LMFDB - Embedded newform 728.2.t.a.9.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [728,2,Mod(9,728)]

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, 4]))

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

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

chi := DirichletCharacter("728.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 728 = 2^{3} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	728.t (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:	$1$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			9.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	728.9
Dual form			728.2.t.a.81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +(-1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} -2.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{3} +(-1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} -2.00000 q^{9} -1.00000 q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.50000 - 2.59808i) q^{15} +(1.00000 - 1.73205i) q^{17} -3.00000 q^{19} +(-0.500000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{25} +5.00000 q^{27} +(4.50000 - 7.79423i) q^{29} +(-0.500000 - 0.866025i) q^{31} +1.00000 q^{33} +(-7.50000 - 2.59808i) q^{35} +(-5.00000 - 8.66025i) q^{37} +(1.00000 + 3.46410i) q^{39} +(-1.50000 + 2.59808i) q^{41} +(-1.50000 - 2.59808i) q^{43} +(3.00000 - 5.19615i) q^{45} +(-5.50000 + 9.52628i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-1.50000 - 2.59808i) q^{53} +(1.50000 - 2.59808i) q^{55} +3.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} -5.00000 q^{61} +(-1.00000 - 5.19615i) q^{63} +(10.5000 + 2.59808i) q^{65} -9.00000 q^{67} +(0.500000 + 0.866025i) q^{71} +(-5.50000 - 9.52628i) q^{73} +(2.00000 + 3.46410i) q^{75} +(-0.500000 - 2.59808i) q^{77} +(4.50000 - 7.79423i) q^{79} +1.00000 q^{81} -8.00000 q^{83} +(3.00000 + 5.19615i) q^{85} +(-4.50000 + 7.79423i) q^{87} +(5.00000 + 8.66025i) q^{89} +(8.50000 - 4.33013i) q^{91} +(0.500000 + 0.866025i) q^{93} +(4.50000 - 7.79423i) q^{95} +(6.50000 + 11.2583i) q^{97} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 3 * q^5 + q^7 - 4 * q^9	$ 2 q - 2 q^{3} - 3 q^{5} + q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} + 3 q^{15} + 2 q^{17} - 6 q^{19} - q^{21} - 4 q^{25} + 10 q^{27} + 9 q^{29} - q^{31} + 2 q^{33} - 15 q^{35} - 10 q^{37} + 2 q^{39} - 3 q^{41} - 3 q^{43} + 6 q^{45} - 11 q^{47} - 13 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{55} + 6 q^{57} - 12 q^{59} - 10 q^{61} - 2 q^{63} + 21 q^{65} - 18 q^{67} + q^{71} - 11 q^{73} + 4 q^{75} - q^{77} + 9 q^{79} + 2 q^{81} - 16 q^{83} + 6 q^{85} - 9 q^{87} + 10 q^{89} + 17 q^{91} + q^{93} + 9 q^{95} + 13 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 3 * q^5 + q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 + 3 * q^15 + 2 * q^17 - 6 * q^19 - q^21 - 4 * q^25 + 10 * q^27 + 9 * q^29 - q^31 + 2 * q^33 - 15 * q^35 - 10 * q^37 + 2 * q^39 - 3 * q^41 - 3 * q^43 + 6 * q^45 - 11 * q^47 - 13 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^55 + 6 * q^57 - 12 * q^59 - 10 * q^61 - 2 * q^63 + 21 * q^65 - 18 * q^67 + q^71 - 11 * q^73 + 4 * q^75 - q^77 + 9 * q^79 + 2 * q^81 - 16 * q^83 + 6 * q^85 - 9 * q^87 + 10 * q^89 + 17 * q^91 + q^93 + 9 * q^95 + 13 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$

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
$3$	−1.00000			−0.577350			−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000			−0.577350			−0.288675	+	0.957427i	$0.593215\pi$
$4$		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
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$	−1.00000			−0.301511			−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000			−0.301511			−0.150756	+	0.988571i	$0.548171\pi$
$12$		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
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	−3.00000			−0.688247			−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000			−0.688247			−0.344124	+	0.938924i	$0.611824\pi$
$20$		0			0
$21$	−0.500000	−	2.59808i	−0.109109	−	0.566947i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		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	−0.0578882	−	0.998323i	$-0.518437\pi$
$29$	4.50000	−	7.79423i	0.835629	−	1.44735i	0.893517	−	0.449029i	$-0.148230\pi$
$30$		0			0
$31$	−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
$33$	1.00000			0.174078
$34$		0			0
$35$	−7.50000	−	2.59808i	−1.26773	−	0.439155i
$36$		0			0
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$	1.00000	+	3.46410i	0.160128	+	0.554700i
$40$		0			0
$41$	−1.50000	+	2.59808i	−0.234261	+	0.405751i	−0.959058	−	0.283211i	$-0.908600\pi$
$41$	−1.50000	+	2.59808i	−0.234261	+	0.405751i	0.724797	+	0.688963i	$0.241934\pi$
$42$		0			0
$43$	−1.50000	−	2.59808i	−0.228748	−	0.396203i	0.728689	−	0.684844i	$-0.240130\pi$
$43$	−1.50000	−	2.59808i	−0.228748	−	0.396203i	−0.957437	+	0.288641i	$0.906796\pi$
$44$		0			0
$45$	3.00000	−	5.19615i	0.447214	−	0.774597i
$46$		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
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		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
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	−5.00000			−0.640184			−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000			−0.640184			−0.320092	+	0.947386i	$0.603714\pi$
$62$		0			0
$63$	−1.00000	−	5.19615i	−0.125988	−	0.654654i
$64$		0			0
$65$	10.5000	+	2.59808i	1.30236	+	0.322252i
$66$		0			0
$67$	−9.00000			−1.09952			−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000			−1.09952			−0.549762	+	0.835321i	$0.685282\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.500000	+	0.866025i	0.0593391	+	0.102778i	0.894169	−	0.447730i	$-0.147767\pi$
$71$	0.500000	+	0.866025i	0.0593391	+	0.102778i	−0.834830	+	0.550508i	$0.814434\pi$
$72$		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
$75$	2.00000	+	3.46410i	0.230940	+	0.400000i
$76$		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
$81$	1.00000			0.111111
$82$		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
$87$	−4.50000	+	7.79423i	−0.482451	+	0.835629i
$88$		0			0
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	0.999388	+	0.0349934i	$0.0111410\pi$
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	−0.469389	+	0.882992i	$0.655526\pi$
$90$		0			0
$91$	8.50000	−	4.33013i	0.891042	−	0.453921i
$92$		0			0
$93$	0.500000	+	0.866025i	0.0518476	+	0.0898027i
$94$		0			0
$95$	4.50000	−	7.79423i	0.461690	−	0.799671i
$96$		0			0
$97$	6.50000	+	11.2583i	0.659975	+	1.14311i	0.980622	+	0.195911i	$0.0627665\pi$
$97$	6.50000	+	11.2583i	0.659975	+	1.14311i	−0.320647	+	0.947199i	$0.603900\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	11.0000			1.09454			0.547270	−	0.836956i	$-0.315667\pi$
$101$	11.0000			1.09454			0.547270	+	0.836956i	$0.315667\pi$
$102$		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
$107$	8.00000	+	13.8564i	0.773389	+	1.33955i	0.935695	+	0.352809i	$0.114773\pi$
$107$	8.00000	+	13.8564i	0.773389	+	1.33955i	−0.162306	+	0.986740i	$0.551893\pi$
$108$		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	0.888585	−	0.458712i	$-0.151689\pi$
$113$	0.500000	+	0.866025i	0.0470360	+	0.0814688i	−0.841549	+	0.540181i	$0.818356\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000	+	6.92820i	0.184900	+	0.640513i
$118$		0			0
$119$	5.00000	+	1.73205i	0.458349	+	0.158777i
$120$		0			0
$121$	−10.0000			−0.909091
$122$		0			0
$123$	1.50000	−	2.59808i	0.135250	−	0.234261i
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	3.50000	−	6.06218i	0.310575	−	0.537931i	−0.667912	−	0.744240i	$-0.732812\pi$
$127$	3.50000	−	6.06218i	0.310575	−	0.537931i	0.978487	+	0.206309i	$0.0661452\pi$
$128$		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
$133$	−1.50000	−	7.79423i	−0.130066	−	0.675845i
$134$		0			0
$135$	−7.50000	+	12.9904i	−0.645497	+	1.11803i
$136$		0			0
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	−1.50000	−	2.59808i	−0.127228	−	0.220366i	0.795373	−	0.606120i	$-0.207275\pi$
$139$	−1.50000	−	2.59808i	−0.127228	−	0.220366i	−0.922602	+	0.385754i	$0.873941\pi$
$140$		0			0
$141$	5.50000	−	9.52628i	0.463184	−	0.802257i
$142$		0			0
$143$	1.00000	+	3.46410i	0.0836242	+	0.289683i
$144$		0			0
$145$	13.5000	+	23.3827i	1.12111	+	1.94183i
$146$		0			0
$147$	6.50000	−	2.59808i	0.536111	−	0.214286i
$148$		0			0
$149$	−9.00000			−0.737309			−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000			−0.737309			−0.368654	+	0.929567i	$0.620181\pi$
$150$		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
$159$	1.50000	+	2.59808i	0.118958	+	0.206041i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	21.0000			1.64485			0.822423	−	0.568876i	$-0.192621\pi$
$163$	21.0000			1.64485			0.822423	+	0.568876i	$0.192621\pi$
$164$		0			0
$165$	−1.50000	+	2.59808i	−0.116775	+	0.202260i
$166$		0			0
$167$	−4.50000	+	7.79423i	−0.348220	+	0.603136i	−0.985933	−	0.167139i	$-0.946547\pi$
$167$	−4.50000	+	7.79423i	−0.348220	+	0.603136i	0.637713	+	0.770274i	$0.279881\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$	6.00000			0.458831
$172$		0			0
$173$	−13.0000			−0.988372			−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000			−0.988372			−0.494186	+	0.869356i	$0.664534\pi$
$174$		0			0
$175$	8.00000	−	6.92820i	0.604743	−	0.523723i
$176$		0			0
$177$	6.00000	−	10.3923i	0.450988	−	0.781133i
$178$		0			0
$179$	−21.0000			−1.56961			−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000			−1.56961			−0.784807	+	0.619740i	$0.787238\pi$
$180$		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
$185$	30.0000			2.20564
$186$		0			0
$187$	−1.00000	+	1.73205i	−0.0731272	+	0.126660i
$188$		0			0
$189$	2.50000	+	12.9904i	0.181848	+	0.944911i
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$	−17.0000			−1.22369			−0.611843	−	0.790979i	$-0.709572\pi$
$193$	−17.0000			−1.22369			−0.611843	+	0.790979i	$0.709572\pi$
$194$		0			0
$195$	−10.5000	−	2.59808i	−0.751921	−	0.186052i
$196$		0			0
$197$	−11.5000	+	19.9186i	−0.819341	+	1.41914i	0.0868274	+	0.996223i	$0.472327\pi$
$197$	−11.5000	+	19.9186i	−0.819341	+	1.41914i	−0.906168	+	0.422917i	$0.861006\pi$
$198$		0			0
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	−0.786389	−	0.617731i	$-0.788052\pi$
$199$	2.00000	−	3.46410i	0.141776	−	0.245564i	0.928166	+	0.372168i	$0.121385\pi$
$200$		0			0
$201$	9.00000			0.634811
$202$		0			0
$203$	22.5000	+	7.79423i	1.57919	+	0.547048i
$204$		0			0
$205$	−4.50000	−	7.79423i	−0.314294	−	0.544373i
$206$		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	−0.939156	−	0.343490i	$-0.888391\pi$
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	0.767049	+	0.641588i	$0.221724\pi$
$212$		0			0
$213$	−0.500000	−	0.866025i	−0.0342594	−	0.0593391i
$214$		0			0
$215$	9.00000			0.613795
$216$		0			0
$217$	2.00000	−	1.73205i	0.135769	−	0.117579i
$218$		0			0
$219$	5.50000	+	9.52628i	0.371656	+	0.643726i
$220$		0			0
$221$	−7.00000	−	1.73205i	−0.470871	−	0.116510i
$222$		0			0
$223$	−10.5000	+	18.1865i	−0.703132	+	1.21786i	0.264229	+	0.964460i	$0.414882\pi$
$223$	−10.5000	+	18.1865i	−0.703132	+	1.21786i	−0.967361	+	0.253401i	$0.918451\pi$
$224$		0			0
$225$	4.00000	+	6.92820i	0.266667	+	0.461880i
$226$		0			0
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	−0.993508	−	0.113761i	$-0.963710\pi$
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	0.595274	+	0.803523i	$0.297043\pi$
$228$		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
$231$	0.500000	+	2.59808i	0.0328976	+	0.170941i
$232$		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
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	−0.658838	−	0.752285i	$-0.728952\pi$
$241$	5.00000	−	8.66025i	0.322078	−	0.557856i	0.980917	+	0.194429i	$0.0622852\pi$
$242$		0			0
$243$	−16.0000			−1.02640
$244$		0			0
$245$	3.00000	−	20.7846i	0.191663	−	1.32788i
$246$		0			0
$247$	3.00000	+	10.3923i	0.190885	+	0.661247i
$248$		0			0
$249$	8.00000			0.506979
$250$		0			0
$251$	−1.50000	−	2.59808i	−0.0946792	−	0.163989i	0.814795	−	0.579748i	$-0.196849\pi$
$251$	−1.50000	−	2.59808i	−0.0946792	−	0.163989i	−0.909475	+	0.415759i	$0.863516\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−3.00000	−	5.19615i	−0.187867	−	0.325396i
$256$		0			0
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	0.757159	−	0.653231i	$-0.226587\pi$
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	−0.944294	+	0.329104i	$0.893253\pi$
$258$		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
$263$	−9.00000			−0.554964			−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000			−0.554964			−0.277482	+	0.960731i	$0.589500\pi$
$264$		0			0
$265$	9.00000			0.552866
$266$		0			0
$267$	−5.00000	−	8.66025i	−0.305995	−	0.529999i
$268$		0			0
$269$	−5.00000	+	8.66025i	−0.304855	+	0.528025i	−0.977229	−	0.212187i	$-0.931941\pi$
$269$	−5.00000	+	8.66025i	−0.304855	+	0.528025i	0.672374	+	0.740212i	$0.265275\pi$
$270$		0			0
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	−0.957328	−	0.289003i	$-0.906676\pi$
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	0.228380	−	0.973572i	$-0.426657\pi$
$272$		0			0
$273$	−8.50000	+	4.33013i	−0.514443	+	0.262071i
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		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
$283$	27.0000			1.60498			0.802492	−	0.596663i	$-0.203507\pi$
$283$	27.0000			1.60498			0.802492	+	0.596663i	$0.203507\pi$
$284$		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
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	−6.50000	−	11.2583i	−0.381037	−	0.659975i
$292$		0			0
$293$	−9.50000	−	16.4545i	−0.554996	−	0.961281i	−0.997904	−	0.0647140i	$-0.979386\pi$
$293$	−9.50000	−	16.4545i	−0.554996	−	0.961281i	0.442908	−	0.896567i	$-0.353947\pi$
$294$		0			0
$295$	−18.0000	−	31.1769i	−1.04800	−	1.81519i
$296$		0			0
$297$	−5.00000			−0.290129
$298$		0			0
$299$		0			0
$300$		0			0
$301$	6.00000	−	5.19615i	0.345834	−	0.299501i
$302$		0			0
$303$	−11.0000			−0.631933
$304$		0			0
$305$	7.50000	−	12.9904i	0.429449	−	0.743827i
$306$		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
$315$	15.0000	+	5.19615i	0.845154	+	0.292770i
$316$		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
$319$	−4.50000	+	7.79423i	−0.251952	+	0.436393i
$320$		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
$325$	−10.0000	+	10.3923i	−0.554700	+	0.576461i
$326$		0			0
$327$	7.50000	+	12.9904i	0.414751	+	0.718370i
$328$		0			0
$329$	−27.5000	−	9.52628i	−1.51612	−	0.525201i
$330$		0			0
$331$	−23.0000			−1.26419			−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000			−1.26419			−0.632097	+	0.774889i	$0.717806\pi$
$332$		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
$341$	0.500000	+	0.866025i	0.0270765	+	0.0468979i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	12.5000	−	21.6506i	0.669110	−	1.15893i	−0.309044	−	0.951048i	$-0.600009\pi$
$349$	12.5000	−	21.6506i	0.669110	−	1.15893i	0.978153	−	0.207884i	$-0.0666577\pi$
$350$		0			0
$351$	−5.00000	−	17.3205i	−0.266880	−	0.924500i
$352$		0			0
$353$	15.0000			0.798369			0.399185	−	0.916871i	$-0.369293\pi$
$353$	15.0000			0.798369			0.399185	+	0.916871i	$0.369293\pi$
$354$		0			0
$355$	−3.00000			−0.159223
$356$		0			0
$357$	−5.00000	−	1.73205i	−0.264628	−	0.0916698i
$358$		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
$361$	−10.0000			−0.526316
$362$		0			0
$363$	10.0000			0.524864
$364$		0			0
$365$	33.0000			1.72730
$366$		0			0
$367$	−29.0000			−1.51379			−0.756894	−	0.653538i	$-0.773284\pi$
$367$	−29.0000			−1.51379			−0.756894	+	0.653538i	$0.773284\pi$
$368$		0			0
$369$	3.00000	−	5.19615i	0.156174	−	0.270501i
$370$		0			0
$371$	6.00000	−	5.19615i	0.311504	−	0.269771i
$372$		0			0
$373$	−17.0000			−0.880227			−0.440113	−	0.897942i	$-0.645062\pi$
$373$	−17.0000			−0.880227			−0.440113	+	0.897942i	$0.645062\pi$
$374$		0			0
$375$	3.00000			0.154919
$376$		0			0
$377$	−31.5000	−	7.79423i	−1.62233	−	0.401423i
$378$		0			0
$379$	5.50000	−	9.52628i	0.282516	−	0.489332i	−0.689488	−	0.724297i	$-0.742164\pi$
$379$	5.50000	−	9.52628i	0.282516	−	0.489332i	0.972004	+	0.234965i	$0.0754976\pi$
$380$		0			0
$381$	−3.50000	+	6.06218i	−0.179310	+	0.310575i
$382$		0			0
$383$	17.0000			0.868659			0.434330	−	0.900754i	$-0.356985\pi$
$383$	17.0000			0.868659			0.434330	+	0.900754i	$0.356985\pi$
$384$		0			0
$385$	7.50000	+	2.59808i	0.382235	+	0.132410i
$386$		0			0
$387$	3.00000	+	5.19615i	0.152499	+	0.264135i
$388$		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
$397$	15.0000			0.752828			0.376414	−	0.926451i	$-0.377157\pi$
$397$	15.0000			0.752828			0.376414	+	0.926451i	$0.377157\pi$
$398$		0			0
$399$	1.50000	+	7.79423i	0.0750939	+	0.390199i
$400$		0			0
$401$	5.00000	+	8.66025i	0.249688	+	0.432472i	0.963439	−	0.267927i	$-0.0863386\pi$
$401$	5.00000	+	8.66025i	0.249688	+	0.432472i	−0.713751	+	0.700399i	$0.753005\pi$
$402$		0			0
$403$	−2.50000	+	2.59808i	−0.124534	+	0.129419i
$404$		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
$409$	−11.0000	+	19.0526i	−0.543915	+	0.942088i	0.454759	+	0.890614i	$0.349725\pi$
$409$	−11.0000	+	19.0526i	−0.543915	+	0.942088i	−0.998674	+	0.0514740i	$0.983608\pi$
$410$		0			0
$411$	−3.00000	+	5.19615i	−0.147979	+	0.256307i
$412$		0			0
$413$	−30.0000	−	10.3923i	−1.47620	−	0.511372i
$414$		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	0.257090	+	0.966388i	$0.417236\pi$
$419$	−14.5000	+	25.1147i	−0.708371	+	1.22694i	−0.965461	+	0.260548i	$0.916097\pi$
$420$		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
$423$	11.0000	−	19.0526i	0.534838	−	0.926367i
$424$		0			0
$425$	−8.00000			−0.388057
$426$		0			0
$427$	−2.50000	−	12.9904i	−0.120983	−	0.628649i
$428$		0			0
$429$	−1.00000	−	3.46410i	−0.0482805	−	0.167248i
$430$		0			0
$431$	19.0000			0.915198			0.457599	−	0.889159i	$-0.348710\pi$
$431$	19.0000			0.915198			0.457599	+	0.889159i	$0.348710\pi$
$432$		0			0
$433$	14.5000	+	25.1147i	0.696826	+	1.20694i	0.969561	+	0.244848i	$0.0787382\pi$
$433$	14.5000	+	25.1147i	0.696826	+	1.20694i	−0.272736	+	0.962089i	$0.587929\pi$
$434$		0			0
$435$	−13.5000	−	23.3827i	−0.647275	−	1.12111i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		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
$445$	−30.0000			−1.42214
$446$		0			0
$447$	9.00000			0.425685
$448$		0			0
$449$	0.500000	+	0.866025i	0.0235965	+	0.0408703i	0.877583	−	0.479426i	$-0.159155\pi$
$449$	0.500000	+	0.866025i	0.0235965	+	0.0408703i	−0.853986	+	0.520296i	$0.825822\pi$
$450$		0			0
$451$	1.50000	−	2.59808i	0.0706322	−	0.122339i
$452$		0			0
$453$	0.500000	+	0.866025i	0.0234920	+	0.0406894i
$454$		0			0
$455$	−1.50000	+	28.5788i	−0.0703211	+	1.33980i
$456$		0			0
$457$	1.00000	+	1.73205i	0.0467780	+	0.0810219i	0.888466	−	0.458942i	$-0.151771\pi$
$457$	1.00000	+	1.73205i	0.0467780	+	0.0810219i	−0.841688	+	0.539964i	$0.818438\pi$
$458$		0			0
$459$	5.00000	−	8.66025i	0.233380	−	0.404226i
$460$		0			0
$461$	−17.5000	−	30.3109i	−0.815056	−	1.41172i	−0.909288	−	0.416169i	$-0.863373\pi$
$461$	−17.5000	−	30.3109i	−0.815056	−	1.41172i	0.0942312	−	0.995550i	$-0.469961\pi$
$462$		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
$465$	−3.00000			−0.139122
$466$		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
$471$	−2.50000	−	4.33013i	−0.115194	−	0.199522i
$472$		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
$479$	−25.0000			−1.14228			−0.571140	−	0.820853i	$-0.693499\pi$
$479$	−25.0000			−1.14228			−0.571140	+	0.820853i	$0.693499\pi$
$480$		0			0
$481$	−25.0000	+	25.9808i	−1.13990	+	1.18462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−39.0000			−1.77090
$486$		0			0
$487$	−8.00000	+	13.8564i	−0.362515	+	0.627894i	−0.988374	−	0.152042i	$-0.951415\pi$
$487$	−8.00000	+	13.8564i	−0.362515	+	0.627894i	0.625859	+	0.779936i	$0.284748\pi$
$488$		0			0
$489$	−21.0000			−0.949653
$490$		0			0
$491$	17.5000	−	30.3109i	0.789764	−	1.36791i	−0.136347	−	0.990661i	$-0.543536\pi$
$491$	17.5000	−	30.3109i	0.789764	−	1.36791i	0.926111	−	0.377250i	$-0.123130\pi$
$492$		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
$497$	−2.00000	+	1.73205i	−0.0897123	+	0.0776931i
$498$		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
$501$	4.50000	−	7.79423i	0.201045	−	0.348220i
$502$		0			0
$503$	−5.50000	−	9.52628i	−0.245233	−	0.424756i	0.716964	−	0.697110i	$-0.245531\pi$
$503$	−5.50000	−	9.52628i	−0.245233	−	0.424756i	−0.962197	+	0.272354i	$0.912198\pi$
$504$		0			0
$505$	−16.5000	+	28.5788i	−0.734240	+	1.27174i
$506$		0			0
$507$	11.0000	−	6.92820i	0.488527	−	0.307692i
$508$		0			0
$509$	17.0000	+	29.4449i	0.753512	+	1.30512i	0.946111	+	0.323843i	$0.104975\pi$
$509$	17.0000	+	29.4449i	0.753512	+	1.30512i	−0.192599	+	0.981278i	$0.561692\pi$
$510$		0			0
$511$	22.0000	−	19.0526i	0.973223	−	0.842836i
$512$		0			0
$513$	−15.0000			−0.662266
$514$		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
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	−0.990873	−	0.134801i	$-0.956961\pi$
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	0.378695	−	0.925521i	$-0.376373\pi$
$524$		0			0
$525$	−8.00000	+	6.92820i	−0.349149	+	0.302372i
$526$		0			0
$527$	−2.00000			−0.0871214
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		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
$535$	−48.0000			−2.07522
$536$		0			0
$537$	21.0000			0.906217
$538$		0			0
$539$	6.50000	−	2.59808i	0.279975	−	0.111907i
$540$		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
$543$	6.00000			0.257485
$544$		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
$549$	10.0000			0.426790
$550$		0			0
$551$	−13.5000	+	23.3827i	−0.575119	+	0.996136i
$552$		0			0
$553$	22.5000	+	7.79423i	0.956797	+	0.331444i
$554$		0			0
$555$	−30.0000			−1.27343
$556$		0			0
$557$	27.0000			1.14403			0.572013	−	0.820244i	$-0.306163\pi$
$557$	27.0000			1.14403			0.572013	+	0.820244i	$0.306163\pi$
$558$		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
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	−0.905088	−	0.425223i	$-0.860196\pi$
$563$	−2.00000	+	3.46410i	−0.0842900	+	0.145994i	0.820798	+	0.571218i	$0.193529\pi$
$564$		0			0
$565$	−3.00000			−0.126211
$566$		0			0
$567$	0.500000	+	2.59808i	0.0209980	+	0.109109i
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		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
$579$	17.0000			0.706496
$580$		0			0
$581$	−4.00000	−	20.7846i	−0.165948	−	0.862291i
$582$		0			0
$583$	1.50000	+	2.59808i	0.0621237	+	0.107601i
$584$		0			0
$585$	−21.0000	−	5.19615i	−0.868243	−	0.214834i
$586$		0			0
$587$	−22.5000	+	38.9711i	−0.928674	+	1.60851i	−0.143132	+	0.989704i	$0.545717\pi$
$587$	−22.5000	+	38.9711i	−0.928674	+	1.60851i	−0.785543	+	0.618808i	$0.787616\pi$
$588$		0			0
$589$	1.50000	+	2.59808i	0.0618064	+	0.107052i
$590$		0			0
$591$	11.5000	−	19.9186i	0.473047	−	0.819341i
$592$		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
$595$	−12.0000	+	10.3923i	−0.491952	+	0.426043i
$596$		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	−0.810522	−	0.585708i	$-0.800816\pi$
$601$	2.50000	−	4.33013i	0.101977	−	0.176630i	0.912499	+	0.409079i	$0.134150\pi$
$602$		0			0
$603$	18.0000			0.733017
$604$		0			0
$605$	15.0000	−	25.9808i	0.609837	−	1.05627i
$606$		0			0
$607$	−15.0000			−0.608831			−0.304416	−	0.952539i	$-0.598461\pi$
$607$	−15.0000			−0.608831			−0.304416	+	0.952539i	$0.598461\pi$
$608$		0			0
$609$	−22.5000	−	7.79423i	−0.911746	−	0.315838i
$610$		0			0
$611$	38.5000	+	9.52628i	1.55754	+	0.385392i
$612$		0			0
$613$	47.0000			1.89831			0.949156	−	0.314806i	$-0.101939\pi$
$613$	47.0000			1.89831			0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$	4.50000	+	7.79423i	0.181458	+	0.314294i
$616$		0			0
$617$	16.5000	+	28.5788i	0.664265	+	1.15054i	0.979484	+	0.201522i	$0.0645887\pi$
$617$	16.5000	+	28.5788i	0.664265	+	1.15054i	−0.315219	+	0.949019i	$0.602078\pi$
$618$		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
$627$	−3.00000			−0.119808
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	−21.5000	−	37.2391i	−0.855901	−	1.48246i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−21.5000	−	37.2391i	−0.855901	−	1.48246i	0.0199047	−	0.999802i	$-0.493664\pi$
$632$		0			0
$633$	2.50000	−	4.33013i	0.0993661	−	0.172107i
$634$		0			0
$635$	10.5000	+	18.1865i	0.416680	+	0.721711i
$636$		0			0
$637$	15.5000	+	19.9186i	0.614132	+	0.789203i
$638$		0			0
$639$	−1.00000	−	1.73205i	−0.0395594	−	0.0685189i
$640$		0			0
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	−0.631721	−	0.775196i	$-0.717651\pi$
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	0.987200	+	0.159489i	$0.0509845\pi$
$642$		0			0
$643$	20.5000	+	35.5070i	0.808441	+	1.40026i	0.913943	+	0.405842i	$0.133022\pi$
$643$	20.5000	+	35.5070i	0.808441	+	1.40026i	−0.105502	+	0.994419i	$0.533645\pi$
$644$		0			0
$645$	−9.00000			−0.354375
$646$		0			0
$647$	3.00000			0.117942			0.0589711	−	0.998260i	$-0.481218\pi$
$647$	3.00000			0.117942			0.0589711	+	0.998260i	$0.481218\pi$
$648$		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
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	0.994623	+	0.103558i	$0.0330227\pi$
$653$	15.0000	+	25.9808i	0.586995	+	1.01671i	−0.407628	+	0.913148i	$0.633644\pi$
$654$		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	0.875600	−	0.483037i	$-0.160466\pi$
$659$	0.500000	+	0.866025i	0.0194772	+	0.0337356i	−0.856123	+	0.516773i	$0.827133\pi$
$660$		0			0
$661$	23.0000			0.894596			0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000			0.894596			0.447298	+	0.894385i	$0.352386\pi$
$662$		0			0
$663$	7.00000	+	1.73205i	0.271857	+	0.0672673i
$664$		0			0
$665$	22.5000	+	7.79423i	0.872513	+	0.302247i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	10.5000	−	18.1865i	0.405953	−	0.703132i
$670$		0			0
$671$	5.00000			0.193023
$672$		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
$675$	−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
$679$	−26.0000	+	22.5167i	−0.997788	+	0.864110i
$680$		0			0
$681$	6.00000	−	10.3923i	0.229920	−	0.398234i
$682$		0			0
$683$	−22.0000	+	38.1051i	−0.841807	+	1.45805i	0.0465592	+	0.998916i	$0.485174\pi$
$683$	−22.0000	+	38.1051i	−0.841807	+	1.45805i	−0.888366	+	0.459136i	$0.848159\pi$
$684$		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
$689$	−7.50000	+	7.79423i	−0.285727	+	0.296936i
$690$		0			0
$691$	−6.00000	−	10.3923i	−0.228251	−	0.395342i	0.729039	−	0.684472i	$-0.239967\pi$
$691$	−6.00000	−	10.3923i	−0.228251	−	0.395342i	−0.957290	+	0.289130i	$0.906634\pi$
$692$		0			0
$693$	1.00000	+	5.19615i	0.0379869	+	0.197386i
$694$		0			0
$695$	9.00000			0.341389
$696$		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
$705$	16.5000	+	28.5788i	0.621426	+	1.07634i
$706$		0			0
$707$	5.50000	+	28.5788i	0.206849	+	1.07482i
$708$		0			0
$709$	27.0000			1.01401			0.507003	−	0.861944i	$-0.330753\pi$
$709$	27.0000			1.01401			0.507003	+	0.861944i	$0.330753\pi$
$710$		0			0
$711$	−9.00000	+	15.5885i	−0.337526	+	0.584613i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−10.5000	−	2.59808i	−0.392678	−	0.0971625i
$716$		0			0
$717$	−12.0000			−0.448148
$718$		0			0
$719$	−3.00000			−0.111881			−0.0559406	−	0.998434i	$-0.517816\pi$
$719$	−3.00000			−0.111881			−0.0559406	+	0.998434i	$0.517816\pi$
$720$		0			0
$721$	−17.5000	−	6.06218i	−0.651734	−	0.225767i
$722$		0			0
$723$	−5.00000	+	8.66025i	−0.185952	+	0.322078i
$724$		0			0
$725$	−36.0000			−1.33701
$726$		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
$731$	−6.00000			−0.221918
$732$		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
$735$	−3.00000	+	20.7846i	−0.110657	+	0.766652i
$736$		0			0
$737$	9.00000			0.331519
$738$		0			0
$739$	19.0000			0.698926			0.349463	−	0.936950i	$-0.386364\pi$
$739$	19.0000			0.698926			0.349463	+	0.936950i	$0.386364\pi$
$740$		0			0
$741$	−3.00000	−	10.3923i	−0.110208	−	0.381771i
$742$		0			0
$743$	−16.5000	+	28.5788i	−0.605326	+	1.04846i	0.386674	+	0.922217i	$0.373624\pi$
$743$	−16.5000	+	28.5788i	−0.605326	+	1.04846i	−0.992000	+	0.126239i	$0.959709\pi$
$744$		0			0
$745$	13.5000	−	23.3827i	0.494602	−	0.856675i
$746$		0			0
$747$	16.0000			0.585409
$748$		0			0
$749$	−32.0000	+	27.7128i	−1.16925	+	1.01260i
$750$		0			0
$751$	18.0000	+	31.1769i	0.656829	+	1.13766i	0.981432	+	0.191811i	$0.0614361\pi$
$751$	18.0000	+	31.1769i	0.656829	+	1.13766i	−0.324603	+	0.945851i	$0.605231\pi$
$752$		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	−0.817011	−	0.576622i	$-0.804370\pi$
$757$	2.50000	−	4.33013i	0.0908640	−	0.157381i	0.907875	+	0.419241i	$0.137704\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−25.0000			−0.906249			−0.453125	−	0.891447i	$-0.649691\pi$
$761$	−25.0000			−0.906249			−0.453125	+	0.891447i	$0.649691\pi$
$762$		0			0
$763$	30.0000	−	25.9808i	1.08607	−	0.940567i
$764$		0			0
$765$	−6.00000	−	10.3923i	−0.216930	−	0.375735i
$766$		0			0
$767$	42.0000	+	10.3923i	1.51653	+	0.375244i
$768$		0			0
$769$	10.5000	−	18.1865i	0.378640	−	0.655823i	−0.612225	−	0.790684i	$-0.709725\pi$
$769$	10.5000	−	18.1865i	0.378640	−	0.655823i	0.990865	+	0.134860i	$0.0430586\pi$
$770$		0			0
$771$	3.00000	+	5.19615i	0.108042	+	0.187135i
$772$		0			0
$773$	7.00000	−	12.1244i	0.251773	−	0.436083i	−0.712241	−	0.701935i	$-0.752320\pi$
$773$	7.00000	−	12.1244i	0.251773	−	0.436083i	0.964014	+	0.265852i	$0.0856532\pi$
$774$		0			0
$775$	−2.00000	+	3.46410i	−0.0718421	+	0.124434i
$776$		0			0
$777$	−20.0000	+	17.3205i	−0.717496	+	0.621370i
$778$		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
$787$	−2.00000	+	3.46410i	−0.0712923	+	0.123482i	−0.899468	−	0.436987i	$-0.856046\pi$
$787$	−2.00000	+	3.46410i	−0.0712923	+	0.123482i	0.828176	+	0.560469i	$0.189379\pi$
$788$		0			0
$789$	9.00000			0.320408

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.t (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +(-1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +(-1.50000 + 2.59808i) q^{5} +(0.500000 + 2.59808i) q^{7} -2.00000 q^{9} -1.00000 q^{11} +(-1.00000 - 3.46410i) q^{13} +(1.50000 - 2.59808i) q^{15} +(1.00000 - 1.73205i) q^{17} -3.00000 q^{19} +(-0.500000 - 2.59808i) q^{21} +(-2.00000 - 3.46410i) q^{25} +5.00000 q^{27} +(4.50000 - 7.79423i) q^{29} +(-0.500000 - 0.866025i) q^{31} +1.00000 q^{33} +(-7.50000 - 2.59808i) q^{35} +(-5.00000 - 8.66025i) q^{37} +(1.00000 + 3.46410i) q^{39} +(-1.50000 + 2.59808i) q^{41} +(-1.50000 - 2.59808i) q^{43} +(3.00000 - 5.19615i) q^{45} +(-5.50000 + 9.52628i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-1.00000 + 1.73205i) q^{51} +(-1.50000 - 2.59808i) q^{53} +(1.50000 - 2.59808i) q^{55} +3.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} -5.00000 q^{61} +(-1.00000 - 5.19615i) q^{63} +(10.5000 + 2.59808i) q^{65} -9.00000 q^{67} +(0.500000 + 0.866025i) q^{71} +(-5.50000 - 9.52628i) q^{73} +(2.00000 + 3.46410i) q^{75} +(-0.500000 - 2.59808i) q^{77} +(4.50000 - 7.79423i) q^{79} +1.00000 q^{81} -8.00000 q^{83} +(3.00000 + 5.19615i) q^{85} +(-4.50000 + 7.79423i) q^{87} +(5.00000 + 8.66025i) q^{89} +(8.50000 - 4.33013i) q^{91} +(0.500000 + 0.866025i) q^{93} +(4.50000 - 7.79423i) q^{95} +(6.50000 + 11.2583i) q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 3 q^{5} + q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 3 * q^5 + q^7 - 4 * q^9	\( 2 q - 2 q^{3} - 3 q^{5} + q^{7} - 4 q^{9} - 2 q^{11} - 2 q^{13} + 3 q^{15} + 2 q^{17} - 6 q^{19} - q^{21} - 4 q^{25} + 10 q^{27} + 9 q^{29} - q^{31} + 2 q^{33} - 15 q^{35} - 10 q^{37} + 2 q^{39} - 3 q^{41} - 3 q^{43} + 6 q^{45} - 11 q^{47} - 13 q^{49} - 2 q^{51} - 3 q^{53} + 3 q^{55} + 6 q^{57} - 12 q^{59} - 10 q^{61} - 2 q^{63} + 21 q^{65} - 18 q^{67} + q^{71} - 11 q^{73} + 4 q^{75} - q^{77} + 9 q^{79} + 2 q^{81} - 16 q^{83} + 6 q^{85} - 9 q^{87} + 10 q^{89} + 17 q^{91} + q^{93} + 9 q^{95} + 13 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 3 * q^5 + q^7 - 4 * q^9 - 2 * q^11 - 2 * q^13 + 3 * q^15 + 2 * q^17 - 6 * q^19 - q^21 - 4 * q^25 + 10 * q^27 + 9 * q^29 - q^31 + 2 * q^33 - 15 * q^35 - 10 * q^37 + 2 * q^39 - 3 * q^41 - 3 * q^43 + 6 * q^45 - 11 * q^47 - 13 * q^49 - 2 * q^51 - 3 * q^53 + 3 * q^55 + 6 * q^57 - 12 * q^59 - 10 * q^61 - 2 * q^63 + 21 * q^65 - 18 * q^67 + q^71 - 11 * q^73 + 4 * q^75 - q^77 + 9 * q^79 + 2 * q^81 - 16 * q^83 + 6 * q^85 - 9 * q^87 + 10 * q^89 + 17 * q^91 + q^93 + 9 * q^95 + 13 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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