LMFDB - Embedded newform 220.2.k.a.153.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [220,2,Mod(153,220)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("220.153"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	220.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.75670884447$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			153.2
Root			$-1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	220.153
Dual form			220.2.k.a.197.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{3} +(-1.00000 + 2.00000i) q^{5} +(3.31662 + 3.31662i) q^{7} -1.00000i q^{9} +3.31662i q^{11} +(3.31662 - 3.31662i) q^{13} +(3.00000 - 1.00000i) q^{15} +(3.31662 + 3.31662i) q^{17} -6.63325i q^{21} +(3.00000 + 3.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(-4.00000 + 4.00000i) q^{27} -6.63325 q^{29} -4.00000 q^{31} +(3.31662 - 3.31662i) q^{33} +(-9.94987 + 3.31662i) q^{35} +(5.00000 - 5.00000i) q^{37} -6.63325 q^{39} +(-3.31662 + 3.31662i) q^{43} +(2.00000 + 1.00000i) q^{45} +(5.00000 - 5.00000i) q^{47} +15.0000i q^{49} -6.63325i q^{51} +(-3.00000 - 3.00000i) q^{53} +(-6.63325 - 3.31662i) q^{55} -10.0000i q^{59} -13.2665i q^{61} +(3.31662 - 3.31662i) q^{63} +(3.31662 + 9.94987i) q^{65} +(-3.00000 + 3.00000i) q^{67} -6.00000i q^{69} -4.00000 q^{71} +(3.31662 - 3.31662i) q^{73} +(-1.00000 + 7.00000i) q^{75} +(-11.0000 + 11.0000i) q^{77} +13.2665 q^{79} +5.00000 q^{81} +(-3.31662 + 3.31662i) q^{83} +(-9.94987 + 3.31662i) q^{85} +(6.63325 + 6.63325i) q^{87} -12.0000i q^{89} +22.0000 q^{91} +(4.00000 + 4.00000i) q^{93} +(5.00000 - 5.00000i) q^{97} +3.31662 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + 12 q^{15} + 12 q^{23} - 12 q^{25} - 16 q^{27} - 16 q^{31} + 20 q^{37} + 8 q^{45} + 20 q^{47} - 12 q^{53} - 12 q^{67} - 16 q^{71} - 4 q^{75} - 44 q^{77} + 20 q^{81} + 88 q^{91}+ \cdots + 20 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + 12 * q^15 + 12 * q^23 - 12 * q^25 - 16 * q^27 - 16 * q^31 + 20 * q^37 + 8 * q^45 + 20 * q^47 - 12 * q^53 - 12 * q^67 - 16 * q^71 - 4 * q^75 - 44 * q^77 + 20 * q^81 + 88 * q^91 + 16 * q^93 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/220\mathbb{Z}\right)^\times$.

$n$	$101$	$111$	$177$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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			0
$2$		0			0
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	0.356822	−	0.934172i	$-0.383860\pi$
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	−0.934172	+	0.356822i	$0.883860\pi$
$4$		0			0
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$	3.31662	+	3.31662i	1.25357	+	1.25357i	0.954110	+	0.299456i	$0.0968053\pi$
$7$	3.31662	+	3.31662i	1.25357	+	1.25357i	0.299456	+	0.954110i	$0.403195\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$		0			0
$13$	3.31662	−	3.31662i	0.919866	−	0.919866i	−0.0771531	−	0.997019i	$-0.524583\pi$
$13$	3.31662	−	3.31662i	0.919866	−	0.919866i	0.997019	+	0.0771531i	$0.0245830\pi$
$14$		0			0
$15$	3.00000	−	1.00000i	0.774597	−	0.258199i
$16$		0			0
$17$	3.31662	+	3.31662i	0.804400	+	0.804400i	0.983780	−	0.179380i	$-0.0574092\pi$
$17$	3.31662	+	3.31662i	0.804400	+	0.804400i	−0.179380	+	0.983780i	$0.557409\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		−	6.63325i		−	1.44749i
$22$		0			0
$23$	3.00000	+	3.00000i	0.625543	+	0.625543i	0.946943	−	0.321400i	$-0.104153\pi$
$23$	3.00000	+	3.00000i	0.625543	+	0.625543i	−0.321400	+	0.946943i	$0.604153\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$	−4.00000	+	4.00000i	−0.769800	+	0.769800i
$28$		0			0
$29$	−6.63325			−1.23176			−0.615882	−	0.787839i	$-0.711200\pi$
$29$	−6.63325			−1.23176			−0.615882	+	0.787839i	$0.711200\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	3.31662	−	3.31662i	0.577350	−	0.577350i
$34$		0			0
$35$	−9.94987	+	3.31662i	−1.68184	+	0.560612i
$36$		0			0
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$	−6.63325			−1.06217
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−3.31662	+	3.31662i	−0.505781	+	0.505781i	−0.913228	−	0.407448i	$-0.866419\pi$
$43$	−3.31662	+	3.31662i	−0.505781	+	0.505781i	0.407448	+	0.913228i	$0.366419\pi$
$44$		0			0
$45$	2.00000	+	1.00000i	0.298142	+	0.149071i
$46$		0			0
$47$	5.00000	−	5.00000i	0.729325	−	0.729325i	−0.241160	−	0.970485i	$-0.577528\pi$
$47$	5.00000	−	5.00000i	0.729325	−	0.729325i	0.970485	+	0.241160i	$0.0775280\pi$
$48$		0			0
$49$			15.0000i			2.14286i
$50$		0			0
$51$		−	6.63325i		−	0.928841i
$52$		0			0
$53$	−3.00000	−	3.00000i	−0.412082	−	0.412082i	0.470381	−	0.882463i	$-0.344116\pi$
$53$	−3.00000	−	3.00000i	−0.412082	−	0.412082i	−0.882463	+	0.470381i	$0.844116\pi$
$54$		0			0
$55$	−6.63325	−	3.31662i	−0.894427	−	0.447214i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	10.0000i		−	1.30189i	−0.759125	−	0.650945i	$-0.774373\pi$
$59$		−	10.0000i		−	1.30189i	0.759125	−	0.650945i	$-0.225627\pi$
$60$		0			0
$61$		−	13.2665i		−	1.69860i	−0.527910	−	0.849301i	$-0.677024\pi$
$61$		−	13.2665i		−	1.69860i	0.527910	−	0.849301i	$-0.322976\pi$
$62$		0			0
$63$	3.31662	−	3.31662i	0.417855	−	0.417855i
$64$		0			0
$65$	3.31662	+	9.94987i	0.411377	+	1.23413i
$66$		0			0
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	−0.866202	−	0.499694i	$-0.833446\pi$
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	0.499694	+	0.866202i	$0.333446\pi$
$68$		0			0
$69$		−	6.00000i		−	0.722315i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	3.31662	−	3.31662i	0.388182	−	0.388182i	−0.485857	−	0.874038i	$-0.661492\pi$
$73$	3.31662	−	3.31662i	0.388182	−	0.388182i	0.874038	+	0.485857i	$0.161492\pi$
$74$		0			0
$75$	−1.00000	+	7.00000i	−0.115470	+	0.808290i
$76$		0			0
$77$	−11.0000	+	11.0000i	−1.25357	+	1.25357i
$78$		0			0
$79$	13.2665			1.49260			0.746299	−	0.665611i	$-0.231829\pi$
$79$	13.2665			1.49260			0.746299	+	0.665611i	$0.231829\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−3.31662	+	3.31662i	−0.364047	+	0.364047i	−0.865300	−	0.501254i	$-0.832872\pi$
$83$	−3.31662	+	3.31662i	−0.364047	+	0.364047i	0.501254	+	0.865300i	$0.332872\pi$
$84$		0			0
$85$	−9.94987	+	3.31662i	−1.07922	+	0.359738i
$86$		0			0
$87$	6.63325	+	6.63325i	0.711159	+	0.711159i
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$	22.0000			2.30623
$92$		0			0
$93$	4.00000	+	4.00000i	0.414781	+	0.414781i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$	3.31662			0.333333
$100$		0			0
$101$			13.2665i			1.32007i	0.751237	+	0.660033i	$0.229458\pi$
$101$			13.2665i			1.32007i	−0.751237	+	0.660033i	$0.770542\pi$
$102$		0			0
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	0.107418	−	0.994214i	$-0.465742\pi$
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	−0.994214	+	0.107418i	$0.965742\pi$
$104$		0			0
$105$	13.2665	+	6.63325i	1.29468	+	0.647339i
$106$		0			0
$107$	3.31662	+	3.31662i	0.320630	+	0.320630i	0.849009	−	0.528379i	$-0.177200\pi$
$107$	3.31662	+	3.31662i	0.320630	+	0.320630i	−0.528379	+	0.849009i	$0.677200\pi$
$108$		0			0
$109$	−6.63325			−0.635350			−0.317675	−	0.948200i	$-0.602902\pi$
$109$	−6.63325			−0.635350			−0.317675	+	0.948200i	$0.602902\pi$
$110$		0			0
$111$	−10.0000			−0.949158
$112$		0			0
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	0.752577	−	0.658505i	$-0.228811\pi$
$113$	1.00000	+	1.00000i	0.0940721	+	0.0940721i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	−9.00000	+	3.00000i	−0.839254	+	0.279751i
$116$		0			0
$117$	−3.31662	−	3.31662i	−0.306622	−	0.306622i
$118$		0			0
$119$			22.0000i			2.01674i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$	3.31662	+	3.31662i	0.294303	+	0.294303i	0.838777	−	0.544475i	$-0.183271\pi$
$127$	3.31662	+	3.31662i	0.294303	+	0.294303i	−0.544475	+	0.838777i	$0.683271\pi$
$128$		0			0
$129$	6.63325			0.584025
$130$		0			0
$131$		−	6.63325i		−	0.579550i	−0.957095	−	0.289775i	$-0.906420\pi$
$131$		−	6.63325i		−	0.579550i	0.957095	−	0.289775i	$-0.0935804\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.00000	−	12.0000i	−0.344265	−	1.03280i
$136$		0			0
$137$	1.00000	−	1.00000i	0.0854358	−	0.0854358i	−0.663097	−	0.748533i	$-0.730758\pi$
$137$	1.00000	−	1.00000i	0.0854358	−	0.0854358i	0.748533	+	0.663097i	$0.230758\pi$
$138$		0			0
$139$	13.2665			1.12525			0.562625	−	0.826712i	$-0.309792\pi$
$139$	13.2665			1.12525			0.562625	+	0.826712i	$0.309792\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	11.0000	+	11.0000i	0.919866	+	0.919866i
$144$		0			0
$145$	6.63325	−	13.2665i	0.550861	−	1.10172i
$146$		0			0
$147$	15.0000	−	15.0000i	1.23718	−	1.23718i
$148$		0			0
$149$	−6.63325			−0.543417			−0.271708	−	0.962380i	$-0.587589\pi$
$149$	−6.63325			−0.543417			−0.271708	+	0.962380i	$0.587589\pi$
$150$		0			0
$151$			6.63325i			0.539806i	0.962887	+	0.269903i	$0.0869917\pi$
$151$			6.63325i			0.539806i	−0.962887	+	0.269903i	$0.913008\pi$
$152$		0			0
$153$	3.31662	−	3.31662i	0.268133	−	0.268133i
$154$		0			0
$155$	4.00000	−	8.00000i	0.321288	−	0.642575i
$156$		0			0
$157$	1.00000	−	1.00000i	0.0798087	−	0.0798087i	−0.666076	−	0.745884i	$-0.732027\pi$
$157$	1.00000	−	1.00000i	0.0798087	−	0.0798087i	0.745884	+	0.666076i	$0.232027\pi$
$158$		0			0
$159$			6.00000i			0.475831i
$160$		0			0
$161$			19.8997i			1.56832i
$162$		0			0
$163$	3.00000	+	3.00000i	0.234978	+	0.234978i	0.814767	−	0.579789i	$-0.196865\pi$
$163$	3.00000	+	3.00000i	0.234978	+	0.234978i	−0.579789	+	0.814767i	$0.696865\pi$
$164$		0			0
$165$	3.31662	+	9.94987i	0.258199	+	0.774597i
$166$		0			0
$167$	−9.94987	−	9.94987i	−0.769944	−	0.769944i	0.208152	−	0.978096i	$-0.433255\pi$
$167$	−9.94987	−	9.94987i	−0.769944	−	0.769944i	−0.978096	+	0.208152i	$0.933255\pi$
$168$		0			0
$169$		−	9.00000i		−	0.692308i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.31662	−	3.31662i	0.252158	−	0.252158i	−0.569697	−	0.821855i	$-0.692939\pi$
$173$	3.31662	−	3.31662i	0.252158	−	0.252158i	0.821855	+	0.569697i	$0.192939\pi$
$174$		0			0
$175$	3.31662	−	23.2164i	0.250713	−	1.75499i
$176$		0			0
$177$	−10.0000	+	10.0000i	−0.751646	+	0.751646i
$178$		0			0
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	−13.2665	+	13.2665i	−0.980688	+	0.980688i
$184$		0			0
$185$	5.00000	+	15.0000i	0.367607	+	1.10282i
$186$		0			0
$187$	−11.0000	+	11.0000i	−0.804400	+	0.804400i
$188$		0			0
$189$	−26.5330			−1.92999
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$	−9.94987	+	9.94987i	−0.716208	+	0.716208i	−0.967826	−	0.251619i	$-0.919037\pi$
$193$	−9.94987	+	9.94987i	−0.716208	+	0.716208i	0.251619	+	0.967826i	$0.419037\pi$
$194$		0			0
$195$	6.63325	−	13.2665i	0.475017	−	0.950034i
$196$		0			0
$197$	−9.94987	−	9.94987i	−0.708899	−	0.708899i	0.257405	−	0.966304i	$-0.417133\pi$
$197$	−9.94987	−	9.94987i	−0.708899	−	0.708899i	−0.966304	+	0.257405i	$0.917133\pi$
$198$		0			0
$199$		−	22.0000i		−	1.55954i	−0.626067	−	0.779769i	$-0.715336\pi$
$199$		−	22.0000i		−	1.55954i	0.626067	−	0.779769i	$-0.284664\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$	−22.0000	−	22.0000i	−1.54410	−	1.54410i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.00000	−	3.00000i	0.208514	−	0.208514i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	6.63325i		−	0.456652i	−0.973585	−	0.228326i	$-0.926675\pi$
$211$		−	6.63325i		−	0.456652i	0.973585	−	0.228326i	$-0.0733252\pi$
$212$		0			0
$213$	4.00000	+	4.00000i	0.274075	+	0.274075i
$214$		0			0
$215$	−3.31662	−	9.94987i	−0.226192	−	0.678576i
$216$		0			0
$217$	−13.2665	−	13.2665i	−0.900589	−	0.900589i
$218$		0			0
$219$	−6.63325			−0.448233
$220$		0			0
$221$	22.0000			1.47988
$222$		0			0
$223$	7.00000	+	7.00000i	0.468755	+	0.468755i	0.901511	−	0.432756i	$-0.142459\pi$
$223$	7.00000	+	7.00000i	0.468755	+	0.468755i	−0.432756	+	0.901511i	$0.642459\pi$
$224$		0			0
$225$	−4.00000	+	3.00000i	−0.266667	+	0.200000i
$226$		0			0
$227$	−9.94987	−	9.94987i	−0.660396	−	0.660396i	0.295077	−	0.955473i	$-0.404655\pi$
$227$	−9.94987	−	9.94987i	−0.660396	−	0.660396i	−0.955473	+	0.295077i	$0.904655\pi$
$228$		0			0
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$	22.0000			1.44749
$232$		0			0
$233$	3.31662	−	3.31662i	0.217279	−	0.217279i	−0.590072	−	0.807351i	$-0.700900\pi$
$233$	3.31662	−	3.31662i	0.217279	−	0.217279i	0.807351	+	0.590072i	$0.200900\pi$
$234$		0			0
$235$	5.00000	+	15.0000i	0.326164	+	0.978492i
$236$		0			0
$237$	−13.2665	−	13.2665i	−0.861752	−	0.861752i
$238$		0			0
$239$	−13.2665			−0.858138			−0.429069	−	0.903272i	$-0.641158\pi$
$239$	−13.2665			−0.858138			−0.429069	+	0.903272i	$0.641158\pi$
$240$		0			0
$241$			26.5330i			1.70914i	0.519336	+	0.854570i	$0.326179\pi$
$241$			26.5330i			1.70914i	−0.519336	+	0.854570i	$0.673821\pi$
$242$		0			0
$243$	7.00000	+	7.00000i	0.449050	+	0.449050i
$244$		0			0
$245$	−30.0000	−	15.0000i	−1.91663	−	0.958315i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	6.63325			0.420365
$250$		0			0
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		0			0
$253$	−9.94987	+	9.94987i	−0.625543	+	0.625543i
$254$		0			0
$255$	13.2665	+	6.63325i	0.830780	+	0.415390i
$256$		0			0
$257$	9.00000	−	9.00000i	0.561405	−	0.561405i	−0.368302	−	0.929706i	$-0.620061\pi$
$257$	9.00000	−	9.00000i	0.561405	−	0.561405i	0.929706	+	0.368302i	$0.120061\pi$
$258$		0			0
$259$	33.1662			2.06085
$260$		0			0
$261$			6.63325i			0.410588i
$262$		0			0
$263$	−3.31662	+	3.31662i	−0.204512	+	0.204512i	−0.801930	−	0.597418i	$-0.796193\pi$
$263$	−3.31662	+	3.31662i	−0.204512	+	0.204512i	0.597418	+	0.801930i	$0.296193\pi$
$264$		0			0
$265$	9.00000	−	3.00000i	0.552866	−	0.184289i
$266$		0			0
$267$	−12.0000	+	12.0000i	−0.734388	+	0.734388i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		−	19.8997i		−	1.20882i	−0.796672	−	0.604412i	$-0.793408\pi$
$271$		−	19.8997i		−	1.20882i	0.796672	−	0.604412i	$-0.206592\pi$
$272$		0			0
$273$	−22.0000	−	22.0000i	−1.33150	−	1.33150i
$274$		0			0
$275$	13.2665	−	9.94987i	0.800000	−	0.600000i
$276$		0			0
$277$	16.5831	+	16.5831i	0.996383	+	0.996383i	0.999993	−	0.00361013i	$-0.00114914\pi$
$277$	16.5831	+	16.5831i	0.996383	+	0.996383i	−0.00361013	+	0.999993i	$0.501149\pi$
$278$		0			0
$279$			4.00000i			0.239474i
$280$		0			0
$281$			26.5330i			1.58283i	0.611282	+	0.791413i	$0.290654\pi$
$281$			26.5330i			1.58283i	−0.611282	+	0.791413i	$0.709346\pi$
$282$		0			0
$283$	−16.5831	+	16.5831i	−0.985764	+	0.985764i	−0.999900	−	0.0141357i	$-0.995500\pi$
$283$	−16.5831	+	16.5831i	−0.985764	+	0.985764i	0.0141357	+	0.999900i	$0.495500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			5.00000i			0.294118i
$290$		0			0
$291$	−10.0000			−0.586210
$292$		0			0
$293$	16.5831	−	16.5831i	0.968796	−	0.968796i	−0.0307312	−	0.999528i	$-0.509784\pi$
$293$	16.5831	−	16.5831i	0.968796	−	0.968796i	0.999528	+	0.0307312i	$0.00978360\pi$
$294$		0			0
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$		0			0
$297$	−13.2665	−	13.2665i	−0.769800	−	0.769800i
$298$		0			0
$299$	19.8997			1.15083
$300$		0			0
$301$	−22.0000			−1.26806
$302$		0			0
$303$	13.2665	−	13.2665i	0.762140	−	0.762140i
$304$		0			0
$305$	26.5330	+	13.2665i	1.51928	+	0.759638i
$306$		0			0
$307$	3.31662	+	3.31662i	0.189290	+	0.189290i	0.795389	−	0.606099i	$-0.207267\pi$
$307$	3.31662	+	3.31662i	0.189290	+	0.189290i	−0.606099	+	0.795389i	$0.707267\pi$
$308$		0			0
$309$			18.0000i			1.02398i
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$	21.0000	+	21.0000i	1.18699	+	1.18699i	0.977895	+	0.209095i	$0.0670517\pi$
$313$	21.0000	+	21.0000i	1.18699	+	1.18699i	0.209095	+	0.977895i	$0.432948\pi$
$314$		0			0
$315$	3.31662	+	9.94987i	0.186871	+	0.560612i
$316$		0			0
$317$	17.0000	−	17.0000i	0.954815	−	0.954815i	−0.0442073	−	0.999022i	$-0.514076\pi$
$317$	17.0000	−	17.0000i	0.954815	−	0.954815i	0.999022	+	0.0442073i	$0.0140762\pi$
$318$		0			0
$319$		−	22.0000i		−	1.23176i
$320$		0			0
$321$		−	6.63325i		−	0.370232i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−23.2164	−	3.31662i	−1.28781	−	0.183973i
$326$		0			0
$327$	6.63325	+	6.63325i	0.366820	+	0.366820i
$328$		0			0
$329$	33.1662			1.82851
$330$		0			0
$331$	−16.0000			−0.879440			−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000			−0.879440			−0.439720	+	0.898135i	$0.644922\pi$
$332$		0			0
$333$	−5.00000	−	5.00000i	−0.273998	−	0.273998i
$334$		0			0
$335$	−3.00000	−	9.00000i	−0.163908	−	0.491723i
$336$		0			0
$337$	3.31662	+	3.31662i	0.180668	+	0.180668i	0.791647	−	0.610979i	$-0.209224\pi$
$337$	3.31662	+	3.31662i	0.180668	+	0.180668i	−0.610979	+	0.791647i	$0.709224\pi$
$338$		0			0
$339$		−	2.00000i		−	0.108625i
$340$		0			0
$341$		−	13.2665i		−	0.718421i
$342$		0			0
$343$	−26.5330	+	26.5330i	−1.43265	+	1.43265i
$344$		0			0
$345$	12.0000	+	6.00000i	0.646058	+	0.323029i
$346$		0			0
$347$	−9.94987	−	9.94987i	−0.534137	−	0.534137i	0.387664	−	0.921801i	$-0.373282\pi$
$347$	−9.94987	−	9.94987i	−0.534137	−	0.534137i	−0.921801	+	0.387664i	$0.873282\pi$
$348$		0			0
$349$	19.8997			1.06521			0.532605	−	0.846364i	$-0.321213\pi$
$349$	19.8997			1.06521			0.532605	+	0.846364i	$0.321213\pi$
$350$		0			0
$351$			26.5330i			1.41623i
$352$		0			0
$353$	9.00000	+	9.00000i	0.479022	+	0.479022i	0.904819	−	0.425797i	$-0.140006\pi$
$353$	9.00000	+	9.00000i	0.479022	+	0.479022i	−0.425797	+	0.904819i	$0.640006\pi$
$354$		0			0
$355$	4.00000	−	8.00000i	0.212298	−	0.424596i
$356$		0			0
$357$	22.0000	−	22.0000i	1.16436	−	1.16436i
$358$		0			0
$359$	−13.2665			−0.700179			−0.350090	−	0.936716i	$-0.613849\pi$
$359$	−13.2665			−0.700179			−0.350090	+	0.936716i	$0.613849\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	11.0000	+	11.0000i	0.577350	+	0.577350i
$364$		0			0
$365$	3.31662	+	9.94987i	0.173600	+	0.520800i
$366$		0			0
$367$	−15.0000	+	15.0000i	−0.782994	+	0.782994i	−0.980335	−	0.197341i	$-0.936769\pi$
$367$	−15.0000	+	15.0000i	−0.782994	+	0.782994i	0.197341	+	0.980335i	$0.436769\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	19.8997i		−	1.03314i
$372$		0			0
$373$	3.31662	−	3.31662i	0.171728	−	0.171728i	−0.616010	−	0.787738i	$-0.711252\pi$
$373$	3.31662	−	3.31662i	0.171728	−	0.171728i	0.787738	+	0.616010i	$0.211252\pi$
$374$		0			0
$375$	−13.0000	−	9.00000i	−0.671317	−	0.464758i
$376$		0			0
$377$	−22.0000	+	22.0000i	−1.13306	+	1.13306i
$378$		0			0
$379$		−	2.00000i		−	0.102733i	−0.998680	−	0.0513665i	$-0.983642\pi$
$379$		−	2.00000i		−	0.102733i	0.998680	−	0.0513665i	$-0.0163577\pi$
$380$		0			0
$381$		−	6.63325i		−	0.339832i
$382$		0			0
$383$	−5.00000	−	5.00000i	−0.255488	−	0.255488i	0.567728	−	0.823216i	$-0.307823\pi$
$383$	−5.00000	−	5.00000i	−0.255488	−	0.255488i	−0.823216	+	0.567728i	$0.807823\pi$
$384$		0			0
$385$	−11.0000	−	33.0000i	−0.560612	−	1.68184i
$386$		0			0
$387$	3.31662	+	3.31662i	0.168594	+	0.168594i
$388$		0			0
$389$			32.0000i			1.62246i	0.584724	+	0.811232i	$0.301203\pi$
$389$			32.0000i			1.62246i	−0.584724	+	0.811232i	$0.698797\pi$
$390$		0			0
$391$			19.8997i			1.00637i
$392$		0			0
$393$	−6.63325	+	6.63325i	−0.334603	+	0.334603i
$394$		0			0
$395$	−13.2665	+	26.5330i	−0.667510	+	1.33502i
$396$		0			0
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.168663	+	0.985674i	$0.553945\pi$
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.985674	+	0.168663i	$0.946055\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$	−13.2665	+	13.2665i	−0.660851	+	0.660851i
$404$		0			0
$405$	−5.00000	+	10.0000i	−0.248452	+	0.496904i
$406$		0			0
$407$	16.5831	+	16.5831i	0.821995	+	0.821995i
$408$		0			0
$409$	−19.8997			−0.983979			−0.491990	−	0.870601i	$-0.663730\pi$
$409$	−19.8997			−0.983979			−0.491990	+	0.870601i	$0.663730\pi$
$410$		0			0
$411$	−2.00000			−0.0986527
$412$		0			0
$413$	33.1662	−	33.1662i	1.63200	−	1.63200i
$414$		0			0
$415$	−3.31662	−	9.94987i	−0.162807	−	0.488420i
$416$		0			0
$417$	−13.2665	−	13.2665i	−0.649663	−	0.649663i
$418$		0			0
$419$		−	2.00000i		−	0.0977064i	−0.998806	−	0.0488532i	$-0.984443\pi$
$419$		−	2.00000i		−	0.0977064i	0.998806	−	0.0488532i	$-0.0155566\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$	−5.00000	−	5.00000i	−0.243108	−	0.243108i
$424$		0			0
$425$	3.31662	−	23.2164i	0.160880	−	1.12616i
$426$		0			0
$427$	44.0000	−	44.0000i	2.12931	−	2.12931i
$428$		0			0
$429$		−	22.0000i		−	1.06217i
$430$		0			0
$431$		−	33.1662i		−	1.59756i	−0.601622	−	0.798781i	$-0.705479\pi$
$431$		−	33.1662i		−	1.59756i	0.601622	−	0.798781i	$-0.294521\pi$
$432$		0			0
$433$	−15.0000	−	15.0000i	−0.720854	−	0.720854i	0.247925	−	0.968779i	$-0.420251\pi$
$433$	−15.0000	−	15.0000i	−0.720854	−	0.720854i	−0.968779	+	0.247925i	$0.920251\pi$
$434$		0			0
$435$	−19.8997	+	6.63325i	−0.954120	+	0.318040i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$		0			0
$443$	15.0000	+	15.0000i	0.712672	+	0.712672i	0.967093	−	0.254422i	$-0.0818852\pi$
$443$	15.0000	+	15.0000i	0.712672	+	0.712672i	−0.254422	+	0.967093i	$0.581885\pi$
$444$		0			0
$445$	24.0000	+	12.0000i	1.13771	+	0.568855i
$446$		0			0
$447$	6.63325	+	6.63325i	0.313742	+	0.313742i
$448$		0			0
$449$		−	8.00000i		−	0.377543i	−0.982021	−	0.188772i	$-0.939549\pi$
$449$		−	8.00000i		−	0.377543i	0.982021	−	0.188772i	$-0.0604506\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	6.63325	−	6.63325i	0.311657	−	0.311657i
$454$		0			0
$455$	−22.0000	+	44.0000i	−1.03138	+	2.06275i
$456$		0			0
$457$	3.31662	+	3.31662i	0.155145	+	0.155145i	0.780411	−	0.625266i	$-0.215010\pi$
$457$	3.31662	+	3.31662i	0.155145	+	0.155145i	−0.625266	+	0.780411i	$0.715010\pi$
$458$		0			0
$459$	−26.5330			−1.23845
$460$		0			0
$461$		−	39.7995i		−	1.85365i	−0.375497	−	0.926824i	$-0.622528\pi$
$461$		−	39.7995i		−	1.85365i	0.375497	−	0.926824i	$-0.377472\pi$
$462$		0			0
$463$	−9.00000	−	9.00000i	−0.418265	−	0.418265i	0.466340	−	0.884606i	$-0.345572\pi$
$463$	−9.00000	−	9.00000i	−0.418265	−	0.418265i	−0.884606	+	0.466340i	$0.845572\pi$
$464$		0			0
$465$	−12.0000	+	4.00000i	−0.556487	+	0.185496i
$466$		0			0
$467$	1.00000	−	1.00000i	0.0462745	−	0.0462745i	−0.683591	−	0.729865i	$-0.739583\pi$
$467$	1.00000	−	1.00000i	0.0462745	−	0.0462745i	0.729865	+	0.683591i	$0.239583\pi$
$468$		0			0
$469$	−19.8997			−0.918885
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		0			0
$473$	−11.0000	−	11.0000i	−0.505781	−	0.505781i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−3.00000	+	3.00000i	−0.137361	+	0.137361i
$478$		0			0
$479$	−13.2665			−0.606162			−0.303081	−	0.952965i	$-0.598015\pi$
$479$	−13.2665			−0.606162			−0.303081	+	0.952965i	$0.598015\pi$
$480$		0			0
$481$		−	33.1662i		−	1.51225i
$482$		0			0
$483$	19.8997	−	19.8997i	0.905470	−	0.905470i
$484$		0			0
$485$	5.00000	+	15.0000i	0.227038	+	0.681115i
$486$		0			0
$487$	−23.0000	+	23.0000i	−1.04223	+	1.04223i	−0.0431614	+	0.999068i	$0.513743\pi$
$487$	−23.0000	+	23.0000i	−1.04223	+	1.04223i	−0.999068	+	0.0431614i	$0.986257\pi$
$488$		0			0
$489$		−	6.00000i		−	0.271329i
$490$		0			0
$491$			33.1662i			1.49677i	0.663264	+	0.748386i	$0.269171\pi$
$491$			33.1662i			1.49677i	−0.663264	+	0.748386i	$0.730829\pi$
$492$		0			0
$493$	−22.0000	−	22.0000i	−0.990830	−	0.990830i
$494$		0			0
$495$	−3.31662	+	6.63325i	−0.149071	+	0.298142i
$496$		0			0
$497$	−13.2665	−	13.2665i	−0.595084	−	0.595084i
$498$		0			0
$499$			22.0000i			0.984855i	0.870353	+	0.492428i	$0.163890\pi$
$499$			22.0000i			0.984855i	−0.870353	+	0.492428i	$0.836110\pi$
$500$		0			0
$501$			19.8997i			0.889055i
$502$		0			0
$503$	9.94987	−	9.94987i	0.443643	−	0.443643i	−0.449591	−	0.893234i	$-0.648430\pi$
$503$	9.94987	−	9.94987i	0.443643	−	0.443643i	0.893234	+	0.449591i	$0.148430\pi$
$504$		0			0
$505$	−26.5330	−	13.2665i	−1.18070	−	0.590351i
$506$		0			0
$507$	−9.00000	+	9.00000i	−0.399704	+	0.399704i
$508$		0			0
$509$		−	36.0000i		−	1.59567i	−0.602875	−	0.797836i	$-0.705978\pi$
$509$		−	36.0000i		−	1.59567i	0.602875	−	0.797836i	$-0.294022\pi$
$510$		0			0
$511$	22.0000			0.973223
$512$		0			0
$513$		0			0
$514$		0			0
$515$	27.0000	−	9.00000i	1.18976	−	0.396587i
$516$		0			0
$517$	16.5831	+	16.5831i	0.729325	+	0.729325i
$518$		0			0
$519$	−6.63325			−0.291167
$520$		0			0
$521$	−34.0000			−1.48957			−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000			−1.48957			−0.744784	+	0.667306i	$0.767447\pi$
$522$		0			0
$523$	−16.5831	+	16.5831i	−0.725129	+	0.725129i	−0.969645	−	0.244516i	$-0.921371\pi$
$523$	−16.5831	+	16.5831i	−0.725129	+	0.725129i	0.244516	+	0.969645i	$0.421371\pi$
$524$		0			0
$525$	−26.5330	+	19.8997i	−1.15799	+	0.868496i
$526$		0			0
$527$	−13.2665	−	13.2665i	−0.577898	−	0.577898i
$528$		0			0
$529$		−	5.00000i		−	0.217391i
$530$		0			0
$531$	−10.0000			−0.433963
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−9.94987	+	3.31662i	−0.430171	+	0.143390i
$536$		0			0
$537$	6.00000	−	6.00000i	0.258919	−	0.258919i
$538$		0			0
$539$	−49.7494			−2.14286
$540$		0			0
$541$		−	13.2665i		−	0.570371i	−0.958472	−	0.285186i	$-0.907945\pi$
$541$		−	13.2665i		−	0.570371i	0.958472	−	0.285186i	$-0.0920553\pi$
$542$		0			0
$543$	−2.00000	−	2.00000i	−0.0858282	−	0.0858282i
$544$		0			0
$545$	6.63325	−	13.2665i	0.284137	−	0.568274i
$546$		0			0
$547$	29.8496	+	29.8496i	1.27628	+	1.27628i	0.942735	+	0.333543i	$0.108244\pi$
$547$	29.8496	+	29.8496i	1.27628	+	1.27628i	0.333543	+	0.942735i	$0.391756\pi$
$548$		0			0
$549$	−13.2665			−0.566200
$550$		0			0
$551$		0			0
$552$		0			0
$553$	44.0000	+	44.0000i	1.87107	+	1.87107i
$554$		0			0
$555$	10.0000	−	20.0000i	0.424476	−	0.848953i
$556$		0			0
$557$	29.8496	+	29.8496i	1.26477	+	1.26477i	0.948754	+	0.316015i	$0.102345\pi$
$557$	29.8496	+	29.8496i	1.26477	+	1.26477i	0.316015	+	0.948754i	$0.397655\pi$
$558$		0			0
$559$			22.0000i			0.930501i
$560$		0			0
$561$	22.0000			0.928841
$562$		0			0
$563$	9.94987	−	9.94987i	0.419337	−	0.419337i	−0.465638	−	0.884975i	$-0.654175\pi$
$563$	9.94987	−	9.94987i	0.419337	−	0.419337i	0.884975	+	0.465638i	$0.154175\pi$
$564$		0			0
$565$	−3.00000	+	1.00000i	−0.126211	+	0.0420703i
$566$		0			0
$567$	16.5831	+	16.5831i	0.696426	+	0.696426i
$568$		0			0
$569$	−19.8997			−0.834241			−0.417120	−	0.908851i	$-0.636961\pi$
$569$	−19.8997			−0.834241			−0.417120	+	0.908851i	$0.636961\pi$
$570$		0			0
$571$		−	19.8997i		−	0.832779i	−0.909186	−	0.416389i	$-0.863295\pi$
$571$		−	19.8997i		−	0.832779i	0.909186	−	0.416389i	$-0.136705\pi$
$572$		0			0
$573$	12.0000	+	12.0000i	0.501307	+	0.501307i
$574$		0			0
$575$	3.00000	−	21.0000i	0.125109	−	0.875761i
$576$		0			0
$577$	21.0000	−	21.0000i	0.874241	−	0.874241i	−0.118690	−	0.992931i	$-0.537869\pi$
$577$	21.0000	−	21.0000i	0.874241	−	0.874241i	0.992931	+	0.118690i	$0.0378694\pi$
$578$		0			0
$579$	19.8997			0.827005
$580$		0			0
$581$	−22.0000			−0.912714
$582$		0			0
$583$	9.94987	−	9.94987i	0.412082	−	0.412082i
$584$		0			0
$585$	9.94987	−	3.31662i	0.411377	−	0.137126i
$586$		0			0
$587$	13.0000	−	13.0000i	0.536567	−	0.536567i	−0.385952	−	0.922519i	$-0.626127\pi$
$587$	13.0000	−	13.0000i	0.536567	−	0.536567i	0.922519	+	0.385952i	$0.126127\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			19.8997i			0.818566i
$592$		0			0
$593$	−9.94987	+	9.94987i	−0.408592	+	0.408592i	−0.881247	−	0.472655i	$-0.843296\pi$
$593$	−9.94987	+	9.94987i	−0.408592	+	0.408592i	0.472655	+	0.881247i	$0.343296\pi$
$594$		0			0
$595$	−44.0000	−	22.0000i	−1.80382	−	0.901912i
$596$		0			0
$597$	−22.0000	+	22.0000i	−0.900400	+	0.900400i
$598$		0			0
$599$		−	30.0000i		−	1.22577i	−0.790173	−	0.612883i	$-0.790010\pi$
$599$		−	30.0000i		−	1.22577i	0.790173	−	0.612883i	$-0.209990\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	3.00000	+	3.00000i	0.122169	+	0.122169i
$604$		0			0
$605$	11.0000	−	22.0000i	0.447214	−	0.894427i
$606$		0			0
$607$	3.31662	+	3.31662i	0.134618	+	0.134618i	0.771205	−	0.636587i	$-0.219654\pi$
$607$	3.31662	+	3.31662i	0.134618	+	0.134618i	−0.636587	+	0.771205i	$0.719654\pi$
$608$		0			0
$609$			44.0000i			1.78297i
$610$		0			0
$611$		−	33.1662i		−	1.34176i
$612$		0			0
$613$	16.5831	−	16.5831i	0.669786	−	0.669786i	−0.287880	−	0.957666i	$-0.592951\pi$
$613$	16.5831	−	16.5831i	0.669786	−	0.669786i	0.957666	+	0.287880i	$0.0929505\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.00000	−	1.00000i	0.0402585	−	0.0402585i	−0.686691	−	0.726949i	$-0.740937\pi$
$617$	1.00000	−	1.00000i	0.0402585	−	0.0402585i	0.726949	+	0.686691i	$0.240937\pi$
$618$		0			0
$619$			38.0000i			1.52735i	0.645601	+	0.763674i	$0.276607\pi$
$619$			38.0000i			1.52735i	−0.645601	+	0.763674i	$0.723393\pi$
$620$		0			0
$621$	−24.0000			−0.963087
$622$		0			0
$623$	39.7995	−	39.7995i	1.59453	−	1.59453i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	33.1662			1.32242
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$	−6.63325	+	6.63325i	−0.263648	+	0.263648i
$634$		0			0
$635$	−9.94987	+	3.31662i	−0.394849	+	0.131616i
$636$		0			0
$637$	49.7494	+	49.7494i	1.97114	+	1.97114i
$638$		0			0
$639$			4.00000i			0.158238i
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	−21.0000	−	21.0000i	−0.828159	−	0.828159i	0.159103	−	0.987262i	$-0.449140\pi$
$643$	−21.0000	−	21.0000i	−0.828159	−	0.828159i	−0.987262	+	0.159103i	$0.949140\pi$
$644$		0			0
$645$	−6.63325	+	13.2665i	−0.261184	+	0.522368i
$646$		0			0
$647$	17.0000	−	17.0000i	0.668339	−	0.668339i	−0.288992	−	0.957331i	$-0.593320\pi$
$647$	17.0000	−	17.0000i	0.668339	−	0.668339i	0.957331	+	0.288992i	$0.0933202\pi$
$648$		0			0
$649$	33.1662			1.30189
$650$		0			0
$651$			26.5330i			1.03991i
$652$		0			0
$653$	5.00000	+	5.00000i	0.195665	+	0.195665i	0.798139	−	0.602474i	$-0.205818\pi$
$653$	5.00000	+	5.00000i	0.195665	+	0.195665i	−0.602474	+	0.798139i	$0.705818\pi$
$654$		0			0
$655$	13.2665	+	6.63325i	0.518365	+	0.259183i
$656$		0			0
$657$	−3.31662	−	3.31662i	−0.129394	−	0.129394i
$658$		0			0
$659$	13.2665			0.516789			0.258395	−	0.966039i	$-0.416806\pi$
$659$	13.2665			0.516789			0.258395	+	0.966039i	$0.416806\pi$
$660$		0			0
$661$	34.0000			1.32245			0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000			1.32245			0.661223	+	0.750189i	$0.270038\pi$
$662$		0			0
$663$	−22.0000	−	22.0000i	−0.854409	−	0.854409i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19.8997	−	19.8997i	−0.770521	−	0.770521i
$668$		0			0
$669$		−	14.0000i		−	0.541271i
$670$		0			0
$671$	44.0000			1.69860
$672$		0			0
$673$	−9.94987	+	9.94987i	−0.383539	+	0.383539i	−0.872376	−	0.488836i	$-0.837422\pi$
$673$	−9.94987	+	9.94987i	−0.383539	+	0.383539i	0.488836	+	0.872376i	$0.337422\pi$
$674$		0			0
$675$	28.0000	+	4.00000i	1.07772	+	0.153960i
$676$		0			0
$677$	29.8496	+	29.8496i	1.14721	+	1.14721i	0.987098	+	0.160116i	$0.0511868\pi$
$677$	29.8496	+	29.8496i	1.14721	+	1.14721i	0.160116	+	0.987098i	$0.448813\pi$
$678$		0			0
$679$	33.1662			1.27280
$680$		0			0
$681$			19.8997i			0.762560i
$682$		0			0
$683$	23.0000	+	23.0000i	0.880071	+	0.880071i	0.993541	−	0.113471i	$-0.0361968\pi$
$683$	23.0000	+	23.0000i	0.880071	+	0.880071i	−0.113471	+	0.993541i	$0.536197\pi$
$684$		0			0
$685$	1.00000	+	3.00000i	0.0382080	+	0.114624i
$686$		0			0
$687$	4.00000	−	4.00000i	0.152610	−	0.152610i
$688$		0			0
$689$	−19.8997			−0.758120
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$	11.0000	+	11.0000i	0.417855	+	0.417855i
$694$		0			0
$695$	−13.2665	+	26.5330i	−0.503227	+	1.00645i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−6.63325			−0.250893
$700$		0			0
$701$			13.2665i			0.501069i	0.968108	+	0.250534i	$0.0806063\pi$
$701$			13.2665i			0.501069i	−0.968108	+	0.250534i	$0.919394\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	10.0000	−	20.0000i	0.376622	−	0.753244i
$706$		0			0
$707$	−44.0000	+	44.0000i	−1.65479	+	1.65479i
$708$		0			0
$709$		−	40.0000i		−	1.50223i	−0.660171	−	0.751116i	$-0.729516\pi$
$709$		−	40.0000i		−	1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$		0			0
$711$		−	13.2665i		−	0.497533i
$712$		0			0
$713$	−12.0000	−	12.0000i	−0.449404	−	0.449404i
$714$		0			0
$715$	−33.0000	+	11.0000i	−1.23413	+	0.411377i
$716$		0			0
$717$	13.2665	+	13.2665i	0.495446	+	0.495446i
$718$		0			0
$719$			2.00000i			0.0745874i	0.999304	+	0.0372937i	$0.0118737\pi$
$719$			2.00000i			0.0745874i	−0.999304	+	0.0372937i	$0.988126\pi$
$720$		0			0
$721$		−	59.6992i		−	2.22332i
$722$		0			0
$723$	26.5330	−	26.5330i	0.986773	−	0.986773i
$724$		0			0
$725$	19.8997	+	26.5330i	0.739058	+	0.985411i
$726$		0			0
$727$	25.0000	−	25.0000i	0.927199	−	0.927199i	−0.0703254	−	0.997524i	$-0.522404\pi$
$727$	25.0000	−	25.0000i	0.927199	−	0.927199i	0.997524	+	0.0703254i	$0.0224038\pi$
$728$		0			0
$729$		−	29.0000i		−	1.07407i
$730$		0			0
$731$	−22.0000			−0.813699
$732$		0			0
$733$	−36.4829	+	36.4829i	−1.34753	+	1.34753i	−0.459185	+	0.888341i	$0.651858\pi$
$733$	−36.4829	+	36.4829i	−1.34753	+	1.34753i	−0.888341	+	0.459185i	$0.848142\pi$
$734$		0			0
$735$	15.0000	+	45.0000i	0.553283	+	1.65985i
$736$		0			0
$737$	−9.94987	−	9.94987i	−0.366508	−	0.366508i
$738$		0			0
$739$	−39.7995			−1.46405			−0.732024	−	0.681279i	$-0.761424\pi$
$739$	−39.7995			−1.46405			−0.732024	+	0.681279i	$0.761424\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.5831	+	16.5831i	−0.608376	+	0.608376i	−0.942521	−	0.334146i	$-0.891552\pi$
$743$	−16.5831	+	16.5831i	−0.608376	+	0.608376i	0.334146	+	0.942521i	$0.391552\pi$
$744$		0			0
$745$	6.63325	−	13.2665i	0.243023	−	0.486047i
$746$		0			0
$747$	3.31662	+	3.31662i	0.121349	+	0.121349i
$748$		0			0
$749$			22.0000i			0.803863i
$750$		0			0
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$	8.00000	+	8.00000i	0.291536	+	0.291536i
$754$		0			0
$755$	−13.2665	−	6.63325i	−0.482817	−	0.241409i
$756$		0			0
$757$	−3.00000	+	3.00000i	−0.109037	+	0.109037i	−0.759520	−	0.650484i	$-0.774566\pi$
$757$	−3.00000	+	3.00000i	−0.109037	+	0.109037i	0.650484	+	0.759520i	$0.274566\pi$
$758$		0			0
$759$	19.8997			0.722315
$760$		0			0
$761$			26.5330i			0.961820i	0.876770	+	0.480910i	$0.159694\pi$
$761$			26.5330i			0.961820i	−0.876770	+	0.480910i	$0.840306\pi$
$762$		0			0
$763$	−22.0000	−	22.0000i	−0.796453	−	0.796453i
$764$		0			0
$765$	3.31662	+	9.94987i	0.119913	+	0.359738i
$766$		0			0
$767$	−33.1662	−	33.1662i	−1.19756	−	1.19756i
$768$		0			0
$769$	−19.8997			−0.717603			−0.358802	−	0.933414i	$-0.616815\pi$
$769$	−19.8997			−0.717603			−0.358802	+	0.933414i	$0.616815\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	5.00000	+	5.00000i	0.179838	+	0.179838i	0.791285	−	0.611448i	$-0.209412\pi$
$773$	5.00000	+	5.00000i	0.179838	+	0.179838i	−0.611448	+	0.791285i	$0.709412\pi$
$774$		0			0
$775$	12.0000	+	16.0000i	0.431053	+	0.574737i
$776$		0			0
$777$	−33.1662	−	33.1662i	−1.18983	−	1.18983i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	13.2665i		−	0.474713i
$782$		0			0
$783$	26.5330	−	26.5330i	0.948212	−	0.948212i
$784$		0			0
$785$	1.00000	+	3.00000i	0.0356915	+	0.107075i
$786$		0			0
$787$	−23.2164	−	23.2164i	−0.827574	−	0.827574i	0.159606	−	0.987181i	$-0.448977\pi$
$787$	−23.2164	−	23.2164i	−0.827574	−	0.827574i	−0.987181	+	0.159606i	$0.948977\pi$
$788$		0			0
$789$	6.63325			0.236150
$790$		0			0
$791$			6.63325i			0.235851i
$792$		0			0
$793$	−44.0000	−	44.0000i	−1.56249	−	1.56249i
$794$		0			0
$795$	−12.0000	−	6.00000i	−0.425596	−	0.212798i
$796$		0			0
$797$	−19.0000	+	19.0000i	−0.673015	+	0.673015i	−0.958410	−	0.285395i	$-0.907875\pi$
$797$	−19.0000	+	19.0000i	−0.673015	+	0.673015i	0.285395	+	0.958410i	$0.407875\pi$
$798$		0			0
$799$	33.1662			1.17334
$800$		0			0
$801$	−12.0000			−0.423999
$802$		0			0
$803$	11.0000	+	11.0000i	0.388182	+	0.388182i
$804$		0			0
$805$	−39.7995	−	19.8997i	−1.40275	−	0.701374i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−19.8997			−0.699638			−0.349819	−	0.936817i	$-0.613757\pi$
$809$	−19.8997			−0.699638			−0.349819	+	0.936817i	$0.613757\pi$
$810$		0			0
$811$			33.1662i			1.16462i	0.812965	+	0.582312i	$0.197852\pi$
$811$			33.1662i			1.16462i	−0.812965	+	0.582312i	$0.802148\pi$
$812$		0			0
$813$	−19.8997	+	19.8997i	−0.697915	+	0.697915i
$814$		0			0
$815$	−9.00000	+	3.00000i	−0.315256	+	0.105085i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	22.0000i		−	0.768742i
$820$		0			0
$821$		−	39.7995i		−	1.38901i	−0.719487	−	0.694506i	$-0.755623\pi$
$821$		−	39.7995i		−	1.38901i	0.719487	−	0.694506i	$-0.244377\pi$
$822$		0			0
$823$	27.0000	+	27.0000i	0.941161	+	0.941161i	0.998363	−	0.0572018i	$-0.0182178\pi$
$823$	27.0000	+	27.0000i	0.941161	+	0.941161i	−0.0572018	+	0.998363i	$0.518218\pi$
$824$		0			0
$825$	−23.2164	−	3.31662i	−0.808290	−	0.115470i
$826$		0			0
$827$	3.31662	+	3.31662i	0.115330	+	0.115330i	0.762417	−	0.647086i	$-0.224013\pi$
$827$	3.31662	+	3.31662i	0.115330	+	0.115330i	−0.647086	+	0.762417i	$0.724013\pi$
$828$		0			0
$829$			20.0000i			0.694629i	0.937749	+	0.347314i	$0.112906\pi$
$829$			20.0000i			0.694629i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		−	33.1662i		−	1.15052i
$832$		0			0
$833$	−49.7494	+	49.7494i	−1.72371	+	1.72371i
$834$		0			0
$835$	29.8496	−	9.94987i	1.03299	−	0.344330i
$836$		0			0
$837$	16.0000	−	16.0000i	0.553041	−	0.553041i
$838$		0			0
$839$		−	30.0000i		−	1.03572i	−0.855467	−	0.517858i	$-0.826730\pi$
$839$		−	30.0000i		−	1.03572i	0.855467	−	0.517858i	$-0.173270\pi$
$840$		0			0
$841$	15.0000			0.517241
$842$		0			0
$843$	26.5330	−	26.5330i	0.913845	−	0.913845i
$844$		0			0
$845$	18.0000	+	9.00000i	0.619219	+	0.309609i
$846$		0			0
$847$	−36.4829	−	36.4829i	−1.25357	−	1.25357i
$848$		0			0
$849$	33.1662			1.13826
$850$		0			0
$851$	30.0000			1.02839
$852$		0			0
$853$	−9.94987	+	9.94987i	−0.340677	+	0.340677i	−0.856622	−	0.515945i	$-0.827441\pi$
$853$	−9.94987	+	9.94987i	−0.340677	+	0.340677i	0.515945	+	0.856622i	$0.327441\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.2164	−	23.2164i	−0.793056	−	0.793056i	0.188934	−	0.981990i	$-0.439497\pi$
$857$	−23.2164	−	23.2164i	−0.793056	−	0.793056i	−0.981990	+	0.188934i	$0.939497\pi$
$858$		0			0
$859$		−	10.0000i		−	0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$		−	10.0000i		−	0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29.0000	−	29.0000i	−0.987171	−	0.987171i	0.0127473	−	0.999919i	$-0.495942\pi$
$863$	−29.0000	−	29.0000i	−0.987171	−	0.987171i	−0.999919	+	0.0127473i	$0.995942\pi$
$864$		0			0
$865$	3.31662	+	9.94987i	0.112769	+	0.338306i
$866$		0			0
$867$	5.00000	−	5.00000i	0.169809	−	0.169809i
$868$		0			0
$869$			44.0000i			1.49260i
$870$		0			0
$871$			19.8997i			0.674277i
$872$		0			0
$873$	−5.00000	−	5.00000i	−0.169224	−	0.169224i
$874$		0			0
$875$	43.1161	+	29.8496i	1.45759	+	1.00910i
$876$		0			0
$877$	3.31662	+	3.31662i	0.111994	+	0.111994i	0.760883	−	0.648889i	$-0.224766\pi$
$877$	3.31662	+	3.31662i	0.111994	+	0.111994i	−0.648889	+	0.760883i	$0.724766\pi$
$878$		0			0
$879$	−33.1662			−1.11867
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$	27.0000	+	27.0000i	0.908622	+	0.908622i	0.996161	−	0.0875388i	$-0.0279002\pi$
$883$	27.0000	+	27.0000i	0.908622	+	0.908622i	−0.0875388	+	0.996161i	$0.527900\pi$
$884$		0			0
$885$	−10.0000	−	30.0000i	−0.336146	−	1.00844i
$886$		0			0
$887$	16.5831	+	16.5831i	0.556807	+	0.556807i	0.928397	−	0.371590i	$-0.121187\pi$
$887$	16.5831	+	16.5831i	0.556807	+	0.556807i	−0.371590	+	0.928397i	$0.621187\pi$
$888$		0			0
$889$			22.0000i			0.737856i
$890$		0			0
$891$			16.5831i			0.555556i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−12.0000	−	6.00000i	−0.401116	−	0.200558i
$896$		0			0
$897$	−19.8997	−	19.8997i	−0.664433	−	0.664433i
$898$		0			0
$899$	26.5330			0.884925
$900$		0			0
$901$		−	19.8997i		−	0.662957i
$902$		0			0
$903$	22.0000	+	22.0000i	0.732114	+	0.732114i
$904$		0			0
$905$	−2.00000	+	4.00000i	−0.0664822	+	0.132964i
$906$		0			0
$907$	−7.00000	+	7.00000i	−0.232431	+	0.232431i	−0.813707	−	0.581276i	$-0.802554\pi$
$907$	−7.00000	+	7.00000i	−0.232431	+	0.232431i	0.581276	+	0.813707i	$0.302554\pi$
$908$		0			0
$909$	13.2665			0.440022
$910$		0			0
$911$	−44.0000			−1.45779			−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000			−1.45779			−0.728893	+	0.684628i	$0.759965\pi$
$912$		0			0
$913$	−11.0000	−	11.0000i	−0.364047	−	0.364047i
$914$		0			0
$915$	−13.2665	−	39.7995i	−0.438577	−	1.31573i
$916$		0			0
$917$	22.0000	−	22.0000i	0.726504	−	0.726504i
$918$		0			0
$919$	−26.5330			−0.875243			−0.437621	−	0.899159i	$-0.644179\pi$
$919$	−26.5330			−0.875243			−0.437621	+	0.899159i	$0.644179\pi$
$920$		0			0
$921$		−	6.63325i		−	0.218573i
$922$		0			0
$923$	−13.2665	+	13.2665i	−0.436672	+	0.436672i
$924$		0			0
$925$	−35.0000	−	5.00000i	−1.15079	−	0.164399i
$926$		0			0
$927$	−9.00000	+	9.00000i	−0.295599	+	0.295599i
$928$		0			0
$929$			44.0000i			1.44359i	0.692105	+	0.721797i	$0.256683\pi$
$929$			44.0000i			1.44359i	−0.692105	+	0.721797i	$0.743317\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−12.0000	−	12.0000i	−0.392862	−	0.392862i
$934$		0			0
$935$	−11.0000	−	33.0000i	−0.359738	−	1.07922i
$936$		0			0
$937$	−9.94987	−	9.94987i	−0.325048	−	0.325048i	0.525652	−	0.850700i	$-0.323822\pi$
$937$	−9.94987	−	9.94987i	−0.325048	−	0.325048i	−0.850700	+	0.525652i	$0.823822\pi$
$938$		0			0
$939$		−	42.0000i		−	1.37062i
$940$		0			0
$941$			39.7995i			1.29743i	0.761033	+	0.648713i	$0.224693\pi$
$941$			39.7995i			1.29743i	−0.761033	+	0.648713i	$0.775307\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	26.5330	−	53.0660i	0.863118	−	1.72624i
$946$		0			0
$947$	−19.0000	+	19.0000i	−0.617417	+	0.617417i	−0.944868	−	0.327451i	$-0.893810\pi$
$947$	−19.0000	+	19.0000i	−0.617417	+	0.617417i	0.327451	+	0.944868i	$0.393810\pi$
$948$		0			0
$949$		−	22.0000i		−	0.714150i
$950$		0			0
$951$	−34.0000			−1.10253
$952$		0			0
$953$	−36.4829	+	36.4829i	−1.18180	+	1.18180i	−0.202518	+	0.979279i	$0.564912\pi$
$953$	−36.4829	+	36.4829i	−1.18180	+	1.18180i	−0.979279	+	0.202518i	$0.935088\pi$
$954$		0			0
$955$	12.0000	−	24.0000i	0.388311	−	0.776622i
$956$		0			0
$957$	−22.0000	+	22.0000i	−0.711159	+	0.711159i
$958$		0			0
$959$	6.63325			0.214199
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	3.31662	−	3.31662i	0.106877	−	0.106877i
$964$		0			0
$965$	−9.94987	−	29.8496i	−0.320298	−	0.960893i
$966$		0			0
$967$	−9.94987	−	9.94987i	−0.319966	−	0.319966i	0.528788	−	0.848754i	$-0.322647\pi$
$967$	−9.94987	−	9.94987i	−0.319966	−	0.319966i	−0.848754	+	0.528788i	$0.822647\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.00000			−0.256732			−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000			−0.256732			−0.128366	+	0.991727i	$0.540973\pi$
$972$		0			0
$973$	44.0000	+	44.0000i	1.41058	+	1.41058i
$974$		0			0
$975$	19.8997	+	26.5330i	0.637302	+	0.849736i
$976$		0			0
$977$	13.0000	−	13.0000i	0.415907	−	0.415907i	−0.467883	−	0.883790i	$-0.654983\pi$
$977$	13.0000	−	13.0000i	0.415907	−	0.415907i	0.883790	+	0.467883i	$0.154983\pi$
$978$		0			0
$979$	39.7995			1.27200
$980$		0			0
$981$			6.63325i			0.211783i
$982$		0			0
$983$	15.0000	+	15.0000i	0.478426	+	0.478426i	0.904628	−	0.426202i	$-0.140149\pi$
$983$	15.0000	+	15.0000i	0.478426	+	0.478426i	−0.426202	+	0.904628i	$0.640149\pi$
$984$		0			0
$985$	29.8496	−	9.94987i	0.951088	−	0.317029i
$986$		0			0
$987$	−33.1662	−	33.1662i	−1.05569	−	1.05569i
$988$		0			0
$989$	−19.8997			−0.632775
$990$		0			0
$991$	−44.0000			−1.39771			−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000			−1.39771			−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$	16.0000	+	16.0000i	0.507745	+	0.507745i
$994$		0			0
$995$	44.0000	+	22.0000i	1.39489	+	0.697447i
$996$		0			0
$997$	16.5831	+	16.5831i	0.525193	+	0.525193i	0.919135	−	0.393942i	$-0.128889\pi$
$997$	16.5831	+	16.5831i	0.525193	+	0.525193i	−0.393942	+	0.919135i	$0.628889\pi$
$998$		0			0
$999$			40.0000i			1.26554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.2.k.a.153.2	yes	4
3.2	odd	2		1980.2.y.a.1693.2		4
4.3	odd	2		880.2.bd.f.593.1		4
5.2	odd	4	inner	220.2.k.a.197.1	yes	4
5.3	odd	4		1100.2.k.a.857.2		4
5.4	even	2		1100.2.k.a.593.1		4
11.10	odd	2	inner	220.2.k.a.153.1	✓	4
15.2	even	4		1980.2.y.a.1297.1		4
20.7	even	4		880.2.bd.f.417.2		4
33.32	even	2		1980.2.y.a.1693.1		4
44.43	even	2		880.2.bd.f.593.2		4
55.32	even	4	inner	220.2.k.a.197.2	yes	4
55.43	even	4		1100.2.k.a.857.1		4
55.54	odd	2		1100.2.k.a.593.2		4
165.32	odd	4		1980.2.y.a.1297.2		4
220.87	odd	4		880.2.bd.f.417.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.k.a.153.1	✓	4	11.10	odd	2	inner
220.2.k.a.153.2	yes	4	1.1	even	1	trivial
220.2.k.a.197.1	yes	4	5.2	odd	4	inner
220.2.k.a.197.2	yes	4	55.32	even	4	inner
880.2.bd.f.417.1		4	220.87	odd	4
880.2.bd.f.417.2		4	20.7	even	4
880.2.bd.f.593.1		4	4.3	odd	2
880.2.bd.f.593.2		4	44.43	even	2
1100.2.k.a.593.1		4	5.4	even	2
1100.2.k.a.593.2		4	55.54	odd	2
1100.2.k.a.857.1		4	55.43	even	4
1100.2.k.a.857.2		4	5.3	odd	4
1980.2.y.a.1297.1		4	15.2	even	4
1980.2.y.a.1297.2		4	165.32	odd	4
1980.2.y.a.1693.1		4	33.32	even	2
1980.2.y.a.1693.2		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	220.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{3} +(-1.00000 + 2.00000i) q^{5} +(3.31662 + 3.31662i) q^{7} -1.00000i q^{9} +3.31662i q^{11} +(3.31662 - 3.31662i) q^{13} +(3.00000 - 1.00000i) q^{15} +(3.31662 + 3.31662i) q^{17} -6.63325i q^{21} +(3.00000 + 3.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(-4.00000 + 4.00000i) q^{27} -6.63325 q^{29} -4.00000 q^{31} +(3.31662 - 3.31662i) q^{33} +(-9.94987 + 3.31662i) q^{35} +(5.00000 - 5.00000i) q^{37} -6.63325 q^{39} +(-3.31662 + 3.31662i) q^{43} +(2.00000 + 1.00000i) q^{45} +(5.00000 - 5.00000i) q^{47} +15.0000i q^{49} -6.63325i q^{51} +(-3.00000 - 3.00000i) q^{53} +(-6.63325 - 3.31662i) q^{55} -10.0000i q^{59} -13.2665i q^{61} +(3.31662 - 3.31662i) q^{63} +(3.31662 + 9.94987i) q^{65} +(-3.00000 + 3.00000i) q^{67} -6.00000i q^{69} -4.00000 q^{71} +(3.31662 - 3.31662i) q^{73} +(-1.00000 + 7.00000i) q^{75} +(-11.0000 + 11.0000i) q^{77} +13.2665 q^{79} +5.00000 q^{81} +(-3.31662 + 3.31662i) q^{83} +(-9.94987 + 3.31662i) q^{85} +(6.63325 + 6.63325i) q^{87} -12.0000i q^{89} +22.0000 q^{91} +(4.00000 + 4.00000i) q^{93} +(5.00000 - 5.00000i) q^{97} +3.31662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + 12 q^{15} + 12 q^{23} - 12 q^{25} - 16 q^{27} - 16 q^{31} + 20 q^{37} + 8 q^{45} + 20 q^{47} - 12 q^{53} - 12 q^{67} - 16 q^{71} - 4 q^{75} - 44 q^{77} + 20 q^{81} + 88 q^{91}+ \cdots + 20 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + 12 * q^15 + 12 * q^23 - 12 * q^25 - 16 * q^27 - 16 * q^31 + 20 * q^37 + 8 * q^45 + 20 * q^47 - 12 * q^53 - 12 * q^67 - 16 * q^71 - 4 * q^75 - 44 * q^77 + 20 * q^81 + 88 * q^91 + 16 * q^93 + 20 * q^97

Introduction

L-functions

Modular forms

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