LMFDB - Embedded newform 52.2.h.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [52,2,Mod(17,52)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(52, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("52.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 52 = 2^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	52.h (of order $6$, 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:	$0.415222090511$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	52.17
Dual form			52.2.h.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +(-4.50000 - 2.59808i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(4.50000 - 2.59808i) q^{19} +1.73205i q^{21} +(-1.50000 + 2.59808i) q^{23} +5.00000 q^{25} +5.00000 q^{27} +(4.50000 - 7.79423i) q^{29} +3.46410i q^{31} +(-4.50000 + 2.59808i) q^{33} +(-4.50000 - 2.59808i) q^{37} +(2.50000 + 2.59808i) q^{39} +(-4.50000 - 2.59808i) q^{41} +(-2.50000 - 4.33013i) q^{43} +10.3923i q^{47} +(-2.00000 + 3.46410i) q^{49} -3.00000 q^{51} -6.00000 q^{53} -5.19615i q^{57} +(4.50000 - 2.59808i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-3.00000 - 1.73205i) q^{63} +(1.50000 + 0.866025i) q^{67} +(1.50000 + 2.59808i) q^{69} +(4.50000 - 2.59808i) q^{71} -6.92820i q^{73} +(2.50000 - 4.33013i) q^{75} +9.00000 q^{77} +4.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +10.3923i q^{83} +(-4.50000 - 7.79423i) q^{87} +(13.5000 + 7.79423i) q^{89} +(-1.50000 - 6.06218i) q^{91} +(3.00000 + 1.73205i) q^{93} +(-10.5000 + 6.06218i) q^{97} -10.3923i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^3 - 3 * q^7 + 2 * q^9	$ 2 q + q^{3} - 3 q^{7} + 2 q^{9} - 9 q^{11} - 2 q^{13} - 3 q^{17} + 9 q^{19} - 3 q^{23} + 10 q^{25} + 10 q^{27} + 9 q^{29} - 9 q^{33} - 9 q^{37} + 5 q^{39} - 9 q^{41} - 5 q^{43} - 4 q^{49} - 6 q^{51} - 12 q^{53} + 9 q^{59} + 5 q^{61} - 6 q^{63} + 3 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{75} + 18 q^{77} + 8 q^{79} - q^{81} - 9 q^{87} + 27 q^{89} - 3 q^{91} + 6 q^{93} - 21 q^{97}+O(q^{100}) $ 2 * q + q^3 - 3 * q^7 + 2 * q^9 - 9 * q^11 - 2 * q^13 - 3 * q^17 + 9 * q^19 - 3 * q^23 + 10 * q^25 + 10 * q^27 + 9 * q^29 - 9 * q^33 - 9 * q^37 + 5 * q^39 - 9 * q^41 - 5 * q^43 - 4 * q^49 - 6 * q^51 - 12 * q^53 + 9 * q^59 + 5 * q^61 - 6 * q^63 + 3 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^75 + 18 * q^77 + 8 * q^79 - q^81 - 9 * q^87 + 27 * q^89 - 3 * q^91 + 6 * q^93 - 21 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/52\mathbb{Z}\right)^\times$.

$n$	$27$	$41$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i	−0.684819	−	0.728714i	$-0.740119\pi$
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i	0.973494	+	0.228714i	$0.0734519\pi$
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−1.50000	+	0.866025i	−0.566947	+	0.327327i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	−4.50000	−	2.59808i	−1.35680	−	0.783349i	−0.367610	−	0.929980i	$-0.619824\pi$
$11$	−4.50000	−	2.59808i	−1.35680	−	0.783349i	−0.989191	+	0.146631i	$0.953157\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	0.624780	−	0.780801i	$-0.285189\pi$
$17$	−1.50000	−	2.59808i	−0.363803	−	0.630126i	−0.988583	+	0.150675i	$0.951855\pi$
$18$		0			0
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$			1.73205i			0.377964i
$22$		0			0
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	−0.978961	−	0.204046i	$-0.934591\pi$
$23$	−1.50000	+	2.59808i	−0.312772	+	0.541736i	0.666190	+	0.745782i	$0.267924\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	4.50000	−	7.79423i	0.835629	−	1.44735i	−0.0578882	−	0.998323i	$-0.518437\pi$
$29$	4.50000	−	7.79423i	0.835629	−	1.44735i	0.893517	−	0.449029i	$-0.148230\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$		0			0
$33$	−4.50000	+	2.59808i	−0.783349	+	0.452267i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.50000	−	2.59808i	−0.739795	−	0.427121i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	−4.50000	−	2.59808i	−0.739795	−	0.427121i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	2.50000	+	2.59808i	0.400320	+	0.416025i
$40$		0			0
$41$	−4.50000	−	2.59808i	−0.702782	−	0.405751i	0.105601	−	0.994409i	$-0.466323\pi$
$41$	−4.50000	−	2.59808i	−0.702782	−	0.405751i	−0.808383	+	0.588657i	$0.799657\pi$
$42$		0			0
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	0.609994	−	0.792406i	$-0.291172\pi$
$43$	−2.50000	−	4.33013i	−0.381246	−	0.660338i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.3923i			1.51587i	0.652328	+	0.757937i	$0.273792\pi$
$47$			10.3923i			1.51587i	−0.652328	+	0.757937i	$0.726208\pi$
$48$		0			0
$49$	−2.00000	+	3.46410i	−0.285714	+	0.494872i
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	5.19615i		−	0.688247i
$58$		0			0
$59$	4.50000	−	2.59808i	0.585850	−	0.338241i	−0.177605	−	0.984102i	$-0.556835\pi$
$59$	4.50000	−	2.59808i	0.585850	−	0.338241i	0.763455	+	0.645861i	$0.223502\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$		0			0
$63$	−3.00000	−	1.73205i	−0.377964	−	0.218218i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.50000	+	0.866025i	0.183254	+	0.105802i	0.588821	−	0.808264i	$-0.299592\pi$
$67$	1.50000	+	0.866025i	0.183254	+	0.105802i	−0.405567	+	0.914066i	$0.632926\pi$
$68$		0			0
$69$	1.50000	+	2.59808i	0.180579	+	0.312772i
$70$		0			0
$71$	4.50000	−	2.59808i	0.534052	−	0.308335i	−0.208613	−	0.977998i	$-0.566895\pi$
$71$	4.50000	−	2.59808i	0.534052	−	0.308335i	0.742665	+	0.669663i	$0.233562\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$	2.50000	−	4.33013i	0.288675	−	0.500000i
$76$		0			0
$77$	9.00000			1.02565
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$			10.3923i			1.14070i	0.821401	+	0.570352i	$0.193193\pi$
$83$			10.3923i			1.14070i	−0.821401	+	0.570352i	$0.806807\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−4.50000	−	7.79423i	−0.482451	−	0.835629i
$88$		0			0
$89$	13.5000	+	7.79423i	1.43100	+	0.826187i	0.997197	−	0.0748225i	$-0.0238390\pi$
$89$	13.5000	+	7.79423i	1.43100	+	0.826187i	0.433800	+	0.901009i	$0.357172\pi$
$90$		0			0
$91$	−1.50000	−	6.06218i	−0.157243	−	0.635489i
$92$		0			0
$93$	3.00000	+	1.73205i	0.311086	+	0.179605i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−10.5000	+	6.06218i	−1.06611	+	0.615521i	−0.927117	−	0.374772i	$-0.877721\pi$
$97$	−10.5000	+	6.06218i	−1.06611	+	0.615521i	−0.138996	+	0.990293i	$0.544388\pi$
$98$		0			0
$99$		−	10.3923i		−	1.04447i
$100$		0			0
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	−0.930953	−	0.365140i	$-0.881021\pi$
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	0.781697	+	0.623658i	$0.214354\pi$
$102$		0			0
$103$	−16.0000			−1.57653			−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000			−1.57653			−0.788263	+	0.615338i	$0.789020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.50000	+	12.9904i	−0.725052	+	1.25583i	0.233900	+	0.972261i	$0.424851\pi$
$107$	−7.50000	+	12.9904i	−0.725052	+	1.25583i	−0.958952	+	0.283567i	$0.908482\pi$
$108$		0			0
$109$		−	13.8564i		−	1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$		−	13.8564i		−	1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$		0			0
$111$	−4.50000	+	2.59808i	−0.427121	+	0.246598i
$112$		0			0
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	0.786806	−	0.617200i	$-0.211733\pi$
$113$	−1.50000	−	2.59808i	−0.141108	−	0.244406i	−0.927914	+	0.372794i	$0.878400\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.00000	+	1.73205i	−0.647150	+	0.160128i
$118$		0			0
$119$	4.50000	+	2.59808i	0.412514	+	0.238165i
$120$		0			0
$121$	8.00000	+	13.8564i	0.727273	+	1.25967i
$122$		0			0
$123$	−4.50000	+	2.59808i	−0.405751	+	0.234261i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.50000	−	14.7224i	0.754253	−	1.30640i	−0.191492	−	0.981494i	$-0.561333\pi$
$127$	8.50000	−	14.7224i	0.754253	−	1.30640i	0.945745	−	0.324910i	$-0.105334\pi$
$128$		0			0
$129$	−5.00000			−0.440225
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−4.50000	+	7.79423i	−0.390199	+	0.675845i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.5000	−	7.79423i	1.15338	−	0.665906i	0.203674	−	0.979039i	$-0.434712\pi$
$137$	13.5000	−	7.79423i	1.15338	−	0.665906i	0.949709	+	0.313133i	$0.101379\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$	9.00000	+	5.19615i	0.757937	+	0.437595i
$142$		0			0
$143$	13.5000	−	12.9904i	1.12893	−	1.08631i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.00000	+	3.46410i	0.164957	+	0.285714i
$148$		0			0
$149$	−4.50000	+	2.59808i	−0.368654	+	0.212843i	−0.672870	−	0.739760i	$-0.734939\pi$
$149$	−4.50000	+	2.59808i	−0.368654	+	0.212843i	0.304216	+	0.952603i	$0.401606\pi$
$150$		0			0
$151$		−	10.3923i		−	0.845714i	−0.906196	−	0.422857i	$-0.861027\pi$
$151$		−	10.3923i		−	0.845714i	0.906196	−	0.422857i	$-0.138973\pi$
$152$		0			0
$153$	3.00000	−	5.19615i	0.242536	−	0.420084i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$	−3.00000	+	5.19615i	−0.237915	+	0.412082i
$160$		0			0
$161$		−	5.19615i		−	0.409514i
$162$		0			0
$163$	10.5000	−	6.06218i	0.822423	−	0.474826i	−0.0288280	−	0.999584i	$-0.509178\pi$
$163$	10.5000	−	6.06218i	0.822423	−	0.474826i	0.851251	+	0.524758i	$0.175844\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.5000	+	7.79423i	1.04466	+	0.603136i	0.921150	−	0.389208i	$-0.127251\pi$
$167$	13.5000	+	7.79423i	1.04466	+	0.603136i	0.123511	+	0.992343i	$0.460584\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$	9.00000	+	5.19615i	0.688247	+	0.397360i
$172$		0			0
$173$	−1.50000	−	2.59808i	−0.114043	−	0.197528i	0.803354	−	0.595502i	$-0.203047\pi$
$173$	−1.50000	−	2.59808i	−0.114043	−	0.197528i	−0.917397	+	0.397974i	$0.869713\pi$
$174$		0			0
$175$	−7.50000	+	4.33013i	−0.566947	+	0.327327i
$176$		0			0
$177$		−	5.19615i		−	0.390567i
$178$		0			0
$179$	−7.50000	+	12.9904i	−0.560576	+	0.970947i	0.436870	+	0.899525i	$0.356087\pi$
$179$	−7.50000	+	12.9904i	−0.560576	+	0.970947i	−0.997446	+	0.0714220i	$0.977246\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	5.00000			0.369611
$184$		0			0
$185$		0			0
$186$		0			0
$187$			15.5885i			1.13994i
$188$		0			0
$189$	−7.50000	+	4.33013i	−0.545545	+	0.314970i
$190$		0			0
$191$	−10.5000	−	18.1865i	−0.759753	−	1.31593i	−0.942976	−	0.332860i	$-0.891986\pi$
$191$	−10.5000	−	18.1865i	−0.759753	−	1.31593i	0.183223	−	0.983071i	$-0.441347\pi$
$192$		0			0
$193$	7.50000	+	4.33013i	0.539862	+	0.311689i	0.745023	−	0.667039i	$-0.232439\pi$
$193$	7.50000	+	4.33013i	0.539862	+	0.311689i	−0.205161	+	0.978728i	$0.565772\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−22.5000	−	12.9904i	−1.60306	−	0.925526i	−0.990871	−	0.134814i	$-0.956956\pi$
$197$	−22.5000	−	12.9904i	−1.60306	−	0.925526i	−0.612188	−	0.790712i	$-0.709710\pi$
$198$		0			0
$199$	−0.500000	−	0.866025i	−0.0354441	−	0.0613909i	0.847759	−	0.530381i	$-0.177951\pi$
$199$	−0.500000	−	0.866025i	−0.0354441	−	0.0613909i	−0.883203	+	0.468990i	$0.844618\pi$
$200$		0			0
$201$	1.50000	−	0.866025i	0.105802	−	0.0610847i
$202$		0			0
$203$			15.5885i			1.09410i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−6.00000			−0.417029
$208$		0			0
$209$	−27.0000			−1.86763
$210$		0			0
$211$	−3.50000	+	6.06218i	−0.240950	+	0.417338i	−0.960985	−	0.276600i	$-0.910792\pi$
$211$	−3.50000	+	6.06218i	−0.240950	+	0.417338i	0.720035	+	0.693938i	$0.244126\pi$
$212$		0			0
$213$		−	5.19615i		−	0.356034i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.00000	−	5.19615i	−0.203653	−	0.352738i
$218$		0			0
$219$	−6.00000	−	3.46410i	−0.405442	−	0.234082i
$220$		0			0
$221$	10.5000	−	2.59808i	0.706306	−	0.174766i
$222$		0			0
$223$	13.5000	+	7.79423i	0.904027	+	0.521940i	0.878504	−	0.477734i	$-0.158542\pi$
$223$	13.5000	+	7.79423i	0.904027	+	0.521940i	0.0255224	+	0.999674i	$0.491875\pi$
$224$		0			0
$225$	5.00000	+	8.66025i	0.333333	+	0.577350i
$226$		0			0
$227$	4.50000	−	2.59808i	0.298675	−	0.172440i	−0.343172	−	0.939272i	$-0.611501\pi$
$227$	4.50000	−	2.59808i	0.298675	−	0.172440i	0.641848	+	0.766832i	$0.278168\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	4.50000	−	7.79423i	0.296078	−	0.512823i
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.00000	−	3.46410i	0.129914	−	0.225018i
$238$		0			0
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$	−4.50000	+	2.59808i	−0.289870	+	0.167357i	−0.637883	−	0.770133i	$-0.720190\pi$
$241$	−4.50000	+	2.59808i	−0.289870	+	0.167357i	0.348013	+	0.937490i	$0.386857\pi$
$242$		0			0
$243$	8.00000	+	13.8564i	0.513200	+	0.888889i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.50000	+	18.1865i	0.286328	+	1.15718i
$248$		0			0
$249$	9.00000	+	5.19615i	0.570352	+	0.329293i
$250$		0			0
$251$	−10.5000	−	18.1865i	−0.662754	−	1.14792i	−0.979889	−	0.199543i	$-0.936054\pi$
$251$	−10.5000	−	18.1865i	−0.662754	−	1.14792i	0.317135	−	0.948380i	$-0.397279\pi$
$252$		0			0
$253$	13.5000	−	7.79423i	0.848738	−	0.490019i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	−0.909010	−	0.416775i	$-0.863160\pi$
$257$	−1.50000	+	2.59808i	−0.0935674	+	0.162064i	0.815442	+	0.578838i	$0.196494\pi$
$258$		0			0
$259$	9.00000			0.559233
$260$		0			0
$261$	18.0000			1.11417
$262$		0			0
$263$	10.5000	−	18.1865i	0.647458	−	1.12143i	−0.336270	−	0.941766i	$-0.609166\pi$
$263$	10.5000	−	18.1865i	0.647458	−	1.12143i	0.983728	−	0.179664i	$-0.0575011\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	13.5000	−	7.79423i	0.826187	−	0.476999i
$268$		0			0
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	0.969976	−	0.243201i	$-0.0781974\pi$
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	−0.695606	+	0.718423i	$0.744864\pi$
$270$		0			0
$271$	−10.5000	−	6.06218i	−0.637830	−	0.368251i	0.145948	−	0.989292i	$-0.453377\pi$
$271$	−10.5000	−	6.06218i	−0.637830	−	0.368251i	−0.783778	+	0.621041i	$0.786710\pi$
$272$		0			0
$273$	−6.00000	−	1.73205i	−0.363137	−	0.104828i
$274$		0			0
$275$	−22.5000	−	12.9904i	−1.35680	−	0.783349i
$276$		0			0
$277$	0.500000	+	0.866025i	0.0300421	+	0.0520344i	0.880656	−	0.473757i	$-0.157103\pi$
$277$	0.500000	+	0.866025i	0.0300421	+	0.0520344i	−0.850613	+	0.525792i	$0.823769\pi$
$278$		0			0
$279$	−6.00000	+	3.46410i	−0.359211	+	0.207390i
$280$		0			0
$281$			20.7846i			1.23991i	0.784639	+	0.619953i	$0.212848\pi$
$281$			20.7846i			1.23991i	−0.784639	+	0.619953i	$0.787152\pi$
$282$		0			0
$283$	−3.50000	+	6.06218i	−0.208053	+	0.360359i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−3.50000	+	6.06218i	−0.208053	+	0.360359i	0.743048	+	0.669238i	$0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.00000			0.531253
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$			12.1244i			0.710742i
$292$		0			0
$293$	−4.50000	+	2.59808i	−0.262893	+	0.151781i	−0.625653	−	0.780101i	$-0.715168\pi$
$293$	−4.50000	+	2.59808i	−0.262893	+	0.151781i	0.362761	+	0.931882i	$0.381834\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−22.5000	−	12.9904i	−1.30558	−	0.753778i
$298$		0			0
$299$	−7.50000	−	7.79423i	−0.433736	−	0.450752i
$300$		0			0
$301$	7.50000	+	4.33013i	0.432293	+	0.249584i
$302$		0			0
$303$	1.50000	+	2.59808i	0.0861727	+	0.149256i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	17.3205i		−	0.988534i	−0.869310	−	0.494267i	$-0.835437\pi$
$307$		−	17.3205i		−	0.988534i	0.869310	−	0.494267i	$-0.164563\pi$
$308$		0			0
$309$	−8.00000	+	13.8564i	−0.455104	+	0.788263i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	22.0000			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			20.7846i			1.16738i	0.811977	+	0.583690i	$0.198392\pi$
$317$			20.7846i			1.16738i	−0.811977	+	0.583690i	$0.801608\pi$
$318$		0			0
$319$	−40.5000	+	23.3827i	−2.26756	+	1.30918i
$320$		0			0
$321$	7.50000	+	12.9904i	0.418609	+	0.725052i
$322$		0			0
$323$	−13.5000	−	7.79423i	−0.751160	−	0.433682i
$324$		0			0
$325$	−5.00000	+	17.3205i	−0.277350	+	0.960769i
$326$		0			0
$327$	−12.0000	−	6.92820i	−0.663602	−	0.383131i
$328$		0			0
$329$	−9.00000	−	15.5885i	−0.496186	−	0.859419i
$330$		0			0
$331$	10.5000	−	6.06218i	0.577132	−	0.333207i	−0.182861	−	0.983139i	$-0.558536\pi$
$331$	10.5000	−	6.06218i	0.577132	−	0.333207i	0.759993	+	0.649931i	$0.225202\pi$
$332$		0			0
$333$		−	10.3923i		−	0.569495i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	−3.00000			−0.162938
$340$		0			0
$341$	9.00000	−	15.5885i	0.487377	−	0.844162i
$342$		0			0
$343$		−	19.0526i		−	1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	0.903475	−	0.428640i	$-0.141007\pi$
$347$	1.50000	+	2.59808i	0.0805242	+	0.139472i	−0.822951	+	0.568112i	$0.807674\pi$
$348$		0			0
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.0115044	−	0.999934i	$-0.503662\pi$
$349$	−16.5000	−	9.52628i	−0.883225	−	0.509930i	−0.871720	+	0.490004i	$0.836995\pi$
$350$		0			0
$351$	−5.00000	+	17.3205i	−0.266880	+	0.924500i
$352$		0			0
$353$	13.5000	+	7.79423i	0.718532	+	0.414845i	0.814212	−	0.580567i	$-0.197169\pi$
$353$	13.5000	+	7.79423i	0.718532	+	0.414845i	−0.0956798	+	0.995412i	$0.530502\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.50000	−	2.59808i	0.238165	−	0.137505i
$358$		0			0
$359$			31.1769i			1.64545i	0.568436	+	0.822727i	$0.307549\pi$
$359$			31.1769i			1.64545i	−0.568436	+	0.822727i	$0.692451\pi$
$360$		0			0
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$	16.0000			0.839782
$364$		0			0
$365$		0			0
$366$		0			0
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	−0.645003	−	0.764180i	$-0.723144\pi$
$367$	6.50000	−	11.2583i	0.339297	−	0.587680i	0.984301	+	0.176500i	$0.0564774\pi$
$368$		0			0
$369$		−	10.3923i		−	0.541002i
$370$		0			0
$371$	9.00000	−	5.19615i	0.467257	−	0.269771i
$372$		0			0
$373$	−3.50000	−	6.06218i	−0.181223	−	0.313888i	0.761074	−	0.648665i	$-0.224672\pi$
$373$	−3.50000	−	6.06218i	−0.181223	−	0.313888i	−0.942297	+	0.334777i	$0.891339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	22.5000	+	23.3827i	1.15881	+	1.20427i
$378$		0			0
$379$	7.50000	+	4.33013i	0.385249	+	0.222424i	0.680100	−	0.733120i	$-0.261937\pi$
$379$	7.50000	+	4.33013i	0.385249	+	0.222424i	−0.294850	+	0.955543i	$0.595270\pi$
$380$		0			0
$381$	−8.50000	−	14.7224i	−0.435468	−	0.754253i
$382$		0			0
$383$	22.5000	−	12.9904i	1.14970	−	0.663777i	0.200883	−	0.979615i	$-0.435619\pi$
$383$	22.5000	−	12.9904i	1.14970	−	0.663777i	0.948813	+	0.315838i	$0.102286\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	5.00000	−	8.66025i	0.254164	−	0.440225i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	9.00000			0.455150
$392$		0			0
$393$	−6.00000	+	10.3923i	−0.302660	+	0.524222i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.50000	−	0.866025i	0.0752828	−	0.0434646i	−0.461886	−	0.886939i	$-0.652827\pi$
$397$	1.50000	−	0.866025i	0.0752828	−	0.0434646i	0.537169	+	0.843475i	$0.319494\pi$
$398$		0			0
$399$	4.50000	+	7.79423i	0.225282	+	0.390199i
$400$		0			0
$401$	−4.50000	−	2.59808i	−0.224719	−	0.129742i	0.383414	−	0.923576i	$-0.374748\pi$
$401$	−4.50000	−	2.59808i	−0.224719	−	0.129742i	−0.608134	+	0.793835i	$0.708081\pi$
$402$		0			0
$403$	−12.0000	−	3.46410i	−0.597763	−	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.5000	+	23.3827i	0.669170	+	1.15904i
$408$		0			0
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	−0.302979	−	0.952997i	$-0.597981\pi$
$409$	7.50000	−	4.33013i	0.370851	−	0.214111i	0.673830	+	0.738886i	$0.264648\pi$
$410$		0			0
$411$		−	15.5885i		−	0.768922i
$412$		0			0
$413$	−4.50000	+	7.79423i	−0.221431	+	0.383529i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.00000			0.342791
$418$		0			0
$419$	10.5000	−	18.1865i	0.512959	−	0.888470i	−0.486928	−	0.873442i	$-0.661883\pi$
$419$	10.5000	−	18.1865i	0.512959	−	0.888470i	0.999887	−	0.0150285i	$-0.00478389\pi$
$420$		0			0
$421$			13.8564i			0.675320i	0.941268	+	0.337660i	$0.109635\pi$
$421$			13.8564i			0.675320i	−0.941268	+	0.337660i	$0.890365\pi$
$422$		0			0
$423$	−18.0000	+	10.3923i	−0.875190	+	0.505291i
$424$		0			0
$425$	−7.50000	−	12.9904i	−0.363803	−	0.630126i
$426$		0			0
$427$	−7.50000	−	4.33013i	−0.362950	−	0.209550i
$428$		0			0
$429$	−4.50000	−	18.1865i	−0.217262	−	0.878054i
$430$		0			0
$431$	−4.50000	−	2.59808i	−0.216757	−	0.125145i	0.387691	−	0.921790i	$-0.373273\pi$
$431$	−4.50000	−	2.59808i	−0.216757	−	0.125145i	−0.604448	+	0.796645i	$0.706606\pi$
$432$		0			0
$433$	−9.50000	−	16.4545i	−0.456541	−	0.790752i	0.542234	−	0.840227i	$-0.317578\pi$
$433$	−9.50000	−	16.4545i	−0.456541	−	0.790752i	−0.998775	+	0.0494752i	$0.984245\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			15.5885i			0.745697i
$438$		0			0
$439$	−11.5000	+	19.9186i	−0.548865	+	0.950662i	0.449488	+	0.893287i	$0.351607\pi$
$439$	−11.5000	+	19.9186i	−0.548865	+	0.950662i	−0.998353	+	0.0573756i	$0.981727\pi$
$440$		0			0
$441$	−8.00000			−0.380952
$442$		0			0
$443$	−12.0000			−0.570137			−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000			−0.570137			−0.285069	+	0.958507i	$0.592016\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			5.19615i			0.245770i
$448$		0			0
$449$	−4.50000	+	2.59808i	−0.212368	+	0.122611i	−0.602411	−	0.798186i	$-0.705793\pi$
$449$	−4.50000	+	2.59808i	−0.212368	+	0.122611i	0.390043	+	0.920796i	$0.372460\pi$
$450$		0			0
$451$	13.5000	+	23.3827i	0.635690	+	1.10105i
$452$		0			0
$453$	−9.00000	−	5.19615i	−0.422857	−	0.244137i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−34.5000	−	19.9186i	−1.61384	−	0.931752i	−0.988469	−	0.151426i	$-0.951613\pi$
$457$	−34.5000	−	19.9186i	−1.61384	−	0.931752i	−0.625373	−	0.780326i	$-0.715053\pi$
$458$		0			0
$459$	−7.50000	−	12.9904i	−0.350070	−	0.606339i
$460$		0			0
$461$	−22.5000	+	12.9904i	−1.04793	+	0.605022i	−0.922069	−	0.387026i	$-0.873503\pi$
$461$	−22.5000	+	12.9904i	−1.04793	+	0.605022i	−0.125860	+	0.992048i	$0.540169\pi$
$462$		0			0
$463$			17.3205i			0.804952i	0.915430	+	0.402476i	$0.131850\pi$
$463$			17.3205i			0.804952i	−0.915430	+	0.402476i	$0.868150\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$		0			0
$469$	−3.00000			−0.138527
$470$		0			0
$471$	1.00000	−	1.73205i	0.0460776	−	0.0798087i
$472$		0			0
$473$			25.9808i			1.19460i
$474$		0			0
$475$	22.5000	−	12.9904i	1.03237	−	0.596040i
$476$		0			0
$477$	−6.00000	−	10.3923i	−0.274721	−	0.475831i
$478$		0			0
$479$	31.5000	+	18.1865i	1.43927	+	0.830964i	0.997799	−	0.0663107i	$-0.0211229\pi$
$479$	31.5000	+	18.1865i	1.43927	+	0.830964i	0.441473	+	0.897275i	$0.354456\pi$
$480$		0			0
$481$	13.5000	−	12.9904i	0.615547	−	0.592310i
$482$		0			0
$483$	−4.50000	−	2.59808i	−0.204757	−	0.118217i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−31.5000	+	18.1865i	−1.42740	+	0.824110i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−31.5000	+	18.1865i	−1.42740	+	0.824110i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$		−	12.1244i		−	0.548282i
$490$		0			0
$491$	−13.5000	+	23.3827i	−0.609246	+	1.05525i	0.382118	+	0.924113i	$0.375195\pi$
$491$	−13.5000	+	23.3827i	−0.609246	+	1.05525i	−0.991365	+	0.131132i	$0.958139\pi$
$492$		0			0
$493$	−27.0000			−1.21602
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.50000	+	7.79423i	−0.201853	+	0.349619i
$498$		0			0
$499$		−	31.1769i		−	1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$		−	31.1769i		−	1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		0			0
$501$	13.5000	−	7.79423i	0.603136	−	0.348220i
$502$		0			0
$503$	−10.5000	−	18.1865i	−0.468172	−	0.810897i	0.531167	−	0.847267i	$-0.321754\pi$
$503$	−10.5000	−	18.1865i	−0.468172	−	0.810897i	−0.999338	+	0.0363700i	$0.988421\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−11.5000	+	6.06218i	−0.510733	+	0.269231i
$508$		0			0
$509$	31.5000	+	18.1865i	1.39621	+	0.806104i	0.993993	−	0.109439i	$-0.0349055\pi$
$509$	31.5000	+	18.1865i	1.39621	+	0.806104i	0.402219	+	0.915543i	$0.368239\pi$
$510$		0			0
$511$	6.00000	+	10.3923i	0.265424	+	0.459728i
$512$		0			0
$513$	22.5000	−	12.9904i	0.993399	−	0.573539i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	27.0000	−	46.7654i	1.18746	−	2.05674i
$518$		0			0
$519$	−3.00000			−0.131685
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	−5.50000	+	9.52628i	−0.240498	+	0.416555i	−0.960856	−	0.277047i	$-0.910644\pi$
$523$	−5.50000	+	9.52628i	−0.240498	+	0.416555i	0.720358	+	0.693602i	$0.243977\pi$
$524$		0			0
$525$			8.66025i			0.377964i
$526$		0			0
$527$	9.00000	−	5.19615i	0.392046	−	0.226348i
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$		0			0
$531$	9.00000	+	5.19615i	0.390567	+	0.225494i
$532$		0			0
$533$	13.5000	−	12.9904i	0.584750	−	0.562676i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	7.50000	+	12.9904i	0.323649	+	0.560576i
$538$		0			0
$539$	18.0000	−	10.3923i	0.775315	−	0.447628i
$540$		0			0
$541$			6.92820i			0.297867i	0.988847	+	0.148933i	$0.0475840\pi$
$541$			6.92820i			0.297867i	−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	1.00000	−	1.73205i	0.0429141	−	0.0743294i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.00000			−0.171028			−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000			−0.171028			−0.0855138	+	0.996337i	$0.527253\pi$
$548$		0			0
$549$	−5.00000	+	8.66025i	−0.213395	+	0.369611i
$550$		0			0
$551$		−	46.7654i		−	1.99227i
$552$		0			0
$553$	−6.00000	+	3.46410i	−0.255146	+	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.5000	+	7.79423i	0.572013	+	0.330252i	0.757953	−	0.652309i	$-0.226200\pi$
$557$	13.5000	+	7.79423i	0.572013	+	0.330252i	−0.185940	+	0.982561i	$0.559533\pi$
$558$		0			0
$559$	17.5000	−	4.33013i	0.740171	−	0.183145i
$560$		0			0
$561$	13.5000	+	7.79423i	0.569970	+	0.329073i
$562$		0			0
$563$	7.50000	+	12.9904i	0.316087	+	0.547479i	0.979668	−	0.200625i	$-0.0642974\pi$
$563$	7.50000	+	12.9904i	0.316087	+	0.547479i	−0.663581	+	0.748105i	$0.730964\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	1.73205i		−	0.0727393i
$568$		0			0
$569$	4.50000	−	7.79423i	0.188650	−	0.326751i	−0.756151	−	0.654398i	$-0.772922\pi$
$569$	4.50000	−	7.79423i	0.188650	−	0.326751i	0.944800	+	0.327647i	$0.106256\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	−21.0000			−0.877288
$574$		0			0
$575$	−7.50000	+	12.9904i	−0.312772	+	0.541736i
$576$		0			0
$577$			20.7846i			0.865275i	0.901568	+	0.432637i	$0.142417\pi$
$577$			20.7846i			0.865275i	−0.901568	+	0.432637i	$0.857583\pi$
$578$		0			0
$579$	7.50000	−	4.33013i	0.311689	−	0.179954i
$580$		0			0
$581$	−9.00000	−	15.5885i	−0.373383	−	0.646718i
$582$		0			0
$583$	27.0000	+	15.5885i	1.11823	+	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.5000	−	12.9904i	−0.928674	−	0.536170i	−0.0422823	−	0.999106i	$-0.513463\pi$
$587$	−22.5000	−	12.9904i	−0.928674	−	0.536170i	−0.886392	+	0.462935i	$0.846796\pi$
$588$		0			0
$589$	9.00000	+	15.5885i	0.370839	+	0.642311i
$590$		0			0
$591$	−22.5000	+	12.9904i	−0.925526	+	0.534353i
$592$		0			0
$593$		−	20.7846i		−	0.853522i	−0.904365	−	0.426761i	$-0.859655\pi$
$593$		−	20.7846i		−	0.853522i	0.904365	−	0.426761i	$-0.140345\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.00000			−0.0409273
$598$		0			0
$599$	−12.0000			−0.490307			−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000			−0.490307			−0.245153	+	0.969484i	$0.578838\pi$
$600$		0			0
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	−0.992114	−	0.125336i	$-0.959999\pi$
$601$	−9.50000	+	16.4545i	−0.387513	+	0.671192i	0.604601	+	0.796528i	$0.293332\pi$
$602$		0			0
$603$			3.46410i			0.141069i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−20.5000	−	35.5070i	−0.832069	−	1.44119i	−0.896394	−	0.443257i	$-0.853823\pi$
$607$	−20.5000	−	35.5070i	−0.832069	−	1.44119i	0.0643251	−	0.997929i	$-0.479511\pi$
$608$		0			0
$609$	13.5000	+	7.79423i	0.547048	+	0.315838i
$610$		0			0
$611$	−36.0000	−	10.3923i	−1.45640	−	0.420428i
$612$		0			0
$613$	19.5000	+	11.2583i	0.787598	+	0.454720i	0.839116	−	0.543952i	$-0.183073\pi$
$613$	19.5000	+	11.2583i	0.787598	+	0.454720i	−0.0515185	+	0.998672i	$0.516406\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.5000	+	12.9904i	−0.905816	+	0.522973i	−0.879083	−	0.476670i	$-0.841844\pi$
$617$	−22.5000	+	12.9904i	−0.905816	+	0.522973i	−0.0267333	+	0.999643i	$0.508510\pi$
$618$		0			0
$619$		−	45.0333i		−	1.81004i	−0.425367	−	0.905021i	$-0.639855\pi$
$619$		−	45.0333i		−	1.81004i	0.425367	−	0.905021i	$-0.360145\pi$
$620$		0			0
$621$	−7.50000	+	12.9904i	−0.300965	+	0.521286i
$622$		0			0
$623$	−27.0000			−1.08173
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−13.5000	+	23.3827i	−0.539138	+	0.933815i
$628$		0			0
$629$			15.5885i			0.621552i
$630$		0			0
$631$	−7.50000	+	4.33013i	−0.298570	+	0.172380i	−0.641800	−	0.766872i	$-0.721812\pi$
$631$	−7.50000	+	4.33013i	−0.298570	+	0.172380i	0.343230	+	0.939251i	$0.388479\pi$
$632$		0			0
$633$	3.50000	+	6.06218i	0.139113	+	0.240950i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−10.0000	−	10.3923i	−0.396214	−	0.411758i
$638$		0			0
$639$	9.00000	+	5.19615i	0.356034	+	0.205557i
$640$		0			0
$641$	−13.5000	−	23.3827i	−0.533218	−	0.923561i	−0.999247	−	0.0387913i	$-0.987649\pi$
$641$	−13.5000	−	23.3827i	−0.533218	−	0.923561i	0.466029	−	0.884769i	$-0.345684\pi$
$642$		0			0
$643$	−37.5000	+	21.6506i	−1.47886	+	0.853818i	−0.999714	−	0.0239198i	$-0.992385\pi$
$643$	−37.5000	+	21.6506i	−1.47886	+	0.853818i	−0.479142	+	0.877738i	$0.659052\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.5000	−	28.5788i	0.648682	−	1.12355i	−0.334756	−	0.942305i	$-0.608654\pi$
$647$	16.5000	−	28.5788i	0.648682	−	1.12355i	0.983438	−	0.181245i	$-0.0580128\pi$
$648$		0			0
$649$	−27.0000			−1.05984
$650$		0			0
$651$	−6.00000			−0.235159
$652$		0			0
$653$	−13.5000	+	23.3827i	−0.528296	+	0.915035i	0.471160	+	0.882048i	$0.343835\pi$
$653$	−13.5000	+	23.3827i	−0.528296	+	0.915035i	−0.999456	+	0.0329874i	$0.989498\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	12.0000	−	6.92820i	0.468165	−	0.270295i
$658$		0			0
$659$	13.5000	+	23.3827i	0.525885	+	0.910860i	0.999545	+	0.0301523i	$0.00959924\pi$
$659$	13.5000	+	23.3827i	0.525885	+	0.910860i	−0.473660	+	0.880708i	$0.657067\pi$
$660$		0			0
$661$	19.5000	+	11.2583i	0.758462	+	0.437898i	0.828743	−	0.559629i	$-0.189056\pi$
$661$	19.5000	+	11.2583i	0.758462	+	0.437898i	−0.0702812	+	0.997527i	$0.522390\pi$
$662$		0			0
$663$	3.00000	−	10.3923i	0.116510	−	0.403604i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	13.5000	+	23.3827i	0.522722	+	0.905381i
$668$		0			0
$669$	13.5000	−	7.79423i	0.521940	−	0.301342i
$670$		0			0
$671$		−	25.9808i		−	1.00298i
$672$		0			0
$673$	−11.5000	+	19.9186i	−0.443292	+	0.767805i	−0.997932	−	0.0642860i	$-0.979523\pi$
$673$	−11.5000	+	19.9186i	−0.443292	+	0.767805i	0.554639	+	0.832091i	$0.312856\pi$
$674$		0			0
$675$	25.0000			0.962250
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$	10.5000	−	18.1865i	0.402953	−	0.697935i
$680$		0			0
$681$		−	5.19615i		−	0.199117i
$682$		0			0
$683$	4.50000	−	2.59808i	0.172188	−	0.0994126i	−0.411429	−	0.911442i	$-0.634970\pi$
$683$	4.50000	−	2.59808i	0.172188	−	0.0994126i	0.583617	+	0.812029i	$0.301637\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.00000	−	20.7846i	0.228582	−	0.791831i
$690$		0			0
$691$	7.50000	+	4.33013i	0.285313	+	0.164726i	0.635826	−	0.771832i	$-0.280659\pi$
$691$	7.50000	+	4.33013i	0.285313	+	0.164726i	−0.350513	+	0.936558i	$0.613993\pi$
$692$		0			0
$693$	9.00000	+	15.5885i	0.341882	+	0.592157i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			15.5885i			0.590455i
$698$		0			0
$699$	3.00000	−	5.19615i	0.113470	−	0.196537i
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	−27.0000			−1.01832
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	5.19615i		−	0.195421i
$708$		0			0
$709$	25.5000	−	14.7224i	0.957673	−	0.552913i	0.0622167	−	0.998063i	$-0.480183\pi$
$709$	25.5000	−	14.7224i	0.957673	−	0.552913i	0.895456	+	0.445150i	$0.146850\pi$
$710$		0			0
$711$	4.00000	+	6.92820i	0.150012	+	0.259828i
$712$		0			0
$713$	−9.00000	−	5.19615i	−0.337053	−	0.194597i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−9.00000	−	5.19615i	−0.336111	−	0.194054i
$718$		0			0
$719$	13.5000	+	23.3827i	0.503465	+	0.872027i	0.999992	+	0.00400572i	$0.00127506\pi$
$719$	13.5000	+	23.3827i	0.503465	+	0.872027i	−0.496527	+	0.868021i	$0.665392\pi$
$720$		0			0
$721$	24.0000	−	13.8564i	0.893807	−	0.516040i
$722$		0			0
$723$			5.19615i			0.193247i
$724$		0			0
$725$	22.5000	−	38.9711i	0.835629	−	1.44735i
$726$		0			0
$727$	40.0000			1.48352			0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000			1.48352			0.741759	+	0.670667i	$0.233992\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−7.50000	+	12.9904i	−0.277398	+	0.480467i
$732$		0			0
$733$		−	34.6410i		−	1.27950i	−0.768585	−	0.639748i	$-0.779039\pi$
$733$		−	34.6410i		−	1.27950i	0.768585	−	0.639748i	$-0.220961\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.50000	−	7.79423i	−0.165760	−	0.287104i
$738$		0			0
$739$	13.5000	+	7.79423i	0.496606	+	0.286715i	0.727311	−	0.686308i	$-0.240770\pi$
$739$	13.5000	+	7.79423i	0.496606	+	0.286715i	−0.230705	+	0.973024i	$0.574103\pi$
$740$		0			0
$741$	18.0000	+	5.19615i	0.661247	+	0.190885i
$742$		0			0
$743$	−4.50000	−	2.59808i	−0.165089	−	0.0953142i	0.415179	−	0.909740i	$-0.363719\pi$
$743$	−4.50000	−	2.59808i	−0.165089	−	0.0953142i	−0.580268	+	0.814426i	$0.697052\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−18.0000	+	10.3923i	−0.658586	+	0.380235i
$748$		0			0
$749$		−	25.9808i		−	0.949316i
$750$		0			0
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	0.959906	+	0.280321i	$0.0904408\pi$
$752$		0			0
$753$	−21.0000			−0.765283
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−21.5000	+	37.2391i	−0.781431	+	1.35348i	0.149677	+	0.988735i	$0.452176\pi$
$757$	−21.5000	+	37.2391i	−0.781431	+	1.35348i	−0.931108	+	0.364743i	$0.881157\pi$
$758$		0			0
$759$		−	15.5885i		−	0.565825i
$760$		0			0
$761$	−4.50000	+	2.59808i	−0.163125	+	0.0941802i	−0.579340	−	0.815086i	$-0.696690\pi$
$761$	−4.50000	+	2.59808i	−0.163125	+	0.0941802i	0.416215	+	0.909266i	$0.363356\pi$
$762$		0			0
$763$	12.0000	+	20.7846i	0.434429	+	0.752453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.50000	+	18.1865i	0.162486	+	0.656678i
$768$		0			0
$769$	31.5000	+	18.1865i	1.13592	+	0.655823i	0.945417	−	0.325864i	$-0.105655\pi$
$769$	31.5000	+	18.1865i	1.13592	+	0.655823i	0.190502	+	0.981687i	$0.438988\pi$
$770$		0			0
$771$	1.50000	+	2.59808i	0.0540212	+	0.0935674i
$772$		0			0
$773$	−22.5000	+	12.9904i	−0.809269	+	0.467232i	−0.846702	−	0.532068i	$-0.821415\pi$
$773$	−22.5000	+	12.9904i	−0.809269	+	0.467232i	0.0374331	+	0.999299i	$0.488082\pi$
$774$		0			0
$775$			17.3205i			0.622171i
$776$		0			0
$777$	4.50000	−	7.79423i	0.161437	−	0.279616i
$778$		0			0
$779$	−27.0000			−0.967375
$780$		0			0
$781$	−27.0000			−0.966136
$782$		0			0
$783$	22.5000	−	38.9711i	0.804084	−	1.39272i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.50000	+	0.866025i	−0.0534692	+	0.0308705i	−0.526496	−	0.850177i	$-0.676495\pi$
$787$	−1.50000	+	0.866025i	−0.0534692	+	0.0308705i	0.473027	+	0.881048i	$0.343161\pi$
$788$		0			0
$789$	−10.5000	−	18.1865i	−0.373810	−	0.647458i
$790$		0			0
$791$	4.50000	+	2.59808i	0.160002	+	0.0923770i
$792$		0			0
$793$	−17.5000	+	4.33013i	−0.621443	+	0.153767i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.5000	−	23.3827i	−0.478195	−	0.828257i	0.521493	−	0.853256i	$-0.325375\pi$
$797$	−13.5000	−	23.3827i	−0.478195	−	0.828257i	−0.999687	+	0.0249984i	$0.992042\pi$
$798$		0			0
$799$	27.0000	−	15.5885i	0.955191	−	0.551480i
$800$		0			0
$801$			31.1769i			1.10158i
$802$		0			0
$803$	−18.0000	+	31.1769i	−0.635206	+	1.10021i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	9.00000			0.316815
$808$		0			0
$809$	10.5000	−	18.1865i	0.369160	−	0.639404i	−0.620274	−	0.784385i	$-0.712979\pi$
$809$	10.5000	−	18.1865i	0.369160	−	0.639404i	0.989434	+	0.144981i	$0.0463120\pi$
$810$		0			0
$811$			45.0333i			1.58133i	0.612247	+	0.790667i	$0.290266\pi$
$811$			45.0333i			1.58133i	−0.612247	+	0.790667i	$0.709734\pi$
$812$		0			0
$813$	−10.5000	+	6.06218i	−0.368251	+	0.212610i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−22.5000	−	12.9904i	−0.787175	−	0.454476i
$818$		0			0
$819$	9.00000	−	8.66025i	0.314485	−	0.302614i
$820$		0			0
$821$	−22.5000	−	12.9904i	−0.785255	−	0.453367i	0.0530342	−	0.998593i	$-0.483111\pi$
$821$	−22.5000	−	12.9904i	−0.785255	−	0.453367i	−0.838290	+	0.545225i	$0.816444\pi$
$822$		0			0
$823$	−20.5000	−	35.5070i	−0.714585	−	1.23770i	−0.963119	−	0.269075i	$-0.913282\pi$
$823$	−20.5000	−	35.5070i	−0.714585	−	1.23770i	0.248534	−	0.968623i	$-0.420051\pi$
$824$		0			0
$825$	−22.5000	+	12.9904i	−0.783349	+	0.452267i
$826$		0			0
$827$			10.3923i			0.361376i	0.983540	+	0.180688i	$0.0578324\pi$
$827$			10.3923i			0.361376i	−0.983540	+	0.180688i	$0.942168\pi$
$828$		0			0
$829$	−27.5000	+	47.6314i	−0.955114	+	1.65431i	−0.221009	+	0.975272i	$0.570935\pi$
$829$	−27.5000	+	47.6314i	−0.955114	+	1.65431i	−0.734106	+	0.679035i	$0.762398\pi$
$830$		0			0
$831$	1.00000			0.0346896
$832$		0			0
$833$	12.0000			0.415775
$834$		0			0
$835$		0			0
$836$		0			0
$837$			17.3205i			0.598684i
$838$		0			0
$839$	−31.5000	+	18.1865i	−1.08750	+	0.627869i	−0.932910	−	0.360110i	$-0.882739\pi$
$839$	−31.5000	+	18.1865i	−1.08750	+	0.627869i	−0.154591	+	0.987979i	$0.549406\pi$
$840$		0			0
$841$	−26.0000	−	45.0333i	−0.896552	−	1.55287i
$842$		0			0
$843$	18.0000	+	10.3923i	0.619953	+	0.357930i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−24.0000	−	13.8564i	−0.824650	−	0.476112i
$848$		0			0
$849$	3.50000	+	6.06218i	0.120120	+	0.208053i
$850$		0			0
$851$	13.5000	−	7.79423i	0.462774	−	0.267183i
$852$		0			0
$853$		−	13.8564i		−	0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$		−	13.8564i		−	0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	54.0000			1.84460			0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000			1.84460			0.922302	+	0.386469i	$0.126305\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	4.50000	−	7.79423i	0.153360	−	0.265627i
$862$		0			0
$863$		−	10.3923i		−	0.353758i	−0.984233	−	0.176879i	$-0.943400\pi$
$863$		−	10.3923i		−	0.353758i	0.984233	−	0.176879i	$-0.0566002\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−4.00000	−	6.92820i	−0.135847	−	0.235294i
$868$		0			0
$869$	−18.0000	−	10.3923i	−0.610608	−	0.352535i
$870$		0			0
$871$	−4.50000	+	4.33013i	−0.152477	+	0.146721i
$872$		0			0
$873$	−21.0000	−	12.1244i	−0.710742	−	0.410347i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.5000	−	18.1865i	1.06368	−	0.614116i	0.137232	−	0.990539i	$-0.456180\pi$
$877$	31.5000	−	18.1865i	1.06368	−	0.614116i	0.926448	+	0.376423i	$0.122846\pi$
$878$		0			0
$879$			5.19615i			0.175262i
$880$		0			0
$881$	−1.50000	+	2.59808i	−0.0505363	+	0.0875314i	−0.890187	−	0.455595i	$-0.849426\pi$
$881$	−1.50000	+	2.59808i	−0.0505363	+	0.0875314i	0.839651	+	0.543127i	$0.182760\pi$
$882$		0			0
$883$	28.0000			0.942275			0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000			0.942275			0.471138	+	0.882060i	$0.343844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7.50000	+	12.9904i	−0.251825	+	0.436174i	−0.964028	−	0.265799i	$-0.914364\pi$
$887$	−7.50000	+	12.9904i	−0.251825	+	0.436174i	0.712203	+	0.701974i	$0.247698\pi$
$888$		0			0
$889$			29.4449i			0.987549i
$890$		0			0
$891$	4.50000	−	2.59808i	0.150756	−	0.0870388i
$892$		0			0
$893$	27.0000	+	46.7654i	0.903521	+	1.56494i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−10.5000	+	2.59808i	−0.350585	+	0.0867472i
$898$		0			0
$899$	27.0000	+	15.5885i	0.900500	+	0.519904i
$900$		0			0
$901$	9.00000	+	15.5885i	0.299833	+	0.519327i
$902$		0			0
$903$	7.50000	−	4.33013i	0.249584	−	0.144098i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8.50000	−	14.7224i	0.282238	−	0.488850i	−0.689698	−	0.724097i	$-0.742257\pi$
$907$	8.50000	−	14.7224i	0.282238	−	0.488850i	0.971936	+	0.235247i	$0.0755899\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	27.0000	−	46.7654i	0.893570	−	1.54771i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.0000	−	10.3923i	0.594412	−	0.343184i
$918$		0			0
$919$	5.50000	+	9.52628i	0.181428	+	0.314243i	0.942367	−	0.334581i	$-0.108595\pi$
$919$	5.50000	+	9.52628i	0.181428	+	0.314243i	−0.760939	+	0.648824i	$0.775261\pi$
$920$		0			0
$921$	−15.0000	−	8.66025i	−0.494267	−	0.285365i
$922$		0			0
$923$	4.50000	+	18.1865i	0.148119	+	0.598617i
$924$		0			0
$925$	−22.5000	−	12.9904i	−0.739795	−	0.427121i
$926$		0			0
$927$	−16.0000	−	27.7128i	−0.525509	−	0.910208i
$928$		0			0
$929$	31.5000	−	18.1865i	1.03348	−	0.596681i	0.115501	−	0.993307i	$-0.463153\pi$
$929$	31.5000	−	18.1865i	1.03348	−	0.596681i	0.917980	+	0.396627i	$0.129819\pi$
$930$		0			0
$931$			20.7846i			0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	11.0000	−	19.0526i	0.358971	−	0.621757i
$940$		0			0
$941$		−	20.7846i		−	0.677559i	−0.940866	−	0.338779i	$-0.889986\pi$
$941$		−	20.7846i		−	0.677559i	0.940866	−	0.338779i	$-0.110014\pi$
$942$		0			0
$943$	13.5000	−	7.79423i	0.439620	−	0.253815i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−40.5000	−	23.3827i	−1.31607	−	0.759835i	−0.332979	−	0.942934i	$-0.608054\pi$
$947$	−40.5000	−	23.3827i	−1.31607	−	0.759835i	−0.983094	+	0.183099i	$0.941387\pi$
$948$		0			0
$949$	24.0000	+	6.92820i	0.779073	+	0.224899i
$950$		0			0
$951$	18.0000	+	10.3923i	0.583690	+	0.336994i
$952$		0			0
$953$	4.50000	+	7.79423i	0.145769	+	0.252480i	0.929660	−	0.368419i	$-0.120101\pi$
$953$	4.50000	+	7.79423i	0.145769	+	0.252480i	−0.783890	+	0.620899i	$0.786768\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			46.7654i			1.51171i
$958$		0			0
$959$	−13.5000	+	23.3827i	−0.435938	+	0.755066i
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	−30.0000			−0.966736
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	10.3923i		−	0.334194i	−0.985940	−	0.167097i	$-0.946561\pi$
$967$		−	10.3923i		−	0.334194i	0.985940	−	0.167097i	$-0.0534393\pi$
$968$		0			0
$969$	−13.5000	+	7.79423i	−0.433682	+	0.250387i
$970$		0			0
$971$	13.5000	+	23.3827i	0.433236	+	0.750386i	0.997150	−	0.0754473i	$-0.0240385\pi$
$971$	13.5000	+	23.3827i	0.433236	+	0.750386i	−0.563914	+	0.825833i	$0.690705\pi$
$972$		0			0
$973$	−10.5000	−	6.06218i	−0.336615	−	0.194344i
$974$		0			0
$975$	12.5000	+	12.9904i	0.400320	+	0.416025i
$976$		0			0
$977$	13.5000	+	7.79423i	0.431903	+	0.249359i	0.700157	−	0.713989i	$-0.253113\pi$
$977$	13.5000	+	7.79423i	0.431903	+	0.249359i	−0.268254	+	0.963348i	$0.586447\pi$
$978$		0			0
$979$	−40.5000	−	70.1481i	−1.29439	−	2.24194i
$980$		0			0
$981$	24.0000	−	13.8564i	0.766261	−	0.442401i
$982$		0			0
$983$		−	10.3923i		−	0.331463i	−0.986171	−	0.165732i	$-0.947001\pi$
$983$		−	10.3923i		−	0.331463i	0.986171	−	0.165732i	$-0.0529985\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−18.0000			−0.572946
$988$		0			0
$989$	15.0000			0.476972
$990$		0			0
$991$	12.5000	−	21.6506i	0.397076	−	0.687755i	−0.596288	−	0.802771i	$-0.703358\pi$
$991$	12.5000	−	21.6506i	0.397076	−	0.687755i	0.993364	+	0.115015i	$0.0366917\pi$
$992$		0			0
$993$		−	12.1244i		−	0.384755i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	6.50000	+	11.2583i	0.205857	+	0.356555i	0.950405	−	0.311014i	$-0.100668\pi$
$997$	6.50000	+	11.2583i	0.205857	+	0.356555i	−0.744548	+	0.667568i	$0.767335\pi$
$998$		0			0
$999$	−22.5000	−	12.9904i	−0.711868	−	0.410997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	52.2.h.a.17.1	✓	2
3.2	odd	2		468.2.t.a.433.1		2
4.3	odd	2		208.2.w.a.17.1		2
5.2	odd	4		1300.2.ba.a.849.1		4
5.3	odd	4		1300.2.ba.a.849.2		4
5.4	even	2		1300.2.y.a.901.1		2
7.2	even	3		2548.2.bb.b.1733.1		2
7.3	odd	6		2548.2.bq.b.1941.1		2
7.4	even	3		2548.2.bq.a.1941.1		2
7.5	odd	6		2548.2.bb.a.1733.1		2
7.6	odd	2		2548.2.u.a.589.1		2
8.3	odd	2		832.2.w.c.641.1		2
8.5	even	2		832.2.w.b.641.1		2
12.11	even	2		1872.2.by.e.433.1		2
13.2	odd	12		676.2.e.e.653.1		4
13.3	even	3		676.2.h.a.361.1		2
13.4	even	6		676.2.d.b.337.2		2
13.5	odd	4		676.2.e.e.529.1		4
13.6	odd	12		676.2.a.f.1.2		2
13.7	odd	12		676.2.a.f.1.1		2
13.8	odd	4		676.2.e.e.529.2		4
13.9	even	3		676.2.d.b.337.1		2
13.10	even	6	inner	52.2.h.a.49.1	yes	2
13.11	odd	12		676.2.e.e.653.2		4
13.12	even	2		676.2.h.a.485.1		2
39.17	odd	6		6084.2.b.d.4393.2		2
39.20	even	12		6084.2.a.t.1.1		2
39.23	odd	6		468.2.t.a.361.1		2
39.32	even	12		6084.2.a.t.1.2		2
39.35	odd	6		6084.2.b.d.4393.1		2
52.7	even	12		2704.2.a.u.1.2		2
52.19	even	12		2704.2.a.u.1.1		2
52.23	odd	6		208.2.w.a.49.1		2
52.35	odd	6		2704.2.f.h.337.2		2
52.43	odd	6		2704.2.f.h.337.1		2
65.23	odd	12		1300.2.ba.a.49.1		4
65.49	even	6		1300.2.y.a.101.1		2
65.62	odd	12		1300.2.ba.a.49.2		4
91.10	odd	6		2548.2.bb.a.569.1		2
91.23	even	6		2548.2.bq.a.361.1		2
91.62	odd	6		2548.2.u.a.1765.1		2
91.75	odd	6		2548.2.bq.b.361.1		2
91.88	even	6		2548.2.bb.b.569.1		2
104.75	odd	6		832.2.w.c.257.1		2
104.101	even	6		832.2.w.b.257.1		2
156.23	even	6		1872.2.by.e.1297.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.h.a.17.1	✓	2	1.1	even	1	trivial
52.2.h.a.49.1	yes	2	13.10	even	6	inner
208.2.w.a.17.1		2	4.3	odd	2
208.2.w.a.49.1		2	52.23	odd	6
468.2.t.a.361.1		2	39.23	odd	6
468.2.t.a.433.1		2	3.2	odd	2
676.2.a.f.1.1		2	13.7	odd	12
676.2.a.f.1.2		2	13.6	odd	12
676.2.d.b.337.1		2	13.9	even	3
676.2.d.b.337.2		2	13.4	even	6
676.2.e.e.529.1		4	13.5	odd	4
676.2.e.e.529.2		4	13.8	odd	4
676.2.e.e.653.1		4	13.2	odd	12
676.2.e.e.653.2		4	13.11	odd	12
676.2.h.a.361.1		2	13.3	even	3
676.2.h.a.485.1		2	13.12	even	2
832.2.w.b.257.1		2	104.101	even	6
832.2.w.b.641.1		2	8.5	even	2
832.2.w.c.257.1		2	104.75	odd	6
832.2.w.c.641.1		2	8.3	odd	2
1300.2.y.a.101.1		2	65.49	even	6
1300.2.y.a.901.1		2	5.4	even	2
1300.2.ba.a.49.1		4	65.23	odd	12
1300.2.ba.a.49.2		4	65.62	odd	12
1300.2.ba.a.849.1		4	5.2	odd	4
1300.2.ba.a.849.2		4	5.3	odd	4
1872.2.by.e.433.1		2	12.11	even	2
1872.2.by.e.1297.1		2	156.23	even	6
2548.2.u.a.589.1		2	7.6	odd	2
2548.2.u.a.1765.1		2	91.62	odd	6
2548.2.bb.a.569.1		2	91.10	odd	6
2548.2.bb.a.1733.1		2	7.5	odd	6
2548.2.bb.b.569.1		2	91.88	even	6
2548.2.bb.b.1733.1		2	7.2	even	3
2548.2.bq.a.361.1		2	91.23	even	6
2548.2.bq.a.1941.1		2	7.4	even	3
2548.2.bq.b.361.1		2	91.75	odd	6
2548.2.bq.b.1941.1		2	7.3	odd	6
2704.2.a.u.1.1		2	52.19	even	12
2704.2.a.u.1.2		2	52.7	even	12
2704.2.f.h.337.1		2	52.43	odd	6
2704.2.f.h.337.2		2	52.35	odd	6
6084.2.a.t.1.1		2	39.20	even	12
6084.2.a.t.1.2		2	39.32	even	12
6084.2.b.d.4393.1		2	39.35	odd	6
6084.2.b.d.4393.2		2	39.17	odd	6

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:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	52.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{9} +(-4.50000 - 2.59808i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 - 2.59808i) q^{17} +(4.50000 - 2.59808i) q^{19} +1.73205i q^{21} +(-1.50000 + 2.59808i) q^{23} +5.00000 q^{25} +5.00000 q^{27} +(4.50000 - 7.79423i) q^{29} +3.46410i q^{31} +(-4.50000 + 2.59808i) q^{33} +(-4.50000 - 2.59808i) q^{37} +(2.50000 + 2.59808i) q^{39} +(-4.50000 - 2.59808i) q^{41} +(-2.50000 - 4.33013i) q^{43} +10.3923i q^{47} +(-2.00000 + 3.46410i) q^{49} -3.00000 q^{51} -6.00000 q^{53} -5.19615i q^{57} +(4.50000 - 2.59808i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-3.00000 - 1.73205i) q^{63} +(1.50000 + 0.866025i) q^{67} +(1.50000 + 2.59808i) q^{69} +(4.50000 - 2.59808i) q^{71} -6.92820i q^{73} +(2.50000 - 4.33013i) q^{75} +9.00000 q^{77} +4.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +10.3923i q^{83} +(-4.50000 - 7.79423i) q^{87} +(13.5000 + 7.79423i) q^{89} +(-1.50000 - 6.06218i) q^{91} +(3.00000 + 1.73205i) q^{93} +(-10.5000 + 6.06218i) q^{97} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^3 - 3 * q^7 + 2 * q^9	\( 2 q + q^{3} - 3 q^{7} + 2 q^{9} - 9 q^{11} - 2 q^{13} - 3 q^{17} + 9 q^{19} - 3 q^{23} + 10 q^{25} + 10 q^{27} + 9 q^{29} - 9 q^{33} - 9 q^{37} + 5 q^{39} - 9 q^{41} - 5 q^{43} - 4 q^{49} - 6 q^{51} - 12 q^{53} + 9 q^{59} + 5 q^{61} - 6 q^{63} + 3 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{75} + 18 q^{77} + 8 q^{79} - q^{81} - 9 q^{87} + 27 q^{89} - 3 q^{91} + 6 q^{93} - 21 q^{97}+O(q^{100}) \) 2 * q + q^3 - 3 * q^7 + 2 * q^9 - 9 * q^11 - 2 * q^13 - 3 * q^17 + 9 * q^19 - 3 * q^23 + 10 * q^25 + 10 * q^27 + 9 * q^29 - 9 * q^33 - 9 * q^37 + 5 * q^39 - 9 * q^41 - 5 * q^43 - 4 * q^49 - 6 * q^51 - 12 * q^53 + 9 * q^59 + 5 * q^61 - 6 * q^63 + 3 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^75 + 18 * q^77 + 8 * q^79 - q^81 - 9 * q^87 + 27 * q^89 - 3 * q^91 + 6 * q^93 - 21 * q^97

Introduction

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