LMFDB - Embedded newform 777.2.r.a.26.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [777,2,Mod(26,777)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("777.26"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(777, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 777 = 3 \cdot 7 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	777.r (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,4,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.20437623705$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			26.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	777.26
Dual form			777.2.r.a.269.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +2.00000 q^{4} +(0.500000 + 2.59808i) q^{7} -3.00000 q^{9} -3.46410i q^{12} +(4.50000 + 2.59808i) q^{13} +4.00000 q^{16} +(4.50000 - 2.59808i) q^{19} +(4.50000 - 0.866025i) q^{21} +(2.50000 - 4.33013i) q^{25} +5.19615i q^{27} +(1.00000 + 5.19615i) q^{28} +(1.50000 - 0.866025i) q^{31} -6.00000 q^{36} +(-5.50000 - 2.59808i) q^{37} +(4.50000 - 7.79423i) q^{39} -8.00000 q^{43} -6.92820i q^{48} +(-6.50000 + 2.59808i) q^{49} +(9.00000 + 5.19615i) q^{52} +(-4.50000 - 7.79423i) q^{57} +(-6.00000 + 3.46410i) q^{61} +(-1.50000 - 7.79423i) q^{63} +8.00000 q^{64} +5.00000 q^{67} +(1.50000 - 0.866025i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(9.00000 - 5.19615i) q^{76} +(-8.50000 - 14.7224i) q^{79} +9.00000 q^{81} +(9.00000 - 1.73205i) q^{84} +(-4.50000 + 12.9904i) q^{91} +(-1.50000 - 2.59808i) q^{93} +19.0526i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + q^{7} - 6 q^{9} + 9 q^{13} + 8 q^{16} + 9 q^{19} + 9 q^{21} + 5 q^{25} + 2 q^{28} + 3 q^{31} - 12 q^{36} - 11 q^{37} + 9 q^{39} - 16 q^{43} - 13 q^{49} + 18 q^{52} - 9 q^{57} - 12 q^{61}+ \cdots - 3 q^{93}+O(q^{100}) $ 2 * q + 4 * q^4 + q^7 - 6 * q^9 + 9 * q^13 + 8 * q^16 + 9 * q^19 + 9 * q^21 + 5 * q^25 + 2 * q^28 + 3 * q^31 - 12 * q^36 - 11 * q^37 + 9 * q^39 - 16 * q^43 - 13 * q^49 + 18 * q^52 - 9 * q^57 - 12 * q^61 - 3 * q^63 + 16 * q^64 + 10 * q^67 + 3 * q^73 - 15 * q^75 + 18 * q^76 - 17 * q^79 + 18 * q^81 + 18 * q^84 - 9 * q^91 - 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/777\mathbb{Z}\right)^\times$.

$n$	$260$	$556$	$631$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{3}\right)$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	2.00000			1.00000
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		−	3.46410i		−	1.00000i
$13$	4.50000	+	2.59808i	1.24808	+	0.720577i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	4.50000	+	2.59808i	1.24808	+	0.720577i	0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	4.50000	−	2.59808i	1.03237	−	0.596040i	0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$	4.50000	−	0.866025i	0.981981	−	0.188982i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$			5.19615i			1.00000i
$28$	1.00000	+	5.19615i	0.188982	+	0.981981i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	1.50000	−	0.866025i	0.269408	−	0.155543i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	1.50000	−	0.866025i	0.269408	−	0.155543i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.00000			−1.00000
$37$	−5.50000	−	2.59808i	−0.904194	−	0.427121i
$37$	−5.50000	−	2.59808i	−0.904194	−	0.427121i
$38$		0			0
$39$	4.50000	−	7.79423i	0.720577	−	1.24808i
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		−	6.92820i		−	1.00000i
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$	9.00000	+	5.19615i	1.24808	+	0.720577i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.50000	−	7.79423i	−0.596040	−	1.03237i
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−6.00000	+	3.46410i	−0.768221	+	0.443533i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−6.00000	+	3.46410i	−0.768221	+	0.443533i	0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$	−1.50000	−	7.79423i	−0.188982	−	0.981981i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	5.00000			0.610847			0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000			0.610847			0.305424	+	0.952217i	$0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$	1.50000	−	0.866025i	0.175562	−	0.101361i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	1.50000	−	0.866025i	0.175562	−	0.101361i	0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	−7.50000	−	4.33013i	−0.866025	−	0.500000i
$76$	9.00000	−	5.19615i	1.03237	−	0.596040i
$77$		0			0
$78$		0			0
$79$	−8.50000	−	14.7224i	−0.956325	−	1.65640i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−8.50000	−	14.7224i	−0.956325	−	1.65640i	−0.225018	−	0.974355i	$-0.572244\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$	9.00000	−	1.73205i	0.981981	−	0.188982i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−4.50000	+	12.9904i	−0.471728	+	1.36176i
$92$		0			0
$93$	−1.50000	−	2.59808i	−0.155543	−	0.269408i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.0526i			1.93449i	0.253837	+	0.967247i	$0.418307\pi$
$97$			19.0526i			1.93449i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	16.5000	+	9.52628i	1.62579	+	0.938652i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	16.5000	+	9.52628i	1.62579	+	0.938652i	0.640464	+	0.767988i	$0.278742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$			10.3923i			1.00000i
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	−4.50000	+	9.52628i	−0.427121	+	0.904194i
$112$	2.00000	+	10.3923i	0.188982	+	0.981981i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.5000	−	7.79423i	−1.24808	−	0.720577i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$		0			0
$126$		0			0
$127$	−10.0000	−	17.3205i	−0.887357	−	1.53695i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−10.0000	−	17.3205i	−0.887357	−	1.53695i	−0.0443678	−	0.999015i	$-0.514127\pi$
$128$		0			0
$129$			13.8564i			1.21999i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	9.00000	+	10.3923i	0.780399	+	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−15.0000	+	8.66025i	−1.27228	+	0.734553i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−15.0000	+	8.66025i	−1.27228	+	0.734553i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	4.50000	+	11.2583i	0.371154	+	0.928571i
$148$	−11.0000	−	5.19615i	−0.904194	−	0.427121i
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−23.0000			−1.87171			−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−23.0000			−1.87171			−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	9.00000	−	15.5885i	0.720577	−	1.24808i
$157$	−1.50000	−	0.866025i	−0.119713	−	0.0691164i	0.438948	−	0.898513i	$-0.355351\pi$
$157$	−1.50000	−	0.866025i	−0.119713	−	0.0691164i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	7.00000	+	12.1244i	0.538462	+	0.932643i
$170$		0			0
$171$	−13.5000	+	7.79423i	−1.03237	+	0.596040i
$172$	−16.0000			−1.21999
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	12.5000	+	4.33013i	0.944911	+	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−16.5000	−	9.52628i	−1.22644	−	0.708083i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−16.5000	−	9.52628i	−1.22644	−	0.708083i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$	6.00000	+	10.3923i	0.443533	+	0.768221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−13.5000	+	2.59808i	−0.981981	+	0.188982i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		−	13.8564i		−	1.00000i
$193$	1.00000	+	1.73205i	0.0719816	+	0.124676i	0.899770	−	0.436365i	$-0.143734\pi$
$193$	1.00000	+	1.73205i	0.0719816	+	0.124676i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		0			0
$195$		0			0
$196$	−13.0000	+	5.19615i	−0.928571	+	0.371154i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	22.5000	−	12.9904i	1.59498	−	0.920864i	0.602549	−	0.798082i	$-0.294152\pi$
$199$	22.5000	−	12.9904i	1.59498	−	0.920864i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$		0			0
$201$		−	8.66025i		−	0.610847i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	18.0000	+	10.3923i	1.24808	+	0.720577i
$209$		0			0
$210$		0			0
$211$	−29.0000			−1.99644			−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−29.0000			−1.99644			−0.998221	+	0.0596196i	$0.981011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.00000	+	3.46410i	0.203653	+	0.235159i
$218$		0			0
$219$	−1.50000	−	2.59808i	−0.101361	−	0.175562i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			19.0526i			1.27585i	0.770097	+	0.637927i	$0.220208\pi$
$223$			19.0526i			1.27585i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$	−7.50000	+	12.9904i	−0.500000	+	0.866025i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−9.00000	−	15.5885i	−0.596040	−	1.03237i
$229$	−18.0000	+	10.3923i	−1.18947	+	0.686743i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−18.0000	+	10.3923i	−1.18947	+	0.686743i	−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−25.5000	+	14.7224i	−1.65640	+	0.956325i
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$			1.73205i			0.111571i	0.998443	+	0.0557856i	$0.0177663\pi$
$241$			1.73205i			0.111571i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		−	15.5885i		−	1.00000i
$244$	−12.0000	+	6.92820i	−0.768221	+	0.443533i
$245$		0			0
$246$		0			0
$247$	27.0000			1.71797
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−3.00000	−	15.5885i	−0.188982	−	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	4.00000	−	15.5885i	0.248548	−	0.968620i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	10.0000			0.610847
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		−	32.9090i		−	1.99908i	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$		−	32.9090i		−	1.99908i	0.0303728	−	0.999539i	$-0.490331\pi$
$272$		0			0
$273$	22.5000	+	7.79423i	1.36176	+	0.471728i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−31.0000			−1.86261			−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−31.0000			−1.86261			−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$	−4.50000	+	2.59808i	−0.269408	+	0.155543i
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		−	22.5167i		−	1.33848i	−0.743048	−	0.669238i	$-0.766621\pi$
$283$		−	22.5167i		−	1.33848i	0.743048	−	0.669238i	$-0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$	33.0000			1.93449
$292$	3.00000	−	1.73205i	0.175562	−	0.101361i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−15.0000	−	8.66025i	−0.866025	−	0.500000i
$301$	−4.00000	−	20.7846i	−0.230556	−	1.19800i
$302$		0			0
$303$		0			0
$304$	18.0000	−	10.3923i	1.03237	−	0.596040i
$305$		0			0
$306$		0			0
$307$			31.1769i			1.77936i	0.456584	+	0.889680i	$0.349073\pi$
$307$			31.1769i			1.77936i	−0.456584	+	0.889680i	$0.650927\pi$
$308$		0			0
$309$	16.5000	−	28.5788i	0.938652	−	1.62579i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$			27.7128i			1.56642i	0.621757	+	0.783210i	$0.286419\pi$
$313$			27.7128i			1.56642i	−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$		0			0
$316$	−17.0000	−	29.4449i	−0.956325	−	1.65640i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	22.5000	−	12.9904i	1.24808	−	0.720577i
$326$		0			0
$327$	−3.00000	−	1.73205i	−0.165900	−	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	31.0000			1.70391			0.851957	−	0.523612i	$-0.175416\pi$
$331$	31.0000			1.70391			0.851957	+	0.523612i	$0.175416\pi$
$332$		0			0
$333$	16.5000	+	7.79423i	0.904194	+	0.427121i
$334$		0			0
$335$		0			0
$336$	18.0000	−	3.46410i	0.981981	−	0.188982i
$337$	−17.0000	+	29.4449i	−0.926049	+	1.60396i	−0.136184	+	0.990684i	$0.543484\pi$
$337$	−17.0000	+	29.4449i	−0.926049	+	1.60396i	−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	30.0000	−	17.3205i	1.60586	−	0.927146i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	30.0000	−	17.3205i	1.60586	−	0.927146i	0.990282	−	0.139072i	$-0.0444119\pi$
$350$		0			0
$351$	−13.5000	+	23.3827i	−0.720577	+	1.24808i
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$	16.5000	+	9.52628i	0.866025	+	0.500000i
$364$	−9.00000	+	25.9808i	−0.471728	+	1.36176i
$365$		0			0
$366$		0			0
$367$		−	22.5167i		−	1.17536i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$		−	22.5167i		−	1.17536i	0.809093	−	0.587680i	$-0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.00000	−	5.19615i	−0.155543	−	0.269408i
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	12.5000	+	21.6506i	0.647225	+	1.12103i	−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−18.5000	−	32.0429i	−0.950281	−	1.64594i	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−18.5000	−	32.0429i	−0.950281	−	1.64594i	−0.205466	−	0.978664i	$-0.565871\pi$
$380$		0			0
$381$	−30.0000	+	17.3205i	−1.53695	+	0.887357i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	24.0000			1.21999
$388$			38.1051i			1.93449i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−34.5000	−	19.9186i	−1.73151	−	0.999685i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	−34.5000	−	19.9186i	−1.73151	−	0.999685i	−0.853206	−	0.521575i	$-0.825345\pi$
$398$		0			0
$399$	18.0000	−	15.5885i	0.901127	−	0.780399i
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	9.00000			0.448322
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.5000	−	12.9904i	−1.11255	−	0.642333i	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−22.5000	−	12.9904i	−1.11255	−	0.642333i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$	33.0000	+	19.0526i	1.62579	+	0.938652i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	15.0000	+	25.9808i	0.734553	+	1.27228i
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.0000	−	13.8564i	−0.580721	−	0.670559i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			20.7846i			1.00000i
$433$			22.5167i			1.08208i	0.840996	+	0.541041i	$0.181970\pi$
$433$			22.5167i			1.08208i	−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$		0			0
$438$		0			0
$439$	34.5000	−	19.9186i	1.64660	−	0.950662i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	34.5000	−	19.9186i	1.64660	−	0.950662i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$	19.5000	−	7.79423i	0.928571	−	0.371154i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−9.00000	+	19.0526i	−0.427121	+	0.904194i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	4.00000	+	20.7846i	0.188982	+	0.981981i
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			39.8372i			1.87171i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.5000	+	35.5070i	0.958950	+	1.66095i	0.725059	+	0.688686i	$0.241812\pi$
$457$	20.5000	+	35.5070i	0.958950	+	1.66095i	0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	21.5000	−	37.2391i	0.999190	−	1.73065i	0.464739	−	0.885448i	$-0.346148\pi$
$463$	21.5000	−	37.2391i	0.999190	−	1.73065i	0.534450	−	0.845200i	$-0.320519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$	−27.0000	−	15.5885i	−1.24808	−	0.720577i
$469$	2.50000	+	12.9904i	0.115439	+	0.599840i
$470$		0			0
$471$	−1.50000	+	2.59808i	−0.0691164	+	0.119713i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	25.9808i		−	1.19208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−18.0000	−	25.9808i	−0.820729	−	1.18462i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−12.5000	−	21.6506i	−0.566429	−	0.981084i	0.430486	−	0.902597i	$-0.358342\pi$
$488$		0			0
$489$		−	13.8564i		−	0.626608i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	6.00000	−	3.46410i	0.269408	−	0.155543i
$497$		0			0
$498$		0			0
$499$	−21.5000	−	37.2391i	−0.962472	−	1.66705i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−21.5000	−	37.2391i	−0.962472	−	1.66705i	−0.246214	−	0.969216i	$-0.579187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	21.0000	−	12.1244i	0.932643	−	0.538462i
$508$	−20.0000	−	34.6410i	−0.887357	−	1.53695i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	3.00000	+	3.46410i	0.132712	+	0.153243i
$512$		0			0
$513$	13.5000	+	23.3827i	0.596040	+	1.03237i
$514$		0			0
$515$		0			0
$516$			27.7128i			1.21999i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	13.5000	−	7.79423i	0.590314	−	0.340818i	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	13.5000	−	7.79423i	0.590314	−	0.340818i	0.765222	+	0.643767i	$0.222629\pi$
$524$		0			0
$525$	7.50000	−	21.6506i	0.327327	−	0.944911i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	18.0000	+	20.7846i	0.780399	+	0.901127i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.623404	−	0.781900i	$-0.285749\pi$
$542$		0			0
$543$	−16.5000	+	28.5788i	−0.708083	+	1.22644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	41.0000			1.75303			0.876517	−	0.481371i	$-0.159861\pi$
$547$	41.0000			1.75303			0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$	18.0000	−	10.3923i	0.768221	−	0.443533i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	34.0000	−	29.4449i	1.44583	−	1.25212i
$554$		0			0
$555$		0			0
$556$	−30.0000	+	17.3205i	−1.27228	+	0.734553i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	−36.0000	−	20.7846i	−1.52264	−	0.879095i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.50000	+	23.3827i	0.188982	+	0.981981i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	−28.5000	−	16.4545i	−1.18647	−	0.685009i	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−28.5000	−	16.4545i	−1.18647	−	0.685009i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	3.00000	−	1.73205i	0.124676	−	0.0719816i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$	9.00000	+	22.5167i	0.371154	+	0.928571i
$589$	4.50000	−	7.79423i	0.185419	−	0.321156i
$590$		0			0
$591$		0			0
$592$	−22.0000	−	10.3923i	−0.904194	−	0.427121i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−22.5000	−	38.9711i	−0.920864	−	1.59498i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−36.0000	+	20.7846i	−1.46847	+	0.847822i	−0.999376	−	0.0353259i	$-0.988753\pi$
$601$	−36.0000	+	20.7846i	−1.46847	+	0.847822i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$	−15.0000			−0.610847
$604$	−46.0000			−1.87171
$605$		0			0
$606$		0			0
$607$			5.19615i			0.210905i	0.994424	+	0.105453i	$0.0336291\pi$
$607$			5.19615i			0.210905i	−0.994424	+	0.105453i	$0.966371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	33.0000	−	19.0526i	1.32638	−	0.765787i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	33.0000	−	19.0526i	1.32638	−	0.765787i	0.984738	+	0.174042i	$0.0556830\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	18.0000	−	31.1769i	0.720577	−	1.24808i
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−3.00000	−	1.73205i	−0.119713	−	0.0691164i
$629$		0			0
$630$		0			0
$631$	21.5000	+	37.2391i	0.855901	+	1.48246i	0.875806	+	0.482663i	$0.160330\pi$
$631$	21.5000	+	37.2391i	0.855901	+	1.48246i	−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$			50.2295i			1.99644i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−36.0000	−	5.19615i	−1.42637	−	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$			31.1769i			1.22950i	0.788723	+	0.614749i	$0.210743\pi$
$643$			31.1769i			1.22950i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	6.00000	−	5.19615i	0.235159	−	0.203653i
$652$	16.0000			0.626608
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−4.50000	+	2.59808i	−0.175562	+	0.101361i
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		−	34.6410i		−	1.34738i	−0.739014	−	0.673690i	$-0.764708\pi$
$661$		−	34.6410i		−	1.34738i	0.739014	−	0.673690i	$-0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	33.0000			1.27585
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−18.5000	+	32.0429i	−0.713123	+	1.23516i	0.250557	+	0.968102i	$0.419386\pi$
$673$	−18.5000	+	32.0429i	−0.713123	+	1.23516i	−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$	22.5000	+	12.9904i	0.866025	+	0.500000i
$676$	14.0000	+	24.2487i	0.538462	+	0.932643i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−49.5000	+	9.52628i	−1.89964	+	0.365585i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−27.0000	+	15.5885i	−1.03237	+	0.596040i
$685$		0			0
$686$		0			0
$687$	18.0000	+	31.1769i	0.686743	+	1.18947i
$688$	−32.0000			−1.21999
$689$		0			0
$690$		0			0
$691$			51.9615i			1.97671i	0.152167	+	0.988355i	$0.451375\pi$
$691$			51.9615i			1.97671i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	25.0000	+	8.66025i	0.944911	+	0.327327i
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$	−31.5000	+	2.59808i	−1.18805	+	0.0979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	0.413114	+	0.910679i	$0.364441\pi$
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$	25.5000	+	44.1673i	0.956325	+	1.65640i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−16.5000	+	47.6314i	−0.614492	+	1.77389i
$722$		0			0
$723$	3.00000			0.111571
$724$	−33.0000	−	19.0526i	−1.22644	−	0.708083i
$725$		0			0
$726$		0			0
$727$	46.5000	+	26.8468i	1.72459	+	0.995692i	0.908655	+	0.417548i	$0.137111\pi$
$727$	46.5000	+	26.8468i	1.72459	+	0.995692i	0.815935	+	0.578144i	$0.196223\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	12.0000	+	20.7846i	0.443533	+	0.768221i
$733$	−28.5000	+	16.4545i	−1.05267	+	0.607760i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−28.5000	+	16.4545i	−1.05267	+	0.607760i	−0.129275	+	0.991609i	$0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−26.5000	−	45.8993i	−0.974818	−	1.68843i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−26.5000	−	45.8993i	−0.974818	−	1.68843i	−0.294285	−	0.955718i	$-0.595081\pi$
$740$		0			0
$741$		−	46.7654i		−	1.71797i
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.0000	+	45.0333i	−0.948753	+	1.64329i	−0.200698	+	0.979653i	$0.564321\pi$
$751$	−26.0000	+	45.0333i	−0.948753	+	1.64329i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−27.0000	+	5.19615i	−0.981981	+	0.188982i
$757$	14.5000	+	25.1147i	0.527011	+	0.912811i	0.999505	+	0.0314762i	$0.0100208\pi$
$757$	14.5000	+	25.1147i	0.527011	+	0.912811i	−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	5.00000	+	1.73205i	0.181012	+	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	27.7128i		−	1.00000i
$769$		−	29.4449i		−	1.06181i	−0.847432	−	0.530904i	$-0.821852\pi$
$769$		−	29.4449i		−	1.06181i	0.847432	−	0.530904i	$-0.178148\pi$
$770$		0			0
$771$		0			0
$772$	2.00000	+	3.46410i	0.0719816	+	0.124676i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		−	8.66025i		−	0.311086i
$776$		0			0
$777$	−27.0000	−	6.92820i	−0.968620	−	0.248548i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$785$		0			0
$786$		0			0
$787$	43.5000	−	25.1147i	1.55061	−	0.895244i	0.552515	−	0.833503i	$-0.313668\pi$
$787$	43.5000	−	25.1147i	1.55061	−	0.895244i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−36.0000			−1.27840
$794$		0			0
$795$		0			0
$796$	45.0000	−	25.9808i	1.59498	−	0.920864i
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	17.3205i		−	0.610847i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−9.00000	−	5.19615i	−0.316033	−	0.182462i	0.333590	−	0.942718i	$-0.391740\pi$
$811$	−9.00000	−	5.19615i	−0.316033	−	0.182462i	−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$	−57.0000			−1.99908
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−36.0000	+	20.7846i	−1.25948	+	0.727161i
$818$		0			0
$819$	13.5000	−	38.9711i	0.471728	−	1.36176i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−23.5000	+	40.7032i	−0.819159	+	1.41882i	0.0871445	+	0.996196i	$0.472226\pi$
$823$	−23.5000	+	40.7032i	−0.819159	+	1.41882i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$827$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$828$		0			0
$829$	19.5000	+	11.2583i	0.677263	+	0.391018i	0.798823	−	0.601566i	$-0.205456\pi$
$829$	19.5000	+	11.2583i	0.677263	+	0.391018i	−0.121560	+	0.992584i	$0.538790\pi$
$830$		0			0
$831$			53.6936i			1.86261i
$832$	36.0000	+	20.7846i	1.24808	+	0.720577i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.50000	+	7.79423i	0.155543	+	0.269408i
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$	−58.0000			−1.99644
$845$		0			0
$846$		0			0
$847$	−27.5000	−	9.52628i	−0.944911	−	0.327327i
$848$		0			0
$849$	−39.0000			−1.33848
$850$		0			0
$851$		0			0
$852$		0			0
$853$	40.5000	−	23.3827i	1.38669	−	0.800608i	0.393753	−	0.919216i	$-0.371177\pi$
$853$	40.5000	−	23.3827i	1.38669	−	0.800608i	0.992941	+	0.118609i	$0.0378434\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	49.5000	+	28.5788i	1.68892	+	0.975097i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	49.5000	+	28.5788i	1.68892	+	0.975097i	0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.5000	−	14.7224i	0.866025	−	0.500000i
$868$	6.00000	+	6.92820i	0.203653	+	0.235159i
$869$		0			0
$870$		0			0
$871$	22.5000	+	12.9904i	0.762383	+	0.440162i
$872$		0			0
$873$		−	57.1577i		−	1.93449i
$874$		0			0
$875$		0			0
$876$	−3.00000	−	5.19615i	−0.101361	−	0.175562i
$877$	12.5000	−	21.6506i	0.422095	−	0.731090i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	12.5000	−	21.6506i	0.422095	−	0.731090i	0.996144	+	0.0877308i	$0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$	23.5000	−	40.7032i	0.790838	−	1.36977i	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	23.5000	−	40.7032i	0.790838	−	1.36977i	0.925449	−	0.378873i	$-0.123688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	40.0000	−	34.6410i	1.34156	−	1.16182i
$890$		0			0
$891$		0			0
$892$			38.1051i			1.27585i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−15.0000	+	25.9808i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−36.0000	+	6.92820i	−1.19800	+	0.230556i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	40.0000			1.32818			0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000			1.32818			0.664089	+	0.747653i	$0.268820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−18.0000	−	31.1769i	−0.596040	−	1.03237i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−36.0000	+	20.7846i	−1.18947	+	0.686743i
$917$		0			0
$918$		0			0
$919$	−26.0000	+	45.0333i	−0.857661	+	1.48551i	0.0164935	+	0.999864i	$0.494750\pi$
$919$	−26.0000	+	45.0333i	−0.857661	+	1.48551i	−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$	54.0000			1.77936
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−25.0000	+	17.3205i	−0.821995	+	0.569495i
$926$		0			0
$927$	−49.5000	−	28.5788i	−1.62579	−	0.938652i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−22.5000	+	28.5788i	−0.737408	+	0.936634i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	48.0000	−	27.7128i	1.56809	−	0.905338i	0.571700	−	0.820463i	$-0.306284\pi$
$937$	48.0000	−	27.7128i	1.56809	−	0.905338i	0.996392	−	0.0848755i	$-0.0270492\pi$
$938$		0			0
$939$	48.0000			1.56642
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	−51.0000	+	29.4449i	−1.65640	+	0.956325i
$949$	9.00000			0.292152
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14.0000	+	24.2487i	−0.451613	+	0.782216i
$962$		0			0
$963$		0			0
$964$			3.46410i			0.111571i
$965$		0			0
$966$		0			0
$967$	−10.0000	+	17.3205i	−0.321578	+	0.556990i	−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−10.0000	+	17.3205i	−0.321578	+	0.556990i	0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		−	31.1769i		−	1.00000i
$973$	−30.0000	−	34.6410i	−0.961756	−	1.11054i
$974$		0			0
$975$	−22.5000	−	38.9711i	−0.720577	−	1.24808i
$976$	−24.0000	+	13.8564i	−0.768221	+	0.443533i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−3.00000	+	5.19615i	−0.0957826	+	0.165900i
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	54.0000			1.71797
$989$		0			0
$990$		0			0
$991$	8.50000	+	14.7224i	0.270011	+	0.467673i	0.968864	−	0.247592i	$-0.0796392\pi$
$991$	8.50000	+	14.7224i	0.270011	+	0.467673i	−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$		−	53.6936i		−	1.70391i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	34.5000	+	19.9186i	1.09263	+	0.630828i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	34.5000	+	19.9186i	1.09263	+	0.630828i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	13.5000	−	28.5788i	0.427121	−	0.904194i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	777.2.r.a.26.1	✓	2
3.2	odd	2	CM	777.2.r.a.26.1	✓	2
7.3	odd	6		777.2.bo.a.248.1	yes	2
21.17	even	6		777.2.bo.a.248.1	yes	2
37.10	even	3		777.2.bo.a.47.1	yes	2
111.47	odd	6		777.2.bo.a.47.1	yes	2
259.10	odd	6	inner	777.2.r.a.269.1	yes	2
777.269	even	6	inner	777.2.r.a.269.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
777.2.r.a.26.1	✓	2	1.1	even	1	trivial
777.2.r.a.26.1	✓	2	3.2	odd	2	CM
777.2.r.a.269.1	yes	2	259.10	odd	6	inner
777.2.r.a.269.1	yes	2	777.269	even	6	inner
777.2.bo.a.47.1	yes	2	37.10	even	3
777.2.bo.a.47.1	yes	2	111.47	odd	6
777.2.bo.a.248.1	yes	2	7.3	odd	6
777.2.bo.a.248.1	yes	2	21.17	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 777 = 3 \cdot 7 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	777.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +2.00000 q^{4} +(0.500000 + 2.59808i) q^{7} -3.00000 q^{9} -3.46410i q^{12} +(4.50000 + 2.59808i) q^{13} +4.00000 q^{16} +(4.50000 - 2.59808i) q^{19} +(4.50000 - 0.866025i) q^{21} +(2.50000 - 4.33013i) q^{25} +5.19615i q^{27} +(1.00000 + 5.19615i) q^{28} +(1.50000 - 0.866025i) q^{31} -6.00000 q^{36} +(-5.50000 - 2.59808i) q^{37} +(4.50000 - 7.79423i) q^{39} -8.00000 q^{43} -6.92820i q^{48} +(-6.50000 + 2.59808i) q^{49} +(9.00000 + 5.19615i) q^{52} +(-4.50000 - 7.79423i) q^{57} +(-6.00000 + 3.46410i) q^{61} +(-1.50000 - 7.79423i) q^{63} +8.00000 q^{64} +5.00000 q^{67} +(1.50000 - 0.866025i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(9.00000 - 5.19615i) q^{76} +(-8.50000 - 14.7224i) q^{79} +9.00000 q^{81} +(9.00000 - 1.73205i) q^{84} +(-4.50000 + 12.9904i) q^{91} +(-1.50000 - 2.59808i) q^{93} +19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + q^{7} - 6 q^{9} + 9 q^{13} + 8 q^{16} + 9 q^{19} + 9 q^{21} + 5 q^{25} + 2 q^{28} + 3 q^{31} - 12 q^{36} - 11 q^{37} + 9 q^{39} - 16 q^{43} - 13 q^{49} + 18 q^{52} - 9 q^{57} - 12 q^{61}+ \cdots - 3 q^{93}+O(q^{100}) \) 2 * q + 4 * q^4 + q^7 - 6 * q^9 + 9 * q^13 + 8 * q^16 + 9 * q^19 + 9 * q^21 + 5 * q^25 + 2 * q^28 + 3 * q^31 - 12 * q^36 - 11 * q^37 + 9 * q^39 - 16 * q^43 - 13 * q^49 + 18 * q^52 - 9 * q^57 - 12 * q^61 - 3 * q^63 + 16 * q^64 + 10 * q^67 + 3 * q^73 - 15 * q^75 + 18 * q^76 - 17 * q^79 + 18 * q^81 + 18 * q^84 - 9 * q^91 - 3 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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