LMFDB - Embedded newform 1520.2.g.d.1519.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.g (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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1519.4
Root			$-1.63746 - 1.52274i$ of defining polynomial
Character	$\chi$	$=$	1520.1519
Dual form			1520.2.g.d.1519.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.13746 + 0.656712i) q^{5} -2.27492 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+(2.13746 + 0.656712i) q^{5} -2.27492 q^{7} +3.00000 q^{9} -2.15068i q^{11} -8.24163i q^{17} -4.35890i q^{19} -4.00000 q^{23} +(4.13746 + 2.80739i) q^{25} +(-4.86254 - 1.49397i) q^{35} +10.8248 q^{43} +(6.41238 + 1.97014i) q^{45} +10.2749 q^{47} -1.82475 q^{49} +(1.41238 - 4.59698i) q^{55} +3.72508 q^{61} -6.82475 q^{63} -2.98793i q^{73} +4.89261i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(5.41238 - 17.6161i) q^{85} +(2.86254 - 9.31697i) q^{95} -6.45203i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} + 6 q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q + q^5 + 6 * q^7 + 12 * q^9	$ 4 q + q^{5} + 6 q^{7} + 12 q^{9} - 16 q^{23} + 9 q^{25} - 27 q^{35} - 2 q^{43} + 3 q^{45} + 26 q^{47} + 38 q^{49} - 17 q^{55} + 30 q^{61} + 18 q^{63} + 36 q^{81} - 64 q^{83} - q^{85} + 19 q^{95}+O(q^{100}) $ 4 * q + q^5 + 6 * q^7 + 12 * q^9 - 16 * q^23 + 9 * q^25 - 27 * q^35 - 2 * q^43 + 3 * q^45 + 26 * q^47 + 38 * q^49 - 17 * q^55 + 30 * q^61 + 18 * q^63 + 36 * q^81 - 64 * q^83 - q^85 + 19 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$		0			0
$5$	2.13746	+	0.656712i	0.955901	+	0.293691i
$5$	2.13746	+	0.656712i	0.955901	+	0.293691i
$6$		0			0
$7$	−2.27492			−0.859838			−0.429919	−	0.902867i	$-0.641458\pi$
$7$	−2.27492			−0.859838			−0.429919	+	0.902867i	$0.641458\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		−	2.15068i		−	0.648454i	−0.945979	−	0.324227i	$-0.894896\pi$
$11$		−	2.15068i		−	0.648454i	0.945979	−	0.324227i	$-0.105104\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	8.24163i		−	1.99889i	−0.0333386	−	0.999444i	$-0.510614\pi$
$17$		−	8.24163i		−	1.99889i	0.0333386	−	0.999444i	$-0.489386\pi$
$18$		0			0
$19$		−	4.35890i		−	1.00000i
$19$		−	4.35890i		−	1.00000i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	4.13746	+	2.80739i	0.827492	+	0.561478i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.86254	−	1.49397i	−0.821920	−	0.252526i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	10.8248			1.65076			0.825380	−	0.564578i	$-0.190961\pi$
$43$	10.8248			1.65076			0.825380	+	0.564578i	$0.190961\pi$
$44$		0			0
$45$	6.41238	+	1.97014i	0.955901	+	0.293691i
$46$		0			0
$47$	10.2749			1.49875			0.749375	−	0.662145i	$-0.230354\pi$
$47$	10.2749			1.49875			0.749375	+	0.662145i	$0.230354\pi$
$48$		0			0
$49$	−1.82475			−0.260679
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	1.41238	−	4.59698i	0.190445	−	0.619857i
$56$		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
$61$	3.72508			0.476948			0.238474	−	0.971149i	$-0.423353\pi$
$61$	3.72508			0.476948			0.238474	+	0.971149i	$0.423353\pi$
$62$		0			0
$63$	−6.82475			−0.859838
$64$		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
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		−	2.98793i		−	0.349711i	−0.984594	−	0.174855i	$-0.944054\pi$
$73$		−	2.98793i		−	0.349711i	0.984594	−	0.174855i	$-0.0559458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			4.89261i			0.557565i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	5.41238	−	17.6161i	0.587055	−	1.91074i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.86254	−	9.31697i	0.293691	−	0.955901i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		−	6.45203i		−	0.648454i
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		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
$115$	−8.54983	−	2.62685i	−0.797276	−	0.244955i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			18.7490i			1.71872i
$120$		0			0
$121$	6.37459			0.579508
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.00000	+	8.71780i	0.626099	+	0.779744i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	16.9594i		−	1.48175i	−0.671642	−	0.740876i	$-0.734411\pi$
$131$		−	16.9594i		−	1.48175i	0.671642	−	0.740876i	$-0.265589\pi$
$132$		0			0
$133$			9.91613i			0.859838i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			22.0980i			1.88796i	0.329999	+	0.943981i	$0.392952\pi$
$137$			22.0980i			1.88796i	−0.329999	+	0.943981i	$0.607048\pi$
$138$		0			0
$139$			3.10302i			0.263195i	0.991303	+	0.131597i	$0.0420106\pi$
$139$			3.10302i			0.263195i	−0.991303	+	0.131597i	$0.957989\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.3746			−1.09569			−0.547844	−	0.836580i	$-0.684551\pi$
$149$	−13.3746			−1.09569			−0.547844	+	0.836580i	$0.684551\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	24.7249i		−	1.99889i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	17.4356i		−	1.39151i	−0.718278	−	0.695756i	$-0.755069\pi$
$157$		−	17.4356i		−	1.39151i	0.718278	−	0.695756i	$-0.244931\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.09967			0.717154
$162$		0			0
$163$	24.0000			1.87983			0.939913	−	0.341415i	$-0.110906\pi$
$163$	24.0000			1.87983			0.939913	+	0.341415i	$0.110906\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		−	13.0767i		−	1.00000i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−9.41238	−	6.38658i	−0.711509	−	0.482780i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		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
$187$	−17.7251			−1.29619
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3.82518i			0.276781i	0.990378	+	0.138390i	$0.0441928\pi$
$191$			3.82518i			0.276781i	−0.990378	+	0.138390i	$0.955807\pi$
$192$		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
$197$		−	17.4356i		−	1.24223i	−0.783718	−	0.621117i	$-0.786679\pi$
$197$		−	17.4356i		−	1.24223i	0.783718	−	0.621117i	$-0.213321\pi$
$198$		0			0
$199$			28.1890i			1.99826i	0.0416556	+	0.999132i	$0.486737\pi$
$199$			28.1890i			1.99826i	−0.0416556	+	0.999132i	$0.513263\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.0000			−0.834058
$208$		0			0
$209$	−9.37459			−0.648454
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	23.1375	+	7.10874i	1.57796	+	0.484812i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	12.4124	+	8.42217i	0.827492	+	0.561478i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−29.3746			−1.94113			−0.970564	−	0.240845i	$-0.922576\pi$
$229$	−29.3746			−1.94113			−0.970564	+	0.240845i	$0.922576\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.1222i			1.05620i	0.849183	+	0.528099i	$0.177095\pi$
$233$			16.1222i			1.05620i	−0.849183	+	0.528099i	$0.822905\pi$
$234$		0			0
$235$	21.9622	+	6.74766i	1.43266	+	0.440169i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			19.5863i			1.26693i	0.773771	+	0.633465i	$0.218368\pi$
$239$			19.5863i			1.26693i	−0.773771	+	0.633465i	$0.781632\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.90033	−	1.19834i	−0.249183	−	0.0765589i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	30.8158i		−	1.94508i	−0.232740	−	0.972539i	$-0.574769\pi$
$251$		−	30.8158i		−	1.94508i	0.232740	−	0.972539i	$-0.425231\pi$
$252$		0			0
$253$			8.60271i			0.540848i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	31.9244			1.96854			0.984272	−	0.176659i	$-0.0565291\pi$
$263$	31.9244			1.96854			0.984272	+	0.176659i	$0.0565291\pi$
$264$		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
$271$			26.1534i			1.58871i	0.607457	+	0.794353i	$0.292190\pi$
$271$			26.1534i			1.58871i	−0.607457	+	0.794353i	$0.707810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.03779	−	8.89834i	0.364092	−	0.536590i
$276$		0			0
$277$			26.3994i			1.58619i	0.609101	+	0.793093i	$0.291530\pi$
$277$			26.3994i			1.58619i	−0.609101	+	0.793093i	$0.708470\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−26.8248			−1.59457			−0.797283	−	0.603606i	$-0.793730\pi$
$283$	−26.8248			−1.59457			−0.797283	+	0.603606i	$0.793730\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−50.9244			−2.99555
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−24.6254			−1.41939
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.96221	+	2.44631i	0.455915	+	0.140075i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			32.4903i			1.84236i	0.389139	+	0.921179i	$0.372773\pi$
$311$			32.4903i			1.84236i	−0.389139	+	0.921179i	$0.627227\pi$
$312$		0			0
$313$			34.8712i			1.97104i	0.169570	+	0.985518i	$0.445762\pi$
$313$			34.8712i			1.97104i	−0.169570	+	0.985518i	$0.554238\pi$
$314$		0			0
$315$	−14.5876	−	4.48190i	−0.821920	−	0.252526i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−35.9244			−1.99889
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−23.3746			−1.28868
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		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
$343$	20.0756			1.08398
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−22.2749			−1.19578			−0.597890	−	0.801578i	$-0.703994\pi$
$347$	−22.2749			−1.19578			−0.597890	+	0.801578i	$0.703994\pi$
$348$		0			0
$349$	−28.8248			−1.54295			−0.771477	−	0.636257i	$-0.780482\pi$
$349$	−28.8248			−1.54295			−0.771477	+	0.636257i	$0.780482\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	34.8712i		−	1.85601i	−0.372572	−	0.928003i	$-0.621524\pi$
$353$		−	34.8712i		−	1.85601i	0.372572	−	0.928003i	$-0.378476\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			37.7440i			1.99205i	0.0890519	+	0.996027i	$0.471616\pi$
$359$			37.7440i			1.99205i	−0.0890519	+	0.996027i	$0.528384\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.96221	−	6.38658i	0.102707	−	0.334289i
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$		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
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$	−3.21304	+	10.4578i	−0.163752	+	0.532977i
$386$		0			0
$387$	32.4743			1.65076
$388$		0			0
$389$	−13.9244			−0.705996			−0.352998	−	0.935624i	$-0.614838\pi$
$389$	−13.9244			−0.705996			−0.352998	+	0.935624i	$0.614838\pi$
$390$		0			0
$391$			32.9665i			1.66719i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			25.6772i			1.28870i	0.764730	+	0.644351i	$0.222873\pi$
$397$			25.6772i			1.28870i	−0.764730	+	0.644351i	$0.777127\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	19.2371	+	5.91041i	0.955901	+	0.293691i
$406$		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
$415$	−34.1993	−	10.5074i	−1.67878	−	0.515788i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	8.71780i		−	0.425892i	−0.977064	−	0.212946i	$-0.931694\pi$
$419$		−	8.71780i		−	0.425892i	0.977064	−	0.212946i	$-0.0683059\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	30.8248			1.49875
$424$		0			0
$425$	23.1375	−	34.0994i	1.12233	−	1.65406i
$426$		0			0
$427$	−8.47425			−0.410098
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			17.4356i			0.834058i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−5.47425			−0.260679
$442$		0			0
$443$	11.9244			0.566546			0.283273	−	0.959039i	$-0.408580\pi$
$443$	11.9244			0.566546			0.283273	+	0.959039i	$0.408580\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	4.89261i		−	0.228867i	−0.993431	−	0.114433i	$-0.963495\pi$
$457$		−	4.89261i		−	0.228867i	0.993431	−	0.114433i	$-0.0365053\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	37.3746			1.74071			0.870354	−	0.492427i	$-0.163890\pi$
$461$	37.3746			1.74071			0.870354	+	0.492427i	$0.163890\pi$
$462$		0			0
$463$	−9.17525			−0.426410			−0.213205	−	0.977007i	$-0.568390\pi$
$463$	−9.17525			−0.426410			−0.213205	+	0.977007i	$0.568390\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.7251			0.820219			0.410110	−	0.912036i	$-0.365490\pi$
$467$	17.7251			0.820219			0.410110	+	0.912036i	$0.365490\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	23.2806i		−	1.07044i
$474$		0			0
$475$	12.2371	−	18.0348i	0.561478	−	0.827492i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	43.5890i		−	1.99163i	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$		−	43.5890i		−	1.99163i	0.0913823	−	0.995816i	$-0.470871\pi$
$480$		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
$491$		−	43.5890i		−	1.96714i	−0.180517	−	0.983572i	$-0.557777\pi$
$491$		−	43.5890i		−	1.96714i	0.180517	−	0.983572i	$-0.442223\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	4.23713	−	13.7910i	0.190445	−	0.619857i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			44.6722i			1.99980i	0.0139987	+	0.999902i	$0.495544\pi$
$499$			44.6722i			1.99980i	−0.0139987	+	0.999902i	$0.504456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−44.0000			−1.96186			−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000			−1.96186			−0.980932	+	0.194354i	$0.937739\pi$
$504$		0			0
$505$	21.3746	+	6.56712i	0.951157	+	0.292233i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$			6.79730i			0.300695i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	22.0980i		−	0.971870i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			3.92445i			0.169038i
$540$		0			0
$541$	−46.4743			−1.99808			−0.999042	−	0.0437584i	$-0.986067\pi$
$541$	−46.4743			−1.99808			−0.999042	+	0.0437584i	$0.986067\pi$
$542$		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
$549$	11.1752			0.476948
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	42.8826i		−	1.81700i	−0.417889	−	0.908498i	$-0.637230\pi$
$557$		−	42.8826i		−	1.81700i	0.417889	−	0.908498i	$-0.362770\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−20.4743			−0.859838
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	26.1534i		−	1.09449i	−0.836974	−	0.547243i	$-0.815677\pi$
$571$		−	26.1534i		−	1.09449i	0.836974	−	0.547243i	$-0.184323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.5498	−	11.2296i	−0.690176	−	0.468305i
$576$		0			0
$577$			21.3759i			0.889889i	0.895558	+	0.444945i	$0.146777\pi$
$577$			21.3759i			0.889889i	−0.895558	+	0.444945i	$0.853223\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	36.3987			1.51007
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	45.0241			1.85834			0.929172	−	0.369649i	$-0.120522\pi$
$587$	45.0241			1.85834			0.929172	+	0.369649i	$0.120522\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			34.8712i			1.43199i	0.698106	+	0.715994i	$0.254026\pi$
$593$			34.8712i			1.43199i	−0.698106	+	0.715994i	$0.745974\pi$
$594$		0			0
$595$	−12.3127	+	40.0753i	−0.504772	+	1.64293i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.6254	+	4.18627i	0.553952	+	0.170196i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	18.5188i		−	0.747969i	−0.927435	−	0.373985i	$-0.877991\pi$
$613$		−	18.5188i		−	0.747969i	0.927435	−	0.373985i	$-0.122009\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	35.2323i		−	1.41840i	−0.705008	−	0.709199i	$-0.749057\pi$
$617$		−	35.2323i		−	1.41840i	0.705008	−	0.709199i	$-0.250943\pi$
$618$		0			0
$619$			43.5890i			1.75199i	0.482321	+	0.875995i	$0.339794\pi$
$619$			43.5890i			1.75199i	−0.482321	+	0.875995i	$0.660206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.23713	+	23.2309i	0.369485	+	0.929237i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	10.9836i		−	0.437249i	−0.975809	−	0.218624i	$-0.929843\pi$
$631$		−	10.9836i		−	0.437249i	0.975809	−	0.218624i	$-0.0701569\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	13.1752			0.519581			0.259791	−	0.965665i	$-0.416346\pi$
$643$	13.1752			0.519581			0.259791	+	0.965665i	$0.416346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−33.0241			−1.29831			−0.649155	−	0.760656i	$-0.724878\pi$
$647$	−33.0241			−1.29831			−0.649155	+	0.760656i	$0.724878\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			50.7632i			1.98652i	0.115920	+	0.993259i	$0.463018\pi$
$653$			50.7632i			1.98652i	−0.115920	+	0.993259i	$0.536982\pi$
$654$		0			0
$655$	11.1375	−	36.2501i	0.435177	−	1.41641i
$656$		0			0
$657$		−	8.96379i		−	0.349711i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.51204	+	21.1953i	−0.252526	+	0.821920i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	8.01145i		−	0.309279i
$672$		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
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−14.5120	+	47.2336i	−0.554477	+	1.80470i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			49.9259i			1.89927i	0.313355	+	0.949636i	$0.398547\pi$
$691$			49.9259i			1.89927i	−0.313355	+	0.949636i	$0.601453\pi$
$692$		0			0
$693$			14.6778i			0.557565i
$694$		0			0
$695$	−2.03779	+	6.63257i	−0.0772978	+	0.251588i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.7492			−0.855571
$708$		0			0
$709$	−10.0000			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	31.5380i		−	1.17617i	−0.808800	−	0.588084i	$-0.799882\pi$
$719$		−	31.5380i		−	1.17617i	0.808800	−	0.588084i	$-0.200118\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	52.4743			1.94616			0.973081	−	0.230463i	$-0.0740239\pi$
$727$	52.4743			1.94616			0.973081	+	0.230463i	$0.0740239\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		−	89.2136i		−	3.29968i
$732$		0			0
$733$		−	52.3068i		−	1.93200i	−0.258551	−	0.965998i	$-0.583245\pi$
$733$		−	52.3068i		−	1.93200i	0.258551	−	0.965998i	$-0.416755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$			54.2273i			1.99478i	0.0721811	+	0.997392i	$0.477004\pi$
$739$			54.2273i			1.99478i	−0.0721811	+	0.997392i	$0.522996\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−28.5876	−	8.78325i	−1.04737	−	0.321793i
$746$		0			0
$747$	−48.0000			−1.75623
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	0.591258i		−	0.0214896i	−0.999942	−	0.0107448i	$-0.996580\pi$
$757$		−	0.591258i		−	0.0214896i	0.999942	−	0.0107448i	$-0.00342025\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	23.7251			0.860034			0.430017	−	0.902821i	$-0.358508\pi$
$761$	23.7251			0.860034			0.430017	+	0.902821i	$0.358508\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	16.2371	−	52.8484i	0.587055	−	1.91074i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	6.62541			0.238919			0.119459	−	0.992839i	$-0.461884\pi$
$769$	6.62541			0.238919			0.119459	+	0.992839i	$0.461884\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.4502	−	37.2679i	0.408674	−	1.33015i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		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
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		−	84.6820i		−	2.99584i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.42608			−0.226771
$804$		0			0
$805$	19.4502	+	5.97586i	0.685528	+	0.210621i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	51.5739			1.81324			0.906621	−	0.421945i	$-0.138653\pi$
$809$	51.5739			1.81324			0.906621	+	0.421945i	$0.138653\pi$
$810$		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
$815$	51.2990	+	15.7611i	1.79693	+	0.552087i
$816$		0			0
$817$		−	47.1840i		−	1.65076i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	45.3746			1.58358			0.791792	−	0.610791i	$-0.209148\pi$
$821$	45.3746			1.58358			0.791792	+	0.610791i	$0.209148\pi$
$822$		0			0
$823$	−53.5739			−1.86747			−0.933735	−	0.357966i	$-0.883471\pi$
$823$	−53.5739			−1.86747			−0.933735	+	0.357966i	$0.883471\pi$
$824$		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
$833$			15.0389i			0.521068i
$834$		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
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−27.7870	−	8.53726i	−0.955901	−	0.293691i
$846$		0			0
$847$	−14.5017			−0.498283
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	52.3068i		−	1.79095i	−0.445112	−	0.895475i	$-0.646836\pi$
$853$		−	52.3068i		−	1.79095i	0.445112	−	0.895475i	$-0.353164\pi$
$854$		0			0
$855$	8.58762	−	27.9509i	0.293691	−	0.955901i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		−	41.3232i		−	1.40993i	−0.709242	−	0.704965i	$-0.750963\pi$
$859$		−	41.3232i		−	1.40993i	0.709242	−	0.704965i	$-0.249037\pi$
$860$		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
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−15.9244	−	19.8323i	−0.538344	−	0.670453i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	59.0241			1.98857			0.994286	−	0.106749i	$-0.0340440\pi$
$881$	59.0241			1.98857			0.994286	+	0.106749i	$0.0340440\pi$
$882$		0			0
$883$	51.9244			1.74740			0.873698	−	0.486469i	$-0.161715\pi$
$883$	51.9244			1.74740			0.873698	+	0.486469i	$0.161715\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	19.3561i		−	0.648454i
$892$		0			0
$893$		−	44.7873i		−	1.49875i
$894$		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
$909$	30.0000			0.995037
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$			34.4108i			1.13883i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			38.5813i			1.27407i
$918$		0			0
$919$			8.71780i			0.287574i	0.989609	+	0.143787i	$0.0459280\pi$
$919$			8.71780i			0.287574i	−0.989609	+	0.143787i	$0.954072\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	50.0000			1.64045			0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000			1.64045			0.820223	+	0.572043i	$0.193849\pi$
$930$		0			0
$931$			7.95391i			0.260679i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−37.8866	−	11.6403i	−1.23903	−	0.380678i
$936$		0			0
$937$			60.3182i			1.97051i	0.171089	+	0.985255i	$0.445271\pi$
$937$			60.3182i			1.97051i	−0.171089	+	0.985255i	$0.554729\pi$
$938$		0			0
$939$		0			0
$940$		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
$947$	8.00000			0.259965			0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000			0.259965			0.129983	+	0.991516i	$0.458508\pi$
$948$		0			0
$949$		0			0
$950$		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
$955$	−2.51204	+	8.17617i	−0.0812879	+	0.264575i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	50.2712i		−	1.62334i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	12.0000			0.385894			0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000			0.385894			0.192947	+	0.981209i	$0.438195\pi$
$968$		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
$973$		−	7.05911i		−	0.226305i
$974$		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
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	11.4502	−	37.2679i	0.364833	−	1.18745i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−43.2990			−1.37683
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−18.5120	+	60.2528i	−0.586871	+	1.91014i
$996$		0			0
$997$			56.7390i			1.79694i	0.439031	+	0.898472i	$0.355322\pi$
$997$			56.7390i			1.79694i	−0.439031	+	0.898472i	$0.644678\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.g.d.1519.4	yes	4
4.3	odd	2		1520.2.g.c.1519.4	yes	4
5.4	even	2		1520.2.g.c.1519.3	✓	4
19.18	odd	2	CM	1520.2.g.d.1519.4	yes	4
20.19	odd	2	inner	1520.2.g.d.1519.3	yes	4
76.75	even	2		1520.2.g.c.1519.4	yes	4
95.94	odd	2		1520.2.g.c.1519.3	✓	4
380.379	even	2	inner	1520.2.g.d.1519.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.g.c.1519.3	✓	4	5.4	even	2
1520.2.g.c.1519.3	✓	4	95.94	odd	2
1520.2.g.c.1519.4	yes	4	4.3	odd	2
1520.2.g.c.1519.4	yes	4	76.75	even	2
1520.2.g.d.1519.3	yes	4	20.19	odd	2	inner
1520.2.g.d.1519.3	yes	4	380.379	even	2	inner
1520.2.g.d.1519.4	yes	4	1.1	even	1	trivial
1520.2.g.d.1519.4	yes	4	19.18	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.13746 + 0.656712i) q^{5} -2.27492 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+(2.13746 + 0.656712i) q^{5} -2.27492 q^{7} +3.00000 q^{9} -2.15068i q^{11} -8.24163i q^{17} -4.35890i q^{19} -4.00000 q^{23} +(4.13746 + 2.80739i) q^{25} +(-4.86254 - 1.49397i) q^{35} +10.8248 q^{43} +(6.41238 + 1.97014i) q^{45} +10.2749 q^{47} -1.82475 q^{49} +(1.41238 - 4.59698i) q^{55} +3.72508 q^{61} -6.82475 q^{63} -2.98793i q^{73} +4.89261i q^{77} +9.00000 q^{81} -16.0000 q^{83} +(5.41238 - 17.6161i) q^{85} +(2.86254 - 9.31697i) q^{95} -6.45203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} + 6 q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q + q^5 + 6 * q^7 + 12 * q^9	\( 4 q + q^{5} + 6 q^{7} + 12 q^{9} - 16 q^{23} + 9 q^{25} - 27 q^{35} - 2 q^{43} + 3 q^{45} + 26 q^{47} + 38 q^{49} - 17 q^{55} + 30 q^{61} + 18 q^{63} + 36 q^{81} - 64 q^{83} - q^{85} + 19 q^{95}+O(q^{100}) \) 4 * q + q^5 + 6 * q^7 + 12 * q^9 - 16 * q^23 + 9 * q^25 - 27 * q^35 - 2 * q^43 + 3 * q^45 + 26 * q^47 + 38 * q^49 - 17 * q^55 + 30 * q^61 + 18 * q^63 + 36 * q^81 - 64 * q^83 - q^85 + 19 * q^95

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