LMFDB - Embedded newform 912.2.q.a.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(49,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("912.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.q (of order $3$, degree $2$, not 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:	$7.28235666434$
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:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	912.577
Dual form			912.2.q.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{3} -1.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{3} -1.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +2.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(2.00000 + 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{21} +(-2.00000 + 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +(4.00000 - 6.92820i) q^{29} +3.00000 q^{31} +(-1.00000 - 1.73205i) q^{33} +3.00000 q^{37} +5.00000 q^{39} +(6.00000 + 10.3923i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(-3.00000 + 5.19615i) q^{47} -6.00000 q^{49} +(2.00000 - 3.46410i) q^{51} +(-2.00000 + 3.46410i) q^{53} +(-0.500000 - 4.33013i) q^{57} +(5.00000 + 8.66025i) q^{59} +(6.50000 - 11.2583i) q^{61} +(0.500000 - 0.866025i) q^{63} +(5.50000 - 9.52628i) q^{67} +4.00000 q^{69} +(3.00000 + 5.19615i) q^{71} +(5.50000 + 9.52628i) q^{73} -5.00000 q^{75} -2.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -8.00000 q^{87} +(3.00000 - 5.19615i) q^{89} +(2.50000 - 4.33013i) q^{91} +(-1.50000 - 2.59808i) q^{93} +(-1.00000 - 1.73205i) q^{97} +(-1.00000 + 1.73205i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{7} - q^{9}+O(q^{10}) $ 2 * q - q^3 - 2 * q^7 - q^9	$ 2 q - q^{3} - 2 q^{7} - q^{9} + 4 q^{11} - 5 q^{13} + 4 q^{17} + 8 q^{19} + q^{21} - 4 q^{23} + 5 q^{25} + 2 q^{27} + 8 q^{29} + 6 q^{31} - 2 q^{33} + 6 q^{37} + 10 q^{39} + 12 q^{41} - q^{43} - 6 q^{47} - 12 q^{49} + 4 q^{51} - 4 q^{53} - q^{57} + 10 q^{59} + 13 q^{61} + q^{63} + 11 q^{67} + 8 q^{69} + 6 q^{71} + 11 q^{73} - 10 q^{75} - 4 q^{77} + q^{79} - q^{81} - 16 q^{87} + 6 q^{89} + 5 q^{91} - 3 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^7 - q^9 + 4 * q^11 - 5 * q^13 + 4 * q^17 + 8 * q^19 + q^21 - 4 * q^23 + 5 * q^25 + 2 * q^27 + 8 * q^29 + 6 * q^31 - 2 * q^33 + 6 * q^37 + 10 * q^39 + 12 * q^41 - q^43 - 6 * q^47 - 12 * q^49 + 4 * q^51 - 4 * q^53 - q^57 + 10 * q^59 + 13 * q^61 + q^63 + 11 * q^67 + 8 * q^69 + 6 * q^71 + 11 * q^73 - 10 * q^75 - 4 * q^77 + q^79 - q^81 - 16 * q^87 + 6 * q^89 + 5 * q^91 - 3 * q^93 - 2 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

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
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$3$	−0.500000	−	0.866025i	−0.288675	−	0.500000i
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−1.00000			−0.377964			−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000			−0.377964			−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	0.277350	+	0.960769i	$0.410544\pi$
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$	0.500000	+	0.866025i	0.109109	+	0.188982i
$22$		0			0
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	4.00000	−	6.92820i	0.742781	−	1.28654i	−0.208443	−	0.978035i	$-0.566840\pi$
$29$	4.00000	−	6.92820i	0.742781	−	1.28654i	0.951224	−	0.308500i	$-0.0998271\pi$
$30$		0			0
$31$	3.00000			0.538816			0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000			0.538816			0.269408	+	0.963026i	$0.413172\pi$
$32$		0			0
$33$	−1.00000	−	1.73205i	−0.174078	−	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$		0			0
$39$	5.00000			0.800641
$40$		0			0
$41$	6.00000	+	10.3923i	0.937043	+	1.62301i	0.770950	+	0.636895i	$0.219782\pi$
$41$	6.00000	+	10.3923i	0.937043	+	1.62301i	0.166092	+	0.986110i	$0.446885\pi$
$42$		0			0
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	$-0.190961\pi$
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	−0.901629	+	0.432511i	$0.857628\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$	2.00000	−	3.46410i	0.280056	−	0.485071i
$52$		0			0
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.500000	−	4.33013i	−0.0662266	−	0.573539i
$58$		0			0
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	0.982894	+	0.184172i	$0.0589603\pi$
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	−0.331949	+	0.943297i	$0.607706\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$	0.500000	−	0.866025i	0.0629941	−	0.109109i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	3.00000	+	5.19615i	0.356034	+	0.616670i	0.987294	−	0.158901i	$-0.0507952\pi$
$71$	3.00000	+	5.19615i	0.356034	+	0.616670i	−0.631260	+	0.775571i	$0.717462\pi$
$72$		0			0
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	0.984594	+	0.174855i	$0.0559458\pi$
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	−0.340868	+	0.940111i	$0.610721\pi$
$74$		0			0
$75$	−5.00000			−0.577350
$76$		0			0
$77$	−2.00000			−0.227921
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−8.00000			−0.857690
$88$		0			0
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	0.980071	+	0.198650i	$0.0636557\pi$
$90$		0			0
$91$	2.50000	−	4.33013i	0.262071	−	0.453921i
$92$		0			0
$93$	−1.50000	−	2.59808i	−0.155543	−	0.269408i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$	−1.00000	+	1.73205i	−0.100504	+	0.174078i
$100$		0			0
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	0.273138	+	0.961975i	$0.411939\pi$
$101$	−7.00000	+	12.1244i	−0.696526	+	1.20642i	−0.969664	+	0.244443i	$0.921395\pi$
$102$		0			0
$103$	−13.0000			−1.28093			−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000			−1.28093			−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	0.909935	−	0.414751i	$-0.136131\pi$
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$	−1.50000	−	2.59808i	−0.142374	−	0.246598i
$112$		0			0
$113$	2.00000			0.188144			0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000			0.188144			0.0940721	+	0.995565i	$0.470012\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.50000	−	4.33013i	−0.231125	−	0.400320i
$118$		0			0
$119$	−2.00000	−	3.46410i	−0.183340	−	0.317554i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$	6.00000	−	10.3923i	0.541002	−	0.937043i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.00000	+	6.92820i	−0.354943	+	0.614779i	−0.987108	−	0.160055i	$-0.948833\pi$
$127$	−4.00000	+	6.92820i	−0.354943	+	0.614779i	0.632166	+	0.774833i	$0.282166\pi$
$128$		0			0
$129$	−0.500000	+	0.866025i	−0.0440225	+	0.0762493i
$130$		0			0
$131$	−7.00000	−	12.1244i	−0.611593	−	1.05931i	−0.990972	−	0.134069i	$-0.957196\pi$
$131$	−7.00000	−	12.1244i	−0.611593	−	1.05931i	0.379379	−	0.925241i	$-0.376138\pi$
$132$		0			0
$133$	−4.00000	−	1.73205i	−0.346844	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	0.169226	+	0.985577i	$0.445873\pi$
$137$	−9.00000	+	15.5885i	−0.768922	+	1.33181i	−0.938148	+	0.346235i	$0.887460\pi$
$138$		0			0
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	−0.740308	−	0.672268i	$-0.765320\pi$
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	0.952355	+	0.304991i	$0.0986536\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$	−5.00000	+	8.66025i	−0.418121	+	0.724207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.00000	+	5.19615i	0.247436	+	0.428571i
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.50000	−	2.59808i	−0.119713	−	0.207349i	0.799941	−	0.600079i	$-0.204864\pi$
$157$	−1.50000	−	2.59808i	−0.119713	−	0.207349i	−0.919654	+	0.392730i	$0.871531\pi$
$158$		0			0
$159$	4.00000			0.317221
$160$		0			0
$161$	2.00000	−	3.46410i	0.157622	−	0.273009i
$162$		0			0
$163$	−3.00000			−0.234978			−0.117489	−	0.993074i	$-0.537485\pi$
$163$	−3.00000			−0.234978			−0.117489	+	0.993074i	$0.537485\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.00000	+	1.73205i	−0.0773823	+	0.134030i	−0.902120	−	0.431486i	$-0.857990\pi$
$167$	−1.00000	+	1.73205i	−0.0773823	+	0.134030i	0.824737	+	0.565516i	$0.191323\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$	−3.50000	+	2.59808i	−0.267652	+	0.198680i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−2.50000	+	4.33013i	−0.188982	+	0.327327i
$176$		0			0
$177$	5.00000	−	8.66025i	0.375823	−	0.650945i
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	1.00000	−	1.73205i	0.0743294	−	0.128742i	−0.826465	−	0.562988i	$-0.809652\pi$
$181$	1.00000	−	1.73205i	0.0743294	−	0.128742i	0.900794	+	0.434246i	$0.142985\pi$
$182$		0			0
$183$	−13.0000			−0.960988
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.00000	+	6.92820i	0.292509	+	0.506640i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	−0.973805	−	0.227387i	$-0.926982\pi$
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	0.289980	−	0.957033i	$-0.406351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−26.0000			−1.85242			−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000			−1.85242			−0.926212	+	0.377004i	$0.876954\pi$
$198$		0			0
$199$	10.5000	−	18.1865i	0.744325	−	1.28921i	−0.206184	−	0.978513i	$-0.566105\pi$
$199$	10.5000	−	18.1865i	0.744325	−	1.28921i	0.950509	−	0.310696i	$-0.100562\pi$
$200$		0			0
$201$	−11.0000			−0.775880
$202$		0			0
$203$	−4.00000	+	6.92820i	−0.280745	+	0.486265i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.00000	−	3.46410i	−0.139010	−	0.240772i
$208$		0			0
$209$	8.00000	+	3.46410i	0.553372	+	0.239617i
$210$		0			0
$211$	−7.50000	−	12.9904i	−0.516321	−	0.894295i	−0.999820	−	0.0189499i	$-0.993968\pi$
$211$	−7.50000	−	12.9904i	−0.516321	−	0.894295i	0.483499	−	0.875345i	$-0.339366\pi$
$212$		0			0
$213$	3.00000	−	5.19615i	0.205557	−	0.356034i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.00000			−0.203653
$218$		0			0
$219$	5.50000	−	9.52628i	0.371656	−	0.643726i
$220$		0			0
$221$	−20.0000			−1.34535
$222$		0			0
$223$	4.50000	+	7.79423i	0.301342	+	0.521940i	0.976440	−	0.215788i	$-0.0692320\pi$
$223$	4.50000	+	7.79423i	0.301342	+	0.521940i	−0.675098	+	0.737728i	$0.735899\pi$
$224$		0			0
$225$	2.50000	+	4.33013i	0.166667	+	0.288675i
$226$		0			0
$227$	24.0000			1.59294			0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000			1.59294			0.796468	+	0.604681i	$0.206699\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$		0			0
$231$	1.00000	+	1.73205i	0.0657952	+	0.113961i
$232$		0			0
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	0.947403	−	0.320043i	$-0.103697\pi$
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	−0.750867	+	0.660454i	$0.770364\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.500000	−	0.866025i	0.0324785	−	0.0562544i
$238$		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
$241$	3.50000	−	6.06218i	0.225455	−	0.390499i	−0.731001	−	0.682376i	$-0.760947\pi$
$241$	3.50000	−	6.06218i	0.225455	−	0.390499i	0.956456	+	0.291877i	$0.0942799\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−17.5000	+	12.9904i	−1.11350	+	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.00000	+	1.73205i	−0.0631194	+	0.109326i	−0.895858	−	0.444340i	$-0.853438\pi$
$251$	−1.00000	+	1.73205i	−0.0631194	+	0.109326i	0.832739	+	0.553666i	$0.186772\pi$
$252$		0			0
$253$	−4.00000	+	6.92820i	−0.251478	+	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.0000	−	17.3205i	0.623783	−	1.08042i	−0.364992	−	0.931011i	$-0.618928\pi$
$257$	10.0000	−	17.3205i	0.623783	−	1.08042i	0.988775	−	0.149413i	$-0.0477384\pi$
$258$		0			0
$259$	−3.00000			−0.186411
$260$		0			0
$261$	4.00000	+	6.92820i	0.247594	+	0.428845i
$262$		0			0
$263$	13.0000	+	22.5167i	0.801614	+	1.38844i	0.918553	+	0.395298i	$0.129359\pi$
$263$	13.0000	+	22.5167i	0.801614	+	1.38844i	−0.116939	+	0.993139i	$0.537308\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−6.00000			−0.367194
$268$		0			0
$269$	−5.00000	−	8.66025i	−0.304855	−	0.528025i	0.672374	−	0.740212i	$-0.265275\pi$
$269$	−5.00000	−	8.66025i	−0.304855	−	0.528025i	−0.977229	+	0.212187i	$0.931941\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$	−5.00000			−0.302614
$274$		0			0
$275$	5.00000	−	8.66025i	0.301511	−	0.522233i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$	−1.50000	+	2.59808i	−0.0898027	+	0.155543i
$280$		0			0
$281$	4.00000	−	6.92820i	0.238620	−	0.413302i	−0.721699	−	0.692207i	$-0.756638\pi$
$281$	4.00000	−	6.92820i	0.238620	−	0.413302i	0.960319	+	0.278906i	$0.0899716\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	−	10.3923i	−0.354169	−	0.613438i
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$	−1.00000	+	1.73205i	−0.0586210	+	0.101535i
$292$		0			0
$293$	14.0000			0.817889			0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000			0.817889			0.408944	+	0.912559i	$0.365897\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.00000			0.116052
$298$		0			0
$299$	−10.0000	−	17.3205i	−0.578315	−	1.00167i
$300$		0			0
$301$	0.500000	+	0.866025i	0.0288195	+	0.0499169i
$302$		0			0
$303$	14.0000			0.804279
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−6.00000	−	10.3923i	−0.342438	−	0.593120i	0.642447	−	0.766330i	$-0.277919\pi$
$307$	−6.00000	−	10.3923i	−0.342438	−	0.593120i	−0.984885	+	0.173210i	$0.944586\pi$
$308$		0			0
$309$	6.50000	+	11.2583i	0.369772	+	0.640464i
$310$		0			0
$311$	−18.0000			−1.02069			−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000			−1.02069			−0.510343	+	0.859971i	$0.670482\pi$
$312$		0			0
$313$	−11.0000	+	19.0526i	−0.621757	+	1.07691i	0.367402	+	0.930062i	$0.380247\pi$
$313$	−11.0000	+	19.0526i	−0.621757	+	1.07691i	−0.989158	+	0.146852i	$0.953086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000	−	3.46410i	0.112331	−	0.194563i	−0.804379	−	0.594117i	$-0.797502\pi$
$317$	2.00000	−	3.46410i	0.112331	−	0.194563i	0.916710	+	0.399554i	$0.130835\pi$
$318$		0			0
$319$	8.00000	−	13.8564i	0.447914	−	0.775810i
$320$		0			0
$321$	−9.00000	−	15.5885i	−0.502331	−	0.870063i
$322$		0			0
$323$	2.00000	+	17.3205i	0.111283	+	0.963739i
$324$		0			0
$325$	12.5000	+	21.6506i	0.693375	+	1.20096i
$326$		0			0
$327$	1.00000	−	1.73205i	0.0553001	−	0.0957826i
$328$		0			0
$329$	3.00000	−	5.19615i	0.165395	−	0.286473i
$330$		0			0
$331$	25.0000			1.37412			0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000			1.37412			0.687062	+	0.726599i	$0.258900\pi$
$332$		0			0
$333$	−1.50000	+	2.59808i	−0.0821995	+	0.142374i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.50000	−	11.2583i	−0.354078	−	0.613280i	0.632882	−	0.774248i	$-0.281872\pi$
$337$	−6.50000	−	11.2583i	−0.354078	−	0.613280i	−0.986960	+	0.160968i	$0.948538\pi$
$338$		0			0
$339$	−1.00000	−	1.73205i	−0.0543125	−	0.0940721i
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	−	13.8564i	−0.429463	−	0.743851i	0.567363	−	0.823468i	$-0.307964\pi$
$347$	−8.00000	−	13.8564i	−0.429463	−	0.743851i	−0.996826	+	0.0796169i	$0.974630\pi$
$348$		0			0
$349$	−21.0000			−1.12410			−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000			−1.12410			−0.562052	+	0.827102i	$0.689988\pi$
$350$		0			0
$351$	−2.50000	+	4.33013i	−0.133440	+	0.231125i
$352$		0			0
$353$	4.00000			0.212899			0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000			0.212899			0.106449	+	0.994318i	$0.466052\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.00000	+	3.46410i	−0.105851	+	0.183340i
$358$		0			0
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	−0.999476	−	0.0323801i	$-0.989691\pi$
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	0.471696	−	0.881761i	$-0.343642\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$	3.50000	+	6.06218i	0.183702	+	0.318182i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	−0.760100	−	0.649806i	$-0.774850\pi$
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	0.942799	+	0.333363i	$0.108183\pi$
$368$		0			0
$369$	−12.0000			−0.624695
$370$		0			0
$371$	2.00000	−	3.46410i	0.103835	−	0.179847i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.0000	+	34.6410i	1.03005	+	1.78410i
$378$		0			0
$379$	5.00000			0.256833			0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000			0.256833			0.128416	+	0.991720i	$0.459011\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$	7.00000	+	12.1244i	0.357683	+	0.619526i	0.987573	−	0.157159i	$-0.0502334\pi$
$383$	7.00000	+	12.1244i	0.357683	+	0.619526i	−0.629890	+	0.776684i	$0.716900\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.00000			0.0508329
$388$		0			0
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	−0.383076	−	0.923717i	$-0.625135\pi$
$389$	12.0000	−	20.7846i	0.608424	−	1.05382i	0.991500	−	0.130105i	$-0.0415314\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$	−7.00000	+	12.1244i	−0.353103	+	0.611593i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	0.764730	−	0.644351i	$-0.222873\pi$
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	−0.940389	+	0.340099i	$0.889539\pi$
$398$		0			0
$399$	0.500000	+	4.33013i	0.0250313	+	0.216777i
$400$		0			0
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$	−7.50000	+	12.9904i	−0.373602	+	0.647097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000			0.297409
$408$		0			0
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	−0.639430	−	0.768849i	$-0.720830\pi$
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	0.985558	+	0.169338i	$0.0541630\pi$
$410$		0			0
$411$	18.0000			0.887875
$412$		0			0
$413$	−5.00000	−	8.66025i	−0.246034	−	0.426143i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−5.00000			−0.244851
$418$		0			0
$419$	14.0000			0.683945			0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000			0.683945			0.341972	+	0.939710i	$0.388905\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$	−3.00000	−	5.19615i	−0.145865	−	0.252646i
$424$		0			0
$425$	20.0000			0.970143
$426$		0			0
$427$	−6.50000	+	11.2583i	−0.314557	+	0.544829i
$428$		0			0
$429$	10.0000			0.482805
$430$		0			0
$431$	−5.00000	+	8.66025i	−0.240842	+	0.417150i	−0.960954	−	0.276707i	$-0.910757\pi$
$431$	−5.00000	+	8.66025i	−0.240842	+	0.417150i	0.720113	+	0.693857i	$0.244090\pi$
$432$		0			0
$433$	4.50000	−	7.79423i	0.216256	−	0.374567i	−0.737404	−	0.675452i	$-0.763949\pi$
$433$	4.50000	−	7.79423i	0.216256	−	0.374567i	0.953660	+	0.300885i	$0.0972820\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.0000	+	10.3923i	−0.669711	+	0.497131i
$438$		0			0
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	0.893843	+	0.448379i	$0.147999\pi$
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	−0.0586141	+	0.998281i	$0.518668\pi$
$440$		0			0
$441$	3.00000	−	5.19615i	0.142857	−	0.247436i
$442$		0			0
$443$	−16.0000	+	27.7128i	−0.760183	+	1.31668i	0.182573	+	0.983192i	$0.441557\pi$
$443$	−16.0000	+	27.7128i	−0.760183	+	1.31668i	−0.942756	+	0.333483i	$0.891776\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	3.00000	−	5.19615i	0.141895	−	0.245770i
$448$		0			0
$449$	−12.0000			−0.566315			−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$		0			0
$453$	6.00000	+	10.3923i	0.281905	+	0.488273i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0000			−0.514558			−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000			−0.514558			−0.257279	+	0.966337i	$0.582826\pi$
$458$		0			0
$459$	2.00000	+	3.46410i	0.0933520	+	0.161690i
$460$		0			0
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	−0.968945	−	0.247276i	$-0.920465\pi$
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	0.270326	−	0.962769i	$-0.412869\pi$
$462$		0			0
$463$	−23.0000			−1.06890			−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000			−1.06890			−0.534450	+	0.845200i	$0.679481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.00000			0.370196			0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000			0.370196			0.185098	+	0.982720i	$0.440740\pi$
$468$		0			0
$469$	−5.50000	+	9.52628i	−0.253966	+	0.439883i
$470$		0			0
$471$	−1.50000	+	2.59808i	−0.0691164	+	0.119713i
$472$		0			0
$473$	−1.00000	−	1.73205i	−0.0459800	−	0.0796398i
$474$		0			0
$475$	17.5000	−	12.9904i	0.802955	−	0.596040i
$476$		0			0
$477$	−2.00000	−	3.46410i	−0.0915737	−	0.158610i
$478$		0			0
$479$	7.00000	−	12.1244i	0.319838	−	0.553976i	−0.660616	−	0.750724i	$-0.729705\pi$
$479$	7.00000	−	12.1244i	0.319838	−	0.553976i	0.980454	+	0.196748i	$0.0630381\pi$
$480$		0			0
$481$	−7.50000	+	12.9904i	−0.341971	+	0.592310i
$482$		0			0
$483$	−4.00000			−0.182006
$484$		0			0
$485$		0			0
$486$		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
$489$	1.50000	+	2.59808i	0.0678323	+	0.117489i
$490$		0			0
$491$	3.00000	+	5.19615i	0.135388	+	0.234499i	0.925746	−	0.378147i	$-0.123439\pi$
$491$	3.00000	+	5.19615i	0.135388	+	0.234499i	−0.790358	+	0.612646i	$0.790105\pi$
$492$		0			0
$493$	32.0000			1.44121
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.00000	−	5.19615i	−0.134568	−	0.233079i
$498$		0			0
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	0.983336	+	0.181797i	$0.0581915\pi$
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	−0.334227	+	0.942493i	$0.608475\pi$
$500$		0			0
$501$	2.00000			0.0893534
$502$		0			0
$503$	20.0000	−	34.6410i	0.891756	−	1.54457i	0.0539870	−	0.998542i	$-0.482807\pi$
$503$	20.0000	−	34.6410i	0.891756	−	1.54457i	0.837769	−	0.546025i	$-0.183860\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6.00000	+	10.3923i	−0.266469	+	0.461538i
$508$		0			0
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	−0.791849	−	0.610718i	$-0.790881\pi$
$509$	3.00000	−	5.19615i	0.132973	−	0.230315i	0.924821	+	0.380402i	$0.124214\pi$
$510$		0			0
$511$	−5.50000	−	9.52628i	−0.243306	−	0.421418i
$512$		0			0
$513$	4.00000	+	1.73205i	0.176604	+	0.0764719i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	−0.352677	−	0.935745i	$-0.614728\pi$
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	0.986718	−	0.162446i	$-0.0519382\pi$
$524$		0			0
$525$	5.00000			0.218218
$526$		0			0
$527$	6.00000	+	10.3923i	0.261364	+	0.452696i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$	−10.0000			−0.433963
$532$		0			0
$533$	−60.0000			−2.59889
$534$		0			0
$535$		0			0
$536$		0			0
$537$	6.00000	+	10.3923i	0.258919	+	0.448461i
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	19.5000	−	33.7750i	0.838370	−	1.45210i	−0.0528859	−	0.998601i	$-0.516842\pi$
$541$	19.5000	−	33.7750i	0.838370	−	1.45210i	0.891256	−	0.453500i	$-0.149825\pi$
$542$		0			0
$543$	−2.00000			−0.0858282
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.5000	+	25.1147i	−0.619975	+	1.07383i	0.369514	+	0.929225i	$0.379524\pi$
$547$	−14.5000	+	25.1147i	−0.619975	+	1.07383i	−0.989490	+	0.144604i	$0.953809\pi$
$548$		0			0
$549$	6.50000	+	11.2583i	0.277413	+	0.480494i
$550$		0			0
$551$	28.0000	−	20.7846i	1.19284	−	0.885454i
$552$		0			0
$553$	−0.500000	−	0.866025i	−0.0212622	−	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.0000	−	32.9090i	0.805056	−	1.39440i	−0.111198	−	0.993798i	$-0.535469\pi$
$557$	19.0000	−	32.9090i	0.805056	−	1.39440i	0.916253	−	0.400599i	$-0.131198\pi$
$558$		0			0
$559$	5.00000			0.211477
$560$		0			0
$561$	4.00000	−	6.92820i	0.168880	−	0.292509i
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.500000	+	0.866025i	0.0209980	+	0.0363696i
$568$		0			0
$569$	24.0000			1.00613			0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000			1.00613			0.503066	+	0.864248i	$0.332205\pi$
$570$		0			0
$571$	−33.0000			−1.38101			−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000			−1.38101			−0.690504	+	0.723329i	$0.742611\pi$
$572$		0			0
$573$	12.0000	+	20.7846i	0.501307	+	0.868290i
$574$		0			0
$575$	10.0000	+	17.3205i	0.417029	+	0.722315i
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$	−9.50000	+	16.4545i	−0.394807	+	0.683825i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.00000	+	6.92820i	−0.165663	+	0.286937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.0000	−	22.5167i	−0.536567	−	0.929362i	−0.999086	−	0.0427523i	$-0.986387\pi$
$587$	−13.0000	−	22.5167i	−0.536567	−	0.929362i	0.462518	−	0.886610i	$-0.346946\pi$
$588$		0			0
$589$	12.0000	+	5.19615i	0.494451	+	0.214104i
$590$		0			0
$591$	13.0000	+	22.5167i	0.534749	+	0.926212i
$592$		0			0
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	−0.797831	−	0.602881i	$-0.794019\pi$
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	0.921026	+	0.389501i	$0.127353\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−21.0000			−0.859473
$598$		0			0
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	−0.621480	−	0.783430i	$-0.713468\pi$
$599$	9.00000	−	15.5885i	0.367730	−	0.636927i	0.989210	+	0.146503i	$0.0468017\pi$
$600$		0			0
$601$	−21.0000			−0.856608			−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000			−0.856608			−0.428304	+	0.903635i	$0.640889\pi$
$602$		0			0
$603$	5.50000	+	9.52628i	0.223977	+	0.387940i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	43.0000			1.74532			0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000			1.74532			0.872658	+	0.488332i	$0.162394\pi$
$608$		0			0
$609$	8.00000			0.324176
$610$		0			0
$611$	−15.0000	−	25.9808i	−0.606835	−	1.05107i
$612$		0			0
$613$	15.0000	+	25.9808i	0.605844	+	1.04935i	0.991917	+	0.126885i	$0.0404979\pi$
$613$	15.0000	+	25.9808i	0.605844	+	1.04935i	−0.386073	+	0.922468i	$0.626169\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	+	5.19615i	−0.120775	+	0.209189i	−0.920074	−	0.391745i	$-0.871871\pi$
$617$	−3.00000	+	5.19615i	−0.120775	+	0.209189i	0.799298	+	0.600935i	$0.205205\pi$
$618$		0			0
$619$	−25.0000			−1.00483			−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000			−1.00483			−0.502417	+	0.864625i	$0.667556\pi$
$620$		0			0
$621$	−2.00000	+	3.46410i	−0.0802572	+	0.139010i
$622$		0			0
$623$	−3.00000	+	5.19615i	−0.120192	+	0.208179i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$	−1.00000	−	8.66025i	−0.0399362	−	0.345857i
$628$		0			0
$629$	6.00000	+	10.3923i	0.239236	+	0.414368i
$630$		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
$633$	−7.50000	+	12.9904i	−0.298098	+	0.516321i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.0000	−	25.9808i	0.594322	−	1.02940i
$638$		0			0
$639$	−6.00000			−0.237356
$640$		0			0
$641$	15.0000	+	25.9808i	0.592464	+	1.02618i	0.993899	+	0.110291i	$0.0351782\pi$
$641$	15.0000	+	25.9808i	0.592464	+	1.02618i	−0.401435	+	0.915888i	$0.631488\pi$
$642$		0			0
$643$	−9.50000	−	16.4545i	−0.374643	−	0.648901i	0.615630	−	0.788035i	$-0.288902\pi$
$643$	−9.50000	−	16.4545i	−0.374643	−	0.648901i	−0.990274	+	0.139134i	$0.955568\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	10.0000	+	17.3205i	0.392534	+	0.679889i
$650$		0			0
$651$	1.50000	+	2.59808i	0.0587896	+	0.101827i
$652$		0			0
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−11.0000			−0.429151
$658$		0			0
$659$	17.0000	−	29.4449i	0.662226	−	1.14701i	−0.317803	−	0.948157i	$-0.602945\pi$
$659$	17.0000	−	29.4449i	0.662226	−	1.14701i	0.980029	−	0.198852i	$-0.0637214\pi$
$660$		0			0
$661$	−21.0000	+	36.3731i	−0.816805	+	1.41475i	0.0912190	+	0.995831i	$0.470924\pi$
$661$	−21.0000	+	36.3731i	−0.816805	+	1.41475i	−0.908024	+	0.418917i	$0.862410\pi$
$662$		0			0
$663$	10.0000	+	17.3205i	0.388368	+	0.672673i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.0000	+	27.7128i	0.619522	+	1.07304i
$668$		0			0
$669$	4.50000	−	7.79423i	0.173980	−	0.301342i
$670$		0			0
$671$	13.0000	−	22.5167i	0.501859	−	0.869246i
$672$		0			0
$673$	−33.0000			−1.27206			−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000			−1.27206			−0.636028	+	0.771666i	$0.719424\pi$
$674$		0			0
$675$	2.50000	−	4.33013i	0.0962250	−	0.166667i
$676$		0			0
$677$	34.0000			1.30673			0.653363	−	0.757045i	$-0.273358\pi$
$677$	34.0000			1.30673			0.653363	+	0.757045i	$0.273358\pi$
$678$		0			0
$679$	1.00000	+	1.73205i	0.0383765	+	0.0664700i
$680$		0			0
$681$	−12.0000	−	20.7846i	−0.459841	−	0.796468i
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−6.50000	−	11.2583i	−0.247990	−	0.429532i
$688$		0			0
$689$	−10.0000	−	17.3205i	−0.380970	−	0.659859i
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$	1.00000	−	1.73205i	0.0379869	−	0.0657952i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−24.0000	+	41.5692i	−0.909065	+	1.57455i
$698$		0			0
$699$	3.00000	−	5.19615i	0.113470	−	0.196537i
$700$		0			0
$701$	−10.0000	−	17.3205i	−0.377695	−	0.654187i	0.613032	−	0.790058i	$-0.289950\pi$
$701$	−10.0000	−	17.3205i	−0.377695	−	0.654187i	−0.990726	+	0.135872i	$0.956616\pi$
$702$		0			0
$703$	12.0000	+	5.19615i	0.452589	+	0.195977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.00000	−	12.1244i	0.263262	−	0.455983i
$708$		0			0
$709$	19.5000	−	33.7750i	0.732338	−	1.26845i	−0.223544	−	0.974694i	$-0.571763\pi$
$709$	19.5000	−	33.7750i	0.732338	−	1.26845i	0.955882	−	0.293752i	$-0.0949041\pi$
$710$		0			0
$711$	−1.00000			−0.0375029
$712$		0			0
$713$	−6.00000	+	10.3923i	−0.224702	+	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−6.00000	−	10.3923i	−0.224074	−	0.388108i
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	13.0000			0.484145
$722$		0			0
$723$	−7.00000			−0.260333
$724$		0			0
$725$	−20.0000	−	34.6410i	−0.742781	−	1.28654i
$726$		0			0
$727$	−23.5000	−	40.7032i	−0.871567	−	1.50960i	−0.860376	−	0.509661i	$-0.829771\pi$
$727$	−23.5000	−	40.7032i	−0.871567	−	1.50960i	−0.0111912	−	0.999937i	$-0.503562\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	2.00000	−	3.46410i	0.0739727	−	0.128124i
$732$		0			0
$733$	−50.0000			−1.84679			−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000			−1.84679			−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	11.0000	−	19.0526i	0.405190	−	0.701810i
$738$		0			0
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	0.986154	−	0.165831i	$-0.0530307\pi$
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	−0.636691	+	0.771119i	$0.719697\pi$
$740$		0			0
$741$	20.0000	+	8.66025i	0.734718	+	0.318142i
$742$		0			0
$743$	−25.0000	−	43.3013i	−0.917161	−	1.58857i	−0.803706	−	0.595026i	$-0.797142\pi$
$743$	−25.0000	−	43.3013i	−0.917161	−	1.58857i	−0.113455	−	0.993543i	$-0.536192\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.0000			−0.657706
$750$		0			0
$751$	22.5000	−	38.9711i	0.821037	−	1.42208i	−0.0838743	−	0.996476i	$-0.526729\pi$
$751$	22.5000	−	38.9711i	0.821037	−	1.42208i	0.904911	−	0.425601i	$-0.139937\pi$
$752$		0			0
$753$	2.00000			0.0728841
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.50000	−	11.2583i	−0.236247	−	0.409191i	0.723388	−	0.690442i	$-0.242584\pi$
$757$	−6.50000	−	11.2583i	−0.236247	−	0.409191i	−0.959634	+	0.281251i	$0.909251\pi$
$758$		0			0
$759$	8.00000			0.290382
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	−1.00000	−	1.73205i	−0.0362024	−	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−50.0000			−1.80540
$768$		0			0
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	−0.311577	−	0.950221i	$-0.600857\pi$
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	0.978704	−	0.205277i	$-0.0658095\pi$
$770$		0			0
$771$	−20.0000			−0.720282
$772$		0			0
$773$	−7.00000	+	12.1244i	−0.251773	+	0.436083i	−0.964014	−	0.265852i	$-0.914347\pi$
$773$	−7.00000	+	12.1244i	−0.251773	+	0.436083i	0.712241	+	0.701935i	$0.247680\pi$
$774$		0			0
$775$	7.50000	−	12.9904i	0.269408	−	0.466628i
$776$		0			0
$777$	1.50000	+	2.59808i	0.0538122	+	0.0932055i
$778$		0			0
$779$	6.00000	+	51.9615i	0.214972	+	1.86171i
$780$		0			0
$781$	6.00000	+	10.3923i	0.214697	+	0.371866i
$782$		0			0
$783$	4.00000	−	6.92820i	0.142948	−	0.247594i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−25.0000			−0.891154			−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000			−0.891154			−0.445577	+	0.895244i	$0.647001\pi$
$788$		0			0
$789$	13.0000	−	22.5167i	0.462812	−	0.801614i
$790$		0			0
$791$	−2.00000			−0.0711118
$792$		0			0
$793$	32.5000	+	56.2917i	1.15411	+	1.99898i

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} -1.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{3} -1.00000 q^{7} +(-0.500000 + 0.866025i) q^{9} +2.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(2.00000 + 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{21} +(-2.00000 + 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +1.00000 q^{27} +(4.00000 - 6.92820i) q^{29} +3.00000 q^{31} +(-1.00000 - 1.73205i) q^{33} +3.00000 q^{37} +5.00000 q^{39} +(6.00000 + 10.3923i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(-3.00000 + 5.19615i) q^{47} -6.00000 q^{49} +(2.00000 - 3.46410i) q^{51} +(-2.00000 + 3.46410i) q^{53} +(-0.500000 - 4.33013i) q^{57} +(5.00000 + 8.66025i) q^{59} +(6.50000 - 11.2583i) q^{61} +(0.500000 - 0.866025i) q^{63} +(5.50000 - 9.52628i) q^{67} +4.00000 q^{69} +(3.00000 + 5.19615i) q^{71} +(5.50000 + 9.52628i) q^{73} -5.00000 q^{75} -2.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} -8.00000 q^{87} +(3.00000 - 5.19615i) q^{89} +(2.50000 - 4.33013i) q^{91} +(-1.50000 - 2.59808i) q^{93} +(-1.00000 - 1.73205i) q^{97} +(-1.00000 + 1.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{7} - q^{9}+O(q^{10}) \) 2 * q - q^3 - 2 * q^7 - q^9	\( 2 q - q^{3} - 2 q^{7} - q^{9} + 4 q^{11} - 5 q^{13} + 4 q^{17} + 8 q^{19} + q^{21} - 4 q^{23} + 5 q^{25} + 2 q^{27} + 8 q^{29} + 6 q^{31} - 2 q^{33} + 6 q^{37} + 10 q^{39} + 12 q^{41} - q^{43} - 6 q^{47} - 12 q^{49} + 4 q^{51} - 4 q^{53} - q^{57} + 10 q^{59} + 13 q^{61} + q^{63} + 11 q^{67} + 8 q^{69} + 6 q^{71} + 11 q^{73} - 10 q^{75} - 4 q^{77} + q^{79} - q^{81} - 16 q^{87} + 6 q^{89} + 5 q^{91} - 3 q^{93} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^7 - q^9 + 4 * q^11 - 5 * q^13 + 4 * q^17 + 8 * q^19 + q^21 - 4 * q^23 + 5 * q^25 + 2 * q^27 + 8 * q^29 + 6 * q^31 - 2 * q^33 + 6 * q^37 + 10 * q^39 + 12 * q^41 - q^43 - 6 * q^47 - 12 * q^49 + 4 * q^51 - 4 * q^53 - q^57 + 10 * q^59 + 13 * q^61 + q^63 + 11 * q^67 + 8 * q^69 + 6 * q^71 + 11 * q^73 - 10 * q^75 - 4 * q^77 + q^79 - q^81 - 16 * q^87 + 6 * q^89 + 5 * q^91 - 3 * q^93 - 2 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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