LMFDB - Embedded newform 980.2.s.b.19.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(19,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("980.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.s (of order $6$, degree $2$, not 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:	$7.82533939809$
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:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			19.2
Root			$-1.01575 + 1.40294i$ of defining polynomial
Character	$\chi$	$=$	980.19
Dual form			980.2.s.b.619.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.707107 + 1.22474i) q^{2} +(2.73861 - 1.58114i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(1.93649 + 1.11803i) q^{5} +4.47214i q^{6} +2.82843 q^{8} +(3.50000 - 6.06218i) q^{9} +(-2.73861 + 1.58114i) q^{10} +(-5.47723 - 3.16228i) q^{12} +7.07107 q^{15} +(-2.00000 + 3.46410i) q^{16} +(4.94975 + 8.57321i) q^{18} -4.47214i q^{20} +(0.707107 - 1.22474i) q^{23} +(7.74597 - 4.47214i) q^{24} +(2.50000 + 4.33013i) q^{25} -12.6491i q^{27} +6.00000 q^{29} +(-5.00000 + 8.66025i) q^{30} +(-2.82843 - 4.89898i) q^{32} -14.0000 q^{36} +(5.47723 + 3.16228i) q^{40} +4.47214i q^{41} -12.7279 q^{43} +(13.5554 - 7.82624i) q^{45} +(1.00000 + 1.73205i) q^{46} +(-8.21584 - 4.74342i) q^{47} +12.6491i q^{48} -7.07107 q^{50} +(15.4919 + 8.94427i) q^{54} +(-4.24264 + 7.34847i) q^{58} +(-7.07107 - 12.2474i) q^{60} +(11.6190 + 6.70820i) q^{61} +8.00000 q^{64} +(-2.12132 - 3.67423i) q^{67} -4.47214i q^{69} +(9.89949 - 17.1464i) q^{72} +(13.6931 + 7.90569i) q^{75} +(-7.74597 + 4.47214i) q^{80} +(-9.50000 - 16.4545i) q^{81} +(-5.47723 - 3.16228i) q^{82} -9.48683i q^{83} +(9.00000 - 15.5885i) q^{86} +(16.4317 - 9.48683i) q^{87} +(-15.4919 - 8.94427i) q^{89} +22.1359i q^{90} -2.82843 q^{92} +(11.6190 - 6.70820i) q^{94} +(-15.4919 - 8.94427i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 28 q^{9} - 16 q^{16} + 20 q^{25} + 48 q^{29} - 40 q^{30} - 112 q^{36} + 8 q^{46} + 64 q^{64} - 76 q^{81} + 72 q^{86}+O(q^{100}) $ 8 * q - 8 * q^4 + 28 * q^9 - 16 * q^16 + 20 * q^25 + 48 * q^29 - 40 * q^30 - 112 * q^36 + 8 * q^46 + 64 * q^64 - 76 * q^81 + 72 * q^86

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−0.707107	+	1.22474i	−0.500000	+	0.866025i
$2$	−0.707107	+	1.22474i	−0.500000	+	0.866025i
$3$	2.73861	−	1.58114i	1.58114	−	0.912871i	0.586445	−	0.809989i	$-0.300527\pi$
$3$	2.73861	−	1.58114i	1.58114	−	0.912871i	0.994694	−	0.102882i	$-0.0328064\pi$
$4$	−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
$6$			4.47214i			1.82574i
$7$		0			0
$7$		0			0
$8$	2.82843			1.00000
$9$	3.50000	−	6.06218i	1.16667	−	2.02073i
$10$	−2.73861	+	1.58114i	−0.866025	+	0.500000i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−5.47723	−	3.16228i	−1.58114	−	0.912871i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	7.07107			1.82574
$16$	−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	$0.833333\pi$
$18$	4.94975	+	8.57321i	1.16667	+	2.02073i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		−	4.47214i		−	1.00000i
$21$		0			0
$22$		0			0
$23$	0.707107	−	1.22474i	0.147442	−	0.255377i	−0.782839	−	0.622224i	$-0.786229\pi$
$23$	0.707107	−	1.22474i	0.147442	−	0.255377i	0.930281	+	0.366847i	$0.119563\pi$
$24$	7.74597	−	4.47214i	1.58114	−	0.912871i
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		−	12.6491i		−	2.43432i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$	−5.00000	+	8.66025i	−0.912871	+	1.58114i
$31$		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			0
$36$	−14.0000			−2.33333
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$	5.47723	+	3.16228i	0.866025	+	0.500000i
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	−12.7279			−1.94099			−0.970495	−	0.241121i	$-0.922485\pi$
$43$	−12.7279			−1.94099			−0.970495	+	0.241121i	$0.922485\pi$
$44$		0			0
$45$	13.5554	−	7.82624i	2.02073	−	1.16667i
$46$	1.00000	+	1.73205i	0.147442	+	0.255377i
$47$	−8.21584	−	4.74342i	−1.19840	−	0.691898i	−0.238204	−	0.971215i	$-0.576559\pi$
$47$	−8.21584	−	4.74342i	−1.19840	−	0.691898i	−0.960199	+	0.279317i	$0.909892\pi$
$48$			12.6491i			1.82574i
$49$		0			0
$50$	−7.07107			−1.00000
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	15.4919	+	8.94427i	2.10819	+	1.21716i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−4.24264	+	7.34847i	−0.557086	+	0.964901i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$	−7.07107	−	12.2474i	−0.912871	−	1.58114i
$61$	11.6190	+	6.70820i	1.48765	+	0.858898i	0.999901	−	0.0140840i	$-0.00448323\pi$
$61$	11.6190	+	6.70820i	1.48765	+	0.858898i	0.487753	+	0.872982i	$0.337817\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−2.12132	−	3.67423i	−0.259161	−	0.448879i	0.706857	−	0.707357i	$-0.250113\pi$
$67$	−2.12132	−	3.67423i	−0.259161	−	0.448879i	−0.966017	+	0.258478i	$0.916779\pi$
$68$		0			0
$69$		−	4.47214i		−	0.538382i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	9.89949	−	17.1464i	1.16667	−	2.02073i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$		0			0
$75$	13.6931	+	7.90569i	1.58114	+	0.912871i
$76$		0			0
$77$		0			0
$78$		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
$81$	−9.50000	−	16.4545i	−1.05556	−	1.82828i
$82$	−5.47723	−	3.16228i	−0.604858	−	0.349215i
$83$		−	9.48683i		−	1.04132i	−0.853766	−	0.520658i	$-0.825687\pi$
$83$		−	9.48683i		−	1.04132i	0.853766	−	0.520658i	$-0.174313\pi$
$84$		0			0
$85$		0			0
$86$	9.00000	−	15.5885i	0.970495	−	1.68095i
$87$	16.4317	−	9.48683i	1.76166	−	1.01710i
$88$		0			0
$89$	−15.4919	−	8.94427i	−1.64214	−	0.948091i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−15.4919	−	8.94427i	−1.64214	−	0.948091i	−0.662071	−	0.749441i	$-0.730322\pi$
$90$			22.1359i			2.33333i
$91$		0			0
$92$	−2.82843			−0.294884
$93$		0			0
$94$	11.6190	−	6.70820i	1.19840	−	0.691898i
$95$		0			0
$96$	−15.4919	−	8.94427i	−1.58114	−	0.912871i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$	7.74597	−	4.47214i	0.770752	−	0.444994i	−0.0623905	−	0.998052i	$-0.519872\pi$
$101$	7.74597	−	4.47214i	0.770752	−	0.444994i	0.833143	+	0.553058i	$0.186539\pi$
$102$		0			0
$103$	13.6931	+	7.90569i	1.34922	+	0.778971i	0.988139	−	0.153564i	$-0.0490751\pi$
$103$	13.6931	+	7.90569i	1.34922	+	0.778971i	0.361079	+	0.932535i	$0.382408\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.19239	+	15.9217i	−0.888662	+	1.53921i	−0.0472033	+	0.998885i	$0.515031\pi$
$107$	−9.19239	+	15.9217i	−0.888662	+	1.53921i	−0.841458	+	0.540322i	$0.818302\pi$
$108$	−21.9089	+	12.6491i	−2.10819	+	1.21716i
$109$	8.00000	+	13.8564i	0.766261	+	1.32720i	0.939577	+	0.342337i	$0.111218\pi$
$109$	8.00000	+	13.8564i	0.766261	+	1.32720i	−0.173316	+	0.984866i	$0.555448\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	2.73861	−	1.58114i	0.255377	−	0.147442i
$116$	−6.00000	−	10.3923i	−0.557086	−	0.964901i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	20.0000			1.82574
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$	−16.4317	+	9.48683i	−1.48765	+	0.858898i
$123$	7.07107	+	12.2474i	0.637577	+	1.10432i
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$	−4.24264			−0.376473			−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264			−0.376473			−0.188237	+	0.982124i	$0.560277\pi$
$128$	−5.65685	+	9.79796i	−0.500000	+	0.866025i
$129$	−34.8569	+	20.1246i	−3.06897	+	1.77187i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$	6.00000			0.518321
$135$	14.1421	−	24.4949i	1.21716	−	2.10819i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$	5.47723	+	3.16228i	0.466252	+	0.269191i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−30.0000			−2.52646
$142$		0			0
$143$		0			0
$144$	14.0000	+	24.2487i	1.16667	+	2.02073i
$145$	11.6190	+	6.70820i	0.964901	+	0.557086i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.0000	+	20.7846i	−0.983078	+	1.70274i	−0.332896	+	0.942964i	$0.608026\pi$
$149$	−12.0000	+	20.7846i	−0.983078	+	1.70274i	−0.650183	+	0.759778i	$0.725308\pi$
$150$	−19.3649	+	11.1803i	−1.58114	+	0.912871i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$		0			0
$159$		0			0
$160$		−	12.6491i		−	1.00000i
$161$		0			0
$162$	26.8701			2.11111
$163$	−6.36396	+	11.0227i	−0.498464	+	0.863365i	−0.999998	−	0.00177283i	$-0.999436\pi$
$163$	−6.36396	+	11.0227i	−0.498464	+	0.863365i	0.501535	+	0.865138i	$0.332769\pi$
$164$	7.74597	−	4.47214i	0.604858	−	0.349215i
$165$		0			0
$166$	11.6190	+	6.70820i	0.901805	+	0.520658i
$167$		−	9.48683i		−	0.734113i	−0.930199	−	0.367057i	$-0.880366\pi$
$167$		−	9.48683i		−	0.734113i	0.930199	−	0.367057i	$-0.119634\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	12.7279	+	22.0454i	0.970495	+	1.68095i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$			26.8328i			2.03419i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	21.9089	−	12.6491i	1.64214	−	0.948091i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	−27.1109	−	15.6525i	−2.02073	−	1.16667i
$181$		−	26.8328i		−	1.99447i	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$		−	26.8328i		−	1.99447i	0.0743294	−	0.997234i	$-0.476318\pi$
$182$		0			0
$183$	42.4264			3.13625
$184$	2.00000	−	3.46410i	0.147442	−	0.255377i
$185$		0			0
$186$		0			0
$187$		0			0
$188$			18.9737i			1.38380i
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	21.9089	−	12.6491i	1.58114	−	0.912871i
$193$		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
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		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
$201$	−11.6190	−	6.70820i	−0.819538	−	0.473160i
$202$			12.6491i			0.889988i
$203$		0			0
$204$		0			0
$205$	−5.00000	+	8.66025i	−0.349215	+	0.604858i
$206$	−19.3649	+	11.1803i	−1.34922	+	0.778971i
$207$	−4.94975	−	8.57321i	−0.344031	−	0.595880i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	−13.0000	−	22.5167i	−0.888662	−	1.53921i
$215$	−24.6475	−	14.2302i	−1.68095	−	0.970495i
$216$		−	35.7771i		−	2.43432i
$217$		0			0
$218$	−22.6274			−1.53252
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	3.16228i		−	0.211762i	−0.994379	−	0.105881i	$-0.966234\pi$
$223$		−	3.16228i		−	0.211762i	0.994379	−	0.105881i	$-0.0337662\pi$
$224$		0			0
$225$	35.0000			2.33333
$226$		0			0
$227$	−24.6475	+	14.2302i	−1.63591	+	0.944495i	−0.653693	+	0.756760i	$0.726781\pi$
$227$	−24.6475	+	14.2302i	−1.63591	+	0.944495i	−0.982220	+	0.187735i	$0.939885\pi$
$228$		0			0
$229$	−23.2379	−	13.4164i	−1.53560	−	0.886581i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−23.2379	−	13.4164i	−1.53560	−	0.886581i	−0.536515	−	0.843891i	$-0.680260\pi$
$230$			4.47214i			0.294884i
$231$		0			0
$232$	16.9706			1.11417
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	−10.6066	−	18.3712i	−0.691898	−	1.19840i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−14.1421	+	24.4949i	−0.912871	+	1.58114i
$241$	11.6190	−	6.70820i	0.748442	−	0.432113i	−0.0766885	−	0.997055i	$-0.524435\pi$
$241$	11.6190	−	6.70820i	0.748442	−	0.432113i	0.825131	+	0.564942i	$0.191101\pi$
$242$	−7.77817	−	13.4722i	−0.500000	−	0.866025i
$243$	−19.1703	−	11.0680i	−1.22977	−	0.710011i
$244$		−	26.8328i		−	1.71780i
$245$		0			0
$246$	−20.0000			−1.27515
$247$		0			0
$248$		0			0
$249$	−15.0000	−	25.9808i	−0.950586	−	1.64646i
$250$	−13.6931	−	7.90569i	−0.866025	−	0.500000i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	3.00000	−	5.19615i	0.188237	−	0.326036i
$255$		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	$0.166667\pi$
$258$		−	56.9210i		−	3.54375i
$259$		0			0
$260$		0			0
$261$	21.0000	−	36.3731i	1.29987	−	2.25144i
$262$		0			0
$263$	7.77817	+	13.4722i	0.479623	+	0.830731i	0.999727	−	0.0233719i	$-0.00744017\pi$
$263$	7.77817	+	13.4722i	0.479623	+	0.830731i	−0.520104	+	0.854103i	$0.674107\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−56.5685			−3.46194
$268$	−4.24264	+	7.34847i	−0.259161	+	0.448879i
$269$	−19.3649	+	11.1803i	−1.18070	+	0.681677i	−0.956176	−	0.292791i	$-0.905416\pi$
$269$	−19.3649	+	11.1803i	−1.18070	+	0.681677i	−0.224523	+	0.974469i	$0.572083\pi$
$270$	20.0000	+	34.6410i	1.21716	+	2.10819i
$271$		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
$275$		0			0
$276$	−7.74597	+	4.47214i	−0.466252	+	0.269191i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$	21.2132	−	36.7423i	1.26323	−	2.18797i
$283$	13.6931	−	7.90569i	0.813968	−	0.469945i	−0.0343638	−	0.999409i	$-0.510941\pi$
$283$	13.6931	−	7.90569i	0.813968	−	0.469945i	0.848332	+	0.529465i	$0.177607\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−39.5980			−2.33333
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$	−16.4317	+	9.48683i	−0.964901	+	0.557086i
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−16.9706	−	29.3939i	−0.983078	−	1.70274i
$299$		0			0
$300$		−	31.6228i		−	1.82574i
$301$		0			0
$302$		0			0
$303$	14.1421	−	24.4949i	0.812444	−	1.40720i
$304$		0			0
$305$	15.0000	+	25.9808i	0.858898	+	1.48765i
$306$		0			0
$307$			34.7851i			1.98529i	0.121070	+	0.992644i	$0.461367\pi$
$307$			34.7851i			1.98529i	−0.121070	+	0.992644i	$0.538633\pi$
$308$		0			0
$309$	50.0000			2.84440
$310$		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	$0.166667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	15.4919	+	8.94427i	0.866025	+	0.500000i
$321$			58.1378i			3.24493i
$322$		0			0
$323$		0			0
$324$	−19.0000	+	32.9090i	−1.05556	+	1.82828i
$325$		0			0
$326$	−9.00000	−	15.5885i	−0.498464	−	0.863365i
$327$	43.8178	+	25.2982i	2.42313	+	1.39899i
$328$			12.6491i			0.698430i
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−16.4317	+	9.48683i	−0.901805	+	0.520658i
$333$		0			0
$334$	11.6190	+	6.70820i	0.635761	+	0.367057i
$335$		−	9.48683i		−	0.518321i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	9.19239	−	15.9217i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−36.0000			−1.94099
$345$	5.00000	−	8.66025i	0.269191	−	0.466252i
$346$		0			0
$347$	−12.0208	−	20.8207i	−0.645311	−	1.11771i	−0.984230	−	0.176896i	$-0.943394\pi$
$347$	−12.0208	−	20.8207i	−0.645311	−	1.11771i	0.338918	−	0.940816i	$-0.389939\pi$
$348$	−32.8634	−	18.9737i	−1.76166	−	1.01710i
$349$		−	26.8328i		−	1.43633i	−0.695874	−	0.718164i	$-0.744983\pi$
$349$		−	26.8328i		−	1.43633i	0.695874	−	0.718164i	$-0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		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
$356$			35.7771i			1.89618i
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	38.3406	−	22.1359i	2.02073	−	1.16667i
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$	32.8634	+	18.9737i	1.72726	+	0.997234i
$363$			34.7851i			1.82574i
$364$		0			0
$365$		0			0
$366$	−30.0000	+	51.9615i	−1.56813	+	2.71607i
$367$	2.73861	−	1.58114i	0.142954	−	0.0825348i	−0.426817	−	0.904338i	$-0.640365\pi$
$367$	2.73861	−	1.58114i	0.142954	−	0.0825348i	0.569771	+	0.821803i	$0.307032\pi$
$368$	2.82843	+	4.89898i	0.147442	+	0.255377i
$369$	27.1109	+	15.6525i	1.41134	+	0.814835i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	17.6777	+	30.6186i	0.912871	+	1.58114i
$376$	−23.2379	−	13.4164i	−1.19840	−	0.691898i
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−11.6190	+	6.70820i	−0.595257	+	0.343672i
$382$		0			0
$383$	−24.6475	−	14.2302i	−1.25943	−	0.727132i	−0.286466	−	0.958091i	$-0.592480\pi$
$383$	−24.6475	−	14.2302i	−1.25943	−	0.727132i	−0.972964	+	0.230959i	$0.925814\pi$
$384$			35.7771i			1.82574i
$385$		0			0
$386$		0			0
$387$	−44.5477	+	77.1589i	−2.26449	+	3.92221i
$388$		0			0
$389$	12.0000	+	20.7846i	0.608424	+	1.05382i	0.991500	+	0.130105i	$0.0415314\pi$
$389$	12.0000	+	20.7846i	0.608424	+	1.05382i	−0.383076	+	0.923717i	$0.625135\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	−0.548911	−	0.835881i	$-0.684957\pi$
$401$	9.00000	−	15.5885i	0.449439	−	0.778450i	0.998350	+	0.0574304i	$0.0182907\pi$
$402$	16.4317	−	9.48683i	0.819538	−	0.473160i
$403$		0			0
$404$	−15.4919	−	8.94427i	−0.770752	−	0.444994i
$405$		−	42.4853i		−	2.11111i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	34.8569	−	20.1246i	1.72356	−	0.995098i	0.812333	−	0.583193i	$-0.198197\pi$
$409$	34.8569	−	20.1246i	1.72356	−	0.995098i	0.911227	−	0.411905i	$-0.135136\pi$
$410$	−7.07107	−	12.2474i	−0.349215	−	0.604858i
$411$		0			0
$412$		−	31.6228i		−	1.55794i
$413$		0			0
$414$	14.0000			0.688062
$415$	10.6066	−	18.3712i	0.520658	−	0.901805i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	−57.5109	+	33.2039i	−2.79627	+	1.61443i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	36.7696			1.77732
$429$		0			0
$430$	34.8569	−	20.1246i	1.68095	−	0.970495i
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	43.8178	+	25.2982i	2.10819	+	1.21716i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	42.4264			2.03419
$436$	16.0000	−	27.7128i	0.766261	−	1.32720i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.5061	−	35.5176i	0.974274	−	1.68749i	0.291964	−	0.956429i	$-0.405691\pi$
$443$	20.5061	−	35.5176i	0.974274	−	1.68749i	0.682310	−	0.731063i	$-0.260975\pi$
$444$		0			0
$445$	−20.0000	−	34.6410i	−0.948091	−	1.64214i
$446$	3.87298	+	2.23607i	0.183391	+	0.105881i
$447$			75.8947i			3.58969i
$448$		0			0
$449$	−36.0000			−1.69895			−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000			−1.69895			−0.849473	+	0.527633i	$0.823080\pi$
$450$	−24.7487	+	42.8661i	−1.16667	+	2.02073i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	40.2492i		−	1.88899i
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	32.8634	−	18.9737i	1.53560	−	0.886581i
$459$		0			0
$460$	−5.47723	−	3.16228i	−0.255377	−	0.147442i
$461$		−	8.94427i		−	0.416576i	−0.978068	−	0.208288i	$-0.933211\pi$
$461$		−	8.94427i		−	0.416576i	0.978068	−	0.208288i	$-0.0667892\pi$
$462$		0			0
$463$	−12.7279			−0.591517			−0.295758	−	0.955263i	$-0.595572\pi$
$463$	−12.7279			−0.591517			−0.295758	+	0.955263i	$0.595572\pi$
$464$	−12.0000	+	20.7846i	−0.557086	+	0.964901i
$465$		0			0
$466$		0			0
$467$	−24.6475	−	14.2302i	−1.14055	−	0.658497i	−0.193984	−	0.981005i	$-0.562141\pi$
$467$	−24.6475	−	14.2302i	−1.14055	−	0.658497i	−0.946567	+	0.322507i	$0.895474\pi$
$468$		0			0
$469$		0			0
$470$	30.0000			1.38380
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	−20.0000	−	34.6410i	−0.912871	−	1.58114i
$481$		0			0
$482$			18.9737i			0.864227i
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$	27.1109	−	15.6525i	1.22977	−	0.710011i
$487$	19.0919	+	33.0681i	0.865136	+	1.49846i	0.866912	+	0.498461i	$0.166101\pi$
$487$	19.0919	+	33.0681i	0.865136	+	1.49846i	−0.00177647	+	0.999998i	$0.500565\pi$
$488$	32.8634	+	18.9737i	1.48765	+	0.858898i
$489$			40.2492i			1.82013i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	14.1421	−	24.4949i	0.637577	−	1.10432i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	42.4264			1.90117
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	19.3649	−	11.1803i	0.866025	−	0.500000i
$501$	−15.0000	−	25.9808i	−0.670151	−	1.16073i
$502$		0			0
$503$		−	9.48683i		−	0.422997i	−0.977378	−	0.211498i	$-0.932166\pi$
$503$		−	9.48683i		−	0.422997i	0.977378	−	0.211498i	$-0.0678343\pi$
$504$		0			0
$505$	20.0000			0.889988
$506$		0			0
$507$	−35.6020	+	20.5548i	−1.58114	+	0.912871i
$508$	4.24264	+	7.34847i	0.188237	+	0.326036i
$509$	38.7298	+	22.3607i	1.71667	+	0.991120i	0.924821	+	0.380402i	$0.124214\pi$
$509$	38.7298	+	22.3607i	1.71667	+	0.991120i	0.791849	+	0.610718i	$0.209119\pi$
$510$		0			0
$511$		0			0
$512$	22.6274			1.00000
$513$		0			0
$514$		0			0
$515$	17.6777	+	30.6186i	0.778971	+	1.34922i
$516$	69.7137	+	40.2492i	3.06897	+	1.77187i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.4919	+	8.94427i	−0.678714	+	0.391856i	−0.799370	−	0.600839i	$-0.794833\pi$
$521$	−15.4919	+	8.94427i	−0.678714	+	0.391856i	0.120656	+	0.992694i	$0.461500\pi$
$522$	29.6985	+	51.4393i	1.29987	+	2.25144i
$523$	30.1247	+	17.3925i	1.31726	+	0.760522i	0.983287	−	0.182060i	$-0.0582764\pi$
$523$	30.1247	+	17.3925i	1.31726	+	0.760522i	0.333975	+	0.942582i	$0.391610\pi$
$524$		0			0
$525$		0			0
$526$	−22.0000			−0.959246
$527$		0			0
$528$		0			0
$529$	10.5000	+	18.1865i	0.456522	+	0.790719i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	40.0000	−	69.2820i	1.73097	−	2.99813i
$535$	−35.6020	+	20.5548i	−1.53921	+	0.888662i
$536$	−6.00000	−	10.3923i	−0.259161	−	0.448879i
$537$		0			0
$538$		−	31.6228i		−	1.36335i
$539$		0			0
$540$	−56.5685			−2.43432
$541$	−19.0000	+	32.9090i	−0.816874	+	1.41487i	0.0911008	+	0.995842i	$0.470961\pi$
$541$	−19.0000	+	32.9090i	−0.816874	+	1.41487i	−0.907975	+	0.419025i	$0.862372\pi$
$542$		0			0
$543$	−42.4264	−	73.4847i	−1.82069	−	3.15353i
$544$		0			0
$545$			35.7771i			1.53252i
$546$		0			0
$547$	46.6690			1.99542			0.997712	−	0.0676046i	$-0.0215356\pi$
$547$	46.6690			1.99542			0.997712	+	0.0676046i	$0.0215356\pi$
$548$		0			0
$549$	81.3327	−	46.9574i	3.47119	−	2.00409i
$550$		0			0
$551$		0			0
$552$		−	12.6491i		−	0.538382i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	8.48528	−	14.6969i	0.357930	−	0.619953i
$563$	41.0792	−	23.7171i	1.73128	−	0.999556i	0.850740	−	0.525586i	$-0.176154\pi$
$563$	41.0792	−	23.7171i	1.73128	−	0.999556i	0.880541	−	0.473970i	$-0.157179\pi$
$564$	30.0000	+	51.9615i	1.26323	+	2.18797i
$565$		0			0
$566$			22.3607i			0.939889i
$567$		0			0
$568$		0			0
$569$	−18.0000	+	31.1769i	−0.754599	+	1.30700i	0.190974	+	0.981595i	$0.438835\pi$
$569$	−18.0000	+	31.1769i	−0.754599	+	1.30700i	−0.945573	+	0.325409i	$0.894498\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.07107			0.294884
$576$	28.0000	−	48.4974i	1.16667	−	2.02073i
$577$		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
$580$		−	26.8328i		−	1.11417i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	47.4342i		−	1.95782i	−0.204298	−	0.978909i	$-0.565491\pi$
$587$		−	47.4342i		−	1.95782i	0.204298	−	0.978909i	$-0.434509\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		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
$596$	48.0000			1.96616
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	38.7298	+	22.3607i	1.58114	+	0.912871i
$601$		−	40.2492i		−	1.64180i	−0.571072	−	0.820900i	$-0.693472\pi$
$601$		−	40.2492i		−	1.64180i	0.571072	−	0.820900i	$-0.306528\pi$
$602$		0			0
$603$	−29.6985			−1.20942
$604$		0			0
$605$	−21.3014	+	12.2984i	−0.866025	+	0.500000i
$606$	20.0000	+	34.6410i	0.812444	+	1.40720i
$607$	13.6931	+	7.90569i	0.555784	+	0.320882i	0.751452	−	0.659788i	$-0.229354\pi$
$607$	13.6931	+	7.90569i	0.555784	+	0.320882i	−0.195667	+	0.980670i	$0.562687\pi$
$608$		0			0
$609$		0			0
$610$	−42.4264			−1.71780
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	−42.6028	−	24.5967i	−1.71931	−	0.992644i
$615$			31.6228i			1.27515i
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−35.3553	+	61.2372i	−1.42220	+	2.46332i
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$	−15.4919	−	8.94427i	−0.621670	−	0.358921i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.21584	−	4.74342i	−0.326036	−	0.188237i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−21.9089	+	12.6491i	−0.866025	+	0.500000i
$641$	−6.00000	−	10.3923i	−0.236986	−	0.410471i	0.722862	−	0.690992i	$-0.242826\pi$
$641$	−6.00000	−	10.3923i	−0.236986	−	0.410471i	−0.959848	+	0.280521i	$0.909493\pi$
$642$	−71.2039	−	41.1096i	−2.81020	−	1.62247i
$643$			41.1096i			1.62120i	0.585597	+	0.810602i	$0.300860\pi$
$643$			41.1096i			1.62120i	−0.585597	+	0.810602i	$0.699140\pi$
$644$		0			0
$645$	−90.0000			−3.54375
$646$		0			0
$647$	41.0792	−	23.7171i	1.61499	−	0.932415i	0.626800	−	0.779180i	$-0.284364\pi$
$647$	41.0792	−	23.7171i	1.61499	−	0.932415i	0.988190	−	0.153234i	$-0.0489689\pi$
$648$	−26.8701	−	46.5403i	−1.05556	−	1.82828i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	25.4558			0.996928
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−61.9677	+	35.7771i	−2.42313	+	1.39899i
$655$		0			0
$656$	−15.4919	−	8.94427i	−0.604858	−	0.349215i
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	34.8569	−	20.1246i	1.35577	−	0.782757i	0.366723	−	0.930330i	$-0.380480\pi$
$661$	34.8569	−	20.1246i	1.35577	−	0.782757i	0.989051	+	0.147573i	$0.0471463\pi$
$662$		0			0
$663$		0			0
$664$		−	26.8328i		−	1.04132i
$665$		0			0
$666$		0			0
$667$	4.24264	−	7.34847i	0.164276	−	0.284534i
$668$	−16.4317	+	9.48683i	−0.635761	+	0.367057i
$669$	−5.00000	−	8.66025i	−0.193311	−	0.334825i
$670$	11.6190	+	6.70820i	0.448879	+	0.259161i
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	54.7723	−	31.6228i	2.10819	−	1.21716i
$676$	13.0000	+	22.5167i	0.500000	+	0.866025i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−45.0000	+	77.9423i	−1.72440	+	2.98675i
$682$		0			0
$683$	−21.9203	−	37.9671i	−0.838757	−	1.45277i	−0.890934	−	0.454132i	$-0.849949\pi$
$683$	−21.9203	−	37.9671i	−0.838757	−	1.45277i	0.0521768	−	0.998638i	$-0.483384\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−84.8528			−3.23734
$688$	25.4558	−	44.0908i	0.970495	−	1.68095i
$689$		0			0
$690$	7.07107	+	12.2474i	0.269191	+	0.466252i
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$		0			0
$694$	34.0000			1.29062
$695$		0			0
$696$	46.4758	−	26.8328i	1.76166	−	1.01710i
$697$		0			0
$698$	32.8634	+	18.9737i	1.24390	+	0.718164i
$699$		0			0
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−58.0948	−	33.5410i	−2.18797	−	1.26323i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	−0.00446726	−	0.999990i	$-0.501422\pi$
$709$	23.0000	−	39.8372i	0.863783	−	1.49612i	0.868250	−	0.496126i	$-0.165245\pi$
$710$		0			0
$711$		0			0
$712$	−43.8178	−	25.2982i	−1.64214	−	0.948091i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$			62.6099i			2.33333i
$721$		0			0
$722$	−26.8701			−1.00000
$723$	21.2132	−	36.7423i	0.788928	−	1.36646i
$724$	−46.4758	+	26.8328i	−1.72726	+	0.997234i
$725$	15.0000	+	25.9808i	0.557086	+	0.964901i
$726$	−42.6028	−	24.5967i	−1.58114	−	0.912871i
$727$		−	53.7587i		−	1.99380i	−0.0786754	−	0.996900i	$-0.525069\pi$
$727$		−	53.7587i		−	1.99380i	0.0786754	−	0.996900i	$-0.474931\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$	−42.4264	−	73.4847i	−1.56813	−	2.71607i
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$			4.47214i			0.165070i
$735$		0			0
$736$	−8.00000			−0.294884
$737$		0			0
$738$	−38.3406	+	22.1359i	−1.41134	+	0.814835i
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.8701			0.985767			0.492883	−	0.870095i	$-0.335943\pi$
$743$	26.8701			0.985767			0.492883	+	0.870095i	$0.335943\pi$
$744$		0			0
$745$	−46.4758	+	26.8328i	−1.70274	+	0.983078i
$746$		0			0
$747$	−57.5109	−	33.2039i	−2.10421	−	1.21487i
$748$		0			0
$749$		0			0
$750$	−50.0000			−1.82574
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	32.8634	−	18.9737i	1.19840	−	0.691898i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	30.9839	+	17.8885i	1.12316	+	0.648459i	0.942207	−	0.335032i	$-0.108747\pi$
$761$	30.9839	+	17.8885i	1.12316	+	0.648459i	0.180957	+	0.983491i	$0.442080\pi$
$762$		−	18.9737i		−	0.687343i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	34.8569	−	20.1246i	1.25943	−	0.727132i
$767$		0			0
$768$	−43.8178	−	25.2982i	−1.58114	−	0.912871i
$769$			53.6656i			1.93523i	0.252426	+	0.967616i	$0.418771\pi$
$769$			53.6656i			1.93523i	−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	−63.0000	−	109.119i	−2.26449	−	3.92221i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−33.9411			−1.21685
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	75.8947i		−	2.71225i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−35.6020	+	20.5548i	−1.26907	+	0.732700i	−0.974813	−	0.223026i	$-0.928407\pi$
$787$	−35.6020	+	20.5548i	−1.26907	+	0.732700i	−0.294260	+	0.955725i	$0.595073\pi$
$788$		0			0
$789$	42.6028	+	24.5967i	1.51670	+	0.875667i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	14.1421	−	24.4949i	0.500000	−	0.866025i
$801$	−108.444	+	62.6099i	−3.83166	+	2.21221i
$802$	12.7279	+	22.0454i	0.449439	+	0.778450i
$803$		0			0
$804$			26.8328i			0.946320i
$805$		0			0
$806$		0			0
$807$	−35.3553	+	61.2372i	−1.24457	+	2.15565i
$808$	21.9089	−	12.6491i	0.770752	−	0.444994i
$809$	−27.0000	−	46.7654i	−0.949269	−	1.64418i	−0.746968	−	0.664860i	$-0.768491\pi$
$809$	−27.0000	−	46.7654i	−0.949269	−	1.64418i	−0.202301	−	0.979323i	$-0.564842\pi$
$810$	52.0336	+	30.0416i	1.82828	+	1.05556i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−24.6475	+	14.2302i	−0.863365	+	0.498464i
$816$		0			0
$817$		0			0
$818$			56.9210i			1.99020i
$819$		0			0
$820$	20.0000			0.698430
$821$	24.0000	−	41.5692i	0.837606	−	1.45078i	−0.0542853	−	0.998525i	$-0.517288\pi$
$821$	24.0000	−	41.5692i	0.837606	−	1.45078i	0.891891	−	0.452250i	$-0.149379\pi$
$822$		0			0
$823$	27.5772	+	47.7650i	0.961280	+	1.66498i	0.719295	+	0.694705i	$0.244465\pi$
$823$	27.5772	+	47.7650i	0.961280	+	1.66498i	0.241985	+	0.970280i	$0.422202\pi$
$824$	38.7298	+	22.3607i	1.34922	+	0.778971i
$825$		0			0
$826$		0			0
$827$	−32.5269			−1.13107			−0.565536	−	0.824724i	$-0.691331\pi$
$827$	−32.5269			−1.13107			−0.565536	+	0.824724i	$0.691331\pi$
$828$	−9.89949	+	17.1464i	−0.344031	+	0.595880i
$829$	11.6190	−	6.70820i	0.403543	−	0.232986i	−0.284469	−	0.958685i	$-0.591817\pi$
$829$	11.6190	−	6.70820i	0.403543	−	0.232986i	0.688012	+	0.725700i	$0.258484\pi$
$830$	15.0000	+	25.9808i	0.520658	+	0.901805i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	10.6066	−	18.3712i	0.367057	−	0.635761i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	5.65685	−	9.79796i	0.194948	−	0.337660i
$843$	−32.8634	+	18.9737i	−1.13187	+	0.653488i
$844$		0			0
$845$	−25.1744	−	14.5344i	−0.866025	−	0.500000i
$846$		−	93.9149i		−	3.22886i
$847$		0			0
$848$		0			0
$849$	25.0000	−	43.3013i	0.857998	−	1.48610i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−26.0000	+	45.0333i	−0.888662	+	1.53921i
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$			56.9210i			1.94099i
$861$		0			0
$862$		0			0
$863$	−28.9914	+	50.2145i	−0.986878	+	1.70932i	−0.353604	+	0.935395i	$0.615044\pi$
$863$	−28.9914	+	50.2145i	−0.986878	+	1.70932i	−0.633274	+	0.773928i	$0.718289\pi$
$864$	−61.9677	+	35.7771i	−2.10819	+	1.21716i
$865$		0			0
$866$		0			0
$867$		−	53.7587i		−	1.82574i
$868$		0			0
$869$		0			0
$870$	−30.0000	+	51.9615i	−1.01710	+	1.76166i
$871$		0			0
$872$	22.6274	+	39.1918i	0.766261	+	1.32720i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	58.1378i		−	1.95871i	−0.202145	−	0.979356i	$-0.564791\pi$
$881$		−	58.1378i		−	1.95871i	0.202145	−	0.979356i	$-0.435209\pi$
$882$		0			0
$883$	55.1543			1.85609			0.928045	−	0.372467i	$-0.121488\pi$
$883$	55.1543			1.85609			0.928045	+	0.372467i	$0.121488\pi$
$884$		0			0
$885$		0			0
$886$	29.0000	+	50.2295i	0.974274	+	1.68749i
$887$	−24.6475	−	14.2302i	−0.827583	−	0.477805i	0.0254417	−	0.999676i	$-0.491901\pi$
$887$	−24.6475	−	14.2302i	−0.827583	−	0.477805i	−0.853024	+	0.521871i	$0.825234\pi$
$888$		0			0
$889$		0			0
$890$	56.5685			1.89618
$891$		0			0
$892$	−5.47723	+	3.16228i	−0.183391	+	0.105881i
$893$		0			0
$894$	−92.9516	−	53.6656i	−3.10877	−	1.79485i
$895$		0			0
$896$		0			0
$897$		0			0
$898$	25.4558	−	44.0908i	0.849473	−	1.47133i
$899$		0			0
$900$	−35.0000	−	60.6218i	−1.16667	−	2.02073i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	30.0000	−	51.9615i	0.997234	−	1.72726i
$906$		0			0
$907$	−2.12132	−	3.67423i	−0.0704373	−	0.122001i	0.828656	−	0.559759i	$-0.189106\pi$
$907$	−2.12132	−	3.67423i	−0.0704373	−	0.122001i	−0.899093	+	0.437758i	$0.855773\pi$
$908$	49.2950	+	28.4605i	1.63591	+	0.944495i
$909$		−	62.6099i		−	2.07664i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	82.1584	+	47.4342i	2.71607	+	1.56813i
$916$			53.6656i			1.77316i
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$	7.74597	−	4.47214i	0.255377	−	0.147442i
$921$	55.0000	+	95.2628i	1.81231	+	3.13902i
$922$	10.9545	+	6.32456i	0.360766	+	0.208288i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	9.00000	−	15.5885i	0.295758	−	0.512268i
$927$	95.8514	−	55.3399i	3.14817	−	1.81760i
$928$	−16.9706	−	29.3939i	−0.557086	−	0.964901i
$929$	−42.6028	−	24.5967i	−1.39775	−	0.806993i	−0.403596	−	0.914937i	$-0.632240\pi$
$929$	−42.6028	−	24.5967i	−1.39775	−	0.806993i	−0.994157	+	0.107944i	$0.965573\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	34.8569	−	20.1246i	1.14055	−	0.658497i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	−21.2132	+	36.7423i	−0.691898	+	1.19840i
$941$	38.7298	−	22.3607i	1.26256	−	0.728937i	0.288988	−	0.957333i	$-0.406681\pi$
$941$	38.7298	−	22.3607i	1.26256	−	0.728937i	0.973568	+	0.228395i	$0.0733479\pi$
$942$		0			0
$943$	5.47723	+	3.16228i	0.178363	+	0.102978i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	30.4056	−	52.6640i	0.988049	−	1.71135i	0.360535	−	0.932746i	$-0.382594\pi$
$947$	30.4056	−	52.6640i	0.988049	−	1.71135i	0.627514	−	0.778605i	$-0.284073\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	56.5685			1.82574
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$	64.3467	+	111.452i	2.07354	+	3.59148i
$964$	−23.2379	−	13.4164i	−0.748442	−	0.432113i
$965$		0			0
$966$		0			0
$967$	46.6690			1.50078			0.750388	−	0.660998i	$-0.229867\pi$
$967$	46.6690			1.50078			0.750388	+	0.660998i	$0.229867\pi$
$968$	−15.5563	+	26.9444i	−0.500000	+	0.866025i
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$			44.2719i			1.42002i
$973$		0			0
$974$	−54.0000			−1.73027
$975$		0			0
$976$	−46.4758	+	26.8328i	−1.48765	+	0.858898i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	−49.2950	−	28.4605i	−1.57628	−	0.910066i
$979$		0			0
$980$		0			0
$981$	112.000			3.57588
$982$		0			0
$983$	41.0792	−	23.7171i	1.31022	−	0.756457i	0.328090	−	0.944646i	$-0.393595\pi$
$983$	41.0792	−	23.7171i	1.31022	−	0.756457i	0.982133	+	0.188189i	$0.0602618\pi$
$984$	20.0000	+	34.6410i	0.637577	+	1.10432i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−9.00000	+	15.5885i	−0.286183	+	0.495684i
$990$		0			0
$991$		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
$996$	−30.0000	+	51.9615i	−0.950586	+	1.64646i
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.s.b.19.2		8
4.3	odd	2	inner	980.2.s.b.19.3		8
5.4	even	2	inner	980.2.s.b.19.3		8
7.2	even	3		140.2.c.a.139.4	yes	4
7.3	odd	6	inner	980.2.s.b.619.2		8
7.4	even	3	inner	980.2.s.b.619.1		8
7.5	odd	6		140.2.c.a.139.3	yes	4
7.6	odd	2	inner	980.2.s.b.19.1		8
20.19	odd	2	CM	980.2.s.b.19.2		8
28.3	even	6	inner	980.2.s.b.619.3		8
28.11	odd	6	inner	980.2.s.b.619.4		8
28.19	even	6		140.2.c.a.139.2	yes	4
28.23	odd	6		140.2.c.a.139.1	✓	4
28.27	even	2	inner	980.2.s.b.19.4		8
35.2	odd	12		700.2.g.d.251.4		4
35.4	even	6	inner	980.2.s.b.619.4		8
35.9	even	6		140.2.c.a.139.1	✓	4
35.12	even	12		700.2.g.d.251.3		4
35.19	odd	6		140.2.c.a.139.2	yes	4
35.23	odd	12		700.2.g.d.251.1		4
35.24	odd	6	inner	980.2.s.b.619.3		8
35.33	even	12		700.2.g.d.251.2		4
35.34	odd	2	inner	980.2.s.b.19.4		8
56.5	odd	6		2240.2.e.a.2239.3		4
56.19	even	6		2240.2.e.a.2239.1		4
56.37	even	6		2240.2.e.a.2239.2		4
56.51	odd	6		2240.2.e.a.2239.4		4
140.19	even	6		140.2.c.a.139.3	yes	4
140.23	even	12		700.2.g.d.251.4		4
140.39	odd	6	inner	980.2.s.b.619.1		8
140.47	odd	12		700.2.g.d.251.2		4
140.59	even	6	inner	980.2.s.b.619.2		8
140.79	odd	6		140.2.c.a.139.4	yes	4
140.103	odd	12		700.2.g.d.251.3		4
140.107	even	12		700.2.g.d.251.1		4
140.139	even	2	inner	980.2.s.b.19.1		8
280.19	even	6		2240.2.e.a.2239.3		4
280.149	even	6		2240.2.e.a.2239.4		4
280.219	odd	6		2240.2.e.a.2239.2		4
280.229	odd	6		2240.2.e.a.2239.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.c.a.139.1	✓	4	28.23	odd	6
140.2.c.a.139.1	✓	4	35.9	even	6
140.2.c.a.139.2	yes	4	28.19	even	6
140.2.c.a.139.2	yes	4	35.19	odd	6
140.2.c.a.139.3	yes	4	7.5	odd	6
140.2.c.a.139.3	yes	4	140.19	even	6
140.2.c.a.139.4	yes	4	7.2	even	3
140.2.c.a.139.4	yes	4	140.79	odd	6
700.2.g.d.251.1		4	35.23	odd	12
700.2.g.d.251.1		4	140.107	even	12
700.2.g.d.251.2		4	35.33	even	12
700.2.g.d.251.2		4	140.47	odd	12
700.2.g.d.251.3		4	35.12	even	12
700.2.g.d.251.3		4	140.103	odd	12
700.2.g.d.251.4		4	35.2	odd	12
700.2.g.d.251.4		4	140.23	even	12
980.2.s.b.19.1		8	7.6	odd	2	inner
980.2.s.b.19.1		8	140.139	even	2	inner
980.2.s.b.19.2		8	1.1	even	1	trivial
980.2.s.b.19.2		8	20.19	odd	2	CM
980.2.s.b.19.3		8	4.3	odd	2	inner
980.2.s.b.19.3		8	5.4	even	2	inner
980.2.s.b.19.4		8	28.27	even	2	inner
980.2.s.b.19.4		8	35.34	odd	2	inner
980.2.s.b.619.1		8	7.4	even	3	inner
980.2.s.b.619.1		8	140.39	odd	6	inner
980.2.s.b.619.2		8	7.3	odd	6	inner
980.2.s.b.619.2		8	140.59	even	6	inner
980.2.s.b.619.3		8	28.3	even	6	inner
980.2.s.b.619.3		8	35.24	odd	6	inner
980.2.s.b.619.4		8	28.11	odd	6	inner
980.2.s.b.619.4		8	35.4	even	6	inner
2240.2.e.a.2239.1		4	56.19	even	6
2240.2.e.a.2239.1		4	280.229	odd	6
2240.2.e.a.2239.2		4	56.37	even	6
2240.2.e.a.2239.2		4	280.219	odd	6
2240.2.e.a.2239.3		4	56.5	odd	6
2240.2.e.a.2239.3		4	280.19	even	6
2240.2.e.a.2239.4		4	56.51	odd	6
2240.2.e.a.2239.4		4	280.149	even	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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(2.73861 - 1.58114i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(1.93649 + 1.11803i) q^{5} +4.47214i q^{6} +2.82843 q^{8} +(3.50000 - 6.06218i) q^{9} +(-2.73861 + 1.58114i) q^{10} +(-5.47723 - 3.16228i) q^{12} +7.07107 q^{15} +(-2.00000 + 3.46410i) q^{16} +(4.94975 + 8.57321i) q^{18} -4.47214i q^{20} +(0.707107 - 1.22474i) q^{23} +(7.74597 - 4.47214i) q^{24} +(2.50000 + 4.33013i) q^{25} -12.6491i q^{27} +6.00000 q^{29} +(-5.00000 + 8.66025i) q^{30} +(-2.82843 - 4.89898i) q^{32} -14.0000 q^{36} +(5.47723 + 3.16228i) q^{40} +4.47214i q^{41} -12.7279 q^{43} +(13.5554 - 7.82624i) q^{45} +(1.00000 + 1.73205i) q^{46} +(-8.21584 - 4.74342i) q^{47} +12.6491i q^{48} -7.07107 q^{50} +(15.4919 + 8.94427i) q^{54} +(-4.24264 + 7.34847i) q^{58} +(-7.07107 - 12.2474i) q^{60} +(11.6190 + 6.70820i) q^{61} +8.00000 q^{64} +(-2.12132 - 3.67423i) q^{67} -4.47214i q^{69} +(9.89949 - 17.1464i) q^{72} +(13.6931 + 7.90569i) q^{75} +(-7.74597 + 4.47214i) q^{80} +(-9.50000 - 16.4545i) q^{81} +(-5.47723 - 3.16228i) q^{82} -9.48683i q^{83} +(9.00000 - 15.5885i) q^{86} +(16.4317 - 9.48683i) q^{87} +(-15.4919 - 8.94427i) q^{89} +22.1359i q^{90} -2.82843 q^{92} +(11.6190 - 6.70820i) q^{94} +(-15.4919 - 8.94427i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 28 q^{9} - 16 q^{16} + 20 q^{25} + 48 q^{29} - 40 q^{30} - 112 q^{36} + 8 q^{46} + 64 q^{64} - 76 q^{81} + 72 q^{86}+O(q^{100}) \) 8 * q - 8 * q^4 + 28 * q^9 - 16 * q^16 + 20 * q^25 + 48 * q^29 - 40 * q^30 - 112 * q^36 + 8 * q^46 + 64 * q^64 - 76 * q^81 + 72 * q^86

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