LMFDB - Embedded newform 980.2.e.d.589.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(589,980)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			589.3
Root			$1.17325i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.2.e.d.589.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17325i q^{3} +2.23607i q^{5} +1.62348 q^{9} +O(q^{10})$	$q+1.17325i q^{3} +2.23607i q^{5} +1.62348 q^{9} +6.62348 q^{11} +5.64539i q^{13} -2.62348 q^{15} -7.99190i q^{17} -5.00000 q^{25} +5.42451i q^{27} +0.623475 q^{29} +7.77102i q^{33} -6.62348 q^{39} +3.63020i q^{45} +12.4640i q^{47} +9.37652 q^{51} +14.8105i q^{55} -12.6235 q^{65} -12.0000 q^{71} -13.4164i q^{73} -5.86627i q^{75} -15.8704 q^{79} -1.49390 q^{81} +8.94427i q^{83} +17.8704 q^{85} +0.731495i q^{87} -12.6849i q^{97} +10.7530 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{9}+O(q^{10}) $ 4 * q - 14 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$	1.17325i	0.677378i	0.940898	+	0.338689i	$0.109984\pi$
$3$	1.17325i	0.677378i	−0.940898	+	0.338689i	$0.890016\pi$
$4$	0	0
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.62348	0.541158
$10$	0	0
$11$	6.62348	1.99705	0.998526	−	0.0542666i	$-0.0172821\pi$
$11$	6.62348	1.99705	0.998526	+	0.0542666i	$0.0172821\pi$
$12$	0	0
$13$	5.64539i	1.56575i	0.622179	+	0.782875i	$0.286247\pi$
$13$	5.64539i	1.56575i	−0.622179	+	0.782875i	$0.713753\pi$
$14$	0	0
$15$	−2.62348	−0.677378
$16$	0	0
$17$	− 7.99190i	− 1.93832i	−0.246433	−	0.969160i	$-0.579258\pi$
$17$	− 7.99190i	− 1.93832i	0.246433	−	0.969160i	$-0.420742\pi$
$18$	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
$27$	5.42451i	1.04395i
$28$	0	0
$29$	0.623475	0.115776	0.0578882	−	0.998323i	$-0.481563\pi$
$29$	0.623475	0.115776	0.0578882	+	0.998323i	$0.481563\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	7.77102i	1.35276i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−6.62348	−1.06060
$40$	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
$45$	3.63020i	0.541158i
$46$	0	0
$47$	12.4640i	1.81807i	0.416724	+	0.909033i	$0.363178\pi$
$47$	12.4640i	1.81807i	−0.416724	+	0.909033i	$0.636822\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	9.37652	1.31298
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	14.8105i	1.99705i
$56$	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
$65$	−12.6235	−1.56575
$66$	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
$75$	− 5.86627i	− 0.677378i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.8704	−1.78556	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−15.8704	−1.78556	−0.892781	+	0.450490i	$0.851249\pi$
$80$	0	0
$81$	−1.49390	−0.165989
$82$	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
$85$	17.8704	1.93832
$86$	0	0
$87$	0.731495i	0.0784245i
$88$	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
$97$	− 12.6849i	− 1.28796i	−0.765043	−	0.643979i	$-0.777282\pi$
$97$	− 12.6849i	− 1.28796i	0.765043	−	0.643979i	$-0.222718\pi$
$98$	0	0
$99$	10.7530	1.08072
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 7.77102i	− 0.765701i	−0.923810	−	0.382851i	$-0.874942\pi$
$103$	− 7.77102i	− 0.765701i	0.923810	−	0.382851i	$-0.125058\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	9.87043	0.945415	0.472708	−	0.881219i	$-0.343277\pi$
$109$	9.87043	0.945415	0.472708	+	0.881219i	$0.343277\pi$
$110$	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
$121$	32.8704	2.98822
$122$	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
$135$	−12.1296	−1.04395
$136$	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
$141$	−14.6235	−1.23152
$142$	0	0
$143$	37.3921i	3.12688i
$144$	0	0
$145$	1.39413i	0.115776i
$146$	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
$151$	23.8704	1.94255	0.971274	−	0.237964i	$-0.0764802\pi$
$151$	23.8704	1.94255	0.971274	+	0.237964i	$0.0764802\pi$
$152$	0	0
$153$	− 12.9746i	− 1.04894i
$154$	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
$165$	−17.3765	−1.35276
$166$	0	0
$167$	− 5.42451i	− 0.419761i	−0.977727	−	0.209881i	$-0.932692\pi$
$167$	− 5.42451i	− 0.419761i	0.977727	−	0.209881i	$-0.0673075\pi$
$168$	0	0
$169$	−18.8704	−1.45157
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15.0314i	− 1.14282i	−0.820666	−	0.571409i	$-0.806397\pi$
$173$	− 15.0314i	− 1.14282i	0.820666	−	0.571409i	$-0.193603\pi$
$174$	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
$187$	− 52.9341i	− 3.87093i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.6235	1.34755	0.673774	−	0.738938i	$-0.264672\pi$
$191$	18.6235	1.34755	0.673774	+	0.738938i	$0.264672\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	− 14.8105i	− 1.06060i
$196$	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
$211$	3.87043	0.266451	0.133226	−	0.991086i	$-0.457467\pi$
$211$	3.87043	0.266451	0.133226	+	0.991086i	$0.457467\pi$
$212$	0	0
$213$	− 14.0790i	− 0.964680i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	15.7409	1.06367
$220$	0	0
$221$	45.1174	3.03492
$222$	0	0
$223$	− 21.8501i	− 1.46319i	−0.681740	−	0.731594i	$-0.738777\pi$
$223$	− 21.8501i	− 1.46319i	0.681740	−	0.731594i	$-0.261223\pi$
$224$	0	0
$225$	−8.11738	−0.541158
$226$	0	0
$227$	21.4083i	1.42092i	0.703738	+	0.710460i	$0.251513\pi$
$227$	21.4083i	1.42092i	−0.703738	+	0.710460i	$0.748487\pi$
$228$	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
$235$	−27.8704	−1.81807
$236$	0	0
$237$	− 18.6200i	− 1.20950i
$238$	0	0
$239$	21.1174	1.36597	0.682985	−	0.730433i	$-0.260682\pi$
$239$	21.1174	1.36597	0.682985	+	0.730433i	$0.260682\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	14.5208i	0.931510i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.4939	−0.665024
$250$	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
$255$	20.9665i	1.31298i
$256$	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
$261$	1.01220	0.0626534
$262$	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
$275$	−33.1174	−1.99705
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.3765	0.678667	0.339333	−	0.940666i	$-0.389799\pi$
$281$	11.3765	0.678667	0.339333	+	0.940666i	$0.389799\pi$
$282$	0	0
$283$	− 19.0618i	− 1.13311i	−0.824025	−	0.566553i	$-0.808277\pi$
$283$	− 19.0618i	− 1.13311i	0.824025	−	0.566553i	$-0.191723\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−46.8704	−2.75708
$290$	0	0
$291$	14.8826	0.872435
$292$	0	0
$293$	− 32.9200i	− 1.92320i	−0.274446	−	0.961602i	$-0.588495\pi$
$293$	− 32.9200i	− 1.92320i	0.274446	−	0.961602i	$-0.411505\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	35.9291i	2.08482i
$298$	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
$307$	33.1408i	1.89145i	0.324971	+	0.945724i	$0.394645\pi$
$307$	33.1408i	1.89145i	−0.324971	+	0.945724i	$0.605355\pi$
$308$	0	0
$309$	9.11738	0.518669
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 35.2665i	− 1.99338i	−0.0813030	−	0.996689i	$-0.525908\pi$
$313$	− 35.2665i	− 1.99338i	0.0813030	−	0.996689i	$-0.474092\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	4.12957	0.231212
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 28.2269i	− 1.56575i
$326$	0	0
$327$	11.5805i	0.640404i
$328$	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
$351$	−30.6235	−1.63456
$352$	0	0
$353$	− 6.08715i	− 0.323986i	−0.986792	−	0.161993i	$-0.948208\pi$
$353$	− 6.08715i	− 0.323986i	0.986792	−	0.161993i	$-0.0517922\pi$
$354$	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
$363$	38.5654i	2.02416i
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	− 0.731495i	− 0.0381837i	−0.999818	−	0.0190919i	$-0.993923\pi$
$367$	− 0.731495i	− 0.0381837i	0.999818	−	0.0190919i	$-0.00607750\pi$
$368$	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
$375$	13.1174	0.677378
$376$	0	0
$377$	3.51976i	0.181277i
$378$	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
$389$	−24.6235	−1.24846	−0.624230	−	0.781241i	$-0.714587\pi$
$389$	−24.6235	−1.24846	−0.624230	+	0.781241i	$0.714587\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 35.4874i	− 1.78556i
$396$	0	0
$397$	19.7244i	0.989941i	0.868910	+	0.494971i	$0.164821\pi$
$397$	19.7244i	0.989941i	−0.868910	+	0.494971i	$0.835179\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.1174	1.95343	0.976714	−	0.214544i	$-0.0688266\pi$
$401$	39.1174	1.95343	0.976714	+	0.214544i	$0.0688266\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 3.34047i	− 0.165989i
$406$	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
$421$	−33.8704	−1.65074	−0.825372	−	0.564590i	$-0.809034\pi$
$421$	−33.8704	−1.65074	−0.825372	+	0.564590i	$0.809034\pi$
$422$	0	0
$423$	20.2351i	0.983862i
$424$	0	0
$425$	39.9595i	1.93832i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−43.8704	−2.11808
$430$	0	0
$431$	−34.3643	−1.65527	−0.827636	−	0.561266i	$-0.810315\pi$
$431$	−34.3643	−1.65527	−0.827636	+	0.561266i	$0.810315\pi$
$432$	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
$435$	−1.63567	−0.0784245
$436$	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
$447$	− 7.03952i	− 0.332958i
$448$	0	0
$449$	−40.3643	−1.90491	−0.952455	−	0.304679i	$-0.901451\pi$
$449$	−40.3643	−1.90491	−0.952455	+	0.304679i	$0.901451\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	28.0061i	1.31584i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	43.3521	2.02350
$460$	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
$467$	14.3688i	0.664908i	0.943119	+	0.332454i	$0.107877\pi$
$467$	14.3688i	0.664908i	−0.943119	+	0.332454i	$0.892123\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	15.7409	0.725300
$472$	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
$485$	28.3643	1.28796
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.11738	−0.411461	−0.205731	−	0.978609i	$-0.565957\pi$
$491$	−9.11738	−0.411461	−0.205731	+	0.978609i	$0.565957\pi$
$492$	0	0
$493$	− 4.98275i	− 0.224412i
$494$	0	0
$495$	24.0445i	1.08072i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	35.8704	1.60578	0.802890	−	0.596127i	$-0.203294\pi$
$499$	35.8704	1.60578	0.802890	+	0.596127i	$0.203294\pi$
$500$	0	0
$501$	6.36433	0.284337
$502$	0	0
$503$	− 1.61501i	− 0.0720099i	−0.999352	−	0.0360049i	$-0.988537\pi$
$503$	− 1.61501i	− 0.0720099i	0.999352	−	0.0360049i	$-0.0114632\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 22.1398i	− 0.983263i
$508$	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
$515$	17.3765	0.765701
$516$	0	0
$517$	82.5552i	3.63077i
$518$	0	0
$519$	17.6357	0.774120
$520$	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
$537$	− 28.1581i	− 1.21511i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.12957	−0.263531	−0.131765	−	0.991281i	$-0.542065\pi$
$541$	−6.12957	−0.263531	−0.131765	+	0.991281i	$0.542065\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.0709i	0.945415i
$546$	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
$561$	62.1052	2.62208
$562$	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
$573$	21.8501i	0.912800i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 46.5573i	− 1.93820i	−0.246661	−	0.969102i	$-0.579333\pi$
$577$	− 46.5573i	− 1.93820i	0.246661	−	0.969102i	$-0.420667\pi$
$578$	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
$593$	41.8642i	1.71916i	0.511003	+	0.859579i	$0.329274\pi$
$593$	41.8642i	1.71916i	−0.511003	+	0.859579i	$0.670726\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.1174	−1.84345	−0.921723	−	0.387849i	$-0.873218\pi$
$599$	−45.1174	−1.84345	−0.921723	+	0.387849i	$0.873218\pi$
$600$	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
$605$	73.5005i	2.98822i
$606$	0	0
$607$	28.8896i	1.17259i	0.810097	+	0.586296i	$0.199414\pi$
$607$	28.8896i	1.17259i	−0.810097	+	0.586296i	$0.800586\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−70.3643	−2.84664
$612$	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
$631$	8.12957	0.323633	0.161817	−	0.986821i	$-0.448265\pi$
$631$	8.12957	0.323633	0.161817	+	0.986821i	$0.448265\pi$
$632$	0	0
$633$	4.54099i	0.180488i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−19.4817	−0.770684
$640$	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
$643$	− 40.1804i	− 1.58456i	−0.610158	−	0.792279i	$-0.708894\pi$
$643$	− 40.1804i	− 1.58456i	0.610158	−	0.792279i	$-0.291106\pi$
$644$	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
$657$	− 21.7812i	− 0.849766i
$658$	0	0
$659$	−30.6235	−1.19292	−0.596461	−	0.802642i	$-0.703427\pi$
$659$	−30.6235	−1.19292	−0.596461	+	0.802642i	$0.703427\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	52.9341i	2.05579i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	25.6357	0.991132
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 27.1226i	− 1.04395i
$676$	0	0
$677$	− 18.8409i	− 0.724115i	−0.932156	−	0.362058i	$-0.882074\pi$
$677$	− 18.8409i	− 0.724115i	0.932156	−	0.362058i	$-0.117926\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−25.1174	−0.962500
$682$	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
$701$	52.3643	1.97777	0.988887	−	0.148671i	$-0.0474996\pi$
$701$	52.3643	1.97777	0.988887	+	0.148671i	$0.0474996\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 32.6991i	− 1.23152i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	45.6113	1.71297	0.856484	−	0.516174i	$-0.172644\pi$
$709$	45.6113	1.71297	0.856484	+	0.516174i	$0.172644\pi$
$710$	0	0
$711$	−25.7652	−0.966272
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−83.6113	−3.12688
$716$	0	0
$717$	24.7760i	0.925278i
$718$	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
$725$	−3.11738	−0.115776
$726$	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
$729$	−21.5183	−0.796974
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 53.5968i	− 1.97964i	−0.142318	−	0.989821i	$-0.545455\pi$
$733$	− 53.5968i	− 1.97964i	0.142318	−	0.989821i	$-0.454545\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−51.6113	−1.89855	−0.949276	−	0.314445i	$-0.898182\pi$
$739$	−51.6113	−1.89855	−0.949276	+	0.314445i	$0.898182\pi$
$740$	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
$747$	14.5208i	0.531288i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−39.6113	−1.44544	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	−39.6113	−1.44544	−0.722718	+	0.691143i	$0.757107\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	53.3759i	1.94255i
$756$	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
$765$	29.0122	1.04894
$766$	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
$771$	−5.24695	−0.188964
$772$	0	0
$773$	46.9990i	1.69044i	0.534421	+	0.845218i	$0.320530\pi$
$773$	46.9990i	1.69044i	−0.534421	+	0.845218i	$0.679470\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−79.4817	−2.84408
$782$	0	0
$783$	3.38205i	0.120865i
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	− 55.7224i	− 1.98629i	−0.116892	−	0.993145i	$-0.537293\pi$
$787$	− 55.7224i	− 1.98629i	0.116892	−	0.993145i	$-0.462707\pi$
$788$	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
$797$	− 34.8247i	− 1.23355i	−0.787138	−	0.616777i	$-0.788438\pi$
$797$	− 34.8247i	− 1.23355i	0.787138	−	0.616777i	$-0.211562\pi$
$798$	0	0
$799$	99.6113	3.52399
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 88.8632i	− 3.13592i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.3643	0.575339	0.287670	−	0.957730i	$-0.407120\pi$
$809$	16.3643	0.575339	0.287670	+	0.957730i	$0.407120\pi$
$810$	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
$821$	−23.3765	−0.815846	−0.407923	−	0.913016i	$-0.633747\pi$
$821$	−23.3765	−0.815846	−0.407923	+	0.913016i	$0.633747\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	− 38.8551i	− 1.35276i
$826$	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
$835$	12.1296	0.419761
$836$	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
$841$	−28.6113	−0.986596
$842$	0	0
$843$	13.3476i	0.459714i
$844$	0	0
$845$	− 42.1956i	− 1.45157i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.3643	0.767542
$850$	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
$865$	33.6113	1.14282
$866$	0	0
$867$	− 54.9909i	− 1.86759i
$868$	0	0
$869$	−105.117	−3.56586
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 20.5936i	− 0.696989i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	38.6235	1.30274
$880$	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
$891$	−9.89482	−0.331489
$892$	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
$913$	59.2422i	1.96063i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	31.6113	1.04276	0.521380	−	0.853325i	$-0.325417\pi$
$919$	31.6113	1.04276	0.521380	+	0.853325i	$0.325417\pi$
$920$	0	0
$921$	−38.8826	−1.28123
$922$	0	0
$923$	− 67.7447i	− 2.22984i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 12.6161i	− 0.414366i
$928$	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
$935$	118.364	3.87093
$936$	0	0
$937$	− 49.3455i	− 1.61205i	−0.591883	−	0.806024i	$-0.701615\pi$
$937$	− 49.3455i	− 1.61205i	0.591883	−	0.806024i	$-0.298385\pi$
$938$	0	0
$939$	41.3765	1.35027
$940$	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
$949$	75.7409	2.45865
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	41.6434i	1.34755i
$956$	0	0
$957$	4.84504i	0.156618i
$958$	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
$975$	33.1174	1.06060
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	16.0244	0.511620
$982$	0	0
$983$	− 33.5826i	− 1.07112i	−0.844498	−	0.535559i	$-0.820101\pi$
$983$	− 33.5826i	− 1.07112i	0.844498	−	0.535559i	$-0.179899\pi$
$984$	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
$993$	9.38603i	0.297857i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.1479i	0.448069i	0.974581	+	0.224034i	$0.0719228\pi$
$997$	14.1479i	0.448069i	−0.974581	+	0.224034i	$0.928077\pi$
$998$	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.589.3	yes	4
5.2	odd	4		4900.2.a.bj.1.3		4
5.3	odd	4		4900.2.a.bj.1.2		4
5.4	even	2	inner	980.2.e.d.589.2	✓	4
7.2	even	3		980.2.q.i.949.2		8
7.3	odd	6		980.2.q.i.569.2		8
7.4	even	3		980.2.q.i.569.3		8
7.5	odd	6		980.2.q.i.949.3		8
7.6	odd	2	inner	980.2.e.d.589.2	✓	4
35.4	even	6		980.2.q.i.569.2		8
35.9	even	6		980.2.q.i.949.3		8
35.13	even	4		4900.2.a.bj.1.3		4
35.19	odd	6		980.2.q.i.949.2		8
35.24	odd	6		980.2.q.i.569.3		8
35.27	even	4		4900.2.a.bj.1.2		4
35.34	odd	2	CM	980.2.e.d.589.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.d.589.2	✓	4	5.4	even	2	inner
980.2.e.d.589.2	✓	4	7.6	odd	2	inner
980.2.e.d.589.3	yes	4	1.1	even	1	trivial
980.2.e.d.589.3	yes	4	35.34	odd	2	CM
980.2.q.i.569.2		8	7.3	odd	6
980.2.q.i.569.2		8	35.4	even	6
980.2.q.i.569.3		8	7.4	even	3
980.2.q.i.569.3		8	35.24	odd	6
980.2.q.i.949.2		8	7.2	even	3
980.2.q.i.949.2		8	35.19	odd	6
980.2.q.i.949.3		8	7.5	odd	6
980.2.q.i.949.3		8	35.9	even	6
4900.2.a.bj.1.2		4	5.3	odd	4
4900.2.a.bj.1.2		4	35.27	even	4
4900.2.a.bj.1.3		4	5.2	odd	4
4900.2.a.bj.1.3		4	35.13	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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.17325i q^{3} +2.23607i q^{5} +1.62348 q^{9} +O(q^{10})\)	\(q+1.17325i q^{3} +2.23607i q^{5} +1.62348 q^{9} +6.62348 q^{11} +5.64539i q^{13} -2.62348 q^{15} -7.99190i q^{17} -5.00000 q^{25} +5.42451i q^{27} +0.623475 q^{29} +7.77102i q^{33} -6.62348 q^{39} +3.63020i q^{45} +12.4640i q^{47} +9.37652 q^{51} +14.8105i q^{55} -12.6235 q^{65} -12.0000 q^{71} -13.4164i q^{73} -5.86627i q^{75} -15.8704 q^{79} -1.49390 q^{81} +8.94427i q^{83} +17.8704 q^{85} +0.731495i q^{87} -12.6849i q^{97} +10.7530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{9}+O(q^{10}) \) 4 * q - 14 * q^9	\( 4 q - 14 q^{9} + 6 q^{11} + 10 q^{15} - 20 q^{25} - 18 q^{29} - 6 q^{39} + 58 q^{51} - 30 q^{65} - 48 q^{71} - 2 q^{79} + 76 q^{81} + 10 q^{85} + 84 q^{99}+O(q^{100}) \) 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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