LMFDB - Embedded newform 1575.2.bk.b.1151.2

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			1151.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1575.1151
Dual form			1575.2.bk.b.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(-1.00000 + 1.73205i) q^{4} +(2.00000 + 1.73205i) q^{7} +(3.67423 + 2.12132i) q^{11} -3.46410i q^{13} +(-2.00000 - 3.46410i) q^{16} +(-3.67423 + 6.36396i) q^{17} +(6.00000 - 3.46410i) q^{19} +(7.34847 - 4.24264i) q^{23} +(-5.00000 + 1.73205i) q^{28} +8.48528i q^{29} +(4.50000 + 2.59808i) q^{31} +(-0.500000 - 0.866025i) q^{37} -7.34847 q^{41} +1.00000 q^{43} +(-7.34847 + 4.24264i) q^{44} +(-3.67423 - 6.36396i) q^{47} +(1.00000 + 6.92820i) q^{49} +(6.00000 + 3.46410i) q^{52} +(-3.67423 - 2.12132i) q^{53} +(-3.67423 + 6.36396i) q^{59} +(-1.50000 + 0.866025i) q^{61} +8.00000 q^{64} +(-1.00000 + 1.73205i) q^{67} +(-7.34847 - 12.7279i) q^{68} +4.24264i q^{71} +(-7.50000 - 4.33013i) q^{73} +13.8564i q^{76} +(3.67423 + 10.6066i) q^{77} +(6.50000 + 11.2583i) q^{79} +7.34847 q^{83} +(3.67423 + 6.36396i) q^{89} +(6.00000 - 6.92820i) q^{91} +16.9706i q^{92} +1.73205i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

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

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	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$		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	$-0.112432\pi$
$11$	3.67423	+	2.12132i	1.10782	+	0.639602i	0.169559	+	0.985520i	$0.445766\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−3.67423	+	6.36396i	−0.891133	+	1.54349i	−0.0526138	+	0.998615i	$0.516755\pi$
$17$	−3.67423	+	6.36396i	−0.891133	+	1.54349i	−0.838519	+	0.544872i	$0.816578\pi$
$18$		0			0
$19$	6.00000	−	3.46410i	1.37649	−	0.794719i	0.384759	−	0.923017i	$-0.374285\pi$
$19$	6.00000	−	3.46410i	1.37649	−	0.794719i	0.991736	+	0.128298i	$0.0409513\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.34847	−	4.24264i	1.53226	−	0.884652i	0.533005	−	0.846112i	$-0.321063\pi$
$23$	7.34847	−	4.24264i	1.53226	−	0.884652i	0.999257	−	0.0385394i	$-0.0122705\pi$
$24$		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	0.846339	−	0.532645i	$-0.178802\pi$
$31$	4.50000	+	2.59808i	0.808224	+	0.466628i	−0.0381148	+	0.999273i	$0.512135\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	0.821995	−	0.569495i	$-0.192861\pi$
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.34847			−1.14764			−0.573819	−	0.818982i	$-0.694539\pi$
$41$	−7.34847			−1.14764			−0.573819	+	0.818982i	$0.694539\pi$
$42$		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	−0.999117	−	0.0420122i	$-0.986623\pi$
$47$	−3.67423	−	6.36396i	−0.535942	−	0.928279i	0.463175	−	0.886267i	$-0.346710\pi$
$48$		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	$-0.427450\pi$
$53$	−3.67423	−	2.12132i	−0.504695	−	0.291386i	−0.730650	+	0.682752i	$0.760783\pi$
$54$		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	$-0.992096\pi$
$59$	−3.67423	+	6.36396i	−0.478345	+	0.828517i	0.521347	+	0.853345i	$0.325430\pi$
$60$		0			0
$61$	−1.50000	+	0.866025i	−0.192055	+	0.110883i	−0.592944	−	0.805243i	$-0.702035\pi$
$61$	−1.50000	+	0.866025i	−0.192055	+	0.110883i	0.400889	+	0.916127i	$0.368701\pi$
$62$		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	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$	−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	−0.00787336	−	0.999969i	$-0.502506\pi$
$73$	−7.50000	−	4.33013i	−0.877809	−	0.506803i	−0.869935	+	0.493166i	$0.835840\pi$
$74$		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	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.34847			0.806599			0.403300	−	0.915068i	$-0.367863\pi$
$83$	7.34847			0.806599			0.403300	+	0.915068i	$0.367863\pi$
$84$		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	$-0.0393255\pi$
$89$	3.67423	+	6.36396i	0.389468	+	0.674579i	−0.602910	+	0.797809i	$0.705992\pi$
$90$		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	$0.0280257\pi$
$97$			1.73205i			0.175863i	−0.996127	+	0.0879316i	$0.971974\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.34847	+	12.7279i	−0.731200	+	1.26648i	0.225171	+	0.974319i	$0.427706\pi$
$101$	−7.34847	+	12.7279i	−0.731200	+	1.26648i	−0.956371	+	0.292156i	$0.905627\pi$
$102$		0			0
$103$	−7.50000	+	4.33013i	−0.738997	+	0.426660i	−0.821705	−	0.569914i	$-0.806977\pi$
$103$	−7.50000	+	4.33013i	−0.738997	+	0.426660i	0.0827075	+	0.996574i	$0.473643\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.34847	−	4.24264i	0.710403	−	0.410152i	−0.100807	−	0.994906i	$-0.532142\pi$
$107$	7.34847	−	4.24264i	0.710403	−	0.410152i	0.811210	+	0.584754i	$0.198809\pi$
$108$		0			0
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	−0.608377	−	0.793648i	$-0.708179\pi$
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	0.991508	+	0.130046i	$0.0415126\pi$
$110$		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	$-0.0626412\pi$
$131$	3.67423	+	6.36396i	0.321019	+	0.556022i	−0.659679	+	0.751547i	$0.729308\pi$
$132$		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	$-0.451401\pi$
$137$	−7.34847	−	4.24264i	−0.627822	−	0.362473i	−0.779908	+	0.625894i	$0.784734\pi$
$138$		0			0
$139$		−	8.66025i		−	0.734553i	−0.930112	−	0.367277i	$-0.880290\pi$
$139$		−	8.66025i		−	0.734553i	0.930112	−	0.367277i	$-0.119710\pi$
$140$		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	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	−0.746190	−	0.665733i	$-0.768119\pi$
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	0.949637	+	0.313353i	$0.101452\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.50000	−	0.866025i	−0.119713	−	0.0691164i	0.438948	−	0.898513i	$-0.355351\pi$
$157$	−1.50000	−	0.866025i	−0.119713	−	0.0691164i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		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	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−8.50000	−	14.7224i	−0.665771	−	1.15315i	0.313304	−	0.949653i	$-0.398564\pi$
$164$	7.34847	−	12.7279i	0.573819	−	0.993884i
$165$		0			0
$166$		0			0
$167$	−14.6969			−1.13728			−0.568642	−	0.822585i	$-0.692531\pi$
$167$	−14.6969			−1.13728			−0.568642	+	0.822585i	$0.692531\pi$
$168$		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	$-0.256785\pi$
$173$	−3.67423	−	6.36396i	−0.279347	−	0.483843i	−0.971223	+	0.238174i	$0.923451\pi$
$174$		0			0
$175$		0			0
$176$		−	16.9706i		−	1.27920i
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		−	15.5885i		−	1.15868i	−0.815086	−	0.579340i	$-0.803310\pi$
$181$		−	15.5885i		−	1.15868i	0.815086	−	0.579340i	$-0.196690\pi$
$182$		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	$-0.567346\pi$
$191$	7.34847	−	4.24264i	0.531717	−	0.306987i	0.741715	+	0.670715i	$0.234013\pi$
$192$		0			0
$193$	−6.50000	+	11.2583i	−0.467880	+	0.810392i	−0.999326	−	0.0366998i	$-0.988315\pi$
$193$	−6.50000	+	11.2583i	−0.467880	+	0.810392i	0.531446	+	0.847092i	$0.321649\pi$
$194$		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	0.552223	−	0.833696i	$-0.313780\pi$
$199$	1.50000	+	0.866025i	0.106332	+	0.0613909i	−0.445891	+	0.895087i	$0.647113\pi$
$200$		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	$-0.779792\pi$
$223$		−	19.0526i		−	1.27585i	0.770097	−	0.637927i	$-0.220208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.67423	−	6.36396i	0.243868	−	0.422391i	−0.717945	−	0.696100i	$-0.754917\pi$
$227$	3.67423	−	6.36396i	0.243868	−	0.422391i	0.961813	+	0.273709i	$0.0882505\pi$
$228$		0			0
$229$	−6.00000	+	3.46410i	−0.396491	+	0.228914i	−0.684969	−	0.728572i	$-0.740184\pi$
$229$	−6.00000	+	3.46410i	−0.396491	+	0.228914i	0.288478	+	0.957487i	$0.406851\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.34847	+	4.24264i	−0.481414	+	0.277945i	−0.721006	−	0.692929i	$-0.756320\pi$
$233$	−7.34847	+	4.24264i	−0.481414	+	0.277945i	0.239591	+	0.970874i	$0.422987\pi$
$234$		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	0.721715	−	0.692191i	$-0.243354\pi$
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	−0.238597	+	0.971119i	$0.576688\pi$
$242$		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	$-0.653526\pi$
$251$	−14.6969			−0.927663			−0.463831	+	0.885924i	$0.653526\pi$
$252$		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	−0.972619	−	0.232404i	$-0.925341\pi$
$257$	−11.0227	−	19.0919i	−0.687577	−	1.19092i	0.285042	−	0.958515i	$-0.407992\pi$
$258$		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	$-0.291577\pi$
$263$	3.67423	+	2.12132i	0.226563	+	0.130806i	−0.382422	+	0.923988i	$0.624910\pi$
$264$		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	0.0636361	−	0.997973i	$-0.479730\pi$
$269$	14.6969	−	25.4558i	0.896088	−	1.55207i	0.832452	−	0.554097i	$-0.186936\pi$
$270$		0			0
$271$	18.0000	−	10.3923i	1.09342	−	0.631288i	0.158937	−	0.987289i	$-0.449193\pi$
$271$	18.0000	−	10.3923i	1.09342	−	0.631288i	0.934485	+	0.356001i	$0.115860\pi$
$272$	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	−0.799666	−	0.600446i	$-0.794990\pi$
$277$	2.00000	−	3.46410i	0.120168	−	0.208138i	0.919834	+	0.392308i	$0.128323\pi$
$278$		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	0.970774	−	0.239994i	$-0.0771455\pi$
$283$	21.0000	+	12.1244i	1.24832	+	0.720718i	0.277546	+	0.960712i	$0.410479\pi$
$284$	−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	$-0.277293\pi$
$293$	22.0454			1.28791			0.643953	+	0.765065i	$0.277293\pi$
$294$		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	$-0.952626\pi$
$307$		−	5.19615i		−	0.296560i	0.988945	−	0.148280i	$-0.0473737\pi$
$308$	−22.0454	−	4.24264i	−1.25615	−	0.241747i
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−13.5000	+	7.79423i	−0.763065	+	0.440556i	−0.830395	−	0.557175i	$-0.811885\pi$
$313$	−13.5000	+	7.79423i	−0.763065	+	0.440556i	0.0673300	+	0.997731i	$0.478552\pi$
$314$		0			0
$315$		0			0
$316$	−26.0000			−1.46261
$317$	3.67423	−	2.12132i	0.206366	−	0.119145i	−0.393256	−	0.919429i	$-0.628651\pi$
$317$	3.67423	−	2.12132i	0.206366	−	0.119145i	0.599621	+	0.800284i	$0.295318\pi$
$318$		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	0.630232	−	0.776407i	$-0.282960\pi$
$331$	−6.50000	−	11.2583i	−0.357272	−	0.618814i	−0.987504	+	0.157593i	$0.949627\pi$
$332$	−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	$-0.140512\pi$
$347$	18.3712	+	10.6066i	0.986216	+	0.569392i	0.0820751	+	0.996626i	$0.473845\pi$
$348$		0			0
$349$		−	20.7846i		−	1.11257i	−0.830990	−	0.556287i	$-0.812225\pi$
$349$		−	20.7846i		−	1.11257i	0.830990	−	0.556287i	$-0.187775\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.34847	+	12.7279i	−0.391120	+	0.677439i	−0.992597	−	0.121450i	$-0.961245\pi$
$353$	−7.34847	+	12.7279i	−0.391120	+	0.677439i	0.601478	+	0.798889i	$0.294579\pi$
$354$		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	$-0.557634\pi$
$359$	11.0227	−	6.36396i	0.581756	−	0.335877i	0.761831	+	0.647776i	$0.224301\pi$
$360$		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	0.104399	−	0.994535i	$-0.466708\pi$
$367$	−13.5000	−	7.79423i	−0.704694	−	0.406855i	−0.809093	+	0.587680i	$0.800041\pi$
$368$	−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	0.851089	+	0.525022i	$0.175943\pi$
$373$	17.0000	+	29.4449i	0.880227	+	1.52460i	0.0291379	+	0.999575i	$0.490724\pi$
$374$		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	$-0.289192\pi$
$383$	−7.34847	−	12.7279i	−0.375489	−	0.650366i	−0.990400	+	0.138230i	$0.955859\pi$
$384$		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	$-0.367636\pi$
$389$	−3.67423	−	2.12132i	−0.186291	−	0.107555i	−0.590245	+	0.807224i	$0.700969\pi$
$390$		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	−0.382807	−	0.923828i	$-0.625043\pi$
$397$	4.50000	−	2.59808i	0.225849	−	0.130394i	0.608655	+	0.793435i	$0.291709\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.0227	+	6.36396i	−0.550448	+	0.317801i	−0.749302	−	0.662228i	$-0.769611\pi$
$401$	−11.0227	+	6.36396i	−0.550448	+	0.317801i	0.198855	+	0.980029i	$0.436278\pi$
$402$		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	0.423993	−	0.905666i	$-0.360628\pi$
$409$	−3.00000	−	1.73205i	−0.148340	−	0.0856444i	−0.572333	+	0.820021i	$0.693962\pi$
$410$		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	$-0.245061\pi$
$419$	29.3939			1.43598			0.717992	+	0.696051i	$0.245061\pi$
$420$		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	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$		−	25.9808i		−	1.24856i	−0.781202	−	0.624278i	$-0.785393\pi$
$433$		−	25.9808i		−	1.24856i	0.781202	−	0.624278i	$-0.214607\pi$
$434$		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	−0.839082	−	0.544004i	$-0.816908\pi$
$439$	−16.5000	+	9.52628i	−0.787502	+	0.454665i	0.0515804	+	0.998669i	$0.483574\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.0227	−	6.36396i	0.523704	−	0.302361i	−0.214745	−	0.976670i	$-0.568892\pi$
$443$	11.0227	−	6.36396i	0.523704	−	0.302361i	0.738449	+	0.674309i	$0.235559\pi$
$444$		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	0.999014	+	0.0443868i	$0.0141334\pi$
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	−0.461067	+	0.887365i	$0.652533\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.34847			−0.342252			−0.171126	−	0.985249i	$-0.554741\pi$
$461$	−7.34847			−0.342252			−0.171126	+	0.985249i	$0.554741\pi$
$462$		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	$-0.221051\pi$
$467$	−3.67423	−	6.36396i	−0.170023	−	0.294489i	−0.938428	+	0.345476i	$0.887718\pi$
$468$		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	$-0.942326\pi$
$479$	−7.34847	+	12.7279i	−0.335760	+	0.581554i	0.647870	+	0.761751i	$0.275660\pi$
$480$		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	0.430486	+	0.902597i	$0.358342\pi$
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		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	0.887540	−	0.460730i	$-0.152412\pi$
$499$	1.00000	+	1.73205i	0.0447661	+	0.0775372i	−0.842774	+	0.538267i	$0.819079\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	29.3939			1.31061			0.655304	−	0.755365i	$-0.272540\pi$
$503$	29.3939			1.31061			0.655304	+	0.755365i	$0.272540\pi$
$504$		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	$-0.218738\pi$
$509$	−3.67423	−	6.36396i	−0.162858	−	0.282078i	−0.935892	+	0.352286i	$0.885404\pi$
$510$		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	$-0.672912\pi$
$521$	11.0227	−	19.0919i	0.482913	−	0.836431i	0.999808	+	0.0196188i	$0.00624525\pi$
$522$		0			0
$523$	22.5000	−	12.9904i	0.983856	−	0.568030i	0.0804241	−	0.996761i	$-0.474373\pi$
$523$	22.5000	−	12.9904i	0.983856	−	0.568030i	0.903432	+	0.428731i	$0.141039\pi$
$524$	−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	0.767136	−	0.641484i	$-0.221681\pi$
$541$	−4.00000	−	6.92820i	−0.171973	−	0.297867i	−0.939110	+	0.343617i	$0.888348\pi$
$542$		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	$-0.420241\pi$
$557$	−11.0227	−	6.36396i	−0.467047	−	0.269650i	−0.715003	+	0.699122i	$0.753575\pi$
$558$		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	−0.160961	−	0.986961i	$-0.551459\pi$
$563$	18.3712	−	31.8198i	0.774253	−	1.34104i	0.935214	−	0.354084i	$-0.115207\pi$
$564$		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	$-0.487512\pi$
$569$	22.0454	−	12.7279i	0.924192	−	0.533582i	0.884970	+	0.465648i	$0.154179\pi$
$570$		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	$0.179991\pi$
$572$	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	−0.329040	−	0.944316i	$-0.606725\pi$
$577$	−31.5000	−	18.1865i	−1.31136	−	0.757115i	−0.982322	+	0.187201i	$0.940059\pi$
$578$		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	$-0.650345\pi$
$587$	−22.0454			−0.909911			−0.454956	+	0.890514i	$0.650345\pi$
$588$		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	−0.820519	−	0.571619i	$-0.806315\pi$
$593$	−22.0454	−	38.1838i	−0.905296	−	1.56802i	−0.0847775	−	0.996400i	$-0.527018\pi$
$594$		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	$-0.507420\pi$
$599$	−22.0454	−	12.7279i	−0.900751	−	0.520049i	−0.877444	+	0.479680i	$0.840753\pi$
$600$		0			0
$601$		−	17.3205i		−	0.706518i	−0.935526	−	0.353259i	$-0.885073\pi$
$601$		−	17.3205i		−	0.706518i	0.935526	−	0.353259i	$-0.114927\pi$
$602$		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	−0.469249	−	0.883066i	$-0.655475\pi$
$607$	1.50000	−	0.866025i	0.0608831	−	0.0351509i	0.530133	+	0.847915i	$0.322142\pi$
$608$		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	−0.658012	−	0.753007i	$-0.728603\pi$
$613$	8.00000	−	13.8564i	0.323117	−	0.559655i	0.981129	+	0.193352i	$0.0619359\pi$
$614$		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	0.587701	−	0.809078i	$-0.300033\pi$
$619$	4.50000	+	2.59808i	0.180870	+	0.104425i	−0.406831	+	0.913503i	$0.633366\pi$
$620$		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	$-0.252455\pi$
$641$	11.0227	+	6.36396i	0.435371	+	0.251361i	−0.266261	+	0.963901i	$0.585788\pi$
$642$		0			0
$643$		−	12.1244i		−	0.478138i	−0.971003	−	0.239069i	$-0.923158\pi$
$643$		−	12.1244i		−	0.478138i	0.971003	−	0.239069i	$-0.0768422\pi$
$644$	−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	$-0.787192\pi$
$647$	3.67423	−	6.36396i	0.144449	−	0.250193i	0.929167	+	0.369660i	$0.120526\pi$
$648$		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	$-0.864042\pi$
$653$	−25.7196	+	14.8492i	−1.00649	+	0.581096i	−0.0963261	+	0.995350i	$0.530709\pi$
$654$		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	0.905822	−	0.423658i	$-0.139254\pi$
$661$	25.5000	+	14.7224i	0.991835	+	0.572636i	0.0860127	+	0.996294i	$0.472587\pi$
$662$		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	0.997064	+	0.0765767i	$0.0243990\pi$
$677$	14.6969	+	25.4558i	0.564849	+	0.978348i	−0.432214	+	0.901771i	$0.642268\pi$
$678$		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	$-0.200309\pi$
$683$	18.3712	+	10.6066i	0.702953	+	0.405850i	−0.105493	+	0.994420i	$0.533642\pi$
$684$		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	0.329942	−	0.944001i	$-0.392971\pi$
$691$	34.5000	−	19.9186i	1.31244	−	0.757739i	0.982500	+	0.186263i	$0.0596375\pi$
$692$	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	0.997036	−	0.0769337i	$-0.0245130\pi$
$709$	11.5000	+	19.9186i	0.431892	+	0.748058i	−0.565145	+	0.824992i	$0.691180\pi$
$710$		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	$-0.255031\pi$
$719$	−7.34847	−	12.7279i	−0.274052	−	0.474671i	−0.969895	+	0.243522i	$0.921697\pi$
$720$		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	$0.355949\pi$
$727$			48.4974i			1.79867i	−0.437260	+	0.899335i	$0.644051\pi$
$728$		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	−0.992757	−	0.120137i	$-0.961667\pi$
$733$	−37.5000	+	21.6506i	−1.38509	+	0.799684i	−0.392337	+	0.919822i	$0.628333\pi$
$734$		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	−0.294285	+	0.955718i	$0.595081\pi$
$739$	−26.5000	+	45.8993i	−0.974818	+	1.68843i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		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	0.874281	+	0.485421i	$0.161334\pi$
$751$	23.5000	+	40.7032i	0.857527	+	1.48528i	−0.0167534	+	0.999860i	$0.505333\pi$
$752$	−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	−0.999268	−	0.0382548i	$-0.987820\pi$
$761$	−14.6969	−	25.4558i	−0.532764	−	0.922774i	0.466504	−	0.884519i	$-0.345513\pi$
$762$		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	$-0.909312\pi$
$769$		−	15.5885i		−	0.562134i	0.959688	−	0.281067i	$-0.0906883\pi$
$770$		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	$-0.748190\pi$
$773$	7.34847	−	12.7279i	0.264306	−	0.457792i	0.967382	+	0.253324i	$0.0815238\pi$
$774$		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	0.627681	−	0.778471i	$-0.284004\pi$
$787$	7.50000	+	4.33013i	0.267346	+	0.154352i	−0.360335	+	0.932823i	$0.617338\pi$
$788$	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	$-0.325711\pi$
$797$	29.3939			1.04118			0.520592	+	0.853805i	$0.325711\pi$
$798$		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	$-0.429762\pi$
$809$	−14.6969	−	8.48528i	−0.516717	−	0.298327i	−0.735590	+	0.677427i	$0.763095\pi$
$810$		0			0
$811$			22.5167i			0.790667i	0.918538	+	0.395333i	$0.129371\pi$
$811$			22.5167i			0.790667i	−0.918538	+	0.395333i	$0.870629\pi$
$812$	−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	$-0.868437\pi$
$821$	−29.3939	+	16.9706i	−1.02585	+	0.592277i	−0.110059	+	0.993925i	$0.535104\pi$
$822$		0			0
$823$	−14.5000	+	25.1147i	−0.505438	+	0.875445i	0.494542	+	0.869154i	$0.335336\pi$
$823$	−14.5000	+	25.1147i	−0.505438	+	0.875445i	−0.999980	+	0.00629095i	$0.997998\pi$
$824$		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	0.624556	−	0.780980i	$-0.285280\pi$
$829$	7.50000	+	4.33013i	0.260486	+	0.150392i	−0.364070	+	0.931371i	$0.618613\pi$
$830$		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	$-0.718688\pi$
$839$	−36.7423			−1.26849			−0.634243	+	0.773133i	$0.718688\pi$
$840$		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	$0.295490\pi$
$853$			46.7654i			1.60122i	−0.599189	+	0.800608i	$0.704510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	18.0000	−	10.3923i	0.614152	−	0.354581i	−0.160437	−	0.987046i	$-0.551290\pi$
$859$	18.0000	−	10.3923i	0.614152	−	0.354581i	0.774589	+	0.632465i	$0.217957\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.3712	−	10.6066i	0.625362	−	0.361053i	−0.153592	−	0.988134i	$-0.549084\pi$
$863$	18.3712	−	10.6066i	0.625362	−	0.361053i	0.778954	+	0.627081i	$0.215751\pi$
$864$		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	−0.923763	−	0.382965i	$-0.874903\pi$
$877$	−23.5000	−	40.7032i	−0.793539	−	1.37445i	0.130224	−	0.991485i	$-0.458430\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.34847			−0.247576			−0.123788	−	0.992309i	$-0.539504\pi$
$881$	−7.34847			−0.247576			−0.123788	+	0.992309i	$0.539504\pi$
$882$		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	−0.990058	−	0.140659i	$-0.955078\pi$
$887$	−18.3712	−	31.8198i	−0.616844	−	1.06840i	0.373214	−	0.927745i	$-0.378256\pi$
$888$		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	−0.821531	−	0.570164i	$-0.806880\pi$
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	0.904542	+	0.426385i	$0.140213\pi$
$908$	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	0.582808	−	0.812610i	$-0.301954\pi$
$919$	−12.5000	−	21.6506i	−0.412337	−	0.714189i	−0.995145	+	0.0984214i	$0.968621\pi$
$920$		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	0.992404	+	0.123018i	$0.0392572\pi$
$929$	18.3712	+	31.8198i	0.602739	+	1.04397i	−0.389666	+	0.920956i	$0.627409\pi$
$930$		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	$0.0270492\pi$
$937$			5.19615i			0.169751i	−0.996392	+	0.0848755i	$0.972951\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.67423	−	6.36396i	0.119777	−	0.207459i	−0.799902	−	0.600130i	$-0.795115\pi$
$941$	3.67423	−	6.36396i	0.119777	−	0.207459i	0.919679	+	0.392671i	$0.128449\pi$
$942$		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	$-0.732972\pi$
$947$	−11.0227	+	6.36396i	−0.358190	+	0.206801i	0.310097	+	0.950705i	$0.399639\pi$
$948$		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	−0.965792	−	0.259316i	$-0.916503\pi$
$971$	−22.0454	−	38.1838i	−0.707471	−	1.22538i	0.258322	−	0.966059i	$-0.416831\pi$
$972$		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	$-0.150646\pi$
$977$	29.3939	+	16.9706i	0.940393	+	0.542936i	0.0503098	+	0.998734i	$0.483979\pi$
$978$		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	$-0.718981\pi$
$983$	11.0227	−	19.0919i	0.351570	−	0.608937i	0.986525	+	0.163613i	$0.0523147\pi$
$984$		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	−0.202988	−	0.979181i	$-0.565065\pi$
$991$	23.5000	−	40.7032i	0.746502	−	1.29298i	0.949490	−	0.313798i	$-0.101602\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	+	11.2583i	0.617571	+	0.356555i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	19.5000	+	11.2583i	0.617571	+	0.356555i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		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.1151.2	yes	4
3.2	odd	2	inner	1575.2.bk.b.1151.1	yes	4
5.2	odd	4		1575.2.bc.b.899.2		8
5.3	odd	4		1575.2.bc.b.899.4		8
5.4	even	2		1575.2.bk.a.1151.2	yes	4
7.5	odd	6	inner	1575.2.bk.b.26.1	yes	4
15.2	even	4		1575.2.bc.b.899.1		8
15.8	even	4		1575.2.bc.b.899.3		8
15.14	odd	2		1575.2.bk.a.1151.1	yes	4
21.5	even	6	inner	1575.2.bk.b.26.2	yes	4
35.12	even	12		1575.2.bc.b.1349.3		8
35.19	odd	6		1575.2.bk.a.26.1	✓	4
35.33	even	12		1575.2.bc.b.1349.1		8
105.47	odd	12		1575.2.bc.b.1349.4		8
105.68	odd	12		1575.2.bc.b.1349.2		8
105.89	even	6		1575.2.bk.a.26.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.bc.b.899.1		8	15.2	even	4
1575.2.bc.b.899.2		8	5.2	odd	4
1575.2.bc.b.899.3		8	15.8	even	4
1575.2.bc.b.899.4		8	5.3	odd	4
1575.2.bc.b.1349.1		8	35.33	even	12
1575.2.bc.b.1349.2		8	105.68	odd	12
1575.2.bc.b.1349.3		8	35.12	even	12
1575.2.bc.b.1349.4		8	105.47	odd	12
1575.2.bk.a.26.1	✓	4	35.19	odd	6
1575.2.bk.a.26.2	yes	4	105.89	even	6
1575.2.bk.a.1151.1	yes	4	15.14	odd	2
1575.2.bk.a.1151.2	yes	4	5.4	even	2
1575.2.bk.b.26.1	yes	4	7.5	odd	6	inner
1575.2.bk.b.26.2	yes	4	21.5	even	6	inner
1575.2.bk.b.1151.1	yes	4	3.2	odd	2	inner
1575.2.bk.b.1151.2	yes	4	1.1	even	1	trivial

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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