LMFDB - Embedded newform 980.2.e.d.589.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(589,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.589"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{-21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 13x^{2} + 16 $ x^4 + 13*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			589.1
Root			$-3.40932i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.2.e.d.589.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40932i q^{3} +2.23607i q^{5} -8.62348 q^{9} -3.62348 q^{11} +1.06281i q^{13} +7.62348 q^{15} +5.75583i q^{17} -5.00000 q^{25} +19.1722i q^{27} -9.62348 q^{29} +12.3536i q^{33} +3.62348 q^{39} -19.2827i q^{45} -1.28369i q^{47} +19.6235 q^{51} -8.10234i q^{55} -2.37652 q^{65} -12.0000 q^{71} -13.4164i q^{73} +17.0466i q^{75} +14.8704 q^{79} +39.4939 q^{81} +8.94427i q^{83} -12.8704 q^{85} +32.8095i q^{87} +19.3931i q^{97} +31.2470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{9} + 6 q^{11} + 10 q^{15} - 20 q^{25} - 18 q^{29} - 6 q^{39} + 58 q^{51} - 30 q^{65} - 48 q^{71} - 2 q^{79} + 76 q^{81} + 10 q^{85} + 84 q^{99}+O(q^{100}) $ 4 * q - 14 * q^9 + 6 * q^11 + 10 * q^15 - 20 * q^25 - 18 * q^29 - 6 * q^39 + 58 * q^51 - 30 * q^65 - 48 * q^71 - 2 * q^79 + 76 * q^81 + 10 * q^85 + 84 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 3.40932i	− 1.96837i	−0.177136	−	0.984186i	$-0.556683\pi$
$3$	− 3.40932i	− 1.96837i	0.177136	−	0.984186i	$-0.443317\pi$
$4$	0	0
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−8.62348	−2.87449
$10$	0	0
$11$	−3.62348	−1.09252	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	−3.62348	−1.09252	−0.546259	+	0.837616i	$0.683949\pi$
$12$	0	0
$13$	1.06281i	0.294772i	0.989079	+	0.147386i	$0.0470859\pi$
$13$	1.06281i	0.294772i	−0.989079	+	0.147386i	$0.952914\pi$
$14$	0	0
$15$	7.62348	1.96837
$16$	0	0
$17$	5.75583i	1.39599i	0.716101	+	0.697997i	$0.245925\pi$
$17$	5.75583i	1.39599i	−0.716101	+	0.697997i	$0.754075\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	19.1722i	3.68970i
$28$	0	0
$29$	−9.62348	−1.78703	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−9.62348	−1.78703	−0.893517	+	0.449029i	$0.851770\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	12.3536i	2.15048i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	3.62348	0.580220
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 19.2827i	− 2.87449i
$46$	0	0
$47$	− 1.28369i	− 0.187246i	−0.995608	−	0.0936230i	$-0.970155\pi$
$47$	− 1.28369i	− 0.187246i	0.995608	−	0.0936230i	$-0.0298448\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	19.6235	2.74784
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 8.10234i	− 1.09252i
$56$	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
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.37652	−0.294772
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	− 13.4164i	− 1.57027i	−0.619324	−	0.785136i	$-0.712593\pi$
$73$	− 13.4164i	− 1.57027i	0.619324	−	0.785136i	$-0.287407\pi$
$74$	0	0
$75$	17.0466i	1.96837i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.8704	1.67305	0.836527	−	0.547926i	$-0.184582\pi$
$79$	14.8704	1.67305	0.836527	+	0.547926i	$0.184582\pi$
$80$	0	0
$81$	39.4939	4.38821
$82$	0	0
$83$	8.94427i	0.981761i	0.871227	+	0.490881i	$0.163325\pi$
$83$	8.94427i	0.981761i	−0.871227	+	0.490881i	$0.836675\pi$
$84$	0	0
$85$	−12.8704	−1.39599
$86$	0	0
$87$	32.8095i	3.51755i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	19.3931i	1.96907i	0.175180	+	0.984536i	$0.443949\pi$
$97$	19.3931i	1.96907i	−0.175180	+	0.984536i	$0.556051\pi$
$98$	0	0
$99$	31.2470	3.14044
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 12.3536i	− 1.21724i	−0.793463	−	0.608618i	$-0.791724\pi$
$103$	− 12.3536i	− 1.21724i	0.793463	−	0.608618i	$-0.208276\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−20.8704	−1.99902	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−20.8704	−1.99902	−0.999512	+	0.0312328i	$0.990057\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9.16515i	− 0.847319i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.12957	0.193598
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−42.8704	−3.68970
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−4.37652	−0.368570
$142$	0	0
$143$	− 3.85108i	− 0.322044i
$144$	0	0
$145$	− 21.5187i	− 1.78703i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−6.87043	−0.559107	−0.279554	−	0.960130i	$-0.590186\pi$
$151$	−6.87043	−0.559107	−0.279554	+	0.960130i	$0.590186\pi$
$152$	0	0
$153$	− 49.6353i	− 4.01277i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.4164i	− 1.07075i	−0.844616	−	0.535373i	$-0.820171\pi$
$157$	− 13.4164i	− 1.07075i	0.844616	−	0.535373i	$-0.179829\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−27.6235	−2.15048
$166$	0	0
$167$	− 19.1722i	− 1.48359i	−0.670625	−	0.741796i	$-0.733974\pi$
$167$	− 19.1722i	− 1.48359i	0.670625	−	0.741796i	$-0.266026\pi$
$168$	0	0
$169$	11.8704	0.913110
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.2118i	1.99284i	0.0845218	+	0.996422i	$0.473064\pi$
$173$	26.2118i	1.99284i	−0.0845218	+	0.996422i	$0.526936\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.8561i	− 1.52515i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.37652	0.606104	0.303052	−	0.952974i	$-0.401994\pi$
$191$	8.37652	0.606104	0.303052	+	0.952974i	$0.401994\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	8.10234i	0.580220i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−26.8704	−1.84984	−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−26.8704	−1.84984	−0.924918	+	0.380166i	$0.875867\pi$
$212$	0	0
$213$	40.9119i	2.80323i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−45.7409	−3.09088
$220$	0	0
$221$	−6.11738	−0.411499
$222$	0	0
$223$	28.5583i	1.91240i	0.292709	+	0.956202i	$0.405443\pi$
$223$	28.5583i	1.91240i	−0.292709	+	0.956202i	$0.594557\pi$
$224$	0	0
$225$	43.1174	2.87449
$226$	0	0
$227$	7.66058i	0.508450i	0.967145	+	0.254225i	$0.0818204\pi$
$227$	7.66058i	0.508450i	−0.967145	+	0.254225i	$0.918180\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	2.87043	0.187246
$236$	0	0
$237$	− 50.6981i	− 3.29319i
$238$	0	0
$239$	−30.1174	−1.94813	−0.974066	−	0.226266i	$-0.927348\pi$
$239$	−30.1174	−1.94813	−0.974066	+	0.226266i	$0.927348\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 77.1307i	− 4.94794i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	30.4939	1.93247
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	43.8794i	2.74784i
$256$	0	0
$257$	4.47214i	0.278964i	0.990225	+	0.139482i	$0.0445438\pi$
$257$	4.47214i	0.278964i	−0.990225	+	0.139482i	$0.955456\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	82.9878	5.13682
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.1174	1.09252
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.6235	1.28995	0.644974	−	0.764204i	$-0.276868\pi$
$281$	21.6235	1.28995	0.644974	+	0.764204i	$0.276868\pi$
$282$	0	0
$283$	− 14.4792i	− 0.860700i	−0.902662	−	0.430350i	$-0.858390\pi$
$283$	− 14.4792i	− 0.860700i	0.902662	−	0.430350i	$-0.141610\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.1296	−0.948798
$290$	0	0
$291$	66.1174	3.87587
$292$	0	0
$293$	8.32322i	0.486248i	0.969995	+	0.243124i	$0.0781721\pi$
$293$	8.32322i	0.486248i	−0.969995	+	0.243124i	$0.921828\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 69.4701i	− 4.03107i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 26.4326i	− 1.50859i	−0.656535	−	0.754295i	$-0.727979\pi$
$307$	− 26.4326i	− 1.50859i	0.656535	−	0.754295i	$-0.272021\pi$
$308$	0	0
$309$	−42.1174	−2.39597
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	15.1419i	0.855869i	0.903810	+	0.427934i	$0.140759\pi$
$313$	15.1419i	0.855869i	−0.903810	+	0.427934i	$0.859241\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	34.8704	1.95237
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 5.31407i	− 0.294772i
$326$	0	0
$327$	71.1540i	3.93483i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−20.3765	−1.08762
$352$	0	0
$353$	35.1560i	1.87117i	0.353106	+	0.935583i	$0.385126\pi$
$353$	35.1560i	1.87117i	−0.353106	+	0.935583i	$0.614874\pi$
$354$	0	0
$355$	− 26.8328i	− 1.42414i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 7.26040i	− 0.381072i
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	− 32.8095i	− 1.71264i	−0.516443	−	0.856322i	$-0.672744\pi$
$367$	− 32.8095i	− 1.71264i	0.516443	−	0.856322i	$-0.327256\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−38.1174	−1.96837
$376$	0	0
$377$	− 10.2280i	− 0.526767i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	35.7771i	1.82812i	0.405575	+	0.914062i	$0.367071\pi$
$383$	35.7771i	1.82812i	−0.405575	+	0.914062i	$0.632929\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.3765	−0.728919	−0.364459	−	0.931219i	$-0.618746\pi$
$389$	−14.3765	−0.728919	−0.364459	+	0.931219i	$0.618746\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.2513i	1.67305i
$396$	0	0
$397$	− 39.8490i	− 1.99997i	−0.00579782	−	0.999983i	$-0.501846\pi$
$397$	− 39.8490i	− 1.99997i	0.00579782	−	0.999983i	$-0.498154\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.1174	−0.605113	−0.302556	−	0.953131i	$-0.597840\pi$
$401$	−12.1174	−0.605113	−0.302556	+	0.953131i	$0.597840\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	88.3110i	4.38821i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−20.0000	−0.981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−3.12957	−0.152526	−0.0762630	−	0.997088i	$-0.524299\pi$
$421$	−3.12957	−0.152526	−0.0762630	+	0.997088i	$0.524299\pi$
$422$	0	0
$423$	11.0699i	0.538237i
$424$	0	0
$425$	− 28.7791i	− 1.39599i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.1296	−0.633902
$430$	0	0
$431$	37.3643	1.79978	0.899888	−	0.436121i	$-0.143648\pi$
$431$	37.3643	1.79978	0.899888	+	0.436121i	$0.143648\pi$
$432$	0	0
$433$	40.2492i	1.93425i	0.254293	+	0.967127i	$0.418157\pi$
$433$	40.2492i	1.93425i	−0.254293	+	0.967127i	$0.581843\pi$
$434$	0	0
$435$	−73.3643	−3.51755
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.4559i	0.967532i
$448$	0	0
$449$	31.3643	1.48017	0.740087	−	0.672511i	$-0.234784\pi$
$449$	31.3643	1.48017	0.740087	+	0.672511i	$0.234784\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	23.4235i	1.10053i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−110.352	−5.15080
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.1165i	1.30108i	0.759473	+	0.650538i	$0.225457\pi$
$467$	28.1165i	1.30108i	−0.759473	+	0.650538i	$0.774543\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−45.7409	−2.10763
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−43.3643	−1.96907
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	42.1174	1.90073	0.950365	−	0.311136i	$-0.100710\pi$
$491$	42.1174	1.90073	0.950365	+	0.311136i	$0.100710\pi$
$492$	0	0
$493$	− 55.3911i	− 2.49469i
$494$	0	0
$495$	69.8703i	3.14044i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.12957	0.229631	0.114816	−	0.993387i	$-0.463372\pi$
$499$	5.12957	0.229631	0.114816	+	0.993387i	$0.463372\pi$
$500$	0	0
$501$	−65.3643	−2.92026
$502$	0	0
$503$	39.6282i	1.76693i	0.468495	+	0.883466i	$0.344797\pi$
$503$	39.6282i	1.76693i	−0.468495	+	0.883466i	$0.655203\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 40.4701i	− 1.79734i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	27.6235	1.21724
$516$	0	0
$517$	4.65143i	0.204570i
$518$	0	0
$519$	89.3643	3.92266
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 26.8328i	− 1.17332i	−0.809834	−	0.586659i	$-0.800443\pi$
$523$	− 26.8328i	− 1.17332i	0.809834	−	0.586659i	$-0.199557\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	81.8237i	3.53095i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−36.8704	−1.58518	−0.792592	−	0.609753i	$-0.791269\pi$
$541$	−36.8704	−1.58518	−0.792592	+	0.609753i	$0.791269\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 46.6677i	− 1.99902i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−71.1052	−3.00206
$562$	0	0
$563$	− 44.7214i	− 1.88478i	−0.334515	−	0.942390i	$-0.608573\pi$
$563$	− 44.7214i	− 1.88478i	0.334515	−	0.942390i	$-0.391427\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	− 28.5583i	− 1.19304i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.0162i	0.541873i	0.962597	+	0.270936i	$0.0873333\pi$
$577$	13.0162i	0.541873i	−0.962597	+	0.270936i	$0.912667\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	20.4939	0.847319
$586$	0	0
$587$	8.94427i	0.369170i	0.982817	+	0.184585i	$0.0590940\pi$
$587$	8.94427i	0.369170i	−0.982817	+	0.184585i	$0.940906\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0.621055i	0.0255037i	0.999919	+	0.0127518i	$0.00405915\pi$
$593$	0.621055i	0.0255037i	−0.999919	+	0.0127518i	$0.995941\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.11738	0.249949	0.124975	−	0.992160i	$-0.460115\pi$
$599$	6.11738	0.249949	0.124975	+	0.992160i	$0.460115\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.76187i	0.193598i
$606$	0	0
$607$	− 49.0142i	− 1.98942i	−0.102699	−	0.994712i	$-0.532748\pi$
$607$	− 49.0142i	− 1.98942i	0.102699	−	0.994712i	$-0.467252\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.36433	0.0551948
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	38.8704	1.54741	0.773704	−	0.633548i	$-0.218402\pi$
$631$	38.8704	1.54741	0.773704	+	0.633548i	$0.218402\pi$
$632$	0	0
$633$	91.6099i	3.64117i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	103.482	4.09367
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	46.8886i	1.84910i	0.381055	+	0.924552i	$0.375561\pi$
$643$	46.8886i	1.84910i	−0.381055	+	0.924552i	$0.624439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.8885i	0.703271i	0.936137	+	0.351636i	$0.114374\pi$
$647$	17.8885i	0.703271i	−0.936137	+	0.351636i	$0.885626\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	115.696i	4.51373i
$658$	0	0
$659$	−20.3765	−0.793757	−0.396878	−	0.917871i	$-0.629907\pi$
$659$	−20.3765	−0.793757	−0.396878	+	0.917871i	$0.629907\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	20.8561i	0.809984i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	97.3643	3.76432
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 95.8612i	− 3.68970i
$676$	0	0
$677$	− 32.5886i	− 1.25248i	−0.779629	−	0.626242i	$-0.784592\pi$
$677$	− 32.5886i	− 1.25248i	0.779629	−	0.626242i	$-0.215408\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	26.1174	1.00082
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	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
$701$	−19.3643	−0.731381	−0.365690	−	0.930737i	$-0.619167\pi$
$701$	−19.3643	−0.731381	−0.365690	+	0.930737i	$0.619167\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 9.78621i	− 0.368570i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−46.6113	−1.75052	−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−46.6113	−1.75052	−0.875262	+	0.483650i	$0.839311\pi$
$710$	0	0
$711$	−128.235	−4.80918
$712$	0	0
$713$	0	0
$714$	0	0
$715$	8.61128	0.322044
$716$	0	0
$717$	102.680i	3.83465i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	48.1174	1.78703
$726$	0	0
$727$	− 53.6656i	− 1.99035i	−0.0981255	−	0.995174i	$-0.531285\pi$
$727$	− 53.6656i	− 1.99035i	0.0981255	−	0.995174i	$-0.468715\pi$
$728$	0	0
$729$	−144.482	−5.35117
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.4722i	1.23632i	0.786051	+	0.618161i	$0.212122\pi$
$733$	33.4722i	1.23632i	−0.786051	+	0.618161i	$0.787878\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	40.6113	1.49391	0.746955	−	0.664875i	$-0.231515\pi$
$739$	40.6113	1.49391	0.746955	+	0.664875i	$0.231515\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	− 13.4164i	− 0.491539i
$746$	0	0
$747$	− 77.1307i	− 2.82207i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	52.6113	1.91981	0.959906	−	0.280321i	$-0.0904408\pi$
$751$	52.6113	1.91981	0.959906	+	0.280321i	$0.0904408\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 15.3627i	− 0.559107i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	110.988	4.01277
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	15.2470	0.549106
$772$	0	0
$773$	− 49.2351i	− 1.77086i	−0.464770	−	0.885431i	$-0.653863\pi$
$773$	− 49.2351i	− 1.77086i	0.464770	−	0.885431i	$-0.346137\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	43.4817	1.55590
$782$	0	0
$783$	− 184.504i	− 6.59362i
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	22.1814i	0.790681i	0.918535	+	0.395340i	$0.129373\pi$
$787$	22.1814i	0.790681i	−0.918535	+	0.395340i	$0.870627\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 21.0770i	− 0.746585i	−0.927714	−	0.373293i	$-0.878229\pi$
$797$	− 21.0770i	− 0.746585i	0.927714	−	0.373293i	$-0.121771\pi$
$798$	0	0
$799$	7.38872	0.261394
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.6140i	1.71555i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−55.3643	−1.94651	−0.973253	−	0.229736i	$-0.926214\pi$
$809$	−55.3643	−1.94651	−0.973253	+	0.229736i	$0.926214\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.6235	−1.17347	−0.586734	−	0.809780i	$-0.699586\pi$
$821$	−33.6235	−1.17347	−0.586734	+	0.809780i	$0.699586\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	− 61.7680i	− 2.15048i
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	42.8704	1.48359
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	63.6113	2.19349
$842$	0	0
$843$	− 73.7214i	− 2.53910i
$844$	0	0
$845$	26.5431i	0.913110i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−49.3643	−1.69418
$850$	0	0
$851$	0	0
$852$	0	0
$853$	40.2492i	1.37811i	0.724710	+	0.689054i	$0.241974\pi$
$853$	40.2492i	1.37811i	−0.724710	+	0.689054i	$0.758026\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1935i	1.68042i	0.542263	+	0.840209i	$0.317568\pi$
$857$	49.1935i	1.68042i	−0.542263	+	0.840209i	$0.682432\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−58.6113	−1.99284
$866$	0	0
$867$	54.9909i	1.86759i
$868$	0	0
$869$	−53.8826	−1.82784
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 167.236i	− 5.66008i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	28.3765	0.957116
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.7771i	1.20128i	0.799521	+	0.600639i	$0.205087\pi$
$887$	35.7771i	1.20128i	−0.799521	+	0.600639i	$0.794913\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−143.105	−4.79420
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 53.6656i	− 1.79384i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	− 32.4093i	− 1.07259i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−60.6113	−1.99938	−0.999691	−	0.0248659i	$-0.992084\pi$
$919$	−60.6113	−1.99938	−0.999691	+	0.0248659i	$0.992084\pi$
$920$	0	0
$921$	−90.1174	−2.96947
$922$	0	0
$923$	− 12.7538i	− 0.419795i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	106.531i	3.49893i
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	46.6357	1.52515
$936$	0	0
$937$	56.0537i	1.83120i	0.402096	+	0.915598i	$0.368282\pi$
$937$	56.0537i	1.83120i	−0.402096	+	0.915598i	$0.631718\pi$
$938$	0	0
$939$	51.6235	1.68467
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	14.2591	0.462872
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	18.7305i	0.606104i
$956$	0	0
$957$	− 118.885i	− 3.84299i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−18.1174	−0.580220
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	179.976	5.74618
$982$	0	0
$983$	62.6515i	1.99827i	0.0415592	+	0.999136i	$0.486767\pi$
$983$	62.6515i	1.99827i	−0.0415592	+	0.999136i	$0.513233\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	− 27.2746i	− 0.865532i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.2259i	1.46399i	0.681310	+	0.731995i	$0.261411\pi$
$997$	46.2259i	1.46399i	−0.681310	+	0.731995i	$0.738589\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.e.d.589.1	✓	4
5.2	odd	4		4900.2.a.bj.1.1		4
5.3	odd	4		4900.2.a.bj.1.4		4
5.4	even	2	inner	980.2.e.d.589.4	yes	4
7.2	even	3		980.2.q.i.949.4		8
7.3	odd	6		980.2.q.i.569.4		8
7.4	even	3		980.2.q.i.569.1		8
7.5	odd	6		980.2.q.i.949.1		8
7.6	odd	2	inner	980.2.e.d.589.4	yes	4
35.4	even	6		980.2.q.i.569.4		8
35.9	even	6		980.2.q.i.949.1		8
35.13	even	4		4900.2.a.bj.1.1		4
35.19	odd	6		980.2.q.i.949.4		8
35.24	odd	6		980.2.q.i.569.1		8
35.27	even	4		4900.2.a.bj.1.4		4
35.34	odd	2	CM	980.2.e.d.589.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.d.589.1	✓	4	1.1	even	1	trivial
980.2.e.d.589.1	✓	4	35.34	odd	2	CM
980.2.e.d.589.4	yes	4	5.4	even	2	inner
980.2.e.d.589.4	yes	4	7.6	odd	2	inner
980.2.q.i.569.1		8	7.4	even	3
980.2.q.i.569.1		8	35.24	odd	6
980.2.q.i.569.4		8	7.3	odd	6
980.2.q.i.569.4		8	35.4	even	6
980.2.q.i.949.1		8	7.5	odd	6
980.2.q.i.949.1		8	35.9	even	6
980.2.q.i.949.4		8	7.2	even	3
980.2.q.i.949.4		8	35.19	odd	6
4900.2.a.bj.1.1		4	5.2	odd	4
4900.2.a.bj.1.1		4	35.13	even	4
4900.2.a.bj.1.4		4	5.3	odd	4
4900.2.a.bj.1.4		4	35.27	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.40932i q^{3} +2.23607i q^{5} -8.62348 q^{9} -3.62348 q^{11} +1.06281i q^{13} +7.62348 q^{15} +5.75583i q^{17} -5.00000 q^{25} +19.1722i q^{27} -9.62348 q^{29} +12.3536i q^{33} +3.62348 q^{39} -19.2827i q^{45} -1.28369i q^{47} +19.6235 q^{51} -8.10234i q^{55} -2.37652 q^{65} -12.0000 q^{71} -13.4164i q^{73} +17.0466i q^{75} +14.8704 q^{79} +39.4939 q^{81} +8.94427i q^{83} -12.8704 q^{85} +32.8095i q^{87} +19.3931i q^{97} +31.2470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{9} + 6 q^{11} + 10 q^{15} - 20 q^{25} - 18 q^{29} - 6 q^{39} + 58 q^{51} - 30 q^{65} - 48 q^{71} - 2 q^{79} + 76 q^{81} + 10 q^{85} + 84 q^{99}+O(q^{100}) \) 4 * q - 14 * q^9 + 6 * q^11 + 10 * q^15 - 20 * q^25 - 18 * q^29 - 6 * q^39 + 58 * q^51 - 30 * q^65 - 48 * q^71 - 2 * q^79 + 76 * q^81 + 10 * q^85 + 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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