LMFDB - Embedded newform 400.2.l.a.301.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,2,Mod(101,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.101"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	400.l (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,-2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$3.19401608085$
Analytic rank:	$1$
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 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			301.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.301
Dual form			400.2.l.a.101.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +(-1.00000 - 1.00000i) q^{3} +2.00000i q^{4} +2.00000i q^{6} +(2.00000 - 2.00000i) q^{8} -1.00000i q^{9} +(-3.00000 + 3.00000i) q^{11} +(2.00000 - 2.00000i) q^{12} +(-3.00000 - 3.00000i) q^{13} -4.00000 q^{16} -4.00000 q^{17} +(-1.00000 + 1.00000i) q^{18} +(1.00000 + 1.00000i) q^{19} +6.00000 q^{22} +8.00000i q^{23} -4.00000 q^{24} +6.00000i q^{26} +(-4.00000 + 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{29} +(4.00000 + 4.00000i) q^{32} +6.00000 q^{33} +(4.00000 + 4.00000i) q^{34} +2.00000 q^{36} +(-3.00000 + 3.00000i) q^{37} -2.00000i q^{38} +6.00000i q^{39} +(3.00000 - 3.00000i) q^{43} +(-6.00000 - 6.00000i) q^{44} +(8.00000 - 8.00000i) q^{46} -2.00000 q^{47} +(4.00000 + 4.00000i) q^{48} +7.00000 q^{49} +(4.00000 + 4.00000i) q^{51} +(6.00000 - 6.00000i) q^{52} +(-9.00000 + 9.00000i) q^{53} +8.00000 q^{54} -2.00000i q^{57} +6.00000i q^{58} +(-9.00000 + 9.00000i) q^{59} +(-5.00000 - 5.00000i) q^{61} -8.00000i q^{64} +(-6.00000 - 6.00000i) q^{66} +(-3.00000 - 3.00000i) q^{67} -8.00000i q^{68} +(8.00000 - 8.00000i) q^{69} -6.00000i q^{71} +(-2.00000 - 2.00000i) q^{72} -6.00000i q^{73} +6.00000 q^{74} +(-2.00000 + 2.00000i) q^{76} +(6.00000 - 6.00000i) q^{78} -8.00000 q^{79} +5.00000 q^{81} +(-9.00000 - 9.00000i) q^{83} -6.00000 q^{86} +6.00000i q^{87} +12.0000i q^{88} -12.0000i q^{89} -16.0000 q^{92} +(2.00000 + 2.00000i) q^{94} -8.00000i q^{96} +12.0000 q^{97} +(-7.00000 - 7.00000i) q^{98} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 4 q^{8} - 6 q^{11} + 4 q^{12} - 6 q^{13} - 8 q^{16} - 8 q^{17} - 2 q^{18} + 2 q^{19} + 12 q^{22} - 8 q^{24} - 8 q^{27} - 6 q^{29} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{36} - 6 q^{37}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^8 - 6 * q^11 + 4 * q^12 - 6 * q^13 - 8 * q^16 - 8 * q^17 - 2 * q^18 + 2 * q^19 + 12 * q^22 - 8 * q^24 - 8 * q^27 - 6 * q^29 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^36 - 6 * q^37 + 6 * q^43 - 12 * q^44 + 16 * q^46 - 4 * q^47 + 8 * q^48 + 14 * q^49 + 8 * q^51 + 12 * q^52 - 18 * q^53 + 16 * q^54 - 18 * q^59 - 10 * q^61 - 12 * q^66 - 6 * q^67 + 16 * q^69 - 4 * q^72 + 12 * q^74 - 4 * q^76 + 12 * q^78 - 16 * q^79 + 10 * q^81 - 18 * q^83 - 12 * q^86 - 32 * q^92 + 4 * q^94 + 24 * q^97 - 14 * q^98 + 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	0.356822	−	0.934172i	$-0.383860\pi$
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	−0.934172	+	0.356822i	$0.883860\pi$
$4$			2.00000i			1.00000i
$5$		0			0
$5$		0			0
$6$			2.00000i			0.816497i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	−3.00000	+	3.00000i	−0.904534	+	0.904534i	−0.995824	−	0.0912903i	$-0.970901\pi$
$11$	−3.00000	+	3.00000i	−0.904534	+	0.904534i	0.0912903	+	0.995824i	$0.470901\pi$
$12$	2.00000	−	2.00000i	0.577350	−	0.577350i
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−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	+	1.00000i	−0.235702	+	0.235702i
$19$	1.00000	+	1.00000i	0.229416	+	0.229416i	0.812449	−	0.583033i	$-0.198134\pi$
$19$	1.00000	+	1.00000i	0.229416	+	0.229416i	−0.583033	+	0.812449i	$0.698134\pi$
$20$		0			0
$21$		0			0
$22$	6.00000			1.27920
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	−4.00000			−0.816497
$25$		0			0
$26$			6.00000i			1.17670i
$27$	−4.00000	+	4.00000i	−0.769800	+	0.769800i
$28$		0			0
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$	6.00000			1.04447
$34$	4.00000	+	4.00000i	0.685994	+	0.685994i
$35$		0			0
$36$	2.00000			0.333333
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$		−	2.00000i		−	0.324443i
$39$			6.00000i			0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	3.00000	−	3.00000i	0.457496	−	0.457496i	−0.440337	−	0.897833i	$-0.645141\pi$
$43$	3.00000	−	3.00000i	0.457496	−	0.457496i	0.897833	+	0.440337i	$0.145141\pi$
$44$	−6.00000	−	6.00000i	−0.904534	−	0.904534i
$45$		0			0
$46$	8.00000	−	8.00000i	1.17954	−	1.17954i
$47$	−2.00000			−0.291730			−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000			−0.291730			−0.145865	+	0.989305i	$0.546597\pi$
$48$	4.00000	+	4.00000i	0.577350	+	0.577350i
$49$	7.00000			1.00000
$50$		0			0
$51$	4.00000	+	4.00000i	0.560112	+	0.560112i
$52$	6.00000	−	6.00000i	0.832050	−	0.832050i
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.274721	+	0.961524i	$0.588586\pi$
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	8.00000			1.08866
$55$		0			0
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$			6.00000i			0.787839i
$59$	−9.00000	+	9.00000i	−1.17170	+	1.17170i	−0.189896	+	0.981804i	$0.560815\pi$
$59$	−9.00000	+	9.00000i	−1.17170	+	1.17170i	−0.981804	+	0.189896i	$0.939185\pi$
$60$		0			0
$61$	−5.00000	−	5.00000i	−0.640184	−	0.640184i	0.310416	−	0.950601i	$-0.399532\pi$
$61$	−5.00000	−	5.00000i	−0.640184	−	0.640184i	−0.950601	+	0.310416i	$0.899532\pi$
$62$		0			0
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$		0			0
$66$	−6.00000	−	6.00000i	−0.738549	−	0.738549i
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	0.499694	−	0.866202i	$-0.333446\pi$
$67$	−3.00000	−	3.00000i	−0.366508	−	0.366508i	−0.866202	+	0.499694i	$0.833446\pi$
$68$		−	8.00000i		−	0.970143i
$69$	8.00000	−	8.00000i	0.963087	−	0.963087i
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$	−2.00000	−	2.00000i	−0.235702	−	0.235702i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	6.00000			0.697486
$75$		0			0
$76$	−2.00000	+	2.00000i	−0.229416	+	0.229416i
$77$		0			0
$78$	6.00000	−	6.00000i	0.679366	−	0.679366i
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−9.00000	−	9.00000i	−0.987878	−	0.987878i	0.0120491	−	0.999927i	$-0.496165\pi$
$83$	−9.00000	−	9.00000i	−0.987878	−	0.987878i	−0.999927	+	0.0120491i	$0.996165\pi$
$84$		0			0
$85$		0			0
$86$	−6.00000			−0.646997
$87$			6.00000i			0.643268i
$88$			12.0000i			1.27920i
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$		0			0
$92$	−16.0000			−1.66812
$93$		0			0
$94$	2.00000	+	2.00000i	0.206284	+	0.206284i
$95$		0			0
$96$		−	8.00000i		−	0.816497i
$97$	12.0000			1.21842			0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000			1.21842			0.609208	+	0.793011i	$0.291488\pi$
$98$	−7.00000	−	7.00000i	−0.707107	−	0.707107i
$99$	3.00000	+	3.00000i	0.301511	+	0.301511i
$100$		0			0
$101$	3.00000	−	3.00000i	0.298511	−	0.298511i	−0.541919	−	0.840431i	$-0.682302\pi$
$101$	3.00000	−	3.00000i	0.298511	−	0.298511i	0.840431	+	0.541919i	$0.182302\pi$
$102$		−	8.00000i		−	0.792118i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−12.0000			−1.17670
$105$		0			0
$106$	18.0000			1.74831
$107$	9.00000	−	9.00000i	0.870063	−	0.870063i	−0.122416	−	0.992479i	$-0.539064\pi$
$107$	9.00000	−	9.00000i	0.870063	−	0.870063i	0.992479	+	0.122416i	$0.0390642\pi$
$108$	−8.00000	−	8.00000i	−0.769800	−	0.769800i
$109$	1.00000	+	1.00000i	0.0957826	+	0.0957826i	0.753374	−	0.657592i	$-0.228425\pi$
$109$	1.00000	+	1.00000i	0.0957826	+	0.0957826i	−0.657592	+	0.753374i	$0.728425\pi$
$110$		0			0
$111$	6.00000			0.569495
$112$		0			0
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$	−2.00000	+	2.00000i	−0.187317	+	0.187317i
$115$		0			0
$116$	6.00000	−	6.00000i	0.557086	−	0.557086i
$117$	−3.00000	+	3.00000i	−0.277350	+	0.277350i
$118$	18.0000			1.65703
$119$		0			0
$120$		0			0
$121$		−	7.00000i		−	0.636364i
$122$			10.0000i			0.905357i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	6.00000			0.532414			0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000			0.532414			0.266207	+	0.963916i	$0.414230\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$	−6.00000			−0.528271
$130$		0			0
$131$	−9.00000	−	9.00000i	−0.786334	−	0.786334i	0.194557	−	0.980891i	$-0.437673\pi$
$131$	−9.00000	−	9.00000i	−0.786334	−	0.786334i	−0.980891	+	0.194557i	$0.937673\pi$
$132$			12.0000i			1.04447i
$133$		0			0
$134$			6.00000i			0.518321i
$135$		0			0
$136$	−8.00000	+	8.00000i	−0.685994	+	0.685994i
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	−16.0000			−1.36201
$139$	7.00000	−	7.00000i	0.593732	−	0.593732i	−0.344905	−	0.938638i	$-0.612089\pi$
$139$	7.00000	−	7.00000i	0.593732	−	0.593732i	0.938638	+	0.344905i	$0.112089\pi$
$140$		0			0
$141$	2.00000	+	2.00000i	0.168430	+	0.168430i
$142$	−6.00000	+	6.00000i	−0.503509	+	0.503509i
$143$	18.0000			1.50524
$144$			4.00000i			0.333333i
$145$		0			0
$146$	−6.00000	+	6.00000i	−0.496564	+	0.496564i
$147$	−7.00000	−	7.00000i	−0.577350	−	0.577350i
$148$	−6.00000	−	6.00000i	−0.493197	−	0.493197i
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$			18.0000i			1.46482i	0.680864	+	0.732410i	$0.261604\pi$
$151$			18.0000i			1.46482i	−0.680864	+	0.732410i	$0.738396\pi$
$152$	4.00000			0.324443
$153$			4.00000i			0.323381i
$154$		0			0
$155$		0			0
$156$	−12.0000			−0.960769
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	0.249974	−	0.968252i	$-0.419578\pi$
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	−0.968252	+	0.249974i	$0.919578\pi$
$158$	8.00000	+	8.00000i	0.636446	+	0.636446i
$159$	18.0000			1.42749
$160$		0			0
$161$		0			0
$162$	−5.00000	−	5.00000i	−0.392837	−	0.392837i
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	0.260531	−	0.965465i	$-0.416102\pi$
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	−0.965465	+	0.260531i	$0.916102\pi$
$164$		0			0
$165$		0			0
$166$			18.0000i			1.39707i
$167$			8.00000i			0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$			8.00000i			0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	1.00000	−	1.00000i	0.0764719	−	0.0764719i
$172$	6.00000	+	6.00000i	0.457496	+	0.457496i
$173$	9.00000	+	9.00000i	0.684257	+	0.684257i	0.960957	−	0.276699i	$-0.0892406\pi$
$173$	9.00000	+	9.00000i	0.684257	+	0.684257i	−0.276699	+	0.960957i	$0.589241\pi$
$174$	6.00000	−	6.00000i	0.454859	−	0.454859i
$175$		0			0
$176$	12.0000	−	12.0000i	0.904534	−	0.904534i
$177$	18.0000			1.35296
$178$	−12.0000	+	12.0000i	−0.899438	+	0.899438i
$179$	−3.00000	−	3.00000i	−0.224231	−	0.224231i	0.586047	−	0.810277i	$-0.300683\pi$
$179$	−3.00000	−	3.00000i	−0.224231	−	0.224231i	−0.810277	+	0.586047i	$0.800683\pi$
$180$		0			0
$181$	−1.00000	+	1.00000i	−0.0743294	+	0.0743294i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−1.00000	+	1.00000i	−0.0743294	+	0.0743294i	0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$			10.0000i			0.739221i
$184$	16.0000	+	16.0000i	1.17954	+	1.17954i
$185$		0			0
$186$		0			0
$187$	12.0000	−	12.0000i	0.877527	−	0.877527i
$188$		−	4.00000i		−	0.291730i
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$	−8.00000	+	8.00000i	−0.577350	+	0.577350i
$193$	12.0000			0.863779			0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000			0.863779			0.431889	+	0.901927i	$0.357847\pi$
$194$	−12.0000	−	12.0000i	−0.861550	−	0.861550i
$195$		0			0
$196$			14.0000i			1.00000i
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		−	6.00000i		−	0.426401i
$199$		−	2.00000i		−	0.141776i	−0.997484	−	0.0708881i	$-0.977417\pi$
$199$		−	2.00000i		−	0.141776i	0.997484	−	0.0708881i	$-0.0225833\pi$
$200$		0			0
$201$			6.00000i			0.423207i
$202$	−6.00000			−0.422159
$203$		0			0
$204$	−8.00000	+	8.00000i	−0.560112	+	0.560112i
$205$		0			0
$206$		0			0
$207$	8.00000			0.556038
$208$	12.0000	+	12.0000i	0.832050	+	0.832050i
$209$	−6.00000			−0.415029
$210$		0			0
$211$	11.0000	+	11.0000i	0.757271	+	0.757271i	0.975825	−	0.218554i	$-0.0701339\pi$
$211$	11.0000	+	11.0000i	0.757271	+	0.757271i	−0.218554	+	0.975825i	$0.570134\pi$
$212$	−18.0000	−	18.0000i	−1.23625	−	1.23625i
$213$	−6.00000	+	6.00000i	−0.411113	+	0.411113i
$214$	−18.0000			−1.23045
$215$		0			0
$216$			16.0000i			1.08866i
$217$		0			0
$218$		−	2.00000i		−	0.135457i
$219$	−6.00000	+	6.00000i	−0.405442	+	0.405442i
$220$		0			0
$221$	12.0000	+	12.0000i	0.807207	+	0.807207i
$222$	−6.00000	−	6.00000i	−0.402694	−	0.402694i
$223$	−6.00000			−0.401790			−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000			−0.401790			−0.200895	+	0.979613i	$0.564385\pi$
$224$		0			0
$225$		0			0
$226$	−8.00000	−	8.00000i	−0.532152	−	0.532152i
$227$	9.00000	+	9.00000i	0.597351	+	0.597351i	0.939607	−	0.342256i	$-0.111191\pi$
$227$	9.00000	+	9.00000i	0.597351	+	0.597351i	−0.342256	+	0.939607i	$0.611191\pi$
$228$	4.00000			0.264906
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	−0.899498	−	0.436925i	$-0.856068\pi$
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	0.436925	+	0.899498i	$0.356068\pi$
$230$		0			0
$231$		0			0
$232$	−12.0000			−0.787839
$233$		−	22.0000i		−	1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$		−	22.0000i		−	1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$	6.00000			0.392232
$235$		0			0
$236$	−18.0000	−	18.0000i	−1.17170	−	1.17170i
$237$	8.00000	+	8.00000i	0.519656	+	0.519656i
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$	−7.00000	+	7.00000i	−0.449977	+	0.449977i
$243$	7.00000	+	7.00000i	0.449050	+	0.449050i
$244$	10.0000	−	10.0000i	0.640184	−	0.640184i
$245$		0			0
$246$		0			0
$247$		−	6.00000i		−	0.381771i
$248$		0			0
$249$			18.0000i			1.14070i
$250$		0			0
$251$	9.00000	−	9.00000i	0.568075	−	0.568075i	−0.363514	−	0.931589i	$-0.618423\pi$
$251$	9.00000	−	9.00000i	0.568075	−	0.568075i	0.931589	+	0.363514i	$0.118423\pi$
$252$		0			0
$253$	−24.0000	−	24.0000i	−1.50887	−	1.50887i
$254$	−6.00000	−	6.00000i	−0.376473	−	0.376473i
$255$		0			0
$256$	16.0000			1.00000
$257$	−8.00000			−0.499026			−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000			−0.499026			−0.249513	+	0.968371i	$0.580271\pi$
$258$	6.00000	+	6.00000i	0.373544	+	0.373544i
$259$		0			0
$260$		0			0
$261$	−3.00000	+	3.00000i	−0.185695	+	0.185695i
$262$			18.0000i			1.11204i
$263$			16.0000i			0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$			16.0000i			0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	12.0000	−	12.0000i	0.738549	−	0.738549i
$265$		0			0
$266$		0			0
$267$	−12.0000	+	12.0000i	−0.734388	+	0.734388i
$268$	6.00000	−	6.00000i	0.366508	−	0.366508i
$269$	9.00000	+	9.00000i	0.548740	+	0.548740i	0.926076	−	0.377337i	$-0.123160\pi$
$269$	9.00000	+	9.00000i	0.548740	+	0.548740i	−0.377337	+	0.926076i	$0.623160\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$	16.0000			0.970143
$273$		0			0
$274$	2.00000	−	2.00000i	0.120824	−	0.120824i
$275$		0			0
$276$	16.0000	+	16.0000i	0.963087	+	0.963087i
$277$	−3.00000	+	3.00000i	−0.180253	+	0.180253i	−0.791466	−	0.611213i	$-0.790682\pi$
$277$	−3.00000	+	3.00000i	−0.180253	+	0.180253i	0.611213	+	0.791466i	$0.290682\pi$
$278$	−14.0000			−0.839664
$279$		0			0
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		−	4.00000i		−	0.238197i
$283$	15.0000	−	15.0000i	0.891657	−	0.891657i	−0.103022	−	0.994679i	$-0.532851\pi$
$283$	15.0000	−	15.0000i	0.891657	−	0.891657i	0.994679	+	0.103022i	$0.0328511\pi$
$284$	12.0000			0.712069
$285$		0			0
$286$	−18.0000	−	18.0000i	−1.06436	−	1.06436i
$287$		0			0
$288$	4.00000	−	4.00000i	0.235702	−	0.235702i
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−12.0000	−	12.0000i	−0.703452	−	0.703452i
$292$	12.0000			0.702247
$293$	−9.00000	+	9.00000i	−0.525786	+	0.525786i	−0.919313	−	0.393527i	$-0.871255\pi$
$293$	−9.00000	+	9.00000i	−0.525786	+	0.525786i	0.393527	+	0.919313i	$0.371255\pi$
$294$			14.0000i			0.816497i
$295$		0			0
$296$			12.0000i			0.697486i
$297$		−	24.0000i		−	1.39262i
$298$	6.00000			0.347571
$299$	24.0000	−	24.0000i	1.38796	−	1.38796i
$300$		0			0
$301$		0			0
$302$	18.0000	−	18.0000i	1.03578	−	1.03578i
$303$	−6.00000			−0.344691
$304$	−4.00000	−	4.00000i	−0.229416	−	0.229416i
$305$		0			0
$306$	4.00000	−	4.00000i	0.228665	−	0.228665i
$307$	−3.00000	−	3.00000i	−0.171219	−	0.171219i	0.616296	−	0.787515i	$-0.288633\pi$
$307$	−3.00000	−	3.00000i	−0.171219	−	0.171219i	−0.787515	+	0.616296i	$0.788633\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	6.00000i		−	0.340229i	−0.985424	−	0.170114i	$-0.945586\pi$
$311$		−	6.00000i		−	0.340229i	0.985424	−	0.170114i	$-0.0544137\pi$
$312$	12.0000	+	12.0000i	0.679366	+	0.679366i
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$			18.0000i			1.01580i
$315$		0			0
$316$		−	16.0000i		−	0.900070i
$317$	7.00000	+	7.00000i	0.393159	+	0.393159i	0.875812	−	0.482653i	$-0.160327\pi$
$317$	7.00000	+	7.00000i	0.393159	+	0.393159i	−0.482653	+	0.875812i	$0.660327\pi$
$318$	−18.0000	−	18.0000i	−1.00939	−	1.00939i
$319$	18.0000			1.00781
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$	−4.00000	−	4.00000i	−0.222566	−	0.222566i
$324$			10.0000i			0.555556i
$325$		0			0
$326$			18.0000i			0.996928i
$327$		−	2.00000i		−	0.110600i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5.00000	−	5.00000i	0.274825	−	0.274825i	−0.556214	−	0.831039i	$-0.687747\pi$
$331$	5.00000	−	5.00000i	0.274825	−	0.274825i	0.831039	+	0.556214i	$0.187747\pi$
$332$	18.0000	−	18.0000i	0.987878	−	0.987878i
$333$	3.00000	+	3.00000i	0.164399	+	0.164399i
$334$	8.00000	−	8.00000i	0.437741	−	0.437741i
$335$		0			0
$336$		0			0
$337$	−24.0000			−1.30736			−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000			−1.30736			−0.653682	+	0.756770i	$0.726776\pi$
$338$	5.00000	−	5.00000i	0.271964	−	0.271964i
$339$	−8.00000	−	8.00000i	−0.434500	−	0.434500i
$340$		0			0
$341$		0			0
$342$	−2.00000			−0.108148
$343$		0			0
$344$		−	12.0000i		−	0.646997i
$345$		0			0
$346$		−	18.0000i		−	0.967686i
$347$	−19.0000	+	19.0000i	−1.01997	+	1.01997i	−0.0201770	+	0.999796i	$0.506423\pi$
$347$	−19.0000	+	19.0000i	−1.01997	+	1.01997i	−0.999796	+	0.0201770i	$0.993577\pi$
$348$	−12.0000			−0.643268
$349$	5.00000	+	5.00000i	0.267644	+	0.267644i	0.828150	−	0.560506i	$-0.189393\pi$
$349$	5.00000	+	5.00000i	0.267644	+	0.267644i	−0.560506	+	0.828150i	$0.689393\pi$
$350$		0			0
$351$	24.0000			1.28103
$352$	−24.0000			−1.27920
$353$	16.0000			0.851594			0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000			0.851594			0.425797	+	0.904819i	$0.359994\pi$
$354$	−18.0000	−	18.0000i	−0.956689	−	0.956689i
$355$		0			0
$356$	24.0000			1.27200
$357$		0			0
$358$			6.00000i			0.317110i
$359$		−	18.0000i		−	0.950004i	−0.879985	−	0.475002i	$-0.842447\pi$
$359$		−	18.0000i		−	0.950004i	0.879985	−	0.475002i	$-0.157553\pi$
$360$		0			0
$361$		−	17.0000i		−	0.894737i
$362$	2.00000			0.105118
$363$	−7.00000	+	7.00000i	−0.367405	+	0.367405i
$364$		0			0
$365$		0			0
$366$	10.0000	−	10.0000i	0.522708	−	0.522708i
$367$	−18.0000			−0.939592			−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000			−0.939592			−0.469796	+	0.882775i	$0.655673\pi$
$368$		−	32.0000i		−	1.66812i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.00000	−	3.00000i	0.155334	−	0.155334i	−0.625161	−	0.780496i	$-0.714967\pi$
$373$	3.00000	−	3.00000i	0.155334	−	0.155334i	0.780496	+	0.625161i	$0.214967\pi$
$374$	−24.0000			−1.24101
$375$		0			0
$376$	−4.00000	+	4.00000i	−0.206284	+	0.206284i
$377$			18.0000i			0.927047i
$378$		0			0
$379$	−1.00000	+	1.00000i	−0.0513665	+	0.0513665i	−0.732323	−	0.680957i	$-0.761564\pi$
$379$	−1.00000	+	1.00000i	−0.0513665	+	0.0513665i	0.680957	+	0.732323i	$0.261564\pi$
$380$		0			0
$381$	−6.00000	−	6.00000i	−0.307389	−	0.307389i
$382$	24.0000	+	24.0000i	1.22795	+	1.22795i
$383$	10.0000			0.510976			0.255488	−	0.966812i	$-0.417764\pi$
$383$	10.0000			0.510976			0.255488	+	0.966812i	$0.417764\pi$
$384$	16.0000			0.816497
$385$		0			0
$386$	−12.0000	−	12.0000i	−0.610784	−	0.610784i
$387$	−3.00000	−	3.00000i	−0.152499	−	0.152499i
$388$			24.0000i			1.21842i
$389$	−15.0000	+	15.0000i	−0.760530	+	0.760530i	−0.976418	−	0.215888i	$-0.930735\pi$
$389$	−15.0000	+	15.0000i	−0.760530	+	0.760530i	0.215888	+	0.976418i	$0.430735\pi$
$390$		0			0
$391$		−	32.0000i		−	1.61831i
$392$	14.0000	−	14.0000i	0.707107	−	0.707107i
$393$			18.0000i			0.907980i
$394$	−10.0000			−0.503793
$395$		0			0
$396$	−6.00000	+	6.00000i	−0.301511	+	0.301511i
$397$	−9.00000	−	9.00000i	−0.451697	−	0.451697i	0.444220	−	0.895918i	$-0.353481\pi$
$397$	−9.00000	−	9.00000i	−0.451697	−	0.451697i	−0.895918	+	0.444220i	$0.853481\pi$
$398$	−2.00000	+	2.00000i	−0.100251	+	0.100251i
$399$		0			0
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$	6.00000	−	6.00000i	0.299253	−	0.299253i
$403$		0			0
$404$	6.00000	+	6.00000i	0.298511	+	0.298511i
$405$		0			0
$406$		0			0
$407$		−	18.0000i		−	0.892227i
$408$	16.0000			0.792118
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	2.00000	−	2.00000i	0.0986527	−	0.0986527i
$412$		0			0
$413$		0			0
$414$	−8.00000	−	8.00000i	−0.393179	−	0.393179i
$415$		0			0
$416$		−	24.0000i		−	1.17670i
$417$	−14.0000			−0.685583
$418$	6.00000	+	6.00000i	0.293470	+	0.293470i
$419$	−15.0000	−	15.0000i	−0.732798	−	0.732798i	0.238375	−	0.971173i	$-0.423385\pi$
$419$	−15.0000	−	15.0000i	−0.732798	−	0.732798i	−0.971173	+	0.238375i	$0.923385\pi$
$420$		0			0
$421$	−5.00000	+	5.00000i	−0.243685	+	0.243685i	−0.818373	−	0.574688i	$-0.805124\pi$
$421$	−5.00000	+	5.00000i	−0.243685	+	0.243685i	0.574688	+	0.818373i	$0.305124\pi$
$422$		−	22.0000i		−	1.07094i
$423$			2.00000i			0.0972433i
$424$			36.0000i			1.74831i
$425$		0			0
$426$	12.0000			0.581402
$427$		0			0
$428$	18.0000	+	18.0000i	0.870063	+	0.870063i
$429$	−18.0000	−	18.0000i	−0.869048	−	0.869048i
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$	16.0000	−	16.0000i	0.769800	−	0.769800i
$433$	−36.0000			−1.73005			−0.865025	−	0.501729i	$-0.832697\pi$
$433$	−36.0000			−1.73005			−0.865025	+	0.501729i	$0.832697\pi$
$434$		0			0
$435$		0			0
$436$	−2.00000	+	2.00000i	−0.0957826	+	0.0957826i
$437$	−8.00000	+	8.00000i	−0.382692	+	0.382692i
$438$	12.0000			0.573382
$439$		−	10.0000i		−	0.477274i	−0.971109	−	0.238637i	$-0.923299\pi$
$439$		−	10.0000i		−	0.477274i	0.971109	−	0.238637i	$-0.0767006\pi$
$440$		0			0
$441$		−	7.00000i		−	0.333333i
$442$		−	24.0000i		−	1.14156i
$443$	−9.00000	+	9.00000i	−0.427603	+	0.427603i	−0.887811	−	0.460208i	$-0.847775\pi$
$443$	−9.00000	+	9.00000i	−0.427603	+	0.427603i	0.460208	+	0.887811i	$0.347775\pi$
$444$			12.0000i			0.569495i
$445$		0			0
$446$	6.00000	+	6.00000i	0.284108	+	0.284108i
$447$	6.00000			0.283790
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$		0			0
$452$			16.0000i			0.752577i
$453$	18.0000	−	18.0000i	0.845714	−	0.845714i
$454$		−	18.0000i		−	0.844782i
$455$		0			0
$456$	−4.00000	−	4.00000i	−0.187317	−	0.187317i
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	14.0000			0.654177
$459$	16.0000	−	16.0000i	0.746816	−	0.746816i
$460$		0			0
$461$	3.00000	+	3.00000i	0.139724	+	0.139724i	0.773509	−	0.633785i	$-0.218500\pi$
$461$	3.00000	+	3.00000i	0.139724	+	0.139724i	−0.633785	+	0.773509i	$0.718500\pi$
$462$		0			0
$463$	−30.0000			−1.39422			−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000			−1.39422			−0.697109	+	0.716965i	$0.745531\pi$
$464$	12.0000	+	12.0000i	0.557086	+	0.557086i
$465$		0			0
$466$	−22.0000	+	22.0000i	−1.01913	+	1.01913i
$467$	5.00000	+	5.00000i	0.231372	+	0.231372i	0.813265	−	0.581893i	$-0.197688\pi$
$467$	5.00000	+	5.00000i	0.231372	+	0.231372i	−0.581893	+	0.813265i	$0.697688\pi$
$468$	−6.00000	−	6.00000i	−0.277350	−	0.277350i
$469$		0			0
$470$		0			0
$471$			18.0000i			0.829396i
$472$			36.0000i			1.65703i
$473$			18.0000i			0.827641i
$474$		−	16.0000i		−	0.734904i
$475$		0			0
$476$		0			0
$477$	9.00000	+	9.00000i	0.412082	+	0.412082i
$478$	24.0000	+	24.0000i	1.09773	+	1.09773i
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	18.0000			0.820729
$482$	18.0000	+	18.0000i	0.819878	+	0.819878i
$483$		0			0
$484$	14.0000			0.636364
$485$		0			0
$486$		−	14.0000i		−	0.635053i
$487$			24.0000i			1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$			24.0000i			1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	−20.0000			−0.905357
$489$			18.0000i			0.813988i
$490$		0			0
$491$	−15.0000	+	15.0000i	−0.676941	+	0.676941i	−0.959307	−	0.282366i	$-0.908881\pi$
$491$	−15.0000	+	15.0000i	−0.676941	+	0.676941i	0.282366	+	0.959307i	$0.408881\pi$
$492$		0			0
$493$	12.0000	+	12.0000i	0.540453	+	0.540453i
$494$	−6.00000	+	6.00000i	−0.269953	+	0.269953i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	18.0000	−	18.0000i	0.806599	−	0.806599i
$499$	29.0000	+	29.0000i	1.29822	+	1.29822i	0.929568	+	0.368650i	$0.120180\pi$
$499$	29.0000	+	29.0000i	1.29822	+	1.29822i	0.368650	+	0.929568i	$0.379820\pi$
$500$		0			0
$501$	8.00000	−	8.00000i	0.357414	−	0.357414i
$502$	−18.0000			−0.803379
$503$		−	32.0000i		−	1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$		−	32.0000i		−	1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$		0			0
$505$		0			0
$506$			48.0000i			2.13386i
$507$	5.00000	−	5.00000i	0.222058	−	0.222058i
$508$			12.0000i			0.532414i
$509$	9.00000	+	9.00000i	0.398918	+	0.398918i	0.877851	−	0.478933i	$-0.158976\pi$
$509$	9.00000	+	9.00000i	0.398918	+	0.398918i	−0.478933	+	0.877851i	$0.658976\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$	−8.00000			−0.353209
$514$	8.00000	+	8.00000i	0.352865	+	0.352865i
$515$		0			0
$516$		−	12.0000i		−	0.528271i
$517$	6.00000	−	6.00000i	0.263880	−	0.263880i
$518$		0			0
$519$		−	18.0000i		−	0.790112i
$520$		0			0
$521$			24.0000i			1.05146i	0.850652	+	0.525730i	$0.176208\pi$
$521$			24.0000i			1.05146i	−0.850652	+	0.525730i	$0.823792\pi$
$522$	6.00000			0.262613
$523$	−9.00000	+	9.00000i	−0.393543	+	0.393543i	−0.875948	−	0.482405i	$-0.839763\pi$
$523$	−9.00000	+	9.00000i	−0.393543	+	0.393543i	0.482405	+	0.875948i	$0.339763\pi$
$524$	18.0000	−	18.0000i	0.786334	−	0.786334i
$525$		0			0
$526$	16.0000	−	16.0000i	0.697633	−	0.697633i
$527$		0			0
$528$	−24.0000			−1.04447
$529$	−41.0000			−1.78261
$530$		0			0
$531$	9.00000	+	9.00000i	0.390567	+	0.390567i
$532$		0			0
$533$		0			0
$534$	24.0000			1.03858
$535$		0			0
$536$	−12.0000			−0.518321
$537$			6.00000i			0.258919i
$538$		−	18.0000i		−	0.776035i
$539$	−21.0000	+	21.0000i	−0.904534	+	0.904534i
$540$		0			0
$541$	−1.00000	−	1.00000i	−0.0429934	−	0.0429934i	0.685283	−	0.728277i	$-0.259678\pi$
$541$	−1.00000	−	1.00000i	−0.0429934	−	0.0429934i	−0.728277	+	0.685283i	$0.759678\pi$
$542$	−16.0000	−	16.0000i	−0.687259	−	0.687259i
$543$	2.00000			0.0858282
$544$	−16.0000	−	16.0000i	−0.685994	−	0.685994i
$545$		0			0
$546$		0			0
$547$	−3.00000	−	3.00000i	−0.128271	−	0.128271i	0.640057	−	0.768328i	$-0.278911\pi$
$547$	−3.00000	−	3.00000i	−0.128271	−	0.128271i	−0.768328	+	0.640057i	$0.778911\pi$
$548$	−4.00000			−0.170872
$549$	−5.00000	+	5.00000i	−0.213395	+	0.213395i
$550$		0			0
$551$		−	6.00000i		−	0.255609i
$552$		−	32.0000i		−	1.36201i
$553$		0			0
$554$	6.00000			0.254916
$555$		0			0
$556$	14.0000	+	14.0000i	0.593732	+	0.593732i
$557$	−9.00000	−	9.00000i	−0.381342	−	0.381342i	0.490243	−	0.871586i	$-0.336908\pi$
$557$	−9.00000	−	9.00000i	−0.381342	−	0.381342i	−0.871586	+	0.490243i	$0.836908\pi$
$558$		0			0
$559$	−18.0000			−0.761319
$560$		0			0
$561$	−24.0000			−1.01328
$562$	12.0000	−	12.0000i	0.506189	−	0.506189i
$563$	19.0000	+	19.0000i	0.800755	+	0.800755i	0.983213	−	0.182459i	$-0.0584057\pi$
$563$	19.0000	+	19.0000i	0.800755	+	0.800755i	−0.182459	+	0.983213i	$0.558406\pi$
$564$	−4.00000	+	4.00000i	−0.168430	+	0.168430i
$565$		0			0
$566$	−30.0000			−1.26099
$567$		0			0
$568$	−12.0000	−	12.0000i	−0.503509	−	0.503509i
$569$			24.0000i			1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$			24.0000i			1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$		0			0
$571$	−11.0000	+	11.0000i	−0.460336	+	0.460336i	−0.898765	−	0.438430i	$-0.855535\pi$
$571$	−11.0000	+	11.0000i	−0.460336	+	0.460336i	0.438430	+	0.898765i	$0.355535\pi$
$572$			36.0000i			1.50524i
$573$	24.0000	+	24.0000i	1.00261	+	1.00261i
$574$		0			0
$575$		0			0
$576$	−8.00000			−0.333333
$577$	24.0000			0.999133			0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000			0.999133			0.499567	+	0.866276i	$0.333493\pi$
$578$	1.00000	+	1.00000i	0.0415945	+	0.0415945i
$579$	−12.0000	−	12.0000i	−0.498703	−	0.498703i
$580$		0			0
$581$		0			0
$582$			24.0000i			0.994832i
$583$		−	54.0000i		−	2.23645i
$584$	−12.0000	−	12.0000i	−0.496564	−	0.496564i
$585$		0			0
$586$	18.0000			0.743573
$587$	9.00000	−	9.00000i	0.371470	−	0.371470i	−0.496543	−	0.868012i	$-0.665397\pi$
$587$	9.00000	−	9.00000i	0.371470	−	0.371470i	0.868012	+	0.496543i	$0.165397\pi$
$588$	14.0000	−	14.0000i	0.577350	−	0.577350i
$589$		0			0
$590$		0			0
$591$	−10.0000			−0.411345
$592$	12.0000	−	12.0000i	0.493197	−	0.493197i
$593$	−32.0000			−1.31408			−0.657041	−	0.753855i	$-0.728192\pi$
$593$	−32.0000			−1.31408			−0.657041	+	0.753855i	$0.728192\pi$
$594$	−24.0000	+	24.0000i	−0.984732	+	0.984732i
$595$		0			0
$596$	−6.00000	−	6.00000i	−0.245770	−	0.245770i
$597$	−2.00000	+	2.00000i	−0.0818546	+	0.0818546i
$598$	−48.0000			−1.96287
$599$			30.0000i			1.22577i	0.790173	+	0.612883i	$0.209990\pi$
$599$			30.0000i			1.22577i	−0.790173	+	0.612883i	$0.790010\pi$
$600$		0			0
$601$			36.0000i			1.46847i	0.678895	+	0.734235i	$0.262459\pi$
$601$			36.0000i			1.46847i	−0.678895	+	0.734235i	$0.737541\pi$
$602$		0			0
$603$	−3.00000	+	3.00000i	−0.122169	+	0.122169i
$604$	−36.0000			−1.46482
$605$		0			0
$606$	6.00000	+	6.00000i	0.243733	+	0.243733i
$607$	−42.0000			−1.70473			−0.852364	−	0.522949i	$-0.824832\pi$
$607$	−42.0000			−1.70473			−0.852364	+	0.522949i	$0.824832\pi$
$608$			8.00000i			0.324443i
$609$		0			0
$610$		0			0
$611$	6.00000	+	6.00000i	0.242734	+	0.242734i
$612$	−8.00000			−0.323381
$613$	27.0000	−	27.0000i	1.09052	−	1.09052i	0.0950469	−	0.995473i	$-0.469700\pi$
$613$	27.0000	−	27.0000i	1.09052	−	1.09052i	0.995473	−	0.0950469i	$-0.0303001\pi$
$614$			6.00000i			0.242140i
$615$		0			0
$616$		0			0
$617$		−	10.0000i		−	0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$		−	10.0000i		−	0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$		0			0
$619$	−13.0000	+	13.0000i	−0.522514	+	0.522514i	−0.918330	−	0.395816i	$-0.870462\pi$
$619$	−13.0000	+	13.0000i	−0.522514	+	0.522514i	0.395816	+	0.918330i	$0.370462\pi$
$620$		0			0
$621$	−32.0000	−	32.0000i	−1.28412	−	1.28412i
$622$	−6.00000	+	6.00000i	−0.240578	+	0.240578i
$623$		0			0
$624$		−	24.0000i		−	0.960769i
$625$		0			0
$626$	6.00000	−	6.00000i	0.239808	−	0.239808i
$627$	6.00000	+	6.00000i	0.239617	+	0.239617i
$628$	18.0000	−	18.0000i	0.718278	−	0.718278i
$629$	12.0000	−	12.0000i	0.478471	−	0.478471i
$630$		0			0
$631$			2.00000i			0.0796187i	0.999207	+	0.0398094i	$0.0126751\pi$
$631$			2.00000i			0.0796187i	−0.999207	+	0.0398094i	$0.987325\pi$
$632$	−16.0000	+	16.0000i	−0.636446	+	0.636446i
$633$		−	22.0000i		−	0.874421i
$634$		−	14.0000i		−	0.556011i
$635$		0			0
$636$			36.0000i			1.42749i
$637$	−21.0000	−	21.0000i	−0.832050	−	0.832050i
$638$	−18.0000	−	18.0000i	−0.712627	−	0.712627i
$639$	−6.00000			−0.237356
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$	18.0000	+	18.0000i	0.710403	+	0.710403i
$643$	27.0000	+	27.0000i	1.06478	+	1.06478i	0.997751	+	0.0670247i	$0.0213506\pi$
$643$	27.0000	+	27.0000i	1.06478	+	1.06478i	0.0670247	+	0.997751i	$0.478649\pi$
$644$		0			0
$645$		0			0
$646$			8.00000i			0.314756i
$647$		−	32.0000i		−	1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$		−	32.0000i		−	1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$	10.0000	−	10.0000i	0.392837	−	0.392837i
$649$		−	54.0000i		−	2.11969i
$650$		0			0
$651$		0			0
$652$	18.0000	−	18.0000i	0.704934	−	0.704934i
$653$	9.00000	+	9.00000i	0.352197	+	0.352197i	0.860927	−	0.508729i	$-0.169885\pi$
$653$	9.00000	+	9.00000i	0.352197	+	0.352197i	−0.508729	+	0.860927i	$0.669885\pi$
$654$	−2.00000	+	2.00000i	−0.0782062	+	0.0782062i
$655$		0			0
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	21.0000	+	21.0000i	0.818044	+	0.818044i	0.985824	−	0.167781i	$-0.0536600\pi$
$659$	21.0000	+	21.0000i	0.818044	+	0.818044i	−0.167781	+	0.985824i	$0.553660\pi$
$660$		0			0
$661$	−29.0000	+	29.0000i	−1.12797	+	1.12797i	−0.137462	+	0.990507i	$0.543895\pi$
$661$	−29.0000	+	29.0000i	−1.12797	+	1.12797i	−0.990507	+	0.137462i	$0.956105\pi$
$662$	−10.0000			−0.388661
$663$		−	24.0000i		−	0.932083i
$664$	−36.0000			−1.39707
$665$		0			0
$666$		−	6.00000i		−	0.232495i
$667$	24.0000	−	24.0000i	0.929284	−	0.929284i
$668$	−16.0000			−0.619059
$669$	6.00000	+	6.00000i	0.231973	+	0.231973i
$670$		0			0
$671$	30.0000			1.15814
$672$		0			0
$673$	−12.0000			−0.462566			−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000			−0.462566			−0.231283	+	0.972887i	$0.574292\pi$
$674$	24.0000	+	24.0000i	0.924445	+	0.924445i
$675$		0			0
$676$	−10.0000			−0.384615
$677$	9.00000	−	9.00000i	0.345898	−	0.345898i	−0.512681	−	0.858579i	$-0.671348\pi$
$677$	9.00000	−	9.00000i	0.345898	−	0.345898i	0.858579	+	0.512681i	$0.171348\pi$
$678$			16.0000i			0.614476i
$679$		0			0
$680$		0			0
$681$		−	18.0000i		−	0.689761i
$682$		0			0
$683$	−13.0000	+	13.0000i	−0.497431	+	0.497431i	−0.910637	−	0.413206i	$-0.864409\pi$
$683$	−13.0000	+	13.0000i	−0.497431	+	0.497431i	0.413206	+	0.910637i	$0.364409\pi$
$684$	2.00000	+	2.00000i	0.0764719	+	0.0764719i
$685$		0			0
$686$		0			0
$687$	14.0000			0.534133
$688$	−12.0000	+	12.0000i	−0.457496	+	0.457496i
$689$	54.0000			2.05724
$690$		0			0
$691$	−5.00000	−	5.00000i	−0.190209	−	0.190209i	0.605577	−	0.795786i	$-0.292942\pi$
$691$	−5.00000	−	5.00000i	−0.190209	−	0.190209i	−0.795786	+	0.605577i	$0.792942\pi$
$692$	−18.0000	+	18.0000i	−0.684257	+	0.684257i
$693$		0			0
$694$	38.0000			1.44246
$695$		0			0
$696$	12.0000	+	12.0000i	0.454859	+	0.454859i
$697$		0			0
$698$		−	10.0000i		−	0.378506i
$699$	−22.0000	+	22.0000i	−0.832116	+	0.832116i
$700$		0			0
$701$	3.00000	+	3.00000i	0.113308	+	0.113308i	0.761488	−	0.648179i	$-0.224469\pi$
$701$	3.00000	+	3.00000i	0.113308	+	0.113308i	−0.648179	+	0.761488i	$0.724469\pi$
$702$	−24.0000	−	24.0000i	−0.905822	−	0.905822i
$703$	−6.00000			−0.226294
$704$	24.0000	+	24.0000i	0.904534	+	0.904534i
$705$		0			0
$706$	−16.0000	−	16.0000i	−0.602168	−	0.602168i
$707$		0			0
$708$			36.0000i			1.35296i
$709$	13.0000	−	13.0000i	0.488225	−	0.488225i	−0.419521	−	0.907746i	$-0.637802\pi$
$709$	13.0000	−	13.0000i	0.488225	−	0.488225i	0.907746	+	0.419521i	$0.137802\pi$
$710$		0			0
$711$			8.00000i			0.300023i
$712$	−24.0000	−	24.0000i	−0.899438	−	0.899438i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	6.00000	−	6.00000i	0.224231	−	0.224231i
$717$	24.0000	+	24.0000i	0.896296	+	0.896296i
$718$	−18.0000	+	18.0000i	−0.671754	+	0.671754i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$	−17.0000	+	17.0000i	−0.632674	+	0.632674i
$723$	18.0000	+	18.0000i	0.669427	+	0.669427i
$724$	−2.00000	−	2.00000i	−0.0743294	−	0.0743294i
$725$		0			0
$726$	14.0000			0.519589
$727$			24.0000i			0.890111i	0.895503	+	0.445055i	$0.146816\pi$
$727$			24.0000i			0.890111i	−0.895503	+	0.445055i	$0.853184\pi$
$728$		0			0
$729$		−	29.0000i		−	1.07407i
$730$		0			0
$731$	−12.0000	+	12.0000i	−0.443836	+	0.443836i
$732$	−20.0000			−0.739221
$733$	−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	$0.774966\pi$
$734$	18.0000	+	18.0000i	0.664392	+	0.664392i
$735$		0			0
$736$	−32.0000	+	32.0000i	−1.17954	+	1.17954i
$737$	18.0000			0.663039
$738$		0			0
$739$	−19.0000	−	19.0000i	−0.698926	−	0.698926i	0.265253	−	0.964179i	$-0.414545\pi$
$739$	−19.0000	−	19.0000i	−0.698926	−	0.698926i	−0.964179	+	0.265253i	$0.914545\pi$
$740$		0			0
$741$	−6.00000	+	6.00000i	−0.220416	+	0.220416i
$742$		0			0
$743$			8.00000i			0.293492i	0.989174	+	0.146746i	$0.0468799\pi$
$743$			8.00000i			0.293492i	−0.989174	+	0.146746i	$0.953120\pi$
$744$		0			0
$745$		0			0
$746$	−6.00000			−0.219676
$747$	−9.00000	+	9.00000i	−0.329293	+	0.329293i
$748$	24.0000	+	24.0000i	0.877527	+	0.877527i
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	8.00000			0.291730
$753$	−18.0000			−0.655956
$754$	18.0000	−	18.0000i	0.655521	−	0.655521i
$755$		0			0
$756$		0			0
$757$	33.0000	−	33.0000i	1.19941	−	1.19941i	0.225061	−	0.974345i	$-0.427742\pi$
$757$	33.0000	−	33.0000i	1.19941	−	1.19941i	0.974345	−	0.225061i	$-0.0722580\pi$
$758$	2.00000			0.0726433
$759$			48.0000i			1.74229i
$760$		0			0
$761$		−	48.0000i		−	1.74000i	−0.493053	−	0.869999i	$-0.664119\pi$
$761$		−	48.0000i		−	1.74000i	0.493053	−	0.869999i	$-0.335881\pi$
$762$			12.0000i			0.434714i
$763$		0			0
$764$		−	48.0000i		−	1.73658i
$765$		0			0
$766$	−10.0000	−	10.0000i	−0.361315	−	0.361315i
$767$	54.0000			1.94983
$768$	−16.0000	−	16.0000i	−0.577350	−	0.577350i
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	8.00000	+	8.00000i	0.288113	+	0.288113i
$772$			24.0000i			0.863779i
$773$	23.0000	−	23.0000i	0.827253	−	0.827253i	−0.159883	−	0.987136i	$-0.551112\pi$
$773$	23.0000	−	23.0000i	0.827253	−	0.827253i	0.987136	+	0.159883i	$0.0511118\pi$
$774$			6.00000i			0.215666i
$775$		0			0
$776$	24.0000	−	24.0000i	0.861550	−	0.861550i
$777$		0			0
$778$	30.0000			1.07555
$779$		0			0
$780$		0			0
$781$	18.0000	+	18.0000i	0.644091	+	0.644091i
$782$	−32.0000	+	32.0000i	−1.14432	+	1.14432i
$783$	24.0000			0.857690
$784$	−28.0000			−1.00000
$785$		0			0
$786$	18.0000	−	18.0000i	0.642039	−	0.642039i
$787$	33.0000	+	33.0000i	1.17632	+	1.17632i	0.980674	+	0.195649i	$0.0626813\pi$
$787$	33.0000	+	33.0000i	1.17632	+	1.17632i	0.195649	+	0.980674i	$0.437319\pi$
$788$	10.0000	+	10.0000i	0.356235	+	0.356235i
$789$	16.0000	−	16.0000i	0.569615	−	0.569615i
$790$		0			0
$791$		0			0
$792$	12.0000			0.426401
$793$			30.0000i			1.06533i
$794$			18.0000i			0.638796i
$795$		0			0
$796$	4.00000			0.141776
$797$	19.0000	+	19.0000i	0.673015	+	0.673015i	0.958410	−	0.285395i	$-0.0921249\pi$
$797$	19.0000	+	19.0000i	0.673015	+	0.673015i	−0.285395	+	0.958410i	$0.592125\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$	−12.0000			−0.423999
$802$	−30.0000	−	30.0000i	−1.05934	−	1.05934i
$803$	18.0000	+	18.0000i	0.635206	+	0.635206i
$804$	−12.0000			−0.423207
$805$		0			0
$806$		0			0
$807$		−	18.0000i		−	0.633630i
$808$		−	12.0000i		−	0.422159i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	37.0000	−	37.0000i	1.29925	−	1.29925i	0.370356	−	0.928890i	$-0.379236\pi$
$811$	37.0000	−	37.0000i	1.29925	−	1.29925i	0.928890	−	0.370356i	$-0.120764\pi$
$812$		0			0
$813$	−16.0000	−	16.0000i	−0.561144	−	0.561144i
$814$	−18.0000	+	18.0000i	−0.630900	+	0.630900i
$815$		0			0
$816$	−16.0000	−	16.0000i	−0.560112	−	0.560112i
$817$	6.00000			0.209913
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.488603	−	0.872506i	$-0.337507\pi$
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.872506	−	0.488603i	$-0.162493\pi$
$822$	−4.00000			−0.139516
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.0000	+	31.0000i	−1.07798	+	1.07798i	−0.0812847	+	0.996691i	$0.525902\pi$
$827$	−31.0000	+	31.0000i	−1.07798	+	1.07798i	−0.996691	+	0.0812847i	$0.974098\pi$
$828$			16.0000i			0.556038i
$829$	−35.0000	−	35.0000i	−1.21560	−	1.21560i	−0.969157	−	0.246443i	$-0.920738\pi$
$829$	−35.0000	−	35.0000i	−1.21560	−	1.21560i	−0.246443	−	0.969157i	$-0.579262\pi$
$830$		0			0
$831$	6.00000			0.208138
$832$	−24.0000	+	24.0000i	−0.832050	+	0.832050i
$833$	−28.0000			−0.970143
$834$	14.0000	+	14.0000i	0.484780	+	0.484780i
$835$		0			0
$836$		−	12.0000i		−	0.415029i
$837$		0			0
$838$			30.0000i			1.03633i
$839$		−	42.0000i		−	1.45000i	−0.688748	−	0.725001i	$-0.741839\pi$
$839$		−	42.0000i		−	1.45000i	0.688748	−	0.725001i	$-0.258161\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$	10.0000			0.344623
$843$	12.0000	−	12.0000i	0.413302	−	0.413302i
$844$	−22.0000	+	22.0000i	−0.757271	+	0.757271i
$845$		0			0
$846$	2.00000	−	2.00000i	0.0687614	−	0.0687614i
$847$		0			0
$848$	36.0000	−	36.0000i	1.23625	−	1.23625i
$849$	−30.0000			−1.02960
$850$		0			0
$851$	−24.0000	−	24.0000i	−0.822709	−	0.822709i
$852$	−12.0000	−	12.0000i	−0.411113	−	0.411113i
$853$	15.0000	−	15.0000i	0.513590	−	0.513590i	−0.402034	−	0.915625i	$-0.631697\pi$
$853$	15.0000	−	15.0000i	0.513590	−	0.513590i	0.915625	+	0.402034i	$0.131697\pi$
$854$		0			0
$855$		0			0
$856$		−	36.0000i		−	1.23045i
$857$		−	38.0000i		−	1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$		−	38.0000i		−	1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$			36.0000i			1.22902i
$859$	7.00000	−	7.00000i	0.238837	−	0.238837i	−0.577531	−	0.816368i	$-0.695984\pi$
$859$	7.00000	−	7.00000i	0.238837	−	0.238837i	0.816368	+	0.577531i	$0.195984\pi$
$860$		0			0
$861$		0			0
$862$	24.0000	+	24.0000i	0.817443	+	0.817443i
$863$	−22.0000			−0.748889			−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000			−0.748889			−0.374444	+	0.927249i	$0.622167\pi$
$864$	−32.0000			−1.08866
$865$		0			0
$866$	36.0000	+	36.0000i	1.22333	+	1.22333i
$867$	1.00000	+	1.00000i	0.0339618	+	0.0339618i
$868$		0			0
$869$	24.0000	−	24.0000i	0.814144	−	0.814144i
$870$		0			0
$871$			18.0000i			0.609907i
$872$	4.00000			0.135457
$873$		−	12.0000i		−	0.406138i
$874$	16.0000			0.541208
$875$		0			0
$876$	−12.0000	−	12.0000i	−0.405442	−	0.405442i
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	0.755942	−	0.654639i	$-0.227179\pi$
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	−0.654639	+	0.755942i	$0.727179\pi$
$878$	−10.0000	+	10.0000i	−0.337484	+	0.337484i
$879$	18.0000			0.607125
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$	−7.00000	+	7.00000i	−0.235702	+	0.235702i
$883$	−21.0000	−	21.0000i	−0.706706	−	0.706706i	0.259135	−	0.965841i	$-0.416563\pi$
$883$	−21.0000	−	21.0000i	−0.706706	−	0.706706i	−0.965841	+	0.259135i	$0.916563\pi$
$884$	−24.0000	+	24.0000i	−0.807207	+	0.807207i
$885$		0			0
$886$	18.0000			0.604722
$887$		−	8.00000i		−	0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$		−	8.00000i		−	0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	12.0000	−	12.0000i	0.402694	−	0.402694i
$889$		0			0
$890$		0			0
$891$	−15.0000	+	15.0000i	−0.502519	+	0.502519i
$892$		−	12.0000i		−	0.401790i
$893$	−2.00000	−	2.00000i	−0.0669274	−	0.0669274i
$894$	−6.00000	−	6.00000i	−0.200670	−	0.200670i
$895$		0			0
$896$		0			0
$897$	−48.0000			−1.60267
$898$	−18.0000	−	18.0000i	−0.600668	−	0.600668i
$899$		0			0
$900$		0			0
$901$	36.0000	−	36.0000i	1.19933	−	1.19933i
$902$		0			0
$903$		0			0
$904$	16.0000	−	16.0000i	0.532152	−	0.532152i
$905$		0			0
$906$	−36.0000			−1.19602
$907$	21.0000	−	21.0000i	0.697294	−	0.697294i	−0.266532	−	0.963826i	$-0.585878\pi$
$907$	21.0000	−	21.0000i	0.697294	−	0.697294i	0.963826	+	0.266532i	$0.0858779\pi$
$908$	−18.0000	+	18.0000i	−0.597351	+	0.597351i
$909$	−3.00000	−	3.00000i	−0.0995037	−	0.0995037i
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$			8.00000i			0.264906i
$913$	54.0000			1.78714
$914$	18.0000	−	18.0000i	0.595387	−	0.595387i
$915$		0			0
$916$	−14.0000	−	14.0000i	−0.462573	−	0.462573i
$917$		0			0
$918$	−32.0000			−1.05616
$919$			54.0000i			1.78130i	0.454694	+	0.890648i	$0.349749\pi$
$919$			54.0000i			1.78130i	−0.454694	+	0.890648i	$0.650251\pi$
$920$		0			0
$921$			6.00000i			0.197707i
$922$		−	6.00000i		−	0.197599i
$923$	−18.0000	+	18.0000i	−0.592477	+	0.592477i
$924$		0			0
$925$		0			0
$926$	30.0000	+	30.0000i	0.985861	+	0.985861i
$927$		0			0
$928$		−	24.0000i		−	0.787839i
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	7.00000	+	7.00000i	0.229416	+	0.229416i
$932$	44.0000			1.44127
$933$	−6.00000	+	6.00000i	−0.196431	+	0.196431i
$934$		−	10.0000i		−	0.327210i
$935$		0			0
$936$			12.0000i			0.392232i
$937$			54.0000i			1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$			54.0000i			1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$		0			0
$939$	6.00000	−	6.00000i	0.195803	−	0.195803i
$940$		0			0
$941$	27.0000	+	27.0000i	0.880175	+	0.880175i	0.993552	−	0.113377i	$-0.0361668\pi$
$941$	27.0000	+	27.0000i	0.880175	+	0.880175i	−0.113377	+	0.993552i	$0.536167\pi$
$942$	18.0000	−	18.0000i	0.586472	−	0.586472i
$943$		0			0
$944$	36.0000	−	36.0000i	1.17170	−	1.17170i
$945$		0			0
$946$	18.0000	−	18.0000i	0.585230	−	0.585230i
$947$	−27.0000	−	27.0000i	−0.877382	−	0.877382i	0.115881	−	0.993263i	$-0.463031\pi$
$947$	−27.0000	−	27.0000i	−0.877382	−	0.877382i	−0.993263	+	0.115881i	$0.963031\pi$
$948$	−16.0000	+	16.0000i	−0.519656	+	0.519656i
$949$	−18.0000	+	18.0000i	−0.584305	+	0.584305i
$950$		0			0
$951$		−	14.0000i		−	0.453981i
$952$		0			0
$953$			22.0000i			0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$			22.0000i			0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$		−	18.0000i		−	0.582772i
$955$		0			0
$956$		−	48.0000i		−	1.55243i
$957$	−18.0000	−	18.0000i	−0.581857	−	0.581857i
$958$	−24.0000	−	24.0000i	−0.775405	−	0.775405i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$	−18.0000	−	18.0000i	−0.580343	−	0.580343i
$963$	−9.00000	−	9.00000i	−0.290021	−	0.290021i
$964$		−	36.0000i		−	1.15948i
$965$		0			0
$966$		0			0
$967$		−	48.0000i		−	1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$		−	48.0000i		−	1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$	−14.0000	−	14.0000i	−0.449977	−	0.449977i
$969$			8.00000i			0.256997i
$970$		0			0
$971$	9.00000	−	9.00000i	0.288824	−	0.288824i	−0.547791	−	0.836615i	$-0.684531\pi$
$971$	9.00000	−	9.00000i	0.288824	−	0.288824i	0.836615	+	0.547791i	$0.184531\pi$
$972$	−14.0000	+	14.0000i	−0.449050	+	0.449050i
$973$		0			0
$974$	24.0000	−	24.0000i	0.769010	−	0.769010i
$975$		0			0
$976$	20.0000	+	20.0000i	0.640184	+	0.640184i
$977$	−28.0000			−0.895799			−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000			−0.895799			−0.447900	+	0.894084i	$0.647828\pi$
$978$	18.0000	−	18.0000i	0.575577	−	0.575577i
$979$	36.0000	+	36.0000i	1.15056	+	1.15056i
$980$		0			0
$981$	1.00000	−	1.00000i	0.0319275	−	0.0319275i
$982$	30.0000			0.957338
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$		0			0
$986$		−	24.0000i		−	0.764316i
$987$		0			0
$988$	12.0000			0.381771
$989$	24.0000	+	24.0000i	0.763156	+	0.763156i
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$	−10.0000			−0.317340
$994$		0			0
$995$		0			0
$996$	−36.0000			−1.14070
$997$	9.00000	−	9.00000i	0.285033	−	0.285033i	−0.550079	−	0.835112i	$-0.685403\pi$
$997$	9.00000	−	9.00000i	0.285033	−	0.285033i	0.835112	+	0.550079i	$0.185403\pi$
$998$		−	58.0000i		−	1.83596i
$999$		−	24.0000i		−	0.759326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.2.l.a.301.1		2
4.3	odd	2		1600.2.l.c.401.1		2
5.2	odd	4		80.2.q.b.29.1	yes	2
5.3	odd	4		80.2.q.a.29.1	✓	2
5.4	even	2		400.2.l.b.301.1		2
15.2	even	4		720.2.bm.a.109.1		2
15.8	even	4		720.2.bm.b.109.1		2
16.5	even	4	inner	400.2.l.a.101.1		2
16.11	odd	4		1600.2.l.c.1201.1		2
20.3	even	4		320.2.q.a.209.1		2
20.7	even	4		320.2.q.b.209.1		2
20.19	odd	2		1600.2.l.b.401.1		2
40.3	even	4		640.2.q.d.289.1		2
40.13	odd	4		640.2.q.b.289.1		2
40.27	even	4		640.2.q.a.289.1		2
40.37	odd	4		640.2.q.c.289.1		2
80.3	even	4		640.2.q.a.609.1		2
80.13	odd	4		640.2.q.c.609.1		2
80.27	even	4		320.2.q.a.49.1		2
80.37	odd	4		80.2.q.a.69.1	yes	2
80.43	even	4		320.2.q.b.49.1		2
80.53	odd	4		80.2.q.b.69.1	yes	2
80.59	odd	4		1600.2.l.b.1201.1		2
80.67	even	4		640.2.q.d.609.1		2
80.69	even	4		400.2.l.b.101.1		2
80.77	odd	4		640.2.q.b.609.1		2
240.53	even	4		720.2.bm.a.469.1		2
240.197	even	4		720.2.bm.b.469.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.q.a.29.1	✓	2	5.3	odd	4
80.2.q.a.69.1	yes	2	80.37	odd	4
80.2.q.b.29.1	yes	2	5.2	odd	4
80.2.q.b.69.1	yes	2	80.53	odd	4
320.2.q.a.49.1		2	80.27	even	4
320.2.q.a.209.1		2	20.3	even	4
320.2.q.b.49.1		2	80.43	even	4
320.2.q.b.209.1		2	20.7	even	4
400.2.l.a.101.1		2	16.5	even	4	inner
400.2.l.a.301.1		2	1.1	even	1	trivial
400.2.l.b.101.1		2	80.69	even	4
400.2.l.b.301.1		2	5.4	even	2
640.2.q.a.289.1		2	40.27	even	4
640.2.q.a.609.1		2	80.3	even	4
640.2.q.b.289.1		2	40.13	odd	4
640.2.q.b.609.1		2	80.77	odd	4
640.2.q.c.289.1		2	40.37	odd	4
640.2.q.c.609.1		2	80.13	odd	4
640.2.q.d.289.1		2	40.3	even	4
640.2.q.d.609.1		2	80.67	even	4
720.2.bm.a.109.1		2	15.2	even	4
720.2.bm.a.469.1		2	240.53	even	4
720.2.bm.b.109.1		2	15.8	even	4
720.2.bm.b.469.1		2	240.197	even	4
1600.2.l.b.401.1		2	20.19	odd	2
1600.2.l.b.1201.1		2	80.59	odd	4
1600.2.l.c.401.1		2	4.3	odd	2
1600.2.l.c.1201.1		2	16.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +(-1.00000 - 1.00000i) q^{3} +2.00000i q^{4} +2.00000i q^{6} +(2.00000 - 2.00000i) q^{8} -1.00000i q^{9} +(-3.00000 + 3.00000i) q^{11} +(2.00000 - 2.00000i) q^{12} +(-3.00000 - 3.00000i) q^{13} -4.00000 q^{16} -4.00000 q^{17} +(-1.00000 + 1.00000i) q^{18} +(1.00000 + 1.00000i) q^{19} +6.00000 q^{22} +8.00000i q^{23} -4.00000 q^{24} +6.00000i q^{26} +(-4.00000 + 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{29} +(4.00000 + 4.00000i) q^{32} +6.00000 q^{33} +(4.00000 + 4.00000i) q^{34} +2.00000 q^{36} +(-3.00000 + 3.00000i) q^{37} -2.00000i q^{38} +6.00000i q^{39} +(3.00000 - 3.00000i) q^{43} +(-6.00000 - 6.00000i) q^{44} +(8.00000 - 8.00000i) q^{46} -2.00000 q^{47} +(4.00000 + 4.00000i) q^{48} +7.00000 q^{49} +(4.00000 + 4.00000i) q^{51} +(6.00000 - 6.00000i) q^{52} +(-9.00000 + 9.00000i) q^{53} +8.00000 q^{54} -2.00000i q^{57} +6.00000i q^{58} +(-9.00000 + 9.00000i) q^{59} +(-5.00000 - 5.00000i) q^{61} -8.00000i q^{64} +(-6.00000 - 6.00000i) q^{66} +(-3.00000 - 3.00000i) q^{67} -8.00000i q^{68} +(8.00000 - 8.00000i) q^{69} -6.00000i q^{71} +(-2.00000 - 2.00000i) q^{72} -6.00000i q^{73} +6.00000 q^{74} +(-2.00000 + 2.00000i) q^{76} +(6.00000 - 6.00000i) q^{78} -8.00000 q^{79} +5.00000 q^{81} +(-9.00000 - 9.00000i) q^{83} -6.00000 q^{86} +6.00000i q^{87} +12.0000i q^{88} -12.0000i q^{89} -16.0000 q^{92} +(2.00000 + 2.00000i) q^{94} -8.00000i q^{96} +12.0000 q^{97} +(-7.00000 - 7.00000i) q^{98} +(3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 4 q^{8} - 6 q^{11} + 4 q^{12} - 6 q^{13} - 8 q^{16} - 8 q^{17} - 2 q^{18} + 2 q^{19} + 12 q^{22} - 8 q^{24} - 8 q^{27} - 6 q^{29} + 8 q^{32} + 12 q^{33} + 8 q^{34} + 4 q^{36} - 6 q^{37}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^8 - 6 * q^11 + 4 * q^12 - 6 * q^13 - 8 * q^16 - 8 * q^17 - 2 * q^18 + 2 * q^19 + 12 * q^22 - 8 * q^24 - 8 * q^27 - 6 * q^29 + 8 * q^32 + 12 * q^33 + 8 * q^34 + 4 * q^36 - 6 * q^37 + 6 * q^43 - 12 * q^44 + 16 * q^46 - 4 * q^47 + 8 * q^48 + 14 * q^49 + 8 * q^51 + 12 * q^52 - 18 * q^53 + 16 * q^54 - 18 * q^59 - 10 * q^61 - 12 * q^66 - 6 * q^67 + 16 * q^69 - 4 * q^72 + 12 * q^74 - 4 * q^76 + 12 * q^78 - 16 * q^79 + 10 * q^81 - 18 * q^83 - 12 * q^86 - 32 * q^92 + 4 * q^94 + 24 * q^97 - 14 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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