LMFDB - Embedded newform 1575.2.bk.d.1151.1

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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.796594176.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 $ x^8 - 2x^7 - 5x^6 - 2x^5 + 63x^4 - 64x^3 + 46x^2 - 16*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1151.1
Root			$-2.12039 + 1.80156i$ of defining polynomial
Character	$\chi$	$=$	1575.1151
Dual form			1575.2.bk.d.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{4} +(-2.29129 + 1.32288i) q^{7} +O(q^{10})$	$q+(-1.00000 + 1.73205i) q^{4} +(-2.29129 + 1.32288i) q^{7} +(-2.44949 - 1.41421i) q^{11} -2.64575i q^{13} +(-2.00000 - 3.46410i) q^{16} +(1.87083 - 3.24037i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(5.61249 - 3.24037i) q^{23} -5.29150i q^{28} +1.41421i q^{29} +(4.50000 + 2.59808i) q^{31} +(-2.29129 - 3.96863i) q^{37} +4.89898 q^{41} +4.58258 q^{43} +(4.89898 - 2.82843i) q^{44} +(1.87083 + 3.24037i) q^{47} +(3.50000 - 6.06218i) q^{49} +(4.58258 + 2.64575i) q^{52} +(11.2250 + 6.48074i) q^{53} +(-3.67423 + 6.36396i) q^{59} +(-9.00000 + 5.19615i) q^{61} +8.00000 q^{64} +(6.87386 - 11.9059i) q^{67} +(3.74166 + 6.48074i) q^{68} +11.3137i q^{71} +(-11.4564 - 6.61438i) q^{73} -3.46410i q^{76} +7.48331 q^{77} +(-3.50000 - 6.06218i) q^{79} +14.9666 q^{83} +(-8.57321 - 14.8492i) q^{89} +(3.50000 + 6.06218i) q^{91} +12.9615i q^{92} -5.29150i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 16 q^{16} - 12 q^{19} + 36 q^{31} + 28 q^{49} - 72 q^{61} + 64 q^{64} - 28 q^{79} + 28 q^{91}+O(q^{100}) $ 8 * q - 8 * q^4 - 16 * q^16 - 12 * q^19 + 36 * q^31 + 28 * q^49 - 72 * q^61 + 64 * q^64 - 28 * q^79 + 28 * 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
$7$	−2.29129	+	1.32288i	−0.866025	+	0.500000i
$7$	−2.29129	+	1.32288i	−0.866025	+	0.500000i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.44949	−	1.41421i	−0.738549	−	0.426401i	0.0829925	−	0.996550i	$-0.473552\pi$
$11$	−2.44949	−	1.41421i	−0.738549	−	0.426401i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$		−	2.64575i		−	0.733799i	−0.930261	−	0.366900i	$-0.880419\pi$
$13$		−	2.64575i		−	0.733799i	0.930261	−	0.366900i	$-0.119581\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	1.87083	−	3.24037i	0.453743	−	0.785905i	−0.544872	−	0.838519i	$-0.683422\pi$
$17$	1.87083	−	3.24037i	0.453743	−	0.785905i	0.998615	+	0.0526138i	$0.0167552\pi$
$18$		0			0
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	−0.662094	−	0.749421i	$-0.730332\pi$
$19$	−1.50000	+	0.866025i	−0.344124	+	0.198680i	0.317970	+	0.948101i	$0.396999\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.61249	−	3.24037i	1.17028	−	0.675664i	0.216537	−	0.976274i	$-0.430524\pi$
$23$	5.61249	−	3.24037i	1.17028	−	0.675664i	0.953747	+	0.300610i	$0.0971904\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		−	5.29150i		−	1.00000i
$29$			1.41421i			0.262613i	0.991342	+	0.131306i	$0.0419172\pi$
$29$			1.41421i			0.262613i	−0.991342	+	0.131306i	$0.958083\pi$
$30$		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
$37$	−2.29129	−	3.96863i	−0.376685	−	0.652438i	0.613892	−	0.789390i	$-0.289603\pi$
$37$	−2.29129	−	3.96863i	−0.376685	−	0.652438i	−0.990578	+	0.136951i	$0.956270\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.89898			0.765092			0.382546	−	0.923936i	$-0.375047\pi$
$41$	4.89898			0.765092			0.382546	+	0.923936i	$0.375047\pi$
$42$		0			0
$43$	4.58258			0.698836			0.349418	−	0.936967i	$-0.386379\pi$
$43$	4.58258			0.698836			0.349418	+	0.936967i	$0.386379\pi$
$44$	4.89898	−	2.82843i	0.738549	−	0.426401i
$45$		0			0
$46$		0			0
$47$	1.87083	+	3.24037i	0.272888	+	0.472657i	0.969600	−	0.244695i	$-0.0786877\pi$
$47$	1.87083	+	3.24037i	0.272888	+	0.472657i	−0.696712	+	0.717351i	$0.745354\pi$
$48$		0			0
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$	4.58258	+	2.64575i	0.635489	+	0.366900i
$53$	11.2250	+	6.48074i	1.54187	+	0.890198i	0.998721	+	0.0505609i	$0.0161009\pi$
$53$	11.2250	+	6.48074i	1.54187	+	0.890198i	0.543148	+	0.839637i	$0.317232\pi$
$54$		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
$61$	−9.00000	+	5.19615i	−1.15233	+	0.665299i	−0.949454	−	0.313905i	$-0.898363\pi$
$61$	−9.00000	+	5.19615i	−1.15233	+	0.665299i	−0.202878	+	0.979204i	$0.565029\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	6.87386	−	11.9059i	0.839776	−	1.45453i	−0.0503056	−	0.998734i	$-0.516020\pi$
$67$	6.87386	−	11.9059i	0.839776	−	1.45453i	0.890082	−	0.455801i	$-0.150647\pi$
$68$	3.74166	+	6.48074i	0.453743	+	0.785905i
$69$		0			0
$70$		0			0
$71$			11.3137i			1.34269i	0.741145	+	0.671345i	$0.234283\pi$
$71$			11.3137i			1.34269i	−0.741145	+	0.671345i	$0.765717\pi$
$72$		0			0
$73$	−11.4564	−	6.61438i	−1.34087	−	0.774154i	−0.353939	−	0.935269i	$-0.615158\pi$
$73$	−11.4564	−	6.61438i	−1.34087	−	0.774154i	−0.986936	+	0.161114i	$0.948491\pi$
$74$		0			0
$75$		0			0
$76$		−	3.46410i		−	0.397360i
$77$	7.48331			0.852803
$78$		0			0
$79$	−3.50000	−	6.06218i	−0.393781	−	0.682048i	0.599164	−	0.800626i	$-0.295500\pi$
$79$	−3.50000	−	6.06218i	−0.393781	−	0.682048i	−0.992945	+	0.118578i	$0.962166\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.9666			1.64280			0.821401	−	0.570352i	$-0.193193\pi$
$83$	14.9666			1.64280			0.821401	+	0.570352i	$0.193193\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.57321	−	14.8492i	−0.908759	−	1.57402i	−0.815791	−	0.578347i	$-0.803698\pi$
$89$	−8.57321	−	14.8492i	−0.908759	−	1.57402i	−0.0929683	−	0.995669i	$-0.529636\pi$
$90$		0			0
$91$	3.50000	+	6.06218i	0.366900	+	0.635489i
$92$			12.9615i			1.35133i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	5.29150i		−	0.537271i	−0.963242	−	0.268635i	$-0.913427\pi$
$97$		−	5.29150i		−	0.537271i	0.963242	−	0.268635i	$-0.0865727\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.89898	−	8.48528i	0.487467	−	0.844317i	−0.512429	−	0.858729i	$-0.671254\pi$
$101$	4.89898	−	8.48528i	0.487467	−	0.844317i	0.999896	+	0.0144123i	$0.00458773\pi$
$102$		0			0
$103$	11.4564	−	6.61438i	1.12884	−	0.651734i	0.185194	−	0.982702i	$-0.440708\pi$
$103$	11.4564	−	6.61438i	1.12884	−	0.651734i	0.943642	+	0.330968i	$0.107375\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.61249	−	3.24037i	0.542580	−	0.313258i	−0.203544	−	0.979066i	$-0.565246\pi$
$107$	5.61249	−	3.24037i	0.542580	−	0.313258i	0.746124	+	0.665807i	$0.231913\pi$
$108$		0			0
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	−0.983531	−	0.180741i	$-0.942150\pi$
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	0.648292	+	0.761392i	$0.275484\pi$
$110$		0			0
$111$		0			0
$112$	9.16515	+	5.29150i	0.866025	+	0.500000i
$113$		−	6.48074i		−	0.609657i	−0.952407	−	0.304828i	$-0.901401\pi$
$113$		−	6.48074i		−	0.609657i	0.952407	−	0.304828i	$-0.0985991\pi$
$114$		0			0
$115$		0			0
$116$	−2.44949	−	1.41421i	−0.227429	−	0.131306i
$117$		0			0
$118$		0			0
$119$			9.89949i			0.907485i
$120$		0			0
$121$	−1.50000	−	2.59808i	−0.136364	−	0.236189i
$122$		0			0
$123$		0			0
$124$	−9.00000	+	5.19615i	−0.808224	+	0.466628i
$125$		0			0
$126$		0			0
$127$	−13.7477			−1.21991			−0.609957	−	0.792435i	$-0.708813\pi$
$127$	−13.7477			−1.21991			−0.609957	+	0.792435i	$0.708813\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.57321	−	14.8492i	−0.749045	−	1.29738i	−0.948281	−	0.317433i	$-0.897179\pi$
$131$	−8.57321	−	14.8492i	−0.749045	−	1.29738i	0.199236	−	0.979952i	$-0.436154\pi$
$132$		0			0
$133$	2.29129	−	3.96863i	0.198680	−	0.344124i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−5.61249	−	3.24037i	−0.479507	−	0.276844i	0.240704	−	0.970599i	$-0.422622\pi$
$137$	−5.61249	−	3.24037i	−0.479507	−	0.276844i	−0.720211	+	0.693755i	$0.755955\pi$
$138$		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
$143$	−3.74166	+	6.48074i	−0.312893	+	0.541947i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	9.16515			0.753371
$149$	12.2474	−	7.07107i	1.00335	−	0.579284i	0.0941123	−	0.995562i	$-0.469999\pi$
$149$	12.2474	−	7.07107i	1.00335	−	0.579284i	0.909238	+	0.416277i	$0.136665\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	4.58258	+	2.64575i	0.365729	+	0.211154i	0.671591	−	0.740922i	$-0.265611\pi$
$157$	4.58258	+	2.64575i	0.365729	+	0.211154i	−0.305862	+	0.952076i	$0.598945\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.57321	+	14.8492i	−0.675664	+	1.17028i
$162$		0			0
$163$	−4.58258	−	7.93725i	−0.358935	−	0.621694i	0.628848	−	0.777528i	$-0.283527\pi$
$163$	−4.58258	−	7.93725i	−0.358935	−	0.621694i	−0.987783	+	0.155834i	$0.950193\pi$
$164$	−4.89898	+	8.48528i	−0.382546	+	0.662589i
$165$		0			0
$166$		0			0
$167$	7.48331			0.579076			0.289538	−	0.957166i	$-0.406498\pi$
$167$	7.48331			0.579076			0.289538	+	0.957166i	$0.406498\pi$
$168$		0			0
$169$	6.00000			0.461538
$170$		0			0
$171$		0			0
$172$	−4.58258	+	7.93725i	−0.349418	+	0.605210i
$173$	−7.48331	−	12.9615i	−0.568946	−	0.985443i	−0.996671	−	0.0815341i	$-0.974018\pi$
$173$	−7.48331	−	12.9615i	−0.568946	−	0.985443i	0.427725	−	0.903909i	$-0.359315\pi$
$174$		0			0
$175$		0			0
$176$			11.3137i			0.852803i
$177$		0			0
$178$		0			0
$179$	6.12372	+	3.53553i	0.457709	+	0.264258i	0.711080	−	0.703111i	$-0.248206\pi$
$179$	6.12372	+	3.53553i	0.457709	+	0.264258i	−0.253372	+	0.967369i	$0.581540\pi$
$180$		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
$187$	−9.16515	+	5.29150i	−0.670222	+	0.386953i
$188$	−7.48331			−0.545777
$189$		0			0
$190$		0			0
$191$	13.4722	−	7.77817i	0.974814	−	0.562809i	0.0741134	−	0.997250i	$-0.476387\pi$
$191$	13.4722	−	7.77817i	0.974814	−	0.562809i	0.900700	+	0.434441i	$0.143054\pi$
$192$		0			0
$193$	−6.87386	+	11.9059i	−0.494792	+	0.857004i	−0.999982	−	0.00600382i	$-0.998089\pi$
$193$	−6.87386	+	11.9059i	−0.494792	+	0.857004i	0.505190	+	0.863008i	$0.331422\pi$
$194$		0			0
$195$		0			0
$196$	7.00000	+	12.1244i	0.500000	+	0.866025i
$197$		−	19.4422i		−	1.38520i	−0.721321	−	0.692600i	$-0.756465\pi$
$197$		−	19.4422i		−	1.38520i	0.721321	−	0.692600i	$-0.243535\pi$
$198$		0			0
$199$	−6.00000	−	3.46410i	−0.425329	−	0.245564i	0.272026	−	0.962290i	$-0.412306\pi$
$199$	−6.00000	−	3.46410i	−0.425329	−	0.245564i	−0.697355	+	0.716726i	$0.745640\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.87083	−	3.24037i	−0.131306	−	0.227429i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−9.16515	+	5.29150i	−0.635489	+	0.366900i
$209$	4.89898			0.338869
$210$		0			0
$211$	14.0000			0.963800			0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000			0.963800			0.481900	+	0.876226i	$0.339947\pi$
$212$	−22.4499	+	12.9615i	−1.54187	+	0.890198i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.7477			−0.933257
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.57321	−	4.94975i	−0.576697	−	0.332956i
$222$		0			0
$223$			5.29150i			0.354345i	0.984180	+	0.177173i	$0.0566951\pi$
$223$			5.29150i			0.354345i	−0.984180	+	0.177173i	$0.943305\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.48331	−	12.9615i	0.496685	−	0.860284i	−0.503308	−	0.864107i	$-0.667884\pi$
$227$	7.48331	−	12.9615i	0.496685	−	0.860284i	0.999993	+	0.00382356i	$0.00121708\pi$
$228$		0			0
$229$	1.50000	−	0.866025i	0.0991228	−	0.0572286i	−0.449619	−	0.893220i	$-0.648440\pi$
$229$	1.50000	−	0.866025i	0.0991228	−	0.0572286i	0.548742	+	0.835992i	$0.315107\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.61249	+	3.24037i	−0.367686	+	0.212284i	−0.672447	−	0.740145i	$-0.734757\pi$
$233$	−5.61249	+	3.24037i	−0.367686	+	0.212284i	0.304761	+	0.952429i	$0.401423\pi$
$234$		0			0
$235$		0			0
$236$	−7.34847	−	12.7279i	−0.478345	−	0.828517i
$237$		0			0
$238$		0			0
$239$			18.3848i			1.18921i	0.804017	+	0.594606i	$0.202692\pi$
$239$			18.3848i			1.18921i	−0.804017	+	0.594606i	$0.797308\pi$
$240$		0			0
$241$	−15.0000	−	8.66025i	−0.966235	−	0.557856i	−0.0681486	−	0.997675i	$-0.521709\pi$
$241$	−15.0000	−	8.66025i	−0.966235	−	0.557856i	−0.898086	+	0.439819i	$0.855043\pi$
$242$		0			0
$243$		0			0
$244$		−	20.7846i		−	1.33060i
$245$		0			0
$246$		0			0
$247$	2.29129	+	3.96863i	0.145791	+	0.252518i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.0454			1.39149			0.695747	−	0.718287i	$-0.255074\pi$
$251$	22.0454			1.39149			0.695747	+	0.718287i	$0.255074\pi$
$252$		0			0
$253$	−18.3303			−1.15242
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−3.74166	−	6.48074i	−0.233398	−	0.404257i	0.725408	−	0.688319i	$-0.241651\pi$
$257$	−3.74166	−	6.48074i	−0.233398	−	0.404257i	−0.958806	+	0.284062i	$0.908318\pi$
$258$		0			0
$259$	10.5000	+	6.06218i	0.652438	+	0.376685i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−11.2250	−	6.48074i	−0.692161	−	0.399620i	0.112260	−	0.993679i	$-0.464191\pi$
$263$	−11.2250	−	6.48074i	−0.692161	−	0.399620i	−0.804421	+	0.594059i	$0.797524\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	13.7477	+	23.8118i	0.839776	+	1.45453i
$269$	8.57321	−	14.8492i	0.522718	−	0.905374i	−0.476932	−	0.878940i	$-0.658251\pi$
$269$	8.57321	−	14.8492i	0.522718	−	0.905374i	0.999651	−	0.0264343i	$-0.00841529\pi$
$270$		0			0
$271$	−12.0000	+	6.92820i	−0.728948	+	0.420858i	−0.818037	−	0.575165i	$-0.804938\pi$
$271$	−12.0000	+	6.92820i	−0.728948	+	0.420858i	0.0890891	+	0.996024i	$0.471604\pi$
$272$	−14.9666			−0.907485
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.29129	+	3.96863i	−0.137670	+	0.238452i	−0.926614	−	0.376013i	$-0.877295\pi$
$277$	−2.29129	+	3.96863i	−0.137670	+	0.238452i	0.788944	+	0.614465i	$0.210628\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			28.2843i			1.68730i	0.536895	+	0.843649i	$0.319597\pi$
$281$			28.2843i			1.68730i	−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	16.0390	+	9.26013i	0.953420	+	0.550458i	0.894142	−	0.447784i	$-0.147787\pi$
$283$	16.0390	+	9.26013i	0.953420	+	0.550458i	0.0592787	+	0.998241i	$0.481120\pi$
$284$	−19.5959	−	11.3137i	−1.16280	−	0.671345i
$285$		0			0
$286$		0			0
$287$	−11.2250	+	6.48074i	−0.662589	+	0.382546i
$288$		0			0
$289$	1.50000	+	2.59808i	0.0882353	+	0.152828i
$290$		0			0
$291$		0			0
$292$	22.9129	−	13.2288i	1.34087	−	0.774154i
$293$	−29.9333			−1.74872			−0.874360	−	0.485278i	$-0.838718\pi$
$293$	−29.9333			−1.74872			−0.874360	+	0.485278i	$0.838718\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.57321	−	14.8492i	−0.495802	−	0.858754i
$300$		0			0
$301$	−10.5000	+	6.06218i	−0.605210	+	0.349418i
$302$		0			0
$303$		0			0
$304$	6.00000	+	3.46410i	0.344124	+	0.198680i
$305$		0			0
$306$		0			0
$307$			2.64575i			0.151001i	0.997146	+	0.0755005i	$0.0240554\pi$
$307$			2.64575i			0.151001i	−0.997146	+	0.0755005i	$0.975945\pi$
$308$	−7.48331	+	12.9615i	−0.426401	+	0.738549i
$309$		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
$313$	−16.0390	+	9.26013i	−0.906579	+	0.523413i	−0.879329	−	0.476215i	$-0.842008\pi$
$313$	−16.0390	+	9.26013i	−0.906579	+	0.523413i	−0.0272499	+	0.999629i	$0.508675\pi$
$314$		0			0
$315$		0			0
$316$	14.0000			0.787562
$317$	16.8375	−	9.72111i	0.945686	−	0.545992i	0.0539477	−	0.998544i	$-0.482820\pi$
$317$	16.8375	−	9.72111i	0.945686	−	0.545992i	0.891738	+	0.452552i	$0.149486\pi$
$318$		0			0
$319$	2.00000	−	3.46410i	0.111979	−	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.48074i			0.360598i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.57321	−	4.94975i	−0.472657	−	0.272888i
$330$		0			0
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	0.946038	−	0.324057i	$-0.105047\pi$
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	−0.753660	+	0.657264i	$0.771714\pi$
$332$	−14.9666	+	25.9230i	−0.821401	+	1.42271i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	32.0780			1.74740			0.873701	−	0.486464i	$-0.161713\pi$
$337$	32.0780			1.74740			0.873701	+	0.486464i	$0.161713\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.34847	−	12.7279i	−0.397942	−	0.689256i
$342$		0			0
$343$			18.5203i			1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		−	3.46410i		−	0.185429i	−0.995693	−	0.0927146i	$-0.970446\pi$
$349$		−	3.46410i		−	0.185429i	0.995693	−	0.0927146i	$-0.0295544\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.74166	−	6.48074i	0.199148	−	0.344935i	−0.749104	−	0.662452i	$-0.769516\pi$
$353$	3.74166	−	6.48074i	0.199148	−	0.344935i	0.948253	+	0.317517i	$0.102849\pi$
$354$		0			0
$355$		0			0
$356$	34.2929			1.81752
$357$		0			0
$358$		0			0
$359$	−13.4722	+	7.77817i	−0.711035	+	0.410516i	−0.811444	−	0.584430i	$-0.801318\pi$
$359$	−13.4722	+	7.77817i	−0.711035	+	0.410516i	0.100409	+	0.994946i	$0.467985\pi$
$360$		0			0
$361$	−8.00000	+	13.8564i	−0.421053	+	0.729285i
$362$		0			0
$363$		0			0
$364$	−14.0000			−0.733799
$365$		0			0
$366$		0			0
$367$	−16.0390	−	9.26013i	−0.837230	−	0.483375i	0.0190919	−	0.999818i	$-0.493923\pi$
$367$	−16.0390	−	9.26013i	−0.837230	−	0.483375i	−0.856322	+	0.516443i	$0.827256\pi$
$368$	−22.4499	−	12.9615i	−1.17028	−	0.675664i
$369$		0			0
$370$		0			0
$371$	−34.2929			−1.78040
$372$		0			0
$373$	−2.29129	−	3.96863i	−0.118638	−	0.205488i	0.800590	−	0.599213i	$-0.204520\pi$
$373$	−2.29129	−	3.96863i	−0.118638	−	0.205488i	−0.919228	+	0.393725i	$0.871186\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.74166			0.192705
$378$		0			0
$379$	−5.00000			−0.256833			−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000			−0.256833			−0.128416	+	0.991720i	$0.540989\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.74166	+	6.48074i	0.191190	+	0.331150i	0.945645	−	0.325201i	$-0.105432\pi$
$383$	3.74166	+	6.48074i	0.191190	+	0.331150i	−0.754455	+	0.656352i	$0.772099\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	9.16515	+	5.29150i	0.465290	+	0.268635i
$389$	2.44949	+	1.41421i	0.124194	+	0.0717035i	0.560810	−	0.827945i	$-0.310490\pi$
$389$	2.44949	+	1.41421i	0.124194	+	0.0717035i	−0.436616	+	0.899648i	$0.643823\pi$
$390$		0			0
$391$		−	24.2487i		−	1.22631i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−25.2042	+	14.5516i	−1.26496	+	0.730325i	−0.974030	−	0.226419i	$-0.927298\pi$
$397$	−25.2042	+	14.5516i	−1.26496	+	0.730325i	−0.290931	+	0.956744i	$0.593965\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.89898	+	2.82843i	−0.244643	+	0.141245i	−0.617309	−	0.786721i	$-0.711777\pi$
$401$	−4.89898	+	2.82843i	−0.244643	+	0.141245i	0.372666	+	0.927966i	$0.378444\pi$
$402$		0			0
$403$	6.87386	−	11.9059i	0.342412	−	0.593074i
$404$	9.79796	+	16.9706i	0.487467	+	0.844317i
$405$		0			0
$406$		0			0
$407$			12.9615i			0.642477i
$408$		0			0
$409$	4.50000	+	2.59808i	0.222511	+	0.128467i	0.607112	−	0.794616i	$-0.292328\pi$
$409$	4.50000	+	2.59808i	0.222511	+	0.128467i	−0.384602	+	0.923083i	$0.625661\pi$
$410$		0			0
$411$		0			0
$412$			26.4575i			1.30347i
$413$		−	19.4422i		−	0.956689i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	17.1464			0.837658			0.418829	−	0.908065i	$-0.362441\pi$
$419$	17.1464			0.837658			0.418829	+	0.908065i	$0.362441\pi$
$420$		0			0
$421$	35.0000			1.70580			0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000			1.70580			0.852898	+	0.522078i	$0.174843\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.7477	−	23.8118i	0.665299	−	1.15233i
$428$			12.9615i			0.626517i
$429$		0			0
$430$		0			0
$431$	−6.12372	−	3.53553i	−0.294969	−	0.170301i	0.345211	−	0.938525i	$-0.387807\pi$
$431$	−6.12372	−	3.53553i	−0.294969	−	0.170301i	−0.640181	+	0.768224i	$0.721141\pi$
$432$		0			0
$433$			13.2288i			0.635733i	0.948135	+	0.317867i	$0.102966\pi$
$433$			13.2288i			0.635733i	−0.948135	+	0.317867i	$0.897034\pi$
$434$		0			0
$435$		0			0
$436$	−7.00000	−	12.1244i	−0.335239	−	0.580651i
$437$	−5.61249	+	9.72111i	−0.268482	+	0.465024i
$438$		0			0
$439$	−9.00000	+	5.19615i	−0.429547	+	0.247999i	−0.699153	−	0.714972i	$-0.746440\pi$
$439$	−9.00000	+	5.19615i	−0.429547	+	0.247999i	0.269607	+	0.962970i	$0.413106\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−33.6749	+	19.4422i	−1.59994	+	0.923728i	−0.608447	+	0.793594i	$0.708207\pi$
$443$	−33.6749	+	19.4422i	−1.59994	+	0.923728i	−0.991496	+	0.130133i	$0.958459\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−18.3303	+	10.5830i	−0.866025	+	0.500000i
$449$			18.3848i			0.867631i	0.901002	+	0.433816i	$0.142833\pi$
$449$			18.3848i			0.867631i	−0.901002	+	0.433816i	$0.857167\pi$
$450$		0			0
$451$	−12.0000	−	6.92820i	−0.565058	−	0.326236i
$452$	11.2250	+	6.48074i	0.527978	+	0.304828i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.87386	+	11.9059i	0.321546	+	0.556934i	0.980807	−	0.194980i	$-0.0624643\pi$
$457$	6.87386	+	11.9059i	0.321546	+	0.556934i	−0.659261	+	0.751914i	$0.729131\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.1464			0.798589			0.399294	−	0.916823i	$-0.369255\pi$
$461$	17.1464			0.798589			0.399294	+	0.916823i	$0.369255\pi$
$462$		0			0
$463$	−4.58258			−0.212970			−0.106485	−	0.994314i	$-0.533960\pi$
$463$	−4.58258			−0.212970			−0.106485	+	0.994314i	$0.533960\pi$
$464$	4.89898	−	2.82843i	0.227429	−	0.131306i
$465$		0			0
$466$		0			0
$467$	1.87083	+	3.24037i	0.0865716	+	0.149946i	0.906060	−	0.423150i	$-0.139076\pi$
$467$	1.87083	+	3.24037i	0.0865716	+	0.149946i	−0.819488	+	0.573096i	$0.805742\pi$
$468$		0			0
$469$			36.3731i			1.67955i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.2250	−	6.48074i	−0.516125	−	0.297985i
$474$		0			0
$475$		0			0
$476$	−17.1464	−	9.89949i	−0.785905	−	0.453743i
$477$		0			0
$478$		0			0
$479$	17.1464	−	29.6985i	0.783440	−	1.35696i	−0.146486	−	0.989213i	$-0.546796\pi$
$479$	17.1464	−	29.6985i	0.783440	−	1.35696i	0.929926	−	0.367746i	$-0.119870\pi$
$480$		0			0
$481$	−10.5000	+	6.06218i	−0.478759	+	0.276412i
$482$		0			0
$483$		0			0
$484$	6.00000			0.272727
$485$		0			0
$486$		0			0
$487$	−11.4564	+	19.8431i	−0.519141	+	0.899178i	0.480612	+	0.876933i	$0.340415\pi$
$487$	−11.4564	+	19.8431i	−0.519141	+	0.899178i	−0.999753	+	0.0222448i	$0.992919\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			11.3137i			0.510581i	0.966864	+	0.255290i	$0.0821710\pi$
$491$			11.3137i			0.510581i	−0.966864	+	0.255290i	$0.917829\pi$
$492$		0			0
$493$	4.58258	+	2.64575i	0.206389	+	0.119159i
$494$		0			0
$495$		0			0
$496$		−	20.7846i		−	0.933257i
$497$	−14.9666	−	25.9230i	−0.671345	−	1.16280i
$498$		0			0
$499$	−6.50000	−	11.2583i	−0.290980	−	0.503992i	0.683062	−	0.730361i	$-0.260648\pi$
$499$	−6.50000	−	11.2583i	−0.290980	−	0.503992i	−0.974042	+	0.226369i	$0.927315\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.74166			0.166832			0.0834161	−	0.996515i	$-0.473417\pi$
$503$	3.74166			0.166832			0.0834161	+	0.996515i	$0.473417\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	13.7477	−	23.8118i	0.609957	−	1.05648i
$509$	8.57321	+	14.8492i	0.380001	+	0.658181i	0.991062	−	0.133402i	$-0.0425903\pi$
$509$	8.57321	+	14.8492i	0.380001	+	0.658181i	−0.611061	+	0.791584i	$0.709257\pi$
$510$		0			0
$511$	35.0000			1.54831
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	10.5830i		−	0.465440i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	17.1464	−	29.6985i	0.751199	−	1.30111i	−0.196043	−	0.980595i	$-0.562809\pi$
$521$	17.1464	−	29.6985i	0.751199	−	1.30111i	0.947242	−	0.320519i	$-0.103857\pi$
$522$		0			0
$523$	11.4564	−	6.61438i	0.500955	−	0.289227i	−0.228153	−	0.973625i	$-0.573269\pi$
$523$	11.4564	−	6.61438i	0.500955	−	0.289227i	0.729108	+	0.684399i	$0.239935\pi$
$524$	34.2929			1.49809
$525$		0			0
$526$		0			0
$527$	16.8375	−	9.72111i	0.733451	−	0.423458i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$	4.58258	+	7.93725i	0.198680	+	0.344124i
$533$		−	12.9615i		−	0.561424i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.1464	+	9.89949i	−0.738549	+	0.426401i
$540$		0			0
$541$	3.50000	+	6.06218i	0.150477	+	0.260633i	0.931403	−	0.363990i	$-0.118586\pi$
$541$	3.50000	+	6.06218i	0.150477	+	0.260633i	−0.780926	+	0.624623i	$0.785252\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−18.3303			−0.783747			−0.391874	−	0.920019i	$-0.628173\pi$
$547$	−18.3303			−0.783747			−0.391874	+	0.920019i	$0.628173\pi$
$548$	11.2250	−	6.48074i	0.479507	−	0.276844i
$549$		0			0
$550$		0			0
$551$	−1.22474	−	2.12132i	−0.0521759	−	0.0903713i
$552$		0			0
$553$	16.0390	+	9.26013i	0.682048	+	0.393781i
$554$		0			0
$555$		0			0
$556$	−15.0000	−	8.66025i	−0.636142	−	0.367277i
$557$	5.61249	+	3.24037i	0.237809	+	0.137299i	0.614169	−	0.789174i	$-0.289491\pi$
$557$	5.61249	+	3.24037i	0.237809	+	0.137299i	−0.376360	+	0.926473i	$0.622825\pi$
$558$		0			0
$559$		−	12.1244i		−	0.512806i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.35414	−	16.2019i	0.394230	−	0.682827i	−0.598772	−	0.800919i	$-0.704345\pi$
$563$	9.35414	−	16.2019i	0.394230	−	0.682827i	0.993003	+	0.118093i	$0.0376780\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−39.1918	+	22.6274i	−1.64301	+	0.948591i	−0.663251	+	0.748397i	$0.730824\pi$
$569$	−39.1918	+	22.6274i	−1.64301	+	0.948591i	−0.979756	+	0.200194i	$0.935843\pi$
$570$		0			0
$571$	−8.50000	+	14.7224i	−0.355714	+	0.616115i	−0.987240	−	0.159240i	$-0.949096\pi$
$571$	−8.50000	+	14.7224i	−0.355714	+	0.616115i	0.631526	+	0.775355i	$0.282429\pi$
$572$	−7.48331	−	12.9615i	−0.312893	−	0.541947i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−29.7867	−	17.1974i	−1.24004	−	0.715936i	−0.270936	−	0.962597i	$-0.587333\pi$
$577$	−29.7867	−	17.1974i	−1.24004	−	0.715936i	−0.969102	+	0.246661i	$0.920667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−34.2929	+	19.7990i	−1.42271	+	0.821401i
$582$		0			0
$583$	−18.3303	−	31.7490i	−0.759164	−	1.31491i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−26.1916			−1.08104			−0.540522	−	0.841330i	$-0.681773\pi$
$587$	−26.1916			−1.08104			−0.540522	+	0.841330i	$0.681773\pi$
$588$		0			0
$589$	−9.00000			−0.370839
$590$		0			0
$591$		0			0
$592$	−9.16515	+	15.8745i	−0.376685	+	0.652438i
$593$	−7.48331	−	12.9615i	−0.307303	−	0.532264i	0.670468	−	0.741938i	$-0.266093\pi$
$593$	−7.48331	−	12.9615i	−0.307303	−	0.532264i	−0.977771	+	0.209674i	$0.932760\pi$
$594$		0			0
$595$		0			0
$596$			28.2843i			1.15857i
$597$		0			0
$598$		0			0
$599$	2.44949	+	1.41421i	0.100083	+	0.0577832i	0.549206	−	0.835687i	$-0.314930\pi$
$599$	2.44949	+	1.41421i	0.100083	+	0.0577832i	−0.449123	+	0.893470i	$0.648263\pi$
$600$		0			0
$601$			8.66025i			0.353259i	0.984277	+	0.176630i	$0.0565195\pi$
$601$			8.66025i			0.353259i	−0.984277	+	0.176630i	$0.943481\pi$
$602$		0			0
$603$		0			0
$604$	10.0000	+	17.3205i	0.406894	+	0.704761i
$605$		0			0
$606$		0			0
$607$	29.7867	−	17.1974i	1.20901	−	0.698020i	0.246464	−	0.969152i	$-0.420731\pi$
$607$	29.7867	−	17.1974i	1.20901	−	0.698020i	0.962542	+	0.271132i	$0.0873979\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.57321	−	4.94975i	0.346835	−	0.200245i
$612$		0			0
$613$	−9.16515	+	15.8745i	−0.370177	+	0.641165i	−0.989593	−	0.143898i	$-0.954036\pi$
$613$	−9.16515	+	15.8745i	−0.370177	+	0.641165i	0.619416	+	0.785063i	$0.287370\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.4037i		−	1.30452i	−0.757994	−	0.652262i	$-0.773820\pi$
$617$		−	32.4037i		−	1.30452i	0.757994	−	0.652262i	$-0.226180\pi$
$618$		0			0
$619$	−25.5000	−	14.7224i	−1.02493	−	0.591744i	−0.109403	−	0.993997i	$-0.534894\pi$
$619$	−25.5000	−	14.7224i	−1.02493	−	0.591744i	−0.915529	+	0.402253i	$0.868227\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	39.2874	+	22.6826i	1.57402	+	0.908759i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−9.16515	+	5.29150i	−0.365729	+	0.211154i
$629$	−17.1464			−0.683673
$630$		0			0
$631$	14.0000			0.557331			0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000			0.557331			0.278666	+	0.960388i	$0.410108\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−16.0390	−	9.26013i	−0.635489	−	0.366900i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.2702	+	13.4350i	0.919116	+	0.530652i	0.883353	−	0.468709i	$-0.155281\pi$
$641$	23.2702	+	13.4350i	0.919116	+	0.530652i	0.0357629	+	0.999360i	$0.488614\pi$
$642$		0			0
$643$		−	2.64575i		−	0.104338i	−0.998638	−	0.0521691i	$-0.983387\pi$
$643$		−	2.64575i		−	0.104338i	0.998638	−	0.0521691i	$-0.0166135\pi$
$644$	−17.1464	−	29.6985i	−0.675664	−	1.17028i
$645$		0			0
$646$		0			0
$647$	7.48331	−	12.9615i	0.294199	−	0.509568i	−0.680599	−	0.732656i	$-0.738280\pi$
$647$	7.48331	−	12.9615i	0.294199	−	0.509568i	0.974798	+	0.223088i	$0.0716137\pi$
$648$		0			0
$649$	18.0000	−	10.3923i	0.706562	−	0.407934i
$650$		0			0
$651$		0			0
$652$	18.3303			0.717870
$653$	22.4499	−	12.9615i	0.878534	−	0.507222i	0.00835938	−	0.999965i	$-0.497339\pi$
$653$	22.4499	−	12.9615i	0.878534	−	0.507222i	0.870175	+	0.492743i	$0.164006\pi$
$654$		0			0
$655$		0			0
$656$	−9.79796	−	16.9706i	−0.382546	−	0.662589i
$657$		0			0
$658$		0			0
$659$		−	28.2843i		−	1.10180i	−0.834572	−	0.550899i	$-0.814285\pi$
$659$		−	28.2843i		−	1.10180i	0.834572	−	0.550899i	$-0.185715\pi$
$660$		0			0
$661$	−4.50000	−	2.59808i	−0.175030	−	0.101053i	0.409926	−	0.912119i	$-0.365555\pi$
$661$	−4.50000	−	2.59808i	−0.175030	−	0.101053i	−0.584955	+	0.811065i	$0.698888\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	4.58258	+	7.93725i	0.177438	+	0.307332i
$668$	−7.48331	+	12.9615i	−0.289538	+	0.501495i
$669$		0			0
$670$		0			0
$671$	29.3939			1.13474
$672$		0			0
$673$	−22.9129			−0.883227			−0.441613	−	0.897206i	$-0.645594\pi$
$673$	−22.9129			−0.883227			−0.441613	+	0.897206i	$0.645594\pi$
$674$		0			0
$675$		0			0
$676$	−6.00000	+	10.3923i	−0.230769	+	0.399704i
$677$	1.87083	+	3.24037i	0.0719018	+	0.124538i	0.899735	−	0.436437i	$-0.143760\pi$
$677$	1.87083	+	3.24037i	0.0719018	+	0.124538i	−0.827833	+	0.560975i	$0.810427\pi$
$678$		0			0
$679$	7.00000	+	12.1244i	0.268635	+	0.465290i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	28.0624	+	16.2019i	1.07378	+	0.619947i	0.929212	−	0.369548i	$-0.120488\pi$
$683$	28.0624	+	16.2019i	1.07378	+	0.619947i	0.144568	+	0.989495i	$0.453821\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−9.16515	−	15.8745i	−0.349418	−	0.605210i
$689$	17.1464	−	29.6985i	0.653227	−	1.13142i
$690$		0			0
$691$	−40.5000	+	23.3827i	−1.54069	+	0.889519i	−0.541897	+	0.840445i	$0.682294\pi$
$691$	−40.5000	+	23.3827i	−1.54069	+	0.889519i	−0.998795	+	0.0490747i	$0.984373\pi$
$692$	29.9333			1.13789
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	9.16515	−	15.8745i	0.347155	−	0.601290i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1.41421i		−	0.0534141i	−0.999643	−	0.0267071i	$-0.991498\pi$
$701$		−	1.41421i		−	0.0534141i	0.999643	−	0.0267071i	$-0.00850213\pi$
$702$		0			0
$703$	6.87386	+	3.96863i	0.259253	+	0.149680i
$704$	−19.5959	−	11.3137i	−0.738549	−	0.426401i
$705$		0			0
$706$		0			0
$707$			25.9230i			0.974933i
$708$		0			0
$709$	4.00000	+	6.92820i	0.150223	+	0.260194i	0.931309	−	0.364229i	$-0.118667\pi$
$709$	4.00000	+	6.92820i	0.150223	+	0.260194i	−0.781086	+	0.624423i	$0.785334\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	33.6749			1.26114
$714$		0			0
$715$		0			0
$716$	−12.2474	+	7.07107i	−0.457709	+	0.264258i
$717$		0			0
$718$		0			0
$719$	−1.22474	−	2.12132i	−0.0456753	−	0.0791119i	0.842284	−	0.539034i	$-0.181211\pi$
$719$	−1.22474	−	2.12132i	−0.0456753	−	0.0791119i	−0.887959	+	0.459922i	$0.847877\pi$
$720$		0			0
$721$	−17.5000	+	30.3109i	−0.651734	+	1.12884i
$722$		0			0
$723$		0			0
$724$	27.0000	+	15.5885i	1.00345	+	0.579340i
$725$		0			0
$726$		0			0
$727$		−	29.1033i		−	1.07938i	−0.841864	−	0.539690i	$-0.818541\pi$
$727$		−	29.1033i		−	1.07938i	0.841864	−	0.539690i	$-0.181459\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.57321	−	14.8492i	0.317092	−	0.549219i
$732$		0			0
$733$	11.4564	−	6.61438i	0.423153	−	0.244308i	−0.273272	−	0.961937i	$-0.588106\pi$
$733$	11.4564	−	6.61438i	0.423153	−	0.244308i	0.696426	+	0.717629i	$0.254773\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−33.6749	+	19.4422i	−1.24043	+	0.716163i
$738$		0			0
$739$	8.50000	−	14.7224i	0.312678	−	0.541573i	−0.666264	−	0.745716i	$-0.732107\pi$
$739$	8.50000	−	14.7224i	0.312678	−	0.541573i	0.978941	+	0.204143i	$0.0654407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			51.8459i			1.90204i	0.309125	+	0.951021i	$0.399964\pi$
$743$			51.8459i			1.90204i	−0.309125	+	0.951021i	$0.600036\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		−	21.1660i		−	0.773906i
$749$	−8.57321	+	14.8492i	−0.313258	+	0.542580i
$750$		0			0
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	0.922792	−	0.385299i	$-0.125902\pi$
$751$	3.50000	+	6.06218i	0.127717	+	0.221212i	−0.795075	+	0.606511i	$0.792568\pi$
$752$	7.48331	−	12.9615i	0.272888	−	0.472657i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.57321	−	14.8492i	−0.310779	−	0.538285i	0.667752	−	0.744383i	$-0.267256\pi$
$761$	−8.57321	−	14.8492i	−0.310779	−	0.538285i	−0.978531	+	0.206099i	$0.933923\pi$
$762$		0			0
$763$		−	18.5203i		−	0.670478i
$764$			31.1127i			1.12562i
$765$		0			0
$766$		0			0
$767$	16.8375	+	9.72111i	0.607965	+	0.351009i
$768$		0			0
$769$		−	32.9090i		−	1.18673i	−0.804934	−	0.593364i	$-0.797800\pi$
$769$		−	32.9090i		−	1.18673i	0.804934	−	0.593364i	$-0.202200\pi$
$770$		0			0
$771$		0			0
$772$	−13.7477	−	23.8118i	−0.494792	−	0.857004i
$773$	14.9666	−	25.9230i	0.538312	−	0.932384i	−0.460683	−	0.887565i	$-0.652395\pi$
$773$	14.9666	−	25.9230i	0.538312	−	0.932384i	0.998995	−	0.0448193i	$-0.0142712\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.34847	+	4.24264i	−0.263286	+	0.152008i
$780$		0			0
$781$	16.0000	−	27.7128i	0.572525	−	0.991642i
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$		0			0
$786$		0			0
$787$	45.8258	+	26.4575i	1.63351	+	0.943108i	0.982998	+	0.183614i	$0.0587795\pi$
$787$	45.8258	+	26.4575i	1.63351	+	0.943108i	0.650513	+	0.759495i	$0.274554\pi$
$788$	33.6749	+	19.4422i	1.19962	+	0.692600i
$789$		0			0
$790$		0			0
$791$	8.57321	+	14.8492i	0.304828	+	0.527978i
$792$		0			0
$793$	13.7477	+	23.8118i	0.488196	+	0.845580i
$794$		0			0
$795$		0			0
$796$	12.0000	−	6.92820i	0.425329	−	0.245564i
$797$	41.1582			1.45790			0.728950	−	0.684567i	$-0.240009\pi$
$797$	41.1582			1.45790			0.728950	+	0.684567i	$0.240009\pi$
$798$		0			0
$799$	14.0000			0.495284
$800$		0			0
$801$		0			0
$802$		0			0
$803$	18.7083	+	32.4037i	0.660201	+	1.14350i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	28.1691	+	16.2635i	0.990374	+	0.571793i	0.905386	−	0.424589i	$-0.139582\pi$
$809$	28.1691	+	16.2635i	0.990374	+	0.571793i	0.0849879	+	0.996382i	$0.472915\pi$
$810$		0			0
$811$		−	20.7846i		−	0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$		−	20.7846i		−	0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$	7.48331			0.262613
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−6.87386	+	3.96863i	−0.240486	+	0.138845i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.89898	+	2.82843i	−0.170976	+	0.0987128i	−0.583046	−	0.812439i	$-0.698139\pi$
$821$	−4.89898	+	2.82843i	−0.170976	+	0.0987128i	0.412070	+	0.911152i	$0.364806\pi$
$822$		0			0
$823$	13.7477	−	23.8118i	0.479216	−	0.830026i	−0.520500	−	0.853862i	$-0.674255\pi$
$823$	13.7477	−	23.8118i	0.479216	−	0.830026i	0.999716	+	0.0238357i	$0.00758785\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			19.4422i			0.676072i	0.941133	+	0.338036i	$0.109763\pi$
$827$			19.4422i			0.676072i	−0.941133	+	0.338036i	$0.890237\pi$
$828$		0			0
$829$	−37.5000	−	21.6506i	−1.30243	−	0.751958i	−0.321609	−	0.946873i	$-0.604224\pi$
$829$	−37.5000	−	21.6506i	−1.30243	−	0.751958i	−0.980820	+	0.194915i	$0.937557\pi$
$830$		0			0
$831$		0			0
$832$		−	21.1660i		−	0.733799i
$833$	−13.0958	−	22.6826i	−0.453743	−	0.785905i
$834$		0			0
$835$		0			0
$836$	−4.89898	+	8.48528i	−0.169435	+	0.293470i
$837$		0			0
$838$		0			0
$839$	12.2474			0.422829			0.211414	−	0.977397i	$-0.432193\pi$
$839$	12.2474			0.422829			0.211414	+	0.977397i	$0.432193\pi$
$840$		0			0
$841$	27.0000			0.931034
$842$		0			0
$843$		0			0
$844$	−14.0000	+	24.2487i	−0.481900	+	0.834675i
$845$		0			0
$846$		0			0
$847$	6.87386	+	3.96863i	0.236189	+	0.136364i
$848$		−	51.8459i		−	1.78040i
$849$		0			0
$850$		0			0
$851$	−25.7196	−	14.8492i	−0.881658	−	0.509025i
$852$		0			0
$853$		−	50.2693i		−	1.72119i	−0.509292	−	0.860594i	$-0.670093\pi$
$853$		−	50.2693i		−	1.72119i	0.509292	−	0.860594i	$-0.329907\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.35414	+	16.2019i	−0.319531	+	0.553445i	−0.980390	−	0.197065i	$-0.936859\pi$
$857$	−9.35414	+	16.2019i	−0.319531	+	0.553445i	0.660859	+	0.750510i	$0.270192\pi$
$858$		0			0
$859$	−12.0000	+	6.92820i	−0.409435	+	0.236387i	−0.690547	−	0.723288i	$-0.742630\pi$
$859$	−12.0000	+	6.92820i	−0.409435	+	0.236387i	0.281112	+	0.959675i	$0.409297\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.0624	−	16.2019i	0.955256	−	0.551517i	0.0605464	−	0.998165i	$-0.480716\pi$
$863$	28.0624	−	16.2019i	0.955256	−	0.551517i	0.894710	+	0.446648i	$0.147382\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	13.7477	−	23.8118i	0.466628	−	0.808224i
$869$			19.7990i			0.671635i
$870$		0			0
$871$	−31.5000	−	18.1865i	−1.06734	−	0.616227i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.58258	−	7.93725i	−0.154743	−	0.268022i	0.778223	−	0.627988i	$-0.216121\pi$
$877$	−4.58258	−	7.93725i	−0.154743	−	0.268022i	−0.932965	+	0.359966i	$0.882788\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	29.3939			0.990305			0.495152	−	0.868806i	$-0.335112\pi$
$881$	29.3939			0.990305			0.495152	+	0.868806i	$0.335112\pi$
$882$		0			0
$883$	41.2432			1.38794			0.693972	−	0.720002i	$-0.255859\pi$
$883$	41.2432			1.38794			0.693972	+	0.720002i	$0.255859\pi$
$884$	17.1464	−	9.89949i	0.576697	−	0.332956i
$885$		0			0
$886$		0			0
$887$	18.7083	+	32.4037i	0.628163	+	1.08801i	0.987920	+	0.154964i	$0.0495261\pi$
$887$	18.7083	+	32.4037i	0.628163	+	1.08801i	−0.359757	+	0.933046i	$0.617141\pi$
$888$		0			0
$889$	31.5000	−	18.1865i	1.05648	−	0.609957i
$890$		0			0
$891$		0			0
$892$	−9.16515	−	5.29150i	−0.306872	−	0.177173i
$893$	−5.61249	−	3.24037i	−0.187815	−	0.108435i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.67423	+	6.36396i	−0.122543	+	0.212250i
$900$		0			0
$901$	42.0000	−	24.2487i	1.39922	−	0.807842i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	11.4564	−	19.8431i	0.380405	−	0.658880i	−0.610715	−	0.791850i	$-0.709118\pi$
$907$	11.4564	−	19.8431i	0.380405	−	0.658880i	0.991120	+	0.132970i	$0.0424513\pi$
$908$	14.9666	+	25.9230i	0.496685	+	0.860284i
$909$		0			0
$910$		0			0
$911$			41.0122i			1.35879i	0.733771	+	0.679397i	$0.237759\pi$
$911$			41.0122i			1.35879i	−0.733771	+	0.679397i	$0.762241\pi$
$912$		0			0
$913$	−36.6606	−	21.1660i	−1.21329	−	0.700493i
$914$		0			0
$915$		0			0
$916$			3.46410i			0.114457i
$917$	39.2874	+	22.6826i	1.29738	+	0.749045i
$918$		0			0
$919$	17.5000	+	30.3109i	0.577272	+	0.999864i	0.995791	+	0.0916559i	$0.0292160\pi$
$919$	17.5000	+	30.3109i	0.577272	+	0.999864i	−0.418519	+	0.908208i	$0.637451\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	29.9333			0.985265
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$			12.1244i			0.397360i
$932$		−	12.9615i		−	0.424567i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			50.2693i			1.64223i	0.570766	+	0.821113i	$0.306646\pi$
$937$			50.2693i			1.64223i	−0.570766	+	0.821113i	$0.693354\pi$
$938$		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
$943$	27.4955	−	15.8745i	0.895375	−	0.516945i
$944$	29.3939			0.956689
$945$		0			0
$946$		0			0
$947$	33.6749	−	19.4422i	1.09429	−	0.631787i	0.159573	−	0.987186i	$-0.448988\pi$
$947$	33.6749	−	19.4422i	1.09429	−	0.631787i	0.934715	+	0.355399i	$0.115655\pi$
$948$		0			0
$949$	−17.5000	+	30.3109i	−0.568074	+	0.983933i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−31.8434	−	18.3848i	−1.02989	−	0.594606i
$957$		0			0
$958$		0			0
$959$	17.1464			0.553687
$960$		0			0
$961$	−2.00000	−	3.46410i	−0.0645161	−	0.111745i
$962$		0			0
$963$		0			0
$964$	30.0000	−	17.3205i	0.966235	−	0.557856i
$965$		0			0
$966$		0			0
$967$	4.58258			0.147366			0.0736828	−	0.997282i	$-0.476525\pi$
$967$	4.58258			0.147366			0.0736828	+	0.997282i	$0.476525\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.79796	−	16.9706i	−0.314431	−	0.544611i	0.664885	−	0.746946i	$-0.268481\pi$
$971$	−9.79796	−	16.9706i	−0.314431	−	0.544611i	−0.979316	+	0.202334i	$0.935147\pi$
$972$		0			0
$973$	−11.4564	−	19.8431i	−0.367277	−	0.636142i
$974$		0			0
$975$		0			0
$976$	36.0000	+	20.7846i	1.15233	+	0.665299i
$977$	−33.6749	−	19.4422i	−1.07736	−	0.622012i	−0.147174	−	0.989111i	$-0.547018\pi$
$977$	−33.6749	−	19.4422i	−1.07736	−	0.622012i	−0.930182	+	0.367099i	$0.880351\pi$
$978$		0			0
$979$			48.4974i			1.54998i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24.3208	+	42.1248i	−0.775712	+	1.34357i	0.158681	+	0.987330i	$0.449276\pi$
$983$	−24.3208	+	42.1248i	−0.775712	+	1.34357i	−0.934393	+	0.356243i	$0.884058\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−9.16515			−0.291582
$989$	25.7196	−	14.8492i	0.817837	−	0.472178i
$990$		0			0
$991$	3.50000	−	6.06218i	0.111181	−	0.192571i	−0.805066	−	0.593186i	$-0.797870\pi$
$991$	3.50000	−	6.06218i	0.111181	−	0.192571i	0.916247	+	0.400614i	$0.131203\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.29129	−	1.32288i	−0.0725658	−	0.0418959i	0.463278	−	0.886213i	$-0.346673\pi$
$997$	−2.29129	−	1.32288i	−0.0725658	−	0.0418959i	−0.535844	+	0.844317i	$0.680006\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.bk.d.1151.1		8
3.2	odd	2	inner	1575.2.bk.d.1151.2		8
5.2	odd	4		315.2.bb.a.269.1	yes	8
5.3	odd	4		315.2.bb.a.269.3	yes	8
5.4	even	2	inner	1575.2.bk.d.1151.3		8
7.5	odd	6	inner	1575.2.bk.d.26.2		8
15.2	even	4		315.2.bb.a.269.4	yes	8
15.8	even	4		315.2.bb.a.269.2	yes	8
15.14	odd	2	inner	1575.2.bk.d.1151.4		8
21.5	even	6	inner	1575.2.bk.d.26.1		8
35.3	even	12		2205.2.g.a.2204.2		8
35.12	even	12		315.2.bb.a.89.2	yes	8
35.17	even	12		2205.2.g.a.2204.4		8
35.18	odd	12		2205.2.g.a.2204.8		8
35.19	odd	6	inner	1575.2.bk.d.26.4		8
35.32	odd	12		2205.2.g.a.2204.6		8
35.33	even	12		315.2.bb.a.89.4	yes	8
105.17	odd	12		2205.2.g.a.2204.5		8
105.32	even	12		2205.2.g.a.2204.3		8
105.38	odd	12		2205.2.g.a.2204.7		8
105.47	odd	12		315.2.bb.a.89.3	yes	8
105.53	even	12		2205.2.g.a.2204.1		8
105.68	odd	12		315.2.bb.a.89.1	✓	8
105.89	even	6	inner	1575.2.bk.d.26.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bb.a.89.1	✓	8	105.68	odd	12
315.2.bb.a.89.2	yes	8	35.12	even	12
315.2.bb.a.89.3	yes	8	105.47	odd	12
315.2.bb.a.89.4	yes	8	35.33	even	12
315.2.bb.a.269.1	yes	8	5.2	odd	4
315.2.bb.a.269.2	yes	8	15.8	even	4
315.2.bb.a.269.3	yes	8	5.3	odd	4
315.2.bb.a.269.4	yes	8	15.2	even	4
1575.2.bk.d.26.1		8	21.5	even	6	inner
1575.2.bk.d.26.2		8	7.5	odd	6	inner
1575.2.bk.d.26.3		8	105.89	even	6	inner
1575.2.bk.d.26.4		8	35.19	odd	6	inner
1575.2.bk.d.1151.1		8	1.1	even	1	trivial
1575.2.bk.d.1151.2		8	3.2	odd	2	inner
1575.2.bk.d.1151.3		8	5.4	even	2	inner
1575.2.bk.d.1151.4		8	15.14	odd	2	inner
2205.2.g.a.2204.1		8	105.53	even	12
2205.2.g.a.2204.2		8	35.3	even	12
2205.2.g.a.2204.3		8	105.32	even	12
2205.2.g.a.2204.4		8	35.17	even	12
2205.2.g.a.2204.5		8	105.17	odd	12
2205.2.g.a.2204.6		8	35.32	odd	12
2205.2.g.a.2204.7		8	105.38	odd	12
2205.2.g.a.2204.8		8	35.18	odd	12

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)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{4} +(-2.29129 + 1.32288i) q^{7} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{4} +(-2.29129 + 1.32288i) q^{7} +(-2.44949 - 1.41421i) q^{11} -2.64575i q^{13} +(-2.00000 - 3.46410i) q^{16} +(1.87083 - 3.24037i) q^{17} +(-1.50000 + 0.866025i) q^{19} +(5.61249 - 3.24037i) q^{23} -5.29150i q^{28} +1.41421i q^{29} +(4.50000 + 2.59808i) q^{31} +(-2.29129 - 3.96863i) q^{37} +4.89898 q^{41} +4.58258 q^{43} +(4.89898 - 2.82843i) q^{44} +(1.87083 + 3.24037i) q^{47} +(3.50000 - 6.06218i) q^{49} +(4.58258 + 2.64575i) q^{52} +(11.2250 + 6.48074i) q^{53} +(-3.67423 + 6.36396i) q^{59} +(-9.00000 + 5.19615i) q^{61} +8.00000 q^{64} +(6.87386 - 11.9059i) q^{67} +(3.74166 + 6.48074i) q^{68} +11.3137i q^{71} +(-11.4564 - 6.61438i) q^{73} -3.46410i q^{76} +7.48331 q^{77} +(-3.50000 - 6.06218i) q^{79} +14.9666 q^{83} +(-8.57321 - 14.8492i) q^{89} +(3.50000 + 6.06218i) q^{91} +12.9615i q^{92} -5.29150i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 16 q^{16} - 12 q^{19} + 36 q^{31} + 28 q^{49} - 72 q^{61} + 64 q^{64} - 28 q^{79} + 28 q^{91}+O(q^{100}) \) 8 * q - 8 * q^4 - 16 * q^16 - 12 * q^19 + 36 * q^31 + 28 * q^49 - 72 * q^61 + 64 * q^64 - 28 * q^79 + 28 * q^91

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