LMFDB - Embedded newform 425.2.e.a.276.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(251,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("425.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.e (of order $4$, degree $2$, 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:	$3.39364208590$
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 85)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			276.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	425.276
Dual form			425.2.e.a.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +(-1.00000 + 1.00000i) q^{3} +1.00000 q^{4} +(1.00000 + 1.00000i) q^{6} +(-3.00000 - 3.00000i) q^{7} -3.00000i q^{8} +1.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(-1.00000 + 1.00000i) q^{12} +(-3.00000 + 3.00000i) q^{14} -1.00000 q^{16} +(4.00000 - 1.00000i) q^{17} +1.00000 q^{18} -6.00000i q^{19} +6.00000 q^{21} +(-3.00000 + 3.00000i) q^{22} +(-1.00000 - 1.00000i) q^{23} +(3.00000 + 3.00000i) q^{24} +(-4.00000 - 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{28} +(3.00000 - 3.00000i) q^{29} +(-1.00000 + 1.00000i) q^{31} -5.00000i q^{32} +6.00000 q^{33} +(-1.00000 - 4.00000i) q^{34} +1.00000i q^{36} +(-3.00000 + 3.00000i) q^{37} -6.00000 q^{38} +(-3.00000 - 3.00000i) q^{41} -6.00000i q^{42} +12.0000i q^{43} +(-3.00000 - 3.00000i) q^{44} +(-1.00000 + 1.00000i) q^{46} +2.00000 q^{47} +(1.00000 - 1.00000i) q^{48} +11.0000i q^{49} +(-3.00000 + 5.00000i) q^{51} -2.00000i q^{53} +(-4.00000 + 4.00000i) q^{54} +(-9.00000 + 9.00000i) q^{56} +(6.00000 + 6.00000i) q^{57} +(-3.00000 - 3.00000i) q^{58} -6.00000i q^{59} +(1.00000 + 1.00000i) q^{61} +(1.00000 + 1.00000i) q^{62} +(3.00000 - 3.00000i) q^{63} -7.00000 q^{64} -6.00000i q^{66} +6.00000 q^{67} +(4.00000 - 1.00000i) q^{68} +2.00000 q^{69} +(3.00000 - 3.00000i) q^{71} +3.00000 q^{72} +(3.00000 - 3.00000i) q^{73} +(3.00000 + 3.00000i) q^{74} -6.00000i q^{76} +18.0000i q^{77} +(7.00000 + 7.00000i) q^{79} +5.00000 q^{81} +(-3.00000 + 3.00000i) q^{82} +4.00000i q^{83} +6.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +(-9.00000 + 9.00000i) q^{88} -6.00000 q^{89} +(-1.00000 - 1.00000i) q^{92} -2.00000i q^{93} -2.00000i q^{94} +(5.00000 + 5.00000i) q^{96} +(-3.00000 + 3.00000i) q^{97} +11.0000 q^{98} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 6 q^{11} - 2 q^{12} - 6 q^{14} - 2 q^{16} + 8 q^{17} + 2 q^{18} + 12 q^{21} - 6 q^{22} - 2 q^{23} + 6 q^{24} - 8 q^{27} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 12 q^{33}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 6 * q^11 - 2 * q^12 - 6 * q^14 - 2 * q^16 + 8 * q^17 + 2 * q^18 + 12 * q^21 - 6 * q^22 - 2 * q^23 + 6 * q^24 - 8 * q^27 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 12 * q^33 - 2 * q^34 - 6 * q^37 - 12 * q^38 - 6 * q^41 - 6 * q^44 - 2 * q^46 + 4 * q^47 + 2 * q^48 - 6 * q^51 - 8 * q^54 - 18 * q^56 + 12 * q^57 - 6 * q^58 + 2 * q^61 + 2 * q^62 + 6 * q^63 - 14 * q^64 + 12 * q^67 + 8 * q^68 + 4 * q^69 + 6 * q^71 + 6 * q^72 + 6 * q^73 + 6 * q^74 + 14 * q^79 + 10 * q^81 - 6 * q^82 + 12 * q^84 + 24 * q^86 - 18 * q^88 - 12 * q^89 - 2 * q^92 + 10 * q^96 - 6 * q^97 + 22 * q^98 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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 $

$2$		−	1.00000i		−	0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$		−	1.00000i		−	0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	0.356822	+	0.934172i	$0.383860\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$	1.00000	+	1.00000i	0.408248	+	0.408248i
$7$	−3.00000	−	3.00000i	−1.13389	−	1.13389i	−0.989524	−	0.144370i	$-0.953885\pi$
$7$	−3.00000	−	3.00000i	−1.13389	−	1.13389i	−0.144370	−	0.989524i	$-0.546115\pi$
$8$		−	3.00000i		−	1.06066i
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	0.0912903	−	0.995824i	$-0.470901\pi$
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	−0.995824	+	0.0912903i	$0.970901\pi$
$12$	−1.00000	+	1.00000i	−0.288675	+	0.288675i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−3.00000	+	3.00000i	−0.801784	+	0.801784i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	4.00000	−	1.00000i	0.970143	−	0.242536i
$17$	4.00000	−	1.00000i	0.970143	−	0.242536i
$18$	1.00000			0.235702
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$	6.00000			1.30931
$22$	−3.00000	+	3.00000i	−0.639602	+	0.639602i
$23$	−1.00000	−	1.00000i	−0.208514	−	0.208514i	0.595121	−	0.803636i	$-0.297104\pi$
$23$	−1.00000	−	1.00000i	−0.208514	−	0.208514i	−0.803636	+	0.595121i	$0.797104\pi$
$24$	3.00000	+	3.00000i	0.612372	+	0.612372i
$25$		0			0
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$	−3.00000	−	3.00000i	−0.566947	−	0.566947i
$29$	3.00000	−	3.00000i	0.557086	−	0.557086i	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	3.00000	−	3.00000i	0.557086	−	0.557086i	0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	−1.00000	+	1.00000i	−0.179605	+	0.179605i	−0.791184	−	0.611578i	$-0.790535\pi$
$31$	−1.00000	+	1.00000i	−0.179605	+	0.179605i	0.611578	+	0.791184i	$0.290535\pi$
$32$		−	5.00000i		−	0.883883i
$33$	6.00000			1.04447
$34$	−1.00000	−	4.00000i	−0.171499	−	0.685994i
$35$		0			0
$36$			1.00000i			0.166667i
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$		0			0
$41$	−3.00000	−	3.00000i	−0.468521	−	0.468521i	0.432914	−	0.901435i	$-0.357485\pi$
$41$	−3.00000	−	3.00000i	−0.468521	−	0.468521i	−0.901435	+	0.432914i	$0.857485\pi$
$42$		−	6.00000i		−	0.925820i
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	−3.00000	−	3.00000i	−0.452267	−	0.452267i
$45$		0			0
$46$	−1.00000	+	1.00000i	−0.147442	+	0.147442i
$47$	2.00000			0.291730			0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000			0.291730			0.145865	+	0.989305i	$0.453403\pi$
$48$	1.00000	−	1.00000i	0.144338	−	0.144338i
$49$			11.0000i			1.57143i
$50$		0			0
$51$	−3.00000	+	5.00000i	−0.420084	+	0.700140i
$52$		0			0
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	−4.00000	+	4.00000i	−0.544331	+	0.544331i
$55$		0			0
$56$	−9.00000	+	9.00000i	−1.20268	+	1.20268i
$57$	6.00000	+	6.00000i	0.794719	+	0.794719i
$58$	−3.00000	−	3.00000i	−0.393919	−	0.393919i
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	0.768221	−	0.640184i	$-0.221142\pi$
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	1.00000	+	1.00000i	0.127000	+	0.127000i
$63$	3.00000	−	3.00000i	0.377964	−	0.377964i
$64$	−7.00000			−0.875000
$65$		0			0
$66$		−	6.00000i		−	0.738549i
$67$	6.00000			0.733017			0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000			0.733017			0.366508	+	0.930415i	$0.380553\pi$
$68$	4.00000	−	1.00000i	0.485071	−	0.121268i
$69$	2.00000			0.240772
$70$		0			0
$71$	3.00000	−	3.00000i	0.356034	−	0.356034i	−0.506314	−	0.862349i	$-0.668992\pi$
$71$	3.00000	−	3.00000i	0.356034	−	0.356034i	0.862349	+	0.506314i	$0.168992\pi$
$72$	3.00000			0.353553
$73$	3.00000	−	3.00000i	0.351123	−	0.351123i	−0.509404	−	0.860527i	$-0.670134\pi$
$73$	3.00000	−	3.00000i	0.351123	−	0.351123i	0.860527	+	0.509404i	$0.170134\pi$
$74$	3.00000	+	3.00000i	0.348743	+	0.348743i
$75$		0			0
$76$		−	6.00000i		−	0.688247i
$77$			18.0000i			2.05129i
$78$		0			0
$79$	7.00000	+	7.00000i	0.787562	+	0.787562i	0.981094	−	0.193532i	$-0.0619944\pi$
$79$	7.00000	+	7.00000i	0.787562	+	0.787562i	−0.193532	+	0.981094i	$0.561994\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$	−3.00000	+	3.00000i	−0.331295	+	0.331295i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	6.00000			0.654654
$85$		0			0
$86$	12.0000			1.29399
$87$			6.00000i			0.643268i
$88$	−9.00000	+	9.00000i	−0.959403	+	0.959403i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$	−1.00000	−	1.00000i	−0.104257	−	0.104257i
$93$		−	2.00000i		−	0.207390i
$94$		−	2.00000i		−	0.206284i
$95$		0			0
$96$	5.00000	+	5.00000i	0.510310	+	0.510310i
$97$	−3.00000	+	3.00000i	−0.304604	+	0.304604i	−0.842812	−	0.538208i	$-0.819101\pi$
$97$	−3.00000	+	3.00000i	−0.304604	+	0.304604i	0.538208	+	0.842812i	$0.319101\pi$
$98$	11.0000			1.11117
$99$	3.00000	−	3.00000i	0.301511	−	0.301511i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	5.00000	+	3.00000i	0.495074	+	0.297044i
$103$	6.00000			0.591198			0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000			0.591198			0.295599	+	0.955312i	$0.404481\pi$
$104$		0			0
$105$		0			0
$106$	−2.00000			−0.194257
$107$	9.00000	−	9.00000i	0.870063	−	0.870063i	−0.122416	−	0.992479i	$-0.539064\pi$
$107$	9.00000	−	9.00000i	0.870063	−	0.870063i	0.992479	+	0.122416i	$0.0390642\pi$
$108$	−4.00000	−	4.00000i	−0.384900	−	0.384900i
$109$	7.00000	+	7.00000i	0.670478	+	0.670478i	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	7.00000	+	7.00000i	0.670478	+	0.670478i	−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$		−	6.00000i		−	0.569495i
$112$	3.00000	+	3.00000i	0.283473	+	0.283473i
$113$	9.00000	+	9.00000i	0.846649	+	0.846649i	0.989713	−	0.143065i	$-0.0456957\pi$
$113$	9.00000	+	9.00000i	0.846649	+	0.846649i	−0.143065	+	0.989713i	$0.545696\pi$
$114$	6.00000	−	6.00000i	0.561951	−	0.561951i
$115$		0			0
$116$	3.00000	−	3.00000i	0.278543	−	0.278543i
$117$		0			0
$118$	−6.00000			−0.552345
$119$	−15.0000	−	9.00000i	−1.37505	−	0.825029i
$120$		0			0
$121$			7.00000i			0.636364i
$122$	1.00000	−	1.00000i	0.0905357	−	0.0905357i
$123$	6.00000			0.541002
$124$	−1.00000	+	1.00000i	−0.0898027	+	0.0898027i
$125$		0			0
$126$	−3.00000	−	3.00000i	−0.267261	−	0.267261i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	3.00000i		−	0.265165i
$129$	−12.0000	−	12.0000i	−1.05654	−	1.05654i
$130$		0			0
$131$	3.00000	−	3.00000i	0.262111	−	0.262111i	−0.563800	−	0.825911i	$-0.690661\pi$
$131$	3.00000	−	3.00000i	0.262111	−	0.262111i	0.825911	+	0.563800i	$0.190661\pi$
$132$	6.00000			0.522233
$133$	−18.0000	+	18.0000i	−1.56080	+	1.56080i
$134$		−	6.00000i		−	0.518321i
$135$		0			0
$136$	−3.00000	−	12.0000i	−0.257248	−	1.02899i
$137$	4.00000			0.341743			0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000			0.341743			0.170872	+	0.985293i	$0.445342\pi$
$138$		−	2.00000i		−	0.170251i
$139$	−7.00000	+	7.00000i	−0.593732	+	0.593732i	−0.938638	−	0.344905i	$-0.887911\pi$
$139$	−7.00000	+	7.00000i	−0.593732	+	0.593732i	0.344905	+	0.938638i	$0.387911\pi$
$140$		0			0
$141$	−2.00000	+	2.00000i	−0.168430	+	0.168430i
$142$	−3.00000	−	3.00000i	−0.251754	−	0.251754i
$143$		0			0
$144$		−	1.00000i		−	0.0833333i
$145$		0			0
$146$	−3.00000	−	3.00000i	−0.248282	−	0.248282i
$147$	−11.0000	−	11.0000i	−0.907265	−	0.907265i
$148$	−3.00000	+	3.00000i	−0.246598	+	0.246598i
$149$	18.0000			1.47462			0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000			1.47462			0.737309	+	0.675556i	$0.236096\pi$
$150$		0			0
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	−18.0000			−1.45999
$153$	1.00000	+	4.00000i	0.0808452	+	0.323381i
$154$	18.0000			1.45048
$155$		0			0
$156$		0			0
$157$	12.0000			0.957704			0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000			0.957704			0.478852	+	0.877896i	$0.341053\pi$
$158$	7.00000	−	7.00000i	0.556890	−	0.556890i
$159$	2.00000	+	2.00000i	0.158610	+	0.158610i
$160$		0			0
$161$			6.00000i			0.472866i
$162$		−	5.00000i		−	0.392837i
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	0.260531	−	0.965465i	$-0.416102\pi$
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	−0.965465	+	0.260531i	$0.916102\pi$
$164$	−3.00000	−	3.00000i	−0.234261	−	0.234261i
$165$		0			0
$166$	4.00000			0.310460
$167$	−3.00000	+	3.00000i	−0.232147	+	0.232147i	−0.813588	−	0.581441i	$-0.802489\pi$
$167$	−3.00000	+	3.00000i	−0.232147	+	0.232147i	0.581441	+	0.813588i	$0.302489\pi$
$168$		−	18.0000i		−	1.38873i
$169$	−13.0000			−1.00000
$170$		0			0
$171$	6.00000			0.458831
$172$			12.0000i			0.914991i
$173$	15.0000	−	15.0000i	1.14043	−	1.14043i	0.152057	−	0.988372i	$-0.451410\pi$
$173$	15.0000	−	15.0000i	1.14043	−	1.14043i	0.988372	−	0.152057i	$-0.0485898\pi$
$174$	6.00000			0.454859
$175$		0			0
$176$	3.00000	+	3.00000i	0.226134	+	0.226134i
$177$	6.00000	+	6.00000i	0.450988	+	0.450988i
$178$			6.00000i			0.449719i
$179$		−	6.00000i		−	0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$		−	6.00000i		−	0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$		0			0
$181$	−11.0000	−	11.0000i	−0.817624	−	0.817624i	0.168140	−	0.985763i	$-0.446224\pi$
$181$	−11.0000	−	11.0000i	−0.817624	−	0.817624i	−0.985763	+	0.168140i	$0.946224\pi$
$182$		0			0
$183$	−2.00000			−0.147844
$184$	−3.00000	+	3.00000i	−0.221163	+	0.221163i
$185$		0			0
$186$	−2.00000			−0.146647
$187$	−15.0000	−	9.00000i	−1.09691	−	0.658145i
$188$	2.00000			0.145865
$189$			24.0000i			1.74574i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	7.00000	−	7.00000i	0.505181	−	0.505181i
$193$	−3.00000	−	3.00000i	−0.215945	−	0.215945i	0.590842	−	0.806787i	$-0.298796\pi$
$193$	−3.00000	−	3.00000i	−0.215945	−	0.215945i	−0.806787	+	0.590842i	$0.798796\pi$
$194$	3.00000	+	3.00000i	0.215387	+	0.215387i
$195$		0			0
$196$			11.0000i			0.785714i
$197$	−13.0000	−	13.0000i	−0.926212	−	0.926212i	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	−13.0000	−	13.0000i	−0.926212	−	0.926212i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	−3.00000	−	3.00000i	−0.213201	−	0.213201i
$199$	1.00000	−	1.00000i	0.0708881	−	0.0708881i	−0.670774	−	0.741662i	$-0.734038\pi$
$199$	1.00000	−	1.00000i	0.0708881	−	0.0708881i	0.741662	+	0.670774i	$0.234038\pi$
$200$		0			0
$201$	−6.00000	+	6.00000i	−0.423207	+	0.423207i
$202$			6.00000i			0.422159i
$203$	−18.0000			−1.26335
$204$	−3.00000	+	5.00000i	−0.210042	+	0.350070i
$205$		0			0
$206$		−	6.00000i		−	0.418040i
$207$	1.00000	−	1.00000i	0.0695048	−	0.0695048i
$208$		0			0
$209$	−18.0000	+	18.0000i	−1.24509	+	1.24509i
$210$		0			0
$211$	17.0000	+	17.0000i	1.17033	+	1.17033i	0.982131	+	0.188197i	$0.0602643\pi$
$211$	17.0000	+	17.0000i	1.17033	+	1.17033i	0.188197	+	0.982131i	$0.439736\pi$
$212$		−	2.00000i		−	0.137361i
$213$			6.00000i			0.411113i
$214$	−9.00000	−	9.00000i	−0.615227	−	0.615227i
$215$		0			0
$216$	−12.0000	+	12.0000i	−0.816497	+	0.816497i
$217$	6.00000			0.407307
$218$	7.00000	−	7.00000i	0.474100	−	0.474100i
$219$			6.00000i			0.405442i
$220$		0			0
$221$		0			0
$222$	−6.00000			−0.402694
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−15.0000	+	15.0000i	−1.00223	+	1.00223i
$225$		0			0
$226$	9.00000	−	9.00000i	0.598671	−	0.598671i
$227$	−15.0000	−	15.0000i	−0.995585	−	0.995585i	0.00440533	−	0.999990i	$-0.498598\pi$
$227$	−15.0000	−	15.0000i	−0.995585	−	0.995585i	−0.999990	+	0.00440533i	$0.998598\pi$
$228$	6.00000	+	6.00000i	0.397360	+	0.397360i
$229$		−	24.0000i		−	1.58596i	−0.609245	−	0.792982i	$-0.708527\pi$
$229$		−	24.0000i		−	1.58596i	0.609245	−	0.792982i	$-0.291473\pi$
$230$		0			0
$231$	−18.0000	−	18.0000i	−1.18431	−	1.18431i
$232$	−9.00000	−	9.00000i	−0.590879	−	0.590879i
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−5.00000	+	5.00000i	−0.327561	+	0.327561i	0.524097	+	0.851658i	$0.324403\pi$
$234$		0			0
$235$		0			0
$236$		−	6.00000i		−	0.390567i
$237$	−14.0000			−0.909398
$238$	−9.00000	+	15.0000i	−0.583383	+	0.972306i
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−19.0000	+	19.0000i	−1.22390	+	1.22390i	−0.257663	+	0.966235i	$0.582952\pi$
$241$	−19.0000	+	19.0000i	−1.22390	+	1.22390i	−0.966235	+	0.257663i	$0.917048\pi$
$242$	7.00000			0.449977
$243$	7.00000	−	7.00000i	0.449050	−	0.449050i
$244$	1.00000	+	1.00000i	0.0640184	+	0.0640184i
$245$		0			0
$246$		−	6.00000i		−	0.382546i
$247$		0			0
$248$	3.00000	+	3.00000i	0.190500	+	0.190500i
$249$	−4.00000	−	4.00000i	−0.253490	−	0.253490i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$	3.00000	−	3.00000i	0.188982	−	0.188982i
$253$			6.00000i			0.377217i
$254$		0			0
$255$		0			0
$256$	−17.0000			−1.06250
$257$		−	14.0000i		−	0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$		−	14.0000i		−	0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	−12.0000	+	12.0000i	−0.747087	+	0.747087i
$259$	18.0000			1.11847
$260$		0			0
$261$	3.00000	+	3.00000i	0.185695	+	0.185695i
$262$	−3.00000	−	3.00000i	−0.185341	−	0.185341i
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		−	18.0000i		−	1.10782i
$265$		0			0
$266$	18.0000	+	18.0000i	1.10365	+	1.10365i
$267$	6.00000	−	6.00000i	0.367194	−	0.367194i
$268$	6.00000			0.366508
$269$	3.00000	−	3.00000i	0.182913	−	0.182913i	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	3.00000	−	3.00000i	0.182913	−	0.182913i	0.792624	+	0.609711i	$0.208714\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$	−4.00000	+	1.00000i	−0.242536	+	0.0606339i
$273$		0			0
$274$		−	4.00000i		−	0.241649i
$275$		0			0
$276$	2.00000			0.120386
$277$	21.0000	−	21.0000i	1.26177	−	1.26177i	0.311532	−	0.950236i	$-0.399158\pi$
$277$	21.0000	−	21.0000i	1.26177	−	1.26177i	0.950236	−	0.311532i	$-0.100842\pi$
$278$	7.00000	+	7.00000i	0.419832	+	0.419832i
$279$	−1.00000	−	1.00000i	−0.0598684	−	0.0598684i
$280$		0			0
$281$		−	24.0000i		−	1.43172i	−0.698244	−	0.715860i	$-0.746035\pi$
$281$		−	24.0000i		−	1.43172i	0.698244	−	0.715860i	$-0.253965\pi$
$282$	2.00000	+	2.00000i	0.119098	+	0.119098i
$283$	−9.00000	−	9.00000i	−0.534994	−	0.534994i	0.387060	−	0.922055i	$-0.373491\pi$
$283$	−9.00000	−	9.00000i	−0.534994	−	0.534994i	−0.922055	+	0.387060i	$0.873491\pi$
$284$	3.00000	−	3.00000i	0.178017	−	0.178017i
$285$		0			0
$286$		0			0
$287$			18.0000i			1.06251i
$288$	5.00000			0.294628
$289$	15.0000	−	8.00000i	0.882353	−	0.470588i
$290$		0			0
$291$		−	6.00000i		−	0.351726i
$292$	3.00000	−	3.00000i	0.175562	−	0.175562i
$293$	4.00000			0.233682			0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000			0.233682			0.116841	+	0.993151i	$0.462723\pi$
$294$	−11.0000	+	11.0000i	−0.641533	+	0.641533i
$295$		0			0
$296$	9.00000	+	9.00000i	0.523114	+	0.523114i
$297$			24.0000i			1.39262i
$298$		−	18.0000i		−	1.04271i
$299$		0			0
$300$		0			0
$301$	36.0000	−	36.0000i	2.07501	−	2.07501i
$302$	10.0000			0.575435
$303$	6.00000	−	6.00000i	0.344691	−	0.344691i
$304$			6.00000i			0.344124i
$305$		0			0
$306$	4.00000	−	1.00000i	0.228665	−	0.0571662i
$307$	−18.0000			−1.02731			−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000			−1.02731			−0.513657	+	0.857996i	$0.671710\pi$
$308$			18.0000i			1.02565i
$309$	−6.00000	+	6.00000i	−0.341328	+	0.341328i
$310$		0			0
$311$	3.00000	−	3.00000i	0.170114	−	0.170114i	−0.616915	−	0.787030i	$-0.711618\pi$
$311$	3.00000	−	3.00000i	0.170114	−	0.170114i	0.787030	+	0.616915i	$0.211618\pi$
$312$		0			0
$313$	−3.00000	−	3.00000i	−0.169570	−	0.169570i	0.617220	−	0.786790i	$-0.288259\pi$
$313$	−3.00000	−	3.00000i	−0.169570	−	0.169570i	−0.786790	+	0.617220i	$0.788259\pi$
$314$		−	12.0000i		−	0.677199i
$315$		0			0
$316$	7.00000	+	7.00000i	0.393781	+	0.393781i
$317$	15.0000	+	15.0000i	0.842484	+	0.842484i	0.989181	−	0.146697i	$-0.0468644\pi$
$317$	15.0000	+	15.0000i	0.842484	+	0.842484i	−0.146697	+	0.989181i	$0.546864\pi$
$318$	2.00000	−	2.00000i	0.112154	−	0.112154i
$319$	−18.0000			−1.00781
$320$		0			0
$321$			18.0000i			1.00466i
$322$	6.00000			0.334367
$323$	−6.00000	−	24.0000i	−0.333849	−	1.33540i
$324$	5.00000			0.277778
$325$		0			0
$326$	−9.00000	+	9.00000i	−0.498464	+	0.498464i
$327$	−14.0000			−0.774202
$328$	−9.00000	+	9.00000i	−0.496942	+	0.496942i
$329$	−6.00000	−	6.00000i	−0.330791	−	0.330791i
$330$		0			0
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$			4.00000i			0.219529i
$333$	−3.00000	−	3.00000i	−0.164399	−	0.164399i
$334$	3.00000	+	3.00000i	0.164153	+	0.164153i
$335$		0			0
$336$	−6.00000			−0.327327
$337$	−15.0000	+	15.0000i	−0.817102	+	0.817102i	−0.985687	−	0.168585i	$-0.946080\pi$
$337$	−15.0000	+	15.0000i	−0.817102	+	0.817102i	0.168585	+	0.985687i	$0.446080\pi$
$338$			13.0000i			0.707107i
$339$	−18.0000			−0.977626
$340$		0			0
$341$	6.00000			0.324918
$342$		−	6.00000i		−	0.324443i
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$	36.0000			1.94099
$345$		0			0
$346$	−15.0000	−	15.0000i	−0.806405	−	0.806405i
$347$	9.00000	+	9.00000i	0.483145	+	0.483145i	0.906135	−	0.422989i	$-0.139019\pi$
$347$	9.00000	+	9.00000i	0.483145	+	0.483145i	−0.422989	+	0.906135i	$0.639019\pi$
$348$			6.00000i			0.321634i
$349$		−	28.0000i		−	1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$		−	28.0000i		−	1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$		0			0
$351$		0			0
$352$	−15.0000	+	15.0000i	−0.799503	+	0.799503i
$353$	−16.0000			−0.851594			−0.425797	−	0.904819i	$-0.640006\pi$
$353$	−16.0000			−0.851594			−0.425797	+	0.904819i	$0.640006\pi$
$354$	6.00000	−	6.00000i	0.318896	−	0.318896i
$355$		0			0
$356$	−6.00000			−0.317999
$357$	24.0000	−	6.00000i	1.27021	−	0.317554i
$358$	−6.00000			−0.317110
$359$			30.0000i			1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$			30.0000i			1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$	−11.0000	+	11.0000i	−0.578147	+	0.578147i
$363$	−7.00000	−	7.00000i	−0.367405	−	0.367405i
$364$		0			0
$365$		0			0
$366$			2.00000i			0.104542i
$367$	−3.00000	−	3.00000i	−0.156599	−	0.156599i	0.624459	−	0.781058i	$-0.285320\pi$
$367$	−3.00000	−	3.00000i	−0.156599	−	0.156599i	−0.781058	+	0.624459i	$0.785320\pi$
$368$	1.00000	+	1.00000i	0.0521286	+	0.0521286i
$369$	3.00000	−	3.00000i	0.156174	−	0.156174i
$370$		0			0
$371$	−6.00000	+	6.00000i	−0.311504	+	0.311504i
$372$		−	2.00000i		−	0.103695i
$373$	24.0000			1.24267			0.621336	−	0.783544i	$-0.286590\pi$
$373$	24.0000			1.24267			0.621336	+	0.783544i	$0.286590\pi$
$374$	−9.00000	+	15.0000i	−0.465379	+	0.775632i
$375$		0			0
$376$		−	6.00000i		−	0.309426i
$377$		0			0
$378$	24.0000			1.23443
$379$	−11.0000	+	11.0000i	−0.565032	+	0.565032i	−0.930733	−	0.365701i	$-0.880829\pi$
$379$	−11.0000	+	11.0000i	−0.565032	+	0.565032i	0.365701	+	0.930733i	$0.380829\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			16.0000i			0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$			16.0000i			0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	3.00000	+	3.00000i	0.153093	+	0.153093i
$385$		0			0
$386$	−3.00000	+	3.00000i	−0.152696	+	0.152696i
$387$	−12.0000			−0.609994
$388$	−3.00000	+	3.00000i	−0.152302	+	0.152302i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−5.00000	−	3.00000i	−0.252861	−	0.151717i
$392$	33.0000			1.66675
$393$			6.00000i			0.302660i
$394$	−13.0000	+	13.0000i	−0.654931	+	0.654931i
$395$		0			0
$396$	3.00000	−	3.00000i	0.150756	−	0.150756i
$397$	27.0000	+	27.0000i	1.35509	+	1.35509i	0.879862	+	0.475229i	$0.157635\pi$
$397$	27.0000	+	27.0000i	1.35509	+	1.35509i	0.475229	+	0.879862i	$0.342365\pi$
$398$	−1.00000	−	1.00000i	−0.0501255	−	0.0501255i
$399$		−	36.0000i		−	1.80225i
$400$		0			0
$401$	9.00000	+	9.00000i	0.449439	+	0.449439i	0.895168	−	0.445729i	$-0.147056\pi$
$401$	9.00000	+	9.00000i	0.449439	+	0.449439i	−0.445729	+	0.895168i	$0.647056\pi$
$402$	6.00000	+	6.00000i	0.299253	+	0.299253i
$403$		0			0
$404$	−6.00000			−0.298511
$405$		0			0
$406$			18.0000i			0.893325i
$407$	18.0000			0.892227
$408$	15.0000	+	9.00000i	0.742611	+	0.445566i
$409$	30.0000			1.48340			0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000			1.48340			0.741702	+	0.670729i	$0.234019\pi$
$410$		0			0
$411$	−4.00000	+	4.00000i	−0.197305	+	0.197305i
$412$	6.00000			0.295599
$413$	−18.0000	+	18.0000i	−0.885722	+	0.885722i
$414$	−1.00000	−	1.00000i	−0.0491473	−	0.0491473i
$415$		0			0
$416$		0			0
$417$		−	14.0000i		−	0.685583i
$418$	18.0000	+	18.0000i	0.880409	+	0.880409i
$419$	15.0000	+	15.0000i	0.732798	+	0.732798i	0.971173	−	0.238375i	$-0.0766148\pi$
$419$	15.0000	+	15.0000i	0.732798	+	0.732798i	−0.238375	+	0.971173i	$0.576615\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$	17.0000	−	17.0000i	0.827547	−	0.827547i
$423$			2.00000i			0.0972433i
$424$	−6.00000			−0.291386
$425$		0			0
$426$	6.00000			0.290701
$427$		−	6.00000i		−	0.290360i
$428$	9.00000	−	9.00000i	0.435031	−	0.435031i
$429$		0			0
$430$		0			0
$431$	9.00000	+	9.00000i	0.433515	+	0.433515i	0.889822	−	0.456307i	$-0.150828\pi$
$431$	9.00000	+	9.00000i	0.433515	+	0.433515i	−0.456307	+	0.889822i	$0.650828\pi$
$432$	4.00000	+	4.00000i	0.192450	+	0.192450i
$433$			30.0000i			1.44171i	0.693087	+	0.720854i	$0.256250\pi$
$433$			30.0000i			1.44171i	−0.693087	+	0.720854i	$0.743750\pi$
$434$		−	6.00000i		−	0.288009i
$435$		0			0
$436$	7.00000	+	7.00000i	0.335239	+	0.335239i
$437$	−6.00000	+	6.00000i	−0.287019	+	0.287019i
$438$	6.00000			0.286691
$439$	−19.0000	+	19.0000i	−0.906821	+	0.906821i	−0.996014	−	0.0891938i	$-0.971571\pi$
$439$	−19.0000	+	19.0000i	−0.906821	+	0.906821i	0.0891938	+	0.996014i	$0.471571\pi$
$440$		0			0
$441$	−11.0000			−0.523810
$442$		0			0
$443$	−22.0000			−1.04525			−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000			−1.04525			−0.522626	+	0.852562i	$0.675047\pi$
$444$		−	6.00000i		−	0.284747i
$445$		0			0
$446$		0			0
$447$	−18.0000	+	18.0000i	−0.851371	+	0.851371i
$448$	21.0000	+	21.0000i	0.992157	+	0.992157i
$449$	15.0000	+	15.0000i	0.707894	+	0.707894i	0.966092	−	0.258198i	$-0.0831288\pi$
$449$	15.0000	+	15.0000i	0.707894	+	0.707894i	−0.258198	+	0.966092i	$0.583129\pi$
$450$		0			0
$451$			18.0000i			0.847587i
$452$	9.00000	+	9.00000i	0.423324	+	0.423324i
$453$	−10.0000	−	10.0000i	−0.469841	−	0.469841i
$454$	−15.0000	+	15.0000i	−0.703985	+	0.703985i
$455$		0			0
$456$	18.0000	−	18.0000i	0.842927	−	0.842927i
$457$		−	6.00000i		−	0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$		−	6.00000i		−	0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	−24.0000			−1.12145
$459$	−20.0000	−	12.0000i	−0.933520	−	0.560112i
$460$		0			0
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	−18.0000	+	18.0000i	−0.837436	+	0.837436i
$463$	−18.0000			−0.836531			−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000			−0.836531			−0.418265	+	0.908325i	$0.637362\pi$
$464$	−3.00000	+	3.00000i	−0.139272	+	0.139272i
$465$		0			0
$466$	5.00000	+	5.00000i	0.231621	+	0.231621i
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$	−18.0000	−	18.0000i	−0.831163	−	0.831163i
$470$		0			0
$471$	−12.0000	+	12.0000i	−0.552931	+	0.552931i
$472$	−18.0000			−0.828517
$473$	36.0000	−	36.0000i	1.65528	−	1.65528i
$474$			14.0000i			0.643041i
$475$		0			0
$476$	−15.0000	−	9.00000i	−0.687524	−	0.412514i
$477$	2.00000			0.0915737
$478$		−	24.0000i		−	1.09773i
$479$	9.00000	−	9.00000i	0.411220	−	0.411220i	−0.470943	−	0.882164i	$-0.656086\pi$
$479$	9.00000	−	9.00000i	0.411220	−	0.411220i	0.882164	+	0.470943i	$0.156086\pi$
$480$		0			0
$481$		0			0
$482$	19.0000	+	19.0000i	0.865426	+	0.865426i
$483$	−6.00000	−	6.00000i	−0.273009	−	0.273009i
$484$			7.00000i			0.318182i
$485$		0			0
$486$	−7.00000	−	7.00000i	−0.317526	−	0.317526i
$487$	−3.00000	−	3.00000i	−0.135943	−	0.135943i	0.635861	−	0.771804i	$-0.280645\pi$
$487$	−3.00000	−	3.00000i	−0.135943	−	0.135943i	−0.771804	+	0.635861i	$0.780645\pi$
$488$	3.00000	−	3.00000i	0.135804	−	0.135804i
$489$	18.0000			0.813988
$490$		0			0
$491$			6.00000i			0.270776i	0.990793	+	0.135388i	$0.0432281\pi$
$491$			6.00000i			0.270776i	−0.990793	+	0.135388i	$0.956772\pi$
$492$	6.00000			0.270501
$493$	9.00000	−	15.0000i	0.405340	−	0.675566i
$494$		0			0
$495$		0			0
$496$	1.00000	−	1.00000i	0.0449013	−	0.0449013i
$497$	−18.0000			−0.807410
$498$	−4.00000	+	4.00000i	−0.179244	+	0.179244i
$499$	23.0000	+	23.0000i	1.02962	+	1.02962i	0.999548	+	0.0300737i	$0.00957421\pi$
$499$	23.0000	+	23.0000i	1.02962	+	1.02962i	0.0300737	+	0.999548i	$0.490426\pi$
$500$		0			0
$501$		−	6.00000i		−	0.268060i
$502$			12.0000i			0.535586i
$503$	3.00000	+	3.00000i	0.133763	+	0.133763i	0.770818	−	0.637055i	$-0.219848\pi$
$503$	3.00000	+	3.00000i	0.133763	+	0.133763i	−0.637055	+	0.770818i	$0.719848\pi$
$504$	−9.00000	−	9.00000i	−0.400892	−	0.400892i
$505$		0			0
$506$	6.00000			0.266733
$507$	13.0000	−	13.0000i	0.577350	−	0.577350i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$			11.0000i			0.486136i
$513$	−24.0000	+	24.0000i	−1.05963	+	1.05963i
$514$	−14.0000			−0.617514
$515$		0			0
$516$	−12.0000	−	12.0000i	−0.528271	−	0.528271i
$517$	−6.00000	−	6.00000i	−0.263880	−	0.263880i
$518$		−	18.0000i		−	0.790875i
$519$			30.0000i			1.31685i
$520$		0			0
$521$	9.00000	+	9.00000i	0.394297	+	0.394297i	0.876216	−	0.481919i	$-0.160060\pi$
$521$	9.00000	+	9.00000i	0.394297	+	0.394297i	−0.481919	+	0.876216i	$0.660060\pi$
$522$	3.00000	−	3.00000i	0.131306	−	0.131306i
$523$	−6.00000			−0.262362			−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000			−0.262362			−0.131181	+	0.991358i	$0.541877\pi$
$524$	3.00000	−	3.00000i	0.131056	−	0.131056i
$525$		0			0
$526$	−16.0000			−0.697633
$527$	−3.00000	+	5.00000i	−0.130682	+	0.217803i
$528$	−6.00000			−0.261116
$529$		−	21.0000i		−	0.913043i
$530$		0			0
$531$	6.00000			0.260378
$532$	−18.0000	+	18.0000i	−0.780399	+	0.780399i
$533$		0			0
$534$	−6.00000	−	6.00000i	−0.259645	−	0.259645i
$535$		0			0
$536$		−	18.0000i		−	0.777482i
$537$	6.00000	+	6.00000i	0.258919	+	0.258919i
$538$	−3.00000	−	3.00000i	−0.129339	−	0.129339i
$539$	33.0000	−	33.0000i	1.42141	−	1.42141i
$540$		0			0
$541$	1.00000	−	1.00000i	0.0429934	−	0.0429934i	−0.685283	−	0.728277i	$-0.740322\pi$
$541$	1.00000	−	1.00000i	0.0429934	−	0.0429934i	0.728277	+	0.685283i	$0.240322\pi$
$542$		−	24.0000i		−	1.03089i
$543$	22.0000			0.944110
$544$	−5.00000	−	20.0000i	−0.214373	−	0.857493i
$545$		0			0
$546$		0			0
$547$	9.00000	−	9.00000i	0.384812	−	0.384812i	−0.488020	−	0.872832i	$-0.662281\pi$
$547$	9.00000	−	9.00000i	0.384812	−	0.384812i	0.872832	+	0.488020i	$0.162281\pi$
$548$	4.00000			0.170872
$549$	−1.00000	+	1.00000i	−0.0426790	+	0.0426790i
$550$		0			0
$551$	−18.0000	−	18.0000i	−0.766826	−	0.766826i
$552$		−	6.00000i		−	0.255377i
$553$		−	42.0000i		−	1.78602i
$554$	−21.0000	−	21.0000i	−0.892205	−	0.892205i
$555$		0			0
$556$	−7.00000	+	7.00000i	−0.296866	+	0.296866i
$557$	−32.0000			−1.35588			−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000			−1.35588			−0.677942	+	0.735116i	$0.737128\pi$
$558$	−1.00000	+	1.00000i	−0.0423334	+	0.0423334i
$559$		0			0
$560$		0			0
$561$	24.0000	−	6.00000i	1.01328	−	0.253320i
$562$	−24.0000			−1.01238
$563$			28.0000i			1.18006i	0.807382	+	0.590030i	$0.200884\pi$
$563$			28.0000i			1.18006i	−0.807382	+	0.590030i	$0.799116\pi$
$564$	−2.00000	+	2.00000i	−0.0842152	+	0.0842152i
$565$		0			0
$566$	−9.00000	+	9.00000i	−0.378298	+	0.378298i
$567$	−15.0000	−	15.0000i	−0.629941	−	0.629941i
$568$	−9.00000	−	9.00000i	−0.377632	−	0.377632i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−23.0000	−	23.0000i	−0.962520	−	0.962520i	0.0368025	−	0.999323i	$-0.488283\pi$
$571$	−23.0000	−	23.0000i	−0.962520	−	0.962520i	−0.999323	+	0.0368025i	$0.988283\pi$
$572$		0			0
$573$		0			0
$574$	18.0000			0.751305
$575$		0			0
$576$		−	7.00000i		−	0.291667i
$577$	36.0000			1.49870			0.749350	−	0.662174i	$-0.230366\pi$
$577$	36.0000			1.49870			0.749350	+	0.662174i	$0.230366\pi$
$578$	−8.00000	−	15.0000i	−0.332756	−	0.623918i
$579$	6.00000			0.249351
$580$		0			0
$581$	12.0000	−	12.0000i	0.497844	−	0.497844i
$582$	−6.00000			−0.248708
$583$	−6.00000	+	6.00000i	−0.248495	+	0.248495i
$584$	−9.00000	−	9.00000i	−0.372423	−	0.372423i
$585$		0			0
$586$		−	4.00000i		−	0.165238i
$587$			4.00000i			0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$			4.00000i			0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	−11.0000	−	11.0000i	−0.453632	−	0.453632i
$589$	6.00000	+	6.00000i	0.247226	+	0.247226i
$590$		0			0
$591$	26.0000			1.06950
$592$	3.00000	−	3.00000i	0.123299	−	0.123299i
$593$			2.00000i			0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$			2.00000i			0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$	24.0000			0.984732
$595$		0			0
$596$	18.0000			0.737309
$597$			2.00000i			0.0818546i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	29.0000	+	29.0000i	1.18293	+	1.18293i	0.978980	+	0.203954i	$0.0653794\pi$
$601$	29.0000	+	29.0000i	1.18293	+	1.18293i	0.203954	+	0.978980i	$0.434621\pi$
$602$	−36.0000	−	36.0000i	−1.46725	−	1.46725i
$603$			6.00000i			0.244339i
$604$			10.0000i			0.406894i
$605$		0			0
$606$	−6.00000	−	6.00000i	−0.243733	−	0.243733i
$607$	33.0000	−	33.0000i	1.33943	−	1.33943i	0.442816	−	0.896612i	$-0.353979\pi$
$607$	33.0000	−	33.0000i	1.33943	−	1.33943i	0.896612	−	0.442816i	$-0.146021\pi$
$608$	−30.0000			−1.21666
$609$	18.0000	−	18.0000i	0.729397	−	0.729397i
$610$		0			0
$611$		0			0
$612$	1.00000	+	4.00000i	0.0404226	+	0.161690i
$613$	36.0000			1.45403			0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000			1.45403			0.727013	+	0.686624i	$0.240908\pi$
$614$			18.0000i			0.726421i
$615$		0			0
$616$	54.0000			2.17572
$617$	−15.0000	+	15.0000i	−0.603877	+	0.603877i	−0.941339	−	0.337462i	$-0.890432\pi$
$617$	−15.0000	+	15.0000i	−0.603877	+	0.603877i	0.337462	+	0.941339i	$0.390432\pi$
$618$	6.00000	+	6.00000i	0.241355	+	0.241355i
$619$	−13.0000	−	13.0000i	−0.522514	−	0.522514i	0.395816	−	0.918330i	$-0.370462\pi$
$619$	−13.0000	−	13.0000i	−0.522514	−	0.522514i	−0.918330	+	0.395816i	$0.870462\pi$
$620$		0			0
$621$			8.00000i			0.321029i
$622$	−3.00000	−	3.00000i	−0.120289	−	0.120289i
$623$	18.0000	+	18.0000i	0.721155	+	0.721155i
$624$		0			0
$625$		0			0
$626$	−3.00000	+	3.00000i	−0.119904	+	0.119904i
$627$		−	36.0000i		−	1.43770i
$628$	12.0000			0.478852
$629$	−9.00000	+	15.0000i	−0.358854	+	0.598089i
$630$		0			0
$631$			34.0000i			1.35352i	0.736204	+	0.676759i	$0.236616\pi$
$631$			34.0000i			1.35352i	−0.736204	+	0.676759i	$0.763384\pi$
$632$	21.0000	−	21.0000i	0.835335	−	0.835335i
$633$	−34.0000			−1.35138
$634$	15.0000	−	15.0000i	0.595726	−	0.595726i
$635$		0			0
$636$	2.00000	+	2.00000i	0.0793052	+	0.0793052i
$637$		0			0
$638$			18.0000i			0.712627i
$639$	3.00000	+	3.00000i	0.118678	+	0.118678i
$640$		0			0
$641$	9.00000	−	9.00000i	0.355479	−	0.355479i	−0.506665	−	0.862143i	$-0.669122\pi$
$641$	9.00000	−	9.00000i	0.355479	−	0.355479i	0.862143	+	0.506665i	$0.169122\pi$
$642$	18.0000			0.710403
$643$	−9.00000	+	9.00000i	−0.354925	+	0.354925i	−0.861938	−	0.507013i	$-0.830750\pi$
$643$	−9.00000	+	9.00000i	−0.354925	+	0.354925i	0.507013	+	0.861938i	$0.330750\pi$
$644$			6.00000i			0.236433i
$645$		0			0
$646$	−24.0000	+	6.00000i	−0.944267	+	0.236067i
$647$	−46.0000			−1.80845			−0.904223	−	0.427060i	$-0.859549\pi$
$647$	−46.0000			−1.80845			−0.904223	+	0.427060i	$0.859549\pi$
$648$		−	15.0000i		−	0.589256i
$649$	−18.0000	+	18.0000i	−0.706562	+	0.706562i
$650$		0			0
$651$	−6.00000	+	6.00000i	−0.235159	+	0.235159i
$652$	−9.00000	−	9.00000i	−0.352467	−	0.352467i
$653$	21.0000	+	21.0000i	0.821794	+	0.821794i	0.986365	−	0.164572i	$-0.0526242\pi$
$653$	21.0000	+	21.0000i	0.821794	+	0.821794i	−0.164572	+	0.986365i	$0.552624\pi$
$654$			14.0000i			0.547443i
$655$		0			0
$656$	3.00000	+	3.00000i	0.117130	+	0.117130i
$657$	3.00000	+	3.00000i	0.117041	+	0.117041i
$658$	−6.00000	+	6.00000i	−0.233904	+	0.233904i
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$		−	36.0000i		−	1.40024i	−0.714026	−	0.700119i	$-0.753130\pi$
$661$		−	36.0000i		−	1.40024i	0.714026	−	0.700119i	$-0.246870\pi$
$662$	−18.0000			−0.699590
$663$		0			0
$664$	12.0000			0.465690
$665$		0			0
$666$	−3.00000	+	3.00000i	−0.116248	+	0.116248i
$667$	−6.00000			−0.232321
$668$	−3.00000	+	3.00000i	−0.116073	+	0.116073i
$669$		0			0
$670$		0			0
$671$		−	6.00000i		−	0.231627i
$672$		−	30.0000i		−	1.15728i
$673$	−15.0000	−	15.0000i	−0.578208	−	0.578208i	0.356202	−	0.934409i	$-0.384072\pi$
$673$	−15.0000	−	15.0000i	−0.578208	−	0.578208i	−0.934409	+	0.356202i	$0.884072\pi$
$674$	15.0000	+	15.0000i	0.577778	+	0.577778i
$675$		0			0
$676$	−13.0000			−0.500000
$677$	−11.0000	+	11.0000i	−0.422764	+	0.422764i	−0.886154	−	0.463390i	$-0.846633\pi$
$677$	−11.0000	+	11.0000i	−0.422764	+	0.422764i	0.463390	+	0.886154i	$0.346633\pi$
$678$			18.0000i			0.691286i
$679$	18.0000			0.690777
$680$		0			0
$681$	30.0000			1.14960
$682$		−	6.00000i		−	0.229752i
$683$	15.0000	−	15.0000i	0.573959	−	0.573959i	−0.359273	−	0.933232i	$-0.616975\pi$
$683$	15.0000	−	15.0000i	0.573959	−	0.573959i	0.933232	+	0.359273i	$0.116975\pi$
$684$	6.00000			0.229416
$685$		0			0
$686$	−12.0000	−	12.0000i	−0.458162	−	0.458162i
$687$	24.0000	+	24.0000i	0.915657	+	0.915657i
$688$		−	12.0000i		−	0.457496i
$689$		0			0
$690$		0			0
$691$	−19.0000	−	19.0000i	−0.722794	−	0.722794i	0.246379	−	0.969173i	$-0.420759\pi$
$691$	−19.0000	−	19.0000i	−0.722794	−	0.722794i	−0.969173	+	0.246379i	$0.920759\pi$
$692$	15.0000	−	15.0000i	0.570214	−	0.570214i
$693$	−18.0000			−0.683763
$694$	9.00000	−	9.00000i	0.341635	−	0.341635i
$695$		0			0
$696$	18.0000			0.682288
$697$	−15.0000	−	9.00000i	−0.568166	−	0.340899i
$698$	−28.0000			−1.05982
$699$		−	10.0000i		−	0.378235i
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	18.0000	+	18.0000i	0.678883	+	0.678883i
$704$	21.0000	+	21.0000i	0.791467	+	0.791467i
$705$		0			0
$706$			16.0000i			0.602168i
$707$	18.0000	+	18.0000i	0.676960	+	0.676960i
$708$	6.00000	+	6.00000i	0.225494	+	0.225494i
$709$	−5.00000	+	5.00000i	−0.187779	+	0.187779i	−0.794735	−	0.606956i	$-0.792390\pi$
$709$	−5.00000	+	5.00000i	−0.187779	+	0.187779i	0.606956	+	0.794735i	$0.292390\pi$
$710$		0			0
$711$	−7.00000	+	7.00000i	−0.262521	+	0.262521i
$712$			18.0000i			0.674579i
$713$	2.00000			0.0749006
$714$	−6.00000	−	24.0000i	−0.224544	−	0.898177i
$715$		0			0
$716$		−	6.00000i		−	0.224231i
$717$	−24.0000	+	24.0000i	−0.896296	+	0.896296i
$718$	30.0000			1.11959
$719$	21.0000	−	21.0000i	0.783168	−	0.783168i	−0.197196	−	0.980364i	$-0.563184\pi$
$719$	21.0000	−	21.0000i	0.783168	−	0.783168i	0.980364	+	0.197196i	$0.0631836\pi$
$720$		0			0
$721$	−18.0000	−	18.0000i	−0.670355	−	0.670355i
$722$			17.0000i			0.632674i
$723$		−	38.0000i		−	1.41324i
$724$	−11.0000	−	11.0000i	−0.408812	−	0.408812i
$725$		0			0
$726$	−7.00000	+	7.00000i	−0.259794	+	0.259794i
$727$	18.0000			0.667583			0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000			0.667583			0.333792	+	0.942647i	$0.391672\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$	12.0000	+	48.0000i	0.443836	+	1.77534i
$732$	−2.00000			−0.0739221
$733$		−	42.0000i		−	1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$		−	42.0000i		−	1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	−3.00000	+	3.00000i	−0.110732	+	0.110732i
$735$		0			0
$736$	−5.00000	+	5.00000i	−0.184302	+	0.184302i
$737$	−18.0000	−	18.0000i	−0.663039	−	0.663039i
$738$	−3.00000	−	3.00000i	−0.110432	−	0.110432i
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$		0			0
$741$		0			0
$742$	6.00000	+	6.00000i	0.220267	+	0.220267i
$743$	−9.00000	+	9.00000i	−0.330178	+	0.330178i	−0.852654	−	0.522476i	$-0.825008\pi$
$743$	−9.00000	+	9.00000i	−0.330178	+	0.330178i	0.522476	+	0.852654i	$0.325008\pi$
$744$	−6.00000			−0.219971
$745$		0			0
$746$		−	24.0000i		−	0.878702i
$747$	−4.00000			−0.146352
$748$	−15.0000	−	9.00000i	−0.548454	−	0.329073i
$749$	−54.0000			−1.97312
$750$		0			0
$751$	19.0000	−	19.0000i	0.693320	−	0.693320i	−0.269641	−	0.962961i	$-0.586905\pi$
$751$	19.0000	−	19.0000i	0.693320	−	0.693320i	0.962961	+	0.269641i	$0.0869050\pi$
$752$	−2.00000			−0.0729325
$753$	12.0000	−	12.0000i	0.437304	−	0.437304i
$754$		0			0
$755$		0			0
$756$			24.0000i			0.872872i
$757$		−	42.0000i		−	1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$		−	42.0000i		−	1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	11.0000	+	11.0000i	0.399538	+	0.399538i
$759$	−6.00000	−	6.00000i	−0.217786	−	0.217786i
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$		−	42.0000i		−	1.52050i
$764$		0			0
$765$		0			0
$766$	16.0000			0.578103
$767$		0			0
$768$	17.0000	−	17.0000i	0.613435	−	0.613435i
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	14.0000	+	14.0000i	0.504198	+	0.504198i
$772$	−3.00000	−	3.00000i	−0.107972	−	0.107972i
$773$			26.0000i			0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$			26.0000i			0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$			12.0000i			0.431331i
$775$		0			0
$776$	9.00000	+	9.00000i	0.323081	+	0.323081i
$777$	−18.0000	+	18.0000i	−0.645746	+	0.645746i
$778$		0			0
$779$	−18.0000	+	18.0000i	−0.644917	+	0.644917i
$780$		0			0
$781$	−18.0000			−0.644091
$782$	−3.00000	+	5.00000i	−0.107280	+	0.178800i
$783$	−24.0000			−0.857690

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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(-1.00000 + 1.00000i) q^{3} +1.00000 q^{4} +(1.00000 + 1.00000i) q^{6} +(-3.00000 - 3.00000i) q^{7} -3.00000i q^{8} +1.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(-1.00000 + 1.00000i) q^{12} +(-3.00000 + 3.00000i) q^{14} -1.00000 q^{16} +(4.00000 - 1.00000i) q^{17} +1.00000 q^{18} -6.00000i q^{19} +6.00000 q^{21} +(-3.00000 + 3.00000i) q^{22} +(-1.00000 - 1.00000i) q^{23} +(3.00000 + 3.00000i) q^{24} +(-4.00000 - 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{28} +(3.00000 - 3.00000i) q^{29} +(-1.00000 + 1.00000i) q^{31} -5.00000i q^{32} +6.00000 q^{33} +(-1.00000 - 4.00000i) q^{34} +1.00000i q^{36} +(-3.00000 + 3.00000i) q^{37} -6.00000 q^{38} +(-3.00000 - 3.00000i) q^{41} -6.00000i q^{42} +12.0000i q^{43} +(-3.00000 - 3.00000i) q^{44} +(-1.00000 + 1.00000i) q^{46} +2.00000 q^{47} +(1.00000 - 1.00000i) q^{48} +11.0000i q^{49} +(-3.00000 + 5.00000i) q^{51} -2.00000i q^{53} +(-4.00000 + 4.00000i) q^{54} +(-9.00000 + 9.00000i) q^{56} +(6.00000 + 6.00000i) q^{57} +(-3.00000 - 3.00000i) q^{58} -6.00000i q^{59} +(1.00000 + 1.00000i) q^{61} +(1.00000 + 1.00000i) q^{62} +(3.00000 - 3.00000i) q^{63} -7.00000 q^{64} -6.00000i q^{66} +6.00000 q^{67} +(4.00000 - 1.00000i) q^{68} +2.00000 q^{69} +(3.00000 - 3.00000i) q^{71} +3.00000 q^{72} +(3.00000 - 3.00000i) q^{73} +(3.00000 + 3.00000i) q^{74} -6.00000i q^{76} +18.0000i q^{77} +(7.00000 + 7.00000i) q^{79} +5.00000 q^{81} +(-3.00000 + 3.00000i) q^{82} +4.00000i q^{83} +6.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +(-9.00000 + 9.00000i) q^{88} -6.00000 q^{89} +(-1.00000 - 1.00000i) q^{92} -2.00000i q^{93} -2.00000i q^{94} +(5.00000 + 5.00000i) q^{96} +(-3.00000 + 3.00000i) q^{97} +11.0000 q^{98} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{7} - 6 q^{11} - 2 q^{12} - 6 q^{14} - 2 q^{16} + 8 q^{17} + 2 q^{18} + 12 q^{21} - 6 q^{22} - 2 q^{23} + 6 q^{24} - 8 q^{27} - 6 q^{28} + 6 q^{29} - 2 q^{31} + 12 q^{33}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^7 - 6 * q^11 - 2 * q^12 - 6 * q^14 - 2 * q^16 + 8 * q^17 + 2 * q^18 + 12 * q^21 - 6 * q^22 - 2 * q^23 + 6 * q^24 - 8 * q^27 - 6 * q^28 + 6 * q^29 - 2 * q^31 + 12 * q^33 - 2 * q^34 - 6 * q^37 - 12 * q^38 - 6 * q^41 - 6 * q^44 - 2 * q^46 + 4 * q^47 + 2 * q^48 - 6 * q^51 - 8 * q^54 - 18 * q^56 + 12 * q^57 - 6 * q^58 + 2 * q^61 + 2 * q^62 + 6 * q^63 - 14 * q^64 + 12 * q^67 + 8 * q^68 + 4 * q^69 + 6 * q^71 + 6 * q^72 + 6 * q^73 + 6 * q^74 + 14 * q^79 + 10 * q^81 - 6 * q^82 + 12 * q^84 + 24 * q^86 - 18 * q^88 - 12 * q^89 - 2 * q^92 + 10 * q^96 - 6 * q^97 + 22 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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