LMFDB - Embedded newform 980.2.x.c.67.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(67,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.67"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 8])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.x (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,2,0,0,-4,0,0,8,0,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			67.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	980.67
Dual form			980.2.x.c.863.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.366025 + 1.36603i) q^{2} +(-1.73205 - 1.00000i) q^{4} +(-1.86603 - 1.23205i) q^{5} +(2.00000 - 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +(2.36603 - 2.09808i) q^{10} +(1.00000 - 1.00000i) q^{13} +(2.00000 + 3.46410i) q^{16} +(4.09808 - 1.09808i) q^{17} +(-1.09808 - 4.09808i) q^{18} +(2.00000 + 4.00000i) q^{20} +(1.96410 + 4.59808i) q^{25} +(1.00000 + 1.73205i) q^{26} +4.00000i q^{29} +(-5.46410 + 1.46410i) q^{32} +6.00000i q^{34} +6.00000 q^{36} +(-2.56218 + 9.56218i) q^{37} +(-6.19615 + 1.26795i) q^{40} +8.00000 q^{41} +(6.69615 + 0.401924i) q^{45} +(-7.00000 + 1.00000i) q^{50} +(-2.73205 + 0.732051i) q^{52} +(3.29423 + 12.2942i) q^{53} +(-5.46410 - 1.46410i) q^{58} +(6.00000 + 10.3923i) q^{61} -8.00000i q^{64} +(-3.09808 + 0.633975i) q^{65} +(-8.19615 - 2.19615i) q^{68} +(-2.19615 + 8.19615i) q^{72} +(4.02628 + 15.0263i) q^{73} +(-12.1244 - 7.00000i) q^{74} +(0.535898 - 8.92820i) q^{80} +(4.50000 - 7.79423i) q^{81} +(-2.92820 + 10.9282i) q^{82} +(-9.00000 - 3.00000i) q^{85} +(13.8564 - 8.00000i) q^{89} +(-3.00000 + 9.00000i) q^{90} +(-13.0000 - 13.0000i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 6 q^{10} + 4 q^{13} + 8 q^{16} + 6 q^{17} + 6 q^{18} + 8 q^{20} - 6 q^{25} + 4 q^{26} - 8 q^{32} + 24 q^{36} + 14 q^{37} - 4 q^{40} + 32 q^{41} + 6 q^{45} - 28 q^{50}+ \cdots - 52 q^{97}+O(q^{100}) $ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 + 6 * q^10 + 4 * q^13 + 8 * q^16 + 6 * q^17 + 6 * q^18 + 8 * q^20 - 6 * q^25 + 4 * q^26 - 8 * q^32 + 24 * q^36 + 14 * q^37 - 4 * q^40 + 32 * q^41 + 6 * q^45 - 28 * q^50 - 4 * q^52 - 18 * q^53 - 8 * q^58 + 24 * q^61 - 2 * q^65 - 12 * q^68 + 12 * q^72 - 22 * q^73 + 16 * q^80 + 18 * q^81 + 16 * q^82 - 36 * q^85 - 12 * q^90 - 52 * q^97

Character values

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

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.366025	+	1.36603i	−0.258819	+	0.965926i
$2$	−0.366025	+	1.36603i	−0.258819	+	0.965926i
$3$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$3$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$4$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$5$	−1.86603	−	1.23205i	−0.834512	−	0.550990i
$5$	−1.86603	−	1.23205i	−0.834512	−	0.550990i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	−2.59808	+	1.50000i	−0.866025	+	0.500000i
$10$	2.36603	−	2.09808i	0.748203	−	0.663470i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	4.09808	−	1.09808i	0.993929	−	0.266323i	0.275029	−	0.961436i	$-0.411312\pi$
$17$	4.09808	−	1.09808i	0.993929	−	0.266323i	0.718900	+	0.695113i	$0.244646\pi$
$18$	−1.09808	−	4.09808i	−0.258819	−	0.965926i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	2.00000	+	4.00000i	0.447214	+	0.894427i
$21$		0			0
$22$		0			0
$23$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$23$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$24$		0			0
$25$	1.96410	+	4.59808i	0.392820	+	0.919615i
$26$	1.00000	+	1.73205i	0.196116	+	0.339683i
$27$		0			0
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−5.46410	+	1.46410i	−0.965926	+	0.258819i
$33$		0			0
$34$			6.00000i			1.02899i
$35$		0			0
$36$	6.00000			1.00000
$37$	−2.56218	+	9.56218i	−0.421219	+	1.57201i	0.350823	+	0.936442i	$0.385902\pi$
$37$	−2.56218	+	9.56218i	−0.421219	+	1.57201i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$	−6.19615	+	1.26795i	−0.979698	+	0.200480i
$41$	8.00000			1.24939			0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000			1.24939			0.624695	+	0.780869i	$0.285223\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	6.69615	+	0.401924i	0.998203	+	0.0599153i
$46$		0			0
$47$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$47$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$48$		0			0
$49$		0			0
$50$	−7.00000	+	1.00000i	−0.989949	+	0.141421i
$51$		0			0
$52$	−2.73205	+	0.732051i	−0.378867	+	0.101517i
$53$	3.29423	+	12.2942i	0.452497	+	1.68874i	0.695344	+	0.718677i	$0.255252\pi$
$53$	3.29423	+	12.2942i	0.452497	+	1.68874i	−0.242846	+	0.970065i	$0.578081\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−5.46410	−	1.46410i	−0.717472	−	0.192246i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	6.00000	+	10.3923i	0.768221	+	1.33060i	0.938527	+	0.345207i	$0.112191\pi$
$61$	6.00000	+	10.3923i	0.768221	+	1.33060i	−0.170305	+	0.985391i	$0.554475\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	−3.09808	+	0.633975i	−0.384269	+	0.0786349i
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$	−8.19615	−	2.19615i	−0.993929	−	0.266323i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−2.19615	+	8.19615i	−0.258819	+	0.965926i
$73$	4.02628	+	15.0263i	0.471240	+	1.75869i	0.635323	+	0.772246i	$0.280867\pi$
$73$	4.02628	+	15.0263i	0.471240	+	1.75869i	−0.164083	+	0.986447i	$0.552466\pi$
$74$	−12.1244	−	7.00000i	−1.40943	−	0.813733i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$79$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$80$	0.535898	−	8.92820i	0.0599153	−	0.998203i
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$	−2.92820	+	10.9282i	−0.323366	+	1.20682i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	−9.00000	−	3.00000i	−0.976187	−	0.325396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.8564	−	8.00000i	1.46878	−	0.847998i	0.469389	−	0.882992i	$-0.344474\pi$
$89$	13.8564	−	8.00000i	1.46878	−	0.847998i	0.999388	+	0.0349934i	$0.0111410\pi$
$90$	−3.00000	+	9.00000i	−0.316228	+	0.948683i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.0000	−	13.0000i	−1.31995	−	1.31995i	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−13.0000	−	13.0000i	−1.31995	−	1.31995i	−0.406138	−	0.913812i	$-0.633125\pi$
$98$		0			0
$99$		0			0
$100$	1.19615	−	9.92820i	0.119615	−	0.992820i
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$103$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$104$		−	4.00000i		−	0.392232i
$105$		0			0
$106$	−18.0000			−1.74831
$107$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$107$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$108$		0			0
$109$	5.19615	+	3.00000i	0.497701	+	0.287348i	0.727764	−	0.685828i	$-0.240560\pi$
$109$	5.19615	+	3.00000i	0.497701	+	0.287348i	−0.230063	+	0.973176i	$0.573893\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	+	1.00000i	−0.0940721	+	0.0940721i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−1.00000	+	1.00000i	−0.0940721	+	0.0940721i	0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$	4.00000	−	6.92820i	0.371391	−	0.643268i
$117$	−1.09808	+	4.09808i	−0.101517	+	0.378867i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$	−16.3923	+	4.39230i	−1.48409	+	0.397661i
$123$		0			0
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	10.9282	+	2.92820i	0.965926	+	0.258819i
$129$		0			0
$130$	0.267949	−	4.46410i	0.0235007	−	0.391528i
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	6.00000	−	10.3923i	0.514496	−	0.891133i
$137$	9.56218	−	2.56218i	0.816952	−	0.218902i	0.173939	−	0.984757i	$-0.444351\pi$
$137$	9.56218	−	2.56218i	0.816952	−	0.218902i	0.643013	+	0.765855i	$0.277684\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−10.3923	−	6.00000i	−0.866025	−	0.500000i
$145$	4.92820	−	7.46410i	0.409265	−	0.619860i
$146$	−22.0000			−1.82073
$147$		0			0
$148$	14.0000	−	14.0000i	1.15079	−	1.15079i
$149$	12.1244	−	7.00000i	0.993266	−	0.573462i	0.0870170	−	0.996207i	$-0.472267\pi$
$149$	12.1244	−	7.00000i	0.993266	−	0.573462i	0.906249	+	0.422744i	$0.138933\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	−9.00000	+	9.00000i	−0.727607	+	0.727607i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−23.2224	+	6.22243i	−1.85335	+	0.496604i	−0.999706	−	0.0242497i	$-0.992280\pi$
$157$	−23.2224	+	6.22243i	−1.85335	+	0.496604i	−0.853646	+	0.520854i	$0.825614\pi$
$158$		0			0
$159$		0			0
$160$	12.0000	+	4.00000i	0.948683	+	0.316228i
$161$		0			0
$162$	9.00000	+	9.00000i	0.707107	+	0.707107i
$163$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$163$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$164$	−13.8564	−	8.00000i	−1.08200	−	0.624695i
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$	7.39230	−	11.1962i	0.566964	−	0.858706i
$171$		0			0
$172$		0			0
$173$	−15.0263	−	4.02628i	−1.14243	−	0.306112i	−0.362500	−	0.931984i	$-0.618077\pi$
$173$	−15.0263	−	4.02628i	−1.14243	−	0.306112i	−0.779926	+	0.625871i	$0.784744\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	5.85641	+	21.8564i	0.438956	+	1.63821i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	−11.1962	−	7.39230i	−0.834512	−	0.550990i
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.5622	−	14.6865i	1.21768	−	1.07978i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	6.95448	+	25.9545i	0.500595	+	1.86824i	0.496119	+	0.868255i	$0.334758\pi$
$193$	6.95448	+	25.9545i	0.500595	+	1.86824i	0.00447566	+	0.999990i	$0.498575\pi$
$194$	22.5167	−	13.0000i	1.61660	−	0.933346i
$195$		0			0
$196$		0			0
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	13.1244	+	5.26795i	0.928032	+	0.372500i
$201$		0			0
$202$	2.00000	+	2.00000i	0.140720	+	0.140720i
$203$		0			0
$204$		0			0
$205$	−14.9282	−	9.85641i	−1.04263	−	0.688401i
$206$		0			0
$207$		0			0
$208$	5.46410	+	1.46410i	0.378867	+	0.101517i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	6.58846	−	24.5885i	0.452497	−	1.68874i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−6.00000	+	6.00000i	−0.406371	+	0.406371i
$219$		0			0
$220$		0			0
$221$	3.00000	−	5.19615i	0.201802	−	0.349531i
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$	−12.0000	−	9.00000i	−0.800000	−	0.600000i
$226$	−1.00000	−	1.73205i	−0.0665190	−	0.115214i
$227$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$227$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$228$		0			0
$229$	−3.46410	+	2.00000i	−0.228914	+	0.132164i	−0.610071	−	0.792347i	$-0.708859\pi$
$229$	−3.46410	+	2.00000i	−0.228914	+	0.132164i	0.381157	+	0.924510i	$0.375526\pi$
$230$		0			0
$231$		0			0
$232$	8.00000	+	8.00000i	0.525226	+	0.525226i
$233$	28.6865	+	7.68653i	1.87932	+	0.503562i	0.999606	+	0.0280525i	$0.00893057\pi$
$233$	28.6865	+	7.68653i	1.87932	+	0.503562i	0.879711	+	0.475509i	$0.157736\pi$
$234$	−5.19615	−	3.00000i	−0.339683	−	0.196116i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−4.00000	+	6.92820i	−0.257663	+	0.446285i	−0.965615	−	0.259975i	$-0.916286\pi$
$241$	−4.00000	+	6.92820i	−0.257663	+	0.446285i	0.707953	+	0.706260i	$0.249619\pi$
$242$	15.0263	−	4.02628i	0.965926	−	0.258819i
$243$		0			0
$244$		−	24.0000i		−	1.53644i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	14.2942	+	6.75833i	0.904046	+	0.427434i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	6.22243	−	23.2224i	0.388145	−	1.44858i	−0.445005	−	0.895528i	$-0.646798\pi$
$257$	6.22243	−	23.2224i	0.388145	−	1.44858i	0.833150	−	0.553047i	$-0.186535\pi$
$258$		0			0
$259$		0			0
$260$	6.00000	+	2.00000i	0.372104	+	0.124035i
$261$	−6.00000	−	10.3923i	−0.371391	−	0.643268i
$262$		0			0
$263$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$263$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$264$		0			0
$265$	9.00000	−	27.0000i	0.552866	−	1.65860i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.5167	−	13.0000i	−1.37287	−	0.792624i	−0.381577	−	0.924337i	$-0.624619\pi$
$269$	−22.5167	−	13.0000i	−1.37287	−	0.792624i	−0.991288	+	0.131713i	$0.957952\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$	12.0000	+	12.0000i	0.727607	+	0.727607i
$273$		0			0
$274$			14.0000i			0.845771i
$275$		0			0
$276$		0			0
$277$	−31.4186	+	8.41858i	−1.88776	+	0.505824i	−0.888899	+	0.458103i	$0.848529\pi$
$277$	−31.4186	+	8.41858i	−1.88776	+	0.505824i	−0.998861	+	0.0477206i	$0.984804\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	32.0000			1.90896			0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000			1.90896			0.954480	+	0.298275i	$0.0964112\pi$
$282$		0			0
$283$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$283$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.0000	−	12.0000i	0.707107	−	0.707107i
$289$	0.866025	−	0.500000i	0.0509427	−	0.0294118i
$290$	8.39230	+	9.46410i	0.492813	+	0.555751i
$291$		0			0
$292$	8.05256	−	30.0526i	0.471240	−	1.75869i
$293$	−19.0000	+	19.0000i	−1.10999	+	1.10999i	−0.116841	+	0.993151i	$0.537277\pi$
$293$	−19.0000	+	19.0000i	−1.10999	+	1.10999i	−0.993151	+	0.116841i	$0.962723\pi$
$294$		0			0
$295$		0			0
$296$	14.0000	+	24.2487i	0.813733	+	1.40943i
$297$		0			0
$298$	5.12436	+	19.1244i	0.296846	+	1.10784i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.60770	−	26.7846i	0.0920564	−	1.53368i
$306$	−9.00000	−	15.5885i	−0.514496	−	0.891133i
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−1.36603	−	0.366025i	−0.0772123	−	0.0206890i	0.220006	−	0.975499i	$-0.429392\pi$
$313$	−1.36603	−	0.366025i	−0.0772123	−	0.0206890i	−0.297218	+	0.954810i	$0.596059\pi$
$314$		−	34.0000i		−	1.91873i
$315$		0			0
$316$		0			0
$317$	1.09808	−	4.09808i	0.0616741	−	0.230171i	−0.928208	−	0.372061i	$-0.878651\pi$
$317$	1.09808	−	4.09808i	0.0616741	−	0.230171i	0.989882	+	0.141890i	$0.0453179\pi$
$318$		0			0
$319$		0			0
$320$	−9.85641	+	14.9282i	−0.550990	+	0.834512i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−15.5885	+	9.00000i	−0.866025	+	0.500000i
$325$	6.56218	+	2.63397i	0.364004	+	0.146107i
$326$		0			0
$327$		0			0
$328$	16.0000	−	16.0000i	0.883452	−	0.883452i
$329$		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
$333$	−7.68653	−	28.6865i	−0.421219	−	1.57201i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.00000	−	7.00000i	−0.381314	−	0.381314i	0.490261	−	0.871576i	$-0.336901\pi$
$337$	−7.00000	−	7.00000i	−0.381314	−	0.381314i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	−15.0263	−	4.02628i	−0.817322	−	0.219001i
$339$		0			0
$340$	12.5885	+	14.1962i	0.682705	+	0.769894i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	11.0000	−	19.0526i	0.591364	−	1.02427i
$347$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$347$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$348$		0			0
$349$			36.0000i			1.92704i	0.267644	+	0.963518i	$0.413755\pi$
$349$			36.0000i			1.92704i	−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.29423	−	12.2942i	−0.175334	−	0.654356i	−0.996495	−	0.0836583i	$-0.973340\pi$
$353$	−3.29423	−	12.2942i	−0.175334	−	0.654356i	0.821160	−	0.570697i	$-0.193327\pi$
$354$		0			0
$355$		0			0
$356$	−32.0000			−1.69600
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$	14.1962	−	12.5885i	0.748203	−	0.663470i
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$	−6.58846	+	24.5885i	−0.346282	+	1.29234i
$363$		0			0
$364$		0			0
$365$	11.0000	−	33.0000i	0.575766	−	1.72730i
$366$		0			0
$367$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$367$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$368$		0			0
$369$	−20.7846	+	12.0000i	−1.08200	+	0.624695i
$370$	14.0000	+	28.0000i	0.727825	+	1.45565i
$371$		0			0
$372$		0			0
$373$	15.0263	+	4.02628i	0.778031	+	0.208473i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	15.0263	+	4.02628i	0.778031	+	0.208473i	0.152115	+	0.988363i	$0.451392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.00000	+	4.00000i	0.206010	+	0.206010i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$383$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$384$		0			0
$385$		0			0
$386$	−38.0000			−1.93415
$387$		0			0
$388$	9.51666	+	35.5167i	0.483135	+	1.80309i
$389$	−29.4449	−	17.0000i	−1.49291	−	0.861934i	−0.492947	−	0.870059i	$-0.664080\pi$
$389$	−29.4449	−	17.0000i	−1.49291	−	0.861934i	−0.999967	+	0.00812520i	$0.997414\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−22.5167	+	13.0000i	−1.13437	+	0.654931i
$395$		0			0
$396$		0			0
$397$	−4.75833	+	17.7583i	−0.238814	+	0.891265i	0.737579	+	0.675261i	$0.235969\pi$
$397$	−4.75833	+	17.7583i	−0.238814	+	0.891265i	−0.976392	+	0.216004i	$0.930698\pi$
$398$		0			0
$399$		0			0
$400$	−12.0000	+	16.0000i	−0.600000	+	0.800000i
$401$	−1.00000	−	1.73205i	−0.0499376	−	0.0864945i	0.839976	−	0.542623i	$-0.182569\pi$
$401$	−1.00000	−	1.73205i	−0.0499376	−	0.0864945i	−0.889914	+	0.456129i	$0.849236\pi$
$402$		0			0
$403$		0			0
$404$	−3.46410	+	2.00000i	−0.172345	+	0.0995037i
$405$	−18.0000	+	9.00000i	−0.894427	+	0.447214i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5.19615	−	3.00000i	−0.256933	−	0.148340i	0.366002	−	0.930614i	$-0.380726\pi$
$409$	−5.19615	−	3.00000i	−0.256933	−	0.148340i	−0.622935	+	0.782274i	$0.714060\pi$
$410$	18.9282	−	16.7846i	0.934797	−	0.828933i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−4.00000	+	6.92820i	−0.196116	+	0.339683i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$	31.1769	+	18.0000i	1.51408	+	0.874157i
$425$	13.0981	+	16.6865i	0.635350	+	0.809416i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−29.0000	+	29.0000i	−1.39365	+	1.39365i	−0.576683	+	0.816968i	$0.695653\pi$
$433$	−29.0000	+	29.0000i	−1.39365	+	1.39365i	−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$	−6.00000	−	10.3923i	−0.287348	−	0.497701i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$	6.00000	+	6.00000i	0.285391	+	0.285391i
$443$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$443$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$444$		0			0
$445$	−35.7128	−	2.14359i	−1.69295	−	0.101616i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			14.0000i			0.660701i	0.943858	+	0.330350i	$0.107167\pi$
$449$			14.0000i			0.660701i	−0.943858	+	0.330350i	$0.892833\pi$
$450$	16.6865	−	13.0981i	0.786611	−	0.617449i
$451$		0			0
$452$	2.73205	−	0.732051i	0.128505	−	0.0344328i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6.22243	+	23.2224i	−0.291073	+	1.08630i	0.653213	+	0.757174i	$0.273421\pi$
$457$	−6.22243	+	23.2224i	−0.291073	+	1.08630i	−0.944286	+	0.329125i	$0.893246\pi$
$458$	−1.46410	−	5.46410i	−0.0684130	−	0.255321i
$459$		0			0
$460$		0			0
$461$	38.0000			1.76984			0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000			1.76984			0.884918	+	0.465746i	$0.154214\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−13.8564	+	8.00000i	−0.643268	+	0.371391i
$465$		0			0
$466$	−21.0000	+	36.3731i	−0.972806	+	1.68495i
$467$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$467$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$468$	6.00000	−	6.00000i	0.277350	−	0.277350i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−27.0000	−	27.0000i	−1.23625	−	1.23625i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	7.00000	+	12.1244i	0.319173	+	0.552823i
$482$	−8.00000	−	8.00000i	−0.364390	−	0.364390i
$483$		0			0
$484$			22.0000i			1.00000i
$485$	8.24167	+	40.2750i	0.374235	+	1.82879i
$486$		0			0
$487$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$487$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$488$	32.7846	+	8.78461i	1.48409	+	0.397661i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	4.39230	+	16.3923i	0.197819	+	0.738272i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$499$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$500$	−14.4641	+	17.0526i	−0.646854	+	0.762614i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−4.00000	+	2.00000i	−0.177998	+	0.0889988i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−38.1051	+	22.0000i	−1.68898	+	0.975133i	−0.733679	+	0.679496i	$0.762199\pi$
$509$	−38.1051	+	22.0000i	−1.68898	+	0.975133i	−0.955300	+	0.295637i	$0.904468\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	29.4449	+	17.0000i	1.29876	+	0.749838i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−4.92820	+	7.46410i	−0.216116	+	0.327323i
$521$	11.0000	−	19.0526i	0.481919	−	0.834708i	−0.517866	−	0.855462i	$-0.673273\pi$
$521$	11.0000	−	19.0526i	0.481919	−	0.834708i	0.999785	+	0.0207541i	$0.00660670\pi$
$522$	16.3923	−	4.39230i	0.717472	−	0.192246i
$523$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$523$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.9186	+	11.5000i	0.866025	+	0.500000i
$530$	33.5885	+	22.1769i	1.45899	+	0.963304i
$531$		0			0
$532$		0			0
$533$	8.00000	−	8.00000i	0.346518	−	0.346518i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	26.0000	−	26.0000i	1.12094	−	1.12094i
$539$		0			0
$540$		0			0
$541$	−21.0000	−	36.3731i	−0.902861	−	1.56380i	−0.823764	−	0.566933i	$-0.808130\pi$
$541$	−21.0000	−	36.3731i	−0.902861	−	1.56380i	−0.0790969	−	0.996867i	$-0.525204\pi$
$542$		0			0
$543$		0			0
$544$	−20.7846	+	12.0000i	−0.891133	+	0.514496i
$545$	−6.00000	−	12.0000i	−0.257012	−	0.514024i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	−19.1244	−	5.12436i	−0.816952	−	0.218902i
$549$	−31.1769	−	18.0000i	−1.33060	−	0.768221i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	46.0000i		−	1.95435i
$555$		0			0
$556$		0			0
$557$	−45.0788	+	12.0788i	−1.91005	+	0.511797i	−0.916253	+	0.400599i	$0.868802\pi$
$557$	−45.0788	+	12.0788i	−1.91005	+	0.511797i	−0.993798	+	0.111198i	$0.964531\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−11.7128	+	43.7128i	−0.494075	+	1.84391i
$563$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$563$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$564$		0			0
$565$	3.09808	−	0.633975i	0.130337	−	0.0266715i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.5167	+	13.0000i	−0.943948	+	0.544988i	−0.891196	−	0.453619i	$-0.850133\pi$
$569$	−22.5167	+	13.0000i	−0.943948	+	0.544988i	−0.0527519	+	0.998608i	$0.516799\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$	31.4186	−	8.41858i	1.30797	−	0.350470i	0.463513	−	0.886090i	$-0.346589\pi$
$577$	31.4186	−	8.41858i	1.30797	−	0.350470i	0.844459	+	0.535620i	$0.179922\pi$
$578$	0.366025	+	1.36603i	0.0152246	+	0.0568192i
$579$		0			0
$580$	−16.0000	+	8.00000i	−0.664364	+	0.332182i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	38.1051	+	22.0000i	1.57680	+	0.910366i
$585$	7.09808	−	6.29423i	0.293469	−	0.260234i
$586$	−19.0000	−	32.9090i	−0.784883	−	1.35946i
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−38.2487	+	10.2487i	−1.57201	+	0.421219i
$593$	−42.3468	−	11.3468i	−1.73897	−	0.465957i	−0.756756	−	0.653698i	$-0.773217\pi$
$593$	−42.3468	−	11.3468i	−1.73897	−	0.465957i	−0.982219	+	0.187741i	$0.939883\pi$
$594$		0			0
$595$		0			0
$596$	−28.0000			−1.14692
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	48.0000			1.95796			0.978980	−	0.203954i	$-0.0653794\pi$
$601$	48.0000			1.95796			0.978980	+	0.203954i	$0.0653794\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.47372	+	24.5526i	−0.0599153	+	0.998203i
$606$		0			0
$607$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$607$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$608$		0			0
$609$		0			0
$610$	36.0000	+	12.0000i	1.45760	+	0.485866i
$611$		0			0
$612$	24.5885	−	6.58846i	0.993929	−	0.266323i
$613$	−0.366025	−	1.36603i	−0.0147836	−	0.0551732i	0.958140	−	0.286300i	$-0.0924254\pi$
$613$	−0.366025	−	1.36603i	−0.0147836	−	0.0551732i	−0.972924	+	0.231127i	$0.925759\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	+	3.00000i	0.120775	+	0.120775i	0.764911	−	0.644136i	$-0.222783\pi$
$617$	3.00000	+	3.00000i	0.120775	+	0.120775i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.2846	+	18.0622i	−0.691384	+	0.722487i
$626$	1.00000	−	1.73205i	0.0399680	−	0.0692267i
$627$		0			0
$628$	46.4449	+	12.4449i	1.85335	+	0.496604i
$629$			42.0000i			1.67465i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	5.19615	+	3.00000i	0.206366	+	0.119145i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−16.7846	−	18.9282i	−0.663470	−	0.748203i
$641$	4.00000	−	6.92820i	0.157991	−	0.273648i	−0.776153	−	0.630544i	$-0.782832\pi$
$641$	4.00000	−	6.92820i	0.157991	−	0.273648i	0.934144	+	0.356897i	$0.116165\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$647$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$648$	−6.58846	−	24.5885i	−0.258819	−	0.965926i
$649$		0			0
$650$	−6.00000	+	8.00000i	−0.235339	+	0.313786i
$651$		0			0
$652$		0			0
$653$	−12.2942	−	3.29423i	−0.481110	−	0.128913i	0.0101092	−	0.999949i	$-0.496782\pi$
$653$	−12.2942	−	3.29423i	−0.481110	−	0.128913i	−0.491220	+	0.871036i	$0.663449\pi$
$654$		0			0
$655$		0			0
$656$	16.0000	+	27.7128i	0.624695	+	1.08200i
$657$	−33.0000	−	33.0000i	−1.28745	−	1.28745i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	6.00000	−	10.3923i	0.233373	−	0.404214i	−0.725426	−	0.688301i	$-0.758357\pi$
$661$	6.00000	−	10.3923i	0.233373	−	0.404214i	0.958799	+	0.284087i	$0.0916904\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	42.0000			1.62747
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−11.0000	+	11.0000i	−0.424019	+	0.424019i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−11.0000	+	11.0000i	−0.424019	+	0.424019i	0.462566	+	0.886585i	$0.346929\pi$
$674$	12.1244	−	7.00000i	0.467013	−	0.269630i
$675$		0			0
$676$	11.0000	−	19.0526i	0.423077	−	0.732791i
$677$	9.88269	−	36.8827i	0.379822	−	1.41752i	−0.466347	−	0.884602i	$-0.654430\pi$
$677$	9.88269	−	36.8827i	0.379822	−	1.41752i	0.846169	−	0.532915i	$-0.178903\pi$
$678$		0			0
$679$		0			0
$680$	−24.0000	+	12.0000i	−0.920358	+	0.460179i
$681$		0			0
$682$		0			0
$683$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$683$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$684$		0			0
$685$	−21.0000	−	7.00000i	−0.802369	−	0.267456i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.5885	+	9.00000i	0.593873	+	0.342873i
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	22.0000	+	22.0000i	0.836315	+	0.836315i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	32.7846	−	8.78461i	1.24181	−	0.332741i
$698$	−49.1769	−	13.1769i	−1.86137	−	0.498754i
$699$		0			0
$700$		0			0
$701$	52.0000			1.96401			0.982006	−	0.188847i	$-0.0604752\pi$
$701$	52.0000			1.96401			0.982006	+	0.188847i	$0.0604752\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	18.0000			0.677439
$707$		0			0
$708$		0			0
$709$	38.1051	−	22.0000i	1.43107	−	0.826227i	0.433865	−	0.900978i	$-0.357149\pi$
$709$	38.1051	−	22.0000i	1.43107	−	0.826227i	0.997202	+	0.0747503i	$0.0238160\pi$
$710$		0			0
$711$		0			0
$712$	11.7128	−	43.7128i	0.438956	−	1.63821i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	12.0000	+	24.0000i	0.447214	+	0.894427i
$721$		0			0
$722$	19.0000	+	19.0000i	0.707107	+	0.707107i
$723$		0			0
$724$	−31.1769	−	18.0000i	−1.15868	−	0.668965i
$725$	−18.3923	+	7.85641i	−0.683073	+	0.291780i
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$	41.0526	+	27.1051i	1.51942	+	1.00321i
$731$		0			0
$732$		0			0
$733$	39.6147	+	10.6147i	1.46320	+	0.392064i	0.900595	−	0.434659i	$-0.143131\pi$
$733$	39.6147	+	10.6147i	1.46320	+	0.392064i	0.562609	+	0.826723i	$0.309798\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−8.78461	−	32.7846i	−0.323366	−	1.20682i
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$	−43.3731	+	8.87564i	−1.59443	+	0.326275i
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−31.2487	−	1.87564i	−1.14486	−	0.0687183i
$746$	−11.0000	+	19.0526i	−0.402739	+	0.697564i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$		0			0
$754$	−6.92820	+	4.00000i	−0.252310	+	0.145671i
$755$		0			0
$756$		0			0
$757$	−17.0000	−	17.0000i	−0.617876	−	0.617876i	0.327111	−	0.944986i	$-0.393925\pi$
$757$	−17.0000	−	17.0000i	−0.617876	−	0.617876i	−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.0000	−	32.9090i	−0.688749	−	1.19295i	−0.972243	−	0.233975i	$-0.924827\pi$
$761$	−19.0000	−	32.9090i	−0.688749	−	1.19295i	0.283493	−	0.958974i	$-0.408507\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	27.8827	−	5.70577i	1.00810	−	0.206293i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$		0			0
$771$		0			0
$772$	13.9090	−	51.9090i	0.500595	−	1.86824i
$773$	−14.2750	−	53.2750i	−0.513436	−	1.91617i	−0.379549	−	0.925172i	$-0.623921\pi$
$773$	−14.2750	−	53.2750i	−0.513436	−	1.91617i	−0.133887	−	0.990997i	$-0.542746\pi$
$774$		0			0
$775$		0			0
$776$	−52.0000			−1.86669
$777$		0			0
$778$	34.0000	−	34.0000i	1.21896	−	1.21896i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	51.0000	+	17.0000i	1.82027	+	0.606756i
$786$		0			0
$787$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$787$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$788$	−9.51666	−	35.5167i	−0.339017	−	1.26523i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	16.3923	+	4.39230i	0.582108	+	0.155975i
$794$	−22.5167	−	13.0000i	−0.799086	−	0.461353i
$795$		0			0
$796$		0			0
$797$	37.0000	+	37.0000i	1.31061	+	1.31061i	0.920967	+	0.389640i	$0.127401\pi$
$797$	37.0000	+	37.0000i	1.31061	+	1.31061i	0.389640	+	0.920967i	$0.372599\pi$
$798$		0			0
$799$		0			0
$800$	−17.4641	−	22.2487i	−0.617449	−	0.786611i
$801$	−24.0000	+	41.5692i	−0.847998	+	1.46878i
$802$	2.73205	−	0.732051i	0.0964721	−	0.0258496i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.46410	−	5.46410i	−0.0515069	−	0.192226i
$809$	48.4974	+	28.0000i	1.70508	+	0.984428i	0.940435	+	0.339975i	$0.110418\pi$
$809$	48.4974	+	28.0000i	1.70508	+	0.984428i	0.764644	+	0.644453i	$0.222915\pi$
$810$	−5.70577	−	27.8827i	−0.200480	−	0.979698i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	6.00000	−	6.00000i	0.209785	−	0.209785i
$819$		0			0
$820$	16.0000	+	32.0000i	0.558744	+	1.11749i
$821$	14.0000	+	24.2487i	0.488603	+	0.846286i	0.999914	−	0.0131101i	$-0.00417319\pi$
$821$	14.0000	+	24.2487i	0.488603	+	0.846286i	−0.511311	+	0.859396i	$0.670840\pi$
$822$		0			0
$823$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$823$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	46.7654	+	27.0000i	1.62423	+	0.937749i	0.985771	+	0.168091i	$0.0537604\pi$
$829$	46.7654	+	27.0000i	1.62423	+	0.937749i	0.638457	+	0.769657i	$0.279573\pi$
$830$		0			0
$831$		0			0
$832$	−8.00000	−	8.00000i	−0.277350	−	0.277350i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		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
$841$	13.0000			0.448276
$842$	10.2487	−	38.2487i	0.353194	−	1.31814i
$843$		0			0
$844$		0			0
$845$	13.5526	−	20.5263i	0.466222	−	0.706125i
$846$		0			0
$847$		0			0
$848$	−36.0000	+	36.0000i	−1.23625	+	1.23625i
$849$		0			0
$850$	−27.5885	+	11.7846i	−0.946276	+	0.404209i
$851$		0			0
$852$		0			0
$853$	41.0000	−	41.0000i	1.40381	−	1.40381i	0.616308	−	0.787505i	$-0.288628\pi$
$853$	41.0000	−	41.0000i	1.40381	−	1.40381i	0.787505	−	0.616308i	$-0.211372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	45.0788	−	12.0788i	1.53986	−	0.412605i	0.613642	−	0.789584i	$-0.289704\pi$
$857$	45.0788	−	12.0788i	1.53986	−	0.412605i	0.926222	+	0.376979i	$0.123037\pi$
$858$		0			0
$859$		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
$862$		0			0
$863$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$863$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$864$		0			0
$865$	23.0788	+	26.0263i	0.784704	+	0.884920i
$866$	−29.0000	−	50.2295i	−0.985460	−	1.70687i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	16.3923	−	4.39230i	0.555113	−	0.148742i
$873$	53.2750	+	14.2750i	1.80309	+	0.483135i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.41858	−	31.4186i	0.284275	−	1.06093i	−0.665092	−	0.746762i	$-0.731608\pi$
$877$	8.41858	−	31.4186i	0.284275	−	1.06093i	0.949367	−	0.314169i	$-0.101726\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.0000			−1.07811			−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000			−1.07811			−0.539054	+	0.842271i	$0.681218\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$	−10.3923	+	6.00000i	−0.349531	+	0.201802i
$885$		0			0
$886$		0			0
$887$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$887$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$888$		0			0
$889$		0			0
$890$	16.0000	−	48.0000i	0.536321	−	1.60896i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−19.1244	−	5.12436i	−0.638188	−	0.171002i
$899$		0			0
$900$	11.7846	+	27.5885i	0.392820	+	0.919615i
$901$	27.0000	+	46.7654i	0.899500	+	1.55798i
$902$		0			0
$903$		0			0
$904$			4.00000i			0.133038i
$905$	−33.5885	−	22.1769i	−1.11652	−	0.737186i
$906$		0			0
$907$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$907$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$908$		0			0
$909$			6.00000i			0.199007i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	−29.4449	−	17.0000i	−0.973950	−	0.562310i
$915$		0			0
$916$	8.00000			0.264327
$917$		0			0
$918$		0			0
$919$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$919$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$920$		0			0
$921$		0			0
$922$	−13.9090	+	51.9090i	−0.458067	+	1.70953i
$923$		0			0
$924$		0			0
$925$	−49.0000	+	7.00000i	−1.61111	+	0.230159i
$926$		0			0
$927$		0			0
$928$	−5.85641	−	21.8564i	−0.192246	−	0.717472i
$929$	39.8372	−	23.0000i	1.30702	−	0.754606i	0.325418	−	0.945570i	$-0.394495\pi$
$929$	39.8372	−	23.0000i	1.30702	−	0.754606i	0.981597	+	0.190965i	$0.0611616\pi$
$930$		0			0
$931$		0			0
$932$	−42.0000	−	42.0000i	−1.37576	−	1.37576i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	6.00000	+	10.3923i	0.196116	+	0.339683i
$937$	−43.0000	−	43.0000i	−1.40475	−	1.40475i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	−43.0000	−	43.0000i	−1.40475	−	1.40475i	−0.620703	−	0.784046i	$-0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−29.0000	+	50.2295i	−0.945373	+	1.63743i	−0.190370	+	0.981712i	$0.560969\pi$
$941$	−29.0000	+	50.2295i	−0.945373	+	1.63743i	−0.755003	+	0.655722i	$0.772364\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$947$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$948$		0			0
$949$	19.0526	+	11.0000i	0.618472	+	0.357075i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.0000	+	41.0000i	−1.32812	+	1.32812i	−0.421111	+	0.907009i	$0.638360\pi$
$953$	−41.0000	+	41.0000i	−1.32812	+	1.32812i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	46.7654	−	27.0000i	1.51408	−	0.874157i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.5000	−	26.8468i	−0.500000	−	0.866025i
$962$	−19.1244	+	5.12436i	−0.616594	+	0.165216i
$963$		0			0
$964$	13.8564	−	8.00000i	0.446285	−	0.257663i
$965$	19.0000	−	57.0000i	0.611632	−	1.83489i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−30.0526	−	8.05256i	−0.965926	−	0.258819i
$969$		0			0
$970$	−58.0333	−	3.48334i	−1.86334	−	0.111843i
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−24.0000	+	41.5692i	−0.768221	+	1.33060i
$977$	36.8827	−	9.88269i	1.17998	−	0.316175i	0.385063	−	0.922890i	$-0.374180\pi$
$977$	36.8827	−	9.88269i	1.17998	−	0.316175i	0.794919	+	0.606715i	$0.207513\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−18.0000			−0.574696
$982$		0			0
$983$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$983$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$984$		0			0
$985$	−8.24167	−	40.2750i	−0.262601	−	1.28327i
$986$	−24.0000			−0.764316
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−50.5429	+	13.5429i	−1.60071	+	0.428909i	−0.945257	−	0.326326i	$-0.894189\pi$
$997$	−50.5429	+	13.5429i	−1.60071	+	0.428909i	−0.655454	+	0.755235i	$0.727523\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.c.67.1		4
4.3	odd	2	CM	980.2.x.c.67.1		4
5.3	odd	4	inner	980.2.x.c.263.1		4
7.2	even	3	inner	980.2.x.c.667.1		4
7.3	odd	6		20.2.e.a.7.1	yes	2
7.4	even	3		980.2.k.a.687.1		2
7.5	odd	6		980.2.x.d.667.1		4
7.6	odd	2		980.2.x.d.67.1		4
20.3	even	4	inner	980.2.x.c.263.1		4
21.17	even	6		180.2.k.c.127.1		2
28.3	even	6		20.2.e.a.7.1	yes	2
28.11	odd	6		980.2.k.a.687.1		2
28.19	even	6		980.2.x.d.667.1		4
28.23	odd	6	inner	980.2.x.c.667.1		4
28.27	even	2		980.2.x.d.67.1		4
35.3	even	12		20.2.e.a.3.1	✓	2
35.13	even	4		980.2.x.d.263.1		4
35.17	even	12		100.2.e.b.43.1		2
35.18	odd	12		980.2.k.a.883.1		2
35.23	odd	12	inner	980.2.x.c.863.1		4
35.24	odd	6		100.2.e.b.7.1		2
35.33	even	12		980.2.x.d.863.1		4
56.3	even	6		320.2.n.e.127.1		2
56.45	odd	6		320.2.n.e.127.1		2
84.59	odd	6		180.2.k.c.127.1		2
105.17	odd	12		900.2.k.c.343.1		2
105.38	odd	12		180.2.k.c.163.1		2
105.59	even	6		900.2.k.c.307.1		2
112.3	even	12		1280.2.o.j.127.1		2
112.45	odd	12		1280.2.o.j.127.1		2
112.59	even	12		1280.2.o.g.127.1		2
112.101	odd	12		1280.2.o.g.127.1		2
140.3	odd	12		20.2.e.a.3.1	✓	2
140.23	even	12	inner	980.2.x.c.863.1		4
140.59	even	6		100.2.e.b.7.1		2
140.83	odd	4		980.2.x.d.263.1		4
140.87	odd	12		100.2.e.b.43.1		2
140.103	odd	12		980.2.x.d.863.1		4
140.123	even	12		980.2.k.a.883.1		2
280.3	odd	12		320.2.n.e.63.1		2
280.59	even	6		1600.2.n.h.1407.1		2
280.157	even	12		1600.2.n.h.1343.1		2
280.213	even	12		320.2.n.e.63.1		2
280.227	odd	12		1600.2.n.h.1343.1		2
280.269	odd	6		1600.2.n.h.1407.1		2
420.59	odd	6		900.2.k.c.307.1		2
420.143	even	12		180.2.k.c.163.1		2
420.227	even	12		900.2.k.c.343.1		2
560.3	odd	12		1280.2.o.g.383.1		2
560.213	even	12		1280.2.o.j.383.1		2
560.283	odd	12		1280.2.o.j.383.1		2
560.493	even	12		1280.2.o.g.383.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.e.a.3.1	✓	2	35.3	even	12
20.2.e.a.3.1	✓	2	140.3	odd	12
20.2.e.a.7.1	yes	2	7.3	odd	6
20.2.e.a.7.1	yes	2	28.3	even	6
100.2.e.b.7.1		2	35.24	odd	6
100.2.e.b.7.1		2	140.59	even	6
100.2.e.b.43.1		2	35.17	even	12
100.2.e.b.43.1		2	140.87	odd	12
180.2.k.c.127.1		2	21.17	even	6
180.2.k.c.127.1		2	84.59	odd	6
180.2.k.c.163.1		2	105.38	odd	12
180.2.k.c.163.1		2	420.143	even	12
320.2.n.e.63.1		2	280.3	odd	12
320.2.n.e.63.1		2	280.213	even	12
320.2.n.e.127.1		2	56.3	even	6
320.2.n.e.127.1		2	56.45	odd	6
900.2.k.c.307.1		2	105.59	even	6
900.2.k.c.307.1		2	420.59	odd	6
900.2.k.c.343.1		2	105.17	odd	12
900.2.k.c.343.1		2	420.227	even	12
980.2.k.a.687.1		2	7.4	even	3
980.2.k.a.687.1		2	28.11	odd	6
980.2.k.a.883.1		2	35.18	odd	12
980.2.k.a.883.1		2	140.123	even	12
980.2.x.c.67.1		4	1.1	even	1	trivial
980.2.x.c.67.1		4	4.3	odd	2	CM
980.2.x.c.263.1		4	5.3	odd	4	inner
980.2.x.c.263.1		4	20.3	even	4	inner
980.2.x.c.667.1		4	7.2	even	3	inner
980.2.x.c.667.1		4	28.23	odd	6	inner
980.2.x.c.863.1		4	35.23	odd	12	inner
980.2.x.c.863.1		4	140.23	even	12	inner
980.2.x.d.67.1		4	7.6	odd	2
980.2.x.d.67.1		4	28.27	even	2
980.2.x.d.263.1		4	35.13	even	4
980.2.x.d.263.1		4	140.83	odd	4
980.2.x.d.667.1		4	7.5	odd	6
980.2.x.d.667.1		4	28.19	even	6
980.2.x.d.863.1		4	35.33	even	12
980.2.x.d.863.1		4	140.103	odd	12
1280.2.o.g.127.1		2	112.59	even	12
1280.2.o.g.127.1		2	112.101	odd	12
1280.2.o.g.383.1		2	560.3	odd	12
1280.2.o.g.383.1		2	560.493	even	12
1280.2.o.j.127.1		2	112.3	even	12
1280.2.o.j.127.1		2	112.45	odd	12
1280.2.o.j.383.1		2	560.213	even	12
1280.2.o.j.383.1		2	560.283	odd	12
1600.2.n.h.1343.1		2	280.157	even	12
1600.2.n.h.1343.1		2	280.227	odd	12
1600.2.n.h.1407.1		2	280.59	even	6
1600.2.n.h.1407.1		2	280.269	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.x (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.366025 + 1.36603i) q^{2} +(-1.73205 - 1.00000i) q^{4} +(-1.86603 - 1.23205i) q^{5} +(2.00000 - 2.00000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +(2.36603 - 2.09808i) q^{10} +(1.00000 - 1.00000i) q^{13} +(2.00000 + 3.46410i) q^{16} +(4.09808 - 1.09808i) q^{17} +(-1.09808 - 4.09808i) q^{18} +(2.00000 + 4.00000i) q^{20} +(1.96410 + 4.59808i) q^{25} +(1.00000 + 1.73205i) q^{26} +4.00000i q^{29} +(-5.46410 + 1.46410i) q^{32} +6.00000i q^{34} +6.00000 q^{36} +(-2.56218 + 9.56218i) q^{37} +(-6.19615 + 1.26795i) q^{40} +8.00000 q^{41} +(6.69615 + 0.401924i) q^{45} +(-7.00000 + 1.00000i) q^{50} +(-2.73205 + 0.732051i) q^{52} +(3.29423 + 12.2942i) q^{53} +(-5.46410 - 1.46410i) q^{58} +(6.00000 + 10.3923i) q^{61} -8.00000i q^{64} +(-3.09808 + 0.633975i) q^{65} +(-8.19615 - 2.19615i) q^{68} +(-2.19615 + 8.19615i) q^{72} +(4.02628 + 15.0263i) q^{73} +(-12.1244 - 7.00000i) q^{74} +(0.535898 - 8.92820i) q^{80} +(4.50000 - 7.79423i) q^{81} +(-2.92820 + 10.9282i) q^{82} +(-9.00000 - 3.00000i) q^{85} +(13.8564 - 8.00000i) q^{89} +(-3.00000 + 9.00000i) q^{90} +(-13.0000 - 13.0000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 6 q^{10} + 4 q^{13} + 8 q^{16} + 6 q^{17} + 6 q^{18} + 8 q^{20} - 6 q^{25} + 4 q^{26} - 8 q^{32} + 24 q^{36} + 14 q^{37} - 4 q^{40} + 32 q^{41} + 6 q^{45} - 28 q^{50}+ \cdots - 52 q^{97}+O(q^{100}) \) 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 + 6 * q^10 + 4 * q^13 + 8 * q^16 + 6 * q^17 + 6 * q^18 + 8 * q^20 - 6 * q^25 + 4 * q^26 - 8 * q^32 + 24 * q^36 + 14 * q^37 - 4 * q^40 + 32 * q^41 + 6 * q^45 - 28 * q^50 - 4 * q^52 - 18 * q^53 - 8 * q^58 + 24 * q^61 - 2 * q^65 - 12 * q^68 + 12 * q^72 - 22 * q^73 + 16 * q^80 + 18 * q^81 + 16 * q^82 - 36 * q^85 - 12 * q^90 - 52 * q^97

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