LMFDB - Embedded newform 2200.1.fr.c.1443.1

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			1443.1
Root			$0.838671 - 0.544639i$ of defining polynomial
Character	$\chi$	$=$	2200.1443
Dual form			2200.1.fr.c.2107.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.891007 + 0.453990i) q^{2} +(-1.46799 - 0.232507i) q^{3} +(0.587785 - 0.809017i) q^{4} +(1.41355 - 0.459289i) q^{6} +(-0.156434 + 0.987688i) q^{8} +(1.14988 + 0.373619i) q^{9} +O(q^{10})$	\(q+(-0.891007 + 0.453990i) q^{2} +(-1.46799 - 0.232507i) q^{3} +(0.587785 - 0.809017i) q^{4} +(1.41355 - 0.459289i) q^{6} +(-0.156434 + 0.987688i) q^{8} +(1.14988 + 0.373619i) q^{9} +(-0.104528 - 0.994522i) q^{11} +(-1.05097 + 1.05097i) q^{12} +(-0.309017 - 0.951057i) q^{16} +(0.829482 - 1.62795i) q^{17} +(-1.19417 + 0.189138i) q^{18} +(-1.60917 + 1.16913i) q^{19} +(0.544639 + 0.838671i) q^{22} +(0.459289 - 1.41355i) q^{24} +(-0.276853 - 0.141064i) q^{27} +(0.707107 + 0.707107i) q^{32} +(-0.0777864 + 1.48425i) q^{33} +1.82709i q^{34} +(0.978148 - 0.710666i) q^{36} +(0.903007 - 1.77225i) q^{38} +(0.244415 + 0.336408i) q^{41} +(0.437016 - 0.437016i) q^{43} +(-0.866025 - 0.500000i) q^{44} +(0.232507 + 1.46799i) q^{48} +(-0.951057 + 0.309017i) q^{49} +(-1.59618 + 2.19696i) q^{51} +0.310719 q^{54} +(2.63408 - 1.34213i) q^{57} +(0.951057 - 1.30902i) q^{59} +(-0.951057 - 0.309017i) q^{64} +(-0.604528 - 1.35779i) q^{66} +(-0.575212 + 0.575212i) q^{67} +(-0.829482 - 1.62795i) q^{68} +(-0.548900 + 1.07728i) q^{72} +(-1.93221 + 0.306032i) q^{73} +1.98904i q^{76} +(-0.604528 - 0.439216i) q^{81} +(-0.370501 - 0.188780i) q^{82} +(-1.74307 - 0.888139i) q^{83} +(-0.190983 + 0.587785i) q^{86} +(0.998630 + 0.0523360i) q^{88} -1.33826i q^{89} +(-0.873619 - 1.20243i) q^{96} +(-0.863541 - 1.69480i) q^{97} +(0.707107 - 0.707107i) q^{98} +(0.251377 - 1.18264i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.46799	−	0.232507i	−1.46799	−	0.232507i	−0.629320	−	0.777146i	$-0.716667\pi$
$3$	−1.46799	−	0.232507i	−1.46799	−	0.232507i	−0.838671	+	0.544639i	$0.816667\pi$
$4$	0.587785	−	0.809017i	0.587785	−	0.809017i
$5$		0			0
$5$		0			0
$6$	1.41355	−	0.459289i	1.41355	−	0.459289i
$7$		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
$9$	1.14988	+	0.373619i	1.14988	+	0.373619i
$10$		0			0
$11$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$11$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$12$	−1.05097	+	1.05097i	−1.05097	+	1.05097i
$13$		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
$17$	0.829482	−	1.62795i	0.829482	−	1.62795i	0.0523360	−	0.998630i	$-0.483333\pi$
$17$	0.829482	−	1.62795i	0.829482	−	1.62795i	0.777146	−	0.629320i	$-0.216667\pi$
$18$	−1.19417	+	0.189138i	−1.19417	+	0.189138i
$19$	−1.60917	+	1.16913i	−1.60917	+	1.16913i	−0.743145	+	0.669131i	$0.766667\pi$
$19$	−1.60917	+	1.16913i	−1.60917	+	1.16913i	−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$	0.544639	+	0.838671i	0.544639	+	0.838671i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$	0.459289	−	1.41355i	0.459289	−	1.41355i
$25$		0			0
$26$		0			0
$27$	−0.276853	−	0.141064i	−0.276853	−	0.141064i
$28$		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
$33$	−0.0777864	+	1.48425i	−0.0777864	+	1.48425i
$34$			1.82709i			1.82709i
$35$		0			0
$36$	0.978148	−	0.710666i	0.978148	−	0.710666i
$37$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$37$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$38$	0.903007	−	1.77225i	0.903007	−	1.77225i
$39$		0			0
$40$		0			0
$41$	0.244415	+	0.336408i	0.244415	+	0.336408i	0.913545	−	0.406737i	$-0.133333\pi$
$41$	0.244415	+	0.336408i	0.244415	+	0.336408i	−0.669131	+	0.743145i	$0.733333\pi$
$42$		0			0
$43$	0.437016	−	0.437016i	0.437016	−	0.437016i	−0.453990	−	0.891007i	$-0.650000\pi$
$43$	0.437016	−	0.437016i	0.437016	−	0.437016i	0.891007	+	0.453990i	$0.150000\pi$
$44$	−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	$0.550000\pi$
$48$	0.232507	+	1.46799i	0.232507	+	1.46799i
$49$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$50$		0			0
$51$	−1.59618	+	2.19696i	−1.59618	+	2.19696i
$52$		0			0
$53$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$53$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$54$	0.310719			0.310719
$55$		0			0
$56$		0			0
$57$	2.63408	−	1.34213i	2.63408	−	1.34213i
$58$		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
$66$	−0.604528	−	1.35779i	−0.604528	−	1.35779i
$67$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	−0.933580	−	0.358368i	$-0.883333\pi$
$67$	−0.575212	+	0.575212i	−0.575212	+	0.575212i	0.358368	+	0.933580i	$0.383333\pi$
$68$	−0.829482	−	1.62795i	−0.829482	−	1.62795i
$69$		0			0
$70$		0			0
$71$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$71$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$72$	−0.548900	+	1.07728i	−0.548900	+	1.07728i
$73$	−1.93221	+	0.306032i	−1.93221	+	0.306032i	−0.998630	−	0.0523360i	$-0.983333\pi$
$73$	−1.93221	+	0.306032i	−1.93221	+	0.306032i	−0.933580	+	0.358368i	$0.883333\pi$
$74$		0			0
$75$		0			0
$76$			1.98904i			1.98904i
$77$		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
$81$	−0.604528	−	0.439216i	−0.604528	−	0.439216i
$82$	−0.370501	−	0.188780i	−0.370501	−	0.188780i
$83$	−1.74307	−	0.888139i	−1.74307	−	0.888139i	−0.965926	−	0.258819i	$-0.916667\pi$
$83$	−1.74307	−	0.888139i	−1.74307	−	0.888139i	−0.777146	−	0.629320i	$-0.783333\pi$
$84$		0			0
$85$		0			0
$86$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$87$		0			0
$88$	0.998630	+	0.0523360i	0.998630	+	0.0523360i
$89$		−	1.33826i		−	1.33826i	−0.743145	−	0.669131i	$-0.766667\pi$
$89$		−	1.33826i		−	1.33826i	0.743145	−	0.669131i	$-0.233333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−0.873619	−	1.20243i	−0.873619	−	1.20243i
$97$	−0.863541	−	1.69480i	−0.863541	−	1.69480i	−0.707107	−	0.707107i	$-0.750000\pi$
$97$	−0.863541	−	1.69480i	−0.863541	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$98$	0.707107	−	0.707107i	0.707107	−	0.707107i
$99$	0.251377	−	1.18264i	0.251377	−	1.18264i
$100$		0			0
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$	0.424811	−	2.68215i	0.424811	−	2.68215i
$103$		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
$107$	−1.32178	−	0.209350i	−1.32178	−	0.209350i	−0.544639	−	0.838671i	$-0.683333\pi$
$107$	−1.32178	−	0.209350i	−1.32178	−	0.209350i	−0.777146	+	0.629320i	$0.783333\pi$
$108$	−0.276853	+	0.141064i	−0.276853	+	0.141064i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.803458	+	0.127255i	0.803458	+	0.127255i	0.544639	−	0.838671i	$-0.316667\pi$
$113$	0.803458	+	0.127255i	0.803458	+	0.127255i	0.258819	+	0.965926i	$0.416667\pi$
$114$	−1.73767	+	2.39169i	−1.73767	+	2.39169i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.253116	+	1.59811i	−0.253116	+	1.59811i
$119$		0			0
$120$		0			0
$121$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$122$		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
$129$	−0.743145	+	0.539926i	−0.743145	+	0.539926i
$130$		0			0
$131$		−	1.17557i		−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$131$		−	1.17557i		−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$132$	1.15506	+	0.935352i	1.15506	+	0.935352i
$133$		0			0
$134$	0.251377	−	0.773659i	0.251377	−	0.773659i
$135$		0			0
$136$	1.47815	+	1.07394i	1.47815	+	1.07394i
$137$	0.370501	+	0.188780i	0.370501	+	0.188780i	0.629320	−	0.777146i	$-0.283333\pi$
$137$	0.370501	+	0.188780i	0.370501	+	0.188780i	−0.258819	+	0.965926i	$0.583333\pi$
$138$		0			0
$139$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.994522	−	0.104528i	$-0.966667\pi$
$139$	−1.40126	−	1.01807i	−1.40126	−	1.01807i	−0.406737	−	0.913545i	$-0.633333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		−	1.20906i		−	1.20906i
$145$		0			0
$146$	1.58268	−	1.14988i	1.58268	−	1.14988i
$147$	1.46799	−	0.232507i	1.46799	−	0.232507i
$148$		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	$0.300000\pi$
$152$	−0.903007	−	1.77225i	−0.903007	−	1.77225i
$153$	1.56204	−	1.56204i	1.56204	−	1.56204i
$154$		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
$162$	0.738039	+	0.116894i	0.738039	+	0.116894i
$163$	1.77225	−	0.903007i	1.77225	−	0.903007i	0.838671	−	0.544639i	$-0.183333\pi$
$163$	1.77225	−	0.903007i	1.77225	−	0.903007i	0.933580	−	0.358368i	$-0.116667\pi$
$164$	0.415823			0.415823
$165$		0			0
$166$	1.95630			1.95630
$167$		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
$171$	−2.28716	+	0.743145i	−2.28716	+	0.743145i
$172$	−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
$176$	−0.913545	+	0.406737i	−0.913545	+	0.406737i
$177$	−1.70050	+	1.70050i	−1.70050	+	1.70050i
$178$	0.607558	+	1.19240i	0.607558	+	1.19240i
$179$	−1.07394	−	1.47815i	−1.07394	−	1.47815i	−0.866025	−	0.500000i	$-0.833333\pi$
$179$	−1.07394	−	1.47815i	−1.07394	−	1.47815i	−0.207912	−	0.978148i	$-0.566667\pi$
$180$		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
$187$	−1.70574	−	0.654771i	−1.70574	−	0.654771i
$188$		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	$0.200000\pi$
$192$	1.32429	+	0.674761i	1.32429	+	0.674761i
$193$	0.891007	+	0.453990i	0.891007	+	0.453990i	0.838671	−	0.544639i	$-0.183333\pi$
$193$	0.891007	+	0.453990i	0.891007	+	0.453990i	0.0523360	+	0.998630i	$0.483333\pi$
$194$	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	$0.250000\pi$
$198$	0.312927	+	1.16786i	0.312927	+	1.16786i
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	0.978148	−	0.710666i	0.978148	−	0.710666i
$202$		0			0
$203$		0			0
$204$	0.839162	+	2.58268i	0.839162	+	2.58268i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.33093	+	1.47815i	1.33093	+	1.47815i
$210$		0			0
$211$	−0.395472	−	0.128496i	−0.395472	−	0.128496i	0.104528	−	0.994522i	$-0.466667\pi$
$211$	−0.395472	−	0.128496i	−0.395472	−	0.128496i	−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$		0			0
$214$	1.27276	−	0.413545i	1.27276	−	0.413545i
$215$		0			0
$216$	0.182636	−	0.251377i	0.182636	−	0.251377i
$217$		0			0
$218$		0			0
$219$	2.90762			2.90762
$220$		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
$226$	−0.773659	+	0.251377i	−0.773659	+	0.251377i
$227$	0.253116	+	1.59811i	0.253116	+	1.59811i	0.707107	+	0.707107i	$0.250000\pi$
$227$	0.253116	+	1.59811i	0.253116	+	1.59811i	−0.453990	+	0.891007i	$0.650000\pi$
$228$	0.462466	−	2.91990i	0.462466	−	2.91990i
$229$		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	0.707107	−	0.707107i	$-0.250000\pi$
$233$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	−0.987688	+	0.156434i	$0.950000\pi$
$234$		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
$241$			0.813473i			0.813473i	0.913545	+	0.406737i	$0.133333\pi$
$241$			0.813473i			0.813473i	−0.913545	+	0.406737i	$0.866667\pi$
$242$	0.777146	−	0.629320i	0.777146	−	0.629320i
$243$	1.00503	+	1.00503i	1.00503	+	1.00503i
$244$		0			0
$245$		0			0
$246$	0.500000	+	0.363271i	0.500000	+	0.363271i
$247$		0			0
$248$		0			0
$249$	2.35232	+	1.70906i	2.35232	+	1.70906i
$250$		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	−0.987688	−	0.156434i	$-0.950000\pi$
$257$	−1.87869	+	0.297556i	−1.87869	+	0.297556i	−0.891007	+	0.453990i	$0.850000\pi$
$258$	0.417025	−	0.818459i	0.417025	−	0.818459i
$259$		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	$0.750000\pi$
$264$	−1.45381	−	0.309017i	−1.45381	−	0.309017i
$265$		0			0
$266$		0			0
$267$	−0.311155	+	1.96456i	−0.311155	+	1.96456i
$268$	0.127255	+	0.803458i	0.127255	+	0.803458i
$269$		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	$0.700000\pi$
$272$	−1.80460	−	0.285820i	−1.80460	−	0.285820i
$273$		0			0
$274$	−0.415823			−0.415823
$275$		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	$0.216667\pi$
$283$	−0.156434	+	0.987688i	−0.156434	+	0.987688i	−0.933580	+	0.358368i	$0.883333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.548900	+	1.07728i	0.548900	+	1.07728i
$289$	−1.37440	−	1.89169i	−1.37440	−	1.89169i
$290$		0			0
$291$	0.873619	+	2.68872i	0.873619	+	2.68872i
$292$	−0.888139	+	1.74307i	−0.888139	+	1.74307i
$293$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$293$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$294$	−1.20243	+	0.873619i	−1.20243	+	0.873619i
$295$		0			0
$296$		0			0
$297$	−0.111352	+	0.290081i	−0.111352	+	0.290081i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	1.60917	+	1.16913i	1.60917	+	1.16913i
$305$		0			0
$306$	−0.682636	+	2.10094i	−0.682636	+	2.10094i
$307$	0.946294	+	0.946294i	0.946294	+	0.946294i	0.998630	−	0.0523360i	$-0.0166667\pi$
$307$	0.946294	+	0.946294i	0.946294	+	0.946294i	−0.0523360	+	0.998630i	$0.516667\pi$
$308$		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	$-0.650000\pi$
$313$	0.533698	−	1.04744i	0.533698	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$314$		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
$321$	1.89169	+	0.614648i	1.89169	+	0.614648i
$322$		0			0
$323$	0.568508	+	3.58942i	0.568508	+	3.58942i
$324$	−0.710666	+	0.230909i	−0.710666	+	0.230909i
$325$		0			0
$326$	−1.16913	+	1.60917i	−1.16913	+	1.60917i
$327$		0			0
$328$	−0.370501	+	0.188780i	−0.370501	+	0.188780i
$329$		0			0
$330$		0			0
$331$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$331$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$332$	−1.74307	+	0.888139i	−1.74307	+	0.888139i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0327037	+	0.206483i	0.0327037	+	0.206483i	0.998630	−	0.0523360i	$-0.0166667\pi$
$337$	0.0327037	+	0.206483i	0.0327037	+	0.206483i	−0.965926	+	0.258819i	$0.916667\pi$
$338$	0.156434	−	0.987688i	0.156434	−	0.987688i
$339$	−1.14988	−	0.373619i	−1.14988	−	0.373619i
$340$		0			0
$341$		0			0
$342$	1.70050	−	1.70050i	1.70050	−	1.70050i
$343$		0			0
$344$	0.363271	+	0.500000i	0.363271	+	0.500000i
$345$		0			0
$346$		0			0
$347$	0.0949099	−	0.186271i	0.0949099	−	0.186271i	−0.838671	−	0.544639i	$-0.816667\pi$
$347$	0.0949099	−	0.186271i	0.0949099	−	0.186271i	0.933580	+	0.358368i	$0.116667\pi$
$348$		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
$352$	0.629320	−	0.777146i	0.629320	−	0.777146i
$353$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$353$	−0.831254	−	0.831254i	−0.831254	−	0.831254i	−0.987688	+	0.156434i	$0.950000\pi$
$354$	0.743145	−	2.28716i	0.743145	−	2.28716i
$355$		0			0
$356$	−1.08268	−	0.786610i	−1.08268	−	0.786610i
$357$		0			0
$358$	1.62795	+	0.829482i	1.62795	+	0.829482i
$359$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$359$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$360$		0			0
$361$	0.913545	−	2.81160i	0.913545	−	2.81160i
$362$		0			0
$363$	1.48425	−	0.0777864i	1.48425	−	0.0777864i
$364$		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
$369$	0.155360	+	0.478148i	0.155360	+	0.478148i
$370$		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	$0.750000\pi$
$374$	1.81708	−	0.190983i	1.81708	−	0.190983i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.86055	−	0.604528i	1.86055	−	0.604528i	0.866025	−	0.500000i	$-0.166667\pi$
$379$	1.86055	−	0.604528i	1.86055	−	0.604528i	0.994522	−	0.104528i	$-0.0333333\pi$
$380$		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	$0.850000\pi$
$384$	−1.48629			−1.48629
$385$		0			0
$386$	−1.00000			−1.00000
$387$	0.665794	−	0.339239i	0.665794	−	0.339239i
$388$	−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
$393$	−0.273328	+	1.72573i	−0.273328	+	1.72573i
$394$		0			0
$395$		0			0
$396$	−0.809017	−	0.898504i	−0.809017	−	0.898504i
$397$		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
$401$	0.604528	+	1.86055i	0.604528	+	1.86055i	0.500000	+	0.866025i	$0.333333\pi$
$401$	0.604528	+	1.86055i	0.604528	+	1.86055i	0.104528	+	0.994522i	$0.466667\pi$
$402$	−0.548900	+	1.07728i	−0.548900	+	1.07728i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.92021	−	1.92021i	−1.92021	−	1.92021i
$409$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	0.207912	+	0.978148i	$0.433333\pi$
$409$	−0.535233	+	1.64728i	−0.535233	+	1.64728i	−0.743145	+	0.669131i	$0.766667\pi$
$410$		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
$417$	1.82033	+	1.82033i	1.82033	+	1.82033i
$418$	−1.85693	−	0.712810i	−1.85693	−	0.712810i
$419$			1.95630i			1.95630i	0.207912	+	0.978148i	$0.433333\pi$
$419$			1.95630i			1.95630i	−0.207912	+	0.978148i	$0.566667\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$	0.410704	−	0.0650491i	0.410704	−	0.0650491i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.946294	+	0.946294i	−0.946294	+	0.946294i
$429$		0			0
$430$		0			0
$431$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$431$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$432$	−0.0486072	+	0.306894i	−0.0486072	+	0.306894i
$433$	−0.311155	−	1.96456i	−0.311155	−	1.96456i	−0.258819	−	0.965926i	$-0.583333\pi$
$433$	−0.311155	−	1.96456i	−0.311155	−	1.96456i	−0.0523360	−	0.998630i	$-0.516667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2.59071	+	1.32003i	−2.59071	+	1.32003i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−1.20906			−1.20906
$442$		0			0
$443$	−0.410704	−	0.0650491i	−0.410704	−	0.0650491i	−0.0523360	−	0.998630i	$-0.516667\pi$
$443$	−0.410704	−	0.0650491i	−0.410704	−	0.0650491i	−0.358368	+	0.933580i	$0.616667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.73767	+	0.564602i	1.73767	+	0.564602i	0.994522	−	0.104528i	$-0.0333333\pi$
$449$	1.73767	+	0.564602i	1.73767	+	0.564602i	0.743145	+	0.669131i	$0.233333\pi$
$450$		0			0
$451$	0.309017	−	0.278240i	0.309017	−	0.278240i
$452$	0.575212	−	0.575212i	0.575212	−	0.575212i
$453$		0			0
$454$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$455$		0			0
$456$	0.913545	+	2.81160i	0.913545	+	2.81160i
$457$	0.607558	−	1.19240i	0.607558	−	1.19240i	−0.358368	−	0.933580i	$-0.616667\pi$
$457$	0.607558	−	1.19240i	0.607558	−	1.19240i	0.965926	−	0.258819i	$-0.0833333\pi$
$458$		0			0
$459$	−0.459289	+	0.333693i	−0.459289	+	0.333693i
$460$		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
$473$	−0.480303	−	0.388941i	−0.480303	−	0.388941i
$474$		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
$482$	−0.369309	−	0.724810i	−0.369309	−	0.724810i
$483$		0			0
$484$	−0.406737	+	0.913545i	−0.406737	+	0.913545i
$485$		0			0
$486$	−1.35177	−	0.439216i	−1.35177	−	0.439216i
$487$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$487$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$488$		0			0
$489$	−2.81160	+	0.913545i	−2.81160	+	0.913545i
$490$		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
$498$	−2.87182	−	0.454852i	−2.87182	−	0.454852i
$499$	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
$507$	1.05097	−	1.05097i	1.05097	−	1.05097i
$508$		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
$516$			0.918578i			0.918578i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.47815	−	1.07394i	−1.47815	−	1.07394i	−0.978148	−	0.207912i	$-0.933333\pi$
$521$	−1.47815	−	1.07394i	−1.47815	−	1.07394i	−0.500000	−	0.866025i	$-0.666667\pi$
$522$		0			0
$523$	−0.186271	−	0.0949099i	−0.186271	−	0.0949099i	0.358368	−	0.933580i	$-0.383333\pi$
$523$	−0.186271	−	0.0949099i	−0.186271	−	0.0949099i	−0.544639	+	0.838671i	$0.683333\pi$
$524$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.43565	−	0.384680i	1.43565	−	0.384680i
$529$			1.00000i			1.00000i
$530$		0			0
$531$	1.58268	−	1.14988i	1.58268	−	1.14988i
$532$		0			0
$533$		0			0
$534$	−0.614648	−	1.89169i	−0.614648	−	1.89169i
$535$		0			0
$536$	−0.478148	−	0.658114i	−0.478148	−	0.658114i
$537$	1.23285	+	2.41961i	1.23285	+	2.41961i
$538$		0			0
$539$	0.406737	+	0.913545i	0.406737	+	0.913545i
$540$		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
$544$	1.73767	−	0.564602i	1.73767	−	0.564602i
$545$		0			0
$546$		0			0
$547$	1.80460	+	0.285820i	1.80460	+	0.285820i	0.965926	−	0.258819i	$-0.0833333\pi$
$547$	1.80460	+	0.285820i	1.80460	+	0.285820i	0.838671	+	0.544639i	$0.183333\pi$
$548$	0.370501	−	0.188780i	0.370501	−	0.188780i
$549$		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
$561$	2.35177	+	1.35779i	2.35177	+	1.35779i
$562$	1.34500	−	1.34500i	1.34500	−	1.34500i
$563$	0.280582	+	0.550672i	0.280582	+	0.550672i	0.987688	−	0.156434i	$-0.0500000\pi$
$563$	0.280582	+	0.550672i	0.280582	+	0.550672i	−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$567$		0			0
$568$		0			0
$569$	1.60917	−	1.16913i	1.60917	−	1.16913i	0.743145	−	0.669131i	$-0.233333\pi$
$569$	1.60917	−	1.16913i	1.60917	−	1.16913i	0.866025	−	0.500000i	$-0.166667\pi$
$570$		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
$576$	−0.978148	−	0.710666i	−0.978148	−	0.710666i
$577$	−1.77225	−	0.903007i	−1.77225	−	0.903007i	−0.933580	−	0.358368i	$-0.883333\pi$
$577$	−1.77225	−	0.903007i	−1.77225	−	0.903007i	−0.838671	−	0.544639i	$-0.816667\pi$
$578$	2.08341	+	1.06155i	2.08341	+	1.06155i
$579$	−1.20243	−	0.873619i	−1.20243	−	0.873619i
$580$		0			0
$581$		0			0
$582$	−1.99906	−	1.99906i	−1.99906	−	1.99906i
$583$		0			0
$584$		−	1.95630i		−	1.95630i
$585$		0			0
$586$		0			0
$587$	0.803458	−	0.127255i	0.803458	−	0.127255i	0.258819	−	0.965926i	$-0.416667\pi$
$587$	0.803458	−	0.127255i	0.803458	−	0.127255i	0.544639	+	0.838671i	$0.316667\pi$
$588$	0.674761	−	1.32429i	0.674761	−	1.32429i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.147826	−	0.147826i	0.147826	−	0.147826i	−0.629320	−	0.777146i	$-0.716667\pi$
$593$	0.147826	−	0.147826i	0.147826	−	0.147826i	0.777146	+	0.629320i	$0.216667\pi$
$594$	−0.0324790	−	0.309017i	−0.0324790	−	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
$601$	−0.244415	+	0.336408i	−0.244415	+	0.336408i	−0.913545	−	0.406737i	$-0.866667\pi$
$601$	−0.244415	+	0.336408i	−0.244415	+	0.336408i	0.669131	+	0.743145i	$0.266667\pi$
$602$		0			0
$603$	−0.876337	+	0.446516i	−0.876337	+	0.446516i
$604$		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	$0.850000\pi$
$608$	−1.96456	−	0.311155i	−1.96456	−	0.311155i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.345573	−	2.18186i	−0.345573	−	2.18186i
$613$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$613$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$614$	−1.27276	−	0.413545i	−1.27276	−	0.413545i
$615$		0			0
$616$		0			0
$617$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$617$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$618$		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
$627$	−1.61011	−	2.47936i	−1.61011	−	2.47936i
$628$		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	$0.200000\pi$
$642$	−1.96456	+	0.311155i	−1.96456	+	0.311155i
$643$	0.863541	−	1.69480i	0.863541	−	1.69480i	0.156434	−	0.987688i	$-0.450000\pi$
$643$	0.863541	−	1.69480i	0.863541	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$644$		0			0
$645$		0			0
$646$	−2.13611	−	2.94010i	−2.13611	−	2.94010i
$647$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$647$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$648$	0.528377	−	0.528377i	0.528377	−	0.528377i
$649$	−1.40126	−	0.809017i	−1.40126	−	0.809017i
$650$		0			0
$651$		0			0
$652$	0.311155	−	1.96456i	0.311155	−	1.96456i
$653$		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
$656$	0.244415	−	0.336408i	0.244415	−	0.336408i
$657$	−2.33615	−	0.370010i	−2.33615	−	0.370010i
$658$		0			0
$659$	0.813473			0.813473			0.406737	−	0.913545i	$-0.366667\pi$
$659$	0.813473			0.813473			0.406737	+	0.913545i	$0.366667\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−1.19240	+	0.607558i	−1.19240	+	0.607558i
$663$		0			0
$664$	1.14988	−	1.58268i	1.14988	−	1.58268i
$665$		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	$0.150000\pi$
$673$	0.734572	+	1.44168i	0.734572	+	1.44168i	−0.156434	+	0.987688i	$0.550000\pi$
$674$	−0.122881	−	0.169131i	−0.122881	−	0.169131i
$675$		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	$0.650000\pi$
$678$	1.19417	−	0.189138i	1.19417	−	0.189138i
$679$		0			0
$680$		0			0
$681$		−	2.40487i		−	2.40487i
$682$		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	$-0.583333\pi$
$684$	−0.743145	+	2.28716i	−0.743145	+	2.28716i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.550672	−	0.280582i	−0.550672	−	0.280582i
$689$		0			0
$690$		0			0
$691$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i	−0.978148	−	0.207912i	$-0.933333\pi$
$691$	−0.0646021	+	0.198825i	−0.0646021	+	0.198825i	0.913545	+	0.406737i	$0.133333\pi$
$692$		0			0
$693$		0			0
$694$			0.209057i			0.209057i
$695$		0			0
$696$		0			0
$697$	0.750393	−	0.118851i	0.750393	−	0.118851i
$698$		0			0
$699$	0.283856	+	0.873619i	0.283856	+	0.873619i
$700$		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
$704$	−0.207912	+	0.978148i	−0.207912	+	0.978148i
$705$		0			0
$706$	1.11803	+	0.363271i	1.11803	+	0.363271i
$707$		0			0
$708$	0.376204	+	2.37526i	0.376204	+	2.37526i
$709$		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
$712$	1.32178	+	0.209350i	1.32178	+	0.209350i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.82709			−1.82709
$717$		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
$722$	0.462466	+	2.91990i	0.462466	+	2.91990i
$723$	0.189138	−	1.19417i	0.189138	−	1.19417i
$724$		0			0
$725$		0			0
$726$	−1.28716	+	0.743145i	−1.28716	+	0.743145i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	−0.802487	−	1.10453i	−0.802487	−	1.10453i
$730$		0			0
$731$	−0.348943	−	1.07394i	−0.348943	−	1.07394i
$732$		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
$737$	0.632187	+	0.511935i	0.632187	+	0.511935i
$738$	−0.355501	−	0.355501i	−0.355501	−	0.355501i
$739$	−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
$747$	−1.67250	−	1.67250i	−1.67250	−	1.67250i
$748$	−1.53233	+	0.995105i	−1.53233	+	0.995105i
$749$		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
$753$	−0.674761	+	1.32429i	−0.674761	+	1.32429i
$754$		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	$0.350000\pi$
$758$	−1.38331	+	1.38331i	−1.38331	+	1.38331i
$759$		0			0
$760$		0			0
$761$	0.773659	+	0.251377i	0.773659	+	0.251377i	0.669131	−	0.743145i	$-0.266667\pi$
$761$	0.773659	+	0.251377i	0.773659	+	0.251377i	0.104528	+	0.994522i	$0.466667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.32429	−	0.674761i	1.32429	−	0.674761i
$769$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$769$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	2.82709			2.82709
$772$	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	$0.0500000\pi$
$774$	−0.439216	+	0.604528i	−0.439216	+	0.604528i
$775$		0			0
$776$	1.80902	−	0.587785i	1.80902	−	0.587785i
$777$		0			0
$778$		0			0
$779$	−0.786610	−	0.255585i	−0.786610	−	0.255585i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.587785	+	0.809017i	0.587785	+	0.809017i
$785$		0			0
$786$	−0.539926	−	1.66172i	−0.539926	−	1.66172i
$787$	−0.734572	+	1.44168i	−0.734572	+	1.44168i	0.156434	+	0.987688i	$0.450000\pi$
$787$	−0.734572	+	1.44168i	−0.734572	+	1.44168i	−0.891007	+	0.453990i	$0.850000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.12875	+	0.433287i	1.12875	+	0.433287i
$793$		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
$802$	−1.38331	−	1.38331i	−1.38331	−	1.38331i
$803$	0.506326	+	1.88964i	0.506326	+	1.88964i
$804$		−	1.20906i		−	1.20906i
$805$		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
$816$	2.58268	+	0.839162i	2.58268	+	0.839162i
$817$	−0.192304	+	1.21416i	−0.192304	+	1.21416i
$818$	−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
$827$	0.186271	−	0.0949099i	0.186271	−	0.0949099i	−0.358368	−	0.933580i	$-0.616667\pi$
$827$	0.186271	−	0.0949099i	0.186271	−	0.0949099i	0.544639	+	0.838671i	$0.316667\pi$
$828$		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
$833$	−0.285820	+	1.80460i	−0.285820	+	1.80460i
$834$	−2.44833	−	0.795511i	−2.44833	−	0.795511i
$835$		0			0
$836$	1.97815	−	0.207912i	1.97815	−	0.207912i
$837$		0			0
$838$	−0.888139	−	1.74307i	−0.888139	−	1.74307i
$839$		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
$843$	2.79228	−	0.442254i	2.79228	−	0.442254i
$844$	−0.336408	+	0.244415i	−0.336408	+	0.244415i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.459289	−	1.41355i	0.459289	−	1.41355i
$850$		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
$856$	0.413545	−	1.27276i	0.413545	−	1.27276i
$857$	−0.946294	−	0.946294i	−0.946294	−	0.946294i	0.0523360	−	0.998630i	$-0.483333\pi$
$857$	−0.946294	−	0.946294i	−0.946294	−	0.946294i	−0.998630	+	0.0523360i	$0.983333\pi$
$858$		0			0
$859$			0.209057i			0.209057i	0.994522	+	0.104528i	$0.0333333\pi$
$859$			0.209057i			0.209057i	−0.994522	+	0.104528i	$0.966667\pi$
$860$		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	$0.650000\pi$
$864$	−0.0960175	−	0.295511i	−0.0960175	−	0.295511i
$865$		0			0
$866$	1.16913	+	1.60917i	1.16913	+	1.60917i
$867$	1.57777	+	3.09654i	1.57777	+	3.09654i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.359762	−	2.27145i	−0.359762	−	2.27145i
$874$		0			0
$875$		0			0
$876$	1.70906	−	2.35232i	1.70906	−	2.35232i
$877$		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	$0.600000\pi$
$882$	1.07728	−	0.548900i	1.07728	−	0.548900i
$883$	0.410704	+	0.0650491i	0.410704	+	0.0650491i	0.358368	−	0.933580i	$-0.383333\pi$
$883$	0.410704	+	0.0650491i	0.410704	+	0.0650491i	0.0523360	+	0.998630i	$0.483333\pi$
$884$		0			0
$885$		0			0
$886$	0.395472	−	0.128496i	0.395472	−	0.128496i
$887$		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
$891$	−0.373619	+	0.647127i	−0.373619	+	0.647127i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.80460	+	0.285820i	−1.80460	+	0.285820i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−0.149018	+	0.388205i	−0.149018	+	0.388205i
$903$		0			0
$904$	−0.251377	+	0.773659i	−0.251377	+	0.773659i
$905$		0			0
$906$		0			0
$907$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$907$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$908$	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	$0.600000\pi$
$912$	−2.09042	−	2.09042i	−2.09042	−	2.09042i
$913$	−0.701074	+	1.82636i	−0.701074	+	1.82636i
$914$			1.33826i			1.33826i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.257736	−	0.505835i	0.257736	−	0.505835i
$919$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$919$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$920$		0			0
$921$	−1.16913	−	1.60917i	−1.16913	−	1.60917i
$922$		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
$931$	1.16913	−	1.60917i	1.16913	−	1.60917i
$932$	−0.610425	−	0.0966818i	−0.610425	−	0.0966818i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.186271	+	0.0949099i	−0.186271	+	0.0949099i	−0.544639	−	0.838671i	$-0.683333\pi$
$937$	−0.186271	+	0.0949099i	−0.186271	+	0.0949099i	0.358368	+	0.933580i	$0.383333\pi$
$938$		0			0
$939$	−1.02700	+	1.41355i	−1.02700	+	1.41355i
$940$		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
$946$	0.604528	+	0.128496i	0.604528	+	0.128496i
$947$	0.831254	−	0.831254i	0.831254	−	0.831254i	−0.156434	−	0.987688i	$-0.550000\pi$
$947$	0.831254	−	0.831254i	0.831254	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.93221	−	0.306032i	1.93221	−	0.306032i	0.933580	−	0.358368i	$-0.116667\pi$
$953$	1.93221	−	0.306032i	1.93221	−	0.306032i	0.998630	+	0.0523360i	$0.0166667\pi$
$954$		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
$964$	0.658114	+	0.478148i	0.658114	+	0.478148i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	−0.0523360	−	0.998630i	−0.0523360	−	0.998630i
$969$		−	5.40142i		−	5.40142i
$970$		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	$0.133333\pi$
$972$	1.40383	−	0.222345i	1.40383	−	0.222345i
$973$		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	$0.516667\pi$
$978$	2.09042	−	2.09042i	2.09042	−	2.09042i
$979$	−1.33093	+	0.139886i	−1.33093	+	0.139886i
$980$		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
$993$	−1.96456	−	0.311155i	−1.96456	−	0.311155i
$994$		0			0
$995$		0			0
$996$	2.76531	−	0.898504i	2.76531	−	0.898504i
$997$		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.1443.1	✓	32
5.2	odd	4	inner	2200.1.fr.c.1707.1	yes	32
5.3	odd	4	inner	2200.1.fr.c.1707.4	yes	32
5.4	even	2	inner	2200.1.fr.c.1443.4	yes	32
8.3	odd	2	CM	2200.1.fr.c.1443.1	✓	32
11.6	odd	10	inner	2200.1.fr.c.1843.1	yes	32
40.3	even	4	inner	2200.1.fr.c.1707.4	yes	32
40.19	odd	2	inner	2200.1.fr.c.1443.4	yes	32
40.27	even	4	inner	2200.1.fr.c.1707.1	yes	32
55.17	even	20	inner	2200.1.fr.c.2107.1	yes	32
55.28	even	20	inner	2200.1.fr.c.2107.4	yes	32
55.39	odd	10	inner	2200.1.fr.c.1843.4	yes	32
88.83	even	10	inner	2200.1.fr.c.1843.1	yes	32
440.83	odd	20	inner	2200.1.fr.c.2107.4	yes	32
440.259	even	10	inner	2200.1.fr.c.1843.4	yes	32
440.347	odd	20	inner	2200.1.fr.c.2107.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.1.fr.c.1443.1	✓	32	1.1	even	1	trivial
2200.1.fr.c.1443.1	✓	32	8.3	odd	2	CM
2200.1.fr.c.1443.4	yes	32	5.4	even	2	inner
2200.1.fr.c.1443.4	yes	32	40.19	odd	2	inner
2200.1.fr.c.1707.1	yes	32	5.2	odd	4	inner
2200.1.fr.c.1707.1	yes	32	40.27	even	4	inner
2200.1.fr.c.1707.4	yes	32	5.3	odd	4	inner
2200.1.fr.c.1707.4	yes	32	40.3	even	4	inner
2200.1.fr.c.1843.1	yes	32	11.6	odd	10	inner
2200.1.fr.c.1843.1	yes	32	88.83	even	10	inner
2200.1.fr.c.1843.4	yes	32	55.39	odd	10	inner
2200.1.fr.c.1843.4	yes	32	440.259	even	10	inner
2200.1.fr.c.2107.1	yes	32	55.17	even	20	inner
2200.1.fr.c.2107.1	yes	32	440.347	odd	20	inner
2200.1.fr.c.2107.4	yes	32	55.28	even	20	inner
2200.1.fr.c.2107.4	yes	32	440.83	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)

\(f(q)\)	\(=\)	\(q+(-0.891007 + 0.453990i) q^{2} +(-1.46799 - 0.232507i) q^{3} +(0.587785 - 0.809017i) q^{4} +(1.41355 - 0.459289i) q^{6} +(-0.156434 + 0.987688i) q^{8} +(1.14988 + 0.373619i) q^{9} +O(q^{10})\)	\(q+(-0.891007 + 0.453990i) q^{2} +(-1.46799 - 0.232507i) q^{3} +(0.587785 - 0.809017i) q^{4} +(1.41355 - 0.459289i) q^{6} +(-0.156434 + 0.987688i) q^{8} +(1.14988 + 0.373619i) q^{9} +(-0.104528 - 0.994522i) q^{11} +(-1.05097 + 1.05097i) q^{12} +(-0.309017 - 0.951057i) q^{16} +(0.829482 - 1.62795i) q^{17} +(-1.19417 + 0.189138i) q^{18} +(-1.60917 + 1.16913i) q^{19} +(0.544639 + 0.838671i) q^{22} +(0.459289 - 1.41355i) q^{24} +(-0.276853 - 0.141064i) q^{27} +(0.707107 + 0.707107i) q^{32} +(-0.0777864 + 1.48425i) q^{33} +1.82709i q^{34} +(0.978148 - 0.710666i) q^{36} +(0.903007 - 1.77225i) q^{38} +(0.244415 + 0.336408i) q^{41} +(0.437016 - 0.437016i) q^{43} +(-0.866025 - 0.500000i) q^{44} +(0.232507 + 1.46799i) q^{48} +(-0.951057 + 0.309017i) q^{49} +(-1.59618 + 2.19696i) q^{51} +0.310719 q^{54} +(2.63408 - 1.34213i) q^{57} +(0.951057 - 1.30902i) q^{59} +(-0.951057 - 0.309017i) q^{64} +(-0.604528 - 1.35779i) q^{66} +(-0.575212 + 0.575212i) q^{67} +(-0.829482 - 1.62795i) q^{68} +(-0.548900 + 1.07728i) q^{72} +(-1.93221 + 0.306032i) q^{73} +1.98904i q^{76} +(-0.604528 - 0.439216i) q^{81} +(-0.370501 - 0.188780i) q^{82} +(-1.74307 - 0.888139i) q^{83} +(-0.190983 + 0.587785i) q^{86} +(0.998630 + 0.0523360i) q^{88} -1.33826i q^{89} +(-0.873619 - 1.20243i) q^{96} +(-0.863541 - 1.69480i) q^{97} +(0.707107 - 0.707107i) q^{98} +(0.251377 - 1.18264i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 20 q^{6}+O(q^{10}) \) 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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