LMFDB - Embedded newform 4032.2.j.e.2591.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(2591,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("4032.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.j (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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.162447943996702457856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} - 15x^{8} - 16x^{4} + 256 $ x^16 - x^12 - 15x^8 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.4
Root			$-1.40721 + 0.140577i$ of defining polynomial
Character	$\chi$	$=$	4032.2591
Dual form			4032.2.j.e.2591.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.09557 q^{5} +1.00000i q^{7} +O(q^{10})$	$q-3.09557 q^{5} +1.00000i q^{7} +0.646084i q^{11} +4.37780i q^{13} -0.295164i q^{17} -5.29150 q^{19} +3.94748 q^{23} +4.58258 q^{25} +5.03383 q^{29} +9.16515i q^{31} -3.09557i q^{35} +2.55040i q^{37} -10.4282i q^{41} -1.63670 q^{43} +5.65685 q^{47} -1.00000 q^{49} -8.64064 q^{53} -2.00000i q^{55} +6.92820i q^{61} -13.5518i q^{65} -2.55040 q^{67} -1.11905 q^{71} -9.58258 q^{73} -0.646084 q^{77} -16.7477i q^{79} -8.50579i q^{83} +0.913701i q^{85} +10.4282i q^{89} -4.37780 q^{91} +16.3802 q^{95} -2.41742 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{49} - 80 q^{73} - 112 q^{97}+O(q^{100}) $ 16 * q - 16 * q^49 - 80 * q^73 - 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$5$	−3.09557	−1.38438	−0.692191	−	0.721714i	$-0.743355\pi$
$5$	−3.09557	−1.38438	−0.692191	+	0.721714i	$0.743355\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.646084i	0.194802i	0.995245	+	0.0974008i	$0.0310529\pi$
$11$	0.646084i	0.194802i	−0.995245	+	0.0974008i	$0.968947\pi$
$12$	0	0
$13$	4.37780i	1.21418i	0.794632	+	0.607092i	$0.207664\pi$
$13$	4.37780i	1.21418i	−0.794632	+	0.607092i	$0.792336\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.295164i	− 0.0715877i	−0.999359	−	0.0357938i	$-0.988604\pi$
$17$	− 0.295164i	− 0.0715877i	0.999359	−	0.0357938i	$-0.0113960\pi$
$18$	0	0
$19$	−5.29150	−1.21395	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	−5.29150	−1.21395	−0.606977	+	0.794719i	$0.707618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.94748	0.823106	0.411553	−	0.911386i	$-0.364987\pi$
$23$	3.94748	0.823106	0.411553	+	0.911386i	$0.364987\pi$
$24$	0	0
$25$	4.58258	0.916515
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.03383	0.934758	0.467379	−	0.884057i	$-0.345198\pi$
$29$	5.03383	0.934758	0.467379	+	0.884057i	$0.345198\pi$
$30$	0	0
$31$	9.16515i	1.64611i	0.567962	+	0.823055i	$0.307732\pi$
$31$	9.16515i	1.64611i	−0.567962	+	0.823055i	$0.692268\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.09557i	− 0.523247i
$36$	0	0
$37$	2.55040i	0.419283i	0.977778	+	0.209642i	$0.0672297\pi$
$37$	2.55040i	0.419283i	−0.977778	+	0.209642i	$0.932770\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.4282i	− 1.62861i	−0.580434	−	0.814307i	$-0.697117\pi$
$41$	− 10.4282i	− 1.62861i	0.580434	−	0.814307i	$-0.302883\pi$
$42$	0	0
$43$	−1.63670	−0.249595	−0.124797	−	0.992182i	$-0.539828\pi$
$43$	−1.63670	−0.249595	−0.124797	+	0.992182i	$0.539828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.64064	−1.18688	−0.593441	−	0.804877i	$-0.702231\pi$
$53$	−8.64064	−1.18688	−0.593441	+	0.804877i	$0.702231\pi$
$54$	0	0
$55$	− 2.00000i	− 0.269680i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	6.92820i	0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.92820i	0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 13.5518i	− 1.68089i
$66$	0	0
$67$	−2.55040	−0.311581	−0.155791	−	0.987790i	$-0.549792\pi$
$67$	−2.55040	−0.311581	−0.155791	+	0.987790i	$0.549792\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.11905	−0.132807	−0.0664034	−	0.997793i	$-0.521152\pi$
$71$	−1.11905	−0.132807	−0.0664034	+	0.997793i	$0.521152\pi$
$72$	0	0
$73$	−9.58258	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$73$	−9.58258	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.646084	−0.0736281
$78$	0	0
$79$	− 16.7477i	− 1.88427i	−0.335239	−	0.942133i	$-0.608817\pi$
$79$	− 16.7477i	− 1.88427i	0.335239	−	0.942133i	$-0.391183\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.50579i	− 0.933632i	−0.884354	−	0.466816i	$-0.845401\pi$
$83$	− 8.50579i	− 0.933632i	0.884354	−	0.466816i	$-0.154599\pi$
$84$	0	0
$85$	0.913701i	0.0991047i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.4282i	1.10539i	0.833384	+	0.552694i	$0.186400\pi$
$89$	10.4282i	1.10539i	−0.833384	+	0.552694i	$0.813600\pi$
$90$	0	0
$91$	−4.37780	−0.458918
$92$	0	0
$93$	0	0
$94$	0	0
$95$	16.3802	1.68058
$96$	0	0
$97$	−2.41742	−0.245452	−0.122726	−	0.992441i	$-0.539164\pi$
$97$	−2.41742	−0.245452	−0.122726	+	0.992441i	$0.539164\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.7925	−1.77042	−0.885211	−	0.465191i	$-0.845986\pi$
$101$	−17.7925	−1.77042	−0.885211	+	0.465191i	$0.845986\pi$
$102$	0	0
$103$	2.00000i	0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$	2.00000i	0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.4440i	− 1.00966i	−0.863218	−	0.504832i	$-0.831554\pi$
$107$	− 10.4440i	− 1.00966i	0.863218	−	0.504832i	$-0.168446\pi$
$108$	0	0
$109$	5.29150i	0.506834i	0.967357	+	0.253417i	$0.0815545\pi$
$109$	5.29150i	0.506834i	−0.967357	+	0.253417i	$0.918446\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 9.89949i	− 0.931266i	−0.884978	−	0.465633i	$-0.845827\pi$
$113$	− 9.89949i	− 0.931266i	0.884978	−	0.465633i	$-0.154173\pi$
$114$	0	0
$115$	−12.2197	−1.13949
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.295164	0.0270576
$120$	0	0
$121$	10.5826	0.962052
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.29217	0.115575
$126$	0	0
$127$	− 13.5826i	− 1.20526i	−0.798021	−	0.602629i	$-0.794120\pi$
$127$	− 13.5826i	− 1.20526i	0.798021	−	0.602629i	$-0.205880\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 3.60681i	− 0.315129i	−0.987509	−	0.157564i	$-0.949636\pi$
$131$	− 3.60681i	− 0.315129i	0.987509	−	0.157564i	$-0.0503642\pi$
$132$	0	0
$133$	− 5.29150i	− 0.458831i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.0325i	1.71150i	0.517393	+	0.855748i	$0.326903\pi$
$137$	20.0325i	1.71150i	−0.517393	+	0.855748i	$0.673097\pi$
$138$	0	0
$139$	−8.75560	−0.742641	−0.371320	−	0.928505i	$-0.621095\pi$
$139$	−8.75560	−0.742641	−0.371320	+	0.928505i	$0.621095\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.82843	−0.236525
$144$	0	0
$145$	−15.5826	−1.29406
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.76413	−0.390293	−0.195147	−	0.980774i	$-0.562518\pi$
$149$	−4.76413	−0.390293	−0.195147	+	0.980774i	$0.562518\pi$
$150$	0	0
$151$	− 4.83485i	− 0.393454i	−0.980458	−	0.196727i	$-0.936969\pi$
$151$	− 4.83485i	− 0.393454i	0.980458	−	0.196727i	$-0.0630313\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 28.3714i	− 2.27885i
$156$	0	0
$157$	5.10080i	0.407088i	0.979066	+	0.203544i	$0.0652461\pi$
$157$	5.10080i	0.407088i	−0.979066	+	0.203544i	$0.934754\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.94748i	0.311105i
$162$	0	0
$163$	23.7164	1.85761	0.928806	−	0.370565i	$-0.120836\pi$
$163$	23.7164	1.85761	0.928806	+	0.370565i	$0.120836\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.06653	−0.392060	−0.196030	−	0.980598i	$-0.562805\pi$
$167$	−5.06653	−0.392060	−0.196030	+	0.980598i	$0.562805\pi$
$168$	0	0
$169$	−6.16515	−0.474242
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.0847	−1.45098	−0.725491	−	0.688232i	$-0.758387\pi$
$173$	−19.0847	−1.45098	−0.725491	+	0.688232i	$0.758387\pi$
$174$	0	0
$175$	4.58258i	0.346410i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 15.3430i	− 1.14679i	−0.819279	−	0.573396i	$-0.805626\pi$
$179$	− 15.3430i	− 1.14679i	0.819279	−	0.573396i	$-0.194374\pi$
$180$	0	0
$181$	− 6.20520i	− 0.461229i	−0.973045	−	0.230615i	$-0.925926\pi$
$181$	− 6.20520i	− 0.461229i	0.973045	−	0.230615i	$-0.0740736\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 7.89495i	− 0.580449i
$186$	0	0
$187$	0.190700	0.0139454
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.8039	−1.79475	−0.897374	−	0.441271i	$-0.854528\pi$
$191$	−24.8039	−1.79475	−0.897374	+	0.441271i	$0.854528\pi$
$192$	0	0
$193$	−20.7477	−1.49345	−0.746727	−	0.665131i	$-0.768376\pi$
$193$	−20.7477	−1.49345	−0.746727	+	0.665131i	$0.768376\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.0005	−1.42497	−0.712487	−	0.701686i	$-0.752431\pi$
$197$	−20.0005	−1.42497	−0.712487	+	0.701686i	$0.752431\pi$
$198$	0	0
$199$	0.834849i	0.0591808i	0.999562	+	0.0295904i	$0.00942030\pi$
$199$	0.834849i	0.0591808i	−0.999562	+	0.0295904i	$0.990580\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.03383i	0.353305i
$204$	0	0
$205$	32.2813i	2.25462i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.41875i	− 0.236480i
$210$	0	0
$211$	−20.9753	−1.44400	−0.722000	−	0.691893i	$-0.756777\pi$
$211$	−20.9753	−1.44400	−0.722000	+	0.691893i	$0.756777\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.06653	0.345534
$216$	0	0
$217$	−9.16515	−0.622171
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.29217	0.0869206
$222$	0	0
$223$	− 20.0000i	− 1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$	− 20.0000i	− 1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.60681i	0.239393i	0.992811	+	0.119696i	$0.0381921\pi$
$227$	3.60681i	0.239393i	−0.992811	+	0.119696i	$0.961808\pi$
$228$	0	0
$229$	− 14.9608i	− 0.988638i	−0.869281	−	0.494319i	$-0.835417\pi$
$229$	− 14.9608i	− 0.988638i	0.869281	−	0.494319i	$-0.164583\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.07107i	− 0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$	− 7.07107i	− 0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$	0	0
$235$	−17.5112	−1.14231
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.4328	0.804208	0.402104	−	0.915594i	$-0.368279\pi$
$239$	12.4328	0.804208	0.402104	+	0.915594i	$0.368279\pi$
$240$	0	0
$241$	−29.5826	−1.90558	−0.952791	−	0.303628i	$-0.901802\pi$
$241$	−29.5826	−1.90558	−0.952791	+	0.303628i	$0.901802\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.09557	0.197769
$246$	0	0
$247$	− 23.1652i	− 1.47396i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 23.4724i	− 1.48157i	−0.671744	−	0.740783i	$-0.734455\pi$
$251$	− 23.4724i	− 1.48157i	0.671744	−	0.740783i	$-0.265545\pi$
$252$	0	0
$253$	2.55040i	0.160342i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.6754i	− 1.04018i	−0.854111	−	0.520091i	$-0.825898\pi$
$257$	− 16.6754i	− 1.04018i	0.854111	−	0.520091i	$-0.174102\pi$
$258$	0	0
$259$	−2.55040	−0.158474
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.4993	1.07905	0.539526	−	0.841969i	$-0.318603\pi$
$263$	17.4993	1.07905	0.539526	+	0.841969i	$0.318603\pi$
$264$	0	0
$265$	26.7477	1.64310
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.3993	−1.30474	−0.652370	−	0.757901i	$-0.726225\pi$
$269$	−21.3993	−1.30474	−0.652370	+	0.757901i	$0.726225\pi$
$270$	0	0
$271$	13.1652i	0.799726i	0.916575	+	0.399863i	$0.130942\pi$
$271$	13.1652i	0.799726i	−0.916575	+	0.399863i	$0.869058\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.96073i	0.178539i
$276$	0	0
$277$	− 11.4967i	− 0.690770i	−0.938461	−	0.345385i	$-0.887748\pi$
$277$	− 11.4967i	− 0.690770i	0.938461	−	0.345385i	$-0.112252\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.66139i	0.457041i	0.973539	+	0.228520i	$0.0733887\pi$
$281$	7.66139i	0.457041i	−0.973539	+	0.228520i	$0.926611\pi$
$282$	0	0
$283$	17.3205	1.02960	0.514799	−	0.857311i	$-0.327867\pi$
$283$	17.3205	1.02960	0.514799	+	0.857311i	$0.327867\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.4282	0.615558
$288$	0	0
$289$	16.9129	0.994875
$290$	0	0
$291$	0	0
$292$	0	0
$293$	11.8711	0.693514	0.346757	−	0.937955i	$-0.387283\pi$
$293$	11.8711	0.693514	0.346757	+	0.937955i	$0.387283\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.2813i	0.999402i
$300$	0	0
$301$	− 1.63670i	− 0.0943379i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 21.4468i	− 1.22804i
$306$	0	0
$307$	8.94630	0.510593	0.255296	−	0.966863i	$-0.417827\pi$
$307$	8.94630	0.510593	0.255296	+	0.966863i	$0.417827\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.4468	1.21613	0.608067	−	0.793886i	$-0.291945\pi$
$311$	21.4468	1.21613	0.608067	+	0.793886i	$0.291945\pi$
$312$	0	0
$313$	−9.16515	−0.518045	−0.259022	−	0.965871i	$-0.583400\pi$
$313$	−9.16515	−0.518045	−0.259022	+	0.965871i	$0.583400\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.8318	−0.833036	−0.416518	−	0.909127i	$-0.636750\pi$
$317$	−14.8318	−0.833036	−0.416518	+	0.909127i	$0.636750\pi$
$318$	0	0
$319$	3.25227i	0.182092i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.56186i	0.0869041i
$324$	0	0
$325$	20.0616i	1.11282i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.65685i	0.311872i
$330$	0	0
$331$	−22.8027	−1.25335	−0.626675	−	0.779281i	$-0.715585\pi$
$331$	−22.8027	−1.25335	−0.626675	+	0.779281i	$0.715585\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.89495	0.431347
$336$	0	0
$337$	14.3303	0.780621	0.390311	−	0.920683i	$-0.372368\pi$
$337$	14.3303	0.780621	0.390311	+	0.920683i	$0.372368\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.92146	−0.320665
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5.27537i	− 0.283197i	−0.989924	−	0.141598i	$-0.954776\pi$
$347$	− 5.27537i	− 0.283197i	0.989924	−	0.141598i	$-0.0452242\pi$
$348$	0	0
$349$	21.1660i	1.13299i	0.824065	+	0.566495i	$0.191701\pi$
$349$	21.1660i	1.13299i	−0.824065	+	0.566495i	$0.808299\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 15.0276i	− 0.799840i	−0.916550	−	0.399920i	$-0.869038\pi$
$353$	− 15.0276i	− 0.799840i	0.916550	−	0.399920i	$-0.130962\pi$
$354$	0	0
$355$	3.46410	0.183855
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.2227	−1.48954	−0.744768	−	0.667324i	$-0.767440\pi$
$359$	−28.2227	−1.48954	−0.744768	+	0.667324i	$0.767440\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	29.6636	1.55266
$366$	0	0
$367$	19.1652i	1.00041i	0.865906	+	0.500206i	$0.166743\pi$
$367$	19.1652i	1.00041i	−0.865906	+	0.500206i	$0.833257\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 8.64064i	− 0.448600i
$372$	0	0
$373$	21.1660i	1.09593i	0.836500	+	0.547967i	$0.184598\pi$
$373$	21.1660i	1.09593i	−0.836500	+	0.547967i	$0.815402\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	22.0371i	1.13497i
$378$	0	0
$379$	29.7309	1.52717	0.763587	−	0.645705i	$-0.223436\pi$
$379$	29.7309	1.52717	0.763587	+	0.645705i	$0.223436\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	22.0371	1.12604	0.563021	−	0.826442i	$-0.309639\pi$
$383$	22.0371	1.12604	0.563021	+	0.826442i	$0.309639\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.8543	0.803843	0.401921	−	0.915674i	$-0.368342\pi$
$389$	15.8543	0.803843	0.401921	+	0.915674i	$0.368342\pi$
$390$	0	0
$391$	− 1.16515i	− 0.0589242i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	51.8438i	2.60855i
$396$	0	0
$397$	1.82740i	0.0917146i	0.998948	+	0.0458573i	$0.0146020\pi$
$397$	1.82740i	0.0917146i	−0.998948	+	0.0458573i	$0.985398\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 32.5269i	− 1.62432i	−0.583437	−	0.812158i	$-0.698293\pi$
$401$	− 32.5269i	− 1.62432i	0.583437	−	0.812158i	$-0.301707\pi$
$402$	0	0
$403$	−40.1232	−1.99868
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.64777	−0.0816771
$408$	0	0
$409$	−1.58258	−0.0782533	−0.0391267	−	0.999234i	$-0.512458\pi$
$409$	−1.58258	−0.0782533	−0.0391267	+	0.999234i	$0.512458\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	26.3303i	1.29250i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 10.0677i	− 0.491837i	−0.969291	−	0.245918i	$-0.920910\pi$
$419$	− 10.0677i	− 0.491837i	0.969291	−	0.245918i	$-0.0790895\pi$
$420$	0	0
$421$	6.01450i	0.293129i	0.989201	+	0.146564i	$0.0468216\pi$
$421$	6.01450i	0.293129i	−0.989201	+	0.146564i	$0.953178\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.35261i	− 0.0656112i
$426$	0	0
$427$	−6.92820	−0.335279
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.6323	−1.33100	−0.665501	−	0.746397i	$-0.731782\pi$
$431$	−27.6323	−1.33100	−0.665501	+	0.746397i	$0.731782\pi$
$432$	0	0
$433$	2.83485	0.136234	0.0681171	−	0.997677i	$-0.478301\pi$
$433$	2.83485	0.136234	0.0681171	+	0.997677i	$0.478301\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.8881	−0.999213
$438$	0	0
$439$	18.3303i	0.874858i	0.899253	+	0.437429i	$0.144111\pi$
$439$	18.3303i	0.874858i	−0.899253	+	0.437429i	$0.855889\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 10.7137i	− 0.509025i	−0.967070	−	0.254512i	$-0.918085\pi$
$443$	− 10.7137i	− 0.509025i	0.967070	−	0.254512i	$-0.0819150\pi$
$444$	0	0
$445$	− 32.2813i	− 1.53028i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.89949i	0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$	9.89949i	0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$	0	0
$451$	6.73750	0.317257
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.5518	0.635319
$456$	0	0
$457$	15.1652	0.709396	0.354698	−	0.934981i	$-0.384584\pi$
$457$	15.1652	0.709396	0.354698	+	0.934981i	$0.384584\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.55641	−0.445086	−0.222543	−	0.974923i	$-0.571436\pi$
$461$	−9.55641	−0.445086	−0.222543	+	0.974923i	$0.571436\pi$
$462$	0	0
$463$	− 2.41742i	− 0.112347i	−0.998421	−	0.0561736i	$-0.982110\pi$
$463$	− 2.41742i	− 0.112347i	0.998421	−	0.0561736i	$-0.0178900\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.8431i	− 0.871956i	−0.899957	−	0.435978i	$-0.856403\pi$
$467$	− 18.8431i	− 0.871956i	0.899957	−	0.435978i	$-0.143597\pi$
$468$	0	0
$469$	− 2.55040i	− 0.117767i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.05745i	− 0.0486214i
$474$	0	0
$475$	−24.2487	−1.11261
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.66594	−0.441648	−0.220824	−	0.975314i	$-0.570875\pi$
$479$	−9.66594	−0.441648	−0.220824	+	0.975314i	$0.570875\pi$
$480$	0	0
$481$	−11.1652	−0.509087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.48331	0.339800
$486$	0	0
$487$	14.3303i	0.649368i	0.945823	+	0.324684i	$0.105258\pi$
$487$	14.3303i	0.649368i	−0.945823	+	0.324684i	$0.894742\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 20.2420i	− 0.913509i	−0.889593	−	0.456754i	$-0.849012\pi$
$491$	− 20.2420i	− 0.913509i	0.889593	−	0.456754i	$-0.150988\pi$
$492$	0	0
$493$	− 1.48580i	− 0.0669171i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.11905i	− 0.0501963i
$498$	0	0
$499$	33.0043	1.47748	0.738738	−	0.673993i	$-0.235422\pi$
$499$	33.0043	1.47748	0.738738	+	0.673993i	$0.235422\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14.7325	0.656888	0.328444	−	0.944523i	$-0.393476\pi$
$503$	14.7325	0.656888	0.328444	+	0.944523i	$0.393476\pi$
$504$	0	0
$505$	55.0780	2.45094
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.07310	−0.0918884	−0.0459442	−	0.998944i	$-0.514630\pi$
$509$	−2.07310	−0.0918884	−0.0459442	+	0.998944i	$0.514630\pi$
$510$	0	0
$511$	− 9.58258i	− 0.423908i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6.19115i	− 0.272815i
$516$	0	0
$517$	3.65480i	0.160738i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.8750i	1.39647i	0.715869	+	0.698234i	$0.246031\pi$
$521$	31.8750i	1.39647i	−0.715869	+	0.698234i	$0.753969\pi$
$522$	0	0
$523$	−33.5764	−1.46819	−0.734097	−	0.679045i	$-0.762394\pi$
$523$	−33.5764	−1.46819	−0.734097	+	0.679045i	$0.762394\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.70522	0.117841
$528$	0	0
$529$	−7.41742	−0.322497
$530$	0	0
$531$	0	0
$532$	0	0
$533$	45.6527	1.97744
$534$	0	0
$535$	32.3303i	1.39776i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 0.646084i	− 0.0278288i
$540$	0	0
$541$	− 20.2523i	− 0.870715i	−0.900258	−	0.435357i	$-0.856622\pi$
$541$	− 20.2523i	− 0.870715i	0.900258	−	0.435357i	$-0.143378\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 16.3802i	− 0.701652i
$546$	0	0
$547$	−44.5010	−1.90273	−0.951363	−	0.308072i	$-0.900316\pi$
$547$	−44.5010	−1.90273	−0.951363	+	0.308072i	$0.900316\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−26.6365	−1.13475
$552$	0	0
$553$	16.7477	0.712186
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.9110	1.77583	0.887913	−	0.460011i	$-0.152154\pi$
$557$	41.9110	1.77583	0.887913	+	0.460011i	$0.152154\pi$
$558$	0	0
$559$	− 7.16515i	− 0.303054i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.3605i	1.86957i	0.355211	+	0.934786i	$0.384409\pi$
$563$	44.3605i	1.86957i	−0.355211	+	0.934786i	$0.615591\pi$
$564$	0	0
$565$	30.6446i	1.28923i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.25172i	0.345930i	0.984928	+	0.172965i	$0.0553348\pi$
$569$	8.25172i	0.345930i	−0.984928	+	0.172965i	$0.944665\pi$
$570$	0	0
$571$	44.8824	1.87827	0.939135	−	0.343547i	$-0.111628\pi$
$571$	44.8824	1.87827	0.939135	+	0.343547i	$0.111628\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.0896	0.754389
$576$	0	0
$577$	−13.1652	−0.548072	−0.274036	−	0.961719i	$-0.588359\pi$
$577$	−13.1652	−0.548072	−0.274036	+	0.961719i	$0.588359\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.50579	0.352880
$582$	0	0
$583$	− 5.58258i	− 0.231207i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.7421i	0.979942i	0.871739	+	0.489971i	$0.162993\pi$
$587$	23.7421i	0.979942i	−0.871739	+	0.489971i	$0.837007\pi$
$588$	0	0
$589$	− 48.4974i	− 1.99830i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 12.1992i	− 0.500961i	−0.968122	−	0.250481i	$-0.919411\pi$
$593$	− 12.1992i	− 0.500961i	0.968122	−	0.250481i	$-0.0805886\pi$
$594$	0	0
$595$	−0.913701	−0.0374581
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.95656	0.325096	0.162548	−	0.986701i	$-0.448029\pi$
$599$	7.95656	0.325096	0.162548	+	0.986701i	$0.448029\pi$
$600$	0	0
$601$	6.83485	0.278799	0.139400	−	0.990236i	$-0.455483\pi$
$601$	6.83485	0.278799	0.139400	+	0.990236i	$0.455483\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−32.7591	−1.33185
$606$	0	0
$607$	− 22.3303i	− 0.906359i	−0.891419	−	0.453180i	$-0.850290\pi$
$607$	− 22.3303i	− 0.906359i	0.891419	−	0.453180i	$-0.149710\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.7646i	1.00187i
$612$	0	0
$613$	0.381401i	0.0154046i	0.999970	+	0.00770232i	$0.00245175\pi$
$613$	0.381401i	0.0154046i	−0.999970	+	0.00770232i	$0.997548\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.0031i	1.48969i	0.667238	+	0.744845i	$0.267476\pi$
$617$	37.0031i	1.48969i	−0.667238	+	0.744845i	$0.732524\pi$
$618$	0	0
$619$	41.9506	1.68614	0.843069	−	0.537806i	$-0.180747\pi$
$619$	41.9506	1.68614	0.843069	+	0.537806i	$0.180747\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.4282	−0.417798
$624$	0	0
$625$	−26.9129	−1.07652
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.752785	0.0300155
$630$	0	0
$631$	− 31.9129i	− 1.27043i	−0.772335	−	0.635216i	$-0.780911\pi$
$631$	− 31.9129i	− 1.27043i	0.772335	−	0.635216i	$-0.219089\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	42.0459i	1.66854i
$636$	0	0
$637$	− 4.37780i	− 0.173455i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.7468i	1.05644i	0.849108	+	0.528219i	$0.177140\pi$
$641$	26.7468i	1.05644i	−0.849108	+	0.528219i	$0.822860\pi$
$642$	0	0
$643$	41.7599	1.64685	0.823425	−	0.567425i	$-0.192060\pi$
$643$	41.7599	1.64685	0.823425	+	0.567425i	$0.192060\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.3508	−1.31116	−0.655578	−	0.755128i	$-0.727575\pi$
$647$	−33.3508	−1.31116	−0.655578	+	0.755128i	$0.727575\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.07878	−0.277014	−0.138507	−	0.990361i	$-0.544230\pi$
$653$	−7.07878	−0.277014	−0.138507	+	0.990361i	$0.544230\pi$
$654$	0	0
$655$	11.1652i	0.436259i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.1185i	0.939524i	0.882793	+	0.469762i	$0.155660\pi$
$659$	24.1185i	0.939524i	−0.882793	+	0.469762i	$0.844340\pi$
$660$	0	0
$661$	19.3386i	0.752185i	0.926582	+	0.376092i	$0.122732\pi$
$661$	19.3386i	0.752185i	−0.926582	+	0.376092i	$0.877268\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.3802i	0.635198i
$666$	0	0
$667$	19.8709	0.769405
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.47620	−0.172802
$672$	0	0
$673$	−39.9129	−1.53853	−0.769264	−	0.638931i	$-0.779377\pi$
$673$	−39.9129	−1.53853	−0.769264	+	0.638931i	$0.779377\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.2534	0.932132	0.466066	−	0.884750i	$-0.345671\pi$
$677$	24.2534	0.932132	0.466066	+	0.884750i	$0.345671\pi$
$678$	0	0
$679$	− 2.41742i	− 0.0927722i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 49.1528i	− 1.88078i	−0.340099	−	0.940390i	$-0.610461\pi$
$683$	− 49.1528i	− 1.88078i	0.340099	−	0.940390i	$-0.389539\pi$
$684$	0	0
$685$	− 62.0122i	− 2.36937i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 37.8270i	− 1.44109i
$690$	0	0
$691$	36.8498	1.40183	0.700917	−	0.713243i	$-0.252775\pi$
$691$	36.8498	1.40183	0.700917	+	0.713243i	$0.252775\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.1036	1.02810
$696$	0	0
$697$	−3.07803	−0.116589
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.37095	0.316166	0.158083	−	0.987426i	$-0.449469\pi$
$701$	8.37095	0.316166	0.158083	+	0.987426i	$0.449469\pi$
$702$	0	0
$703$	− 13.4955i	− 0.508991i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.7925i	− 0.669156i
$708$	0	0
$709$	− 2.01810i	− 0.0757914i	−0.999282	−	0.0378957i	$-0.987935\pi$
$709$	− 2.01810i	− 0.0757914i	0.999282	−	0.0378957i	$-0.0120655\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.1792i	1.35492i
$714$	0	0
$715$	8.75560	0.327441
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.3032	−1.57764	−0.788822	−	0.614622i	$-0.789308\pi$
$719$	−42.3032	−1.57764	−0.788822	+	0.614622i	$0.789308\pi$
$720$	0	0
$721$	−2.00000	−0.0744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	23.0679	0.856720
$726$	0	0
$727$	− 35.4955i	− 1.31645i	−0.752820	−	0.658227i	$-0.771307\pi$
$727$	− 35.4955i	− 1.31645i	0.752820	−	0.658227i	$-0.228693\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.483094i	0.0178679i
$732$	0	0
$733$	− 28.8172i	− 1.06439i	−0.846622	−	0.532194i	$-0.821368\pi$
$733$	− 28.8172i	− 1.06439i	0.846622	−	0.532194i	$-0.178632\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.64777i	− 0.0606965i
$738$	0	0
$739$	2.55040	0.0938180	0.0469090	−	0.998899i	$-0.485063\pi$
$739$	2.55040	0.0938180	0.0469090	+	0.998899i	$0.485063\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.17650	0.0798479	0.0399239	−	0.999203i	$-0.487288\pi$
$743$	2.17650	0.0798479	0.0399239	+	0.999203i	$0.487288\pi$
$744$	0	0
$745$	14.7477	0.540315
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.4440	0.381617
$750$	0	0
$751$	16.8348i	0.614312i	0.951659	+	0.307156i	$0.0993774\pi$
$751$	16.8348i	0.614312i	−0.951659	+	0.307156i	$0.900623\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.9666i	0.544691i
$756$	0	0
$757$	− 20.5939i	− 0.748498i	−0.927328	−	0.374249i	$-0.877900\pi$
$757$	− 20.5939i	− 0.748498i	0.927328	−	0.374249i	$-0.122100\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	40.3603i	1.46306i	0.681810	+	0.731529i	$0.261193\pi$
$761$	40.3603i	1.46306i	−0.681810	+	0.731529i	$0.738807\pi$
$762$	0	0
$763$	−5.29150	−0.191565
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28.6129	−1.02914	−0.514568	−	0.857450i	$-0.672048\pi$
$773$	−28.6129	−1.02914	−0.514568	+	0.857450i	$0.672048\pi$
$774$	0	0
$775$	42.0000i	1.50868i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	55.1809i	1.97706i
$780$	0	0
$781$	− 0.723000i	− 0.0258710i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 15.7899i	− 0.563566i
$786$	0	0
$787$	22.2306	0.792436	0.396218	−	0.918157i	$-0.370323\pi$
$787$	22.2306	0.792436	0.396218	+	0.918157i	$0.370323\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.89949	0.351986
$792$	0	0
$793$	−30.3303	−1.07706
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.0404	1.77252	0.886261	−	0.463186i	$-0.153294\pi$
$797$	50.0404	1.77252	0.886261	+	0.463186i	$0.153294\pi$
$798$	0	0
$799$	− 1.66970i	− 0.0590696i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 6.19115i	− 0.218481i
$804$	0	0
$805$	− 12.2197i	− 0.430688i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.84205i	0.310870i	0.987846	+	0.155435i	$0.0496779\pi$
$809$	8.84205i	0.310870i	−0.987846	+	0.155435i	$0.950322\pi$
$810$	0	0
$811$	−26.2668	−0.922353	−0.461176	−	0.887309i	$-0.652572\pi$
$811$	−26.2668	−0.922353	−0.461176	+	0.887309i	$0.652572\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−73.4159	−2.57165
$816$	0	0
$817$	8.66061	0.302996
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.74166	−0.130585	−0.0652924	−	0.997866i	$-0.520798\pi$
$821$	−3.74166	−0.130585	−0.0652924	+	0.997866i	$0.520798\pi$
$822$	0	0
$823$	− 43.9129i	− 1.53071i	−0.643610	−	0.765353i	$-0.722564\pi$
$823$	− 43.9129i	− 1.53071i	0.643610	−	0.765353i	$-0.277436\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 47.8606i	− 1.66428i	−0.554568	−	0.832138i	$-0.687117\pi$
$827$	− 47.8606i	− 1.66428i	0.554568	−	0.832138i	$-0.312883\pi$
$828$	0	0
$829$	27.3712i	0.950642i	0.879813	+	0.475321i	$0.157668\pi$
$829$	27.3712i	0.950642i	−0.879813	+	0.475321i	$0.842332\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.295164i	0.0102268i
$834$	0	0
$835$	15.6838	0.542761
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.0462	−0.899214	−0.449607	−	0.893227i	$-0.648436\pi$
$839$	−26.0462	−0.899214	−0.449607	+	0.893227i	$0.648436\pi$
$840$	0	0
$841$	−3.66061	−0.126228
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.0847	0.656533
$846$	0	0
$847$	10.5826i	0.363622i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.0677i	0.345115i
$852$	0	0
$853$	− 7.30960i	− 0.250276i	−0.992139	−	0.125138i	$-0.960063\pi$
$853$	− 7.30960i	− 0.250276i	0.992139	−	0.125138i	$-0.0399374\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 19.5038i	− 0.666238i	−0.942885	−	0.333119i	$-0.891899\pi$
$857$	− 19.5038i	− 0.666238i	0.942885	−	0.333119i	$-0.108101\pi$
$858$	0	0
$859$	27.5221	0.939042	0.469521	−	0.882921i	$-0.344427\pi$
$859$	27.5221	0.939042	0.469521	+	0.882921i	$0.344427\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.1947	−0.347030	−0.173515	−	0.984831i	$-0.555513\pi$
$863$	−10.1947	−0.347030	−0.173515	+	0.984831i	$0.555513\pi$
$864$	0	0
$865$	59.0780	2.00871
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.8204	0.367058
$870$	0	0
$871$	− 11.1652i	− 0.378317i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.29217i	0.0436832i
$876$	0	0
$877$	− 35.0224i	− 1.18262i	−0.806443	−	0.591311i	$-0.798610\pi$
$877$	− 35.0224i	− 1.18262i	0.806443	−	0.591311i	$-0.201390\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.6089i	0.391113i	0.980692	+	0.195556i	$0.0626513\pi$
$881$	11.6089i	0.391113i	−0.980692	+	0.195556i	$0.937349\pi$
$882$	0	0
$883$	−24.2487	−0.816034	−0.408017	−	0.912974i	$-0.633780\pi$
$883$	−24.2487	−0.816034	−0.408017	+	0.912974i	$0.633780\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.3032	−1.42040	−0.710201	−	0.703999i	$-0.751396\pi$
$887$	−42.3032	−1.42040	−0.710201	+	0.703999i	$0.751396\pi$
$888$	0	0
$889$	13.5826	0.455545
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−29.9333	−1.00168
$894$	0	0
$895$	47.4955i	1.58760i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46.1358i	1.53871i
$900$	0	0
$901$	2.55040i	0.0849662i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.2087i	0.638518i
$906$	0	0
$907$	−34.8317	−1.15657	−0.578284	−	0.815835i	$-0.696277\pi$
$907$	−34.8317	−1.15657	−0.578284	+	0.815835i	$0.696277\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.18558	0.204937	0.102469	−	0.994736i	$-0.467326\pi$
$911$	6.18558	0.204937	0.102469	+	0.994736i	$0.467326\pi$
$912$	0	0
$913$	5.49545	0.181873
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.60681	0.119107
$918$	0	0
$919$	31.1652i	1.02804i	0.857777	+	0.514022i	$0.171845\pi$
$919$	31.1652i	1.02804i	−0.857777	+	0.514022i	$0.828155\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 4.89898i	− 0.161252i
$924$	0	0
$925$	11.6874i	0.384280i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.36169i	0.175911i	0.996124	+	0.0879557i	$0.0280334\pi$
$929$	5.36169i	0.175911i	−0.996124	+	0.0879557i	$0.971967\pi$
$930$	0	0
$931$	5.29150	0.173422
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.590327	−0.0193058
$936$	0	0
$937$	−15.4955	−0.506214	−0.253107	−	0.967438i	$-0.581453\pi$
$937$	−15.4955	−0.506214	−0.253107	+	0.967438i	$0.581453\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−23.9837	−0.781845	−0.390922	−	0.920424i	$-0.627844\pi$
$941$	−23.9837	−0.781845	−0.390922	+	0.920424i	$0.627844\pi$
$942$	0	0
$943$	− 41.1652i	− 1.34052i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.7821i	1.74768i	0.486211	+	0.873841i	$0.338379\pi$
$947$	53.7821i	1.74768i	−0.486211	+	0.873841i	$0.661621\pi$
$948$	0	0
$949$	− 41.9506i	− 1.36177i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	40.4219i	1.30939i	0.755892	+	0.654696i	$0.227203\pi$
$953$	40.4219i	1.30939i	−0.755892	+	0.654696i	$0.772797\pi$
$954$	0	0
$955$	76.7823	2.48462
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.0325	−0.646885
$960$	0	0
$961$	−53.0000	−1.70968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	64.2261	2.06751
$966$	0	0
$967$	51.0780i	1.64256i	0.570526	+	0.821279i	$0.306739\pi$
$967$	51.0780i	1.64256i	−0.570526	+	0.821279i	$0.693261\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.4615i	1.26638i	0.773996	+	0.633190i	$0.218255\pi$
$971$	39.4615i	1.26638i	−0.773996	+	0.633190i	$0.781745\pi$
$972$	0	0
$973$	− 8.75560i	− 0.280692i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.8981i	1.43642i	0.695828	+	0.718208i	$0.255037\pi$
$977$	44.8981i	1.43642i	−0.695828	+	0.718208i	$0.744963\pi$
$978$	0	0
$979$	−6.73750	−0.215332
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−50.1982	−1.60107	−0.800536	−	0.599284i	$-0.795452\pi$
$983$	−50.1982	−1.60107	−0.800536	+	0.599284i	$0.795452\pi$
$984$	0	0
$985$	61.9129	1.97271
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.46084	−0.205443
$990$	0	0
$991$	− 20.0000i	− 0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$	− 20.0000i	− 0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.58434i	− 0.0819289i
$996$	0	0
$997$	48.8788i	1.54801i	0.633181	+	0.774004i	$0.281749\pi$
$997$	48.8788i	1.54801i	−0.633181	+	0.774004i	$0.718251\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.j.e.2591.4	yes	16
3.2	odd	2	inner	4032.2.j.e.2591.16	yes	16
4.3	odd	2	inner	4032.2.j.e.2591.2	yes	16
8.3	odd	2	inner	4032.2.j.e.2591.13	yes	16
8.5	even	2	inner	4032.2.j.e.2591.15	yes	16
12.11	even	2	inner	4032.2.j.e.2591.14	yes	16
24.5	odd	2	inner	4032.2.j.e.2591.3	yes	16
24.11	even	2	inner	4032.2.j.e.2591.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4032.2.j.e.2591.1	✓	16	24.11	even	2	inner
4032.2.j.e.2591.2	yes	16	4.3	odd	2	inner
4032.2.j.e.2591.3	yes	16	24.5	odd	2	inner
4032.2.j.e.2591.4	yes	16	1.1	even	1	trivial
4032.2.j.e.2591.13	yes	16	8.3	odd	2	inner
4032.2.j.e.2591.14	yes	16	12.11	even	2	inner
4032.2.j.e.2591.15	yes	16	8.5	even	2	inner
4032.2.j.e.2591.16	yes	16	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.09557 q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-3.09557 q^{5} +1.00000i q^{7} +0.646084i q^{11} +4.37780i q^{13} -0.295164i q^{17} -5.29150 q^{19} +3.94748 q^{23} +4.58258 q^{25} +5.03383 q^{29} +9.16515i q^{31} -3.09557i q^{35} +2.55040i q^{37} -10.4282i q^{41} -1.63670 q^{43} +5.65685 q^{47} -1.00000 q^{49} -8.64064 q^{53} -2.00000i q^{55} +6.92820i q^{61} -13.5518i q^{65} -2.55040 q^{67} -1.11905 q^{71} -9.58258 q^{73} -0.646084 q^{77} -16.7477i q^{79} -8.50579i q^{83} +0.913701i q^{85} +10.4282i q^{89} -4.37780 q^{91} +16.3802 q^{95} -2.41742 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{49} - 80 q^{73} - 112 q^{97}+O(q^{100}) \) 16 * q - 16 * q^49 - 80 * q^73 - 112 * q^97

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