LMFDB - Embedded newform 425.2.e.b.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.b.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	$0.115027\pi$
$2$			1.00000i			0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	0.934172	+	0.356822i	$0.116140\pi$
$4$	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	$0.0461154\pi$
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.144370	+	0.989524i	$0.453885\pi$
$8$			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	0.803636	−	0.595121i	$-0.202896\pi$
$23$	1.00000	+	1.00000i	0.208514	+	0.208514i	−0.595121	+	0.803636i	$0.702896\pi$
$24$	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	−0.416115	−	0.909312i	$-0.636609\pi$
$37$	3.00000	−	3.00000i	0.493197	−	0.493197i	0.909312	+	0.416115i	$0.136609\pi$
$38$	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	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	−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	$-0.546597\pi$
$47$	−2.00000			−0.291730			−0.145865	+	0.989305i	$0.546597\pi$
$48$	−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	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	−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	$-0.619447\pi$
$67$	−6.00000			−0.733017			−0.366508	+	0.930415i	$0.619447\pi$
$68$	−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	−0.860527	−	0.509404i	$-0.829866\pi$
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	0.509404	+	0.860527i	$0.329866\pi$
$74$	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	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	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	−0.538208	−	0.842812i	$-0.680899\pi$
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	0.842812	+	0.538208i	$0.180899\pi$
$98$	−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	$-0.595519\pi$
$103$	−6.00000			−0.591198			−0.295599	+	0.955312i	$0.595519\pi$
$104$		0			0
$105$		0			0
$106$	−2.00000			−0.194257
$107$	−9.00000	+	9.00000i	−0.870063	+	0.870063i	−0.992479	−	0.122416i	$-0.960936\pi$
$107$	−9.00000	+	9.00000i	−0.870063	+	0.870063i	0.122416	+	0.992479i	$0.460936\pi$
$108$	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	0.143065	−	0.989713i	$-0.454304\pi$
$113$	−9.00000	−	9.00000i	−0.846649	−	0.846649i	−0.989713	+	0.143065i	$0.954304\pi$
$114$	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	$-0.554658\pi$
$137$	−4.00000			−0.341743			−0.170872	+	0.985293i	$0.554658\pi$
$138$			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	$-0.658947\pi$
$157$	−12.0000			−0.957704			−0.478852	+	0.877896i	$0.658947\pi$
$158$	−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	0.965465	−	0.260531i	$-0.0838976\pi$
$163$	9.00000	+	9.00000i	0.704934	+	0.704934i	−0.260531	+	0.965465i	$0.583898\pi$
$164$	−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	−0.581441	−	0.813588i	$-0.697511\pi$
$167$	3.00000	−	3.00000i	0.232147	−	0.232147i	0.813588	+	0.581441i	$0.197511\pi$
$168$			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	$0.548590\pi$
$173$	−15.0000	+	15.0000i	−1.14043	+	1.14043i	−0.988372	+	0.152057i	$0.951410\pi$
$174$	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	0.806787	−	0.590842i	$-0.201204\pi$
$193$	3.00000	+	3.00000i	0.215945	+	0.215945i	−0.590842	+	0.806787i	$0.701204\pi$
$194$	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	0.997459	−	0.0712470i	$-0.0226979\pi$
$197$	13.0000	+	13.0000i	0.926212	+	0.926212i	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	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	0.999990	−	0.00440533i	$-0.00140226\pi$
$227$	15.0000	+	15.0000i	0.995585	+	0.995585i	−0.00440533	+	0.999990i	$0.501402\pi$
$228$	−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	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	5.00000	−	5.00000i	0.327561	−	0.327561i	0.851658	+	0.524097i	$0.175597\pi$
$234$		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	$0.143834\pi$
$257$			14.0000i			0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	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	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$		−	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	$0.600842\pi$
$277$	−21.0000	+	21.0000i	−1.26177	+	1.26177i	−0.950236	+	0.311532i	$0.899158\pi$
$278$	−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	0.922055	−	0.387060i	$-0.126509\pi$
$283$	9.00000	+	9.00000i	0.534994	+	0.534994i	−0.387060	+	0.922055i	$0.626509\pi$
$284$	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	$-0.537277\pi$
$293$	−4.00000			−0.233682			−0.116841	+	0.993151i	$0.537277\pi$
$294$	−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	$-0.328290\pi$
$307$	18.0000			1.02731			0.513657	+	0.857996i	$0.328290\pi$
$308$		−	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	0.786790	−	0.617220i	$-0.211741\pi$
$313$	3.00000	+	3.00000i	0.169570	+	0.169570i	−0.617220	+	0.786790i	$0.711741\pi$
$314$		−	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	0.146697	−	0.989181i	$-0.453136\pi$
$317$	−15.0000	−	15.0000i	−0.842484	−	0.842484i	−0.989181	+	0.146697i	$0.953136\pi$
$318$	−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	−0.168585	−	0.985687i	$-0.553920\pi$
$337$	15.0000	−	15.0000i	0.817102	−	0.817102i	0.985687	+	0.168585i	$0.0539198\pi$
$338$		−	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	0.422989	−	0.906135i	$-0.360981\pi$
$347$	−9.00000	−	9.00000i	−0.483145	−	0.483145i	−0.906135	+	0.422989i	$0.860981\pi$
$348$		−	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	$-0.359994\pi$
$353$	16.0000			0.851594			0.425797	+	0.904819i	$0.359994\pi$
$354$	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	0.781058	−	0.624459i	$-0.214680\pi$
$367$	3.00000	+	3.00000i	0.156599	+	0.156599i	−0.624459	+	0.781058i	$0.714680\pi$
$368$	−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	$-0.713410\pi$
$373$	−24.0000			−1.24267			−0.621336	+	0.783544i	$0.713410\pi$
$374$	−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	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	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	$-0.842365\pi$
$397$	−27.0000	−	27.0000i	−1.35509	−	1.35509i	−0.475229	−	0.879862i	$-0.657635\pi$
$398$	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	$-0.743750\pi$
$433$		−	30.0000i		−	1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$		−	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	$-0.324953\pi$
$443$	22.0000			1.04525			0.522626	+	0.852562i	$0.324953\pi$
$444$		−	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	$0.0448177\pi$
$457$			6.00000i			0.280668i	−0.990104	+	0.140334i	$0.955182\pi$
$458$	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	$-0.362638\pi$
$463$	18.0000			0.836531			0.418265	+	0.908325i	$0.362638\pi$
$464$	−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	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$		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	0.771804	−	0.635861i	$-0.219355\pi$
$487$	3.00000	+	3.00000i	0.135943	+	0.135943i	−0.635861	+	0.771804i	$0.719355\pi$
$488$	−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	0.637055	−	0.770818i	$-0.280152\pi$
$503$	−3.00000	−	3.00000i	−0.133763	−	0.133763i	−0.770818	+	0.637055i	$0.780152\pi$
$504$	−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	$-0.458123\pi$
$523$	6.00000			0.262362			0.131181	+	0.991358i	$0.458123\pi$
$524$	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	−0.872832	−	0.488020i	$-0.837719\pi$
$547$	−9.00000	+	9.00000i	−0.384812	+	0.384812i	0.488020	+	0.872832i	$0.337719\pi$
$548$	−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	$-0.262872\pi$
$557$	32.0000			1.35588			0.677942	+	0.735116i	$0.262872\pi$
$558$	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	$-0.799116\pi$
$563$		−	28.0000i		−	1.18006i	0.807382	−	0.590030i	$-0.200884\pi$
$564$	−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	$-0.769634\pi$
$577$	−36.0000			−1.49870			−0.749350	+	0.662174i	$0.769634\pi$
$578$	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	$-0.973694\pi$
$587$		−	4.00000i		−	0.165098i	0.996587	−	0.0825488i	$-0.0263060\pi$
$588$	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	$-0.986925\pi$
$593$		−	2.00000i		−	0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	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	$0.646021\pi$
$607$	−33.0000	+	33.0000i	−1.33943	+	1.33943i	−0.896612	+	0.442816i	$0.853979\pi$
$608$	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	$-0.759092\pi$
$613$	−36.0000			−1.45403			−0.727013	+	0.686624i	$0.759092\pi$
$614$			18.0000i			0.726421i
$615$		0			0
$616$	54.0000			2.17572
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	−0.337462	−	0.941339i	$-0.609568\pi$
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	0.941339	+	0.337462i	$0.109568\pi$
$618$	−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	−0.507013	−	0.861938i	$-0.669250\pi$
$643$	9.00000	−	9.00000i	0.354925	−	0.354925i	0.861938	+	0.507013i	$0.169250\pi$
$644$			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	$-0.140451\pi$
$647$	46.0000			1.80845			0.904223	+	0.427060i	$0.140451\pi$
$648$			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	0.164572	−	0.986365i	$-0.447376\pi$
$653$	−21.0000	−	21.0000i	−0.821794	−	0.821794i	−0.986365	+	0.164572i	$0.947376\pi$
$654$			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	0.934409	−	0.356202i	$-0.115928\pi$
$673$	15.0000	+	15.0000i	0.578208	+	0.578208i	−0.356202	+	0.934409i	$0.615928\pi$
$674$	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	−0.463390	−	0.886154i	$-0.653367\pi$
$677$	11.0000	−	11.0000i	0.422764	−	0.422764i	0.886154	+	0.463390i	$0.153367\pi$
$678$		−	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	−0.933232	−	0.359273i	$-0.883025\pi$
$683$	−15.0000	+	15.0000i	−0.573959	+	0.573959i	0.359273	+	0.933232i	$0.383025\pi$
$684$	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	$-0.608328\pi$
$727$	−18.0000			−0.667583			−0.333792	+	0.942647i	$0.608328\pi$
$728$		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	$0.282579\pi$
$733$			42.0000i			1.55131i	−0.631160	+	0.775653i	$0.717421\pi$
$734$	−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	−0.522476	−	0.852654i	$-0.674992\pi$
$743$	9.00000	−	9.00000i	0.330178	−	0.330178i	0.852654	+	0.522476i	$0.174992\pi$
$744$	−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	$0.276401\pi$
$757$			42.0000i			1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	−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	$-0.845127\pi$
$773$		−	26.0000i		−	0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$			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
$784$		−	11.0000i		−	0.392857i

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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