LMFDB - Embedded newform 2240.2.e.d.2239.8

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:	$8$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2239.8
Root			$-0.862555 + 0.141174i$ of defining polynomial
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.d.2239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.25937i q^{3} +2.23607 q^{5} -2.64575i q^{7} -7.62348 q^{9} +O(q^{10})$	$q+3.25937i q^{3} +2.23607 q^{5} -2.64575i q^{7} -7.62348 q^{9} -0.359964i q^{11} -5.64539 q^{13} +7.28817i q^{15} -7.99190 q^{17} +8.62348 q^{21} +5.00000 q^{25} -15.0696i q^{27} +0.623475 q^{29} +1.17325 q^{33} -5.91608i q^{35} -18.4004i q^{39} -17.0466 q^{45} -5.71383i q^{47} -7.00000 q^{49} -26.0485i q^{51} -0.804903i q^{55} +20.1698i q^{63} -12.6235 q^{65} -11.8322i q^{71} -13.4164 q^{73} +16.2968i q^{75} -0.952374 q^{77} -8.00809i q^{79} +26.2470 q^{81} +15.8745i q^{83} -17.8704 q^{85} +2.03214i q^{87} +14.9363i q^{91} -12.6849 q^{97} +2.74417i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 20 q^{9}+O(q^{10}) $ 8 * q - 20 * q^9	$ 8 q - 20 q^{9} + 28 q^{21} + 40 q^{25} - 36 q^{29} - 56 q^{49} - 60 q^{65} + 128 q^{81} - 20 q^{85}+O(q^{100}) $ 8 * q - 20 * q^9 + 28 * q^21 + 40 * q^25 - 36 * q^29 - 56 * q^49 - 60 * q^65 + 128 * 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
$3$	3.25937i	1.88180i	0.338689	+	0.940898i	$0.390016\pi$
$3$	3.25937i	1.88180i	−0.338689	+	0.940898i	$0.609984\pi$
$4$	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
$9$	−7.62348	−2.54116
$10$	0	0
$11$	− 0.359964i	− 0.108533i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	− 0.359964i	− 0.108533i	0.998526	−	0.0542666i	$-0.0172821\pi$
$12$	0	0
$13$	−5.64539	−1.56575	−0.782875	−	0.622179i	$-0.786247\pi$
$13$	−5.64539	−1.56575	−0.782875	+	0.622179i	$0.786247\pi$
$14$	0	0
$15$	7.28817i	1.88180i
$16$	0	0
$17$	−7.99190	−1.93832	−0.969160	−	0.246433i	$-0.920742\pi$
$17$	−7.99190	−1.93832	−0.969160	+	0.246433i	$0.920742\pi$
$18$	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		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 15.0696i	− 2.90015i
$28$	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
$33$	1.17325	0.204237
$34$	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
$39$	− 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		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−17.0466	−2.54116
$46$	0	0
$47$	− 5.71383i	− 0.833448i	−0.909033	−	0.416724i	$-0.863178\pi$
$47$	− 5.71383i	− 0.833448i	0.909033	−	0.416724i	$-0.136822\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	− 26.0485i	− 3.64752i
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 0.804903i	− 0.108533i
$56$	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
$63$	20.1698i	2.54116i
$64$	0	0
$65$	−12.6235	−1.56575
$66$	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	$-0.752242\pi$
$71$	− 11.8322i	− 1.40422i	0.712069	−	0.702109i	$-0.247758\pi$
$72$	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
$75$	16.2968i	1.88180i
$76$	0	0
$77$	−0.952374	−0.108533
$78$	0	0
$79$	− 8.00809i	− 0.900981i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	− 8.00809i	− 0.900981i	0.892781	−	0.450490i	$-0.148751\pi$
$80$	0	0
$81$	26.2470	2.91633
$82$	0	0
$83$	15.8745i	1.74245i	0.490881	+	0.871227i	$0.336675\pi$
$83$	15.8745i	1.74245i	−0.490881	+	0.871227i	$0.663325\pi$
$84$	0	0
$85$	−17.8704	−1.93832
$86$	0	0
$87$	2.03214i	0.217868i
$88$	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
$97$	−12.6849	−1.28796	−0.643979	−	0.765043i	$-0.722718\pi$
$97$	−12.6849	−1.28796	−0.643979	+	0.765043i	$0.722718\pi$
$98$	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
$103$	− 18.7513i	− 1.84762i	−0.382851	−	0.923810i	$-0.625058\pi$
$103$	− 18.7513i	− 1.84762i	0.382851	−	0.923810i	$-0.374942\pi$
$104$	0	0
$105$	19.2827	1.88180
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	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	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	43.0375	3.97882
$118$	0	0
$119$	21.1446i	1.93832i
$120$	0	0
$121$	10.8704	0.988221
$122$	0	0
$123$	0	0
$124$	0	0
$125$	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
$135$	− 33.6967i	− 2.90015i
$136$	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
$141$	18.6235	1.56838
$142$	0	0
$143$	2.03214i	0.169936i
$144$	0	0
$145$	1.39413	0.115776
$146$	0	0
$147$	− 22.8156i	− 1.88180i
$148$	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
$153$	60.9260	4.92558
$154$	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
$165$	2.62348	0.204237
$166$	0	0
$167$	25.2700i	1.95545i	0.209881	+	0.977727i	$0.432692\pi$
$167$	25.2700i	1.95545i	−0.209881	+	0.977727i	$0.567308\pi$
$168$	0	0
$169$	18.8704	1.45157
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.0314	−1.14282	−0.571409	−	0.820666i	$-0.693603\pi$
$173$	−15.0314	−1.14282	−0.571409	+	0.820666i	$0.693603\pi$
$174$	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	$-0.854202\pi$
$179$	− 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
$187$	2.87679i	0.210372i
$188$	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	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	− 41.1445i	− 2.94642i
$196$	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
$203$	− 1.64956i	− 0.115776i
$204$	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	$0.457467\pi$
$211$	28.7927i	1.98217i	−0.133226	+	0.991086i	$0.542533\pi$
$212$	0	0
$213$	38.5654	2.64245
$214$	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
$223$	20.3611i	1.36348i	0.731594	+	0.681740i	$0.238777\pi$
$223$	20.3611i	1.36348i	−0.731594	+	0.681740i	$0.761223\pi$
$224$	0	0
$225$	−38.1174	−2.54116
$226$	0	0
$227$	21.2058i	1.40748i	0.710460	+	0.703738i	$0.248487\pi$
$227$	21.2058i	1.40748i	−0.710460	+	0.703738i	$0.751513\pi$
$228$	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	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	− 12.7765i	− 0.833448i
$236$	0	0
$237$	26.1013	1.69546
$238$	0	0
$239$	22.5844i	1.46087i	0.682985	+	0.730433i	$0.260682\pi$
$239$	22.5844i	1.46087i	−0.682985	+	0.730433i	$0.739318\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	40.3396i	2.58779i
$244$	0	0
$245$	−15.6525	−1.00000
$246$	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
$255$	− 58.2463i	− 3.64752i
$256$	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
$261$	−4.75305	−0.294206
$262$	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
$273$	−48.6829	−2.94642
$274$	0	0
$275$	− 1.79982i	− 0.108533i
$276$	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
$283$	− 27.7245i	− 1.64805i	−0.566553	−	0.824025i	$-0.691723\pi$
$283$	− 27.7245i	− 1.64805i	0.566553	−	0.824025i	$-0.308277\pi$
$284$	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
$293$	−32.9200	−1.92320	−0.961602	−	0.274446i	$-0.911505\pi$
$293$	−32.9200	−1.92320	−0.961602	+	0.274446i	$0.911505\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.42451	−0.314762
$298$	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
$307$	11.3879i	0.649942i	0.945724	+	0.324971i	$0.105355\pi$
$307$	11.3879i	0.649942i	−0.945724	+	0.324971i	$0.894645\pi$
$308$	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
$313$	−35.2665	−1.99338	−0.996689	−	0.0813030i	$-0.974092\pi$
$313$	−35.2665	−1.99338	−0.996689	+	0.0813030i	$0.974092\pi$
$314$	0	0
$315$	45.1011i	2.54116i
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	− 0.224428i	− 0.0125656i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−28.2269	−1.56575
$326$	0	0
$327$	− 32.1713i	− 1.77908i
$328$	0	0
$329$	−15.1174	−0.833448
$330$	0	0
$331$	35.4965i	1.95106i	0.219860	+	0.975531i	$0.429440\pi$
$331$	35.4965i	1.95106i	−0.219860	+	0.975531i	$0.570560\pi$
$332$	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
$351$	85.0738i	4.54090i
$352$	0	0
$353$	6.08715	0.323986	0.161993	−	0.986792i	$-0.448208\pi$
$353$	6.08715	0.323986	0.161993	+	0.986792i	$0.448208\pi$
$354$	0	0
$355$	− 26.4575i	− 1.40422i
$356$	0	0
$357$	−68.9179	−3.64752
$358$	0	0
$359$	− 11.8322i	− 0.624477i	−0.950004	−	0.312239i	$-0.898921\pi$
$359$	− 11.8322i	− 0.624477i	0.950004	−	0.312239i	$-0.101079\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	35.4307i	1.85963i
$364$	0	0
$365$	−30.0000	−1.57027
$366$	0	0
$367$	38.3075i	1.99964i	0.0190919	+	0.999818i	$0.493923\pi$
$367$	38.3075i	1.99964i	−0.0190919	+	0.999818i	$0.506077\pi$
$368$	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
$375$	36.4408i	1.88180i
$376$	0	0
$377$	−3.51976	−0.181277
$378$	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
$383$	15.8745i	0.811149i	0.914062	+	0.405575i	$0.132929\pi$
$383$	15.8745i	0.811149i	−0.914062	+	0.405575i	$0.867071\pi$
$384$	0	0
$385$	−2.12957	−0.108533
$386$	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
$395$	− 17.9066i	− 0.900981i
$396$	0	0
$397$	−19.7244	−0.989941	−0.494971	−	0.868910i	$-0.664821\pi$
$397$	−19.7244	−0.989941	−0.494971	+	0.868910i	$0.664821\pi$
$398$	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
$405$	58.6900	2.91633
$406$	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
$421$	−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
$423$	43.5593i	2.11792i
$424$	0	0
$425$	−39.9595	−1.93832
$426$	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
$433$	40.2492	1.93425	0.967127	−	0.254293i	$-0.0818429\pi$
$433$	40.2492	1.93425	0.967127	+	0.254293i	$0.0818429\pi$
$434$	0	0
$435$	4.54399i	0.217868i
$436$	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		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 19.5562i	− 0.924977i
$448$	0	0
$449$	40.3643	1.90491	0.952455	−	0.304679i	$-0.0985491\pi$
$449$	40.3643	1.90491	0.952455	+	0.304679i	$0.0985491\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−19.0618	−0.895601
$454$	0	0
$455$	33.3986i	1.56575i
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	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		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 40.7620i	− 1.88624i	−0.332454	−	0.943119i	$-0.607877\pi$
$467$	− 40.7620i	− 1.88624i	0.332454	−	0.943119i	$-0.392123\pi$
$468$	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
$485$	−28.3643	−1.28796
$486$	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
$491$	− 43.3690i	− 1.95722i	−0.205731	−	0.978609i	$-0.565957\pi$
$491$	− 43.3690i	− 1.95722i	0.205731	−	0.978609i	$-0.434043\pi$
$492$	0	0
$493$	−4.98275	−0.224412
$494$	0	0
$495$	6.13616i	0.275800i
$496$	0	0
$497$	−31.3050	−1.40422
$498$	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
$503$	− 44.8262i	− 1.99870i	−0.0360049	−	0.999352i	$-0.511463\pi$
$503$	− 44.8262i	− 1.99870i	0.0360049	−	0.999352i	$-0.488537\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	61.5057i	2.73156i
$508$	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
$515$	− 41.9292i	− 1.84762i
$516$	0	0
$517$	−2.05677	−0.0904567
$518$	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
$523$	37.0405i	1.61967i	0.586659	+	0.809834i	$0.300443\pi$
$523$	37.0405i	1.61967i	−0.586659	+	0.809834i	$0.699557\pi$
$524$	0	0
$525$	43.1174	1.88180
$526$	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
$537$	38.5654	1.66422
$538$	0	0
$539$	2.51975i	0.108533i
$540$	0	0
$541$	6.12957	0.263531	0.131765	−	0.991281i	$-0.457935\pi$
$541$	6.12957	0.263531	0.131765	+	0.991281i	$0.457935\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−22.0709	−0.945415
$546$	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
$553$	−21.1874	−0.900981
$554$	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
$561$	−9.37652	−0.395877
$562$	0	0
$563$	15.8745i	0.669031i	0.942390	+	0.334515i	$0.108573\pi$
$563$	15.8745i	0.669031i	−0.942390	+	0.334515i	$0.891427\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 69.4429i	− 2.91633i
$568$	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	$0.266474\pi$
$571$	35.4965i	1.48548i	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	66.5714	2.78106
$574$	0	0
$575$	0	0
$576$	0	0
$577$	46.5573	1.93820	0.969102	−	0.246661i	$-0.0793334\pi$
$577$	46.5573	1.93820	0.969102	+	0.246661i	$0.0793334\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	42.0000	1.74245
$582$	0	0
$583$	0	0
$584$	0	0
$585$	96.2348	3.97882
$586$	0	0
$587$	− 47.6235i	− 1.96563i	−0.184585	−	0.982817i	$-0.559094\pi$
$587$	− 47.6235i	− 1.96563i	0.184585	−	0.982817i	$-0.440906\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.8642	1.71916	0.859579	−	0.511003i	$-0.170726\pi$
$593$	41.8642	1.71916	0.859579	+	0.511003i	$0.170726\pi$
$594$	0	0
$595$	47.2807i	1.93832i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 18.9848i	− 0.775698i	−0.921723	−	0.387849i	$-0.873218\pi$
$599$	− 18.9848i	− 0.775698i	0.921723	−	0.387849i	$-0.126782\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24.3070	0.988221
$606$	0	0
$607$	− 39.9173i	− 1.62019i	−0.586296	−	0.810097i	$-0.699414\pi$
$607$	− 39.9173i	− 1.62019i	0.586296	−	0.810097i	$-0.300586\pi$
$608$	0	0
$609$	5.37652	0.217868
$610$	0	0
$611$	32.2568i	1.30497i
$612$	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
$631$	− 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
$633$	−93.8460	−3.73004
$634$	0	0
$635$	0	0
$636$	0	0
$637$	39.5177	1.56575
$638$	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
$643$	− 30.9441i	− 1.22032i	−0.792279	−	0.610158i	$-0.791106\pi$
$643$	− 30.9441i	− 1.22032i	0.792279	−	0.610158i	$-0.208894\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 47.6235i	− 1.87227i	−0.351636	−	0.936137i	$-0.614374\pi$
$647$	− 47.6235i	− 1.87227i	0.351636	−	0.936137i	$-0.385626\pi$
$648$	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
$657$	102.280	3.99031
$658$	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
$663$	147.054i	5.71111i
$664$	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	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	− 75.3480i	− 2.90015i
$676$	0	0
$677$	−18.8409	−0.724115	−0.362058	−	0.932156i	$-0.617926\pi$
$677$	−18.8409	−0.724115	−0.362058	+	0.932156i	$0.617926\pi$
$678$	0	0
$679$	33.5611i	1.28796i
$680$	0	0
$681$	−69.1174	−2.64858
$682$	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
$693$	7.26040	0.275800
$694$	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	$-0.952500\pi$
$701$	−52.3643	−1.97777	−0.988887	+	0.148671i	$0.952500\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	41.6434	1.56838
$706$	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
$715$	4.54399i	0.169936i
$716$	0	0
$717$	−73.6109	−2.74905
$718$	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
$725$	3.11738	0.115776
$726$	0	0
$727$	− 5.29150i	− 0.196251i	−0.995174	−	0.0981255i	$-0.968715\pi$
$727$	− 5.29150i	− 0.196251i	0.995174	−	0.0981255i	$-0.0312847\pi$
$728$	0	0
$729$	−52.7409	−1.95336
$730$	0	0
$731$	0	0
$732$	0	0
$733$	53.5968	1.97964	0.989821	−	0.142318i	$-0.0454555\pi$
$733$	53.5968	1.97964	0.989821	+	0.142318i	$0.0454555\pi$
$734$	0	0
$735$	− 51.0172i	− 1.88180i
$736$	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		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−13.4164	−0.491539
$746$	0	0
$747$	− 121.019i	− 4.42785i
$748$	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
$755$	13.0772i	0.475929i
$756$	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
$763$	26.1147i	0.945415i
$764$	0	0
$765$	136.235	4.92558
$766$	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
$773$	−46.9990	−1.69044	−0.845218	−	0.534421i	$-0.820530\pi$
$773$	−46.9990	−1.69044	−0.845218	+	0.534421i	$0.820530\pi$
$774$	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
$783$	− 9.39553i	− 0.335769i
$784$	0	0
$785$	−30.0000	−1.07075
$786$	0	0
$787$	− 6.55849i	− 0.233785i	−0.993145	−	0.116892i	$-0.962707\pi$
$787$	− 6.55849i	− 0.233785i	0.993145	−	0.116892i	$-0.0372933\pi$
$788$	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
$797$	34.8247	1.23355	0.616777	−	0.787138i	$-0.288438\pi$
$797$	34.8247	1.23355	0.616777	+	0.787138i	$0.288438\pi$
$798$	0	0
$799$	45.6644i	1.61549i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.82942i	0.170427i
$804$	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
$819$	− 113.866i	− 3.97882i
$820$	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		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	5.86627	0.204237
$826$	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
$833$	55.9433	1.93832
$834$	0	0
$835$	56.5055i	1.95545i
$836$	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
$843$	37.0803i	1.27711i
$844$	0	0
$845$	42.1956	1.45157
$846$	0	0
$847$	− 28.7604i	− 0.988221i
$848$	0	0
$849$	90.3643	3.10130
$850$	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
$865$	−33.6113	−1.14282
$866$	0	0
$867$	152.768i	5.18827i
$868$	0	0
$869$	−2.88262	−0.0977863
$870$	0	0
$871$	0	0
$872$	0	0
$873$	96.7031	3.27290
$874$	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
$879$	− 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		−	1.00000i	$-0.5\pi$
$883$	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	$-0.705087\pi$
$887$	− 47.6235i	− 1.59904i	0.600639	−	0.799521i	$-0.294913\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 9.44795i	− 0.316518i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 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	$-0.563701\pi$
$911$	− 59.1608i	− 1.96008i	0.198789	−	0.980042i	$-0.436299\pi$
$912$	0	0
$913$	5.71425	0.189114
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	51.7371i	1.70665i	0.521380	+	0.853325i	$0.325417\pi$
$919$	51.7371i	1.70665i	−0.521380	+	0.853325i	$0.674583\pi$
$920$	0	0
$921$	−37.1174	−1.22306
$922$	0	0
$923$	66.7972i	2.19866i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	142.950i	4.69510i
$928$	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
$935$	6.43270i	0.210372i
$936$	0	0
$937$	−49.3455	−1.61205	−0.806024	−	0.591883i	$-0.798385\pi$
$937$	−49.3455	−1.61205	−0.806024	+	0.591883i	$0.798385\pi$
$938$	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
$945$	−89.1530	−2.90015
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	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
$955$	− 45.6709i	− 1.47788i
$956$	0	0
$957$	0.731495	0.0236459
$958$	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
$975$	− 92.0020i	− 2.94642i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	75.2470	2.40245
$982$	0	0
$983$	− 52.9548i	− 1.68900i	−0.535559	−	0.844498i	$-0.679899\pi$
$983$	− 52.9548i	− 1.68900i	0.535559	−	0.844498i	$-0.320101\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 49.2731i	− 1.56838i
$988$	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
$993$	−115.696	−3.67150
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−14.1479	−0.448069	−0.224034	−	0.974581i	$-0.571923\pi$
$997$	−14.1479	−0.448069	−0.224034	+	0.974581i	$0.571923\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.d.2239.8		8
4.3	odd	2	inner	2240.2.e.d.2239.2		8
5.4	even	2	inner	2240.2.e.d.2239.1		8
7.6	odd	2	inner	2240.2.e.d.2239.1		8
8.3	odd	2		560.2.e.c.559.7	yes	8
8.5	even	2		560.2.e.c.559.1	✓	8
20.19	odd	2	inner	2240.2.e.d.2239.7		8
28.27	even	2	inner	2240.2.e.d.2239.7		8
35.34	odd	2	CM	2240.2.e.d.2239.8		8
40.3	even	4		2800.2.k.p.2351.1		8
40.13	odd	4		2800.2.k.p.2351.8		8
40.19	odd	2		560.2.e.c.559.2	yes	8
40.27	even	4		2800.2.k.p.2351.7		8
40.29	even	2		560.2.e.c.559.8	yes	8
40.37	odd	4		2800.2.k.p.2351.2		8
56.13	odd	2		560.2.e.c.559.8	yes	8
56.27	even	2		560.2.e.c.559.2	yes	8
140.139	even	2	inner	2240.2.e.d.2239.2		8
280.13	even	4		2800.2.k.p.2351.2		8
280.27	odd	4		2800.2.k.p.2351.1		8
280.69	odd	2		560.2.e.c.559.1	✓	8
280.83	odd	4		2800.2.k.p.2351.7		8
280.139	even	2		560.2.e.c.559.7	yes	8
280.237	even	4		2800.2.k.p.2351.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.e.c.559.1	✓	8	8.5	even	2
560.2.e.c.559.1	✓	8	280.69	odd	2
560.2.e.c.559.2	yes	8	40.19	odd	2
560.2.e.c.559.2	yes	8	56.27	even	2
560.2.e.c.559.7	yes	8	8.3	odd	2
560.2.e.c.559.7	yes	8	280.139	even	2
560.2.e.c.559.8	yes	8	40.29	even	2
560.2.e.c.559.8	yes	8	56.13	odd	2
2240.2.e.d.2239.1		8	5.4	even	2	inner
2240.2.e.d.2239.1		8	7.6	odd	2	inner
2240.2.e.d.2239.2		8	4.3	odd	2	inner
2240.2.e.d.2239.2		8	140.139	even	2	inner
2240.2.e.d.2239.7		8	20.19	odd	2	inner
2240.2.e.d.2239.7		8	28.27	even	2	inner
2240.2.e.d.2239.8		8	1.1	even	1	trivial
2240.2.e.d.2239.8		8	35.34	odd	2	CM
2800.2.k.p.2351.1		8	40.3	even	4
2800.2.k.p.2351.1		8	280.27	odd	4
2800.2.k.p.2351.2		8	40.37	odd	4
2800.2.k.p.2351.2		8	280.13	even	4
2800.2.k.p.2351.7		8	40.27	even	4
2800.2.k.p.2351.7		8	280.83	odd	4
2800.2.k.p.2351.8		8	40.13	odd	4
2800.2.k.p.2351.8		8	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)

\(f(q)\)	\(=\)	\(q+3.25937i q^{3} +2.23607 q^{5} -2.64575i q^{7} -7.62348 q^{9} +O(q^{10})\)	\(q+3.25937i q^{3} +2.23607 q^{5} -2.64575i q^{7} -7.62348 q^{9} -0.359964i q^{11} -5.64539 q^{13} +7.28817i q^{15} -7.99190 q^{17} +8.62348 q^{21} +5.00000 q^{25} -15.0696i q^{27} +0.623475 q^{29} +1.17325 q^{33} -5.91608i q^{35} -18.4004i q^{39} -17.0466 q^{45} -5.71383i q^{47} -7.00000 q^{49} -26.0485i q^{51} -0.804903i q^{55} +20.1698i q^{63} -12.6235 q^{65} -11.8322i q^{71} -13.4164 q^{73} +16.2968i q^{75} -0.952374 q^{77} -8.00809i q^{79} +26.2470 q^{81} +15.8745i q^{83} -17.8704 q^{85} +2.03214i q^{87} +14.9363i q^{91} -12.6849 q^{97} +2.74417i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 20 q^{9}+O(q^{10}) \) 8 * q - 20 * q^9	\( 8 q - 20 q^{9} + 28 q^{21} + 40 q^{25} - 36 q^{29} - 56 q^{49} - 60 q^{65} + 128 q^{81} - 20 q^{85}+O(q^{100}) \) 8 * q - 20 * q^9 + 28 * q^21 + 40 * q^25 - 36 * q^29 - 56 * q^49 - 60 * q^65 + 128 * q^81 - 20 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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