LMFDB - Embedded newform 504.2.i.a.125.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(125,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("504.125");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.i (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:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			125.4
Root			$-0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	504.125
Dual form			504.2.i.a.125.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} -2.64575 q^{7} +(2.70318 - 0.832353i) q^{8} +O(q^{10})$	$q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} -2.64575 q^{7} +(2.70318 - 0.832353i) q^{8} +6.57008 q^{11} +(1.53946 - 3.41029i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-3.82288 + 8.46863i) q^{22} +9.39851i q^{23} +5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} +8.89753 q^{29} +(-4.82450 - 2.95367i) q^{32} -6.00000i q^{37} +12.0000i q^{43} +(-8.69140 - 9.85513i) q^{44} +(-12.1144 - 5.46863i) q^{46} +7.00000 q^{49} +(-2.90930 + 6.44484i) q^{50} +0.412247 q^{53} +(-7.15195 + 2.20220i) q^{56} +(-5.17712 + 11.4686i) q^{58} +(6.61438 - 4.50000i) q^{64} -15.8745i q^{67} +7.57205i q^{71} +(7.73381 + 3.49117i) q^{74} -17.3828 q^{77} -8.00000 q^{79} +(-15.4676 - 6.98233i) q^{86} +(17.7601 - 5.46863i) q^{88} +(14.0978 - 12.4331i) q^{92} +(-4.07303 + 9.02277i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 4 q^{16} - 20 q^{22} + 40 q^{25} + 28 q^{28} - 44 q^{46} + 56 q^{49} - 52 q^{58} - 64 q^{79} + 68 q^{88}+O(q^{100}) $ 8 * q - 4 * q^16 - 20 * q^22 + 40 * q^25 + 28 * q^28 - 44 * q^46 + 56 * q^49 - 52 * q^58 - 64 * q^79 + 68 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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 $

$2$	−0.581861	+	1.28897i	−0.411438	+	0.911438i
$2$	−0.581861	+	1.28897i	−0.411438	+	0.911438i
$3$		0			0
$3$		0			0
$4$	−1.32288	−	1.50000i	−0.661438	−	0.750000i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−2.64575			−1.00000
$7$	−2.64575			−1.00000
$8$	2.70318	−	0.832353i	0.955719	−	0.294281i
$9$		0			0
$10$		0			0
$11$	6.57008			1.98096			0.990478	−	0.137675i	$-0.0439628\pi$
$11$	6.57008			1.98096			0.990478	+	0.137675i	$0.0439628\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	1.53946	−	3.41029i	0.411438	−	0.911438i
$15$		0			0
$16$	−0.500000	+	3.96863i	−0.125000	+	0.992157i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	−3.82288	+	8.46863i	−0.815040	+	1.80552i
$23$			9.39851i			1.95973i	0.199673	+	0.979863i	$0.436012\pi$
$23$			9.39851i			1.95973i	−0.199673	+	0.979863i	$0.563988\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$	3.50000	+	3.96863i	0.661438	+	0.750000i
$29$	8.89753			1.65223			0.826115	−	0.563502i	$-0.190546\pi$
$29$	8.89753			1.65223			0.826115	+	0.563502i	$0.190546\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.82450	−	2.95367i	−0.852859	−	0.522141i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	−8.69140	−	9.85513i	−1.31028	−	1.48572i
$45$		0			0
$46$	−12.1144	−	5.46863i	−1.78617	−	0.806305i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	−2.90930	+	6.44484i	−0.411438	+	0.911438i
$51$		0			0
$52$		0			0
$53$	0.412247			0.0566265			0.0283132	−	0.999599i	$-0.490986\pi$
$53$	0.412247			0.0566265			0.0283132	+	0.999599i	$0.490986\pi$
$54$		0			0
$55$		0			0
$56$	−7.15195	+	2.20220i	−0.955719	+	0.294281i
$57$		0			0
$58$	−5.17712	+	11.4686i	−0.679790	+	1.50590i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	6.61438	−	4.50000i	0.826797	−	0.562500i
$65$		0			0
$66$		0			0
$67$		−	15.8745i		−	1.93938i	−0.244339	−	0.969690i	$-0.578571\pi$
$67$		−	15.8745i		−	1.93938i	0.244339	−	0.969690i	$-0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.57205i			0.898637i	0.893372	+	0.449319i	$0.148333\pi$
$71$			7.57205i			0.898637i	−0.893372	+	0.449319i	$0.851667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	7.73381	+	3.49117i	0.899037	+	0.405840i
$75$		0			0
$76$		0			0
$77$	−17.3828			−1.98096
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	−15.4676	−	6.98233i	−1.66792	−	0.752924i
$87$		0			0
$88$	17.7601	−	5.46863i	1.89324	−	0.582958i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	14.0978	−	12.4331i	1.46979	−	1.29624i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−4.07303	+	9.02277i	−0.411438	+	0.911438i
$99$		0			0
$100$	−6.61438	−	7.50000i	−0.661438	−	0.750000i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−0.239870	+	0.531373i	−0.0232983	+	0.0516115i
$107$	10.4005			1.00545			0.502726	−	0.864446i	$-0.332330\pi$
$107$	10.4005			1.00545			0.502726	+	0.864446i	$0.332330\pi$
$108$		0			0
$109$		−	18.0000i		−	1.72409i	−0.506834	−	0.862044i	$-0.669184\pi$
$109$		−	18.0000i		−	1.72409i	0.506834	−	0.862044i	$-0.330816\pi$
$110$		0			0
$111$		0			0
$112$	1.32288	−	10.5000i	0.125000	−	0.992157i
$113$		−	16.3808i		−	1.54098i	−0.637452	−	0.770490i	$-0.720012\pi$
$113$		−	16.3808i		−	1.54098i	0.637452	−	0.770490i	$-0.279988\pi$
$114$		0			0
$115$		0			0
$116$	−11.7703	−	13.3463i	−1.09285	−	1.23917i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	32.1660			2.92418
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.95171	+	11.1441i	0.172508	+	0.985008i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	20.4617	+	9.23676i	1.76762	+	0.797934i
$135$		0			0
$136$		0			0
$137$			7.89556i			0.674563i	0.941404	+	0.337282i	$0.109507\pi$
$137$			7.89556i			0.674563i	−0.941404	+	0.337282i	$0.890493\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−9.76013	−	4.40588i	−0.819052	−	0.369733i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−9.00000	+	7.93725i	−0.739795	+	0.652438i
$149$	8.07303			0.661369			0.330684	−	0.943741i	$-0.392720\pi$
$149$	8.07303			0.661369			0.330684	+	0.943741i	$0.392720\pi$
$150$		0			0
$151$	5.29150			0.430616			0.215308	−	0.976546i	$-0.430924\pi$
$151$	5.29150			0.430616			0.215308	+	0.976546i	$0.430924\pi$
$152$		0			0
$153$		0			0
$154$	10.1144	−	22.4059i	0.815040	−	1.80552i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	4.65489	−	10.3117i	0.370323	−	0.820358i
$159$		0			0
$160$		0			0
$161$		−	24.8661i		−	1.95973i
$162$		0			0
$163$			15.8745i			1.24339i	0.783260	+	0.621694i	$0.213555\pi$
$163$			15.8745i			1.24339i	−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	18.0000	−	15.8745i	1.37249	−	1.21042i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	−13.2288			−1.00000
$176$	−3.28504	+	26.0742i	−0.247619	+	1.96542i
$177$		0			0
$178$		0			0
$179$	−15.8799			−1.18692			−0.593458	−	0.804865i	$-0.702238\pi$
$179$	−15.8799			−1.18692			−0.593458	+	0.804865i	$0.702238\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	7.82288	+	25.4059i	0.576710	+	1.87295i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			13.0514i			0.944369i	0.881500	+	0.472184i	$0.156534\pi$
$191$			13.0514i			0.944369i	−0.881500	+	0.472184i	$0.843466\pi$
$192$		0			0
$193$	−21.1660			−1.52356			−0.761781	−	0.647834i	$-0.775675\pi$
$193$	−21.1660			−1.52356			−0.761781	+	0.647834i	$0.775675\pi$
$194$		0			0
$195$		0			0
$196$	−9.26013	−	10.5000i	−0.661438	−	0.750000i
$197$	−25.8681			−1.84303			−0.921513	−	0.388348i	$-0.873046\pi$
$197$	−25.8681			−1.84303			−0.921513	+	0.388348i	$0.873046\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	13.5159	−	4.16176i	0.955719	−	0.294281i
$201$		0			0
$202$		0			0
$203$	−23.5406			−1.65223
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	−0.545351	−	0.618370i	−0.0374549	−	0.0424699i
$213$		0			0
$214$	−6.05163	+	13.4059i	−0.413681	+	0.916407i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	23.2014	+	10.4735i	1.57140	+	0.709355i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	12.7644	+	7.81468i	0.852859	+	0.522141i
$225$		0			0
$226$	21.1144	+	9.53137i	1.40451	+	0.634018i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	24.0516	−	7.40588i	1.57907	−	0.486220i
$233$		−	0.589720i		−	0.0386338i	−0.999813	−	0.0193169i	$-0.993851\pi$
$233$		−	0.589720i		−	0.0386338i	0.999813	−	0.0193169i	$-0.00614915\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	30.0220i		−	1.94196i	−0.239158	−	0.970981i	$-0.576871\pi$
$239$		−	30.0220i		−	1.94196i	0.239158	−	0.970981i	$-0.423129\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−18.7161	+	41.4609i	−1.20312	+	2.66521i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$			61.7490i			3.88213i
$254$	9.30978	−	20.6235i	0.584147	−	1.29403i
$255$		0			0
$256$	−15.5000	−	3.96863i	−0.968750	−	0.248039i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$			15.8745i			0.986394i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	26.3691i		−	1.62599i	−0.582273	−	0.812993i	$-0.697836\pi$
$263$		−	26.3691i		−	1.62599i	0.582273	−	0.812993i	$-0.302164\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−23.8118	+	21.0000i	−1.45453	+	1.28278i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	−10.1771	−	4.59412i	−0.614823	−	0.277541i
$275$	32.8504			1.98096
$276$		0			0
$277$			31.7490i			1.90761i	0.300421	+	0.953807i	$0.402873\pi$
$277$			31.7490i			1.90761i	−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			33.3514i			1.98958i	0.101955	+	0.994789i	$0.467490\pi$
$281$			33.3514i			1.98958i	−0.101955	+	0.994789i	$0.532510\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	11.3581	−	10.0169i	0.673978	−	0.594393i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	−4.99412	−	16.2191i	−0.290277	−	0.942715i
$297$		0			0
$298$	−4.69738	+	10.4059i	−0.272112	+	0.602796i
$299$		0			0
$300$		0			0
$301$		−	31.7490i		−	1.82998i
$302$	−3.07892	+	6.82058i	−0.177172	+	0.392480i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	22.9953	+	26.0742i	1.31028	+	1.48572i
$309$		0			0
$310$		0			0
$311$		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
$316$	10.5830	+	12.0000i	0.595341	+	0.675053i
$317$	16.5583			0.930008			0.465004	−	0.885309i	$-0.346053\pi$
$317$	16.5583			0.930008			0.465004	+	0.885309i	$0.346053\pi$
$318$		0			0
$319$	58.4575			3.27299
$320$		0			0
$321$		0			0
$322$	32.0516	+	14.4686i	1.78617	+	0.806305i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−20.4617	−	9.23676i	−1.13327	−	0.511577i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	36.0000i		−	1.97874i	−0.145424	−	0.989369i	$-0.546455\pi$
$331$		−	36.0000i		−	1.97874i	0.145424	−	0.989369i	$-0.453545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−21.1660			−1.15299			−0.576493	−	0.817102i	$-0.695579\pi$
$337$	−21.1660			−1.15299			−0.576493	+	0.817102i	$0.695579\pi$
$338$	7.56419	−	16.7566i	0.411438	−	0.911438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−18.5203			−1.00000
$344$	9.98823	+	32.4382i	0.538529	+	1.74895i
$345$		0			0
$346$		0			0
$347$	−29.0200			−1.55788			−0.778938	−	0.627100i	$-0.784242\pi$
$347$	−29.0200			−1.55788			−0.778938	+	0.627100i	$0.784242\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	7.69730	−	17.0514i	0.411438	−	0.911438i
$351$		0			0
$352$	−31.6974	−	19.4059i	−1.68948	−	1.03434i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	9.23987	−	20.4686i	0.488342	−	1.08180i
$359$			31.8485i			1.68090i	0.541891	+	0.840449i	$0.317708\pi$
$359$			31.8485i			1.68090i	−0.541891	+	0.840449i	$0.682292\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−37.2992	−	4.69926i	−1.94435	−	0.244966i
$369$		0			0
$370$		0			0
$371$	−1.09070			−0.0566265
$372$		0			0
$373$		−	31.7490i		−	1.64390i	−0.569558	−	0.821951i	$-0.692886\pi$
$373$		−	31.7490i		−	1.64390i	0.569558	−	0.821951i	$-0.307114\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$		0			0
$381$		0			0
$382$	−16.8229	−	7.59412i	−0.860733	−	0.388549i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	12.3157	−	27.2823i	0.626851	−	1.38863i
$387$		0			0
$388$		0			0
$389$	34.3534			1.74179			0.870893	−	0.491473i	$-0.163542\pi$
$389$	34.3534			1.74179			0.870893	+	0.491473i	$0.163542\pi$
$390$		0			0
$391$		0			0
$392$	18.9223	−	5.82647i	0.955719	−	0.294281i
$393$		0			0
$394$	15.0516	−	33.3431i	0.758290	−	1.67980i
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−2.50000	+	19.8431i	−0.125000	+	0.992157i
$401$			9.07500i			0.453184i	0.973990	+	0.226592i	$0.0727584\pi$
$401$			9.07500i			0.453184i	−0.973990	+	0.226592i	$0.927242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	13.6974	−	30.3431i	0.679790	−	1.50590i
$407$		−	39.4205i		−	1.95400i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		−	31.7490i		−	1.54735i	−0.633581	−	0.773676i	$-0.718416\pi$
$421$		−	31.7490i		−	1.54735i	0.633581	−	0.773676i	$-0.281584\pi$
$422$	−15.4676	−	6.98233i	−0.752952	−	0.339895i
$423$		0			0
$424$	1.11438	−	0.343135i	0.0541190	−	0.0166641i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−13.7585	−	15.6007i	−0.665044	−	0.754089i
$429$		0			0
$430$		0			0
$431$			3.91913i			0.188778i	0.995535	+	0.0943889i	$0.0300897\pi$
$431$			3.91913i			0.188778i	−0.995535	+	0.0943889i	$0.969910\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−27.0000	+	23.8118i	−1.29307	+	1.14038i
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.0495			0.572487			0.286244	−	0.958157i	$-0.407593\pi$
$443$	12.0495			0.572487			0.286244	+	0.958157i	$0.407593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−17.5000	+	11.9059i	−0.826797	+	0.562500i
$449$		−	28.5190i		−	1.34590i	−0.739689	−	0.672948i	$-0.765028\pi$
$449$		−	28.5190i		−	1.34590i	0.739689	−	0.672948i	$-0.234972\pi$
$450$		0			0
$451$		0			0
$452$	−24.5713	+	21.6698i	−1.15574	+	1.01926i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	42.3320			1.98021			0.990104	−	0.140334i	$-0.0448177\pi$
$457$	42.3320			1.98021			0.990104	+	0.140334i	$0.0448177\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	40.0000			1.85896			0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000			1.85896			0.929479	+	0.368875i	$0.120257\pi$
$464$	−4.44876	+	35.3110i	−0.206529	+	1.63927i
$465$		0			0
$466$	0.760130	+	0.343135i	0.0352123	+	0.0158954i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$			42.0000i			1.93938i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			78.8410i			3.62511i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	38.6974	+	17.4686i	1.76998	+	0.798996i
$479$		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
$484$	−42.5516	−	48.2490i	−1.93417	−	2.19314i
$485$		0			0
$486$		0			0
$487$	37.0405			1.67847			0.839233	−	0.543772i	$-0.183004\pi$
$487$	37.0405			1.67847			0.839233	+	0.543772i	$0.183004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−27.3710			−1.23524			−0.617619	−	0.786478i	$-0.711903\pi$
$491$	−27.3710			−1.23524			−0.617619	+	0.786478i	$0.711903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	20.0338i		−	0.898637i
$498$		0			0
$499$		−	36.0000i		−	1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$		−	36.0000i		−	1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	−79.5925	−	35.9293i	−3.53832	−	1.59725i
$507$		0			0
$508$	21.1660	+	24.0000i	0.939090	+	1.06483i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	14.1343	−	17.6698i	0.624653	−	0.780903i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−20.4617	−	9.23676i	−0.899037	−	0.405840i
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	33.9889	+	15.3431i	1.48199	+	0.668992i
$527$		0			0
$528$		0			0
$529$	−65.3320			−2.84052
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−13.2132	−	42.9117i	−0.570723	−	1.85350i
$537$		0			0
$538$		0			0
$539$	45.9906			1.98096
$540$		0			0
$541$		−	31.7490i		−	1.36500i	−0.730887	−	0.682498i	$-0.760893\pi$
$541$		−	31.7490i		−	1.36500i	0.730887	−	0.682498i	$-0.239107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	15.8745i		−	0.678745i	−0.940652	−	0.339372i	$-0.889785\pi$
$547$		−	15.8745i		−	0.678745i	0.940652	−	0.339372i	$-0.110215\pi$
$548$	11.8433	−	10.4448i	0.505923	−	0.446182i
$549$		0			0
$550$	−19.1144	+	42.3431i	−0.815040	+	1.80552i
$551$		0			0
$552$		0			0
$553$	21.1660			0.900070
$554$	−40.9235	−	18.4735i	−1.73867	−	0.784864i
$555$		0			0
$556$		0			0
$557$	−25.0436			−1.06113			−0.530566	−	0.847644i	$-0.678020\pi$
$557$	−25.0436			−1.06113			−0.530566	+	0.847644i	$0.678020\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−42.9889	−	19.4059i	−1.81338	−	0.818588i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	6.30262	+	20.4686i	0.264452	+	0.858845i
$569$			45.4896i			1.90702i	0.301356	+	0.953512i	$0.402561\pi$
$569$			45.4896i			1.90702i	−0.301356	+	0.953512i	$0.597439\pi$
$570$		0			0
$571$		−	47.6235i		−	1.99298i	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$		−	47.6235i		−	1.99298i	0.0836974	−	0.996491i	$-0.473327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			46.9926i			1.95973i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	9.89164	−	21.9125i	0.411438	−	0.911438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.70850			0.112174
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	23.8118	+	3.00000i	0.978657	+	0.123299i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−10.6796	−	12.1096i	−0.437454	−	0.496027i
$597$		0			0
$598$		0			0
$599$		−	48.8190i		−	1.99469i	−0.0728143	−	0.997346i	$-0.523198\pi$
$599$		−	48.8190i		−	1.99469i	0.0728143	−	0.997346i	$-0.476802\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	40.9235	+	18.4735i	1.66792	+	0.752924i
$603$		0			0
$604$	−7.00000	−	7.93725i	−0.284826	−	0.322962i
$605$		0			0
$606$		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
$613$			31.7490i			1.28233i	0.767403	+	0.641165i	$0.221549\pi$
$613$			31.7490i			1.28233i	−0.767403	+	0.641165i	$0.778451\pi$
$614$		0			0
$615$		0			0
$616$	−46.9889	+	14.4686i	−1.89324	+	0.582958i
$617$			11.5485i			0.464924i	0.972605	+	0.232462i	$0.0746782\pi$
$617$			11.5485i			0.464924i	−0.972605	+	0.232462i	$0.925322\pi$
$618$		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
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	−21.6255	+	6.65882i	−0.860214	+	0.264874i
$633$		0			0
$634$	−9.63464	+	21.3431i	−0.382640	+	0.847644i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−34.0141	+	75.3498i	−1.34663	+	2.98313i
$639$		0			0
$640$		0			0
$641$			17.5603i			0.693589i	0.937941	+	0.346795i	$0.112730\pi$
$641$			17.5603i			0.693589i	−0.937941	+	0.346795i	$0.887270\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	−37.2992	+	32.8948i	−1.46979	+	1.29624i
$645$		0			0
$646$		0			0
$647$		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
$652$	23.8118	−	21.0000i	0.932541	−	0.822423i
$653$	42.8387			1.67641			0.838203	−	0.545358i	$-0.183606\pi$
$653$	42.8387			1.67641			0.838203	+	0.545358i	$0.183606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−49.8210			−1.94075			−0.970375	−	0.241604i	$-0.922327\pi$
$659$	−49.8210			−1.94075			−0.970375	+	0.241604i	$0.922327\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	46.4028	+	20.9470i	1.80350	+	0.814128i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			83.6235i			3.23792i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−42.3320			−1.63178			−0.815890	−	0.578208i	$-0.803752\pi$
$673$	−42.3320			−1.63178			−0.815890	+	0.578208i	$0.803752\pi$
$674$	12.3157	−	27.2823i	0.474382	−	1.05088i
$675$		0			0
$676$	17.1974	+	19.5000i	0.661438	+	0.750000i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	40.5112			1.55012			0.775059	−	0.631889i	$-0.217720\pi$
$683$	40.5112			1.55012			0.775059	+	0.631889i	$0.217720\pi$
$684$		0			0
$685$		0			0
$686$	10.7762	−	23.8720i	0.411438	−	0.911438i
$687$		0			0
$688$	−47.6235	−	6.00000i	−1.81563	−	0.228748i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	16.8856	−	37.4059i	0.640969	−	1.41991i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	17.5000	+	19.8431i	0.661438	+	0.750000i
$701$	36.0024			1.35979			0.679895	−	0.733309i	$-0.262025\pi$
$701$	36.0024			1.35979			0.679895	+	0.733309i	$0.262025\pi$
$702$		0			0
$703$		0			0
$704$	43.4570	−	29.5654i	1.63785	−	1.11429i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	21.0071	+	23.8198i	0.785071	+	0.890187i
$717$		0			0
$718$	−41.0516	−	18.5314i	−1.53203	−	0.691585i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	11.0554	−	24.4904i	0.411438	−	0.911438i
$723$		0			0
$724$		0			0
$725$	44.4876			1.65223
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	27.7601	−	45.3431i	1.02325	−	1.67137i
$737$		−	104.297i		−	3.84182i
$738$		0			0
$739$		−	15.8745i		−	0.583953i	−0.956425	−	0.291977i	$-0.905687\pi$
$739$		−	15.8745i		−	0.583953i	0.956425	−	0.291977i	$-0.0943129\pi$
$740$		0			0
$741$		0			0
$742$	0.634637	−	1.40588i	0.0232983	−	0.0516115i
$743$		−	2.09267i		−	0.0767726i	−0.999263	−	0.0383863i	$-0.987778\pi$
$743$		−	2.09267i		−	0.0767726i	0.999263	−	0.0383863i	$-0.0122217\pi$
$744$		0			0
$745$		0			0
$746$	40.9235	+	18.4735i	1.49831	+	0.676363i
$747$		0			0
$748$		0			0
$749$	−27.5171			−1.00545
$750$		0			0
$751$	−26.4575			−0.965448			−0.482724	−	0.875772i	$-0.660353\pi$
$751$	−26.4575			−0.965448			−0.482724	+	0.875772i	$0.660353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	54.0000i		−	1.96266i	−0.192323	−	0.981332i	$-0.561602\pi$
$757$		−	54.0000i		−	1.96266i	0.192323	−	0.981332i	$-0.438398\pi$
$758$	15.4676	+	6.98233i	0.561809	+	0.253610i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$			47.6235i			1.72409i
$764$	19.5771	−	17.2654i	0.708276	−	0.624641i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	28.0000	+	31.7490i	1.00774	+	1.14267i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−19.9889	+	44.2804i	−0.716636	+	1.58753i
$779$		0			0
$780$		0			0
$781$			49.7490i			1.78016i
$782$		0			0
$783$		0			0
$784$	−3.50000	+	27.7804i	−0.125000	+	0.992157i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	34.2203	+	38.8021i	1.21905	+	1.38227i
$789$		0			0
$790$		0			0
$791$			43.3396i			1.54098i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−24.1225	−	14.7684i	−0.852859	−	0.522141i
$801$		0			0
$802$	−11.6974	−	5.28039i	−0.413049	−	0.186457i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			3.06320i			0.107696i	0.998549	+	0.0538482i	$0.0171487\pi$
$809$			3.06320i			0.107696i	−0.998549	+	0.0538482i	$0.982851\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	31.1413	+	35.3110i	1.09285	+	1.23917i
$813$		0			0
$814$	50.8118	+	22.9373i	1.78095	+	0.803950i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−52.9729			−1.84877			−0.924384	−	0.381464i	$-0.875420\pi$
$821$	−52.9729			−1.84877			−0.924384	+	0.381464i	$0.875420\pi$
$822$		0			0
$823$	−32.0000			−1.11545			−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000			−1.11545			−0.557725	+	0.830026i	$0.688326\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.92110			0.171123			0.0855616	−	0.996333i	$-0.472732\pi$
$827$	4.92110			0.171123			0.0855616	+	0.996333i	$0.472732\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		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
$841$	50.1660			1.72986
$842$	40.9235	+	18.4735i	1.41032	+	0.636639i
$843$		0			0
$844$	18.0000	−	15.8745i	0.619586	−	0.546423i
$845$		0			0
$846$		0			0
$847$	−85.1033			−2.92418
$848$	−0.206123	+	1.63605i	−0.00707831	+	0.0561823i
$849$		0			0
$850$		0			0
$851$	56.3911			1.93306
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	28.1144	−	8.65687i	0.960930	−	0.295886i
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−5.05163	−	2.28039i	−0.172059	−	0.0776703i
$863$			35.5014i			1.20848i	0.796802	+	0.604240i	$0.206523\pi$
$863$			35.5014i			1.20848i	−0.796802	+	0.604240i	$0.793477\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−52.5607			−1.78300
$870$		0			0
$871$		0			0
$872$	−14.9823	−	48.6573i	−0.507366	−	1.64774i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			31.7490i			1.07209i	0.844190	+	0.536044i	$0.180082\pi$
$877$			31.7490i			1.07209i	−0.844190	+	0.536044i	$0.819918\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	12.0000i		−	0.403832i	−0.979403	−	0.201916i	$-0.935283\pi$
$883$		−	12.0000i		−	0.403832i	0.979403	−	0.201916i	$-0.0647168\pi$
$884$		0			0
$885$		0			0
$886$	−7.01111	+	15.5314i	−0.235543	+	0.521787i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	42.3320			1.41977
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−5.16373	−	29.4845i	−0.172508	−	0.985008i
$897$		0			0
$898$	36.7601	+	16.5941i	1.22670	+	0.553753i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−13.6346	−	44.2804i	−0.453481	−	1.47274i
$905$		0			0
$906$		0			0
$907$			60.0000i			1.99227i	0.0878507	+	0.996134i	$0.472000\pi$
$907$			60.0000i			1.99227i	−0.0878507	+	0.996134i	$0.528000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	52.4719i		−	1.73847i	−0.494397	−	0.869236i	$-0.664611\pi$
$911$		−	52.4719i		−	1.73847i	0.494397	−	0.869236i	$-0.335389\pi$
$912$		0			0
$913$		0			0
$914$	−24.6314	+	54.5646i	−0.814733	+	1.80484i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−37.0405			−1.22185			−0.610927	−	0.791687i	$-0.709203\pi$
$919$	−37.0405			−1.22185			−0.610927	+	0.791687i	$0.709203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	30.0000i		−	0.986394i
$926$	−23.2744	+	51.5587i	−0.764845	+	1.69432i
$927$		0			0
$928$	−42.9261	−	26.2804i	−1.40912	−	0.862696i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$	−0.884579	+	0.780126i	−0.0289754	+	0.0255539i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−54.1366	−	24.4382i	−1.76762	−	0.797934i
$939$		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
$946$	−101.624	−	45.8745i	−3.30407	−	1.49151i
$947$	−55.3004			−1.79702			−0.898510	−	0.438953i	$-0.855350\pi$
$947$	−55.3004			−1.79702			−0.898510	+	0.438953i	$0.855350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.0456i		−	0.843699i	−0.906666	−	0.421849i	$-0.861381\pi$
$953$		−	26.0456i		−	0.843699i	0.906666	−	0.421849i	$-0.138619\pi$
$954$		0			0
$955$		0			0
$956$	−45.0330	+	39.7154i	−1.45647	+	1.28449i
$957$		0			0
$958$		0			0
$959$		−	20.8897i		−	0.674563i
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−40.0000			−1.28631			−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000			−1.28631			−0.643157	+	0.765735i	$0.722376\pi$
$968$	86.9506	−	26.7735i	2.79470	−	0.860532i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$	−21.5524	+	47.7440i	−0.690584	+	1.52982i
$975$		0			0
$976$		0			0
$977$		−	62.4602i		−	1.99828i	−0.0414892	−	0.999139i	$-0.513210\pi$
$977$		−	62.4602i		−	1.99828i	0.0414892	−	0.999139i	$-0.486790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	15.9261	−	35.2804i	0.508224	−	1.12584i
$983$		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
$989$	−112.782			−3.58626
$990$		0			0
$991$	58.2065			1.84899			0.924496	−	0.381193i	$-0.124487\pi$
$991$	58.2065			1.84899			0.924496	+	0.381193i	$0.124487\pi$
$992$		0			0
$993$		0			0
$994$	25.8229	+	11.6569i	0.819052	+	0.369733i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	46.4028	+	20.9470i	1.46886	+	0.663066i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.i.a.125.4	yes	8
3.2	odd	2	inner	504.2.i.a.125.5	yes	8
4.3	odd	2		2016.2.i.a.881.5		8
7.6	odd	2	CM	504.2.i.a.125.4	yes	8
8.3	odd	2		2016.2.i.a.881.7		8
8.5	even	2	inner	504.2.i.a.125.6	yes	8
12.11	even	2		2016.2.i.a.881.8		8
21.20	even	2	inner	504.2.i.a.125.5	yes	8
24.5	odd	2	inner	504.2.i.a.125.3	✓	8
24.11	even	2		2016.2.i.a.881.6		8
28.27	even	2		2016.2.i.a.881.5		8
56.13	odd	2	inner	504.2.i.a.125.6	yes	8
56.27	even	2		2016.2.i.a.881.7		8
84.83	odd	2		2016.2.i.a.881.8		8
168.83	odd	2		2016.2.i.a.881.6		8
168.125	even	2	inner	504.2.i.a.125.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.i.a.125.3	✓	8	24.5	odd	2	inner
504.2.i.a.125.3	✓	8	168.125	even	2	inner
504.2.i.a.125.4	yes	8	1.1	even	1	trivial
504.2.i.a.125.4	yes	8	7.6	odd	2	CM
504.2.i.a.125.5	yes	8	3.2	odd	2	inner
504.2.i.a.125.5	yes	8	21.20	even	2	inner
504.2.i.a.125.6	yes	8	8.5	even	2	inner
504.2.i.a.125.6	yes	8	56.13	odd	2	inner
2016.2.i.a.881.5		8	4.3	odd	2
2016.2.i.a.881.5		8	28.27	even	2
2016.2.i.a.881.6		8	24.11	even	2
2016.2.i.a.881.6		8	168.83	odd	2
2016.2.i.a.881.7		8	8.3	odd	2
2016.2.i.a.881.7		8	56.27	even	2
2016.2.i.a.881.8		8	12.11	even	2
2016.2.i.a.881.8		8	84.83	odd	2

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} -2.64575 q^{7} +(2.70318 - 0.832353i) q^{8} +O(q^{10})\)	\(q+(-0.581861 + 1.28897i) q^{2} +(-1.32288 - 1.50000i) q^{4} -2.64575 q^{7} +(2.70318 - 0.832353i) q^{8} +6.57008 q^{11} +(1.53946 - 3.41029i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-3.82288 + 8.46863i) q^{22} +9.39851i q^{23} +5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} +8.89753 q^{29} +(-4.82450 - 2.95367i) q^{32} -6.00000i q^{37} +12.0000i q^{43} +(-8.69140 - 9.85513i) q^{44} +(-12.1144 - 5.46863i) q^{46} +7.00000 q^{49} +(-2.90930 + 6.44484i) q^{50} +0.412247 q^{53} +(-7.15195 + 2.20220i) q^{56} +(-5.17712 + 11.4686i) q^{58} +(6.61438 - 4.50000i) q^{64} -15.8745i q^{67} +7.57205i q^{71} +(7.73381 + 3.49117i) q^{74} -17.3828 q^{77} -8.00000 q^{79} +(-15.4676 - 6.98233i) q^{86} +(17.7601 - 5.46863i) q^{88} +(14.0978 - 12.4331i) q^{92} +(-4.07303 + 9.02277i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 4 q^{16} - 20 q^{22} + 40 q^{25} + 28 q^{28} - 44 q^{46} + 56 q^{49} - 52 q^{58} - 64 q^{79} + 68 q^{88}+O(q^{100}) \) 8 * q - 4 * q^16 - 20 * q^22 + 40 * q^25 + 28 * q^28 - 44 * q^46 + 56 * q^49 - 52 * q^58 - 64 * q^79 + 68 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more