LMFDB - Embedded newform 784.2.i.m.753.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(177,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("784.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 98)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			753.1
Root			$-0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	784.753
Dual form			784.2.i.m.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.22474i) q^{3} +(-1.41421 - 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(-0.707107 + 1.22474i) q^{3} +(-1.41421 - 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +4.00000 q^{15} +(0.707107 - 1.22474i) q^{17} +(-3.53553 - 6.12372i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-1.50000 + 2.59808i) q^{25} -5.65685 q^{27} +2.00000 q^{29} +(4.24264 - 7.34847i) q^{31} +(-1.41421 - 2.44949i) q^{33} +(-5.00000 - 8.66025i) q^{37} -9.89949 q^{41} -2.00000 q^{43} +(1.41421 - 2.44949i) q^{45} +(1.41421 + 2.44949i) q^{47} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +5.65685 q^{55} +10.0000 q^{57} +(-0.707107 + 1.22474i) q^{59} +(-1.41421 - 2.44949i) q^{61} +(6.00000 - 10.3923i) q^{67} +5.65685 q^{69} +12.0000 q^{71} +(0.707107 - 1.22474i) q^{73} +(-2.12132 - 3.67423i) q^{75} +(-2.00000 - 3.46410i) q^{79} +(2.50000 - 4.33013i) q^{81} -9.89949 q^{83} -4.00000 q^{85} +(-1.41421 + 2.44949i) q^{87} +(3.53553 + 6.12372i) q^{89} +(6.00000 + 10.3923i) q^{93} +(-10.0000 + 17.3205i) q^{95} +9.89949 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} - 4 q^{11} + 16 q^{15} - 8 q^{23} - 6 q^{25} + 8 q^{29} - 20 q^{37} - 8 q^{43} + 4 q^{51} + 4 q^{53} + 40 q^{57} + 24 q^{67} + 48 q^{71} - 8 q^{79} + 10 q^{81} - 16 q^{85} + 24 q^{93} - 40 q^{95} - 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 - 4 * q^11 + 16 * q^15 - 8 * q^23 - 6 * q^25 + 8 * q^29 - 20 * q^37 - 8 * q^43 + 4 * q^51 + 4 * q^53 + 40 * q^57 + 24 * q^67 + 48 * q^71 - 8 * q^79 + 10 * q^81 - 16 * q^85 + 24 * q^93 - 40 * q^95 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−0.707107	+	1.22474i	−0.408248	+	0.707107i	−0.994694	−	0.102882i	$-0.967194\pi$
$3$	−0.707107	+	1.22474i	−0.408248	+	0.707107i	0.586445	+	0.809989i	$0.300527\pi$
$4$		0			0
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	−0.987048	−	0.160424i	$-0.948714\pi$
$5$	−1.41421	−	2.44949i	−0.632456	−	1.09545i	0.354593	−	0.935021i	$-0.384620\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.500000	+	0.866025i	0.166667	+	0.288675i
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	4.00000			1.03280
$16$		0			0
$17$	0.707107	−	1.22474i	0.171499	−	0.297044i	−0.767445	−	0.641114i	$-0.778472\pi$
$17$	0.707107	−	1.22474i	0.171499	−	0.297044i	0.938944	+	0.344070i	$0.111806\pi$
$18$		0			0
$19$	−3.53553	−	6.12372i	−0.811107	−	1.40488i	−0.912090	−	0.409991i	$-0.865532\pi$
$19$	−3.53553	−	6.12372i	−0.811107	−	1.40488i	0.100983	−	0.994888i	$-0.467801\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−1.50000	+	2.59808i	−0.300000	+	0.519615i
$26$		0			0
$27$	−5.65685			−1.08866
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	4.24264	−	7.34847i	0.762001	−	1.31982i	−0.179817	−	0.983700i	$-0.557551\pi$
$31$	4.24264	−	7.34847i	0.762001	−	1.31982i	0.941818	−	0.336124i	$-0.109116\pi$
$32$		0			0
$33$	−1.41421	−	2.44949i	−0.246183	−	0.426401i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.89949			−1.54604			−0.773021	−	0.634381i	$-0.781255\pi$
$41$	−9.89949			−1.54604			−0.773021	+	0.634381i	$0.781255\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$	1.41421	−	2.44949i	0.210819	−	0.365148i
$46$		0			0
$47$	1.41421	+	2.44949i	0.206284	+	0.357295i	0.950541	−	0.310599i	$-0.100530\pi$
$47$	1.41421	+	2.44949i	0.206284	+	0.357295i	−0.744257	+	0.667893i	$0.767196\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.00000	+	1.73205i	0.140028	+	0.242536i
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$	5.65685			0.762770
$56$		0			0
$57$	10.0000			1.32453
$58$		0			0
$59$	−0.707107	+	1.22474i	−0.0920575	+	0.159448i	−0.908377	−	0.418153i	$-0.862678\pi$
$59$	−0.707107	+	1.22474i	−0.0920575	+	0.159448i	0.816319	+	0.577601i	$0.196011\pi$
$60$		0			0
$61$	−1.41421	−	2.44949i	−0.181071	−	0.313625i	0.761174	−	0.648547i	$-0.224623\pi$
$61$	−1.41421	−	2.44949i	−0.181071	−	0.313625i	−0.942246	+	0.334922i	$0.891290\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	−0.222571	−	0.974916i	$-0.571445\pi$
$67$	6.00000	−	10.3923i	0.733017	−	1.26962i	0.955588	−	0.294706i	$-0.0952216\pi$
$68$		0			0
$69$	5.65685			0.681005
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	0.707107	−	1.22474i	0.0827606	−	0.143346i	−0.821674	−	0.569958i	$-0.806960\pi$
$73$	0.707107	−	1.22474i	0.0827606	−	0.143346i	0.904435	+	0.426612i	$0.140293\pi$
$74$		0			0
$75$	−2.12132	−	3.67423i	−0.244949	−	0.424264i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	2.50000	−	4.33013i	0.277778	−	0.481125i
$82$		0			0
$83$	−9.89949			−1.08661			−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949			−1.08661			−0.543305	+	0.839535i	$0.682827\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$	−1.41421	+	2.44949i	−0.151620	+	0.262613i
$88$		0			0
$89$	3.53553	+	6.12372i	0.374766	+	0.649113i	0.990292	−	0.139003i	$-0.0443898\pi$
$89$	3.53553	+	6.12372i	0.374766	+	0.649113i	−0.615526	+	0.788116i	$0.711056\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.00000	+	10.3923i	0.622171	+	1.07763i
$94$		0			0
$95$	−10.0000	+	17.3205i	−1.02598	+	1.77705i
$96$		0			0
$97$	9.89949			1.00514			0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949			1.00514			0.502571	+	0.864536i	$0.332388\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	−4.24264	+	7.34847i	−0.422159	+	0.731200i	−0.996150	−	0.0876610i	$-0.972061\pi$
$101$	−4.24264	+	7.34847i	−0.422159	+	0.731200i	0.573992	+	0.818861i	$0.305394\pi$
$102$		0			0
$103$	1.41421	+	2.44949i	0.139347	+	0.241355i	0.927249	−	0.374444i	$-0.122166\pi$
$103$	1.41421	+	2.44949i	0.139347	+	0.241355i	−0.787903	+	0.615800i	$0.788833\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	0.753010	−	0.658009i	$-0.228601\pi$
$107$	−2.00000	−	3.46410i	−0.193347	−	0.334887i	−0.946357	+	0.323122i	$0.895268\pi$
$108$		0			0
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	14.1421			1.34231
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$		0			0
$115$	−5.65685	+	9.79796i	−0.527504	+	0.913664i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$	7.00000	−	12.1244i	0.631169	−	1.09322i
$124$		0			0
$125$	−5.65685			−0.505964
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	1.41421	−	2.44949i	0.124515	−	0.215666i
$130$		0			0
$131$	6.36396	+	11.0227i	0.556022	+	0.963058i	0.997823	+	0.0659452i	$0.0210063\pi$
$131$	6.36396	+	11.0227i	0.556022	+	0.963058i	−0.441801	+	0.897113i	$0.645660\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	8.00000	+	13.8564i	0.688530	+	1.19257i
$136$		0			0
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	−9.89949			−0.839664			−0.419832	−	0.907602i	$-0.637911\pi$
$139$	−9.89949			−0.839664			−0.419832	+	0.907602i	$0.637911\pi$
$140$		0			0
$141$	−4.00000			−0.336861
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−2.82843	−	4.89898i	−0.234888	−	0.406838i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	0.585231	−	0.810867i	$-0.301004\pi$
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	−0.994847	+	0.101391i	$0.967671\pi$
$150$		0			0
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	0.331842	+	0.943335i	$0.392330\pi$
$151$	−8.00000	+	13.8564i	−0.651031	+	1.12762i	−0.982873	+	0.184284i	$0.941004\pi$
$152$		0			0
$153$	1.41421			0.114332
$154$		0			0
$155$	−24.0000			−1.92773
$156$		0			0
$157$	5.65685	−	9.79796i	0.451466	−	0.781962i	−0.547011	−	0.837125i	$-0.684235\pi$
$157$	5.65685	−	9.79796i	0.451466	−	0.781962i	0.998477	+	0.0551630i	$0.0175678\pi$
$158$		0			0
$159$	1.41421	+	2.44949i	0.112154	+	0.194257i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.00000	+	8.66025i	0.391630	+	0.678323i	0.992665	−	0.120900i	$-0.0385779\pi$
$163$	5.00000	+	8.66025i	0.391630	+	0.678323i	−0.601035	+	0.799223i	$0.705245\pi$
$164$		0			0
$165$	−4.00000	+	6.92820i	−0.311400	+	0.539360i
$166$		0			0
$167$	19.7990			1.53209			0.766046	−	0.642786i	$-0.222221\pi$
$167$	19.7990			1.53209			0.766046	+	0.642786i	$0.222221\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	3.53553	−	6.12372i	0.270369	−	0.468293i
$172$		0			0
$173$	8.48528	+	14.6969i	0.645124	+	1.11739i	0.984273	+	0.176655i	$0.0565276\pi$
$173$	8.48528	+	14.6969i	0.645124	+	1.11739i	−0.339149	+	0.940733i	$0.610139\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.00000	−	1.73205i	−0.0751646	−	0.130189i
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$	4.00000			0.295689
$184$		0			0
$185$	−14.1421	+	24.4949i	−1.03975	+	1.80090i
$186$		0			0
$187$	1.41421	+	2.44949i	0.103418	+	0.179124i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	0.784552	−	0.620063i	$-0.212893\pi$
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	−0.929267	+	0.369410i	$0.879560\pi$
$192$		0			0
$193$	8.00000	−	13.8564i	0.575853	−	0.997406i	−0.420096	−	0.907480i	$-0.638004\pi$
$193$	8.00000	−	13.8564i	0.575853	−	0.997406i	0.995948	−	0.0899262i	$-0.0286631\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	4.24264	−	7.34847i	0.300753	−	0.520919i	−0.675554	−	0.737311i	$-0.736095\pi$
$199$	4.24264	−	7.34847i	0.300753	−	0.520919i	0.976307	+	0.216391i	$0.0694287\pi$
$200$		0			0
$201$	8.48528	+	14.6969i	0.598506	+	1.03664i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	14.0000	+	24.2487i	0.977802	+	1.69360i
$206$		0			0
$207$	2.00000	−	3.46410i	0.139010	−	0.240772i
$208$		0			0
$209$	14.1421			0.978232
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$	−8.48528	+	14.6969i	−0.581402	+	1.00702i
$214$		0			0
$215$	2.82843	+	4.89898i	0.192897	+	0.334108i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.00000	+	1.73205i	0.0675737	+	0.117041i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	−10.6066	+	18.3712i	−0.703985	+	1.21934i	0.263072	+	0.964776i	$0.415264\pi$
$227$	−10.6066	+	18.3712i	−0.703985	+	1.21934i	−0.967057	+	0.254561i	$0.918069\pi$
$228$		0			0
$229$	8.48528	+	14.6969i	0.560723	+	0.971201i	0.997434	+	0.0715988i	$0.0228101\pi$
$229$	8.48528	+	14.6969i	0.560723	+	0.971201i	−0.436710	+	0.899602i	$0.643857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	−0.928312	−	0.371802i	$-0.878740\pi$
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	0.142166	−	0.989843i	$-0.454593\pi$
$234$		0			0
$235$	4.00000	−	6.92820i	0.260931	−	0.451946i
$236$		0			0
$237$	5.65685			0.367452
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	10.6066	−	18.3712i	0.683231	−	1.18339i	−0.290758	−	0.956797i	$-0.593907\pi$
$241$	10.6066	−	18.3712i	0.683231	−	1.18339i	0.973989	−	0.226595i	$-0.0727593\pi$
$242$		0			0
$243$	−4.94975	−	8.57321i	−0.317526	−	0.549972i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	7.00000	−	12.1244i	0.443607	−	0.768350i
$250$		0			0
$251$	−9.89949			−0.624851			−0.312425	−	0.949942i	$-0.601141\pi$
$251$	−9.89949			−0.624851			−0.312425	+	0.949942i	$0.601141\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$		0			0
$255$	2.82843	−	4.89898i	0.177123	−	0.306786i
$256$		0			0
$257$	−6.36396	−	11.0227i	−0.396973	−	0.687577i	0.596378	−	0.802704i	$-0.296606\pi$
$257$	−6.36396	−	11.0227i	−0.396973	−	0.687577i	−0.993351	+	0.115126i	$0.963273\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	1.00000	+	1.73205i	0.0618984	+	0.107211i
$262$		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
$265$	−5.65685			−0.347498
$266$		0			0
$267$	−10.0000			−0.611990
$268$		0			0
$269$	5.65685	−	9.79796i	0.344904	−	0.597392i	−0.640432	−	0.768015i	$-0.721245\pi$
$269$	5.65685	−	9.79796i	0.344904	−	0.597392i	0.985336	+	0.170623i	$0.0545780\pi$
$270$		0			0
$271$	11.3137	+	19.5959i	0.687259	+	1.19037i	0.972721	+	0.231977i	$0.0745195\pi$
$271$	11.3137	+	19.5959i	0.687259	+	1.19037i	−0.285462	+	0.958390i	$0.592147\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	5.19615i	−0.180907	−	0.313340i
$276$		0			0
$277$	1.00000	−	1.73205i	0.0600842	−	0.104069i	−0.834419	−	0.551131i	$-0.814196\pi$
$277$	1.00000	−	1.73205i	0.0600842	−	0.104069i	0.894503	+	0.447062i	$0.147530\pi$
$278$		0			0
$279$	8.48528			0.508001
$280$		0			0
$281$	16.0000			0.954480			0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000			0.954480			0.477240	+	0.878773i	$0.341637\pi$
$282$		0			0
$283$	−0.707107	+	1.22474i	−0.0420331	+	0.0728035i	−0.886277	−	0.463156i	$-0.846717\pi$
$283$	−0.707107	+	1.22474i	−0.0420331	+	0.0728035i	0.844243	+	0.535960i	$0.180050\pi$
$284$		0			0
$285$	−14.1421	−	24.4949i	−0.837708	−	1.45095i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.50000	+	12.9904i	0.441176	+	0.764140i
$290$		0			0
$291$	−7.00000	+	12.1244i	−0.410347	+	0.710742i
$292$		0			0
$293$	−19.7990			−1.15667			−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990			−1.15667			−0.578335	+	0.815800i	$0.696297\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$	5.65685	−	9.79796i	0.328244	−	0.568535i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−6.00000	−	10.3923i	−0.344691	−	0.597022i
$304$		0			0
$305$	−4.00000	+	6.92820i	−0.229039	+	0.396708i
$306$		0			0
$307$	9.89949			0.564994			0.282497	−	0.959268i	$-0.408837\pi$
$307$	9.89949			0.564994			0.282497	+	0.959268i	$0.408837\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	−0.980647	−	0.195783i	$-0.937275\pi$
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	0.659877	+	0.751374i	$0.270609\pi$
$312$		0			0
$313$	−6.36396	−	11.0227i	−0.359712	−	0.623040i	0.628200	−	0.778052i	$-0.283792\pi$
$313$	−6.36396	−	11.0227i	−0.359712	−	0.623040i	−0.987913	+	0.155012i	$0.950459\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.00000	−	8.66025i	−0.280828	−	0.486408i	0.690761	−	0.723083i	$-0.257276\pi$
$317$	−5.00000	−	8.66025i	−0.280828	−	0.486408i	−0.971589	+	0.236675i	$0.923942\pi$
$318$		0			0
$319$	−2.00000	+	3.46410i	−0.111979	+	0.193952i
$320$		0			0
$321$	5.65685			0.315735
$322$		0			0
$323$	−10.0000			−0.556415
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.41421	+	2.44949i	0.0782062	+	0.135457i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5.00000	+	8.66025i	0.274825	+	0.476011i	0.970091	−	0.242742i	$-0.0780468\pi$
$331$	5.00000	+	8.66025i	0.274825	+	0.476011i	−0.695266	+	0.718752i	$0.744713\pi$
$332$		0			0
$333$	5.00000	−	8.66025i	0.273998	−	0.474579i
$334$		0			0
$335$	−33.9411			−1.85440
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$	8.48528	−	14.6969i	0.460857	−	0.798228i
$340$		0			0
$341$	8.48528	+	14.6969i	0.459504	+	0.795884i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.00000	−	13.8564i	−0.430706	−	0.746004i
$346$		0			0
$347$	−15.0000	+	25.9808i	−0.805242	+	1.39472i	0.110885	+	0.993833i	$0.464631\pi$
$347$	−15.0000	+	25.9808i	−0.805242	+	1.39472i	−0.916127	+	0.400887i	$0.868702\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.707107	−	1.22474i	0.0376355	−	0.0651866i	−0.846594	−	0.532239i	$-0.821351\pi$
$353$	0.707107	−	1.22474i	0.0376355	−	0.0651866i	0.884230	+	0.467052i	$0.154684\pi$
$354$		0			0
$355$	−16.9706	−	29.3939i	−0.900704	−	1.56007i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.0000	−	27.7128i	−0.844448	−	1.46263i	−0.886100	−	0.463494i	$-0.846596\pi$
$359$	−16.0000	−	27.7128i	−0.844448	−	1.46263i	0.0416523	−	0.999132i	$-0.486738\pi$
$360$		0			0
$361$	−15.5000	+	26.8468i	−0.815789	+	1.41299i
$362$		0			0
$363$	−9.89949			−0.519589
$364$		0			0
$365$	−4.00000			−0.209370
$366$		0			0
$367$	14.1421	−	24.4949i	0.738213	−	1.27862i	−0.215086	−	0.976595i	$-0.569003\pi$
$367$	14.1421	−	24.4949i	0.738213	−	1.27862i	0.953299	−	0.302028i	$-0.0976636\pi$
$368$		0			0
$369$	−4.94975	−	8.57321i	−0.257674	−	0.446304i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	0.707055	−	0.707159i	$-0.250023\pi$
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	−0.965945	+	0.258748i	$0.916690\pi$
$374$		0			0
$375$	4.00000	−	6.92820i	0.206559	−	0.357771i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	26.0000			1.33553			0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000			1.33553			0.667765	+	0.744372i	$0.267251\pi$
$380$		0			0
$381$	11.3137	−	19.5959i	0.579619	−	1.00393i
$382$		0			0
$383$	−18.3848	−	31.8434i	−0.939418	−	1.62712i	−0.766559	−	0.642173i	$-0.778033\pi$
$383$	−18.3848	−	31.8434i	−0.939418	−	1.62712i	−0.172859	−	0.984947i	$-0.555300\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.00000	−	1.73205i	−0.0508329	−	0.0880451i
$388$		0			0
$389$	−13.0000	+	22.5167i	−0.659126	+	1.14164i	0.321716	+	0.946836i	$0.395740\pi$
$389$	−13.0000	+	22.5167i	−0.659126	+	1.14164i	−0.980842	+	0.194804i	$0.937593\pi$
$390$		0			0
$391$	−5.65685			−0.286079
$392$		0			0
$393$	−18.0000			−0.907980
$394$		0			0
$395$	−5.65685	+	9.79796i	−0.284627	+	0.492989i
$396$		0			0
$397$	−11.3137	−	19.5959i	−0.567819	−	0.983491i	−0.996781	−	0.0801687i	$-0.974454\pi$
$397$	−11.3137	−	19.5959i	−0.567819	−	0.983491i	0.428963	−	0.903322i	$-0.358879\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	0.998350	−	0.0574304i	$-0.0182907\pi$
$401$	9.00000	+	15.5885i	0.449439	+	0.778450i	−0.548911	+	0.835881i	$0.684957\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−14.1421			−0.702728
$406$		0			0
$407$	20.0000			0.991363
$408$		0			0
$409$	−19.0919	+	33.0681i	−0.944033	+	1.63511i	−0.186357	+	0.982482i	$0.559668\pi$
$409$	−19.0919	+	33.0681i	−0.944033	+	1.63511i	−0.757676	+	0.652631i	$0.773665\pi$
$410$		0			0
$411$	−8.48528	−	14.6969i	−0.418548	−	0.724947i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	14.0000	+	24.2487i	0.687233	+	1.19032i
$416$		0			0
$417$	7.00000	−	12.1244i	0.342791	−	0.593732i
$418$		0			0
$419$	9.89949			0.483622			0.241811	−	0.970323i	$-0.422259\pi$
$419$	9.89949			0.483622			0.241811	+	0.970323i	$0.422259\pi$
$420$		0			0
$421$	30.0000			1.46211			0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000			1.46211			0.731055	+	0.682318i	$0.239028\pi$
$422$		0			0
$423$	−1.41421	+	2.44949i	−0.0687614	+	0.119098i
$424$		0			0
$425$	2.12132	+	3.67423i	0.102899	+	0.178227i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	−0.684564	−	0.728953i	$-0.740007\pi$
$431$	6.00000	−	10.3923i	0.289010	−	0.500580i	0.973574	+	0.228373i	$0.0733406\pi$
$432$		0			0
$433$	−29.6985			−1.42722			−0.713609	−	0.700544i	$-0.752941\pi$
$433$	−29.6985			−1.42722			−0.713609	+	0.700544i	$0.752941\pi$
$434$		0			0
$435$	8.00000			0.383571
$436$		0			0
$437$	−14.1421	+	24.4949i	−0.676510	+	1.17175i
$438$		0			0
$439$	−8.48528	−	14.6969i	−0.404980	−	0.701447i	0.589339	−	0.807886i	$-0.299388\pi$
$439$	−8.48528	−	14.6969i	−0.404980	−	0.701447i	−0.994319	+	0.106439i	$0.966055\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	0.814595	−	0.580030i	$-0.196959\pi$
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	−0.909618	+	0.415445i	$0.863626\pi$
$444$		0			0
$445$	10.0000	−	17.3205i	0.474045	−	0.821071i
$446$		0			0
$447$	14.1421			0.668900
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$	9.89949	−	17.1464i	0.466149	−	0.807394i
$452$		0			0
$453$	−11.3137	−	19.5959i	−0.531564	−	0.920697i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−12.0000	−	20.7846i	−0.561336	−	0.972263i	−0.997380	−	0.0723376i	$-0.976954\pi$
$457$	−12.0000	−	20.7846i	−0.561336	−	0.972263i	0.436044	−	0.899925i	$-0.356379\pi$
$458$		0			0
$459$	−4.00000	+	6.92820i	−0.186704	+	0.323381i
$460$		0			0
$461$	39.5980			1.84426			0.922131	−	0.386878i	$-0.126447\pi$
$461$	39.5980			1.84426			0.922131	+	0.386878i	$0.126447\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$	16.9706	−	29.3939i	0.786991	−	1.36311i
$466$		0			0
$467$	16.2635	+	28.1691i	0.752583	+	1.30351i	0.946567	+	0.322507i	$0.104526\pi$
$467$	16.2635	+	28.1691i	0.752583	+	1.30351i	−0.193984	+	0.981005i	$0.562141\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	8.00000	+	13.8564i	0.368621	+	0.638470i
$472$		0			0
$473$	2.00000	−	3.46410i	0.0919601	−	0.159280i
$474$		0			0
$475$	21.2132			0.973329
$476$		0			0
$477$	2.00000			0.0915737
$478$		0			0
$479$	−15.5563	+	26.9444i	−0.710788	+	1.23112i	0.253774	+	0.967264i	$0.418328\pi$
$479$	−15.5563	+	26.9444i	−0.710788	+	1.23112i	−0.964562	+	0.263857i	$0.915005\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−14.0000	−	24.2487i	−0.635707	−	1.10108i
$486$		0			0
$487$	6.00000	−	10.3923i	0.271886	−	0.470920i	−0.697459	−	0.716625i	$-0.745686\pi$
$487$	6.00000	−	10.3923i	0.271886	−	0.470920i	0.969345	+	0.245705i	$0.0790193\pi$
$488$		0			0
$489$	−14.1421			−0.639529
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	1.41421	−	2.44949i	0.0636930	−	0.110319i
$494$		0			0
$495$	2.82843	+	4.89898i	0.127128	+	0.220193i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	0.817781	−	0.575529i	$-0.195204\pi$
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	−0.907314	+	0.420455i	$0.861871\pi$
$500$		0			0
$501$	−14.0000	+	24.2487i	−0.625474	+	1.08335i
$502$		0			0
$503$	−39.5980			−1.76559			−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980			−1.76559			−0.882793	+	0.469762i	$0.844340\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$	9.19239	−	15.9217i	0.408248	−	0.707107i
$508$		0			0
$509$	−11.3137	−	19.5959i	−0.501471	−	0.868574i	−0.999999	−	0.00169976i	$-0.999459\pi$
$509$	−11.3137	−	19.5959i	−0.501471	−	0.868574i	0.498527	−	0.866874i	$-0.333874\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	20.0000	+	34.6410i	0.883022	+	1.52944i
$514$		0			0
$515$	4.00000	−	6.92820i	0.176261	−	0.305293i
$516$		0			0
$517$	−5.65685			−0.248788
$518$		0			0
$519$	−24.0000			−1.05348
$520$		0			0
$521$	0.707107	−	1.22474i	0.0309789	−	0.0536570i	−0.850120	−	0.526589i	$-0.823471\pi$
$521$	0.707107	−	1.22474i	0.0309789	−	0.0536570i	0.881099	+	0.472931i	$0.156804\pi$
$522$		0			0
$523$	6.36396	+	11.0227i	0.278277	+	0.481989i	0.970957	−	0.239256i	$-0.0769035\pi$
$523$	6.36396	+	11.0227i	0.278277	+	0.481989i	−0.692680	+	0.721245i	$0.743570\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	−	10.3923i	−0.261364	−	0.452696i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−1.41421			−0.0613716
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−5.65685	+	9.79796i	−0.244567	+	0.423603i
$536$		0			0
$537$	8.48528	+	14.6969i	0.366167	+	0.634220i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	0.738296	−	0.674477i	$-0.235631\pi$
$541$	−5.00000	−	8.66025i	−0.214967	−	0.372333i	−0.953262	+	0.302144i	$0.902298\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.65685			−0.242313
$546$		0			0
$547$	26.0000			1.11168			0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000			1.11168			0.555840	+	0.831289i	$0.312397\pi$
$548$		0			0
$549$	1.41421	−	2.44949i	0.0603572	−	0.104542i
$550$		0			0
$551$	−7.07107	−	12.2474i	−0.301238	−	0.521759i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−20.0000	−	34.6410i	−0.848953	−	1.47043i
$556$		0			0
$557$	15.0000	−	25.9808i	0.635570	−	1.10084i	−0.350824	−	0.936442i	$-0.614098\pi$
$557$	15.0000	−	25.9808i	0.635570	−	1.10084i	0.986394	−	0.164399i	$-0.0525683\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−4.00000			−0.168880
$562$		0			0
$563$	−0.707107	+	1.22474i	−0.0298010	+	0.0516168i	−0.880541	−	0.473970i	$-0.842821\pi$
$563$	−0.707107	+	1.22474i	−0.0298010	+	0.0516168i	0.850740	+	0.525586i	$0.176154\pi$
$564$		0			0
$565$	16.9706	+	29.3939i	0.713957	+	1.23661i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	0.741981	−	0.670421i	$-0.233886\pi$
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	−0.951592	+	0.307364i	$0.900553\pi$
$570$		0			0
$571$	−1.00000	+	1.73205i	−0.0418487	+	0.0724841i	−0.886191	−	0.463320i	$-0.846658\pi$
$571$	−1.00000	+	1.73205i	−0.0418487	+	0.0724841i	0.844342	+	0.535804i	$0.179991\pi$
$572$		0			0
$573$	5.65685			0.236318
$574$		0			0
$575$	12.0000			0.500435
$576$		0			0
$577$	10.6066	−	18.3712i	0.441559	−	0.764802i	−0.556247	−	0.831017i	$-0.687759\pi$
$577$	10.6066	−	18.3712i	0.441559	−	0.764802i	0.997805	+	0.0662152i	$0.0210924\pi$
$578$		0			0
$579$	11.3137	+	19.5959i	0.470182	+	0.814379i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.00000	+	3.46410i	0.0828315	+	0.143468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	29.6985			1.22579			0.612894	−	0.790165i	$-0.290005\pi$
$587$	29.6985			1.22579			0.612894	+	0.790165i	$0.290005\pi$
$588$		0			0
$589$	−60.0000			−2.47226
$590$		0			0
$591$	−1.41421	+	2.44949i	−0.0581730	+	0.100759i
$592$		0			0
$593$	3.53553	+	6.12372i	0.145187	+	0.251471i	0.929443	−	0.368967i	$-0.120288\pi$
$593$	3.53553	+	6.12372i	0.145187	+	0.251471i	−0.784256	+	0.620438i	$0.786955\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.00000	+	10.3923i	0.245564	+	0.425329i
$598$		0			0
$599$	−8.00000	+	13.8564i	−0.326871	+	0.566157i	−0.981889	−	0.189456i	$-0.939328\pi$
$599$	−8.00000	+	13.8564i	−0.326871	+	0.566157i	0.655018	+	0.755613i	$0.272661\pi$
$600$		0			0
$601$	29.6985			1.21143			0.605713	−	0.795683i	$-0.292888\pi$
$601$	29.6985			1.21143			0.605713	+	0.795683i	$0.292888\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	9.89949	−	17.1464i	0.402472	−	0.697101i
$606$		0			0
$607$	−8.48528	−	14.6969i	−0.344407	−	0.596530i	0.640839	−	0.767675i	$-0.278587\pi$
$607$	−8.48528	−	14.6969i	−0.344407	−	0.596530i	−0.985246	+	0.171145i	$0.945253\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	−0.386073	−	0.922468i	$-0.626169\pi$
$613$	15.0000	−	25.9808i	0.605844	−	1.04935i	0.991917	−	0.126885i	$-0.0404979\pi$
$614$		0			0
$615$	−39.5980			−1.59674
$616$		0			0
$617$	−26.0000			−1.04672			−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000			−1.04672			−0.523360	+	0.852111i	$0.675322\pi$
$618$		0			0
$619$	9.19239	−	15.9217i	0.369473	−	0.639946i	−0.620010	−	0.784594i	$-0.712871\pi$
$619$	9.19239	−	15.9217i	0.369473	−	0.639946i	0.989483	+	0.144647i	$0.0462048\pi$
$620$		0			0
$621$	11.3137	+	19.5959i	0.454003	+	0.786357i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.5000	+	26.8468i	0.620000	+	1.07387i
$626$		0			0
$627$	−10.0000	+	17.3205i	−0.399362	+	0.691714i
$628$		0			0
$629$	−14.1421			−0.563884
$630$		0			0
$631$	−44.0000			−1.75161			−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000			−1.75161			−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−8.48528	+	14.6969i	−0.337260	+	0.584151i
$634$		0			0
$635$	22.6274	+	39.1918i	0.897942	+	1.55528i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	6.00000	+	10.3923i	0.237356	+	0.411113i
$640$		0			0
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	0.486409	+	0.873731i	$0.338307\pi$
$641$	−13.0000	+	22.5167i	−0.513469	+	0.889355i	−0.999878	+	0.0156233i	$0.995027\pi$
$642$		0			0
$643$	−9.89949			−0.390398			−0.195199	−	0.980764i	$-0.562535\pi$
$643$	−9.89949			−0.390398			−0.195199	+	0.980764i	$0.562535\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		0			0
$647$	4.24264	−	7.34847i	0.166795	−	0.288898i	−0.770496	−	0.637445i	$-0.779991\pi$
$647$	4.24264	−	7.34847i	0.166795	−	0.288898i	0.937291	+	0.348547i	$0.113325\pi$
$648$		0			0
$649$	−1.41421	−	2.44949i	−0.0555127	−	0.0961509i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$	18.0000	−	31.1769i	0.703318	−	1.21818i
$656$		0			0
$657$	1.41421			0.0551737
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−4.24264	+	7.34847i	−0.165020	+	0.285822i	−0.936662	−	0.350234i	$-0.886102\pi$
$661$	−4.24264	+	7.34847i	−0.165020	+	0.285822i	0.771643	+	0.636056i	$0.219435\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.00000	−	6.92820i	−0.154881	−	0.268261i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.65685			0.218380
$672$		0			0
$673$	−12.0000			−0.462566			−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000			−0.462566			−0.231283	+	0.972887i	$0.574292\pi$
$674$		0			0
$675$	8.48528	−	14.6969i	0.326599	−	0.565685i
$676$		0			0
$677$	8.48528	+	14.6969i	0.326116	+	0.564849i	0.981738	−	0.190240i	$-0.0609267\pi$
$677$	8.48528	+	14.6969i	0.326116	+	0.564849i	−0.655622	+	0.755090i	$0.727593\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−15.0000	−	25.9808i	−0.574801	−	0.995585i
$682$		0			0
$683$	6.00000	−	10.3923i	0.229584	−	0.397650i	−0.728101	−	0.685470i	$-0.759597\pi$
$683$	6.00000	−	10.3923i	0.229584	−	0.397650i	0.957685	+	0.287819i	$0.0929302\pi$
$684$		0			0
$685$	33.9411			1.29682
$686$		0			0
$687$	−24.0000			−0.915657
$688$		0			0
$689$		0			0
$690$		0			0
$691$	6.36396	+	11.0227i	0.242096	+	0.419323i	0.961311	−	0.275464i	$-0.0888316\pi$
$691$	6.36396	+	11.0227i	0.242096	+	0.419323i	−0.719215	+	0.694788i	$0.755498\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.0000	+	24.2487i	0.531050	+	0.919806i
$696$		0			0
$697$	−7.00000	+	12.1244i	−0.265144	+	0.459243i
$698$		0			0
$699$	33.9411			1.28377
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	−35.3553	+	61.2372i	−1.33345	+	2.30961i
$704$		0			0
$705$	5.65685	+	9.79796i	0.213049	+	0.369012i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.00000	−	8.66025i	−0.187779	−	0.325243i	0.756730	−	0.653727i	$-0.226796\pi$
$709$	−5.00000	−	8.66025i	−0.187779	−	0.325243i	−0.944509	+	0.328484i	$0.893462\pi$
$710$		0			0
$711$	2.00000	−	3.46410i	0.0750059	−	0.129914i
$712$		0			0
$713$	−33.9411			−1.27111
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.48528	+	14.6969i	−0.316889	+	0.548867i
$718$		0			0
$719$	1.41421	+	2.44949i	0.0527413	+	0.0913506i	0.891191	−	0.453629i	$-0.149871\pi$
$719$	1.41421	+	2.44949i	0.0527413	+	0.0913506i	−0.838449	+	0.544979i	$0.816537\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	15.0000	+	25.9808i	0.557856	+	0.966235i
$724$		0			0
$725$	−3.00000	+	5.19615i	−0.111417	+	0.192980i
$726$		0			0
$727$	19.7990			0.734304			0.367152	−	0.930161i	$-0.380333\pi$
$727$	19.7990			0.734304			0.367152	+	0.930161i	$0.380333\pi$
$728$		0			0
$729$	29.0000			1.07407
$730$		0			0
$731$	−1.41421	+	2.44949i	−0.0523066	+	0.0905977i
$732$		0			0
$733$	−21.2132	−	36.7423i	−0.783528	−	1.35711i	−0.929875	−	0.367876i	$-0.880085\pi$
$733$	−21.2132	−	36.7423i	−0.783528	−	1.35711i	0.146347	−	0.989233i	$-0.453248\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.0000	+	20.7846i	0.442026	+	0.765611i
$738$		0			0
$739$	−15.0000	+	25.9808i	−0.551784	+	0.955718i	0.446362	+	0.894852i	$0.352719\pi$
$739$	−15.0000	+	25.9808i	−0.551784	+	0.955718i	−0.998146	+	0.0608653i	$0.980614\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.0000			−0.586983			−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000			−0.586983			−0.293492	+	0.955962i	$0.594817\pi$
$744$		0			0
$745$	−14.1421	+	24.4949i	−0.518128	+	0.897424i
$746$		0			0
$747$	−4.94975	−	8.57321i	−0.181102	−	0.313678i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.00000	−	3.46410i	−0.0729810	−	0.126407i	0.827225	−	0.561870i	$-0.189918\pi$
$751$	−2.00000	−	3.46410i	−0.0729810	−	0.126407i	−0.900207	+	0.435463i	$0.856585\pi$
$752$		0			0
$753$	7.00000	−	12.1244i	0.255094	−	0.441836i
$754$		0			0
$755$	45.2548			1.64699
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$	−5.65685	+	9.79796i	−0.205331	+	0.355643i
$760$		0			0
$761$	3.53553	+	6.12372i	0.128163	+	0.221985i	0.922965	−	0.384884i	$-0.125759\pi$
$761$	3.53553	+	6.12372i	0.128163	+	0.221985i	−0.794802	+	0.606869i	$0.792425\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.00000	−	3.46410i	−0.0723102	−	0.125245i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	29.6985			1.07095			0.535477	−	0.844550i	$-0.320132\pi$
$769$	29.6985			1.07095			0.535477	+	0.844550i	$0.320132\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$	−24.0416	+	41.6413i	−0.864717	+	1.49773i	0.00261021	+	0.999997i	$0.499169\pi$
$773$	−24.0416	+	41.6413i	−0.864717	+	1.49773i	−0.867328	+	0.497738i	$0.834164\pi$
$774$		0			0
$775$	12.7279	+	22.0454i	0.457200	+	0.791894i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	35.0000	+	60.6218i	1.25401	+	2.17200i
$780$		0			0
$781$	−12.0000	+	20.7846i	−0.429394	+	0.743732i
$782$		0			0
$783$	−11.3137			−0.404319
$784$		0			0
$785$	−32.0000			−1.14213
$786$		0			0
$787$	−0.707107	+	1.22474i	−0.0252056	+	0.0436574i	−0.878353	−	0.478012i	$-0.841357\pi$
$787$	−0.707107	+	1.22474i	−0.0252056	+	0.0436574i	0.853147	+	0.521670i	$0.174691\pi$
$788$		0			0
$789$	8.48528	+	14.6969i	0.302084	+	0.523225i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	4.00000	−	6.92820i	0.141865	−	0.245718i
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$		0			0
$801$	−3.53553	+	6.12372i	−0.124922	+	0.216371i
$802$		0			0
$803$	1.41421	+	2.44949i	0.0499065	+	0.0864406i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	8.00000	+	13.8564i	0.281613	+	0.487769i
$808$		0			0
$809$	8.00000	−	13.8564i	0.281265	−	0.487165i	−0.690432	−	0.723398i	$-0.742579\pi$
$809$	8.00000	−	13.8564i	0.281265	−	0.487165i	0.971697	+	0.236232i	$0.0759127\pi$
$810$		0			0
$811$	29.6985			1.04285			0.521427	−	0.853296i	$-0.325400\pi$
$811$	29.6985			1.04285			0.521427	+	0.853296i	$0.325400\pi$
$812$		0			0
$813$	−32.0000			−1.12229
$814$		0			0
$815$	14.1421	−	24.4949i	0.495377	−	0.858019i
$816$		0			0
$817$	7.07107	+	12.2474i	0.247385	+	0.428484i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	0.979246	−	0.202674i	$-0.0649632\pi$
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	−0.665144	+	0.746715i	$0.731630\pi$
$822$		0			0
$823$	20.0000	−	34.6410i	0.697156	−	1.20751i	−0.272292	−	0.962215i	$-0.587782\pi$
$823$	20.0000	−	34.6410i	0.697156	−	1.20751i	0.969448	−	0.245295i	$-0.0788849\pi$
$824$		0			0
$825$	8.48528			0.295420
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	15.5563	−	26.9444i	0.540294	−	0.935817i	−0.458593	−	0.888647i	$-0.651646\pi$
$829$	15.5563	−	26.9444i	0.540294	−	0.935817i	0.998887	−	0.0471706i	$-0.0150204\pi$
$830$		0			0
$831$	1.41421	+	2.44949i	0.0490585	+	0.0849719i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−28.0000	−	48.4974i	−0.968980	−	1.67832i
$836$		0			0
$837$	−24.0000	+	41.5692i	−0.829561	+	1.43684i
$838$		0			0
$839$	−19.7990			−0.683537			−0.341769	−	0.939784i	$-0.611026\pi$
$839$	−19.7990			−0.683537			−0.341769	+	0.939784i	$0.611026\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−11.3137	+	19.5959i	−0.389665	+	0.674919i
$844$		0			0
$845$	18.3848	+	31.8434i	0.632456	+	1.09545i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.00000	−	1.73205i	−0.0343199	−	0.0594438i
$850$		0			0
$851$	−20.0000	+	34.6410i	−0.685591	+	1.18748i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$	−20.0000			−0.683986
$856$		0			0
$857$	−9.19239	+	15.9217i	−0.314006	+	0.543874i	−0.979226	−	0.202773i	$-0.935005\pi$
$857$	−9.19239	+	15.9217i	−0.314006	+	0.543874i	0.665220	+	0.746648i	$0.268338\pi$
$858$		0			0
$859$	−13.4350	−	23.2702i	−0.458397	−	0.793967i	0.540479	−	0.841357i	$-0.318243\pi$
$859$	−13.4350	−	23.2702i	−0.458397	−	0.793967i	−0.998876	+	0.0473900i	$0.984910\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.00000	−	3.46410i	−0.0680808	−	0.117919i	0.829976	−	0.557800i	$-0.188354\pi$
$863$	−2.00000	−	3.46410i	−0.0680808	−	0.117919i	−0.898056	+	0.439880i	$0.855021\pi$
$864$		0			0
$865$	24.0000	−	41.5692i	0.816024	−	1.41340i
$866$		0			0
$867$	−21.2132			−0.720438
$868$		0			0
$869$	8.00000			0.271381
$870$		0			0
$871$		0			0
$872$		0			0
$873$	4.94975	+	8.57321i	0.167524	+	0.290159i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.0000	+	39.8372i	0.776655	+	1.34521i	0.933860	+	0.357640i	$0.116418\pi$
$877$	23.0000	+	39.8372i	0.776655	+	1.34521i	−0.157205	+	0.987566i	$0.550248\pi$
$878$		0			0
$879$	14.0000	−	24.2487i	0.472208	−	0.817889i
$880$		0			0
$881$	29.6985			1.00057			0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985			1.00057			0.500284	+	0.865862i	$0.333229\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$		0			0
$885$	−2.82843	+	4.89898i	−0.0950765	+	0.164677i
$886$		0			0
$887$	−18.3848	−	31.8434i	−0.617300	−	1.06920i	−0.989976	−	0.141234i	$-0.954893\pi$
$887$	−18.3848	−	31.8434i	−0.617300	−	1.06920i	0.372676	−	0.927962i	$-0.378440\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	5.00000	+	8.66025i	0.167506	+	0.290129i
$892$		0			0
$893$	10.0000	−	17.3205i	0.334637	−	0.579609i
$894$		0			0
$895$	−33.9411			−1.13453
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.48528	−	14.6969i	0.283000	−	0.490170i
$900$		0			0
$901$	−1.41421	−	2.44949i	−0.0471143	−	0.0816043i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.0000	+	38.1051i	−0.730498	+	1.26526i	0.226173	+	0.974087i	$0.427379\pi$
$907$	−22.0000	+	38.1051i	−0.730498	+	1.26526i	−0.956671	+	0.291172i	$0.905955\pi$
$908$		0			0
$909$	−8.48528			−0.281439
$910$		0			0
$911$	40.0000			1.32526			0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000			1.32526			0.662630	+	0.748947i	$0.269440\pi$
$912$		0			0
$913$	9.89949	−	17.1464i	0.327625	−	0.567464i
$914$		0			0
$915$	−5.65685	−	9.79796i	−0.187010	−	0.323911i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−16.0000	−	27.7128i	−0.527791	−	0.914161i	−0.999475	−	0.0323936i	$-0.989687\pi$
$919$	−16.0000	−	27.7128i	−0.527791	−	0.914161i	0.471684	−	0.881768i	$-0.343646\pi$
$920$		0			0
$921$	−7.00000	+	12.1244i	−0.230658	+	0.399511i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	30.0000			0.986394
$926$		0			0
$927$	−1.41421	+	2.44949i	−0.0464489	+	0.0804518i
$928$		0			0
$929$	−16.2635	−	28.1691i	−0.533587	−	0.924199i	−0.999230	−	0.0392269i	$-0.987510\pi$
$929$	−16.2635	−	28.1691i	−0.533587	−	0.924199i	0.465644	−	0.884972i	$-0.345823\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−8.00000	−	13.8564i	−0.261908	−	0.453638i
$934$		0			0
$935$	4.00000	−	6.92820i	0.130814	−	0.226576i
$936$		0			0
$937$	9.89949			0.323402			0.161701	−	0.986840i	$-0.448302\pi$
$937$	9.89949			0.323402			0.161701	+	0.986840i	$0.448302\pi$
$938$		0			0
$939$	18.0000			0.587408
$940$		0			0
$941$	15.5563	−	26.9444i	0.507122	−	0.878362i	−0.492844	−	0.870118i	$-0.664043\pi$
$941$	15.5563	−	26.9444i	0.507122	−	0.878362i	0.999966	−	0.00824396i	$-0.00262416\pi$
$942$		0			0
$943$	19.7990	+	34.2929i	0.644744	+	1.11673i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	0.681930	−	0.731417i	$-0.261141\pi$
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	−0.974391	+	0.224860i	$0.927807\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	14.1421			0.458590
$952$		0			0
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$		0			0
$955$	−5.65685	+	9.79796i	−0.183052	+	0.317055i
$956$		0			0
$957$	−2.82843	−	4.89898i	−0.0914301	−	0.158362i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.5000	−	35.5070i	−0.661290	−	1.14539i
$962$		0			0
$963$	2.00000	−	3.46410i	0.0644491	−	0.111629i
$964$		0			0
$965$	−45.2548			−1.45680
$966$		0			0
$967$	12.0000			0.385894			0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000			0.385894			0.192947	+	0.981209i	$0.438195\pi$
$968$		0			0
$969$	7.07107	−	12.2474i	0.227155	−	0.393445i
$970$		0			0
$971$	16.2635	+	28.1691i	0.521919	+	0.903990i	0.999675	+	0.0254978i	$0.00811707\pi$
$971$	16.2635	+	28.1691i	0.521919	+	0.903990i	−0.477756	+	0.878493i	$0.658550\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−6.00000	+	10.3923i	−0.191957	+	0.332479i	−0.945899	−	0.324462i	$-0.894817\pi$
$977$	−6.00000	+	10.3923i	−0.191957	+	0.332479i	0.753942	+	0.656941i	$0.228150\pi$
$978$		0			0
$979$	−14.1421			−0.451985
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$	24.0416	−	41.6413i	0.766809	−	1.32815i	−0.172476	−	0.985014i	$-0.555177\pi$
$983$	24.0416	−	41.6413i	0.766809	−	1.32815i	0.939285	−	0.343138i	$-0.111490\pi$
$984$		0			0
$985$	−2.82843	−	4.89898i	−0.0901212	−	0.156094i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4.00000	+	6.92820i	0.127193	+	0.220304i
$990$		0			0
$991$	−8.00000	+	13.8564i	−0.254128	+	0.440163i	−0.964658	−	0.263504i	$-0.915122\pi$
$991$	−8.00000	+	13.8564i	−0.254128	+	0.440163i	0.710530	+	0.703667i	$0.248455\pi$
$992$		0			0
$993$	−14.1421			−0.448787
$994$		0			0
$995$	−24.0000			−0.760851
$996$		0			0
$997$	15.5563	−	26.9444i	0.492675	−	0.853337i	−0.507290	−	0.861775i	$-0.669353\pi$
$997$	15.5563	−	26.9444i	0.492675	−	0.853337i	0.999964	+	0.00843818i	$0.00268599\pi$
$998$		0			0
$999$	28.2843	+	48.9898i	0.894875	+	1.54997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.i.m.753.1		4
4.3	odd	2		98.2.c.c.67.2		4
7.2	even	3	inner	784.2.i.m.177.1		4
7.3	odd	6		784.2.a.l.1.1		2
7.4	even	3		784.2.a.l.1.2		2
7.5	odd	6	inner	784.2.i.m.177.2		4
7.6	odd	2	inner	784.2.i.m.753.2		4
12.11	even	2		882.2.g.l.361.2		4
21.11	odd	6		7056.2.a.cl.1.1		2
21.17	even	6		7056.2.a.cl.1.2		2
28.3	even	6		98.2.a.b.1.2	yes	2
28.11	odd	6		98.2.a.b.1.1	✓	2
28.19	even	6		98.2.c.c.79.1		4
28.23	odd	6		98.2.c.c.79.2		4
28.27	even	2		98.2.c.c.67.1		4
56.3	even	6		3136.2.a.bn.1.1		2
56.11	odd	6		3136.2.a.bn.1.2		2
56.45	odd	6		3136.2.a.bm.1.2		2
56.53	even	6		3136.2.a.bm.1.1		2
84.11	even	6		882.2.a.n.1.1		2
84.23	even	6		882.2.g.l.667.2		4
84.47	odd	6		882.2.g.l.667.1		4
84.59	odd	6		882.2.a.n.1.2		2
84.83	odd	2		882.2.g.l.361.1		4
140.3	odd	12		2450.2.c.v.99.2		4
140.39	odd	6		2450.2.a.bj.1.2		2
140.59	even	6		2450.2.a.bj.1.1		2
140.67	even	12		2450.2.c.v.99.4		4
140.87	odd	12		2450.2.c.v.99.3		4
140.123	even	12		2450.2.c.v.99.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.2.a.b.1.1	✓	2	28.11	odd	6
98.2.a.b.1.2	yes	2	28.3	even	6
98.2.c.c.67.1		4	28.27	even	2
98.2.c.c.67.2		4	4.3	odd	2
98.2.c.c.79.1		4	28.19	even	6
98.2.c.c.79.2		4	28.23	odd	6
784.2.a.l.1.1		2	7.3	odd	6
784.2.a.l.1.2		2	7.4	even	3
784.2.i.m.177.1		4	7.2	even	3	inner
784.2.i.m.177.2		4	7.5	odd	6	inner
784.2.i.m.753.1		4	1.1	even	1	trivial
784.2.i.m.753.2		4	7.6	odd	2	inner
882.2.a.n.1.1		2	84.11	even	6
882.2.a.n.1.2		2	84.59	odd	6
882.2.g.l.361.1		4	84.83	odd	2
882.2.g.l.361.2		4	12.11	even	2
882.2.g.l.667.1		4	84.47	odd	6
882.2.g.l.667.2		4	84.23	even	6
2450.2.a.bj.1.1		2	140.59	even	6
2450.2.a.bj.1.2		2	140.39	odd	6
2450.2.c.v.99.1		4	140.123	even	12
2450.2.c.v.99.2		4	140.3	odd	12
2450.2.c.v.99.3		4	140.87	odd	12
2450.2.c.v.99.4		4	140.67	even	12
3136.2.a.bm.1.1		2	56.53	even	6
3136.2.a.bm.1.2		2	56.45	odd	6
3136.2.a.bn.1.1		2	56.3	even	6
3136.2.a.bn.1.2		2	56.11	odd	6
7056.2.a.cl.1.1		2	21.11	odd	6
7056.2.a.cl.1.2		2	21.17	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{3} +(-1.41421 - 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 1.22474i) q^{3} +(-1.41421 - 2.44949i) q^{5} +(0.500000 + 0.866025i) q^{9} +(-1.00000 + 1.73205i) q^{11} +4.00000 q^{15} +(0.707107 - 1.22474i) q^{17} +(-3.53553 - 6.12372i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-1.50000 + 2.59808i) q^{25} -5.65685 q^{27} +2.00000 q^{29} +(4.24264 - 7.34847i) q^{31} +(-1.41421 - 2.44949i) q^{33} +(-5.00000 - 8.66025i) q^{37} -9.89949 q^{41} -2.00000 q^{43} +(1.41421 - 2.44949i) q^{45} +(1.41421 + 2.44949i) q^{47} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +5.65685 q^{55} +10.0000 q^{57} +(-0.707107 + 1.22474i) q^{59} +(-1.41421 - 2.44949i) q^{61} +(6.00000 - 10.3923i) q^{67} +5.65685 q^{69} +12.0000 q^{71} +(0.707107 - 1.22474i) q^{73} +(-2.12132 - 3.67423i) q^{75} +(-2.00000 - 3.46410i) q^{79} +(2.50000 - 4.33013i) q^{81} -9.89949 q^{83} -4.00000 q^{85} +(-1.41421 + 2.44949i) q^{87} +(3.53553 + 6.12372i) q^{89} +(6.00000 + 10.3923i) q^{93} +(-10.0000 + 17.3205i) q^{95} +9.89949 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} - 4 q^{11} + 16 q^{15} - 8 q^{23} - 6 q^{25} + 8 q^{29} - 20 q^{37} - 8 q^{43} + 4 q^{51} + 4 q^{53} + 40 q^{57} + 24 q^{67} + 48 q^{71} - 8 q^{79} + 10 q^{81} - 16 q^{85} + 24 q^{93} - 40 q^{95} - 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 - 4 * q^11 + 16 * q^15 - 8 * q^23 - 6 * q^25 + 8 * q^29 - 20 * q^37 - 8 * q^43 + 4 * q^51 + 4 * q^53 + 40 * q^57 + 24 * q^67 + 48 * q^71 - 8 * q^79 + 10 * q^81 - 16 * q^85 + 24 * q^93 - 40 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more