LMFDB - Embedded newform 75.7.d.a.74.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,7,Mod(74,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("75.74");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	75.d (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:	$17.2540562715$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			74.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.74
Dual form			75.7.d.a.74.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} -64.0000 q^{4} +286.000i q^{7} -729.000 q^{9} +O(q^{10})$	$q-27.0000i q^{3} -64.0000 q^{4} +286.000i q^{7} -729.000 q^{9} +1728.00i q^{12} +506.000i q^{13} +4096.00 q^{16} +10582.0 q^{19} +7722.00 q^{21} +19683.0i q^{27} -18304.0i q^{28} +35282.0 q^{31} +46656.0 q^{36} +89206.0i q^{37} +13662.0 q^{39} +111386. i q^{43} -110592. i q^{48} +35853.0 q^{49} -32384.0i q^{52} -285714. i q^{57} -420838. q^{61} -208494. i q^{63} -262144. q^{64} -172874. i q^{67} +638066. i q^{73} -677248. q^{76} +204622. q^{79} +531441. q^{81} -494208. q^{84} -144716. q^{91} -952614. i q^{93} +56446.0i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 1458 q^{9}+O(q^{10}) $ 2 * q - 128 * q^4 - 1458 * q^9	$ 2 q - 128 q^{4} - 1458 q^{9} + 8192 q^{16} + 21164 q^{19} + 15444 q^{21} + 70564 q^{31} + 93312 q^{36} + 27324 q^{39} + 71706 q^{49} - 841676 q^{61} - 524288 q^{64} - 1354496 q^{76} + 409244 q^{79} + 1062882 q^{81} - 988416 q^{84} - 289432 q^{91}+O(q^{100}) $ 2 * q - 128 * q^4 - 1458 * q^9 + 8192 * q^16 + 21164 * q^19 + 15444 * q^21 + 70564 * q^31 + 93312 * q^36 + 27324 * q^39 + 71706 * q^49 - 841676 * q^61 - 524288 * q^64 - 1354496 * q^76 + 409244 * q^79 + 1062882 * q^81 - 988416 * q^84 - 289432 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$-1$	$-1$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	− 27.0000i	− 1.00000i
$3$	− 27.0000i	− 1.00000i
$4$	−64.0000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	286.000i	0.833819i	0.908948	+	0.416910i	$0.136887\pi$
$7$	286.000i	0.833819i	−0.908948	+	0.416910i	$0.863113\pi$
$8$	0	0
$9$	−729.000	−1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	1728.00i	1.00000i
$13$	506.000i	0.230314i	0.993347	+	0.115157i	$0.0367371\pi$
$13$	506.000i	0.230314i	−0.993347	+	0.115157i	$0.963263\pi$
$14$	0	0
$15$	0	0
$16$	4096.00	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	10582.0	1.54279	0.771395	−	0.636356i	$-0.219559\pi$
$19$	10582.0	1.54279	0.771395	+	0.636356i	$0.219559\pi$
$20$	0	0
$21$	7722.00	0.833819
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	19683.0i	1.00000i
$28$	− 18304.0i	− 0.833819i
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	35282.0	1.18432	0.592159	−	0.805821i	$-0.298276\pi$
$31$	35282.0	1.18432	0.592159	+	0.805821i	$0.298276\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	46656.0	1.00000
$37$	89206.0i	1.76112i	0.473935	+	0.880560i	$0.342833\pi$
$37$	89206.0i	1.76112i	−0.473935	+	0.880560i	$0.657167\pi$
$38$	0	0
$39$	13662.0	0.230314
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	111386.i	1.40096i	0.713673	+	0.700479i	$0.247030\pi$
$43$	111386.i	1.40096i	−0.713673	+	0.700479i	$0.752970\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 110592.i	− 1.00000i
$49$	35853.0	0.304745
$50$	0	0
$51$	0	0
$52$	− 32384.0i	− 0.230314i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 285714.i	− 1.54279i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−420838.	−1.85407	−0.927034	−	0.374978i	$-0.877650\pi$
$61$	−420838.	−1.85407	−0.927034	+	0.374978i	$0.877650\pi$
$62$	0	0
$63$	− 208494.i	− 0.833819i
$64$	−262144.	−1.00000
$65$	0	0
$66$	0	0
$67$	− 172874.i	− 0.574785i	−0.957813	−	0.287392i	$-0.907212\pi$
$67$	− 172874.i	− 0.574785i	0.957813	−	0.287392i	$-0.0927884\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	638066.i	1.64020i	0.572220	+	0.820100i	$0.306082\pi$
$73$	638066.i	1.64020i	−0.572220	+	0.820100i	$0.693918\pi$
$74$	0	0
$75$	0	0
$76$	−677248.	−1.54279
$77$	0	0
$78$	0	0
$79$	204622.	0.415022	0.207511	−	0.978233i	$-0.433464\pi$
$79$	204622.	0.415022	0.207511	+	0.978233i	$0.433464\pi$
$80$	0	0
$81$	531441.	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−494208.	−0.833819
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−144716.	−0.192040
$92$	0	0
$93$	− 952614.i	− 1.18432i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	56446.0i	0.0618469i	0.999522	+	0.0309235i	$0.00984481\pi$
$97$	56446.0i	0.0618469i	−0.999522	+	0.0309235i	$0.990155\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1.12695e6i	1.03132i	0.856795	+	0.515658i	$0.172452\pi$
$103$	1.12695e6i	1.03132i	−0.856795	+	0.515658i	$0.827548\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 1.25971e6i	− 1.00000i
$109$	2.17274e6	1.67776	0.838878	−	0.544320i	$-0.183212\pi$
$109$	2.17274e6	1.67776	0.838878	+	0.544320i	$0.183212\pi$
$110$	0	0
$111$	2.40856e6	1.76112
$112$	1.17146e6i	0.833819i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 368874.i	− 0.230314i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.77156e6	1.00000
$122$	0	0
$123$	0	0
$124$	−2.25805e6	−1.18432
$125$	0	0
$126$	0	0
$127$	3.95237e6i	1.92951i	0.263158	+	0.964753i	$0.415236\pi$
$127$	3.95237e6i	1.92951i	−0.263158	+	0.964753i	$0.584764\pi$
$128$	0	0
$129$	3.00742e6	1.40096
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	3.02645e6i	1.28641i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	−1.26454e6	−0.470855	−0.235428	−	0.971892i	$-0.575649\pi$
$139$	−1.26454e6	−0.470855	−0.235428	+	0.971892i	$0.575649\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	−2.98598e6	−1.00000
$145$	0	0
$146$	0	0
$147$	− 968031.i	− 0.304745i
$148$	− 5.70918e6i	− 1.76112i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−3.83040e6	−1.11253	−0.556267	−	0.831004i	$-0.687767\pi$
$151$	−3.83040e6	−1.11253	−0.556267	+	0.831004i	$0.687767\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−874368.	−0.230314
$157$	− 7.08271e6i	− 1.83021i	−0.403216	−	0.915105i	$-0.632108\pi$
$157$	− 7.08271e6i	− 1.83021i	0.403216	−	0.915105i	$-0.367892\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.89851e6i	0.669285i	0.942345	+	0.334643i	$0.108616\pi$
$163$	2.89851e6i	0.669285i	−0.942345	+	0.334643i	$0.891384\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	4.57077e6	0.946955
$170$	0	0
$171$	−7.71428e6	−1.54279
$172$	− 7.12870e6i	− 1.40096i
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	98282.0	0.0165744	0.00828721	−	0.999966i	$-0.497362\pi$
$181$	98282.0	0.0165744	0.00828721	+	0.999966i	$0.497362\pi$
$182$	0	0
$183$	1.13626e7i	1.85407i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−5.62934e6	−0.833819
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	7.07789e6i	1.00000i
$193$	− 1.30556e7i	− 1.81604i	−0.418927	−	0.908020i	$-0.637594\pi$
$193$	− 1.30556e7i	− 1.81604i	0.418927	−	0.908020i	$-0.362406\pi$
$194$	0	0
$195$	0	0
$196$	−2.29459e6	−0.304745
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−1.16545e7	−1.47888	−0.739442	−	0.673220i	$-0.764911\pi$
$199$	−1.16545e7	−1.47888	−0.739442	+	0.673220i	$0.764911\pi$
$200$	0	0
$201$	−4.66760e6	−0.574785
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	2.07258e6i	0.230314i
$209$	0	0
$210$	0	0
$211$	1.75972e7	1.87325	0.936624	−	0.350336i	$-0.113933\pi$
$211$	1.75972e7	1.87325	0.936624	+	0.350336i	$0.113933\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.00907e7i	0.987507i
$218$	0	0
$219$	1.72278e7	1.64020
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 1.18107e7i	− 1.06503i	−0.846420	−	0.532516i	$-0.821247\pi$
$223$	− 1.18107e7i	− 1.06503i	0.846420	−	0.532516i	$-0.178753\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	1.82857e7i	1.54279i
$229$	4.07282e6	0.339148	0.169574	−	0.985517i	$-0.445761\pi$
$229$	4.07282e6	0.339148	0.169574	+	0.985517i	$0.445761\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 5.52479e6i	− 0.415022i
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	2.64398e7	1.88889	0.944447	−	0.328663i	$-0.106598\pi$
$241$	2.64398e7	1.88889	0.944447	+	0.328663i	$0.106598\pi$
$242$	0	0
$243$	− 1.43489e7i	− 1.00000i
$244$	2.69336e7	1.85407
$245$	0	0
$246$	0	0
$247$	5.35449e6i	0.355326i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	1.33436e7i	0.833819i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.67772e7	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	−2.55129e7	−1.46846
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	1.10639e7i	0.574785i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−3.91457e7	−1.96687	−0.983436	−	0.181258i	$-0.941983\pi$
$271$	−3.91457e7	−1.96687	−0.983436	+	0.181258i	$0.941983\pi$
$272$	0	0
$273$	3.90733e6i	0.192040i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.62670e7i	1.23586i	0.786232	+	0.617932i	$0.212029\pi$
$277$	2.62670e7i	1.23586i	−0.786232	+	0.617932i	$0.787971\pi$
$278$	0	0
$279$	−2.57206e7	−1.18432
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 1.39704e7i	− 0.616380i	−0.951325	−	0.308190i	$-0.900277\pi$
$283$	− 1.39704e7i	− 0.616380i	0.951325	−	0.308190i	$-0.0997233\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.41376e7	−1.00000
$290$	0	0
$291$	1.52404e6	0.0618469
$292$	− 4.08362e7i	− 1.64020i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.18564e7	−1.16815
$302$	0	0
$303$	0	0
$304$	4.33439e7	1.54279
$305$	0	0
$306$	0	0
$307$	− 5.53407e7i	− 1.91262i	−0.292348	−	0.956312i	$-0.594436\pi$
$307$	− 5.53407e7i	− 1.91262i	0.292348	−	0.956312i	$-0.405564\pi$
$308$	0	0
$309$	3.04275e7	1.03132
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	3.88715e7i	1.26765i	0.773478	+	0.633824i	$0.218515\pi$
$313$	3.88715e7i	1.26765i	−0.773478	+	0.633824i	$0.781485\pi$
$314$	0	0
$315$	0	0
$316$	−1.30958e7	−0.415022
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−3.40122e7	−1.00000
$325$	0	0
$326$	0	0
$327$	− 5.86640e7i	− 1.67776i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.15453e7	−1.97286	−0.986432	−	0.164169i	$-0.947506\pi$
$331$	−7.15453e7	−1.97286	−0.986432	+	0.164169i	$0.947506\pi$
$332$	0	0
$333$	− 6.50312e7i	− 1.76112i
$334$	0	0
$335$	0	0
$336$	3.16293e7	0.833819
$337$	5.22406e7i	1.36496i	0.730906	+	0.682478i	$0.239098\pi$
$337$	5.22406e7i	1.36496i	−0.730906	+	0.682478i	$0.760902\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.39016e7i	1.08792i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−5.69263e7	−1.33917	−0.669586	−	0.742734i	$-0.733529\pi$
$349$	−5.69263e7	−1.33917	−0.669586	+	0.742734i	$0.733529\pi$
$350$	0	0
$351$	−9.95960e6	−0.230314
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	6.49328e7	1.38020
$362$	0	0
$363$	− 4.78321e7i	− 1.00000i
$364$	9.26182e6	0.192040
$365$	0	0
$366$	0	0
$367$	8.00261e7i	1.61895i	0.587154	+	0.809475i	$0.300248\pi$
$367$	8.00261e7i	1.61895i	−0.587154	+	0.809475i	$0.699752\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	6.09673e7i	1.18432i
$373$	4.87323e7i	0.939053i	0.882918	+	0.469527i	$0.155575\pi$
$373$	4.87323e7i	0.939053i	−0.882918	+	0.469527i	$0.844425\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.51948e7	0.646489	0.323245	−	0.946315i	$-0.395226\pi$
$379$	3.51948e7	0.646489	0.323245	+	0.946315i	$0.395226\pi$
$380$	0	0
$381$	1.06714e8	1.92951
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.12004e7i	− 1.40096i
$388$	− 3.61254e6i	− 0.0618469i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.23725e8i	1.97737i	0.150013	+	0.988684i	$0.452068\pi$
$397$	1.23725e8i	1.97737i	−0.150013	+	0.988684i	$0.547932\pi$
$398$	0	0
$399$	8.17142e7	1.28641
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	1.78527e7i	0.272765i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.88393e7	1.00616	0.503080	−	0.864240i	$-0.332200\pi$
$409$	6.88393e7	1.00616	0.503080	+	0.864240i	$0.332200\pi$
$410$	0	0
$411$	0	0
$412$	− 7.21245e7i	− 1.03132i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.41425e7i	0.470855i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	1.44474e8	1.93617	0.968086	−	0.250620i	$-0.0806343\pi$
$421$	1.44474e8	1.93617	0.968086	+	0.250620i	$0.0806343\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.20360e8i	− 1.54596i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	8.06216e7i	1.00000i
$433$	− 1.55657e8i	− 1.91737i	−0.284469	−	0.958685i	$-0.591817\pi$
$433$	− 1.55657e8i	− 1.91737i	0.284469	−	0.958685i	$-0.408183\pi$
$434$	0	0
$435$	0	0
$436$	−1.39055e8	−1.67776
$437$	0	0
$438$	0	0
$439$	−5.35167e7	−0.632552	−0.316276	−	0.948667i	$-0.602433\pi$
$439$	−5.35167e7	−0.632552	−0.316276	+	0.948667i	$0.602433\pi$
$440$	0	0
$441$	−2.61368e7	−0.304745
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−1.54148e8	−1.76112
$445$	0	0
$446$	0	0
$447$	0	0
$448$	− 7.49732e7i	− 0.833819i
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	1.03421e8i	1.11253i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.93439e7i	0.307446i	0.988114	+	0.153723i	$0.0491264\pi$
$457$	2.93439e7i	0.307446i	−0.988114	+	0.153723i	$0.950874\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	1.92743e8i	1.94194i	0.239209	+	0.970968i	$0.423112\pi$
$463$	1.92743e8i	1.94194i	−0.239209	+	0.970968i	$0.576888\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	2.36079e7i	0.230314i
$469$	4.94420e7	0.479267
$470$	0	0
$471$	−1.91233e8	−1.83021
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	−4.51382e7	−0.405611
$482$	0	0
$483$	0	0
$484$	−1.13380e8	−1.00000
$485$	0	0
$486$	0	0
$487$	− 2.05807e8i	− 1.78186i	−0.454139	−	0.890931i	$-0.650053\pi$
$487$	− 2.05807e8i	− 1.78186i	0.454139	−	0.890931i	$-0.349947\pi$
$488$	0	0
$489$	7.82597e7	0.669285
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	1.44515e8	1.18432
$497$	0	0
$498$	0	0
$499$	1.94045e7	0.156171	0.0780856	−	0.996947i	$-0.475119\pi$
$499$	1.94045e7	0.156171	0.0780856	+	0.996947i	$0.475119\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.23411e8i	− 0.946955i
$508$	− 2.52951e8i	− 1.92951i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−1.82487e8	−1.36763
$512$	0	0
$513$	2.08286e8i	1.54279i
$514$	0	0
$515$	0	0
$516$	−1.92475e8	−1.40096
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 1.41150e8i	− 0.986677i	−0.869837	−	0.493338i	$-0.835776\pi$
$523$	− 1.41150e8i	− 0.986677i	0.869837	−	0.493338i	$-0.164224\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−1.48036e8	−1.00000
$530$	0	0
$531$	0	0
$532$	− 1.93693e8i	− 1.28641i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.97611e8	1.24801	0.624006	−	0.781419i	$-0.285504\pi$
$541$	1.97611e8	1.24801	0.624006	+	0.781419i	$0.285504\pi$
$542$	0	0
$543$	− 2.65361e6i	− 0.0165744i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.24645e8i	1.98357i	0.127929	+	0.991783i	$0.459167\pi$
$547$	3.24645e8i	1.98357i	−0.127929	+	0.991783i	$0.540833\pi$
$548$	0	0
$549$	3.06791e8	1.85407
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5.85219e7i	0.346053i
$554$	0	0
$555$	0	0
$556$	8.09304e7	0.470855
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−5.63613e7	−0.322660
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.51992e8i	0.833819i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	1.71111e8	0.919112	0.459556	−	0.888149i	$-0.348009\pi$
$571$	1.71111e8	0.919112	0.459556	+	0.888149i	$0.348009\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.91103e8	1.00000
$577$	7.05560e7i	0.367288i	0.982993	+	0.183644i	$0.0587893\pi$
$577$	7.05560e7i	0.367288i	−0.982993	+	0.183644i	$0.941211\pi$
$578$	0	0
$579$	−3.52502e8	−1.81604
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	6.19540e7i	0.304745i
$589$	3.73354e8	1.82715
$590$	0	0
$591$	0	0
$592$	3.65388e8i	1.76112i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.14671e8i	1.47888i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	4.24444e8	1.95522	0.977612	−	0.210415i	$-0.0674816\pi$
$601$	4.24444e8	1.95522	0.977612	+	0.210415i	$0.0674816\pi$
$602$	0	0
$603$	1.26025e8i	0.574785i
$604$	2.45145e8	1.11253
$605$	0	0
$606$	0	0
$607$	− 3.60399e8i	− 1.61145i	−0.592287	−	0.805727i	$-0.701775\pi$
$607$	− 3.60399e8i	− 1.61145i	0.592287	−	0.805727i	$-0.298225\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 1.58281e8i	− 0.687142i	−0.939127	−	0.343571i	$-0.888363\pi$
$613$	− 1.58281e8i	− 0.687142i	0.939127	−	0.343571i	$-0.111637\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−2.36189e8	−0.995836	−0.497918	−	0.867224i	$-0.665902\pi$
$619$	−2.36189e8	−0.995836	−0.497918	+	0.867224i	$0.665902\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	5.59596e7	0.230314
$625$	0	0
$626$	0	0
$627$	0	0
$628$	4.53294e8i	1.83021i
$629$	0	0
$630$	0	0
$631$	−4.98900e8	−1.98575	−0.992876	−	0.119152i	$-0.961982\pi$
$631$	−4.98900e8	−1.98575	−0.992876	+	0.119152i	$0.961982\pi$
$632$	0	0
$633$	− 4.75123e8i	− 1.87325i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.81416e7i	0.0701872i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 3.58510e8i	− 1.34855i	−0.738479	−	0.674277i	$-0.764456\pi$
$643$	− 3.58510e8i	− 1.34855i	0.738479	−	0.674277i	$-0.235544\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.72448e8	0.987507
$652$	− 1.85504e8i	− 0.669285i
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.65150e8i	− 1.64020i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1.58097e8	−0.547419	−0.273710	−	0.961812i	$-0.588251\pi$
$661$	−1.58097e8	−0.547419	−0.273710	+	0.961812i	$0.588251\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−3.18890e8	−1.06503
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 3.12399e7i	− 0.102486i	−0.998686	−	0.0512430i	$-0.983682\pi$
$673$	− 3.12399e7i	− 0.102486i	0.998686	−	0.0512430i	$-0.0163183\pi$
$674$	0	0
$675$	0	0
$676$	−2.92529e8	−0.946955
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−1.61436e7	−0.0515691
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	4.93714e8	1.54279
$685$	0	0
$686$	0	0
$687$	− 1.09966e8i	− 0.339148i
$688$	4.56237e8i	1.40096i
$689$	0	0
$690$	0	0
$691$	−4.01570e8	−1.21710	−0.608551	−	0.793515i	$-0.708249\pi$
$691$	−4.01570e8	−1.21710	−0.608551	+	0.793515i	$0.708249\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	9.43978e8i	2.71704i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.93732e8	−1.66591	−0.832955	−	0.553341i	$-0.813353\pi$
$709$	−5.93732e8	−1.66591	−0.832955	+	0.553341i	$0.813353\pi$
$710$	0	0
$711$	−1.49169e8	−0.415022
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−3.22307e8	−0.859930
$722$	0	0
$723$	− 7.13876e8i	− 1.88889i
$724$	−6.29005e6	−0.0165744
$725$	0	0
$726$	0	0
$727$	6.52273e8i	1.69756i	0.528743	+	0.848782i	$0.322663\pi$
$727$	6.52273e8i	1.69756i	−0.528743	+	0.848782i	$0.677337\pi$
$728$	0	0
$729$	−3.87420e8	−1.00000
$730$	0	0
$731$	0	0
$732$	− 7.27208e8i	− 1.85407i
$733$	− 5.61163e8i	− 1.42488i	−0.701735	−	0.712438i	$-0.747591\pi$
$733$	− 5.61163e8i	− 1.42488i	0.701735	−	0.712438i	$-0.252409\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1.77287e8	−0.439281	−0.219641	−	0.975581i	$-0.570488\pi$
$739$	−1.77287e8	−0.439281	−0.219641	+	0.975581i	$0.570488\pi$
$740$	0	0
$741$	1.44571e8	0.355326
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.97133e8	−0.701506	−0.350753	−	0.936468i	$-0.614074\pi$
$751$	−2.97133e8	−0.701506	−0.350753	+	0.936468i	$0.614074\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	3.60278e8	0.833819
$757$	− 8.52165e8i	− 1.96443i	−0.187767	−	0.982214i	$-0.560125\pi$
$757$	− 8.52165e8i	− 1.96443i	0.187767	−	0.982214i	$-0.439875\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	6.21404e8i	1.39894i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 4.52985e8i	− 1.00000i
$769$	8.88298e8	1.95335	0.976674	−	0.214727i	$-0.0688862\pi$
$769$	8.88298e8	1.95335	0.976674	+	0.214727i	$0.0688862\pi$
$770$	0	0
$771$	0	0
$772$	8.35559e8i	1.81604i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.88849e8i	1.46846i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.46854e8	0.304745
$785$	0	0
$786$	0	0
$787$	− 9.08673e8i	− 1.86416i	−0.362251	−	0.932081i	$-0.617992\pi$
$787$	− 9.08673e8i	− 1.86416i	0.362251	−	0.932081i	$-0.382008\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 2.12944e8i	− 0.427018i
$794$	0	0
$795$	0	0
$796$	7.45888e8	1.47888
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	2.98726e8	0.574785
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	4.83016e8	0.905522	0.452761	−	0.891632i	$-0.350439\pi$
$811$	4.83016e8	0.905522	0.452761	+	0.891632i	$0.350439\pi$
$812$	0	0
$813$	1.05693e9i	1.96687i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.17869e9i	2.16139i
$818$	0	0
$819$	1.05498e8	0.192040
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 9.65555e8i	− 1.73212i	−0.499941	−	0.866059i	$-0.666645\pi$
$823$	− 9.65555e8i	− 1.73212i	0.499941	−	0.866059i	$-0.333355\pi$
$824$	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
$829$	8.48197e8	1.48879	0.744395	−	0.667740i	$-0.232738\pi$
$829$	8.48197e8	1.48879	0.744395	+	0.667740i	$0.232738\pi$
$830$	0	0
$831$	7.09208e8	1.23586
$832$	− 1.32645e8i	− 0.230314i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.94456e8i	1.18432i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	5.94823e8	1.00000
$842$	0	0
$843$	0	0
$844$	−1.12622e9	−1.87325
$845$	0	0
$846$	0	0
$847$	5.06666e8i	0.833819i
$848$	0	0
$849$	−3.77200e8	−0.616380
$850$	0	0
$851$	0	0
$852$	0	0
$853$	9.38655e8i	1.51237i	0.654356	+	0.756187i	$0.272940\pi$
$853$	9.38655e8i	1.51237i	−0.654356	+	0.756187i	$0.727060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	2.87739e8	0.453962	0.226981	−	0.973899i	$-0.427114\pi$
$859$	2.87739e8	0.453962	0.226981	+	0.973899i	$0.427114\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.51714e8i	1.00000i
$868$	− 6.45802e8i	− 0.987507i
$869$	0	0
$870$	0	0
$871$	8.74742e7	0.132381
$872$	0	0
$873$	− 4.11491e7i	− 0.0618469i
$874$	0	0
$875$	0	0
$876$	−1.10258e9	−1.64020
$877$	− 1.16597e9i	− 1.72858i	−0.502997	−	0.864288i	$-0.667769\pi$
$877$	− 1.16597e9i	− 1.72858i	0.502997	−	0.864288i	$-0.332231\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 9.49268e8i	− 1.37882i	−0.724372	−	0.689409i	$-0.757870\pi$
$883$	− 9.49268e8i	− 1.37882i	0.724372	−	0.689409i	$-0.242130\pi$
$884$	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
$889$	−1.13038e9	−1.60886
$890$	0	0
$891$	0	0
$892$	7.55887e8i	1.06503i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	8.60123e8i	1.16815i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 5.18340e8i	− 0.694693i	−0.937737	−	0.347347i	$-0.887083\pi$
$907$	− 5.18340e8i	− 0.694693i	0.937737	−	0.347347i	$-0.112917\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 1.17028e9i	− 1.54279i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	−2.60661e8	−0.339148
$917$	0	0
$918$	0	0
$919$	1.54471e9	1.99021	0.995107	−	0.0988039i	$-0.0315017\pi$
$919$	1.54471e9	1.99021	0.995107	+	0.0988039i	$0.0315017\pi$
$920$	0	0
$921$	−1.49420e9	−1.91262
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.21544e8i	− 1.03132i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	3.79396e8	0.470158
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.43605e9i	− 1.74562i	−0.488060	−	0.872810i	$-0.662295\pi$
$937$	− 1.43605e9i	− 1.74562i	0.488060	−	0.872810i	$-0.337705\pi$
$938$	0	0
$939$	1.04953e9	1.26765
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	3.53587e8i	0.415022i
$949$	−3.22861e8	−0.377761
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.57316e8	0.402608
$962$	0	0
$963$	0	0
$964$	−1.69215e9	−1.88889
$965$	0	0
$966$	0	0
$967$	− 6.93538e8i	− 0.766992i	−0.923542	−	0.383496i	$-0.874720\pi$
$967$	− 6.93538e8i	− 0.766992i	0.923542	−	0.383496i	$-0.125280\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	9.18330e8i	1.00000i
$973$	− 3.61658e8i	− 0.392608i
$974$	0	0
$975$	0	0
$976$	−1.72375e9	−1.85407
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−1.58393e9	−1.67776
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 3.42687e8i	− 0.355326i
$989$	0	0
$990$	0	0
$991$	1.35430e8	0.139153	0.0695766	−	0.997577i	$-0.477835\pi$
$991$	1.35430e8	0.139153	0.0695766	+	0.997577i	$0.477835\pi$
$992$	0	0
$993$	1.93172e9i	1.97286i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.14627e9i	1.15664i	0.815808	+	0.578322i	$0.196292\pi$
$997$	1.14627e9i	1.15664i	−0.815808	+	0.578322i	$0.803708\pi$
$998$	0	0
$999$	−1.75584e9	−1.76112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.7.d.a.74.1		2
3.2	odd	2	CM	75.7.d.a.74.1		2
5.2	odd	4		3.7.b.a.2.1	✓	1
5.3	odd	4		75.7.c.a.26.1		1
5.4	even	2	inner	75.7.d.a.74.2		2
15.2	even	4		3.7.b.a.2.1	✓	1
15.8	even	4		75.7.c.a.26.1		1
15.14	odd	2	inner	75.7.d.a.74.2		2
20.7	even	4		48.7.e.a.17.1		1
35.27	even	4		147.7.b.a.50.1		1
40.27	even	4		192.7.e.a.65.1		1
40.37	odd	4		192.7.e.b.65.1		1
45.2	even	12		81.7.d.a.53.1		2
45.7	odd	12		81.7.d.a.53.1		2
45.22	odd	12		81.7.d.a.26.1		2
45.32	even	12		81.7.d.a.26.1		2
60.47	odd	4		48.7.e.a.17.1		1
105.62	odd	4		147.7.b.a.50.1		1
120.77	even	4		192.7.e.b.65.1		1
120.107	odd	4		192.7.e.a.65.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3.7.b.a.2.1	✓	1	5.2	odd	4
3.7.b.a.2.1	✓	1	15.2	even	4
48.7.e.a.17.1		1	20.7	even	4
48.7.e.a.17.1		1	60.47	odd	4
75.7.c.a.26.1		1	5.3	odd	4
75.7.c.a.26.1		1	15.8	even	4
75.7.d.a.74.1		2	1.1	even	1	trivial
75.7.d.a.74.1		2	3.2	odd	2	CM
75.7.d.a.74.2		2	5.4	even	2	inner
75.7.d.a.74.2		2	15.14	odd	2	inner
81.7.d.a.26.1		2	45.22	odd	12
81.7.d.a.26.1		2	45.32	even	12
81.7.d.a.53.1		2	45.2	even	12
81.7.d.a.53.1		2	45.7	odd	12
147.7.b.a.50.1		1	35.27	even	4
147.7.b.a.50.1		1	105.62	odd	4
192.7.e.a.65.1		1	40.27	even	4
192.7.e.a.65.1		1	120.107	odd	4
192.7.e.b.65.1		1	40.37	odd	4
192.7.e.b.65.1		1	120.77	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-27.0000i q^{3} -64.0000 q^{4} +286.000i q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-27.0000i q^{3} -64.0000 q^{4} +286.000i q^{7} -729.000 q^{9} +1728.00i q^{12} +506.000i q^{13} +4096.00 q^{16} +10582.0 q^{19} +7722.00 q^{21} +19683.0i q^{27} -18304.0i q^{28} +35282.0 q^{31} +46656.0 q^{36} +89206.0i q^{37} +13662.0 q^{39} +111386. i q^{43} -110592. i q^{48} +35853.0 q^{49} -32384.0i q^{52} -285714. i q^{57} -420838. q^{61} -208494. i q^{63} -262144. q^{64} -172874. i q^{67} +638066. i q^{73} -677248. q^{76} +204622. q^{79} +531441. q^{81} -494208. q^{84} -144716. q^{91} -952614. i q^{93} +56446.0i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 1458 q^{9}+O(q^{10}) \) 2 * q - 128 * q^4 - 1458 * q^9	\( 2 q - 128 q^{4} - 1458 q^{9} + 8192 q^{16} + 21164 q^{19} + 15444 q^{21} + 70564 q^{31} + 93312 q^{36} + 27324 q^{39} + 71706 q^{49} - 841676 q^{61} - 524288 q^{64} - 1354496 q^{76} + 409244 q^{79} + 1062882 q^{81} - 988416 q^{84} - 289432 q^{91}+O(q^{100}) \) 2 * q - 128 * q^4 - 1458 * q^9 + 8192 * q^16 + 21164 * q^19 + 15444 * q^21 + 70564 * q^31 + 93312 * q^36 + 27324 * q^39 + 71706 * q^49 - 841676 * q^61 - 524288 * q^64 - 1354496 * q^76 + 409244 * q^79 + 1062882 * q^81 - 988416 * q^84 - 289432 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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