LMFDB - Embedded newform 504.2.i.a.125.8

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			125.8
Root			$1.28897 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	504.125
Dual form			504.2.i.a.125.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.28897 + 0.581861i) q^{2} +(1.32288 + 1.50000i) q^{4} +2.64575 q^{7} +(0.832353 + 2.70318i) q^{8} +O(q^{10})$	$q+(1.28897 + 0.581861i) q^{2} +(1.32288 + 1.50000i) q^{4} +2.64575 q^{7} +(0.832353 + 2.70318i) q^{8} -0.913230 q^{11} +(3.41029 + 1.53946i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-1.17712 - 0.531373i) q^{22} -1.91520i q^{23} +5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} -6.06910 q^{29} +(-2.95367 + 4.82450i) q^{32} +6.00000i q^{37} -12.0000i q^{43} +(-1.20809 - 1.36985i) q^{44} +(1.11438 - 2.46863i) q^{46} +7.00000 q^{49} +(6.44484 + 2.90930i) q^{50} -14.5544 q^{53} +(2.20220 + 7.15195i) q^{56} +(-7.82288 - 3.53137i) q^{58} +(-6.61438 + 4.50000i) q^{64} -15.8745i q^{67} -15.0554i q^{71} +(-3.49117 + 7.73381i) q^{74} -2.41618 q^{77} -8.00000 q^{79} +(6.98233 - 15.4676i) q^{86} +(-0.760130 - 2.46863i) q^{88} +(2.87280 - 2.53357i) q^{92} +(9.02277 + 4.07303i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.28897	+	0.581861i	0.911438	+	0.411438i
$2$	1.28897	+	0.581861i	0.911438	+	0.411438i
$3$		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
$8$	0.832353	+	2.70318i	0.294281	+	0.955719i
$9$		0			0
$10$		0			0
$11$	−0.913230			−0.275349			−0.137675	−	0.990478i	$-0.543963\pi$
$11$	−0.913230			−0.275349			−0.137675	+	0.990478i	$0.543963\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	3.41029	+	1.53946i	0.911438	+	0.411438i
$15$		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
$22$	−1.17712	−	0.531373i	−0.250964	−	0.113289i
$23$		−	1.91520i		−	0.399346i	−0.979863	−	0.199673i	$-0.936012\pi$
$23$		−	1.91520i		−	0.399346i	0.979863	−	0.199673i	$-0.0639880\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$	3.50000	+	3.96863i	0.661438	+	0.750000i
$29$	−6.06910			−1.12700			−0.563502	−	0.826115i	$-0.690546\pi$
$29$	−6.06910			−1.12700			−0.563502	+	0.826115i	$0.690546\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−2.95367	+	4.82450i	−0.522141	+	0.852859i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		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	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	−1.20809	−	1.36985i	−0.182126	−	0.206512i
$45$		0			0
$46$	1.11438	−	2.46863i	0.164306	−	0.363979i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	6.44484	+	2.90930i	0.911438	+	0.411438i
$51$		0			0
$52$		0			0
$53$	−14.5544			−1.99920			−0.999599	−	0.0283132i	$-0.990986\pi$
$53$	−14.5544			−1.99920			−0.999599	+	0.0283132i	$0.990986\pi$
$54$		0			0
$55$		0			0
$56$	2.20220	+	7.15195i	0.294281	+	0.955719i
$57$		0			0
$58$	−7.82288	−	3.53137i	−1.02719	−	0.463692i
$59$		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
$71$		−	15.0554i		−	1.78674i	−0.449319	−	0.893372i	$-0.648333\pi$
$71$		−	15.0554i		−	1.78674i	0.449319	−	0.893372i	$-0.351667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−3.49117	+	7.73381i	−0.405840	+	0.899037i
$75$		0			0
$76$		0			0
$77$	−2.41618			−0.275349
$78$		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
$86$	6.98233	−	15.4676i	0.752924	−	1.66792i
$87$		0			0
$88$	−0.760130	−	2.46863i	−0.0810301	−	0.263157i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	2.87280	−	2.53357i	0.299510	−	0.264143i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	9.02277	+	4.07303i	0.911438	+	0.411438i
$99$		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
$106$	−18.7601	−	8.46863i	−1.82214	−	0.822546i
$107$	17.8838			1.72889			0.864446	−	0.502726i	$-0.167670\pi$
$107$	17.8838			1.72889			0.864446	+	0.502726i	$0.167670\pi$
$108$		0			0
$109$			18.0000i			1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$			18.0000i			1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$		0			0
$111$		0			0
$112$	−1.32288	+	10.5000i	−0.125000	+	0.992157i
$113$		−	13.5524i		−	1.27490i	−0.770490	−	0.637452i	$-0.779988\pi$
$113$		−	13.5524i		−	1.27490i	0.770490	−	0.637452i	$-0.220012\pi$
$114$		0			0
$115$		0			0
$116$	−8.02867	−	9.10365i	−0.745443	−	0.845253i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.1660			−0.924183
$122$		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	$0.751253\pi$
$128$	−11.1441	+	1.95171i	−0.985008	+	0.172508i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	9.23676	−	20.4617i	0.797934	−	1.76762i
$135$		0			0
$136$		0			0
$137$			22.0377i			1.88281i	0.337282	+	0.941404i	$0.390493\pi$
$137$			22.0377i			1.88281i	−0.337282	+	0.941404i	$0.609507\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	8.76013	−	19.4059i	0.735134	−	1.62851i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−9.00000	+	7.93725i	−0.739795	+	0.652438i
$149$	23.0397			1.88748			0.943741	−	0.330684i	$-0.107280\pi$
$149$	23.0397			1.88748			0.943741	+	0.330684i	$0.107280\pi$
$150$		0			0
$151$	−5.29150			−0.430616			−0.215308	−	0.976546i	$-0.569076\pi$
$151$	−5.29150			−0.430616			−0.215308	+	0.976546i	$0.569076\pi$
$152$		0			0
$153$		0			0
$154$	−3.11438	−	1.40588i	−0.250964	−	0.113289i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−10.3117	−	4.65489i	−0.820358	−	0.370323i
$159$		0			0
$160$		0			0
$161$		−	5.06713i		−	0.399346i
$162$		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
$176$	0.456615	−	3.62427i	0.0344187	−	0.273190i
$177$		0			0
$178$		0			0
$179$	21.5367			1.60973			0.804865	−	0.593458i	$-0.202238\pi$
$179$	21.5367			1.60973			0.804865	+	0.593458i	$0.202238\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	5.17712	−	1.59412i	0.381663	−	0.117520i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			24.3651i			1.76300i	0.472184	+	0.881500i	$0.343466\pi$
$191$			24.3651i			1.76300i	−0.472184	+	0.881500i	$0.656534\pi$
$192$		0			0
$193$	21.1660			1.52356			0.761781	−	0.647834i	$-0.224325\pi$
$193$	21.1660			1.52356			0.761781	+	0.647834i	$0.224325\pi$
$194$		0			0
$195$		0			0
$196$	9.26013	+	10.5000i	0.661438	+	0.750000i
$197$	−10.9015			−0.776697			−0.388348	−	0.921513i	$-0.626954\pi$
$197$	−10.9015			−0.776697			−0.388348	+	0.921513i	$0.626954\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	4.16176	+	13.5159i	0.294281	+	0.955719i
$201$		0			0
$202$		0			0
$203$	−16.0573			−1.12700
$204$		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	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	−19.2536	−	21.8316i	−1.32235	−	1.49940i
$213$		0			0
$214$	23.0516	+	10.4059i	1.57578	+	0.711331i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−10.4735	+	23.2014i	−0.709355	+	1.57140i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−7.81468	+	12.7644i	−0.522141	+	0.852859i
$225$		0			0
$226$	7.88562	−	17.4686i	0.524544	−	1.16200i
$227$		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
$232$	−5.05163	−	16.4059i	−0.331656	−	1.07710i
$233$			30.5230i			1.99963i	0.0193169	+	0.999813i	$0.493851\pi$
$233$			30.5230i			1.99963i	−0.0193169	+	0.999813i	$0.506149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.39458i		−	0.478316i	−0.970981	−	0.239158i	$-0.923129\pi$
$239$		−	7.39458i		−	0.478316i	0.970981	−	0.239158i	$-0.0768713\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−13.1037	−	5.91520i	−0.842335	−	0.380244i
$243$		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
$253$			1.74902i			0.109960i
$254$	−20.6235	−	9.30978i	−1.29403	−	0.584147i
$255$		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
$263$			18.8858i			1.16455i	0.812993	+	0.582273i	$0.197836\pi$
$263$			18.8858i			1.16455i	−0.812993	+	0.582273i	$0.802164\pi$
$264$		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
$274$	−12.8229	+	28.4059i	−0.774658	+	1.71606i
$275$	−4.56615			−0.275349
$276$		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
$281$		−	3.41815i		−	0.203910i	−0.994789	−	0.101955i	$-0.967490\pi$
$281$		−	3.41815i		−	0.203910i	0.994789	−	0.101955i	$-0.0325097\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	22.5830	−	19.9164i	1.34006	−	1.18182i
$285$		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
$296$	−16.2191	+	4.99412i	−0.942715	+	0.290277i
$297$		0			0
$298$	29.6974	+	13.4059i	1.72032	+	0.776582i
$299$		0			0
$300$		0			0
$301$		−	31.7490i		−	1.82998i
$302$	−6.82058	−	3.07892i	−0.392480	−	0.177172i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−3.19631	−	3.62427i	−0.182126	−	0.206512i
$309$		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
$317$	31.5249			1.77062			0.885309	−	0.465004i	$-0.153947\pi$
$317$	31.5249			1.77062			0.885309	+	0.465004i	$0.153947\pi$
$318$		0			0
$319$	5.54249			0.310320
$320$		0			0
$321$		0			0
$322$	2.94837	−	6.53137i	0.164306	−	0.363979i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−9.23676	+	20.4617i	−0.511577	+	1.13327i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			36.0000i			1.97874i	0.145424	+	0.989369i	$0.453545\pi$
$331$			36.0000i			1.97874i	−0.145424	+	0.989369i	$0.546455\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	21.1660			1.15299			0.576493	−	0.817102i	$-0.304421\pi$
$337$	21.1660			1.15299			0.576493	+	0.817102i	$0.304421\pi$
$338$	−16.7566	−	7.56419i	−0.911438	−	0.411438i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$	32.4382	−	9.98823i	1.74895	−	0.538529i
$345$		0			0
$346$		0			0
$347$	23.3632			1.25420			0.627100	−	0.778938i	$-0.284242\pi$
$347$	23.3632			1.25420			0.627100	+	0.778938i	$0.284242\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	17.0514	+	7.69730i	0.911438	+	0.411438i
$351$		0			0
$352$	2.69738	−	4.40588i	0.143771	−	0.234834i
$353$		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
$358$	27.7601	+	12.5314i	1.46717	+	0.662304i
$359$			20.5347i			1.08378i	0.840449	+	0.541891i	$0.182292\pi$
$359$			20.5347i			1.08378i	−0.840449	+	0.541891i	$0.817708\pi$
$360$		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			$\pi$
$368$	7.60070	+	0.957598i	0.396214	+	0.0499183i
$369$		0			0
$370$		0			0
$371$	−38.5073			−1.99920
$372$		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	$0.0997264\pi$
$379$			12.0000i			0.616399i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	−14.1771	+	31.4059i	−0.725365	+	1.60686i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	27.2823	+	12.3157i	1.38863	+	0.626851i
$387$		0			0
$388$		0			0
$389$	19.3867			0.982947			0.491473	−	0.870893i	$-0.336458\pi$
$389$	19.3867			0.982947			0.491473	+	0.870893i	$0.336458\pi$
$390$		0			0
$391$		0			0
$392$	5.82647	+	18.9223i	0.294281	+	0.955719i
$393$		0			0
$394$	−14.0516	−	6.34313i	−0.707911	−	0.319563i
$395$		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
$401$		−	39.0083i		−	1.94798i	−0.226592	−	0.973990i	$-0.572758\pi$
$401$		−	39.0083i		−	1.94798i	0.226592	−	0.973990i	$-0.427242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−20.6974	−	9.34313i	−1.02719	−	0.463692i
$407$		−	5.47938i		−	0.271603i
$408$		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	$-0.281584\pi$
$422$	6.98233	−	15.4676i	0.339895	−	0.752952i
$423$		0			0
$424$	−12.1144	−	39.3431i	−0.588326	−	1.91067i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	23.6580	+	26.8257i	1.14355	+	1.29667i
$429$		0			0
$430$		0			0
$431$		−	41.3357i		−	1.99107i	−0.0943889	−	0.995535i	$-0.530090\pi$
$431$		−	41.3357i		−	1.99107i	0.0943889	−	0.995535i	$-0.469910\pi$
$432$		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
$443$	−40.3337			−1.91631			−0.958157	−	0.286244i	$-0.907593\pi$
$443$	−40.3337			−1.91631			−0.958157	+	0.286244i	$0.907593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−17.5000	+	11.9059i	−0.826797	+	0.562500i
$449$		−	31.3475i		−	1.47938i	−0.672948	−	0.739689i	$-0.734972\pi$
$449$		−	31.3475i		−	1.47938i	0.672948	−	0.739689i	$-0.265028\pi$
$450$		0			0
$451$		0			0
$452$	20.3286	−	17.9282i	0.956178	−	0.843270i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−42.3320			−1.98021			−0.990104	−	0.140334i	$-0.955182\pi$
$457$	−42.3320			−1.98021			−0.990104	+	0.140334i	$0.955182\pi$
$458$		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	$0.120257\pi$
$464$	3.03455	−	24.0860i	0.140875	−	1.11816i
$465$		0			0
$466$	−17.7601	+	39.3431i	−0.822722	+	1.82254i
$467$		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
$473$			10.9588i			0.503884i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	4.30262	−	9.53137i	0.196797	−	0.435955i
$479$		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
$484$	−13.4484	−	15.2490i	−0.611289	−	0.693137i
$485$		0			0
$486$		0			0
$487$	−37.0405			−1.67847			−0.839233	−	0.543772i	$-0.816996\pi$
$487$	−37.0405			−1.67847			−0.839233	+	0.543772i	$0.816996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−34.8544			−1.57296			−0.786478	−	0.617619i	$-0.788097\pi$
$491$	−34.8544			−1.57296			−0.786478	+	0.617619i	$0.788097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	39.8328i		−	1.78674i
$498$		0			0
$499$			36.0000i			1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$			36.0000i			1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$		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
$506$	−1.01768	+	2.25442i	−0.0452416	+	0.100221i
$507$		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
$512$	−17.6698	−	14.1343i	−0.780903	−	0.624653i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−9.23676	+	20.4617i	−0.405840	+	0.899037i
$519$		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
$526$	−10.9889	+	24.3431i	−0.479138	+	1.06141i
$527$		0			0
$528$		0			0
$529$	19.3320			0.840523
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	42.9117	−	13.2132i	1.85350	−	0.570723i
$537$		0			0
$538$		0			0
$539$	−6.39261			−0.275349
$540$		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	$-0.110215\pi$
$548$	−33.0565	+	29.1531i	−1.41211	+	1.24536i
$549$		0			0
$550$	−5.88562	−	2.65687i	−0.250964	−	0.113289i
$551$		0			0
$552$		0			0
$553$	−21.1660			−0.900070
$554$	−18.4735	+	40.9235i	−0.784864	+	1.73867i
$555$		0			0
$556$		0			0
$557$	−40.0102			−1.69529			−0.847644	−	0.530566i	$-0.821980\pi$
$557$	−40.0102			−1.69529			−0.847644	+	0.530566i	$0.821980\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.98889	−	4.40588i	0.0838961	−	0.185851i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	40.6974	−	12.5314i	1.70762	−	0.525805i
$569$			14.3769i			0.602711i	0.953512	+	0.301356i	$0.0974392\pi$
$569$			14.3769i			0.602711i	−0.953512	+	0.301356i	$0.902561\pi$
$570$		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
$575$		−	9.57598i		−	0.399346i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−21.9125	−	9.89164i	−0.911438	−	0.411438i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	13.2915			0.550478
$584$		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
$596$	30.4786	+	34.5595i	1.24845	+	1.41561i
$597$		0			0
$598$		0			0
$599$		−	3.56418i		−	0.145629i	−0.997346	−	0.0728143i	$-0.976802\pi$
$599$		−	3.56418i		−	0.145629i	0.997346	−	0.0728143i	$-0.0231980\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	18.4735	−	40.9235i	0.752924	−	1.66792i
$603$		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
$616$	−2.01111	−	6.53137i	−0.0810301	−	0.263157i
$617$			48.3180i			1.94521i	0.232462	+	0.972605i	$0.425322\pi$
$617$			48.3180i			1.94521i	−0.232462	+	0.972605i	$0.574678\pi$
$618$		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	$0.396829\pi$
$632$	−6.65882	−	21.6255i	−0.264874	−	0.860214i
$633$		0			0
$634$	40.6346	+	18.3431i	1.61381	+	0.728499i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	7.14409	+	3.22496i	0.282837	+	0.127677i
$639$		0			0
$640$		0			0
$641$		−	47.4935i		−	1.87588i	−0.346795	−	0.937941i	$-0.612730\pi$
$641$		−	47.4935i		−	1.87588i	0.346795	−	0.937941i	$-0.387270\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	7.60070	−	6.70319i	0.299510	−	0.264143i
$645$		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
$653$	27.8720			1.09072			0.545358	−	0.838203i	$-0.316394\pi$
$653$	27.8720			1.09072			0.545358	+	0.838203i	$0.316394\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.4044			−0.483207			−0.241604	−	0.970375i	$-0.577673\pi$
$659$	−12.4044			−0.483207			−0.241604	+	0.970375i	$0.577673\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−20.9470	+	46.4028i	−0.814128	+	1.80350i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			11.6235i			0.450065i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	42.3320			1.63178			0.815890	−	0.578208i	$-0.196248\pi$
$673$	42.3320			1.63178			0.815890	+	0.578208i	$0.196248\pi$
$674$	27.2823	+	12.3157i	1.05088	+	0.474382i
$675$		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
$683$	33.0279			1.26378			0.631889	−	0.775059i	$-0.282280\pi$
$683$	33.0279			1.26378			0.631889	+	0.775059i	$0.282280\pi$
$684$		0			0
$685$		0			0
$686$	23.8720	+	10.7762i	0.911438	+	0.411438i
$687$		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
$694$	30.1144	+	13.5941i	1.14313	+	0.516026i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	17.5000	+	19.8431i	0.661438	+	0.750000i
$701$	−38.8308			−1.46662			−0.733309	−	0.679895i	$-0.762025\pi$
$701$	−38.8308			−1.46662			−0.733309	+	0.679895i	$0.762025\pi$
$702$		0			0
$703$		0			0
$704$	6.04045	−	4.10954i	0.227658	−	0.154884i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	28.4904	+	32.3051i	1.06474	+	1.20730i
$717$		0			0
$718$	−11.9484	+	26.4686i	−0.445909	+	0.987800i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−24.4904	−	11.0554i	−0.911438	−	0.411438i
$723$		0			0
$724$		0			0
$725$	−30.3455			−1.12700
$726$		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
$736$	9.23987	+	5.65687i	0.340586	+	0.208515i
$737$			14.4971i			0.534007i
$738$		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
$742$	−49.6346	−	22.4059i	−1.82214	−	0.822546i
$743$			54.4759i			1.99853i	0.0383863	+	0.999263i	$0.487778\pi$
$743$			54.4759i			1.99853i	−0.0383863	+	0.999263i	$0.512222\pi$
$744$		0			0
$745$		0			0
$746$	18.4735	−	40.9235i	0.676363	−	1.49831i
$747$		0			0
$748$		0			0
$749$	47.3161			1.72889
$750$		0			0
$751$	26.4575			0.965448			0.482724	−	0.875772i	$-0.339647\pi$
$751$	26.4575			0.965448			0.482724	+	0.875772i	$0.339647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			54.0000i			1.96266i	0.192323	+	0.981332i	$0.438398\pi$
$757$			54.0000i			1.96266i	−0.192323	+	0.981332i	$0.561602\pi$
$758$	−6.98233	+	15.4676i	−0.253610	+	0.561809i
$759$		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
$764$	−36.5477	+	32.2321i	−1.32225	+	1.16611i
$765$		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
$778$	24.9889	+	11.2804i	0.895895	+	0.404422i
$779$		0			0
$780$		0			0
$781$			13.7490i			0.491978i
$782$		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	$0.5\pi$
$788$	−14.4213	−	16.3522i	−0.513737	−	0.582523i
$789$		0			0
$790$		0			0
$791$		−	35.8563i		−	1.27490i
$792$		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
$800$	−14.7684	+	24.1225i	−0.522141	+	0.852859i
$801$		0			0
$802$	22.6974	−	50.2804i	0.801472	−	1.77546i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			56.8033i			1.99710i	0.0538482	+	0.998549i	$0.482851\pi$
$809$			56.8033i			1.99710i	−0.0538482	+	0.998549i	$0.517149\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−21.2419	−	24.0860i	−0.745443	−	0.845253i
$813$		0			0
$814$	3.18824	−	7.06275i	0.111748	−	0.247549i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.8602			0.762927			0.381464	−	0.924384i	$-0.375420\pi$
$821$	21.8602			0.762927			0.381464	+	0.924384i	$0.375420\pi$
$822$		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
$827$	57.3043			1.99267			0.996333	−	0.0855616i	$-0.0272685\pi$
$827$	57.3043			1.99267			0.996333	+	0.0855616i	$0.0272685\pi$
$828$		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
$841$	7.83399			0.270138
$842$	18.4735	−	40.9235i	0.636639	−	1.41032i
$843$		0			0
$844$	18.0000	−	15.8745i	0.619586	−	0.546423i
$845$		0			0
$846$		0			0
$847$	−26.8967			−0.924183
$848$	7.27719	−	57.7609i	0.249900	−	1.98352i
$849$		0			0
$850$		0			0
$851$	11.4912			0.393913
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	14.8856	+	48.3431i	0.508780	+	1.65233i
$857$		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
$862$	24.0516	−	53.2804i	0.819202	−	1.81474i
$863$			46.8151i			1.59360i	0.604240	+	0.796802i	$0.293477\pi$
$863$			46.8151i			1.59360i	−0.604240	+	0.796802i	$0.706523\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.30584			0.247834
$870$		0			0
$871$		0			0
$872$	−48.6573	+	14.9823i	−1.64774	+	0.507366i
$873$		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	$0.0647168\pi$
$883$			12.0000i			0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$		0			0
$885$		0			0
$886$	−51.9889	−	23.4686i	−1.74660	−	0.788444i
$887$		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
$896$	−29.4845	+	5.16373i	−0.985008	+	0.172508i
$897$		0			0
$898$	18.2399	−	40.4059i	0.608672	−	1.34836i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	36.6346	−	11.2804i	1.21845	−	0.375180i
$905$		0			0
$906$		0			0
$907$		−	60.0000i		−	1.99227i	−0.0878507	−	0.996134i	$-0.528000\pi$
$907$		−	60.0000i		−	1.99227i	0.0878507	−	0.996134i	$-0.472000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	29.8445i		−	0.988793i	−0.869236	−	0.494397i	$-0.835389\pi$
$911$		−	29.8445i		−	0.988793i	0.869236	−	0.494397i	$-0.164611\pi$
$912$		0			0
$913$		0			0
$914$	−54.5646	−	24.6314i	−1.80484	−	0.814733i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	37.0405			1.22185			0.610927	−	0.791687i	$-0.290797\pi$
$919$	37.0405			1.22185			0.610927	+	0.791687i	$0.290797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			30.0000i			0.986394i
$926$	51.5587	+	23.2744i	1.69432	+	0.764845i
$927$		0			0
$928$	17.9261	−	29.2804i	0.588454	−	0.961176i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$	−45.7845	+	40.3781i	−1.49972	+	1.32263i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	24.4382	−	54.1366i	0.797934	−	1.76762i
$939$		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
$946$	−6.37648	+	14.1255i	−0.207317	+	0.459259i
$947$	27.0161			0.877905			0.438953	−	0.898510i	$-0.355350\pi$
$947$	27.0161			0.877905			0.438953	+	0.898510i	$0.355350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			55.9788i			1.81333i	0.421849	+	0.906666i	$0.361381\pi$
$953$			55.9788i			1.81333i	−0.421849	+	0.906666i	$0.638619\pi$
$954$		0			0
$955$		0			0
$956$	11.0919	−	9.78211i	0.358737	−	0.316376i
$957$		0			0
$958$		0			0
$959$			58.3063i			1.88281i
$960$		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	$0.722376\pi$
$968$	−8.46171	−	27.4806i	−0.271970	−	0.883259i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$	−47.7440	−	21.5524i	−1.52982	−	0.690584i
$975$		0			0
$976$		0			0
$977$			2.59365i			0.0829783i	0.999139	+	0.0414892i	$0.0132102\pi$
$977$			2.59365i			0.0829783i	−0.999139	+	0.0414892i	$0.986790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−44.9261	−	20.2804i	−1.43365	−	0.647173i
$983$		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
$989$	−22.9824			−0.730797
$990$		0			0
$991$	−58.2065			−1.84899			−0.924496	−	0.381193i	$-0.875513\pi$
$991$	−58.2065			−1.84899			−0.924496	+	0.381193i	$0.875513\pi$
$992$		0			0
$993$		0			0
$994$	23.1771	−	51.3431i	0.735134	−	1.62851i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−20.9470	+	46.4028i	−0.663066	+	1.46886i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.i.a.125.8	yes	8
3.2	odd	2	inner	504.2.i.a.125.1	✓	8
4.3	odd	2		2016.2.i.a.881.4		8
7.6	odd	2	CM	504.2.i.a.125.8	yes	8
8.3	odd	2		2016.2.i.a.881.2		8
8.5	even	2	inner	504.2.i.a.125.2	yes	8
12.11	even	2		2016.2.i.a.881.1		8
21.20	even	2	inner	504.2.i.a.125.1	✓	8
24.5	odd	2	inner	504.2.i.a.125.7	yes	8
24.11	even	2		2016.2.i.a.881.3		8
28.27	even	2		2016.2.i.a.881.4		8
56.13	odd	2	inner	504.2.i.a.125.2	yes	8
56.27	even	2		2016.2.i.a.881.2		8
84.83	odd	2		2016.2.i.a.881.1		8
168.83	odd	2		2016.2.i.a.881.3		8
168.125	even	2	inner	504.2.i.a.125.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.i.a.125.1	✓	8	3.2	odd	2	inner
504.2.i.a.125.1	✓	8	21.20	even	2	inner
504.2.i.a.125.2	yes	8	8.5	even	2	inner
504.2.i.a.125.2	yes	8	56.13	odd	2	inner
504.2.i.a.125.7	yes	8	24.5	odd	2	inner
504.2.i.a.125.7	yes	8	168.125	even	2	inner
504.2.i.a.125.8	yes	8	1.1	even	1	trivial
504.2.i.a.125.8	yes	8	7.6	odd	2	CM
2016.2.i.a.881.1		8	12.11	even	2
2016.2.i.a.881.1		8	84.83	odd	2
2016.2.i.a.881.2		8	8.3	odd	2
2016.2.i.a.881.2		8	56.27	even	2
2016.2.i.a.881.3		8	24.11	even	2
2016.2.i.a.881.3		8	168.83	odd	2
2016.2.i.a.881.4		8	4.3	odd	2
2016.2.i.a.881.4		8	28.27	even	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)

\(f(q)\)	\(=\)	\(q+(1.28897 + 0.581861i) q^{2} +(1.32288 + 1.50000i) q^{4} +2.64575 q^{7} +(0.832353 + 2.70318i) q^{8} +O(q^{10})\)	\(q+(1.28897 + 0.581861i) q^{2} +(1.32288 + 1.50000i) q^{4} +2.64575 q^{7} +(0.832353 + 2.70318i) q^{8} -0.913230 q^{11} +(3.41029 + 1.53946i) q^{14} +(-0.500000 + 3.96863i) q^{16} +(-1.17712 - 0.531373i) q^{22} -1.91520i q^{23} +5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} -6.06910 q^{29} +(-2.95367 + 4.82450i) q^{32} +6.00000i q^{37} -12.0000i q^{43} +(-1.20809 - 1.36985i) q^{44} +(1.11438 - 2.46863i) q^{46} +7.00000 q^{49} +(6.44484 + 2.90930i) q^{50} -14.5544 q^{53} +(2.20220 + 7.15195i) q^{56} +(-7.82288 - 3.53137i) q^{58} +(-6.61438 + 4.50000i) q^{64} -15.8745i q^{67} -15.0554i q^{71} +(-3.49117 + 7.73381i) q^{74} -2.41618 q^{77} -8.00000 q^{79} +(6.98233 - 15.4676i) q^{86} +(-0.760130 - 2.46863i) q^{88} +(2.87280 - 2.53357i) q^{92} +(9.02277 + 4.07303i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

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