LMFDB - Embedded newform 476.2.q.a.339.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,2,Mod(271,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("476.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.q (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:	$3.80087913621$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 16x^{12} + 175x^{8} - 1296x^{4} + 6561 $ x^16 - 16x^12 + 175x^8 - 1296*x^4 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			339.8
Root			$0.246547 + 1.71441i$ of defining polynomial
Character	$\chi$	$=$	476.339
Dual form			476.2.q.a.271.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 + 1.22474i) q^{2} +(2.96945 + 1.71441i) q^{3} +(-1.00000 + 1.73205i) q^{4} +4.84909i q^{6} +(0.563932 - 2.58495i) q^{7} -2.82843 q^{8} +(4.37843 + 7.58366i) q^{9} +(-1.25933 + 2.18122i) q^{11} +(-5.93890 + 3.42883i) q^{12} -6.58971i q^{13} +(3.56467 - 1.13716i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-3.57071 - 2.06155i) q^{17} +(-6.19203 + 10.7249i) q^{18} +(6.10625 - 6.70908i) q^{21} -3.56192 q^{22} +(2.48932 + 4.31162i) q^{23} +(-8.39888 - 4.84909i) q^{24} +(2.50000 - 4.33013i) q^{25} +(8.07071 - 4.65963i) q^{26} +19.7393i q^{27} +(3.91334 + 3.56171i) q^{28} +(-3.39090 - 1.95774i) q^{31} +(2.82843 - 4.89898i) q^{32} +(-7.47904 + 4.31802i) q^{33} -5.83095i q^{34} -17.5137 q^{36} +(11.2975 - 19.5678i) q^{39} +(12.5347 + 2.73456i) q^{42} +(-2.51866 - 4.36245i) q^{44} +(-3.52043 + 6.09756i) q^{46} -13.7153i q^{48} +(-6.36396 - 2.91548i) q^{49} +7.07107 q^{50} +(-7.06871 - 12.2434i) q^{51} +(11.4137 + 6.58971i) q^{52} +(-0.570714 + 0.988506i) q^{53} +(-24.1756 + 13.9578i) q^{54} +(-1.59504 + 7.31135i) q^{56} -5.53732i q^{62} +(22.0725 - 7.04136i) q^{63} +8.00000 q^{64} +(-10.5770 - 6.10661i) q^{66} +(7.14143 - 4.12311i) q^{68} +17.0709i q^{69} -7.01357 q^{71} +(-12.3841 - 21.4498i) q^{72} +(14.8473 - 8.57207i) q^{75} +(4.92818 + 4.48537i) q^{77} +31.9541 q^{78} +(8.73733 + 15.1335i) q^{79} +(-20.7060 + 35.8638i) q^{81} +(5.51422 + 17.2854i) q^{84} +(3.56192 - 6.16943i) q^{88} +(2.27133 - 1.31135i) q^{89} +(-17.0341 - 3.71615i) q^{91} -9.95727 q^{92} +(-6.71275 - 11.6268i) q^{93} +(16.7978 - 9.69819i) q^{96} +(-0.929286 - 9.85578i) q^{98} -22.0555 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} + 24 q^{9} - 32 q^{16} - 8 q^{18} - 16 q^{21} + 40 q^{25} + 72 q^{26} - 96 q^{33} - 96 q^{36} + 40 q^{42} + 48 q^{53} + 128 q^{64} - 16 q^{72} - 48 q^{77} - 136 q^{81} + 64 q^{84} + 40 q^{93}+ \cdots - 72 q^{98}+O(q^{100}) $ 16 * q - 16 * q^4 + 24 * q^9 - 32 * q^16 - 8 * q^18 - 16 * q^21 + 40 * q^25 + 72 * q^26 - 96 * q^33 - 96 * q^36 + 40 * q^42 + 48 * q^53 + 128 * q^64 - 16 * q^72 - 48 * q^77 - 136 * q^81 + 64 * q^84 + 40 * q^93 - 72 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$.

$n$	$239$	$309$	$409$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	2.96945	+	1.71441i	1.71441	+	0.989817i	0.928379	+	0.371635i	$0.121203\pi$
$3$	2.96945	+	1.71441i	1.71441	+	0.989817i	0.786035	+	0.618182i	$0.212130\pi$
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$			4.84909i			1.97963i
$7$	0.563932	−	2.58495i	0.213146	−	0.977020i
$7$	0.563932	−	2.58495i	0.213146	−	0.977020i
$8$	−2.82843			−1.00000
$9$	4.37843	+	7.58366i	1.45948	+	2.52789i
$10$		0			0
$11$	−1.25933	+	2.18122i	−0.379702	+	0.657664i	−0.991019	−	0.133723i	$-0.957307\pi$
$11$	−1.25933	+	2.18122i	−0.379702	+	0.657664i	0.611317	+	0.791386i	$0.290640\pi$
$12$	−5.93890	+	3.42883i	−1.71441	+	0.989817i
$13$		−	6.58971i		−	1.82766i	−0.406100	−	0.913828i	$-0.633112\pi$
$13$		−	6.58971i		−	1.82766i	0.406100	−	0.913828i	$-0.366888\pi$
$14$	3.56467	−	1.13716i	0.952698	−	0.303920i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−3.57071	−	2.06155i	−0.866025	−	0.500000i
$17$	−3.57071	−	2.06155i	−0.866025	−	0.500000i
$18$	−6.19203	+	10.7249i	−1.45948	+	2.52789i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$	6.10625	−	6.70908i	1.33249	−	1.46404i
$22$	−3.56192			−0.759404
$23$	2.48932	+	4.31162i	0.519058	+	0.899036i	0.999755	+	0.0221485i	$0.00705065\pi$
$23$	2.48932	+	4.31162i	0.519058	+	0.899036i	−0.480696	+	0.876887i	$0.659616\pi$
$24$	−8.39888	−	4.84909i	−1.71441	−	0.989817i
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$	8.07071	−	4.65963i	1.58280	−	0.913828i
$27$			19.7393i			3.79883i
$28$	3.91334	+	3.56171i	0.739551	+	0.673100i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−3.39090	−	1.95774i	−0.609024	−	0.351620i	0.163559	−	0.986533i	$-0.447702\pi$
$31$	−3.39090	−	1.95774i	−0.609024	−	0.351620i	−0.772583	+	0.634913i	$0.781036\pi$
$32$	2.82843	−	4.89898i	0.500000	−	0.866025i
$33$	−7.47904	+	4.31802i	−1.30193	+	0.751672i
$34$		−	5.83095i		−	1.00000i
$35$		0			0
$36$	−17.5137			−2.91895
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$	11.2975	−	19.5678i	1.80905	−	3.13336i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	12.5347	+	2.73456i	1.93414	+	0.421952i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−2.51866	−	4.36245i	−0.379702	−	0.657664i
$45$		0			0
$46$	−3.52043	+	6.09756i	−0.519058	+	0.899036i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		−	13.7153i		−	1.97963i
$49$	−6.36396	−	2.91548i	−0.909137	−	0.416497i
$50$	7.07107			1.00000
$51$	−7.06871	−	12.2434i	−0.989817	−	1.71441i
$52$	11.4137	+	6.58971i	1.58280	+	0.913828i
$53$	−0.570714	+	0.988506i	−0.0783936	+	0.135782i	−0.902557	−	0.430570i	$-0.858312\pi$
$53$	−0.570714	+	0.988506i	−0.0783936	+	0.135782i	0.824163	+	0.566352i	$0.191646\pi$
$54$	−24.1756	+	13.9578i	−3.28988	+	1.89941i
$55$		0			0
$56$	−1.59504	+	7.31135i	−0.213146	+	0.977020i
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		−	5.53732i		−	0.703240i
$63$	22.0725	−	7.04136i	2.78088	−	0.887128i
$64$	8.00000			1.00000
$65$		0			0
$66$	−10.5770	−	6.10661i	−1.30193	−	0.751672i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	7.14143	−	4.12311i	0.866025	−	0.500000i
$69$			17.0709i			2.05509i
$70$		0			0
$71$	−7.01357			−0.832358			−0.416179	−	0.909283i	$-0.636631\pi$
$71$	−7.01357			−0.832358			−0.416179	+	0.909283i	$0.636631\pi$
$72$	−12.3841	−	21.4498i	−1.45948	−	2.52789i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$	14.8473	−	8.57207i	1.71441	−	0.989817i
$76$		0			0
$77$	4.92818	+	4.48537i	0.561618	+	0.511155i
$78$	31.9541			3.61809
$79$	8.73733	+	15.1335i	0.983027	+	1.70265i	0.650395	+	0.759596i	$0.274603\pi$
$79$	8.73733	+	15.1335i	0.983027	+	1.70265i	0.332632	+	0.943057i	$0.392063\pi$
$80$		0			0
$81$	−20.7060	+	35.8638i	−2.30067	+	3.98487i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	5.51422	+	17.2854i	0.601651	+	1.88599i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	3.56192	−	6.16943i	0.379702	−	0.657664i
$89$	2.27133	−	1.31135i	0.240760	−	0.139003i	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	2.27133	−	1.31135i	0.240760	−	0.139003i	0.615526	+	0.788116i	$0.288944\pi$
$90$		0			0
$91$	−17.0341	−	3.71615i	−1.78566	−	0.389558i
$92$	−9.95727			−1.03812
$93$	−6.71275	−	11.6268i	−0.696079	−	1.20564i
$94$		0			0
$95$		0			0
$96$	16.7978	−	9.69819i	1.71441	−	0.989817i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−0.929286	−	9.85578i	−0.0938720	−	0.995584i
$99$	−22.0555			−2.21667
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$	−15.1493	−	8.74643i	−1.50741	−	0.870302i	−0.999963	−	0.00861771i	$-0.997257\pi$
$101$	−15.1493	−	8.74643i	−1.50741	−	0.870302i	−0.507445	−	0.861684i	$-0.669410\pi$
$102$	9.99666	−	17.3147i	0.989817	−	1.71441i
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$			18.6385i			1.82766i
$105$		0			0
$106$	−1.61422			−0.156787
$107$	8.84967	+	15.3281i	0.855530	+	1.48182i	0.876153	+	0.482034i	$0.160102\pi$
$107$	8.84967	+	15.3281i	0.855530	+	1.48182i	−0.0206226	+	0.999787i	$0.506565\pi$
$108$	−34.1894	−	19.7393i	−3.28988	−	1.89941i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	−10.0824	+	3.21639i	−0.952698	+	0.303920i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	49.9741	−	28.8526i	4.62011	−	2.66742i
$118$		0			0
$119$	−7.34266	+	8.06755i	−0.673100	+	0.739551i
$120$		0			0
$121$	2.32818	+	4.03252i	0.211652	+	0.366593i
$122$		0			0
$123$		0			0
$124$	6.78180	−	3.91548i	0.609024	−	0.351620i
$125$		0			0
$126$	24.2315	+	22.0542i	2.15872	+	1.96475i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	5.65685	+	9.79796i	0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$	19.7636	−	11.4105i	1.72676	−	0.996943i	0.824311	−	0.566137i	$-0.191563\pi$
$131$	19.7636	−	11.4105i	1.72676	−	0.996943i	0.902445	−	0.430806i	$-0.141771\pi$
$132$		−	17.2721i		−	1.50334i
$133$		0			0
$134$		0			0
$135$		0			0
$136$	10.0995	+	5.83095i	0.866025	+	0.500000i
$137$	−3.68554	+	6.38354i	−0.314877	+	0.545383i	−0.979411	−	0.201875i	$-0.935297\pi$
$137$	−3.68554	+	6.38354i	−0.314877	+	0.545383i	0.664534	+	0.747258i	$0.268630\pi$
$138$	−20.9075	+	12.0709i	−1.77976	+	1.02755i
$139$		−	4.16258i		−	0.353066i	−0.984295	−	0.176533i	$-0.943512\pi$
$139$		−	4.16258i		−	0.353066i	0.984295	−	0.176533i	$-0.0564882\pi$
$140$		0			0
$141$		0			0
$142$	−4.95935	−	8.58984i	−0.416179	−	0.720843i
$143$	14.3736	+	8.29862i	1.20198	+	0.693965i
$144$	17.5137	−	30.3347i	1.45948	−	2.52789i
$145$		0			0
$146$		0			0
$147$	−13.8991	−	19.5678i	−1.14638	−	1.61393i
$148$		0			0
$149$	−11.6414	−	20.1635i	−0.953703	−	1.65186i	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−11.6414	−	20.1635i	−0.953703	−	1.65186i	−0.216394	−	0.976306i	$-0.569430\pi$
$150$	20.9972	+	12.1227i	1.71441	+	0.989817i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		−	36.1055i		−	2.91895i
$154$	−2.00868	+	9.20740i	−0.161864	+	0.741954i
$155$		0			0
$156$	22.5950	+	39.1357i	1.80905	+	3.13336i
$157$	13.7121	+	7.91671i	1.09435	+	0.631822i	0.934731	−	0.355357i	$-0.115641\pi$
$157$	13.7121	+	7.91671i	1.09435	+	0.631822i	0.159617	+	0.987179i	$0.448974\pi$
$158$	−12.3565	+	21.4020i	−0.983027	+	1.70265i
$159$	−3.38942	+	1.95688i	−0.268798	+	0.155191i
$160$		0			0
$161$	12.5491	−	4.00330i	0.989011	−	0.315505i
$162$	−58.5654			−4.60133
$163$	−10.9884	−	19.0324i	−0.860676	−	1.49073i	−0.871277	−	0.490791i	$-0.836708\pi$
$163$	−10.9884	−	19.0324i	−0.860676	−	1.49073i	0.0106013	−	0.999944i	$-0.496625\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			13.8608i			1.07258i	0.844034	+	0.536289i	$0.180174\pi$
$167$			13.8608i			1.07258i	−0.844034	+	0.536289i	$0.819826\pi$
$168$	−17.2711	+	18.9761i	−1.33249	+	1.46404i
$169$	−30.4243			−2.34033
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−9.78334	−	8.90428i	−0.739551	−	0.673100i
$176$	10.0746			0.759404
$177$		0			0
$178$	3.21214	+	1.85453i	0.240760	+	0.139003i
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	−7.49359	−	23.4901i	−0.555462	−	1.74120i
$183$		0			0
$184$	−7.04085	−	12.1951i	−0.519058	−	0.899036i
$185$		0			0
$186$	9.49326	−	16.4428i	0.696079	−	1.20564i
$187$	8.99341	−	5.19235i	0.657664	−	0.379702i
$188$		0			0
$189$	51.0251	+	11.1316i	3.71153	+	0.809706i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	23.7556	+	13.7153i	1.71441	+	0.989817i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	11.4137	−	8.10723i	0.815265	−	0.579088i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−15.5956	−	27.0124i	−1.10833	−	1.91969i
$199$	−16.2032	−	9.35493i	−1.14862	−	0.663154i	−0.200066	−	0.979782i	$-0.564116\pi$
$199$	−16.2032	−	9.35493i	−1.14862	−	0.663154i	−0.948550	+	0.316629i	$0.897449\pi$
$200$	−7.07107	+	12.2474i	−0.500000	+	0.866025i
$201$		0			0
$202$		−	24.7386i		−	1.74060i
$203$		0			0
$204$	28.2748			1.97963
$205$		0			0
$206$		0			0
$207$	−21.7986	+	37.7563i	−1.51511	+	2.62424i
$208$	−22.8274	+	13.1794i	−1.58280	+	0.913828i
$209$		0			0
$210$		0			0
$211$	−29.0176			−1.99765			−0.998827	−	0.0484178i	$-0.984582\pi$
$211$	−29.0176			−1.99765			−0.998827	+	0.0484178i	$0.984582\pi$
$212$	−1.14143	−	1.97701i	−0.0783936	−	0.135782i
$213$	−20.8265	−	12.0242i	−1.42701	−	0.823882i
$214$	−12.5153	+	21.6772i	−0.855530	+	1.48182i
$215$		0			0
$216$		−	55.8311i		−	3.79883i
$217$	−6.97290	+	7.66129i	−0.473351	+	0.520082i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−13.5850	+	23.5300i	−0.913828	+	1.58280i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−11.0686	−	10.0740i	−0.739551	−	0.673100i
$225$	43.7843			2.91895
$226$		0			0
$227$	13.1929	+	7.61690i	0.875640	+	0.505551i	0.869218	−	0.494428i	$-0.164623\pi$
$227$	13.1929	+	7.61690i	0.875640	+	0.505551i	0.00642189	+	0.999979i	$0.497956\pi$
$228$		0			0
$229$	−5.04975	+	2.91548i	−0.333697	+	0.192660i	−0.657481	−	0.753471i	$-0.728378\pi$
$229$	−5.04975	+	2.91548i	−0.333697	+	0.192660i	0.323784	+	0.946131i	$0.395045\pi$
$230$		0			0
$231$	6.94422	+	21.7680i	0.456896	+	1.43223i
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$	70.6741	+	40.8037i	4.62011	+	2.66742i
$235$		0			0
$236$		0			0
$237$			59.9176i			3.89207i
$238$	−15.0727	−	3.28826i	−0.977020	−	0.213146i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	−3.29254	+	5.70285i	−0.211652	+	0.366593i
$243$	−71.6868	+	41.3884i	−4.59871	+	2.65507i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	9.59092	+	5.53732i	0.609024	+	0.351620i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−9.87655	+	45.2721i	−0.622164	+	2.85188i
$253$	−12.5395			−0.788351
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−12.5779	+	7.26186i	−0.784589	+	0.452983i	−0.838054	−	0.545587i	$-0.816307\pi$
$257$	−12.5779	+	7.26186i	−0.784589	+	0.452983i	0.0534653	+	0.998570i	$0.482973\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	27.9500	+	16.1369i	1.72676	+	0.996943i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	21.1539	−	12.2132i	1.30193	−	0.751672i
$265$		0			0
$266$		0			0
$267$	8.99280			0.550350
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$			16.4924i			1.00000i
$273$	−44.2109	−	40.2384i	−2.67576	−	2.43534i
$274$	−10.4243			−0.629754
$275$	6.29665	+	10.9061i	0.379702	+	0.657664i
$276$	−29.5676	−	17.0709i	−1.77976	−	1.02755i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	5.09810	−	2.94339i	0.305764	−	0.176533i
$279$		−	34.2873i		−	2.05273i
$280$		0			0
$281$	25.5657			1.52512			0.762561	−	0.646916i	$-0.223942\pi$
$281$	25.5657			1.52512			0.762561	+	0.646916i	$0.223942\pi$
$282$		0			0
$283$	−8.06756	−	4.65781i	−0.479566	−	0.276878i	0.240669	−	0.970607i	$-0.422633\pi$
$283$	−8.06756	−	4.65781i	−0.479566	−	0.276878i	−0.720236	+	0.693729i	$0.755966\pi$
$284$	7.01357	−	12.1479i	0.416179	−	0.720843i
$285$		0			0
$286$			23.4720i			1.38793i
$287$		0			0
$288$	49.5363			2.91895
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	17.7346i		−	1.03606i	−0.855361	−	0.518032i	$-0.826665\pi$
$293$		−	17.7346i		−	1.03606i	0.855361	−	0.518032i	$-0.173335\pi$
$294$	14.1374	−	30.8594i	0.824511	−	1.79976i
$295$		0			0
$296$		0			0
$297$	−43.0558	−	24.8583i	−2.49835	−	1.44242i
$298$	16.4635	−	28.5156i	0.953703	−	1.65186i
$299$	28.4124	−	16.4039i	1.64313	−	0.948661i
$300$			34.2883i			1.97963i
$301$		0			0
$302$		0			0
$303$	−29.9900	−	51.9442i	−1.72288	−	2.98412i
$304$		0			0
$305$		0			0
$306$	44.2200	−	25.5304i	2.52789	−	1.45948i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−12.6971	+	4.05049i	−0.723483	+	0.230798i
$309$		0			0
$310$		0			0
$311$	14.0065	+	8.08663i	0.794233	+	0.458551i	0.841451	−	0.540334i	$-0.181702\pi$
$311$	14.0065	+	8.08663i	0.794233	+	0.458551i	−0.0472173	+	0.998885i	$0.515035\pi$
$312$	−31.9541	+	55.3462i	−1.80905	+	3.13336i
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$			22.3918i			1.26364i
$315$		0			0
$316$	−34.9493			−1.96605
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$	−4.79336	−	2.76745i	−0.268798	−	0.155191i
$319$		0			0
$320$		0			0
$321$			60.6880i			3.38727i
$322$	13.7766	+	12.5387i	0.767741	+	0.698757i
$323$		0			0
$324$	−41.4120	−	71.7277i	−2.30067	−	3.98487i
$325$	−28.5343	−	16.4743i	−1.58280	−	0.913828i
$326$	15.5399	−	26.9159i	0.860676	−	1.49073i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	−16.9759	+	9.80105i	−0.928881	+	0.536289i
$335$		0			0
$336$	−35.4534	−	7.73450i	−1.93414	−	0.421952i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−21.5132	−	37.2620i	−1.17016	−	2.02678i
$339$		0			0
$340$		0			0
$341$	8.54053	−	4.93088i	0.462495	−	0.267022i
$342$		0			0
$343$	−11.1252	+	14.8064i	−0.600705	+	0.799471i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.0506	+	22.6043i	−0.700593	+	1.21346i	0.267666	+	0.963512i	$0.413748\pi$
$347$	−13.0506	+	22.6043i	−0.700593	+	1.21346i	−0.968259	+	0.249950i	$0.919586\pi$
$348$		0			0
$349$		−	24.7386i		−	1.32423i	−0.749403	−	0.662114i	$-0.769659\pi$
$349$		−	24.7386i		−	1.32423i	0.749403	−	0.662114i	$-0.230341\pi$
$350$	3.98760	−	18.2784i	0.213146	−	0.977020i
$351$	130.076			6.94295
$352$	7.12384	+	12.3389i	0.379702	+	0.657664i
$353$	−0.782857	−	0.451983i	−0.0416673	−	0.0240566i	0.479022	−	0.877803i	$-0.340992\pi$
$353$	−0.782857	−	0.451983i	−0.0416673	−	0.0240566i	−0.520689	+	0.853746i	$0.674325\pi$
$354$		0			0
$355$		0			0
$356$			5.24541i			0.278006i
$357$	−35.6348	+	11.3679i	−1.88599	+	0.601651i
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$			15.9658i			0.837989i
$364$	23.4707	−	25.7878i	1.23020	−	1.35165i
$365$		0			0
$366$		0			0
$367$	28.8015	+	16.6286i	1.50343	+	0.868005i	0.999992	+	0.00397207i	$0.00126435\pi$
$367$	28.8015	+	16.6286i	1.50343	+	0.868005i	0.503436	+	0.864033i	$0.332069\pi$
$368$	9.95727	−	17.2465i	0.519058	−	0.899036i
$369$		0			0
$370$		0			0
$371$	2.23340	+	2.03272i	0.115952	+	0.105534i
$372$	26.8510			1.39216
$373$	−7.02107	−	12.1608i	−0.363537	−	0.629665i	0.625003	−	0.780622i	$-0.285098\pi$
$373$	−7.02107	−	12.1608i	−0.363537	−	0.629665i	−0.988540	+	0.150958i	$0.951764\pi$
$374$	12.7186	+	7.34309i	0.657664	+	0.379702i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	22.4468	+	70.3640i	1.15454	+	3.61913i
$379$	−38.8503			−1.99561			−0.997804	−	0.0662388i	$-0.978900\pi$
$379$	−38.8503			−1.99561			−0.997804	+	0.0662388i	$0.978900\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$			38.7928i			1.97963i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.26421	−	2.18967i	0.0640978	−	0.111021i	−0.832196	−	0.554482i	$-0.812916\pi$
$389$	1.26421	−	2.18967i	0.0640978	−	0.111021i	0.896293	+	0.443461i	$0.146250\pi$
$390$		0			0
$391$		−	20.5274i		−	1.03812i
$392$	18.0000	+	8.24621i	0.909137	+	0.416497i
$393$	78.2495			3.94717
$394$		0			0
$395$		0			0
$396$	22.0555	−	38.2013i	1.10833	−	1.91969i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		−	26.4597i		−	1.32631i
$399$		0			0
$400$	−20.0000			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−12.9009	+	22.3451i	−0.642641	+	1.11309i
$404$	30.2985	−	17.4929i	1.50741	−	0.870302i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	19.9933	+	34.6295i	0.989817	+	1.71441i
$409$	34.7121	+	20.0411i	1.71640	+	0.990967i	0.925256	+	0.379344i	$0.123850\pi$
$409$	34.7121	+	20.0411i	1.71640	+	0.990967i	0.791149	+	0.611623i	$0.209483\pi$
$410$		0			0
$411$	−21.8881	+	12.6371i	−1.07966	+	0.623342i
$412$		0			0
$413$		0			0
$414$	−61.6557			−3.03021
$415$		0			0
$416$	−32.2829	−	18.6385i	−1.58280	−	0.913828i
$417$	7.13639	−	12.3606i	0.349471	−	0.605301i
$418$		0			0
$419$			29.7811i			1.45490i	0.686160	+	0.727450i	$0.259295\pi$
$419$			29.7811i			1.45490i	−0.686160	+	0.727450i	$0.740705\pi$
$420$		0			0
$421$	4.24264			0.206774			0.103387	−	0.994641i	$-0.467032\pi$
$421$	4.24264			0.206774			0.103387	+	0.994641i	$0.467032\pi$
$422$	−20.5185	−	35.5392i	−0.998827	−	1.73002i
$423$		0			0
$424$	1.61422	−	2.79592i	0.0783936	−	0.135782i
$425$	−17.8536	+	10.3078i	−0.866025	+	0.500000i
$426$		−	34.0095i		−	1.64776i
$427$		0			0
$428$	−35.3987			−1.71106
$429$	28.4545	+	49.2847i	1.37380	+	2.37949i
$430$		0			0
$431$	1.83610	−	3.18021i	0.0884417	−	0.153185i	−0.818411	−	0.574633i	$-0.805145\pi$
$431$	1.83610	−	3.18021i	0.0884417	−	0.153185i	0.906853	+	0.421448i	$0.138478\pi$
$432$	68.3789	−	39.4786i	3.28988	−	1.89941i
$433$			29.1548i			1.40109i	0.713609	+	0.700544i	$0.247059\pi$
$433$			29.1548i			1.40109i	−0.713609	+	0.700544i	$0.752941\pi$
$434$	−14.3137	−	3.12267i	−0.687080	−	0.149893i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	25.8843	−	14.9443i	1.23539	−	0.713252i	0.267240	−	0.963630i	$-0.413888\pi$
$439$	25.8843	−	14.9443i	1.23539	−	0.713252i	0.968148	+	0.250378i	$0.0805549\pi$
$440$		0			0
$441$	−5.75417	−	61.0273i	−0.274008	−	2.90606i
$442$	−38.4243			−1.82766
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	79.8329i		−	3.77597i
$448$	4.51146	−	20.6796i	0.213146	−	0.977020i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	30.9602	+	53.6246i	1.45948	+	2.52789i
$451$		0			0
$452$		0			0
$453$		0			0
$454$			21.5438i			1.01110i
$455$		0			0
$456$		0			0
$457$	6.36396	+	11.0227i	0.297694	+	0.515620i	0.975608	−	0.219521i	$-0.0704493\pi$
$457$	6.36396	+	11.0227i	0.297694	+	0.515620i	−0.677914	+	0.735141i	$0.737116\pi$
$458$	−7.14143	−	4.12311i	−0.333697	−	0.192660i
$459$	40.6936	−	70.4833i	1.89941	−	3.28988i
$460$		0			0
$461$		−	10.2232i		−	0.476143i	−0.971248	−	0.238071i	$-0.923485\pi$
$461$		−	10.2232i		−	0.476143i	0.971248	−	0.238071i	$-0.0765153\pi$
$462$	−21.7500	+	23.8972i	−1.01190	+	1.11180i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$			115.410i			5.33484i
$469$		0			0
$470$		0			0
$471$	27.1450	+	47.0166i	1.25078	+	2.16641i
$472$		0			0
$473$		0			0
$474$	−73.3838	+	42.3682i	−3.37063	+	1.94603i
$475$		0			0
$476$	−6.63075	−	20.7854i	−0.303920	−	0.952698i
$477$	−9.99533			−0.457655
$478$		0			0
$479$	−29.3545	−	16.9479i	−1.34124	−	0.774367i	−0.354253	−	0.935149i	$-0.615265\pi$
$479$	−29.3545	−	16.9479i	−1.34124	−	0.774367i	−0.986990	+	0.160782i	$0.948598\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	44.1274	+	9.62682i	2.00787	+	0.438035i
$484$	−9.31271			−0.423305
$485$		0			0
$486$	−101.380	−	58.5320i	−4.59871	−	2.65507i
$487$	21.2527	−	36.8107i	0.963050	−	1.66805i	0.248284	−	0.968687i	$-0.420133\pi$
$487$	21.2527	−	36.8107i	0.963050	−	1.66805i	0.714766	−	0.699364i	$-0.246533\pi$
$488$		0			0
$489$		−	75.3545i		−	3.40765i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$			15.6619i			0.703240i
$497$	−3.95518	+	18.1298i	−0.177414	+	0.813231i
$498$		0			0
$499$	12.3013	+	21.3065i	0.550683	+	0.953811i	0.998225	+	0.0595480i	$0.0189659\pi$
$499$	12.3013	+	21.3065i	0.550683	+	0.953811i	−0.447543	+	0.894263i	$0.647701\pi$
$500$		0			0
$501$	−23.7631	+	41.1589i	−1.06166	+	1.83884i
$502$		0			0
$503$		−	24.9727i		−	1.11348i	−0.830688	−	0.556738i	$-0.812053\pi$
$503$		−	24.9727i		−	1.11348i	0.830688	−	0.556738i	$-0.187947\pi$
$504$	−62.4306	+	19.9160i	−2.78088	+	0.887128i
$505$		0			0
$506$	−8.86675	−	15.3577i	−0.394175	−	0.682732i
$507$	−90.3435	−	52.1598i	−4.01229	−	2.31650i
$508$		0			0
$509$	14.2829	−	8.24621i	0.633076	−	0.365507i	−0.148866	−	0.988857i	$-0.547562\pi$
$509$	14.2829	−	8.24621i	0.633076	−	0.365507i	0.781943	+	0.623350i	$0.214229\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$		0			0
$514$	−17.7879	−	10.2698i	−0.784589	−	0.452983i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$			45.6421i			1.99389i
$525$	−13.7855	−	43.2135i	−0.601651	−	1.88599i
$526$		0			0
$527$	8.07196	+	13.9810i	0.351620	+	0.609024i
$528$	29.9162	+	17.2721i	1.30193	+	0.751672i
$529$	−0.893398	+	1.54741i	−0.0388434	+	0.0672787i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	6.35887	+	11.0139i	0.275175	+	0.476617i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14.3736	−	10.2097i	0.619116	−	0.439762i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$	−20.1990	+	11.6619i	−0.866025	+	0.500000i
$545$		0			0
$546$	18.0200	−	82.5999i	0.771183	−	3.53495i
$547$	32.5866			1.39330			0.696652	−	0.717409i	$-0.254672\pi$
$547$	32.5866			1.39330			0.696652	+	0.717409i	$0.254672\pi$
$548$	−7.37108	−	12.7671i	−0.314877	−	0.545383i
$549$		0			0
$550$	−8.90481	+	15.4236i	−0.379702	+	0.657664i
$551$		0			0
$552$		−	48.2837i		−	2.05509i
$553$	44.0467	−	14.0513i	1.87305	−	0.597523i
$554$		0			0
$555$		0			0
$556$	7.20981	+	4.16258i	0.305764	+	0.176533i
$557$	−18.5348	+	32.1032i	−0.785344	+	1.36026i	0.143449	+	0.989658i	$0.454181\pi$
$557$	−18.5348	+	32.1032i	−0.785344	+	1.36026i	−0.928793	+	0.370598i	$0.879153\pi$
$558$	41.9932	−	24.2448i	1.77771	−	1.02636i
$559$		0			0
$560$		0			0
$561$	35.6073			1.50334
$562$	18.0777	+	31.3115i	0.762561	+	1.32079i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		−	13.1743i		−	0.553756i
$567$	81.0296	+	73.7488i	3.40292	+	3.09716i
$568$	19.8374			0.832358
$569$	23.8536	+	41.3156i	0.999994	+	1.73204i	0.503066	+	0.864248i	$0.332205\pi$
$569$	23.8536	+	41.3156i	0.999994	+	1.73204i	0.496928	+	0.867792i	$0.334461\pi$
$570$		0			0
$571$	0.427099	−	0.739758i	0.0178735	−	0.0309579i	−0.856950	−	0.515399i	$-0.827644\pi$
$571$	0.427099	−	0.739758i	0.0178735	−	0.0309579i	0.874824	+	0.484441i	$0.160977\pi$
$572$	−28.7473	+	16.5972i	−1.20198	+	0.693965i
$573$		0			0
$574$		0			0
$575$	24.8932			1.03812
$576$	35.0274	+	60.6693i	1.45948	+	2.52789i
$577$	16.0134	+	9.24537i	0.666648	+	0.384890i	0.794805	−	0.606864i	$-0.207573\pi$
$577$	16.0134	+	9.24537i	0.666648	+	0.384890i	−0.128157	+	0.991754i	$0.540906\pi$
$578$	−12.0208	+	20.8207i	−0.500000	+	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.43743	−	2.48971i	−0.0595325	−	0.103113i
$584$		0			0
$585$		0			0
$586$	21.7203	−	12.5402i	0.897257	−	0.518032i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	47.7916	−	4.50619i	1.97089	−	0.185832i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	39.5707	−	22.8462i	1.62497	−	0.938179i	0.639413	−	0.768864i	$-0.279178\pi$
$593$	39.5707	−	22.8462i	1.62497	−	0.938179i	0.985562	−	0.169316i	$-0.0541557\pi$
$594$		−	70.3098i		−	2.88485i
$595$		0			0
$596$	46.5657			1.90741
$597$	−32.0765	−	55.5581i	−1.31280	−	2.27384i
$598$	40.1811	+	23.1986i	1.64313	+	0.948661i
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−41.9944	+	24.2455i	−1.71441	+	0.989817i
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	42.4123	−	73.4602i	1.72288	−	2.98412i
$607$	−34.1900	+	19.7396i	−1.38773	+	0.801207i	−0.993059	−	0.117615i	$-0.962475\pi$
$607$	−34.1900	+	19.7396i	−1.38773	+	0.801207i	−0.394672	+	0.918822i	$0.629142\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	62.5365	+	36.1055i	2.52789	+	1.45948i
$613$	20.9243	−	36.2419i	0.845124	−	1.46380i	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	20.9243	−	36.2419i	0.845124	−	1.46380i	0.885514	−	0.464614i	$-0.153807\pi$
$614$		0			0
$615$		0			0
$616$	−13.9390	−	12.6865i	−0.561618	−	0.511155i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−27.5659	−	15.9152i	−1.10797	−	0.639684i	−0.169664	−	0.985502i	$-0.554268\pi$
$619$	−27.5659	−	15.9152i	−1.10797	−	0.639684i	−0.938302	+	0.345818i	$0.887602\pi$
$620$		0			0
$621$	−85.1083	+	49.1373i	−3.41528	+	1.97181i
$622$			22.8725i			0.917102i
$623$	−2.10891	−	6.61079i	−0.0844916	−	0.264856i
$624$	−90.3799			−3.61809
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−27.4243	+	15.8334i	−1.09435	+	0.631822i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−24.7129	−	42.8040i	−0.983027	−	1.70265i
$633$	−86.1664	−	49.7482i	−3.42481	−	1.97731i
$634$		0			0
$635$		0			0
$636$		−	7.82752i		−	0.310381i
$637$	−19.2121	+	41.9367i	−0.761213	+	1.66159i
$638$		0			0
$639$	−30.7084	−	53.1886i	−1.21481	−	2.10411i
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	−74.3273	+	42.9129i	−2.93346	+	1.69364i
$643$			47.8044i			1.88522i	0.333892	+	0.942611i	$0.391638\pi$
$643$			47.8044i			1.88522i	−0.333892	+	0.942611i	$0.608362\pi$
$644$	−5.61522	+	25.7391i	−0.221271	+	1.01426i
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	58.5654	−	101.438i	2.30067	−	3.98487i
$649$		0			0
$650$		−	46.5963i		−	1.82766i
$651$	−33.8403	+	10.7954i	−1.32631	+	0.423105i
$652$	43.9535			1.72135
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	42.8486	+	24.7386i	1.66662	+	0.962221i	0.969442	+	0.245319i	$0.0788928\pi$
$661$	42.8486	+	24.7386i	1.66662	+	0.962221i	0.697174	+	0.716902i	$0.254441\pi$
$662$		0			0
$663$	−80.6802	+	46.5807i	−3.13336	+	1.80905i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−24.0076	−	13.8608i	−0.928881	−	0.536289i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−15.5966	−	48.8905i	−0.601651	−	1.88599i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	85.4736	+	49.3482i	3.28988	+	1.89941i
$676$	30.4243	−	52.6964i	1.17016	−	2.02678i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	26.1170	+	45.2360i	1.00081	+	1.73345i
$682$	12.0781	+	6.97331i	0.462495	+	0.267022i
$683$	23.6933	−	41.0381i	0.906601	−	1.57028i	0.0878466	−	0.996134i	$-0.472001\pi$
$683$	23.6933	−	41.0381i	0.906601	−	1.57028i	0.818754	−	0.574144i	$-0.194665\pi$
$684$		0			0
$685$		0			0
$686$	−26.0008	−	3.15583i	−0.992715	−	0.120490i
$687$	−19.9933			−0.762793
$688$		0			0
$689$	6.51397	+	3.76084i	0.248162	+	0.143277i
$690$		0			0
$691$	38.1227	−	22.0101i	1.45026	−	0.837305i	0.451760	−	0.892140i	$-0.350796\pi$
$691$	38.1227	−	22.0101i	1.45026	−	0.837305i	0.998495	+	0.0548341i	$0.0174630\pi$
$692$		0			0
$693$	−12.4378	+	57.0125i	−0.472474	+	2.16573i
$694$	−36.9127			−1.40119
$695$		0			0
$696$		0			0
$697$		0			0
$698$	30.2985	−	17.4929i	1.14682	−	0.662114i
$699$		0			0
$700$	25.2060	−	8.04097i	0.952698	−	0.303920i
$701$	−41.9122			−1.58300			−0.791502	−	0.611167i	$-0.790700\pi$
$701$	−41.9122			−1.58300			−0.791502	+	0.611167i	$0.790700\pi$
$702$	91.9777	+	159.310i	3.47148	+	6.01277i
$703$		0			0
$704$	−10.0746	+	17.4498i	−0.379702	+	0.657664i
$705$		0			0
$706$		−	1.27840i		−	0.0481132i
$707$	−31.1523	+	34.2277i	−1.17160	+	1.28727i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$	−76.5116	+	132.522i	−2.86941	+	4.96996i
$712$	−6.42429	+	3.70906i	−0.240760	+	0.139003i
$713$		−	19.4937i		−	0.730046i
$714$	−39.1203	−	35.6052i	−1.46404	−	1.33249i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	38.3895	−	22.1642i	1.43169	−	0.826584i	0.434437	−	0.900702i	$-0.356947\pi$
$719$	38.3895	−	22.1642i	1.43169	−	0.826584i	0.997249	+	0.0741179i	$0.0236141\pi$
$720$		0			0
$721$		0			0
$722$	26.8701			1.00000
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−19.5541	+	11.2896i	−0.725720	+	0.418995i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	48.1797	+	10.5109i	1.78566	+	0.389558i
$729$	−159.591			−5.91079
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−37.9243	+	21.8956i	−1.40077	+	0.808732i	−0.994471	−	0.105010i	$-0.966513\pi$
$733$	−37.9243	+	21.8956i	−1.40077	+	0.808732i	−0.406294	+	0.913742i	$0.633179\pi$
$734$			47.0327i			1.73601i
$735$		0			0
$736$	28.1634			1.03812
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$	−0.910313	+	4.17269i	−0.0334186	+	0.153184i
$743$	42.3494			1.55365			0.776824	−	0.629718i	$-0.216830\pi$
$743$	42.3494			1.55365			0.776824	+	0.629718i	$0.216830\pi$
$744$	18.9865	+	32.8856i	0.696079	+	1.20564i
$745$		0			0
$746$	9.92929	−	17.1980i	0.363537	−	0.629665i
$747$		0			0
$748$			20.7694i			0.759404i
$749$	44.6130	−	14.2320i	1.63012	−	0.520025i
$750$		0			0
$751$	24.7681	+	42.8995i	0.903799	+	1.56543i	0.822521	+	0.568734i	$0.192567\pi$
$751$	24.7681	+	42.8995i	0.903799	+	1.56543i	0.0812777	+	0.996691i	$0.474100\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−70.3056	+	77.2465i	−2.55699	+	2.80943i
$757$	48.5833			1.76579			0.882895	−	0.469571i	$-0.155591\pi$
$757$	48.5833			1.76579			0.882895	+	0.469571i	$0.155591\pi$
$758$	−27.4713	−	47.5817i	−0.997804	−	1.72825i
$759$	−37.2354	−	21.4979i	−1.35156	−	0.780323i
$760$		0			0
$761$	46.8191	−	27.0310i	1.69719	−	0.979873i	0.748783	−	0.662815i	$-0.230638\pi$
$761$	46.8191	−	27.0310i	1.69719	−	0.979873i	0.948406	−	0.317058i	$-0.102695\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−47.5112	+	27.4306i	−1.71441	+	0.989817i
$769$			44.8496i			1.61732i	0.588278	+	0.808659i	$0.299806\pi$
$769$			44.8496i			1.61732i	−0.588278	+	0.808659i	$0.700194\pi$
$770$		0			0
$771$	−49.7993			−1.79348
$772$		0			0
$773$	36.5125	+	21.0805i	1.31326	+	0.758212i	0.982635	−	0.185549i	$-0.0594064\pi$
$773$	36.5125	+	21.0805i	1.31326	+	0.758212i	0.330627	+	0.943761i	$0.392740\pi$
$774$		0			0
$775$	−16.9545	+	9.78869i	−0.609024	+	0.351620i
$776$		0			0
$777$		0			0
$778$	3.57571			0.128196
$779$		0			0
$780$		0			0
$781$	8.83240	−	15.2982i	0.316048	−	0.547412i
$782$	25.1409	−	14.5151i	0.899036	−	0.519058i
$783$		0			0
$784$	2.62842	+	27.8764i	0.0938720	+	0.995584i
$785$		0			0
$786$	55.3308	+	95.8357i	1.97358	+	3.41835i
$787$	−10.9955	−	6.34824i	−0.391946	−	0.226290i	0.291057	−	0.956706i	$-0.405993\pi$
$787$	−10.9955	−	6.34824i	−0.391946	−	0.226290i	−0.683003	+	0.730415i	$0.739326\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	62.3825			2.21667
$793$		0			0
$794$		0			0
$795$		0			0
$796$	32.4064	−	18.7099i	1.14862	−	0.663154i
$797$			55.0116i			1.94861i	0.225232	+	0.974305i	$0.427686\pi$
$797$			55.0116i			1.94861i	−0.225232	+	0.974305i	$0.572314\pi$
$798$		0			0
$799$		0			0
$800$	−14.1421	−	24.4949i	−0.500000	−	0.866025i
$801$	19.8897	+	11.4833i	0.702768	+	0.405743i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−36.4893			−1.28528
$807$		0			0
$808$	42.8486	+	24.7386i	1.50741	+	0.870302i
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		−	50.4615i		−	1.77194i	−0.463738	−	0.885972i	$-0.653492\pi$
$811$		−	50.4615i		−	1.77194i	0.463738	−	0.885972i	$-0.346508\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−28.2748	+	48.9735i	−0.989817	+	1.71441i
$817$		0			0
$818$			56.6847i			1.98193i
$819$	−46.4005	−	145.452i	−1.62137	−	5.08249i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−30.9544	−	17.8715i	−1.07966	−	0.623342i
$823$	8.73940	−	15.1371i	0.304636	−	0.527646i	−0.672544	−	0.740057i	$-0.734798\pi$
$823$	8.73940	−	15.1371i	0.304636	−	0.527646i	0.977180	+	0.212412i	$0.0681318\pi$
$824$		0			0
$825$			43.1802i			1.50334i
$826$		0			0
$827$	−57.4613			−1.99813			−0.999063	−	0.0432777i	$-0.986220\pi$
$827$	−57.4613			−1.99813			−0.999063	+	0.0432777i	$0.986220\pi$
$828$	−43.5972	−	75.5126i	−1.51511	−	2.62424i
$829$	−28.2879	−	16.3320i	−0.982478	−	0.567234i	−0.0794606	−	0.996838i	$-0.525320\pi$
$829$	−28.2879	−	16.3320i	−0.982478	−	0.567234i	−0.903017	+	0.429604i	$0.858653\pi$
$830$		0			0
$831$		0			0
$832$		−	52.7177i		−	1.82766i
$833$	16.7135	+	23.5300i	0.579088	+	0.815265i
$834$	20.1848			0.698941
$835$		0			0
$836$		0			0
$837$	38.6443	−	66.9340i	1.33574	−	2.31358i
$838$	−36.4742	+	21.0584i	−1.25998	+	0.727450i
$839$			39.4330i			1.36138i	0.732572	+	0.680690i	$0.238320\pi$
$839$			39.4330i			1.36138i	−0.732572	+	0.680690i	$0.761680\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	3.00000	+	5.19615i	0.103387	+	0.179071i
$843$	75.9162	+	43.8302i	2.61469	+	1.50959i
$844$	29.0176	−	50.2600i	0.998827	−	1.73002i
$845$		0			0
$846$		0			0
$847$	11.7368	−	3.74416i	0.403282	−	0.128651i
$848$	4.56571			0.156787
$849$	−15.9708	−	27.6623i	−0.548117	−	0.949366i
$850$	−25.2488	−	14.5774i	−0.866025	−	0.500000i
$851$		0			0
$852$	41.6529	−	24.0483i	1.42701	−	0.823882i
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−25.0306	−	43.3544i	−0.855530	−	1.48182i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$	−40.2408	+	69.6991i	−1.37380	+	2.37949i
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	5.19327			0.176883
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$	96.7023	+	55.8311i	3.28988	+	1.89941i
$865$		0			0
$866$	−35.7071	+	20.6155i	−1.21338	+	0.700544i
$867$			58.2901i			1.97963i
$868$	−6.29684	−	19.7387i	−0.213729	−	0.669975i
$869$	−44.0127			−1.49303
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	36.6059	+	21.1344i	1.23539	+	0.713252i
$879$	30.4044	−	52.6619i	1.02551	−	1.77624i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	70.6741	−	50.2002i	2.37972	−	1.69033i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−27.1701	−	47.0599i	−0.913828	−	1.58280i
$885$		0			0
$886$		0			0
$887$	−49.7987	+	28.7513i	−1.67208	+	0.965375i	−0.705605	+	0.708605i	$0.749325\pi$
$887$	−49.7987	+	28.7513i	−1.67208	+	0.965375i	−0.966473	+	0.256769i	$0.917342\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−52.1514	−	90.3288i	−1.74714	−	3.02613i
$892$		0			0
$893$		0			0
$894$	97.7749	−	56.4504i	3.27008	−	1.88798i
$895$		0			0
$896$	28.5173	−	9.09732i	0.952698	−	0.303920i
$897$	112.492			3.75600
$898$		0			0
$899$		0			0
$900$	−43.7843	+	75.8366i	−1.45948	+	2.52789i
$901$	4.07571	−	2.35311i	0.135782	−	0.0783936i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	25.0701	−	43.4227i	0.832438	−	1.44183i	−0.0636608	−	0.997972i	$-0.520278\pi$
$907$	25.0701	−	43.4227i	0.832438	−	1.44183i	0.896099	−	0.443854i	$-0.146389\pi$
$908$	−26.3857	+	15.2338i	−0.875640	+	0.505551i
$909$		−	153.182i		−	5.08074i
$910$		0			0
$911$	−55.9730			−1.85447			−0.927234	−	0.374482i	$-0.877820\pi$
$911$	−55.9730			−1.85447			−0.927234	+	0.374482i	$0.877820\pi$
$912$		0			0
$913$		0			0
$914$	−9.00000	+	15.5885i	−0.297694	+	0.515620i
$915$		0			0
$916$		−	11.6619i		−	0.385320i
$917$	−18.3503	−	57.5228i	−0.605982	−	1.89957i
$918$	115.099			3.79883
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$	12.5208	−	7.22891i	0.412352	−	0.238071i
$923$			46.2174i			1.52126i
$924$	−44.6476	−	9.74029i	−1.46880	−	0.320432i
$925$		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
$931$		0			0
$932$		0			0
$933$	27.7277	+	48.0257i	0.907763	+	1.57229i
$934$		0			0
$935$		0			0
$936$	−141.348	+	81.6074i	−4.62011	+	2.66742i
$937$		−	24.7386i		−	0.808176i	−0.914720	−	0.404088i	$-0.867589\pi$
$937$		−	24.7386i		−	0.808176i	0.914720	−	0.404088i	$-0.132411\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$	−38.3889	+	66.4915i	−1.25078	+	2.16641i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	28.2197	+	48.8780i	0.917017	+	1.58832i	0.803921	+	0.594736i	$0.202743\pi$
$947$	28.2197	+	48.8780i	0.917017	+	1.58832i	0.113096	+	0.993584i	$0.463923\pi$
$948$	−103.780	−	59.9176i	−3.37063	−	1.94603i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	20.7682	−	22.8185i	0.673100	−	0.739551i
$953$	−61.7112			−1.99902			−0.999511	−	0.0312570i	$-0.990049\pi$
$953$	−61.7112			−1.99902			−0.999511	+	0.0312570i	$0.990049\pi$
$954$	−7.06776	−	12.2417i	−0.228827	−	0.396341i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		−	47.9358i		−	1.54873i
$959$	14.4228	+	13.1268i	0.465735	+	0.423888i
$960$		0			0
$961$	−7.83452	−	13.5698i	−0.252727	−	0.437735i
$962$		0			0
$963$	−77.4953	+	134.226i	−2.49725	+	4.32537i
$964$		0			0
$965$		0			0
$966$	19.4124	+	60.8520i	0.624584	+	1.95788i
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−6.58508	−	11.4057i	−0.211652	−	0.366593i
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		−	165.554i		−	5.31013i
$973$	−10.7601	−	2.34742i	−0.344952	−	0.0752547i
$974$	60.1116			1.92610
$975$	−56.4875	−	97.8391i	−1.80905	−	3.13336i
$976$		0			0
$977$	12.0000	−	20.7846i	0.383914	−	0.664959i	−0.607704	−	0.794164i	$-0.707909\pi$
$977$	12.0000	−	20.7846i	0.383914	−	0.664959i	0.991618	+	0.129205i	$0.0412426\pi$
$978$	92.2900	−	53.2837i	2.95111	−	1.70382i
$979$			6.60570i			0.211119i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−49.6399	−	28.6596i	−1.58327	−	0.914099i	−0.994378	−	0.105885i	$-0.966233\pi$
$983$	−49.6399	−	28.6596i	−1.58327	−	0.914099i	−0.588888	−	0.808215i	$-0.700434\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−6.14070	+	10.6360i	−0.195066	+	0.337864i	−0.946922	−	0.321463i	$-0.895825\pi$
$991$	−6.14070	+	10.6360i	−0.195066	+	0.337864i	0.751856	+	0.659327i	$0.229159\pi$
$992$	−19.1818	+	11.0746i	−0.609024	+	0.351620i
$993$		0			0
$994$	−25.0011	+	7.97559i	−0.792985	+	0.252970i
$995$		0			0
$996$		0			0
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$	−17.3967	+	30.1320i	−0.550683	+	0.953811i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.2.q.a.339.8	yes	16
4.3	odd	2	inner	476.2.q.a.339.5	yes	16
7.5	odd	6	inner	476.2.q.a.271.8	yes	16
17.16	even	2	inner	476.2.q.a.339.5	yes	16
28.19	even	6	inner	476.2.q.a.271.5	✓	16
68.67	odd	2	CM	476.2.q.a.339.8	yes	16
119.33	odd	6	inner	476.2.q.a.271.5	✓	16
476.271	even	6	inner	476.2.q.a.271.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.q.a.271.5	✓	16	28.19	even	6	inner
476.2.q.a.271.5	✓	16	119.33	odd	6	inner
476.2.q.a.271.8	yes	16	7.5	odd	6	inner
476.2.q.a.271.8	yes	16	476.271	even	6	inner
476.2.q.a.339.5	yes	16	4.3	odd	2	inner
476.2.q.a.339.5	yes	16	17.16	even	2	inner
476.2.q.a.339.8	yes	16	1.1	even	1	trivial
476.2.q.a.339.8	yes	16	68.67	odd	2	CM

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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 1.22474i) q^{2} +(2.96945 + 1.71441i) q^{3} +(-1.00000 + 1.73205i) q^{4} +4.84909i q^{6} +(0.563932 - 2.58495i) q^{7} -2.82843 q^{8} +(4.37843 + 7.58366i) q^{9} +(-1.25933 + 2.18122i) q^{11} +(-5.93890 + 3.42883i) q^{12} -6.58971i q^{13} +(3.56467 - 1.13716i) q^{14} +(-2.00000 - 3.46410i) q^{16} +(-3.57071 - 2.06155i) q^{17} +(-6.19203 + 10.7249i) q^{18} +(6.10625 - 6.70908i) q^{21} -3.56192 q^{22} +(2.48932 + 4.31162i) q^{23} +(-8.39888 - 4.84909i) q^{24} +(2.50000 - 4.33013i) q^{25} +(8.07071 - 4.65963i) q^{26} +19.7393i q^{27} +(3.91334 + 3.56171i) q^{28} +(-3.39090 - 1.95774i) q^{31} +(2.82843 - 4.89898i) q^{32} +(-7.47904 + 4.31802i) q^{33} -5.83095i q^{34} -17.5137 q^{36} +(11.2975 - 19.5678i) q^{39} +(12.5347 + 2.73456i) q^{42} +(-2.51866 - 4.36245i) q^{44} +(-3.52043 + 6.09756i) q^{46} -13.7153i q^{48} +(-6.36396 - 2.91548i) q^{49} +7.07107 q^{50} +(-7.06871 - 12.2434i) q^{51} +(11.4137 + 6.58971i) q^{52} +(-0.570714 + 0.988506i) q^{53} +(-24.1756 + 13.9578i) q^{54} +(-1.59504 + 7.31135i) q^{56} -5.53732i q^{62} +(22.0725 - 7.04136i) q^{63} +8.00000 q^{64} +(-10.5770 - 6.10661i) q^{66} +(7.14143 - 4.12311i) q^{68} +17.0709i q^{69} -7.01357 q^{71} +(-12.3841 - 21.4498i) q^{72} +(14.8473 - 8.57207i) q^{75} +(4.92818 + 4.48537i) q^{77} +31.9541 q^{78} +(8.73733 + 15.1335i) q^{79} +(-20.7060 + 35.8638i) q^{81} +(5.51422 + 17.2854i) q^{84} +(3.56192 - 6.16943i) q^{88} +(2.27133 - 1.31135i) q^{89} +(-17.0341 - 3.71615i) q^{91} -9.95727 q^{92} +(-6.71275 - 11.6268i) q^{93} +(16.7978 - 9.69819i) q^{96} +(-0.929286 - 9.85578i) q^{98} -22.0555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 24 q^{9} - 32 q^{16} - 8 q^{18} - 16 q^{21} + 40 q^{25} + 72 q^{26} - 96 q^{33} - 96 q^{36} + 40 q^{42} + 48 q^{53} + 128 q^{64} - 16 q^{72} - 48 q^{77} - 136 q^{81} + 64 q^{84} + 40 q^{93}+ \cdots - 72 q^{98}+O(q^{100}) \) 16 * q - 16 * q^4 + 24 * q^9 - 32 * q^16 - 8 * q^18 - 16 * q^21 + 40 * q^25 + 72 * q^26 - 96 * q^33 - 96 * q^36 + 40 * q^42 + 48 * q^53 + 128 * q^64 - 16 * q^72 - 48 * q^77 - 136 * q^81 + 64 * q^84 + 40 * q^93 - 72 * q^98

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