LMFDB - Embedded newform 600.2.k.a.301.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(301,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("600.301");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.k (of order $2$, degree $1$, 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:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			301.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.301
Dual form			600.2.k.a.301.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +(1.00000 - 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +(1.00000 - 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} -4.00000i q^{11} -2.00000 q^{12} +(2.00000 + 2.00000i) q^{14} -4.00000 q^{16} +6.00000 q^{17} +(1.00000 + 1.00000i) q^{18} -4.00000i q^{19} -2.00000i q^{21} +(-4.00000 + 4.00000i) q^{22} +4.00000 q^{23} +(2.00000 + 2.00000i) q^{24} -1.00000i q^{27} -4.00000i q^{28} +6.00000i q^{29} +10.0000 q^{31} +(4.00000 + 4.00000i) q^{32} +4.00000 q^{33} +(-6.00000 - 6.00000i) q^{34} -2.00000i q^{36} +4.00000i q^{37} +(-4.00000 + 4.00000i) q^{38} +10.0000 q^{41} +(-2.00000 + 2.00000i) q^{42} -4.00000i q^{43} +8.00000 q^{44} +(-4.00000 - 4.00000i) q^{46} +4.00000 q^{47} -4.00000i q^{48} -3.00000 q^{49} +6.00000i q^{51} -10.0000i q^{53} +(-1.00000 + 1.00000i) q^{54} +(-4.00000 + 4.00000i) q^{56} +4.00000 q^{57} +(6.00000 - 6.00000i) q^{58} -8.00000i q^{59} +8.00000i q^{61} +(-10.0000 - 10.0000i) q^{62} +2.00000 q^{63} -8.00000i q^{64} +(-4.00000 - 4.00000i) q^{66} -12.0000i q^{67} +12.0000i q^{68} +4.00000i q^{69} -4.00000 q^{71} +(-2.00000 + 2.00000i) q^{72} -10.0000 q^{73} +(4.00000 - 4.00000i) q^{74} +8.00000 q^{76} +8.00000i q^{77} -14.0000 q^{79} +1.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} +4.00000 q^{84} +(-4.00000 + 4.00000i) q^{86} -6.00000 q^{87} +(-8.00000 - 8.00000i) q^{88} +14.0000 q^{89} +8.00000i q^{92} +10.0000i q^{93} +(-4.00000 - 4.00000i) q^{94} +(-4.00000 + 4.00000i) q^{96} +10.0000 q^{97} +(3.00000 + 3.00000i) q^{98} +4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9} - 4 q^{12} + 4 q^{14} - 8 q^{16} + 12 q^{17} + 2 q^{18} - 8 q^{22} + 8 q^{23} + 4 q^{24} + 20 q^{31} + 8 q^{32} + 8 q^{33} - 12 q^{34} - 8 q^{38} + 20 q^{41} - 4 q^{42} + 16 q^{44} - 8 q^{46} + 8 q^{47} - 6 q^{49} - 2 q^{54} - 8 q^{56} + 8 q^{57} + 12 q^{58} - 20 q^{62} + 4 q^{63} - 8 q^{66} - 8 q^{71} - 4 q^{72} - 20 q^{73} + 8 q^{74} + 16 q^{76} - 28 q^{79} + 2 q^{81} - 20 q^{82} + 8 q^{84} - 8 q^{86} - 12 q^{87} - 16 q^{88} + 28 q^{89} - 8 q^{94} - 8 q^{96} + 20 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9 - 4 * q^12 + 4 * q^14 - 8 * q^16 + 12 * q^17 + 2 * q^18 - 8 * q^22 + 8 * q^23 + 4 * q^24 + 20 * q^31 + 8 * q^32 + 8 * q^33 - 12 * q^34 - 8 * q^38 + 20 * q^41 - 4 * q^42 + 16 * q^44 - 8 * q^46 + 8 * q^47 - 6 * q^49 - 2 * q^54 - 8 * q^56 + 8 * q^57 + 12 * q^58 - 20 * q^62 + 4 * q^63 - 8 * q^66 - 8 * q^71 - 4 * q^72 - 20 * q^73 + 8 * q^74 + 16 * q^76 - 28 * q^79 + 2 * q^81 - 20 * q^82 + 8 * q^84 - 8 * q^86 - 12 * q^87 - 16 * q^88 + 28 * q^89 - 8 * q^94 - 8 * q^96 + 20 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$			2.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$	1.00000	−	1.00000i	0.408248	−	0.408248i
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	−2.00000			−0.577350
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	2.00000	+	2.00000i	0.534522	+	0.534522i
$15$		0			0
$16$	−4.00000			−1.00000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	+	1.00000i	0.235702	+	0.235702i
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$	−4.00000	+	4.00000i	−0.852803	+	0.852803i
$23$	4.00000			0.834058			0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000			0.834058			0.417029	+	0.908893i	$0.363071\pi$
$24$	2.00000	+	2.00000i	0.408248	+	0.408248i
$25$		0			0
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		−	4.00000i		−	0.755929i
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$	10.0000			1.79605			0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000			1.79605			0.898027	+	0.439941i	$0.145001\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$	4.00000			0.696311
$34$	−6.00000	−	6.00000i	−1.02899	−	1.02899i
$35$		0			0
$36$		−	2.00000i		−	0.333333i
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	−4.00000	+	4.00000i	−0.648886	+	0.648886i
$39$		0			0
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$	−2.00000	+	2.00000i	−0.308607	+	0.308607i
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	8.00000			1.20605
$45$		0			0
$46$	−4.00000	−	4.00000i	−0.589768	−	0.589768i
$47$	4.00000			0.583460			0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000			0.583460			0.291730	+	0.956501i	$0.405769\pi$
$48$		−	4.00000i		−	0.577350i
$49$	−3.00000			−0.428571
$50$		0			0
$51$			6.00000i			0.840168i
$52$		0			0
$53$		−	10.0000i		−	1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$		−	10.0000i		−	1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	−1.00000	+	1.00000i	−0.136083	+	0.136083i
$55$		0			0
$56$	−4.00000	+	4.00000i	−0.534522	+	0.534522i
$57$	4.00000			0.529813
$58$	6.00000	−	6.00000i	0.787839	−	0.787839i
$59$		−	8.00000i		−	1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$		−	8.00000i		−	1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$		0			0
$61$			8.00000i			1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$			8.00000i			1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$	−10.0000	−	10.0000i	−1.27000	−	1.27000i
$63$	2.00000			0.251976
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$	−4.00000	−	4.00000i	−0.492366	−	0.492366i
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$			12.0000i			1.45521i
$69$			4.00000i			0.481543i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$	−2.00000	+	2.00000i	−0.235702	+	0.235702i
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$	4.00000	−	4.00000i	0.464991	−	0.464991i
$75$		0			0
$76$	8.00000			0.917663
$77$			8.00000i			0.911685i
$78$		0			0
$79$	−14.0000			−1.57512			−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000			−1.57512			−0.787562	+	0.616236i	$0.788657\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	−10.0000	−	10.0000i	−1.10432	−	1.10432i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	4.00000			0.436436
$85$		0			0
$86$	−4.00000	+	4.00000i	−0.431331	+	0.431331i
$87$	−6.00000			−0.643268
$88$	−8.00000	−	8.00000i	−0.852803	−	0.852803i
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$		0			0
$92$			8.00000i			0.834058i
$93$			10.0000i			1.03695i
$94$	−4.00000	−	4.00000i	−0.412568	−	0.412568i
$95$		0			0
$96$	−4.00000	+	4.00000i	−0.408248	+	0.408248i
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$	3.00000	+	3.00000i	0.303046	+	0.303046i
$99$			4.00000i			0.402015i
$100$		0			0
$101$		−	14.0000i		−	1.39305i	−0.717532	−	0.696526i	$-0.754728\pi$
$101$		−	14.0000i		−	1.39305i	0.717532	−	0.696526i	$-0.245272\pi$
$102$	6.00000	−	6.00000i	0.594089	−	0.594089i
$103$	2.00000			0.197066			0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000			0.197066			0.0985329	+	0.995134i	$0.468585\pi$
$104$		0			0
$105$		0			0
$106$	−10.0000	+	10.0000i	−0.971286	+	0.971286i
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	2.00000			0.192450
$109$		−	4.00000i		−	0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$		−	4.00000i		−	0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$		0			0
$111$	−4.00000			−0.379663
$112$	8.00000			0.755929
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−	4.00000i	−0.374634	−	0.374634i
$115$		0			0
$116$	−12.0000			−1.11417
$117$		0			0
$118$	−8.00000	+	8.00000i	−0.736460	+	0.736460i
$119$	−12.0000			−1.10004
$120$		0			0
$121$	−5.00000			−0.454545
$122$	8.00000	−	8.00000i	0.724286	−	0.724286i
$123$			10.0000i			0.901670i
$124$			20.0000i			1.79605i
$125$		0			0
$126$	−2.00000	−	2.00000i	−0.178174	−	0.178174i
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$	4.00000			0.352180
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$			8.00000i			0.696311i
$133$			8.00000i			0.693688i
$134$	−12.0000	+	12.0000i	−1.03664	+	1.03664i
$135$		0			0
$136$	12.0000	−	12.0000i	1.02899	−	1.02899i
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$	4.00000	−	4.00000i	0.340503	−	0.340503i
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$		0			0
$141$			4.00000i			0.336861i
$142$	4.00000	+	4.00000i	0.335673	+	0.335673i
$143$		0			0
$144$	4.00000			0.333333
$145$		0			0
$146$	10.0000	+	10.0000i	0.827606	+	0.827606i
$147$		−	3.00000i		−	0.247436i
$148$	−8.00000			−0.657596
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−8.00000	−	8.00000i	−0.648886	−	0.648886i
$153$	−6.00000			−0.485071
$154$	8.00000	−	8.00000i	0.644658	−	0.644658i
$155$		0			0
$156$		0			0
$157$			20.0000i			1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$157$			20.0000i			1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$158$	14.0000	+	14.0000i	1.11378	+	1.11378i
$159$	10.0000			0.793052
$160$		0			0
$161$	−8.00000			−0.630488
$162$	−1.00000	−	1.00000i	−0.0785674	−	0.0785674i
$163$			20.0000i			1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$			20.0000i			1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$			20.0000i			1.56174i
$165$		0			0
$166$		0			0
$167$	−24.0000			−1.85718			−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000			−1.85718			−0.928588	+	0.371113i	$0.878976\pi$
$168$	−4.00000	−	4.00000i	−0.308607	−	0.308607i
$169$	13.0000			1.00000
$170$		0			0
$171$			4.00000i			0.305888i
$172$	8.00000			0.609994
$173$		−	10.0000i		−	0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$		−	10.0000i		−	0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	6.00000	+	6.00000i	0.454859	+	0.454859i
$175$		0			0
$176$			16.0000i			1.20605i
$177$	8.00000			0.601317
$178$	−14.0000	−	14.0000i	−1.04934	−	1.04934i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			20.0000i			1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$			20.0000i			1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$	−8.00000			−0.591377
$184$	8.00000	−	8.00000i	0.589768	−	0.589768i
$185$		0			0
$186$	10.0000	−	10.0000i	0.733236	−	0.733236i
$187$		−	24.0000i		−	1.75505i
$188$			8.00000i			0.583460i
$189$			2.00000i			0.145479i
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$	8.00000			0.577350
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$	−10.0000	−	10.0000i	−0.717958	−	0.717958i
$195$		0			0
$196$		−	6.00000i		−	0.428571i
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	4.00000	−	4.00000i	0.284268	−	0.284268i
$199$	−6.00000			−0.425329			−0.212664	−	0.977125i	$-0.568214\pi$
$199$	−6.00000			−0.425329			−0.212664	+	0.977125i	$0.568214\pi$
$200$		0			0
$201$	12.0000			0.846415
$202$	−14.0000	+	14.0000i	−0.985037	+	0.985037i
$203$		−	12.0000i		−	0.842235i
$204$	−12.0000			−0.840168
$205$		0			0
$206$	−2.00000	−	2.00000i	−0.139347	−	0.139347i
$207$	−4.00000			−0.278019
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	20.0000			1.37361
$213$		−	4.00000i		−	0.274075i
$214$	4.00000	−	4.00000i	0.273434	−	0.273434i
$215$		0			0
$216$	−2.00000	−	2.00000i	−0.136083	−	0.136083i
$217$	−20.0000			−1.35769
$218$	−4.00000	+	4.00000i	−0.270914	+	0.270914i
$219$		−	10.0000i		−	0.675737i
$220$		0			0
$221$		0			0
$222$	4.00000	+	4.00000i	0.268462	+	0.268462i
$223$	−10.0000			−0.669650			−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000			−0.669650			−0.334825	+	0.942280i	$0.608677\pi$
$224$	−8.00000	−	8.00000i	−0.534522	−	0.534522i
$225$		0			0
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$			8.00000i			0.529813i
$229$			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
$231$	−8.00000			−0.526361
$232$	12.0000	+	12.0000i	0.787839	+	0.787839i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$	16.0000			1.04151
$237$		−	14.0000i		−	0.909398i
$238$	12.0000	+	12.0000i	0.777844	+	0.777844i
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$	5.00000	+	5.00000i	0.321412	+	0.321412i
$243$			1.00000i			0.0641500i
$244$	−16.0000			−1.02430
$245$		0			0
$246$	10.0000	−	10.0000i	0.637577	−	0.637577i
$247$		0			0
$248$	20.0000	−	20.0000i	1.27000	−	1.27000i
$249$		0			0
$250$		0			0
$251$		−	12.0000i		−	0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$		−	12.0000i		−	0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$			4.00000i			0.251976i
$253$		−	16.0000i		−	1.00591i
$254$	−6.00000	−	6.00000i	−0.376473	−	0.376473i
$255$		0			0
$256$	16.0000			1.00000
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$	−4.00000	−	4.00000i	−0.249029	−	0.249029i
$259$		−	8.00000i		−	0.497096i
$260$		0			0
$261$		−	6.00000i		−	0.371391i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	8.00000	−	8.00000i	0.492366	−	0.492366i
$265$		0			0
$266$	8.00000	−	8.00000i	0.490511	−	0.490511i
$267$			14.0000i			0.856786i
$268$	24.0000			1.46603
$269$		−	10.0000i		−	0.609711i	−0.952399	−	0.304855i	$-0.901392\pi$
$269$		−	10.0000i		−	0.609711i	0.952399	−	0.304855i	$-0.0986081\pi$
$270$		0			0
$271$	−2.00000			−0.121491			−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000			−0.121491			−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−24.0000			−1.45521
$273$		0			0
$274$	−2.00000	−	2.00000i	−0.120824	−	0.120824i
$275$		0			0
$276$	−8.00000			−0.481543
$277$		−	16.0000i		−	0.961347i	−0.876900	−	0.480673i	$-0.840392\pi$
$277$		−	16.0000i		−	0.961347i	0.876900	−	0.480673i	$-0.159608\pi$
$278$	4.00000	−	4.00000i	0.239904	−	0.239904i
$279$	−10.0000			−0.598684
$280$		0			0
$281$	−10.0000			−0.596550			−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000			−0.596550			−0.298275	+	0.954480i	$0.596411\pi$
$282$	4.00000	−	4.00000i	0.238197	−	0.238197i
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		−	8.00000i		−	0.474713i
$285$		0			0
$286$		0			0
$287$	−20.0000			−1.18056
$288$	−4.00000	−	4.00000i	−0.235702	−	0.235702i
$289$	19.0000			1.11765
$290$		0			0
$291$			10.0000i			0.586210i
$292$		−	20.0000i		−	1.17041i
$293$		−	2.00000i		−	0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$		−	2.00000i		−	0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	−3.00000	+	3.00000i	−0.174964	+	0.174964i
$295$		0			0
$296$	8.00000	+	8.00000i	0.464991	+	0.464991i
$297$	−4.00000			−0.232104
$298$	6.00000	−	6.00000i	0.347571	−	0.347571i
$299$		0			0
$300$		0			0
$301$			8.00000i			0.461112i
$302$	2.00000	+	2.00000i	0.115087	+	0.115087i
$303$	14.0000			0.804279
$304$			16.0000i			0.917663i
$305$		0			0
$306$	6.00000	+	6.00000i	0.342997	+	0.342997i
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	−16.0000			−0.911685
$309$			2.00000i			0.113776i
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$	20.0000	−	20.0000i	1.12867	−	1.12867i
$315$		0			0
$316$		−	28.0000i		−	1.57512i
$317$			18.0000i			1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$			18.0000i			1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	−10.0000	−	10.0000i	−0.560772	−	0.560772i
$319$	24.0000			1.34374
$320$		0			0
$321$	−4.00000			−0.223258
$322$	8.00000	+	8.00000i	0.445823	+	0.445823i
$323$		−	24.0000i		−	1.33540i
$324$			2.00000i			0.111111i
$325$		0			0
$326$	20.0000	−	20.0000i	1.10770	−	1.10770i
$327$	4.00000			0.221201
$328$	20.0000	−	20.0000i	1.10432	−	1.10432i
$329$	−8.00000			−0.441054
$330$		0			0
$331$		−	4.00000i		−	0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$331$		−	4.00000i		−	0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$332$		0			0
$333$		−	4.00000i		−	0.219199i
$334$	24.0000	+	24.0000i	1.31322	+	1.31322i
$335$		0			0
$336$			8.00000i			0.436436i
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$	−13.0000	−	13.0000i	−0.707107	−	0.707107i
$339$		−	6.00000i		−	0.325875i
$340$		0			0
$341$		−	40.0000i		−	2.16612i
$342$	4.00000	−	4.00000i	0.216295	−	0.216295i
$343$	20.0000			1.07990
$344$	−8.00000	−	8.00000i	−0.431331	−	0.431331i
$345$		0			0
$346$	−10.0000	+	10.0000i	−0.537603	+	0.537603i
$347$			16.0000i			0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$			16.0000i			0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$		−	12.0000i		−	0.643268i
$349$		−	8.00000i		−	0.428230i	−0.976808	−	0.214115i	$-0.931313\pi$
$349$		−	8.00000i		−	0.428230i	0.976808	−	0.214115i	$-0.0686868\pi$
$350$		0			0
$351$		0			0
$352$	16.0000	−	16.0000i	0.852803	−	0.852803i
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$	−8.00000	−	8.00000i	−0.425195	−	0.425195i
$355$		0			0
$356$			28.0000i			1.48400i
$357$		−	12.0000i		−	0.635107i
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	20.0000	−	20.0000i	1.05118	−	1.05118i
$363$		−	5.00000i		−	0.262432i
$364$		0			0
$365$		0			0
$366$	8.00000	+	8.00000i	0.418167	+	0.418167i
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$	−16.0000			−0.834058
$369$	−10.0000			−0.520579
$370$		0			0
$371$			20.0000i			1.03835i
$372$	−20.0000			−1.03695
$373$			20.0000i			1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$			20.0000i			1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$	−24.0000	+	24.0000i	−1.24101	+	1.24101i
$375$		0			0
$376$	8.00000	−	8.00000i	0.412568	−	0.412568i
$377$		0			0
$378$	2.00000	−	2.00000i	0.102869	−	0.102869i
$379$			36.0000i			1.84920i	0.380945	+	0.924598i	$0.375599\pi$
$379$			36.0000i			1.84920i	−0.380945	+	0.924598i	$0.624401\pi$
$380$		0			0
$381$			6.00000i			0.307389i
$382$	8.00000	+	8.00000i	0.409316	+	0.409316i
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$	−8.00000	−	8.00000i	−0.408248	−	0.408248i
$385$		0			0
$386$	−14.0000	−	14.0000i	−0.712581	−	0.712581i
$387$			4.00000i			0.203331i
$388$			20.0000i			1.01535i
$389$		−	10.0000i		−	0.507020i	−0.967333	−	0.253510i	$-0.918415\pi$
$389$		−	10.0000i		−	0.507020i	0.967333	−	0.253510i	$-0.0815851\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$	−6.00000	+	6.00000i	−0.303046	+	0.303046i
$393$		0			0
$394$	−10.0000	+	10.0000i	−0.503793	+	0.503793i
$395$		0			0
$396$	−8.00000			−0.402015
$397$			20.0000i			1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$			20.0000i			1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$	6.00000	+	6.00000i	0.300753	+	0.300753i
$399$	−8.00000			−0.400501
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$	−12.0000	−	12.0000i	−0.598506	−	0.598506i
$403$		0			0
$404$	28.0000			1.39305
$405$		0			0
$406$	−12.0000	+	12.0000i	−0.595550	+	0.595550i
$407$	16.0000			0.793091
$408$	12.0000	+	12.0000i	0.594089	+	0.594089i
$409$	−6.00000			−0.296681			−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000			−0.296681			−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$			2.00000i			0.0986527i
$412$			4.00000i			0.197066i
$413$			16.0000i			0.787309i
$414$	4.00000	+	4.00000i	0.196589	+	0.196589i
$415$		0			0
$416$		0			0
$417$	−4.00000			−0.195881
$418$	16.0000	+	16.0000i	0.782586	+	0.782586i
$419$			4.00000i			0.195413i	0.995215	+	0.0977064i	$0.0311506\pi$
$419$			4.00000i			0.195413i	−0.995215	+	0.0977064i	$0.968849\pi$
$420$		0			0
$421$		−	20.0000i		−	0.974740i	−0.873195	−	0.487370i	$-0.837956\pi$
$421$		−	20.0000i		−	0.974740i	0.873195	−	0.487370i	$-0.162044\pi$
$422$	−12.0000	+	12.0000i	−0.584151	+	0.584151i
$423$	−4.00000			−0.194487
$424$	−20.0000	−	20.0000i	−0.971286	−	0.971286i
$425$		0			0
$426$	−4.00000	+	4.00000i	−0.193801	+	0.193801i
$427$		−	16.0000i		−	0.774294i
$428$	−8.00000			−0.386695
$429$		0			0
$430$		0			0
$431$	−36.0000			−1.73406			−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000			−1.73406			−0.867029	+	0.498257i	$0.833974\pi$
$432$			4.00000i			0.192450i
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$	20.0000	+	20.0000i	0.960031	+	0.960031i
$435$		0			0
$436$	8.00000			0.383131
$437$		−	16.0000i		−	0.765384i
$438$	−10.0000	+	10.0000i	−0.477818	+	0.477818i
$439$	2.00000			0.0954548			0.0477274	−	0.998860i	$-0.484802\pi$
$439$	2.00000			0.0954548			0.0477274	+	0.998860i	$0.484802\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$			16.0000i			0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$			16.0000i			0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$		−	8.00000i		−	0.379663i
$445$		0			0
$446$	10.0000	+	10.0000i	0.473514	+	0.473514i
$447$	−6.00000			−0.283790
$448$			16.0000i			0.755929i
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		−	40.0000i		−	1.88353i
$452$		−	12.0000i		−	0.564433i
$453$		−	2.00000i		−	0.0939682i
$454$		0			0
$455$		0			0
$456$	8.00000	−	8.00000i	0.374634	−	0.374634i
$457$	−18.0000			−0.842004			−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000			−0.842004			−0.421002	+	0.907060i	$0.638322\pi$
$458$	4.00000	−	4.00000i	0.186908	−	0.186908i
$459$		−	6.00000i		−	0.280056i
$460$		0			0
$461$			34.0000i			1.58354i	0.610821	+	0.791769i	$0.290840\pi$
$461$			34.0000i			1.58354i	−0.610821	+	0.791769i	$0.709160\pi$
$462$	8.00000	+	8.00000i	0.372194	+	0.372194i
$463$	22.0000			1.02243			0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000			1.02243			0.511213	+	0.859454i	$0.329196\pi$
$464$		−	24.0000i		−	1.11417i
$465$		0			0
$466$	−6.00000	−	6.00000i	−0.277945	−	0.277945i
$467$		−	8.00000i		−	0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$		−	8.00000i		−	0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$		0			0
$469$			24.0000i			1.10822i
$470$		0			0
$471$	−20.0000			−0.921551
$472$	−16.0000	−	16.0000i	−0.736460	−	0.736460i
$473$	−16.0000			−0.735681
$474$	−14.0000	+	14.0000i	−0.643041	+	0.643041i
$475$		0			0
$476$		−	24.0000i		−	1.10004i
$477$			10.0000i			0.457869i
$478$	−16.0000	−	16.0000i	−0.731823	−	0.731823i
$479$	4.00000			0.182765			0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000			0.182765			0.0913823	+	0.995816i	$0.470871\pi$
$480$		0			0
$481$		0			0
$482$	30.0000	+	30.0000i	1.36646	+	1.36646i
$483$		−	8.00000i		−	0.364013i
$484$		−	10.0000i		−	0.454545i
$485$		0			0
$486$	1.00000	−	1.00000i	0.0453609	−	0.0453609i
$487$	30.0000			1.35943			0.679715	−	0.733476i	$-0.262104\pi$
$487$	30.0000			1.35943			0.679715	+	0.733476i	$0.262104\pi$
$488$	16.0000	+	16.0000i	0.724286	+	0.724286i
$489$	−20.0000			−0.904431
$490$		0			0
$491$			40.0000i			1.80517i	0.430507	+	0.902587i	$0.358335\pi$
$491$			40.0000i			1.80517i	−0.430507	+	0.902587i	$0.641665\pi$
$492$	−20.0000			−0.901670
$493$			36.0000i			1.62136i
$494$		0			0
$495$		0			0
$496$	−40.0000			−1.79605
$497$	8.00000			0.358849
$498$		0			0
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$		0			0
$501$		−	24.0000i		−	1.07224i
$502$	−12.0000	+	12.0000i	−0.535586	+	0.535586i
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$	4.00000	−	4.00000i	0.178174	−	0.178174i
$505$		0			0
$506$	−16.0000	+	16.0000i	−0.711287	+	0.711287i
$507$			13.0000i			0.577350i
$508$			12.0000i			0.532414i
$509$		−	18.0000i		−	0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$		−	18.0000i		−	0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$		0			0
$511$	20.0000			0.884748
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$	−4.00000			−0.176604
$514$	−2.00000	−	2.00000i	−0.0882162	−	0.0882162i
$515$		0			0
$516$			8.00000i			0.352180i
$517$		−	16.0000i		−	0.703679i
$518$	−8.00000	+	8.00000i	−0.351500	+	0.351500i
$519$	10.0000			0.438951
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$	−6.00000	+	6.00000i	−0.262613	+	0.262613i
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	60.0000			2.61364
$528$	−16.0000			−0.696311
$529$	−7.00000			−0.304348
$530$		0			0
$531$			8.00000i			0.347170i
$532$	−16.0000			−0.693688
$533$		0			0
$534$	14.0000	−	14.0000i	0.605839	−	0.605839i
$535$		0			0
$536$	−24.0000	−	24.0000i	−1.03664	−	1.03664i
$537$		0			0
$538$	−10.0000	+	10.0000i	−0.431131	+	0.431131i
$539$			12.0000i			0.516877i
$540$		0			0
$541$			44.0000i			1.89171i	0.324593	+	0.945854i	$0.394773\pi$
$541$			44.0000i			1.89171i	−0.324593	+	0.945854i	$0.605227\pi$
$542$	2.00000	+	2.00000i	0.0859074	+	0.0859074i
$543$	−20.0000			−0.858282
$544$	24.0000	+	24.0000i	1.02899	+	1.02899i
$545$		0			0
$546$		0			0
$547$		−	4.00000i		−	0.171028i	−0.996337	−	0.0855138i	$-0.972747\pi$
$547$		−	4.00000i		−	0.171028i	0.996337	−	0.0855138i	$-0.0272532\pi$
$548$			4.00000i			0.170872i
$549$		−	8.00000i		−	0.341432i
$550$		0			0
$551$	24.0000			1.02243
$552$	8.00000	+	8.00000i	0.340503	+	0.340503i
$553$	28.0000			1.19068
$554$	−16.0000	+	16.0000i	−0.679775	+	0.679775i
$555$		0			0
$556$	−8.00000			−0.339276
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$	10.0000	+	10.0000i	0.423334	+	0.423334i
$559$		0			0
$560$		0			0
$561$	24.0000			1.01328
$562$	10.0000	+	10.0000i	0.421825	+	0.421825i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	−8.00000			−0.336861
$565$		0			0
$566$	−4.00000	+	4.00000i	−0.168133	+	0.168133i
$567$	−2.00000			−0.0839921
$568$	−8.00000	+	8.00000i	−0.335673	+	0.335673i
$569$	2.00000			0.0838444			0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000			0.0838444			0.0419222	+	0.999121i	$0.486652\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$		−	8.00000i		−	0.334205i
$574$	20.0000	+	20.0000i	0.834784	+	0.834784i
$575$		0			0
$576$			8.00000i			0.333333i
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−19.0000	−	19.0000i	−0.790296	−	0.790296i
$579$			14.0000i			0.581820i
$580$		0			0
$581$		0			0
$582$	10.0000	−	10.0000i	0.414513	−	0.414513i
$583$	−40.0000			−1.65663
$584$	−20.0000	+	20.0000i	−0.827606	+	0.827606i
$585$		0			0
$586$	−2.00000	+	2.00000i	−0.0826192	+	0.0826192i
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	6.00000			0.247436
$589$		−	40.0000i		−	1.64817i
$590$		0			0
$591$	10.0000			0.411345
$592$		−	16.0000i		−	0.657596i
$593$	−22.0000			−0.903432			−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000			−0.903432			−0.451716	+	0.892162i	$0.649188\pi$
$594$	4.00000	+	4.00000i	0.164122	+	0.164122i
$595$		0			0
$596$	−12.0000			−0.491539
$597$		−	6.00000i		−	0.245564i
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$	8.00000	−	8.00000i	0.326056	−	0.326056i
$603$			12.0000i			0.488678i
$604$		−	4.00000i		−	0.162758i
$605$		0			0
$606$	−14.0000	−	14.0000i	−0.568711	−	0.568711i
$607$	−2.00000			−0.0811775			−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000			−0.0811775			−0.0405887	+	0.999176i	$0.512923\pi$
$608$	16.0000	−	16.0000i	0.648886	−	0.648886i
$609$	12.0000			0.486265
$610$		0			0
$611$		0			0
$612$		−	12.0000i		−	0.485071i
$613$		−	20.0000i		−	0.807792i	−0.914805	−	0.403896i	$-0.867656\pi$
$613$		−	20.0000i		−	0.807792i	0.914805	−	0.403896i	$-0.132344\pi$
$614$	28.0000	−	28.0000i	1.12999	−	1.12999i
$615$		0			0
$616$	16.0000	+	16.0000i	0.644658	+	0.644658i
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$	2.00000	−	2.00000i	0.0804518	−	0.0804518i
$619$		−	20.0000i		−	0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$		−	20.0000i		−	0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$		0			0
$621$		−	4.00000i		−	0.160514i
$622$	−8.00000	−	8.00000i	−0.320771	−	0.320771i
$623$	−28.0000			−1.12180
$624$		0			0
$625$		0			0
$626$	6.00000	+	6.00000i	0.239808	+	0.239808i
$627$		−	16.0000i		−	0.638978i
$628$	−40.0000			−1.59617
$629$			24.0000i			0.956943i
$630$		0			0
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$	−28.0000	+	28.0000i	−1.11378	+	1.11378i
$633$	12.0000			0.476957
$634$	18.0000	−	18.0000i	0.714871	−	0.714871i
$635$		0			0
$636$			20.0000i			0.793052i
$637$		0			0
$638$	−24.0000	−	24.0000i	−0.950169	−	0.950169i
$639$	4.00000			0.158238
$640$		0			0
$641$	14.0000			0.552967			0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000			0.552967			0.276483	+	0.961019i	$0.410831\pi$
$642$	4.00000	+	4.00000i	0.157867	+	0.157867i
$643$		−	12.0000i		−	0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$		−	12.0000i		−	0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$		−	16.0000i		−	0.630488i
$645$		0			0
$646$	−24.0000	+	24.0000i	−0.944267	+	0.944267i
$647$	−12.0000			−0.471769			−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000			−0.471769			−0.235884	+	0.971781i	$0.575799\pi$
$648$	2.00000	−	2.00000i	0.0785674	−	0.0785674i
$649$	−32.0000			−1.25611
$650$		0			0
$651$		−	20.0000i		−	0.783862i
$652$	−40.0000			−1.56652
$653$			30.0000i			1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$			30.0000i			1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−4.00000	−	4.00000i	−0.156412	−	0.156412i
$655$		0			0
$656$	−40.0000			−1.56174
$657$	10.0000			0.390137
$658$	8.00000	+	8.00000i	0.311872	+	0.311872i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	32.0000i		−	1.24466i	−0.782757	−	0.622328i	$-0.786187\pi$
$661$		−	32.0000i		−	1.24466i	0.782757	−	0.622328i	$-0.213813\pi$
$662$	−4.00000	+	4.00000i	−0.155464	+	0.155464i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−4.00000	+	4.00000i	−0.154997	+	0.154997i
$667$			24.0000i			0.929284i
$668$		−	48.0000i		−	1.85718i
$669$		−	10.0000i		−	0.386622i
$670$		0			0
$671$	32.0000			1.23535
$672$	8.00000	−	8.00000i	0.308607	−	0.308607i
$673$	2.00000			0.0770943			0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000			0.0770943			0.0385472	+	0.999257i	$0.487727\pi$
$674$	30.0000	+	30.0000i	1.15556	+	1.15556i
$675$		0			0
$676$			26.0000i			1.00000i
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	−6.00000	+	6.00000i	−0.230429	+	0.230429i
$679$	−20.0000			−0.767530
$680$		0			0
$681$		0			0
$682$	−40.0000	+	40.0000i	−1.53168	+	1.53168i
$683$			48.0000i			1.83667i	0.395805	+	0.918334i	$0.370466\pi$
$683$			48.0000i			1.83667i	−0.395805	+	0.918334i	$0.629534\pi$
$684$	−8.00000			−0.305888
$685$		0			0
$686$	−20.0000	−	20.0000i	−0.763604	−	0.763604i
$687$	−4.00000			−0.152610
$688$			16.0000i			0.609994i
$689$		0			0
$690$		0			0
$691$		−	12.0000i		−	0.456502i	−0.973602	−	0.228251i	$-0.926699\pi$
$691$		−	12.0000i		−	0.456502i	0.973602	−	0.228251i	$-0.0733006\pi$
$692$	20.0000			0.760286
$693$		−	8.00000i		−	0.303895i
$694$	16.0000	−	16.0000i	0.607352	−	0.607352i
$695$		0			0
$696$	−12.0000	+	12.0000i	−0.454859	+	0.454859i
$697$	60.0000			2.27266
$698$	−8.00000	+	8.00000i	−0.302804	+	0.302804i
$699$			6.00000i			0.226941i
$700$		0			0
$701$			22.0000i			0.830929i	0.909610	+	0.415464i	$0.136381\pi$
$701$			22.0000i			0.830929i	−0.909610	+	0.415464i	$0.863619\pi$
$702$		0			0
$703$	16.0000			0.603451
$704$	−32.0000			−1.20605
$705$		0			0
$706$	−18.0000	−	18.0000i	−0.677439	−	0.677439i
$707$			28.0000i			1.05305i
$708$			16.0000i			0.601317i
$709$		−	44.0000i		−	1.65245i	−0.563337	−	0.826227i	$-0.690483\pi$
$709$		−	44.0000i		−	1.65245i	0.563337	−	0.826227i	$-0.309517\pi$
$710$		0			0
$711$	14.0000			0.525041
$712$	28.0000	−	28.0000i	1.04934	−	1.04934i
$713$	40.0000			1.49801
$714$	−12.0000	+	12.0000i	−0.449089	+	0.449089i
$715$		0			0
$716$		0			0
$717$			16.0000i			0.597531i
$718$	−12.0000	−	12.0000i	−0.447836	−	0.447836i
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$	−4.00000			−0.148968
$722$	−3.00000	−	3.00000i	−0.111648	−	0.111648i
$723$		−	30.0000i		−	1.11571i
$724$	−40.0000			−1.48659
$725$		0			0
$726$	−5.00000	+	5.00000i	−0.185567	+	0.185567i
$727$	14.0000			0.519231			0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000			0.519231			0.259616	+	0.965712i	$0.416404\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	24.0000i		−	0.887672i
$732$		−	16.0000i		−	0.591377i
$733$		−	24.0000i		−	0.886460i	−0.896408	−	0.443230i	$-0.853832\pi$
$733$		−	24.0000i		−	0.886460i	0.896408	−	0.443230i	$-0.146168\pi$
$734$	−18.0000	−	18.0000i	−0.664392	−	0.664392i
$735$		0			0
$736$	16.0000	+	16.0000i	0.589768	+	0.589768i
$737$	−48.0000			−1.76810
$738$	10.0000	+	10.0000i	0.368105	+	0.368105i
$739$		−	12.0000i		−	0.441427i	−0.975339	−	0.220714i	$-0.929161\pi$
$739$		−	12.0000i		−	0.441427i	0.975339	−	0.220714i	$-0.0708386\pi$
$740$		0			0
$741$		0			0
$742$	20.0000	−	20.0000i	0.734223	−	0.734223i
$743$	48.0000			1.76095			0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000			1.76095			0.880475	+	0.474093i	$0.157224\pi$
$744$	20.0000	+	20.0000i	0.733236	+	0.733236i
$745$		0			0
$746$	20.0000	−	20.0000i	0.732252	−	0.732252i
$747$		0			0
$748$	48.0000			1.75505
$749$		−	8.00000i		−	0.292314i
$750$		0			0
$751$	26.0000			0.948753			0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000			0.948753			0.474377	+	0.880322i	$0.342673\pi$
$752$	−16.0000			−0.583460
$753$	12.0000			0.437304
$754$		0			0
$755$		0			0
$756$	−4.00000			−0.145479
$757$		−	8.00000i		−	0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$		−	8.00000i		−	0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	36.0000	−	36.0000i	1.30758	−	1.30758i
$759$	16.0000			0.580763
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$	6.00000	−	6.00000i	0.217357	−	0.217357i
$763$			8.00000i			0.289619i
$764$		−	16.0000i		−	0.578860i
$765$		0			0
$766$	24.0000	+	24.0000i	0.867155	+	0.867155i
$767$		0			0
$768$			16.0000i			0.577350i
$769$	−6.00000			−0.216366			−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000			−0.216366			−0.108183	+	0.994131i	$0.534503\pi$
$770$		0			0
$771$			2.00000i			0.0720282i
$772$			28.0000i			1.00774i
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	4.00000	−	4.00000i	0.143777	−	0.143777i
$775$		0			0
$776$	20.0000	−	20.0000i	0.717958	−	0.717958i
$777$	8.00000			0.286998
$778$	−10.0000	+	10.0000i	−0.358517	+	0.358517i
$779$		−	40.0000i		−	1.43315i
$780$		0			0
$781$			16.0000i			0.572525i
$782$	−24.0000	−	24.0000i	−0.858238	−	0.858238i
$783$	6.00000			0.214423
$784$	12.0000			0.428571
$785$		0			0
$786$		0			0

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +(1.00000 - 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.00000i) q^{2} +1.00000i q^{3} +2.00000i q^{4} +(1.00000 - 1.00000i) q^{6} -2.00000 q^{7} +(2.00000 - 2.00000i) q^{8} -1.00000 q^{9} -4.00000i q^{11} -2.00000 q^{12} +(2.00000 + 2.00000i) q^{14} -4.00000 q^{16} +6.00000 q^{17} +(1.00000 + 1.00000i) q^{18} -4.00000i q^{19} -2.00000i q^{21} +(-4.00000 + 4.00000i) q^{22} +4.00000 q^{23} +(2.00000 + 2.00000i) q^{24} -1.00000i q^{27} -4.00000i q^{28} +6.00000i q^{29} +10.0000 q^{31} +(4.00000 + 4.00000i) q^{32} +4.00000 q^{33} +(-6.00000 - 6.00000i) q^{34} -2.00000i q^{36} +4.00000i q^{37} +(-4.00000 + 4.00000i) q^{38} +10.0000 q^{41} +(-2.00000 + 2.00000i) q^{42} -4.00000i q^{43} +8.00000 q^{44} +(-4.00000 - 4.00000i) q^{46} +4.00000 q^{47} -4.00000i q^{48} -3.00000 q^{49} +6.00000i q^{51} -10.0000i q^{53} +(-1.00000 + 1.00000i) q^{54} +(-4.00000 + 4.00000i) q^{56} +4.00000 q^{57} +(6.00000 - 6.00000i) q^{58} -8.00000i q^{59} +8.00000i q^{61} +(-10.0000 - 10.0000i) q^{62} +2.00000 q^{63} -8.00000i q^{64} +(-4.00000 - 4.00000i) q^{66} -12.0000i q^{67} +12.0000i q^{68} +4.00000i q^{69} -4.00000 q^{71} +(-2.00000 + 2.00000i) q^{72} -10.0000 q^{73} +(4.00000 - 4.00000i) q^{74} +8.00000 q^{76} +8.00000i q^{77} -14.0000 q^{79} +1.00000 q^{81} +(-10.0000 - 10.0000i) q^{82} +4.00000 q^{84} +(-4.00000 + 4.00000i) q^{86} -6.00000 q^{87} +(-8.00000 - 8.00000i) q^{88} +14.0000 q^{89} +8.00000i q^{92} +10.0000i q^{93} +(-4.00000 - 4.00000i) q^{94} +(-4.00000 + 4.00000i) q^{96} +10.0000 q^{97} +(3.00000 + 3.00000i) q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{6} - 4 q^{7} + 4 q^{8} - 2 q^{9} - 4 q^{12} + 4 q^{14} - 8 q^{16} + 12 q^{17} + 2 q^{18} - 8 q^{22} + 8 q^{23} + 4 q^{24} + 20 q^{31} + 8 q^{32} + 8 q^{33} - 12 q^{34} - 8 q^{38} + 20 q^{41} - 4 q^{42} + 16 q^{44} - 8 q^{46} + 8 q^{47} - 6 q^{49} - 2 q^{54} - 8 q^{56} + 8 q^{57} + 12 q^{58} - 20 q^{62} + 4 q^{63} - 8 q^{66} - 8 q^{71} - 4 q^{72} - 20 q^{73} + 8 q^{74} + 16 q^{76} - 28 q^{79} + 2 q^{81} - 20 q^{82} + 8 q^{84} - 8 q^{86} - 12 q^{87} - 16 q^{88} + 28 q^{89} - 8 q^{94} - 8 q^{96} + 20 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^6 - 4 * q^7 + 4 * q^8 - 2 * q^9 - 4 * q^12 + 4 * q^14 - 8 * q^16 + 12 * q^17 + 2 * q^18 - 8 * q^22 + 8 * q^23 + 4 * q^24 + 20 * q^31 + 8 * q^32 + 8 * q^33 - 12 * q^34 - 8 * q^38 + 20 * q^41 - 4 * q^42 + 16 * q^44 - 8 * q^46 + 8 * q^47 - 6 * q^49 - 2 * q^54 - 8 * q^56 + 8 * q^57 + 12 * q^58 - 20 * q^62 + 4 * q^63 - 8 * q^66 - 8 * q^71 - 4 * q^72 - 20 * q^73 + 8 * q^74 + 16 * q^76 - 28 * q^79 + 2 * q^81 - 20 * q^82 + 8 * q^84 - 8 * q^86 - 12 * q^87 - 16 * q^88 + 28 * q^89 - 8 * q^94 - 8 * q^96 + 20 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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