LMFDB - Embedded newform 980.2.x.g.667.2

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 = [8,-4,0,0,0,0,0,-16,0,0,0] 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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			667.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	980.667
Dual form			980.2.x.g.263.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.36603 + 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(0.448288 - 2.19067i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +(0.189469 + 3.15660i) q^{10} +(-4.24264 + 4.24264i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-2.07055 + 7.72741i) q^{17} +(-4.09808 - 1.09808i) q^{18} +(-1.41421 - 4.24264i) q^{20} +(-4.59808 - 1.96410i) q^{25} +(4.24264 - 7.34847i) q^{26} +10.0000i q^{29} +(-1.46410 + 5.46410i) q^{32} -11.3137i q^{34} +6.00000 q^{36} +(9.56218 - 2.56218i) q^{37} +(3.48477 + 5.27792i) q^{40} +1.41421 q^{41} +(4.45069 - 5.01910i) q^{45} +(7.00000 + 1.00000i) q^{50} +(-3.10583 + 11.5911i) q^{52} +(6.83013 + 1.83013i) q^{53} +(-3.66025 - 13.6603i) q^{58} +(0.707107 - 1.22474i) q^{61} -8.00000i q^{64} +(7.39230 + 11.1962i) q^{65} +(4.14110 + 15.4548i) q^{68} +(-8.19615 + 2.19615i) q^{72} +(-5.79555 - 1.55291i) q^{73} +(-12.1244 + 7.00000i) q^{74} +(-6.69213 - 5.93426i) q^{80} +(4.50000 + 7.79423i) q^{81} +(-1.93185 + 0.517638i) q^{82} +(16.0000 + 8.00000i) q^{85} +(-15.9217 - 9.19239i) q^{89} +(-4.24264 + 8.48528i) q^{90} +(5.65685 + 5.65685i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} - 16 q^{8} + 16 q^{16} - 12 q^{18} - 16 q^{25} + 16 q^{32} + 48 q^{36} + 28 q^{37} + 56 q^{50} + 20 q^{53} + 40 q^{58} - 24 q^{65} - 24 q^{72} + 36 q^{81} + 128 q^{85}+O(q^{100}) $ 8 * q - 4 * q^2 - 16 * q^8 + 16 * q^16 - 12 * q^18 - 16 * q^25 + 16 * q^32 + 48 * q^36 + 28 * q^37 + 56 * q^50 + 20 * q^53 + 40 * q^58 - 24 * q^65 - 24 * q^72 + 36 * q^81 + 128 * q^85

Character values

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

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

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 $

$2$	−1.36603	+	0.366025i	−0.965926	+	0.258819i
$2$	−1.36603	+	0.366025i	−0.965926	+	0.258819i
$3$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$3$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$	0.448288	−	2.19067i	0.200480	−	0.979698i
$5$	0.448288	−	2.19067i	0.200480	−	0.979698i
$6$		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
$10$	0.189469	+	3.15660i	0.0599153	+	0.998203i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−4.24264	+	4.24264i	−1.17670	+	1.17670i	−0.196116	+	0.980581i	$0.562833\pi$
$13$	−4.24264	+	4.24264i	−1.17670	+	1.17670i	−0.980581	+	0.196116i	$0.937167\pi$
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	−2.07055	+	7.72741i	−0.502183	+	1.87417i	−0.0168199	+	0.999859i	$0.505354\pi$
$17$	−2.07055	+	7.72741i	−0.502183	+	1.87417i	−0.485363	+	0.874313i	$0.661312\pi$
$18$	−4.09808	−	1.09808i	−0.965926	−	0.258819i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	−1.41421	−	4.24264i	−0.316228	−	0.948683i
$21$		0			0
$22$		0			0
$23$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$23$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$24$		0			0
$25$	−4.59808	−	1.96410i	−0.919615	−	0.392820i
$26$	4.24264	−	7.34847i	0.832050	−	1.44115i
$27$		0			0
$28$		0			0
$29$			10.0000i			1.85695i	0.371391	+	0.928477i	$0.378881\pi$
$29$			10.0000i			1.85695i	−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−1.46410	+	5.46410i	−0.258819	+	0.965926i
$33$		0			0
$34$		−	11.3137i		−	1.94029i
$35$		0			0
$36$	6.00000			1.00000
$37$	9.56218	−	2.56218i	1.57201	−	0.421219i	0.635571	−	0.772043i	$-0.280765\pi$
$37$	9.56218	−	2.56218i	1.57201	−	0.421219i	0.936442	+	0.350823i	$0.114098\pi$
$38$		0			0
$39$		0			0
$40$	3.48477	+	5.27792i	0.550990	+	0.834512i
$41$	1.41421			0.220863			0.110432	−	0.993884i	$-0.464777\pi$
$41$	1.41421			0.220863			0.110432	+	0.993884i	$0.464777\pi$
$42$		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
$45$	4.45069	−	5.01910i	0.663470	−	0.748203i
$46$		0			0
$47$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$47$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$48$		0			0
$49$		0			0
$50$	7.00000	+	1.00000i	0.989949	+	0.141421i
$51$		0			0
$52$	−3.10583	+	11.5911i	−0.430701	+	1.60740i
$53$	6.83013	+	1.83013i	0.938190	+	0.251387i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	6.83013	+	1.83013i	0.938190	+	0.251387i	0.242846	+	0.970065i	$0.421919\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−3.66025	−	13.6603i	−0.480615	−	1.79368i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	0.707107	−	1.22474i	0.0905357	−	0.156813i	−0.817201	−	0.576353i	$-0.804475\pi$
$61$	0.707107	−	1.22474i	0.0905357	−	0.156813i	0.907737	+	0.419540i	$0.137809\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	7.39230	+	11.1962i	0.916903	+	1.38871i
$66$		0			0
$67$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$67$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$68$	4.14110	+	15.4548i	0.502183	+	1.87417i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−8.19615	+	2.19615i	−0.965926	+	0.258819i
$73$	−5.79555	−	1.55291i	−0.678318	−	0.181755i	−0.0968194	−	0.995302i	$-0.530867\pi$
$73$	−5.79555	−	1.55291i	−0.678318	−	0.181755i	−0.581499	+	0.813547i	$0.697534\pi$
$74$	−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	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$	−6.69213	−	5.93426i	−0.748203	−	0.663470i
$81$	4.50000	+	7.79423i	0.500000	+	0.866025i
$82$	−1.93185	+	0.517638i	−0.213337	+	0.0571636i
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	16.0000	+	8.00000i	1.73544	+	0.867722i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−15.9217	−	9.19239i	−1.68770	−	0.974391i	−0.956277	−	0.292462i	$-0.905526\pi$
$89$	−15.9217	−	9.19239i	−1.68770	−	0.974391i	−0.731418	−	0.681930i	$-0.761141\pi$
$90$	−4.24264	+	8.48528i	−0.447214	+	0.894427i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.65685	+	5.65685i	0.574367	+	0.574367i	0.933346	−	0.358979i	$-0.116875\pi$
$97$	5.65685	+	5.65685i	0.574367	+	0.574367i	−0.358979	+	0.933346i	$0.616875\pi$
$98$		0			0
$99$		0			0
$100$	−9.92820	+	1.19615i	−0.992820	+	0.119615i
$101$	6.36396	+	11.0227i	0.633238	+	1.09680i	0.986886	+	0.161421i	$0.0516078\pi$
$101$	6.36396	+	11.0227i	0.633238	+	1.09680i	−0.353648	+	0.935379i	$0.615059\pi$
$102$		0			0
$103$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$103$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$104$		−	16.9706i		−	1.66410i
$105$		0			0
$106$	−10.0000			−0.971286
$107$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$107$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$108$		0			0
$109$	17.3205	−	10.0000i	1.65900	−	0.957826i	0.685828	−	0.727764i	$-0.259440\pi$
$109$	17.3205	−	10.0000i	1.65900	−	0.957826i	0.973176	−	0.230063i	$-0.0738931\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−15.0000	+	15.0000i	−1.41108	+	1.41108i	−0.658505	+	0.752577i	$0.728811\pi$
$113$	−15.0000	+	15.0000i	−1.41108	+	1.41108i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$	10.0000	+	17.3205i	0.928477	+	1.60817i
$117$	−17.3867	+	4.65874i	−1.60740	+	0.430701i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$	−0.517638	+	1.93185i	−0.0468648	+	0.174902i
$123$		0			0
$124$		0			0
$125$	−6.36396	+	9.19239i	−0.569210	+	0.822192i
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$	2.92820	+	10.9282i	0.258819	+	0.965926i
$129$		0			0
$130$	−14.1962	−	12.5885i	−1.24508	−	1.10408i
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−11.3137	−	19.5959i	−0.970143	−	1.68034i
$137$	−2.56218	+	9.56218i	−0.218902	+	0.816952i	0.765855	+	0.643013i	$0.222316\pi$
$137$	−2.56218	+	9.56218i	−0.218902	+	0.816952i	−0.984757	+	0.173939i	$0.944351\pi$
$138$		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
$145$	21.9067	+	4.48288i	1.81925	+	0.372283i
$146$	8.48528			0.702247
$147$		0			0
$148$	14.0000	−	14.0000i	1.15079	−	1.15079i
$149$	12.1244	+	7.00000i	0.993266	+	0.573462i	0.906249	−	0.422744i	$-0.138933\pi$
$149$	12.1244	+	7.00000i	0.993266	+	0.573462i	0.0870170	+	0.996207i	$0.472267\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−16.9706	+	16.9706i	−1.37199	+	1.37199i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.69402	−	21.2504i	0.454432	−	1.69596i	−0.235320	−	0.971918i	$-0.575614\pi$
$157$	5.69402	−	21.2504i	0.454432	−	1.69596i	0.689752	−	0.724046i	$-0.257720\pi$
$158$		0			0
$159$		0			0
$160$	11.3137	+	5.65685i	0.894427	+	0.447214i
$161$		0			0
$162$	−9.00000	−	9.00000i	−0.707107	−	0.707107i
$163$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$163$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$164$	2.44949	−	1.41421i	0.191273	−	0.110432i
$165$		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
$169$		−	23.0000i		−	1.76923i
$170$	−24.7846	−	5.07180i	−1.90089	−	0.388989i
$171$		0			0
$172$		0			0
$173$	1.03528	+	3.86370i	0.0787106	+	0.293752i	0.994049	−	0.108933i	$-0.0347435\pi$
$173$	1.03528	+	3.86370i	0.0787106	+	0.293752i	−0.915338	+	0.402685i	$0.868077\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	25.1141	+	6.72930i	1.88238	+	0.504382i
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$	2.68973	−	13.1440i	0.200480	−	0.979698i
$181$	26.8701			1.99724			0.998618	−	0.0525588i	$-0.0167377\pi$
$181$	26.8701			1.99724			0.998618	+	0.0525588i	$0.0167377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.32628	−	22.0962i	−0.0975101	−	1.62454i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−6.83013	−	1.83013i	−0.491643	−	0.131735i	0.00447566	−	0.999990i	$-0.498575\pi$
$193$	−6.83013	−	1.83013i	−0.491643	−	0.131735i	−0.496119	+	0.868255i	$0.665242\pi$
$194$	−9.79796	−	5.65685i	−0.703452	−	0.406138i
$195$		0			0
$196$		0			0
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−15.0000	−	15.0000i	−1.06871	−	1.06871i	−0.0712470	−	0.997459i	$-0.522698\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	13.1244	−	5.26795i	0.928032	−	0.372500i
$201$		0			0
$202$	−12.7279	−	12.7279i	−0.895533	−	0.895533i
$203$		0			0
$204$		0			0
$205$	0.633975	−	3.09808i	0.0442787	−	0.216379i
$206$		0			0
$207$		0			0
$208$	6.21166	+	23.1822i	0.430701	+	1.60740i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	13.6603	−	3.66025i	0.938190	−	0.251387i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−20.0000	+	20.0000i	−1.35457	+	1.35457i
$219$		0			0
$220$		0			0
$221$	−24.0000	−	41.5692i	−1.61441	−	2.79625i
$222$		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
$225$	−9.00000	−	12.0000i	−0.600000	−	0.800000i
$226$	15.0000	−	25.9808i	0.997785	−	1.72821i
$227$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$227$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$	−20.8207	−	12.0208i	−1.37587	−	0.794358i	−0.384209	−	0.923246i	$-0.625526\pi$
$229$	−20.8207	−	12.0208i	−1.37587	−	0.794358i	−0.991659	+	0.128888i	$0.958859\pi$
$230$		0			0
$231$		0			0
$232$	−20.0000	−	20.0000i	−1.31306	−	1.31306i
$233$	−7.68653	−	28.6865i	−0.503562	−	1.87932i	−0.475509	−	0.879711i	$-0.657736\pi$
$233$	−7.68653	−	28.6865i	−0.503562	−	1.87932i	−0.0280525	−	0.999606i	$-0.508931\pi$
$234$	22.0454	−	12.7279i	1.44115	−	0.832050i
$235$		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
$241$	7.77817	+	13.4722i	0.501036	+	0.867820i	0.999999	+	0.00119700i	$0.000381016\pi$
$241$	7.77817	+	13.4722i	0.501036	+	0.867820i	−0.498963	+	0.866623i	$0.666286\pi$
$242$	4.02628	−	15.0263i	0.258819	−	0.965926i
$243$		0			0
$244$		−	2.82843i		−	0.181071i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	5.32868	−	14.8864i	0.337016	−	0.941499i
$251$		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
$257$	30.9096	−	8.28221i	1.92809	−	0.516630i	0.947900	−	0.318568i	$-0.103202\pi$
$257$	30.9096	−	8.28221i	1.92809	−	0.516630i	0.980189	−	0.198062i	$-0.0634648\pi$
$258$		0			0
$259$		0			0
$260$	24.0000	+	12.0000i	1.48842	+	0.744208i
$261$	−15.0000	+	25.9808i	−0.928477	+	1.60817i
$262$		0			0
$263$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$263$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$264$		0			0
$265$	7.07107	−	14.1421i	0.434372	−	0.868744i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−28.1691	+	16.2635i	−1.71750	+	0.991600i	−0.794082	+	0.607811i	$0.792048\pi$
$269$	−28.1691	+	16.2635i	−1.71750	+	0.991600i	−0.923421	+	0.383789i	$0.874619\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	22.6274	+	22.6274i	1.37199	+	1.37199i
$273$		0			0
$274$		−	14.0000i		−	0.845771i
$275$		0			0
$276$		0			0
$277$	−1.83013	+	6.83013i	−0.109962	+	0.410383i	−0.998861	−	0.0477206i	$-0.984804\pi$
$277$	−1.83013	+	6.83013i	−0.109962	+	0.410383i	0.888899	+	0.458103i	$0.151471\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$283$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−12.0000	+	12.0000i	−0.707107	+	0.707107i
$289$	−40.7032	−	23.5000i	−2.39431	−	1.38235i
$290$	−31.5660	+	1.89469i	−1.85362	+	0.111260i
$291$		0			0
$292$	−11.5911	+	3.10583i	−0.678318	+	0.181755i
$293$	−2.82843	+	2.82843i	−0.165238	+	0.165238i	−0.784883	−	0.619644i	$-0.787277\pi$
$293$	−2.82843	+	2.82843i	−0.165238	+	0.165238i	0.619644	+	0.784883i	$0.287277\pi$
$294$		0			0
$295$		0			0
$296$	−14.0000	+	24.2487i	−0.813733	+	1.40943i
$297$		0			0
$298$	−19.1244	−	5.12436i	−1.10784	−	0.296846i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.36603	−	2.09808i	−0.135478	−	0.120135i
$306$	16.9706	−	29.3939i	0.970143	−	1.68034i
$307$		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	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	6.72930	+	25.1141i	0.380362	+	1.41953i	0.845349	+	0.534214i	$0.179392\pi$
$313$	6.72930	+	25.1141i	0.380362	+	1.41953i	−0.464987	+	0.885317i	$0.653941\pi$
$314$			31.1127i			1.75579i
$315$		0			0
$316$		0			0
$317$	34.1506	−	9.15064i	1.91809	−	0.513951i	0.928208	−	0.372061i	$-0.121349\pi$
$317$	34.1506	−	9.15064i	1.91809	−	0.513951i	0.989882	−	0.141890i	$-0.0453179\pi$
$318$		0			0
$319$		0			0
$320$	−17.5254	−	3.58630i	−0.979698	−	0.200480i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	15.5885	+	9.00000i	0.866025	+	0.500000i
$325$	27.8410	−	11.1750i	1.54434	−	0.619878i
$326$		0			0
$327$		0			0
$328$	−2.82843	+	2.82843i	−0.156174	+	0.156174i
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	28.6865	+	7.68653i	1.57201	+	0.421219i
$334$		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	$0.836901\pi$
$338$	8.41858	+	31.4186i	0.457911	+	1.70895i
$339$		0			0
$340$	35.7128	−	2.14359i	1.93680	−	0.116253i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−2.82843	−	4.89898i	−0.152057	−	0.263371i
$347$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$347$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$348$		0			0
$349$		−	18.3848i		−	0.984115i	−0.870563	−	0.492057i	$-0.836245\pi$
$349$		−	18.3848i		−	0.984115i	0.870563	−	0.492057i	$-0.163755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.4548	+	4.14110i	0.822577	+	0.220409i	0.645473	−	0.763783i	$-0.276660\pi$
$353$	15.4548	+	4.14110i	0.822577	+	0.220409i	0.177104	+	0.984192i	$0.443327\pi$
$354$		0			0
$355$		0			0
$356$	−36.7696			−1.94878
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$	1.13681	+	18.9396i	0.0599153	+	0.998203i
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$	−36.7052	+	9.83512i	−1.92918	+	0.516923i
$363$		0			0
$364$		0			0
$365$	−6.00000	+	12.0000i	−0.314054	+	0.628109i
$366$		0			0
$367$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$367$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$368$		0			0
$369$	3.67423	+	2.12132i	0.191273	+	0.110432i
$370$	9.89949	+	29.6985i	0.514650	+	1.54395i
$371$		0			0
$372$		0			0
$373$	−9.15064	−	34.1506i	−0.473802	−	1.76825i	−0.625917	−	0.779890i	$-0.715275\pi$
$373$	−9.15064	−	34.1506i	−0.473802	−	1.76825i	0.152115	−	0.988363i	$-0.451392\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−42.4264	−	42.4264i	−2.18507	−	2.18507i
$378$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$383$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		0			0
$388$	15.4548	+	4.14110i	0.784599	+	0.210233i
$389$	−17.3205	+	10.0000i	−0.878185	+	0.507020i	−0.870059	−	0.492947i	$-0.835920\pi$
$389$	−17.3205	+	10.0000i	−0.878185	+	0.507020i	−0.00812520	+	0.999967i	$0.502586\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	25.9808	+	15.0000i	1.30889	+	0.755689i
$395$		0			0
$396$		0			0
$397$	−11.5911	+	3.10583i	−0.581741	+	0.155877i	−0.537676	−	0.843152i	$-0.680698\pi$
$397$	−11.5911	+	3.10583i	−0.581741	+	0.155877i	−0.0440652	+	0.999029i	$0.514031\pi$
$398$		0			0
$399$		0			0
$400$	−16.0000	+	12.0000i	−0.800000	+	0.600000i
$401$	−20.0000	+	34.6410i	−0.998752	+	1.72989i	−0.456129	+	0.889914i	$0.650764\pi$
$401$	−20.0000	+	34.6410i	−0.998752	+	1.72989i	−0.542623	+	0.839976i	$0.682569\pi$
$402$		0			0
$403$		0			0
$404$	22.0454	+	12.7279i	1.09680	+	0.633238i
$405$	19.0919	−	6.36396i	0.948683	−	0.316228i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−20.8207	+	12.0208i	−1.02952	+	0.594391i	−0.916846	−	0.399241i	$-0.869274\pi$
$409$	−20.8207	+	12.0208i	−1.02952	+	0.594391i	−0.112670	+	0.993632i	$0.535940\pi$
$410$	0.267949	+	4.46410i	0.0132331	+	0.220466i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−16.9706	−	29.3939i	−0.832050	−	1.44115i
$417$		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	$-0.260972\pi$
$421$	28.0000			1.36464			0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$		0			0
$424$	−17.3205	+	10.0000i	−0.841158	+	0.485643i
$425$	24.6980	−	31.4644i	1.19803	−	1.52625i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	24.0416	−	24.0416i	1.15537	−	1.15537i	0.169907	−	0.985460i	$-0.445653\pi$
$433$	24.0416	−	24.0416i	1.15537	−	1.15537i	0.985460	−	0.169907i	$-0.0543467\pi$
$434$		0			0
$435$		0			0
$436$	20.0000	−	34.6410i	0.957826	−	1.65900i
$437$		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
$442$	48.0000	+	48.0000i	2.28313	+	2.28313i
$443$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$443$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$444$		0			0
$445$	−27.2750	+	30.7583i	−1.29296	+	1.45808i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14.0000i		−	0.660701i	−0.943858	−	0.330350i	$-0.892833\pi$
$449$		−	14.0000i		−	0.660701i	0.943858	−	0.330350i	$-0.107167\pi$
$450$	16.6865	+	13.0981i	0.786611	+	0.617449i
$451$		0			0
$452$	−10.9808	+	40.9808i	−0.516492	+	1.92757i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−34.1506	+	9.15064i	−1.59750	+	0.428049i	−0.944286	−	0.329125i	$-0.893246\pi$
$457$	−34.1506	+	9.15064i	−1.59750	+	0.428049i	−0.653213	+	0.757174i	$0.726579\pi$
$458$	32.8415	+	8.79985i	1.53458	+	0.411190i
$459$		0			0
$460$		0			0
$461$	12.7279			0.592798			0.296399	−	0.955064i	$-0.404214\pi$
$461$	12.7279			0.592798			0.296399	+	0.955064i	$0.404214\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	34.6410	+	20.0000i	1.60817	+	0.928477i
$465$		0			0
$466$	21.0000	+	36.3731i	0.972806	+	1.68495i
$467$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$467$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$468$	−25.4558	+	25.4558i	−1.17670	+	1.17670i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.0000	+	15.0000i	0.686803	+	0.686803i
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−29.6985	+	51.4393i	−1.35413	+	2.34543i
$482$	−15.5563	−	15.5563i	−0.708572	−	0.708572i
$483$		0			0
$484$			22.0000i			1.00000i
$485$	14.9282	−	9.85641i	0.677855	−	0.447556i
$486$		0			0
$487$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$487$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$488$	1.03528	+	3.86370i	0.0468648	+	0.174902i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	−77.2741	−	20.7055i	−3.48025	−	0.932530i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$499$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$500$	−1.83032	+	22.2856i	−0.0818542	+	0.996644i
$501$		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
$505$	27.0000	−	9.00000i	1.20148	−	0.400495i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	33.0681	+	19.0919i	1.46572	+	0.846233i	0.999266	−	0.0383134i	$-0.0121985\pi$
$509$	33.0681	+	19.0919i	1.46572	+	0.846233i	0.466453	+	0.884546i	$0.345532\pi$
$510$		0			0
$511$		0			0
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	−39.1918	+	22.6274i	−1.72868	+	0.998053i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−37.1769	−	7.60770i	−1.63032	−	0.333620i
$521$	21.9203	+	37.9671i	0.960346	+	1.66337i	0.721630	+	0.692279i	$0.243393\pi$
$521$	21.9203	+	37.9671i	0.960346	+	1.66337i	0.238716	+	0.971089i	$0.423273\pi$
$522$	10.9808	−	40.9808i	0.480615	−	1.79368i
$523$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$523$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−19.9186	+	11.5000i	−0.866025	+	0.500000i
$530$	−4.48288	+	21.9067i	−0.194724	+	0.951567i
$531$		0			0
$532$		0			0
$533$	−6.00000	+	6.00000i	−0.259889	+	0.259889i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	32.5269	−	32.5269i	1.40233	−	1.40233i
$539$		0			0
$540$		0			0
$541$	21.0000	−	36.3731i	0.902861	−	1.56380i	0.0790969	−	0.996867i	$-0.474796\pi$
$541$	21.0000	−	36.3731i	0.902861	−	1.56380i	0.823764	−	0.566933i	$-0.191870\pi$
$542$		0			0
$543$		0			0
$544$	−39.1918	−	22.6274i	−1.68034	−	0.970143i
$545$	−14.1421	−	42.4264i	−0.605783	−	1.81735i
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$	5.12436	+	19.1244i	0.218902	+	0.816952i
$549$	3.67423	−	2.12132i	0.156813	−	0.0905357i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	10.0000i		−	0.424859i
$555$		0			0
$556$		0			0
$557$	1.83013	−	6.83013i	0.0775450	−	0.289402i	−0.916253	−	0.400599i	$-0.868802\pi$
$557$	1.83013	−	6.83013i	0.0775450	−	0.289402i	0.993798	+	0.111198i	$0.0354686\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−13.6603	+	3.66025i	−0.576223	+	0.154398i
$563$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$563$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$564$		0			0
$565$	26.1357	+	39.5844i	1.09954	+	1.66533i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−34.6410	−	20.0000i	−1.45223	−	0.838444i	−0.453619	−	0.891196i	$-0.649867\pi$
$569$	−34.6410	−	20.0000i	−1.45223	−	0.838444i	−0.998608	+	0.0527519i	$0.983201\pi$
$570$		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
$575$		0			0
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$	−0.517638	+	1.93185i	−0.0215496	+	0.0804240i	−0.975863	−	0.218383i	$-0.929922\pi$
$577$	−0.517638	+	1.93185i	−0.0215496	+	0.0804240i	0.954314	+	0.298807i	$0.0965886\pi$
$578$	64.2032	+	17.2032i	2.67050	+	0.715559i
$579$		0			0
$580$	42.4264	−	14.1421i	1.76166	−	0.587220i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	14.6969	−	8.48528i	0.608164	−	0.351123i
$585$	2.41154	+	40.1769i	0.0997050	+	1.66111i
$586$	2.82843	−	4.89898i	0.116841	−	0.202375i
$587$		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
$592$	10.2487	−	38.2487i	0.421219	−	1.57201i
$593$	−11.9057	−	44.4326i	−0.488908	−	1.82463i	−0.561780	−	0.827286i	$-0.689883\pi$
$593$	−11.9057	−	44.4326i	−0.488908	−	1.82463i	0.0728728	−	0.997341i	$-0.476783\pi$
$594$		0			0
$595$		0			0
$596$	28.0000			1.14692
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−41.0122			−1.67292			−0.836461	−	0.548026i	$-0.815379\pi$
$601$	−41.0122			−1.67292			−0.836461	+	0.548026i	$0.815379\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	18.4034	+	16.3192i	0.748203	+	0.663470i
$606$		0			0
$607$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$607$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$608$		0			0
$609$		0			0
$610$	4.00000	+	2.00000i	0.161955	+	0.0809776i
$611$		0			0
$612$	−12.4233	+	46.3644i	−0.502183	+	1.87417i
$613$	−1.36603	−	0.366025i	−0.0551732	−	0.0147836i	0.231127	−	0.972924i	$-0.425759\pi$
$613$	−1.36603	−	0.366025i	−0.0551732	−	0.0147836i	−0.286300	+	0.958140i	$0.592425\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	−	3.00000i	−0.120775	−	0.120775i	0.644136	−	0.764911i	$-0.277217\pi$
$617$	−3.00000	−	3.00000i	−0.120775	−	0.120775i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.2846	+	18.0622i	0.691384	+	0.722487i
$626$	−18.3848	−	31.8434i	−0.734803	−	1.27272i
$627$		0			0
$628$	−11.3880	−	42.5007i	−0.454432	−	1.69596i
$629$			79.1960i			3.15775i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−43.3013	+	25.0000i	−1.71971	+	0.992877i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	25.2528	−	1.51575i	0.998203	−	0.0599153i
$641$	−25.0000	−	43.3013i	−0.987441	−	1.71030i	−0.630544	−	0.776153i	$-0.717168\pi$
$641$	−25.0000	−	43.3013i	−0.987441	−	1.71030i	−0.356897	−	0.934144i	$-0.616165\pi$
$642$		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		0.258819	−	0.965926i	$-0.416667\pi$
$647$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$648$	−24.5885	−	6.58846i	−0.965926	−	0.258819i
$649$		0			0
$650$	−33.9411	+	25.4558i	−1.33128	+	0.998460i
$651$		0			0
$652$		0			0
$653$	−3.29423	−	12.2942i	−0.128913	−	0.481110i	0.871036	−	0.491220i	$-0.163449\pi$
$653$	−3.29423	−	12.2942i	−0.128913	−	0.481110i	−0.999949	+	0.0101092i	$0.996782\pi$
$654$		0			0
$655$		0			0
$656$	2.82843	−	4.89898i	0.110432	−	0.191273i
$657$	−12.7279	−	12.7279i	−0.496564	−	0.496564i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	13.4350	+	23.2702i	0.522562	+	0.905104i	0.999655	+	0.0262514i	$0.00835704\pi$
$661$	13.4350	+	23.2702i	0.522562	+	0.905104i	−0.477093	+	0.878853i	$0.658310\pi$
$662$		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	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	11.0000	−	11.0000i	0.424019	−	0.424019i	0.886585	+	0.462566i	$0.153071\pi$
$674$	12.1244	+	7.00000i	0.467013	+	0.269630i
$675$		0			0
$676$	−23.0000	−	39.8372i	−0.884615	−	1.53220i
$677$	−1.93185	+	0.517638i	−0.0742471	+	0.0198944i	−0.295751	−	0.955265i	$-0.595570\pi$
$677$	−1.93185	+	0.517638i	−0.0742471	+	0.0198944i	0.221504	+	0.975159i	$0.428903\pi$
$678$		0			0
$679$		0			0
$680$	−48.0000	+	16.0000i	−1.84072	+	0.613572i
$681$		0			0
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$	19.7990	+	9.89949i	0.756481	+	0.378240i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.7423	+	21.2132i	−1.39977	+	0.808159i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	5.65685	+	5.65685i	0.215041	+	0.215041i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.92820	+	10.9282i	−0.110914	+	0.413935i
$698$	6.72930	+	25.1141i	0.254708	+	0.950582i
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−22.6274			−0.851594
$707$		0			0
$708$		0			0
$709$	25.9808	+	15.0000i	0.975728	+	0.563337i	0.900978	−	0.433865i	$-0.142851\pi$
$709$	25.9808	+	15.0000i	0.975728	+	0.563337i	0.0747503	+	0.997202i	$0.476184\pi$
$710$		0			0
$711$		0			0
$712$	50.2281	−	13.4586i	1.88238	−	0.504382i
$713$		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	$0.166667\pi$
$720$	−8.48528	−	25.4558i	−0.316228	−	0.948683i
$721$		0			0
$722$	−19.0000	−	19.0000i	−0.707107	−	0.707107i
$723$		0			0
$724$	46.5403	−	26.8701i	1.72966	−	0.998618i
$725$	19.6410	−	45.9808i	0.729449	−	1.70768i
$726$		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
$730$	3.80385	−	18.5885i	0.140787	−	0.687990i
$731$		0			0
$732$		0			0
$733$	−13.9762	−	52.1600i	−0.516224	−	1.92657i	−0.329466	−	0.944167i	$-0.606869\pi$
$733$	−13.9762	−	52.1600i	−0.516224	−	1.92657i	−0.186757	−	0.982406i	$-0.559798\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−5.79555	−	1.55291i	−0.213337	−	0.0571636i
$739$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$739$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$740$	−24.3934	−	36.9454i	−0.896718	−	1.35814i
$741$		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
$745$	20.7699	−	23.4225i	0.760950	−	0.858132i
$746$	25.0000	+	43.3013i	0.915315	+	1.58537i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$	73.4847	+	42.4264i	2.67615	+	1.54508i
$755$		0			0
$756$		0			0
$757$	17.0000	+	17.0000i	0.617876	+	0.617876i	0.944986	−	0.327111i	$-0.106075\pi$
$757$	17.0000	+	17.0000i	0.617876	+	0.617876i	−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.5772	−	47.7650i	0.999671	−	1.73148i	0.477637	−	0.878557i	$-0.341493\pi$
$761$	27.5772	−	47.7650i	0.999671	−	1.73148i	0.522034	−	0.852925i	$-0.325173\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	29.5692	+	44.7846i	1.06908	+	1.61919i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	52.3259i		−	1.88692i	−0.331486	−	0.943460i	$-0.607550\pi$
$769$		−	52.3259i		−	1.88692i	0.331486	−	0.943460i	$-0.392450\pi$
$770$		0			0
$771$		0			0
$772$	−13.6603	+	3.66025i	−0.491643	+	0.131735i
$773$	32.8415	+	8.79985i	1.18123	+	0.316509i	0.795413	−	0.606068i	$-0.207254\pi$
$773$	32.8415	+	8.79985i	1.18123	+	0.316509i	0.385813	+	0.922577i	$0.373921\pi$
$774$		0			0
$775$		0			0
$776$	−22.6274			−0.812277
$777$		0			0
$778$	20.0000	−	20.0000i	0.717035	−	0.717035i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−44.0000	−	22.0000i	−1.57043	−	0.785214i
$786$		0			0
$787$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$787$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$788$	−40.9808	−	10.9808i	−1.45988	−	0.391173i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.19615	+	8.19615i	0.0779877	+	0.291054i
$794$	14.6969	−	8.48528i	0.521575	−	0.301131i
$795$		0			0
$796$		0			0
$797$	36.7696	+	36.7696i	1.30244	+	1.30244i	0.926739	+	0.375705i	$0.122599\pi$
$797$	36.7696	+	36.7696i	1.30244	+	1.30244i	0.375705	+	0.926739i	$0.377401\pi$
$798$		0			0
$799$		0			0
$800$	17.4641	−	22.2487i	0.617449	−	0.786611i
$801$	−27.5772	−	47.7650i	−0.974391	−	1.68770i
$802$	14.6410	−	54.6410i	0.516992	−	1.92944i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−34.7733	−	9.31749i	−1.22332	−	0.327788i
$809$	48.4974	−	28.0000i	1.70508	−	0.984428i	0.764644	−	0.644453i	$-0.222915\pi$
$809$	48.4974	−	28.0000i	1.70508	−	0.984428i	0.940435	−	0.339975i	$-0.110418\pi$
$810$	−23.7506	+	15.6814i	−0.834512	+	0.550990i
$811$		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
$818$	24.0416	−	24.0416i	0.840596	−	0.840596i
$819$		0			0
$820$	−2.00000	−	6.00000i	−0.0698430	−	0.209529i
$821$	−14.0000	+	24.2487i	−0.488603	+	0.846286i	−0.999914	−	0.0131101i	$-0.995827\pi$
$821$	−14.0000	+	24.2487i	−0.488603	+	0.846286i	0.511311	+	0.859396i	$0.329160\pi$
$822$		0			0
$823$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$823$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$824$		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
$829$	45.3156	−	26.1630i	1.57387	−	0.908677i	0.578187	−	0.815904i	$-0.303760\pi$
$829$	45.3156	−	26.1630i	1.57387	−	0.908677i	0.995687	−	0.0927727i	$-0.0295730\pi$
$830$		0			0
$831$		0			0
$832$	33.9411	+	33.9411i	1.17670	+	1.17670i
$833$		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
$841$	−71.0000			−2.44828
$842$	−38.2487	+	10.2487i	−1.31814	+	0.353194i
$843$		0			0
$844$		0			0
$845$	−50.3854	−	10.3106i	−1.73331	−	0.354696i
$846$		0			0
$847$		0			0
$848$	20.0000	−	20.0000i	0.686803	−	0.686803i
$849$		0			0
$850$	−22.2213	+	52.0213i	−0.762183	+	1.78432i
$851$		0			0
$852$		0			0
$853$	−25.4558	+	25.4558i	−0.871592	+	0.871592i	−0.992646	−	0.121054i	$-0.961372\pi$
$853$	−25.4558	+	25.4558i	−0.871592	+	0.871592i	0.121054	+	0.992646i	$0.461372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.0115	−	56.0237i	0.512783	−	1.91373i	0.124410	−	0.992231i	$-0.460296\pi$
$857$	15.0115	−	56.0237i	0.512783	−	1.91373i	0.388373	−	0.921502i	$-0.373037\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$863$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$864$		0			0
$865$	8.92820	−	0.535898i	0.303568	−	0.0182211i
$866$	−24.0416	+	41.6413i	−0.816968	+	1.41503i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−14.6410	+	54.6410i	−0.495807	+	1.85038i
$873$	6.21166	+	23.1822i	0.210233	+	0.784599i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.4186	−	8.41858i	1.06093	−	0.284275i	0.314169	−	0.949367i	$-0.398274\pi$
$877$	31.4186	−	8.41858i	1.06093	−	0.284275i	0.746762	+	0.665092i	$0.231608\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.7279			0.428815			0.214407	−	0.976744i	$-0.431218\pi$
$881$	12.7279			0.428815			0.214407	+	0.976744i	$0.431218\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$	−83.1384	−	48.0000i	−2.79625	−	1.61441i
$885$		0			0
$886$		0			0
$887$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$887$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$888$		0			0
$889$		0			0
$890$	26.0000	−	52.0000i	0.871522	−	1.74304i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	5.12436	+	19.1244i	0.171002	+	0.638188i
$899$		0			0
$900$	−27.5885	−	11.7846i	−0.919615	−	0.392820i
$901$	−28.2843	+	48.9898i	−0.942286	+	1.63209i
$902$		0			0
$903$		0			0
$904$		−	60.0000i		−	1.99557i
$905$	12.0455	−	58.8634i	0.400407	−	1.95669i
$906$		0			0
$907$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$907$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$908$		0			0
$909$			38.1838i			1.26648i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	43.3013	−	25.0000i	1.43228	−	0.826927i
$915$		0			0
$916$	−48.0833			−1.58872
$917$		0			0
$918$		0			0
$919$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$919$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$920$		0			0
$921$		0			0
$922$	−17.3867	+	4.65874i	−0.572599	+	0.153428i
$923$		0			0
$924$		0			0
$925$	−49.0000	−	7.00000i	−1.61111	−	0.230159i
$926$		0			0
$927$		0			0
$928$	−54.6410	−	14.6410i	−1.79368	−	0.480615i
$929$	3.67423	+	2.12132i	0.120548	+	0.0695983i	0.559061	−	0.829126i	$-0.311162\pi$
$929$	3.67423	+	2.12132i	0.120548	+	0.0695983i	−0.438514	+	0.898725i	$0.644495\pi$
$930$		0			0
$931$		0			0
$932$	−42.0000	−	42.0000i	−1.37576	−	1.37576i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	25.4558	−	44.0908i	0.832050	−	1.44115i
$937$	33.9411	+	33.9411i	1.10881	+	1.10881i	0.993307	+	0.115501i	$0.0368473\pi$
$937$	33.9411	+	33.9411i	1.10881	+	1.10881i	0.115501	+	0.993307i	$0.463153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.4350	+	23.2702i	0.437969	+	0.758585i	0.997533	−	0.0702023i	$-0.0223645\pi$
$941$	13.4350	+	23.2702i	0.437969	+	0.758585i	−0.559563	+	0.828788i	$0.689031\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$947$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$948$		0			0
$949$	31.1769	−	18.0000i	1.01205	−	0.584305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.0000	−	15.0000i	0.485898	−	0.485898i	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	15.0000	−	15.0000i	0.485898	−	0.485898i	0.907009	+	0.421111i	$0.138360\pi$
$954$	−25.9808	−	15.0000i	−0.841158	−	0.485643i
$955$		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
$962$	21.7408	−	81.1378i	0.700952	−	2.61599i
$963$		0			0
$964$	26.9444	+	15.5563i	0.867820	+	0.501036i
$965$	−7.07107	+	14.1421i	−0.227626	+	0.455251i
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−8.05256	−	30.0526i	−0.258819	−	0.965926i
$969$		0			0
$970$	−16.7846	+	18.9282i	−0.538921	+	0.607748i
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−2.82843	−	4.89898i	−0.0905357	−	0.156813i
$977$	9.88269	−	36.8827i	0.316175	−	1.17998i	−0.606715	−	0.794919i	$-0.707513\pi$
$977$	9.88269	−	36.8827i	0.316175	−	1.17998i	0.922890	−	0.385063i	$-0.125820\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	60.0000			1.91565
$982$		0			0
$983$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$983$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$984$		0			0
$985$	−39.5844	+	26.1357i	−1.26126	+	0.832754i
$986$	113.137			3.60302
$987$		0			0
$988$		0			0
$989$		0			0
$990$		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
$996$		0			0
$997$	−3.10583	+	11.5911i	−0.0983626	+	0.367094i	−0.997508	−	0.0705563i	$-0.977523\pi$
$997$	−3.10583	+	11.5911i	−0.0983626	+	0.367094i	0.899145	+	0.437650i	$0.144189\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.g.667.2		8
4.3	odd	2	CM	980.2.x.g.667.2		8
5.3	odd	4	inner	980.2.x.g.863.2		8
7.2	even	3		980.2.k.f.687.1	✓	4
7.3	odd	6	inner	980.2.x.g.67.1		8
7.4	even	3	inner	980.2.x.g.67.2		8
7.5	odd	6		980.2.k.f.687.2	yes	4
7.6	odd	2	inner	980.2.x.g.667.1		8
20.3	even	4	inner	980.2.x.g.863.2		8
28.3	even	6	inner	980.2.x.g.67.1		8
28.11	odd	6	inner	980.2.x.g.67.2		8
28.19	even	6		980.2.k.f.687.2	yes	4
28.23	odd	6		980.2.k.f.687.1	✓	4
28.27	even	2	inner	980.2.x.g.667.1		8
35.3	even	12	inner	980.2.x.g.263.1		8
35.13	even	4	inner	980.2.x.g.863.1		8
35.18	odd	12	inner	980.2.x.g.263.2		8
35.23	odd	12		980.2.k.f.883.1	yes	4
35.33	even	12		980.2.k.f.883.2	yes	4
140.3	odd	12	inner	980.2.x.g.263.1		8
140.23	even	12		980.2.k.f.883.1	yes	4
140.83	odd	4	inner	980.2.x.g.863.1		8
140.103	odd	12		980.2.k.f.883.2	yes	4
140.123	even	12	inner	980.2.x.g.263.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.k.f.687.1	✓	4	7.2	even	3
980.2.k.f.687.1	✓	4	28.23	odd	6
980.2.k.f.687.2	yes	4	7.5	odd	6
980.2.k.f.687.2	yes	4	28.19	even	6
980.2.k.f.883.1	yes	4	35.23	odd	12
980.2.k.f.883.1	yes	4	140.23	even	12
980.2.k.f.883.2	yes	4	35.33	even	12
980.2.k.f.883.2	yes	4	140.103	odd	12
980.2.x.g.67.1		8	7.3	odd	6	inner
980.2.x.g.67.1		8	28.3	even	6	inner
980.2.x.g.67.2		8	7.4	even	3	inner
980.2.x.g.67.2		8	28.11	odd	6	inner
980.2.x.g.263.1		8	35.3	even	12	inner
980.2.x.g.263.1		8	140.3	odd	12	inner
980.2.x.g.263.2		8	35.18	odd	12	inner
980.2.x.g.263.2		8	140.123	even	12	inner
980.2.x.g.667.1		8	7.6	odd	2	inner
980.2.x.g.667.1		8	28.27	even	2	inner
980.2.x.g.667.2		8	1.1	even	1	trivial
980.2.x.g.667.2		8	4.3	odd	2	CM
980.2.x.g.863.1		8	35.13	even	4	inner
980.2.x.g.863.1		8	140.83	odd	4	inner
980.2.x.g.863.2		8	5.3	odd	4	inner
980.2.x.g.863.2		8	20.3	even	4	inner

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)

\(f(q)\)	\(=\)	\(q+(-1.36603 + 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(0.448288 - 2.19067i) q^{5} +(-2.00000 + 2.00000i) q^{8} +(2.59808 + 1.50000i) q^{9} +(0.189469 + 3.15660i) q^{10} +(-4.24264 + 4.24264i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-2.07055 + 7.72741i) q^{17} +(-4.09808 - 1.09808i) q^{18} +(-1.41421 - 4.24264i) q^{20} +(-4.59808 - 1.96410i) q^{25} +(4.24264 - 7.34847i) q^{26} +10.0000i q^{29} +(-1.46410 + 5.46410i) q^{32} -11.3137i q^{34} +6.00000 q^{36} +(9.56218 - 2.56218i) q^{37} +(3.48477 + 5.27792i) q^{40} +1.41421 q^{41} +(4.45069 - 5.01910i) q^{45} +(7.00000 + 1.00000i) q^{50} +(-3.10583 + 11.5911i) q^{52} +(6.83013 + 1.83013i) q^{53} +(-3.66025 - 13.6603i) q^{58} +(0.707107 - 1.22474i) q^{61} -8.00000i q^{64} +(7.39230 + 11.1962i) q^{65} +(4.14110 + 15.4548i) q^{68} +(-8.19615 + 2.19615i) q^{72} +(-5.79555 - 1.55291i) q^{73} +(-12.1244 + 7.00000i) q^{74} +(-6.69213 - 5.93426i) q^{80} +(4.50000 + 7.79423i) q^{81} +(-1.93185 + 0.517638i) q^{82} +(16.0000 + 8.00000i) q^{85} +(-15.9217 - 9.19239i) q^{89} +(-4.24264 + 8.48528i) q^{90} +(5.65685 + 5.65685i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} - 16 q^{8} + 16 q^{16} - 12 q^{18} - 16 q^{25} + 16 q^{32} + 48 q^{36} + 28 q^{37} + 56 q^{50} + 20 q^{53} + 40 q^{58} - 24 q^{65} - 24 q^{72} + 36 q^{81} + 128 q^{85}+O(q^{100}) \) 8 * q - 4 * q^2 - 16 * q^8 + 16 * q^16 - 12 * q^18 - 16 * q^25 + 16 * q^32 + 48 * q^36 + 28 * q^37 + 56 * q^50 + 20 * q^53 + 40 * q^58 - 24 * q^65 - 24 * q^72 + 36 * q^81 + 128 * q^85

Introduction

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