LMFDB - Embedded newform 1575.2.bk.b.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(26,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

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

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

chi := DirichletCharacter("1575.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.bk (of order $6$, degree $2$, 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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1575.26
Dual form			1575.2.bk.b.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{4} +(2.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{4} +(2.00000 - 1.73205i) q^{7} +(-3.67423 + 2.12132i) q^{11} +3.46410i q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.67423 + 6.36396i) q^{17} +(6.00000 + 3.46410i) q^{19} +(-7.34847 - 4.24264i) q^{23} +(-5.00000 - 1.73205i) q^{28} +8.48528i q^{29} +(4.50000 - 2.59808i) q^{31} +(-0.500000 + 0.866025i) q^{37} +7.34847 q^{41} +1.00000 q^{43} +(7.34847 + 4.24264i) q^{44} +(3.67423 - 6.36396i) q^{47} +(1.00000 - 6.92820i) q^{49} +(6.00000 - 3.46410i) q^{52} +(3.67423 - 2.12132i) q^{53} +(3.67423 + 6.36396i) q^{59} +(-1.50000 - 0.866025i) q^{61} +8.00000 q^{64} +(-1.00000 - 1.73205i) q^{67} +(7.34847 - 12.7279i) q^{68} +4.24264i q^{71} +(-7.50000 + 4.33013i) q^{73} -13.8564i q^{76} +(-3.67423 + 10.6066i) q^{77} +(6.50000 - 11.2583i) q^{79} -7.34847 q^{83} +(-3.67423 + 6.36396i) q^{89} +(6.00000 + 6.92820i) q^{91} +16.9706i q^{92} -1.73205i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^7	$ 4 q - 4 q^{4} + 8 q^{7} - 8 q^{16} + 24 q^{19} - 20 q^{28} + 18 q^{31} - 2 q^{37} + 4 q^{43} + 4 q^{49} + 24 q^{52} - 6 q^{61} + 32 q^{64} - 4 q^{67} - 30 q^{73} + 26 q^{79} + 24 q^{91}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^7 - 8 * q^16 + 24 * q^19 - 20 * q^28 + 18 * q^31 - 2 * q^37 + 4 * q^43 + 4 * q^49 + 24 * q^52 - 6 * q^61 + 32 * q^64 - 4 * q^67 - 30 * q^73 + 26 * q^79 + 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$-1$

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		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.67423	+	2.12132i	−1.10782	+	0.639602i	−0.938265	−	0.345918i	$-0.887568\pi$
$11$	−3.67423	+	2.12132i	−1.10782	+	0.639602i	−0.169559	+	0.985520i	$0.554234\pi$
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	3.67423	+	6.36396i	0.891133	+	1.54349i	0.838519	+	0.544872i	$0.183422\pi$
$17$	3.67423	+	6.36396i	0.891133	+	1.54349i	0.0526138	+	0.998615i	$0.483245\pi$
$18$		0			0
$19$	6.00000	+	3.46410i	1.37649	+	0.794719i	0.991736	−	0.128298i	$-0.0409513\pi$
$19$	6.00000	+	3.46410i	1.37649	+	0.794719i	0.384759	+	0.923017i	$0.374285\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.34847	−	4.24264i	−1.53226	−	0.884652i	−0.999257	−	0.0385394i	$-0.987729\pi$
$23$	−7.34847	−	4.24264i	−1.53226	−	0.884652i	−0.533005	−	0.846112i	$-0.678937\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$	−5.00000	−	1.73205i	−0.944911	−	0.327327i
$29$			8.48528i			1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$			8.48528i			1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$		0			0
$31$	4.50000	−	2.59808i	0.808224	−	0.466628i	−0.0381148	−	0.999273i	$-0.512135\pi$
$31$	4.50000	−	2.59808i	0.808224	−	0.466628i	0.846339	+	0.532645i	$0.178802\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.34847			1.14764			0.573819	−	0.818982i	$-0.305461\pi$
$41$	7.34847			1.14764			0.573819	+	0.818982i	$0.305461\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$	7.34847	+	4.24264i	1.10782	+	0.639602i
$45$		0			0
$46$		0			0
$47$	3.67423	−	6.36396i	0.535942	−	0.928279i	−0.463175	−	0.886267i	$-0.653290\pi$
$47$	3.67423	−	6.36396i	0.535942	−	0.928279i	0.999117	−	0.0420122i	$-0.0133768\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$	6.00000	−	3.46410i	0.832050	−	0.480384i
$53$	3.67423	−	2.12132i	0.504695	−	0.291386i	−0.225955	−	0.974138i	$-0.572550\pi$
$53$	3.67423	−	2.12132i	0.504695	−	0.291386i	0.730650	+	0.682752i	$0.239217\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.67423	+	6.36396i	0.478345	+	0.828517i	0.999692	−	0.0248275i	$-0.00790366\pi$
$59$	3.67423	+	6.36396i	0.478345	+	0.828517i	−0.521347	+	0.853345i	$0.674570\pi$
$60$		0			0
$61$	−1.50000	−	0.866025i	−0.192055	−	0.110883i	0.400889	−	0.916127i	$-0.368701\pi$
$61$	−1.50000	−	0.866025i	−0.192055	−	0.110883i	−0.592944	+	0.805243i	$0.702035\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	0.798454	−	0.602056i	$-0.205652\pi$
$67$	−1.00000	−	1.73205i	−0.122169	−	0.211604i	−0.920623	+	0.390453i	$0.872318\pi$
$68$	7.34847	−	12.7279i	0.891133	−	1.54349i
$69$		0			0
$70$		0			0
$71$			4.24264i			0.503509i	0.967791	+	0.251754i	$0.0810075\pi$
$71$			4.24264i			0.503509i	−0.967791	+	0.251754i	$0.918992\pi$
$72$		0			0
$73$	−7.50000	+	4.33013i	−0.877809	+	0.506803i	−0.869935	−	0.493166i	$-0.835840\pi$
$73$	−7.50000	+	4.33013i	−0.877809	+	0.506803i	−0.00787336	+	0.999969i	$0.502506\pi$
$74$		0			0
$75$		0			0
$76$		−	13.8564i		−	1.58944i
$77$	−3.67423	+	10.6066i	−0.418718	+	1.20873i
$78$		0			0
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.34847			−0.806599			−0.403300	−	0.915068i	$-0.632137\pi$
$83$	−7.34847			−0.806599			−0.403300	+	0.915068i	$0.632137\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.67423	+	6.36396i	−0.389468	+	0.674579i	−0.992378	−	0.123231i	$-0.960674\pi$
$89$	−3.67423	+	6.36396i	−0.389468	+	0.674579i	0.602910	+	0.797809i	$0.294008\pi$
$90$		0			0
$91$	6.00000	+	6.92820i	0.628971	+	0.726273i
$92$			16.9706i			1.76930i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.73205i		−	0.175863i	−0.996127	−	0.0879316i	$-0.971974\pi$
$97$		−	1.73205i		−	0.175863i	0.996127	−	0.0879316i	$-0.0280257\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.34847	+	12.7279i	0.731200	+	1.26648i	0.956371	+	0.292156i	$0.0943727\pi$
$101$	7.34847	+	12.7279i	0.731200	+	1.26648i	−0.225171	+	0.974319i	$0.572294\pi$
$102$		0			0
$103$	−7.50000	−	4.33013i	−0.738997	−	0.426660i	0.0827075	−	0.996574i	$-0.473643\pi$
$103$	−7.50000	−	4.33013i	−0.738997	−	0.426660i	−0.821705	+	0.569914i	$0.806977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.34847	−	4.24264i	−0.710403	−	0.410152i	0.100807	−	0.994906i	$-0.467858\pi$
$107$	−7.34847	−	4.24264i	−0.710403	−	0.410152i	−0.811210	+	0.584754i	$0.801191\pi$
$108$		0			0
$109$	4.00000	+	6.92820i	0.383131	+	0.663602i	0.991508	−	0.130046i	$-0.0415126\pi$
$109$	4.00000	+	6.92820i	0.383131	+	0.663602i	−0.608377	+	0.793648i	$0.708179\pi$
$110$		0			0
$111$		0			0
$112$	2.00000	+	10.3923i	0.188982	+	0.981981i
$113$			12.7279i			1.19734i	0.800995	+	0.598671i	$0.204304\pi$
$113$			12.7279i			1.19734i	−0.800995	+	0.598671i	$0.795696\pi$
$114$		0			0
$115$		0			0
$116$	14.6969	−	8.48528i	1.36458	−	0.787839i
$117$		0			0
$118$		0			0
$119$	18.3712	+	6.36396i	1.68408	+	0.583383i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$	−9.00000	−	5.19615i	−0.808224	−	0.466628i
$125$		0			0
$126$		0			0
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.67423	+	6.36396i	−0.321019	+	0.556022i	−0.980699	−	0.195525i	$-0.937359\pi$
$131$	−3.67423	+	6.36396i	−0.321019	+	0.556022i	0.659679	+	0.751547i	$0.270692\pi$
$132$		0			0
$133$	18.0000	−	3.46410i	1.56080	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.34847	−	4.24264i	0.627822	−	0.362473i	−0.152086	−	0.988367i	$-0.548599\pi$
$137$	7.34847	−	4.24264i	0.627822	−	0.362473i	0.779908	+	0.625894i	$0.215266\pi$
$138$		0			0
$139$			8.66025i			0.734553i	0.930112	+	0.367277i	$0.119710\pi$
$139$			8.66025i			0.734553i	−0.930112	+	0.367277i	$0.880290\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.34847	−	12.7279i	−0.614510	−	1.06436i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	2.00000			0.164399
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.50000	+	0.866025i	−0.119713	+	0.0691164i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−1.50000	+	0.866025i	−0.119713	+	0.0691164i	0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−22.0454	+	4.24264i	−1.73742	+	0.334367i
$162$		0			0
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	0.313304	+	0.949653i	$0.398564\pi$
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	−0.979076	+	0.203497i	$0.934769\pi$
$164$	−7.34847	−	12.7279i	−0.573819	−	0.993884i
$165$		0			0
$166$		0			0
$167$	14.6969			1.13728			0.568642	−	0.822585i	$-0.307469\pi$
$167$	14.6969			1.13728			0.568642	+	0.822585i	$0.307469\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$	−1.00000	−	1.73205i	−0.0762493	−	0.132068i
$173$	3.67423	−	6.36396i	0.279347	−	0.483843i	−0.691876	−	0.722017i	$-0.743215\pi$
$173$	3.67423	−	6.36396i	0.279347	−	0.483843i	0.971223	+	0.238174i	$0.0765487\pi$
$174$		0			0
$175$		0			0
$176$		−	16.9706i		−	1.27920i
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$			15.5885i			1.15868i	0.815086	+	0.579340i	$0.196690\pi$
$181$			15.5885i			1.15868i	−0.815086	+	0.579340i	$0.803310\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−27.0000	−	15.5885i	−1.97444	−	1.13994i
$188$	−14.6969			−1.07188
$189$		0			0
$190$		0			0
$191$	−7.34847	−	4.24264i	−0.531717	−	0.306987i	0.209999	−	0.977702i	$-0.432654\pi$
$191$	−7.34847	−	4.24264i	−0.531717	−	0.306987i	−0.741715	+	0.670715i	$0.765987\pi$
$192$		0			0
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	0.531446	−	0.847092i	$-0.321649\pi$
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	−0.999326	+	0.0366998i	$0.988315\pi$
$194$		0			0
$195$		0			0
$196$	−13.0000	+	5.19615i	−0.928571	+	0.371154i
$197$		−	4.24264i		−	0.302276i	−0.988513	−	0.151138i	$-0.951706\pi$
$197$		−	4.24264i		−	0.302276i	0.988513	−	0.151138i	$-0.0482937\pi$
$198$		0			0
$199$	1.50000	−	0.866025i	0.106332	−	0.0613909i	−0.445891	−	0.895087i	$-0.647113\pi$
$199$	1.50000	−	0.866025i	0.106332	−	0.0613909i	0.552223	+	0.833696i	$0.313780\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.6969	+	16.9706i	1.03152	+	1.19110i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−12.0000	−	6.92820i	−0.832050	−	0.480384i
$209$	−29.3939			−2.03322
$210$		0			0
$211$	−11.0000			−0.757271			−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000			−0.757271			−0.378636	+	0.925546i	$0.623607\pi$
$212$	−7.34847	−	4.24264i	−0.504695	−	0.291386i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.50000	−	12.9904i	0.305480	−	0.881845i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.0454	+	12.7279i	−1.48293	+	0.856173i
$222$		0			0
$223$			19.0526i			1.27585i	0.770097	+	0.637927i	$0.220208\pi$
$223$			19.0526i			1.27585i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.67423	−	6.36396i	−0.243868	−	0.422391i	0.717945	−	0.696100i	$-0.245083\pi$
$227$	−3.67423	−	6.36396i	−0.243868	−	0.422391i	−0.961813	+	0.273709i	$0.911750\pi$
$228$		0			0
$229$	−6.00000	−	3.46410i	−0.396491	−	0.228914i	0.288478	−	0.957487i	$-0.406851\pi$
$229$	−6.00000	−	3.46410i	−0.396491	−	0.228914i	−0.684969	+	0.728572i	$0.740184\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.34847	+	4.24264i	0.481414	+	0.277945i	0.721006	−	0.692929i	$-0.243680\pi$
$233$	7.34847	+	4.24264i	0.481414	+	0.277945i	−0.239591	+	0.970874i	$0.577013\pi$
$234$		0			0
$235$		0			0
$236$	7.34847	−	12.7279i	0.478345	−	0.828517i
$237$		0			0
$238$		0			0
$239$			4.24264i			0.274434i	0.990541	+	0.137217i	$0.0438157\pi$
$239$			4.24264i			0.274434i	−0.990541	+	0.137217i	$0.956184\pi$
$240$		0			0
$241$	7.50000	−	4.33013i	0.483117	−	0.278928i	−0.238597	−	0.971119i	$-0.576688\pi$
$241$	7.50000	−	4.33013i	0.483117	−	0.278928i	0.721715	+	0.692191i	$0.243354\pi$
$242$		0			0
$243$		0			0
$244$			3.46410i			0.221766i
$245$		0			0
$246$		0			0
$247$	−12.0000	+	20.7846i	−0.763542	+	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.6969			0.927663			0.463831	−	0.885924i	$-0.346474\pi$
$251$	14.6969			0.927663			0.463831	+	0.885924i	$0.346474\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	11.0227	−	19.0919i	0.687577	−	1.19092i	−0.285042	−	0.958515i	$-0.592008\pi$
$257$	11.0227	−	19.0919i	0.687577	−	1.19092i	0.972619	−	0.232404i	$-0.0746591\pi$
$258$		0			0
$259$	0.500000	+	2.59808i	0.0310685	+	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.67423	+	2.12132i	−0.226563	+	0.130806i	−0.608985	−	0.793181i	$-0.708423\pi$
$263$	−3.67423	+	2.12132i	−0.226563	+	0.130806i	0.382422	+	0.923988i	$0.375090\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2.00000	+	3.46410i	−0.122169	+	0.211604i
$269$	−14.6969	−	25.4558i	−0.896088	−	1.55207i	−0.832452	−	0.554097i	$-0.813064\pi$
$269$	−14.6969	−	25.4558i	−0.896088	−	1.55207i	−0.0636361	−	0.997973i	$-0.520270\pi$
$270$		0			0
$271$	18.0000	+	10.3923i	1.09342	+	0.631288i	0.934485	−	0.356001i	$-0.115860\pi$
$271$	18.0000	+	10.3923i	1.09342	+	0.631288i	0.158937	+	0.987289i	$0.449193\pi$
$272$	−29.3939			−1.78227
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.00000	+	3.46410i	0.120168	+	0.208138i	0.919834	−	0.392308i	$-0.128323\pi$
$277$	2.00000	+	3.46410i	0.120168	+	0.208138i	−0.799666	+	0.600446i	$0.794990\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	21.2132i		−	1.26547i	−0.774367	−	0.632737i	$-0.781932\pi$
$281$		−	21.2132i		−	1.26547i	0.774367	−	0.632737i	$-0.218068\pi$
$282$		0			0
$283$	21.0000	−	12.1244i	1.24832	−	0.720718i	0.277546	−	0.960712i	$-0.410479\pi$
$283$	21.0000	−	12.1244i	1.24832	−	0.720718i	0.970774	+	0.239994i	$0.0771455\pi$
$284$	7.34847	−	4.24264i	0.436051	−	0.251754i
$285$		0			0
$286$		0			0
$287$	14.6969	−	12.7279i	0.867533	−	0.751305i
$288$		0			0
$289$	−18.5000	+	32.0429i	−1.08824	+	1.88488i
$290$		0			0
$291$		0			0
$292$	15.0000	+	8.66025i	0.877809	+	0.506803i
$293$	−22.0454			−1.28791			−0.643953	−	0.765065i	$-0.722707\pi$
$293$	−22.0454			−1.28791			−0.643953	+	0.765065i	$0.722707\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.6969	−	25.4558i	0.849946	−	1.47215i
$300$		0			0
$301$	2.00000	−	1.73205i	0.115278	−	0.0998337i
$302$		0			0
$303$		0			0
$304$	−24.0000	+	13.8564i	−1.37649	+	0.794719i
$305$		0			0
$306$		0			0
$307$			5.19615i			0.296560i	0.988945	+	0.148280i	$0.0473737\pi$
$307$			5.19615i			0.296560i	−0.988945	+	0.148280i	$0.952626\pi$
$308$	22.0454	−	4.24264i	1.25615	−	0.241747i
$309$		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
$313$	−13.5000	−	7.79423i	−0.763065	−	0.440556i	0.0673300	−	0.997731i	$-0.478552\pi$
$313$	−13.5000	−	7.79423i	−0.763065	−	0.440556i	−0.830395	+	0.557175i	$0.811885\pi$
$314$		0			0
$315$		0			0
$316$	−26.0000			−1.46261
$317$	−3.67423	−	2.12132i	−0.206366	−	0.119145i	0.393256	−	0.919429i	$-0.371349\pi$
$317$	−3.67423	−	2.12132i	−0.206366	−	0.119145i	−0.599621	+	0.800284i	$0.704682\pi$
$318$		0			0
$319$	−18.0000	−	31.1769i	−1.00781	−	1.74557i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			50.9117i			2.83280i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.67423	−	19.0919i	−0.202567	−	1.05257i
$330$		0			0
$331$	−6.50000	+	11.2583i	−0.357272	+	0.618814i	−0.987504	−	0.157593i	$-0.949627\pi$
$331$	−6.50000	+	11.2583i	−0.357272	+	0.618814i	0.630232	+	0.776407i	$0.282960\pi$
$332$	7.34847	+	12.7279i	0.403300	+	0.698535i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−11.0227	+	19.0919i	−0.596913	+	1.03388i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.3712	+	10.6066i	−0.986216	+	0.569392i	−0.904141	−	0.427234i	$-0.859488\pi$
$347$	−18.3712	+	10.6066i	−0.986216	+	0.569392i	−0.0820751	+	0.996626i	$0.526155\pi$
$348$		0			0
$349$			20.7846i			1.11257i	0.830990	+	0.556287i	$0.187775\pi$
$349$			20.7846i			1.11257i	−0.830990	+	0.556287i	$0.812225\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.34847	+	12.7279i	0.391120	+	0.677439i	0.992597	−	0.121450i	$-0.0387546\pi$
$353$	7.34847	+	12.7279i	0.391120	+	0.677439i	−0.601478	+	0.798889i	$0.705421\pi$
$354$		0			0
$355$		0			0
$356$	14.6969			0.778936
$357$		0			0
$358$		0			0
$359$	−11.0227	−	6.36396i	−0.581756	−	0.335877i	0.180075	−	0.983653i	$-0.442366\pi$
$359$	−11.0227	−	6.36396i	−0.581756	−	0.335877i	−0.761831	+	0.647776i	$0.775699\pi$
$360$		0			0
$361$	14.5000	+	25.1147i	0.763158	+	1.32183i
$362$		0			0
$363$		0			0
$364$	6.00000	−	17.3205i	0.314485	−	0.907841i
$365$		0			0
$366$		0			0
$367$	−13.5000	+	7.79423i	−0.704694	+	0.406855i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	−13.5000	+	7.79423i	−0.704694	+	0.406855i	0.104399	+	0.994535i	$0.466708\pi$
$368$	29.3939	−	16.9706i	1.53226	−	0.884652i
$369$		0			0
$370$		0			0
$371$	3.67423	−	10.6066i	0.190757	−	0.550667i
$372$		0			0
$373$	17.0000	−	29.4449i	0.880227	−	1.52460i	0.0291379	−	0.999575i	$-0.490724\pi$
$373$	17.0000	−	29.4449i	0.880227	−	1.52460i	0.851089	−	0.525022i	$-0.175943\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−29.3939			−1.51386
$378$		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
$383$	7.34847	−	12.7279i	0.375489	−	0.650366i	−0.614911	−	0.788597i	$-0.710808\pi$
$383$	7.34847	−	12.7279i	0.375489	−	0.650366i	0.990400	+	0.138230i	$0.0441414\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−3.00000	+	1.73205i	−0.152302	+	0.0879316i
$389$	3.67423	−	2.12132i	0.186291	−	0.107555i	−0.403954	−	0.914779i	$-0.632364\pi$
$389$	3.67423	−	2.12132i	0.186291	−	0.107555i	0.590245	+	0.807224i	$0.299031\pi$
$390$		0			0
$391$		−	62.3538i		−	3.15337i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.50000	+	2.59808i	0.225849	+	0.130394i	0.608655	−	0.793435i	$-0.291709\pi$
$397$	4.50000	+	2.59808i	0.225849	+	0.130394i	−0.382807	+	0.923828i	$0.625043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.0227	+	6.36396i	0.550448	+	0.317801i	0.749302	−	0.662228i	$-0.230389\pi$
$401$	11.0227	+	6.36396i	0.550448	+	0.317801i	−0.198855	+	0.980029i	$0.563722\pi$
$402$		0			0
$403$	9.00000	+	15.5885i	0.448322	+	0.776516i
$404$	14.6969	−	25.4558i	0.731200	−	1.26648i
$405$		0			0
$406$		0			0
$407$		−	4.24264i		−	0.210300i
$408$		0			0
$409$	−3.00000	+	1.73205i	−0.148340	+	0.0856444i	−0.572333	−	0.820021i	$-0.693962\pi$
$409$	−3.00000	+	1.73205i	−0.148340	+	0.0856444i	0.423993	+	0.905666i	$0.360628\pi$
$410$		0			0
$411$		0			0
$412$			17.3205i			0.853320i
$413$	18.3712	+	6.36396i	0.903986	+	0.313150i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−29.3939			−1.43598			−0.717992	−	0.696051i	$-0.754939\pi$
$419$	−29.3939			−1.43598			−0.717992	+	0.696051i	$0.754939\pi$
$420$		0			0
$421$	5.00000			0.243685			0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000			0.243685			0.121843	+	0.992549i	$0.461120\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−4.50000	+	0.866025i	−0.217770	+	0.0419099i
$428$			16.9706i			0.820303i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$			25.9808i			1.24856i	0.781202	+	0.624278i	$0.214607\pi$
$433$			25.9808i			1.24856i	−0.781202	+	0.624278i	$0.785393\pi$
$434$		0			0
$435$		0			0
$436$	8.00000	−	13.8564i	0.383131	−	0.663602i
$437$	−29.3939	−	50.9117i	−1.40610	−	2.43544i
$438$		0			0
$439$	−16.5000	−	9.52628i	−0.787502	−	0.454665i	0.0515804	−	0.998669i	$-0.483574\pi$
$439$	−16.5000	−	9.52628i	−0.787502	−	0.454665i	−0.839082	+	0.544004i	$0.816908\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.0227	−	6.36396i	−0.523704	−	0.302361i	0.214745	−	0.976670i	$-0.431108\pi$
$443$	−11.0227	−	6.36396i	−0.523704	−	0.302361i	−0.738449	+	0.674309i	$0.764441\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	16.0000	−	13.8564i	0.755929	−	0.654654i
$449$			25.4558i			1.20134i	0.799499	+	0.600668i	$0.205099\pi$
$449$			25.4558i			1.20134i	−0.799499	+	0.600668i	$0.794901\pi$
$450$		0			0
$451$	−27.0000	+	15.5885i	−1.27138	+	0.734032i
$452$	22.0454	−	12.7279i	1.03693	−	0.598671i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	−0.461067	−	0.887365i	$-0.652533\pi$
$457$	11.5000	−	19.9186i	0.537947	−	0.931752i	0.999014	−	0.0443868i	$-0.0141334\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.34847			0.342252			0.171126	−	0.985249i	$-0.445259\pi$
$461$	7.34847			0.342252			0.171126	+	0.985249i	$0.445259\pi$
$462$		0			0
$463$	29.0000			1.34774			0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000			1.34774			0.673872	+	0.738848i	$0.264630\pi$
$464$	−29.3939	−	16.9706i	−1.36458	−	0.787839i
$465$		0			0
$466$		0			0
$467$	3.67423	−	6.36396i	0.170023	−	0.294489i	−0.768404	−	0.639965i	$-0.778949\pi$
$467$	3.67423	−	6.36396i	0.170023	−	0.294489i	0.938428	+	0.345476i	$0.112282\pi$
$468$		0			0
$469$	−5.00000	−	1.73205i	−0.230879	−	0.0799787i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.67423	+	2.12132i	−0.168941	+	0.0975384i
$474$		0			0
$475$		0			0
$476$	−7.34847	−	38.1838i	−0.336817	−	1.75015i
$477$		0			0
$478$		0			0
$479$	7.34847	+	12.7279i	0.335760	+	0.581554i	0.983631	−	0.180197i	$-0.0576735\pi$
$479$	7.34847	+	12.7279i	0.335760	+	0.581554i	−0.647870	+	0.761751i	$0.724340\pi$
$480$		0			0
$481$	−3.00000	−	1.73205i	−0.136788	−	0.0789747i
$482$		0			0
$483$		0			0
$484$	−14.0000			−0.636364
$485$		0			0
$486$		0			0
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	0.430486	−	0.902597i	$-0.358342\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	16.9706i		−	0.765871i	−0.923775	−	0.382935i	$-0.874913\pi$
$491$		−	16.9706i		−	0.765871i	0.923775	−	0.382935i	$-0.125087\pi$
$492$		0			0
$493$	−54.0000	+	31.1769i	−2.43204	+	1.40414i
$494$		0			0
$495$		0			0
$496$			20.7846i			0.933257i
$497$	7.34847	+	8.48528i	0.329624	+	0.380617i
$498$		0			0
$499$	1.00000	−	1.73205i	0.0447661	−	0.0775372i	−0.842774	−	0.538267i	$-0.819079\pi$
$499$	1.00000	−	1.73205i	0.0447661	−	0.0775372i	0.887540	+	0.460730i	$0.152412\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−29.3939			−1.31061			−0.655304	−	0.755365i	$-0.727460\pi$
$503$	−29.3939			−1.31061			−0.655304	+	0.755365i	$0.727460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−17.0000	−	29.4449i	−0.754253	−	1.30640i
$509$	3.67423	−	6.36396i	0.162858	−	0.282078i	−0.773035	−	0.634364i	$-0.781262\pi$
$509$	3.67423	−	6.36396i	0.162858	−	0.282078i	0.935892	+	0.352286i	$0.114596\pi$
$510$		0			0
$511$	−7.50000	+	21.6506i	−0.331780	+	0.957768i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			31.1769i			1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.0227	−	19.0919i	−0.482913	−	0.836431i	0.516894	−	0.856049i	$-0.327088\pi$
$521$	−11.0227	−	19.0919i	−0.482913	−	0.836431i	−0.999808	+	0.0196188i	$0.993755\pi$
$522$		0			0
$523$	22.5000	+	12.9904i	0.983856	+	0.568030i	0.903432	−	0.428731i	$-0.141039\pi$
$523$	22.5000	+	12.9904i	0.983856	+	0.568030i	0.0804241	+	0.996761i	$0.474373\pi$
$524$	14.6969			0.642039
$525$		0			0
$526$		0			0
$527$	33.0681	+	19.0919i	1.44047	+	0.831655i
$528$		0			0
$529$	24.5000	+	42.4352i	1.06522	+	1.84501i
$530$		0			0
$531$		0			0
$532$	−24.0000	−	27.7128i	−1.04053	−	1.20150i
$533$			25.4558i			1.10262i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11.0227	+	27.5772i	0.474781	+	1.18783i
$540$		0			0
$541$	−4.00000	+	6.92820i	−0.171973	+	0.297867i	−0.939110	−	0.343617i	$-0.888348\pi$
$541$	−4.00000	+	6.92820i	−0.171973	+	0.297867i	0.767136	+	0.641484i	$0.221681\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	26.0000			1.11168			0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000			1.11168			0.555840	+	0.831289i	$0.312397\pi$
$548$	−14.6969	−	8.48528i	−0.627822	−	0.362473i
$549$		0			0
$550$		0			0
$551$	−29.3939	+	50.9117i	−1.25222	+	2.16891i
$552$		0			0
$553$	−6.50000	−	33.7750i	−0.276408	−	1.43626i
$554$		0			0
$555$		0			0
$556$	15.0000	−	8.66025i	0.636142	−	0.367277i
$557$	11.0227	−	6.36396i	0.467047	−	0.269650i	−0.247956	−	0.968771i	$-0.579759\pi$
$557$	11.0227	−	6.36396i	0.467047	−	0.269650i	0.715003	+	0.699122i	$0.246425\pi$
$558$		0			0
$559$			3.46410i			0.146516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.3712	−	31.8198i	−0.774253	−	1.34104i	−0.935214	−	0.354084i	$-0.884793\pi$
$563$	−18.3712	−	31.8198i	−0.774253	−	1.34104i	0.160961	−	0.986961i	$-0.448541\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.0454	−	12.7279i	−0.924192	−	0.533582i	−0.0392217	−	0.999231i	$-0.512488\pi$
$569$	−22.0454	−	12.7279i	−0.924192	−	0.533582i	−0.884970	+	0.465648i	$0.845821\pi$
$570$		0			0
$571$	−1.00000	−	1.73205i	−0.0418487	−	0.0724841i	0.844342	−	0.535804i	$-0.179991\pi$
$571$	−1.00000	−	1.73205i	−0.0418487	−	0.0724841i	−0.886191	+	0.463320i	$0.846658\pi$
$572$	−14.6969	+	25.4558i	−0.614510	+	1.06436i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−31.5000	+	18.1865i	−1.31136	+	0.757115i	−0.982322	−	0.187201i	$-0.940059\pi$
$577$	−31.5000	+	18.1865i	−1.31136	+	0.757115i	−0.329040	+	0.944316i	$0.606725\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−14.6969	+	12.7279i	−0.609732	+	0.528043i
$582$		0			0
$583$	−9.00000	+	15.5885i	−0.372742	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	22.0454			0.909911			0.454956	−	0.890514i	$-0.349655\pi$
$587$	22.0454			0.909911			0.454956	+	0.890514i	$0.349655\pi$
$588$		0			0
$589$	36.0000			1.48335
$590$		0			0
$591$		0			0
$592$	−2.00000	−	3.46410i	−0.0821995	−	0.142374i
$593$	22.0454	−	38.1838i	0.905296	−	1.56802i	0.0847775	−	0.996400i	$-0.472982\pi$
$593$	22.0454	−	38.1838i	0.905296	−	1.56802i	0.820519	−	0.571619i	$-0.193685\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.0454	−	12.7279i	0.900751	−	0.520049i	0.0233072	−	0.999728i	$-0.492580\pi$
$599$	22.0454	−	12.7279i	0.900751	−	0.520049i	0.877444	+	0.479680i	$0.159247\pi$
$600$		0			0
$601$			17.3205i			0.706518i	0.935526	+	0.353259i	$0.114927\pi$
$601$			17.3205i			0.706518i	−0.935526	+	0.353259i	$0.885073\pi$
$602$		0			0
$603$		0			0
$604$	5.00000	−	8.66025i	0.203447	−	0.352381i
$605$		0			0
$606$		0			0
$607$	1.50000	+	0.866025i	0.0608831	+	0.0351509i	0.530133	−	0.847915i	$-0.322142\pi$
$607$	1.50000	+	0.866025i	0.0608831	+	0.0351509i	−0.469249	+	0.883066i	$0.655475\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22.0454	+	12.7279i	0.891862	+	0.514917i
$612$		0			0
$613$	8.00000	+	13.8564i	0.323117	+	0.559655i	0.981129	−	0.193352i	$-0.0619359\pi$
$613$	8.00000	+	13.8564i	0.323117	+	0.559655i	−0.658012	+	0.753007i	$0.728603\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21.2132i			0.854011i	0.904249	+	0.427006i	$0.140432\pi$
$617$			21.2132i			0.854011i	−0.904249	+	0.427006i	$0.859568\pi$
$618$		0			0
$619$	4.50000	−	2.59808i	0.180870	−	0.104425i	−0.406831	−	0.913503i	$-0.633366\pi$
$619$	4.50000	−	2.59808i	0.180870	−	0.104425i	0.587701	+	0.809078i	$0.300033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.67423	+	19.0919i	0.147205	+	0.764900i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	3.00000	+	1.73205i	0.119713	+	0.0691164i
$629$	−7.34847			−0.293003
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	24.0000	+	3.46410i	0.950915	+	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.0227	+	6.36396i	−0.435371	+	0.251361i	−0.701632	−	0.712540i	$-0.747545\pi$
$641$	−11.0227	+	6.36396i	−0.435371	+	0.251361i	0.266261	+	0.963901i	$0.414212\pi$
$642$		0			0
$643$			12.1244i			0.478138i	0.971003	+	0.239069i	$0.0768422\pi$
$643$			12.1244i			0.478138i	−0.971003	+	0.239069i	$0.923158\pi$
$644$	29.3939	+	33.9411i	1.15828	+	1.33747i
$645$		0			0
$646$		0			0
$647$	−3.67423	−	6.36396i	−0.144449	−	0.250193i	0.784718	−	0.619853i	$-0.212808\pi$
$647$	−3.67423	−	6.36396i	−0.144449	−	0.250193i	−0.929167	+	0.369660i	$0.879474\pi$
$648$		0			0
$649$	−27.0000	−	15.5885i	−1.05984	−	0.611900i
$650$		0			0
$651$		0			0
$652$	34.0000			1.33154
$653$	25.7196	+	14.8492i	1.00649	+	0.581096i	0.910161	−	0.414254i	$-0.135958\pi$
$653$	25.7196	+	14.8492i	1.00649	+	0.581096i	0.0963261	+	0.995350i	$0.469291\pi$
$654$		0			0
$655$		0			0
$656$	−14.6969	+	25.4558i	−0.573819	+	0.993884i
$657$		0			0
$658$		0			0
$659$			21.2132i			0.826349i	0.910652	+	0.413175i	$0.135580\pi$
$659$			21.2132i			0.826349i	−0.910652	+	0.413175i	$0.864420\pi$
$660$		0			0
$661$	25.5000	−	14.7224i	0.991835	−	0.572636i	0.0860127	−	0.996294i	$-0.472587\pi$
$661$	25.5000	−	14.7224i	0.991835	−	0.572636i	0.905822	+	0.423658i	$0.139254\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	36.0000	−	62.3538i	1.39393	−	2.41435i
$668$	−14.6969	−	25.4558i	−0.568642	−	0.984916i
$669$		0			0
$670$		0			0
$671$	7.34847			0.283685
$672$		0			0
$673$	−5.00000			−0.192736			−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000			−0.192736			−0.0963679	+	0.995346i	$0.530723\pi$
$674$		0			0
$675$		0			0
$676$	−1.00000	−	1.73205i	−0.0384615	−	0.0666173i
$677$	−14.6969	+	25.4558i	−0.564849	+	0.978348i	0.432214	+	0.901771i	$0.357732\pi$
$677$	−14.6969	+	25.4558i	−0.564849	+	0.978348i	−0.997064	+	0.0765767i	$0.975601\pi$
$678$		0			0
$679$	−3.00000	−	3.46410i	−0.115129	−	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.3712	+	10.6066i	−0.702953	+	0.405850i	−0.808446	−	0.588570i	$-0.799691\pi$
$683$	−18.3712	+	10.6066i	−0.702953	+	0.405850i	0.105493	+	0.994420i	$0.466358\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−2.00000	+	3.46410i	−0.0762493	+	0.132068i
$689$	7.34847	+	12.7279i	0.279954	+	0.484895i
$690$		0			0
$691$	34.5000	+	19.9186i	1.31244	+	0.757739i	0.982500	−	0.186263i	$-0.0596375\pi$
$691$	34.5000	+	19.9186i	1.31244	+	0.757739i	0.329942	+	0.944001i	$0.392971\pi$
$692$	−14.6969			−0.558694
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	27.0000	+	46.7654i	1.02270	+	1.77136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	29.6985i		−	1.12170i	−0.827919	−	0.560848i	$-0.810475\pi$
$701$		−	29.6985i		−	1.12170i	0.827919	−	0.560848i	$-0.189525\pi$
$702$		0			0
$703$	−6.00000	+	3.46410i	−0.226294	+	0.130651i
$704$	−29.3939	+	16.9706i	−1.10782	+	0.639602i
$705$		0			0
$706$		0			0
$707$	36.7423	+	12.7279i	1.38184	+	0.478683i
$708$		0			0
$709$	11.5000	−	19.9186i	0.431892	−	0.748058i	−0.565145	−	0.824992i	$-0.691180\pi$
$709$	11.5000	−	19.9186i	0.431892	−	0.748058i	0.997036	+	0.0769337i	$0.0245130\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−44.0908			−1.65121
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.34847	−	12.7279i	0.274052	−	0.474671i	−0.695844	−	0.718193i	$-0.744969\pi$
$719$	7.34847	−	12.7279i	0.274052	−	0.474671i	0.969895	+	0.243522i	$0.0783027\pi$
$720$		0			0
$721$	−22.5000	+	4.33013i	−0.837944	+	0.161262i
$722$		0			0
$723$		0			0
$724$	27.0000	−	15.5885i	1.00345	−	0.579340i
$725$		0			0
$726$		0			0
$727$		−	48.4974i		−	1.79867i	−0.437260	−	0.899335i	$-0.644051\pi$
$727$		−	48.4974i		−	1.79867i	0.437260	−	0.899335i	$-0.355949\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.67423	+	6.36396i	0.135896	+	0.235380i
$732$		0			0
$733$	−37.5000	−	21.6506i	−1.38509	−	0.799684i	−0.392337	−	0.919822i	$-0.628333\pi$
$733$	−37.5000	−	21.6506i	−1.38509	−	0.799684i	−0.992757	+	0.120137i	$0.961667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.34847	+	4.24264i	0.270684	+	0.156280i
$738$		0			0
$739$	−26.5000	−	45.8993i	−0.974818	−	1.68843i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−26.5000	−	45.8993i	−0.974818	−	1.68843i	−0.294285	−	0.955718i	$-0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	16.9706i		−	0.622590i	−0.950313	−	0.311295i	$-0.899237\pi$
$743$		−	16.9706i		−	0.622590i	0.950313	−	0.311295i	$-0.100763\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$			62.3538i			2.27988i
$749$	−22.0454	+	4.24264i	−0.805522	+	0.155023i
$750$		0			0
$751$	23.5000	−	40.7032i	0.857527	−	1.48528i	−0.0167534	−	0.999860i	$-0.505333\pi$
$751$	23.5000	−	40.7032i	0.857527	−	1.48528i	0.874281	−	0.485421i	$-0.161334\pi$
$752$	14.6969	+	25.4558i	0.535942	+	0.928279i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−35.0000			−1.27210			−0.636048	−	0.771649i	$-0.719432\pi$
$757$	−35.0000			−1.27210			−0.636048	+	0.771649i	$0.719432\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14.6969	−	25.4558i	0.532764	−	0.922774i	−0.466504	−	0.884519i	$-0.654487\pi$
$761$	14.6969	−	25.4558i	0.532764	−	0.922774i	0.999268	−	0.0382548i	$-0.0121798\pi$
$762$		0			0
$763$	20.0000	+	6.92820i	0.724049	+	0.250818i
$764$			16.9706i			0.613973i
$765$		0			0
$766$		0			0
$767$	−22.0454	+	12.7279i	−0.796014	+	0.459579i
$768$		0			0
$769$			15.5885i			0.562134i	0.959688	+	0.281067i	$0.0906883\pi$
$769$			15.5885i			0.562134i	−0.959688	+	0.281067i	$0.909312\pi$
$770$		0			0
$771$		0			0
$772$	−13.0000	+	22.5167i	−0.467880	+	0.810392i
$773$	−7.34847	−	12.7279i	−0.264306	−	0.457792i	0.703076	−	0.711115i	$-0.251810\pi$
$773$	−7.34847	−	12.7279i	−0.264306	−	0.457792i	−0.967382	+	0.253324i	$0.918476\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	44.0908	+	25.4558i	1.57972	+	0.912050i
$780$		0			0
$781$	−9.00000	−	15.5885i	−0.322045	−	0.557799i
$782$		0			0
$783$		0			0
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$		0			0
$786$		0			0
$787$	7.50000	−	4.33013i	0.267346	−	0.154352i	−0.360335	−	0.932823i	$-0.617338\pi$
$787$	7.50000	−	4.33013i	0.267346	−	0.154352i	0.627681	+	0.778471i	$0.284004\pi$
$788$	−7.34847	+	4.24264i	−0.261778	+	0.151138i
$789$		0			0
$790$		0			0
$791$	22.0454	+	25.4558i	0.783844	+	0.905106i
$792$		0			0
$793$	3.00000	−	5.19615i	0.106533	−	0.184521i
$794$		0			0
$795$		0			0
$796$	−3.00000	−	1.73205i	−0.106332	−	0.0613909i
$797$	−29.3939			−1.04118			−0.520592	−	0.853805i	$-0.674289\pi$
$797$	−29.3939			−1.04118			−0.520592	+	0.853805i	$0.674289\pi$
$798$		0			0
$799$	54.0000			1.91038
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18.3712	−	31.8198i	0.648305	−	1.12290i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.6969	−	8.48528i	0.516717	−	0.298327i	−0.218874	−	0.975753i	$-0.570238\pi$
$809$	14.6969	−	8.48528i	0.516717	−	0.298327i	0.735590	+	0.677427i	$0.236905\pi$
$810$		0			0
$811$		−	22.5167i		−	0.790667i	−0.918538	−	0.395333i	$-0.870629\pi$
$811$		−	22.5167i		−	0.790667i	0.918538	−	0.395333i	$-0.129371\pi$
$812$	14.6969	−	42.4264i	0.515761	−	1.48888i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.00000	+	3.46410i	0.209913	+	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.3939	+	16.9706i	1.02585	+	0.592277i	0.915794	−	0.401648i	$-0.131563\pi$
$821$	29.3939	+	16.9706i	1.02585	+	0.592277i	0.110059	+	0.993925i	$0.464896\pi$
$822$		0			0
$823$	−14.5000	−	25.1147i	−0.505438	−	0.875445i	−0.999980	−	0.00629095i	$-0.997998\pi$
$823$	−14.5000	−	25.1147i	−0.505438	−	0.875445i	0.494542	−	0.869154i	$-0.335336\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			46.6690i			1.62284i	0.584462	+	0.811421i	$0.301305\pi$
$827$			46.6690i			1.62284i	−0.584462	+	0.811421i	$0.698695\pi$
$828$		0			0
$829$	7.50000	−	4.33013i	0.260486	−	0.150392i	−0.364070	−	0.931371i	$-0.618613\pi$
$829$	7.50000	−	4.33013i	0.260486	−	0.150392i	0.624556	+	0.780980i	$0.285280\pi$
$830$		0			0
$831$		0			0
$832$			27.7128i			0.960769i
$833$	47.7650	−	19.0919i	1.65496	−	0.661495i
$834$		0			0
$835$		0			0
$836$	29.3939	+	50.9117i	1.01661	+	1.76082i
$837$		0			0
$838$		0			0
$839$	36.7423			1.26849			0.634243	−	0.773133i	$-0.281312\pi$
$839$	36.7423			1.26849			0.634243	+	0.773133i	$0.281312\pi$
$840$		0			0
$841$	−43.0000			−1.48276
$842$		0			0
$843$		0			0
$844$	11.0000	+	19.0526i	0.378636	+	0.655816i
$845$		0			0
$846$		0			0
$847$	−3.50000	−	18.1865i	−0.120261	−	0.624897i
$848$			16.9706i			0.582772i
$849$		0			0
$850$		0			0
$851$	7.34847	−	4.24264i	0.251902	−	0.145436i
$852$		0			0
$853$		−	46.7654i		−	1.60122i	−0.599189	−	0.800608i	$-0.704510\pi$
$853$		−	46.7654i		−	1.60122i	0.599189	−	0.800608i	$-0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	18.0000	+	10.3923i	0.614152	+	0.354581i	0.774589	−	0.632465i	$-0.217957\pi$
$859$	18.0000	+	10.3923i	0.614152	+	0.354581i	−0.160437	+	0.987046i	$0.551290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.3712	−	10.6066i	−0.625362	−	0.361053i	0.153592	−	0.988134i	$-0.450916\pi$
$863$	−18.3712	−	10.6066i	−0.625362	−	0.361053i	−0.778954	+	0.627081i	$0.784249\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−27.0000	+	5.19615i	−0.916440	+	0.176369i
$869$			55.1543i			1.87098i
$870$		0			0
$871$	6.00000	−	3.46410i	0.203302	−	0.117377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.5000	+	40.7032i	−0.793539	+	1.37445i	0.130224	+	0.991485i	$0.458430\pi$
$877$	−23.5000	+	40.7032i	−0.793539	+	1.37445i	−0.923763	+	0.382965i	$0.874903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.34847			0.247576			0.123788	−	0.992309i	$-0.460496\pi$
$881$	7.34847			0.247576			0.123788	+	0.992309i	$0.460496\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$	44.0908	+	25.4558i	1.48293	+	0.856173i
$885$		0			0
$886$		0			0
$887$	18.3712	−	31.8198i	0.616844	−	1.06840i	−0.373214	−	0.927745i	$-0.621744\pi$
$887$	18.3712	−	31.8198i	0.616844	−	1.06840i	0.990058	−	0.140659i	$-0.0449222\pi$
$888$		0			0
$889$	34.0000	−	29.4449i	1.14032	−	0.987549i
$890$		0			0
$891$		0			0
$892$	33.0000	−	19.0526i	1.10492	−	0.637927i
$893$	44.0908	−	25.4558i	1.47544	−	0.851847i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	22.0454	+	38.1838i	0.735256	+	1.27350i
$900$		0			0
$901$	27.0000	+	15.5885i	0.899500	+	0.519327i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.50000	+	4.33013i	0.0830111	+	0.143780i	0.904542	−	0.426385i	$-0.140213\pi$
$907$	2.50000	+	4.33013i	0.0830111	+	0.143780i	−0.821531	+	0.570164i	$0.806880\pi$
$908$	−7.34847	+	12.7279i	−0.243868	+	0.422391i
$909$		0			0
$910$		0			0
$911$		−	8.48528i		−	0.281130i	−0.990071	−	0.140565i	$-0.955108\pi$
$911$		−	8.48528i		−	0.281130i	0.990071	−	0.140565i	$-0.0448919\pi$
$912$		0			0
$913$	27.0000	−	15.5885i	0.893570	−	0.515903i
$914$		0			0
$915$		0			0
$916$			13.8564i			0.457829i
$917$	3.67423	+	19.0919i	0.121334	+	0.630470i
$918$		0			0
$919$	−12.5000	+	21.6506i	−0.412337	+	0.714189i	−0.995145	−	0.0984214i	$-0.968621\pi$
$919$	−12.5000	+	21.6506i	−0.412337	+	0.714189i	0.582808	+	0.812610i	$0.301954\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.6969			−0.483756
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18.3712	+	31.8198i	−0.602739	+	1.04397i	0.389666	+	0.920956i	$0.372591\pi$
$929$	−18.3712	+	31.8198i	−0.602739	+	1.04397i	−0.992404	+	0.123018i	$0.960743\pi$
$930$		0			0
$931$	30.0000	−	38.1051i	0.983210	−	1.24884i
$932$		−	16.9706i		−	0.555889i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	5.19615i		−	0.169751i	−0.996392	−	0.0848755i	$-0.972951\pi$
$937$		−	5.19615i		−	0.169751i	0.996392	−	0.0848755i	$-0.0270492\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.67423	−	6.36396i	−0.119777	−	0.207459i	0.799902	−	0.600130i	$-0.204885\pi$
$941$	−3.67423	−	6.36396i	−0.119777	−	0.207459i	−0.919679	+	0.392671i	$0.871551\pi$
$942$		0			0
$943$	−54.0000	−	31.1769i	−1.75848	−	1.01526i
$944$	−29.3939			−0.956689
$945$		0			0
$946$		0			0
$947$	11.0227	+	6.36396i	0.358190	+	0.206801i	0.668286	−	0.743904i	$-0.267028\pi$
$947$	11.0227	+	6.36396i	0.358190	+	0.206801i	−0.310097	+	0.950705i	$0.600361\pi$
$948$		0			0
$949$	−15.0000	−	25.9808i	−0.486921	−	0.843371i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	42.4264i		−	1.37433i	−0.726503	−	0.687163i	$-0.758856\pi$
$953$		−	42.4264i		−	1.37433i	0.726503	−	0.687163i	$-0.241144\pi$
$954$		0			0
$955$		0			0
$956$	7.34847	−	4.24264i	0.237666	−	0.137217i
$957$		0			0
$958$		0			0
$959$	7.34847	−	21.2132i	0.237294	−	0.685010i
$960$		0			0
$961$	−2.00000	+	3.46410i	−0.0645161	+	0.111745i
$962$		0			0
$963$		0			0
$964$	−15.0000	−	8.66025i	−0.483117	−	0.278928i
$965$		0			0
$966$		0			0
$967$	11.0000			0.353736			0.176868	−	0.984235i	$-0.443403\pi$
$967$	11.0000			0.353736			0.176868	+	0.984235i	$0.443403\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22.0454	−	38.1838i	0.707471	−	1.22538i	−0.258322	−	0.966059i	$-0.583169\pi$
$971$	22.0454	−	38.1838i	0.707471	−	1.22538i	0.965792	−	0.259316i	$-0.0834972\pi$
$972$		0			0
$973$	15.0000	+	17.3205i	0.480878	+	0.555270i
$974$		0			0
$975$		0			0
$976$	6.00000	−	3.46410i	0.192055	−	0.110883i
$977$	−29.3939	+	16.9706i	−0.940393	+	0.542936i	−0.890084	−	0.455797i	$-0.849354\pi$
$977$	−29.3939	+	16.9706i	−0.940393	+	0.542936i	−0.0503098	+	0.998734i	$0.516021\pi$
$978$		0			0
$979$		−	31.1769i		−	0.996419i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.0227	−	19.0919i	−0.351570	−	0.608937i	0.634955	−	0.772549i	$-0.281019\pi$
$983$	−11.0227	−	19.0919i	−0.351570	−	0.608937i	−0.986525	+	0.163613i	$0.947685\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	48.0000			1.52708
$989$	−7.34847	−	4.24264i	−0.233668	−	0.134908i
$990$		0			0
$991$	23.5000	+	40.7032i	0.746502	+	1.29298i	0.949490	+	0.313798i	$0.101602\pi$
$991$	23.5000	+	40.7032i	0.746502	+	1.29298i	−0.202988	+	0.979181i	$0.565065\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.bk.b.26.1	yes	4
3.2	odd	2	inner	1575.2.bk.b.26.2	yes	4
5.2	odd	4		1575.2.bc.b.1349.3		8
5.3	odd	4		1575.2.bc.b.1349.1		8
5.4	even	2		1575.2.bk.a.26.1	✓	4
7.3	odd	6	inner	1575.2.bk.b.1151.2	yes	4
15.2	even	4		1575.2.bc.b.1349.4		8
15.8	even	4		1575.2.bc.b.1349.2		8
15.14	odd	2		1575.2.bk.a.26.2	yes	4
21.17	even	6	inner	1575.2.bk.b.1151.1	yes	4
35.3	even	12		1575.2.bc.b.899.4		8
35.17	even	12		1575.2.bc.b.899.2		8
35.24	odd	6		1575.2.bk.a.1151.2	yes	4
105.17	odd	12		1575.2.bc.b.899.1		8
105.38	odd	12		1575.2.bc.b.899.3		8
105.59	even	6		1575.2.bk.a.1151.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.bc.b.899.1		8	105.17	odd	12
1575.2.bc.b.899.2		8	35.17	even	12
1575.2.bc.b.899.3		8	105.38	odd	12
1575.2.bc.b.899.4		8	35.3	even	12
1575.2.bc.b.1349.1		8	5.3	odd	4
1575.2.bc.b.1349.2		8	15.8	even	4
1575.2.bc.b.1349.3		8	5.2	odd	4
1575.2.bc.b.1349.4		8	15.2	even	4
1575.2.bk.a.26.1	✓	4	5.4	even	2
1575.2.bk.a.26.2	yes	4	15.14	odd	2
1575.2.bk.a.1151.1	yes	4	105.59	even	6
1575.2.bk.a.1151.2	yes	4	35.24	odd	6
1575.2.bk.b.26.1	yes	4	1.1	even	1	trivial
1575.2.bk.b.26.2	yes	4	3.2	odd	2	inner
1575.2.bk.b.1151.1	yes	4	21.17	even	6	inner
1575.2.bk.b.1151.2	yes	4	7.3	odd	6	inner

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{4} +(2.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{4} +(2.00000 - 1.73205i) q^{7} +(-3.67423 + 2.12132i) q^{11} +3.46410i q^{13} +(-2.00000 + 3.46410i) q^{16} +(3.67423 + 6.36396i) q^{17} +(6.00000 + 3.46410i) q^{19} +(-7.34847 - 4.24264i) q^{23} +(-5.00000 - 1.73205i) q^{28} +8.48528i q^{29} +(4.50000 - 2.59808i) q^{31} +(-0.500000 + 0.866025i) q^{37} +7.34847 q^{41} +1.00000 q^{43} +(7.34847 + 4.24264i) q^{44} +(3.67423 - 6.36396i) q^{47} +(1.00000 - 6.92820i) q^{49} +(6.00000 - 3.46410i) q^{52} +(3.67423 - 2.12132i) q^{53} +(3.67423 + 6.36396i) q^{59} +(-1.50000 - 0.866025i) q^{61} +8.00000 q^{64} +(-1.00000 - 1.73205i) q^{67} +(7.34847 - 12.7279i) q^{68} +4.24264i q^{71} +(-7.50000 + 4.33013i) q^{73} -13.8564i q^{76} +(-3.67423 + 10.6066i) q^{77} +(6.50000 - 11.2583i) q^{79} -7.34847 q^{83} +(-3.67423 + 6.36396i) q^{89} +(6.00000 + 6.92820i) q^{91} +16.9706i q^{92} -1.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^7	\( 4 q - 4 q^{4} + 8 q^{7} - 8 q^{16} + 24 q^{19} - 20 q^{28} + 18 q^{31} - 2 q^{37} + 4 q^{43} + 4 q^{49} + 24 q^{52} - 6 q^{61} + 32 q^{64} - 4 q^{67} - 30 q^{73} + 26 q^{79} + 24 q^{91}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^7 - 8 * q^16 + 24 * q^19 - 20 * q^28 + 18 * q^31 - 2 * q^37 + 4 * q^43 + 4 * q^49 + 24 * q^52 - 6 * q^61 + 32 * q^64 - 4 * q^67 - 30 * q^73 + 26 * q^79 + 24 * q^91

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