LMFDB - Embedded newform 504.2.p.a.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,2,Mod(307,504)]

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([1, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("504.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	504.p (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:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			307.1
Root			$0.500000 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	504.307
Dual form			504.2.p.a.307.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 1.32288i) q^{2} +(-1.50000 + 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 + 1.32288i) q^{8} +O(q^{10})$	$q+(-0.500000 - 1.32288i) q^{2} +(-1.50000 + 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 + 1.32288i) q^{8} +4.00000 q^{11} +(-3.50000 + 1.32288i) q^{14} +(0.500000 - 3.96863i) q^{16} +(-2.00000 - 5.29150i) q^{22} -5.29150i q^{23} -5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} -10.5830i q^{29} +(-5.50000 + 1.32288i) q^{32} -10.5830i q^{37} +12.0000 q^{43} +(-6.00000 + 5.29150i) q^{44} +(-7.00000 + 2.64575i) q^{46} -7.00000 q^{49} +(2.50000 + 6.61438i) q^{50} +10.5830i q^{53} +(3.50000 - 6.61438i) q^{56} +(-14.0000 + 5.29150i) q^{58} +(4.50000 + 6.61438i) q^{64} +4.00000 q^{67} +5.29150i q^{71} +(-14.0000 + 5.29150i) q^{74} -10.5830i q^{77} +15.8745i q^{79} +(-6.00000 - 15.8745i) q^{86} +(10.0000 + 5.29150i) q^{88} +(7.00000 + 7.93725i) q^{92} +(3.50000 + 9.26013i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 3 q^{4} + 5 q^{8}+O(q^{10}) $ 2 * q - q^2 - 3 * q^4 + 5 * q^8	$ 2 q - q^{2} - 3 q^{4} + 5 q^{8} + 8 q^{11} - 7 q^{14} + q^{16} - 4 q^{22} - 10 q^{25} + 7 q^{28} - 11 q^{32} + 24 q^{43} - 12 q^{44} - 14 q^{46} - 14 q^{49} + 5 q^{50} + 7 q^{56} - 28 q^{58} + 9 q^{64} + 8 q^{67} - 28 q^{74} - 12 q^{86} + 20 q^{88} + 14 q^{92} + 7 q^{98}+O(q^{100}) $ 2 * q - q^2 - 3 * q^4 + 5 * q^8 + 8 * q^11 - 7 * q^14 + q^16 - 4 * q^22 - 10 * q^25 + 7 * q^28 - 11 * q^32 + 24 * q^43 - 12 * q^44 - 14 * q^46 - 14 * q^49 + 5 * q^50 + 7 * q^56 - 28 * q^58 + 9 * q^64 + 8 * q^67 - 28 * q^74 - 12 * q^86 + 20 * q^88 + 14 * q^92 + 7 * q^98

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 $

$2$	−0.500000	−	1.32288i	−0.353553	−	0.935414i
$2$	−0.500000	−	1.32288i	−0.353553	−	0.935414i
$3$		0			0
$3$		0			0
$4$	−1.50000	+	1.32288i	−0.750000	+	0.661438i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$	2.50000	+	1.32288i	0.883883	+	0.467707i
$9$		0			0
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−3.50000	+	1.32288i	−0.935414	+	0.353553i
$15$		0			0
$16$	0.500000	−	3.96863i	0.125000	−	0.992157i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$	−2.00000	−	5.29150i	−0.426401	−	1.12815i
$23$		−	5.29150i		−	1.10335i	−0.834058	−	0.551677i	$-0.813988\pi$
$23$		−	5.29150i		−	1.10335i	0.834058	−	0.551677i	$-0.186012\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$	3.50000	+	3.96863i	0.661438	+	0.750000i
$29$		−	10.5830i		−	1.96521i	−0.185695	−	0.982607i	$-0.559454\pi$
$29$		−	10.5830i		−	1.96521i	0.185695	−	0.982607i	$-0.440546\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.50000	+	1.32288i	−0.972272	+	0.233854i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	10.5830i		−	1.73984i	−0.493197	−	0.869918i	$-0.664172\pi$
$37$		−	10.5830i		−	1.73984i	0.493197	−	0.869918i	$-0.335828\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$	−6.00000	+	5.29150i	−0.904534	+	0.797724i
$45$		0			0
$46$	−7.00000	+	2.64575i	−1.03209	+	0.390095i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	2.50000	+	6.61438i	0.353553	+	0.935414i
$51$		0			0
$52$		0			0
$53$			10.5830i			1.45369i	0.686803	+	0.726844i	$0.259014\pi$
$53$			10.5830i			1.45369i	−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$		0			0
$56$	3.50000	−	6.61438i	0.467707	−	0.883883i
$57$		0			0
$58$	−14.0000	+	5.29150i	−1.83829	+	0.694808i
$59$		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
$64$	4.50000	+	6.61438i	0.562500	+	0.826797i
$65$		0			0
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.29150i			0.627986i	0.949425	+	0.313993i	$0.101667\pi$
$71$			5.29150i			0.627986i	−0.949425	+	0.313993i	$0.898333\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−14.0000	+	5.29150i	−1.62747	+	0.615125i
$75$		0			0
$76$		0			0
$77$		−	10.5830i		−	1.20605i
$78$		0			0
$79$			15.8745i			1.78602i	0.450035	+	0.893011i	$0.351411\pi$
$79$			15.8745i			1.78602i	−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
$86$	−6.00000	−	15.8745i	−0.646997	−	1.71179i
$87$		0			0
$88$	10.0000	+	5.29150i	1.06600	+	0.564076i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	7.00000	+	7.93725i	0.729800	+	0.827516i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	3.50000	+	9.26013i	0.353553	+	0.935414i
$99$		0			0
$100$	7.50000	−	6.61438i	0.750000	−	0.661438i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	14.0000	−	5.29150i	1.35980	−	0.513956i
$107$	20.0000			1.93347			0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000			1.93347			0.966736	+	0.255774i	$0.0823304\pi$
$108$		0			0
$109$			10.5830i			1.01367i	0.862044	+	0.506834i	$0.169184\pi$
$109$			10.5830i			1.01367i	−0.862044	+	0.506834i	$0.830816\pi$
$110$		0			0
$111$		0			0
$112$	−10.5000	−	1.32288i	−0.992157	−	0.125000i
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$		0			0
$116$	14.0000	+	15.8745i	1.29987	+	1.47391i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	15.8745i		−	1.40863i	−0.709885	−	0.704317i	$-0.751253\pi$
$127$		−	15.8745i		−	1.40863i	0.709885	−	0.704317i	$-0.248747\pi$
$128$	6.50000	−	9.26013i	0.574524	−	0.818488i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−2.00000	−	5.29150i	−0.172774	−	0.457116i
$135$		0			0
$136$		0			0
$137$	−10.0000			−0.854358			−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000			−0.854358			−0.427179	+	0.904167i	$0.640493\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	7.00000	−	2.64575i	0.587427	−	0.222027i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	14.0000	+	15.8745i	1.15079	+	1.30488i
$149$			10.5830i			0.866994i	0.901155	+	0.433497i	$0.142720\pi$
$149$			10.5830i			0.866994i	−0.901155	+	0.433497i	$0.857280\pi$
$150$		0			0
$151$			5.29150i			0.430616i	0.976546	+	0.215308i	$0.0690756\pi$
$151$			5.29150i			0.430616i	−0.976546	+	0.215308i	$0.930924\pi$
$152$		0			0
$153$		0			0
$154$	−14.0000	+	5.29150i	−1.12815	+	0.426401i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	21.0000	−	7.93725i	1.67067	−	0.631454i
$159$		0			0
$160$		0			0
$161$	−14.0000			−1.10335
$162$		0			0
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$		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			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$			13.2288i			1.00000i
$176$	2.00000	−	15.8745i	0.150756	−	1.19659i
$177$		0			0
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	7.00000	−	13.2288i	0.516047	−	0.975237i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			26.4575i			1.91440i	0.289430	+	0.957199i	$0.406534\pi$
$191$			26.4575i			1.91440i	−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$	−18.0000			−1.29567			−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000			−1.29567			−0.647834	+	0.761781i	$0.724325\pi$
$194$		0			0
$195$		0			0
$196$	10.5000	−	9.26013i	0.750000	−	0.661438i
$197$			10.5830i			0.754008i	0.926212	+	0.377004i	$0.123046\pi$
$197$			10.5830i			0.754008i	−0.926212	+	0.377004i	$0.876954\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−12.5000	−	6.61438i	−0.883883	−	0.467707i
$201$		0			0
$202$		0			0
$203$	−28.0000			−1.96521
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$	−14.0000	−	15.8745i	−0.961524	−	1.09027i
$213$		0			0
$214$	−10.0000	−	26.4575i	−0.683586	−	1.80860i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	14.0000	−	5.29150i	0.948200	−	0.358386i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	3.50000	+	14.5516i	0.233854	+	0.972272i
$225$		0			0
$226$	−1.00000	−	2.64575i	−0.0665190	−	0.175993i
$227$		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
$232$	14.0000	−	26.4575i	0.919145	−	1.73702i
$233$	22.0000			1.44127			0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000			1.44127			0.720634	+	0.693316i	$0.243851\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	26.4575i		−	1.71139i	−0.517477	−	0.855697i	$-0.673129\pi$
$239$		−	26.4575i		−	1.71139i	0.517477	−	0.855697i	$-0.326871\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−2.50000	−	6.61438i	−0.160706	−	0.425188i
$243$		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
$253$		−	21.1660i		−	1.33070i
$254$	−21.0000	+	7.93725i	−1.31766	+	0.498028i
$255$		0			0
$256$	−15.5000	−	3.96863i	−0.968750	−	0.248039i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−28.0000			−1.73984
$260$		0			0
$261$		0			0
$262$		0			0
$263$			5.29150i			0.326288i	0.986602	+	0.163144i	$0.0521635\pi$
$263$			5.29150i			0.326288i	−0.986602	+	0.163144i	$0.947836\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−6.00000	+	5.29150i	−0.366508	+	0.323230i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	5.00000	+	13.2288i	0.302061	+	0.799178i
$275$	−20.0000			−1.20605
$276$		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
$281$	−26.0000			−1.55103			−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000			−1.55103			−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−7.00000	−	7.93725i	−0.415374	−	0.470989i
$285$		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			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	14.0000	−	26.4575i	0.813733	−	1.53781i
$297$		0			0
$298$	14.0000	−	5.29150i	0.810998	−	0.306529i
$299$		0			0
$300$		0			0
$301$		−	31.7490i		−	1.82998i
$302$	7.00000	−	2.64575i	0.402805	−	0.152246i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	14.0000	+	15.8745i	0.797724	+	0.904534i
$309$		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
$316$	−21.0000	−	23.8118i	−1.18134	−	1.33952i
$317$		−	10.5830i		−	0.594401i	−0.954815	−	0.297200i	$-0.903947\pi$
$317$		−	10.5830i		−	0.594401i	0.954815	−	0.297200i	$-0.0960529\pi$
$318$		0			0
$319$		−	42.3320i		−	2.37014i
$320$		0			0
$321$		0			0
$322$	7.00000	+	18.5203i	0.390095	+	1.03209i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−10.0000	−	26.4575i	−0.553849	−	1.46535i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−36.0000			−1.97874			−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000			−1.97874			−0.989369	+	0.145424i	$0.953545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	30.0000			1.63420			0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000			1.63420			0.817102	+	0.576493i	$0.195579\pi$
$338$	6.50000	+	17.1974i	0.353553	+	0.935414i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			18.5203i			1.00000i
$344$	30.0000	+	15.8745i	1.61749	+	0.855896i
$345$		0			0
$346$		0			0
$347$	4.00000			0.214731			0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000			0.214731			0.107366	+	0.994220i	$0.465758\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	17.5000	−	6.61438i	0.935414	−	0.353553i
$351$		0			0
$352$	−22.0000	+	5.29150i	−1.17260	+	0.282038i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	2.00000	+	5.29150i	0.105703	+	0.279665i
$359$			37.0405i			1.95492i	0.211112	+	0.977462i	$0.432292\pi$
$359$			37.0405i			1.95492i	−0.211112	+	0.977462i	$0.567708\pi$
$360$		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			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−21.0000	−	2.64575i	−1.09470	−	0.137919i
$369$		0			0
$370$		0			0
$371$	28.0000			1.45369
$372$		0			0
$373$			31.7490i			1.64390i	0.569558	+	0.821951i	$0.307114\pi$
$373$			31.7490i			1.64390i	−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$	35.0000	−	13.2288i	1.79076	−	0.676842i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	9.00000	+	23.8118i	0.458088	+	1.21199i
$387$		0			0
$388$		0			0
$389$			10.5830i			0.536580i	0.963338	+	0.268290i	$0.0864585\pi$
$389$			10.5830i			0.536580i	−0.963338	+	0.268290i	$0.913542\pi$
$390$		0			0
$391$		0			0
$392$	−17.5000	−	9.26013i	−0.883883	−	0.467707i
$393$		0			0
$394$	14.0000	−	5.29150i	0.705310	−	0.266582i
$395$		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
$401$	34.0000			1.69788			0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000			1.69788			0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	14.0000	+	37.0405i	0.694808	+	1.83829i
$407$		−	42.3320i		−	2.09832i
$408$		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	$0.281584\pi$
$421$			31.7490i			1.54735i	−0.633581	+	0.773676i	$0.718416\pi$
$422$	6.00000	+	15.8745i	0.292075	+	0.772759i
$423$		0			0
$424$	−14.0000	+	26.4575i	−0.679900	+	1.28489i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−30.0000	+	26.4575i	−1.45010	+	1.27887i
$429$		0			0
$430$		0			0
$431$		−	26.4575i		−	1.27441i	−0.770693	−	0.637207i	$-0.780090\pi$
$431$		−	26.4575i		−	1.27441i	0.770693	−	0.637207i	$-0.219910\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−14.0000	−	15.8745i	−0.670478	−	0.760251i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	17.5000	−	11.9059i	0.826797	−	0.562500i
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$		0			0
$452$	−3.00000	+	2.64575i	−0.141108	+	0.124446i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.00000			0.280668			0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000			0.280668			0.140334	+	0.990104i	$0.455182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			15.8745i			0.737751i	0.929479	+	0.368875i	$0.120257\pi$
$463$			15.8745i			0.737751i	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−42.0000	−	5.29150i	−1.94980	−	0.245652i
$465$		0			0
$466$	−11.0000	−	29.1033i	−0.509565	−	1.34818i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		−	10.5830i		−	0.488678i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	48.0000			2.20704
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−35.0000	+	13.2288i	−1.60086	+	0.605069i
$479$		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
$484$	−7.50000	+	6.61438i	−0.340909	+	0.300654i
$485$		0			0
$486$		0			0
$487$		−	37.0405i		−	1.67847i	−0.543772	−	0.839233i	$-0.683004\pi$
$487$		−	37.0405i		−	1.67847i	0.543772	−	0.839233i	$-0.316996\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−44.0000			−1.98569			−0.992846	−	0.119401i	$-0.961903\pi$
$491$	−44.0000			−1.98569			−0.992846	+	0.119401i	$0.961903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.0000			0.627986
$498$		0			0
$499$	36.0000			1.61158			0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000			1.61158			0.805791	+	0.592200i	$0.201741\pi$
$500$		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
$506$	−28.0000	+	10.5830i	−1.24475	+	0.470472i
$507$		0			0
$508$	21.0000	+	23.8118i	0.931724	+	1.05648i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	2.50000	+	22.4889i	0.110485	+	0.993878i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	14.0000	+	37.0405i	0.615125	+	1.62747i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	7.00000	−	2.64575i	0.305215	−	0.115360i
$527$		0			0
$528$		0			0
$529$	−5.00000			−0.217391
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	10.0000	+	5.29150i	0.431934	+	0.228558i
$537$		0			0
$538$		0			0
$539$	−28.0000			−1.20605
$540$		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
$547$	−44.0000			−1.88130			−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000			−1.88130			−0.940652	+	0.339372i	$0.889785\pi$
$548$	15.0000	−	13.2288i	0.640768	−	0.565104i
$549$		0			0
$550$	10.0000	+	26.4575i	0.426401	+	1.12815i
$551$		0			0
$552$		0			0
$553$	42.0000			1.78602
$554$	42.0000	−	15.8745i	1.78441	−	0.674443i
$555$		0			0
$556$		0			0
$557$		−	10.5830i		−	0.448416i	−0.974541	−	0.224208i	$-0.928020\pi$
$557$		−	10.5830i		−	0.448416i	0.974541	−	0.224208i	$-0.0719796\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	13.0000	+	34.3948i	0.548372	+	1.45086i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−7.00000	+	13.2288i	−0.293713	+	0.555066i
$569$	−22.0000			−0.922288			−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000			−0.922288			−0.461144	+	0.887325i	$0.652561\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			26.4575i			1.10335i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−8.50000	−	22.4889i	−0.353553	−	0.935414i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			42.3320i			1.75321i
$584$		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
$592$	−42.0000	−	5.29150i	−1.72619	−	0.217479i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	−14.0000	−	15.8745i	−0.573462	−	0.650245i
$597$		0			0
$598$		0			0
$599$		−	37.0405i		−	1.51343i	−0.653742	−	0.756717i	$-0.726802\pi$
$599$		−	37.0405i		−	1.51343i	0.653742	−	0.756717i	$-0.273198\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−42.0000	+	15.8745i	−1.71179	+	0.646997i
$603$		0			0
$604$	−7.00000	−	7.93725i	−0.284826	−	0.322962i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		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
$616$	14.0000	−	26.4575i	0.564076	−	1.06600i
$617$	26.0000			1.04672			0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000			1.04672			0.523360	+	0.852111i	$0.324678\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		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
$631$		−	47.6235i		−	1.89586i	−0.318475	−	0.947931i	$-0.603171\pi$
$631$		−	47.6235i		−	1.89586i	0.318475	−	0.947931i	$-0.396829\pi$
$632$	−21.0000	+	39.6863i	−0.835335	+	1.57864i
$633$		0			0
$634$	−14.0000	+	5.29150i	−0.556011	+	0.210152i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−56.0000	+	21.1660i	−2.21706	+	0.837970i
$639$		0			0
$640$		0			0
$641$	−46.0000			−1.81689			−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000			−1.81689			−0.908445	+	0.418004i	$0.862730\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	21.0000	−	18.5203i	0.827516	−	0.729800i
$645$		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
$652$	−30.0000	+	26.4575i	−1.17489	+	1.03616i
$653$		−	10.5830i		−	0.414145i	−0.978326	−	0.207072i	$-0.933606\pi$
$653$		−	10.5830i		−	0.414145i	0.978326	−	0.207072i	$-0.0663936\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.0000			1.71400			0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000			1.71400			0.856998	+	0.515319i	$0.172327\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	18.0000	+	47.6235i	0.699590	+	1.85094i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−56.0000			−2.16833
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−30.0000			−1.15642			−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000			−1.15642			−0.578208	+	0.815890i	$0.696248\pi$
$674$	−15.0000	−	39.6863i	−0.577778	−	1.52866i
$675$		0			0
$676$	19.5000	−	17.1974i	0.750000	−	0.661438i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	52.0000			1.98972			0.994862	−	0.101237i	$-0.0322800\pi$
$683$	52.0000			1.98972			0.994862	+	0.101237i	$0.0322800\pi$
$684$		0			0
$685$		0			0
$686$	24.5000	−	9.26013i	0.935414	−	0.353553i
$687$		0			0
$688$	6.00000	−	47.6235i	0.228748	−	1.81563i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−2.00000	−	5.29150i	−0.0759190	−	0.200863i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	−	19.8431i	−0.661438	−	0.750000i
$701$		−	52.9150i		−	1.99857i	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$		−	52.9150i		−	1.99857i	0.0377695	−	0.999286i	$-0.487975\pi$
$702$		0			0
$703$		0			0
$704$	18.0000	+	26.4575i	0.678401	+	0.997155i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	52.9150i		−	1.98727i	−0.112667	−	0.993633i	$-0.535939\pi$
$709$		−	52.9150i		−	1.98727i	0.112667	−	0.993633i	$-0.464061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	6.00000	−	5.29150i	0.224231	−	0.197753i
$717$		0			0
$718$	49.0000	−	18.5203i	1.82866	−	0.691170i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−9.50000	−	25.1346i	−0.353553	−	0.935414i
$723$		0			0
$724$		0			0
$725$			52.9150i			1.96521i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		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
$736$	7.00000	+	29.1033i	0.258023	+	1.07276i
$737$	16.0000			0.589368
$738$		0			0
$739$	52.0000			1.91285			0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000			1.91285			0.956425	+	0.291977i	$0.0943129\pi$
$740$		0			0
$741$		0			0
$742$	−14.0000	−	37.0405i	−0.513956	−	1.35980i
$743$			37.0405i			1.35888i	0.733729	+	0.679442i	$0.237778\pi$
$743$			37.0405i			1.35888i	−0.733729	+	0.679442i	$0.762222\pi$
$744$		0			0
$745$		0			0
$746$	42.0000	−	15.8745i	1.53773	−	0.581207i
$747$		0			0
$748$		0			0
$749$		−	52.9150i		−	1.93347i
$750$		0			0
$751$			26.4575i			0.965448i	0.875772	+	0.482724i	$0.160353\pi$
$751$			26.4575i			0.965448i	−0.875772	+	0.482724i	$0.839647\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	10.5830i		−	0.384646i	−0.981332	−	0.192323i	$-0.938398\pi$
$757$		−	10.5830i		−	0.384646i	0.981332	−	0.192323i	$-0.0616021\pi$
$758$	−6.00000	−	15.8745i	−0.217930	−	0.576588i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	28.0000			1.01367
$764$	−35.0000	−	39.6863i	−1.26626	−	1.43580i
$765$		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
$772$	27.0000	−	23.8118i	0.971751	−	0.857004i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	14.0000	−	5.29150i	0.501924	−	0.189710i
$779$		0			0
$780$		0			0
$781$			21.1660i			0.757379i
$782$		0			0
$783$		0			0
$784$	−3.50000	+	27.7804i	−0.125000	+	0.992157i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−14.0000	−	15.8745i	−0.498729	−	0.565506i
$789$		0			0
$790$		0			0
$791$		−	5.29150i		−	0.188144i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	27.5000	−	6.61438i	0.972272	−	0.233854i
$801$		0			0
$802$	−17.0000	−	44.9778i	−0.600291	−	1.58822i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−38.0000			−1.33601			−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000			−1.33601			−0.668004	+	0.744157i	$0.732851\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	42.0000	−	37.0405i	1.47391	−	1.29987i
$813$		0			0
$814$	−56.0000	+	21.1660i	−1.96280	+	0.741868i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			52.9150i			1.84675i	0.383903	+	0.923374i	$0.374580\pi$
$821$			52.9150i			1.84675i	−0.383903	+	0.923374i	$0.625420\pi$
$822$		0			0
$823$		−	47.6235i		−	1.66005i	−0.557725	−	0.830026i	$-0.688326\pi$
$823$		−	47.6235i		−	1.66005i	0.557725	−	0.830026i	$-0.311674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−44.0000			−1.53003			−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000			−1.53003			−0.765015	+	0.644013i	$0.777268\pi$
$828$		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
$841$	−83.0000			−2.86207
$842$	42.0000	−	15.8745i	1.44742	−	0.547072i
$843$		0			0
$844$	18.0000	−	15.8745i	0.619586	−	0.546423i
$845$		0			0
$846$		0			0
$847$		−	13.2288i		−	0.454545i
$848$	42.0000	+	5.29150i	1.44229	+	0.181711i
$849$		0			0
$850$		0			0
$851$	−56.0000			−1.91966
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	50.0000	+	26.4575i	1.70896	+	0.904299i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	−35.0000	+	13.2288i	−1.19210	+	0.450573i
$863$		−	58.2065i		−	1.98137i	−0.136162	−	0.990687i	$-0.543477\pi$
$863$		−	58.2065i		−	1.98137i	0.136162	−	0.990687i	$-0.456523\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			63.4980i			2.15402i
$870$		0			0
$871$		0			0
$872$	−14.0000	+	26.4575i	−0.474100	+	0.895964i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	31.7490i		−	1.07209i	−0.844190	−	0.536044i	$-0.819918\pi$
$877$		−	31.7490i		−	1.07209i	0.844190	−	0.536044i	$-0.180082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−12.0000			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$		0			0
$886$	−10.0000	−	26.4575i	−0.335957	−	0.888858i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−42.0000			−1.40863
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−24.5000	−	17.1974i	−0.818488	−	0.574524i
$897$		0			0
$898$	1.00000	+	2.64575i	0.0333704	+	0.0882899i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	5.00000	+	2.64575i	0.166298	+	0.0879964i
$905$		0			0
$906$		0			0
$907$	60.0000			1.99227			0.996134	−	0.0878507i	$-0.0279999\pi$
$907$	60.0000			1.99227			0.996134	+	0.0878507i	$0.0279999\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			58.2065i			1.92847i	0.265052	+	0.964234i	$0.414611\pi$
$911$			58.2065i			1.92847i	−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		0			0
$914$	−3.00000	−	7.93725i	−0.0992312	−	0.262541i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			37.0405i			1.22185i	0.791687	+	0.610927i	$0.209203\pi$
$919$			37.0405i			1.22185i	−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			52.9150i			1.73984i
$926$	21.0000	−	7.93725i	0.690103	−	0.260834i
$927$		0			0
$928$	14.0000	+	58.2065i	0.459573	+	1.91072i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−33.0000	+	29.1033i	−1.08095	+	0.953309i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−14.0000	+	5.29150i	−0.457116	+	0.172774i
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−24.0000	−	63.4980i	−0.780307	−	2.06450i
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−58.0000			−1.87880			−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−58.0000			−1.87880			−0.939402	+	0.342817i	$0.888619\pi$
$954$		0			0
$955$		0			0
$956$	35.0000	+	39.6863i	1.13198	+	1.28355i
$957$		0			0
$958$		0			0
$959$			26.4575i			0.854358i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			47.6235i			1.53147i	0.643157	+	0.765735i	$0.277624\pi$
$967$			47.6235i			1.53147i	−0.643157	+	0.765735i	$0.722376\pi$
$968$	12.5000	+	6.61438i	0.401765	+	0.212594i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$	−49.0000	+	18.5203i	−1.57006	+	0.593427i
$975$		0			0
$976$		0			0
$977$	46.0000			1.47167			0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000			1.47167			0.735835	+	0.677161i	$0.236790\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	22.0000	+	58.2065i	0.702048	+	1.85744i
$983$		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
$989$		−	63.4980i		−	2.01912i
$990$		0			0
$991$			58.2065i			1.84899i	0.381193	+	0.924496i	$0.375513\pi$
$991$			58.2065i			1.84899i	−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		0			0
$994$	−7.00000	−	18.5203i	−0.222027	−	0.587427i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	−18.0000	−	47.6235i	−0.569780	−	1.50750i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.p.a.307.1		2
3.2	odd	2		56.2.e.a.27.2	yes	2
4.3	odd	2		2016.2.p.a.559.2		2
7.6	odd	2	CM	504.2.p.a.307.1		2
8.3	odd	2	inner	504.2.p.a.307.2		2
8.5	even	2		2016.2.p.a.559.1		2
12.11	even	2		224.2.e.a.111.2		2
21.2	odd	6		392.2.m.a.227.2		4
21.5	even	6		392.2.m.a.227.2		4
21.11	odd	6		392.2.m.a.19.1		4
21.17	even	6		392.2.m.a.19.1		4
21.20	even	2		56.2.e.a.27.2	yes	2
24.5	odd	2		224.2.e.a.111.1		2
24.11	even	2		56.2.e.a.27.1	✓	2
28.27	even	2		2016.2.p.a.559.2		2
48.5	odd	4		1792.2.f.d.1791.3		4
48.11	even	4		1792.2.f.d.1791.2		4
48.29	odd	4		1792.2.f.d.1791.4		4
48.35	even	4		1792.2.f.d.1791.1		4
56.13	odd	2		2016.2.p.a.559.1		2
56.27	even	2	inner	504.2.p.a.307.2		2
84.11	even	6		1568.2.q.a.1391.2		4
84.23	even	6		1568.2.q.a.815.1		4
84.47	odd	6		1568.2.q.a.815.1		4
84.59	odd	6		1568.2.q.a.1391.2		4
84.83	odd	2		224.2.e.a.111.2		2
168.5	even	6		1568.2.q.a.815.2		4
168.11	even	6		392.2.m.a.19.2		4
168.53	odd	6		1568.2.q.a.1391.1		4
168.59	odd	6		392.2.m.a.19.2		4
168.83	odd	2		56.2.e.a.27.1	✓	2
168.101	even	6		1568.2.q.a.1391.1		4
168.107	even	6		392.2.m.a.227.1		4
168.125	even	2		224.2.e.a.111.1		2
168.131	odd	6		392.2.m.a.227.1		4
168.149	odd	6		1568.2.q.a.815.2		4
336.83	odd	4		1792.2.f.d.1791.1		4
336.125	even	4		1792.2.f.d.1791.4		4
336.251	odd	4		1792.2.f.d.1791.2		4
336.293	even	4		1792.2.f.d.1791.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.e.a.27.1	✓	2	24.11	even	2
56.2.e.a.27.1	✓	2	168.83	odd	2
56.2.e.a.27.2	yes	2	3.2	odd	2
56.2.e.a.27.2	yes	2	21.20	even	2
224.2.e.a.111.1		2	24.5	odd	2
224.2.e.a.111.1		2	168.125	even	2
224.2.e.a.111.2		2	12.11	even	2
224.2.e.a.111.2		2	84.83	odd	2
392.2.m.a.19.1		4	21.11	odd	6
392.2.m.a.19.1		4	21.17	even	6
392.2.m.a.19.2		4	168.11	even	6
392.2.m.a.19.2		4	168.59	odd	6
392.2.m.a.227.1		4	168.107	even	6
392.2.m.a.227.1		4	168.131	odd	6
392.2.m.a.227.2		4	21.2	odd	6
392.2.m.a.227.2		4	21.5	even	6
504.2.p.a.307.1		2	1.1	even	1	trivial
504.2.p.a.307.1		2	7.6	odd	2	CM
504.2.p.a.307.2		2	8.3	odd	2	inner
504.2.p.a.307.2		2	56.27	even	2	inner
1568.2.q.a.815.1		4	84.23	even	6
1568.2.q.a.815.1		4	84.47	odd	6
1568.2.q.a.815.2		4	168.5	even	6
1568.2.q.a.815.2		4	168.149	odd	6
1568.2.q.a.1391.1		4	168.53	odd	6
1568.2.q.a.1391.1		4	168.101	even	6
1568.2.q.a.1391.2		4	84.11	even	6
1568.2.q.a.1391.2		4	84.59	odd	6
1792.2.f.d.1791.1		4	48.35	even	4
1792.2.f.d.1791.1		4	336.83	odd	4
1792.2.f.d.1791.2		4	48.11	even	4
1792.2.f.d.1791.2		4	336.251	odd	4
1792.2.f.d.1791.3		4	48.5	odd	4
1792.2.f.d.1791.3		4	336.293	even	4
1792.2.f.d.1791.4		4	48.29	odd	4
1792.2.f.d.1791.4		4	336.125	even	4
2016.2.p.a.559.1		2	8.5	even	2
2016.2.p.a.559.1		2	56.13	odd	2
2016.2.p.a.559.2		2	4.3	odd	2
2016.2.p.a.559.2		2	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.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 1.32288i) q^{2} +(-1.50000 + 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 + 1.32288i) q^{8} +O(q^{10})\)	\(q+(-0.500000 - 1.32288i) q^{2} +(-1.50000 + 1.32288i) q^{4} -2.64575i q^{7} +(2.50000 + 1.32288i) q^{8} +4.00000 q^{11} +(-3.50000 + 1.32288i) q^{14} +(0.500000 - 3.96863i) q^{16} +(-2.00000 - 5.29150i) q^{22} -5.29150i q^{23} -5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} -10.5830i q^{29} +(-5.50000 + 1.32288i) q^{32} -10.5830i q^{37} +12.0000 q^{43} +(-6.00000 + 5.29150i) q^{44} +(-7.00000 + 2.64575i) q^{46} -7.00000 q^{49} +(2.50000 + 6.61438i) q^{50} +10.5830i q^{53} +(3.50000 - 6.61438i) q^{56} +(-14.0000 + 5.29150i) q^{58} +(4.50000 + 6.61438i) q^{64} +4.00000 q^{67} +5.29150i q^{71} +(-14.0000 + 5.29150i) q^{74} -10.5830i q^{77} +15.8745i q^{79} +(-6.00000 - 15.8745i) q^{86} +(10.0000 + 5.29150i) q^{88} +(7.00000 + 7.93725i) q^{92} +(3.50000 + 9.26013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 3 q^{4} + 5 q^{8}+O(q^{10}) \) 2 * q - q^2 - 3 * q^4 + 5 * q^8	\( 2 q - q^{2} - 3 q^{4} + 5 q^{8} + 8 q^{11} - 7 q^{14} + q^{16} - 4 q^{22} - 10 q^{25} + 7 q^{28} - 11 q^{32} + 24 q^{43} - 12 q^{44} - 14 q^{46} - 14 q^{49} + 5 q^{50} + 7 q^{56} - 28 q^{58} + 9 q^{64} + 8 q^{67} - 28 q^{74} - 12 q^{86} + 20 q^{88} + 14 q^{92} + 7 q^{98}+O(q^{100}) \) 2 * q - q^2 - 3 * q^4 + 5 * q^8 + 8 * q^11 - 7 * q^14 + q^16 - 4 * q^22 - 10 * q^25 + 7 * q^28 - 11 * q^32 + 24 * q^43 - 12 * q^44 - 14 * q^46 - 14 * q^49 + 5 * q^50 + 7 * q^56 - 28 * q^58 + 9 * q^64 + 8 * q^67 - 28 * q^74 - 12 * q^86 + 20 * q^88 + 14 * q^92 + 7 * q^98

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