LMFDB - Embedded newform 2800.2.k.p.2351.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2800,2,Mod(2351,2800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("2800.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2800 = 2^{4} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2800.k (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:	$22.3581125660$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.31116960000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 $ x^8 + x^6 - 8x^4 + 9x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2351.8
Root			$1.62968 + 0.586627i$ of defining polynomial
Character	$\chi$	$=$	2800.2351
Dual form			2800.2.k.p.2351.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.25937 q^{3} -2.64575 q^{7} +7.62348 q^{9} +O(q^{10})$	$q+3.25937 q^{3} -2.64575 q^{7} +7.62348 q^{9} +0.359964i q^{11} +5.64539i q^{13} +7.99190i q^{17} -8.62348 q^{21} +15.0696 q^{27} +0.623475 q^{29} +1.17325i q^{33} +18.4004i q^{39} -5.71383 q^{47} +7.00000 q^{49} +26.0485i q^{51} -20.1698 q^{63} -11.8322i q^{71} -13.4164i q^{73} -0.952374i q^{77} +8.00809i q^{79} +26.2470 q^{81} +15.8745 q^{83} +2.03214 q^{87} -14.9363i q^{91} +12.6849i q^{97} +2.74417i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 20 q^{9}+O(q^{10}) $ 8 * q + 20 * q^9	$ 8 q + 20 q^{9} - 28 q^{21} - 36 q^{29} + 56 q^{49} + 128 q^{81}+O(q^{100}) $ 8 * q + 20 * q^9 - 28 * q^21 - 36 * q^29 + 56 * q^49 + 128 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$.

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.25937	1.88180	0.940898	−	0.338689i	$-0.109984\pi$
$3$	3.25937	1.88180	0.940898	+	0.338689i	$0.109984\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.64575	−1.00000
$7$	−2.64575	−1.00000
$8$	0	0
$9$	7.62348	2.54116
$10$	0	0
$11$	0.359964i	0.108533i	0.998526	+	0.0542666i	$0.0172821\pi$
$11$	0.359964i	0.108533i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	7.99190i	1.93832i	0.246433	+	0.969160i	$0.420742\pi$
$17$	7.99190i	1.93832i	−0.246433	+	0.969160i	$0.579258\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−8.62348	−1.88180
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	15.0696	2.90015
$28$	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
$33$	1.17325i	0.204237i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	18.4004i	2.94642i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.71383	−0.833448	−0.416724	−	0.909033i	$-0.636822\pi$
$47$	−5.71383	−0.833448	−0.416724	+	0.909033i	$0.636822\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	26.0485i	3.64752i
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	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	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−20.1698	−2.54116
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 11.8322i	− 1.40422i	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	− 11.8322i	− 1.40422i	0.712069	−	0.702109i	$-0.247758\pi$
$72$	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
$75$	0	0
$76$	0	0
$77$	− 0.952374i	− 0.108533i
$78$	0	0
$79$	8.00809i	0.900981i	0.892781	+	0.450490i	$0.148751\pi$
$79$	8.00809i	0.900981i	−0.892781	+	0.450490i	$0.851249\pi$
$80$	0	0
$81$	26.2470	2.91633
$82$	0	0
$83$	15.8745	1.74245	0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.03214	0.217868
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 14.9363i	− 1.56575i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.6849i	1.28796i	0.765043	+	0.643979i	$0.222718\pi$
$97$	12.6849i	1.28796i	−0.765043	+	0.643979i	$0.777282\pi$
$98$	0	0
$99$	2.74417i	0.275800i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	18.7513	1.84762	0.923810	−	0.382851i	$-0.125058\pi$
$103$	18.7513	1.84762	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	$-0.656723\pi$
$109$	−9.87043	−0.945415	−0.472708	+	0.881219i	$0.656723\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	43.0375i	3.97882i
$118$	0	0
$119$	− 21.1446i	− 1.93832i
$120$	0	0
$121$	10.8704	0.988221
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	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
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−18.6235	−1.56838
$142$	0	0
$143$	−2.03214	−0.169936
$144$	0	0
$145$	0	0
$146$	0	0
$147$	22.8156	1.88180
$148$	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
$151$	5.84831i	0.475929i	0.971274	+	0.237964i	$0.0764802\pi$
$151$	5.84831i	0.475929i	−0.971274	+	0.237964i	$0.923520\pi$
$152$	0	0
$153$	60.9260i	4.92558i
$154$	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
$165$	0	0
$166$	0	0
$167$	25.2700	1.95545	0.977727	−	0.209881i	$-0.0673075\pi$
$167$	25.2700	1.95545	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	$0.193603\pi$
$173$	15.0314i	1.14282i	−0.820666	+	0.571409i	$0.806397\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 11.8322i	− 0.884377i	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	− 11.8322i	− 0.884377i	0.896922	−	0.442189i	$-0.145798\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.87679	−0.210372
$188$	0	0
$189$	−39.8704	−2.90015
$190$	0	0
$191$	− 20.4246i	− 1.47788i	−0.673774	−	0.738938i	$-0.735328\pi$
$191$	− 20.4246i	− 1.47788i	0.673774	−	0.738938i	$-0.264672\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	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
$203$	−1.64956	−0.115776
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 28.7927i	− 1.98217i	−0.133226	−	0.991086i	$-0.542533\pi$
$211$	− 28.7927i	− 1.98217i	0.133226	−	0.991086i	$-0.457467\pi$
$212$	0	0
$213$	− 38.5654i	− 2.64245i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 43.7290i	− 2.95493i
$220$	0	0
$221$	−45.1174	−3.03492
$222$	0	0
$223$	−20.3611	−1.36348	−0.681740	−	0.731594i	$-0.738777\pi$
$223$	−20.3611	−1.36348	−0.681740	+	0.731594i	$0.738777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.2058	−1.40748	−0.703738	−	0.710460i	$-0.748487\pi$
$227$	−21.2058	−1.40748	−0.703738	+	0.710460i	$0.748487\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	− 3.10414i	− 0.204237i
$232$	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
$237$	26.1013i	1.69546i
$238$	0	0
$239$	− 22.5844i	− 1.46087i	−0.682985	−	0.730433i	$-0.739318\pi$
$239$	− 22.5844i	− 1.46087i	0.682985	−	0.730433i	$-0.260682\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	40.3396	2.58779
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	51.7409	3.27894
$250$	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
$255$	0	0
$256$	0	0
$257$	− 4.47214i	− 0.278964i	−0.990225	−	0.139482i	$-0.955456\pi$
$257$	− 4.47214i	− 0.278964i	0.990225	−	0.139482i	$-0.0445438\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.75305	0.294206
$262$	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	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	− 48.6829i	− 2.94642i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	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
$283$	−27.7245	−1.64805	−0.824025	−	0.566553i	$-0.808277\pi$
$283$	−27.7245	−1.64805	−0.824025	+	0.566553i	$0.808277\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−46.8704	−2.75708
$290$	0	0
$291$	41.3448i	2.42367i
$292$	0	0
$293$	32.9200i	1.92320i	0.274446	+	0.961602i	$0.411505\pi$
$293$	32.9200i	1.92320i	−0.274446	+	0.961602i	$0.588495\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.42451i	0.314762i
$298$	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
$307$	−11.3879	−0.649942	−0.324971	−	0.945724i	$-0.605355\pi$
$307$	−11.3879	−0.649942	−0.324971	+	0.945724i	$0.605355\pi$
$308$	0	0
$309$	61.1174	3.47685
$310$	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		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0.224428i	0.0125656i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−32.1713	−1.77908
$328$	0	0
$329$	15.1174	0.833448
$330$	0	0
$331$	− 35.4965i	− 1.95106i	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	− 35.4965i	− 1.95106i	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		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.5203	−1.00000
$344$	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	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	85.0738i	4.54090i
$352$	0	0
$353$	6.08715i	0.323986i	0.986792	+	0.161993i	$0.0517922\pi$
$353$	6.08715i	0.323986i	−0.986792	+	0.161993i	$0.948208\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 68.9179i	− 3.64752i
$358$	0	0
$359$	11.8322i	0.624477i	0.950004	+	0.312239i	$0.101079\pi$
$359$	11.8322i	0.624477i	−0.950004	+	0.312239i	$0.898921\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	35.4307	1.85963
$364$	0	0
$365$	0	0
$366$	0	0
$367$	38.3075	1.99964	0.999818	−	0.0190919i	$-0.00607750\pi$
$367$	38.3075	1.99964	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		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.51976i	0.181277i
$378$	0	0
$379$	35.4965i	1.82333i	0.410932	+	0.911666i	$0.365203\pi$
$379$	35.4965i	1.82333i	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.8745	−0.811149	−0.405575	−	0.914062i	$-0.632929\pi$
$383$	−15.8745	−0.811149	−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	$-0.285413\pi$
$389$	24.6235	1.24846	0.624230	+	0.781241i	$0.285413\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 19.7244i	− 0.989941i	−0.868910	−	0.494971i	$-0.835179\pi$
$397$	− 19.7244i	− 0.989941i	0.868910	−	0.494971i	$-0.164821\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−39.1174	−1.95343	−0.976714	−	0.214544i	$-0.931173\pi$
$401$	−39.1174	−1.95343	−0.976714	+	0.214544i	$0.931173\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	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	$-0.190966\pi$
$421$	33.8704	1.65074	0.825372	+	0.564590i	$0.190966\pi$
$422$	0	0
$423$	−43.5593	−2.11792
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.62348	−0.319784
$430$	0	0
$431$	− 23.3044i	− 1.12253i	−0.827636	−	0.561266i	$-0.810315\pi$
$431$	− 23.3044i	− 1.12253i	0.827636	−	0.561266i	$-0.189685\pi$
$432$	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
$435$	0	0
$436$	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
$441$	53.3643	2.54116
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−19.5562	−0.924977
$448$	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
$453$	19.0618i	0.895601i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	120.435i	5.62141i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	40.7620	1.88624	0.943119	−	0.332454i	$-0.107877\pi$
$467$	40.7620	1.88624	0.943119	+	0.332454i	$0.107877\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 43.7290i	− 2.01493i
$472$	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
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	43.3690i	1.95722i	0.205731	+	0.978609i	$0.434043\pi$
$491$	43.3690i	1.95722i	−0.205731	+	0.978609i	$0.565957\pi$
$492$	0	0
$493$	4.98275i	0.224412i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.3050i	1.40422i
$498$	0	0
$499$	− 26.6329i	− 1.19225i	−0.802890	−	0.596127i	$-0.796706\pi$
$499$	− 26.6329i	− 1.19225i	0.802890	−	0.596127i	$-0.203294\pi$
$500$	0	0
$501$	82.3643	3.67977
$502$	0	0
$503$	44.8262	1.99870	0.999352	−	0.0360049i	$-0.0114632\pi$
$503$	44.8262	1.99870	0.999352	+	0.0360049i	$0.0114632\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−61.5057	−2.73156
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	35.4965i	1.57027i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.05677i	− 0.0904567i
$518$	0	0
$519$	48.9929i	2.15055i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	37.0405	1.61967	0.809834	−	0.586659i	$-0.199557\pi$
$523$	37.0405	1.61967	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
$537$	− 38.5654i	− 1.66422i
$538$	0	0
$539$	2.51975i	0.108533i
$540$	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
$545$	0	0
$546$	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
$553$	− 21.1874i	− 0.900981i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9.37652	−0.395877
$562$	0	0
$563$	15.8745	0.669031	0.334515	−	0.942390i	$-0.391427\pi$
$563$	15.8745	0.669031	0.334515	+	0.942390i	$0.391427\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−69.4429	−2.91633
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	− 35.4965i	− 1.48548i	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	− 35.4965i	− 1.48548i	0.669579	−	0.742741i	$-0.266474\pi$
$572$	0	0
$573$	− 66.5714i	− 2.78106i
$574$	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
$581$	−42.0000	−1.74245
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.6235	1.96563	0.982817	−	0.184585i	$-0.0590940\pi$
$587$	47.6235	1.96563	0.982817	+	0.184585i	$0.0590940\pi$
$588$	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
$599$	18.9848i	0.775698i	0.921723	+	0.387849i	$0.126782\pi$
$599$	18.9848i	0.775698i	−0.921723	+	0.387849i	$0.873218\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−39.9173	−1.62019	−0.810097	−	0.586296i	$-0.800586\pi$
$607$	−39.9173	−1.62019	−0.810097	+	0.586296i	$0.800586\pi$
$608$	0	0
$609$	−5.37652	−0.217868
$610$	0	0
$611$	− 32.2568i	− 1.30497i
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	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
$619$	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
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 49.5773i	− 1.97364i	−0.161817	−	0.986821i	$-0.551735\pi$
$631$	− 49.5773i	− 1.97364i	0.161817	−	0.986821i	$-0.448265\pi$
$632$	0	0
$633$	− 93.8460i	− 3.73004i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	39.5177i	1.56575i
$638$	0	0
$639$	− 90.2022i	− 3.56834i
$640$	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
$643$	−30.9441	−1.22032	−0.610158	−	0.792279i	$-0.708894\pi$
$643$	−30.9441	−1.22032	−0.610158	+	0.792279i	$0.708894\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.6235	−1.87227	−0.936137	−	0.351636i	$-0.885626\pi$
$647$	−47.6235	−1.87227	−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		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 102.280i	− 3.99031i
$658$	0	0
$659$	41.2093i	1.60528i	0.596461	+	0.802642i	$0.296573\pi$
$659$	41.2093i	1.60528i	−0.596461	+	0.802642i	$0.703427\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	−147.054	−5.71111
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−66.3643	−2.56579
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	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
$679$	− 33.5611i	− 1.28796i
$680$	0	0
$681$	−69.1174	−2.64858
$682$	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
$693$	− 7.26040i	− 0.275800i
$694$	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
$705$	0	0
$706$	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
$711$	61.0495i	2.28954i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 73.6109i	− 2.74905i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−49.6113	−1.84762
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−5.29150	−0.196251	−0.0981255	−	0.995174i	$-0.531285\pi$
$727$	−5.29150	−0.196251	−0.0981255	+	0.995174i	$0.531285\pi$
$728$	0	0
$729$	52.7409	1.95336
$730$	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
$739$	− 17.0961i	− 0.628889i	−0.949276	−	0.314445i	$-0.898182\pi$
$739$	− 17.0961i	− 0.628889i	0.949276	−	0.314445i	$-0.101818\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	121.019	4.42785
$748$	0	0
$749$	0	0
$750$	0	0
$751$	37.8807i	1.38229i	0.722718	+	0.691143i	$0.242893\pi$
$751$	37.8807i	1.38229i	−0.722718	+	0.691143i	$0.757107\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	26.1147	0.945415
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	− 14.5763i	− 0.524954i
$772$	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
$781$	4.25915	0.152404
$782$	0	0
$783$	9.39553	0.335769
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.55849	0.233785	0.116892	−	0.993145i	$-0.462707\pi$
$787$	6.55849	0.233785	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	$0.211562\pi$
$797$	34.8247i	1.23355i	−0.787138	+	0.616777i	$0.788438\pi$
$798$	0	0
$799$	− 45.6644i	− 1.61549i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.82942	0.170427
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.3643	−0.575339	−0.287670	−	0.957730i	$-0.592880\pi$
$809$	−16.3643	−0.575339	−0.287670	+	0.957730i	$0.592880\pi$
$810$	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
$819$	− 113.866i	− 3.97882i
$820$	0	0
$821$	23.3765	0.815846	0.407923	−	0.913016i	$-0.366253\pi$
$821$	23.3765	0.815846	0.407923	+	0.913016i	$0.366253\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	55.9433i	1.93832i
$834$	0	0
$835$	0	0
$836$	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
$843$	37.0803	1.27711
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−28.7604	−0.988221
$848$	0	0
$849$	−90.3643	−3.10130
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 40.2492i	− 1.37811i	−0.724710	−	0.689054i	$-0.758026\pi$
$853$	− 40.2492i	− 1.37811i	0.724710	−	0.689054i	$-0.241974\pi$
$854$	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
$865$	0	0
$866$	0	0
$867$	−152.768	−5.18827
$868$	0	0
$869$	−2.88262	−0.0977863
$870$	0	0
$871$	0	0
$872$	0	0
$873$	96.7031i	3.27290i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	107.298i	3.61908i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.6235	−1.59904	−0.799521	−	0.600639i	$-0.794913\pi$
$887$	−47.6235	−1.59904	−0.799521	+	0.600639i	$0.794913\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.44795i	0.316518i
$892$	0	0
$893$	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
$903$	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
$911$	− 59.1608i	− 1.96008i	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	− 59.1608i	− 1.96008i	0.198789	−	0.980042i	$-0.436299\pi$
$912$	0	0
$913$	5.71425i	0.189114i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 51.7371i	− 1.70665i	−0.521380	−	0.853325i	$-0.674583\pi$
$919$	− 51.7371i	− 1.70665i	0.521380	−	0.853325i	$-0.325417\pi$
$920$	0	0
$921$	−37.1174	−1.22306
$922$	0	0
$923$	66.7972	2.19866
$924$	0	0
$925$	0	0
$926$	0	0
$927$	142.950	4.69510
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.3455i	1.61205i	0.591883	+	0.806024i	$0.298385\pi$
$937$	49.3455i	1.61205i	−0.591883	+	0.806024i	$0.701615\pi$
$938$	0	0
$939$	− 114.946i	− 3.75113i
$940$	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	1.00000			$0$
$947$	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		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.731495i	0.0236459i
$958$	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
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−75.2470	−2.40245
$982$	0	0
$983$	52.9548	1.68900	0.844498	−	0.535559i	$-0.179899\pi$
$983$	52.9548	1.68900	0.844498	+	0.535559i	$0.179899\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	49.2731	1.56838
$988$	0	0
$989$	0	0
$990$	0	0
$991$	35.4965i	1.12758i	0.825917	+	0.563791i	$0.190658\pi$
$991$	35.4965i	1.12758i	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	− 115.696i	− 3.67150i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 14.1479i	− 0.448069i	−0.974581	−	0.224034i	$-0.928077\pi$
$997$	− 14.1479i	− 0.448069i	0.974581	−	0.224034i	$-0.0719228\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2800.2.k.p.2351.8		8
4.3	odd	2	inner	2800.2.k.p.2351.1		8
5.2	odd	4		560.2.e.c.559.1	✓	8
5.3	odd	4		560.2.e.c.559.8	yes	8
5.4	even	2	inner	2800.2.k.p.2351.2		8
7.6	odd	2	inner	2800.2.k.p.2351.2		8
20.3	even	4		560.2.e.c.559.2	yes	8
20.7	even	4		560.2.e.c.559.7	yes	8
20.19	odd	2	inner	2800.2.k.p.2351.7		8
28.27	even	2	inner	2800.2.k.p.2351.7		8
35.13	even	4		560.2.e.c.559.1	✓	8
35.27	even	4		560.2.e.c.559.8	yes	8
35.34	odd	2	CM	2800.2.k.p.2351.8		8
40.3	even	4		2240.2.e.d.2239.7		8
40.13	odd	4		2240.2.e.d.2239.1		8
40.27	even	4		2240.2.e.d.2239.2		8
40.37	odd	4		2240.2.e.d.2239.8		8
140.27	odd	4		560.2.e.c.559.2	yes	8
140.83	odd	4		560.2.e.c.559.7	yes	8
140.139	even	2	inner	2800.2.k.p.2351.1		8
280.13	even	4		2240.2.e.d.2239.8		8
280.27	odd	4		2240.2.e.d.2239.7		8
280.83	odd	4		2240.2.e.d.2239.2		8
280.237	even	4		2240.2.e.d.2239.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.e.c.559.1	✓	8	5.2	odd	4
560.2.e.c.559.1	✓	8	35.13	even	4
560.2.e.c.559.2	yes	8	20.3	even	4
560.2.e.c.559.2	yes	8	140.27	odd	4
560.2.e.c.559.7	yes	8	20.7	even	4
560.2.e.c.559.7	yes	8	140.83	odd	4
560.2.e.c.559.8	yes	8	5.3	odd	4
560.2.e.c.559.8	yes	8	35.27	even	4
2240.2.e.d.2239.1		8	40.13	odd	4
2240.2.e.d.2239.1		8	280.237	even	4
2240.2.e.d.2239.2		8	40.27	even	4
2240.2.e.d.2239.2		8	280.83	odd	4
2240.2.e.d.2239.7		8	40.3	even	4
2240.2.e.d.2239.7		8	280.27	odd	4
2240.2.e.d.2239.8		8	40.37	odd	4
2240.2.e.d.2239.8		8	280.13	even	4
2800.2.k.p.2351.1		8	4.3	odd	2	inner
2800.2.k.p.2351.1		8	140.139	even	2	inner
2800.2.k.p.2351.2		8	5.4	even	2	inner
2800.2.k.p.2351.2		8	7.6	odd	2	inner
2800.2.k.p.2351.7		8	20.19	odd	2	inner
2800.2.k.p.2351.7		8	28.27	even	2	inner
2800.2.k.p.2351.8		8	1.1	even	1	trivial
2800.2.k.p.2351.8		8	35.34	odd	2	CM

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:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2800.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.25937 q^{3} -2.64575 q^{7} +7.62348 q^{9} +O(q^{10})\)	\(q+3.25937 q^{3} -2.64575 q^{7} +7.62348 q^{9} +0.359964i q^{11} +5.64539i q^{13} +7.99190i q^{17} -8.62348 q^{21} +15.0696 q^{27} +0.623475 q^{29} +1.17325i q^{33} +18.4004i q^{39} -5.71383 q^{47} +7.00000 q^{49} +26.0485i q^{51} -20.1698 q^{63} -11.8322i q^{71} -13.4164i q^{73} -0.952374i q^{77} +8.00809i q^{79} +26.2470 q^{81} +15.8745 q^{83} +2.03214 q^{87} -14.9363i q^{91} +12.6849i q^{97} +2.74417i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 20 q^{9}+O(q^{10}) \) 8 * q + 20 * q^9	\( 8 q + 20 q^{9} - 28 q^{21} - 36 q^{29} + 56 q^{49} + 128 q^{81}+O(q^{100}) \) 8 * q + 20 * q^9 - 28 * q^21 - 36 * q^29 + 56 * q^49 + 128 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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