LMFDB - Embedded newform 2793.1.ew.a.1370.1

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			1370.1
Root			$-0.583744 + 0.811938i$ of defining polynomial
Character	$\chi$	$=$	2793.1370
Dual form			2793.1.ew.a.1739.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.878222 - 0.478254i) q^{3} +(0.583744 + 0.811938i) q^{4} +(0.456211 - 0.889872i) q^{7} +(0.542546 - 0.840026i) q^{9} +O(q^{10})$	\(q+(0.878222 - 0.478254i) q^{3} +(0.583744 + 0.811938i) q^{4} +(0.456211 - 0.889872i) q^{7} +(0.542546 - 0.840026i) q^{9} +(0.900969 + 0.433884i) q^{12} +(-1.05490 - 0.799058i) q^{13} +(-0.318487 + 0.947927i) q^{16} +(0.955573 - 0.294755i) q^{19} +(-0.0249307 - 0.999689i) q^{21} +(-0.542546 + 0.840026i) q^{25} +(0.0747301 - 0.997204i) q^{27} +(0.988831 - 0.149042i) q^{28} +(0.698237 - 1.20938i) q^{31} +(0.998757 - 0.0498459i) q^{36} +(0.278007 + 0.407761i) q^{37} +(-1.30859 - 0.197238i) q^{39} +(-0.507752 + 1.51125i) q^{43} +(0.173648 + 0.984808i) q^{48} +(-0.583744 - 0.811938i) q^{49} +(0.0329926 - 1.32296i) q^{52} +(0.698237 - 0.715867i) q^{57} +(0.353291 + 1.38746i) q^{61} +(-0.500000 - 0.866025i) q^{63} +(-0.955573 + 0.294755i) q^{64} +(-0.963900 - 1.14873i) q^{67} +(-0.747834 + 1.77407i) q^{73} +(-0.0747301 + 0.997204i) q^{75} +(0.797133 + 0.603804i) q^{76} +(-0.0681084 - 0.187126i) q^{79} +(-0.411287 - 0.911506i) q^{81} +(0.797133 - 0.603804i) q^{84} +(-1.19232 + 0.574189i) q^{91} +(0.0348151 - 1.39604i) q^{93} +(-0.254586 + 1.44383i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times$.

$n$	$932$	$2110$	$2206$
$\chi(n)$	$-1$	$e\left(\frac{5}{42}\right)$	$e\left(\frac{1}{18}\right)$

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		−0.889872	−	0.456211i	$-0.849206\pi$
$2$		0			0		0.889872	+	0.456211i	$0.150794\pi$
$3$	0.878222	−	0.478254i	0.878222	−	0.478254i
$3$	0.878222	−	0.478254i	0.878222	−	0.478254i
$4$	0.583744	+	0.811938i	0.583744	+	0.811938i
$5$		0			0		−0.478254	−	0.878222i	$-0.658730\pi$
$5$		0			0		0.478254	+	0.878222i	$0.341270\pi$
$6$		0			0
$7$	0.456211	−	0.889872i	0.456211	−	0.889872i
$7$	0.456211	−	0.889872i	0.456211	−	0.889872i
$8$		0			0
$9$	0.542546	−	0.840026i	0.542546	−	0.840026i
$10$		0			0
$11$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$11$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$12$	0.900969	+	0.433884i	0.900969	+	0.433884i
$13$	−1.05490	−	0.799058i	−1.05490	−	0.799058i	−0.0747301	−	0.997204i	$-0.523810\pi$
$13$	−1.05490	−	0.799058i	−1.05490	−	0.799058i	−0.980172	+	0.198146i	$0.936508\pi$
$14$		0			0
$15$		0			0
$16$	−0.318487	+	0.947927i	−0.318487	+	0.947927i
$17$		0			0		0.0995678	−	0.995031i	$-0.468254\pi$
$17$		0			0		−0.0995678	+	0.995031i	$0.531746\pi$
$18$		0			0
$19$	0.955573	−	0.294755i	0.955573	−	0.294755i
$19$	0.955573	−	0.294755i	0.955573	−	0.294755i
$20$		0			0
$21$	−0.0249307	−	0.999689i	−0.0249307	−	0.999689i
$22$		0			0
$23$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$23$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$24$		0			0
$25$	−0.542546	+	0.840026i	−0.542546	+	0.840026i
$26$		0			0
$27$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$28$	0.988831	−	0.149042i	0.988831	−	0.149042i
$29$		0			0		−0.962624	−	0.270840i	$-0.912698\pi$
$29$		0			0		0.962624	+	0.270840i	$0.0873016\pi$
$30$		0			0
$31$	0.698237	−	1.20938i	0.698237	−	1.20938i	−0.270840	−	0.962624i	$-0.587302\pi$
$31$	0.698237	−	1.20938i	0.698237	−	1.20938i	0.969077	−	0.246757i	$-0.0793651\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.998757	−	0.0498459i	0.998757	−	0.0498459i
$37$	0.278007	+	0.407761i	0.278007	+	0.407761i	0.939693	−	0.342020i	$-0.111111\pi$
$37$	0.278007	+	0.407761i	0.278007	+	0.407761i	−0.661686	+	0.749781i	$0.730159\pi$
$38$		0			0
$39$	−1.30859	−	0.197238i	−1.30859	−	0.197238i
$40$		0			0
$41$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$41$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$42$		0			0
$43$	−0.507752	+	1.51125i	−0.507752	+	1.51125i	0.318487	+	0.947927i	$0.396825\pi$
$43$	−0.507752	+	1.51125i	−0.507752	+	1.51125i	−0.826239	+	0.563320i	$0.809524\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.992239	−	0.124344i	$-0.960317\pi$
$47$		0			0		0.992239	+	0.124344i	$0.0396825\pi$
$48$	0.173648	+	0.984808i	0.173648	+	0.984808i
$49$	−0.583744	−	0.811938i	−0.583744	−	0.811938i
$50$		0			0
$51$		0			0
$52$	0.0329926	−	1.32296i	0.0329926	−	1.32296i
$53$		0			0		0.811938	−	0.583744i	$-0.198413\pi$
$53$		0			0		−0.811938	+	0.583744i	$0.801587\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.698237	−	0.715867i	0.698237	−	0.715867i
$58$		0			0
$59$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$59$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$60$		0			0
$61$	0.353291	+	1.38746i	0.353291	+	1.38746i	0.853291	+	0.521435i	$0.174603\pi$
$61$	0.353291	+	1.38746i	0.353291	+	1.38746i	−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$64$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$65$		0			0
$66$		0			0
$67$	−0.963900	−	1.14873i	−0.963900	−	1.14873i	−0.988831	−	0.149042i	$-0.952381\pi$
$67$	−0.963900	−	1.14873i	−0.963900	−	1.14873i	0.0249307	−	0.999689i	$-0.492063\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.715867	−	0.698237i	$-0.753968\pi$
$71$		0			0		0.715867	+	0.698237i	$0.246032\pi$
$72$		0			0
$73$	−0.747834	+	1.77407i	−0.747834	+	1.77407i	−0.124344	+	0.992239i	$0.539683\pi$
$73$	−0.747834	+	1.77407i	−0.747834	+	1.77407i	−0.623490	+	0.781831i	$0.714286\pi$
$74$		0			0
$75$	−0.0747301	+	0.997204i	−0.0747301	+	0.997204i
$76$	0.797133	+	0.603804i	0.797133	+	0.603804i
$77$		0			0
$78$		0			0
$79$	−0.0681084	−	0.187126i	−0.0681084	−	0.187126i	0.900969	−	0.433884i	$-0.142857\pi$
$79$	−0.0681084	−	0.187126i	−0.0681084	−	0.187126i	−0.969077	+	0.246757i	$0.920635\pi$
$80$		0			0
$81$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$82$		0			0
$83$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$83$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$84$	0.797133	−	0.603804i	0.797133	−	0.603804i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.998757	−	0.0498459i	$-0.984127\pi$
$89$		0			0		0.998757	+	0.0498459i	$0.0158730\pi$
$90$		0			0
$91$	−1.19232	+	0.574189i	−1.19232	+	0.574189i
$92$		0			0
$93$	0.0348151	−	1.39604i	0.0348151	−	1.39604i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.254586	+	1.44383i	−0.254586	+	1.44383i	0.542546	+	0.840026i	$0.317460\pi$
$97$	−0.254586	+	1.44383i	−0.254586	+	1.44383i	−0.797133	+	0.603804i	$0.793651\pi$
$98$		0			0
$99$		0			0
$100$	−0.998757	+	0.0498459i	−0.998757	+	0.0498459i
$101$		0			0		0.947927	−	0.318487i	$-0.103175\pi$
$101$		0			0		−0.947927	+	0.318487i	$0.896825\pi$
$102$		0			0
$103$	0.333345	−	0.849349i	0.333345	−	0.849349i	−0.661686	−	0.749781i	$-0.730159\pi$
$103$	0.333345	−	0.849349i	0.333345	−	0.849349i	0.995031	−	0.0995678i	$-0.0317460\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$107$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$108$	0.853291	−	0.521435i	0.853291	−	0.521435i
$109$	1.65672	+	0.849349i	1.65672	+	0.849349i	0.995031	+	0.0995678i	$0.0317460\pi$
$109$	1.65672	+	0.849349i	1.65672	+	0.849349i	0.661686	+	0.749781i	$0.269841\pi$
$110$		0			0
$111$	0.439165	+	0.225147i	0.439165	+	0.225147i
$112$	0.698237	+	0.715867i	0.698237	+	0.715867i
$113$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$113$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.24356	+	0.452620i	−1.24356	+	0.452620i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$122$		0			0
$123$		0			0
$124$	1.38953	−	0.139044i	1.38953	−	0.139044i
$125$		0			0
$126$		0			0
$127$	0.481141	+	1.88956i	0.481141	+	1.88956i	0.456211	+	0.889872i	$0.349206\pi$
$127$	0.481141	+	1.88956i	0.481141	+	1.88956i	0.0249307	+	0.999689i	$0.492063\pi$
$128$		0			0
$129$	0.276841	+	1.57004i	0.276841	+	1.57004i
$130$		0			0
$131$		0			0		−0.198146	−	0.980172i	$-0.563492\pi$
$131$		0			0		0.198146	+	0.980172i	$0.436508\pi$
$132$		0			0
$133$	0.173648	−	0.984808i	0.173648	−	0.984808i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.878222	−	0.478254i	$-0.841270\pi$
$137$		0			0		0.878222	+	0.478254i	$0.158730\pi$
$138$		0			0
$139$	−0.0785238	+	0.388435i	−0.0785238	+	0.388435i	0.921476	+	0.388435i	$0.126984\pi$
$139$	−0.0785238	+	0.388435i	−0.0785238	+	0.388435i	−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.623490	+	0.781831i	0.623490	+	0.781831i
$145$		0			0
$146$		0			0
$147$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$148$	−0.168792	+	0.463752i	−0.168792	+	0.463752i
$149$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$149$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$150$		0			0
$151$	0.332083	−	0.487076i	0.332083	−	0.487076i	−0.623490	−	0.781831i	$-0.714286\pi$
$151$	0.332083	−	0.487076i	0.332083	−	0.487076i	0.955573	+	0.294755i	$0.0952381\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.603736	−	1.17763i	−0.603736	−	1.17763i
$157$	−0.396169	−	1.95974i	−0.396169	−	1.95974i	−0.222521	−	0.974928i	$-0.571429\pi$
$157$	−0.396169	−	1.95974i	−0.396169	−	1.95974i	−0.173648	−	0.984808i	$-0.555556\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.68752	−	0.254353i	−1.68752	−	0.254353i	−0.766044	−	0.642788i	$-0.777778\pi$
$163$	−1.68752	−	0.254353i	−1.68752	−	0.254353i	−0.921476	+	0.388435i	$0.873016\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$167$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$168$		0			0
$169$	0.203486	+	0.723232i	0.203486	+	0.723232i
$170$		0			0
$171$	0.270840	−	0.962624i	0.270840	−	0.962624i
$172$	−1.52344	+	0.469918i	−1.52344	+	0.469918i
$173$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$173$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.500000	+	0.866025i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$179$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$180$		0			0
$181$	−0.729774	−	0.0364215i	−0.729774	−	0.0364215i	−0.318487	−	0.947927i	$-0.603175\pi$
$181$	−0.729774	−	0.0364215i	−0.729774	−	0.0364215i	−0.411287	+	0.911506i	$0.634921\pi$
$182$		0			0
$183$	0.973826	+	1.04954i	0.973826	+	1.04954i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$190$		0			0
$191$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$191$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$192$	−0.698237	+	0.715867i	−0.698237	+	0.715867i
$193$	−0.498757	−	0.915871i	−0.498757	−	0.915871i	−0.998757	−	0.0498459i	$-0.984127\pi$
$193$	−0.498757	−	0.915871i	−0.498757	−	0.915871i	0.500000	−	0.866025i	$-0.333333\pi$
$194$		0			0
$195$		0			0
$196$	0.318487	−	0.947927i	0.318487	−	0.947927i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	0.153934	−	0.761466i	0.153934	−	0.761466i	−0.826239	−	0.563320i	$-0.809524\pi$
$199$	0.153934	−	0.761466i	0.153934	−	0.761466i	0.980172	−	0.198146i	$-0.0634921\pi$
$200$		0			0
$201$	−1.39590	−	0.547852i	−1.39590	−	0.547852i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.09342	−	0.745482i	1.09342	−	0.745482i
$209$		0			0
$210$		0			0
$211$	−0.630770	+	1.49636i	−0.630770	+	1.49636i	0.222521	+	0.974928i	$0.428571\pi$
$211$	−0.630770	+	1.49636i	−0.630770	+	1.49636i	−0.853291	+	0.521435i	$0.825397\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.757652	−	1.17307i	−0.757652	−	1.17307i
$218$		0			0
$219$	0.191693	+	1.91568i	0.191693	+	1.91568i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.574352	−	0.588854i	0.574352	−	0.588854i	−0.365341	−	0.930874i	$-0.619048\pi$
$223$	0.574352	−	0.588854i	0.574352	−	0.588854i	0.939693	+	0.342020i	$0.111111\pi$
$224$		0			0
$225$	0.411287	+	0.911506i	0.411287	+	0.911506i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	0.988831	+	0.149042i	0.988831	+	0.149042i
$229$	1.56392	−	0.613792i	1.56392	−	0.613792i	0.583744	−	0.811938i	$-0.301587\pi$
$229$	1.56392	−	0.613792i	1.56392	−	0.613792i	0.980172	+	0.198146i	$0.0634921\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.583744	−	0.811938i	$-0.301587\pi$
$233$		0			0		−0.583744	+	0.811938i	$0.698413\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.149308	−	0.131765i	−0.149308	−	0.131765i
$238$		0			0
$239$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$239$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$240$		0			0
$241$	−0.0496136	+	0.00496459i	−0.0496136	+	0.00496459i	−0.124344	−	0.992239i	$-0.539683\pi$
$241$	−0.0496136	+	0.00496459i	−0.0496136	+	0.00496459i	0.0747301	+	0.997204i	$0.476190\pi$
$242$		0			0
$243$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$244$	−0.920301	+	1.09677i	−0.920301	+	1.09677i
$245$		0			0
$246$		0			0
$247$	−1.24356	−	0.452620i	−1.24356	−	0.452620i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.521435	−	0.853291i	$-0.674603\pi$
$251$		0			0		0.521435	+	0.853291i	$0.325397\pi$
$252$	0.411287	−	0.911506i	0.411287	−	0.911506i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$257$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$257$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$258$		0			0
$259$	0.489685	−	0.0613655i	0.489685	−	0.0613655i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$263$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.370028	−	1.45319i	0.370028	−	1.45319i
$269$		0			0		−0.921476	−	0.388435i	$-0.873016\pi$
$269$		0			0		0.921476	+	0.388435i	$0.126984\pi$
$270$		0			0
$271$	0.286943	−	0.0807334i	0.286943	−	0.0807334i	−0.124344	−	0.992239i	$-0.539683\pi$
$271$	0.286943	−	0.0807334i	0.286943	−	0.0807334i	0.411287	+	0.911506i	$0.365079\pi$
$272$		0			0
$273$	−0.772510	+	1.07450i	−0.772510	+	1.07450i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.78181	−	0.858075i	−1.78181	−	0.858075i	−0.955573	−	0.294755i	$-0.904762\pi$
$277$	−1.78181	−	0.858075i	−1.78181	−	0.858075i	−0.826239	−	0.563320i	$-0.809524\pi$
$278$		0			0
$279$	−0.637086	−	1.24268i	−0.637086	−	1.24268i
$280$		0			0
$281$		0			0		−0.992239	−	0.124344i	$-0.960317\pi$
$281$		0			0		0.992239	+	0.124344i	$0.0396825\pi$
$282$		0			0
$283$	−0.728947	+	0.470804i	−0.728947	+	0.470804i	−0.853291	−	0.521435i	$-0.825397\pi$
$283$	−0.728947	+	0.470804i	−0.728947	+	0.470804i	0.124344	+	0.992239i	$0.460317\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.980172	−	0.198146i	−0.980172	−	0.198146i
$290$		0			0
$291$	0.466934	+	1.38976i	0.466934	+	1.38976i
$292$	−1.87698	+	0.428408i	−1.87698	+	0.428408i
$293$		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
$300$	−0.853291	+	0.521435i	−0.853291	+	0.521435i
$301$	1.11317	+	1.14128i	1.11317	+	1.14128i
$302$		0			0
$303$		0			0
$304$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$305$		0			0
$306$		0			0
$307$	−0.871885	+	1.70067i	−0.871885	+	1.70067i	−0.173648	+	0.984808i	$0.555556\pi$
$307$	−0.871885	+	1.70067i	−0.871885	+	1.70067i	−0.698237	+	0.715867i	$0.746032\pi$
$308$		0			0
$309$	−0.113454	−	0.905340i	−0.113454	−	0.905340i
$310$		0			0
$311$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$311$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$312$		0			0
$313$	0.682128	−	1.87413i	0.682128	−	1.87413i	0.270840	−	0.962624i	$-0.412698\pi$
$313$	0.682128	−	1.87413i	0.682128	−	1.87413i	0.411287	−	0.911506i	$-0.365079\pi$
$314$		0			0
$315$		0			0
$316$	0.112177	−	0.164534i	0.112177	−	0.164534i
$317$		0			0		0.811938	−	0.583744i	$-0.198413\pi$
$317$		0			0		−0.811938	+	0.583744i	$0.801587\pi$
$318$		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
$325$	1.24356	−	0.452620i	1.24356	−	0.452620i
$326$		0			0
$327$	1.86117	−	0.0464147i	1.86117	−	0.0464147i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.97893	−	0.148300i	1.97893	−	0.148300i	0.980172	−	0.198146i	$-0.0634921\pi$
$331$	1.97893	−	0.148300i	1.97893	−	0.148300i	0.998757	+	0.0498459i	$0.0158730\pi$
$332$		0			0
$333$	0.493361	−	0.0123037i	0.493361	−	0.0123037i
$334$		0			0
$335$		0			0
$336$	0.955573	+	0.294755i	0.955573	+	0.294755i
$337$	−0.906700	−	0.304635i	−0.906700	−	0.304635i	−0.173648	−	0.984808i	$-0.555556\pi$
$337$	−0.906700	−	0.304635i	−0.906700	−	0.304635i	−0.733052	+	0.680173i	$0.761905\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.988831	+	0.149042i	−0.988831	+	0.149042i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.969077	−	0.246757i	$-0.920635\pi$
$347$		0			0		0.969077	+	0.246757i	$0.0793651\pi$
$348$		0			0
$349$	−0.0148583	+	0.0985783i	−0.0148583	+	0.0985783i	−0.995031	−	0.0995678i	$-0.968254\pi$
$349$	−0.0148583	+	0.0985783i	−0.0148583	+	0.0985783i	0.980172	+	0.198146i	$0.0634921\pi$
$350$		0			0
$351$	−0.875656	+	0.992239i	−0.875656	+	0.992239i
$352$		0			0
$353$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$353$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.980172	−	0.198146i	$-0.0634921\pi$
$359$		0			0		−0.980172	+	0.198146i	$0.936508\pi$
$360$		0			0
$361$	0.826239	−	0.563320i	0.826239	−	0.563320i
$362$		0			0
$363$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$364$	−1.16221	−	0.632908i	−1.16221	−	0.632908i
$365$		0			0
$366$		0			0
$367$	1.59257	+	1.02859i	1.59257	+	1.02859i	0.969077	+	0.246757i	$0.0793651\pi$
$367$	1.59257	+	1.02859i	1.59257	+	1.02859i	0.623490	+	0.781831i	$0.285714\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.15382	−	0.786662i	1.15382	−	0.786662i
$373$			1.36035i			1.36035i	0.733052	+	0.680173i	$0.238095\pi$
$373$			1.36035i			1.36035i	−0.733052	+	0.680173i	$0.761905\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.17733	−	0.268718i	1.17733	−	0.268718i	0.411287	−	0.911506i	$-0.365079\pi$
$379$	1.17733	−	0.268718i	1.17733	−	0.268718i	0.766044	+	0.642788i	$0.222222\pi$
$380$		0			0
$381$	1.32624	+	1.42935i	1.32624	+	1.42935i
$382$		0			0
$383$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$383$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.994008	+	1.24645i	0.994008	+	1.24645i
$388$	−1.32091	+	0.636119i	−1.32091	+	0.636119i
$389$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$389$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.82245	−	0.0454489i	−1.82245	−	0.0454489i	−0.900969	−	0.433884i	$-0.857143\pi$
$397$	−1.82245	−	0.0454489i	−1.82245	−	0.0454489i	−0.921476	+	0.388435i	$0.873016\pi$
$398$		0			0
$399$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$400$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$401$		0			0		−0.603804	−	0.797133i	$-0.706349\pi$
$401$		0			0		0.603804	+	0.797133i	$0.293651\pi$
$402$		0			0
$403$	−1.70294	+	0.717848i	−1.70294	+	0.717848i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.60138	−	0.407761i	1.60138	−	0.407761i	0.661686	−	0.749781i	$-0.269841\pi$
$409$	1.60138	−	0.407761i	1.60138	−	0.407761i	0.939693	+	0.342020i	$0.111111\pi$
$410$		0			0
$411$		0			0
$412$	0.884207	−	0.225147i	0.884207	−	0.225147i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.116809	+	0.378686i	0.116809	+	0.378686i
$418$		0			0
$419$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$419$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$420$		0			0
$421$	−1.42529	+	0.643114i	−1.42529	+	0.643114i	−0.969077	−	0.246757i	$-0.920635\pi$
$421$	−1.42529	+	0.643114i	−1.42529	+	0.643114i	−0.456211	+	0.889872i	$0.650794\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.39584	+	0.318591i	1.39584	+	0.318591i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.521435	−	0.853291i	$-0.674603\pi$
$431$		0			0		0.521435	+	0.853291i	$0.325397\pi$
$432$	0.921476	+	0.388435i	0.921476	+	0.388435i
$433$	−0.345587	−	1.02859i	−0.345587	−	1.02859i	−0.969077	−	0.246757i	$-0.920635\pi$
$433$	−0.345587	−	1.02859i	−0.345587	−	1.02859i	0.623490	−	0.781831i	$-0.285714\pi$
$434$		0			0
$435$		0			0
$436$	0.277479	+	1.84095i	0.277479	+	1.84095i
$437$		0			0
$438$		0			0
$439$	0.290594	−	0.566825i	0.290594	−	0.566825i	−0.698237	−	0.715867i	$-0.746032\pi$
$439$	0.290594	−	0.566825i	0.290594	−	0.566825i	0.988831	+	0.149042i	$0.0476190\pi$
$440$		0			0
$441$	−0.998757	+	0.0498459i	−0.998757	+	0.0498459i
$442$		0			0
$443$		0			0		0.921476	−	0.388435i	$-0.126984\pi$
$443$		0			0		−0.921476	+	0.388435i	$0.873016\pi$
$444$	0.0735546	+	0.488003i	0.0735546	+	0.488003i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$449$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$449$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.0586963	−	0.586581i	0.0586963	−	0.586581i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.197898	−	0.504237i	−0.197898	−	0.504237i	0.797133	−	0.603804i	$-0.206349\pi$
$457$	−0.197898	−	0.504237i	−0.197898	−	0.504237i	−0.995031	+	0.0995678i	$0.968254\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.0995678	−	0.995031i	$-0.468254\pi$
$461$		0			0		−0.0995678	+	0.995031i	$0.531746\pi$
$462$		0			0
$463$	1.61971	+	1.10430i	1.61971	+	1.10430i	0.921476	+	0.388435i	$0.126984\pi$
$463$	1.61971	+	1.10430i	1.61971	+	1.10430i	0.698237	+	0.715867i	$0.253968\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$467$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$468$	−1.09342	−	0.745482i	−1.09342	−	0.745482i
$469$	−1.46197	+	0.333684i	−1.46197	+	0.333684i
$470$		0			0
$471$	−1.28518	−	1.53161i	−1.28518	−	1.53161i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.270840	+	0.962624i	−0.270840	+	0.962624i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.388435	−	0.921476i	$-0.626984\pi$
$479$		0			0		0.388435	+	0.921476i	$0.373016\pi$
$480$		0			0
$481$	0.0325545	−	0.652292i	0.0325545	−	0.652292i
$482$		0			0
$483$		0			0
$484$	−0.878222	−	0.478254i	−0.878222	−	0.478254i
$485$		0			0
$486$		0			0
$487$	−0.167917	−	1.11406i	−0.167917	−	1.11406i	−0.900969	−	0.433884i	$-0.857143\pi$
$487$	−0.167917	−	1.11406i	−0.167917	−	1.11406i	0.733052	−	0.680173i	$-0.238095\pi$
$488$		0			0
$489$	−1.60366	+	0.583685i	−1.60366	+	0.583685i
$490$		0			0
$491$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$491$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.924027	+	1.04705i	0.924027	+	1.04705i
$497$		0			0
$498$		0			0
$499$	−1.78596	−	0.752847i	−1.78596	−	0.752847i	−0.988831	−	0.149042i	$-0.952381\pi$
$499$	−1.78596	−	0.752847i	−1.78596	−	0.752847i	−0.797133	−	0.603804i	$-0.793651\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.388435	−	0.921476i	$-0.626984\pi$
$503$		0			0		0.388435	+	0.921476i	$0.373016\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.524594	+	0.537840i	0.524594	+	0.537840i
$508$	−1.25334	+	1.49368i	−1.25334	+	1.49368i
$509$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$509$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$510$		0			0
$511$	1.23753	+	1.47483i	1.23753	+	1.47483i
$512$		0			0
$513$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$514$		0			0
$515$		0			0
$516$	−1.11317	+	1.14128i	−1.11317	+	1.14128i
$517$		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
$523$	−1.65381	+	1.01062i	−1.65381	+	1.01062i	−0.698237	+	0.715867i	$0.746032\pi$
$523$	−1.65381	+	1.01062i	−1.65381	+	1.01062i	−0.955573	+	0.294755i	$0.904762\pi$
$524$		0			0
$525$	0.853291	+	0.521435i	0.853291	+	0.521435i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$530$		0			0
$531$		0			0
$532$	0.900969	−	0.433884i	0.900969	−	0.433884i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.229160	+	1.82865i	−0.229160	+	1.82865i	0.270840	+	0.962624i	$0.412698\pi$
$541$	−0.229160	+	1.82865i	−0.229160	+	1.82865i	−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	−0.658322	+	0.317031i	−0.658322	+	0.317031i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.72808	−	0.580605i	1.72808	−	0.580605i	0.733052	−	0.680173i	$-0.238095\pi$
$547$	1.72808	−	0.580605i	1.72808	−	0.580605i	0.995031	+	0.0995678i	$0.0317460\pi$
$548$		0			0
$549$	1.35718	+	0.455988i	1.35718	+	0.455988i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.197590	−	0.0247613i	−0.197590	−	0.0247613i
$554$		0			0
$555$		0			0
$556$	−0.361223	+	0.162990i	−0.361223	+	0.162990i
$557$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$557$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$558$		0			0
$559$	1.74320	−	1.18850i	1.74320	−	1.18850i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$563$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	0.148717	+	1.98450i	0.148717	+	1.98450i	0.173648	+	0.984808i	$0.444444\pi$
$571$	0.148717	+	1.98450i	0.148717	+	1.98450i	−0.0249307	+	0.999689i	$0.507937\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.270840	+	0.962624i	−0.270840	+	0.962624i
$577$	−0.460898	+	1.49419i	−0.460898	+	1.49419i	0.365341	+	0.930874i	$0.380952\pi$
$577$	−0.460898	+	1.49419i	−0.460898	+	1.49419i	−0.826239	+	0.563320i	$0.809524\pi$
$578$		0			0
$579$	−0.876038	−	0.565805i	−0.876038	−	0.565805i
$580$		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	$-0.944444\pi$
$587$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$588$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$589$	0.310745	−	1.36146i	0.310745	−	1.36146i
$590$		0			0
$591$		0			0
$592$	−0.475069	+	0.133664i	−0.475069	+	0.133664i
$593$		0			0		0.198146	−	0.980172i	$-0.436508\pi$
$593$		0			0		−0.198146	+	0.980172i	$0.563492\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.228986	−	0.742355i	−0.228986	−	0.742355i
$598$		0			0
$599$		0			0		0.603804	−	0.797133i	$-0.293651\pi$
$599$		0			0		−0.603804	+	0.797133i	$0.706349\pi$
$600$		0			0
$601$	−1.12310	+	1.04209i	−1.12310	+	1.04209i	−0.124344	+	0.992239i	$0.539683\pi$
$601$	−1.12310	+	1.04209i	−1.12310	+	1.04209i	−0.998757	+	0.0498459i	$0.984127\pi$
$602$		0			0
$603$	−1.48792	+	0.186461i	−1.48792	+	0.186461i
$604$	0.589327	−	0.0146969i	0.589327	−	0.0146969i
$605$		0			0
$606$		0			0
$607$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$607$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.31724	−	0.997773i	1.31724	−	0.997773i	0.318487	−	0.947927i	$-0.396825\pi$
$613$	1.31724	−	0.997773i	1.31724	−	0.997773i	0.998757	−	0.0498459i	$-0.0158730\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.270840	−	0.962624i	$-0.412698\pi$
$617$		0			0		−0.270840	+	0.962624i	$0.587302\pi$
$618$		0			0
$619$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$619$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.603736	−	1.17763i	0.603736	−	1.17763i
$625$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$626$		0			0
$627$		0			0
$628$	1.35992	−	1.46565i	1.35992	−	1.46565i
$629$		0			0
$630$		0			0
$631$	−0.996206	+	0.608769i	−0.996206	+	0.608769i	−0.921476	−	0.388435i	$-0.873016\pi$
$631$	−0.996206	+	0.608769i	−0.996206	+	0.608769i	−0.0747301	+	0.997204i	$0.523810\pi$
$632$		0			0
$633$	0.161686	+	1.61581i	0.161686	+	1.61581i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.0329926	+	1.32296i	−0.0329926	+	1.32296i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.246757	−	0.969077i	$-0.420635\pi$
$641$		0			0		−0.246757	+	0.969077i	$0.579365\pi$
$642$		0			0
$643$	0.189528	+	0.937543i	0.189528	+	0.937543i	0.955573	+	0.294755i	$0.0952381\pi$
$643$	0.189528	+	0.937543i	0.189528	+	0.937543i	−0.766044	+	0.642788i	$0.777778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$647$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.22641	−	0.667869i	−1.22641	−	0.667869i
$652$	−0.778561	−	1.51864i	−0.778561	−	1.51864i
$653$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$653$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.08453	+	1.59071i	1.08453	+	1.59071i
$658$		0			0
$659$		0			0		−0.911506	−	0.411287i	$-0.865079\pi$
$659$		0			0		0.911506	+	0.411287i	$0.134921\pi$
$660$		0			0
$661$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i	−0.456211	−	0.889872i	$-0.650794\pi$
$661$	−0.998757	−	0.0498459i	−0.998757	−	0.0498459i	−0.542546	+	0.840026i	$0.682540\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.222786	−	0.791830i	0.222786	−	0.791830i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.558813	−	1.81163i	0.558813	−	1.81163i	−0.0249307	−	0.999689i	$-0.507937\pi$
$673$	0.558813	−	1.81163i	0.558813	−	1.81163i	0.583744	−	0.811938i	$-0.301587\pi$
$674$		0			0
$675$	0.797133	+	0.603804i	0.797133	+	0.603804i
$676$	−0.468436	+	0.587400i	−0.468436	+	0.587400i
$677$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$677$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$678$		0			0
$679$	1.16868	+	0.885240i	1.16868	+	0.885240i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$683$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$684$	0.939693	−	0.342020i	0.939693	−	0.342020i
$685$		0			0
$686$		0			0
$687$	1.07992	−	1.28699i	1.07992	−	1.28699i
$688$	−1.27084	−	0.962624i	−1.27084	−	0.962624i
$689$		0			0
$690$		0			0
$691$	1.35417	−	1.07992i	1.35417	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$691$	1.35417	−	1.07992i	1.35417	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.411287	+	0.911506i	−0.411287	+	0.911506i
$701$		0			0		−0.921476	−	0.388435i	$-0.873016\pi$
$701$		0			0		0.921476	+	0.388435i	$0.126984\pi$
$702$		0			0
$703$	0.385845	+	0.307701i	0.385845	+	0.307701i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.633416	+	0.881028i	0.633416	+	0.881028i	0.998757	−	0.0498459i	$-0.0158730\pi$
$709$	0.633416	+	0.881028i	0.633416	+	0.881028i	−0.365341	+	0.930874i	$0.619048\pi$
$710$		0			0
$711$	−0.194143	−	0.0443119i	−0.194143	−	0.0443119i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.947927	−	0.318487i	$-0.896825\pi$
$719$		0			0		0.947927	+	0.318487i	$0.103175\pi$
$720$		0			0
$721$	−0.603736	−	0.684116i	−0.603736	−	0.684116i
$722$		0			0
$723$	−0.0411974	+	0.0280879i	−0.0411974	+	0.0280879i
$724$	−0.396429	−	0.613792i	−0.396429	−	0.613792i
$725$		0			0
$726$		0			0
$727$	−1.02703	+	1.68065i	−1.02703	+	1.68065i	−0.365341	+	0.930874i	$0.619048\pi$
$727$	−1.02703	+	1.68065i	−1.02703	+	1.68065i	−0.661686	+	0.749781i	$0.730159\pi$
$728$		0			0
$729$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$730$		0			0
$731$		0			0
$732$	−0.283693	+	1.40335i	−0.283693	+	1.40335i
$733$	−0.925270	−	0.997204i	−0.925270	−	0.997204i	0.0747301	−	0.997204i	$-0.476190\pi$
$733$	−0.925270	−	0.997204i	−0.925270	−	0.997204i	−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.444758	−	0.455988i	0.444758	−	0.455988i	−0.456211	−	0.889872i	$-0.650794\pi$
$739$	0.444758	−	0.455988i	0.444758	−	0.455988i	0.900969	+	0.433884i	$0.142857\pi$
$740$		0			0
$741$	−1.30859	+	0.197238i	−1.30859	+	0.197238i
$742$		0			0
$743$		0			0		−0.478254	−	0.878222i	$-0.658730\pi$
$743$		0			0		0.478254	+	0.878222i	$0.341270\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.928021	−	1.51864i	−0.928021	−	1.51864i	−0.853291	−	0.521435i	$-0.825397\pi$
$751$	−0.928021	−	1.51864i	−0.928021	−	1.51864i	−0.0747301	−	0.997204i	$-0.523810\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.0747301	−	0.997204i	−0.0747301	−	0.997204i
$757$	−1.70213	−	0.433415i	−1.70213	−	0.433415i	−0.733052	−	0.680173i	$-0.761905\pi$
$757$	−1.70213	−	0.433415i	−1.70213	−	0.433415i	−0.969077	+	0.246757i	$0.920635\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$761$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$762$		0			0
$763$	1.51162	−	1.08678i	1.51162	−	1.08678i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$769$	0.0640806	−	1.28398i	0.0640806	−	1.28398i	−0.733052	−	0.680173i	$-0.761905\pi$
$769$	0.0640806	−	1.28398i	0.0640806	−	1.28398i	0.797133	−	0.603804i	$-0.206349\pi$
$770$		0			0
$771$		0			0
$772$	0.452485	−	0.939594i	0.452485	−	0.939594i
$773$		0			0		0.124344	−	0.992239i	$-0.460317\pi$
$773$		0			0		−0.124344	+	0.992239i	$0.539683\pi$
$774$		0			0
$775$	0.637086	+	1.24268i	0.637086	+	1.24268i
$776$		0			0
$777$	0.400703	−	0.288086i	0.400703	−	0.288086i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.955573	−	0.294755i	0.955573	−	0.294755i
$785$		0			0
$786$		0			0
$787$	−1.24078	−	1.55589i	−1.24078	−	1.55589i	−0.698237	−	0.715867i	$-0.746032\pi$
$787$	−1.24078	−	1.55589i	−1.24078	−	1.55589i	−0.542546	−	0.840026i	$-0.682540\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.735974	−	1.74594i	0.735974	−	1.74594i
$794$		0			0
$795$		0			0
$796$	0.708121	−	0.319516i	0.708121	−	0.319516i
$797$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$797$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.370028	−	1.45319i	−0.370028	−	1.45319i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$809$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$810$		0			0
$811$	0.813387	−	1.80265i	0.813387	−	1.80265i	0.270840	−	0.962624i	$-0.412698\pi$
$811$	0.813387	−	1.80265i	0.813387	−	1.80265i	0.542546	−	0.840026i	$-0.317460\pi$
$812$		0			0
$813$	0.213389	−	0.208134i	0.213389	−	0.208134i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.0397461	+	1.59377i	−0.0397461	+	1.59377i
$818$		0			0
$819$	−0.164553	+	1.31310i	−0.164553	+	1.31310i
$820$		0			0
$821$		0			0		0.969077	−	0.246757i	$-0.0793651\pi$
$821$		0			0		−0.969077	+	0.246757i	$0.920635\pi$
$822$		0			0
$823$	1.63076	−	0.996539i	1.63076	−	0.996539i	0.661686	−	0.749781i	$-0.269841\pi$
$823$	1.63076	−	0.996539i	1.63076	−	0.996539i	0.969077	−	0.246757i	$-0.0793651\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.478254	−	0.878222i	$-0.341270\pi$
$827$		0			0		−0.478254	+	0.878222i	$0.658730\pi$
$828$		0			0
$829$	0.148717	−	1.98450i	0.148717	−	1.98450i	−0.0249307	−	0.999689i	$-0.507937\pi$
$829$	0.148717	−	1.98450i	0.148717	−	1.98450i	0.173648	−	0.984808i	$-0.444444\pi$
$830$		0			0
$831$	−1.97520	+	0.0985783i	−1.97520	+	0.0985783i
$832$	1.24356	+	0.452620i	1.24356	+	0.452620i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.15382	−	0.786662i	−1.15382	−	0.786662i
$838$		0			0
$839$		0			0		0.980172	−	0.198146i	$-0.0634921\pi$
$839$		0			0		−0.980172	+	0.198146i	$0.936508\pi$
$840$		0			0
$841$	0.853291	+	0.521435i	0.853291	+	0.521435i
$842$		0			0
$843$		0			0
$844$	−1.58316	+	0.361346i	−1.58316	+	0.361346i
$845$		0			0
$846$		0			0
$847$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$848$		0			0
$849$	−0.415013	+	0.762092i	−0.415013	+	0.762092i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.980503	+	0.704934i	0.980503	+	0.704934i	0.955573	−	0.294755i	$-0.0952381\pi$
$853$	0.980503	+	0.704934i	0.980503	+	0.704934i	0.0249307	+	0.999689i	$0.492063\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$857$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$858$		0			0
$859$	1.10074	+	0.496674i	1.10074	+	0.496674i	0.878222	−	0.478254i	$-0.158730\pi$
$859$	1.10074	+	0.496674i	1.10074	+	0.496674i	0.222521	+	0.974928i	$0.428571\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.955573	+	0.294755i	−0.955573	+	0.294755i
$868$	0.510189	−	1.29994i	0.510189	−	1.29994i
$869$		0			0
$870$		0			0
$871$	0.0989181	+	1.98201i	0.0989181	+	1.98201i
$872$		0			0
$873$	1.07473	+	0.997204i	1.07473	+	0.997204i
$874$		0			0
$875$		0			0
$876$	−1.44352	+	1.27391i	−1.44352	+	1.27391i
$877$	0.928021	−	1.51864i	0.928021	−	1.51864i	0.0747301	−	0.997204i	$-0.476190\pi$
$877$	0.928021	−	1.51864i	0.928021	−	1.51864i	0.853291	−	0.521435i	$-0.174603\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		0			0
$883$	0.305003	+	1.72976i	0.305003	+	1.72976i	0.623490	+	0.781831i	$0.285714\pi$
$883$	0.305003	+	1.72976i	0.305003	+	1.72976i	−0.318487	+	0.947927i	$0.603175\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$887$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$888$		0			0
$889$	1.90097	+	0.433884i	1.90097	+	0.433884i
$890$		0			0
$891$		0			0
$892$	0.813387	+	0.122598i	0.813387	+	0.122598i
$893$		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
$903$	1.52344	+	0.469918i	1.52344	+	0.469918i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.179985	+	0.237613i	0.179985	+	0.237613i	0.878222	−	0.478254i	$-0.158730\pi$
$907$	0.179985	+	0.237613i	0.179985	+	0.237613i	−0.698237	+	0.715867i	$0.746032\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$911$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$912$	0.456211	+	0.889872i	0.456211	+	0.889872i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.41129	+	0.911506i	1.41129	+	0.911506i
$917$		0			0
$918$		0			0
$919$	0.573893	+	0.276372i	0.573893	+	0.276372i	0.698237	−	0.715867i	$-0.253968\pi$
$919$	0.573893	+	0.276372i	0.573893	+	0.276372i	−0.124344	+	0.992239i	$0.539683\pi$
$920$		0			0
$921$	0.0476462	+	1.91055i	0.0476462	+	1.91055i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.493361	+	0.0123037i	−0.493361	+	0.0123037i
$926$		0			0
$927$	−0.532620	−	0.740830i	−0.532620	−	0.740830i
$928$		0			0
$929$		0			0		−0.889872	−	0.456211i	$-0.849206\pi$
$929$		0			0		0.889872	+	0.456211i	$0.150794\pi$
$930$		0			0
$931$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.98386	−	0.0494744i	1.98386	−	0.0494744i	0.988831	−	0.149042i	$-0.0476190\pi$
$937$	1.98386	−	0.0494744i	1.98386	−	0.0494744i	0.995031	+	0.0995678i	$0.0317460\pi$
$938$		0			0
$939$	−0.297251	−	1.97213i	−0.297251	−	1.97213i
$940$		0			0
$941$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$941$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$947$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$948$	0.0198275	−	0.198146i	0.0198275	−	0.198146i
$949$	2.20648	−	1.27391i	2.20648	−	1.27391i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.246757	−	0.969077i	$-0.420635\pi$
$953$		0			0		−0.246757	+	0.969077i	$0.579365\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.475069	−	0.822844i	−0.475069	−	0.822844i
$962$		0			0
$963$		0			0
$964$	−0.0329926	−	0.0373851i	−0.0329926	−	0.0373851i
$965$		0			0
$966$		0			0
$967$	1.08374	−	0.0540874i	1.08374	−	0.0540874i	0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.08374	−	0.0540874i	1.08374	−	0.0540874i	0.583744	+	0.811938i	$0.301587\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.797133	−	0.603804i	$-0.206349\pi$
$971$		0			0		−0.797133	+	0.603804i	$0.793651\pi$
$972$	0.0249307	−	0.999689i	0.0249307	−	0.999689i
$973$	0.309834	+	0.247084i	0.309834	+	0.247084i
$974$		0			0
$975$	0.875656	−	0.992239i	0.875656	−	0.992239i
$976$	−1.42773	−	0.106994i	−1.42773	−	0.106994i
$977$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$977$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.61232	−	0.930874i	1.61232	−	0.930874i
$982$		0			0
$983$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$983$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.358423	−	1.27391i	−0.358423	−	1.27391i
$989$		0			0
$990$		0			0
$991$	0.00496922	+	0.0995678i	0.00496922	+	0.0995678i	1.00000			$0$
$991$	0.00496922	+	0.0995678i	0.00496922	+	0.0995678i	−0.995031	+	0.0995678i	$0.968254\pi$
$992$		0			0
$993$	1.66701	−	1.07667i	1.66701	−	1.07667i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.317225	+	1.24582i	0.317225	+	1.24582i	0.900969	+	0.433884i	$0.142857\pi$
$997$	0.317225	+	1.24582i	0.317225	+	1.24582i	−0.583744	+	0.811938i	$0.698413\pi$
$998$		0			0
$999$	0.427396	−	0.246757i	0.427396	−	0.246757i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.1.ew.a.1370.1	yes	36
3.2	odd	2	CM	2793.1.ew.a.1370.1	yes	36
19.10	odd	18		2793.1.er.a.2252.1	yes	36
49.24	odd	42		2793.1.er.a.857.1	✓	36
57.29	even	18		2793.1.er.a.2252.1	yes	36
147.122	even	42		2793.1.er.a.857.1	✓	36
931.808	even	126	inner	2793.1.ew.a.1739.1	yes	36
2793.1739	odd	126	inner	2793.1.ew.a.1739.1	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.1.er.a.857.1	✓	36	49.24	odd	42
2793.1.er.a.857.1	✓	36	147.122	even	42
2793.1.er.a.2252.1	yes	36	19.10	odd	18
2793.1.er.a.2252.1	yes	36	57.29	even	18
2793.1.ew.a.1370.1	yes	36	1.1	even	1	trivial
2793.1.ew.a.1370.1	yes	36	3.2	odd	2	CM
2793.1.ew.a.1739.1	yes	36	931.808	even	126	inner
2793.1.ew.a.1739.1	yes	36	2793.1739	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)

\(f(q)\)	\(=\)	\(q+(0.878222 - 0.478254i) q^{3} +(0.583744 + 0.811938i) q^{4} +(0.456211 - 0.889872i) q^{7} +(0.542546 - 0.840026i) q^{9} +O(q^{10})\)	\(q+(0.878222 - 0.478254i) q^{3} +(0.583744 + 0.811938i) q^{4} +(0.456211 - 0.889872i) q^{7} +(0.542546 - 0.840026i) q^{9} +(0.900969 + 0.433884i) q^{12} +(-1.05490 - 0.799058i) q^{13} +(-0.318487 + 0.947927i) q^{16} +(0.955573 - 0.294755i) q^{19} +(-0.0249307 - 0.999689i) q^{21} +(-0.542546 + 0.840026i) q^{25} +(0.0747301 - 0.997204i) q^{27} +(0.988831 - 0.149042i) q^{28} +(0.698237 - 1.20938i) q^{31} +(0.998757 - 0.0498459i) q^{36} +(0.278007 + 0.407761i) q^{37} +(-1.30859 - 0.197238i) q^{39} +(-0.507752 + 1.51125i) q^{43} +(0.173648 + 0.984808i) q^{48} +(-0.583744 - 0.811938i) q^{49} +(0.0329926 - 1.32296i) q^{52} +(0.698237 - 0.715867i) q^{57} +(0.353291 + 1.38746i) q^{61} +(-0.500000 - 0.866025i) q^{63} +(-0.955573 + 0.294755i) q^{64} +(-0.963900 - 1.14873i) q^{67} +(-0.747834 + 1.77407i) q^{73} +(-0.0747301 + 0.997204i) q^{75} +(0.797133 + 0.603804i) q^{76} +(-0.0681084 - 0.187126i) q^{79} +(-0.411287 - 0.911506i) q^{81} +(0.797133 - 0.603804i) q^{84} +(-1.19232 + 0.574189i) q^{91} +(0.0348151 - 1.39604i) q^{93} +(-0.254586 + 1.44383i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 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

Introduction

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