LMFDB - Embedded newform 925.2.f.b.882.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [925,2,Mod(43,925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(925, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3, 3]))

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

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

chi := DirichletCharacter("925.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.f (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:	$7.38616218697$
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 185)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			882.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	925.882
Dual form			925.2.f.b.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} +1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} +(3.00000 + 3.00000i) q^{7} +3.00000i q^{8} -1.00000i q^{9} +2.00000i q^{11} +(1.00000 + 1.00000i) q^{12} +2.00000i q^{13} +(-3.00000 + 3.00000i) q^{14} -1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +(-3.00000 - 3.00000i) q^{19} +6.00000i q^{21} -2.00000 q^{22} -8.00000i q^{23} +(-3.00000 + 3.00000i) q^{24} -2.00000 q^{26} +(4.00000 - 4.00000i) q^{27} +(3.00000 + 3.00000i) q^{28} +(7.00000 - 7.00000i) q^{29} +(3.00000 + 3.00000i) q^{31} +5.00000i q^{32} +(-2.00000 + 2.00000i) q^{33} -4.00000i q^{34} -1.00000i q^{36} +(-6.00000 + 1.00000i) q^{37} +(3.00000 - 3.00000i) q^{38} +(-2.00000 + 2.00000i) q^{39} -6.00000 q^{42} +12.0000i q^{43} +2.00000i q^{44} +8.00000 q^{46} +(-5.00000 - 5.00000i) q^{47} +(-1.00000 - 1.00000i) q^{48} +11.0000i q^{49} +(-4.00000 - 4.00000i) q^{51} +2.00000i q^{52} +(3.00000 - 3.00000i) q^{53} +(4.00000 + 4.00000i) q^{54} +(-9.00000 + 9.00000i) q^{56} -6.00000i q^{57} +(7.00000 + 7.00000i) q^{58} +(-7.00000 - 7.00000i) q^{59} +(1.00000 + 1.00000i) q^{61} +(-3.00000 + 3.00000i) q^{62} +(3.00000 - 3.00000i) q^{63} -7.00000 q^{64} +(-2.00000 - 2.00000i) q^{66} +(3.00000 - 3.00000i) q^{67} -4.00000 q^{68} +(8.00000 - 8.00000i) q^{69} +8.00000 q^{71} +3.00000 q^{72} +(1.00000 + 1.00000i) q^{73} +(-1.00000 - 6.00000i) q^{74} +(-3.00000 - 3.00000i) q^{76} +(-6.00000 + 6.00000i) q^{77} +(-2.00000 - 2.00000i) q^{78} +(-3.00000 - 3.00000i) q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +6.00000i q^{84} -12.0000 q^{86} +14.0000 q^{87} -6.00000 q^{88} +(-5.00000 + 5.00000i) q^{89} +(-6.00000 + 6.00000i) q^{91} -8.00000i q^{92} +6.00000i q^{93} +(5.00000 - 5.00000i) q^{94} +(-5.00000 + 5.00000i) q^{96} +8.00000 q^{97} -11.0000 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{12} - 6 q^{14} - 2 q^{16} - 8 q^{17} + 2 q^{18} - 6 q^{19} - 4 q^{22} - 6 q^{24} - 4 q^{26} + 8 q^{27} + 6 q^{28} + 14 q^{29} + 6 q^{31} - 4 q^{33} - 12 q^{37}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 + 2 * q^12 - 6 * q^14 - 2 * q^16 - 8 * q^17 + 2 * q^18 - 6 * q^19 - 4 * q^22 - 6 * q^24 - 4 * q^26 + 8 * q^27 + 6 * q^28 + 14 * q^29 + 6 * q^31 - 4 * q^33 - 12 * q^37 + 6 * q^38 - 4 * q^39 - 12 * q^42 + 16 * q^46 - 10 * q^47 - 2 * q^48 - 8 * q^51 + 6 * q^53 + 8 * q^54 - 18 * q^56 + 14 * q^58 - 14 * q^59 + 2 * q^61 - 6 * q^62 + 6 * q^63 - 14 * q^64 - 4 * q^66 + 6 * q^67 - 8 * q^68 + 16 * q^69 + 16 * q^71 + 6 * q^72 + 2 * q^73 - 2 * q^74 - 6 * q^76 - 12 * q^77 - 4 * q^78 - 6 * q^79 + 10 * q^81 + 10 * q^83 - 24 * q^86 + 28 * q^87 - 12 * q^88 - 10 * q^89 - 12 * q^91 + 10 * q^94 - 10 * q^96 + 16 * q^97 - 22 * q^98 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/925\mathbb{Z}\right)^\times$.

$n$	$76$	$852$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{1}{4}\right)$

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	0.934172	−	0.356822i	$-0.116140\pi$
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	−0.356822	+	0.934172i	$0.616140\pi$
$4$	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
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	1.00000	+	1.00000i	0.288675	+	0.288675i
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	−3.00000	+	3.00000i	−0.801784	+	0.801784i
$15$		0			0
$16$	−1.00000			−0.250000
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000			0.235702
$19$	−3.00000	−	3.00000i	−0.688247	−	0.688247i	0.273597	−	0.961844i	$-0.411786\pi$
$19$	−3.00000	−	3.00000i	−0.688247	−	0.688247i	−0.961844	+	0.273597i	$0.911786\pi$
$20$		0			0
$21$			6.00000i			1.30931i
$22$	−2.00000			−0.426401
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	−3.00000	+	3.00000i	−0.612372	+	0.612372i
$25$		0			0
$26$	−2.00000			−0.392232
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$	3.00000	+	3.00000i	0.566947	+	0.566947i
$29$	7.00000	−	7.00000i	1.29987	−	1.29987i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	7.00000	−	7.00000i	1.29987	−	1.29987i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$	3.00000	+	3.00000i	0.538816	+	0.538816i	0.923181	−	0.384365i	$-0.125580\pi$
$31$	3.00000	+	3.00000i	0.538816	+	0.538816i	−0.384365	+	0.923181i	$0.625580\pi$
$32$			5.00000i			0.883883i
$33$	−2.00000	+	2.00000i	−0.348155	+	0.348155i
$34$		−	4.00000i		−	0.685994i
$35$		0			0
$36$		−	1.00000i		−	0.166667i
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$37$	−6.00000	+	1.00000i	−0.986394	+	0.164399i
$38$	3.00000	−	3.00000i	0.486664	−	0.486664i
$39$	−2.00000	+	2.00000i	−0.320256	+	0.320256i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−6.00000			−0.925820
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$			2.00000i			0.301511i
$45$		0			0
$46$	8.00000			1.17954
$47$	−5.00000	−	5.00000i	−0.729325	−	0.729325i	0.241160	−	0.970485i	$-0.422472\pi$
$47$	−5.00000	−	5.00000i	−0.729325	−	0.729325i	−0.970485	+	0.241160i	$0.922472\pi$
$48$	−1.00000	−	1.00000i	−0.144338	−	0.144338i
$49$			11.0000i			1.57143i
$50$		0			0
$51$	−4.00000	−	4.00000i	−0.560112	−	0.560112i
$52$			2.00000i			0.277350i
$53$	3.00000	−	3.00000i	0.412082	−	0.412082i	−0.470381	−	0.882463i	$-0.655884\pi$
$53$	3.00000	−	3.00000i	0.412082	−	0.412082i	0.882463	+	0.470381i	$0.155884\pi$
$54$	4.00000	+	4.00000i	0.544331	+	0.544331i
$55$		0			0
$56$	−9.00000	+	9.00000i	−1.20268	+	1.20268i
$57$		−	6.00000i		−	0.794719i
$58$	7.00000	+	7.00000i	0.919145	+	0.919145i
$59$	−7.00000	−	7.00000i	−0.911322	−	0.911322i	0.0850540	−	0.996376i	$-0.472894\pi$
$59$	−7.00000	−	7.00000i	−0.911322	−	0.911322i	−0.996376	+	0.0850540i	$0.972894\pi$
$60$		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	$0.721142\pi$
$62$	−3.00000	+	3.00000i	−0.381000	+	0.381000i
$63$	3.00000	−	3.00000i	0.377964	−	0.377964i
$64$	−7.00000			−0.875000
$65$		0			0
$66$	−2.00000	−	2.00000i	−0.246183	−	0.246183i
$67$	3.00000	−	3.00000i	0.366508	−	0.366508i	−0.499694	−	0.866202i	$-0.666554\pi$
$67$	3.00000	−	3.00000i	0.366508	−	0.366508i	0.866202	+	0.499694i	$0.166554\pi$
$68$	−4.00000			−0.485071
$69$	8.00000	−	8.00000i	0.963087	−	0.963087i
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$	3.00000			0.353553
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	0.763202	−	0.646160i	$-0.223626\pi$
$73$	1.00000	+	1.00000i	0.117041	+	0.117041i	−0.646160	+	0.763202i	$0.723626\pi$
$74$	−1.00000	−	6.00000i	−0.116248	−	0.697486i
$75$		0			0
$76$	−3.00000	−	3.00000i	−0.344124	−	0.344124i
$77$	−6.00000	+	6.00000i	−0.683763	+	0.683763i
$78$	−2.00000	−	2.00000i	−0.226455	−	0.226455i
$79$	−3.00000	−	3.00000i	−0.337526	−	0.337526i	0.517909	−	0.855436i	$-0.326710\pi$
$79$	−3.00000	−	3.00000i	−0.337526	−	0.337526i	−0.855436	+	0.517909i	$0.826710\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	−0.377279	−	0.926100i	$-0.623140\pi$
$83$	5.00000	−	5.00000i	0.548821	−	0.548821i	0.926100	+	0.377279i	$0.123140\pi$
$84$			6.00000i			0.654654i
$85$		0			0
$86$	−12.0000			−1.29399
$87$	14.0000			1.50096
$88$	−6.00000			−0.639602
$89$	−5.00000	+	5.00000i	−0.529999	+	0.529999i	−0.920572	−	0.390573i	$-0.872277\pi$
$89$	−5.00000	+	5.00000i	−0.529999	+	0.529999i	0.390573	+	0.920572i	$0.372277\pi$
$90$		0			0
$91$	−6.00000	+	6.00000i	−0.628971	+	0.628971i
$92$		−	8.00000i		−	0.834058i
$93$			6.00000i			0.622171i
$94$	5.00000	−	5.00000i	0.515711	−	0.515711i
$95$		0			0
$96$	−5.00000	+	5.00000i	−0.510310	+	0.510310i
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$	−11.0000			−1.11117
$99$	2.00000			0.201008
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	4.00000	−	4.00000i	0.396059	−	0.396059i
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$	3.00000	+	3.00000i	0.291386	+	0.291386i
$107$	3.00000	+	3.00000i	0.290021	+	0.290021i	0.837088	−	0.547068i	$-0.184256\pi$
$107$	3.00000	+	3.00000i	0.290021	+	0.290021i	−0.547068	+	0.837088i	$0.684256\pi$
$108$	4.00000	−	4.00000i	0.384900	−	0.384900i
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	0.836031	−	0.548683i	$-0.184871\pi$
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	−0.548683	+	0.836031i	$0.684871\pi$
$110$		0			0
$111$	−7.00000	−	5.00000i	−0.664411	−	0.474579i
$112$	−3.00000	−	3.00000i	−0.283473	−	0.283473i
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$	6.00000			0.561951
$115$		0			0
$116$	7.00000	−	7.00000i	0.649934	−	0.649934i
$117$	2.00000			0.184900
$118$	7.00000	−	7.00000i	0.644402	−	0.644402i
$119$	−12.0000	−	12.0000i	−1.10004	−	1.10004i
$120$		0			0
$121$	7.00000			0.636364
$122$	−1.00000	+	1.00000i	−0.0905357	+	0.0905357i
$123$		0			0
$124$	3.00000	+	3.00000i	0.269408	+	0.269408i
$125$		0			0
$126$	3.00000	+	3.00000i	0.267261	+	0.267261i
$127$	−9.00000	−	9.00000i	−0.798621	−	0.798621i	0.184257	−	0.982878i	$-0.441012\pi$
$127$	−9.00000	−	9.00000i	−0.798621	−	0.798621i	−0.982878	+	0.184257i	$0.941012\pi$
$128$			3.00000i			0.265165i
$129$	−12.0000	+	12.0000i	−1.05654	+	1.05654i
$130$		0			0
$131$	−5.00000	−	5.00000i	−0.436852	−	0.436852i	0.454099	−	0.890951i	$-0.349961\pi$
$131$	−5.00000	−	5.00000i	−0.436852	−	0.436852i	−0.890951	+	0.454099i	$0.849961\pi$
$132$	−2.00000	+	2.00000i	−0.174078	+	0.174078i
$133$		−	18.0000i		−	1.56080i
$134$	3.00000	+	3.00000i	0.259161	+	0.259161i
$135$		0			0
$136$		−	12.0000i		−	1.02899i
$137$	7.00000	+	7.00000i	0.598050	+	0.598050i	0.939793	−	0.341743i	$-0.111017\pi$
$137$	7.00000	+	7.00000i	0.598050	+	0.598050i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	8.00000	+	8.00000i	0.681005	+	0.681005i
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		−	10.0000i		−	0.842152i
$142$			8.00000i			0.671345i
$143$	−4.00000			−0.334497
$144$			1.00000i			0.0833333i
$145$		0			0
$146$	−1.00000	+	1.00000i	−0.0827606	+	0.0827606i
$147$	−11.0000	+	11.0000i	−0.907265	+	0.907265i
$148$	−6.00000	+	1.00000i	−0.493197	+	0.0821995i
$149$			16.0000i			1.31077i	0.755295	+	0.655386i	$0.227494\pi$
$149$			16.0000i			1.31077i	−0.755295	+	0.655386i	$0.772506\pi$
$150$		0			0
$151$		−	2.00000i		−	0.162758i	−0.996683	−	0.0813788i	$-0.974068\pi$
$151$		−	2.00000i		−	0.162758i	0.996683	−	0.0813788i	$-0.0259324\pi$
$152$	9.00000	−	9.00000i	0.729996	−	0.729996i
$153$			4.00000i			0.323381i
$154$	−6.00000	−	6.00000i	−0.483494	−	0.483494i
$155$		0			0
$156$	−2.00000	+	2.00000i	−0.160128	+	0.160128i
$157$	3.00000	+	3.00000i	0.239426	+	0.239426i	0.816612	−	0.577186i	$-0.195849\pi$
$157$	3.00000	+	3.00000i	0.239426	+	0.239426i	−0.577186	+	0.816612i	$0.695849\pi$
$158$	3.00000	−	3.00000i	0.238667	−	0.238667i
$159$	6.00000			0.475831
$160$		0			0
$161$	24.0000	−	24.0000i	1.89146	−	1.89146i
$162$			5.00000i			0.392837i
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$		0			0
$165$		0			0
$166$	5.00000	+	5.00000i	0.388075	+	0.388075i
$167$	6.00000			0.464294			0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000			0.464294			0.232147	+	0.972681i	$0.425425\pi$
$168$	−18.0000			−1.38873
$169$	9.00000			0.692308
$170$		0			0
$171$	−3.00000	+	3.00000i	−0.229416	+	0.229416i
$172$			12.0000i			0.914991i
$173$	1.00000	+	1.00000i	0.0760286	+	0.0760286i	0.744099	−	0.668070i	$-0.232879\pi$
$173$	1.00000	+	1.00000i	0.0760286	+	0.0760286i	−0.668070	+	0.744099i	$0.732879\pi$
$174$			14.0000i			1.06134i
$175$		0			0
$176$		−	2.00000i		−	0.150756i
$177$		−	14.0000i		−	1.05230i
$178$	−5.00000	−	5.00000i	−0.374766	−	0.374766i
$179$	7.00000	−	7.00000i	0.523205	−	0.523205i	−0.395333	−	0.918538i	$-0.629371\pi$
$179$	7.00000	−	7.00000i	0.523205	−	0.523205i	0.918538	+	0.395333i	$0.129371\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$	−6.00000	−	6.00000i	−0.444750	−	0.444750i
$183$			2.00000i			0.147844i
$184$	24.0000			1.76930
$185$		0			0
$186$	−6.00000			−0.439941
$187$		−	8.00000i		−	0.585018i
$188$	−5.00000	−	5.00000i	−0.364662	−	0.364662i
$189$	24.0000			1.74574
$190$		0			0
$191$	−11.0000	+	11.0000i	−0.795932	+	0.795932i	−0.982451	−	0.186519i	$-0.940279\pi$
$191$	−11.0000	+	11.0000i	−0.795932	+	0.795932i	0.186519	+	0.982451i	$0.440279\pi$
$192$	−7.00000	−	7.00000i	−0.505181	−	0.505181i
$193$			2.00000i			0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$			2.00000i			0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$			8.00000i			0.574367i
$195$		0			0
$196$			11.0000i			0.785714i
$197$	−1.00000	−	1.00000i	−0.0712470	−	0.0712470i	0.670585	−	0.741832i	$-0.266043\pi$
$197$	−1.00000	−	1.00000i	−0.0712470	−	0.0712470i	−0.741832	+	0.670585i	$0.766043\pi$
$198$			2.00000i			0.142134i
$199$	15.0000	−	15.0000i	1.06332	−	1.06332i	0.0654671	−	0.997855i	$-0.479146\pi$
$199$	15.0000	−	15.0000i	1.06332	−	1.06332i	0.997855	−	0.0654671i	$-0.0208537\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$	42.0000			2.94782
$204$	−4.00000	−	4.00000i	−0.280056	−	0.280056i
$205$		0			0
$206$		−	14.0000i		−	0.975426i
$207$	−8.00000			−0.556038
$208$		−	2.00000i		−	0.138675i
$209$	6.00000	−	6.00000i	0.415029	−	0.415029i
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$	3.00000	−	3.00000i	0.206041	−	0.206041i
$213$	8.00000	+	8.00000i	0.548151	+	0.548151i
$214$	−3.00000	+	3.00000i	−0.205076	+	0.205076i
$215$		0			0
$216$	12.0000	+	12.0000i	0.816497	+	0.816497i
$217$			18.0000i			1.22192i
$218$	−3.00000	+	3.00000i	−0.203186	+	0.203186i
$219$			2.00000i			0.135147i
$220$		0			0
$221$		−	8.00000i		−	0.538138i
$222$	5.00000	−	7.00000i	0.335578	−	0.469809i
$223$	1.00000	−	1.00000i	0.0669650	−	0.0669650i	−0.672831	−	0.739796i	$-0.734922\pi$
$223$	1.00000	−	1.00000i	0.0669650	−	0.0669650i	0.739796	+	0.672831i	$0.234922\pi$
$224$	−15.0000	+	15.0000i	−1.00223	+	1.00223i
$225$		0			0
$226$		−	4.00000i		−	0.266076i
$227$	2.00000			0.132745			0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000			0.132745			0.0663723	+	0.997795i	$0.478857\pi$
$228$		−	6.00000i		−	0.397360i
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$	−12.0000			−0.789542
$232$	21.0000	+	21.0000i	1.37872	+	1.37872i
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	0.739104	−	0.673592i	$-0.235249\pi$
$233$	1.00000	+	1.00000i	0.0655122	+	0.0655122i	−0.673592	+	0.739104i	$0.735249\pi$
$234$			2.00000i			0.130744i
$235$		0			0
$236$	−7.00000	−	7.00000i	−0.455661	−	0.455661i
$237$		−	6.00000i		−	0.389742i
$238$	12.0000	−	12.0000i	0.777844	−	0.777844i
$239$	−7.00000	−	7.00000i	−0.452792	−	0.452792i	0.443488	−	0.896280i	$-0.353741\pi$
$239$	−7.00000	−	7.00000i	−0.452792	−	0.452792i	−0.896280	+	0.443488i	$0.853741\pi$
$240$		0			0
$241$	1.00000	−	1.00000i	0.0644157	−	0.0644157i	−0.674165	−	0.738581i	$-0.735496\pi$
$241$	1.00000	−	1.00000i	0.0644157	−	0.0644157i	0.738581	+	0.674165i	$0.235496\pi$
$242$			7.00000i			0.449977i
$243$	−7.00000	−	7.00000i	−0.449050	−	0.449050i
$244$	1.00000	+	1.00000i	0.0640184	+	0.0640184i
$245$		0			0
$246$		0			0
$247$	6.00000	−	6.00000i	0.381771	−	0.381771i
$248$	−9.00000	+	9.00000i	−0.571501	+	0.571501i
$249$	10.0000			0.633724
$250$		0			0
$251$	−5.00000	−	5.00000i	−0.315597	−	0.315597i	0.531476	−	0.847073i	$-0.321638\pi$
$251$	−5.00000	−	5.00000i	−0.315597	−	0.315597i	−0.847073	+	0.531476i	$0.821638\pi$
$252$	3.00000	−	3.00000i	0.188982	−	0.188982i
$253$	16.0000			1.00591
$254$	9.00000	−	9.00000i	0.564710	−	0.564710i
$255$		0			0
$256$	−17.0000			−1.06250
$257$	−28.0000			−1.74659			−0.873296	−	0.487190i	$-0.838022\pi$
$257$	−28.0000			−1.74659			−0.873296	+	0.487190i	$0.838022\pi$
$258$	−12.0000	−	12.0000i	−0.747087	−	0.747087i
$259$	−21.0000	−	15.0000i	−1.30488	−	0.932055i
$260$		0			0
$261$	−7.00000	−	7.00000i	−0.433289	−	0.433289i
$262$	5.00000	−	5.00000i	0.308901	−	0.308901i
$263$	−7.00000	−	7.00000i	−0.431638	−	0.431638i	0.457547	−	0.889185i	$-0.348728\pi$
$263$	−7.00000	−	7.00000i	−0.431638	−	0.431638i	−0.889185	+	0.457547i	$0.848728\pi$
$264$	−6.00000	−	6.00000i	−0.369274	−	0.369274i
$265$		0			0
$266$	18.0000			1.10365
$267$	−10.0000			−0.611990
$268$	3.00000	−	3.00000i	0.183254	−	0.183254i
$269$		−	16.0000i		−	0.975537i	−0.872973	−	0.487769i	$-0.837811\pi$
$269$		−	16.0000i		−	0.975537i	0.872973	−	0.487769i	$-0.162189\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	4.00000			0.242536
$273$	−12.0000			−0.726273
$274$	−7.00000	+	7.00000i	−0.422885	+	0.422885i
$275$		0			0
$276$	8.00000	−	8.00000i	0.481543	−	0.481543i
$277$		−	26.0000i		−	1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$		−	26.0000i		−	1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$		−	4.00000i		−	0.239904i
$279$	3.00000	−	3.00000i	0.179605	−	0.179605i
$280$		0			0
$281$	9.00000	−	9.00000i	0.536895	−	0.536895i	−0.385721	−	0.922616i	$-0.626047\pi$
$281$	9.00000	−	9.00000i	0.536895	−	0.536895i	0.922616	+	0.385721i	$0.126047\pi$
$282$	10.0000			0.595491
$283$	22.0000			1.30776			0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000			1.30776			0.653882	+	0.756596i	$0.273139\pi$
$284$	8.00000			0.474713
$285$		0			0
$286$		−	4.00000i		−	0.236525i
$287$		0			0
$288$	5.00000			0.294628
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	8.00000	+	8.00000i	0.468968	+	0.468968i
$292$	1.00000	+	1.00000i	0.0585206	+	0.0585206i
$293$	−1.00000	+	1.00000i	−0.0584206	+	0.0584206i	−0.735714	−	0.677293i	$-0.763153\pi$
$293$	−1.00000	+	1.00000i	−0.0584206	+	0.0584206i	0.677293	+	0.735714i	$0.263153\pi$
$294$	−11.0000	−	11.0000i	−0.641533	−	0.641533i
$295$		0			0
$296$	−3.00000	−	18.0000i	−0.174371	−	1.04623i
$297$	8.00000	+	8.00000i	0.464207	+	0.464207i
$298$	−16.0000			−0.926855
$299$	16.0000			0.925304
$300$		0			0
$301$	−36.0000	+	36.0000i	−2.07501	+	2.07501i
$302$	2.00000			0.115087
$303$		0			0
$304$	3.00000	+	3.00000i	0.172062	+	0.172062i
$305$		0			0
$306$	−4.00000			−0.228665
$307$	−9.00000	+	9.00000i	−0.513657	+	0.513657i	−0.915645	−	0.401988i	$-0.868319\pi$
$307$	−9.00000	+	9.00000i	−0.513657	+	0.513657i	0.401988	+	0.915645i	$0.368319\pi$
$308$	−6.00000	+	6.00000i	−0.341882	+	0.341882i
$309$	−14.0000	−	14.0000i	−0.796432	−	0.796432i
$310$		0			0
$311$	3.00000	+	3.00000i	0.170114	+	0.170114i	0.787030	−	0.616915i	$-0.211618\pi$
$311$	3.00000	+	3.00000i	0.170114	+	0.170114i	−0.616915	+	0.787030i	$0.711618\pi$
$312$	−6.00000	−	6.00000i	−0.339683	−	0.339683i
$313$		−	30.0000i		−	1.69570i	−0.530236	−	0.847850i	$-0.677897\pi$
$313$		−	30.0000i		−	1.69570i	0.530236	−	0.847850i	$-0.322103\pi$
$314$	−3.00000	+	3.00000i	−0.169300	+	0.169300i
$315$		0			0
$316$	−3.00000	−	3.00000i	−0.168763	−	0.168763i
$317$	17.0000	−	17.0000i	0.954815	−	0.954815i	−0.0442073	−	0.999022i	$-0.514076\pi$
$317$	17.0000	−	17.0000i	0.954815	−	0.954815i	0.999022	+	0.0442073i	$0.0140762\pi$
$318$			6.00000i			0.336463i
$319$	14.0000	+	14.0000i	0.783850	+	0.783850i
$320$		0			0
$321$			6.00000i			0.334887i
$322$	24.0000	+	24.0000i	1.33747	+	1.33747i
$323$	12.0000	+	12.0000i	0.667698	+	0.667698i
$324$	5.00000			0.277778
$325$		0			0
$326$		−	10.0000i		−	0.553849i
$327$			6.00000i			0.331801i
$328$		0			0
$329$		−	30.0000i		−	1.65395i
$330$		0			0
$331$	−7.00000	+	7.00000i	−0.384755	+	0.384755i	−0.872812	−	0.488057i	$-0.837706\pi$
$331$	−7.00000	+	7.00000i	−0.384755	+	0.384755i	0.488057	+	0.872812i	$0.337706\pi$
$332$	5.00000	−	5.00000i	0.274411	−	0.274411i
$333$	1.00000	+	6.00000i	0.0547997	+	0.328798i
$334$			6.00000i			0.328305i
$335$		0			0
$336$		−	6.00000i		−	0.327327i
$337$	−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	$0.446080\pi$
$338$			9.00000i			0.489535i
$339$	−4.00000	−	4.00000i	−0.217250	−	0.217250i
$340$		0			0
$341$	−6.00000	+	6.00000i	−0.324918	+	0.324918i
$342$	−3.00000	−	3.00000i	−0.162221	−	0.162221i
$343$	−12.0000	+	12.0000i	−0.647939	+	0.647939i
$344$	−36.0000			−1.94099
$345$		0			0
$346$	−1.00000	+	1.00000i	−0.0537603	+	0.0537603i
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	14.0000			0.750479
$349$			32.0000i			1.71292i	0.516213	+	0.856460i	$0.327341\pi$
$349$			32.0000i			1.71292i	−0.516213	+	0.856460i	$0.672659\pi$
$350$		0			0
$351$	8.00000	+	8.00000i	0.427008	+	0.427008i
$352$	−10.0000			−0.533002
$353$	−32.0000			−1.70319			−0.851594	−	0.524202i	$-0.824364\pi$
$353$	−32.0000			−1.70319			−0.851594	+	0.524202i	$0.824364\pi$
$354$	14.0000			0.744092
$355$		0			0
$356$	−5.00000	+	5.00000i	−0.264999	+	0.264999i
$357$		−	24.0000i		−	1.27021i
$358$	7.00000	+	7.00000i	0.369961	+	0.369961i
$359$			34.0000i			1.79445i	0.441572	+	0.897226i	$0.354421\pi$
$359$			34.0000i			1.79445i	−0.441572	+	0.897226i	$0.645579\pi$
$360$		0			0
$361$		−	1.00000i		−	0.0526316i
$362$			10.0000i			0.525588i
$363$	7.00000	+	7.00000i	0.367405	+	0.367405i
$364$	−6.00000	+	6.00000i	−0.314485	+	0.314485i
$365$		0			0
$366$	−2.00000			−0.104542
$367$	−17.0000	−	17.0000i	−0.887393	−	0.887393i	0.106879	−	0.994272i	$-0.465914\pi$
$367$	−17.0000	−	17.0000i	−0.887393	−	0.887393i	−0.994272	+	0.106879i	$0.965914\pi$
$368$			8.00000i			0.417029i
$369$		0			0
$370$		0			0
$371$	18.0000			0.934513
$372$			6.00000i			0.311086i
$373$	−3.00000	−	3.00000i	−0.155334	−	0.155334i	0.625161	−	0.780496i	$-0.285033\pi$
$373$	−3.00000	−	3.00000i	−0.155334	−	0.155334i	−0.780496	+	0.625161i	$0.785033\pi$
$374$	8.00000			0.413670
$375$		0			0
$376$	15.0000	−	15.0000i	0.773566	−	0.773566i
$377$	14.0000	+	14.0000i	0.721037	+	0.721037i
$378$			24.0000i			1.23443i
$379$			14.0000i			0.719132i	0.933120	+	0.359566i	$0.117075\pi$
$379$			14.0000i			0.719132i	−0.933120	+	0.359566i	$0.882925\pi$
$380$		0			0
$381$		−	18.0000i		−	0.922168i
$382$	−11.0000	−	11.0000i	−0.562809	−	0.562809i
$383$			8.00000i			0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$			8.00000i			0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	−3.00000	+	3.00000i	−0.153093	+	0.153093i
$385$		0			0
$386$	−2.00000			−0.101797
$387$	12.0000			0.609994
$388$	8.00000			0.406138
$389$	−5.00000	−	5.00000i	−0.253510	−	0.253510i	0.568898	−	0.822408i	$-0.307370\pi$
$389$	−5.00000	−	5.00000i	−0.253510	−	0.253510i	−0.822408	+	0.568898i	$0.807370\pi$
$390$		0			0
$391$			32.0000i			1.61831i
$392$	−33.0000			−1.66675
$393$		−	10.0000i		−	0.504433i
$394$	1.00000	−	1.00000i	0.0503793	−	0.0503793i
$395$		0			0
$396$	2.00000			0.100504
$397$	5.00000	−	5.00000i	0.250943	−	0.250943i	−0.570414	−	0.821357i	$-0.693217\pi$
$397$	5.00000	−	5.00000i	0.250943	−	0.250943i	0.821357	+	0.570414i	$0.193217\pi$
$398$	15.0000	+	15.0000i	0.751882	+	0.751882i
$399$	18.0000	−	18.0000i	0.901127	−	0.901127i
$400$		0			0
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	0.731635	−	0.681697i	$-0.238758\pi$
$401$	1.00000	+	1.00000i	0.0499376	+	0.0499376i	−0.681697	+	0.731635i	$0.738758\pi$
$402$			6.00000i			0.299253i
$403$	−6.00000	+	6.00000i	−0.298881	+	0.298881i
$404$		0			0
$405$		0			0
$406$			42.0000i			2.08443i
$407$	−2.00000	−	12.0000i	−0.0991363	−	0.594818i
$408$	12.0000	−	12.0000i	0.594089	−	0.594089i
$409$	3.00000	−	3.00000i	0.148340	−	0.148340i	−0.629036	−	0.777376i	$-0.716550\pi$
$409$	3.00000	−	3.00000i	0.148340	−	0.148340i	0.777376	+	0.629036i	$0.216550\pi$
$410$		0			0
$411$			14.0000i			0.690569i
$412$	−14.0000			−0.689730
$413$		−	42.0000i		−	2.06668i
$414$		−	8.00000i		−	0.393179i
$415$		0			0
$416$	−10.0000			−0.490290
$417$	−4.00000	−	4.00000i	−0.195881	−	0.195881i
$418$	6.00000	+	6.00000i	0.293470	+	0.293470i
$419$			22.0000i			1.07477i	0.843337	+	0.537385i	$0.180588\pi$
$419$			22.0000i			1.07477i	−0.843337	+	0.537385i	$0.819412\pi$
$420$		0			0
$421$	−27.0000	−	27.0000i	−1.31590	−	1.31590i	−0.916991	−	0.398909i	$-0.869389\pi$
$421$	−27.0000	−	27.0000i	−1.31590	−	1.31590i	−0.398909	−	0.916991i	$-0.630611\pi$
$422$			20.0000i			0.973585i
$423$	−5.00000	+	5.00000i	−0.243108	+	0.243108i
$424$	9.00000	+	9.00000i	0.437079	+	0.437079i
$425$		0			0
$426$	−8.00000	+	8.00000i	−0.387601	+	0.387601i
$427$			6.00000i			0.290360i
$428$	3.00000	+	3.00000i	0.145010	+	0.145010i
$429$	−4.00000	−	4.00000i	−0.193122	−	0.193122i
$430$		0			0
$431$	11.0000	+	11.0000i	0.529851	+	0.529851i	0.920528	−	0.390677i	$-0.127759\pi$
$431$	11.0000	+	11.0000i	0.529851	+	0.529851i	−0.390677	+	0.920528i	$0.627759\pi$
$432$	−4.00000	+	4.00000i	−0.192450	+	0.192450i
$433$	−5.00000	+	5.00000i	−0.240285	+	0.240285i	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−5.00000	+	5.00000i	−0.240285	+	0.240285i	0.576683	+	0.816968i	$0.304347\pi$
$434$	−18.0000			−0.864028
$435$		0			0
$436$	3.00000	+	3.00000i	0.143674	+	0.143674i
$437$	−24.0000	+	24.0000i	−1.14808	+	1.14808i
$438$	−2.00000			−0.0955637
$439$	−17.0000	+	17.0000i	−0.811366	+	0.811366i	−0.984839	−	0.173473i	$-0.944501\pi$
$439$	−17.0000	+	17.0000i	−0.811366	+	0.811366i	0.173473	+	0.984839i	$0.444501\pi$
$440$		0			0
$441$	11.0000			0.523810
$442$	8.00000			0.380521
$443$	−11.0000	−	11.0000i	−0.522626	−	0.522626i	0.395738	−	0.918364i	$-0.370489\pi$
$443$	−11.0000	−	11.0000i	−0.522626	−	0.522626i	−0.918364	+	0.395738i	$0.870489\pi$
$444$	−7.00000	−	5.00000i	−0.332205	−	0.237289i
$445$		0			0
$446$	1.00000	+	1.00000i	0.0473514	+	0.0473514i
$447$	−16.0000	+	16.0000i	−0.756774	+	0.756774i
$448$	−21.0000	−	21.0000i	−0.992157	−	0.992157i
$449$	−5.00000	−	5.00000i	−0.235965	−	0.235965i	0.579212	−	0.815177i	$-0.303360\pi$
$449$	−5.00000	−	5.00000i	−0.235965	−	0.235965i	−0.815177	+	0.579212i	$0.803360\pi$
$450$		0			0
$451$		0			0
$452$	−4.00000			−0.188144
$453$	2.00000	−	2.00000i	0.0939682	−	0.0939682i
$454$			2.00000i			0.0938647i
$455$		0			0
$456$	18.0000			0.842927
$457$	32.0000			1.49690			0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000			1.49690			0.748448	+	0.663193i	$0.230799\pi$
$458$	−4.00000			−0.186908
$459$	−16.0000	+	16.0000i	−0.746816	+	0.746816i
$460$		0			0
$461$	21.0000	−	21.0000i	0.978068	−	0.978068i	−0.0216971	−	0.999765i	$-0.506907\pi$
$461$	21.0000	−	21.0000i	0.978068	−	0.978068i	0.999765	+	0.0216971i	$0.00690694\pi$
$462$		−	12.0000i		−	0.558291i
$463$			40.0000i			1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$			40.0000i			1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	−7.00000	+	7.00000i	−0.324967	+	0.324967i
$465$		0			0
$466$	−1.00000	+	1.00000i	−0.0463241	+	0.0463241i
$467$	−38.0000			−1.75843			−0.879215	−	0.476425i	$-0.841932\pi$
$467$	−38.0000			−1.75843			−0.879215	+	0.476425i	$0.841932\pi$
$468$	2.00000			0.0924500
$469$	18.0000			0.831163
$470$		0			0
$471$			6.00000i			0.276465i
$472$	21.0000	−	21.0000i	0.966603	−	0.966603i
$473$	−24.0000			−1.10352
$474$	6.00000			0.275589
$475$		0			0
$476$	−12.0000	−	12.0000i	−0.550019	−	0.550019i
$477$	−3.00000	−	3.00000i	−0.137361	−	0.137361i
$478$	7.00000	−	7.00000i	0.320173	−	0.320173i
$479$	−23.0000	−	23.0000i	−1.05090	−	1.05090i	−0.998633	−	0.0522635i	$-0.983356\pi$
$479$	−23.0000	−	23.0000i	−1.05090	−	1.05090i	−0.0522635	−	0.998633i	$-0.516644\pi$
$480$		0			0
$481$	−2.00000	−	12.0000i	−0.0911922	−	0.547153i
$482$	1.00000	+	1.00000i	0.0455488	+	0.0455488i
$483$	48.0000			2.18408
$484$	7.00000			0.318182
$485$		0			0
$486$	7.00000	−	7.00000i	0.317526	−	0.317526i
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−3.00000	+	3.00000i	−0.135804	+	0.135804i
$489$	−10.0000	−	10.0000i	−0.452216	−	0.452216i
$490$		0			0
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$		0			0
$493$	−28.0000	+	28.0000i	−1.26106	+	1.26106i
$494$	6.00000	+	6.00000i	0.269953	+	0.269953i
$495$		0			0
$496$	−3.00000	−	3.00000i	−0.134704	−	0.134704i
$497$	24.0000	+	24.0000i	1.07655	+	1.07655i
$498$			10.0000i			0.448111i
$499$	27.0000	−	27.0000i	1.20869	−	1.20869i	0.237233	−	0.971453i	$-0.423759\pi$
$499$	27.0000	−	27.0000i	1.20869	−	1.20869i	0.971453	−	0.237233i	$-0.0762406\pi$
$500$		0			0
$501$	6.00000	+	6.00000i	0.268060	+	0.268060i
$502$	5.00000	−	5.00000i	0.223161	−	0.223161i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	9.00000	+	9.00000i	0.400892	+	0.400892i
$505$		0			0
$506$			16.0000i			0.711287i
$507$	9.00000	+	9.00000i	0.399704	+	0.399704i
$508$	−9.00000	−	9.00000i	−0.399310	−	0.399310i
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$			6.00000i			0.265424i
$512$		−	11.0000i		−	0.486136i
$513$	−24.0000			−1.05963
$514$		−	28.0000i		−	1.23503i
$515$		0			0
$516$	−12.0000	+	12.0000i	−0.528271	+	0.528271i
$517$	10.0000	−	10.0000i	0.439799	−	0.439799i
$518$	15.0000	−	21.0000i	0.659062	−	0.922687i
$519$			2.00000i			0.0877903i
$520$		0			0
$521$			44.0000i			1.92767i	0.266491	+	0.963837i	$0.414136\pi$
$521$			44.0000i			1.92767i	−0.266491	+	0.963837i	$0.585864\pi$
$522$	7.00000	−	7.00000i	0.306382	−	0.306382i
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−5.00000	−	5.00000i	−0.218426	−	0.218426i
$525$		0			0
$526$	7.00000	−	7.00000i	0.305215	−	0.305215i
$527$	−12.0000	−	12.0000i	−0.522728	−	0.522728i
$528$	2.00000	−	2.00000i	0.0870388	−	0.0870388i
$529$	−41.0000			−1.78261
$530$		0			0
$531$	−7.00000	+	7.00000i	−0.303774	+	0.303774i
$532$		−	18.0000i		−	0.780399i
$533$		0			0
$534$		−	10.0000i		−	0.432742i
$535$		0			0
$536$	9.00000	+	9.00000i	0.388741	+	0.388741i
$537$	14.0000			0.604145
$538$	16.0000			0.689809
$539$	−22.0000			−0.947607
$540$		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	$0.240322\pi$
$542$		−	8.00000i		−	0.343629i
$543$	10.0000	+	10.0000i	0.429141	+	0.429141i
$544$		−	20.0000i		−	0.857493i
$545$		0			0
$546$		−	12.0000i		−	0.513553i
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	7.00000	+	7.00000i	0.299025	+	0.299025i
$549$	1.00000	−	1.00000i	0.0426790	−	0.0426790i
$550$		0			0
$551$	−42.0000			−1.78926
$552$	24.0000	+	24.0000i	1.02151	+	1.02151i
$553$		−	18.0000i		−	0.765438i
$554$	26.0000			1.10463
$555$		0			0
$556$	−4.00000			−0.169638
$557$			14.0000i			0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$			14.0000i			0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	3.00000	+	3.00000i	0.127000	+	0.127000i
$559$	−24.0000			−1.01509
$560$		0			0
$561$	8.00000	−	8.00000i	0.337760	−	0.337760i
$562$	9.00000	+	9.00000i	0.379642	+	0.379642i
$563$		−	12.0000i		−	0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$		−	12.0000i		−	0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$		−	10.0000i		−	0.421076i
$565$		0			0
$566$			22.0000i			0.924729i
$567$	15.0000	+	15.0000i	0.629941	+	0.629941i
$568$			24.0000i			1.00702i
$569$	−25.0000	+	25.0000i	−1.04805	+	1.04805i	−0.0492690	+	0.998786i	$0.515689\pi$
$569$	−25.0000	+	25.0000i	−1.04805	+	1.04805i	−0.998786	+	0.0492690i	$0.984311\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$	−4.00000			−0.167248
$573$	−22.0000			−0.919063
$574$		0			0
$575$		0			0
$576$			7.00000i			0.291667i
$577$	−12.0000			−0.499567			−0.249783	−	0.968302i	$-0.580359\pi$
$577$	−12.0000			−0.499567			−0.249783	+	0.968302i	$0.580359\pi$
$578$		−	1.00000i		−	0.0415945i
$579$	−2.00000	+	2.00000i	−0.0831172	+	0.0831172i
$580$		0			0
$581$	30.0000			1.24461
$582$	−8.00000	+	8.00000i	−0.331611	+	0.331611i
$583$	6.00000	+	6.00000i	0.248495	+	0.248495i
$584$	−3.00000	+	3.00000i	−0.124141	+	0.124141i
$585$		0			0
$586$	−1.00000	−	1.00000i	−0.0413096	−	0.0413096i
$587$			36.0000i			1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$			36.0000i			1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	−11.0000	+	11.0000i	−0.453632	+	0.453632i
$589$		−	18.0000i		−	0.741677i
$590$		0			0
$591$		−	2.00000i		−	0.0822690i
$592$	6.00000	−	1.00000i	0.246598	−	0.0410997i
$593$	3.00000	−	3.00000i	0.123195	−	0.123195i	−0.642821	−	0.766016i	$-0.722236\pi$
$593$	3.00000	−	3.00000i	0.123195	−	0.123195i	0.766016	+	0.642821i	$0.222236\pi$
$594$	−8.00000	+	8.00000i	−0.328244	+	0.328244i
$595$		0			0
$596$			16.0000i			0.655386i
$597$	30.0000			1.22782
$598$			16.0000i			0.654289i
$599$		−	6.00000i		−	0.245153i	−0.992459	−	0.122577i	$-0.960884\pi$
$599$		−	6.00000i		−	0.245153i	0.992459	−	0.122577i	$-0.0391157\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$	−36.0000	−	36.0000i	−1.46725	−	1.46725i
$603$	−3.00000	−	3.00000i	−0.122169	−	0.122169i
$604$		−	2.00000i		−	0.0813788i
$605$		0			0
$606$		0			0
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	15.0000	−	15.0000i	0.608330	−	0.608330i
$609$	42.0000	+	42.0000i	1.70193	+	1.70193i
$610$		0			0
$611$	10.0000	−	10.0000i	0.404557	−	0.404557i
$612$			4.00000i			0.161690i
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	0.989906	−	0.141724i	$-0.0452646\pi$
$613$	21.0000	+	21.0000i	0.848182	+	0.848182i	−0.141724	+	0.989906i	$0.545265\pi$
$614$	−9.00000	−	9.00000i	−0.363210	−	0.363210i
$615$		0			0
$616$	−18.0000	−	18.0000i	−0.725241	−	0.725241i
$617$	17.0000	−	17.0000i	0.684394	−	0.684394i	−0.276593	−	0.960987i	$-0.589205\pi$
$617$	17.0000	−	17.0000i	0.684394	−	0.684394i	0.960987	+	0.276593i	$0.0892054\pi$
$618$	14.0000	−	14.0000i	0.563163	−	0.563163i
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	−32.0000	−	32.0000i	−1.28412	−	1.28412i
$622$	−3.00000	+	3.00000i	−0.120289	+	0.120289i
$623$	−30.0000			−1.20192
$624$	2.00000	−	2.00000i	0.0800641	−	0.0800641i
$625$		0			0
$626$	30.0000			1.19904
$627$	12.0000			0.479234
$628$	3.00000	+	3.00000i	0.119713	+	0.119713i
$629$	24.0000	−	4.00000i	0.956943	−	0.159490i
$630$		0			0
$631$	23.0000	+	23.0000i	0.915616	+	0.915616i	0.996707	−	0.0810911i	$-0.0258405\pi$
$631$	23.0000	+	23.0000i	0.915616	+	0.915616i	−0.0810911	+	0.996707i	$0.525840\pi$
$632$	9.00000	−	9.00000i	0.358001	−	0.358001i
$633$	20.0000	+	20.0000i	0.794929	+	0.794929i
$634$	17.0000	+	17.0000i	0.675156	+	0.675156i
$635$		0			0
$636$	6.00000			0.237915
$637$	−22.0000			−0.871672
$638$	−14.0000	+	14.0000i	−0.554265	+	0.554265i
$639$		−	8.00000i		−	0.316475i
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$	−6.00000			−0.236801
$643$	−10.0000			−0.394362			−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000			−0.394362			−0.197181	+	0.980367i	$0.563179\pi$
$644$	24.0000	−	24.0000i	0.945732	−	0.945732i
$645$		0			0
$646$	−12.0000	+	12.0000i	−0.472134	+	0.472134i
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$			15.0000i			0.589256i
$649$	14.0000	−	14.0000i	0.549548	−	0.549548i
$650$		0			0
$651$	−18.0000	+	18.0000i	−0.705476	+	0.705476i
$652$	−10.0000			−0.391630
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$	−6.00000			−0.234619
$655$		0			0
$656$		0			0
$657$	1.00000	−	1.00000i	0.0390137	−	0.0390137i
$658$	30.0000			1.16952
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$	33.0000	+	33.0000i	1.28355	+	1.28355i	0.938633	+	0.344919i	$0.112094\pi$
$661$	33.0000	+	33.0000i	1.28355	+	1.28355i	0.344919	+	0.938633i	$0.387906\pi$
$662$	−7.00000	−	7.00000i	−0.272063	−	0.272063i
$663$	8.00000	−	8.00000i	0.310694	−	0.310694i
$664$	15.0000	+	15.0000i	0.582113	+	0.582113i
$665$		0			0
$666$	−6.00000	+	1.00000i	−0.232495	+	0.0387492i
$667$	−56.0000	−	56.0000i	−2.16833	−	2.16833i
$668$	6.00000			0.232147
$669$	2.00000			0.0773245
$670$		0			0
$671$	−2.00000	+	2.00000i	−0.0772091	+	0.0772091i
$672$	−30.0000			−1.15728
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	−0.356202	−	0.934409i	$-0.615928\pi$
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	0.934409	+	0.356202i	$0.115928\pi$
$674$	−15.0000	−	15.0000i	−0.577778	−	0.577778i
$675$		0			0
$676$	9.00000			0.346154
$677$	−23.0000	+	23.0000i	−0.883962	+	0.883962i	−0.993935	−	0.109973i	$-0.964924\pi$
$677$	−23.0000	+	23.0000i	−0.883962	+	0.883962i	0.109973	+	0.993935i	$0.464924\pi$
$678$	4.00000	−	4.00000i	0.153619	−	0.153619i
$679$	24.0000	+	24.0000i	0.921035	+	0.921035i
$680$		0			0
$681$	2.00000	+	2.00000i	0.0766402	+	0.0766402i
$682$	−6.00000	−	6.00000i	−0.229752	−	0.229752i
$683$			4.00000i			0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$			4.00000i			0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	−3.00000	+	3.00000i	−0.114708	+	0.114708i
$685$		0			0
$686$	−12.0000	−	12.0000i	−0.458162	−	0.458162i
$687$	−4.00000	+	4.00000i	−0.152610	+	0.152610i
$688$		−	12.0000i		−	0.457496i
$689$	6.00000	+	6.00000i	0.228582	+	0.228582i
$690$		0			0
$691$		−	46.0000i		−	1.74992i	−0.484193	−	0.874961i	$-0.660887\pi$
$691$		−	46.0000i		−	1.74992i	0.484193	−	0.874961i	$-0.339113\pi$
$692$	1.00000	+	1.00000i	0.0380143	+	0.0380143i
$693$	6.00000	+	6.00000i	0.227921	+	0.227921i
$694$	4.00000			0.151838
$695$		0			0
$696$			42.0000i			1.59201i
$697$		0			0
$698$	−32.0000			−1.21122
$699$			2.00000i			0.0756469i
$700$		0			0
$701$	9.00000	−	9.00000i	0.339925	−	0.339925i	−0.516414	−	0.856339i	$-0.672733\pi$
$701$	9.00000	−	9.00000i	0.339925	−	0.339925i	0.856339	+	0.516414i	$0.172733\pi$
$702$	−8.00000	+	8.00000i	−0.301941	+	0.301941i
$703$	21.0000	+	15.0000i	0.792030	+	0.565736i
$704$		−	14.0000i		−	0.527645i
$705$		0			0
$706$		−	32.0000i		−	1.20434i
$707$		0			0
$708$		−	14.0000i		−	0.526152i
$709$	15.0000	+	15.0000i	0.563337	+	0.563337i	0.930254	−	0.366917i	$-0.119587\pi$
$709$	15.0000	+	15.0000i	0.563337	+	0.563337i	−0.366917	+	0.930254i	$0.619587\pi$
$710$		0			0
$711$	−3.00000	+	3.00000i	−0.112509	+	0.112509i
$712$	−15.0000	−	15.0000i	−0.562149	−	0.562149i
$713$	24.0000	−	24.0000i	0.898807	−	0.898807i
$714$	24.0000			0.898177
$715$		0			0
$716$	7.00000	−	7.00000i	0.261602	−	0.261602i
$717$		−	14.0000i		−	0.522840i
$718$	−34.0000			−1.26887
$719$			26.0000i			0.969636i	0.874615	+	0.484818i	$0.161114\pi$
$719$			26.0000i			0.969636i	−0.874615	+	0.484818i	$0.838886\pi$
$720$		0			0
$721$	−42.0000	−	42.0000i	−1.56416	−	1.56416i
$722$	1.00000			0.0372161
$723$	2.00000			0.0743808
$724$	10.0000			0.371647
$725$		0			0
$726$	−7.00000	+	7.00000i	−0.259794	+	0.259794i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	−18.0000	−	18.0000i	−0.667124	−	0.667124i
$729$		−	29.0000i		−	1.07407i
$730$		0			0
$731$		−	48.0000i		−	1.77534i
$732$			2.00000i			0.0739221i
$733$	−3.00000	−	3.00000i	−0.110808	−	0.110808i	0.649529	−	0.760337i	$-0.274966\pi$
$733$	−3.00000	−	3.00000i	−0.110808	−	0.110808i	−0.760337	+	0.649529i	$0.774966\pi$
$734$	17.0000	−	17.0000i	0.627481	−	0.627481i
$735$		0			0
$736$	40.0000			1.47442
$737$	6.00000	+	6.00000i	0.221013	+	0.221013i
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$		0			0
$741$	12.0000			0.440831
$742$			18.0000i			0.660801i
$743$	5.00000	+	5.00000i	0.183432	+	0.183432i	0.792850	−	0.609417i	$-0.208597\pi$
$743$	5.00000	+	5.00000i	0.183432	+	0.183432i	−0.609417	+	0.792850i	$0.708597\pi$
$744$	−18.0000			−0.659912
$745$		0			0
$746$	3.00000	−	3.00000i	0.109838	−	0.109838i
$747$	−5.00000	−	5.00000i	−0.182940	−	0.182940i
$748$		−	8.00000i		−	0.292509i
$749$			18.0000i			0.657706i
$750$		0			0
$751$			22.0000i			0.802791i	0.915905	+	0.401396i	$0.131475\pi$
$751$			22.0000i			0.802791i	−0.915905	+	0.401396i	$0.868525\pi$
$752$	5.00000	+	5.00000i	0.182331	+	0.182331i
$753$		−	10.0000i		−	0.364420i
$754$	−14.0000	+	14.0000i	−0.509850	+	0.509850i
$755$		0			0
$756$	24.0000			0.872872
$757$	36.0000			1.30844			0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000			1.30844			0.654221	+	0.756303i	$0.272997\pi$
$758$	−14.0000			−0.508503
$759$	16.0000	+	16.0000i	0.580763	+	0.580763i
$760$		0			0
$761$		−	8.00000i		−	0.290000i	−0.989432	−	0.145000i	$-0.953682\pi$
$761$		−	8.00000i		−	0.290000i	0.989432	−	0.145000i	$-0.0463182\pi$
$762$	18.0000			0.652071
$763$			18.0000i			0.651644i
$764$	−11.0000	+	11.0000i	−0.397966	+	0.397966i
$765$		0			0
$766$	−8.00000			−0.289052
$767$	14.0000	−	14.0000i	0.505511	−	0.505511i
$768$	−17.0000	−	17.0000i	−0.613435	−	0.613435i
$769$	15.0000	−	15.0000i	0.540914	−	0.540914i	−0.382883	−	0.923797i	$-0.625069\pi$
$769$	15.0000	−	15.0000i	0.540914	−	0.540914i	0.923797	+	0.382883i	$0.125069\pi$
$770$		0			0
$771$	−28.0000	−	28.0000i	−1.00840	−	1.00840i
$772$			2.00000i			0.0719816i
$773$	11.0000	−	11.0000i	0.395643	−	0.395643i	−0.481050	−	0.876693i	$-0.659745\pi$
$773$	11.0000	−	11.0000i	0.395643	−	0.395643i	0.876693	+	0.481050i	$0.159745\pi$
$774$			12.0000i			0.431331i
$775$		0			0
$776$			24.0000i			0.861550i
$777$	−6.00000	−	36.0000i	−0.215249	−	1.29149i
$778$	5.00000	−	5.00000i	0.179259	−	0.179259i
$779$		0			0
$780$		0			0
$781$			16.0000i			0.572525i
$782$	−32.0000			−1.14432
$783$		−	56.0000i		−	2.00128i
$784$		−	11.0000i		−	0.392857i
$785$		0			0
$786$	10.0000			0.356688
$787$	27.0000	+	27.0000i	0.962446	+	0.962446i	0.999320	−	0.0368739i	$-0.0117400\pi$
$787$	27.0000	+	27.0000i	0.962446	+	0.962446i	−0.0368739	+	0.999320i	$0.511740\pi$
$788$	−1.00000	−	1.00000i	−0.0356235	−	0.0356235i
$789$		−	14.0000i		−	0.498413i
$790$		0			0
$791$	−12.0000	−	12.0000i	−0.426671	−	0.426671i
$792$			6.00000i			0.213201i
$793$	−2.00000	+	2.00000i	−0.0710221	+	0.0710221i
$794$	5.00000	+	5.00000i	0.177443	+	0.177443i
$795$		0			0
$796$	15.0000	−	15.0000i	0.531661	−	0.531661i
$797$			46.0000i			1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$			46.0000i			1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	18.0000	+	18.0000i	0.637193	+	0.637193i
$799$	20.0000	+	20.0000i	0.707549	+	0.707549i
$800$		0			0
$801$	5.00000	+	5.00000i	0.176666	+	0.176666i
$802$	−1.00000	+	1.00000i	−0.0353112	+	0.0353112i
$803$	−2.00000	+	2.00000i	−0.0705785	+	0.0705785i
$804$	6.00000			0.211604
$805$		0			0
$806$	−6.00000	−	6.00000i	−0.211341	−	0.211341i
$807$	16.0000	−	16.0000i	0.563227	−	0.563227i
$808$		0			0
$809$	−25.0000	+	25.0000i	−0.878953	+	0.878953i	−0.993426	−	0.114473i	$-0.963482\pi$
$809$	−25.0000	+	25.0000i	−0.878953	+	0.878953i	0.114473	+	0.993426i	$0.463482\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$	42.0000			1.47391
$813$	−8.00000	−	8.00000i	−0.280572	−	0.280572i
$814$	12.0000	−	2.00000i	0.420600	−	0.0701000i
$815$		0			0
$816$	4.00000	+	4.00000i	0.140028	+	0.140028i
$817$	36.0000	−	36.0000i	1.25948	−	1.25948i
$818$	3.00000	+	3.00000i	0.104893	+	0.104893i
$819$	6.00000	+	6.00000i	0.209657	+	0.209657i
$820$		0			0
$821$	−18.0000			−0.628204			−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000			−0.628204			−0.314102	+	0.949389i	$0.601703\pi$
$822$	−14.0000			−0.488306
$823$	−11.0000	+	11.0000i	−0.383436	+	0.383436i	−0.872338	−	0.488903i	$-0.837397\pi$
$823$	−11.0000	+	11.0000i	−0.383436	+	0.383436i	0.488903	+	0.872338i	$0.337397\pi$
$824$		−	42.0000i		−	1.46314i
$825$		0			0
$826$	42.0000			1.46137
$827$	−38.0000			−1.32139			−0.660695	−	0.750655i	$-0.729738\pi$
$827$	−38.0000			−1.32139			−0.660695	+	0.750655i	$0.729738\pi$
$828$	−8.00000			−0.278019
$829$	−21.0000	+	21.0000i	−0.729360	+	0.729360i	−0.970492	−	0.241132i	$-0.922481\pi$
$829$	−21.0000	+	21.0000i	−0.729360	+	0.729360i	0.241132	+	0.970492i	$0.422481\pi$
$830$		0			0
$831$	26.0000	−	26.0000i	0.901930	−	0.901930i
$832$		−	14.0000i		−	0.485363i
$833$		−	44.0000i		−	1.52451i
$834$	4.00000	−	4.00000i	0.138509	−	0.138509i
$835$		0			0
$836$	6.00000	−	6.00000i	0.207514	−	0.207514i
$837$	24.0000			0.829561
$838$	−22.0000			−0.759977
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$		−	69.0000i		−	2.37931i
$842$	27.0000	−	27.0000i	0.930481	−	0.930481i
$843$	18.0000			0.619953
$844$	20.0000			0.688428
$845$		0			0
$846$	−5.00000	−	5.00000i	−0.171904	−	0.171904i
$847$	21.0000	+	21.0000i	0.721569	+	0.721569i
$848$	−3.00000	+	3.00000i	−0.103020	+	0.103020i
$849$	22.0000	+	22.0000i	0.755038	+	0.755038i
$850$		0			0
$851$	8.00000	+	48.0000i	0.274236	+	1.64542i
$852$	8.00000	+	8.00000i	0.274075	+	0.274075i
$853$	−44.0000			−1.50653			−0.753266	−	0.657716i	$-0.771523\pi$
$853$	−44.0000			−1.50653			−0.753266	+	0.657716i	$0.771523\pi$
$854$	−6.00000			−0.205316
$855$		0			0
$856$	−9.00000	+	9.00000i	−0.307614	+	0.307614i
$857$	32.0000			1.09310			0.546550	−	0.837427i	$-0.315941\pi$
$857$	32.0000			1.09310			0.546550	+	0.837427i	$0.315941\pi$
$858$	4.00000	−	4.00000i	0.136558	−	0.136558i
$859$	−7.00000	−	7.00000i	−0.238837	−	0.238837i	0.577531	−	0.816368i	$-0.304016\pi$
$859$	−7.00000	−	7.00000i	−0.238837	−	0.238837i	−0.816368	+	0.577531i	$0.804016\pi$
$860$		0			0
$861$		0			0
$862$	−11.0000	+	11.0000i	−0.374661	+	0.374661i
$863$	−15.0000	+	15.0000i	−0.510606	+	0.510606i	−0.914712	−	0.404106i	$-0.867583\pi$
$863$	−15.0000	+	15.0000i	−0.510606	+	0.510606i	0.404106	+	0.914712i	$0.367583\pi$
$864$	20.0000	+	20.0000i	0.680414	+	0.680414i
$865$		0			0
$866$	−5.00000	−	5.00000i	−0.169907	−	0.169907i
$867$	−1.00000	−	1.00000i	−0.0339618	−	0.0339618i
$868$			18.0000i			0.610960i
$869$	6.00000	−	6.00000i	0.203536	−	0.203536i
$870$		0			0
$871$	6.00000	+	6.00000i	0.203302	+	0.203302i
$872$	−9.00000	+	9.00000i	−0.304778	+	0.304778i
$873$		−	8.00000i		−	0.270759i
$874$	−24.0000	−	24.0000i	−0.811812	−	0.811812i
$875$		0			0
$876$			2.00000i			0.0675737i
$877$	−37.0000	−	37.0000i	−1.24940	−	1.24940i	−0.955985	−	0.293417i	$-0.905208\pi$
$877$	−37.0000	−	37.0000i	−1.24940	−	1.24940i	−0.293417	−	0.955985i	$-0.594792\pi$
$878$	−17.0000	−	17.0000i	−0.573722	−	0.573722i
$879$	−2.00000			−0.0674583
$880$		0			0
$881$		−	12.0000i		−	0.404290i	−0.979356	−	0.202145i	$-0.935209\pi$
$881$		−	12.0000i		−	0.404290i	0.979356	−	0.202145i	$-0.0647913\pi$
$882$			11.0000i			0.370389i
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$		−	8.00000i		−	0.269069i
$885$		0			0
$886$	11.0000	−	11.0000i	0.369552	−	0.369552i
$887$	−5.00000	+	5.00000i	−0.167884	+	0.167884i	−0.786048	−	0.618165i	$-0.787876\pi$
$887$	−5.00000	+	5.00000i	−0.167884	+	0.167884i	0.618165	+	0.786048i	$0.287876\pi$
$888$	15.0000	−	21.0000i	0.503367	−	0.704714i
$889$		−	54.0000i		−	1.81110i
$890$		0			0
$891$			10.0000i			0.335013i
$892$	1.00000	−	1.00000i	0.0334825	−	0.0334825i
$893$			30.0000i			1.00391i
$894$	−16.0000	−	16.0000i	−0.535120	−	0.535120i
$895$		0			0
$896$	−9.00000	+	9.00000i	−0.300669	+	0.300669i
$897$	16.0000	+	16.0000i	0.534224	+	0.534224i
$898$	5.00000	−	5.00000i	0.166852	−	0.166852i
$899$	42.0000			1.40078
$900$		0			0
$901$	−12.0000	+	12.0000i	−0.399778	+	0.399778i
$902$		0			0
$903$	−72.0000			−2.39601
$904$		−	12.0000i		−	0.399114i
$905$		0			0
$906$	2.00000	+	2.00000i	0.0664455	+	0.0664455i
$907$	−14.0000			−0.464862			−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000			−0.464862			−0.232431	+	0.972613i	$0.574668\pi$
$908$	2.00000			0.0663723
$909$		0			0
$910$		0			0
$911$	−11.0000	+	11.0000i	−0.364446	+	0.364446i	−0.865447	−	0.501001i	$-0.832965\pi$
$911$	−11.0000	+	11.0000i	−0.364446	+	0.364446i	0.501001	+	0.865447i	$0.332965\pi$
$912$			6.00000i			0.198680i
$913$	10.0000	+	10.0000i	0.330952	+	0.330952i
$914$			32.0000i			1.05847i
$915$		0			0
$916$			4.00000i			0.132164i
$917$		−	30.0000i		−	0.990687i
$918$	−16.0000	−	16.0000i	−0.528079	−	0.528079i
$919$	3.00000	−	3.00000i	0.0989609	−	0.0989609i	−0.655893	−	0.754854i	$-0.727708\pi$
$919$	3.00000	−	3.00000i	0.0989609	−	0.0989609i	0.754854	+	0.655893i	$0.227708\pi$
$920$		0			0
$921$	−18.0000			−0.593120
$922$	21.0000	+	21.0000i	0.691598	+	0.691598i
$923$			16.0000i			0.526646i
$924$	−12.0000			−0.394771
$925$		0			0
$926$	−40.0000			−1.31448
$927$			14.0000i			0.459820i
$928$	35.0000	+	35.0000i	1.14893	+	1.14893i
$929$	−34.0000			−1.11550			−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000			−1.11550			−0.557752	+	0.830008i	$0.688336\pi$
$930$		0			0
$931$	33.0000	−	33.0000i	1.08153	−	1.08153i
$932$	1.00000	+	1.00000i	0.0327561	+	0.0327561i
$933$			6.00000i			0.196431i
$934$		−	38.0000i		−	1.24340i
$935$		0			0
$936$			6.00000i			0.196116i
$937$	35.0000	+	35.0000i	1.14340	+	1.14340i	0.987824	+	0.155576i	$0.0497234\pi$
$937$	35.0000	+	35.0000i	1.14340	+	1.14340i	0.155576	+	0.987824i	$0.450277\pi$
$938$			18.0000i			0.587721i
$939$	30.0000	−	30.0000i	0.979013	−	0.979013i
$940$		0			0
$941$	−30.0000			−0.977972			−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000			−0.977972			−0.488986	+	0.872292i	$0.662633\pi$
$942$	−6.00000			−0.195491
$943$		0			0
$944$	7.00000	+	7.00000i	0.227831	+	0.227831i
$945$		0			0
$946$		−	24.0000i		−	0.780307i
$947$	18.0000			0.584921			0.292461	−	0.956278i	$-0.405526\pi$
$947$	18.0000			0.584921			0.292461	+	0.956278i	$0.405526\pi$
$948$		−	6.00000i		−	0.194871i
$949$	−2.00000	+	2.00000i	−0.0649227	+	0.0649227i
$950$		0			0
$951$	34.0000			1.10253
$952$	36.0000	−	36.0000i	1.16677	−	1.16677i
$953$	−7.00000	−	7.00000i	−0.226752	−	0.226752i	0.584582	−	0.811334i	$-0.301258\pi$
$953$	−7.00000	−	7.00000i	−0.226752	−	0.226752i	−0.811334	+	0.584582i	$0.801258\pi$
$954$	3.00000	−	3.00000i	0.0971286	−	0.0971286i
$955$		0			0
$956$	−7.00000	−	7.00000i	−0.226396	−	0.226396i
$957$			28.0000i			0.905111i
$958$	23.0000	−	23.0000i	0.743096	−	0.743096i
$959$			42.0000i			1.35625i
$960$		0			0
$961$		−	13.0000i		−	0.419355i
$962$	12.0000	−	2.00000i	0.386896	−	0.0644826i
$963$	3.00000	−	3.00000i	0.0966736	−	0.0966736i
$964$	1.00000	−	1.00000i	0.0322078	−	0.0322078i
$965$		0			0
$966$			48.0000i			1.54437i
$967$	−2.00000			−0.0643157			−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000			−0.0643157			−0.0321578	+	0.999483i	$0.510238\pi$
$968$			21.0000i			0.674966i
$969$			24.0000i			0.770991i
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$	−7.00000	−	7.00000i	−0.224525	−	0.224525i
$973$	−12.0000	−	12.0000i	−0.384702	−	0.384702i
$974$		−	2.00000i		−	0.0640841i
$975$		0			0
$976$	−1.00000	−	1.00000i	−0.0320092	−	0.0320092i
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	10.0000	−	10.0000i	0.319765	−	0.319765i
$979$	−10.0000	−	10.0000i	−0.319601	−	0.319601i
$980$		0			0
$981$	3.00000	−	3.00000i	0.0957826	−	0.0957826i
$982$		−	4.00000i		−	0.127645i
$983$	−15.0000	−	15.0000i	−0.478426	−	0.478426i	0.426202	−	0.904628i	$-0.359851\pi$
$983$	−15.0000	−	15.0000i	−0.478426	−	0.478426i	−0.904628	+	0.426202i	$0.859851\pi$
$984$		0			0
$985$		0			0
$986$	−28.0000	−	28.0000i	−0.891702	−	0.891702i
$987$	30.0000	−	30.0000i	0.954911	−	0.954911i
$988$	6.00000	−	6.00000i	0.190885	−	0.190885i
$989$	96.0000			3.05262
$990$		0			0
$991$	39.0000	+	39.0000i	1.23888	+	1.23888i	0.960461	+	0.278415i	$0.0898090\pi$
$991$	39.0000	+	39.0000i	1.23888	+	1.23888i	0.278415	+	0.960461i	$0.410191\pi$
$992$	−15.0000	+	15.0000i	−0.476250	+	0.476250i
$993$	−14.0000			−0.444277
$994$	−24.0000	+	24.0000i	−0.761234	+	0.761234i
$995$		0			0
$996$	10.0000			0.316862
$997$	−28.0000			−0.886769			−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000			−0.886769			−0.443384	+	0.896332i	$0.646222\pi$
$998$	27.0000	+	27.0000i	0.854670	+	0.854670i
$999$	−20.0000	+	28.0000i	−0.632772	+	0.885881i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.f.b.882.1		2
5.2	odd	4		185.2.k.b.68.1	yes	2
5.3	odd	4		925.2.k.a.68.1		2
5.4	even	2		185.2.f.a.142.1	yes	2
37.6	odd	4		925.2.k.a.857.1		2
185.43	even	4	inner	925.2.f.b.43.1		2
185.117	even	4		185.2.f.a.43.1	✓	2
185.154	odd	4		185.2.k.b.117.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.f.a.43.1	✓	2	185.117	even	4
185.2.f.a.142.1	yes	2	5.4	even	2
185.2.k.b.68.1	yes	2	5.2	odd	4
185.2.k.b.117.1	yes	2	185.154	odd	4
925.2.f.b.43.1		2	185.43	even	4	inner
925.2.f.b.882.1		2	1.1	even	1	trivial
925.2.k.a.68.1		2	5.3	odd	4
925.2.k.a.857.1		2	37.6	odd	4

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:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +(1.00000 + 1.00000i) q^{3} +1.00000 q^{4} +(-1.00000 + 1.00000i) q^{6} +(3.00000 + 3.00000i) q^{7} +3.00000i q^{8} -1.00000i q^{9} +2.00000i q^{11} +(1.00000 + 1.00000i) q^{12} +2.00000i q^{13} +(-3.00000 + 3.00000i) q^{14} -1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +(-3.00000 - 3.00000i) q^{19} +6.00000i q^{21} -2.00000 q^{22} -8.00000i q^{23} +(-3.00000 + 3.00000i) q^{24} -2.00000 q^{26} +(4.00000 - 4.00000i) q^{27} +(3.00000 + 3.00000i) q^{28} +(7.00000 - 7.00000i) q^{29} +(3.00000 + 3.00000i) q^{31} +5.00000i q^{32} +(-2.00000 + 2.00000i) q^{33} -4.00000i q^{34} -1.00000i q^{36} +(-6.00000 + 1.00000i) q^{37} +(3.00000 - 3.00000i) q^{38} +(-2.00000 + 2.00000i) q^{39} -6.00000 q^{42} +12.0000i q^{43} +2.00000i q^{44} +8.00000 q^{46} +(-5.00000 - 5.00000i) q^{47} +(-1.00000 - 1.00000i) q^{48} +11.0000i q^{49} +(-4.00000 - 4.00000i) q^{51} +2.00000i q^{52} +(3.00000 - 3.00000i) q^{53} +(4.00000 + 4.00000i) q^{54} +(-9.00000 + 9.00000i) q^{56} -6.00000i q^{57} +(7.00000 + 7.00000i) q^{58} +(-7.00000 - 7.00000i) q^{59} +(1.00000 + 1.00000i) q^{61} +(-3.00000 + 3.00000i) q^{62} +(3.00000 - 3.00000i) q^{63} -7.00000 q^{64} +(-2.00000 - 2.00000i) q^{66} +(3.00000 - 3.00000i) q^{67} -4.00000 q^{68} +(8.00000 - 8.00000i) q^{69} +8.00000 q^{71} +3.00000 q^{72} +(1.00000 + 1.00000i) q^{73} +(-1.00000 - 6.00000i) q^{74} +(-3.00000 - 3.00000i) q^{76} +(-6.00000 + 6.00000i) q^{77} +(-2.00000 - 2.00000i) q^{78} +(-3.00000 - 3.00000i) q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +6.00000i q^{84} -12.0000 q^{86} +14.0000 q^{87} -6.00000 q^{88} +(-5.00000 + 5.00000i) q^{89} +(-6.00000 + 6.00000i) q^{91} -8.00000i q^{92} +6.00000i q^{93} +(5.00000 - 5.00000i) q^{94} +(-5.00000 + 5.00000i) q^{96} +8.00000 q^{97} -11.0000 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{7} + 2 q^{12} - 6 q^{14} - 2 q^{16} - 8 q^{17} + 2 q^{18} - 6 q^{19} - 4 q^{22} - 6 q^{24} - 4 q^{26} + 8 q^{27} + 6 q^{28} + 14 q^{29} + 6 q^{31} - 4 q^{33} - 12 q^{37}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^4 - 2 * q^6 + 6 * q^7 + 2 * q^12 - 6 * q^14 - 2 * q^16 - 8 * q^17 + 2 * q^18 - 6 * q^19 - 4 * q^22 - 6 * q^24 - 4 * q^26 + 8 * q^27 + 6 * q^28 + 14 * q^29 + 6 * q^31 - 4 * q^33 - 12 * q^37 + 6 * q^38 - 4 * q^39 - 12 * q^42 + 16 * q^46 - 10 * q^47 - 2 * q^48 - 8 * q^51 + 6 * q^53 + 8 * q^54 - 18 * q^56 + 14 * q^58 - 14 * q^59 + 2 * q^61 - 6 * q^62 + 6 * q^63 - 14 * q^64 - 4 * q^66 + 6 * q^67 - 8 * q^68 + 16 * q^69 + 16 * q^71 + 6 * q^72 + 2 * q^73 - 2 * q^74 - 6 * q^76 - 12 * q^77 - 4 * q^78 - 6 * q^79 + 10 * q^81 + 10 * q^83 - 24 * q^86 + 28 * q^87 - 12 * q^88 - 10 * q^89 - 12 * q^91 + 10 * q^94 - 10 * q^96 + 16 * q^97 - 22 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more