LMFDB - Embedded newform 2793.1.ew.a.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2793,1,Mod(143,2793)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2793, base_ring=CyclotomicField(126))

chi = DirichletCharacter(H, H._module([63, 93, 119]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2793.143");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2793.ew (of order $126$, degree $36$, 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:	$1.39388858028$
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:	$D_{126}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{126} - \cdots)$

Embedding invariants

Embedding label			143.1
Root			$-0.661686 + 0.749781i$ of defining polynomial
Character	$\chi$	$=$	2793.143
Dual form			2793.1.ew.a.2168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.998757 - 0.0498459i) q^{3} +(0.661686 + 0.749781i) q^{4} +(-0.411287 + 0.911506i) q^{7} +(0.995031 + 0.0995678i) q^{9} +O(q^{10})$	\(q+(-0.998757 - 0.0498459i) q^{3} +(0.661686 + 0.749781i) q^{4} +(-0.411287 + 0.911506i) q^{7} +(0.995031 + 0.0995678i) q^{9} +(-0.623490 - 0.781831i) q^{12} +(1.78596 + 0.454762i) q^{13} +(-0.124344 + 0.992239i) q^{16} +(0.826239 - 0.563320i) q^{19} +(0.456211 - 0.889872i) q^{21} +(-0.995031 - 0.0995678i) q^{25} +(-0.988831 - 0.149042i) q^{27} +(-0.955573 + 0.294755i) q^{28} +(-0.853291 - 1.47794i) q^{31} +(0.583744 + 0.811938i) q^{36} +(1.86117 + 0.730455i) q^{37} +(-1.76108 - 0.543220i) q^{39} +(-0.240997 + 1.92311i) q^{43} +(0.173648 - 0.984808i) q^{48} +(-0.661686 - 0.749781i) q^{49} +(0.840775 + 1.63999i) q^{52} +(-0.853291 + 0.521435i) q^{57} +(-1.04255 - 0.0259995i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.826239 + 0.563320i) q^{64} +(0.499362 - 0.595117i) q^{67} +(0.920758 + 0.259061i) q^{73} +(0.988831 + 0.149042i) q^{75} +(0.969077 + 0.246757i) q^{76} +(-0.648420 + 1.78152i) q^{79} +(0.980172 + 0.198146i) q^{81} +(0.969077 - 0.246757i) q^{84} +(-1.14906 + 1.44088i) q^{91} +(0.778561 + 1.51864i) q^{93} +(0.0259535 + 0.147190i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q+O(q^{10}) $ 36 * 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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$932$	$2110$	$2206$
$\chi(n)$	$-1$	$e\left(\frac{31}{42}\right)$	$e\left(\frac{17}{18}\right)$

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		−0.911506	−	0.411287i	$-0.865079\pi$
$2$		0			0		0.911506	+	0.411287i	$0.134921\pi$
$3$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i
$3$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i
$4$	0.661686	+	0.749781i	0.661686	+	0.749781i
$5$		0			0		0.0498459	−	0.998757i	$-0.484127\pi$
$5$		0			0		−0.0498459	+	0.998757i	$0.515873\pi$
$6$		0			0
$7$	−0.411287	+	0.911506i	−0.411287	+	0.911506i
$7$	−0.411287	+	0.911506i	−0.411287	+	0.911506i
$8$		0			0
$9$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$10$		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
$13$	1.78596	+	0.454762i	1.78596	+	0.454762i	0.988831	−	0.149042i	$-0.0476190\pi$
$13$	1.78596	+	0.454762i	1.78596	+	0.454762i	0.797133	+	0.603804i	$0.206349\pi$
$14$		0			0
$15$		0			0
$16$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$17$		0			0		0.947927	−	0.318487i	$-0.103175\pi$
$17$		0			0		−0.947927	+	0.318487i	$0.896825\pi$
$18$		0			0
$19$	0.826239	−	0.563320i	0.826239	−	0.563320i
$19$	0.826239	−	0.563320i	0.826239	−	0.563320i
$20$		0			0
$21$	0.456211	−	0.889872i	0.456211	−	0.889872i
$22$		0			0
$23$		0			0		0.980172	−	0.198146i	$-0.0634921\pi$
$23$		0			0		−0.980172	+	0.198146i	$0.936508\pi$
$24$		0			0
$25$	−0.995031	−	0.0995678i	−0.995031	−	0.0995678i
$26$		0			0
$27$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$28$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$29$		0			0		−0.478254	−	0.878222i	$-0.658730\pi$
$29$		0			0		0.478254	+	0.878222i	$0.341270\pi$
$30$		0			0
$31$	−0.853291	−	1.47794i	−0.853291	−	1.47794i	−0.878222	−	0.478254i	$-0.841270\pi$
$31$	−0.853291	−	1.47794i	−0.853291	−	1.47794i	0.0249307	−	0.999689i	$-0.492063\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.583744	+	0.811938i	0.583744	+	0.811938i
$37$	1.86117	+	0.730455i	1.86117	+	0.730455i	0.939693	+	0.342020i	$0.111111\pi$
$37$	1.86117	+	0.730455i	1.86117	+	0.730455i	0.921476	+	0.388435i	$0.126984\pi$
$38$		0			0
$39$	−1.76108	−	0.543220i	−1.76108	−	0.543220i
$40$		0			0
$41$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$41$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$42$		0			0
$43$	−0.240997	+	1.92311i	−0.240997	+	1.92311i	0.124344	+	0.992239i	$0.460317\pi$
$43$	−0.240997	+	1.92311i	−0.240997	+	1.92311i	−0.365341	+	0.930874i	$0.619048\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.715867	−	0.698237i	$-0.753968\pi$
$47$		0			0		0.715867	+	0.698237i	$0.246032\pi$
$48$	0.173648	−	0.984808i	0.173648	−	0.984808i
$49$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$50$		0			0
$51$		0			0
$52$	0.840775	+	1.63999i	0.840775	+	1.63999i
$53$		0			0		0.749781	−	0.661686i	$-0.230159\pi$
$53$		0			0		−0.749781	+	0.661686i	$0.769841\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$58$		0			0
$59$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$59$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$60$		0			0
$61$	−1.04255	−	0.0259995i	−1.04255	−	0.0259995i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.04255	−	0.0259995i	−1.04255	−	0.0259995i	−0.542546	+	0.840026i	$0.682540\pi$
$62$		0			0
$63$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$64$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$65$		0			0
$66$		0			0
$67$	0.499362	−	0.595117i	0.499362	−	0.595117i	−0.456211	−	0.889872i	$-0.650794\pi$
$67$	0.499362	−	0.595117i	0.499362	−	0.595117i	0.955573	+	0.294755i	$0.0952381\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.521435	−	0.853291i	$-0.674603\pi$
$71$		0			0		0.521435	+	0.853291i	$0.325397\pi$
$72$		0			0
$73$	0.920758	+	0.259061i	0.920758	+	0.259061i	0.698237	−	0.715867i	$-0.253968\pi$
$73$	0.920758	+	0.259061i	0.920758	+	0.259061i	0.222521	+	0.974928i	$0.428571\pi$
$74$		0			0
$75$	0.988831	+	0.149042i	0.988831	+	0.149042i
$76$	0.969077	+	0.246757i	0.969077	+	0.246757i
$77$		0			0
$78$		0			0
$79$	−0.648420	+	1.78152i	−0.648420	+	1.78152i	−0.0249307	+	0.999689i	$0.507937\pi$
$79$	−0.648420	+	1.78152i	−0.648420	+	1.78152i	−0.623490	+	0.781831i	$0.714286\pi$
$80$		0			0
$81$	0.980172	+	0.198146i	0.980172	+	0.198146i
$82$		0			0
$83$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$83$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$84$	0.969077	−	0.246757i	0.969077	−	0.246757i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$89$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$90$		0			0
$91$	−1.14906	+	1.44088i	−1.14906	+	1.44088i
$92$		0			0
$93$	0.778561	+	1.51864i	0.778561	+	1.51864i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.0259535	+	0.147190i	0.0259535	+	0.147190i	0.995031	−	0.0995678i	$-0.0317460\pi$
$97$	0.0259535	+	0.147190i	0.0259535	+	0.147190i	−0.969077	+	0.246757i	$0.920635\pi$
$98$		0			0
$99$		0			0
$100$	−0.583744	−	0.811938i	−0.583744	−	0.811938i
$101$		0			0		0.992239	−	0.124344i	$-0.0396825\pi$
$101$		0			0		−0.992239	+	0.124344i	$0.960317\pi$
$102$		0			0
$103$	0.602990	+	0.559493i	0.602990	+	0.559493i	0.921476	−	0.388435i	$-0.126984\pi$
$103$	0.602990	+	0.559493i	0.602990	+	0.559493i	−0.318487	+	0.947927i	$0.603175\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$107$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$108$	−0.542546	−	0.840026i	−0.542546	−	0.840026i
$109$	−1.23996	−	0.559493i	−1.23996	−	0.559493i	−0.318487	−	0.947927i	$-0.603175\pi$
$109$	−1.23996	−	0.559493i	−1.23996	−	0.559493i	−0.921476	+	0.388435i	$0.873016\pi$
$110$		0			0
$111$	−1.82245	−	0.822319i	−1.82245	−	0.822319i
$112$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$113$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$113$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.73181	+	0.630327i	1.73181	+	0.630327i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.623490	−	0.781831i	0.623490	−	0.781831i
$122$		0			0
$123$		0			0
$124$	0.543524	−	1.61772i	0.543524	−	1.61772i
$125$		0			0
$126$		0			0
$127$	−0.867498	−	0.0216340i	−0.867498	−	0.0216340i	−0.411287	−	0.911506i	$-0.634921\pi$
$127$	−0.867498	−	0.0216340i	−0.867498	−	0.0216340i	−0.456211	+	0.889872i	$0.650794\pi$
$128$		0			0
$129$	0.336557	−	1.90871i	0.336557	−	1.90871i
$130$		0			0
$131$		0			0		0.603804	−	0.797133i	$-0.293651\pi$
$131$		0			0		−0.603804	+	0.797133i	$0.706349\pi$
$132$		0			0
$133$	0.173648	+	0.984808i	0.173648	+	0.984808i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$137$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$138$		0			0
$139$	−0.729160	−	0.962624i	−0.729160	−	0.962624i	0.270840	−	0.962624i	$-0.412698\pi$
$139$	−0.729160	−	0.962624i	−0.729160	−	0.962624i	−1.00000			$\pi$
$140$		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
$148$	0.683828	+	1.87880i	0.683828	+	1.87880i
$149$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$149$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$150$		0			0
$151$	1.04876	−	0.411608i	1.04876	−	0.411608i	0.222521	−	0.974928i	$-0.428571\pi$
$151$	1.04876	−	0.411608i	1.04876	−	0.411608i	0.826239	+	0.563320i	$0.190476\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.757983	−	1.67986i	−0.757983	−	1.67986i
$157$	−1.07462	+	1.41869i	−1.07462	+	1.41869i	−0.173648	+	0.984808i	$0.555556\pi$
$157$	−1.07462	+	1.41869i	−1.07462	+	1.41869i	−0.900969	+	0.433884i	$0.857143\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.03688	−	0.319837i	−1.03688	−	0.319837i	−0.270840	−	0.962624i	$-0.587302\pi$
$163$	−1.03688	−	0.319837i	−1.03688	−	0.319837i	−0.766044	+	0.642788i	$0.777778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$167$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$168$		0			0
$169$	2.10463	+	1.14612i	2.10463	+	1.14612i
$170$		0			0
$171$	0.878222	−	0.478254i	0.878222	−	0.478254i
$172$	−1.60138	+	1.09180i	−1.60138	+	1.09180i
$173$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$173$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$174$		0			0
$175$	0.500000	−	0.866025i	0.500000	−	0.866025i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$179$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$180$		0			0
$181$	0.855829	−	1.19039i	0.855829	−	1.19039i	−0.124344	−	0.992239i	$-0.539683\pi$
$181$	0.855829	−	1.19039i	0.855829	−	1.19039i	0.980172	−	0.198146i	$-0.0634921\pi$
$182$		0			0
$183$	1.03995	+	0.0779338i	1.03995	+	0.0779338i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.542546	−	0.840026i	0.542546	−	0.840026i
$190$		0			0
$191$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$191$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$192$	0.853291	−	0.521435i	0.853291	−	0.521435i
$193$	−0.0837437	+	1.67796i	−0.0837437	+	1.67796i	0.500000	+	0.866025i	$0.333333\pi$
$193$	−0.0837437	+	1.67796i	−0.0837437	+	1.67796i	−0.583744	+	0.811938i	$0.698413\pi$
$194$		0			0
$195$		0			0
$196$	0.124344	−	0.992239i	0.124344	−	0.992239i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	−1.16247	−	1.53468i	−1.16247	−	1.53468i	−0.797133	−	0.603804i	$-0.793651\pi$
$199$	−1.16247	−	1.53468i	−1.16247	−	1.53468i	−0.365341	−	0.930874i	$-0.619048\pi$
$200$		0			0
$201$	−0.528406	+	0.569486i	−0.528406	+	0.569486i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.673306	+	1.71556i	−0.673306	+	1.71556i
$209$		0			0
$210$		0			0
$211$	1.44352	+	0.406142i	1.44352	+	0.406142i	0.900969	−	0.433884i	$-0.142857\pi$
$211$	1.44352	+	0.406142i	1.44352	+	0.406142i	0.542546	+	0.840026i	$0.317460\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.69810	−	0.169921i	1.69810	−	0.169921i
$218$		0			0
$219$	−0.906700	−	0.304635i	−0.906700	−	0.304635i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.67274	−	1.02219i	1.67274	−	1.02219i	0.733052	−	0.680173i	$-0.238095\pi$
$223$	1.67274	−	1.02219i	1.67274	−	1.02219i	0.939693	−	0.342020i	$-0.111111\pi$
$224$		0			0
$225$	−0.980172	−	0.198146i	−0.980172	−	0.198146i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$229$	−0.135447	−	0.145977i	−0.135447	−	0.145977i	0.661686	−	0.749781i	$-0.269841\pi$
$229$	−0.135447	−	0.145977i	−0.135447	−	0.145977i	−0.797133	+	0.603804i	$0.793651\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$233$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.736416	−	1.74699i	0.736416	−	1.74699i
$238$		0			0
$239$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$239$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$240$		0			0
$241$	−0.290594	+	0.864909i	−0.290594	+	0.864909i	0.698237	+	0.715867i	$0.253968\pi$
$241$	−0.290594	+	0.864909i	−0.290594	+	0.864909i	−0.988831	+	0.149042i	$0.952381\pi$
$242$		0			0
$243$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$244$	−0.670344	−	0.798885i	−0.670344	−	0.798885i
$245$		0			0
$246$		0			0
$247$	1.73181	−	0.630327i	1.73181	−	0.630327i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.840026	−	0.542546i	$-0.182540\pi$
$251$		0			0		−0.840026	+	0.542546i	$0.817460\pi$
$252$	−0.980172	+	0.198146i	−0.980172	+	0.198146i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$257$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$257$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$258$		0			0
$259$	−1.43129	+	1.39604i	−1.43129	+	1.39604i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$263$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.776628	−	0.0193679i	0.776628	−	0.0193679i
$269$		0			0		0.270840	−	0.962624i	$-0.412698\pi$
$269$		0			0		−0.270840	+	0.962624i	$0.587302\pi$
$270$		0			0
$271$	−0.281936	+	0.517721i	−0.281936	+	0.517721i	−0.980172	−	0.198146i	$-0.936508\pi$
$271$	−0.281936	+	0.517721i	−0.281936	+	0.517721i	0.698237	+	0.715867i	$0.253968\pi$
$272$		0			0
$273$	1.21946	−	1.38181i	1.21946	−	1.38181i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.19158	−	1.49419i	−1.19158	−	1.49419i	−0.826239	−	0.563320i	$-0.809524\pi$
$277$	−1.19158	−	1.49419i	−1.19158	−	1.49419i	−0.365341	−	0.930874i	$-0.619048\pi$
$278$		0			0
$279$	−0.701895	−	1.55556i	−0.701895	−	1.55556i
$280$		0			0
$281$		0			0		−0.715867	−	0.698237i	$-0.753968\pi$
$281$		0			0		0.715867	+	0.698237i	$0.246032\pi$
$282$		0			0
$283$	−0.155691	−	1.55589i	−0.155691	−	1.55589i	−0.698237	−	0.715867i	$-0.746032\pi$
$283$	−0.155691	−	1.55589i	−0.155691	−	1.55589i	0.542546	−	0.840026i	$-0.317460\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.797133	−	0.603804i	0.797133	−	0.603804i
$290$		0			0
$291$	−0.0185844	−	0.148300i	−0.0185844	−	0.148300i
$292$	0.415013	+	0.861784i	0.415013	+	0.861784i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.542546	+	0.840026i	0.542546	+	0.840026i
$301$	−1.65381	−	1.01062i	−1.65381	−	1.01062i
$302$		0			0
$303$		0			0
$304$	0.456211	+	0.889872i	0.456211	+	0.889872i
$305$		0			0
$306$		0			0
$307$	0.679643	−	1.50624i	0.679643	−	1.50624i	−0.173648	−	0.984808i	$-0.555556\pi$
$307$	0.679643	−	1.50624i	0.679643	−	1.50624i	0.853291	−	0.521435i	$-0.174603\pi$
$308$		0			0
$309$	−0.574352	−	0.588854i	−0.574352	−	0.588854i
$310$		0			0
$311$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$311$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$312$		0			0
$313$	−0.101951	−	0.280108i	−0.101951	−	0.280108i	0.878222	−	0.478254i	$-0.158730\pi$
$313$	−0.101951	−	0.280108i	−0.101951	−	0.280108i	−0.980172	+	0.198146i	$0.936508\pi$
$314$		0			0
$315$		0			0
$316$	−1.76480	+	0.692633i	−1.76480	+	0.692633i
$317$		0			0		0.749781	−	0.661686i	$-0.230159\pi$
$317$		0			0		−0.749781	+	0.661686i	$0.769841\pi$
$318$		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
$325$	−1.73181	−	0.630327i	−1.73181	−	0.630327i
$326$		0			0
$327$	1.21053	+	0.620604i	1.21053	+	0.620604i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.213389	−	1.41574i	−0.213389	−	1.41574i	−0.797133	−	0.603804i	$-0.793651\pi$
$331$	−0.213389	−	1.41574i	−0.213389	−	1.41574i	0.583744	−	0.811938i	$-0.301587\pi$
$332$		0			0
$333$	1.77919	+	0.912138i	1.77919	+	0.912138i
$334$		0			0
$335$		0			0
$336$	0.826239	+	0.563320i	0.826239	+	0.563320i
$337$	−0.0989181	−	0.0123960i	−0.0989181	−	0.0123960i	0.0747301	−	0.997204i	$-0.476190\pi$
$337$	−0.0989181	−	0.0123960i	−0.0989181	−	0.0123960i	−0.173648	+	0.984808i	$0.555556\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.955573	−	0.294755i	0.955573	−	0.294755i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$347$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$348$		0			0
$349$	−0.478646	+	1.55173i	−0.478646	+	1.55173i	0.318487	+	0.947927i	$0.396825\pi$
$349$	−0.478646	+	1.55173i	−0.478646	+	1.55173i	−0.797133	+	0.603804i	$0.793651\pi$
$350$		0			0
$351$	−1.69824	−	0.715867i	−1.69824	−	0.715867i
$352$		0			0
$353$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$353$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$359$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$360$		0			0
$361$	0.365341	−	0.930874i	0.365341	−	0.930874i
$362$		0			0
$363$	−0.661686	+	0.749781i	−0.661686	+	0.749781i
$364$	−1.84066	+	0.0918636i	−1.84066	+	0.0918636i
$365$		0			0
$366$		0			0
$367$	−0.197590	+	1.97462i	−0.197590	+	1.97462i	0.0249307	+	0.999689i	$0.492063\pi$
$367$	−0.197590	+	1.97462i	−0.197590	+	1.97462i	−0.222521	+	0.974928i	$0.571429\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.623484	+	1.58861i	−0.623484	+	1.58861i
$373$		−	1.99441i		−	1.99441i	−0.0747301	−	0.997204i	$-0.523810\pi$
$373$		−	1.99441i		−	1.99441i	0.0747301	−	0.997204i	$-0.476190\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.214128	−	0.444641i	−0.214128	−	0.444641i	0.766044	−	0.642788i	$-0.222222\pi$
$379$	−0.214128	−	0.444641i	−0.214128	−	0.444641i	−0.980172	+	0.198146i	$0.936508\pi$
$380$		0			0
$381$	0.865341	+	0.0648483i	0.865341	+	0.0648483i
$382$		0			0
$383$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$383$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.431280	+	1.88956i	−0.431280	+	1.88956i
$388$	−0.0931869	+	0.116853i	−0.0931869	+	0.116853i
$389$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$389$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.352649	−	0.180793i	0.352649	−	0.180793i	−0.270840	−	0.962624i	$-0.587302\pi$
$397$	0.352649	−	0.180793i	0.352649	−	0.180793i	0.623490	+	0.781831i	$0.285714\pi$
$398$		0			0
$399$	−0.124344	−	0.992239i	−0.124344	−	0.992239i
$400$	0.222521	−	0.974928i	0.222521	−	0.974928i
$401$		0			0		−0.246757	−	0.969077i	$-0.579365\pi$
$401$		0			0		0.246757	+	0.969077i	$0.420635\pi$
$402$		0			0
$403$	−0.851834	−	3.02760i	−0.851834	−	3.02760i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.0182164	−	0.730455i	0.0182164	−	0.730455i	−0.921476	−	0.388435i	$-0.873016\pi$
$409$	0.0182164	−	0.730455i	0.0182164	−	0.730455i	0.939693	−	0.342020i	$-0.111111\pi$
$410$		0			0
$411$		0			0
$412$	−0.0205073	+	0.822319i	−0.0205073	+	0.822319i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.680270	+	0.997773i	0.680270	+	0.997773i
$418$		0			0
$419$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$419$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$420$		0			0
$421$	0.386356	−	1.91120i	0.386356	−	1.91120i	−0.0249307	−	0.999689i	$-0.507937\pi$
$421$	0.386356	−	1.91120i	0.386356	−	1.91120i	0.411287	−	0.911506i	$-0.365079\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.452485	−	0.939594i	0.452485	−	0.939594i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.840026	−	0.542546i	$-0.182540\pi$
$431$		0			0		−0.840026	+	0.542546i	$0.817460\pi$
$432$	0.270840	−	0.962624i	0.270840	−	0.962624i
$433$	−0.247452	−	1.97462i	−0.247452	−	1.97462i	−0.222521	−	0.974928i	$-0.571429\pi$
$433$	−0.247452	−	1.97462i	−0.247452	−	1.97462i	−0.0249307	−	0.999689i	$-0.507937\pi$
$434$		0			0
$435$		0			0
$436$	−0.400969	−	1.29991i	−0.400969	−	1.29991i
$437$		0			0
$438$		0			0
$439$	−0.102282	+	0.226680i	−0.102282	+	0.226680i	−0.955573	−	0.294755i	$-0.904762\pi$
$439$	−0.102282	+	0.226680i	−0.102282	+	0.226680i	0.853291	+	0.521435i	$0.174603\pi$
$440$		0			0
$441$	−0.583744	−	0.811938i	−0.583744	−	0.811938i
$442$		0			0
$443$		0			0		−0.270840	−	0.962624i	$-0.587302\pi$
$443$		0			0		0.270840	+	0.962624i	$0.412698\pi$
$444$	−0.589327	−	1.91055i	−0.589327	−	1.91055i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$449$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$449$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.06797	+	0.358820i	−1.06797	+	0.358820i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.28756	−	1.19468i	1.28756	−	1.19468i	0.318487	−	0.947927i	$-0.396825\pi$
$457$	1.28756	−	1.19468i	1.28756	−	1.19468i	0.969077	−	0.246757i	$-0.0793651\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.947927	−	0.318487i	$-0.103175\pi$
$461$		0			0		−0.947927	+	0.318487i	$0.896825\pi$
$462$		0			0
$463$	−0.582450	−	1.48406i	−0.582450	−	1.48406i	−0.853291	−	0.521435i	$-0.825397\pi$
$463$	−0.582450	−	1.48406i	−0.582450	−	1.48406i	0.270840	−	0.962624i	$-0.412698\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$467$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$468$	0.673306	+	1.71556i	0.673306	+	1.71556i
$469$	0.337071	+	0.699935i	0.337071	+	0.699935i
$470$		0			0
$471$	1.14400	−	1.36336i	1.14400	−	1.36336i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.878222	+	0.478254i	−0.878222	+	0.478254i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.962624	−	0.270840i	$-0.0873016\pi$
$479$		0			0		−0.962624	+	0.270840i	$0.912698\pi$
$480$		0			0
$481$	2.99180	+	2.15095i	2.99180	+	2.15095i
$482$		0			0
$483$		0			0
$484$	0.998757	−	0.0498459i	0.998757	−	0.0498459i
$485$		0			0
$486$		0			0
$487$	0.548760	+	1.77904i	0.548760	+	1.77904i	0.623490	+	0.781831i	$0.285714\pi$
$487$	0.548760	+	1.77904i	0.548760	+	1.77904i	−0.0747301	+	0.997204i	$0.523810\pi$
$488$		0			0
$489$	1.01965	+	0.371124i	1.01965	+	0.371124i
$490$		0			0
$491$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$491$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.57257	−	0.662896i	1.57257	−	0.662896i
$497$		0			0
$498$		0			0
$499$	−0.0135045	+	0.0479978i	−0.0135045	+	0.0479978i	−0.969077	−	0.246757i	$-0.920635\pi$
$499$	−0.0135045	+	0.0479978i	−0.0135045	+	0.0479978i	0.955573	+	0.294755i	$0.0952381\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.962624	−	0.270840i	$-0.0873016\pi$
$503$		0			0		−0.962624	+	0.270840i	$0.912698\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.04489	−	1.24961i	−2.04489	−	1.24961i
$508$	−0.557790	−	0.664748i	−0.557790	−	0.664748i
$509$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$509$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$510$		0			0
$511$	−0.614831	+	0.732728i	−0.614831	+	0.732728i
$512$		0			0
$513$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$514$		0			0
$515$		0			0
$516$	1.65381	−	1.01062i	1.65381	−	1.01062i
$517$		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
$523$	0.0270521	+	0.0418849i	0.0270521	+	0.0418849i	0.853291	−	0.521435i	$-0.174603\pi$
$523$	0.0270521	+	0.0418849i	0.0270521	+	0.0418849i	−0.826239	+	0.563320i	$0.809524\pi$
$524$		0			0
$525$	−0.542546	+	0.840026i	−0.542546	+	0.840026i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.921476	−	0.388435i	0.921476	−	0.388435i
$530$		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
$541$	0.378222	−	0.387771i	0.378222	−	0.387771i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	0.378222	−	0.387771i	0.378222	−	0.387771i	0.878222	+	0.478254i	$0.158730\pi$
$542$		0			0
$543$	−0.914101	+	1.14625i	−0.914101	+	1.14625i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.393217	+	0.0492765i	−0.393217	+	0.0492765i	−0.318487	−	0.947927i	$-0.603175\pi$
$547$	−0.393217	+	0.0492765i	−0.393217	+	0.0492765i	−0.0747301	+	0.997204i	$0.523810\pi$
$548$		0			0
$549$	−1.03478	−	0.129674i	−1.03478	−	0.129674i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.35718	−	1.32376i	−1.35718	−	1.32376i
$554$		0			0
$555$		0			0
$556$	0.239283	−	1.18366i	0.239283	−	1.18366i
$557$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$557$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$558$		0			0
$559$	−1.30497	+	3.32501i	−1.30497	+	3.32501i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$563$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	0.629859	−	0.0949359i	0.629859	−	0.0949359i	0.173648	−	0.984808i	$-0.444444\pi$
$571$	0.629859	−	0.0949359i	0.629859	−	0.0949359i	0.456211	+	0.889872i	$0.349206\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.878222	+	0.478254i	−0.878222	+	0.478254i
$577$	−1.09839	+	1.61105i	−1.09839	+	1.61105i	−0.365341	+	0.930874i	$0.619048\pi$
$577$	−1.09839	+	1.61105i	−1.09839	+	1.61105i	−0.733052	+	0.680173i	$0.761905\pi$
$578$		0			0
$579$	0.167279	−	1.67170i	0.167279	−	1.67170i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$587$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$588$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$589$	−1.53758	−	0.740458i	−1.53758	−	0.740458i
$590$		0			0
$591$		0			0
$592$	−0.956211	+	1.75590i	−0.956211	+	1.75590i
$593$		0			0		−0.603804	−	0.797133i	$-0.706349\pi$
$593$		0			0		0.603804	+	0.797133i	$0.293651\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.08453	+	1.59071i	1.08453	+	1.59071i
$598$		0			0
$599$		0			0		0.246757	−	0.969077i	$-0.420635\pi$
$599$		0			0		−0.246757	+	0.969077i	$0.579365\pi$
$600$		0			0
$601$	0.114493	−	1.52780i	0.114493	−	1.52780i	−0.583744	−	0.811938i	$-0.698413\pi$
$601$	0.114493	−	1.52780i	0.114493	−	1.52780i	0.698237	−	0.715867i	$-0.253968\pi$
$602$		0			0
$603$	0.556135	−	0.542439i	0.556135	−	0.542439i
$604$	1.00257	+	0.513985i	1.00257	+	0.513985i
$605$		0			0
$606$		0			0
$607$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$607$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.708087	−	0.180301i	0.708087	−	0.180301i	0.124344	−	0.992239i	$-0.460317\pi$
$613$	0.708087	−	0.180301i	0.708087	−	0.180301i	0.583744	+	0.811938i	$0.301587\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$617$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$618$		0			0
$619$			0.684040i			0.684040i	0.939693	+	0.342020i	$0.111111\pi$
$619$			0.684040i			0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.757983	−	1.67986i	0.757983	−	1.67986i
$625$	0.980172	+	0.198146i	0.980172	+	0.198146i
$626$		0			0
$627$		0			0
$628$	−1.77477	+	0.133000i	−1.77477	+	0.133000i
$629$		0			0
$630$		0			0
$631$	0.717990	+	1.11167i	0.717990	+	1.11167i	0.988831	+	0.149042i	$0.0476190\pi$
$631$	0.717990	+	1.11167i	0.717990	+	1.11167i	−0.270840	+	0.962624i	$0.587302\pi$
$632$		0			0
$633$	−1.42148	−	0.477591i	−1.42148	−	0.477591i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.840775	−	1.63999i	−0.840775	−	1.63999i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.999689	−	0.0249307i	$-0.00793651\pi$
$641$		0			0		−0.999689	+	0.0249307i	$0.992063\pi$
$642$		0			0
$643$	0.0601943	−	0.0794676i	0.0601943	−	0.0794676i	−0.766044	−	0.642788i	$-0.777778\pi$
$643$	0.0601943	−	0.0794676i	0.0601943	−	0.0794676i	0.826239	+	0.563320i	$0.190476\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$647$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.70446	+	0.0850661i	−1.70446	+	0.0850661i
$652$	−0.446285	−	0.989068i	−0.446285	−	0.989068i
$653$		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
$657$	0.890388	+	0.349452i	0.890388	+	0.349452i
$658$		0			0
$659$		0			0		−0.198146	−	0.980172i	$-0.563492\pi$
$659$		0			0		0.198146	+	0.980172i	$0.436508\pi$
$660$		0			0
$661$	−0.583744	+	0.811938i	−0.583744	+	0.811938i	−0.995031	−	0.0995678i	$-0.968254\pi$
$661$	−0.583744	+	0.811938i	−0.583744	+	0.811938i	0.411287	+	0.911506i	$0.365079\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.72162	+	0.937543i	−1.72162	+	0.937543i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.11790	−	1.63965i	1.11790	−	1.63965i	0.456211	−	0.889872i	$-0.349206\pi$
$673$	1.11790	−	1.63965i	1.11790	−	1.63965i	0.661686	−	0.749781i	$-0.269841\pi$
$674$		0			0
$675$	0.969077	+	0.246757i	0.969077	+	0.246757i
$676$	0.533266	+	2.33639i	0.533266	+	2.33639i
$677$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$677$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$678$		0			0
$679$	−0.144838	−	0.0368804i	−0.144838	−	0.0368804i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.680173	−	0.733052i	$-0.261905\pi$
$683$		0			0		−0.680173	+	0.733052i	$0.738095\pi$
$684$	0.939693	+	0.342020i	0.939693	+	0.342020i
$685$		0			0
$686$		0			0
$687$	0.128002	+	0.152547i	0.128002	+	0.152547i
$688$	−1.87822	−	0.478254i	−1.87822	−	0.478254i
$689$		0			0
$690$		0			0
$691$	−1.68862	−	0.385418i	−1.68862	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$691$	−1.68862	−	0.385418i	−1.68862	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.980172	−	0.198146i	0.980172	−	0.198146i
$701$		0			0		0.270840	−	0.962624i	$-0.412698\pi$
$701$		0			0		−0.270840	+	0.962624i	$0.587302\pi$
$702$		0			0
$703$	1.94925	−	0.444904i	1.94925	−	0.444904i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.31680	+	1.49211i	1.31680	+	1.49211i	0.733052	+	0.680173i	$0.238095\pi$
$709$	1.31680	+	1.49211i	1.31680	+	1.49211i	0.583744	+	0.811938i	$0.301587\pi$
$710$		0			0
$711$	−0.822581	+	1.70811i	−0.822581	+	1.70811i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.992239	−	0.124344i	$-0.960317\pi$
$719$		0			0		0.992239	+	0.124344i	$0.0396825\pi$
$720$		0			0
$721$	−0.757983	+	0.319516i	−0.757983	+	0.319516i
$722$		0			0
$723$	0.333345	−	0.849349i	0.333345	−	0.849349i
$724$	1.45882	−	0.145977i	1.45882	−	0.145977i
$725$		0			0
$726$		0			0
$727$	1.65453	+	1.06861i	1.65453	+	1.06861i	0.921476	+	0.388435i	$0.126984\pi$
$727$	1.65453	+	1.06861i	1.65453	+	1.06861i	0.733052	+	0.680173i	$0.238095\pi$
$728$		0			0
$729$	0.955573	+	0.294755i	0.955573	+	0.294755i
$730$		0			0
$731$		0			0
$732$	0.629690	+	0.831306i	0.629690	+	0.831306i
$733$	−1.98883	−	0.149042i	−1.98883	−	0.149042i	−0.988831	−	0.149042i	$-0.952381\pi$
$733$	−1.98883	−	0.149042i	−1.98883	−	0.149042i	−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.212203	+	0.129674i	−0.212203	+	0.129674i	−0.623490	−	0.781831i	$-0.714286\pi$
$739$	−0.212203	+	0.129674i	−0.212203	+	0.129674i	0.411287	+	0.911506i	$0.365079\pi$
$740$		0			0
$741$	−1.76108	+	0.543220i	−1.76108	+	0.543220i
$742$		0			0
$743$		0			0		0.0498459	−	0.998757i	$-0.484127\pi$
$743$		0			0		−0.0498459	+	0.998757i	$0.515873\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.53138	−	0.989068i	1.53138	−	0.989068i	0.542546	−	0.840026i	$-0.317460\pi$
$751$	1.53138	−	0.989068i	1.53138	−	0.989068i	0.988831	−	0.149042i	$-0.0476190\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.988831	−	0.149042i	0.988831	−	0.149042i
$757$	0.0497994	+	1.99689i	0.0497994	+	1.99689i	0.0747301	+	0.997204i	$0.476190\pi$
$757$	0.0497994	+	1.99689i	0.0497994	+	1.99689i	−0.0249307	+	0.999689i	$0.507937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$761$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$762$		0			0
$763$	1.01996	−	0.900121i	1.01996	−	0.900121i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.955573	+	0.294755i	0.955573	+	0.294755i
$769$	1.04381	+	0.750446i	1.04381	+	0.750446i	0.969077	−	0.246757i	$-0.0793651\pi$
$769$	1.04381	+	0.750446i	1.04381	+	0.750446i	0.0747301	+	0.997204i	$0.476190\pi$
$770$		0			0
$771$		0			0
$772$	−1.31352	+	1.04750i	−1.31352	+	1.04750i
$773$		0			0		0.698237	−	0.715867i	$-0.253968\pi$
$773$		0			0		−0.698237	+	0.715867i	$0.746032\pi$
$774$		0			0
$775$	0.701895	+	1.55556i	0.701895	+	1.55556i
$776$		0			0
$777$	1.49910	−	1.32296i	1.49910	−	1.32296i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.826239	−	0.563320i	0.826239	−	0.563320i
$785$		0			0
$786$		0			0
$787$	−0.141740	+	0.621003i	−0.141740	+	0.621003i	0.853291	+	0.521435i	$0.174603\pi$
$787$	−0.141740	+	0.621003i	−0.141740	+	0.621003i	−0.995031	+	0.0995678i	$0.968254\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.85013	−	0.520545i	−1.85013	−	0.520545i
$794$		0			0
$795$		0			0
$796$	0.381481	−	1.88708i	0.381481	−	1.88708i
$797$		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
$804$	−0.776628	−	0.0193679i	−0.776628	−	0.0193679i
$805$		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
$811$	1.87325	−	0.378686i	1.87325	−	0.378686i	0.878222	−	0.478254i	$-0.158730\pi$
$811$	1.87325	−	0.378686i	1.87325	−	0.378686i	0.995031	+	0.0995678i	$0.0317460\pi$
$812$		0			0
$813$	0.307391	−	0.503024i	0.307391	−	0.503024i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.884207	+	1.72471i	0.884207	+	1.72471i
$818$		0			0
$819$	−1.28682	+	1.31931i	−1.28682	+	1.31931i
$820$		0			0
$821$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$821$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$822$		0			0
$823$	−0.896546	−	1.38812i	−0.896546	−	1.38812i	−0.921476	−	0.388435i	$-0.873016\pi$
$823$	−0.896546	−	1.38812i	−0.896546	−	1.38812i	0.0249307	−	0.999689i	$-0.492063\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.0498459	−	0.998757i	$-0.515873\pi$
$827$		0			0		0.0498459	+	0.998757i	$0.484127\pi$
$828$		0			0
$829$	0.629859	+	0.0949359i	0.629859	+	0.0949359i	0.456211	−	0.889872i	$-0.349206\pi$
$829$	0.629859	+	0.0949359i	0.629859	+	0.0949359i	0.173648	+	0.984808i	$0.444444\pi$
$830$		0			0
$831$	1.11562	+	1.55173i	1.11562	+	1.55173i
$832$	−1.73181	+	0.630327i	−1.73181	+	0.630327i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.623484	+	1.58861i	0.623484	+	1.58861i
$838$		0			0
$839$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$839$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$840$		0			0
$841$	−0.542546	+	0.840026i	−0.542546	+	0.840026i
$842$		0			0
$843$		0			0
$844$	0.650636	+	1.35106i	0.650636	+	1.35106i
$845$		0			0
$846$		0			0
$847$	0.456211	+	0.889872i	0.456211	+	0.889872i
$848$		0			0
$849$	0.0779422	+	1.56172i	0.0779422	+	1.56172i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.370028	+	0.326552i	0.370028	+	0.326552i	0.826239	−	0.563320i	$-0.190476\pi$
$853$	0.370028	+	0.326552i	0.370028	+	0.326552i	−0.456211	+	0.889872i	$0.650794\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$857$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$858$		0			0
$859$	−0.0977881	−	0.483730i	−0.0977881	−	0.483730i	−0.998757	−	0.0498459i	$-0.984127\pi$
$859$	−0.0977881	−	0.483730i	−0.0977881	−	0.483730i	0.900969	−	0.433884i	$-0.142857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$868$	1.25101	+	1.16077i	1.25101	+	1.16077i
$869$		0			0
$870$		0			0
$871$	1.16248	−	0.835765i	1.16248	−	0.835765i
$872$		0			0
$873$	0.0111692	+	0.149042i	0.0111692	+	0.149042i
$874$		0			0
$875$		0			0
$876$	−0.371541	−	0.881399i	−0.371541	−	0.881399i
$877$	−1.53138	−	0.989068i	−1.53138	−	0.989068i	−0.988831	−	0.149042i	$-0.952381\pi$
$877$	−1.53138	−	0.989068i	−1.53138	−	0.989068i	−0.542546	−	0.840026i	$-0.682540\pi$
$878$		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
$883$	−0.346865	+	1.96717i	−0.346865	+	1.96717i	−0.124344	+	0.992239i	$0.539683\pi$
$883$	−0.346865	+	1.96717i	−0.346865	+	1.96717i	−0.222521	+	0.974928i	$0.571429\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.921476	−	0.388435i	$-0.873016\pi$
$887$		0			0		0.921476	+	0.388435i	$0.126984\pi$
$888$		0			0
$889$	0.376510	−	0.781831i	0.376510	−	0.781831i
$890$		0			0
$891$		0			0
$892$	1.87325	+	0.577822i	1.87325	+	0.577822i
$893$		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
$903$	1.60138	+	1.09180i	1.60138	+	1.09180i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.145466	−	0.571281i	−0.145466	−	0.571281i	−0.998757	−	0.0498459i	$-0.984127\pi$
$907$	−0.145466	−	0.571281i	−0.145466	−	0.571281i	0.853291	−	0.521435i	$-0.174603\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$911$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$912$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0198275	−	0.198146i	0.0198275	−	0.198146i
$917$		0			0
$918$		0			0
$919$	−0.155054	−	0.194432i	−0.155054	−	0.194432i	0.698237	−	0.715867i	$-0.253968\pi$
$919$	−0.155054	−	0.194432i	−0.155054	−	0.194432i	−0.853291	+	0.521435i	$0.825397\pi$
$920$		0			0
$921$	−0.753878	+	1.47049i	−0.753878	+	1.47049i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.77919	−	0.912138i	−1.77919	−	0.912138i
$926$		0			0
$927$	0.544286	+	0.616751i	0.544286	+	0.616751i
$928$		0			0
$929$		0			0		−0.911506	−	0.411287i	$-0.865079\pi$
$929$		0			0		0.911506	+	0.411287i	$0.134921\pi$
$930$		0			0
$931$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.27406	−	0.653172i	−1.27406	−	0.653172i	−0.318487	−	0.947927i	$-0.603175\pi$
$937$	−1.27406	−	0.653172i	−1.27406	−	0.653172i	−0.955573	+	0.294755i	$0.904762\pi$
$938$		0			0
$939$	0.0878620	+	0.284841i	0.0878620	+	0.284841i
$940$		0			0
$941$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$941$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$947$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$948$	1.79713	−	0.603804i	1.79713	−	0.603804i
$949$	1.52663	+	0.881399i	1.52663	+	0.881399i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.999689	−	0.0249307i	$-0.00793651\pi$
$953$		0			0		−0.999689	+	0.0249307i	$0.992063\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.956211	+	1.65621i	−0.956211	+	1.65621i
$962$		0			0
$963$		0			0
$964$	−0.840775	+	0.354416i	−0.840775	+	0.354416i
$965$		0			0
$966$		0			0
$967$	1.16169	+	1.61581i	1.16169	+	1.61581i	0.661686	+	0.749781i	$0.269841\pi$
$967$	1.16169	+	1.61581i	1.16169	+	1.61581i	0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$971$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$972$	−0.456211	−	0.889872i	−0.456211	−	0.889872i
$973$	1.17733	−	0.268718i	1.17733	−	0.268718i
$974$		0			0
$975$	1.69824	+	0.715867i	1.69824	+	0.715867i
$976$	0.155432	−	1.03122i	0.155432	−	1.03122i
$977$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$977$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.17809	−	0.680173i	−1.17809	−	0.680173i
$982$		0			0
$983$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$983$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.61852	+	0.881399i	1.61852	+	0.881399i
$989$		0			0
$990$		0			0
$991$	1.31849	−	0.947927i	1.31849	−	0.947927i	0.318487	−	0.947927i	$-0.396825\pi$
$991$	1.31849	−	0.947927i	1.31849	−	0.947927i	1.00000			$0$
$992$		0			0
$993$	0.142555	+	1.42462i	0.142555	+	1.42462i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.28518	−	0.0320503i	−1.28518	−	0.0320503i	−0.623490	−	0.781831i	$-0.714286\pi$
$997$	−1.28518	−	0.0320503i	−1.28518	−	0.0320503i	−0.661686	+	0.749781i	$0.730159\pi$
$998$		0			0
$999$	−1.73151	−	0.999689i	−1.73151	−	0.999689i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.1.ew.a.143.1	yes	36
3.2	odd	2	CM	2793.1.ew.a.143.1	yes	36
19.2	odd	18		2793.1.er.a.2054.1	yes	36
49.12	odd	42		2793.1.er.a.257.1	✓	36
57.2	even	18		2793.1.er.a.2054.1	yes	36
147.110	even	42		2793.1.er.a.257.1	✓	36
931.306	even	126	inner	2793.1.ew.a.2168.1	yes	36
2793.2168	odd	126	inner	2793.1.ew.a.2168.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.1.er.a.257.1	✓	36	49.12	odd	42
2793.1.er.a.257.1	✓	36	147.110	even	42
2793.1.er.a.2054.1	yes	36	19.2	odd	18
2793.1.er.a.2054.1	yes	36	57.2	even	18
2793.1.ew.a.143.1	yes	36	1.1	even	1	trivial
2793.1.ew.a.143.1	yes	36	3.2	odd	2	CM
2793.1.ew.a.2168.1	yes	36	931.306	even	126	inner
2793.1.ew.a.2168.1	yes	36	2793.2168	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-0.998757 - 0.0498459i) q^{3} +(0.661686 + 0.749781i) q^{4} +(-0.411287 + 0.911506i) q^{7} +(0.995031 + 0.0995678i) q^{9} +O(q^{10})\)	\(q+(-0.998757 - 0.0498459i) q^{3} +(0.661686 + 0.749781i) q^{4} +(-0.411287 + 0.911506i) q^{7} +(0.995031 + 0.0995678i) q^{9} +(-0.623490 - 0.781831i) q^{12} +(1.78596 + 0.454762i) q^{13} +(-0.124344 + 0.992239i) q^{16} +(0.826239 - 0.563320i) q^{19} +(0.456211 - 0.889872i) q^{21} +(-0.995031 - 0.0995678i) q^{25} +(-0.988831 - 0.149042i) q^{27} +(-0.955573 + 0.294755i) q^{28} +(-0.853291 - 1.47794i) q^{31} +(0.583744 + 0.811938i) q^{36} +(1.86117 + 0.730455i) q^{37} +(-1.76108 - 0.543220i) q^{39} +(-0.240997 + 1.92311i) q^{43} +(0.173648 - 0.984808i) q^{48} +(-0.661686 - 0.749781i) q^{49} +(0.840775 + 1.63999i) q^{52} +(-0.853291 + 0.521435i) q^{57} +(-1.04255 - 0.0259995i) q^{61} +(-0.500000 + 0.866025i) q^{63} +(-0.826239 + 0.563320i) q^{64} +(0.499362 - 0.595117i) q^{67} +(0.920758 + 0.259061i) q^{73} +(0.988831 + 0.149042i) q^{75} +(0.969077 + 0.246757i) q^{76} +(-0.648420 + 1.78152i) q^{79} +(0.980172 + 0.198146i) q^{81} +(0.969077 - 0.246757i) q^{84} +(-1.14906 + 1.44088i) q^{91} +(0.778561 + 1.51864i) q^{93} +(0.0259535 + 0.147190i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 36 * 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

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