LMFDB - Embedded newform 520.2.q.e.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,2,Mod(81,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("520.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.q (of order $3$, 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:	$4.15222090511$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	520.81
Dual form			520.2.q.e.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 2.59808i) q^{3} -1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{9} +O(q^{10})$	\(q+(1.50000 + 2.59808i) q^{3} -1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{9} +(-2.50000 - 4.33013i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 - 2.59808i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(2.50000 - 4.33013i) q^{19} -9.00000 q^{21} +(1.50000 + 2.59808i) q^{23} +1.00000 q^{25} -9.00000 q^{27} +(2.50000 + 4.33013i) q^{29} +8.00000 q^{31} +(7.50000 - 12.9904i) q^{33} +(1.50000 - 2.59808i) q^{35} +(4.50000 + 7.79423i) q^{37} +(-10.5000 + 2.59808i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(3.00000 - 5.19615i) q^{45} -12.0000 q^{47} +(-1.00000 - 1.73205i) q^{49} -9.00000 q^{51} +2.00000 q^{53} +(2.50000 + 4.33013i) q^{55} +15.0000 q^{57} +(2.50000 - 4.33013i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-9.00000 - 15.5885i) q^{63} +(1.00000 - 3.46410i) q^{65} +(7.50000 + 12.9904i) q^{67} +(-4.50000 + 7.79423i) q^{69} +(0.500000 - 0.866025i) q^{71} -10.0000 q^{73} +(1.50000 + 2.59808i) q^{75} +15.0000 q^{77} -4.00000 q^{79} +(-4.50000 - 7.79423i) q^{81} +12.0000 q^{83} +(1.50000 - 2.59808i) q^{85} +(-7.50000 + 12.9904i) q^{87} +(0.500000 + 0.866025i) q^{89} +(-7.50000 - 7.79423i) q^{91} +(12.0000 + 20.7846i) q^{93} +(-2.50000 + 4.33013i) q^{95} +(-3.50000 + 6.06218i) q^{97} +30.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 2 q^{5} - 3 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 - 2 * q^5 - 3 * q^7 - 6 * q^9	$ 2 q + 3 q^{3} - 2 q^{5} - 3 q^{7} - 6 q^{9} - 5 q^{11} - 2 q^{13} - 3 q^{15} - 3 q^{17} + 5 q^{19} - 18 q^{21} + 3 q^{23} + 2 q^{25} - 18 q^{27} + 5 q^{29} + 16 q^{31} + 15 q^{33} + 3 q^{35} + 9 q^{37} - 21 q^{39} - 3 q^{41} + q^{43} + 6 q^{45} - 24 q^{47} - 2 q^{49} - 18 q^{51} + 4 q^{53} + 5 q^{55} + 30 q^{57} + 5 q^{59} + q^{61} - 18 q^{63} + 2 q^{65} + 15 q^{67} - 9 q^{69} + q^{71} - 20 q^{73} + 3 q^{75} + 30 q^{77} - 8 q^{79} - 9 q^{81} + 24 q^{83} + 3 q^{85} - 15 q^{87} + q^{89} - 15 q^{91} + 24 q^{93} - 5 q^{95} - 7 q^{97} + 60 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 2 * q^5 - 3 * q^7 - 6 * q^9 - 5 * q^11 - 2 * q^13 - 3 * q^15 - 3 * q^17 + 5 * q^19 - 18 * q^21 + 3 * q^23 + 2 * q^25 - 18 * q^27 + 5 * q^29 + 16 * q^31 + 15 * q^33 + 3 * q^35 + 9 * q^37 - 21 * q^39 - 3 * q^41 + q^43 + 6 * q^45 - 24 * q^47 - 2 * q^49 - 18 * q^51 + 4 * q^53 + 5 * q^55 + 30 * q^57 + 5 * q^59 + q^61 - 18 * q^63 + 2 * q^65 + 15 * q^67 - 9 * q^69 + q^71 - 20 * q^73 + 3 * q^75 + 30 * q^77 - 8 * q^79 - 9 * q^81 + 24 * q^83 + 3 * q^85 - 15 * q^87 + q^89 - 15 * q^91 + 24 * q^93 - 5 * q^95 - 7 * q^97 + 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.50000	+	2.59808i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$3$	1.50000	+	2.59808i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	0.429919	+	0.902867i	$0.358542\pi$
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	−0.996866	+	0.0791130i	$0.974791\pi$
$8$		0			0
$9$	−3.00000	+	5.19615i	−1.00000	+	1.73205i
$10$		0			0
$11$	−2.50000	−	4.33013i	−0.753778	−	1.30558i	−0.945979	−	0.324227i	$-0.894896\pi$
$11$	−2.50000	−	4.33013i	−0.753778	−	1.30558i	0.192201	−	0.981356i	$-0.438437\pi$
$12$		0			0
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$13$	−1.00000	+	3.46410i	−0.277350	+	0.960769i
$14$		0			0
$15$	−1.50000	−	2.59808i	−0.387298	−	0.670820i
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	−0.422659	−	0.906289i	$-0.638903\pi$
$19$	2.50000	−	4.33013i	0.573539	−	0.993399i	0.996199	−	0.0871106i	$-0.0277634\pi$
$20$		0			0
$21$	−9.00000			−1.96396
$22$		0			0
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−9.00000			−1.73205
$28$		0			0
$29$	2.50000	+	4.33013i	0.464238	+	0.804084i	0.999167	−	0.0408130i	$-0.0129948\pi$
$29$	2.50000	+	4.33013i	0.464238	+	0.804084i	−0.534928	+	0.844897i	$0.679661\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$	7.50000	−	12.9904i	1.30558	−	2.26134i
$34$		0			0
$35$	1.50000	−	2.59808i	0.253546	−	0.439155i
$36$		0			0
$37$	4.50000	+	7.79423i	0.739795	+	1.28136i	0.952587	+	0.304266i	$0.0984111\pi$
$37$	4.50000	+	7.79423i	0.739795	+	1.28136i	−0.212792	+	0.977098i	$0.568256\pi$
$38$		0			0
$39$	−10.5000	+	2.59808i	−1.68135	+	0.416025i
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$		0			0
$45$	3.00000	−	5.19615i	0.447214	−	0.774597i
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$		0			0
$51$	−9.00000			−1.26025
$52$		0			0
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$		0			0
$55$	2.50000	+	4.33013i	0.337100	+	0.583874i
$56$		0			0
$57$	15.0000			1.98680
$58$		0			0
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	−0.656136	−	0.754643i	$-0.727810\pi$
$59$	2.50000	−	4.33013i	0.325472	−	0.563735i	0.981608	+	0.190909i	$0.0611434\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	−9.00000	−	15.5885i	−1.13389	−	1.96396i
$64$		0			0
$65$	1.00000	−	3.46410i	0.124035	−	0.429669i
$66$		0			0
$67$	7.50000	+	12.9904i	0.916271	+	1.58703i	0.805030	+	0.593234i	$0.202149\pi$
$67$	7.50000	+	12.9904i	0.916271	+	1.58703i	0.111241	+	0.993793i	$0.464517\pi$
$68$		0			0
$69$	−4.50000	+	7.79423i	−0.541736	+	0.938315i
$70$		0			0
$71$	0.500000	−	0.866025i	0.0593391	−	0.102778i	−0.834830	−	0.550508i	$-0.814434\pi$
$71$	0.500000	−	0.866025i	0.0593391	−	0.102778i	0.894169	+	0.447730i	$0.147767\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$	1.50000	+	2.59808i	0.173205	+	0.300000i
$76$		0			0
$77$	15.0000			1.70941
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	1.50000	−	2.59808i	0.162698	−	0.281801i
$86$		0			0
$87$	−7.50000	+	12.9904i	−0.804084	+	1.39272i
$88$		0			0
$89$	0.500000	+	0.866025i	0.0529999	+	0.0917985i	0.891308	−	0.453398i	$-0.149788\pi$
$89$	0.500000	+	0.866025i	0.0529999	+	0.0917985i	−0.838308	+	0.545197i	$0.816455\pi$
$90$		0			0
$91$	−7.50000	−	7.79423i	−0.786214	−	0.817057i
$92$		0			0
$93$	12.0000	+	20.7846i	1.24434	+	2.15526i
$94$		0			0
$95$	−2.50000	+	4.33013i	−0.256495	+	0.444262i
$96$		0			0
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	−0.987181	−	0.159602i	$-0.948979\pi$
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	0.631810	+	0.775123i	$0.282312\pi$
$98$		0			0
$99$	30.0000			3.01511
$100$		0			0
$101$	8.50000	+	14.7224i	0.845782	+	1.46494i	0.884941	+	0.465704i	$0.154199\pi$
$101$	8.50000	+	14.7224i	0.845782	+	1.46494i	−0.0391591	+	0.999233i	$0.512468\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$		0			0
$105$	9.00000			0.878310
$106$		0			0
$107$	−6.50000	−	11.2583i	−0.628379	−	1.08838i	−0.987877	−	0.155238i	$-0.950386\pi$
$107$	−6.50000	−	11.2583i	−0.628379	−	1.08838i	0.359498	−	0.933146i	$-0.382948\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	−13.5000	+	23.3827i	−1.28136	+	2.21939i
$112$		0			0
$113$	10.5000	−	18.1865i	0.987757	−	1.71085i	0.358778	−	0.933423i	$-0.383194\pi$
$113$	10.5000	−	18.1865i	0.987757	−	1.71085i	0.628979	−	0.777422i	$-0.283473\pi$
$114$		0			0
$115$	−1.50000	−	2.59808i	−0.139876	−	0.242272i
$116$		0			0
$117$	−15.0000	−	15.5885i	−1.38675	−	1.44115i
$118$		0			0
$119$	−4.50000	−	7.79423i	−0.412514	−	0.714496i
$120$		0			0
$121$	−7.00000	+	12.1244i	−0.636364	+	1.10221i
$122$		0			0
$123$	4.50000	−	7.79423i	0.405751	−	0.702782i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−8.50000	−	14.7224i	−0.754253	−	1.30640i	−0.945745	−	0.324910i	$-0.894666\pi$
$127$	−8.50000	−	14.7224i	−0.754253	−	1.30640i	0.191492	−	0.981494i	$-0.438667\pi$
$128$		0			0
$129$	3.00000			0.264135
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	7.50000	+	12.9904i	0.650332	+	1.12641i
$134$		0			0
$135$	9.00000			0.774597
$136$		0			0
$137$	8.50000	−	14.7224i	0.726204	−	1.25782i	−0.232273	−	0.972651i	$-0.574616\pi$
$137$	8.50000	−	14.7224i	0.726204	−	1.25782i	0.958477	−	0.285171i	$-0.0920506\pi$
$138$		0			0
$139$	−5.50000	+	9.52628i	−0.466504	+	0.808008i	−0.999268	−	0.0382553i	$-0.987820\pi$
$139$	−5.50000	+	9.52628i	−0.466504	+	0.808008i	0.532764	+	0.846264i	$0.321153\pi$
$140$		0			0
$141$	−18.0000	−	31.1769i	−1.51587	−	2.62557i
$142$		0			0
$143$	17.5000	−	4.33013i	1.46342	−	0.362103i
$144$		0			0
$145$	−2.50000	−	4.33013i	−0.207614	−	0.359597i
$146$		0			0
$147$	3.00000	−	5.19615i	0.247436	−	0.428571i
$148$		0			0
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	−0.620701	−	0.784047i	$-0.713152\pi$
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	0.989355	+	0.145519i	$0.0464853\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$	−9.00000	−	15.5885i	−0.727607	−	1.26025i
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	3.00000	+	5.19615i	0.237915	+	0.412082i
$160$		0			0
$161$	−9.00000			−0.709299
$162$		0			0
$163$	8.50000	−	14.7224i	0.665771	−	1.15315i	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	8.50000	−	14.7224i	0.665771	−	1.15315i	0.979076	−	0.203497i	$-0.0652307\pi$
$164$		0			0
$165$	−7.50000	+	12.9904i	−0.583874	+	1.01130i
$166$		0			0
$167$	−10.5000	−	18.1865i	−0.812514	−	1.40732i	−0.911099	−	0.412188i	$-0.864765\pi$
$167$	−10.5000	−	18.1865i	−0.812514	−	1.40732i	0.0985846	−	0.995129i	$-0.468568\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$	15.0000	+	25.9808i	1.14708	+	1.98680i
$172$		0			0
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	−0.967851	−	0.251523i	$-0.919068\pi$
$173$	−3.50000	+	6.06218i	−0.266100	+	0.460899i	0.701751	+	0.712422i	$0.252402\pi$
$174$		0			0
$175$	−1.50000	+	2.59808i	−0.113389	+	0.196396i
$176$		0			0
$177$	15.0000			1.12747
$178$		0			0
$179$	1.50000	+	2.59808i	0.112115	+	0.194189i	0.916623	−	0.399753i	$-0.130904\pi$
$179$	1.50000	+	2.59808i	0.112115	+	0.194189i	−0.804508	+	0.593942i	$0.797571\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	3.00000			0.221766
$184$		0			0
$185$	−4.50000	−	7.79423i	−0.330847	−	0.573043i
$186$		0			0
$187$	15.0000			1.09691
$188$		0			0
$189$	13.5000	−	23.3827i	0.981981	−	1.70084i
$190$		0			0
$191$	−1.50000	+	2.59808i	−0.108536	+	0.187990i	−0.915177	−	0.403051i	$-0.867950\pi$
$191$	−1.50000	+	2.59808i	−0.108536	+	0.187990i	0.806641	+	0.591041i	$0.201283\pi$
$192$		0			0
$193$	10.5000	+	18.1865i	0.755807	+	1.30910i	0.944972	+	0.327150i	$0.106088\pi$
$193$	10.5000	+	18.1865i	0.755807	+	1.30910i	−0.189166	+	0.981945i	$0.560578\pi$
$194$		0			0
$195$	10.5000	−	2.59808i	0.751921	−	0.186052i
$196$		0			0
$197$	−1.50000	−	2.59808i	−0.106871	−	0.185105i	0.807630	−	0.589689i	$-0.200750\pi$
$197$	−1.50000	−	2.59808i	−0.106871	−	0.185105i	−0.914501	+	0.404584i	$0.867416\pi$
$198$		0			0
$199$	4.50000	−	7.79423i	0.318997	−	0.552518i	−0.661282	−	0.750137i	$-0.729987\pi$
$199$	4.50000	−	7.79423i	0.318997	−	0.552518i	0.980279	+	0.197619i	$0.0633208\pi$
$200$		0			0
$201$	−22.5000	+	38.9711i	−1.58703	+	2.74881i
$202$		0			0
$203$	−15.0000			−1.05279
$204$		0			0
$205$	1.50000	+	2.59808i	0.104765	+	0.181458i
$206$		0			0
$207$	−18.0000			−1.25109
$208$		0			0
$209$	−25.0000			−1.72929
$210$		0			0
$211$	5.50000	+	9.52628i	0.378636	+	0.655816i	0.990864	−	0.134865i	$-0.0430600\pi$
$211$	5.50000	+	9.52628i	0.378636	+	0.655816i	−0.612228	+	0.790681i	$0.709727\pi$
$212$		0			0
$213$	3.00000			0.205557
$214$		0			0
$215$	−0.500000	+	0.866025i	−0.0340997	+	0.0590624i
$216$		0			0
$217$	−12.0000	+	20.7846i	−0.814613	+	1.41095i
$218$		0			0
$219$	−15.0000	−	25.9808i	−1.01361	−	1.75562i
$220$		0			0
$221$	−7.50000	−	7.79423i	−0.504505	−	0.524297i
$222$		0			0
$223$	−6.50000	−	11.2583i	−0.435272	−	0.753914i	0.562046	−	0.827106i	$-0.310015\pi$
$223$	−6.50000	−	11.2583i	−0.435272	−	0.753914i	−0.997318	+	0.0731927i	$0.976681\pi$
$224$		0			0
$225$	−3.00000	+	5.19615i	−0.200000	+	0.346410i
$226$		0			0
$227$	2.50000	−	4.33013i	0.165931	−	0.287401i	−0.771055	−	0.636769i	$-0.780270\pi$
$227$	2.50000	−	4.33013i	0.165931	−	0.287401i	0.936985	+	0.349368i	$0.113604\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$	22.5000	+	38.9711i	1.48039	+	2.56411i
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$	−6.00000	−	10.3923i	−0.389742	−	0.675053i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	0.378963	+	0.925412i	$0.376281\pi$
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	−0.990912	+	0.134515i	$0.957053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	+	1.73205i	0.0638877	+	0.110657i
$246$		0			0
$247$	12.5000	+	12.9904i	0.795356	+	0.826558i
$248$		0			0
$249$	18.0000	+	31.1769i	1.14070	+	1.97576i
$250$		0			0
$251$	0.500000	−	0.866025i	0.0315597	−	0.0546630i	−0.849814	−	0.527082i	$-0.823286\pi$
$251$	0.500000	−	0.866025i	0.0315597	−	0.0546630i	0.881374	+	0.472419i	$0.156619\pi$
$252$		0			0
$253$	7.50000	−	12.9904i	0.471521	−	0.816698i
$254$		0			0
$255$	9.00000			0.563602
$256$		0			0
$257$	6.50000	+	11.2583i	0.405459	+	0.702275i	0.994375	−	0.105919i	$-0.0337784\pi$
$257$	6.50000	+	11.2583i	0.405459	+	0.702275i	−0.588916	+	0.808194i	$0.700445\pi$
$258$		0			0
$259$	−27.0000			−1.67770
$260$		0			0
$261$	−30.0000			−1.85695
$262$		0			0
$263$	9.50000	+	16.4545i	0.585795	+	1.01463i	0.994776	+	0.102084i	$0.0325510\pi$
$263$	9.50000	+	16.4545i	0.585795	+	1.01463i	−0.408981	+	0.912543i	$0.634116\pi$
$264$		0			0
$265$	−2.00000			−0.122859
$266$		0			0
$267$	−1.50000	+	2.59808i	−0.0917985	+	0.159000i
$268$		0			0
$269$	2.50000	−	4.33013i	0.152428	−	0.264013i	−0.779692	−	0.626164i	$-0.784624\pi$
$269$	2.50000	−	4.33013i	0.152428	−	0.264013i	0.932119	+	0.362151i	$0.117958\pi$
$270$		0			0
$271$	−6.50000	−	11.2583i	−0.394847	−	0.683895i	0.598235	−	0.801321i	$-0.295869\pi$
$271$	−6.50000	−	11.2583i	−0.394847	−	0.683895i	−0.993082	+	0.117426i	$0.962536\pi$
$272$		0			0
$273$	9.00000	−	31.1769i	0.544705	−	1.88691i
$274$		0			0
$275$	−2.50000	−	4.33013i	−0.150756	−	0.261116i
$276$		0			0
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	−0.601975	−	0.798515i	$-0.705619\pi$
$277$	6.50000	−	11.2583i	0.390547	−	0.676448i	0.992522	+	0.122068i	$0.0389525\pi$
$278$		0			0
$279$	−24.0000	+	41.5692i	−1.43684	+	2.48868i
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	5.50000	+	9.52628i	0.326941	+	0.566279i	0.981903	−	0.189383i	$-0.0606488\pi$
$283$	5.50000	+	9.52628i	0.326941	+	0.566279i	−0.654962	+	0.755662i	$0.727315\pi$
$284$		0			0
$285$	−15.0000			−0.888523
$286$		0			0
$287$	9.00000			0.531253
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	−21.0000			−1.23104
$292$		0			0
$293$	4.50000	−	7.79423i	0.262893	−	0.455344i	−0.704117	−	0.710084i	$-0.748657\pi$
$293$	4.50000	−	7.79423i	0.262893	−	0.455344i	0.967009	+	0.254741i	$0.0819901\pi$
$294$		0			0
$295$	−2.50000	+	4.33013i	−0.145556	+	0.252110i
$296$		0			0
$297$	22.5000	+	38.9711i	1.30558	+	2.26134i
$298$		0			0
$299$	−10.5000	+	2.59808i	−0.607231	+	0.150251i
$300$		0			0
$301$	1.50000	+	2.59808i	0.0864586	+	0.149751i
$302$		0			0
$303$	−25.5000	+	44.1673i	−1.46494	+	2.53734i
$304$		0			0
$305$	−0.500000	+	0.866025i	−0.0286299	+	0.0495885i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$	12.0000	+	20.7846i	0.682656	+	1.18240i
$310$		0			0
$311$	−8.00000			−0.453638			−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000			−0.453638			−0.226819	+	0.973937i	$0.572833\pi$
$312$		0			0
$313$	6.00000			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$	9.00000	+	15.5885i	0.507093	+	0.878310i
$316$		0			0
$317$	−2.00000			−0.112331			−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000			−0.112331			−0.0561656	+	0.998421i	$0.517887\pi$
$318$		0			0
$319$	12.5000	−	21.6506i	0.699866	−	1.21220i
$320$		0			0
$321$	19.5000	−	33.7750i	1.08838	−	1.88514i
$322$		0			0
$323$	7.50000	+	12.9904i	0.417311	+	0.722804i
$324$		0			0
$325$	−1.00000	+	3.46410i	−0.0554700	+	0.192154i
$326$		0			0
$327$	3.00000	+	5.19615i	0.165900	+	0.287348i
$328$		0			0
$329$	18.0000	−	31.1769i	0.992372	−	1.71884i
$330$		0			0
$331$	12.5000	−	21.6506i	0.687062	−	1.19003i	−0.285722	−	0.958313i	$-0.592233\pi$
$331$	12.5000	−	21.6506i	0.687062	−	1.19003i	0.972784	−	0.231714i	$-0.0744333\pi$
$332$		0			0
$333$	−54.0000			−2.95918
$334$		0			0
$335$	−7.50000	−	12.9904i	−0.409769	−	0.709740i
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$	63.0000			3.42169
$340$		0			0
$341$	−20.0000	−	34.6410i	−1.08306	−	1.87592i
$342$		0			0
$343$	−15.0000			−0.809924
$344$		0			0
$345$	4.50000	−	7.79423i	0.242272	−	0.419627i
$346$		0			0
$347$	8.50000	−	14.7224i	0.456304	−	0.790342i	−0.542458	−	0.840083i	$-0.682506\pi$
$347$	8.50000	−	14.7224i	0.456304	−	0.790342i	0.998762	+	0.0497412i	$0.0158397\pi$
$348$		0			0
$349$	−3.50000	−	6.06218i	−0.187351	−	0.324501i	0.757015	−	0.653397i	$-0.226657\pi$
$349$	−3.50000	−	6.06218i	−0.187351	−	0.324501i	−0.944366	+	0.328896i	$0.893323\pi$
$350$		0			0
$351$	9.00000	−	31.1769i	0.480384	−	1.66410i
$352$		0			0
$353$	−7.50000	−	12.9904i	−0.399185	−	0.691408i	0.594441	−	0.804139i	$-0.297373\pi$
$353$	−7.50000	−	12.9904i	−0.399185	−	0.691408i	−0.993626	+	0.112731i	$0.964040\pi$
$354$		0			0
$355$	−0.500000	+	0.866025i	−0.0265372	+	0.0459639i
$356$		0			0
$357$	13.5000	−	23.3827i	0.714496	−	1.23754i
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−3.00000	−	5.19615i	−0.157895	−	0.273482i
$362$		0			0
$363$	−42.0000			−2.20443
$364$		0			0
$365$	10.0000			0.523424
$366$		0			0
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	0.902516	−	0.430656i	$-0.141718\pi$
$367$	1.50000	+	2.59808i	0.0782994	+	0.135618i	−0.824217	+	0.566274i	$0.808384\pi$
$368$		0			0
$369$	18.0000			0.937043
$370$		0			0
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	0.983783	+	0.179364i	$0.0574041\pi$
$374$		0			0
$375$	−1.50000	−	2.59808i	−0.0774597	−	0.134164i
$376$		0			0
$377$	−17.5000	+	4.33013i	−0.901296	+	0.223013i
$378$		0			0
$379$	−10.5000	−	18.1865i	−0.539349	−	0.934179i	−0.998939	−	0.0460485i	$-0.985337\pi$
$379$	−10.5000	−	18.1865i	−0.539349	−	0.934179i	0.459590	−	0.888131i	$-0.347996\pi$
$380$		0			0
$381$	25.5000	−	44.1673i	1.30640	−	2.26276i
$382$		0			0
$383$	−9.50000	+	16.4545i	−0.485427	+	0.840785i	−0.999860	−	0.0167461i	$-0.994669\pi$
$383$	−9.50000	+	16.4545i	−0.485427	+	0.840785i	0.514432	+	0.857531i	$0.328003\pi$
$384$		0			0
$385$	−15.0000			−0.764471
$386$		0			0
$387$	3.00000	+	5.19615i	0.152499	+	0.264135i
$388$		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
$393$	18.0000	+	31.1769i	0.907980	+	1.57267i
$394$		0			0
$395$	4.00000			0.201262
$396$		0			0
$397$	−13.5000	+	23.3827i	−0.677546	+	1.17354i	0.298172	+	0.954512i	$0.403623\pi$
$397$	−13.5000	+	23.3827i	−0.677546	+	1.17354i	−0.975718	+	0.219031i	$0.929710\pi$
$398$		0			0
$399$	−22.5000	+	38.9711i	−1.12641	+	1.95100i
$400$		0			0
$401$	14.5000	+	25.1147i	0.724095	+	1.25417i	0.959345	+	0.282235i	$0.0910758\pi$
$401$	14.5000	+	25.1147i	0.724095	+	1.25417i	−0.235250	+	0.971935i	$0.575591\pi$
$402$		0			0
$403$	−8.00000	+	27.7128i	−0.398508	+	1.38047i
$404$		0			0
$405$	4.50000	+	7.79423i	0.223607	+	0.387298i
$406$		0			0
$407$	22.5000	−	38.9711i	1.11528	−	1.93173i
$408$		0			0
$409$	2.50000	−	4.33013i	0.123617	−	0.214111i	−0.797574	−	0.603220i	$-0.793884\pi$
$409$	2.50000	−	4.33013i	0.123617	−	0.214111i	0.921192	+	0.389109i	$0.127217\pi$
$410$		0			0
$411$	51.0000			2.51564
$412$		0			0
$413$	7.50000	+	12.9904i	0.369051	+	0.639215i
$414$		0			0
$415$	−12.0000			−0.589057
$416$		0			0
$417$	−33.0000			−1.61602
$418$		0			0
$419$	−0.500000	−	0.866025i	−0.0244266	−	0.0423081i	0.853554	−	0.521005i	$-0.174443\pi$
$419$	−0.500000	−	0.866025i	−0.0244266	−	0.0423081i	−0.877980	+	0.478697i	$0.841109\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$	36.0000	−	62.3538i	1.75038	−	3.03175i
$424$		0			0
$425$	−1.50000	+	2.59808i	−0.0727607	+	0.126025i
$426$		0			0
$427$	1.50000	+	2.59808i	0.0725901	+	0.125730i
$428$		0			0
$429$	37.5000	+	38.9711i	1.81052	+	1.88154i
$430$		0			0
$431$	−16.5000	−	28.5788i	−0.794777	−	1.37659i	−0.922981	−	0.384846i	$-0.874254\pi$
$431$	−16.5000	−	28.5788i	−0.794777	−	1.37659i	0.128204	−	0.991748i	$-0.459079\pi$
$432$		0			0
$433$	14.5000	−	25.1147i	0.696826	−	1.20694i	−0.272736	−	0.962089i	$-0.587929\pi$
$433$	14.5000	−	25.1147i	0.696826	−	1.20694i	0.969561	−	0.244848i	$-0.0787382\pi$
$434$		0			0
$435$	7.50000	−	12.9904i	0.359597	−	0.622841i
$436$		0			0
$437$	15.0000			0.717547
$438$		0			0
$439$	−4.50000	−	7.79423i	−0.214773	−	0.371998i	0.738429	−	0.674331i	$-0.235568\pi$
$439$	−4.50000	−	7.79423i	−0.214773	−	0.371998i	−0.953202	+	0.302333i	$0.902235\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$	−20.0000			−0.950229			−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000			−0.950229			−0.475114	+	0.879924i	$0.657593\pi$
$444$		0			0
$445$	−0.500000	−	0.866025i	−0.0237023	−	0.0410535i
$446$		0			0
$447$	27.0000			1.27706
$448$		0			0
$449$	−9.50000	+	16.4545i	−0.448333	+	0.776535i	−0.998278	−	0.0586659i	$-0.981315\pi$
$449$	−9.50000	+	16.4545i	−0.448333	+	0.776535i	0.549945	+	0.835201i	$0.314649\pi$
$450$		0			0
$451$	−7.50000	+	12.9904i	−0.353161	+	0.611693i
$452$		0			0
$453$	24.0000	+	41.5692i	1.12762	+	1.95309i
$454$		0			0
$455$	7.50000	+	7.79423i	0.351605	+	0.365399i
$456$		0			0
$457$	0.500000	+	0.866025i	0.0233890	+	0.0405110i	0.877483	−	0.479608i	$-0.159221\pi$
$457$	0.500000	+	0.866025i	0.0233890	+	0.0405110i	−0.854094	+	0.520119i	$0.825888\pi$
$458$		0			0
$459$	13.5000	−	23.3827i	0.630126	−	1.09141i
$460$		0			0
$461$	−9.50000	+	16.4545i	−0.442459	+	0.766362i	−0.997871	−	0.0652135i	$-0.979227\pi$
$461$	−9.50000	+	16.4545i	−0.442459	+	0.766362i	0.555412	+	0.831575i	$0.312560\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		0			0
$465$	−12.0000	−	20.7846i	−0.556487	−	0.963863i
$466$		0			0
$467$	−16.0000			−0.740392			−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000			−0.740392			−0.370196	+	0.928954i	$0.620709\pi$
$468$		0			0
$469$	−45.0000			−2.07791
$470$		0			0
$471$	−21.0000	−	36.3731i	−0.967629	−	1.67598i
$472$		0			0
$473$	−5.00000			−0.229900
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$	−6.00000	+	10.3923i	−0.274721	+	0.475831i
$478$		0			0
$479$	7.50000	+	12.9904i	0.342684	+	0.593546i	0.984930	−	0.172953i	$-0.0553307\pi$
$479$	7.50000	+	12.9904i	0.342684	+	0.593546i	−0.642246	+	0.766498i	$0.721997\pi$
$480$		0			0
$481$	−31.5000	+	7.79423i	−1.43628	+	0.355386i
$482$		0			0
$483$	−13.5000	−	23.3827i	−0.614271	−	1.06395i
$484$		0			0
$485$	3.50000	−	6.06218i	0.158927	−	0.275269i
$486$		0			0
$487$	−15.5000	+	26.8468i	−0.702372	+	1.21654i	0.265260	+	0.964177i	$0.414542\pi$
$487$	−15.5000	+	26.8468i	−0.702372	+	1.21654i	−0.967632	+	0.252367i	$0.918791\pi$
$488$		0			0
$489$	51.0000			2.30630
$490$		0			0
$491$	3.50000	+	6.06218i	0.157953	+	0.273582i	0.934130	−	0.356932i	$-0.116177\pi$
$491$	3.50000	+	6.06218i	0.157953	+	0.273582i	−0.776178	+	0.630514i	$0.782844\pi$
$492$		0			0
$493$	−15.0000			−0.675566
$494$		0			0
$495$	−30.0000			−1.34840
$496$		0			0
$497$	1.50000	+	2.59808i	0.0672842	+	0.116540i
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	31.5000	−	54.5596i	1.40732	−	2.43754i
$502$		0			0
$503$	10.5000	−	18.1865i	0.468172	−	0.810897i	−0.531167	−	0.847267i	$-0.678246\pi$
$503$	10.5000	−	18.1865i	0.468172	−	0.810897i	0.999338	+	0.0363700i	$0.0115795\pi$
$504$		0			0
$505$	−8.50000	−	14.7224i	−0.378245	−	0.655140i
$506$		0			0
$507$	1.50000	−	38.9711i	0.0666173	−	1.73077i
$508$		0			0
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	0.650556	−	0.759458i	$-0.274536\pi$
$509$	−7.50000	−	12.9904i	−0.332432	−	0.575789i	−0.982988	+	0.183669i	$0.941202\pi$
$510$		0			0
$511$	15.0000	−	25.9808i	0.663561	−	1.14932i
$512$		0			0
$513$	−22.5000	+	38.9711i	−0.993399	+	1.72062i
$514$		0			0
$515$	−8.00000			−0.352522
$516$		0			0
$517$	30.0000	+	51.9615i	1.31940	+	2.28527i
$518$		0			0
$519$	−21.0000			−0.921798
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	15.5000	+	26.8468i	0.677768	+	1.17393i	0.975652	+	0.219326i	$0.0703858\pi$
$523$	15.5000	+	26.8468i	0.677768	+	1.17393i	−0.297884	+	0.954602i	$0.596281\pi$
$524$		0			0
$525$	−9.00000			−0.392792
$526$		0			0
$527$	−12.0000	+	20.7846i	−0.522728	+	0.905392i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$	15.0000	+	25.9808i	0.650945	+	1.12747i
$532$		0			0
$533$	10.5000	−	2.59808i	0.454805	−	0.112535i
$534$		0			0
$535$	6.50000	+	11.2583i	0.281020	+	0.486740i
$536$		0			0
$537$	−4.50000	+	7.79423i	−0.194189	+	0.336346i
$538$		0			0
$539$	−5.00000	+	8.66025i	−0.215365	+	0.373024i
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	15.0000	+	25.9808i	0.643712	+	1.11494i
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		0			0
$549$	3.00000	+	5.19615i	0.128037	+	0.221766i
$550$		0			0
$551$	25.0000			1.06504
$552$		0			0
$553$	6.00000	−	10.3923i	0.255146	−	0.441926i
$554$		0			0
$555$	13.5000	−	23.3827i	0.573043	−	0.992540i
$556$		0			0
$557$	−1.50000	−	2.59808i	−0.0635570	−	0.110084i	0.832496	−	0.554031i	$-0.186911\pi$
$557$	−1.50000	−	2.59808i	−0.0635570	−	0.110084i	−0.896053	+	0.443947i	$0.853578\pi$
$558$		0			0
$559$	2.50000	+	2.59808i	0.105739	+	0.109887i
$560$		0			0
$561$	22.5000	+	38.9711i	0.949951	+	1.64536i
$562$		0			0
$563$	−1.50000	+	2.59808i	−0.0632175	+	0.109496i	−0.895902	−	0.444252i	$-0.853470\pi$
$563$	−1.50000	+	2.59808i	−0.0632175	+	0.109496i	0.832684	+	0.553748i	$0.186803\pi$
$564$		0			0
$565$	−10.5000	+	18.1865i	−0.441738	+	0.765113i
$566$		0			0
$567$	27.0000			1.13389
$568$		0			0
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	0.664897	−	0.746935i	$-0.268475\pi$
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	−0.979313	+	0.202350i	$0.935142\pi$
$570$		0			0
$571$	28.0000			1.17176			0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000			1.17176			0.585882	+	0.810397i	$0.300748\pi$
$572$		0			0
$573$	−9.00000			−0.375980
$574$		0			0
$575$	1.50000	+	2.59808i	0.0625543	+	0.108347i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$	−31.5000	+	54.5596i	−1.30910	+	2.26742i
$580$		0			0
$581$	−18.0000	+	31.1769i	−0.746766	+	1.29344i
$582$		0			0
$583$	−5.00000	−	8.66025i	−0.207079	−	0.358671i
$584$		0			0
$585$	15.0000	+	15.5885i	0.620174	+	0.644503i
$586$		0			0
$587$	19.5000	+	33.7750i	0.804851	+	1.39404i	0.916392	+	0.400283i	$0.131088\pi$
$587$	19.5000	+	33.7750i	0.804851	+	1.39404i	−0.111540	+	0.993760i	$0.535578\pi$
$588$		0			0
$589$	20.0000	−	34.6410i	0.824086	−	1.42736i
$590$		0			0
$591$	4.50000	−	7.79423i	0.185105	−	0.320612i
$592$		0			0
$593$	−10.0000			−0.410651			−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000			−0.410651			−0.205325	+	0.978694i	$0.565825\pi$
$594$		0			0
$595$	4.50000	+	7.79423i	0.184482	+	0.319532i
$596$		0			0
$597$	27.0000			1.10504
$598$		0			0
$599$	−20.0000			−0.817178			−0.408589	−	0.912719i	$-0.633979\pi$
$599$	−20.0000			−0.817178			−0.408589	+	0.912719i	$0.633979\pi$
$600$		0			0
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	−0.963405	−	0.268049i	$-0.913621\pi$
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	0.249565	−	0.968358i	$-0.419712\pi$
$602$		0			0
$603$	−90.0000			−3.66508
$604$		0			0
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	16.5000	−	28.5788i	0.669714	−	1.15998i	−0.308270	−	0.951299i	$-0.599750\pi$
$607$	16.5000	−	28.5788i	0.669714	−	1.15998i	0.977984	−	0.208680i	$-0.0669168\pi$
$608$		0			0
$609$	−22.5000	−	38.9711i	−0.911746	−	1.57919i
$610$		0			0
$611$	12.0000	−	41.5692i	0.485468	−	1.68171i
$612$		0			0
$613$	0.500000	+	0.866025i	0.0201948	+	0.0349784i	0.875946	−	0.482409i	$-0.160238\pi$
$613$	0.500000	+	0.866025i	0.0201948	+	0.0349784i	−0.855751	+	0.517387i	$0.826905\pi$
$614$		0			0
$615$	−4.50000	+	7.79423i	−0.181458	+	0.314294i
$616$		0			0
$617$	−11.5000	+	19.9186i	−0.462973	+	0.801892i	−0.999107	−	0.0422403i	$-0.986550\pi$
$617$	−11.5000	+	19.9186i	−0.462973	+	0.801892i	0.536135	+	0.844132i	$0.319884\pi$
$618$		0			0
$619$	44.0000			1.76851			0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000			1.76851			0.884255	+	0.467005i	$0.154667\pi$
$620$		0			0
$621$	−13.5000	−	23.3827i	−0.541736	−	0.938315i
$622$		0			0
$623$	−3.00000			−0.120192
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−37.5000	−	64.9519i	−1.49761	−	2.59393i
$628$		0			0
$629$	−27.0000			−1.07656
$630$		0			0
$631$	−7.50000	+	12.9904i	−0.298570	+	0.517139i	−0.975809	−	0.218624i	$-0.929843\pi$
$631$	−7.50000	+	12.9904i	−0.298570	+	0.517139i	0.677239	+	0.735763i	$0.263176\pi$
$632$		0			0
$633$	−16.5000	+	28.5788i	−0.655816	+	1.13591i
$634$		0			0
$635$	8.50000	+	14.7224i	0.337312	+	0.584242i
$636$		0			0
$637$	7.00000	−	1.73205i	0.277350	−	0.0686264i
$638$		0			0
$639$	3.00000	+	5.19615i	0.118678	+	0.205557i
$640$		0			0
$641$	22.5000	−	38.9711i	0.888697	−	1.53927i	0.0472793	−	0.998882i	$-0.484945\pi$
$641$	22.5000	−	38.9711i	0.888697	−	1.53927i	0.841417	−	0.540386i	$-0.181722\pi$
$642$		0			0
$643$	−3.50000	+	6.06218i	−0.138027	+	0.239069i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$	−3.50000	+	6.06218i	−0.138027	+	0.239069i	0.788723	+	0.614749i	$0.210743\pi$
$644$		0			0
$645$	−3.00000			−0.118125
$646$		0			0
$647$	23.5000	+	40.7032i	0.923880	+	1.60021i	0.793352	+	0.608763i	$0.208334\pi$
$647$	23.5000	+	40.7032i	0.923880	+	1.60021i	0.130528	+	0.991445i	$0.458333\pi$
$648$		0			0
$649$	−25.0000			−0.981336
$650$		0			0
$651$	−72.0000			−2.82190
$652$		0			0
$653$	0.500000	+	0.866025i	0.0195665	+	0.0338902i	0.875643	−	0.482959i	$-0.160438\pi$
$653$	0.500000	+	0.866025i	0.0195665	+	0.0338902i	−0.856076	+	0.516849i	$0.827105\pi$
$654$		0			0
$655$	−12.0000			−0.468879
$656$		0			0
$657$	30.0000	−	51.9615i	1.17041	−	2.02721i
$658$		0			0
$659$	8.50000	−	14.7224i	0.331113	−	0.573505i	−0.651617	−	0.758548i	$-0.725909\pi$
$659$	8.50000	−	14.7224i	0.331113	−	0.573505i	0.982730	+	0.185043i	$0.0592425\pi$
$660$		0			0
$661$	12.5000	+	21.6506i	0.486194	+	0.842112i	0.999874	−	0.0158695i	$-0.00505163\pi$
$661$	12.5000	+	21.6506i	0.486194	+	0.842112i	−0.513680	+	0.857982i	$0.671718\pi$
$662$		0			0
$663$	9.00000	−	31.1769i	0.349531	−	1.21081i
$664$		0			0
$665$	−7.50000	−	12.9904i	−0.290838	−	0.503745i
$666$		0			0
$667$	−7.50000	+	12.9904i	−0.290401	+	0.502990i
$668$		0			0
$669$	19.5000	−	33.7750i	0.753914	−	1.30582i
$670$		0			0
$671$	−5.00000			−0.193023
$672$		0			0
$673$	−23.5000	−	40.7032i	−0.905858	−	1.56899i	−0.819761	−	0.572706i	$-0.805894\pi$
$673$	−23.5000	−	40.7032i	−0.905858	−	1.56899i	−0.0860977	−	0.996287i	$-0.527440\pi$
$674$		0			0
$675$	−9.00000			−0.346410
$676$		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
$681$	15.0000			0.574801
$682$		0			0
$683$	6.50000	−	11.2583i	0.248716	−	0.430788i	−0.714454	−	0.699682i	$-0.753325\pi$
$683$	6.50000	−	11.2583i	0.248716	−	0.430788i	0.963170	+	0.268894i	$0.0866582\pi$
$684$		0			0
$685$	−8.50000	+	14.7224i	−0.324768	+	0.562515i
$686$		0			0
$687$	−21.0000	−	36.3731i	−0.801200	−	1.38772i
$688$		0			0
$689$	−2.00000	+	6.92820i	−0.0761939	+	0.263944i
$690$		0			0
$691$	−6.50000	−	11.2583i	−0.247272	−	0.428287i	0.715496	−	0.698617i	$-0.246201\pi$
$691$	−6.50000	−	11.2583i	−0.247272	−	0.428287i	−0.962768	+	0.270330i	$0.912867\pi$
$692$		0			0
$693$	−45.0000	+	77.9423i	−1.70941	+	2.96078i
$694$		0			0
$695$	5.50000	−	9.52628i	0.208627	−	0.361352i
$696$		0			0
$697$	9.00000			0.340899
$698$		0			0
$699$	−27.0000	−	46.7654i	−1.02123	−	1.76883i
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	45.0000			1.69721
$704$		0			0
$705$	18.0000	+	31.1769i	0.677919	+	1.17419i
$706$		0			0
$707$	−51.0000			−1.91805
$708$		0			0
$709$	−17.5000	+	30.3109i	−0.657226	+	1.13835i	0.324104	+	0.946021i	$0.394937\pi$
$709$	−17.5000	+	30.3109i	−0.657226	+	1.13835i	−0.981331	+	0.192328i	$0.938396\pi$
$710$		0			0
$711$	12.0000	−	20.7846i	0.450035	−	0.779484i
$712$		0			0
$713$	12.0000	+	20.7846i	0.449404	+	0.778390i
$714$		0			0
$715$	−17.5000	+	4.33013i	−0.654463	+	0.161938i
$716$		0			0
$717$	−12.0000	−	20.7846i	−0.448148	−	0.776215i
$718$		0			0
$719$	−17.5000	+	30.3109i	−0.652640	+	1.13041i	0.329840	+	0.944037i	$0.393005\pi$
$719$	−17.5000	+	30.3109i	−0.652640	+	1.13041i	−0.982480	+	0.186369i	$0.940328\pi$
$720$		0			0
$721$	−12.0000	+	20.7846i	−0.446903	+	0.774059i
$722$		0			0
$723$	−57.0000			−2.11985
$724$		0			0
$725$	2.50000	+	4.33013i	0.0928477	+	0.160817i
$726$		0			0
$727$	24.0000			0.890111			0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000			0.890111			0.445055	+	0.895503i	$0.353184\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	1.50000	+	2.59808i	0.0554795	+	0.0960933i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$		0			0
$735$	−3.00000	+	5.19615i	−0.110657	+	0.191663i
$736$		0			0
$737$	37.5000	−	64.9519i	1.38133	−	2.39253i
$738$		0			0
$739$	−16.5000	−	28.5788i	−0.606962	−	1.05129i	−0.991738	−	0.128279i	$-0.959055\pi$
$739$	−16.5000	−	28.5788i	−0.606962	−	1.05129i	0.384776	−	0.923010i	$-0.374279\pi$
$740$		0			0
$741$	−15.0000	+	51.9615i	−0.551039	+	1.90885i
$742$		0			0
$743$	7.50000	+	12.9904i	0.275148	+	0.476571i	0.970173	−	0.242415i	$-0.0779397\pi$
$743$	7.50000	+	12.9904i	0.275148	+	0.476571i	−0.695024	+	0.718986i	$0.744606\pi$
$744$		0			0
$745$	−4.50000	+	7.79423i	−0.164867	+	0.285558i
$746$		0			0
$747$	−36.0000	+	62.3538i	−1.31717	+	2.28141i
$748$		0			0
$749$	39.0000			1.42503
$750$		0			0
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	0.999964	−	0.00849853i	$-0.00270520\pi$
$751$	13.5000	+	23.3827i	0.492622	+	0.853246i	−0.507342	+	0.861745i	$0.669372\pi$
$752$		0			0
$753$	3.00000			0.109326
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	−3.50000	−	6.06218i	−0.127210	−	0.220334i	0.795385	−	0.606105i	$-0.207269\pi$
$757$	−3.50000	−	6.06218i	−0.127210	−	0.220334i	−0.922595	+	0.385771i	$0.873935\pi$
$758$		0			0
$759$	45.0000			1.63340
$760$		0			0
$761$	2.50000	−	4.33013i	0.0906249	−	0.156967i	−0.817149	−	0.576426i	$-0.804447\pi$
$761$	2.50000	−	4.33013i	0.0906249	−	0.156967i	0.907774	+	0.419459i	$0.137780\pi$
$762$		0			0
$763$	−3.00000	+	5.19615i	−0.108607	+	0.188113i
$764$		0			0
$765$	9.00000	+	15.5885i	0.325396	+	0.563602i
$766$		0			0
$767$	12.5000	+	12.9904i	0.451349	+	0.469055i
$768$		0			0
$769$	6.50000	+	11.2583i	0.234396	+	0.405986i	0.959097	−	0.283078i	$-0.0913554\pi$
$769$	6.50000	+	11.2583i	0.234396	+	0.405986i	−0.724701	+	0.689063i	$0.758022\pi$
$770$		0			0
$771$	−19.5000	+	33.7750i	−0.702275	+	1.21638i
$772$		0			0
$773$	18.5000	−	32.0429i	0.665399	−	1.15250i	−0.313778	−	0.949496i	$-0.601595\pi$
$773$	18.5000	−	32.0429i	0.665399	−	1.15250i	0.979177	−	0.203008i	$-0.0650718\pi$
$774$		0			0
$775$	8.00000			0.287368
$776$		0			0
$777$	−40.5000	−	70.1481i	−1.45293	−	2.51655i
$778$		0			0
$779$	−15.0000			−0.537431
$780$		0			0
$781$	−5.00000			−0.178914
$782$		0			0
$783$	−22.5000	−	38.9711i	−0.804084	−	1.39272i
$784$		0			0
$785$	14.0000			0.499681
$786$		0			0
$787$	−15.5000	+	26.8468i	−0.552515	+	0.956985i	0.445577	+	0.895244i	$0.352999\pi$
$787$	−15.5000	+	26.8468i	−0.552515	+	0.956985i	−0.998092	+	0.0617409i	$0.980335\pi$
$788$		0			0
$789$	−28.5000	+	49.3634i	−1.01463	+	1.75739i

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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 2.59808i) q^{3} -1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 2.59808i) q^{3} -1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} +(-3.00000 + 5.19615i) q^{9} +(-2.50000 - 4.33013i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(-1.50000 - 2.59808i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(2.50000 - 4.33013i) q^{19} -9.00000 q^{21} +(1.50000 + 2.59808i) q^{23} +1.00000 q^{25} -9.00000 q^{27} +(2.50000 + 4.33013i) q^{29} +8.00000 q^{31} +(7.50000 - 12.9904i) q^{33} +(1.50000 - 2.59808i) q^{35} +(4.50000 + 7.79423i) q^{37} +(-10.5000 + 2.59808i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(0.500000 - 0.866025i) q^{43} +(3.00000 - 5.19615i) q^{45} -12.0000 q^{47} +(-1.00000 - 1.73205i) q^{49} -9.00000 q^{51} +2.00000 q^{53} +(2.50000 + 4.33013i) q^{55} +15.0000 q^{57} +(2.50000 - 4.33013i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-9.00000 - 15.5885i) q^{63} +(1.00000 - 3.46410i) q^{65} +(7.50000 + 12.9904i) q^{67} +(-4.50000 + 7.79423i) q^{69} +(0.500000 - 0.866025i) q^{71} -10.0000 q^{73} +(1.50000 + 2.59808i) q^{75} +15.0000 q^{77} -4.00000 q^{79} +(-4.50000 - 7.79423i) q^{81} +12.0000 q^{83} +(1.50000 - 2.59808i) q^{85} +(-7.50000 + 12.9904i) q^{87} +(0.500000 + 0.866025i) q^{89} +(-7.50000 - 7.79423i) q^{91} +(12.0000 + 20.7846i) q^{93} +(-2.50000 + 4.33013i) q^{95} +(-3.50000 + 6.06218i) q^{97} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 2 q^{5} - 3 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 - 2 * q^5 - 3 * q^7 - 6 * q^9	\( 2 q + 3 q^{3} - 2 q^{5} - 3 q^{7} - 6 q^{9} - 5 q^{11} - 2 q^{13} - 3 q^{15} - 3 q^{17} + 5 q^{19} - 18 q^{21} + 3 q^{23} + 2 q^{25} - 18 q^{27} + 5 q^{29} + 16 q^{31} + 15 q^{33} + 3 q^{35} + 9 q^{37} - 21 q^{39} - 3 q^{41} + q^{43} + 6 q^{45} - 24 q^{47} - 2 q^{49} - 18 q^{51} + 4 q^{53} + 5 q^{55} + 30 q^{57} + 5 q^{59} + q^{61} - 18 q^{63} + 2 q^{65} + 15 q^{67} - 9 q^{69} + q^{71} - 20 q^{73} + 3 q^{75} + 30 q^{77} - 8 q^{79} - 9 q^{81} + 24 q^{83} + 3 q^{85} - 15 q^{87} + q^{89} - 15 q^{91} + 24 q^{93} - 5 q^{95} - 7 q^{97} + 60 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 2 * q^5 - 3 * q^7 - 6 * q^9 - 5 * q^11 - 2 * q^13 - 3 * q^15 - 3 * q^17 + 5 * q^19 - 18 * q^21 + 3 * q^23 + 2 * q^25 - 18 * q^27 + 5 * q^29 + 16 * q^31 + 15 * q^33 + 3 * q^35 + 9 * q^37 - 21 * q^39 - 3 * q^41 + q^43 + 6 * q^45 - 24 * q^47 - 2 * q^49 - 18 * q^51 + 4 * q^53 + 5 * q^55 + 30 * q^57 + 5 * q^59 + q^61 - 18 * q^63 + 2 * q^65 + 15 * q^67 - 9 * q^69 + q^71 - 20 * q^73 + 3 * q^75 + 30 * q^77 - 8 * q^79 - 9 * q^81 + 24 * q^83 + 3 * q^85 - 15 * q^87 + q^89 - 15 * q^91 + 24 * q^93 - 5 * q^95 - 7 * q^97 + 60 * q^99

Introduction

L-functions

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