LMFDB - Embedded newform 504.2.p.d.307.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,2,Mod(307,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,6,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.02446026187$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} +2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} -5.29150 q^{11} +(-1.32288 - 3.50000i) q^{14} +(0.500000 - 3.96863i) q^{16} +(7.00000 - 2.64575i) q^{22} +8.00000i q^{23} -5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} -2.00000i q^{29} +(1.32288 + 5.50000i) q^{32} +10.5830i q^{37} -12.0000 q^{43} +(-7.93725 + 7.00000i) q^{44} +(-4.00000 - 10.5830i) q^{46} -7.00000 q^{49} +(6.61438 - 2.50000i) q^{50} -10.0000i q^{53} +(-6.61438 - 3.50000i) q^{56} +(1.00000 + 2.64575i) q^{58} +(-4.50000 - 6.61438i) q^{64} +4.00000 q^{67} +16.0000i q^{71} +(-5.29150 - 14.0000i) q^{74} -14.0000i q^{77} +15.8745i q^{79} +(15.8745 - 6.00000i) q^{86} +(7.00000 - 13.2288i) q^{88} +(10.5830 + 12.0000i) q^{92} +(9.26013 - 3.50000i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 2 q^{16} + 28 q^{22} - 20 q^{25} + 14 q^{28} - 48 q^{43} - 16 q^{46} - 28 q^{49} + 4 q^{58} - 18 q^{64} + 16 q^{67} + 28 q^{88}+O(q^{100}) $ 4 * q + 6 * q^4 + 2 * q^16 + 28 * q^22 - 20 * q^25 + 14 * q^28 - 48 * q^43 - 16 * q^46 - 28 * q^49 + 4 * q^58 - 18 * q^64 + 16 * q^67 + 28 * q^88

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 $

$2$	−1.32288	+	0.500000i	−0.935414	+	0.353553i
$2$	−1.32288	+	0.500000i	−0.935414	+	0.353553i
$3$		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
$8$	−1.32288	+	2.50000i	−0.467707	+	0.883883i
$9$		0			0
$10$		0			0
$11$	−5.29150			−1.59545			−0.797724	−	0.603023i	$-0.793963\pi$
$11$	−5.29150			−1.59545			−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−1.32288	−	3.50000i	−0.353553	−	0.935414i
$15$		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
$22$	7.00000	−	2.64575i	1.49241	−	0.564076i
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		0			0
$28$	3.50000	+	3.96863i	0.661438	+	0.750000i
$29$		−	2.00000i		−	0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$		−	2.00000i		−	0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	1.32288	+	5.50000i	0.233854	+	0.972272i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.5830i			1.73984i	0.493197	+	0.869918i	$0.335828\pi$
$37$			10.5830i			1.73984i	−0.493197	+	0.869918i	$0.664172\pi$
$38$		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	$-0.867803\pi$
$43$	−12.0000			−1.82998			−0.914991	+	0.403473i	$0.867803\pi$
$44$	−7.93725	+	7.00000i	−1.19659	+	1.05529i
$45$		0			0
$46$	−4.00000	−	10.5830i	−0.589768	−	1.56038i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	6.61438	−	2.50000i	0.935414	−	0.353553i
$51$		0			0
$52$		0			0
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$		0			0
$55$		0			0
$56$	−6.61438	−	3.50000i	−0.883883	−	0.467707i
$57$		0			0
$58$	1.00000	+	2.64575i	0.131306	+	0.347404i
$59$		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
$71$			16.0000i			1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$			16.0000i			1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−5.29150	−	14.0000i	−0.615125	−	1.62747i
$75$		0			0
$76$		0			0
$77$		−	14.0000i		−	1.59545i
$78$		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
$86$	15.8745	−	6.00000i	1.71179	−	0.646997i
$87$		0			0
$88$	7.00000	−	13.2288i	0.746203	−	1.41019i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	10.5830	+	12.0000i	1.10335	+	1.25109i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	9.26013	−	3.50000i	0.935414	−	0.353553i
$99$		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
$106$	5.00000	+	13.2288i	0.485643	+	1.28489i
$107$	5.29150			0.511549			0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150			0.511549			0.255774	+	0.966736i	$0.417670\pi$
$108$		0			0
$109$		−	10.5830i		−	1.01367i	−0.862044	−	0.506834i	$-0.830816\pi$
$109$		−	10.5830i		−	1.01367i	0.862044	−	0.506834i	$-0.169184\pi$
$110$		0			0
$111$		0			0
$112$	10.5000	+	1.32288i	0.992157	+	0.125000i
$113$	21.1660			1.99113			0.995565	−	0.0940721i	$-0.0299884\pi$
$113$	21.1660			1.99113			0.995565	+	0.0940721i	$0.0299884\pi$
$114$		0			0
$115$		0			0
$116$	−2.64575	−	3.00000i	−0.245652	−	0.278543i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	17.0000			1.54545
$122$		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	$-0.248747\pi$
$128$	9.26013	+	6.50000i	0.818488	+	0.574524i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−5.29150	+	2.00000i	−0.457116	+	0.172774i
$135$		0			0
$136$		0			0
$137$	21.1660			1.80833			0.904167	−	0.427179i	$-0.140493\pi$
$137$	21.1660			1.80833			0.904167	+	0.427179i	$0.140493\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−8.00000	−	21.1660i	−0.671345	−	1.77621i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	14.0000	+	15.8745i	1.15079	+	1.30488i
$149$		−	22.0000i		−	1.80231i	−0.433497	−	0.901155i	$-0.642720\pi$
$149$		−	22.0000i		−	1.80231i	0.433497	−	0.901155i	$-0.357280\pi$
$150$		0			0
$151$		−	5.29150i		−	0.430616i	−0.976546	−	0.215308i	$-0.930924\pi$
$151$		−	5.29150i		−	0.430616i	0.976546	−	0.215308i	$-0.0690756\pi$
$152$		0			0
$153$		0			0
$154$	7.00000	+	18.5203i	0.564076	+	1.49241i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−7.93725	−	21.0000i	−0.631454	−	1.67067i
$159$		0			0
$160$		0			0
$161$	−21.1660			−1.66812
$162$		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
$176$	−2.64575	+	21.0000i	−0.199431	+	1.58293i
$177$		0			0
$178$		0			0
$179$	−26.4575			−1.97753			−0.988764	−	0.149487i	$-0.952238\pi$
$179$	−26.4575			−1.97753			−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	−20.0000	−	10.5830i	−1.47442	−	0.780189i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.00000i			0.578860i	0.957199	+	0.289430i	$0.0934657\pi$
$191$			8.00000i			0.578860i	−0.957199	+	0.289430i	$0.906534\pi$
$192$		0			0
$193$	18.0000			1.29567			0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000			1.29567			0.647834	+	0.761781i	$0.275675\pi$
$194$		0			0
$195$		0			0
$196$	−10.5000	+	9.26013i	−0.750000	+	0.661438i
$197$			26.0000i			1.85242i	0.377004	+	0.926212i	$0.376954\pi$
$197$			26.0000i			1.85242i	−0.377004	+	0.926212i	$0.623046\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	6.61438	−	12.5000i	0.467707	−	0.883883i
$201$		0			0
$202$		0			0
$203$	5.29150			0.371391
$204$		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	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$	−13.2288	−	15.0000i	−0.908555	−	1.03020i
$213$		0			0
$214$	−7.00000	+	2.64575i	−0.478510	+	0.180860i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	5.29150	+	14.0000i	0.358386	+	0.948200i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−14.5516	+	3.50000i	−0.972272	+	0.233854i
$225$		0			0
$226$	−28.0000	+	10.5830i	−1.86253	+	0.703971i
$227$		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
$232$	5.00000	+	2.64575i	0.328266	+	0.173702i
$233$	−21.1660			−1.38663			−0.693316	−	0.720634i	$-0.743851\pi$
$233$	−21.1660			−1.38663			−0.693316	+	0.720634i	$0.743851\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			16.0000i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$239$			16.0000i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−22.4889	+	8.50000i	−1.44564	+	0.546401i
$243$		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
$253$		−	42.3320i		−	2.66139i
$254$	7.93725	+	21.0000i	0.498028	+	1.31766i
$255$		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
$263$		−	32.0000i		−	1.97320i	−0.163144	−	0.986602i	$-0.552164\pi$
$263$		−	32.0000i		−	1.97320i	0.163144	−	0.986602i	$-0.447836\pi$
$264$		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
$274$	−28.0000	+	10.5830i	−1.69154	+	0.639343i
$275$	26.4575			1.59545
$276$		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
$281$	−21.1660			−1.26266			−0.631329	−	0.775515i	$-0.717490\pi$
$281$	−21.1660			−1.26266			−0.631329	+	0.775515i	$0.717490\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	21.1660	+	24.0000i	1.25597	+	1.42414i
$285$		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
$296$	−26.4575	−	14.0000i	−1.53781	−	0.813733i
$297$		0			0
$298$	11.0000	+	29.1033i	0.637213	+	1.68591i
$299$		0			0
$300$		0			0
$301$		−	31.7490i		−	1.82998i
$302$	2.64575	+	7.00000i	0.152246	+	0.402805i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−18.5203	−	21.0000i	−1.05529	−	1.19659i
$309$		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
$317$			34.0000i			1.90963i	0.297200	+	0.954815i	$0.403947\pi$
$317$			34.0000i			1.90963i	−0.297200	+	0.954815i	$0.596053\pi$
$318$		0			0
$319$			10.5830i			0.592535i
$320$		0			0
$321$		0			0
$322$	28.0000	−	10.5830i	1.56038	−	0.589768i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	−26.4575	+	10.0000i	−1.46535	+	0.553849i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	36.0000			1.97874			0.989369	−	0.145424i	$-0.0464545\pi$
$331$	36.0000			1.97874			0.989369	+	0.145424i	$0.0464545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$	17.1974	−	6.50000i	0.935414	−	0.353553i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	18.5203i		−	1.00000i
$344$	15.8745	−	30.0000i	0.855896	−	1.61749i
$345$		0			0
$346$		0			0
$347$	−37.0405			−1.98844			−0.994220	−	0.107366i	$-0.965758\pi$
$347$	−37.0405			−1.98844			−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	6.61438	+	17.5000i	0.353553	+	0.935414i
$351$		0			0
$352$	−7.00000	−	29.1033i	−0.373101	−	1.55121i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	35.0000	−	13.2288i	1.84981	−	0.699162i
$359$		−	8.00000i		−	0.422224i	−0.977462	−	0.211112i	$-0.932292\pi$
$359$		−	8.00000i		−	0.422224i	0.977462	−	0.211112i	$-0.0677085\pi$
$360$		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	$0.5\pi$
$368$	31.7490	+	4.00000i	1.65503	+	0.208514i
$369$		0			0
$370$		0			0
$371$	26.4575			1.37361
$372$		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	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$		0			0
$381$		0			0
$382$	−4.00000	−	10.5830i	−0.204658	−	0.541474i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	−23.8118	+	9.00000i	−1.21199	+	0.458088i
$387$		0			0
$388$		0			0
$389$			38.0000i			1.92668i	0.268290	+	0.963338i	$0.413542\pi$
$389$			38.0000i			1.92668i	−0.268290	+	0.963338i	$0.586458\pi$
$390$		0			0
$391$		0			0
$392$	9.26013	−	17.5000i	0.467707	−	0.883883i
$393$		0			0
$394$	−13.0000	−	34.3948i	−0.654931	−	1.73278i
$395$		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
$401$	−21.1660			−1.05698			−0.528490	−	0.848939i	$-0.677242\pi$
$401$	−21.1660			−1.05698			−0.528490	+	0.848939i	$0.677242\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−7.00000	+	2.64575i	−0.347404	+	0.131306i
$407$		−	56.0000i		−	2.77582i
$408$		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	$0.718416\pi$
$422$	−15.8745	+	6.00000i	−0.772759	+	0.292075i
$423$		0			0
$424$	25.0000	+	13.2288i	1.21411	+	0.642445i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	7.93725	−	7.00000i	0.383662	−	0.338358i
$429$		0			0
$430$		0			0
$431$		−	32.0000i		−	1.54139i	−0.637207	−	0.770693i	$-0.719910\pi$
$431$		−	32.0000i		−	1.54139i	0.637207	−	0.770693i	$-0.280090\pi$
$432$		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
$443$	37.0405			1.75985			0.879924	−	0.475114i	$-0.157593\pi$
$443$	37.0405			1.75985			0.879924	+	0.475114i	$0.157593\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	17.5000	−	11.9059i	0.826797	−	0.562500i
$449$	42.3320			1.99777			0.998886	−	0.0471929i	$-0.0150276\pi$
$449$	42.3320			1.99777			0.998886	+	0.0471929i	$0.0150276\pi$
$450$		0			0
$451$		0			0
$452$	31.7490	−	28.0000i	1.49335	−	1.31701i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.00000			−0.280668			−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000			−0.280668			−0.140334	+	0.990104i	$0.544818\pi$
$458$		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	$0.879743\pi$
$464$	−7.93725	−	1.00000i	−0.368478	−	0.0464238i
$465$		0			0
$466$	28.0000	−	10.5830i	1.29707	−	0.490248i
$467$		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
$473$	63.4980			2.91964
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−8.00000	−	21.1660i	−0.365911	−	0.968111i
$479$		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
$484$	25.5000	−	22.4889i	1.15909	−	1.02222i
$485$		0			0
$486$		0			0
$487$			37.0405i			1.67847i	0.543772	+	0.839233i	$0.316996\pi$
$487$			37.0405i			1.67847i	−0.543772	+	0.839233i	$0.683004\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.29150			−0.238802			−0.119401	−	0.992846i	$-0.538097\pi$
$491$	−5.29150			−0.238802			−0.119401	+	0.992846i	$0.538097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−42.3320			−1.89885
$498$		0			0
$499$	−36.0000			−1.61158			−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000			−1.61158			−0.805791	+	0.592200i	$0.798259\pi$
$500$		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
$506$	21.1660	+	56.0000i	0.940944	+	2.48950i
$507$		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
$512$	22.4889	−	2.50000i	0.993878	−	0.110485i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	37.0405	−	14.0000i	1.62747	−	0.615125i
$519$		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
$526$	16.0000	+	42.3320i	0.697633	+	1.84576i
$527$		0			0
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−5.29150	+	10.0000i	−0.228558	+	0.431934i
$537$		0			0
$538$		0			0
$539$	37.0405			1.59545
$540$		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	$0.889785\pi$
$548$	31.7490	−	28.0000i	1.35625	−	1.19610i
$549$		0			0
$550$	−35.0000	+	13.2288i	−1.49241	+	0.564076i
$551$		0			0
$552$		0			0
$553$	−42.0000			−1.78602
$554$	−15.8745	−	42.0000i	−0.674443	−	1.78441i
$555$		0			0
$556$		0			0
$557$			46.0000i			1.94908i	0.224208	+	0.974541i	$0.428020\pi$
$557$			46.0000i			1.94908i	−0.224208	+	0.974541i	$0.571980\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	28.0000	−	10.5830i	1.18111	−	0.446417i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−40.0000	−	21.1660i	−1.67836	−	0.888106i
$569$	−42.3320			−1.77465			−0.887325	−	0.461144i	$-0.847439\pi$
$569$	−42.3320			−1.77465			−0.887325	+	0.461144i	$0.847439\pi$
$570$		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
$575$		−	40.0000i		−	1.66812i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−22.4889	+	8.50000i	−0.935414	+	0.353553i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			52.9150i			2.19152i
$584$		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
$596$	−29.1033	−	33.0000i	−1.19212	−	1.35173i
$597$		0			0
$598$		0			0
$599$			32.0000i			1.30748i	0.756717	+	0.653742i	$0.226802\pi$
$599$			32.0000i			1.30748i	−0.756717	+	0.653742i	$0.773198\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	15.8745	+	42.0000i	0.646997	+	1.71179i
$603$		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
$616$	35.0000	+	18.5203i	1.41019	+	0.746203i
$617$	−42.3320			−1.70422			−0.852111	−	0.523360i	$-0.824678\pi$
$617$	−42.3320			−1.70422			−0.852111	+	0.523360i	$0.824678\pi$
$618$		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	$-0.396829\pi$
$632$	−39.6863	−	21.0000i	−1.57864	−	0.835335i
$633$		0			0
$634$	−17.0000	−	44.9778i	−0.675156	−	1.78630i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−5.29150	−	14.0000i	−0.209493	−	0.554265i
$639$		0			0
$640$		0			0
$641$	21.1660			0.836007			0.418004	−	0.908445i	$-0.362730\pi$
$641$	21.1660			0.836007			0.418004	+	0.908445i	$0.362730\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−31.7490	+	28.0000i	−1.25109	+	1.10335i
$645$		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
$653$		−	50.0000i		−	1.95665i	−0.207072	−	0.978326i	$-0.566394\pi$
$653$		−	50.0000i		−	1.95665i	0.207072	−	0.978326i	$-0.433606\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−26.4575			−1.03064			−0.515319	−	0.856998i	$-0.672327\pi$
$659$	−26.4575			−1.03064			−0.515319	+	0.856998i	$0.672327\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−47.6235	+	18.0000i	−1.85094	+	0.699590i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.0000			0.619522
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	30.0000			1.15642			0.578208	−	0.815890i	$-0.303752\pi$
$673$	30.0000			1.15642			0.578208	+	0.815890i	$0.303752\pi$
$674$	39.6863	−	15.0000i	1.52866	−	0.577778i
$675$		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
$683$	−5.29150			−0.202474			−0.101237	−	0.994862i	$-0.532280\pi$
$683$	−5.29150			−0.202474			−0.101237	+	0.994862i	$0.532280\pi$
$684$		0			0
$685$		0			0
$686$	9.26013	+	24.5000i	0.353553	+	0.935414i
$687$		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
$694$	49.0000	−	18.5203i	1.86001	−	0.703019i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−17.5000	−	19.8431i	−0.661438	−	0.750000i
$701$			2.00000i			0.0755390i	0.999286	+	0.0377695i	$0.0120253\pi$
$701$			2.00000i			0.0755390i	−0.999286	+	0.0377695i	$0.987975\pi$
$702$		0			0
$703$		0			0
$704$	23.8118	+	35.0000i	0.897440	+	1.31911i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			52.9150i			1.98727i	0.112667	+	0.993633i	$0.464061\pi$
$709$			52.9150i			1.98727i	−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−39.6863	+	35.0000i	−1.48315	+	1.30801i
$717$		0			0
$718$	4.00000	+	10.5830i	0.149279	+	0.394954i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−25.1346	+	9.50000i	−0.935414	+	0.353553i
$723$		0			0
$724$		0			0
$725$			10.0000i			0.371391i
$726$		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
$736$	−44.0000	+	10.5830i	−1.62186	+	0.390095i
$737$	−21.1660			−0.779660
$738$		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
$742$	−35.0000	+	13.2288i	−1.28489	+	0.485643i
$743$			40.0000i			1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$			40.0000i			1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$		0			0
$745$		0			0
$746$	−15.8745	−	42.0000i	−0.581207	−	1.53773i
$747$		0			0
$748$		0			0
$749$			14.0000i			0.511549i
$750$		0			0
$751$		−	26.4575i		−	0.965448i	−0.875772	−	0.482724i	$-0.839647\pi$
$751$		−	26.4575i		−	0.965448i	0.875772	−	0.482724i	$-0.160353\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			10.5830i			0.384646i	0.981332	+	0.192323i	$0.0616021\pi$
$757$			10.5830i			0.384646i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	15.8745	−	6.00000i	0.576588	−	0.217930i
$759$		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
$764$	10.5830	+	12.0000i	0.382880	+	0.434145i
$765$		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
$778$	−19.0000	−	50.2693i	−0.681183	−	1.80224i
$779$		0			0
$780$		0			0
$781$		−	84.6640i		−	3.02952i
$782$		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			$\pi$
$788$	34.3948	+	39.0000i	1.22526	+	1.38932i
$789$		0			0
$790$		0			0
$791$			56.0000i			1.99113i
$792$		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
$800$	−6.61438	−	27.5000i	−0.233854	−	0.972272i
$801$		0			0
$802$	28.0000	−	10.5830i	0.988714	−	0.373699i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.3320			1.48831			0.744157	−	0.668004i	$-0.232851\pi$
$809$	42.3320			1.48831			0.744157	+	0.668004i	$0.232851\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	7.93725	−	7.00000i	0.278543	−	0.245652i
$813$		0			0
$814$	28.0000	+	74.0810i	0.981399	+	2.59654i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			22.0000i			0.767805i	0.923374	+	0.383903i	$0.125420\pi$
$821$			22.0000i			0.767805i	−0.923374	+	0.383903i	$0.874580\pi$
$822$		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
$827$	−37.0405			−1.28803			−0.644013	−	0.765015i	$-0.722732\pi$
$827$	−37.0405			−1.28803			−0.644013	+	0.765015i	$0.722732\pi$
$828$		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
$841$	25.0000			0.862069
$842$	−15.8745	−	42.0000i	−0.547072	−	1.44742i
$843$		0			0
$844$	18.0000	−	15.8745i	0.619586	−	0.546423i
$845$		0			0
$846$		0			0
$847$			44.9778i			1.54545i
$848$	−39.6863	−	5.00000i	−1.36283	−	0.171701i
$849$		0			0
$850$		0			0
$851$	−84.6640			−2.90225
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−7.00000	+	13.2288i	−0.239255	+	0.452150i
$857$		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
$862$	16.0000	+	42.3320i	0.544962	+	1.44183i
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	84.0000i		−	2.84950i
$870$		0			0
$871$		0			0
$872$	26.4575	+	14.0000i	0.895964	+	0.474100i
$873$		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	$-0.435283\pi$
$883$	12.0000			0.403832			0.201916	+	0.979403i	$0.435283\pi$
$884$		0			0
$885$		0			0
$886$	−49.0000	+	18.5203i	−1.64619	+	0.622200i
$887$		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
$896$	−17.1974	+	24.5000i	−0.574524	+	0.818488i
$897$		0			0
$898$	−56.0000	+	21.1660i	−1.86874	+	0.706319i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−28.0000	+	52.9150i	−0.931266	+	1.75993i
$905$		0			0
$906$		0			0
$907$	−60.0000			−1.99227			−0.996134	−	0.0878507i	$-0.972000\pi$
$907$	−60.0000			−1.99227			−0.996134	+	0.0878507i	$0.972000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	16.0000i		−	0.530104i	−0.964234	−	0.265052i	$-0.914611\pi$
$911$		−	16.0000i		−	0.530104i	0.964234	−	0.265052i	$-0.0853891\pi$
$912$		0			0
$913$		0			0
$914$	7.93725	−	3.00000i	0.262541	−	0.0992312i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	37.0405i		−	1.22185i	−0.791687	−	0.610927i	$-0.790797\pi$
$919$		−	37.0405i		−	1.22185i	0.791687	−	0.610927i	$-0.209203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	52.9150i		−	1.73984i
$926$	−7.93725	−	21.0000i	−0.260834	−	0.690103i
$927$		0			0
$928$	11.0000	−	2.64575i	0.361093	−	0.0868510i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−31.7490	+	28.0000i	−1.03997	+	0.917170i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−5.29150	−	14.0000i	−0.172774	−	0.457116i
$939$		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
$946$	−84.0000	+	31.7490i	−2.73108	+	1.03225i
$947$	58.2065			1.89146			0.945729	−	0.324956i	$-0.105350\pi$
$947$	58.2065			1.89146			0.945729	+	0.324956i	$0.105350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	21.1660			0.685634			0.342817	−	0.939402i	$-0.388619\pi$
$953$	21.1660			0.685634			0.342817	+	0.939402i	$0.388619\pi$
$954$		0			0
$955$		0			0
$956$	21.1660	+	24.0000i	0.684558	+	0.776215i
$957$		0			0
$958$		0			0
$959$			56.0000i			1.80833i
$960$		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	$0.722376\pi$
$968$	−22.4889	+	42.5000i	−0.722820	+	1.36600i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$	−18.5203	−	49.0000i	−0.593427	−	1.57006i
$975$		0			0
$976$		0			0
$977$	42.3320			1.35432			0.677161	−	0.735835i	$-0.263210\pi$
$977$	42.3320			1.35432			0.677161	+	0.735835i	$0.263210\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	7.00000	−	2.64575i	0.223379	−	0.0844293i
$983$		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
$989$		−	96.0000i		−	3.05262i
$990$		0			0
$991$		−	58.2065i		−	1.84899i	−0.381193	−	0.924496i	$-0.624487\pi$
$991$		−	58.2065i		−	1.84899i	0.381193	−	0.924496i	$-0.375513\pi$
$992$		0			0
$993$		0			0
$994$	56.0000	−	21.1660i	1.77621	−	0.671345i
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	47.6235	−	18.0000i	1.50750	−	0.569780i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.2.p.d.307.2	yes	4
3.2	odd	2	inner	504.2.p.d.307.3	yes	4
4.3	odd	2		2016.2.p.d.559.2		4
7.6	odd	2	CM	504.2.p.d.307.2	yes	4
8.3	odd	2	inner	504.2.p.d.307.1	✓	4
8.5	even	2		2016.2.p.d.559.4		4
12.11	even	2		2016.2.p.d.559.1		4
21.20	even	2	inner	504.2.p.d.307.3	yes	4
24.5	odd	2		2016.2.p.d.559.3		4
24.11	even	2	inner	504.2.p.d.307.4	yes	4
28.27	even	2		2016.2.p.d.559.2		4
56.13	odd	2		2016.2.p.d.559.4		4
56.27	even	2	inner	504.2.p.d.307.1	✓	4
84.83	odd	2		2016.2.p.d.559.1		4
168.83	odd	2	inner	504.2.p.d.307.4	yes	4
168.125	even	2		2016.2.p.d.559.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.p.d.307.1	✓	4	8.3	odd	2	inner
504.2.p.d.307.1	✓	4	56.27	even	2	inner
504.2.p.d.307.2	yes	4	1.1	even	1	trivial
504.2.p.d.307.2	yes	4	7.6	odd	2	CM
504.2.p.d.307.3	yes	4	3.2	odd	2	inner
504.2.p.d.307.3	yes	4	21.20	even	2	inner
504.2.p.d.307.4	yes	4	24.11	even	2	inner
504.2.p.d.307.4	yes	4	168.83	odd	2	inner
2016.2.p.d.559.1		4	12.11	even	2
2016.2.p.d.559.1		4	84.83	odd	2
2016.2.p.d.559.2		4	4.3	odd	2
2016.2.p.d.559.2		4	28.27	even	2
2016.2.p.d.559.3		4	24.5	odd	2
2016.2.p.d.559.3		4	168.125	even	2
2016.2.p.d.559.4		4	8.5	even	2
2016.2.p.d.559.4		4	56.13	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.32288 + 0.500000i) q^{2} +(1.50000 - 1.32288i) q^{4} +2.64575i q^{7} +(-1.32288 + 2.50000i) q^{8} -5.29150 q^{11} +(-1.32288 - 3.50000i) q^{14} +(0.500000 - 3.96863i) q^{16} +(7.00000 - 2.64575i) q^{22} +8.00000i q^{23} -5.00000 q^{25} +(3.50000 + 3.96863i) q^{28} -2.00000i q^{29} +(1.32288 + 5.50000i) q^{32} +10.5830i q^{37} -12.0000 q^{43} +(-7.93725 + 7.00000i) q^{44} +(-4.00000 - 10.5830i) q^{46} -7.00000 q^{49} +(6.61438 - 2.50000i) q^{50} -10.0000i q^{53} +(-6.61438 - 3.50000i) q^{56} +(1.00000 + 2.64575i) q^{58} +(-4.50000 - 6.61438i) q^{64} +4.00000 q^{67} +16.0000i q^{71} +(-5.29150 - 14.0000i) q^{74} -14.0000i q^{77} +15.8745i q^{79} +(15.8745 - 6.00000i) q^{86} +(7.00000 - 13.2288i) q^{88} +(10.5830 + 12.0000i) q^{92} +(9.26013 - 3.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 2 q^{16} + 28 q^{22} - 20 q^{25} + 14 q^{28} - 48 q^{43} - 16 q^{46} - 28 q^{49} + 4 q^{58} - 18 q^{64} + 16 q^{67} + 28 q^{88}+O(q^{100}) \) 4 * q + 6 * q^4 + 2 * q^16 + 28 * q^22 - 20 * q^25 + 14 * q^28 - 48 * q^43 - 16 * q^46 - 28 * q^49 + 4 * q^58 - 18 * q^64 + 16 * q^67 + 28 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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