LMFDB - Embedded newform 2200.1.fr.c.1443.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,1,Mod(107,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 10, 5, 6]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2200.107");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2200.fr (of order $20$, degree $8$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.09794302779$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$4$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{120})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 $ x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{30}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{30} - \cdots)$

Embedding invariants

Embedding label			1443.3
Root			$-0.0523360 - 0.998630i$ of defining polynomial
Character	$\chi$	$=$	2200.1443
Dual form			2200.1.fr.c.2107.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.891007 - 0.453990i) q^{2} +(0.410704 + 0.0650491i) q^{3} +(0.587785 - 0.809017i) q^{4} +(0.395472 - 0.128496i) q^{6} +(0.156434 - 0.987688i) q^{8} +(-0.786610 - 0.255585i) q^{9} +O(q^{10})$	\(q+(0.891007 - 0.453990i) q^{2} +(0.410704 + 0.0650491i) q^{3} +(0.587785 - 0.809017i) q^{4} +(0.395472 - 0.128496i) q^{6} +(0.156434 - 0.987688i) q^{8} +(-0.786610 - 0.255585i) q^{9} +(0.913545 + 0.406737i) q^{11} +(0.294032 - 0.294032i) q^{12} +(-0.309017 - 0.951057i) q^{16} +(0.0949099 - 0.186271i) q^{17} +(-0.816908 + 0.129386i) q^{18} +(0.658114 - 0.478148i) q^{19} +(0.998630 - 0.0523360i) q^{22} +(0.128496 - 0.395472i) q^{24} +(-0.676940 - 0.344918i) q^{27} +(-0.707107 - 0.707107i) q^{32} +(0.348739 + 0.226474i) q^{33} -0.209057i q^{34} +(-0.669131 + 0.486152i) q^{36} +(0.369309 - 0.724810i) q^{38} +(0.873619 + 1.20243i) q^{41} +(-0.437016 + 0.437016i) q^{43} +(0.866025 - 0.500000i) q^{44} +(-0.0650491 - 0.410704i) q^{48} +(-0.951057 + 0.309017i) q^{49} +(0.0510966 - 0.0703285i) q^{51} -0.759747 q^{54} +(0.301393 - 0.153567i) q^{57} +(0.951057 - 1.30902i) q^{59} +(-0.951057 - 0.309017i) q^{64} +(0.413545 + 0.0434654i) q^{66} +(-1.40647 + 1.40647i) q^{67} +(-0.0949099 - 0.186271i) q^{68} +(-0.375491 + 0.736943i) q^{72} +(-1.32178 + 0.209350i) q^{73} -0.813473i q^{76} +(0.413545 + 0.300458i) q^{81} +(1.32429 + 0.674761i) q^{82} +(-1.19240 - 0.607558i) q^{83} +(-0.190983 + 0.587785i) q^{86} +(0.544639 - 0.838671i) q^{88} +1.95630i q^{89} +(-0.244415 - 0.336408i) q^{96} +(0.863541 + 1.69480i) q^{97} +(-0.707107 + 0.707107i) q^{98} +(-0.614648 - 0.553432i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 20 q^{6}+O(q^{10}) $ 32 * q + 20 * q^6	$ 32 q + 20 q^{6} + 4 q^{11} + 8 q^{16} - 4 q^{36} + 20 q^{51} - 12 q^{66} - 12 q^{81} - 24 q^{86}+O(q^{100}) $ 32 * q + 20 * q^6 + 4 * q^11 + 8 * q^16 - 4 * q^36 + 20 * q^51 - 12 * q^66 - 12 * q^81 - 24 * q^86

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$	$e\left(\frac{1}{10}\right)$

Coefficient data

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

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

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

$2$	0.891007	−	0.453990i	0.891007	−	0.453990i
$2$	0.891007	−	0.453990i	0.891007	−	0.453990i
$3$	0.410704	+	0.0650491i	0.410704	+	0.0650491i	0.358368	−	0.933580i	$-0.383333\pi$
$3$	0.410704	+	0.0650491i	0.410704	+	0.0650491i	0.0523360	+	0.998630i	$0.483333\pi$
$4$	0.587785	−	0.809017i	0.587785	−	0.809017i
$5$		0			0
$5$		0			0
$6$	0.395472	−	0.128496i	0.395472	−	0.128496i
$7$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$7$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$8$	0.156434	−	0.987688i	0.156434	−	0.987688i
$9$	−0.786610	−	0.255585i	−0.786610	−	0.255585i
$10$		0			0
$11$	0.913545	+	0.406737i	0.913545	+	0.406737i
$11$	0.913545	+	0.406737i	0.913545	+	0.406737i
$12$	0.294032	−	0.294032i	0.294032	−	0.294032i
$13$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$13$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$14$		0			0
$15$		0			0
$16$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$17$	0.0949099	−	0.186271i	0.0949099	−	0.186271i	−0.838671	−	0.544639i	$-0.816667\pi$
$17$	0.0949099	−	0.186271i	0.0949099	−	0.186271i	0.933580	+	0.358368i	$0.116667\pi$
$18$	−0.816908	+	0.129386i	−0.816908	+	0.129386i
$19$	0.658114	−	0.478148i	0.658114	−	0.478148i	−0.207912	−	0.978148i	$-0.566667\pi$
$19$	0.658114	−	0.478148i	0.658114	−	0.478148i	0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		0			0
$22$	0.998630	−	0.0523360i	0.998630	−	0.0523360i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$	0.128496	−	0.395472i	0.128496	−	0.395472i
$25$		0			0
$26$		0			0
$27$	−0.676940	−	0.344918i	−0.676940	−	0.344918i
$28$		0			0
$29$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$29$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$30$		0			0
$31$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$31$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$32$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$33$	0.348739	+	0.226474i	0.348739	+	0.226474i
$34$		−	0.209057i		−	0.209057i
$35$		0			0
$36$	−0.669131	+	0.486152i	−0.669131	+	0.486152i
$37$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$37$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$38$	0.369309	−	0.724810i	0.369309	−	0.724810i
$39$		0			0
$40$		0			0
$41$	0.873619	+	1.20243i	0.873619	+	1.20243i	0.978148	+	0.207912i	$0.0666667\pi$
$41$	0.873619	+	1.20243i	0.873619	+	1.20243i	−0.104528	+	0.994522i	$0.533333\pi$
$42$		0			0
$43$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	−0.891007	−	0.453990i	$-0.850000\pi$
$43$	−0.437016	+	0.437016i	−0.437016	+	0.437016i	0.453990	+	0.891007i	$0.350000\pi$
$44$	0.866025	−	0.500000i	0.866025	−	0.500000i
$45$		0			0
$46$		0			0
$47$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$47$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$48$	−0.0650491	−	0.410704i	−0.0650491	−	0.410704i
$49$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$50$		0			0
$51$	0.0510966	−	0.0703285i	0.0510966	−	0.0703285i
$52$		0			0
$53$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$53$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$54$	−0.759747			−0.759747
$55$		0			0
$56$		0			0
$57$	0.301393	−	0.153567i	0.301393	−	0.153567i
$58$		0			0
$59$	0.951057	−	1.30902i	0.951057	−	1.30902i		−	1.00000i	$-0.5\pi$
$59$	0.951057	−	1.30902i	0.951057	−	1.30902i	0.951057	−	0.309017i	$-0.100000\pi$
$60$		0			0
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$65$		0			0
$66$	0.413545	+	0.0434654i	0.413545	+	0.0434654i
$67$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.629320	+	0.777146i	$0.716667\pi$
$67$	−1.40647	+	1.40647i	−1.40647	+	1.40647i	−0.777146	+	0.629320i	$0.783333\pi$
$68$	−0.0949099	−	0.186271i	−0.0949099	−	0.186271i
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$	−0.375491	+	0.736943i	−0.375491	+	0.736943i
$73$	−1.32178	+	0.209350i	−1.32178	+	0.209350i	−0.777146	−	0.629320i	$-0.783333\pi$
$73$	−1.32178	+	0.209350i	−1.32178	+	0.209350i	−0.544639	+	0.838671i	$0.683333\pi$
$74$		0			0
$75$		0			0
$76$		−	0.813473i		−	0.813473i
$77$		0			0
$78$		0			0
$79$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$79$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$80$		0			0
$81$	0.413545	+	0.300458i	0.413545	+	0.300458i
$82$	1.32429	+	0.674761i	1.32429	+	0.674761i
$83$	−1.19240	−	0.607558i	−1.19240	−	0.607558i	−0.258819	−	0.965926i	$-0.583333\pi$
$83$	−1.19240	−	0.607558i	−1.19240	−	0.607558i	−0.933580	+	0.358368i	$0.883333\pi$
$84$		0			0
$85$		0			0
$86$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$87$		0			0
$88$	0.544639	−	0.838671i	0.544639	−	0.838671i
$89$			1.95630i			1.95630i	0.207912	+	0.978148i	$0.433333\pi$
$89$			1.95630i			1.95630i	−0.207912	+	0.978148i	$0.566667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.244415	−	0.336408i	−0.244415	−	0.336408i
$97$	0.863541	+	1.69480i	0.863541	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$97$	0.863541	+	1.69480i	0.863541	+	1.69480i	0.156434	+	0.987688i	$0.450000\pi$
$98$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$99$	−0.614648	−	0.553432i	−0.614648	−	0.553432i
$100$		0			0
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$	0.0135990	−	0.0858605i	0.0135990	−	0.0858605i
$103$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$103$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.93221	−	0.306032i	−1.93221	−	0.306032i	−0.933580	−	0.358368i	$-0.883333\pi$
$107$	−1.93221	−	0.306032i	−1.93221	−	0.306032i	−0.998630	+	0.0523360i	$0.983333\pi$
$108$	−0.676940	+	0.344918i	−0.676940	+	0.344918i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.96456	+	0.311155i	1.96456	+	0.311155i	0.998630	+	0.0523360i	$0.0166667\pi$
$113$	1.96456	+	0.311155i	1.96456	+	0.311155i	0.965926	+	0.258819i	$0.0833333\pi$
$114$	0.198825	−	0.273659i	0.198825	−	0.273659i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.253116	−	1.59811i	0.253116	−	1.59811i
$119$		0			0
$120$		0			0
$121$	0.669131	+	0.743145i	0.669131	+	0.743145i
$122$		0			0
$123$	0.280582	+	0.550672i	0.280582	+	0.550672i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$127$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$128$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$129$	−0.207912	+	0.151057i	−0.207912	+	0.151057i
$130$		0			0
$131$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$131$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$132$	0.388205	−	0.149018i	0.388205	−	0.149018i
$133$		0			0
$134$	−0.614648	+	1.89169i	−0.614648	+	1.89169i
$135$		0			0
$136$	−0.169131	−	0.122881i	−0.169131	−	0.122881i
$137$	−1.32429	−	0.674761i	−1.32429	−	0.674761i	−0.358368	−	0.933580i	$-0.616667\pi$
$137$	−1.32429	−	0.674761i	−1.32429	−	0.674761i	−0.965926	+	0.258819i	$0.916667\pi$
$138$		0			0
$139$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.994522	+	0.104528i	$0.0333333\pi$
$139$	1.40126	+	1.01807i	1.40126	+	1.01807i	0.406737	+	0.913545i	$0.366667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			0.827091i			0.827091i
$145$		0			0
$146$	−1.08268	+	0.786610i	−1.08268	+	0.786610i
$147$	−0.410704	+	0.0650491i	−0.410704	+	0.0650491i
$148$		0			0
$149$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$149$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$150$		0			0
$151$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$151$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$152$	−0.369309	−	0.724810i	−0.369309	−	0.724810i
$153$	−0.122265	+	0.122265i	−0.122265	+	0.122265i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$157$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.504877	+	0.0799647i	0.504877	+	0.0799647i
$163$	0.724810	−	0.369309i	0.724810	−	0.369309i	−0.0523360	−	0.998630i	$-0.516667\pi$
$163$	0.724810	−	0.369309i	0.724810	−	0.369309i	0.777146	+	0.629320i	$0.216667\pi$
$164$	1.48629			1.48629
$165$		0			0
$166$	−1.33826			−1.33826
$167$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$167$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$168$		0			0
$169$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$170$		0			0
$171$	−0.639886	+	0.207912i	−0.639886	+	0.207912i
$172$	0.0966818	+	0.610425i	0.0966818	+	0.610425i
$173$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$173$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$174$		0			0
$175$		0			0
$176$	0.104528	−	0.994522i	0.104528	−	0.994522i
$177$	0.475753	−	0.475753i	0.475753	−	0.475753i
$178$	0.888139	+	1.74307i	0.888139	+	1.74307i
$179$	0.122881	+	0.169131i	0.122881	+	0.169131i	0.866025	−	0.500000i	$-0.166667\pi$
$179$	0.122881	+	0.169131i	0.122881	+	0.169131i	−0.743145	+	0.669131i	$0.766667\pi$
$180$		0			0
$181$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$181$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.162468	−	0.131564i	0.162468	−	0.131564i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$191$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$192$	−0.370501	−	0.188780i	−0.370501	−	0.188780i
$193$	−0.891007	−	0.453990i	−0.891007	−	0.453990i	−0.0523360	−	0.998630i	$-0.516667\pi$
$193$	−0.891007	−	0.453990i	−0.891007	−	0.453990i	−0.838671	+	0.544639i	$0.816667\pi$
$194$	1.53884	+	1.11803i	1.53884	+	1.11803i
$195$		0			0
$196$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$	−0.798908	−	0.214067i	−0.798908	−	0.214067i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	−0.669131	+	0.486152i	−0.669131	+	0.486152i
$202$		0			0
$203$		0			0
$204$	−0.0268631	−	0.0826761i	−0.0268631	−	0.0826761i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.795697	−	0.169131i	0.795697	−	0.169131i
$210$		0			0
$211$	−1.41355	−	0.459289i	−1.41355	−	0.459289i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−1.41355	−	0.459289i	−1.41355	−	0.459289i	−0.913545	+	0.406737i	$0.866667\pi$
$212$		0			0
$213$		0			0
$214$	−1.86055	+	0.604528i	−1.86055	+	0.604528i
$215$		0			0
$216$	−0.446568	+	0.614648i	−0.446568	+	0.614648i
$217$		0			0
$218$		0			0
$219$	−0.556480			−0.556480
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$223$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$224$		0			0
$225$		0			0
$226$	1.89169	−	0.614648i	1.89169	−	0.614648i
$227$	−0.253116	−	1.59811i	−0.253116	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$227$	−0.253116	−	1.59811i	−0.253116	−	1.59811i	0.453990	−	0.891007i	$-0.350000\pi$
$228$	0.0529157	−	0.334097i	0.0529157	−	0.334097i
$229$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$229$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.280582	+	0.550672i	0.280582	+	0.550672i	0.987688	−	0.156434i	$-0.0500000\pi$
$233$	0.280582	+	0.550672i	0.280582	+	0.550672i	−0.707107	+	0.707107i	$0.750000\pi$
$234$		0			0
$235$		0			0
$236$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$237$		0			0
$238$		0			0
$239$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$239$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$240$		0			0
$241$		−	1.98904i		−	1.98904i	−0.104528	−	0.994522i	$-0.533333\pi$
$241$		−	1.98904i		−	1.98904i	0.104528	−	0.994522i	$-0.466667\pi$
$242$	0.933580	+	0.358368i	0.933580	+	0.358368i
$243$	0.687523	+	0.687523i	0.687523	+	0.687523i
$244$		0			0
$245$		0			0
$246$	0.500000	+	0.363271i	0.500000	+	0.363271i
$247$		0			0
$248$		0			0
$249$	−0.450202	−	0.327091i	−0.450202	−	0.327091i
$250$		0			0
$251$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$251$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$257$	1.87869	−	0.297556i	1.87869	−	0.297556i	0.891007	−	0.453990i	$-0.150000\pi$
$257$	1.87869	−	0.297556i	1.87869	−	0.297556i	0.987688	+	0.156434i	$0.0500000\pi$
$258$	−0.116672	+	0.228982i	−0.116672	+	0.228982i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.533698	−	1.04744i	−0.533698	−	1.04744i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$	0.278240	−	0.309017i	0.278240	−	0.309017i
$265$		0			0
$266$		0			0
$267$	−0.127255	+	0.803458i	−0.127255	+	0.803458i
$268$	0.311155	+	1.96456i	0.311155	+	1.96456i
$269$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$269$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$	−0.206483	−	0.0327037i	−0.206483	−	0.0327037i
$273$		0			0
$274$	−1.48629			−1.48629
$275$		0			0
$276$		0			0
$277$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$277$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$278$	1.71073	+	0.270952i	1.71073	+	0.270952i
$279$		0			0
$280$		0			0
$281$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$281$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−1.00000			$\pi$
$282$		0			0
$283$	0.156434	−	0.987688i	0.156434	−	0.987688i	−0.777146	−	0.629320i	$-0.783333\pi$
$283$	0.156434	−	0.987688i	0.156434	−	0.987688i	0.933580	−	0.358368i	$-0.116667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.375491	+	0.736943i	0.375491	+	0.736943i
$289$	0.562096	+	0.773659i	0.562096	+	0.773659i
$290$		0			0
$291$	0.244415	+	0.752232i	0.244415	+	0.752232i
$292$	−0.607558	+	1.19240i	−0.607558	+	1.19240i
$293$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$293$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$294$	−0.336408	+	0.244415i	−0.336408	+	0.244415i
$295$		0			0
$296$		0			0
$297$	−0.478124	−	0.590434i	−0.478124	−	0.590434i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.658114	−	0.478148i	−0.658114	−	0.478148i
$305$		0			0
$306$	−0.0534318	+	0.164446i	−0.0534318	+	0.164446i
$307$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.838671	+	0.544639i	$0.183333\pi$
$307$	1.38331	+	1.38331i	1.38331	+	1.38331i	0.544639	+	0.838671i	$0.316667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$311$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$312$		0			0
$313$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	0.453990	+	0.891007i	$0.350000\pi$
$313$	−0.533698	+	1.04744i	−0.533698	+	1.04744i	−0.987688	+	0.156434i	$0.950000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$317$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.773659	−	0.251377i	−0.773659	−	0.251377i
$322$		0			0
$323$	−0.0266036	−	0.167968i	−0.0266036	−	0.167968i
$324$	0.486152	−	0.157960i	0.486152	−	0.157960i
$325$		0			0
$326$	0.478148	−	0.658114i	0.478148	−	0.658114i
$327$		0			0
$328$	1.32429	−	0.674761i	1.32429	−	0.674761i
$329$		0			0
$330$		0			0
$331$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$331$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$332$	−1.19240	+	0.607558i	−1.19240	+	0.607558i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.285820	+	1.80460i	0.285820	+	1.80460i	0.544639	+	0.838671i	$0.316667\pi$
$337$	0.285820	+	1.80460i	0.285820	+	1.80460i	−0.258819	+	0.965926i	$0.583333\pi$
$338$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$339$	0.786610	+	0.255585i	0.786610	+	0.255585i
$340$		0			0
$341$		0			0
$342$	−0.475753	+	0.475753i	−0.475753	+	0.475753i
$343$		0			0
$344$	0.363271	+	0.500000i	0.363271	+	0.500000i
$345$		0			0
$346$		0			0
$347$	0.829482	−	1.62795i	0.829482	−	1.62795i	0.0523360	−	0.998630i	$-0.483333\pi$
$347$	0.829482	−	1.62795i	0.829482	−	1.62795i	0.777146	−	0.629320i	$-0.216667\pi$
$348$		0			0
$349$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$349$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$350$		0			0
$351$		0			0
$352$	−0.358368	−	0.933580i	−0.358368	−	0.933580i
$353$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$353$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$354$	0.207912	−	0.639886i	0.207912	−	0.639886i
$355$		0			0
$356$	1.58268	+	1.14988i	1.58268	+	1.14988i
$357$		0			0
$358$	0.186271	+	0.0949099i	0.186271	+	0.0949099i
$359$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$359$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$360$		0			0
$361$	−0.104528	+	0.321706i	−0.104528	+	0.321706i
$362$		0			0
$363$	0.226474	+	0.348739i	0.226474	+	0.348739i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$367$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$368$		0			0
$369$	−0.379874	−	1.16913i	−0.379874	−	1.16913i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$373$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$374$	0.0850311	−	0.190983i	0.0850311	−	0.190983i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.27276	+	0.413545i	−1.27276	+	0.413545i	−0.866025	−	0.500000i	$-0.833333\pi$
$379$	−1.27276	+	0.413545i	−1.27276	+	0.413545i	−0.406737	+	0.913545i	$0.633333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$383$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$384$	−0.415823			−0.415823
$385$		0			0
$386$	−1.00000			−1.00000
$387$	0.455456	−	0.232066i	0.455456	−	0.232066i
$388$	1.87869	+	0.297556i	1.87869	+	0.297556i
$389$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$389$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$390$		0			0
$391$		0			0
$392$	0.156434	+	0.987688i	0.156434	+	0.987688i
$393$	0.0764698	−	0.482811i	0.0764698	−	0.482811i
$394$		0			0
$395$		0			0
$396$	−0.809017	+	0.171962i	−0.809017	+	0.171962i
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.413545	−	1.27276i	−0.413545	−	1.27276i	−0.913545	−	0.406737i	$-0.866667\pi$
$401$	−0.413545	−	1.27276i	−0.413545	−	1.27276i	0.500000	−	0.866025i	$-0.333333\pi$
$402$	−0.375491	+	0.736943i	−0.375491	+	0.736943i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.0614693	−	0.0614693i	−0.0614693	−	0.0614693i
$409$	0.535233	−	1.64728i	0.535233	−	1.64728i	−0.207912	−	0.978148i	$-0.566667\pi$
$409$	0.535233	−	1.64728i	0.535233	−	1.64728i	0.743145	−	0.669131i	$-0.233333\pi$
$410$		0			0
$411$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.509278	+	0.509278i	0.509278	+	0.509278i
$418$	0.632187	−	0.511935i	0.632187	−	0.511935i
$419$		−	1.33826i		−	1.33826i	−0.743145	−	0.669131i	$-0.766667\pi$
$419$		−	1.33826i		−	1.33826i	0.743145	−	0.669131i	$-0.233333\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$	−1.46799	+	0.232507i	−1.46799	+	0.232507i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.38331	+	1.38331i	−1.38331	+	1.38331i
$429$		0			0
$430$		0			0
$431$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$431$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$432$	−0.118851	+	0.750393i	−0.118851	+	0.750393i
$433$	−0.127255	−	0.803458i	−0.127255	−	0.803458i	−0.965926	−	0.258819i	$-0.916667\pi$
$433$	−0.127255	−	0.803458i	−0.127255	−	0.803458i	0.838671	−	0.544639i	$-0.183333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.495828	+	0.252637i	−0.495828	+	0.252637i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	0.827091			0.827091
$442$		0			0
$443$	1.46799	+	0.232507i	1.46799	+	0.232507i	0.838671	−	0.544639i	$-0.183333\pi$
$443$	1.46799	+	0.232507i	1.46799	+	0.232507i	0.629320	+	0.777146i	$0.283333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.198825	−	0.0646021i	−0.198825	−	0.0646021i	0.207912	−	0.978148i	$-0.433333\pi$
$449$	−0.198825	−	0.0646021i	−0.198825	−	0.0646021i	−0.406737	+	0.913545i	$0.633333\pi$
$450$		0			0
$451$	0.309017	+	1.45381i	0.309017	+	1.45381i
$452$	1.40647	−	1.40647i	1.40647	−	1.40647i
$453$		0			0
$454$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$455$		0			0
$456$	−0.104528	−	0.321706i	−0.104528	−	0.321706i
$457$	0.888139	−	1.74307i	0.888139	−	1.74307i	0.258819	−	0.965926i	$-0.416667\pi$
$457$	0.888139	−	1.74307i	0.888139	−	1.74307i	0.629320	−	0.777146i	$-0.283333\pi$
$458$		0			0
$459$	−0.128496	+	0.0933582i	−0.128496	+	0.0933582i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		0			0
$465$		0			0
$466$	0.500000	+	0.363271i	0.500000	+	0.363271i
$467$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$467$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.14412	−	1.14412i	−1.14412	−	1.14412i
$473$	−0.576984	+	0.221484i	−0.576984	+	0.221484i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$479$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$480$		0			0
$481$		0			0
$482$	−0.903007	−	1.77225i	−0.903007	−	1.77225i
$483$		0			0
$484$	0.994522	−	0.104528i	0.994522	−	0.104528i
$485$		0			0
$486$	0.924716	+	0.300458i	0.924716	+	0.300458i
$487$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$487$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$488$		0			0
$489$	0.321706	−	0.104528i	0.321706	−	0.104528i
$490$		0			0
$491$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$491$	1.11803	−	1.53884i	1.11803	−	1.53884i	0.809017	−	0.587785i	$-0.200000\pi$
$492$	0.610425	+	0.0966818i	0.610425	+	0.0966818i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.549629	−	0.0870527i	−0.549629	−	0.0870527i
$499$	0.363271	−	0.500000i	0.363271	−	0.500000i	−0.587785	−	0.809017i	$-0.700000\pi$
$499$	0.363271	−	0.500000i	0.363271	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$500$		0			0
$501$		0			0
$502$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$503$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$503$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.294032	+	0.294032i	−0.294032	+	0.294032i
$508$		0			0
$509$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$509$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$510$		0			0
$511$		0			0
$512$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$513$	−0.610425	+	0.0966818i	−0.610425	+	0.0966818i
$514$	1.53884	−	1.11803i	1.53884	−	1.11803i
$515$		0			0
$516$			0.256993i			0.256993i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.169131	+	0.122881i	0.169131	+	0.122881i	0.669131	−	0.743145i	$-0.266667\pi$
$521$	0.169131	+	0.122881i	0.169131	+	0.122881i	−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−1.62795	−	0.829482i	−1.62795	−	0.829482i	−0.998630	−	0.0523360i	$-0.983333\pi$
$523$	−1.62795	−	0.829482i	−1.62795	−	0.829482i	−0.629320	−	0.777146i	$-0.716667\pi$
$524$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.107623	−	0.401655i	0.107623	−	0.401655i
$529$			1.00000i			1.00000i
$530$		0			0
$531$	−1.08268	+	0.786610i	−1.08268	+	0.786610i
$532$		0			0
$533$		0			0
$534$	0.251377	+	0.773659i	0.251377	+	0.773659i
$535$		0			0
$536$	1.16913	+	1.60917i	1.16913	+	1.60917i
$537$	0.0394657	+	0.0774559i	0.0394657	+	0.0774559i
$538$		0			0
$539$	−0.994522	−	0.104528i	−0.994522	−	0.104528i
$540$		0			0
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$		0			0
$543$		0			0
$544$	−0.198825	+	0.0646021i	−0.198825	+	0.0646021i
$545$		0			0
$546$		0			0
$547$	0.206483	+	0.0327037i	0.206483	+	0.0327037i	0.258819	−	0.965926i	$-0.416667\pi$
$547$	0.206483	+	0.0327037i	0.206483	+	0.0327037i	−0.0523360	+	0.998630i	$0.516667\pi$
$548$	−1.32429	+	0.674761i	−1.32429	+	0.674761i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.64728	−	0.535233i	1.64728	−	0.535233i
$557$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$557$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.0752842	−	0.0434654i	0.0752842	−	0.0434654i
$562$	−1.34500	+	1.34500i	−1.34500	+	1.34500i
$563$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$563$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	−0.987688	+	0.156434i	$0.950000\pi$
$564$		0			0
$565$		0			0
$566$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$567$		0			0
$568$		0			0
$569$	−0.658114	+	0.478148i	−0.658114	+	0.478148i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$	−0.658114	+	0.478148i	−0.658114	+	0.478148i	0.207912	+	0.978148i	$0.433333\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.669131	+	0.486152i	0.669131	+	0.486152i
$577$	−0.724810	−	0.369309i	−0.724810	−	0.369309i	0.0523360	−	0.998630i	$-0.483333\pi$
$577$	−0.724810	−	0.369309i	−0.724810	−	0.369309i	−0.777146	+	0.629320i	$0.783333\pi$
$578$	0.852065	+	0.434149i	0.852065	+	0.434149i
$579$	−0.336408	−	0.244415i	−0.336408	−	0.244415i
$580$		0			0
$581$		0			0
$582$	0.559281	+	0.559281i	0.559281	+	0.559281i
$583$		0			0
$584$			1.33826i			1.33826i
$585$		0			0
$586$		0			0
$587$	1.96456	−	0.311155i	1.96456	−	0.311155i	0.965926	−	0.258819i	$-0.0833333\pi$
$587$	1.96456	−	0.311155i	1.96456	−	0.311155i	0.998630	−	0.0523360i	$-0.0166667\pi$
$588$	−0.188780	+	0.370501i	−0.188780	+	0.370501i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.358368	−	0.933580i	$-0.383333\pi$
$593$	1.29195	−	1.29195i	1.29195	−	1.29195i	0.933580	−	0.358368i	$-0.116667\pi$
$594$	−0.694064	−	0.309017i	−0.694064	−	0.309017i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$599$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$600$		0			0
$601$	−0.873619	+	1.20243i	−0.873619	+	1.20243i	0.104528	+	0.994522i	$0.466667\pi$
$601$	−0.873619	+	1.20243i	−0.873619	+	1.20243i	−0.978148	+	0.207912i	$0.933333\pi$
$602$		0			0
$603$	1.46581	−	0.746869i	1.46581	−	0.746869i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$607$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$608$	−0.803458	−	0.127255i	−0.803458	−	0.127255i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.0270489	+	0.170780i	0.0270489	+	0.170780i
$613$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$613$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$614$	1.86055	+	0.604528i	1.86055	+	0.604528i
$615$		0			0
$616$		0			0
$617$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$617$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$618$		0			0
$619$	0.951057	+	1.30902i	0.951057	+	1.30902i	0.951057	+	0.309017i	$0.100000\pi$
$619$	0.951057	+	1.30902i	0.951057	+	1.30902i			1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$			1.17557i			1.17557i
$627$	0.337798	−	0.0177032i	0.337798	−	0.0177032i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$		0			0
$633$	−0.550672	−	0.280582i	−0.550672	−	0.280582i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.500000	−	0.363271i	0.500000	−	0.363271i	−0.309017	−	0.951057i	$-0.600000\pi$
$641$	0.500000	−	0.363271i	0.500000	−	0.363271i	0.809017	+	0.587785i	$0.200000\pi$
$642$	−0.803458	+	0.127255i	−0.803458	+	0.127255i
$643$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$643$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$	−0.0999601	−	0.137583i	−0.0999601	−	0.137583i
$647$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$647$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$648$	0.361452	−	0.361452i	0.361452	−	0.361452i
$649$	1.40126	−	0.809017i	1.40126	−	0.809017i
$650$		0			0
$651$		0			0
$652$	0.127255	−	0.803458i	0.127255	−	0.803458i
$653$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$653$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$654$		0			0
$655$		0			0
$656$	0.873619	−	1.20243i	0.873619	−	1.20243i
$657$	1.09324	+	0.173152i	1.09324	+	0.173152i
$658$		0			0
$659$	−1.98904			−1.98904			−0.994522	−	0.104528i	$-0.966667\pi$
$659$	−1.98904			−1.98904			−0.994522	+	0.104528i	$0.966667\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−1.74307	+	0.888139i	−1.74307	+	0.888139i
$663$		0			0
$664$	−0.786610	+	1.08268i	−0.786610	+	1.08268i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	−0.891007	−	0.453990i	$-0.850000\pi$
$673$	−0.734572	−	1.44168i	−0.734572	−	1.44168i	0.156434	−	0.987688i	$-0.450000\pi$
$674$	1.07394	+	1.47815i	1.07394	+	1.47815i
$675$		0			0
$676$	0.309017	+	0.951057i	0.309017	+	0.951057i
$677$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$677$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$678$	0.816908	−	0.129386i	0.816908	−	0.129386i
$679$		0			0
$680$		0			0
$681$		−	0.672816i		−	0.672816i
$682$		0			0
$683$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$683$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$684$	−0.207912	+	0.639886i	−0.207912	+	0.639886i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.550672	+	0.280582i	0.550672	+	0.280582i
$689$		0			0
$690$		0			0
$691$	0.564602	−	1.73767i	0.564602	−	1.73767i	−0.104528	−	0.994522i	$-0.533333\pi$
$691$	0.564602	−	1.73767i	0.564602	−	1.73767i	0.669131	−	0.743145i	$-0.266667\pi$
$692$		0			0
$693$		0			0
$694$		−	1.82709i		−	1.82709i
$695$		0			0
$696$		0			0
$697$	0.306894	−	0.0486072i	0.306894	−	0.0486072i
$698$		0			0
$699$	0.0794152	+	0.244415i	0.0794152	+	0.244415i
$700$		0			0
$701$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$701$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$702$		0			0
$703$		0			0
$704$	−0.743145	−	0.669131i	−0.743145	−	0.669131i
$705$		0			0
$706$	1.11803	+	0.363271i	1.11803	+	0.363271i
$707$		0			0
$708$	−0.105252	−	0.664533i	−0.105252	−	0.664533i
$709$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$709$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$710$		0			0
$711$		0			0
$712$	1.93221	+	0.306032i	1.93221	+	0.306032i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.209057			0.209057
$717$		0			0
$718$		0			0
$719$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$719$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$720$		0			0
$721$		0			0
$722$	0.0529157	+	0.334097i	0.0529157	+	0.334097i
$723$	0.129386	−	0.816908i	0.129386	−	0.816908i
$724$		0			0
$725$		0			0
$726$	0.360114	+	0.207912i	0.360114	+	0.207912i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−0.0628129	−	0.0864545i	−0.0628129	−	0.0864545i
$730$		0			0
$731$	0.0399263	+	0.122881i	0.0399263	+	0.122881i
$732$		0			0
$733$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$733$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.85693	+	0.712810i	−1.85693	+	0.712810i
$738$	−0.869244	−	0.869244i	−0.869244	−	0.869244i
$739$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$739$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$743$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.782671	+	0.782671i	0.782671	+	0.782671i
$748$	−0.0109412	−	0.208770i	−0.0109412	−	0.208770i
$749$		0			0
$750$		0			0
$751$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$751$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$	0.188780	−	0.370501i	0.188780	−	0.370501i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$757$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$758$	−0.946294	+	0.946294i	−0.946294	+	0.946294i
$759$		0			0
$760$		0			0
$761$	−1.89169	−	0.614648i	−1.89169	−	0.614648i	−0.978148	−	0.207912i	$-0.933333\pi$
$761$	−1.89169	−	0.614648i	−1.89169	−	0.614648i	−0.913545	−	0.406737i	$-0.866667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.370501	+	0.188780i	−0.370501	+	0.188780i
$769$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$769$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$	0.790943			0.790943
$772$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$773$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$773$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$774$	0.300458	−	0.413545i	0.300458	−	0.413545i
$775$		0			0
$776$	1.80902	−	0.587785i	1.80902	−	0.587785i
$777$		0			0
$778$		0			0
$779$	1.14988	+	0.373619i	1.14988	+	0.373619i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.587785	+	0.809017i	0.587785	+	0.809017i
$785$		0			0
$786$	−0.151057	−	0.464905i	−0.151057	−	0.464905i
$787$	0.734572	−	1.44168i	0.734572	−	1.44168i	−0.156434	−	0.987688i	$-0.550000\pi$
$787$	0.734572	−	1.44168i	0.734572	−	1.44168i	0.891007	−	0.453990i	$-0.150000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.642770	+	0.520505i	−0.642770	+	0.520505i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$797$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.500000	−	1.53884i	0.500000	−	1.53884i
$802$	−0.946294	−	0.946294i	−0.946294	−	0.946294i
$803$	−1.29266	−	0.346367i	−1.29266	−	0.346367i
$804$			0.827091i			0.827091i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	−0.951057	−	0.309017i	$-0.900000\pi$
$809$	−0.363271	−	1.11803i	−0.363271	−	1.11803i	0.587785	−	0.809017i	$-0.300000\pi$
$810$		0			0
$811$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$811$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.0826761	−	0.0268631i	−0.0826761	−	0.0268631i
$817$	−0.0786481	+	0.496564i	−0.0786481	+	0.496564i
$818$	−0.270952	−	1.71073i	−0.270952	−	1.71073i
$819$		0			0
$820$		0			0
$821$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$821$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$822$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i
$823$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$823$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.62795	−	0.829482i	1.62795	−	0.829482i	0.629320	−	0.777146i	$-0.283333\pi$
$827$	1.62795	−	0.829482i	1.62795	−	0.829482i	0.998630	−	0.0523360i	$-0.0166667\pi$
$828$		0			0
$829$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$829$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.0327037	+	0.206483i	−0.0327037	+	0.206483i
$834$	0.684977	+	0.222562i	0.684977	+	0.222562i
$835$		0			0
$836$	0.330869	−	0.743145i	0.330869	−	0.743145i
$837$		0			0
$838$	−0.607558	−	1.19240i	−0.607558	−	1.19240i
$839$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$839$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$840$		0			0
$841$	0.309017	+	0.951057i	0.309017	+	0.951057i
$842$		0			0
$843$	−0.781205	+	0.123731i	−0.781205	+	0.123731i
$844$	−1.20243	+	0.873619i	−1.20243	+	0.873619i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.128496	−	0.395472i	0.128496	−	0.395472i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$853$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$854$		0			0
$855$		0			0
$856$	−0.604528	+	1.86055i	−0.604528	+	1.86055i
$857$	−1.38331	−	1.38331i	−1.38331	−	1.38331i	−0.838671	−	0.544639i	$-0.816667\pi$
$857$	−1.38331	−	1.38331i	−1.38331	−	1.38331i	−0.544639	−	0.838671i	$-0.683333\pi$
$858$		0			0
$859$		−	1.82709i		−	1.82709i	−0.406737	−	0.913545i	$-0.633333\pi$
$859$		−	1.82709i		−	1.82709i	0.406737	−	0.913545i	$-0.366667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$863$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$864$	0.234775	+	0.722562i	0.234775	+	0.722562i
$865$		0			0
$866$	−0.478148	−	0.658114i	−0.478148	−	0.658114i
$867$	0.180529	+	0.354309i	0.180529	+	0.354309i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.246106	−	1.55385i	−0.246106	−	1.55385i
$874$		0			0
$875$		0			0
$876$	−0.327091	+	0.450202i	−0.327091	+	0.450202i
$877$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$877$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.618034			−0.618034			−0.309017	−	0.951057i	$-0.600000\pi$
$881$	−0.618034			−0.618034			−0.309017	+	0.951057i	$0.600000\pi$
$882$	0.736943	−	0.375491i	0.736943	−	0.375491i
$883$	−1.46799	−	0.232507i	−1.46799	−	0.232507i	−0.629320	−	0.777146i	$-0.716667\pi$
$883$	−1.46799	−	0.232507i	−1.46799	−	0.232507i	−0.838671	+	0.544639i	$0.816667\pi$
$884$		0			0
$885$		0			0
$886$	1.41355	−	0.459289i	1.41355	−	0.459289i
$887$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$887$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.255585	+	0.442686i	0.255585	+	0.442686i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.206483	+	0.0327037i	−0.206483	+	0.0327037i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	0.935352	+	1.15506i	0.935352	+	1.15506i
$903$		0			0
$904$	0.614648	−	1.89169i	0.614648	−	1.89169i
$905$		0			0
$906$		0			0
$907$	1.04744	+	0.533698i	1.04744	+	0.533698i	0.891007	−	0.453990i	$-0.150000\pi$
$907$	1.04744	+	0.533698i	1.04744	+	0.533698i	0.156434	+	0.987688i	$0.450000\pi$
$908$	−1.44168	−	0.734572i	−1.44168	−	0.734572i
$909$		0			0
$910$		0			0
$911$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$911$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$912$	−0.239187	−	0.239187i	−0.239187	−	0.239187i
$913$	−0.842195	−	1.04002i	−0.842195	−	1.04002i
$914$		−	1.95630i		−	1.95630i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.0721075	+	0.141519i	−0.0721075	+	0.141519i
$919$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$919$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$920$		0			0
$921$	0.478148	+	0.658114i	0.478148	+	0.658114i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.587785	−	0.809017i	$-0.300000\pi$
$929$	1.53884	−	0.500000i	1.53884	−	0.500000i	0.951057	+	0.309017i	$0.100000\pi$
$930$		0			0
$931$	−0.478148	+	0.658114i	−0.478148	+	0.658114i
$932$	0.610425	+	0.0966818i	0.610425	+	0.0966818i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.62795	+	0.829482i	−1.62795	+	0.829482i	−0.629320	+	0.777146i	$0.716667\pi$
$937$	−1.62795	+	0.829482i	−1.62795	+	0.829482i	−0.998630	+	0.0523360i	$0.983333\pi$
$938$		0			0
$939$	−0.287327	+	0.395472i	−0.287327	+	0.395472i
$940$		0			0
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$945$		0			0
$946$	−0.413545	+	0.459289i	−0.413545	+	0.459289i
$947$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$947$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.32178	−	0.209350i	1.32178	−	0.209350i	0.544639	−	0.838671i	$-0.316667\pi$
$953$	1.32178	−	0.209350i	1.32178	−	0.209350i	0.777146	+	0.629320i	$0.216667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$962$		0			0
$963$	1.44168	+	0.734572i	1.44168	+	0.734572i
$964$	−1.60917	−	1.16913i	−1.60917	−	1.16913i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	0.838671	−	0.544639i	0.838671	−	0.544639i
$969$		−	0.0707158i		−	0.0707158i
$970$		0			0
$971$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$971$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$972$	0.960333	−	0.152102i	0.960333	−	0.152102i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.786335	+	1.54327i	0.786335	+	1.54327i	0.838671	+	0.544639i	$0.183333\pi$
$977$	0.786335	+	1.54327i	0.786335	+	1.54327i	−0.0523360	+	0.998630i	$0.516667\pi$
$978$	0.239187	−	0.239187i	0.239187	−	0.239187i
$979$	−0.795697	+	1.78716i	−0.795697	+	1.78716i
$980$		0			0
$981$		0			0
$982$	0.297556	−	1.87869i	0.297556	−	1.87869i
$983$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$983$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$984$	0.587785	−	0.190983i	0.587785	−	0.190983i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−0.803458	−	0.127255i	−0.803458	−	0.127255i
$994$		0			0
$995$		0			0
$996$	−0.529244	+	0.171962i	−0.529244	+	0.171962i
$997$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$997$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$998$	0.0966818	−	0.610425i	0.0966818	−	0.610425i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.1.fr.c.1443.3	yes	32
5.2	odd	4	inner	2200.1.fr.c.1707.3	yes	32
5.3	odd	4	inner	2200.1.fr.c.1707.2	yes	32
5.4	even	2	inner	2200.1.fr.c.1443.2	✓	32
8.3	odd	2	CM	2200.1.fr.c.1443.3	yes	32
11.6	odd	10	inner	2200.1.fr.c.1843.3	yes	32
40.3	even	4	inner	2200.1.fr.c.1707.2	yes	32
40.19	odd	2	inner	2200.1.fr.c.1443.2	✓	32
40.27	even	4	inner	2200.1.fr.c.1707.3	yes	32
55.17	even	20	inner	2200.1.fr.c.2107.3	yes	32
55.28	even	20	inner	2200.1.fr.c.2107.2	yes	32
55.39	odd	10	inner	2200.1.fr.c.1843.2	yes	32
88.83	even	10	inner	2200.1.fr.c.1843.3	yes	32
440.83	odd	20	inner	2200.1.fr.c.2107.2	yes	32
440.259	even	10	inner	2200.1.fr.c.1843.2	yes	32
440.347	odd	20	inner	2200.1.fr.c.2107.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.1.fr.c.1443.2	✓	32	5.4	even	2	inner
2200.1.fr.c.1443.2	✓	32	40.19	odd	2	inner
2200.1.fr.c.1443.3	yes	32	1.1	even	1	trivial
2200.1.fr.c.1443.3	yes	32	8.3	odd	2	CM
2200.1.fr.c.1707.2	yes	32	5.3	odd	4	inner
2200.1.fr.c.1707.2	yes	32	40.3	even	4	inner
2200.1.fr.c.1707.3	yes	32	5.2	odd	4	inner
2200.1.fr.c.1707.3	yes	32	40.27	even	4	inner
2200.1.fr.c.1843.2	yes	32	55.39	odd	10	inner
2200.1.fr.c.1843.2	yes	32	440.259	even	10	inner
2200.1.fr.c.1843.3	yes	32	11.6	odd	10	inner
2200.1.fr.c.1843.3	yes	32	88.83	even	10	inner
2200.1.fr.c.2107.2	yes	32	55.28	even	20	inner
2200.1.fr.c.2107.2	yes	32	440.83	odd	20	inner
2200.1.fr.c.2107.3	yes	32	55.17	even	20	inner
2200.1.fr.c.2107.3	yes	32	440.347	odd	20	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2200.fr (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.891007 - 0.453990i) q^{2} +(0.410704 + 0.0650491i) q^{3} +(0.587785 - 0.809017i) q^{4} +(0.395472 - 0.128496i) q^{6} +(0.156434 - 0.987688i) q^{8} +(-0.786610 - 0.255585i) q^{9} +O(q^{10})\)	\(q+(0.891007 - 0.453990i) q^{2} +(0.410704 + 0.0650491i) q^{3} +(0.587785 - 0.809017i) q^{4} +(0.395472 - 0.128496i) q^{6} +(0.156434 - 0.987688i) q^{8} +(-0.786610 - 0.255585i) q^{9} +(0.913545 + 0.406737i) q^{11} +(0.294032 - 0.294032i) q^{12} +(-0.309017 - 0.951057i) q^{16} +(0.0949099 - 0.186271i) q^{17} +(-0.816908 + 0.129386i) q^{18} +(0.658114 - 0.478148i) q^{19} +(0.998630 - 0.0523360i) q^{22} +(0.128496 - 0.395472i) q^{24} +(-0.676940 - 0.344918i) q^{27} +(-0.707107 - 0.707107i) q^{32} +(0.348739 + 0.226474i) q^{33} -0.209057i q^{34} +(-0.669131 + 0.486152i) q^{36} +(0.369309 - 0.724810i) q^{38} +(0.873619 + 1.20243i) q^{41} +(-0.437016 + 0.437016i) q^{43} +(0.866025 - 0.500000i) q^{44} +(-0.0650491 - 0.410704i) q^{48} +(-0.951057 + 0.309017i) q^{49} +(0.0510966 - 0.0703285i) q^{51} -0.759747 q^{54} +(0.301393 - 0.153567i) q^{57} +(0.951057 - 1.30902i) q^{59} +(-0.951057 - 0.309017i) q^{64} +(0.413545 + 0.0434654i) q^{66} +(-1.40647 + 1.40647i) q^{67} +(-0.0949099 - 0.186271i) q^{68} +(-0.375491 + 0.736943i) q^{72} +(-1.32178 + 0.209350i) q^{73} -0.813473i q^{76} +(0.413545 + 0.300458i) q^{81} +(1.32429 + 0.674761i) q^{82} +(-1.19240 - 0.607558i) q^{83} +(-0.190983 + 0.587785i) q^{86} +(0.544639 - 0.838671i) q^{88} +1.95630i q^{89} +(-0.244415 - 0.336408i) q^{96} +(0.863541 + 1.69480i) q^{97} +(-0.707107 + 0.707107i) q^{98} +(-0.614648 - 0.553432i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 20 q^{6}+O(q^{10}) \) 32 * q + 20 * q^6	\( 32 q + 20 q^{6} + 4 q^{11} + 8 q^{16} - 4 q^{36} + 20 q^{51} - 12 q^{66} - 12 q^{81} - 24 q^{86}+O(q^{100}) \) 32 * q + 20 * q^6 + 4 * q^11 + 8 * q^16 - 4 * q^36 + 20 * q^51 - 12 * q^66 - 12 * q^81 - 24 * q^86

Introduction

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