.435796	+	0.900046i	$0.356467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38.1838i	− 1.30433i	−0.758076	−	0.652166i	$-0.773860\pi$
$857$	− 38.1838i	− 1.30433i	0.758076	−	0.652166i	$-0.226140\pi$
$858$	0	0
$859$	− 36.0000i	− 1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$	− 36.0000i	− 1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−44.0000	−1.49604
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.0000i	0.678454i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16.9706	0.573710
$876$	0	0
$877$	42.4264	1.43264	0.716319	−	0.697773i	$-0.245826\pi$
$877$	42.4264	1.43264	0.716319	+	0.697773i	$0.245826\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	55.1543i	1.85820i	0.369833	+	0.929098i	$0.379415\pi$
$881$	55.1543i	1.85820i	−0.369833	+	0.929098i	$0.620585\pi$
$882$	0	0
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 33.9411i	− 1.13453i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.82843	−0.0943333
$900$	0	0
$901$	−8.48528	−0.282686
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33.9411i	1.12824i
$906$	0	0
$907$	36.0000i	1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$	36.0000i	1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24.0000i	0.792550i
$918$	0	0
$919$	− 32.5269i	− 1.07296i	−0.843912	−	0.536482i	$-0.819753\pi$
$919$	− 32.5269i	− 1.07296i	0.843912	−	0.536482i	$-0.180247\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−8.48528	−0.278994
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29.6985i	0.974376i	0.873297	+	0.487188i	$0.161977\pi$
$929$	29.6985i	0.974376i	−0.873297	+	0.487188i	$0.838023\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	0	0
$943$	− 25.4558i	− 0.828956i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.9411	−1.10294	−0.551469	−	0.834195i	$-0.685933\pi$
$947$	−33.9411	−1.10294	−0.551469	+	0.834195i	$0.685933\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	46.6690i	1.51176i	0.654711	+	0.755879i	$0.272790\pi$
$953$	46.6690i	1.51176i	−0.654711	+	0.755879i	$0.727210\pi$
$954$	0	0
$955$	− 48.0000i	− 1.55324i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	0	0
$961$	29.0000	0.935484
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 24.0000i	− 0.772587i
$966$	0	0
$967$	− 1.41421i	− 0.0454780i	−0.999741	−	0.0227390i	$-0.992761\pi$
$967$	− 1.41421i	− 0.0454780i	0.999741	−	0.0227390i	$-0.00723868\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.7990	−0.635380	−0.317690	−	0.948195i	$-0.602907\pi$
$971$	−19.7990	−0.635380	−0.317690	+	0.948195i	$0.602907\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 29.6985i	− 0.950139i	−0.879948	−	0.475069i	$-0.842423\pi$
$977$	− 29.6985i	− 0.950139i	0.879948	−	0.475069i	$-0.157577\pi$
$978$	0	0
$979$	12.0000i	0.383522i
$980$	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
$985$	20.0000	0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	72.0000i	2.28947i
$990$	0	0
$991$	− 15.5563i	− 0.494164i	−0.968995	−	0.247082i	$-0.920528\pi$
$991$	− 15.5563i	− 0.494164i	0.968995	−	0.247082i	$-0.0794717\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−31.1127	−0.986339
$996$	0	0
$997$	59.3970	1.88112	0.940560	−	0.339626i	$-0.110301\pi$
$997$	59.3970	1.88112	0.940560	+	0.339626i	$0.110301\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.c.n.4607.2	yes	4
3.2	odd	2		4608.2.c.k.4607.4	yes	4
4.3	odd	2		4608.2.c.k.4607.1	✓	4
8.3	odd	2		4608.2.c.k.4607.3	yes	4
8.5	even	2	inner	4608.2.c.n.4607.4	yes	4
12.11	even	2	inner	4608.2.c.n.4607.3	yes	4
16.3	odd	4		4608.2.f.c.2303.2		2
16.5	even	4		4608.2.f.f.2303.1		2
16.11	odd	4		4608.2.f.g.2303.2		2
16.13	even	4		4608.2.f.b.2303.1		2
24.5	odd	2		4608.2.c.k.4607.2	yes	4
24.11	even	2	inner	4608.2.c.n.4607.1	yes	4
48.5	odd	4		4608.2.f.c.2303.1		2
48.11	even	4		4608.2.f.b.2303.2		2
48.29	odd	4		4608.2.f.g.2303.1		2
48.35	even	4		4608.2.f.f.2303.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.c.k.4607.1	✓	4	4.3	odd	2
4608.2.c.k.4607.2	yes	4	24.5	odd	2
4608.2.c.k.4607.3	yes	4	8.3	odd	2
4608.2.c.k.4607.4	yes	4	3.2	odd	2
4608.2.c.n.4607.1	yes	4	24.11	even	2	inner
4608.2.c.n.4607.2	yes	4	1.1	even	1	trivial
4608.2.c.n.4607.3	yes	4	12.11	even	2	inner
4608.2.c.n.4607.4	yes	4	8.5	even	2	inner
4608.2.f.b.2303.1		2	16.13	even	4
4608.2.f.b.2303.2		2	48.11	even	4
4608.2.f.c.2303.1		2	48.5	odd	4
4608.2.f.c.2303.2		2	16.3	odd	4
4608.2.f.f.2303.1		2	16.5	even	4
4608.2.f.f.2303.2		2	48.35	even	4
4608.2.f.g.2303.1		2	48.29	odd	4
4608.2.f.g.2303.2		2	16.11	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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{5} +1.41421i q^{7} +O(q^{10})\)	\(q-2.00000i q^{5} +1.41421i q^{7} -2.82843 q^{11} -4.24264i q^{17} +6.00000 q^{23} +1.00000 q^{25} -2.00000i q^{29} -1.41421i q^{31} +2.82843 q^{35} -8.48528 q^{37} -4.24264i q^{41} +12.0000i q^{43} -6.00000 q^{47} +5.00000 q^{49} -2.00000i q^{53} +5.65685i q^{55} -8.48528 q^{61} -12.0000i q^{67} -6.00000 q^{71} +6.00000 q^{73} -4.00000i q^{77} -7.07107i q^{79} +2.82843 q^{83} -8.48528 q^{85} -4.24264i q^{89} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 24 q^{23} + 4 q^{25} - 24 q^{47} + 20 q^{49} - 24 q^{71} + 24 q^{73} - 32 q^{97}+O(q^{100}) \) 4 * q + 24 * q^23 + 4 * q^25 - 24 * q^47 + 20 * q^49 - 24 * q^71 + 24 * q^73 - 32 * q^97

Introduction

L-functions

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