LMFDB - Embedded newform 2793.1.eo.a.1346.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2793,1,Mod(137,2793)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2793, base_ring=CyclotomicField(126)) chi = DirichletCharacter(H, H._module([63, 102, 14])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2793.137"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$2793 = 3 \cdot 7^{2} \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2793.eo (of order $126$ , degree $36$ , minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.39388858028$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{63})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1$ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{63}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{63} - \cdots)$

Embedding invariants

Embedding label			1346.1
Root			$0.921476 - 0.388435i$ of defining polynomial
Character	$\chi$	$=$	2793.1346
Dual form			2793.1.eo.a.1355.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.797133 + 0.603804i) q^{3} +(-0.0249307 - 0.999689i) q^{4} +(0.698237 - 0.715867i) q^{7} +(0.270840 - 0.962624i) q^{9} +(0.623490 + 0.781831i) q^{12} +(0.907887 - 0.0908478i) q^{13} +(-0.998757 + 0.0498459i) q^{16} +(0.0747301 + 0.997204i) q^{19} +(-0.124344 + 0.992239i) q^{21} +(0.270840 - 0.962624i) q^{25} +(0.365341 + 0.930874i) q^{27} +(-0.733052 - 0.680173i) q^{28} +(0.661686 - 1.14607i) q^{31} +(-0.969077 - 0.246757i) q^{36} +(0.629859 - 0.0949359i) q^{37} +(-0.668852 + 0.620604i) q^{39} +(-1.98759 + 0.0991964i) q^{43} +(0.766044 - 0.642788i) q^{48} +(-0.0249307 - 0.999689i) q^{49} +(-0.113454 - 0.905340i) q^{52} +(-0.661686 - 0.749781i) q^{57} +(0.421476 + 1.25446i) q^{61} +(-0.500000 - 0.866025i) q^{63} +(0.0747301 + 0.997204i) q^{64} +(-0.857396 - 0.312066i) q^{67} +(-0.806265 - 1.78687i) q^{73} +(0.365341 + 0.930874i) q^{75} +(0.995031 - 0.0995678i) q^{76} +(0.305003 - 1.72976i) q^{79} +(-0.853291 - 0.521435i) q^{81} +(0.995031 + 0.0995678i) q^{84} +(0.568885 - 0.713360i) q^{91} +(0.164553 + 1.31310i) q^{93} +(1.26587 + 1.06219i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$36 q - 6 q^{12} + 3 q^{13} + 3 q^{19} + 3 q^{27} + 3 q^{28} + 3 q^{43} + 3 q^{52} - 18 q^{61} - 18 q^{63} + 3 q^{64} + 3 q^{67} - 6 q^{73} + 3 q^{75} - 6 q^{79} + 3 q^{93}+O(q^{100})$ 36 * q - 6 * q^12 + 3 * q^13 + 3 * q^19 + 3 * q^27 + 3 * q^28 + 3 * q^43 + 3 * q^52 - 18 * q^61 - 18 * q^63 + 3 * q^64 + 3 * q^67 - 6 * q^73 + 3 * q^75 - 6 * q^79 + 3 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times$ .

$n$	$932$	$2110$	$2206$
$\chi(n)$	$-1$	$e\left(\frac{19}{21}\right)$	$e\left(\frac{2}{9}\right)$

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		0.698237	−	0.715867i	$-0.253968\pi$
$2$		0			0		−0.698237	+	0.715867i	$0.746032\pi$
$3$	−0.797133	+	0.603804i	−0.797133	+	0.603804i
$3$	−0.797133	+	0.603804i	−0.797133	+	0.603804i
$4$	−0.0249307	−	0.999689i	−0.0249307	−	0.999689i
$5$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$5$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$6$		0			0
$7$	0.698237	−	0.715867i	0.698237	−	0.715867i
$7$	0.698237	−	0.715867i	0.698237	−	0.715867i
$8$		0			0
$9$	0.270840	−	0.962624i	0.270840	−	0.962624i
$10$		0			0
$11$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$11$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$12$	0.623490	+	0.781831i	0.623490	+	0.781831i
$13$	0.907887	−	0.0908478i	0.907887	−	0.0908478i	0.365341	−	0.930874i	$-0.380952\pi$
$13$	0.907887	−	0.0908478i	0.907887	−	0.0908478i	0.542546	+	0.840026i	$0.317460\pi$
$14$		0			0
$15$		0			0
$16$	−0.998757	+	0.0498459i	−0.998757	+	0.0498459i
$17$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$17$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$18$		0			0
$19$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$19$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$20$		0			0
$21$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$22$		0			0
$23$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$23$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$24$		0			0
$25$	0.270840	−	0.962624i	0.270840	−	0.962624i
$26$		0			0
$27$	0.365341	+	0.930874i	0.365341	+	0.930874i
$28$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$29$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$29$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$30$		0			0
$31$	0.661686	−	1.14607i	0.661686	−	1.14607i	−0.318487	−	0.947927i	$-0.603175\pi$
$31$	0.661686	−	1.14607i	0.661686	−	1.14607i	0.980172	−	0.198146i	$-0.0634921\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$37$	0.629859	−	0.0949359i	0.629859	−	0.0949359i	0.173648	−	0.984808i	$-0.444444\pi$
$37$	0.629859	−	0.0949359i	0.629859	−	0.0949359i	0.456211	+	0.889872i	$0.349206\pi$
$38$		0			0
$39$	−0.668852	+	0.620604i	−0.668852	+	0.620604i
$40$		0			0
$41$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$41$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$42$		0			0
$43$	−1.98759	+	0.0991964i	−1.98759	+	0.0991964i	−0.998757	−	0.0498459i	$-0.984127\pi$
$43$	−1.98759	+	0.0991964i	−1.98759	+	0.0991964i	−0.988831	+	0.149042i	$0.952381\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.583744	−	0.811938i	$-0.698413\pi$
$47$		0			0		0.583744	+	0.811938i	$0.301587\pi$
$48$	0.766044	−	0.642788i	0.766044	−	0.642788i
$49$	−0.0249307	−	0.999689i	−0.0249307	−	0.999689i
$50$		0			0
$51$		0			0
$52$	−0.113454	−	0.905340i	−0.113454	−	0.905340i
$53$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$53$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$58$		0			0
$59$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$59$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$60$		0			0
$61$	0.421476	+	1.25446i	0.421476	+	1.25446i	0.921476	+	0.388435i	$0.126984\pi$
$61$	0.421476	+	1.25446i	0.421476	+	1.25446i	−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$64$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$65$		0			0
$66$		0			0
$67$	−0.857396	−	0.312066i	−0.857396	−	0.312066i	−0.124344	−	0.992239i	$-0.539683\pi$
$67$	−0.857396	−	0.312066i	−0.857396	−	0.312066i	−0.733052	+	0.680173i	$0.761905\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.661686	−	0.749781i	$-0.730159\pi$
$71$		0			0		0.661686	+	0.749781i	$0.269841\pi$
$72$		0			0
$73$	−0.806265	−	1.78687i	−0.806265	−	1.78687i	−0.583744	−	0.811938i	$-0.698413\pi$
$73$	−0.806265	−	1.78687i	−0.806265	−	1.78687i	−0.222521	−	0.974928i	$-0.571429\pi$
$74$		0			0
$75$	0.365341	+	0.930874i	0.365341	+	0.930874i
$76$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$77$		0			0
$78$		0			0
$79$	0.305003	−	1.72976i	0.305003	−	1.72976i	−0.318487	−	0.947927i	$-0.603175\pi$
$79$	0.305003	−	1.72976i	0.305003	−	1.72976i	0.623490	−	0.781831i	$-0.285714\pi$
$80$		0			0
$81$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$82$		0			0
$83$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$83$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$84$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$89$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$90$		0			0
$91$	0.568885	−	0.713360i	0.568885	−	0.713360i
$92$		0			0
$93$	0.164553	+	1.31310i	0.164553	+	1.31310i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.26587	+	1.06219i	1.26587	+	1.06219i	0.995031	+	0.0995678i	$0.0317460\pi$
$97$	1.26587	+	1.06219i	1.26587	+	1.06219i	0.270840	+	0.962624i	$0.412698\pi$
$98$		0			0
$99$		0			0
$100$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$101$		0			0		−0.998757	−	0.0498459i	$-0.984127\pi$
$101$		0			0		0.998757	+	0.0498459i	$0.0158730\pi$
$102$		0			0
$103$	1.33443	−	0.411618i	1.33443	−	0.411618i	0.456211	−	0.889872i	$-0.349206\pi$
$103$	1.33443	−	0.411618i	1.33443	−	0.411618i	0.878222	+	0.478254i	$0.158730\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$107$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$108$	0.921476	−	0.388435i	0.921476	−	0.388435i
$109$	1.33443	−	1.36813i	1.33443	−	1.36813i	0.456211	−	0.889872i	$-0.349206\pi$
$109$	1.33443	−	1.36813i	1.33443	−	1.36813i	0.878222	−	0.478254i	$-0.158730\pi$
$110$		0			0
$111$	−0.444758	+	0.455988i	−0.444758	+	0.455988i
$112$	−0.661686	+	0.749781i	−0.661686	+	0.749781i
$113$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$113$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.158440	−	0.898560i	0.158440	−	0.898560i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.623490	−	0.781831i	0.623490	−	0.781831i
$122$		0			0
$123$		0			0
$124$	−1.16221	−	0.632908i	−1.16221	−	0.632908i
$125$		0			0
$126$		0			0
$127$	0.573893	+	1.70811i	0.573893	+	1.70811i	0.698237	+	0.715867i	$0.253968\pi$
$127$	0.573893	+	1.70811i	0.573893	+	1.70811i	−0.124344	+	0.992239i	$0.539683\pi$
$128$		0			0
$129$	1.52448	−	1.27919i	1.52448	−	1.27919i
$130$		0			0
$131$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$131$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$132$		0			0
$133$	0.766044	+	0.642788i	0.766044	+	0.642788i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$137$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$138$		0			0
$139$	0.588713	−	0.911506i	0.588713	−	0.911506i	−0.411287	−	0.911506i	$-0.634921\pi$
$139$	0.588713	−	0.911506i	0.588713	−	0.911506i	1.00000			$0$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$145$		0			0
$146$		0			0
$147$	0.623490	+	0.781831i	0.623490	+	0.781831i
$148$	−0.110609	−	0.627296i	−0.110609	−	0.627296i
$149$		0			0		−0.270840	−	0.962624i	$-0.587302\pi$
$149$		0			0		0.270840	+	0.962624i	$0.412698\pi$
$150$		0			0
$151$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i	0.0747301	−	0.997204i	$-0.476190\pi$
$151$	−0.147791	−	0.0222759i	−0.147791	−	0.0222759i	−0.222521	+	0.974928i	$0.571429\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.637086	+	0.653172i	0.637086	+	0.653172i
$157$	−0.134924	−	0.208904i	−0.134924	−	0.208904i	0.766044	−	0.642788i	$-0.222222\pi$
$157$	−0.134924	−	0.208904i	−0.134924	−	0.208904i	−0.900969	+	0.433884i	$0.857143\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.35098	+	1.25353i	−1.35098	+	1.25353i	−0.411287	+	0.911506i	$0.634921\pi$
$163$	−1.35098	+	1.25353i	−1.35098	+	1.25353i	−0.939693	+	0.342020i	$0.888889\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$167$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$168$		0			0
$169$	−0.164166	+	0.0331870i	−0.164166	+	0.0331870i
$170$		0			0
$171$	0.980172	+	0.198146i	0.980172	+	0.198146i
$172$	0.148717	+	1.98450i	0.148717	+	1.98450i
$173$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$173$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$174$		0			0
$175$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$179$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$180$		0			0
$181$	−1.85205	+	0.471589i	−1.85205	+	0.471589i	−0.998757	−	0.0498459i	$-0.984127\pi$
$181$	−1.85205	+	0.471589i	−1.85205	+	0.471589i	−0.853291	+	0.521435i	$0.825397\pi$
$182$		0			0
$183$	−1.09342	−	0.745482i	−1.09342	−	0.745482i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.921476	+	0.388435i	0.921476	+	0.388435i
$190$		0			0
$191$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$191$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$192$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$193$	−1.46908	+	1.11278i	−1.46908	+	1.11278i	−0.500000	+	0.866025i	$0.666667\pi$
$193$	−1.46908	+	1.11278i	−1.46908	+	1.11278i	−0.969077	+	0.246757i	$0.920635\pi$
$194$		0			0
$195$		0			0
$196$	−0.998757	+	0.0498459i	−0.998757	+	0.0498459i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−0.446285	+	0.690984i	−0.446285	+	0.690984i	−0.988831	−	0.149042i	$-0.952381\pi$
$199$	−0.446285	+	0.690984i	−0.446285	+	0.690984i	0.542546	+	0.840026i	$0.317460\pi$
$200$		0			0
$201$	0.871885	−	0.268941i	0.871885	−	0.268941i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.902230	+	0.135989i	−0.902230	+	0.135989i
$209$		0			0
$210$		0			0
$211$	0.0205073	+	0.0454489i	0.0205073	+	0.0454489i	0.921476	−	0.388435i	$-0.126984\pi$
$211$	0.0205073	+	0.0454489i	0.0205073	+	0.0454489i	−0.900969	+	0.433884i	$0.857143\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.358423	−	1.27391i	−0.358423	−	1.27391i
$218$		0			0
$219$	1.72162	+	0.937543i	1.72162	+	0.937543i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.12922	+	1.27956i	1.12922	+	1.27956i	0.955573	+	0.294755i	$0.0952381\pi$
$223$	1.12922	+	1.27956i	1.12922	+	1.27956i	0.173648	+	0.984808i	$0.444444\pi$
$224$		0			0
$225$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$229$	0.517616	+	0.159663i	0.517616	+	0.159663i	0.542546	−	0.840026i	$-0.317460\pi$
$229$	0.517616	+	0.159663i	0.517616	+	0.159663i	−0.0249307	+	0.999689i	$0.507937\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$233$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.801308	+	1.56301i	0.801308	+	1.56301i
$238$		0			0
$239$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$239$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$240$		0			0
$241$	−0.218403	−	0.118936i	−0.218403	−	0.118936i	0.365341	−	0.930874i	$-0.380952\pi$
$241$	−0.218403	−	0.118936i	−0.218403	−	0.118936i	−0.583744	+	0.811938i	$0.698413\pi$
$242$		0			0
$243$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$244$	1.24356	−	0.452620i	1.24356	−	0.452620i
$245$		0			0
$246$		0			0
$247$	0.158440	+	0.898560i	0.158440	+	0.898560i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$251$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$252$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$257$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$257$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$258$		0			0
$259$	0.371829	−	0.517183i	0.371829	−	0.517183i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.290594	+	0.864909i	−0.290594	+	0.864909i
$269$		0			0		−0.411287	−	0.911506i	$-0.634921\pi$
$269$		0			0		0.411287	+	0.911506i	$0.365079\pi$
$270$		0			0
$271$	−1.43703	+	0.290503i	−1.43703	+	0.290503i	−0.853291	−	0.521435i	$-0.825397\pi$
$271$	−1.43703	+	0.290503i	−1.43703	+	0.290503i	−0.583744	+	0.811938i	$0.698413\pi$
$272$		0			0
$273$	−0.0227473	+	0.912138i	−0.0227473	+	0.912138i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	−0.988831	−	0.149042i	$-0.952381\pi$
$277$	−0.914101	−	1.14625i	−0.914101	−	1.14625i	0.0747301	−	0.997204i	$-0.476190\pi$
$278$		0			0
$279$	−0.924027	−	0.947358i	−0.924027	−	0.947358i
$280$		0			0
$281$		0			0		−0.583744	−	0.811938i	$-0.698413\pi$
$281$		0			0		0.583744	+	0.811938i	$0.301587\pi$
$282$		0			0
$283$	0.337733	+	1.20037i	0.337733	+	1.20037i	0.921476	+	0.388435i	$0.126984\pi$
$283$	0.337733	+	1.20037i	0.337733	+	1.20037i	−0.583744	+	0.811938i	$0.698413\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.542546	−	0.840026i	0.542546	−	0.840026i
$290$		0			0
$291$	−1.65042	−	0.0823692i	−1.65042	−	0.0823692i
$292$	−1.76621	+	0.850562i	−1.76621	+	0.850562i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.921476	−	0.388435i	0.921476	−	0.388435i
$301$	−1.31680	+	1.49211i	−1.31680	+	1.49211i
$302$		0			0
$303$		0			0
$304$	−0.124344	−	0.992239i	−0.124344	−	0.992239i
$305$		0			0
$306$		0			0
$307$	0.104359	−	0.106994i	0.104359	−	0.106994i	−0.661686	−	0.749781i	$-0.730159\pi$
$307$	0.104359	−	0.106994i	0.104359	−	0.106994i	0.766044	+	0.642788i	$0.222222\pi$
$308$		0			0
$309$	−0.815183	+	1.13385i	−0.815183	+	1.13385i
$310$		0			0
$311$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$311$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$312$		0			0
$313$	0.126882	+	0.719581i	0.126882	+	0.719581i	0.980172	+	0.198146i	$0.0634921\pi$
$313$	0.126882	+	0.719581i	0.126882	+	0.719581i	−0.853291	+	0.521435i	$0.825397\pi$
$314$		0			0
$315$		0			0
$316$	−1.73683	−	0.261784i	−1.73683	−	0.261784i
$317$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$317$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$	0.158440	−	0.898560i	0.158440	−	0.898560i
$326$		0			0
$327$	−0.237639	+	1.89631i	−0.237639	+	1.89631i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.426531	+	1.08678i	−0.426531	+	1.08678i	0.542546	+	0.840026i	$0.317460\pi$
$331$	−0.426531	+	1.08678i	−0.426531	+	1.08678i	−0.969077	+	0.246757i	$0.920635\pi$
$332$		0			0
$333$	0.0792036	−	0.632030i	0.0792036	−	0.632030i
$334$		0			0
$335$		0			0
$336$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$337$	1.59228	−	0.0794676i	1.59228	−	0.0794676i	0.766044	−	0.642788i	$-0.222222\pi$
$337$	1.59228	−	0.0794676i	1.59228	−	0.0794676i	0.826239	+	0.563320i	$0.190476\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$347$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$348$		0			0
$349$	1.42077	−	1.31828i	1.42077	−	1.31828i	0.542546	−	0.840026i	$-0.317460\pi$
$349$	1.42077	−	1.31828i	1.42077	−	1.31828i	0.878222	−	0.478254i	$-0.158730\pi$
$350$		0			0
$351$	0.416256	+	0.811938i	0.416256	+	0.811938i
$352$		0			0
$353$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$353$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$359$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$360$		0			0
$361$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$362$		0			0
$363$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$364$	−0.727321	−	0.550924i	−0.727321	−	0.550924i
$365$		0			0
$366$		0			0
$367$	−0.541008	+	1.92286i	−0.541008	+	1.92286i	−0.222521	+	0.974928i	$0.571429\pi$
$367$	−0.541008	+	1.92286i	−0.541008	+	1.92286i	−0.318487	+	0.947927i	$0.603175\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.30859	−	0.197238i	1.30859	−	0.197238i
$373$	1.65248			1.65248			0.826239	−	0.563320i	$-0.190476\pi$
$373$	1.65248			1.65248			0.826239	+	0.563320i	$0.190476\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.79298	+	0.863455i	−1.79298	+	0.863455i	−0.853291	+	0.521435i	$0.825397\pi$
$379$	−1.79298	+	0.863455i	−1.79298	+	0.863455i	−0.939693	+	0.342020i	$0.888889\pi$
$380$		0			0
$381$	−1.48883	−	1.01507i	−1.48883	−	1.01507i
$382$		0			0
$383$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$383$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.442830	+	1.94017i	−0.442830	+	1.94017i
$388$	1.03030	−	1.29196i	1.03030	−	1.29196i
$389$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$389$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.212203	+	1.69334i	0.212203	+	1.69334i	0.623490	+	0.781831i	$0.285714\pi$
$397$	0.212203	+	1.69334i	0.212203	+	1.69334i	−0.411287	+	0.911506i	$0.634921\pi$
$398$		0			0
$399$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i
$400$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$401$		0			0		−0.995031	−	0.0995678i	$-0.968254\pi$
$401$		0			0		0.995031	+	0.0995678i	$0.0317460\pi$
$402$		0			0
$403$	0.496618	−	1.10062i	0.496618	−	1.10062i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.629859	+	1.87468i	0.629859	+	1.87468i	0.456211	+	0.889872i	$0.349206\pi$
$409$	0.629859	+	1.87468i	0.629859	+	1.87468i	0.173648	+	0.984808i	$0.444444\pi$
$410$		0			0
$411$		0			0
$412$	−0.444758	−	1.32376i	−0.444758	−	1.32376i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.0810891	+	1.08206i	0.0810891	+	1.08206i
$418$		0			0
$419$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$419$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$420$		0			0
$421$	0.379750	+	0.232060i	0.379750	+	0.232060i	0.698237	−	0.715867i	$-0.253968\pi$
$421$	0.379750	+	0.232060i	0.379750	+	0.232060i	−0.318487	+	0.947927i	$0.603175\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.19232	+	0.574189i	1.19232	+	0.574189i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$431$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$432$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$433$	−0.541008	−	0.0270006i	−0.541008	−	0.0270006i	−0.222521	−	0.974928i	$-0.571429\pi$
$433$	−0.541008	−	0.0270006i	−0.541008	−	0.0270006i	−0.318487	+	0.947927i	$0.603175\pi$
$434$		0			0
$435$		0			0
$436$	−1.40097	−	1.29991i	−1.40097	−	1.29991i
$437$		0			0
$438$		0			0
$439$	−1.39474	+	1.42995i	−1.39474	+	1.42995i	−0.661686	+	0.749781i	$0.730159\pi$
$439$	−1.39474	+	1.42995i	−1.39474	+	1.42995i	−0.733052	+	0.680173i	$0.761905\pi$
$440$		0			0
$441$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$442$		0			0
$443$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$443$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$444$	0.466934	+	0.433252i	0.466934	+	0.433252i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.766044	+	0.642788i	0.766044	+	0.642788i
$449$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$449$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.131259	−	0.0714799i	0.131259	−	0.0714799i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.87325	+	0.577822i	1.87325	+	0.577822i	0.995031	+	0.0995678i	$0.0317460\pi$
$457$	1.87325	+	0.577822i	1.87325	+	0.577822i	0.878222	+	0.478254i	$0.158730\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$461$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$462$		0			0
$463$	−1.07297	−	0.161725i	−1.07297	−	0.161725i	−0.411287	−	0.911506i	$-0.634921\pi$
$463$	−1.07297	−	0.161725i	−1.07297	−	0.161725i	−0.661686	+	0.749781i	$0.730159\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$467$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$468$	−0.902230	−	0.135989i	−0.902230	−	0.135989i
$469$	−0.822063	+	0.395885i	−0.822063	+	0.395885i
$470$		0			0
$471$	0.233690	+	0.0850561i	0.233690	+	0.0850561i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.980172	+	0.198146i	0.980172	+	0.198146i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$479$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$480$		0			0
$481$	0.563216	−	0.143412i	0.563216	−	0.143412i
$482$		0			0
$483$		0			0
$484$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$485$		0			0
$486$		0			0
$487$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.826239	+	0.563320i	$0.190476\pi$
$487$	1.44973	+	1.34515i	1.44973	+	1.34515i	0.623490	+	0.781831i	$0.285714\pi$
$488$		0			0
$489$	0.320025	−	1.81495i	0.320025	−	1.81495i
$490$		0			0
$491$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$491$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.603736	+	1.17763i	−0.603736	+	1.17763i
$497$		0			0
$498$		0			0
$499$	0.261979	+	0.580605i	0.261979	+	0.580605i	0.995031	−	0.0995678i	$-0.0317460\pi$
$499$	0.261979	+	0.580605i	0.261979	+	0.580605i	−0.733052	+	0.680173i	$0.761905\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$503$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.110824	−	0.125579i	0.110824	−	0.125579i
$508$	1.69327	−	0.616299i	1.69327	−	0.616299i
$509$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$509$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$	−1.84212	−	0.670477i	−1.84212	−	0.670477i
$512$		0			0
$513$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$514$		0			0
$515$		0			0
$516$	−1.31680	−	1.49211i	−1.31680	−	1.49211i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−0.586956	+	0.247423i	−0.586956	+	0.247423i	−0.661686	−	0.749781i	$-0.730159\pi$
$523$	−0.586956	+	0.247423i	−0.586956	+	0.247423i	0.0747301	+	0.997204i	$0.476190\pi$
$524$		0			0
$525$	0.921476	+	0.388435i	0.921476	+	0.388435i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.456211	−	0.889872i	0.456211	−	0.889872i
$530$		0			0
$531$		0			0
$532$	0.623490	−	0.781831i	0.623490	−	0.781831i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.480172	+	0.667879i	0.480172	+	0.667879i	0.980172	−	0.198146i	$-0.0634921\pi$
$541$	0.480172	+	0.667879i	0.480172	+	0.667879i	−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	1.19158	−	1.49419i	1.19158	−	1.49419i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.70446	+	0.0850661i	1.70446	+	0.0850661i	0.878222	−	0.478254i	$-0.158730\pi$
$547$	1.70446	+	0.0850661i	1.70446	+	0.0850661i	0.826239	+	0.563320i	$0.190476\pi$
$548$		0			0
$549$	1.32173	−	0.0659646i	1.32173	−	0.0659646i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.02531	−	1.42612i	−1.02531	−	1.42612i
$554$		0			0
$555$		0			0
$556$	−0.925900	−	0.565805i	−0.925900	−	0.565805i
$557$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$557$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$558$		0			0
$559$	−1.79549	+	0.270627i	−1.79549	+	0.270627i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$563$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.969077	+	0.246757i	−0.969077	+	0.246757i
$568$		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
$571$	0.641701	−	1.63503i	0.641701	−	1.63503i	−0.124344	−	0.992239i	$-0.539683\pi$
$571$	0.641701	−	1.63503i	0.641701	−	1.63503i	0.766044	−	0.642788i	$-0.222222\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.980172	+	0.198146i	0.980172	+	0.198146i
$577$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i	0.955573	+	0.294755i	$0.0952381\pi$
$577$	−0.0332580	+	0.443797i	−0.0332580	+	0.443797i	−0.988831	+	0.149042i	$0.952381\pi$
$578$		0			0
$579$	0.499146	−	1.77407i	0.499146	−	1.77407i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$587$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$588$	0.766044	−	0.642788i	0.766044	−	0.642788i
$589$	1.19232	+	0.574189i	1.19232	+	0.574189i
$590$		0			0
$591$		0			0
$592$	−0.624344	+	0.126214i	−0.624344	+	0.126214i
$593$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$593$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.0614710	−	0.820274i	−0.0614710	−	0.820274i
$598$		0			0
$599$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$599$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$600$		0			0
$601$	−1.55282	−	1.05870i	−1.55282	−	1.05870i	−0.969077	−	0.246757i	$-0.920635\pi$
$601$	−1.55282	−	1.05870i	−1.55282	−	1.05870i	−0.583744	−	0.811938i	$-0.698413\pi$
$602$		0			0
$603$	−0.532620	+	0.740830i	−0.532620	+	0.740830i
$604$	−0.0185844	+	0.148300i	−0.0185844	+	0.148300i
$605$		0			0
$606$		0			0
$607$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$607$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.96783	−	0.196912i	−1.96783	−	0.196912i	−0.969077	−	0.246757i	$-0.920635\pi$
$613$	−1.96783	−	0.196912i	−1.96783	−	0.196912i	−0.998757	+	0.0498459i	$0.984127\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$617$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$618$		0			0
$619$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$619$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.637086	−	0.653172i	0.637086	−	0.653172i
$625$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$626$		0			0
$627$		0			0
$628$	−0.205475	+	0.140091i	−0.205475	+	0.140091i
$629$		0			0
$630$		0			0
$631$	−0.0459461	+	0.0193679i	−0.0459461	+	0.0193679i	−0.411287	−	0.911506i	$-0.634921\pi$
$631$	−0.0459461	+	0.0193679i	−0.0459461	+	0.0193679i	0.365341	+	0.930874i	$0.380952\pi$
$632$		0			0
$633$	−0.0437893	−	0.0238464i	−0.0437893	−	0.0238464i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.113454	−	0.905340i	−0.113454	−	0.905340i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$641$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$642$		0			0
$643$	−0.864963	−	1.33922i	−0.864963	−	1.33922i	−0.939693	−	0.342020i	$-0.888889\pi$
$643$	−0.864963	−	1.33922i	−0.864963	−	1.33922i	0.0747301	−	0.997204i	$-0.476190\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$647$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.05490	+	0.799058i	1.05490	+	0.799058i
$652$	1.28682	+	1.31931i	1.28682	+	1.31931i
$653$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$653$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.93845	+	0.292174i	−1.93845	+	0.292174i
$658$		0			0
$659$		0			0		0.853291	−	0.521435i	$-0.174603\pi$
$659$		0			0		−0.853291	+	0.521435i	$0.825397\pi$
$660$		0			0
$661$	0.969077	−	0.246757i	0.969077	−	0.246757i	0.270840	−	0.962624i	$-0.412698\pi$
$661$	0.969077	−	0.246757i	0.969077	−	0.246757i	0.698237	+	0.715867i	$0.253968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.67274	−	0.338153i	−1.67274	−	0.338153i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.149274	+	1.99193i	−0.149274	+	1.99193i	−0.0249307	+	0.999689i	$0.507937\pi$
$673$	−0.149274	+	1.99193i	−0.149274	+	1.99193i	−0.124344	+	0.992239i	$0.539683\pi$
$674$		0			0
$675$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$676$	0.0372694	+	0.163288i	0.0372694	+	0.163288i
$677$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$677$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$678$		0			0
$679$	1.64427	−	0.164534i	1.64427	−	0.164534i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$683$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$684$	0.173648	−	0.984808i	0.173648	−	0.984808i
$685$		0			0
$686$		0			0
$687$	−0.509014	+	0.185266i	−0.509014	+	0.185266i
$688$	1.98017	−	0.198146i	1.98017	−	0.198146i
$689$		0			0
$690$		0			0
$691$	0.222521	−	0.974928i	0.222521	−	0.974928i	−0.733052	−	0.680173i	$-0.761905\pi$
$691$	0.222521	−	0.974928i	0.222521	−	0.974928i	0.955573	−	0.294755i	$-0.0952381\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$701$		0			0		−0.411287	−	0.911506i	$-0.634921\pi$
$701$		0			0		0.411287	+	0.911506i	$0.365079\pi$
$702$		0			0
$703$	0.141740	+	0.621003i	0.141740	+	0.621003i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.0135045	−	0.541513i	−0.0135045	−	0.541513i	−0.969077	−	0.246757i	$-0.920635\pi$
$709$	−0.0135045	−	0.541513i	−0.0135045	−	0.541513i	0.955573	−	0.294755i	$-0.0952381\pi$
$710$		0			0
$711$	−1.58250	−	0.762092i	−1.58250	−	0.762092i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$719$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$720$		0			0
$721$	0.637086	−	1.24268i	0.637086	−	1.24268i
$722$		0			0
$723$	0.245910	−	0.0370649i	0.245910	−	0.0370649i
$724$	0.517616	+	1.83972i	0.517616	+	1.83972i
$725$		0			0
$726$		0			0
$727$	1.41178	+	0.595117i	1.41178	+	0.595117i	0.955573	−	0.294755i	$-0.0952381\pi$
$727$	1.41178	+	0.595117i	1.41178	+	0.595117i	0.456211	+	0.889872i	$0.349206\pi$
$728$		0			0
$729$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$730$		0			0
$731$		0			0
$732$	−0.717990	+	1.11167i	−0.717990	+	1.11167i
$733$	1.36534	+	0.930874i	1.36534	+	0.930874i	1.00000			$0$
$733$	1.36534	+	0.930874i	1.36534	+	0.930874i	0.365341	+	0.930874i	$0.380952\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.32173	+	1.49770i	1.32173	+	1.49770i	0.698237	+	0.715867i	$0.253968\pi$
$739$	1.32173	+	1.49770i	1.32173	+	1.49770i	0.623490	+	0.781831i	$0.285714\pi$
$740$		0			0
$741$	−0.668852	−	0.620604i	−0.668852	−	0.620604i
$742$		0			0
$743$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$743$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.28682	−	0.542439i	1.28682	−	0.542439i	0.365341	−	0.930874i	$-0.380952\pi$
$751$	1.28682	−	0.542439i	1.28682	−	0.542439i	0.921476	+	0.388435i	$0.126984\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.365341	−	0.930874i	0.365341	−	0.930874i
$757$	0.507752	−	1.51125i	0.507752	−	1.51125i	−0.318487	−	0.947927i	$-0.603175\pi$
$757$	0.507752	−	1.51125i	0.507752	−	1.51125i	0.826239	−	0.563320i	$-0.190476\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$761$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$762$		0			0
$763$	−0.0476462	−	1.91055i	−0.0476462	−	1.91055i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$769$	1.82127	−	0.463752i	1.82127	−	0.463752i	0.826239	−	0.563320i	$-0.190476\pi$
$769$	1.82127	−	0.463752i	1.82127	−	0.463752i	0.995031	+	0.0995678i	$0.0317460\pi$
$770$		0			0
$771$		0			0
$772$	1.14906	+	1.44088i	1.14906	+	1.44088i
$773$		0			0		−0.583744	−	0.811938i	$-0.698413\pi$
$773$		0			0		0.583744	+	0.811938i	$0.301587\pi$
$774$		0			0
$775$	−0.924027	−	0.947358i	−0.924027	−	0.947358i
$776$		0			0
$777$	0.0158802	+	0.636775i	0.0158802	+	0.636775i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$785$		0			0
$786$		0			0
$787$	−0.390845	+	1.71241i	−0.390845	+	1.71241i	0.270840	+	0.962624i	$0.412698\pi$
$787$	−0.390845	+	1.71241i	−0.390845	+	1.71241i	−0.661686	+	0.749781i	$0.730159\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.496618	+	1.10062i	0.496618	+	1.10062i
$794$		0			0
$795$		0			0
$796$	0.701895	+	0.428919i	0.701895	+	0.428919i
$797$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$797$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.290594	−	0.864909i	−0.290594	−	0.864909i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$809$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$810$		0			0
$811$	1.25101	−	0.764478i	1.25101	−	0.764478i	0.270840	−	0.962624i	$-0.412698\pi$
$811$	1.25101	−	0.764478i	1.25101	−	0.764478i	0.980172	+	0.198146i	$0.0634921\pi$
$812$		0			0
$813$	0.970100	−	1.09926i	0.970100	−	1.09926i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.247452	−	1.97462i	−0.247452	−	1.97462i
$818$		0			0
$819$	−0.532620	−	0.740830i	−0.532620	−	0.740830i
$820$		0			0
$821$		0			0		−0.318487	−	0.947927i	$-0.603175\pi$
$821$		0			0		0.318487	+	0.947927i	$0.396825\pi$
$822$		0			0
$823$	0.137724	−	0.0580555i	0.137724	−	0.0580555i	−0.318487	−	0.947927i	$-0.603175\pi$
$823$	0.137724	−	0.0580555i	0.137724	−	0.0580555i	0.456211	+	0.889872i	$0.349206\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$827$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$828$		0			0
$829$	0.641701	+	1.63503i	0.641701	+	1.63503i	0.766044	+	0.642788i	$0.222222\pi$
$829$	0.641701	+	1.63503i	0.641701	+	1.63503i	−0.124344	+	0.992239i	$0.539683\pi$
$830$		0			0
$831$	1.42077	+	0.361772i	1.42077	+	0.361772i
$832$	0.158440	+	0.898560i	0.158440	+	0.898560i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.30859	+	0.197238i	1.30859	+	0.197238i
$838$		0			0
$839$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$839$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$840$		0			0
$841$	0.921476	+	0.388435i	0.921476	+	0.388435i
$842$		0			0
$843$		0			0
$844$	0.0449236	−	0.0216340i	0.0449236	−	0.0216340i
$845$		0			0
$846$		0			0
$847$	−0.124344	−	0.992239i	−0.124344	−	0.992239i
$848$		0			0
$849$	−0.994008	−	0.752932i	−0.994008	−	0.752932i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.0496136	+	1.98944i	−0.0496136	+	1.98944i	0.0747301	+	0.997204i	$0.476190\pi$
$853$	−0.0496136	+	1.98944i	−0.0496136	+	1.98944i	−0.124344	+	0.992239i	$0.539683\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$857$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$858$		0			0
$859$	−1.69810	+	1.03769i	−1.69810	+	1.03769i	−0.797133	+	0.603804i	$0.793651\pi$
$859$	−1.69810	+	1.03769i	−1.69810	+	1.03769i	−0.900969	+	0.433884i	$0.857143\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$868$	−1.26458	+	0.390071i	−1.26458	+	0.390071i
$869$		0			0
$870$		0			0
$871$	−0.806769	−	0.205429i	−0.806769	−	0.205429i
$872$		0			0
$873$	1.36534	−	0.930874i	1.36534	−	0.930874i
$874$		0			0
$875$		0			0
$876$	0.894330	−	1.74446i	0.894330	−	1.74446i
$877$	1.28682	+	0.542439i	1.28682	+	0.542439i	0.921476	−	0.388435i	$-0.126984\pi$
$877$	1.28682	+	0.542439i	1.28682	+	0.542439i	0.365341	+	0.930874i	$0.380952\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	−1.22128	+	1.02477i	−1.22128	+	1.02477i	−0.222521	+	0.974928i	$0.571429\pi$
$883$	−1.22128	+	1.02477i	−1.22128	+	1.02477i	−0.998757	+	0.0498459i	$0.984127\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$887$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$888$		0			0
$889$	1.62349	+	0.781831i	1.62349	+	0.781831i
$890$		0			0
$891$		0			0
$892$	1.25101	−	1.16077i	1.25101	−	1.16077i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	0.148717	−	1.98450i	0.148717	−	1.98450i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.45882	−	0.145977i	−1.45882	−	0.145977i	−0.661686	−	0.749781i	$-0.730159\pi$
$907$	−1.45882	−	0.145977i	−1.45882	−	0.145977i	−0.797133	+	0.603804i	$0.793651\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$911$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$912$	0.698237	+	0.715867i	0.698237	+	0.715867i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.146709	−	0.521435i	0.146709	−	0.521435i
$917$		0			0
$918$		0			0
$919$	−1.24543	−	1.56172i	−1.24543	−	1.56172i	−0.661686	−	0.749781i	$-0.730159\pi$
$919$	−1.24543	−	1.56172i	−1.24543	−	1.56172i	−0.583744	−	0.811938i	$-0.698413\pi$
$920$		0			0
$921$	−0.0185844	+	0.148300i	−0.0185844	+	0.148300i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.0792036	−	0.632030i	0.0792036	−	0.632030i
$926$		0			0
$927$	−0.0348151	−	1.39604i	−0.0348151	−	1.39604i
$928$		0			0
$929$		0			0		0.698237	−	0.715867i	$-0.253968\pi$
$929$		0			0		−0.698237	+	0.715867i	$0.746032\pi$
$930$		0			0
$931$	0.995031	−	0.0995678i	0.995031	−	0.0995678i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.145170	−	1.15843i	0.145170	−	1.15843i	−0.733052	−	0.680173i	$-0.761905\pi$
$937$	0.145170	−	1.15843i	0.145170	−	1.15843i	0.878222	−	0.478254i	$-0.158730\pi$
$938$		0			0
$939$	−0.535628	−	0.496990i	−0.535628	−	0.496990i
$940$		0			0
$941$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$941$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$947$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$948$	1.54255	−	0.840026i	1.54255	−	0.840026i
$949$	−0.894330	−	1.54903i	−0.894330	−	1.54903i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$953$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.375656	−	0.650656i	−0.375656	−	0.650656i
$962$		0			0
$963$		0			0
$964$	−0.113454	+	0.221300i	−0.113454	+	0.221300i
$965$		0			0
$966$		0			0
$967$	−0.524931	−	0.133664i	−0.524931	−	0.133664i	−0.0249307	−	0.999689i	$-0.507937\pi$
$967$	−0.524931	−	0.133664i	−0.524931	−	0.133664i	−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.995031	−	0.0995678i	$-0.968254\pi$
$971$		0			0		0.995031	+	0.0995678i	$0.0317460\pi$
$972$	−0.124344	−	0.992239i	−0.124344	−	0.992239i
$973$	−0.241456	−	1.05789i	−0.241456	−	1.05789i
$974$		0			0
$975$	0.416256	+	0.811938i	0.416256	+	0.811938i
$976$	−0.483482	−	1.23189i	−0.483482	−	1.23189i
$977$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$977$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.955573	−	1.65510i	−0.955573	−	1.65510i
$982$		0			0
$983$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$983$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.894330	−	0.180793i	0.894330	−	0.180793i
$989$		0			0
$990$		0			0
$991$	1.87822	+	0.478254i	1.87822	+	0.478254i	1.00000			$0$
$991$	1.87822	+	0.478254i	1.87822	+	0.478254i	0.878222	+	0.478254i	$0.158730\pi$
$992$		0			0
$993$	−0.316203	−	1.12385i	−0.316203	−	1.12385i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.598559	+	1.78152i	0.598559	+	1.78152i	0.623490	+	0.781831i	$0.285714\pi$
$997$	0.598559	+	1.78152i	0.598559	+	1.78152i	−0.0249307	+	0.999689i	$0.507937\pi$
$998$		0			0
$999$	0.318487	+	0.551635i	0.318487	+	0.551635i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.1.eo.a.1346.1	yes	36
3.2	odd	2	CM	2793.1.eo.a.1346.1	yes	36
19.6	even	9		2793.1.ei.a.1640.1	yes	36
49.32	even	21		2793.1.ei.a.1061.1	✓	36
57.44	odd	18		2793.1.ei.a.1640.1	yes	36
147.32	odd	42		2793.1.ei.a.1061.1	✓	36
931.424	even	63	inner	2793.1.eo.a.1355.1	yes	36
2793.1355	odd	126	inner	2793.1.eo.a.1355.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.1.ei.a.1061.1	✓	36	49.32	even	21
2793.1.ei.a.1061.1	✓	36	147.32	odd	42
2793.1.ei.a.1640.1	yes	36	19.6	even	9
2793.1.ei.a.1640.1	yes	36	57.44	odd	18
2793.1.eo.a.1346.1	yes	36	1.1	even	1	trivial
2793.1.eo.a.1346.1	yes	36	3.2	odd	2	CM
2793.1.eo.a.1355.1	yes	36	931.424	even	63	inner
2793.1.eo.a.1355.1	yes	36	2793.1355	odd	126	inner

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2793.1.eo.a.1346.1

Level	$2793$
Weight	$1$
Character	2793.1346
Analytic conductor	$1.394$
Analytic rank	$0$
Dimension	$36$
Projective image	$D_{63}$
CM discriminant	-3
Inner twists	$4$