LMFDB - Embedded newform 728.2.q.a.289.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [728,2,Mod(289,728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(728, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 728 = 2^{3} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	728.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:	$5.81310926715$
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:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	728.289
Dual form			728.2.q.a.529.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(0.500000 - 0.866025i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{15} -2.00000 q^{17} +(1.50000 + 2.59808i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{25} +5.00000 q^{27} +(4.50000 + 7.79423i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(-0.500000 - 0.866025i) q^{33} +(1.50000 + 7.79423i) q^{35} +10.0000 q^{37} +(2.50000 + 2.59808i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(-1.50000 + 2.59808i) q^{43} -6.00000 q^{45} +(-5.50000 + 9.52628i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-1.50000 - 2.59808i) q^{53} +(1.50000 + 2.59808i) q^{55} +3.00000 q^{57} +12.0000 q^{59} +(2.50000 + 4.33013i) q^{61} +(5.00000 + 1.73205i) q^{63} +(-7.50000 - 7.79423i) q^{65} +(4.50000 - 7.79423i) q^{67} +(0.500000 - 0.866025i) q^{71} +(-5.50000 - 9.52628i) q^{73} -4.00000 q^{75} +(-0.500000 - 2.59808i) q^{77} +(4.50000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} -8.00000 q^{83} +(3.00000 - 5.19615i) q^{85} +9.00000 q^{87} -10.0000 q^{89} +(4.00000 + 8.66025i) q^{91} -1.00000 q^{93} -9.00000 q^{95} +(6.50000 - 11.2583i) q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^3 - 3 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} - 4 q^{17} + 3 q^{19} - q^{21} - 4 q^{25} + 10 q^{27} + 9 q^{29} - q^{31} - q^{33} + 3 q^{35} + 20 q^{37} + 5 q^{39} - 3 q^{41} - 3 q^{43} - 12 q^{45} - 11 q^{47} + 2 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{55} + 6 q^{57} + 24 q^{59} + 5 q^{61} + 10 q^{63} - 15 q^{65} + 9 q^{67} + q^{71} - 11 q^{73} - 8 q^{75} - q^{77} + 9 q^{79} - q^{81} - 16 q^{83} + 6 q^{85} + 18 q^{87} - 20 q^{89} + 8 q^{91} - 2 q^{93} - 18 q^{95} + 13 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + q^3 - 3 * q^5 + 4 * q^7 + 2 * q^9 + q^11 - 2 * q^13 + 3 * q^15 - 4 * q^17 + 3 * q^19 - q^21 - 4 * q^25 + 10 * q^27 + 9 * q^29 - q^31 - q^33 + 3 * q^35 + 20 * q^37 + 5 * q^39 - 3 * q^41 - 3 * q^43 - 12 * q^45 - 11 * q^47 + 2 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^55 + 6 * q^57 + 24 * q^59 + 5 * q^61 + 10 * q^63 - 15 * q^65 + 9 * q^67 + q^71 - 11 * q^73 - 8 * q^75 - q^77 + 9 * q^79 - q^81 - 16 * q^83 + 6 * q^85 + 18 * q^87 - 20 * q^89 + 8 * q^91 - 2 * q^93 - 18 * q^95 + 13 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/728\mathbb{Z}\right)^\times$.

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i	−0.780750	−	0.624844i	$-0.785163\pi$
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i	0.931505	+	0.363727i	$0.118496\pi$
$12$		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
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	1.50000	+	2.59808i	0.344124	+	0.596040i	0.985194	−	0.171442i	$-0.0548427\pi$
$19$	1.50000	+	2.59808i	0.344124	+	0.596040i	−0.641071	+	0.767482i	$0.721509\pi$
$20$		0			0
$21$	−0.500000	−	2.59808i	−0.109109	−	0.566947i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	0.893517	+	0.449029i	$0.148230\pi$
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	−0.0578882	+	0.998323i	$0.518437\pi$
$30$		0			0
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	0.817625	−	0.575751i	$-0.195290\pi$
$31$	−0.500000	−	0.866025i	−0.0898027	−	0.155543i	−0.907428	+	0.420208i	$0.861957\pi$
$32$		0			0
$33$	−0.500000	−	0.866025i	−0.0870388	−	0.150756i
$34$		0			0
$35$	1.50000	+	7.79423i	0.253546	+	1.31747i
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$	2.50000	+	2.59808i	0.400320	+	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
$43$	−1.50000	+	2.59808i	−0.228748	+	0.396203i	−0.957437	−	0.288641i	$-0.906796\pi$
$43$	−1.50000	+	2.59808i	−0.228748	+	0.396203i	0.728689	+	0.684844i	$0.240130\pi$
$44$		0			0
$45$	−6.00000			−0.894427
$46$		0			0
$47$	−5.50000	+	9.52628i	−0.802257	+	1.38955i	0.115870	+	0.993264i	$0.463035\pi$
$47$	−5.50000	+	9.52628i	−0.802257	+	1.38955i	−0.918127	+	0.396286i	$0.870299\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	−1.00000	+	1.73205i	−0.140028	+	0.242536i
$52$		0			0
$53$	−1.50000	−	2.59808i	−0.206041	−	0.356873i	0.744423	−	0.667708i	$-0.232725\pi$
$53$	−1.50000	−	2.59808i	−0.206041	−	0.356873i	−0.950464	+	0.310835i	$0.899391\pi$
$54$		0			0
$55$	1.50000	+	2.59808i	0.202260	+	0.350325i
$56$		0			0
$57$	3.00000			0.397360
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		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
$63$	5.00000	+	1.73205i	0.629941	+	0.218218i
$64$		0			0
$65$	−7.50000	−	7.79423i	−0.930261	−	0.966755i
$66$		0			0
$67$	4.50000	−	7.79423i	0.549762	−	0.952217i	−0.448528	−	0.893769i	$-0.648052\pi$
$67$	4.50000	−	7.79423i	0.549762	−	0.952217i	0.998290	−	0.0584478i	$-0.0186151\pi$
$68$		0			0
$69$		0			0
$70$		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
$73$	−5.50000	−	9.52628i	−0.643726	−	1.11497i	−0.984594	−	0.174855i	$-0.944054\pi$
$73$	−5.50000	−	9.52628i	−0.643726	−	1.11497i	0.340868	−	0.940111i	$-0.389279\pi$
$74$		0			0
$75$	−4.00000			−0.461880
$76$		0			0
$77$	−0.500000	−	2.59808i	−0.0569803	−	0.296078i
$78$		0			0
$79$	4.50000	−	7.79423i	0.506290	−	0.876919i	−0.493684	−	0.869641i	$-0.664350\pi$
$79$	4.50000	−	7.79423i	0.506290	−	0.876919i	0.999974	−	0.00727784i	$-0.00231663\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$		0			0
$87$	9.00000			0.964901
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$	4.00000	+	8.66025i	0.419314	+	0.907841i
$92$		0			0
$93$	−1.00000			−0.103695
$94$		0			0
$95$	−9.00000			−0.923381
$96$		0			0
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	−0.320647	−	0.947199i	$-0.603900\pi$
$97$	6.50000	−	11.2583i	0.659975	−	1.14311i	0.980622	−	0.195911i	$-0.0627665\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	−5.50000	+	9.52628i	−0.547270	+	0.947900i	0.451190	+	0.892428i	$0.351000\pi$
$101$	−5.50000	+	9.52628i	−0.547270	+	0.947900i	−0.998460	+	0.0554722i	$0.982334\pi$
$102$		0			0
$103$	−3.50000	+	6.06218i	−0.344865	+	0.597324i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−3.50000	+	6.06218i	−0.344865	+	0.597324i	0.640464	+	0.767988i	$0.278742\pi$
$104$		0			0
$105$	7.50000	+	2.59808i	0.731925	+	0.253546i
$106$		0			0
$107$	−16.0000			−1.54678			−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000			−1.54678			−0.773389	+	0.633932i	$0.781440\pi$
$108$		0			0
$109$	−7.50000	−	12.9904i	−0.718370	−	1.24425i	−0.961645	−	0.274296i	$-0.911555\pi$
$109$	−7.50000	−	12.9904i	−0.718370	−	1.24425i	0.243276	−	0.969957i	$-0.421778\pi$
$110$		0			0
$111$	5.00000	−	8.66025i	0.474579	−	0.821995i
$112$		0			0
$113$	0.500000	−	0.866025i	0.0470360	−	0.0814688i	−0.841549	−	0.540181i	$-0.818356\pi$
$113$	0.500000	−	0.866025i	0.0470360	−	0.0814688i	0.888585	+	0.458712i	$0.151689\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.00000	+	1.73205i	−0.647150	+	0.160128i
$118$		0			0
$119$	−4.00000	+	3.46410i	−0.366679	+	0.317554i
$120$		0			0
$121$	5.00000	+	8.66025i	0.454545	+	0.787296i
$122$		0			0
$123$	−3.00000			−0.270501
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	3.50000	+	6.06218i	0.310575	+	0.537931i	0.978487	−	0.206309i	$-0.0661452\pi$
$127$	3.50000	+	6.06218i	0.310575	+	0.537931i	−0.667912	+	0.744240i	$0.732812\pi$
$128$		0			0
$129$	1.50000	+	2.59808i	0.132068	+	0.228748i
$130$		0			0
$131$	8.50000	−	14.7224i	0.742648	−	1.28630i	−0.208637	−	0.977993i	$-0.566903\pi$
$131$	8.50000	−	14.7224i	0.742648	−	1.28630i	0.951285	−	0.308312i	$-0.0997640\pi$
$132$		0			0
$133$	7.50000	+	2.59808i	0.650332	+	0.225282i
$134$		0			0
$135$	−7.50000	+	12.9904i	−0.645497	+	1.11803i
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$	−1.50000	+	2.59808i	−0.127228	+	0.220366i	−0.922602	−	0.385754i	$-0.873941\pi$
$139$	−1.50000	+	2.59808i	−0.127228	+	0.220366i	0.795373	+	0.606120i	$0.207275\pi$
$140$		0			0
$141$	5.50000	+	9.52628i	0.463184	+	0.802257i
$142$		0			0
$143$	2.50000	+	2.59808i	0.209061	+	0.217262i
$144$		0			0
$145$	−27.0000			−2.24223
$146$		0			0
$147$	−5.50000	−	4.33013i	−0.453632	−	0.357143i
$148$		0			0
$149$	4.50000	+	7.79423i	0.368654	+	0.638528i	0.989355	−	0.145519i	$-0.0464853\pi$
$149$	4.50000	+	7.79423i	0.368654	+	0.638528i	−0.620701	+	0.784047i	$0.713152\pi$
$150$		0			0
$151$	−0.500000	−	0.866025i	−0.0406894	−	0.0704761i	0.844963	−	0.534824i	$-0.179622\pi$
$151$	−0.500000	−	0.866025i	−0.0406894	−	0.0704761i	−0.885653	+	0.464348i	$0.846289\pi$
$152$		0			0
$153$	−2.00000	−	3.46410i	−0.161690	−	0.280056i
$154$		0			0
$155$	3.00000			0.240966
$156$		0			0
$157$	2.50000	+	4.33013i	0.199522	+	0.345582i	0.948373	−	0.317156i	$-0.102728\pi$
$157$	2.50000	+	4.33013i	0.199522	+	0.345582i	−0.748852	+	0.662738i	$0.769394\pi$
$158$		0			0
$159$	−3.00000			−0.237915
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−10.5000	−	18.1865i	−0.822423	−	1.42448i	−0.903873	−	0.427802i	$-0.859288\pi$
$163$	−10.5000	−	18.1865i	−0.822423	−	1.42448i	0.0814491	−	0.996678i	$-0.474045\pi$
$164$		0			0
$165$	3.00000			0.233550
$166$		0			0
$167$	−4.50000	−	7.79423i	−0.348220	−	0.603136i	0.637713	−	0.770274i	$-0.279881\pi$
$167$	−4.50000	−	7.79423i	−0.348220	−	0.603136i	−0.985933	+	0.167139i	$0.946547\pi$
$168$		0			0
$169$	−11.0000	−	6.92820i	−0.846154	−	0.532939i
$170$		0			0
$171$	−3.00000	+	5.19615i	−0.229416	+	0.397360i
$172$		0			0
$173$	6.50000	+	11.2583i	0.494186	+	0.855955i	0.999978	−	0.00670064i	$-0.00213290\pi$
$173$	6.50000	+	11.2583i	0.494186	+	0.855955i	−0.505792	+	0.862656i	$0.668800\pi$
$174$		0			0
$175$	−10.0000	−	3.46410i	−0.755929	−	0.261861i
$176$		0			0
$177$	6.00000	−	10.3923i	0.450988	−	0.781133i
$178$		0			0
$179$	10.5000	−	18.1865i	0.784807	−	1.35933i	−0.144308	−	0.989533i	$-0.546095\pi$
$179$	10.5000	−	18.1865i	0.784807	−	1.35933i	0.929114	−	0.369792i	$-0.120571\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$	5.00000			0.369611
$184$		0			0
$185$	−15.0000	+	25.9808i	−1.10282	+	1.91014i
$186$		0			0
$187$	−1.00000	+	1.73205i	−0.0731272	+	0.126660i
$188$		0			0
$189$	10.0000	−	8.66025i	0.727393	−	0.629941i
$190$		0			0
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	0.915177	−	0.403051i	$-0.132050\pi$
$191$	1.50000	+	2.59808i	0.108536	+	0.187990i	−0.806641	+	0.591041i	$0.798717\pi$
$192$		0			0
$193$	8.50000	−	14.7224i	0.611843	−	1.05974i	−0.379086	−	0.925361i	$-0.623762\pi$
$193$	8.50000	−	14.7224i	0.611843	−	1.05974i	0.990930	−	0.134382i	$-0.0429051\pi$
$194$		0			0
$195$	−10.5000	+	2.59808i	−0.751921	+	0.186052i
$196$		0			0
$197$	−11.5000	−	19.9186i	−0.819341	−	1.41914i	−0.906168	−	0.422917i	$-0.861006\pi$
$197$	−11.5000	−	19.9186i	−0.819341	−	1.41914i	0.0868274	−	0.996223i	$-0.472327\pi$
$198$		0			0
$199$	−4.00000			−0.283552			−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000			−0.283552			−0.141776	+	0.989899i	$0.545281\pi$
$200$		0			0
$201$	−4.50000	−	7.79423i	−0.317406	−	0.549762i
$202$		0			0
$203$	22.5000	+	7.79423i	1.57919	+	0.547048i
$204$		0			0
$205$	9.00000			0.628587
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.00000			0.207514
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$		0			0
$213$	−0.500000	−	0.866025i	−0.0342594	−	0.0593391i
$214$		0			0
$215$	−4.50000	−	7.79423i	−0.306897	−	0.531562i
$216$		0			0
$217$	−2.50000	−	0.866025i	−0.169711	−	0.0587896i
$218$		0			0
$219$	−11.0000			−0.743311
$220$		0			0
$221$	2.00000	−	6.92820i	0.134535	−	0.466041i
$222$		0			0
$223$	−10.5000	−	18.1865i	−0.703132	−	1.21786i	−0.967361	−	0.253401i	$-0.918451\pi$
$223$	−10.5000	−	18.1865i	−0.703132	−	1.21786i	0.264229	−	0.964460i	$-0.414882\pi$
$224$		0			0
$225$	4.00000	−	6.92820i	0.266667	−	0.461880i
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	6.50000	−	11.2583i	0.429532	−	0.743971i	−0.567300	−	0.823511i	$-0.692012\pi$
$229$	6.50000	−	11.2583i	0.429532	−	0.743971i	0.996832	+	0.0795401i	$0.0253452\pi$
$230$		0			0
$231$	−2.50000	−	0.866025i	−0.164488	−	0.0569803i
$232$		0			0
$233$	−1.50000	+	2.59808i	−0.0982683	+	0.170206i	−0.910968	−	0.412477i	$-0.864664\pi$
$233$	−1.50000	+	2.59808i	−0.0982683	+	0.170206i	0.812700	+	0.582683i	$0.197997\pi$
$234$		0			0
$235$	−16.5000	−	28.5788i	−1.07634	−	1.86428i
$236$		0			0
$237$	−4.50000	−	7.79423i	−0.292306	−	0.506290i
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−10.0000			−0.644157			−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000			−0.644157			−0.322078	+	0.946713i	$0.604381\pi$
$242$		0			0
$243$	8.00000	+	13.8564i	0.513200	+	0.888889i
$244$		0			0
$245$	16.5000	+	12.9904i	1.05415	+	0.829925i
$246$		0			0
$247$	−10.5000	+	2.59808i	−0.668099	+	0.165312i
$248$		0			0
$249$	−4.00000	+	6.92820i	−0.253490	+	0.439057i
$250$		0			0
$251$	−1.50000	+	2.59808i	−0.0946792	+	0.163989i	−0.909475	−	0.415759i	$-0.863516\pi$
$251$	−1.50000	+	2.59808i	−0.0946792	+	0.163989i	0.814795	+	0.579748i	$0.196849\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−3.00000	−	5.19615i	−0.187867	−	0.325396i
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	20.0000	−	17.3205i	1.24274	−	1.07624i
$260$		0			0
$261$	−9.00000	+	15.5885i	−0.557086	+	0.964901i
$262$		0			0
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	−0.693276	−	0.720672i	$-0.743833\pi$
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	0.970758	+	0.240059i	$0.0771668\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$	−5.00000	+	8.66025i	−0.305995	+	0.529999i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$	9.50000	+	0.866025i	0.574966	+	0.0524142i
$274$		0			0
$275$	−4.00000			−0.241209
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$	1.00000	−	1.73205i	0.0598684	−	0.103695i
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$	−13.5000	+	23.3827i	−0.802492	+	1.38996i	0.115480	+	0.993310i	$0.463159\pi$
$283$	−13.5000	+	23.3827i	−0.802492	+	1.38996i	−0.917971	+	0.396647i	$0.870174\pi$
$284$		0			0
$285$	−4.50000	+	7.79423i	−0.266557	+	0.461690i
$286$		0			0
$287$	−7.50000	−	2.59808i	−0.442711	−	0.153360i
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$	−6.50000	−	11.2583i	−0.381037	−	0.659975i
$292$		0			0
$293$	−9.50000	+	16.4545i	−0.554996	+	0.961281i	0.442908	+	0.896567i	$0.353947\pi$
$293$	−9.50000	+	16.4545i	−0.554996	+	0.961281i	−0.997904	+	0.0647140i	$0.979386\pi$
$294$		0			0
$295$	−18.0000	+	31.1769i	−1.04800	+	1.81519i
$296$		0			0
$297$	2.50000	−	4.33013i	0.145065	−	0.251259i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.50000	+	7.79423i	0.0864586	+	0.449252i
$302$		0			0
$303$	5.50000	+	9.52628i	0.315967	+	0.547270i
$304$		0			0
$305$	−15.0000			−0.858898
$306$		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
$309$	3.50000	+	6.06218i	0.199108	+	0.344865i
$310$		0			0
$311$	−2.50000	−	4.33013i	−0.141762	−	0.245539i	0.786398	−	0.617720i	$-0.211943\pi$
$311$	−2.50000	−	4.33013i	−0.141762	−	0.245539i	−0.928160	+	0.372181i	$0.878610\pi$
$312$		0			0
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	−0.978553	−	0.205996i	$-0.933957\pi$
$313$	−5.50000	+	9.52628i	−0.310878	+	0.538457i	0.667674	+	0.744453i	$0.267290\pi$
$314$		0			0
$315$	−12.0000	+	10.3923i	−0.676123	+	0.585540i
$316$		0			0
$317$	0.500000	−	0.866025i	0.0280828	−	0.0486408i	−0.851642	−	0.524123i	$-0.824393\pi$
$317$	0.500000	−	0.866025i	0.0280828	−	0.0486408i	0.879725	+	0.475482i	$0.157726\pi$
$318$		0			0
$319$	9.00000			0.503903
$320$		0			0
$321$	−8.00000	+	13.8564i	−0.446516	+	0.773389i
$322$		0			0
$323$	−3.00000	−	5.19615i	−0.166924	−	0.289122i
$324$		0			0
$325$	14.0000	−	3.46410i	0.776580	−	0.192154i
$326$		0			0
$327$	−15.0000			−0.829502
$328$		0			0
$329$	5.50000	+	28.5788i	0.303225	+	1.57560i
$330$		0			0
$331$	11.5000	+	19.9186i	0.632097	+	1.09482i	0.987122	+	0.159968i	$0.0511390\pi$
$331$	11.5000	+	19.9186i	0.632097	+	1.09482i	−0.355025	+	0.934857i	$0.615528\pi$
$332$		0			0
$333$	10.0000	+	17.3205i	0.547997	+	0.949158i
$334$		0			0
$335$	13.5000	+	23.3827i	0.737584	+	1.27753i
$336$		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
$339$	−0.500000	−	0.866025i	−0.0271563	−	0.0470360i
$340$		0			0
$341$	−1.00000			−0.0541530
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	16.0000			0.858925			0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000			0.858925			0.429463	+	0.903085i	$0.358703\pi$
$348$		0			0
$349$	12.5000	+	21.6506i	0.669110	+	1.15893i	0.978153	+	0.207884i	$0.0666577\pi$
$349$	12.5000	+	21.6506i	0.669110	+	1.15893i	−0.309044	+	0.951048i	$0.600009\pi$
$350$		0			0
$351$	−5.00000	+	17.3205i	−0.266880	+	0.924500i
$352$		0			0
$353$	−7.50000	+	12.9904i	−0.399185	+	0.691408i	−0.993626	−	0.112731i	$-0.964040\pi$
$353$	−7.50000	+	12.9904i	−0.399185	+	0.691408i	0.594441	+	0.804139i	$0.297373\pi$
$354$		0			0
$355$	1.50000	+	2.59808i	0.0796117	+	0.137892i
$356$		0			0
$357$	1.00000	+	5.19615i	0.0529256	+	0.275010i
$358$		0			0
$359$	14.5000	−	25.1147i	0.765281	−	1.32551i	−0.174817	−	0.984601i	$-0.555933\pi$
$359$	14.5000	−	25.1147i	0.765281	−	1.32551i	0.940098	−	0.340904i	$-0.110733\pi$
$360$		0			0
$361$	5.00000	−	8.66025i	0.263158	−	0.455803i
$362$		0			0
$363$	10.0000			0.524864
$364$		0			0
$365$	33.0000			1.72730
$366$		0			0
$367$	14.5000	−	25.1147i	0.756894	−	1.31098i	−0.187533	−	0.982258i	$-0.560049\pi$
$367$	14.5000	−	25.1147i	0.756894	−	1.31098i	0.944427	−	0.328720i	$-0.106617\pi$
$368$		0			0
$369$	3.00000	−	5.19615i	0.156174	−	0.270501i
$370$		0			0
$371$	−7.50000	−	2.59808i	−0.389381	−	0.134885i
$372$		0			0
$373$	8.50000	+	14.7224i	0.440113	+	0.762299i	0.997697	−	0.0678218i	$-0.0216049\pi$
$373$	8.50000	+	14.7224i	0.440113	+	0.762299i	−0.557584	+	0.830120i	$0.688272\pi$
$374$		0			0
$375$	−1.50000	+	2.59808i	−0.0774597	+	0.134164i
$376$		0			0
$377$	−31.5000	+	7.79423i	−1.62233	+	0.401423i
$378$		0			0
$379$	5.50000	+	9.52628i	0.282516	+	0.489332i	0.972004	−	0.234965i	$-0.0754976\pi$
$379$	5.50000	+	9.52628i	0.282516	+	0.489332i	−0.689488	+	0.724297i	$0.742164\pi$
$380$		0			0
$381$	7.00000			0.358621
$382$		0			0
$383$	−8.50000	−	14.7224i	−0.434330	−	0.752281i	0.562911	−	0.826518i	$-0.309681\pi$
$383$	−8.50000	−	14.7224i	−0.434330	−	0.752281i	−0.997241	+	0.0742364i	$0.976348\pi$
$384$		0			0
$385$	7.50000	+	2.59808i	0.382235	+	0.132410i
$386$		0			0
$387$	−6.00000			−0.304997
$388$		0			0
$389$	4.50000	+	7.79423i	0.228159	+	0.395183i	0.957263	−	0.289220i	$-0.0933960\pi$
$389$	4.50000	+	7.79423i	0.228159	+	0.395183i	−0.729103	+	0.684403i	$0.760063\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−8.50000	−	14.7224i	−0.428768	−	0.742648i
$394$		0			0
$395$	13.5000	+	23.3827i	0.679259	+	1.17651i
$396$		0			0
$397$	−7.50000	−	12.9904i	−0.376414	−	0.651969i	0.614123	−	0.789210i	$-0.289510\pi$
$397$	−7.50000	−	12.9904i	−0.376414	−	0.651969i	−0.990538	+	0.137241i	$0.956176\pi$
$398$		0			0
$399$	6.00000	−	5.19615i	0.300376	−	0.260133i
$400$		0			0
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$	3.50000	−	0.866025i	0.174347	−	0.0431398i
$404$		0			0
$405$	−1.50000	−	2.59808i	−0.0745356	−	0.129099i
$406$		0			0
$407$	5.00000	−	8.66025i	0.247841	−	0.429273i
$408$		0			0
$409$	22.0000			1.08783			0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000			1.08783			0.543915	+	0.839140i	$0.316941\pi$
$410$		0			0
$411$	−3.00000	+	5.19615i	−0.147979	+	0.256307i
$412$		0			0
$413$	24.0000	−	20.7846i	1.18096	−	1.02274i
$414$		0			0
$415$	12.0000	−	20.7846i	0.589057	−	1.02028i
$416$		0			0
$417$	1.50000	+	2.59808i	0.0734553	+	0.127228i
$418$		0			0
$419$	−14.5000	−	25.1147i	−0.708371	−	1.22694i	−0.965461	−	0.260548i	$-0.916097\pi$
$419$	−14.5000	−	25.1147i	−0.708371	−	1.22694i	0.257090	−	0.966388i	$-0.417236\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		0			0
$423$	−22.0000			−1.06968
$424$		0			0
$425$	4.00000	+	6.92820i	0.194029	+	0.336067i
$426$		0			0
$427$	12.5000	+	4.33013i	0.604917	+	0.209550i
$428$		0			0
$429$	3.50000	−	0.866025i	0.168982	−	0.0418121i
$430$		0			0
$431$	−9.50000	+	16.4545i	−0.457599	+	0.792585i	−0.998833	−	0.0482871i	$-0.984624\pi$
$431$	−9.50000	+	16.4545i	−0.457599	+	0.792585i	0.541235	+	0.840872i	$0.317957\pi$
$432$		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
$435$	−13.5000	+	23.3827i	−0.647275	+	1.12111i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$		0			0
$441$	13.0000	−	5.19615i	0.619048	−	0.247436i
$442$		0			0
$443$	16.5000	−	28.5788i	0.783939	−	1.35782i	−0.145692	−	0.989330i	$-0.546541\pi$
$443$	16.5000	−	28.5788i	0.783939	−	1.35782i	0.929631	−	0.368492i	$-0.120126\pi$
$444$		0			0
$445$	15.0000	−	25.9808i	0.711068	−	1.23161i
$446$		0			0
$447$	9.00000			0.425685
$448$		0			0
$449$	0.500000	−	0.866025i	0.0235965	−	0.0408703i	−0.853986	−	0.520296i	$-0.825822\pi$
$449$	0.500000	−	0.866025i	0.0235965	−	0.0408703i	0.877583	+	0.479426i	$0.159155\pi$
$450$		0			0
$451$	−3.00000			−0.141264
$452$		0			0
$453$	−1.00000			−0.0469841
$454$		0			0
$455$	−28.5000	−	2.59808i	−1.33610	−	0.121800i
$456$		0			0
$457$	−2.00000			−0.0935561			−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000			−0.0935561			−0.0467780	+	0.998905i	$0.514895\pi$
$458$		0			0
$459$	−10.0000			−0.466760
$460$		0			0
$461$	−17.5000	+	30.3109i	−0.815056	+	1.41172i	0.0942312	+	0.995550i	$0.469961\pi$
$461$	−17.5000	+	30.3109i	−0.815056	+	1.41172i	−0.909288	+	0.416169i	$0.863373\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$		0			0
$465$	1.50000	−	2.59808i	0.0695608	−	0.120483i
$466$		0			0
$467$	−13.5000	+	23.3827i	−0.624705	+	1.08202i	0.363892	+	0.931441i	$0.381448\pi$
$467$	−13.5000	+	23.3827i	−0.624705	+	1.08202i	−0.988598	+	0.150581i	$0.951886\pi$
$468$		0			0
$469$	−4.50000	−	23.3827i	−0.207791	−	1.07971i
$470$		0			0
$471$	5.00000			0.230388
$472$		0			0
$473$	1.50000	+	2.59808i	0.0689701	+	0.119460i
$474$		0			0
$475$	6.00000	−	10.3923i	0.275299	−	0.476832i
$476$		0			0
$477$	3.00000	−	5.19615i	0.137361	−	0.237915i
$478$		0			0
$479$	12.5000	−	21.6506i	0.571140	−	0.989243i	−0.425310	−	0.905048i	$-0.639835\pi$
$479$	12.5000	−	21.6506i	0.571140	−	0.989243i	0.996449	−	0.0841949i	$-0.0268318\pi$
$480$		0			0
$481$	−10.0000	+	34.6410i	−0.455961	+	1.57949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19.5000	+	33.7750i	0.885449	+	1.53364i
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$	−21.0000			−0.949653
$490$		0			0
$491$	17.5000	+	30.3109i	0.789764	+	1.36791i	0.926111	+	0.377250i	$0.123130\pi$
$491$	17.5000	+	30.3109i	0.789764	+	1.36791i	−0.136347	+	0.990661i	$0.543536\pi$
$492$		0			0
$493$	−9.00000	−	15.5885i	−0.405340	−	0.702069i
$494$		0			0
$495$	−3.00000	+	5.19615i	−0.134840	+	0.233550i
$496$		0			0
$497$	−0.500000	−	2.59808i	−0.0224281	−	0.116540i
$498$		0			0
$499$	−21.5000	+	37.2391i	−0.962472	+	1.66705i	−0.246214	+	0.969216i	$0.579187\pi$
$499$	−21.5000	+	37.2391i	−0.962472	+	1.66705i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$	−9.00000			−0.402090
$502$		0			0
$503$	−5.50000	+	9.52628i	−0.245233	+	0.424756i	−0.962197	−	0.272354i	$-0.912198\pi$
$503$	−5.50000	+	9.52628i	−0.245233	+	0.424756i	0.716964	+	0.697110i	$0.245531\pi$
$504$		0			0
$505$	−16.5000	−	28.5788i	−0.734240	−	1.27174i
$506$		0			0
$507$	−11.5000	+	6.06218i	−0.510733	+	0.269231i
$508$		0			0
$509$	−34.0000			−1.50702			−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000			−1.50702			−0.753512	+	0.657434i	$0.771642\pi$
$510$		0			0
$511$	−27.5000	−	9.52628i	−1.21653	−	0.421418i
$512$		0			0
$513$	7.50000	+	12.9904i	0.331133	+	0.573539i
$514$		0			0
$515$	−10.5000	−	18.1865i	−0.462685	−	0.801394i
$516$		0			0
$517$	5.50000	+	9.52628i	0.241890	+	0.418965i
$518$		0			0
$519$	13.0000			0.570637
$520$		0			0
$521$	8.50000	+	14.7224i	0.372392	+	0.645001i	0.989933	−	0.141537i	$-0.0452044\pi$
$521$	8.50000	+	14.7224i	0.372392	+	0.645001i	−0.617541	+	0.786539i	$0.711871\pi$
$522$		0			0
$523$	28.0000			1.22435			0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000			1.22435			0.612177	+	0.790721i	$0.290294\pi$
$524$		0			0
$525$	−8.00000	+	6.92820i	−0.349149	+	0.302372i
$526$		0			0
$527$	1.00000	+	1.73205i	0.0435607	+	0.0754493i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	12.0000	+	20.7846i	0.520756	+	0.901975i
$532$		0			0
$533$	10.5000	−	2.59808i	0.454805	−	0.112535i
$534$		0			0
$535$	24.0000	−	41.5692i	1.03761	−	1.79719i
$536$		0			0
$537$	−10.5000	−	18.1865i	−0.453108	−	0.784807i
$538$		0			0
$539$	−5.50000	−	4.33013i	−0.236902	−	0.186512i
$540$		0			0
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	−0.127222	−	0.991874i	$-0.540606\pi$
$541$	18.5000	−	32.0429i	0.795377	−	1.37763i	0.922599	−	0.385759i	$-0.126061\pi$
$542$		0			0
$543$	−3.00000	+	5.19615i	−0.128742	+	0.222988i
$544$		0			0
$545$	45.0000			1.92759
$546$		0			0
$547$	4.00000			0.171028			0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000			0.171028			0.0855138	+	0.996337i	$0.472747\pi$
$548$		0			0
$549$	−5.00000	+	8.66025i	−0.213395	+	0.369611i
$550$		0			0
$551$	−13.5000	+	23.3827i	−0.575119	+	0.996136i
$552$		0			0
$553$	−4.50000	−	23.3827i	−0.191359	−	0.994333i
$554$		0			0
$555$	15.0000	+	25.9808i	0.636715	+	1.10282i
$556$		0			0
$557$	−13.5000	+	23.3827i	−0.572013	+	0.990756i	0.424346	+	0.905500i	$0.360504\pi$
$557$	−13.5000	+	23.3827i	−0.572013	+	0.990756i	−0.996359	+	0.0852559i	$0.972829\pi$
$558$		0			0
$559$	−7.50000	−	7.79423i	−0.317216	−	0.329661i
$560$		0			0
$561$	1.00000	+	1.73205i	0.0422200	+	0.0731272i
$562$		0			0
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		0			0
$565$	1.50000	+	2.59808i	0.0631055	+	0.109302i
$566$		0			0
$567$	0.500000	+	2.59808i	0.0209980	+	0.109109i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	7.50000	+	12.9904i	0.313865	+	0.543631i	0.979196	−	0.202919i	$-0.0650427\pi$
$571$	7.50000	+	12.9904i	0.313865	+	0.543631i	−0.665330	+	0.746549i	$0.731709\pi$
$572$		0			0
$573$	3.00000			0.125327
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−11.5000	−	19.9186i	−0.478751	−	0.829222i	0.520952	−	0.853586i	$-0.325577\pi$
$577$	−11.5000	−	19.9186i	−0.478751	−	0.829222i	−0.999703	+	0.0243645i	$0.992244\pi$
$578$		0			0
$579$	−8.50000	−	14.7224i	−0.353248	−	0.611843i
$580$		0			0
$581$	−16.0000	+	13.8564i	−0.663792	+	0.574861i
$582$		0			0
$583$	−3.00000			−0.124247
$584$		0			0
$585$	6.00000	−	20.7846i	0.248069	−	0.859338i
$586$		0			0
$587$	−22.5000	−	38.9711i	−0.928674	−	1.60851i	−0.785543	−	0.618808i	$-0.787616\pi$
$587$	−22.5000	−	38.9711i	−0.928674	−	1.60851i	−0.143132	−	0.989704i	$-0.545717\pi$
$588$		0			0
$589$	1.50000	−	2.59808i	0.0618064	−	0.107052i
$590$		0			0
$591$	−23.0000			−0.946094
$592$		0			0
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	−0.565792	−	0.824548i	$-0.691430\pi$
$593$	10.5000	−	18.1865i	0.431183	−	0.746831i	0.996976	+	0.0777165i	$0.0247629\pi$
$594$		0			0
$595$	−3.00000	−	15.5885i	−0.122988	−	0.639064i
$596$		0			0
$597$	−2.00000	+	3.46410i	−0.0818546	+	0.141776i
$598$		0			0
$599$	−2.50000	−	4.33013i	−0.102147	−	0.176924i	0.810422	−	0.585847i	$-0.199238\pi$
$599$	−2.50000	−	4.33013i	−0.102147	−	0.176924i	−0.912569	+	0.408923i	$0.865905\pi$
$600$		0			0
$601$	2.50000	+	4.33013i	0.101977	+	0.176630i	0.912499	−	0.409079i	$-0.134150\pi$
$601$	2.50000	+	4.33013i	0.101977	+	0.176630i	−0.810522	+	0.585708i	$0.800816\pi$
$602$		0			0
$603$	18.0000			0.733017
$604$		0			0
$605$	−30.0000			−1.21967
$606$		0			0
$607$	7.50000	+	12.9904i	0.304416	+	0.527263i	0.977131	−	0.212638i	$-0.0682055\pi$
$607$	7.50000	+	12.9904i	0.304416	+	0.527263i	−0.672715	+	0.739901i	$0.734872\pi$
$608$		0			0
$609$	18.0000	−	15.5885i	0.729397	−	0.631676i
$610$		0			0
$611$	−27.5000	−	28.5788i	−1.11253	−	1.15618i
$612$		0			0
$613$	−23.5000	+	40.7032i	−0.949156	+	1.64399i	−0.201948	+	0.979396i	$0.564727\pi$
$613$	−23.5000	+	40.7032i	−0.949156	+	1.64399i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$	4.50000	−	7.79423i	0.181458	−	0.314294i
$616$		0			0
$617$	16.5000	−	28.5788i	0.664265	−	1.15054i	−0.315219	−	0.949019i	$-0.602078\pi$
$617$	16.5000	−	28.5788i	0.664265	−	1.15054i	0.979484	−	0.201522i	$-0.0645887\pi$
$618$		0			0
$619$	3.50000	+	6.06218i	0.140677	+	0.243659i	0.927752	−	0.373198i	$-0.121739\pi$
$619$	3.50000	+	6.06218i	0.140677	+	0.243659i	−0.787075	+	0.616858i	$0.788405\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−20.0000	+	17.3205i	−0.801283	+	0.693932i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$	1.50000	−	2.59808i	0.0599042	−	0.103757i
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	−21.5000	+	37.2391i	−0.855901	+	1.48246i	0.0199047	+	0.999802i	$0.493664\pi$
$631$	−21.5000	+	37.2391i	−0.855901	+	1.48246i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−5.00000			−0.198732
$634$		0			0
$635$	−21.0000			−0.833360
$636$		0			0
$637$	23.0000	+	10.3923i	0.911293	+	0.411758i
$638$		0			0
$639$	2.00000			0.0791188
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$	20.5000	−	35.5070i	0.808441	−	1.40026i	−0.105502	−	0.994419i	$-0.533645\pi$
$643$	20.5000	−	35.5070i	0.808441	−	1.40026i	0.913943	−	0.405842i	$-0.133022\pi$
$644$		0			0
$645$	−9.00000			−0.354375
$646$		0			0
$647$	−1.50000	+	2.59808i	−0.0589711	+	0.102141i	−0.894004	−	0.448059i	$-0.852115\pi$
$647$	−1.50000	+	2.59808i	−0.0589711	+	0.102141i	0.835033	+	0.550200i	$0.185449\pi$
$648$		0			0
$649$	6.00000	−	10.3923i	0.235521	−	0.407934i
$650$		0			0
$651$	−2.00000	+	1.73205i	−0.0783862	+	0.0678844i
$652$		0			0
$653$	−30.0000			−1.17399			−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000			−1.17399			−0.586995	+	0.809590i	$0.699689\pi$
$654$		0			0
$655$	25.5000	+	44.1673i	0.996367	+	1.72576i
$656$		0			0
$657$	11.0000	−	19.0526i	0.429151	−	0.743311i
$658$		0			0
$659$	0.500000	−	0.866025i	0.0194772	−	0.0337356i	−0.856123	−	0.516773i	$-0.827133\pi$
$659$	0.500000	−	0.866025i	0.0194772	−	0.0337356i	0.875600	+	0.483037i	$0.160466\pi$
$660$		0			0
$661$	−11.5000	+	19.9186i	−0.447298	+	0.774743i	−0.998209	−	0.0598209i	$-0.980947\pi$
$661$	−11.5000	+	19.9186i	−0.447298	+	0.774743i	0.550911	+	0.834564i	$0.314280\pi$
$662$		0			0
$663$	−5.00000	−	5.19615i	−0.194184	−	0.201802i
$664$		0			0
$665$	−18.0000	+	15.5885i	−0.698010	+	0.604494i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−21.0000			−0.811907
$670$		0			0
$671$	5.00000			0.193023
$672$		0			0
$673$	−11.5000	−	19.9186i	−0.443292	−	0.767805i	0.554639	−	0.832091i	$-0.312856\pi$
$673$	−11.5000	−	19.9186i	−0.443292	−	0.767805i	−0.997932	+	0.0642860i	$0.979523\pi$
$674$		0			0
$675$	−10.0000	−	17.3205i	−0.384900	−	0.666667i
$676$		0			0
$677$	24.5000	−	42.4352i	0.941611	−	1.63092i	0.179214	−	0.983810i	$-0.442645\pi$
$677$	24.5000	−	42.4352i	0.941611	−	1.63092i	0.762398	−	0.647109i	$-0.224022\pi$
$678$		0			0
$679$	−6.50000	−	33.7750i	−0.249447	−	1.29617i
$680$		0			0
$681$	6.00000	−	10.3923i	0.229920	−	0.398234i
$682$		0			0
$683$	44.0000			1.68361			0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000			1.68361			0.841807	+	0.539779i	$0.181492\pi$
$684$		0			0
$685$	9.00000	−	15.5885i	0.343872	−	0.595604i
$686$		0			0
$687$	−6.50000	−	11.2583i	−0.247990	−	0.429532i
$688$		0			0
$689$	10.5000	−	2.59808i	0.400018	−	0.0989788i
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$	4.00000	−	3.46410i	0.151947	−	0.131590i
$694$		0			0
$695$	−4.50000	−	7.79423i	−0.170695	−	0.295652i
$696$		0			0
$697$	3.00000	+	5.19615i	0.113633	+	0.196818i
$698$		0			0
$699$	1.50000	+	2.59808i	0.0567352	+	0.0982683i
$700$		0			0
$701$	−46.0000			−1.73740			−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000			−1.73740			−0.868698	+	0.495342i	$0.835043\pi$
$702$		0			0
$703$	15.0000	+	25.9808i	0.565736	+	0.979883i
$704$		0			0
$705$	−33.0000			−1.24285
$706$		0			0
$707$	5.50000	+	28.5788i	0.206849	+	1.07482i
$708$		0			0
$709$	−13.5000	−	23.3827i	−0.507003	−	0.878155i	−0.999967	−	0.00810550i	$-0.997420\pi$
$709$	−13.5000	−	23.3827i	−0.507003	−	0.878155i	0.492964	−	0.870050i	$-0.335913\pi$
$710$		0			0
$711$	18.0000			0.675053
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−10.5000	+	2.59808i	−0.392678	+	0.0971625i
$716$		0			0
$717$	6.00000	−	10.3923i	0.224074	−	0.388108i
$718$		0			0
$719$	1.50000	+	2.59808i	0.0559406	+	0.0968919i	0.892640	−	0.450771i	$-0.148851\pi$
$719$	1.50000	+	2.59808i	0.0559406	+	0.0968919i	−0.836699	+	0.547663i	$0.815518\pi$
$720$		0			0
$721$	3.50000	+	18.1865i	0.130347	+	0.677302i
$722$		0			0
$723$	−5.00000	+	8.66025i	−0.185952	+	0.322078i
$724$		0			0
$725$	18.0000	−	31.1769i	0.668503	−	1.15788i
$726$		0			0
$727$	−16.0000			−0.593407			−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000			−0.593407			−0.296704	+	0.954970i	$0.595887\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	3.00000	−	5.19615i	0.110959	−	0.192187i
$732$		0			0
$733$	0.500000	−	0.866025i	0.0184679	−	0.0319874i	−0.856644	−	0.515908i	$-0.827454\pi$
$733$	0.500000	−	0.866025i	0.0184679	−	0.0319874i	0.875112	+	0.483921i	$0.160788\pi$
$734$		0			0
$735$	19.5000	−	7.79423i	0.719268	−	0.287494i
$736$		0			0
$737$	−4.50000	−	7.79423i	−0.165760	−	0.287104i
$738$		0			0
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	−0.986154	−	0.165831i	$-0.946969\pi$
$739$	−9.50000	+	16.4545i	−0.349463	+	0.605288i	0.636691	+	0.771119i	$0.280303\pi$
$740$		0			0
$741$	−3.00000	+	10.3923i	−0.110208	+	0.381771i
$742$		0			0
$743$	−16.5000	−	28.5788i	−0.605326	−	1.04846i	−0.992000	−	0.126239i	$-0.959709\pi$
$743$	−16.5000	−	28.5788i	−0.605326	−	1.04846i	0.386674	−	0.922217i	$-0.373624\pi$
$744$		0			0
$745$	−27.0000			−0.989203
$746$		0			0
$747$	−8.00000	−	13.8564i	−0.292705	−	0.506979i
$748$		0			0
$749$	−32.0000	+	27.7128i	−1.16925	+	1.01260i
$750$		0			0
$751$	−36.0000			−1.31366			−0.656829	−	0.754039i	$-0.728103\pi$
$751$	−36.0000			−1.31366			−0.656829	+	0.754039i	$0.728103\pi$
$752$		0			0
$753$	1.50000	+	2.59808i	0.0546630	+	0.0946792i
$754$		0			0
$755$	3.00000			0.109181
$756$		0			0
$757$	2.50000	+	4.33013i	0.0908640	+	0.157381i	0.907875	−	0.419241i	$-0.137704\pi$
$757$	2.50000	+	4.33013i	0.0908640	+	0.157381i	−0.817011	+	0.576622i	$0.804370\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.5000	+	21.6506i	0.453125	+	0.784835i	0.998578	−	0.0533061i	$-0.0169759\pi$
$761$	12.5000	+	21.6506i	0.453125	+	0.784835i	−0.545454	+	0.838141i	$0.683643\pi$
$762$		0			0
$763$	−37.5000	−	12.9904i	−1.35759	−	0.470283i
$764$		0			0
$765$	12.0000			0.433861
$766$		0			0
$767$	−12.0000	+	41.5692i	−0.433295	+	1.50098i
$768$		0			0
$769$	10.5000	+	18.1865i	0.378640	+	0.655823i	0.990865	−	0.134860i	$-0.0430586\pi$
$769$	10.5000	+	18.1865i	0.378640	+	0.655823i	−0.612225	+	0.790684i	$0.709725\pi$
$770$		0			0
$771$	3.00000	−	5.19615i	0.108042	−	0.187135i
$772$		0			0
$773$	−14.0000			−0.503545			−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000			−0.503545			−0.251773	+	0.967786i	$0.581013\pi$
$774$		0			0
$775$	−2.00000	+	3.46410i	−0.0718421	+	0.124434i
$776$		0			0
$777$	−5.00000	−	25.9808i	−0.179374	−	0.932055i
$778$		0			0
$779$	4.50000	−	7.79423i	0.161229	−	0.279257i
$780$		0			0
$781$	−0.500000	−	0.866025i	−0.0178914	−	0.0309888i
$782$		0			0
$783$	22.5000	+	38.9711i	0.804084	+	1.39272i
$784$		0			0
$785$	−15.0000			−0.535373
$786$		0			0
$787$	4.00000			0.142585			0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000			0.142585			0.0712923	+	0.997455i	$0.477288\pi$
$788$		0			0
$789$	−4.50000	−	7.79423i	−0.160204	−	0.277482i

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:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	728.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{5} +(2.00000 - 1.73205i) q^{7} +(1.00000 + 1.73205i) q^{9} +(0.500000 - 0.866025i) q^{11} +(-1.00000 + 3.46410i) q^{13} +(1.50000 + 2.59808i) q^{15} -2.00000 q^{17} +(1.50000 + 2.59808i) q^{19} +(-0.500000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{25} +5.00000 q^{27} +(4.50000 + 7.79423i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(-0.500000 - 0.866025i) q^{33} +(1.50000 + 7.79423i) q^{35} +10.0000 q^{37} +(2.50000 + 2.59808i) q^{39} +(-1.50000 - 2.59808i) q^{41} +(-1.50000 + 2.59808i) q^{43} -6.00000 q^{45} +(-5.50000 + 9.52628i) q^{47} +(1.00000 - 6.92820i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-1.50000 - 2.59808i) q^{53} +(1.50000 + 2.59808i) q^{55} +3.00000 q^{57} +12.0000 q^{59} +(2.50000 + 4.33013i) q^{61} +(5.00000 + 1.73205i) q^{63} +(-7.50000 - 7.79423i) q^{65} +(4.50000 - 7.79423i) q^{67} +(0.500000 - 0.866025i) q^{71} +(-5.50000 - 9.52628i) q^{73} -4.00000 q^{75} +(-0.500000 - 2.59808i) q^{77} +(4.50000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} -8.00000 q^{83} +(3.00000 - 5.19615i) q^{85} +9.00000 q^{87} -10.0000 q^{89} +(4.00000 + 8.66025i) q^{91} -1.00000 q^{93} -9.00000 q^{95} +(6.50000 - 11.2583i) q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^3 - 3 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + q^{3} - 3 q^{5} + 4 q^{7} + 2 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} - 4 q^{17} + 3 q^{19} - q^{21} - 4 q^{25} + 10 q^{27} + 9 q^{29} - q^{31} - q^{33} + 3 q^{35} + 20 q^{37} + 5 q^{39} - 3 q^{41} - 3 q^{43} - 12 q^{45} - 11 q^{47} + 2 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{55} + 6 q^{57} + 24 q^{59} + 5 q^{61} + 10 q^{63} - 15 q^{65} + 9 q^{67} + q^{71} - 11 q^{73} - 8 q^{75} - q^{77} + 9 q^{79} - q^{81} - 16 q^{83} + 6 q^{85} + 18 q^{87} - 20 q^{89} + 8 q^{91} - 2 q^{93} - 18 q^{95} + 13 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + q^3 - 3 * q^5 + 4 * q^7 + 2 * q^9 + q^11 - 2 * q^13 + 3 * q^15 - 4 * q^17 + 3 * q^19 - q^21 - 4 * q^25 + 10 * q^27 + 9 * q^29 - q^31 - q^33 + 3 * q^35 + 20 * q^37 + 5 * q^39 - 3 * q^41 - 3 * q^43 - 12 * q^45 - 11 * q^47 + 2 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^55 + 6 * q^57 + 24 * q^59 + 5 * q^61 + 10 * q^63 - 15 * q^65 + 9 * q^67 + q^71 - 11 * q^73 - 8 * q^75 - q^77 + 9 * q^79 - q^81 - 16 * q^83 + 6 * q^85 + 18 * q^87 - 20 * q^89 + 8 * q^91 - 2 * q^93 - 18 * q^95 + 13 * q^97 + 4 * q^99

Introduction

L-functions

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