LMFDB - Embedded newform 4212.2.i.d.2809.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4212,2,Mod(1405,4212)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4212, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("4212.1405");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.i (of order $3$, degree $2$, not 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:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2809.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	4212.2809
Dual form			4212.2.i.d.1405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{7} +(1.00000 - 1.73205i) q^{11} +(0.500000 + 0.866025i) q^{13} +6.00000 q^{17} -6.00000 q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{29} +(-5.00000 - 8.66025i) q^{31} -4.00000 q^{35} -6.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(1.00000 - 1.73205i) q^{47} +(1.50000 + 2.59808i) q^{49} +6.00000 q^{53} -4.00000 q^{55} +(5.00000 + 8.66025i) q^{59} +(1.00000 - 1.73205i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.0000 q^{71} +2.00000 q^{73} +(-2.00000 - 3.46410i) q^{77} +(2.00000 - 3.46410i) q^{79} +(3.00000 - 5.19615i) q^{83} +(-6.00000 - 10.3923i) q^{85} -6.00000 q^{89} +2.00000 q^{91} +(6.00000 + 10.3923i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} + 2 q^{11} + q^{13} + 12 q^{17} - 12 q^{19} - 8 q^{23} + q^{25} - 2 q^{29} - 10 q^{31} - 8 q^{35} - 12 q^{37} + 6 q^{41} - 4 q^{43} + 2 q^{47} + 3 q^{49} + 12 q^{53} - 8 q^{55} + 10 q^{59} + 2 q^{61} + 2 q^{65} - 10 q^{67} + 20 q^{71} + 4 q^{73} - 4 q^{77} + 4 q^{79} + 6 q^{83} - 12 q^{85} - 12 q^{89} + 4 q^{91} + 12 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 + q^13 + 12 * q^17 - 12 * q^19 - 8 * q^23 + q^25 - 2 * q^29 - 10 * q^31 - 8 * q^35 - 12 * q^37 + 6 * q^41 - 4 * q^43 + 2 * q^47 + 3 * q^49 + 12 * q^53 - 8 * q^55 + 10 * q^59 + 2 * q^61 + 2 * q^65 - 10 * q^67 + 20 * q^71 + 4 * q^73 - 4 * q^77 + 4 * q^79 + 6 * q^83 - 12 * q^85 - 12 * q^89 + 4 * q^91 + 12 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times$.

$n$	$2107$	$3485$	$3889$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	−0.612801	−	0.790237i	$-0.709957\pi$
$7$	1.00000	−	1.73205i	0.377964	−	0.654654i	0.990766	+	0.135583i	$0.0432908\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.00000	−	6.92820i	−0.834058	−	1.44463i	0.0607377	−	0.998154i	$-0.480655\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000	+	1.73205i	−0.185695	+	0.321634i	−0.943811	−	0.330487i	$-0.892787\pi$
$29$	−1.00000	+	1.73205i	−0.185695	+	0.321634i	0.758115	+	0.652121i	$0.226120\pi$
$30$		0			0
$31$	−5.00000	−	8.66025i	−0.898027	−	1.55543i	−0.830014	−	0.557743i	$-0.811667\pi$
$31$	−5.00000	−	8.66025i	−0.898027	−	1.55543i	−0.0680129	−	0.997684i	$-0.521666\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	−0.783830	−	0.620975i	$-0.786737\pi$
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	0.929695	+	0.368329i	$0.120070\pi$
$48$		0			0
$49$	1.50000	+	2.59808i	0.214286	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−4.00000			−0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	0.982894	+	0.184172i	$0.0589603\pi$
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	−0.331949	+	0.943297i	$0.607706\pi$
$60$		0			0
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	0.922916	+	0.385002i	$0.125799\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000	−	1.73205i	0.124035	−	0.214834i
$66$		0			0
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	−0.991098	−	0.133135i	$-0.957496\pi$
$67$	−5.00000	−	8.66025i	−0.610847	−	1.05802i	0.380251	−	0.924883i	$-0.375838\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.0000			1.18678			0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000			1.18678			0.593391	+	0.804914i	$0.297789\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	−	3.46410i	−0.227921	−	0.394771i
$78$		0			0
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	2.00000	−	3.46410i	0.225018	−	0.389742i	0.956325	+	0.292306i	$0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		0			0
$85$	−6.00000	−	10.3923i	−0.650791	−	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	+	10.3923i	0.615587	+	1.06623i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.0000			−1.54678			−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000			−1.54678			−0.773389	+	0.633932i	$0.781440\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.00000	−	12.1244i	−0.658505	−	1.14056i	−0.981003	−	0.193993i	$-0.937856\pi$
$113$	−7.00000	−	12.1244i	−0.658505	−	1.14056i	0.322498	−	0.946570i	$-0.395477\pi$
$114$		0			0
$115$	−8.00000	+	13.8564i	−0.746004	+	1.29212i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	−	10.3923i	0.550019	−	0.952661i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00000	+	13.8564i	0.698963	+	1.21064i	0.968826	+	0.247741i	$0.0796882\pi$
$131$	8.00000	+	13.8564i	0.698963	+	1.21064i	−0.269863	+	0.962899i	$0.586978\pi$
$132$		0			0
$133$	−6.00000	+	10.3923i	−0.520266	+	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	0.169226	+	0.985577i	$0.445873\pi$
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	−0.938148	+	0.346235i	$0.887460\pi$
$138$		0			0
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000			0.167248
$144$		0			0
$145$	4.00000			0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	−0.953703	−	0.300750i	$-0.902763\pi$
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	0.216394	−	0.976306i	$-0.430570\pi$
$150$		0			0
$151$	−3.00000	+	5.19615i	−0.244137	+	0.422857i	−0.961888	−	0.273442i	$-0.911838\pi$
$151$	−3.00000	+	5.19615i	−0.244137	+	0.422857i	0.717752	+	0.696299i	$0.245171\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.0000	+	17.3205i	−0.803219	+	1.39122i
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.0000			−1.26098
$162$		0			0
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	0.726293	−	0.687386i	$-0.241242\pi$
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	−0.958440	+	0.285295i	$0.907908\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.0384615	+	0.0666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.00000	−	8.66025i	0.380143	−	0.658427i	−0.610939	−	0.791677i	$-0.709208\pi$
$173$	5.00000	−	8.66025i	0.380143	−	0.658427i	0.991082	+	0.133250i	$0.0425415\pi$
$174$		0			0
$175$	−1.00000	−	1.73205i	−0.0755929	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000	+	10.3923i	0.441129	+	0.764057i
$186$		0			0
$187$	6.00000	−	10.3923i	0.438763	−	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	−0.929267	−	0.369410i	$-0.879560\pi$
$191$	−2.00000	+	3.46410i	−0.144715	+	0.250654i	0.784552	+	0.620063i	$0.212893\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.00000	+	3.46410i	0.140372	+	0.243132i
$204$		0			0
$205$	6.00000	−	10.3923i	0.419058	−	0.725830i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000	+	10.3923i	−0.415029	+	0.718851i
$210$		0			0
$211$	−4.00000	−	6.92820i	−0.275371	−	0.476957i	0.694857	−	0.719148i	$-0.255467\pi$
$211$	−4.00000	−	6.92820i	−0.275371	−	0.476957i	−0.970229	+	0.242190i	$0.922134\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	−20.0000			−1.35769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000	+	5.19615i	0.201802	+	0.349531i
$222$		0			0
$223$	3.00000	−	5.19615i	0.200895	−	0.347960i	−0.747922	−	0.663786i	$-0.768948\pi$
$223$	3.00000	−	5.19615i	0.200895	−	0.347960i	0.948817	+	0.315826i	$0.102282\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	0.395860	+	0.918311i	$0.370447\pi$
$227$	−9.00000	+	15.5885i	−0.597351	+	1.03464i	−0.993210	+	0.116331i	$0.962887\pi$
$228$		0			0
$229$	−9.00000	−	15.5885i	−0.594737	−	1.03011i	−0.993584	−	0.113097i	$-0.963923\pi$
$229$	−9.00000	−	15.5885i	−0.594737	−	1.03011i	0.398847	−	0.917017i	$-0.369410\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.0000			0.655122			0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000			0.655122			0.327561	+	0.944830i	$0.393773\pi$
$234$		0			0
$235$	−4.00000			−0.260931
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−9.00000	−	15.5885i	−0.582162	−	1.00833i	−0.995223	−	0.0976302i	$-0.968874\pi$
$239$	−9.00000	−	15.5885i	−0.582162	−	1.00833i	0.413061	−	0.910703i	$-0.364460\pi$
$240$		0			0
$241$	3.00000	−	5.19615i	0.193247	−	0.334714i	−0.753077	−	0.657932i	$-0.771431\pi$
$241$	3.00000	−	5.19615i	0.193247	−	0.334714i	0.946324	+	0.323218i	$0.104765\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.00000	−	5.19615i	0.191663	−	0.331970i
$246$		0			0
$247$	−3.00000	−	5.19615i	−0.190885	−	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	−0.997374	−	0.0724199i	$-0.976928\pi$
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	0.435970	−	0.899961i	$-0.356405\pi$
$258$		0			0
$259$	−6.00000	+	10.3923i	−0.372822	+	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	−0.619586	−	0.784929i	$-0.712699\pi$
$263$	6.00000	−	10.3923i	0.369976	−	0.640817i	0.989561	+	0.144112i	$0.0460326\pi$
$264$		0			0
$265$	−6.00000	−	10.3923i	−0.368577	−	0.638394i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	−15.0000	+	25.9808i	−0.901263	+	1.56103i	−0.0754058	+	0.997153i	$0.524025\pi$
$277$	−15.0000	+	25.9808i	−0.901263	+	1.56103i	−0.825857	+	0.563880i	$0.809308\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.00000	−	12.1244i	0.417585	−	0.723278i	−0.578111	−	0.815958i	$-0.696210\pi$
$281$	7.00000	−	12.1244i	0.417585	−	0.723278i	0.995696	+	0.0926797i	$0.0295433\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.00000	+	12.1244i	0.408944	+	0.708312i	0.994772	−	0.102123i	$-0.0325637\pi$
$293$	7.00000	+	12.1244i	0.408944	+	0.708312i	−0.585827	+	0.810436i	$0.699230\pi$
$294$		0			0
$295$	10.0000	−	17.3205i	0.582223	−	1.00844i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.00000	−	6.92820i	0.231326	−	0.400668i
$300$		0			0
$301$	4.00000	+	6.92820i	0.230556	+	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000			−0.229039
$306$		0			0
$307$	−34.0000			−1.94048			−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000			−1.94048			−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	−0.689412	−	0.724370i	$-0.742131\pi$
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	0.972028	+	0.234863i	$0.0754642\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	2.00000	+	3.46410i	0.111979	+	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−36.0000			−2.00309
$324$		0			0
$325$	1.00000			0.0554700
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.00000	−	3.46410i	−0.110264	−	0.190982i
$330$		0			0
$331$	11.0000	−	19.0526i	0.604615	−	1.04722i	−0.387498	−	0.921871i	$-0.626660\pi$
$331$	11.0000	−	19.0526i	0.604615	−	1.04722i	0.992112	−	0.125353i	$-0.0400062\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.0000	+	17.3205i	−0.546358	+	0.946320i
$336$		0			0
$337$	5.00000	+	8.66025i	0.272367	+	0.471754i	0.969468	−	0.245220i	$-0.0788601\pi$
$337$	5.00000	+	8.66025i	0.272367	+	0.471754i	−0.697100	+	0.716974i	$0.745527\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−20.0000			−1.08306
$342$		0			0
$343$	20.0000			1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	−5.00000	+	8.66025i	−0.267644	+	0.463573i	−0.968253	−	0.249973i	$-0.919578\pi$
$349$	−5.00000	+	8.66025i	−0.267644	+	0.463573i	0.700609	+	0.713545i	$0.252912\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	−0.617388	−	0.786659i	$-0.711809\pi$
$353$	7.00000	−	12.1244i	0.372572	−	0.645314i	0.989960	+	0.141344i	$0.0451425\pi$
$354$		0			0
$355$	−10.0000	−	17.3205i	−0.530745	−	0.919277i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	30.0000			1.58334			0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000			1.58334			0.791670	+	0.610949i	$0.209212\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.00000	−	3.46410i	−0.104685	−	0.181319i
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000	−	10.3923i	0.311504	−	0.539542i
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.00000			−0.103005
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	0.539076	−	0.842257i	$-0.318774\pi$
$383$	−9.00000	−	15.5885i	−0.459879	−	0.796533i	−0.998954	+	0.0457244i	$0.985440\pi$
$384$		0			0
$385$	−4.00000	+	6.92820i	−0.203859	+	0.353094i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$	−24.0000	−	41.5692i	−1.21373	−	2.10225i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.00000			−0.402524
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$	5.00000	−	8.66025i	0.249068	−	0.431398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.00000	+	10.3923i	−0.297409	+	0.515127i
$408$		0			0
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	0.985558	−	0.169338i	$-0.0541630\pi$
$409$	7.00000	+	12.1244i	0.346128	+	0.599511i	−0.639430	+	0.768849i	$0.720830\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	20.0000			0.984136
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000	−	20.7846i	−0.586238	−	1.01539i	−0.994720	−	0.102628i	$-0.967275\pi$
$419$	−12.0000	−	20.7846i	−0.586238	−	1.01539i	0.408481	−	0.912767i	$-0.366058\pi$
$420$		0			0
$421$	−5.00000	+	8.66025i	−0.243685	+	0.422075i	−0.961761	−	0.273890i	$-0.911690\pi$
$421$	−5.00000	+	8.66025i	−0.243685	+	0.422075i	0.718076	+	0.695965i	$0.245023\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.00000	−	5.19615i	0.145521	−	0.252050i
$426$		0			0
$427$	−2.00000	−	3.46410i	−0.0967868	−	0.167640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−30.0000			−1.44505			−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000			−1.44505			−0.722525	+	0.691345i	$0.757018\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.0000	+	41.5692i	1.14808	+	1.98853i
$438$		0			0
$439$	8.00000	−	13.8564i	0.381819	−	0.661330i	−0.609503	−	0.792784i	$-0.708631\pi$
$439$	8.00000	−	13.8564i	0.381819	−	0.661330i	0.991322	+	0.131453i	$0.0419644\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	−0.426414	−	0.904528i	$-0.640223\pi$
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	0.996551	−	0.0829786i	$-0.0264433\pi$
$444$		0			0
$445$	6.00000	+	10.3923i	0.284427	+	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−38.0000			−1.79333			−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000			−1.79333			−0.896665	+	0.442709i	$0.854018\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.00000	−	3.46410i	−0.0937614	−	0.162400i
$456$		0			0
$457$	−17.0000	+	29.4449i	−0.795226	+	1.37737i	0.127469	+	0.991843i	$0.459315\pi$
$457$	−17.0000	+	29.4449i	−0.795226	+	1.37737i	−0.922695	+	0.385530i	$0.874019\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	−0.787668	−	0.616100i	$-0.788712\pi$
$461$	3.00000	−	5.19615i	0.139724	−	0.242009i	0.927392	+	0.374091i	$0.122045\pi$
$462$		0			0
$463$	−11.0000	−	19.0526i	−0.511213	−	0.885448i	−0.999916	−	0.0129968i	$-0.995863\pi$
$463$	−11.0000	−	19.0526i	−0.511213	−	0.885448i	0.488702	−	0.872451i	$-0.337470\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.00000	+	6.92820i	0.183920	+	0.318559i
$474$		0			0
$475$	−3.00000	+	5.19615i	−0.137649	+	0.238416i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	−0.583803	−	0.811895i	$-0.698436\pi$
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	0.995023	+	0.0996406i	$0.0317693\pi$
$480$		0			0
$481$	−3.00000	−	5.19615i	−0.136788	−	0.236924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000			0.181631
$486$		0			0
$487$	2.00000			0.0906287			0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000			0.0906287			0.0453143	+	0.998973i	$0.485571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.0000	+	17.3205i	0.451294	+	0.781664i	0.998467	−	0.0553560i	$-0.0176294\pi$
$491$	10.0000	+	17.3205i	0.451294	+	0.781664i	−0.547173	+	0.837020i	$0.684296\pi$
$492$		0			0
$493$	−6.00000	+	10.3923i	−0.270226	+	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	10.0000	−	17.3205i	0.448561	−	0.776931i
$498$		0			0
$499$	7.00000	+	12.1244i	0.313363	+	0.542761i	0.979088	−	0.203436i	$-0.0652110\pi$
$499$	7.00000	+	12.1244i	0.313363	+	0.542761i	−0.665725	+	0.746197i	$0.731878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	28.0000			1.24846			0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000			1.24846			0.624229	+	0.781241i	$0.285413\pi$
$504$		0			0
$505$	−4.00000			−0.177998
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	0.978421	−	0.206623i	$-0.0662474\pi$
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	−0.668151	+	0.744026i	$0.732914\pi$
$510$		0			0
$511$	2.00000	−	3.46410i	0.0884748	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$	−2.00000	−	3.46410i	−0.0879599	−	0.152351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	24.0000			1.04945			0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000			1.04945			0.524723	+	0.851273i	$0.324169\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−30.0000	−	51.9615i	−1.30682	−	2.26348i
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.00000	+	5.19615i	−0.129944	+	0.225070i
$534$		0			0
$535$	16.0000	+	27.7128i	0.691740	+	1.19813i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	26.0000			1.11783			0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000			1.11783			0.558914	+	0.829226i	$0.311218\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000	+	24.2487i	0.599694	+	1.03870i
$546$		0			0
$547$	16.0000	−	27.7128i	0.684111	−	1.18491i	−0.289605	−	0.957146i	$-0.593524\pi$
$547$	16.0000	−	27.7128i	0.684111	−	1.18491i	0.973715	−	0.227768i	$-0.0731428\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.00000	−	10.3923i	0.255609	−	0.442727i
$552$		0			0
$553$	−4.00000	−	6.92820i	−0.170097	−	0.294617i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	42.0000			1.77960			0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000			1.77960			0.889799	+	0.456354i	$0.150845\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	0.905088	−	0.425223i	$-0.139804\pi$
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	−0.820798	+	0.571218i	$0.806471\pi$
$564$		0			0
$565$	−14.0000	+	24.2487i	−0.588984	+	1.02015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	−0.951592	−	0.307364i	$-0.900553\pi$
$569$	−5.00000	+	8.66025i	−0.209611	+	0.363057i	0.741981	+	0.670421i	$0.233886\pi$
$570$		0			0
$571$	6.00000	+	10.3923i	0.251092	+	0.434904i	0.963827	−	0.266529i	$-0.0858769\pi$
$571$	6.00000	+	10.3923i	0.251092	+	0.434904i	−0.712735	+	0.701434i	$0.752544\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.00000			−0.333623
$576$		0			0
$577$	−6.00000			−0.249783			−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000			−0.249783			−0.124892	+	0.992170i	$0.539858\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.00000	−	10.3923i	−0.248922	−	0.431145i
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.0000	+	22.5167i	−0.536567	+	0.929362i	0.462518	+	0.886610i	$0.346946\pi$
$587$	−13.0000	+	22.5167i	−0.536567	+	0.929362i	−0.999086	+	0.0427523i	$0.986387\pi$
$588$		0			0
$589$	30.0000	+	51.9615i	1.23613	+	2.14104i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26.0000			1.06769			0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000			1.06769			0.533846	+	0.845582i	$0.320746\pi$
$594$		0			0
$595$	−24.0000			−0.983904
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.00000	−	3.46410i	−0.0817178	−	0.141539i	0.822270	−	0.569097i	$-0.192707\pi$
$599$	−2.00000	−	3.46410i	−0.0817178	−	0.141539i	−0.903988	+	0.427558i	$0.859374\pi$
$600$		0			0
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	6.00000	+	10.3923i	0.243532	+	0.421811i	0.961718	−	0.274041i	$-0.0883604\pi$
$607$	6.00000	+	10.3923i	0.243532	+	0.421811i	−0.718186	+	0.695852i	$0.755027\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.00000			0.0809113
$612$		0			0
$613$	26.0000			1.05013			0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000			1.05013			0.525065	+	0.851062i	$0.324041\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17.0000	−	29.4449i	−0.684394	−	1.18541i	−0.973627	−	0.228147i	$-0.926733\pi$
$617$	−17.0000	−	29.4449i	−0.684394	−	1.18541i	0.289233	−	0.957259i	$-0.406600\pi$
$618$		0			0
$619$	1.00000	−	1.73205i	0.0401934	−	0.0696170i	−0.845229	−	0.534404i	$-0.820536\pi$
$619$	1.00000	−	1.73205i	0.0401934	−	0.0696170i	0.885422	+	0.464787i	$0.153869\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.00000	+	10.3923i	−0.240385	+	0.416359i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−36.0000			−1.43541
$630$		0			0
$631$	22.0000			0.875806			0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000			0.875806			0.437903	+	0.899022i	$0.355721\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	+	13.8564i	0.317470	+	0.549875i
$636$		0			0
$637$	−1.50000	+	2.59808i	−0.0594322	+	0.102940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	−0.486409	−	0.873731i	$-0.661693\pi$
$641$	13.0000	−	22.5167i	0.513469	−	0.889355i	0.999878	−	0.0156233i	$-0.00497325\pi$
$642$		0			0
$643$	−1.00000	−	1.73205i	−0.0394362	−	0.0683054i	0.845634	−	0.533764i	$-0.179223\pi$
$643$	−1.00000	−	1.73205i	−0.0394362	−	0.0683054i	−0.885070	+	0.465458i	$0.845890\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−32.0000			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$648$		0			0
$649$	20.0000			0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.00000	−	15.5885i	−0.352197	−	0.610023i	0.634437	−	0.772975i	$-0.281232\pi$
$653$	−9.00000	−	15.5885i	−0.352197	−	0.610023i	−0.986634	+	0.162951i	$0.947899\pi$
$654$		0			0
$655$	16.0000	−	27.7128i	0.625172	−	1.08283i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	0.845922	−	0.533306i	$-0.179051\pi$
$661$	−1.00000	−	1.73205i	−0.0388955	−	0.0673690i	−0.884818	+	0.465937i	$0.845717\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	24.0000			0.930680
$666$		0			0
$667$	16.0000			0.619522
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.00000	−	3.46410i	−0.0772091	−	0.133730i
$672$		0			0
$673$	13.0000	−	22.5167i	0.501113	−	0.867953i	−0.498886	−	0.866668i	$-0.666257\pi$
$673$	13.0000	−	22.5167i	0.501113	−	0.867953i	0.999999	−	0.00128586i	$-0.000409302\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.00000	+	5.19615i	−0.115299	+	0.199704i	−0.917899	−	0.396813i	$-0.870116\pi$
$677$	−3.00000	+	5.19615i	−0.115299	+	0.199704i	0.802600	+	0.596518i	$0.203449\pi$
$678$		0			0
$679$	2.00000	+	3.46410i	0.0767530	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.00000			−0.0765279			−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000			−0.0765279			−0.0382639	+	0.999268i	$0.512183\pi$
$684$		0			0
$685$	36.0000			1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	−17.0000	+	29.4449i	−0.646710	+	1.12014i	0.337193	+	0.941435i	$0.390522\pi$
$691$	−17.0000	+	29.4449i	−0.646710	+	1.12014i	−0.983904	+	0.178700i	$0.942811\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−16.0000	+	27.7128i	−0.606915	+	1.05121i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	36.0000			1.35777
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.00000	−	3.46410i	−0.0752177	−	0.130281i
$708$		0			0
$709$	11.0000	−	19.0526i	0.413114	−	0.715534i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	11.0000	−	19.0526i	0.413114	−	0.715534i	0.995228	+	0.0975728i	$0.0311079\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−40.0000	+	69.2820i	−1.49801	+	2.59463i
$714$		0			0
$715$	−2.00000	−	3.46410i	−0.0747958	−	0.129550i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.0000			−1.49175			−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000			−1.49175			−0.745874	+	0.666087i	$0.767968\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	16.0000	−	27.7128i	0.593407	−	1.02781i	−0.400362	−	0.916357i	$-0.631116\pi$
$727$	16.0000	−	27.7128i	0.593407	−	1.02781i	0.993770	−	0.111454i	$-0.0355509\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$		0			0
$733$	−13.0000	−	22.5167i	−0.480166	−	0.831672i	0.519575	−	0.854425i	$-0.326090\pi$
$733$	−13.0000	−	22.5167i	−0.480166	−	0.831672i	−0.999741	+	0.0227529i	$0.992757\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−20.0000			−0.736709
$738$		0			0
$739$	6.00000			0.220714			0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000			0.220714			0.110357	+	0.993892i	$0.464801\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.00000	+	12.1244i	0.256805	+	0.444799i	0.965384	−	0.260832i	$-0.0839968\pi$
$743$	7.00000	+	12.1244i	0.256805	+	0.444799i	−0.708579	+	0.705631i	$0.750663\pi$
$744$		0			0
$745$	−18.0000	+	31.1769i	−0.659469	+	1.14223i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16.0000	+	27.7128i	−0.584627	+	1.01260i
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	0.655515	−	0.755182i	$-0.272452\pi$
$761$	−9.00000	−	15.5885i	−0.326250	−	0.565081i	−0.981764	+	0.190101i	$0.939118\pi$
$762$		0			0
$763$	−14.0000	+	24.2487i	−0.506834	+	0.877862i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.00000	+	8.66025i	−0.180540	+	0.312704i
$768$		0			0
$769$	27.0000	+	46.7654i	0.973645	+	1.68640i	0.684336	+	0.729167i	$0.260092\pi$
$769$	27.0000	+	46.7654i	0.973645	+	1.68640i	0.289309	+	0.957236i	$0.406575\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−46.0000			−1.65451			−0.827253	−	0.561830i	$-0.810097\pi$
$773$	−46.0000			−1.65451			−0.827253	+	0.561830i	$0.810097\pi$
$774$		0			0
$775$	−10.0000			−0.359211
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.0000	−	31.1769i	−0.644917	−	1.11703i
$780$		0			0
$781$	10.0000	−	17.3205i	0.357828	−	0.619777i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.00000	+	3.46410i	−0.0713831	+	0.123639i
$786$		0			0
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	−0.975798	−	0.218675i	$-0.929827\pi$
$787$	−19.0000	−	32.9090i	−0.677277	−	1.17308i	0.298521	−	0.954403i	$-0.403507\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−28.0000			−0.995565
$792$		0			0
$793$	2.00000			0.0710221
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	0.661441	−	0.749997i	$-0.269945\pi$
$797$	−9.00000	−	15.5885i	−0.318796	−	0.552171i	−0.980237	+	0.197826i	$0.936612\pi$
$798$		0			0
$799$	6.00000	−	10.3923i	0.212265	−	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000	−	3.46410i	0.0705785	−	0.122245i
$804$		0			0
$805$	16.0000	+	27.7128i	0.563926	+	0.976748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.0000	+	17.3205i	0.350285	+	0.606711i
$816$		0			0
$817$	12.0000	−	20.7846i	0.419827	−	0.727161i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	−0.882948	−	0.469471i	$-0.844445\pi$
$821$	−1.00000	+	1.73205i	−0.0349002	+	0.0604490i	0.848048	+	0.529920i	$0.177778\pi$
$822$		0			0
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	0.995768	−	0.0919029i	$-0.0292950\pi$
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	−0.577474	+	0.816409i	$0.695962\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.00000			0.208640			0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000			0.208640			0.104320	+	0.994544i	$0.466733\pi$
$828$		0			0
$829$	22.0000			0.764092			0.382046	−	0.924143i	$-0.375220\pi$
$829$	22.0000			0.764092			0.382046	+	0.924143i	$0.375220\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.00000	+	15.5885i	0.311832	+	0.540108i
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.00000	−	8.66025i	0.172619	−	0.298985i	−0.766716	−	0.641987i	$-0.778110\pi$
$839$	5.00000	−	8.66025i	0.172619	−	0.298985i	0.939335	+	0.343002i	$0.111444\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.00000			0.0688021
$846$		0			0
$847$	14.0000			0.481046
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24.0000	+	41.5692i	0.822709	+	1.42497i
$852$		0			0
$853$	19.0000	−	32.9090i	0.650548	−	1.12678i	−0.332443	−	0.943123i	$-0.607873\pi$
$853$	19.0000	−	32.9090i	0.650548	−	1.12678i	0.982990	−	0.183658i	$-0.0587939\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.0000	−	32.9090i	0.649028	−	1.12415i	−0.334328	−	0.942457i	$-0.608509\pi$
$857$	19.0000	−	32.9090i	0.649028	−	1.12415i	0.983355	−	0.181692i	$-0.0581574\pi$
$858$		0			0
$859$	−4.00000	−	6.92820i	−0.136478	−	0.236387i	0.789683	−	0.613515i	$-0.210245\pi$
$859$	−4.00000	−	6.92820i	−0.136478	−	0.236387i	−0.926161	+	0.377128i	$0.876912\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.0000			1.15737			0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000			1.15737			0.578687	+	0.815550i	$0.303565\pi$
$864$		0			0
$865$	−20.0000			−0.680020
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.00000	−	6.92820i	−0.135691	−	0.235023i
$870$		0			0
$871$	5.00000	−	8.66025i	0.169419	−	0.293442i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.0000	+	20.7846i	−0.405674	+	0.702648i
$876$		0			0
$877$	−9.00000	−	15.5885i	−0.303908	−	0.526385i	0.673109	−	0.739543i	$-0.264958\pi$
$877$	−9.00000	−	15.5885i	−0.303908	−	0.526385i	−0.977018	+	0.213158i	$0.931625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2.00000	−	3.46410i	−0.0671534	−	0.116313i	0.830494	−	0.557028i	$-0.188058\pi$
$887$	−2.00000	−	3.46410i	−0.0671534	−	0.116313i	−0.897647	+	0.440715i	$0.854725\pi$
$888$		0			0
$889$	−8.00000	+	13.8564i	−0.268311	+	0.464729i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.00000	+	10.3923i	−0.200782	+	0.347765i
$894$		0			0
$895$	−12.0000	−	20.7846i	−0.401116	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	20.0000			0.667037
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.00000	+	10.3923i	0.199447	+	0.345452i
$906$		0			0
$907$	−18.0000	+	31.1769i	−0.597680	+	1.03521i	0.395482	+	0.918474i	$0.370577\pi$
$907$	−18.0000	+	31.1769i	−0.597680	+	1.03521i	−0.993163	+	0.116739i	$0.962756\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.0000	−	31.1769i	0.596367	−	1.03294i	−0.396986	−	0.917825i	$-0.629944\pi$
$911$	18.0000	−	31.1769i	0.596367	−	1.03294i	0.993352	−	0.115113i	$-0.0367229\pi$
$912$		0			0
$913$	−6.00000	−	10.3923i	−0.198571	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	32.0000			1.05673
$918$		0			0
$919$	−12.0000			−0.395843			−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000			−0.395843			−0.197922	+	0.980218i	$0.563419\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5.00000	+	8.66025i	0.164577	+	0.285056i
$924$		0			0
$925$	−3.00000	+	5.19615i	−0.0986394	+	0.170848i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	27.0000	−	46.7654i	0.885841	−	1.53432i	0.0410949	−	0.999155i	$-0.486915\pi$
$929$	27.0000	−	46.7654i	0.885841	−	1.53432i	0.844746	−	0.535167i	$-0.179751\pi$
$930$		0			0
$931$	−9.00000	−	15.5885i	−0.294963	−	0.510891i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−24.0000			−0.784884
$936$		0			0
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−5.00000	−	8.66025i	−0.162995	−	0.282316i	0.772946	−	0.634472i	$-0.218782\pi$
$941$	−5.00000	−	8.66025i	−0.162995	−	0.282316i	−0.935942	+	0.352155i	$0.885449\pi$
$942$		0			0
$943$	24.0000	−	41.5692i	0.781548	−	1.35368i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21.0000	−	36.3731i	0.682408	−	1.18197i	−0.291835	−	0.956469i	$-0.594266\pi$
$947$	21.0000	−	36.3731i	0.682408	−	1.18197i	0.974244	−	0.225497i	$-0.0724007\pi$
$948$		0			0
$949$	1.00000	+	1.73205i	0.0324614	+	0.0562247i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	8.00000			0.258874
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.0000	+	31.1769i	0.581250	+	1.00676i
$960$		0			0
$961$	−34.5000	+	59.7558i	−1.11290	+	1.92760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.00000	+	3.46410i	−0.0643823	+	0.111513i
$966$		0			0
$967$	−25.0000	−	43.3013i	−0.803946	−	1.39247i	−0.917000	−	0.398886i	$-0.869397\pi$
$967$	−25.0000	−	43.3013i	−0.803946	−	1.39247i	0.113055	−	0.993589i	$-0.463936\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−48.0000			−1.54039			−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000			−1.54039			−0.770197	+	0.637806i	$0.779842\pi$
$972$		0			0
$973$	−32.0000			−1.02587
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	0.685381	−	0.728184i	$-0.259636\pi$
$977$	−9.00000	−	15.5885i	−0.287936	−	0.498719i	−0.973317	+	0.229465i	$0.926302\pi$
$978$		0			0
$979$	−6.00000	+	10.3923i	−0.191761	+	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.00000	+	12.1244i	−0.223265	+	0.386707i	−0.955798	−	0.294025i	$-0.905005\pi$
$983$	−7.00000	+	12.1244i	−0.223265	+	0.386707i	0.732532	+	0.680732i	$0.238338\pi$
$984$		0			0
$985$	6.00000	+	10.3923i	0.191176	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	12.0000			0.381193			0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000			0.381193			0.190596	+	0.981669i	$0.438958\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.0000	+	27.7128i	0.507234	+	0.878555i
$996$		0			0
$997$	29.0000	−	50.2295i	0.918439	−	1.59078i	0.116653	−	0.993173i	$-0.462784\pi$
$997$	29.0000	−	50.2295i	0.918439	−	1.59078i	0.801786	−	0.597611i	$-0.203883\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.i.d.2809.1		2
3.2	odd	2		4212.2.i.i.2809.1		2
9.2	odd	6		468.2.a.b.1.1		1
9.4	even	3	inner	4212.2.i.d.1405.1		2
9.5	odd	6		4212.2.i.i.1405.1		2
9.7	even	3		52.2.a.a.1.1	✓	1
36.7	odd	6		208.2.a.c.1.1		1
36.11	even	6		1872.2.a.f.1.1		1
45.7	odd	12		1300.2.c.c.1249.1		2
45.34	even	6		1300.2.a.d.1.1		1
45.43	odd	12		1300.2.c.c.1249.2		2
63.16	even	3		2548.2.j.e.1145.1		2
63.25	even	3		2548.2.j.e.1353.1		2
63.34	odd	6		2548.2.a.e.1.1		1
63.52	odd	6		2548.2.j.f.1353.1		2
63.61	odd	6		2548.2.j.f.1145.1		2
72.11	even	6		7488.2.a.bw.1.1		1
72.29	odd	6		7488.2.a.bn.1.1		1
72.43	odd	6		832.2.a.f.1.1		1
72.61	even	6		832.2.a.e.1.1		1
99.43	odd	6		6292.2.a.g.1.1		1
117.7	odd	12		676.2.h.c.361.2		4
117.16	even	3		676.2.e.c.529.1		2
117.25	even	6		676.2.a.c.1.1		1
117.34	odd	12		676.2.d.c.337.2		2
117.38	odd	6		6084.2.a.m.1.1		1
117.43	even	6		676.2.e.b.653.1		2
117.47	even	12		6084.2.b.m.4393.1		2
117.61	even	3		676.2.e.c.653.1		2
117.70	odd	12		676.2.d.c.337.1		2
117.83	even	12		6084.2.b.m.4393.2		2
117.88	even	6		676.2.e.b.529.1		2
117.97	odd	12		676.2.h.c.361.1		4
117.106	odd	12		676.2.h.c.485.1		4
117.115	odd	12		676.2.h.c.485.2		4
144.43	odd	12		3328.2.b.e.1665.2		2
144.61	even	12		3328.2.b.q.1665.1		2
144.115	odd	12		3328.2.b.e.1665.1		2
144.133	even	12		3328.2.b.q.1665.2		2
180.79	odd	6		5200.2.a.q.1.1		1
468.151	even	12		2704.2.f.f.337.2		2
468.187	even	12		2704.2.f.f.337.1		2
468.259	odd	6		2704.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.a.a.1.1	✓	1	9.7	even	3
208.2.a.c.1.1		1	36.7	odd	6
468.2.a.b.1.1		1	9.2	odd	6
676.2.a.c.1.1		1	117.25	even	6
676.2.d.c.337.1		2	117.70	odd	12
676.2.d.c.337.2		2	117.34	odd	12
676.2.e.b.529.1		2	117.88	even	6
676.2.e.b.653.1		2	117.43	even	6
676.2.e.c.529.1		2	117.16	even	3
676.2.e.c.653.1		2	117.61	even	3
676.2.h.c.361.1		4	117.97	odd	12
676.2.h.c.361.2		4	117.7	odd	12
676.2.h.c.485.1		4	117.106	odd	12
676.2.h.c.485.2		4	117.115	odd	12
832.2.a.e.1.1		1	72.61	even	6
832.2.a.f.1.1		1	72.43	odd	6
1300.2.a.d.1.1		1	45.34	even	6
1300.2.c.c.1249.1		2	45.7	odd	12
1300.2.c.c.1249.2		2	45.43	odd	12
1872.2.a.f.1.1		1	36.11	even	6
2548.2.a.e.1.1		1	63.34	odd	6
2548.2.j.e.1145.1		2	63.16	even	3
2548.2.j.e.1353.1		2	63.25	even	3
2548.2.j.f.1145.1		2	63.61	odd	6
2548.2.j.f.1353.1		2	63.52	odd	6
2704.2.a.g.1.1		1	468.259	odd	6
2704.2.f.f.337.1		2	468.187	even	12
2704.2.f.f.337.2		2	468.151	even	12
3328.2.b.e.1665.1		2	144.115	odd	12
3328.2.b.e.1665.2		2	144.43	odd	12
3328.2.b.q.1665.1		2	144.61	even	12
3328.2.b.q.1665.2		2	144.133	even	12
4212.2.i.d.1405.1		2	9.4	even	3	inner
4212.2.i.d.2809.1		2	1.1	even	1	trivial
4212.2.i.i.1405.1		2	9.5	odd	6
4212.2.i.i.2809.1		2	3.2	odd	2
5200.2.a.q.1.1		1	180.79	odd	6
6084.2.a.m.1.1		1	117.38	odd	6
6084.2.b.m.4393.1		2	117.47	even	12
6084.2.b.m.4393.2		2	117.83	even	12
6292.2.a.g.1.1		1	99.43	odd	6
7488.2.a.bn.1.1		1	72.29	odd	6
7488.2.a.bw.1.1		1	72.11	even	6

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:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{5} +(1.00000 - 1.73205i) q^{7} +(1.00000 - 1.73205i) q^{11} +(0.500000 + 0.866025i) q^{13} +6.00000 q^{17} -6.00000 q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{29} +(-5.00000 - 8.66025i) q^{31} -4.00000 q^{35} -6.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(1.00000 - 1.73205i) q^{47} +(1.50000 + 2.59808i) q^{49} +6.00000 q^{53} -4.00000 q^{55} +(5.00000 + 8.66025i) q^{59} +(1.00000 - 1.73205i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-5.00000 - 8.66025i) q^{67} +10.0000 q^{71} +2.00000 q^{73} +(-2.00000 - 3.46410i) q^{77} +(2.00000 - 3.46410i) q^{79} +(3.00000 - 5.19615i) q^{83} +(-6.00000 - 10.3923i) q^{85} -6.00000 q^{89} +2.00000 q^{91} +(6.00000 + 10.3923i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + 2 q^{11} + q^{13} + 12 q^{17} - 12 q^{19} - 8 q^{23} + q^{25} - 2 q^{29} - 10 q^{31} - 8 q^{35} - 12 q^{37} + 6 q^{41} - 4 q^{43} + 2 q^{47} + 3 q^{49} + 12 q^{53} - 8 q^{55} + 10 q^{59} + 2 q^{61} + 2 q^{65} - 10 q^{67} + 20 q^{71} + 4 q^{73} - 4 q^{77} + 4 q^{79} + 6 q^{83} - 12 q^{85} - 12 q^{89} + 4 q^{91} + 12 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 2 * q^11 + q^13 + 12 * q^17 - 12 * q^19 - 8 * q^23 + q^25 - 2 * q^29 - 10 * q^31 - 8 * q^35 - 12 * q^37 + 6 * q^41 - 4 * q^43 + 2 * q^47 + 3 * q^49 + 12 * q^53 - 8 * q^55 + 10 * q^59 + 2 * q^61 + 2 * q^65 - 10 * q^67 + 20 * q^71 + 4 * q^73 - 4 * q^77 + 4 * q^79 + 6 * q^83 - 12 * q^85 - 12 * q^89 + 4 * q^91 + 12 * q^95 - 2 * q^97

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