LMFDB - Embedded newform 252.4.b.b.55.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,4,Mod(55,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("252.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	252.b (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:	$14.8684813214$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			55.4
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	252.55
Dual form			252.4.b.b.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843i q^{2} -8.00000 q^{4} +3.74166i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})$	$q+2.82843i q^{2} -8.00000 q^{4} +3.74166i q^{5} +18.5203 q^{7} -22.6274i q^{8} -10.5830 q^{10} +43.8406i q^{11} +52.3832i q^{14} +64.0000 q^{16} +138.441i q^{17} -153.454 q^{19} -29.9333i q^{20} -124.000 q^{22} -134.350i q^{23} +111.000 q^{25} -148.162 q^{28} -201.077 q^{31} +181.019i q^{32} -391.571 q^{34} +69.2965i q^{35} -376.000 q^{37} -434.032i q^{38} +84.6640 q^{40} +407.841i q^{41} -350.725i q^{44} +380.000 q^{46} +343.000 q^{49} +313.955i q^{50} -164.037 q^{55} -419.066i q^{56} -568.732i q^{62} -512.000 q^{64} -1107.53i q^{68} -196.000 q^{70} +451.134i q^{71} -1063.49i q^{74} +1227.63 q^{76} +811.940i q^{77} +239.466i q^{80} -1153.55 q^{82} -518.000 q^{85} +992.000 q^{88} +1552.79i q^{89} +1074.80i q^{92} -574.171i q^{95} +970.151i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4}+O(q^{10}) $ 4 * q - 32 * q^4	$ 4 q - 32 q^{4} + 256 q^{16} - 496 q^{22} + 444 q^{25} - 1504 q^{37} + 1520 q^{46} + 1372 q^{49} - 2048 q^{64} - 784 q^{70} - 2072 q^{85} + 3968 q^{88}+O(q^{100}) $ 4 * q - 32 * q^4 + 256 * q^16 - 496 * q^22 + 444 * q^25 - 1504 * q^37 + 1520 * q^46 + 1372 * q^49 - 2048 * q^64 - 784 * q^70 - 2072 * q^85 + 3968 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	2.82843i	1.00000i
$2$	2.82843i	1.00000i
$3$	0	0
$3$	0	0
$4$	−8.00000	−1.00000
$5$	3.74166i	0.334664i	0.985901	+	0.167332i	$0.0535152\pi$
$5$	3.74166i	0.334664i	−0.985901	+	0.167332i	$0.946485\pi$
$6$	0	0
$7$	18.5203	1.00000
$7$	18.5203	1.00000
$8$	− 22.6274i	− 1.00000i
$9$	0	0
$10$	−10.5830	−0.334664
$11$	43.8406i	1.20168i	0.799371	+	0.600838i	$0.205166\pi$
$11$	43.8406i	1.20168i	−0.799371	+	0.600838i	$0.794834\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	52.3832i	1.00000i
$15$	0	0
$16$	64.0000	1.00000
$17$	138.441i	1.97511i	0.157259	+	0.987557i	$0.449734\pi$
$17$	138.441i	1.97511i	−0.157259	+	0.987557i	$0.550266\pi$
$18$	0	0
$19$	−153.454	−1.85288	−0.926439	−	0.376446i	$-0.877146\pi$
$19$	−153.454	−1.85288	−0.926439	+	0.376446i	$0.877146\pi$
$20$	− 29.9333i	− 0.334664i
$21$	0	0
$22$	−124.000	−1.20168
$23$	− 134.350i	− 1.21800i	−0.793171	−	0.608999i	$-0.791571\pi$
$23$	− 134.350i	− 1.21800i	0.793171	−	0.608999i	$-0.208429\pi$
$24$	0	0
$25$	111.000	0.888000
$26$	0	0
$27$	0	0
$28$	−148.162	−1.00000
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−201.077	−1.16498	−0.582492	−	0.812836i	$-0.697922\pi$
$31$	−201.077	−1.16498	−0.582492	+	0.812836i	$0.697922\pi$
$32$	181.019i	1.00000i
$33$	0	0
$34$	−391.571	−1.97511
$35$	69.2965i	0.334664i
$36$	0	0
$37$	−376.000	−1.67065	−0.835325	−	0.549757i	$-0.814720\pi$
$37$	−376.000	−1.67065	−0.835325	+	0.549757i	$0.814720\pi$
$38$	− 434.032i	− 1.85288i
$39$	0	0
$40$	84.6640	0.334664
$41$	407.841i	1.55351i	0.629801	+	0.776756i	$0.283136\pi$
$41$	407.841i	1.55351i	−0.629801	+	0.776756i	$0.716864\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	− 350.725i	− 1.20168i
$45$	0	0
$46$	380.000	1.21800
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	313.955i	0.888000i
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−164.037	−0.402158
$56$	− 419.066i	− 1.00000i
$57$	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			$\pi$
$62$	− 568.732i	− 1.16498i
$63$	0	0
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 1107.53i	− 1.97511i
$69$	0	0
$70$	−196.000	−0.334664
$71$	451.134i	0.754081i	0.926197	+	0.377041i	$0.123058\pi$
$71$	451.134i	0.754081i	−0.926197	+	0.377041i	$0.876942\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	− 1063.49i	− 1.67065i
$75$	0	0
$76$	1227.63	1.85288
$77$	811.940i	1.20168i
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	239.466i	0.334664i
$81$	0	0
$82$	−1153.55	−1.55351
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−518.000	−0.661000
$86$	0	0
$87$	0	0
$88$	992.000	1.20168
$89$	1552.79i	1.84938i	0.380716	+	0.924692i	$0.375678\pi$
$89$	1552.79i	1.84938i	−0.380716	+	0.924692i	$0.624322\pi$
$90$	0	0
$91$	0	0
$92$	1074.80i	1.21800i
$93$	0	0
$94$	0	0
$95$	− 574.171i	− 0.620091i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	970.151i	1.00000i
$99$	0	0
$100$	−888.000	−0.888000
$101$	879.289i	0.866263i	0.901331	+	0.433132i	$0.142591\pi$
$101$	879.289i	0.866263i	−0.901331	+	0.433132i	$0.857409\pi$
$102$	0	0
$103$	2084.85	1.99443	0.997216	−	0.0745620i	$-0.0237559\pi$
$103$	2084.85	1.99443	0.997216	+	0.0745620i	$0.0237559\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1916.26i	− 1.73133i	−0.500628	−	0.865663i	$-0.666898\pi$
$107$	− 1916.26i	− 1.73133i	0.500628	−	0.865663i	$-0.333102\pi$
$108$	0	0
$109$	2270.00	1.99474	0.997370	−	0.0724840i	$-0.0230926\pi$
$109$	2270.00	1.99474	0.997370	+	0.0724840i	$0.0230926\pi$
$110$	− 463.966i	− 0.402158i
$111$	0	0
$112$	1185.30	1.00000
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	502.693	0.407620
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2563.97i	1.97511i
$120$	0	0
$121$	−591.000	−0.444027
$122$	0	0
$123$	0	0
$124$	1608.62	1.16498
$125$	883.031i	0.631846i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1448.15i	− 1.00000i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−2842.00	−1.85288
$134$	0	0
$135$	0	0
$136$	3132.57	1.97511
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	656.146	0.400386	0.200193	−	0.979757i	$-0.435843\pi$
$139$	656.146	0.400386	0.200193	+	0.979757i	$0.435843\pi$
$140$	− 554.372i	− 0.334664i
$141$	0	0
$142$	−1276.00	−0.754081
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	3008.00	1.67065
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	3472.26i	1.85288i
$153$	0	0
$154$	−2296.51	−1.20168
$155$	− 752.362i	− 0.389878i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	−677.312	−0.334664
$161$	− 2488.20i	− 1.21800i
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	− 3262.73i	− 1.55351i
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	− 1465.13i	− 0.661000i
$171$	0	0
$172$	0	0
$173$	− 4374.00i	− 1.92225i	−0.276117	−	0.961124i	$-0.589048\pi$
$173$	− 4374.00i	− 1.92225i	0.276117	−	0.961124i	$-0.410952\pi$
$174$	0	0
$175$	2055.75	0.888000
$176$	2805.80i	1.20168i
$177$	0	0
$178$	−4391.95	−1.84938
$179$	3658.57i	1.52768i	0.645407	+	0.763839i	$0.276688\pi$
$179$	3658.57i	1.52768i	−0.645407	+	0.763839i	$0.723312\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−3040.00	−1.21800
$185$	− 1406.86i	− 0.559106i
$186$	0	0
$187$	−6069.35	−2.37345
$188$	0	0
$189$	0	0
$190$	1624.00	0.620091
$191$	− 4003.64i	− 1.51672i	−0.651837	−	0.758359i	$-0.726001\pi$
$191$	− 4003.64i	− 1.51672i	0.651837	−	0.758359i	$-0.273999\pi$
$192$	0	0
$193$	2252.00	0.839909	0.419955	−	0.907545i	$-0.362046\pi$
$193$	2252.00	0.839909	0.419955	+	0.907545i	$0.362046\pi$
$194$	0	0
$195$	0	0
$196$	−2744.00	−1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−3010.86	−1.07254	−0.536268	−	0.844048i	$-0.680166\pi$
$199$	−3010.86	−1.07254	−0.536268	+	0.844048i	$0.680166\pi$
$200$	− 2511.64i	− 0.888000i
$201$	0	0
$202$	−2487.01	−0.866263
$203$	0	0
$204$	0	0
$205$	−1526.00	−0.519905
$206$	5896.85i	1.99443i
$207$	0	0
$208$	0	0
$209$	− 6727.50i	− 2.22656i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	5420.00	1.73133
$215$	0	0
$216$	0	0
$217$	−3724.00	−1.16498
$218$	6420.53i	1.99474i
$219$	0	0
$220$	1312.29	0.402158
$221$	0	0
$222$	0	0
$223$	−820.183	−0.246294	−0.123147	−	0.992388i	$-0.539299\pi$
$223$	−820.183	−0.246294	−0.123147	+	0.992388i	$0.539299\pi$
$224$	3352.53i	1.00000i
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	1421.83i	0.407620i
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−7252.00	−1.97511
$239$	− 3341.79i	− 0.904444i	−0.891905	−	0.452222i	$-0.850631\pi$
$239$	− 3341.79i	− 0.904444i	0.891905	−	0.452222i	$-0.149369\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	− 1671.60i	− 0.444027i
$243$	0	0
$244$	0	0
$245$	1283.39i	0.334664i
$246$	0	0
$247$	0	0
$248$	4549.86i	1.16498i
$249$	0	0
$250$	−2497.59	−0.631846
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	5890.00	1.46364
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	2832.43i	0.687480i	0.939065	+	0.343740i	$0.111694\pi$
$257$	2832.43i	0.687480i	−0.939065	+	0.343740i	$0.888306\pi$
$258$	0	0
$259$	−6963.62	−1.67065
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1112.99i	0.260949i	0.991452	+	0.130475i	$0.0416501\pi$
$263$	1112.99i	0.260949i	−0.991452	+	0.130475i	$0.958350\pi$
$264$	0	0
$265$	0	0
$266$	− 8038.39i	− 1.85288i
$267$	0	0
$268$	0	0
$269$	− 8549.69i	− 1.93786i	−0.247340	−	0.968929i	$-0.579557\pi$
$269$	− 8549.69i	− 1.93786i	0.247340	−	0.968929i	$-0.420443\pi$
$270$	0	0
$271$	7418.69	1.66293	0.831463	−	0.555580i	$-0.187504\pi$
$271$	7418.69	1.66293	0.831463	+	0.555580i	$0.187504\pi$
$272$	8860.24i	1.97511i
$273$	0	0
$274$	0	0
$275$	4866.31i	1.06709i
$276$	0	0
$277$	6176.00	1.33964	0.669819	−	0.742524i	$-0.266372\pi$
$277$	6176.00	1.33964	0.669819	+	0.742524i	$0.266372\pi$
$278$	1855.86i	0.400386i
$279$	0	0
$280$	1568.00	0.334664
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	3942.17	0.828048	0.414024	−	0.910266i	$-0.364123\pi$
$283$	3942.17	0.828048	0.414024	+	0.910266i	$0.364123\pi$
$284$	− 3609.07i	− 0.754081i
$285$	0	0
$286$	0	0
$287$	7553.31i	1.55351i
$288$	0	0
$289$	−14253.0	−2.90108
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5054.98i	1.00790i	0.863732	+	0.503951i	$0.168121\pi$
$293$	5054.98i	1.00790i	−0.863732	+	0.503951i	$0.831879\pi$
$294$	0	0
$295$	0	0
$296$	8507.91i	1.67065i
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−9821.03	−1.85288
$305$	0	0
$306$	0	0
$307$	−4725.31	−0.878462	−0.439231	−	0.898374i	$-0.644749\pi$
$307$	−4725.31	−0.878462	−0.439231	+	0.898374i	$0.644749\pi$
$308$	− 6495.52i	− 1.20168i
$309$	0	0
$310$	2128.00	0.389878
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	− 1915.73i	− 0.334664i
$321$	0	0
$322$	7037.70	1.21800
$323$	− 21244.3i	− 3.65965i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	9228.38	1.55351
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2014.00	−0.325548	−0.162774	−	0.986663i	$-0.552044\pi$
$337$	−2014.00	−0.325548	−0.162774	+	0.986663i	$0.552044\pi$
$338$	6214.05i	1.00000i
$339$	0	0
$340$	4144.00	0.661000
$341$	− 8815.34i	− 1.39993i
$342$	0	0
$343$	6352.45	1.00000
$344$	0	0
$345$	0	0
$346$	12371.5	1.92225
$347$	12924.5i	1.99949i	0.0225586	+	0.999746i	$0.492819\pi$
$347$	12924.5i	1.99949i	−0.0225586	+	0.999746i	$0.507181\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	5814.54i	0.888000i
$351$	0	0
$352$	−7936.00	−1.20168
$353$	9769.47i	1.47302i	0.676426	+	0.736510i	$0.263528\pi$
$353$	9769.47i	1.47302i	−0.676426	+	0.736510i	$0.736472\pi$
$354$	0	0
$355$	−1687.99	−0.252364
$356$	− 12422.3i	− 1.84938i
$357$	0	0
$358$	−10348.0	−1.52768
$359$	− 618.011i	− 0.0908562i	−0.998968	−	0.0454281i	$-0.985535\pi$
$359$	− 618.011i	− 0.0908562i	0.998968	−	0.0454281i	$-0.0144652\pi$
$360$	0	0
$361$	16689.0	2.43315
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10609.5	1.50902	0.754509	−	0.656290i	$-0.227875\pi$
$367$	10609.5	1.50902	0.754509	+	0.656290i	$0.227875\pi$
$368$	− 8598.42i	− 1.21800i
$369$	0	0
$370$	3979.21	0.559106
$371$	0	0
$372$	0	0
$373$	−1258.00	−0.174629	−0.0873147	−	0.996181i	$-0.527829\pi$
$373$	−1258.00	−0.174629	−0.0873147	+	0.996181i	$0.527829\pi$
$374$	− 17166.7i	− 2.37345i
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	4593.37i	0.620091i
$381$	0	0
$382$	11324.0	1.51672
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−3038.00	−0.402158
$386$	6369.62i	0.839909i
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	18599.6	2.40569
$392$	− 7761.20i	− 1.00000i
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	− 8516.01i	− 1.07254i
$399$	0	0
$400$	7104.00	0.888000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	− 7034.32i	− 0.866263i
$405$	0	0
$406$	0	0
$407$	− 16484.1i	− 2.00758i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	− 4316.18i	− 0.519905i
$411$	0	0
$412$	−16678.8	−1.99443
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	19028.2	2.22656
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−13480.0	−1.56051	−0.780255	−	0.625461i	$-0.784911\pi$
$421$	−13480.0	−1.56051	−0.780255	+	0.625461i	$0.784911\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15367.0i	1.75390i
$426$	0	0
$427$	0	0
$428$	15330.1i	1.73133i
$429$	0	0
$430$	0	0
$431$	15953.7i	1.78298i	0.453039	+	0.891490i	$0.350340\pi$
$431$	15953.7i	1.78298i	−0.453039	+	0.891490i	$0.649660\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	− 10533.1i	− 1.16498i
$435$	0	0
$436$	−18160.0	−1.99474
$437$	20616.5i	2.25680i
$438$	0	0
$439$	−6820.75	−0.741541	−0.370770	−	0.928725i	$-0.620906\pi$
$439$	−6820.75	−0.741541	−0.370770	+	0.928725i	$0.620906\pi$
$440$	3711.72i	0.402158i
$441$	0	0
$442$	0	0
$443$	16310.1i	1.74925i	0.484801	+	0.874625i	$0.338892\pi$
$443$	16310.1i	1.74925i	−0.484801	+	0.874625i	$0.661108\pi$
$444$	0	0
$445$	−5810.00	−0.618922
$446$	− 2319.83i	− 0.246294i
$447$	0	0
$448$	−9482.37	−1.00000
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−17880.0	−1.86682
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19514.0	1.99743	0.998716	−	0.0506593i	$-0.0161323\pi$
$457$	19514.0	1.99743	0.998716	+	0.0506593i	$0.0161323\pi$
$458$	0	0
$459$	0	0
$460$	−4021.54	−0.407620
$461$	12800.2i	1.29320i	0.762829	+	0.646600i	$0.223810\pi$
$461$	12800.2i	1.29320i	−0.762829	+	0.646600i	$0.776190\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−17033.3	−1.64535
$476$	− 20511.8i	− 1.97511i
$477$	0	0
$478$	9452.00	0.904444
$479$	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
$484$	4728.00	0.444027
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	−3629.97	−0.334664
$491$	− 19684.4i	− 1.80926i	−0.426199	−	0.904630i	$-0.640148\pi$
$491$	− 19684.4i	− 1.80926i	0.426199	−	0.904630i	$-0.359852\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−12868.9	−1.16498
$497$	8355.12i	0.754081i
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 7064.25i	− 0.631846i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−3290.00	−0.289907
$506$	16659.4i	1.46364i
$507$	0	0
$508$	0	0
$509$	22027.1i	1.91814i	0.283161	+	0.959072i	$0.408617\pi$
$509$	22027.1i	1.91814i	−0.283161	+	0.959072i	$0.591383\pi$
$510$	0	0
$511$	0	0
$512$	11585.2i	1.00000i
$513$	0	0
$514$	−8011.33	−0.687480
$515$	7800.80i	0.667465i
$516$	0	0
$517$	0	0
$518$	− 19696.1i	− 1.67065i
$519$	0	0
$520$	0	0
$521$	− 22693.2i	− 1.90826i	−0.299388	−	0.954132i	$-0.596782\pi$
$521$	− 22693.2i	− 1.90826i	0.299388	−	0.954132i	$-0.403218\pi$
$522$	0	0
$523$	−9440.04	−0.789262	−0.394631	−	0.918840i	$-0.629128\pi$
$523$	−9440.04	−0.789262	−0.394631	+	0.918840i	$0.629128\pi$
$524$	0	0
$525$	0	0
$526$	−3148.00	−0.260949
$527$	− 27837.4i	− 2.30098i
$528$	0	0
$529$	−5883.00	−0.483521
$530$	0	0
$531$	0	0
$532$	22736.0	1.85288
$533$	0	0
$534$	0	0
$535$	7169.99	0.579412
$536$	0	0
$537$	0	0
$538$	24182.2	1.93786
$539$	15037.3i	1.20168i
$540$	0	0
$541$	−12472.0	−0.991152	−0.495576	−	0.868565i	$-0.665043\pi$
$541$	−12472.0	−0.991152	−0.495576	+	0.868565i	$0.665043\pi$
$542$	20983.2i	1.66293i
$543$	0	0
$544$	−25060.6	−1.97511
$545$	8493.56i	0.667567i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−13764.0	−1.06709
$551$	0	0
$552$	0	0
$553$	0	0
$554$	17468.4i	1.33964i
$555$	0	0
$556$	−5249.17	−0.400386
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	4434.97i	0.334664i
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	11150.1i	0.828048i
$567$	0	0
$568$	10208.0	0.754081
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	−21364.0	−1.55351
$575$	− 14912.9i	− 1.08158i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	− 40313.6i	− 2.90108i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−14297.6	−1.00790
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	30856.0	2.15857
$590$	0	0
$591$	0	0
$592$	−24064.0	−1.67065
$593$	− 28552.6i	− 1.97726i	−0.150374	−	0.988629i	$-0.548048\pi$
$593$	− 28552.6i	− 1.97726i	0.150374	−	0.988629i	$-0.451952\pi$
$594$	0	0
$595$	−9593.49	−0.661000
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26823.4i	1.82967i	0.403825	+	0.914836i	$0.367681\pi$
$599$	26823.4i	1.82967i	−0.403825	+	0.914836i	$0.632319\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2211.32i	− 0.148600i
$606$	0	0
$607$	29658.9	1.98322	0.991612	−	0.129253i	$-0.0412580\pi$
$607$	29658.9	1.98322	0.991612	+	0.129253i	$0.0412580\pi$
$608$	− 27778.1i	− 1.85288i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−12742.0	−0.839551	−0.419775	−	0.907628i	$-0.637891\pi$
$613$	−12742.0	−0.839551	−0.419775	+	0.907628i	$0.637891\pi$
$614$	− 13365.2i	− 0.878462i
$615$	0	0
$616$	18372.1	1.20168
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−29822.9	−1.93648	−0.968242	−	0.250014i	$-0.919565\pi$
$619$	−29822.9	−1.93648	−0.968242	+	0.250014i	$0.919565\pi$
$620$	6018.89i	0.389878i
$621$	0	0
$622$	0	0
$623$	28758.0i	1.84938i
$624$	0	0
$625$	10571.0	0.676544
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 52053.9i	− 3.29972i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	5418.50	0.334664
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−20631.6	−1.26536	−0.632682	−	0.774411i	$-0.718046\pi$
$643$	−20631.6	−1.26536	−0.632682	+	0.774411i	$0.718046\pi$
$644$	19905.6i	1.21800i
$645$	0	0
$646$	60088.0	3.65965
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	26101.8i	1.55351i
$657$	0	0
$658$	0	0
$659$	− 33634.2i	− 1.98817i	−0.108605	−	0.994085i	$-0.534638\pi$
$659$	− 33634.2i	− 1.98817i	0.108605	−	0.994085i	$-0.465362\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 10633.8i	− 0.620091i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−3652.00	−0.209174	−0.104587	−	0.994516i	$-0.533352\pi$
$673$	−3652.00	−0.209174	−0.104587	+	0.994516i	$0.533352\pi$
$674$	− 5696.45i	− 0.325548i
$675$	0	0
$676$	−17576.0	−1.00000
$677$	− 13870.3i	− 0.787415i	−0.919236	−	0.393707i	$-0.871192\pi$
$677$	− 13870.3i	− 0.787415i	0.919236	−	0.393707i	$-0.128808\pi$
$678$	0	0
$679$	0	0
$680$	11721.0i	0.661000i
$681$	0	0
$682$	24933.6	1.39993
$683$	− 15051.5i	− 0.843234i	−0.906774	−	0.421617i	$-0.861463\pi$
$683$	− 15051.5i	− 0.843234i	0.906774	−	0.421617i	$-0.138537\pi$
$684$	0	0
$685$	0	0
$686$	17967.4i	1.00000i
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	11895.3	0.654875	0.327437	−	0.944873i	$-0.393815\pi$
$691$	11895.3	0.654875	0.327437	+	0.944873i	$0.393815\pi$
$692$	34992.0i	1.92225i
$693$	0	0
$694$	−36556.0	−1.99949
$695$	2455.07i	0.133995i
$696$	0	0
$697$	−56462.0	−3.06837
$698$	0	0
$699$	0	0
$700$	−16446.0	−0.888000
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	57698.5	3.09551
$704$	− 22446.4i	− 1.20168i
$705$	0	0
$706$	−27632.2	−1.47302
$707$	16284.7i	0.866263i
$708$	0	0
$709$	18650.0	0.987892	0.493946	−	0.869493i	$-0.335554\pi$
$709$	18650.0	0.987892	0.493946	+	0.869493i	$0.335554\pi$
$710$	− 4774.35i	− 0.252364i
$711$	0	0
$712$	35135.6	1.84938
$713$	27014.8i	1.41895i
$714$	0	0
$715$	0	0
$716$	− 29268.6i	− 1.52768i
$717$	0	0
$718$	1748.00	0.0908562
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	38612.0	1.99443
$722$	47203.6i	2.43315i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32754.4	1.67097	0.835484	−	0.549515i	$-0.185187\pi$
$727$	32754.4	1.67097	0.835484	+	0.549515i	$0.185187\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	30008.1i	1.50902i
$735$	0	0
$736$	24320.0	1.21800
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	11254.9i	0.559106i
$741$	0	0
$742$	0	0
$743$	13459.1i	0.664556i	0.943181	+	0.332278i	$0.107817\pi$
$743$	13459.1i	0.664556i	−0.943181	+	0.332278i	$0.892183\pi$
$744$	0	0
$745$	0	0
$746$	− 3558.16i	− 0.174629i
$747$	0	0
$748$	48554.8	2.37345
$749$	− 35489.6i	− 1.73133i
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4534.00	−0.217690	−0.108845	−	0.994059i	$-0.534715\pi$
$757$	−4534.00	−0.217690	−0.108845	+	0.994059i	$0.534715\pi$
$758$	0	0
$759$	0	0
$760$	−12992.0	−0.620091
$761$	− 24242.2i	− 1.15477i	−0.816472	−	0.577384i	$-0.804073\pi$
$761$	− 24242.2i	− 1.15477i	0.816472	−	0.577384i	$-0.195927\pi$
$762$	0	0
$763$	42041.0	1.99474
$764$	32029.1i	1.51672i
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	− 8592.76i	− 0.402158i
$771$	0	0
$772$	−18016.0	−0.839909
$773$	− 2286.15i	− 0.106374i	−0.998585	−	0.0531871i	$-0.983062\pi$
$773$	− 2286.15i	− 0.106374i	0.998585	−	0.0531871i	$-0.0169380\pi$
$774$	0	0
$775$	−22319.6	−1.03451
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 62584.6i	− 2.87847i
$780$	0	0
$781$	−19778.0	−0.906162
$782$	52607.7i	2.40569i
$783$	0	0
$784$	21952.0	1.00000
$785$	0	0
$786$	0	0
$787$	−40490.6	−1.83397	−0.916984	−	0.398923i	$-0.869384\pi$
$787$	−40490.6	−1.83397	−0.916984	+	0.398923i	$0.869384\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	24086.9	1.07254
$797$	8893.92i	0.395281i	0.980275	+	0.197640i	$0.0633278\pi$
$797$	8893.92i	0.395281i	−0.980275	+	0.197640i	$0.936672\pi$
$798$	0	0
$799$	0	0
$800$	20093.1i	0.888000i
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	9310.00	0.407620
$806$	0	0
$807$	0	0
$808$	19896.0	0.866263
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	27134.8	1.17489	0.587443	−	0.809266i	$-0.300135\pi$
$811$	27134.8	1.17489	0.587443	+	0.809266i	$0.300135\pi$
$812$	0	0
$813$	0	0
$814$	46624.0	2.00758
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	12208.0	0.519905
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	− 47174.8i	− 1.99443i
$825$	0	0
$826$	0	0
$827$	43522.4i	1.83002i	0.403436	+	0.915008i	$0.367816\pi$
$827$	43522.4i	1.83002i	−0.403436	+	0.915008i	$0.632184\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	47485.4i	1.97511i
$834$	0	0
$835$	0	0
$836$	53820.0i	2.22656i
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	− 38127.2i	− 1.56051i
$843$	0	0
$844$	0	0
$845$	8220.42i	0.334664i
$846$	0	0
$847$	−10945.5	−0.444027
$848$	0	0
$849$	0	0
$850$	−43464.4	−1.75390
$851$	50515.7i	2.03485i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	−43360.0	−1.73133
$857$	− 36095.8i	− 1.43875i	−0.694622	−	0.719375i	$-0.744429\pi$
$857$	− 36095.8i	− 1.43875i	0.694622	−	0.719375i	$-0.255571\pi$
$858$	0	0
$859$	−47205.5	−1.87501	−0.937503	−	0.347976i	$-0.886869\pi$
$859$	−47205.5	−1.87501	−0.937503	+	0.347976i	$0.886869\pi$
$860$	0	0
$861$	0	0
$862$	−45124.0	−1.78298
$863$	− 17953.4i	− 0.708160i	−0.935215	−	0.354080i	$-0.884794\pi$
$863$	− 17953.4i	− 0.708160i	0.935215	−	0.354080i	$-0.115206\pi$
$864$	0	0
$865$	16366.0	0.643307
$866$	0	0
$867$	0	0
$868$	29792.0	1.16498
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 51364.2i	− 1.99474i
$873$	0	0
$874$	−58312.4	−2.25680
$875$	16354.0i	0.631846i
$876$	0	0
$877$	47234.0	1.81868	0.909338	−	0.416058i	$-0.136589\pi$
$877$	47234.0	1.81868	0.909338	+	0.416058i	$0.136589\pi$
$878$	− 19292.0i	− 0.741541i
$879$	0	0
$880$	−10498.3	−0.402158
$881$	− 42898.1i	− 1.64049i	−0.572011	−	0.820246i	$-0.693836\pi$
$881$	− 42898.1i	− 1.64049i	0.572011	−	0.820246i	$-0.306164\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−46132.0	−1.74925
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	− 16433.2i	− 0.618922i
$891$	0	0
$892$	6561.46	0.246294
$893$	0	0
$894$	0	0
$895$	−13689.1	−0.511259
$896$	− 26820.2i	− 1.00000i
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	− 50572.2i	− 1.86682i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 38750.9i	− 1.40930i	−0.709554	−	0.704651i	$-0.751104\pi$
$911$	− 38750.9i	− 1.40930i	0.709554	−	0.704651i	$-0.248896\pi$
$912$	0	0
$913$	0	0
$914$	55193.9i	1.99743i
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	− 11374.6i	− 0.407620i
$921$	0	0
$922$	−36204.5	−1.29320
$923$	0	0
$924$	0	0
$925$	−41736.0	−1.48354
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20478.1i	0.723212i	0.932331	+	0.361606i	$0.117772\pi$
$929$	20478.1i	0.723212i	−0.932331	+	0.361606i	$0.882228\pi$
$930$	0	0
$931$	−52634.6	−1.85288
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 22709.4i	− 0.794308i
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10510.3i	0.364109i	0.983288	+	0.182054i	$0.0582747\pi$
$941$	10510.3i	0.364109i	−0.983288	+	0.182054i	$0.941725\pi$
$942$	0	0
$943$	54793.5	1.89218
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5389.57i	0.184939i	0.995716	+	0.0924696i	$0.0294761\pi$
$947$	5389.57i	0.184939i	−0.995716	+	0.0924696i	$0.970524\pi$
$948$	0	0
$949$	0	0
$950$	− 48177.6i	− 1.64535i
$951$	0	0
$952$	58016.0	1.97511
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	14980.2	0.507591
$956$	26734.3i	0.904444i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10641.0	0.357188
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8426.21i	0.281087i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	13372.8i	0.444027i
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	12152.0	0.400386
$974$	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
$979$	−68075.2	−2.22236
$980$	− 10267.1i	− 0.334664i
$981$	0	0
$982$	55676.0	1.80926
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	− 36398.8i	− 1.16498i
$993$	0	0
$994$	−23631.9	−0.754081
$995$	− 11265.6i	− 0.358939i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.b.b.55.4	yes	4
3.2	odd	2	inner	252.4.b.b.55.1	✓	4
4.3	odd	2	inner	252.4.b.b.55.2	yes	4
7.6	odd	2	inner	252.4.b.b.55.3	yes	4
12.11	even	2	inner	252.4.b.b.55.3	yes	4
21.20	even	2	inner	252.4.b.b.55.2	yes	4
28.27	even	2	inner	252.4.b.b.55.1	✓	4
84.83	odd	2	CM	252.4.b.b.55.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.b.b.55.1	✓	4	3.2	odd	2	inner
252.4.b.b.55.1	✓	4	28.27	even	2	inner
252.4.b.b.55.2	yes	4	4.3	odd	2	inner
252.4.b.b.55.2	yes	4	21.20	even	2	inner
252.4.b.b.55.3	yes	4	7.6	odd	2	inner
252.4.b.b.55.3	yes	4	12.11	even	2	inner
252.4.b.b.55.4	yes	4	1.1	even	1	trivial
252.4.b.b.55.4	yes	4	84.83	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843i q^{2} -8.00000 q^{4} +3.74166i q^{5} +18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\)	\(q+2.82843i q^{2} -8.00000 q^{4} +3.74166i q^{5} +18.5203 q^{7} -22.6274i q^{8} -10.5830 q^{10} +43.8406i q^{11} +52.3832i q^{14} +64.0000 q^{16} +138.441i q^{17} -153.454 q^{19} -29.9333i q^{20} -124.000 q^{22} -134.350i q^{23} +111.000 q^{25} -148.162 q^{28} -201.077 q^{31} +181.019i q^{32} -391.571 q^{34} +69.2965i q^{35} -376.000 q^{37} -434.032i q^{38} +84.6640 q^{40} +407.841i q^{41} -350.725i q^{44} +380.000 q^{46} +343.000 q^{49} +313.955i q^{50} -164.037 q^{55} -419.066i q^{56} -568.732i q^{62} -512.000 q^{64} -1107.53i q^{68} -196.000 q^{70} +451.134i q^{71} -1063.49i q^{74} +1227.63 q^{76} +811.940i q^{77} +239.466i q^{80} -1153.55 q^{82} -518.000 q^{85} +992.000 q^{88} +1552.79i q^{89} +1074.80i q^{92} -574.171i q^{95} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4}+O(q^{10}) \) 4 * q - 32 * q^4	\( 4 q - 32 q^{4} + 256 q^{16} - 496 q^{22} + 444 q^{25} - 1504 q^{37} + 1520 q^{46} + 1372 q^{49} - 2048 q^{64} - 784 q^{70} - 2072 q^{85} + 3968 q^{88}+O(q^{100}) \) 4 * q - 32 * q^4 + 256 * q^16 - 496 * q^22 + 444 * q^25 - 1504 * q^37 + 1520 * q^46 + 1372 * q^49 - 2048 * q^64 - 784 * q^70 - 2072 * q^85 + 3968 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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