LMFDB - Embedded newform 280.2.ba.a.19.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,2,Mod(19,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("280.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	280.ba (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:	$2.23581125660$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.3317760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			19.4
Root			$-1.72286 + 0.178197i$ of defining polynomial
Character	$\chi$	$=$	280.19
Dual form			280.2.ba.a.59.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-1.41421 - 2.23607i) q^{7} -2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-1.41421 - 2.23607i) q^{7} -2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +(2.73861 - 1.58114i) q^{10} +(2.23861 + 3.87739i) q^{11} -7.13505i q^{13} +(-3.73861 + 0.150912i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(-2.12132 - 3.67423i) q^{18} +(1.76139 + 1.01694i) q^{19} -4.47214i q^{20} +6.33175 q^{22} +(-4.05781 + 7.02834i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-8.73861 - 5.04524i) q^{26} +(-2.45877 + 4.68556i) q^{28} +(2.82843 + 4.89898i) q^{32} +(-0.238613 - 5.91127i) q^{35} -6.00000 q^{36} +(0.891935 - 1.54488i) q^{37} +(2.49098 - 1.43817i) q^{38} +(-5.47723 - 3.16228i) q^{40} +8.05661i q^{41} +(4.47723 - 7.75478i) q^{44} +(5.80948 - 3.35410i) q^{45} +(5.73861 + 9.93957i) q^{46} +(-3.68815 - 2.12936i) q^{47} +(-3.00000 + 6.32456i) q^{49} +7.07107 q^{50} +(-12.3583 + 7.13505i) q^{52} +(7.22369 + 12.5118i) q^{53} +10.0114i q^{55} +(4.00000 + 6.32456i) q^{56} +(5.47723 - 3.16228i) q^{59} +(-7.93080 + 0.320133i) q^{63} +8.00000 q^{64} +(7.97723 - 13.8170i) q^{65} +(-7.40852 - 3.88766i) q^{70} +(-4.24264 + 7.34847i) q^{72} +(-1.26139 - 2.18479i) q^{74} -4.06775i q^{76} +(5.50423 - 10.4891i) q^{77} +(-7.74597 + 4.47214i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(9.86729 + 5.69688i) q^{82} +(-6.33175 - 10.9669i) q^{88} +(-10.9545 - 6.32456i) q^{89} -9.48683i q^{90} +(-15.9545 + 10.0905i) q^{91} +16.2312 q^{92} +(-5.21584 + 3.01137i) q^{94} +(2.27394 + 3.93858i) q^{95} +(5.62465 + 8.14637i) q^{98} +13.4317 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 + 12 * q^9	$ 8 q - 8 q^{4} + 12 q^{9} - 4 q^{11} - 8 q^{14} - 16 q^{16} + 36 q^{19} + 20 q^{25} - 48 q^{26} + 20 q^{35} - 48 q^{36} - 8 q^{44} + 24 q^{46} - 24 q^{49} + 32 q^{56} + 64 q^{64} + 20 q^{65} - 32 q^{74} - 36 q^{81} - 40 q^{91} + 24 q^{94} - 24 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 + 12 * q^9 - 4 * q^11 - 8 * q^14 - 16 * q^16 + 36 * q^19 + 20 * q^25 - 48 * q^26 + 20 * q^35 - 48 * q^36 - 8 * q^44 + 24 * q^46 - 24 * q^49 + 32 * q^56 + 64 * q^64 + 20 * q^65 - 32 * q^74 - 36 * q^81 - 40 * q^91 + 24 * q^94 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.707107	−	1.22474i	0.500000	−	0.866025i
$2$	0.707107	−	1.22474i	0.500000	−	0.866025i
$3$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$	1.93649	+	1.11803i	0.866025	+	0.500000i
$5$	1.93649	+	1.11803i	0.866025	+	0.500000i
$6$		0			0
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$7$	−1.41421	−	2.23607i	−0.534522	−	0.845154i
$8$	−2.82843			−1.00000
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$	2.73861	−	1.58114i	0.866025	−	0.500000i
$11$	2.23861	+	3.87739i	0.674967	+	1.16908i	0.976478	+	0.215615i	$0.0691756\pi$
$11$	2.23861	+	3.87739i	0.674967	+	1.16908i	−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	7.13505i		−	1.97891i	−0.144854	−	0.989453i	$-0.546271\pi$
$13$		−	7.13505i		−	1.97891i	0.144854	−	0.989453i	$-0.453729\pi$
$14$	−3.73861	+	0.150912i	−0.999186	+	0.0403329i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	−2.12132	−	3.67423i	−0.500000	−	0.866025i
$19$	1.76139	+	1.01694i	0.404090	+	0.233301i	0.688247	−	0.725476i	$-0.258380\pi$
$19$	1.76139	+	1.01694i	0.404090	+	0.233301i	−0.284157	+	0.958778i	$0.591714\pi$
$20$		−	4.47214i		−	1.00000i
$21$		0			0
$22$	6.33175			1.34993
$23$	−4.05781	+	7.02834i	−0.846112	+	1.46551i	0.0385394	+	0.999257i	$0.487729\pi$
$23$	−4.05781	+	7.02834i	−0.846112	+	1.46551i	−0.884652	+	0.466252i	$0.845604\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$	−8.73861	−	5.04524i	−1.71378	−	0.989453i
$27$		0			0
$28$	−2.45877	+	4.68556i	−0.464664	+	0.885487i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	2.82843	+	4.89898i	0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$	−0.238613	−	5.91127i	−0.0403329	−	0.999186i
$36$	−6.00000			−1.00000
$37$	0.891935	−	1.54488i	0.146633	−	0.253976i	−0.783348	−	0.621584i	$-0.786490\pi$
$37$	0.891935	−	1.54488i	0.146633	−	0.253976i	0.929981	+	0.367607i	$0.119823\pi$
$38$	2.49098	−	1.43817i	0.404090	−	0.233301i
$39$		0			0
$40$	−5.47723	−	3.16228i	−0.866025	−	0.500000i
$41$			8.05661i			1.25823i	0.777312	+	0.629115i	$0.216583\pi$
$41$			8.05661i			1.25823i	−0.777312	+	0.629115i	$0.783417\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	4.47723	−	7.75478i	0.674967	−	1.16908i
$45$	5.80948	−	3.35410i	0.866025	−	0.500000i
$46$	5.73861	+	9.93957i	0.846112	+	1.46551i
$47$	−3.68815	−	2.12936i	−0.537973	−	0.310599i	0.206284	−	0.978492i	$-0.433863\pi$
$47$	−3.68815	−	2.12936i	−0.537973	−	0.310599i	−0.744257	+	0.667893i	$0.767196\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$	7.07107			1.00000
$51$		0			0
$52$	−12.3583	+	7.13505i	−1.71378	+	0.989453i
$53$	7.22369	+	12.5118i	0.992250	+	1.71863i	0.603736	+	0.797185i	$0.293678\pi$
$53$	7.22369	+	12.5118i	0.992250	+	1.71863i	0.388514	+	0.921443i	$0.372988\pi$
$54$		0			0
$55$			10.0114i			1.34993i
$56$	4.00000	+	6.32456i	0.534522	+	0.845154i
$57$		0			0
$58$		0			0
$59$	5.47723	−	3.16228i	0.713074	−	0.411693i	−0.0991242	−	0.995075i	$-0.531604\pi$
$59$	5.47723	−	3.16228i	0.713074	−	0.411693i	0.812198	+	0.583382i	$0.198271\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$	−7.93080	+	0.320133i	−0.999186	+	0.0403329i
$64$	8.00000			1.00000
$65$	7.97723	−	13.8170i	0.989453	−	1.71378i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	−7.40852	−	3.88766i	−0.885487	−	0.464664i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−4.24264	+	7.34847i	−0.500000	+	0.866025i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$	−1.26139	−	2.18479i	−0.146633	−	0.253976i
$75$		0			0
$76$		−	4.06775i		−	0.466603i
$77$	5.50423	−	10.4891i	0.627266	−	1.19535i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−7.74597	+	4.47214i	−0.866025	+	0.500000i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$	9.86729	+	5.69688i	1.08966	+	0.629115i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−6.33175	−	10.9669i	−0.674967	−	1.16908i
$89$	−10.9545	−	6.32456i	−1.16117	−	0.670402i	−0.209585	−	0.977790i	$-0.567211\pi$
$89$	−10.9545	−	6.32456i	−1.16117	−	0.670402i	−0.951584	+	0.307389i	$0.900545\pi$
$90$		−	9.48683i		−	1.00000i
$91$	−15.9545	+	10.0905i	−1.67248	+	1.05777i
$92$	16.2312			1.69222
$93$		0			0
$94$	−5.21584	+	3.01137i	−0.537973	+	0.310599i
$95$	2.27394	+	3.93858i	0.233301	+	0.404090i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	5.62465	+	8.14637i	0.568175	+	0.822908i
$99$	13.4317			1.34993
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	3.87298	+	2.23607i	0.381616	+	0.220326i	0.678521	−	0.734581i	$-0.262621\pi$
$103$	3.87298	+	2.23607i	0.381616	+	0.220326i	−0.296905	+	0.954907i	$0.595954\pi$
$104$			20.1810i			1.97891i
$105$		0			0
$106$	20.4317			1.98450
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	12.2614	+	7.07912i	1.16908	+	0.674967i
$111$		0			0
$112$	10.5744	−	0.426844i	0.999186	−	0.0403329i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−15.7158	+	9.07354i	−1.46551	+	0.846112i
$116$		0			0
$117$	−18.5374	−	10.7026i	−1.71378	−	0.989453i
$118$		−	8.94427i		−	0.823387i
$119$		0			0
$120$		0			0
$121$	−4.52277	+	7.83368i	−0.411161	+	0.712152i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$	−5.21584	+	9.93957i	−0.464664	+	0.885487i
$127$	−20.7791			−1.84385			−0.921925	−	0.387369i	$-0.873384\pi$
$127$	−20.7791			−1.84385			−0.921925	+	0.387369i	$0.873384\pi$
$128$	5.65685	−	9.79796i	0.500000	−	0.866025i
$129$		0			0
$130$	−11.2815	−	19.5401i	−0.989453	−	1.71378i
$131$	−19.2386	−	11.1074i	−1.68089	−	0.970460i	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−19.2386	−	11.1074i	−1.68089	−	0.970460i	−0.719811	−	0.694170i	$-0.755772\pi$
$132$		0			0
$133$	−0.217037	−	5.37675i	−0.0188195	−	0.466223i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		−	18.9737i		−	1.60933i	−0.593732	−	0.804663i	$-0.702346\pi$
$139$		−	18.9737i		−	1.60933i	0.593732	−	0.804663i	$-0.297654\pi$
$140$	−10.0000	+	6.32456i	−0.845154	+	0.534522i
$141$		0			0
$142$		0			0
$143$	27.6654	−	15.9726i	2.31349	−	1.33570i
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−3.56774			−0.293267
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−4.98196	−	2.87633i	−0.404090	−	0.233301i
$153$		0			0
$154$	−8.95445	−	14.1582i	−0.721570	−	1.14090i
$155$		0			0
$156$		0			0
$157$	1.19718	−	0.691190i	0.0955451	−	0.0551630i	−0.451466	−	0.892288i	$-0.649099\pi$
$157$	1.19718	−	0.691190i	0.0955451	−	0.0551630i	0.547011	+	0.837125i	$0.315765\pi$
$158$		0			0
$159$		0			0
$160$			12.6491i			1.00000i
$161$	21.4545	−	0.866025i	1.69085	−	0.0682524i
$162$	−12.7279			−1.00000
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$	13.9545	−	8.05661i	1.08966	−	0.629115i
$165$		0			0
$166$		0			0
$167$		−	24.2815i		−	1.87896i	−0.342608	−	0.939478i	$-0.611310\pi$
$167$		−	24.2815i		−	1.87896i	0.342608	−	0.939478i	$-0.388690\pi$
$168$		0			0
$169$	−37.9089			−2.91607
$170$		0			0
$171$	5.28416	−	3.05081i	0.404090	−	0.233301i
$172$		0			0
$173$	11.1611	+	6.44386i	0.848562	+	0.489917i	0.860165	−	0.510015i	$-0.170360\pi$
$173$	11.1611	+	6.44386i	0.848562	+	0.489917i	−0.0116035	+	0.999933i	$0.503694\pi$
$174$		0			0
$175$	6.14692	−	11.7139i	0.464664	−	0.885487i
$176$	−17.9089			−1.34993
$177$		0			0
$178$	−15.4919	+	8.94427i	−1.16117	+	0.670402i
$179$	9.23861	+	16.0017i	0.690526	+	1.19603i	0.971666	+	0.236360i	$0.0759544\pi$
$179$	9.23861	+	16.0017i	0.690526	+	1.19603i	−0.281139	+	0.959667i	$0.590712\pi$
$180$	−11.6190	−	6.70820i	−0.866025	−	0.500000i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	1.07676	+	26.6752i	0.0798151	+	1.97730i
$183$		0			0
$184$	11.4772	−	19.8791i	0.846112	−	1.46551i
$185$	3.45445	−	1.99443i	0.253976	−	0.146633i
$186$		0			0
$187$		0			0
$188$			8.51743i			0.621197i
$189$		0			0
$190$	6.43168			0.466603
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	13.9545	−	1.12840i	0.996747	−	0.0806002i
$197$	−10.8796			−0.775142			−0.387571	−	0.921840i	$-0.626686\pi$
$197$	−10.8796			−0.775142			−0.387571	+	0.921840i	$0.626686\pi$
$198$	9.49763	−	16.4504i	0.674967	−	1.16908i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−7.07107	−	12.2474i	−0.500000	−	0.866025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−9.00756	+	15.6016i	−0.629115	+	1.08966i
$206$	5.47723	−	3.16228i	0.381616	−	0.220326i
$207$	12.1734	+	21.0850i	0.846112	+	1.46551i
$208$	24.7165	+	14.2701i	1.71378	+	0.989453i
$209$			9.10612i			0.629883i
$210$		0			0
$211$	27.4317			1.88847			0.944237	−	0.329266i	$-0.106801\pi$
$211$	27.4317			1.88847			0.944237	+	0.329266i	$0.106801\pi$
$212$	14.4474	−	25.0236i	0.992250	−	1.71863i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	17.3402	−	10.0114i	1.16908	−	0.674967i
$221$		0			0
$222$		0			0
$223$			22.3607i			1.49738i	0.662919	+	0.748691i	$0.269317\pi$
$223$			22.3607i			1.49738i	−0.662919	+	0.748691i	$0.730683\pi$
$224$	6.95445	−	13.2528i	0.464664	−	0.885487i
$225$	15.0000			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$			25.6639i			1.69222i
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$	−26.2158	+	15.1357i	−1.71378	+	0.989453i
$235$	−4.76139	−	8.24696i	−0.310599	−	0.537973i
$236$	−10.9545	−	6.32456i	−0.713074	−	0.411693i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	24.4545	−	14.1188i	1.57525	−	0.909471i	0.579741	−	0.814801i	$-0.303154\pi$
$241$	24.4545	−	14.1188i	1.57525	−	0.909471i	0.995509	−	0.0946700i	$-0.0301796\pi$
$242$	6.39617	+	11.0785i	0.411161	+	0.712152i
$243$		0			0
$244$		0			0
$245$	−12.8805	+	8.89335i	−0.822908	+	0.568175i
$246$		0			0
$247$	7.25590	−	12.5676i	0.461682	−	0.799656i
$248$		0			0
$249$		0			0
$250$	13.6931	+	7.90569i	0.866025	+	0.500000i
$251$		−	14.0793i		−	0.888680i	−0.895858	−	0.444340i	$-0.853438\pi$
$251$		−	14.0793i		−	0.888680i	0.895858	−	0.444340i	$-0.146562\pi$
$252$	8.48528	+	13.4164i	0.534522	+	0.845154i
$253$	−36.3355			−2.28439
$254$	−14.6931	+	25.4491i	−0.921925	+	1.59682i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−4.71584	+	0.190358i	−0.293028	+	0.0118283i
$260$	−31.9089			−1.97891
$261$		0			0
$262$	−27.2075	+	15.7083i	−1.68089	+	0.970460i
$263$	−4.24264	−	7.34847i	−0.261612	−	0.453126i	0.705058	−	0.709150i	$-0.250921\pi$
$263$	−4.24264	−	7.34847i	−0.261612	−	0.453126i	−0.966671	+	0.256023i	$0.917588\pi$
$264$		0			0
$265$			32.3053i			1.98450i
$266$	−6.73861	−	3.53612i	−0.413171	−	0.216813i
$267$		0			0
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.1931	+	19.3870i	−0.674967	+	1.16908i
$276$		0			0
$277$	5.65685	+	9.79796i	0.339887	+	0.588702i	0.984411	−	0.175882i	$-0.0562777\pi$
$277$	5.65685	+	9.79796i	0.339887	+	0.588702i	−0.644524	+	0.764584i	$0.722944\pi$
$278$	−23.2379	−	13.4164i	−1.39372	−	0.804663i
$279$		0			0
$280$	0.674899	+	16.7196i	0.0403329	+	0.999186i
$281$	−10.9089			−0.650771			−0.325385	−	0.945582i	$-0.605494\pi$
$281$	−10.9089			−0.650771			−0.325385	+	0.945582i	$0.605494\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$		−	45.1774i		−	2.67139i
$287$	18.0151	−	11.3938i	1.06340	−	0.672552i
$288$	16.9706			1.00000
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			27.1578i			1.58658i	0.608846	+	0.793288i	$0.291633\pi$
$293$			27.1578i			1.58658i	−0.608846	+	0.793288i	$0.708367\pi$
$294$		0			0
$295$	14.1421			0.823387
$296$	−2.52277	+	4.36957i	−0.146633	+	0.253976i
$297$		0			0
$298$		0			0
$299$	50.1475	+	28.9527i	2.90011	+	1.67438i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−7.04555	+	4.06775i	−0.404090	+	0.233301i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−23.6720	+	0.955537i	−1.34884	+	0.0544468i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$		−	1.95498i		−	0.110326i
$315$	−15.7158	−	8.24696i	−0.885487	−	0.464664i
$316$		0			0
$317$	−8.48528	+	14.6969i	−0.476581	+	0.825462i	−0.999640	−	0.0268342i	$-0.991457\pi$
$317$	−8.48528	+	14.6969i	−0.476581	+	0.825462i	0.523059	+	0.852296i	$0.324791\pi$
$318$		0			0
$319$		0			0
$320$	15.4919	+	8.94427i	0.866025	+	0.500000i
$321$		0			0
$322$	14.1099	−	26.8886i	0.786316	−	1.49844i
$323$		0			0
$324$	−9.00000	+	15.5885i	−0.500000	+	0.866025i
$325$	30.8957	−	17.8376i	1.71378	−	0.989453i
$326$		0			0
$327$		0			0
$328$		−	22.7875i		−	1.25823i
$329$	0.454451	+	11.2583i	0.0250547	+	0.620692i
$330$		0			0
$331$	−18.1931	+	31.5113i	−0.999981	+	1.73202i	−0.494685	+	0.869072i	$0.664716\pi$
$331$	−18.1931	+	31.5113i	−0.999981	+	1.73202i	−0.505296	+	0.862946i	$0.668617\pi$
$332$		0			0
$333$	−2.67581	−	4.63463i	−0.146633	−	0.253976i
$334$	−29.7386	−	17.1696i	−1.62722	−	0.939478i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−26.8056	+	46.4287i	−1.45803	+	2.52539i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		−	8.62900i		−	0.466603i
$343$	18.3848	−	2.23607i	0.992685	−	0.120736i
$344$		0			0
$345$		0			0
$346$	15.7842	−	9.11299i	0.848562	−	0.489917i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−10.0000	−	15.8114i	−0.534522	−	0.845154i
$351$		0			0
$352$	−12.6635	+	21.9338i	−0.674967	+	1.16908i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$			25.2982i			1.34080i
$357$		0			0
$358$	26.1307			1.38105
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	−16.4317	+	9.48683i	−0.866025	+	0.500000i
$361$	−7.43168	−	12.8720i	−0.391141	−	0.677476i
$362$		0			0
$363$		0			0
$364$	33.4317	+	17.5434i	1.75230	+	0.919526i
$365$		0			0
$366$		0			0
$367$	−13.6521	+	7.88202i	−0.712632	+	0.411438i	−0.812035	−	0.583609i	$-0.801640\pi$
$367$	−13.6521	+	7.88202i	−0.712632	+	0.411438i	0.0994028	+	0.995047i	$0.468307\pi$
$368$	−16.2312	−	28.1133i	−0.846112	−	1.46551i
$369$	20.9317	+	12.0849i	1.08966	+	0.629115i
$370$		−	5.64110i		−	0.293267i
$371$	17.7614	−	33.8470i	0.922125	−	1.75725i
$372$		0			0
$373$	11.3137	−	19.5959i	0.585802	−	1.01464i	−0.408973	−	0.912546i	$-0.634113\pi$
$373$	11.3137	−	19.5959i	0.585802	−	1.01464i	0.994775	−	0.102092i	$-0.0325536\pi$
$374$		0			0
$375$		0			0
$376$	10.4317	+	6.02273i	0.537973	+	0.310599i
$377$		0			0
$378$		0			0
$379$	−0.568323			−0.0291928			−0.0145964	−	0.999893i	$-0.504646\pi$
$379$	−0.568323			−0.0291928			−0.0145964	+	0.999893i	$0.504646\pi$
$380$	4.54788	−	7.87716i	0.233301	−	0.404090i
$381$		0			0
$382$		0			0
$383$	−33.3866	−	19.2758i	−1.70598	−	0.984947i	−0.939418	−	0.342773i	$-0.888634\pi$
$383$	−33.3866	−	19.2758i	−1.70598	−	0.984947i	−0.766559	−	0.642173i	$-0.778033\pi$
$384$		0			0
$385$	22.3861	−	14.1582i	1.14090	−	0.721570i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	8.48528	−	17.8885i	0.428571	−	0.903508i
$393$		0			0
$394$	−7.69306	+	13.3248i	−0.387571	+	0.671293i
$395$		0			0
$396$	−13.4317	−	23.2643i	−0.674967	−	1.16908i
$397$	3.87298	+	2.23607i	0.194379	+	0.112225i	0.594031	−	0.804442i	$-0.297536\pi$
$397$	3.87298	+	2.23607i	0.194379	+	0.112225i	−0.399652	+	0.916667i	$0.630869\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	14.9772	−	25.9413i	0.747927	−	1.29545i	−0.200888	−	0.979614i	$-0.564383\pi$
$401$	14.9772	−	25.9413i	0.747927	−	1.29545i	0.948815	−	0.315833i	$-0.102284\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	20.1246i		−	1.00000i
$406$		0			0
$407$	7.98679			0.395891
$408$		0			0
$409$	−32.8634	+	18.9737i	−1.62499	+	0.938187i	−0.639430	+	0.768849i	$0.720830\pi$
$409$	−32.8634	+	18.9737i	−1.62499	+	0.938187i	−0.985558	+	0.169338i	$0.945837\pi$
$410$	12.7386	+	22.0639i	0.629115	+	1.08966i
$411$		0			0
$412$		−	8.94427i		−	0.440653i
$413$	−14.8170	−	7.77531i	−0.729099	−	0.382598i
$414$	34.4317			1.69222
$415$		0			0
$416$	34.9545	−	20.1810i	1.71378	−	0.989453i
$417$		0			0
$418$	11.1527	+	6.43900i	0.545495	+	0.314942i
$419$		−	38.3280i		−	1.87245i	−0.351404	−	0.936224i	$-0.614296\pi$
$419$		−	38.3280i		−	1.87245i	0.351404	−	0.936224i	$-0.385704\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	19.3971	−	33.5968i	0.944237	−	1.63547i
$423$	−11.0645	+	6.38807i	−0.537973	+	0.310599i
$424$	−20.4317	−	35.3887i	−0.992250	−	1.71863i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.2948	+	8.25308i	−0.683811	+	0.394798i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		−	28.3165i		−	1.34993i
$441$	11.9317	+	17.2811i	0.568175	+	0.822908i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	−14.1421	−	24.4949i	−0.670402	−	1.16117i
$446$	27.3861	+	15.8114i	1.29677	+	0.748691i
$447$		0			0
$448$	−11.3137	−	17.8885i	−0.534522	−	0.845154i
$449$	−38.9089			−1.83622			−0.918112	−	0.396320i	$-0.870287\pi$
$449$	−38.9089			−1.83622			−0.918112	+	0.396320i	$0.870287\pi$
$450$	10.6066	−	18.3712i	0.500000	−	0.866025i
$451$	−31.2386	+	18.0356i	−1.47097	+	0.849264i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−42.1772	+	1.70251i	−1.97730	+	0.0798151i
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$	31.4317	+	18.1471i	1.46551	+	0.846112i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−0.980140			−0.0455510			−0.0227755	−	0.999741i	$-0.507250\pi$
$463$	−0.980140			−0.0455510			−0.0227755	+	0.999741i	$0.507250\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$			42.8103i			1.97891i
$469$		0			0
$470$	−13.4672			−0.621197
$471$		0			0
$472$	−15.4919	+	8.94427i	−0.713074	+	0.411693i
$473$		0			0
$474$		0			0
$475$			10.1694i			0.466603i
$476$		0			0
$477$	43.3421			1.98450
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−11.0228	−	6.36400i	−0.502595	−	0.290174i
$482$		−	39.9340i		−	1.81894i
$483$		0			0
$484$	18.0911			0.822323
$485$		0			0
$486$		0			0
$487$	15.5563	+	26.9444i	0.704925	+	1.22097i	0.966718	+	0.255843i	$0.0823529\pi$
$487$	15.5563	+	26.9444i	0.704925	+	1.22097i	−0.261793	+	0.965124i	$0.584314\pi$
$488$		0			0
$489$		0			0
$490$	1.78416	+	22.0639i	0.0806002	+	0.996747i
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$		0			0
$494$	−10.2614	−	17.7732i	−0.461682	−	0.799656i
$495$	26.0103	+	15.0171i	1.16908	+	0.674967i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3.00000	−	5.19615i	0.134298	−	0.232612i	−0.791031	−	0.611776i	$-0.790455\pi$
$499$	3.00000	−	5.19615i	0.134298	−	0.232612i	0.925329	+	0.379165i	$0.123789\pi$
$500$	19.3649	−	11.1803i	0.866025	−	0.500000i
$501$		0			0
$502$	−17.2436	−	9.95560i	−0.769619	−	0.444340i
$503$		−	40.2492i		−	1.79462i	−0.441397	−	0.897312i	$-0.645517\pi$
$503$		−	40.2492i		−	1.79462i	0.441397	−	0.897312i	$-0.354483\pi$
$504$	22.4317	−	0.905472i	0.999186	−	0.0403329i
$505$		0			0
$506$	−25.6931	+	44.5017i	−1.14220	+	1.97834i
$507$		0			0
$508$	20.7791	+	35.9905i	0.921925	+	1.59682i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$		0			0
$514$		0			0
$515$	5.00000	+	8.66025i	0.220326	+	0.381616i
$516$		0			0
$517$		−	19.0672i		−	0.838576i
$518$	−3.10146	+	5.91030i	−0.136270	+	0.259684i
$519$		0			0
$520$	−22.5630	+	39.0803i	−0.989453	+	1.71378i
$521$	−17.5455	+	10.1299i	−0.768684	+	0.443800i	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−17.5455	+	10.1299i	−0.768684	+	0.443800i	0.0637207	+	0.997968i	$0.479703\pi$
$522$		0			0
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$			44.4297i			1.94092i
$525$		0			0
$526$	−12.0000			−0.523225
$527$		0			0
$528$		0			0
$529$	−21.4317	−	37.1208i	−0.931812	−	1.61395i
$530$	39.5658	+	22.8433i	1.71863	+	0.992250i
$531$		−	18.9737i		−	0.823387i
$532$	−9.09576	+	5.75267i	−0.394351	+	0.249410i
$533$	57.4843			2.48992
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−31.2386	+	2.52606i	−1.34554	+	0.108805i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	15.8294	+	27.4173i	0.674967	+	1.16908i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	16.0000			0.679775
$555$		0			0
$556$	−32.8634	+	18.9737i	−1.39372	+	0.804663i
$557$	17.1232	+	29.6582i	0.725533	+	1.25666i	0.958754	+	0.284236i	$0.0917398\pi$
$557$	17.1232	+	29.6582i	0.725533	+	1.25666i	−0.233222	+	0.972424i	$0.574927\pi$
$558$		0			0
$559$		0			0
$560$	20.9545	+	10.9960i	0.885487	+	0.464664i
$561$		0			0
$562$	−7.71376	+	13.3606i	−0.325385	+	0.563584i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−11.0645	+	21.0850i	−0.464664	+	0.885487i
$568$		0			0
$569$	−6.02277	+	10.4318i	−0.252488	+	0.437322i	−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−6.02277	+	10.4318i	−0.252488	+	0.437322i	0.711722	+	0.702461i	$0.247915\pi$
$570$		0			0
$571$	−9.00000	−	15.5885i	−0.376638	−	0.652357i	0.613933	−	0.789359i	$-0.289587\pi$
$571$	−9.00000	−	15.5885i	−0.376638	−	0.652357i	−0.990571	+	0.137002i	$0.956253\pi$
$572$	−55.3307	−	31.9452i	−2.31349	−	1.33570i
$573$		0			0
$574$	−1.21584	−	30.1205i	−0.0507481	−	1.25721i
$575$	−40.5781			−1.69222
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	−12.0208	−	20.8207i	−0.500000	−	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−32.3421	+	56.0181i	−1.33947	+	2.32003i
$584$		0			0
$585$	−23.9317	−	41.4509i	−0.989453	−	1.71378i
$586$	33.2614	+	19.2035i	1.37402	+	0.793288i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$	10.0000	−	17.3205i	0.411693	−	0.713074i
$591$		0			0
$592$	3.56774	+	6.17951i	0.146633	+	0.253976i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	70.9193	−	40.9453i	2.90011	−	1.67438i
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$			25.2982i			1.03194i	0.856608	+	0.515968i	$0.172568\pi$
$601$			25.2982i			1.03194i	−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.5166	+	10.1132i	−0.712152	+	0.411161i
$606$		0			0
$607$	23.4227	+	13.5231i	0.950699	+	0.548886i	0.893298	−	0.449465i	$-0.148385\pi$
$607$	23.4227	+	13.5231i	0.950699	+	0.548886i	0.0574012	+	0.998351i	$0.481719\pi$
$608$			11.5053i			0.466603i
$609$		0			0
$610$		0			0
$611$	−15.1931	+	26.3152i	−0.614646	+	1.06460i
$612$		0			0
$613$	−19.8872	−	34.4456i	−0.803236	−	1.39125i	−0.917475	−	0.397793i	$-0.869776\pi$
$613$	−19.8872	−	34.4456i	−0.803236	−	1.39125i	0.114239	−	0.993453i	$-0.463557\pi$
$614$		0			0
$615$		0			0
$616$	−15.5683	+	29.6678i	−0.627266	+	1.19535i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	26.2842	−	15.1752i	1.05645	−	0.609941i	0.132002	−	0.991250i	$-0.457860\pi$
$619$	26.2842	−	15.1752i	1.05645	−	0.609941i	0.924448	+	0.381308i	$0.124526\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.34980	+	33.4392i	0.0540785	+	1.33971i
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−2.39435	−	1.38238i	−0.0955451	−	0.0551630i
$629$		0			0
$630$	−21.2132	+	13.4164i	−0.845154	+	0.534522i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	12.0000	+	20.7846i	0.476581	+	0.825462i
$635$	−40.2386	−	23.2318i	−1.59682	−	0.921925i
$636$		0			0
$637$	45.1260	+	21.4051i	1.78796	+	0.848103i
$638$		0			0
$639$		0			0
$640$	21.9089	−	12.6491i	0.866025	−	0.500000i
$641$	21.4089	+	37.0813i	0.845601	+	1.46462i	0.885098	+	0.465404i	$0.154091\pi$
$641$	21.4089	+	37.0813i	0.845601	+	1.46462i	−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	−22.9545	−	36.2942i	−0.904532	−	1.43019i
$645$		0			0
$646$		0			0
$647$	−40.7629	+	23.5345i	−1.60256	+	0.925237i	−0.611583	+	0.791180i	$0.709467\pi$
$647$	−40.7629	+	23.5345i	−1.60256	+	0.925237i	−0.990974	+	0.134057i	$0.957200\pi$
$648$	12.7279	+	22.0454i	0.500000	+	0.866025i
$649$	24.5228	+	14.1582i	0.962603	+	0.555759i
$650$		−	50.4524i		−	1.97891i
$651$		0			0
$652$		0			0
$653$	10.7914	−	18.6913i	0.422301	−	0.731447i	−0.573863	−	0.818951i	$-0.694556\pi$
$653$	10.7914	−	18.6913i	0.422301	−	0.731447i	0.996164	+	0.0875041i	$0.0278891\pi$
$654$		0			0
$655$	−24.8369	−	43.0188i	−0.970460	−	1.68089i
$656$	−27.9089	−	16.1132i	−1.08966	−	0.629115i
$657$		0			0
$658$	14.1099	+	7.40425i	0.550062	+	0.288648i
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	25.7289	+	44.5637i	0.999981	+	1.73202i
$663$		0			0
$664$		0			0
$665$	5.59110	−	10.6547i	0.216813	−	0.413171i
$666$	−7.56832			−0.293267
$667$		0			0
$668$	−42.0567	+	24.2815i	−1.62722	+	0.939478i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	37.9089	+	65.6601i	1.45803	+	2.52539i
$677$	8.57349	+	4.94990i	0.329506	+	0.190240i	0.655622	−	0.755090i	$-0.272407\pi$
$677$	8.57349	+	4.94990i	0.329506	+	0.190240i	−0.326116	+	0.945330i	$0.605740\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	−10.5683	−	6.10162i	−0.404090	−	0.233301i
$685$		0			0
$686$	10.2614	−	24.0978i	0.391782	−	0.920058i
$687$		0			0
$688$		0			0
$689$	89.2723	−	51.5414i	3.40100	−	1.96357i
$690$		0			0
$691$	27.3861	+	15.8114i	1.04182	+	0.601494i	0.920348	−	0.391102i	$-0.127906\pi$
$691$	27.3861	+	15.8114i	1.04182	+	0.601494i	0.121470	+	0.992595i	$0.461239\pi$
$692$		−	25.7754i		−	0.979835i
$693$	−18.9953	−	30.0341i	−0.721570	−	1.14090i
$694$		0			0
$695$	21.2132	−	36.7423i	0.804663	−	1.39372i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−26.4360	+	1.06711i	−0.999186	+	0.0403329i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	3.14209	−	1.81409i	0.118506	−	0.0684195i
$704$	17.9089	+	31.0191i	0.674967	+	1.16908i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	30.9839	+	17.8885i	1.16117	+	0.670402i
$713$		0			0
$714$		0			0
$715$	71.4317			2.67139
$716$	18.4772	−	32.0035i	0.690526	−	1.19603i
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$			26.8328i			1.00000i
$721$	−0.477226	−	11.8225i	−0.0177728	−	0.440294i
$722$	−21.0200			−0.782282
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			44.3042i			1.64315i	0.570098	+	0.821577i	$0.306905\pi$
$727$			44.3042i			1.64315i	−0.570098	+	0.821577i	$0.693095\pi$
$728$	45.1260	−	28.5402i	1.67248	−	1.05777i
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	38.2720	+	22.0963i	1.41361	+	0.816147i	0.995726	−	0.0923549i	$-0.0294394\pi$
$733$	38.2720	+	22.0963i	1.41361	+	0.816147i	0.417881	+	0.908502i	$0.362773\pi$
$734$			22.2937i			0.822877i
$735$		0			0
$736$	−45.9089			−1.69222
$737$		0			0
$738$	29.6019	−	17.0906i	1.08966	−	0.629115i
$739$	16.2386	+	28.1261i	0.597347	+	1.03464i	0.993211	+	0.116326i	$0.0371118\pi$
$739$	16.2386	+	28.1261i	0.597347	+	1.03464i	−0.395864	+	0.918309i	$0.629555\pi$
$740$	−6.90890	−	3.98886i	−0.253976	−	0.146633i
$741$		0			0
$742$	−28.8948	−	45.6866i	−1.06076	−	1.67721i
$743$	53.2416			1.95325			0.976623	−	0.214960i	$-0.0689620\pi$
$743$	53.2416			1.95325			0.976623	+	0.214960i	$0.0689620\pi$
$744$		0			0
$745$		0			0
$746$	−16.0000	−	27.7128i	−0.585802	−	1.01464i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	14.7526	−	8.51743i	0.537973	−	0.310599i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−45.2548			−1.64481			−0.822407	−	0.568899i	$-0.807370\pi$
$757$	−45.2548			−1.64481			−0.822407	+	0.568899i	$0.807370\pi$
$758$	−0.401865	+	0.696051i	−0.0145964	+	0.0252817i
$759$		0			0
$760$	−6.43168	−	11.1400i	−0.233301	−	0.404090i
$761$	−38.4089	−	22.1754i	−1.39232	−	0.803857i	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−38.4089	−	22.1754i	−1.39232	−	0.803857i	−0.993572	+	0.113203i	$0.963889\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−47.2158	+	27.2601i	−1.70598	+	0.984947i
$767$	−22.5630	−	39.0803i	−0.814703	−	1.41111i
$768$		0			0
$769$		−	40.4408i		−	1.45833i	−0.684336	−	0.729167i	$-0.739908\pi$
$769$		−	40.4408i		−	1.45833i	0.684336	−	0.729167i	$-0.260092\pi$
$770$	−1.51084	−	37.4287i	−0.0544468	−	1.34884i
$771$		0			0
$772$		0			0
$773$	−28.5013	+	16.4552i	−1.02512	+	0.591854i	−0.915583	−	0.402129i	$-0.868270\pi$
$773$	−28.5013	+	16.4552i	−1.02512	+	0.591854i	−0.109538	+	0.993983i	$0.534937\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.19306	+	14.1908i	−0.293547	+	0.508438i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−15.9089	−	23.0414i	−0.568175	−	0.822908i
$785$	3.09110			0.110326
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	10.8796	+	18.8441i	0.387571	+	0.671293i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−37.9905			−1.34993
$793$		0			0
$794$	5.47723	−	3.16228i	0.194379	−	0.112225i
$795$		0			0
$796$		0			0
$797$		−	40.2492i		−	1.42570i	−0.701316	−	0.712850i	$-0.747404\pi$
$797$		−	40.2492i		−	1.42570i	0.701316	−	0.712850i	$-0.252596\pi$
$798$		0			0
$799$		0			0
$800$	−14.1421	+	24.4949i	−0.500000	+	0.866025i
$801$	−32.8634	+	18.9737i	−1.16117	+	0.670402i
$802$	−21.1810	−	36.6866i	−0.747927	−	1.29545i
$803$		0			0
$804$		0			0
$805$	42.5146	+	22.3098i	1.49844	+	0.786316i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	28.4089	+	49.2057i	0.998804	+	1.72998i	0.541748	+	0.840541i	$0.317763\pi$
$809$	28.4089	+	49.2057i	0.998804	+	1.72998i	0.457056	+	0.889438i	$0.348904\pi$
$810$	−24.6475	−	14.2302i	−0.866025	−	0.500000i
$811$			30.1925i			1.06020i	0.847934	+	0.530102i	$0.177846\pi$
$811$			30.1925i			1.06020i	−0.847934	+	0.530102i	$0.822154\pi$
$812$		0			0
$813$		0			0
$814$	5.64752	−	9.78178i	0.197945	−	0.342851i
$815$		0			0
$816$		0			0
$817$		0			0
$818$			53.6656i			1.87637i
$819$	2.28416	+	56.5866i	0.0798151	+	1.97730i
$820$	36.0302			1.25823
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−24.0416	−	41.6413i	−0.838039	−	1.45153i	−0.891532	−	0.452957i	$-0.850369\pi$
$823$	−24.0416	−	41.6413i	−0.838039	−	1.45153i	0.0534937	−	0.998568i	$-0.482964\pi$
$824$	−10.9545	−	6.32456i	−0.381616	−	0.220326i
$825$		0			0
$826$	−20.0000	+	12.6491i	−0.695889	+	0.440119i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	24.3469	−	42.1700i	0.846112	−	1.46551i
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$		0			0
$832$		−	57.0804i		−	1.97891i
$833$		0			0
$834$		0			0
$835$	27.1475	−	47.0209i	0.939478	−	1.62722i
$836$	15.7723	−	9.10612i	0.545495	−	0.314942i
$837$		0			0
$838$	−46.9421	−	27.1020i	−1.62159	−	0.936224i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$	−27.4317	−	47.5131i	−0.944237	−	1.63547i
$845$	−73.4103	−	42.3834i	−2.52539	−	1.45803i
$846$			18.0682i			0.621197i
$847$	23.9128	−	0.965259i	0.821653	−	0.0331667i
$848$	−57.7895			−1.98450
$849$		0			0
$850$		0			0
$851$	7.23861	+	12.5376i	0.248136	+	0.429785i
$852$		0			0
$853$			24.1699i			0.827562i	0.910377	+	0.413781i	$0.135792\pi$
$853$			24.1699i			0.827562i	−0.910377	+	0.413781i	$0.864208\pi$
$854$		0			0
$855$	13.6436			0.466603
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−49.2950	−	28.4605i	−1.68192	−	0.971060i	−0.960381	−	0.278691i	$-0.910099\pi$
$859$	−49.2950	−	28.4605i	−1.68192	−	0.971060i	−0.721544	−	0.692369i	$-0.756567\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	23.0531	−	39.9291i	0.784736	−	1.35920i	−0.144421	−	0.989516i	$-0.546132\pi$
$863$	23.0531	−	39.9291i	0.784736	−	1.35920i	0.929157	−	0.369686i	$-0.120535\pi$
$864$		0			0
$865$	14.4089	+	24.9570i	0.489917	+	0.848562i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$			23.3432i			0.789597i
$875$	25.0000	−	15.8114i	0.845154	−	0.534522i
$876$		0			0
$877$	28.0028	−	48.5023i	0.945588	−	1.63781i	0.191018	−	0.981586i	$-0.438821\pi$
$877$	28.0028	−	48.5023i	0.945588	−	1.63781i	0.754570	−	0.656220i	$-0.227846\pi$
$878$		0			0
$879$		0			0
$880$	−34.6804	−	20.0228i	−1.16908	−	0.674967i
$881$			56.5540i			1.90535i	0.303985	+	0.952677i	$0.401683\pi$
$881$			56.5540i			1.90535i	−0.303985	+	0.952677i	$0.598317\pi$
$882$	29.6019	−	2.39370i	0.996747	−	0.0806002i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.87298	+	2.23607i	0.130042	+	0.0750798i	0.563610	−	0.826041i	$-0.309412\pi$
$887$	3.87298	+	2.23607i	0.130042	+	0.0750798i	−0.433568	+	0.901121i	$0.642746\pi$
$888$		0			0
$889$	29.3861	+	46.4635i	0.985579	+	1.55834i
$890$	−40.0000			−1.34080
$891$	20.1475	−	34.8965i	0.674967	−	1.16908i
$892$	38.7298	−	22.3607i	1.29677	−	0.748691i
$893$	−4.33085	−	7.50124i	−0.144926	−	0.251020i
$894$		0			0
$895$			41.3163i			1.38105i
$896$	−29.9089	+	1.20730i	−0.999186	+	0.0403329i
$897$		0			0
$898$	−27.5127	+	47.6535i	−0.918112	+	1.59022i
$899$		0			0
$900$	−15.0000	−	25.9808i	−0.500000	−	0.866025i
$901$		0			0
$902$			51.0124i			1.69853i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$		0			0
$910$	−27.7386	+	52.8601i	−0.919526	+	1.75230i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.37056	+	58.7271i	0.0782829	+	1.93934i
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$	44.4511	−	25.6639i	1.46551	−	0.846112i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	8.91935			0.293267
$926$	−0.693064	+	1.20042i	−0.0227755	+	0.0394483i
$927$	11.6190	−	6.70820i	0.381616	−	0.220326i
$928$		0			0
$929$	3.59110	+	2.07332i	0.117820	+	0.0680235i	0.557752	−	0.830008i	$-0.311664\pi$
$929$	3.59110	+	2.07332i	0.117820	+	0.0680235i	−0.439932	+	0.898031i	$0.644997\pi$
$930$		0			0
$931$	−11.7158	+	8.08918i	−0.383971	+	0.265112i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	52.4317	+	30.2714i	1.71378	+	0.989453i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	−9.52277	+	16.4939i	−0.310599	+	0.537973i
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$	−56.6245	−	32.6922i	−1.84395	−	1.06460i
$944$			25.2982i			0.823387i
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$	12.4549	+	7.19083i	0.404090	+	0.233301i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	30.6475	−	53.0831i	0.992250	−	1.71863i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$	−15.5886	+	9.00006i	−0.502595	+	0.290174i
$963$		0			0
$964$	−48.9089	−	28.2376i	−1.57525	−	0.909471i
$965$		0			0
$966$		0			0
$967$	53.7401			1.72817			0.864083	−	0.503350i	$-0.167899\pi$
$967$	53.7401			1.72817			0.864083	+	0.503350i	$0.167899\pi$
$968$	12.7923	−	22.1570i	0.411161	−	0.712152i
$969$		0			0
$970$		0			0
$971$	43.7614	+	25.2656i	1.40437	+	0.810813i	0.994837	−	0.101482i	$-0.0323585\pi$
$971$	43.7614	+	25.2656i	1.40437	+	0.810813i	0.409532	+	0.912296i	$0.365692\pi$
$972$		0			0
$973$	−42.4264	+	26.8328i	−1.36013	+	0.860221i
$974$	44.0000			1.40985
$975$		0			0
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		−	56.6329i		−	1.81000i
$980$	28.2843	+	13.4164i	0.903508	+	0.428571i
$981$		0			0
$982$	1.41421	−	2.44949i	0.0451294	−	0.0781664i
$983$	18.6340	−	10.7584i	0.594333	−	0.343138i	−0.172476	−	0.985014i	$-0.555177\pi$
$983$	18.6340	−	10.7584i	0.594333	−	0.343138i	0.766809	+	0.641875i	$0.221843\pi$
$984$		0			0
$985$	−21.0683	−	12.1638i	−0.671293	−	0.387571i
$986$		0			0
$987$		0			0
$988$	−29.0236			−0.923363
$989$		0			0
$990$	36.7842	−	21.2373i	1.16908	−	0.674967i
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−42.6028	+	24.5967i	−1.34924	+	0.778987i	−0.988143	−	0.153539i	$-0.950933\pi$
$997$	−42.6028	+	24.5967i	−1.34924	+	0.778987i	−0.361102	+	0.932526i	$0.617599\pi$
$998$	−4.24264	−	7.34847i	−0.134298	−	0.232612i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.ba.a.19.4	yes	8
4.3	odd	2		1120.2.bq.a.719.4		8
5.4	even	2	inner	280.2.ba.a.19.1	✓	8
7.3	odd	6	inner	280.2.ba.a.59.4	yes	8
8.3	odd	2	inner	280.2.ba.a.19.1	✓	8
8.5	even	2		1120.2.bq.a.719.1		8
20.19	odd	2		1120.2.bq.a.719.1		8
28.3	even	6		1120.2.bq.a.1039.4		8
35.24	odd	6	inner	280.2.ba.a.59.1	yes	8
40.19	odd	2	CM	280.2.ba.a.19.4	yes	8
40.29	even	2		1120.2.bq.a.719.4		8
56.3	even	6	inner	280.2.ba.a.59.1	yes	8
56.45	odd	6		1120.2.bq.a.1039.1		8
140.59	even	6		1120.2.bq.a.1039.1		8
280.59	even	6	inner	280.2.ba.a.59.4	yes	8
280.269	odd	6		1120.2.bq.a.1039.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.ba.a.19.1	✓	8	5.4	even	2	inner
280.2.ba.a.19.1	✓	8	8.3	odd	2	inner
280.2.ba.a.19.4	yes	8	1.1	even	1	trivial
280.2.ba.a.19.4	yes	8	40.19	odd	2	CM
280.2.ba.a.59.1	yes	8	35.24	odd	6	inner
280.2.ba.a.59.1	yes	8	56.3	even	6	inner
280.2.ba.a.59.4	yes	8	7.3	odd	6	inner
280.2.ba.a.59.4	yes	8	280.59	even	6	inner
1120.2.bq.a.719.1		8	8.5	even	2
1120.2.bq.a.719.1		8	20.19	odd	2
1120.2.bq.a.719.4		8	4.3	odd	2
1120.2.bq.a.719.4		8	40.29	even	2
1120.2.bq.a.1039.1		8	56.45	odd	6
1120.2.bq.a.1039.1		8	140.59	even	6
1120.2.bq.a.1039.4		8	28.3	even	6
1120.2.bq.a.1039.4		8	280.269	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.ba (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-1.41421 - 2.23607i) q^{7} -2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(1.93649 + 1.11803i) q^{5} +(-1.41421 - 2.23607i) q^{7} -2.82843 q^{8} +(1.50000 - 2.59808i) q^{9} +(2.73861 - 1.58114i) q^{10} +(2.23861 + 3.87739i) q^{11} -7.13505i q^{13} +(-3.73861 + 0.150912i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(-2.12132 - 3.67423i) q^{18} +(1.76139 + 1.01694i) q^{19} -4.47214i q^{20} +6.33175 q^{22} +(-4.05781 + 7.02834i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-8.73861 - 5.04524i) q^{26} +(-2.45877 + 4.68556i) q^{28} +(2.82843 + 4.89898i) q^{32} +(-0.238613 - 5.91127i) q^{35} -6.00000 q^{36} +(0.891935 - 1.54488i) q^{37} +(2.49098 - 1.43817i) q^{38} +(-5.47723 - 3.16228i) q^{40} +8.05661i q^{41} +(4.47723 - 7.75478i) q^{44} +(5.80948 - 3.35410i) q^{45} +(5.73861 + 9.93957i) q^{46} +(-3.68815 - 2.12936i) q^{47} +(-3.00000 + 6.32456i) q^{49} +7.07107 q^{50} +(-12.3583 + 7.13505i) q^{52} +(7.22369 + 12.5118i) q^{53} +10.0114i q^{55} +(4.00000 + 6.32456i) q^{56} +(5.47723 - 3.16228i) q^{59} +(-7.93080 + 0.320133i) q^{63} +8.00000 q^{64} +(7.97723 - 13.8170i) q^{65} +(-7.40852 - 3.88766i) q^{70} +(-4.24264 + 7.34847i) q^{72} +(-1.26139 - 2.18479i) q^{74} -4.06775i q^{76} +(5.50423 - 10.4891i) q^{77} +(-7.74597 + 4.47214i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(9.86729 + 5.69688i) q^{82} +(-6.33175 - 10.9669i) q^{88} +(-10.9545 - 6.32456i) q^{89} -9.48683i q^{90} +(-15.9545 + 10.0905i) q^{91} +16.2312 q^{92} +(-5.21584 + 3.01137i) q^{94} +(2.27394 + 3.93858i) q^{95} +(5.62465 + 8.14637i) q^{98} +13.4317 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^4 + 12 * q^9	\( 8 q - 8 q^{4} + 12 q^{9} - 4 q^{11} - 8 q^{14} - 16 q^{16} + 36 q^{19} + 20 q^{25} - 48 q^{26} + 20 q^{35} - 48 q^{36} - 8 q^{44} + 24 q^{46} - 24 q^{49} + 32 q^{56} + 64 q^{64} + 20 q^{65} - 32 q^{74} - 36 q^{81} - 40 q^{91} + 24 q^{94} - 24 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 + 12 * q^9 - 4 * q^11 - 8 * q^14 - 16 * q^16 + 36 * q^19 + 20 * q^25 - 48 * q^26 + 20 * q^35 - 48 * q^36 - 8 * q^44 + 24 * q^46 - 24 * q^49 + 32 * q^56 + 64 * q^64 + 20 * q^65 - 32 * q^74 - 36 * q^81 - 40 * q^91 + 24 * q^94 - 24 * q^99

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