LMFDB - Embedded newform 2240.2.e.b.2239.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 5x^{2} + 2x + 4 $ x^4 - x^3 + 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2239.4
Root			$-0.309017 - 0.817582i$ of defining polynomial
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.b.2239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575i q^{3} +2.23607 q^{5} +2.64575i q^{7} -4.00000 q^{9} +O(q^{10})$	$q+2.64575i q^{3} +2.23607 q^{5} +2.64575i q^{7} -4.00000 q^{9} -5.91608i q^{11} +6.70820 q^{13} +5.91608i q^{15} +2.23607 q^{17} -7.00000 q^{21} +5.00000 q^{25} -2.64575i q^{27} +9.00000 q^{29} +15.6525 q^{33} +5.91608i q^{35} +17.7482i q^{39} -8.94427 q^{45} +7.93725i q^{47} -7.00000 q^{49} +5.91608i q^{51} -13.2288i q^{55} -10.5830i q^{63} +15.0000 q^{65} +11.8322i q^{71} -13.4164 q^{73} +13.2288i q^{75} +15.6525 q^{77} -17.7482i q^{79} -5.00000 q^{81} -15.8745i q^{83} +5.00000 q^{85} +23.8118i q^{87} +17.7482i q^{91} -6.70820 q^{97} +23.6643i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{9}+O(q^{10}) $ 4 * q - 16 * q^9	$ 4 q - 16 q^{9} - 28 q^{21} + 20 q^{25} + 36 q^{29} - 28 q^{49} + 60 q^{65} - 20 q^{81} + 20 q^{85}+O(q^{100}) $ 4 * q - 16 * q^9 - 28 * q^21 + 20 * q^25 + 36 * q^29 - 28 * q^49 + 60 * q^65 - 20 * q^81 + 20 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	2.64575i	1.52753i	0.645497	+	0.763763i	$0.276650\pi$
$3$	2.64575i	1.52753i	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	2.23607	1.00000
$5$	2.23607	1.00000
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	0	0
$9$	−4.00000	−1.33333
$10$	0	0
$11$	− 5.91608i	− 1.78377i	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	− 5.91608i	− 1.78377i	0.452267	−	0.891883i	$-0.350615\pi$
$12$	0	0
$13$	6.70820	1.86052	0.930261	−	0.366900i	$-0.119581\pi$
$13$	6.70820	1.86052	0.930261	+	0.366900i	$0.119581\pi$
$14$	0	0
$15$	5.91608i	1.52753i
$16$	0	0
$17$	2.23607	0.542326	0.271163	−	0.962533i	$-0.412592\pi$
$17$	2.23607	0.542326	0.271163	+	0.962533i	$0.412592\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−7.00000	−1.52753
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 2.64575i	− 0.509175i
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	15.6525	2.72475
$34$	0	0
$35$	5.91608i	1.00000i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	17.7482i	2.84199i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−8.94427	−1.33333
$46$	0	0
$47$	7.93725i	1.15777i	0.815410	+	0.578884i	$0.196511\pi$
$47$	7.93725i	1.15777i	−0.815410	+	0.578884i	$0.803489\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	5.91608i	0.828417i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 13.2288i	− 1.78377i
$56$	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
$63$	− 10.5830i	− 1.33333i
$64$	0	0
$65$	15.0000	1.86052
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.8322i	1.40422i	0.712069	+	0.702109i	$0.247758\pi$
$71$	11.8322i	1.40422i	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−13.4164	−1.57027	−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164	−1.57027	−0.785136	+	0.619324i	$0.787407\pi$
$74$	0	0
$75$	13.2288i	1.52753i
$76$	0	0
$77$	15.6525	1.78377
$78$	0	0
$79$	− 17.7482i	− 1.99683i	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	− 17.7482i	− 1.99683i	0.0562544	−	0.998416i	$-0.482084\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	− 15.8745i	− 1.74245i	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	− 15.8745i	− 1.74245i	0.490881	−	0.871227i	$-0.336675\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	23.8118i	2.55289i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	17.7482i	1.86052i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.70820	−0.681115	−0.340557	−	0.940224i	$-0.610616\pi$
$97$	−6.70820	−0.681115	−0.340557	+	0.940224i	$0.610616\pi$
$98$	0	0
$99$	23.6643i	2.37835i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 2.64575i	− 0.260694i	−0.991468	−	0.130347i	$-0.958391\pi$
$103$	− 2.64575i	− 0.260694i	0.991468	−	0.130347i	$-0.0416091\pi$
$104$	0	0
$105$	−15.6525	−1.52753
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	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
$117$	−26.8328	−2.48069
$118$	0	0
$119$	5.91608i	0.542326i
$120$	0	0
$121$	−24.0000	−2.18182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	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
$135$	− 5.91608i	− 0.509175i
$136$	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
$141$	−21.0000	−1.76852
$142$	0	0
$143$	− 39.6863i	− 3.31873i
$144$	0	0
$145$	20.1246	1.67126
$146$	0	0
$147$	− 18.5203i	− 1.52753i
$148$	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
$151$	− 17.7482i	− 1.44433i	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	− 17.7482i	− 1.44433i	0.691720	−	0.722166i	$-0.256853\pi$
$152$	0	0
$153$	−8.94427	−0.723102
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13.4164	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	35.0000	2.72475
$166$	0	0
$167$	7.93725i	0.614203i	0.951677	+	0.307102i	$0.0993591\pi$
$167$	7.93725i	0.614203i	−0.951677	+	0.307102i	$0.900641\pi$
$168$	0	0
$169$	32.0000	2.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.1803	−0.850026	−0.425013	−	0.905187i	$-0.639730\pi$
$173$	−11.1803	−0.850026	−0.425013	+	0.905187i	$0.639730\pi$
$174$	0	0
$175$	13.2288i	1.00000i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.8322i	0.884377i	0.896922	+	0.442189i	$0.145798\pi$
$179$	11.8322i	0.884377i	−0.896922	+	0.442189i	$0.854202\pi$
$180$	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
$187$	− 13.2288i	− 0.967382i
$188$	0	0
$189$	7.00000	0.509175
$190$	0	0
$191$	5.91608i	0.428073i	0.976826	+	0.214036i	$0.0686611\pi$
$191$	5.91608i	0.428073i	−0.976826	+	0.214036i	$0.931339\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	39.6863i	2.84199i
$196$	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
$203$	23.8118i	1.67126i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	17.7482i	1.22184i	0.791693	+	0.610920i	$0.209200\pi$
$211$	17.7482i	1.22184i	−0.791693	+	0.610920i	$0.790800\pi$
$212$	0	0
$213$	−31.3050	−2.14498
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 35.4965i	− 2.39863i
$220$	0	0
$221$	15.0000	1.00901
$222$	0	0
$223$	29.1033i	1.94890i	0.224607	+	0.974449i	$0.427890\pi$
$223$	29.1033i	1.94890i	−0.224607	+	0.974449i	$0.572110\pi$
$224$	0	0
$225$	−20.0000	−1.33333
$226$	0	0
$227$	− 7.93725i	− 0.526814i	−0.964685	−	0.263407i	$-0.915154\pi$
$227$	− 7.93725i	− 0.526814i	0.964685	−	0.263407i	$-0.0848462\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	41.4126i	2.72475i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	17.7482i	1.15777i
$236$	0	0
$237$	46.9574	3.05021
$238$	0	0
$239$	29.5804i	1.91340i	0.291081	+	0.956698i	$0.405985\pi$
$239$	29.5804i	1.91340i	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	− 21.1660i	− 1.35780i
$244$	0	0
$245$	−15.6525	−1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	42.0000	2.66164
$250$	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
$255$	13.2288i	0.828417i
$256$	0	0
$257$	4.47214	0.278964	0.139482	−	0.990225i	$-0.455456\pi$
$257$	4.47214	0.278964	0.139482	+	0.990225i	$0.455456\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−36.0000	−2.22834
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	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
$273$	−46.9574	−2.84199
$274$	0	0
$275$	− 29.5804i	− 1.78377i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−33.0000	−1.96861	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000	−1.96861	−0.984307	+	0.176462i	$0.943535\pi$
$282$	0	0
$283$	2.64575i	0.157274i	0.996903	+	0.0786368i	$0.0250567\pi$
$283$	2.64575i	0.157274i	−0.996903	+	0.0786368i	$0.974943\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−12.0000	−0.705882
$290$	0	0
$291$	− 17.7482i	− 1.04042i
$292$	0	0
$293$	24.5967	1.43696	0.718479	−	0.695549i	$-0.244839\pi$
$293$	24.5967	1.43696	0.718479	+	0.695549i	$0.244839\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.6525	−0.908249
$298$	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
$307$	34.3948i	1.96301i	0.191429	+	0.981507i	$0.438688\pi$
$307$	34.3948i	1.96301i	−0.191429	+	0.981507i	$0.561312\pi$
$308$	0	0
$309$	7.00000	0.398216
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	20.1246	1.13751	0.568755	−	0.822507i	$-0.307425\pi$
$313$	20.1246	1.13751	0.568755	+	0.822507i	$0.307425\pi$
$314$	0	0
$315$	− 23.6643i	− 1.33333i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	− 53.2447i	− 2.98113i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	33.5410	1.86052
$326$	0	0
$327$	− 29.1033i	− 1.60941i
$328$	0	0
$329$	−21.0000	−1.15777
$330$	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	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 18.5203i	− 1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	− 17.7482i	− 0.947331i
$352$	0	0
$353$	29.0689	1.54718	0.773590	−	0.633686i	$-0.218459\pi$
$353$	29.0689	1.54718	0.773590	+	0.633686i	$0.218459\pi$
$354$	0	0
$355$	26.4575i	1.40422i
$356$	0	0
$357$	−15.6525	−0.828417
$358$	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
$363$	− 63.4980i	− 3.33278i
$364$	0	0
$365$	−30.0000	−1.57027
$366$	0	0
$367$	18.5203i	0.966750i	0.875413	+	0.483375i	$0.160589\pi$
$367$	18.5203i	0.966750i	−0.875413	+	0.483375i	$0.839411\pi$
$368$	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
$375$	29.5804i	1.52753i
$376$	0	0
$377$	60.3738	3.10941
$378$	0	0
$379$	− 35.4965i	− 1.82333i	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	− 35.4965i	− 1.82333i	0.410932	−	0.911666i	$-0.365203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 15.8745i	− 0.811149i	−0.914062	−	0.405575i	$-0.867071\pi$
$383$	− 15.8745i	− 0.811149i	0.914062	−	0.405575i	$-0.132929\pi$
$384$	0	0
$385$	35.0000	1.78377
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−39.0000	−1.97738	−0.988689	−	0.149979i	$-0.952080\pi$
$389$	−39.0000	−1.97738	−0.988689	+	0.149979i	$0.952080\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 39.6863i	− 1.99683i
$396$	0	0
$397$	−20.1246	−1.01003	−0.505013	−	0.863112i	$-0.668512\pi$
$397$	−20.1246	−1.01003	−0.505013	+	0.863112i	$0.668512\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−11.1803	−0.555556
$406$	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
$415$	− 35.4965i	− 1.74245i
$416$	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
$421$	37.0000	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$421$	37.0000	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$422$	0	0
$423$	− 31.7490i	− 1.54369i
$424$	0	0
$425$	11.1803	0.542326
$426$	0	0
$427$	0	0
$428$	0	0
$429$	105.000	5.06945
$430$	0	0
$431$	− 41.4126i	− 1.99477i	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	− 41.4126i	− 1.99477i	0.0722525	−	0.997386i	$-0.476981\pi$
$432$	0	0
$433$	40.2492	1.93425	0.967127	−	0.254293i	$-0.0818429\pi$
$433$	40.2492	1.93425	0.967127	+	0.254293i	$0.0818429\pi$
$434$	0	0
$435$	53.2447i	2.55289i
$436$	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
$441$	28.0000	1.33333
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 15.8745i	− 0.750838i
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	46.9574	2.20625
$454$	0	0
$455$	39.6863i	1.86052i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	− 5.91608i	− 0.276139i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.93725i	− 0.367292i	−0.982992	−	0.183646i	$-0.941210\pi$
$467$	− 7.93725i	− 0.367292i	0.982992	−	0.183646i	$-0.0587901\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 35.4965i	− 1.63559i
$472$	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
$485$	−15.0000	−0.681115
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 29.5804i	− 1.33494i	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	− 29.5804i	− 1.33494i	0.744635	−	0.667472i	$-0.232624\pi$
$492$	0	0
$493$	20.1246	0.906367
$494$	0	0
$495$	52.9150i	2.37835i
$496$	0	0
$497$	−31.3050	−1.40422
$498$	0	0
$499$	17.7482i	0.794520i	0.917706	+	0.397260i	$0.130039\pi$
$499$	17.7482i	0.794520i	−0.917706	+	0.397260i	$0.869961\pi$
$500$	0	0
$501$	−21.0000	−0.938211
$502$	0	0
$503$	− 23.8118i	− 1.06171i	−0.847461	−	0.530857i	$-0.821870\pi$
$503$	− 23.8118i	− 1.06171i	0.847461	−	0.530857i	$-0.178130\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	84.6640i	3.76006i
$508$	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
$515$	− 5.91608i	− 0.260694i
$516$	0	0
$517$	46.9574	2.06519
$518$	0	0
$519$	− 29.5804i	− 1.29844i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 37.0405i	− 1.61967i	−0.586659	−	0.809834i	$-0.699557\pi$
$523$	− 37.0405i	− 1.61967i	0.586659	−	0.809834i	$-0.300443\pi$
$524$	0	0
$525$	−35.0000	−1.52753
$526$	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
$537$	−31.3050	−1.35091
$538$	0	0
$539$	41.4126i	1.78377i
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−24.5967	−1.05361
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	46.9574	1.99683
$554$	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
$561$	35.0000	1.47770
$562$	0	0
$563$	− 15.8745i	− 0.669031i	−0.942390	−	0.334515i	$-0.891427\pi$
$563$	− 15.8745i	− 0.669031i	0.942390	−	0.334515i	$-0.108573\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 13.2288i	− 0.555556i
$568$	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
$571$	− 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
$573$	−15.6525	−0.653892
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−33.5410	−1.39633	−0.698165	−	0.715936i	$-0.746000\pi$
$577$	−33.5410	−1.39633	−0.698165	+	0.715936i	$0.746000\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	42.0000	1.74245
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−60.0000	−2.48069
$586$	0	0
$587$	47.6235i	1.96563i	0.184585	+	0.982817i	$0.440906\pi$
$587$	47.6235i	1.96563i	−0.184585	+	0.982817i	$0.559094\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−42.4853	−1.74466	−0.872331	−	0.488916i	$-0.837392\pi$
$593$	−42.4853	−1.74466	−0.872331	+	0.488916i	$0.837392\pi$
$594$	0	0
$595$	13.2288i	0.542326i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.5804i	1.20862i	0.796748	+	0.604311i	$0.206552\pi$
$599$	29.5804i	1.20862i	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−53.6656	−2.18182
$606$	0	0
$607$	− 44.9778i	− 1.82559i	−0.408416	−	0.912796i	$-0.633919\pi$
$607$	− 44.9778i	− 1.82559i	0.408416	−	0.912796i	$-0.366081\pi$
$608$	0	0
$609$	−63.0000	−2.55289
$610$	0	0
$611$	53.2447i	2.15405i
$612$	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
$631$	− 17.7482i	− 0.706546i	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	− 17.7482i	− 0.706546i	0.935520	−	0.353273i	$-0.114931\pi$
$632$	0	0
$633$	−46.9574	−1.86639
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−46.9574	−1.86052
$638$	0	0
$639$	− 47.3286i	− 1.87229i
$640$	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
$643$	− 50.2693i	− 1.98243i	−0.132273	−	0.991213i	$-0.542228\pi$
$643$	− 50.2693i	− 1.98243i	0.132273	−	0.991213i	$-0.457772\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	47.6235i	1.87227i	0.351636	+	0.936137i	$0.385626\pi$
$647$	47.6235i	1.87227i	−0.351636	+	0.936137i	$0.614374\pi$
$648$	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
$657$	53.6656	2.09370
$658$	0	0
$659$	− 5.91608i	− 0.230458i	−0.993339	−	0.115229i	$-0.963240\pi$
$659$	− 5.91608i	− 0.230458i	0.993339	−	0.115229i	$-0.0367601\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	39.6863i	1.54129i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−77.0000	−2.97699
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 13.2288i	− 0.509175i
$676$	0	0
$677$	51.4296	1.97660	0.988299	−	0.152527i	$-0.0487410\pi$
$677$	51.4296	1.97660	0.988299	+	0.152527i	$0.0487410\pi$
$678$	0	0
$679$	− 17.7482i	− 0.681115i
$680$	0	0
$681$	21.0000	0.804722
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	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
$693$	−62.6099	−2.37835
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.0000	1.24639	0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000	1.24639	0.623196	+	0.782065i	$0.285834\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−46.9574	−1.76852
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	70.9930i	2.66244i
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 88.7412i	− 3.31873i
$716$	0	0
$717$	−78.2624	−2.92276
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	0	0
$723$	0	0
$724$	0	0
$725$	45.0000	1.67126
$726$	0	0
$727$	5.29150i	0.196251i	0.995174	+	0.0981255i	$0.0312847\pi$
$727$	5.29150i	0.196251i	−0.995174	+	0.0981255i	$0.968715\pi$
$728$	0	0
$729$	41.0000	1.51852
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−20.1246	−0.743319	−0.371660	−	0.928369i	$-0.621211\pi$
$733$	−20.1246	−0.743319	−0.371660	+	0.928369i	$0.621211\pi$
$734$	0	0
$735$	− 41.4126i	− 1.52753i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 53.2447i	− 1.95864i	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	− 53.2447i	− 1.95864i	0.202321	−	0.979319i	$-0.435152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−13.4164	−0.491539
$746$	0	0
$747$	63.4980i	2.32327i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	53.2447i	1.94293i	0.237188	+	0.971464i	$0.423774\pi$
$751$	53.2447i	1.94293i	−0.237188	+	0.971464i	$0.576226\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 39.6863i	− 1.44433i
$756$	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	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 29.1033i	− 1.05361i
$764$	0	0
$765$	−20.0000	−0.723102
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	11.8322i	0.426125i
$772$	0	0
$773$	−2.23607	−0.0804258	−0.0402129	−	0.999191i	$-0.512804\pi$
$773$	−2.23607	−0.0804258	−0.0402129	+	0.999191i	$0.512804\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	70.0000	2.50480
$782$	0	0
$783$	− 23.8118i	− 0.850963i
$784$	0	0
$785$	−30.0000	−1.07075
$786$	0	0
$787$	44.9778i	1.60328i	0.597804	+	0.801642i	$0.296040\pi$
$787$	44.9778i	1.60328i	−0.597804	+	0.801642i	$0.703960\pi$
$788$	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
$797$	−55.9017	−1.98014	−0.990070	−	0.140576i	$-0.955105\pi$
$797$	−55.9017	−1.98014	−0.990070	+	0.140576i	$0.955105\pi$
$798$	0	0
$799$	17.7482i	0.627888i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	79.3725i	2.80100i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	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
$819$	− 70.9930i	− 2.48069i
$820$	0	0
$821$	57.0000	1.98931	0.994657	−	0.103236i	$-0.0329198\pi$
$821$	57.0000	1.98931	0.994657	+	0.103236i	$0.0329198\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	78.2624	2.72475
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15.6525	−0.542326
$834$	0	0
$835$	17.7482i	0.614203i
$836$	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
$841$	52.0000	1.79310
$842$	0	0
$843$	− 87.3098i	− 3.00711i
$844$	0	0
$845$	71.5542	2.46154
$846$	0	0
$847$	− 63.4980i	− 2.18182i
$848$	0	0
$849$	−7.00000	−0.240239
$850$	0	0
$851$	0	0
$852$	0	0
$853$	40.2492	1.37811	0.689054	−	0.724710i	$-0.258026\pi$
$853$	40.2492	1.37811	0.689054	+	0.724710i	$0.258026\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.1935	−1.68042	−0.840209	−	0.542263i	$-0.817568\pi$
$857$	−49.1935	−1.68042	−0.840209	+	0.542263i	$0.817568\pi$
$858$	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		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−25.0000	−0.850026
$866$	0	0
$867$	− 31.7490i	− 1.07825i
$868$	0	0
$869$	−105.000	−3.56188
$870$	0	0
$871$	0	0
$872$	0	0
$873$	26.8328	0.908153
$874$	0	0
$875$	29.5804i	1.00000i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	65.0769i	2.19499i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.6235i	1.59904i	0.600639	+	0.799521i	$0.294913\pi$
$887$	47.6235i	1.59904i	−0.600639	+	0.799521i	$0.705087\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	29.5804i	0.990981i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	26.4575i	0.884377i
$896$	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		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	59.1608i	1.96008i	0.198789	+	0.980042i	$0.436299\pi$
$911$	59.1608i	1.96008i	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−93.9149	−3.10813
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	53.2447i	1.75638i	0.478311	+	0.878191i	$0.341249\pi$
$919$	53.2447i	1.75638i	−0.478311	+	0.878191i	$0.658751\pi$
$920$	0	0
$921$	−91.0000	−2.99855
$922$	0	0
$923$	79.3725i	2.61258i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.5830i	0.347591i
$928$	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
$935$	− 29.5804i	− 0.967382i
$936$	0	0
$937$	−6.70820	−0.219147	−0.109574	−	0.993979i	$-0.534949\pi$
$937$	−6.70820	−0.219147	−0.109574	+	0.993979i	$0.534949\pi$
$938$	0	0
$939$	53.2447i	1.73758i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	15.6525	0.509175
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−90.0000	−2.92152
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	13.2288i	0.428073i
$956$	0	0
$957$	140.872	4.55375
$958$	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		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	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
$975$	88.7412i	2.84199i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	44.0000	1.40481
$982$	0	0
$983$	− 55.5608i	− 1.77211i	−0.463577	−	0.886057i	$-0.653434\pi$
$983$	− 55.5608i	− 1.77211i	0.463577	−	0.886057i	$-0.346566\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 55.5608i	− 1.76852i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 35.4965i	− 1.12758i	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	− 35.4965i	− 1.12758i	0.825917	−	0.563791i	$-0.190658\pi$
$992$	0	0
$993$	93.9149	2.98030
$994$	0	0
$995$	0	0
$996$	0	0
$997$	60.3738	1.91206	0.956029	−	0.293271i	$-0.0947439\pi$
$997$	60.3738	1.91206	0.956029	+	0.293271i	$0.0947439\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.b.2239.4		4
4.3	odd	2	inner	2240.2.e.b.2239.2		4
5.4	even	2	inner	2240.2.e.b.2239.1		4
7.6	odd	2	inner	2240.2.e.b.2239.1		4
8.3	odd	2		560.2.e.a.559.3	yes	4
8.5	even	2		560.2.e.a.559.1	✓	4
20.19	odd	2	inner	2240.2.e.b.2239.3		4
28.27	even	2	inner	2240.2.e.b.2239.3		4
35.34	odd	2	CM	2240.2.e.b.2239.4		4
40.3	even	4		2800.2.k.j.2351.1		4
40.13	odd	4		2800.2.k.j.2351.4		4
40.19	odd	2		560.2.e.a.559.2	yes	4
40.27	even	4		2800.2.k.j.2351.3		4
40.29	even	2		560.2.e.a.559.4	yes	4
40.37	odd	4		2800.2.k.j.2351.2		4
56.13	odd	2		560.2.e.a.559.4	yes	4
56.27	even	2		560.2.e.a.559.2	yes	4
140.139	even	2	inner	2240.2.e.b.2239.2		4
280.13	even	4		2800.2.k.j.2351.2		4
280.27	odd	4		2800.2.k.j.2351.1		4
280.69	odd	2		560.2.e.a.559.1	✓	4
280.83	odd	4		2800.2.k.j.2351.3		4
280.139	even	2		560.2.e.a.559.3	yes	4
280.237	even	4		2800.2.k.j.2351.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.e.a.559.1	✓	4	8.5	even	2
560.2.e.a.559.1	✓	4	280.69	odd	2
560.2.e.a.559.2	yes	4	40.19	odd	2
560.2.e.a.559.2	yes	4	56.27	even	2
560.2.e.a.559.3	yes	4	8.3	odd	2
560.2.e.a.559.3	yes	4	280.139	even	2
560.2.e.a.559.4	yes	4	40.29	even	2
560.2.e.a.559.4	yes	4	56.13	odd	2
2240.2.e.b.2239.1		4	5.4	even	2	inner
2240.2.e.b.2239.1		4	7.6	odd	2	inner
2240.2.e.b.2239.2		4	4.3	odd	2	inner
2240.2.e.b.2239.2		4	140.139	even	2	inner
2240.2.e.b.2239.3		4	20.19	odd	2	inner
2240.2.e.b.2239.3		4	28.27	even	2	inner
2240.2.e.b.2239.4		4	1.1	even	1	trivial
2240.2.e.b.2239.4		4	35.34	odd	2	CM
2800.2.k.j.2351.1		4	40.3	even	4
2800.2.k.j.2351.1		4	280.27	odd	4
2800.2.k.j.2351.2		4	40.37	odd	4
2800.2.k.j.2351.2		4	280.13	even	4
2800.2.k.j.2351.3		4	40.27	even	4
2800.2.k.j.2351.3		4	280.83	odd	4
2800.2.k.j.2351.4		4	40.13	odd	4
2800.2.k.j.2351.4		4	280.237	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.64575i q^{3} +2.23607 q^{5} +2.64575i q^{7} -4.00000 q^{9} +O(q^{10})\)	\(q+2.64575i q^{3} +2.23607 q^{5} +2.64575i q^{7} -4.00000 q^{9} -5.91608i q^{11} +6.70820 q^{13} +5.91608i q^{15} +2.23607 q^{17} -7.00000 q^{21} +5.00000 q^{25} -2.64575i q^{27} +9.00000 q^{29} +15.6525 q^{33} +5.91608i q^{35} +17.7482i q^{39} -8.94427 q^{45} +7.93725i q^{47} -7.00000 q^{49} +5.91608i q^{51} -13.2288i q^{55} -10.5830i q^{63} +15.0000 q^{65} +11.8322i q^{71} -13.4164 q^{73} +13.2288i q^{75} +15.6525 q^{77} -17.7482i q^{79} -5.00000 q^{81} -15.8745i q^{83} +5.00000 q^{85} +23.8118i q^{87} +17.7482i q^{91} -6.70820 q^{97} +23.6643i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{9}+O(q^{10}) \) 4 * q - 16 * q^9	\( 4 q - 16 q^{9} - 28 q^{21} + 20 q^{25} + 36 q^{29} - 28 q^{49} + 60 q^{65} - 20 q^{81} + 20 q^{85}+O(q^{100}) \) 4 * q - 16 * q^9 - 28 * q^21 + 20 * q^25 + 36 * q^29 - 28 * q^49 + 60 * q^65 - 20 * q^81 + 20 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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