LMFDB - Embedded newform 1044.1.bi.a.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1044,1,Mod(143,1044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1044, base_ring=CyclotomicField(28))

chi = DirichletCharacter(H, H._module([14, 14, 15]))

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

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

chi := DirichletCharacter("1044.143");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1044 = 2^{2} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1044.bi (of order $28$, degree $12$, 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:	$0.521023873189$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{28})$
Coefficient field:	$\Q(\zeta_{56})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} - \cdots)$

Embedding invariants

Embedding label			143.1
Root			$0.532032 + 0.846724i$ of defining polynomial
Character	$\chi$	$=$	1044.143
Dual form			1044.1.bi.a.971.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.993712 - 0.111964i) q^{2} +(0.974928 + 0.222521i) q^{4} +(1.17700 + 1.47592i) q^{5} +(-0.943883 - 0.330279i) q^{8} +O(q^{10})$	\(q+(-0.993712 - 0.111964i) q^{2} +(0.974928 + 0.222521i) q^{4} +(1.17700 + 1.47592i) q^{5} +(-0.943883 - 0.330279i) q^{8} +(-1.00435 - 1.59842i) q^{10} +(0.846011 - 1.75676i) q^{13} +(0.900969 + 0.433884i) q^{16} +(0.881748 - 0.881748i) q^{17} +(0.819071 + 1.70082i) q^{20} +(-0.570469 + 2.49939i) q^{25} +(-1.03739 + 1.65099i) q^{26} +(-0.943883 + 0.330279i) q^{29} +(-0.846724 - 0.532032i) q^{32} +(-0.974928 + 0.777479i) q^{34} +(-0.351438 + 1.00435i) q^{37} +(-0.623490 - 1.78183i) q^{40} +(-0.314692 - 0.314692i) q^{41} +(-0.900969 + 0.433884i) q^{49} +(0.846724 - 2.41980i) q^{50} +(1.21572 - 1.52446i) q^{52} +(-0.175075 + 0.139617i) q^{53} +(0.974928 - 0.222521i) q^{58} +(-1.05737 + 1.68280i) q^{61} +(0.781831 + 0.623490i) q^{64} +(3.58859 - 0.819071i) q^{65} +(1.05585 - 0.663433i) q^{68} +(-0.0250721 - 0.222521i) q^{73} +(0.461680 - 0.958689i) q^{74} +(0.420068 + 1.84044i) q^{80} +(0.277479 + 0.347948i) q^{82} +(2.33921 + 0.263565i) q^{85} +(-0.862311 - 0.0971591i) q^{89} +(-0.900969 - 1.43388i) q^{97} +(0.943883 - 0.330279i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 4 q^{10} + 4 q^{16} - 4 q^{25} - 4 q^{37} + 4 q^{40} - 4 q^{49} + 4 q^{61} - 24 q^{73} + 8 q^{82} + 8 q^{85} - 4 q^{97}+O(q^{100}) $ 24 * q - 4 * q^10 + 4 * q^16 - 4 * q^25 - 4 * q^37 + 4 * q^40 - 4 * q^49 + 4 * q^61 - 24 * q^73 + 8 * q^82 + 8 * q^85 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1044\mathbb{Z}\right)^\times$.

$n$	$523$	$901$	$929$
$\chi(n)$	$-1$	$e\left(\frac{15}{28}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.993712	−	0.111964i	−0.993712	−	0.111964i
$2$	−0.993712	−	0.111964i	−0.993712	−	0.111964i
$3$		0			0
$3$		0			0
$4$	0.974928	+	0.222521i	0.974928	+	0.222521i
$5$	1.17700	+	1.47592i	1.17700	+	1.47592i	0.846724	+	0.532032i	$0.178571\pi$
$5$	1.17700	+	1.47592i	1.17700	+	1.47592i	0.330279	+	0.943883i	$0.392857\pi$
$6$		0			0
$7$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$7$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$8$	−0.943883	−	0.330279i	−0.943883	−	0.330279i
$9$		0			0
$10$	−1.00435	−	1.59842i	−1.00435	−	1.59842i
$11$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$11$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$12$		0			0
$13$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$13$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$14$		0			0
$15$		0			0
$16$	0.900969	+	0.433884i	0.900969	+	0.433884i
$17$	0.881748	−	0.881748i	0.881748	−	0.881748i	−0.111964	−	0.993712i	$-0.535714\pi$
$17$	0.881748	−	0.881748i	0.881748	−	0.881748i	0.993712	+	0.111964i	$0.0357143\pi$
$18$		0			0
$19$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$19$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$20$	0.819071	+	1.70082i	0.819071	+	1.70082i
$21$		0			0
$22$		0			0
$23$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$23$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$24$		0			0
$25$	−0.570469	+	2.49939i	−0.570469	+	2.49939i
$26$	−1.03739	+	1.65099i	−1.03739	+	1.65099i
$27$		0			0
$28$		0			0
$29$	−0.943883	+	0.330279i	−0.943883	+	0.330279i
$29$	−0.943883	+	0.330279i	−0.943883	+	0.330279i
$30$		0			0
$31$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$31$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$32$	−0.846724	−	0.532032i	−0.846724	−	0.532032i
$33$		0			0
$34$	−0.974928	+	0.777479i	−0.974928	+	0.777479i
$35$		0			0
$36$		0			0
$37$	−0.351438	+	1.00435i	−0.351438	+	1.00435i	0.623490	+	0.781831i	$0.285714\pi$
$37$	−0.351438	+	1.00435i	−0.351438	+	1.00435i	−0.974928	+	0.222521i	$0.928571\pi$
$38$		0			0
$39$		0			0
$40$	−0.623490	−	1.78183i	−0.623490	−	1.78183i
$41$	−0.314692	−	0.314692i	−0.314692	−	0.314692i	0.532032	−	0.846724i	$-0.321429\pi$
$41$	−0.314692	−	0.314692i	−0.314692	−	0.314692i	−0.846724	+	0.532032i	$0.821429\pi$
$42$		0			0
$43$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$43$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$47$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$48$		0			0
$49$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$50$	0.846724	−	2.41980i	0.846724	−	2.41980i
$51$		0			0
$52$	1.21572	−	1.52446i	1.21572	−	1.52446i
$53$	−0.175075	+	0.139617i	−0.175075	+	0.139617i	−0.707107	−	0.707107i	$-0.750000\pi$
$53$	−0.175075	+	0.139617i	−0.175075	+	0.139617i	0.532032	+	0.846724i	$0.321429\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.974928	−	0.222521i	0.974928	−	0.222521i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−1.05737	+	1.68280i	−1.05737	+	1.68280i	−0.433884	+	0.900969i	$0.642857\pi$
$61$	−1.05737	+	1.68280i	−1.05737	+	1.68280i	−0.623490	+	0.781831i	$0.714286\pi$
$62$		0			0
$63$		0			0
$64$	0.781831	+	0.623490i	0.781831	+	0.623490i
$65$	3.58859	−	0.819071i	3.58859	−	0.819071i
$66$		0			0
$67$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$67$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$68$	1.05585	−	0.663433i	1.05585	−	0.663433i
$69$		0			0
$70$		0			0
$71$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$71$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$72$		0			0
$73$	−0.0250721	−	0.222521i	−0.0250721	−	0.222521i	0.974928	−	0.222521i	$-0.0714286\pi$
$73$	−0.0250721	−	0.222521i	−0.0250721	−	0.222521i	−1.00000			$\pi$
$74$	0.461680	−	0.958689i	0.461680	−	0.958689i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$79$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$80$	0.420068	+	1.84044i	0.420068	+	1.84044i
$81$		0			0
$82$	0.277479	+	0.347948i	0.277479	+	0.347948i
$83$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$83$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$84$		0			0
$85$	2.33921	+	0.263565i	2.33921	+	0.263565i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.862311	−	0.0971591i	−0.862311	−	0.0971591i	−0.330279	−	0.943883i	$-0.607143\pi$
$89$	−0.862311	−	0.0971591i	−0.862311	−	0.0971591i	−0.532032	+	0.846724i	$0.678571\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.900969	−	1.43388i	−0.900969	−	1.43388i	−0.900969	−	0.433884i	$-0.857143\pi$
$97$	−0.900969	−	1.43388i	−0.900969	−	1.43388i		−	1.00000i	$-0.5\pi$
$98$	0.943883	−	0.330279i	0.943883	−	0.330279i
$99$		0			0
$100$	−1.11233	+	2.30978i	−1.11233	+	2.30978i
$101$	−0.201753	−	1.79061i	−0.201753	−	1.79061i	−0.532032	−	0.846724i	$-0.678571\pi$
$101$	−0.201753	−	1.79061i	−0.201753	−	1.79061i	0.330279	−	0.943883i	$-0.392857\pi$
$102$		0			0
$103$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$103$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$104$	−1.37876	+	1.37876i	−1.37876	+	1.37876i
$105$		0			0
$106$	0.189606	−	0.119137i	0.189606	−	0.119137i
$107$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$107$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$108$		0			0
$109$	−0.433884	+	0.0990311i	−0.433884	+	0.0990311i	−0.433884	−	0.900969i	$-0.642857\pi$
$109$	−0.433884	+	0.0990311i	−0.433884	+	0.0990311i			1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.958689	−	1.52574i	0.958689	−	1.52574i	0.111964	−	0.993712i	$-0.464286\pi$
$113$	0.958689	−	1.52574i	0.958689	−	1.52574i	0.846724	−	0.532032i	$-0.178571\pi$
$114$		0			0
$115$		0			0
$116$	−0.993712	+	0.111964i	−0.993712	+	0.111964i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.781831	−	0.623490i	0.781831	−	0.623490i
$122$	1.23914	−	1.55383i	1.23914	−	1.55383i
$123$		0			0
$124$		0			0
$125$	−2.65951	+	1.28075i	−2.65951	+	1.28075i
$126$		0			0
$127$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$127$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$128$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$129$		0			0
$130$	−3.65773	+	0.412127i	−3.65773	+	0.412127i
$131$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$131$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.12349	+	0.541044i	−1.12349	+	0.541044i
$137$	−0.286605	+	0.819071i	−0.286605	+	0.819071i	0.707107	+	0.707107i	$0.250000\pi$
$137$	−0.286605	+	0.819071i	−0.286605	+	0.819071i	−0.993712	+	0.111964i	$0.964286\pi$
$138$		0			0
$139$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$139$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−1.59842	−	1.00435i	−1.59842	−	1.00435i
$146$			0.223929i			0.223929i
$147$		0			0
$148$	−0.566116	+	0.900969i	−0.566116	+	0.900969i
$149$	−0.442244	+	1.93760i	−0.442244	+	1.93760i	−0.111964	+	0.993712i	$0.535714\pi$
$149$	−0.442244	+	1.93760i	−0.442244	+	1.93760i	−0.330279	+	0.943883i	$0.607143\pi$
$150$		0			0
$151$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$151$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.467085	−	0.467085i	0.467085	−	0.467085i	−0.433884	−	0.900969i	$-0.642857\pi$
$157$	0.467085	−	0.467085i	0.467085	−	0.467085i	0.900969	+	0.433884i	$0.142857\pi$
$158$		0			0
$159$		0			0
$160$	−0.211363	−	1.87590i	−0.211363	−	1.87590i
$161$		0			0
$162$		0			0
$163$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$163$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$164$	−0.236777	−	0.376828i	−0.236777	−	0.376828i
$165$		0			0
$166$		0			0
$167$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$167$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$168$		0			0
$169$	−1.74698	−	2.19064i	−1.74698	−	2.19064i
$170$	−2.29499	−	0.523816i	−2.29499	−	0.523816i
$171$		0			0
$172$		0			0
$173$	0.223929			0.223929			0.111964	−	0.993712i	$-0.464286\pi$
$173$	0.223929			0.223929			0.111964	+	0.993712i	$0.464286\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.846011	+	0.193096i	0.846011	+	0.193096i
$179$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$179$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$180$		0			0
$181$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	−0.974928	−	0.222521i	$-0.928571\pi$
$181$	−0.193096	−	0.846011i	−0.193096	−	0.846011i	0.781831	−	0.623490i	$-0.214286\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.89598	+	0.663433i	−1.89598	+	0.663433i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	−0.559311	+	0.351438i	−0.559311	+	0.351438i	−0.781831	−	0.623490i	$-0.785714\pi$
$193$	−0.559311	+	0.351438i	−0.559311	+	0.351438i	0.222521	+	0.974928i	$0.428571\pi$
$194$	0.734760	+	1.52574i	0.734760	+	1.52574i
$195$		0			0
$196$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$197$	0.516445	+	0.411851i	0.516445	+	0.411851i	0.846724	−	0.532032i	$-0.178571\pi$
$197$	0.516445	+	0.411851i	0.516445	+	0.411851i	−0.330279	+	0.943883i	$0.607143\pi$
$198$		0			0
$199$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$199$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$200$	1.36395	−	2.17072i	1.36395	−	2.17072i
$201$		0			0
$202$			1.80194i			1.80194i
$203$		0			0
$204$		0			0
$205$	0.0940653	−	0.834853i	0.0940653	−	0.834853i
$206$		0			0
$207$		0			0
$208$	1.52446	−	1.21572i	1.52446	−	1.21572i
$209$		0			0
$210$		0			0
$211$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$211$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$212$	−0.201753	+	0.0971591i	−0.201753	+	0.0971591i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.442244	−	0.0498289i	0.442244	−	0.0498289i
$219$		0			0
$220$		0			0
$221$	−0.803051	−	2.29499i	−0.803051	−	2.29499i
$222$		0			0
$223$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$223$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$224$		0			0
$225$		0			0
$226$	−1.12349	+	1.40881i	−1.12349	+	1.40881i
$227$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$227$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$228$		0			0
$229$	1.68280	+	1.05737i	1.68280	+	1.05737i	0.900969	+	0.433884i	$0.142857\pi$
$229$	1.68280	+	1.05737i	1.68280	+	1.05737i	0.781831	+	0.623490i	$0.214286\pi$
$230$		0			0
$231$		0			0
$232$	1.00000			1.00000
$233$		−	0.660558i		−	0.660558i	−0.943883	−	0.330279i	$-0.892857\pi$
$233$		−	0.660558i		−	0.660558i	0.943883	−	0.330279i	$-0.107143\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$239$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$240$		0			0
$241$	0.541044	+	1.12349i	0.541044	+	1.12349i	0.974928	+	0.222521i	$0.0714286\pi$
$241$	0.541044	+	1.12349i	0.541044	+	1.12349i	−0.433884	+	0.900969i	$0.642857\pi$
$242$	−0.846724	+	0.532032i	−0.846724	+	0.532032i
$243$		0			0
$244$	−1.40532	+	1.40532i	−1.40532	+	1.40532i
$245$	−1.70082	−	0.819071i	−1.70082	−	0.819071i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	2.78618	−	0.974928i	2.78618	−	0.974928i
$251$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$251$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.623490	+	0.781831i	0.623490	+	0.781831i
$257$	−1.03739	−	0.236777i	−1.03739	−	0.236777i	−0.330279	−	0.943883i	$-0.607143\pi$
$257$	−1.03739	−	0.236777i	−1.03739	−	0.236777i	−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$	3.68087			3.68087
$261$		0			0
$262$		0			0
$263$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$263$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$264$		0			0
$265$	−0.412127	−	0.0940653i	−0.412127	−	0.0940653i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.70082	+	0.595142i	1.70082	+	0.595142i	0.993712	−	0.111964i	$-0.0357143\pi$
$269$	1.70082	+	0.595142i	1.70082	+	0.595142i	0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$271$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$272$	1.17700	−	0.411851i	1.17700	−	0.411851i
$273$		0			0
$274$	0.376510	−	0.781831i	0.376510	−	0.781831i
$275$		0			0
$276$		0			0
$277$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.222521	−	0.974928i	$-0.571429\pi$
$277$	−1.12349	−	0.541044i	−1.12349	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.461680	−	0.958689i	−0.461680	−	0.958689i	−0.993712	−	0.111964i	$-0.964286\pi$
$281$	−0.461680	−	0.958689i	−0.461680	−	0.958689i	0.532032	−	0.846724i	$-0.321429\pi$
$282$		0			0
$283$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$283$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	0.554958i		−	0.554958i
$290$	1.47592	+	1.17700i	1.47592	+	1.17700i
$291$		0			0
$292$	0.0250721	−	0.222521i	0.0250721	−	0.222521i
$293$	1.65099	+	1.03739i	1.65099	+	1.03739i	0.943883	+	0.330279i	$0.107143\pi$
$293$	1.65099	+	1.03739i	1.65099	+	1.03739i	0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$	0.663433	−	0.831919i	0.663433	−	0.831919i
$297$		0			0
$298$	0.656405	−	1.87590i	0.656405	−	1.87590i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.72820	+	0.420068i	−3.72820	+	0.420068i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$311$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$312$		0			0
$313$	0.974928	−	1.22252i	0.974928	−	1.22252i		−	1.00000i	$-0.5\pi$
$313$	0.974928	−	1.22252i	0.974928	−	1.22252i	0.974928	−	0.222521i	$-0.0714286\pi$
$314$	−0.516445	+	0.411851i	−0.516445	+	0.411851i
$315$		0			0
$316$		0			0
$317$	−0.175075	+	1.55383i	−0.175075	+	1.55383i	0.532032	+	0.846724i	$0.321429\pi$
$317$	−0.175075	+	1.55383i	−0.175075	+	1.55383i	−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$			1.88777i			1.88777i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	3.90820	+	3.11668i	3.90820	+	3.11668i
$326$		0			0
$327$		0			0
$328$	0.193096	+	0.400969i	0.193096	+	0.400969i
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.00435	+	0.351438i	−1.00435	+	0.351438i	−0.781831	−	0.623490i	$-0.785714\pi$
$337$	−1.00435	+	0.351438i	−1.00435	+	0.351438i	−0.222521	+	0.974928i	$0.571429\pi$
$338$	1.49072	+	2.37247i	1.49072	+	2.37247i
$339$		0			0
$340$	2.22191	+	0.777479i	2.22191	+	0.777479i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.222521	−	0.0250721i	−0.222521	−	0.0250721i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−1.56366			−1.56366			−0.781831	−	0.623490i	$-0.785714\pi$
$349$	−1.56366			−1.56366			−0.781831	+	0.623490i	$0.785714\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.05585	−	1.32399i	−1.05585	−	1.32399i	−0.943883	−	0.330279i	$-0.892857\pi$
$353$	−1.05585	−	1.32399i	−1.05585	−	1.32399i	−0.111964	−	0.993712i	$-0.535714\pi$
$354$		0			0
$355$		0			0
$356$	−0.819071	−	0.286605i	−0.819071	−	0.286605i
$357$		0			0
$358$		0			0
$359$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$359$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$360$		0			0
$361$	0.433884	−	0.900969i	0.433884	−	0.900969i
$362$	0.0971591	+	0.862311i	0.0971591	+	0.862311i
$363$		0			0
$364$		0			0
$365$	0.298912	−	0.298912i	0.298912	−	0.298912i
$366$		0			0
$367$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$367$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$368$		0			0
$369$		0			0
$370$	1.95834	−	0.446979i	1.95834	−	0.446979i
$371$		0			0
$372$		0			0
$373$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$373$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.218315	+	1.93760i	−0.218315	+	1.93760i
$378$		0			0
$379$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$379$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$383$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$384$		0			0
$385$		0			0
$386$	0.595142	−	0.286605i	0.595142	−	0.286605i
$387$		0			0
$388$	−0.559311	−	1.59842i	−0.559311	−	1.59842i
$389$	1.10568	+	1.10568i	1.10568	+	1.10568i	0.993712	+	0.111964i	$0.0357143\pi$
$389$	1.10568	+	1.10568i	1.10568	+	1.10568i	0.111964	+	0.993712i	$0.464286\pi$
$390$		0			0
$391$		0			0
$392$	0.993712	−	0.111964i	0.993712	−	0.111964i
$393$		0			0
$394$	−0.467085	−	0.467085i	−0.467085	−	0.467085i
$395$		0			0
$396$		0			0
$397$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.222521	−	0.974928i	$-0.428571\pi$
$397$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.900969	+	0.433884i	$0.142857\pi$
$398$		0			0
$399$		0			0
$400$	−1.59842	+	2.00435i	−1.59842	+	2.00435i
$401$	1.32399	−	1.05585i	1.32399	−	1.05585i	0.330279	−	0.943883i	$-0.392857\pi$
$401$	1.32399	−	1.05585i	1.32399	−	1.05585i	0.993712	−	0.111964i	$-0.0357143\pi$
$402$		0			0
$403$		0			0
$404$	0.201753	−	1.79061i	0.201753	−	1.79061i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.351438	+	0.559311i	−0.351438	+	0.559311i	−0.974928	−	0.222521i	$-0.928571\pi$
$409$	−0.351438	+	0.559311i	−0.351438	+	0.559311i	0.623490	+	0.781831i	$0.285714\pi$
$410$	−0.186948	+	0.819071i	−0.186948	+	0.819071i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.65099	+	1.03739i	−1.65099	+	1.03739i
$417$		0			0
$418$		0			0
$419$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$419$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$420$		0			0
$421$	0.158342	+	1.40532i	0.158342	+	1.40532i	0.781831	+	0.623490i	$0.214286\pi$
$421$	0.158342	+	1.40532i	0.158342	+	1.40532i	−0.623490	+	0.781831i	$0.714286\pi$
$422$		0			0
$423$		0			0
$424$	0.211363	−	0.0739590i	0.211363	−	0.0739590i
$425$	1.70082	+	2.70684i	1.70082	+	2.70684i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$431$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$432$		0			0
$433$	−1.40532	−	0.158342i	−1.40532	−	0.158342i	−0.623490	−	0.781831i	$-0.714286\pi$
$433$	−1.40532	−	0.158342i	−1.40532	−	0.158342i	−0.781831	+	0.623490i	$0.785714\pi$
$434$		0			0
$435$		0			0
$436$	−0.445042			−0.445042
$437$		0			0
$438$		0			0
$439$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$439$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$440$		0			0
$441$		0			0
$442$	0.541044	+	2.37047i	0.541044	+	2.37047i
$443$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$443$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$444$		0			0
$445$	−0.871544	−	1.38705i	−0.871544	−	1.38705i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.139617	+	1.23914i	0.139617	+	1.23914i	0.846724	+	0.532032i	$0.178571\pi$
$449$	0.139617	+	1.23914i	0.139617	+	1.23914i	−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$		0			0
$452$	1.27416	−	1.27416i	1.27416	−	1.27416i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.846011	+	0.193096i	−0.846011	+	0.193096i	−0.623490	−	0.781831i	$-0.714286\pi$
$457$	−0.846011	+	0.193096i	−0.846011	+	0.193096i	−0.222521	+	0.974928i	$0.571429\pi$
$458$	−1.55383	−	1.23914i	−1.55383	−	1.23914i
$459$		0			0
$460$		0			0
$461$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$461$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	−0.993712	−	0.111964i	−0.993712	−	0.111964i
$465$		0			0
$466$	−0.0739590	+	0.656405i	−0.0739590	+	0.656405i
$467$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$467$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$479$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$480$		0			0
$481$	1.46709	+	1.46709i	1.46709	+	1.46709i
$482$	−0.411851	−	1.17700i	−0.411851	−	1.17700i
$483$		0			0
$484$	0.900969	−	0.433884i	0.900969	−	0.433884i
$485$	1.05585	−	3.01744i	1.05585	−	3.01744i
$486$		0			0
$487$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$487$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$488$	1.55383	−	1.23914i	1.55383	−	1.23914i
$489$		0			0
$490$	1.59842	+	1.00435i	1.59842	+	1.00435i
$491$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$491$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$492$		0			0
$493$	−0.541044	+	1.12349i	−0.541044	+	1.12349i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$499$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$500$	−2.87782	+	0.656844i	−2.87782	+	0.656844i
$501$		0			0
$502$		0			0
$503$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$503$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$504$		0			0
$505$	2.40532	−	2.40532i	2.40532	−	2.40532i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.734760	−	1.52574i	0.734760	−	1.52574i	−0.111964	−	0.993712i	$-0.535714\pi$
$509$	0.734760	−	1.52574i	0.734760	−	1.52574i	0.846724	−	0.532032i	$-0.178571\pi$
$510$		0			0
$511$		0			0
$512$	−0.532032	−	0.846724i	−0.532032	−	0.846724i
$513$		0			0
$514$	1.00435	+	0.351438i	1.00435	+	0.351438i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−3.65773	−	0.412127i	−3.65773	−	0.412127i
$521$	0.660558			0.660558			0.330279	−	0.943883i	$-0.392857\pi$
$521$	0.660558			0.660558			0.330279	+	0.943883i	$0.392857\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.222521	+	0.974928i	0.222521	+	0.974928i
$530$	0.399004	+	0.139617i	0.399004	+	0.139617i
$531$		0			0
$532$		0			0
$533$	−0.819071	+	0.286605i	−0.819071	+	0.286605i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.62349	−	0.781831i	−1.62349	−	0.781831i
$539$		0			0
$540$		0			0
$541$	1.43388	−	0.900969i	1.43388	−	0.900969i	0.433884	−	0.900969i	$-0.357143\pi$
$541$	1.43388	−	0.900969i	1.43388	−	0.900969i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$	−1.21572	+	0.277479i	−1.21572	+	0.277479i
$545$	−0.656844	−	0.523816i	−0.656844	−	0.523816i
$546$		0			0
$547$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$547$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$548$	−0.461680	+	0.734760i	−0.461680	+	0.734760i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.05585	+	0.663433i	1.05585	+	0.663433i
$555$		0			0
$556$		0			0
$557$	−1.05585	+	1.32399i	−1.05585	+	1.32399i	−0.111964	+	0.993712i	$0.535714\pi$
$557$	−1.05585	+	1.32399i	−1.05585	+	1.32399i	−0.943883	+	0.330279i	$0.892857\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.351438	+	1.00435i	0.351438	+	1.00435i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	3.38025	−	0.380863i	3.38025	−	0.380863i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.643997	−	1.84044i	−0.643997	−	1.84044i	−0.532032	−	0.846724i	$-0.678571\pi$
$569$	−0.643997	−	1.84044i	−0.643997	−	1.84044i	−0.111964	−	0.993712i	$-0.535714\pi$
$570$		0			0
$571$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$571$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.43388	−	0.900969i	−1.43388	−	0.900969i	−0.433884	−	0.900969i	$-0.642857\pi$
$577$	−1.43388	−	0.900969i	−1.43388	−	0.900969i	−1.00000			$\pi$
$578$	−0.0621356	+	0.551469i	−0.0621356	+	0.551469i
$579$		0			0
$580$	−1.33485	−	1.33485i	−1.33485	−	1.33485i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.0498289	+	0.218315i	−0.0498289	+	0.218315i
$585$		0			0
$586$	−1.52446	−	1.21572i	−1.52446	−	1.21572i
$587$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$587$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.752407	+	0.752407i	−0.752407	+	0.752407i
$593$	1.79061	+	0.862311i	1.79061	+	0.862311i	0.943883	+	0.330279i	$0.107143\pi$
$593$	1.79061	+	0.862311i	1.79061	+	0.862311i	0.846724	+	0.532032i	$0.178571\pi$
$594$		0			0
$595$		0			0
$596$	−0.862311	+	1.79061i	−0.862311	+	1.79061i
$597$		0			0
$598$		0			0
$599$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$599$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$600$		0			0
$601$	1.33485	+	0.467085i	1.33485	+	0.467085i	0.900969	−	0.433884i	$-0.142857\pi$
$601$	1.33485	+	0.467085i	1.33485	+	0.467085i	0.433884	+	0.900969i	$0.357143\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.84044	+	0.420068i	1.84044	+	0.420068i
$606$		0			0
$607$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$607$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$608$		0			0
$609$		0			0
$610$	3.75179			3.75179
$611$		0			0
$612$		0			0
$613$	−1.90097	−	0.433884i	−1.90097	−	0.433884i	−0.900969	−	0.433884i	$-0.857143\pi$
$613$	−1.90097	−	0.433884i	−1.90097	−	0.433884i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.70082	−	0.595142i	−1.70082	−	0.595142i	−0.707107	−	0.707107i	$-0.750000\pi$
$617$	−1.70082	−	0.595142i	−1.70082	−	0.595142i	−0.993712	+	0.111964i	$0.964286\pi$
$618$		0			0
$619$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$619$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.71075	−	1.30543i	−2.71075	−	1.30543i
$626$	−1.10568	+	1.10568i	−1.10568	+	1.10568i
$627$		0			0
$628$	0.559311	−	0.351438i	0.559311	−	0.351438i
$629$	0.575706	+	1.19547i	0.575706	+	1.19547i
$630$		0			0
$631$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$631$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$632$		0			0
$633$		0			0
$634$	0.347948	−	1.52446i	0.347948	−	1.52446i
$635$		0			0
$636$		0			0
$637$			1.94986i			1.94986i
$638$		0			0
$639$		0			0
$640$	0.211363	−	1.87590i	0.211363	−	1.87590i
$641$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$641$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$642$		0			0
$643$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$643$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$647$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$648$		0			0
$649$		0			0
$650$	−3.53467	−	3.53467i	−3.53467	−	3.53467i
$651$		0			0
$652$		0			0
$653$	1.93760	−	0.218315i	1.93760	−	0.218315i	0.943883	−	0.330279i	$-0.107143\pi$
$653$	1.93760	−	0.218315i	1.93760	−	0.218315i	0.993712	+	0.111964i	$0.0357143\pi$
$654$		0			0
$655$		0			0
$656$	−0.146988	−	0.420068i	−0.146988	−	0.420068i
$657$		0			0
$658$		0			0
$659$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$659$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$660$		0			0
$661$	−1.21572	+	1.52446i	−1.21572	+	1.52446i	−0.433884	+	0.900969i	$0.642857\pi$
$661$	−1.21572	+	1.52446i	−1.21572	+	1.52446i	−0.781831	+	0.623490i	$0.785714\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.678448	−	0.541044i	−0.678448	−	0.541044i	0.222521	−	0.974928i	$-0.428571\pi$
$673$	−0.678448	−	0.541044i	−0.678448	−	0.541044i	−0.900969	+	0.433884i	$0.857143\pi$
$674$	1.03739	−	0.236777i	1.03739	−	0.236777i
$675$		0			0
$676$	−1.21572	−	2.52446i	−1.21572	−	2.52446i
$677$	−1.52574	+	0.958689i	−1.52574	+	0.958689i	−0.532032	+	0.846724i	$0.678571\pi$
$677$	−1.52574	+	0.958689i	−1.52574	+	0.958689i	−0.993712	+	0.111964i	$0.964286\pi$
$678$		0			0
$679$		0			0
$680$	−2.12089	−	1.02137i	−2.12089	−	1.02137i
$681$		0			0
$682$		0			0
$683$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$683$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$684$		0			0
$685$	−1.54622	+	0.541044i	−1.54622	+	0.541044i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.0971591	+	0.425682i	0.0971591	+	0.425682i
$690$		0			0
$691$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$691$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$692$	0.218315	+	0.0498289i	0.218315	+	0.0498289i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.554958			−0.554958
$698$	1.55383	+	0.175075i	1.55383	+	0.175075i
$699$		0			0
$700$		0			0
$701$	−0.139617	−	0.175075i	−0.139617	−	0.175075i	0.707107	−	0.707107i	$-0.250000\pi$
$701$	−0.139617	−	0.175075i	−0.139617	−	0.175075i	−0.846724	+	0.532032i	$0.821429\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.900969	+	1.43388i	0.900969	+	1.43388i
$707$		0			0
$708$		0			0
$709$	−0.376510	+	0.781831i	−0.376510	+	0.781831i	0.623490	+	0.781831i	$0.285714\pi$
$709$	−0.376510	+	0.781831i	−0.376510	+	0.781831i	−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	0.781831	+	0.376510i	0.781831	+	0.376510i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$719$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$720$		0			0
$721$		0			0
$722$	−0.532032	+	0.846724i	−0.532032	+	0.846724i
$723$		0			0
$724$		−	0.867767i		−	0.867767i
$725$	−0.287039	−	2.54754i	−0.287039	−	2.54754i
$726$		0			0
$727$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$727$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$728$		0			0
$729$		0			0
$730$	−0.330500	+	0.263565i	−0.330500	+	0.263565i
$731$		0			0
$732$		0			0
$733$	0.467085	−	1.33485i	0.467085	−	1.33485i	−0.433884	−	0.900969i	$-0.642857\pi$
$733$	0.467085	−	1.33485i	0.467085	−	1.33485i	0.900969	−	0.433884i	$-0.142857\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$739$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$740$	−1.99607	+	0.224904i	−1.99607	+	0.224904i
$741$		0			0
$742$		0			0
$743$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$743$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$744$		0			0
$745$	−3.38025	+	1.62784i	−3.38025	+	1.62784i
$746$	−0.595142	+	1.70082i	−0.595142	+	1.70082i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$751$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$752$		0			0
$753$		0			0
$754$	0.433884	−	1.90097i	0.433884	−	1.90097i
$755$		0			0
$756$		0			0
$757$	−1.00435	+	1.59842i	−1.00435	+	1.59842i	−0.222521	+	0.974928i	$0.571429\pi$
$757$	−1.00435	+	1.59842i	−1.00435	+	1.59842i	−0.781831	+	0.623490i	$0.785714\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.643997	+	0.146988i	−0.643997	+	0.146988i	−0.532032	−	0.846724i	$-0.678571\pi$
$761$	−0.643997	+	0.146988i	−0.643997	+	0.146988i	−0.111964	+	0.993712i	$0.535714\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.211363	+	1.87590i	0.211363	+	1.87590i	0.433884	+	0.900969i	$0.357143\pi$
$769$	0.211363	+	1.87590i	0.211363	+	1.87590i	−0.222521	+	0.974928i	$0.571429\pi$
$770$		0			0
$771$		0			0
$772$	−0.623490	+	0.218169i	−0.623490	+	0.218169i
$773$	0.461680	+	0.734760i	0.461680	+	0.734760i	0.993712	−	0.111964i	$-0.0357143\pi$
$773$	0.461680	+	0.734760i	0.461680	+	0.734760i	−0.532032	+	0.846724i	$0.678571\pi$
$774$		0			0
$775$		0			0
$776$	0.376828	+	1.65099i	0.376828	+	1.65099i
$777$		0			0
$778$	−0.974928	−	1.22252i	−0.974928	−	1.22252i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$	1.23914	+	0.139617i	1.23914	+	0.139617i
$786$		0			0
$787$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$787$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$788$	0.411851	+	0.516445i	0.411851	+	0.516445i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.06173	+	3.28122i	2.06173	+	3.28122i
$794$	−1.17700	+	0.411851i	−1.17700	+	0.411851i
$795$		0			0
$796$		0			0
$797$	0.0498289	+	0.442244i	0.0498289	+	0.442244i	0.993712	+	0.111964i	$0.0357143\pi$
$797$	0.0498289	+	0.442244i	0.0498289	+	0.442244i	−0.943883	+	0.330279i	$0.892857\pi$
$798$		0			0
$799$		0			0
$800$	1.81278	−	1.81278i	1.81278	−	1.81278i
$801$		0			0
$802$	−1.43388	+	0.900969i	−1.43388	+	0.900969i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.400969	+	1.75676i	−0.400969	+	1.75676i
$809$	0.236777	−	0.376828i	0.236777	−	0.376828i	−0.707107	−	0.707107i	$-0.750000\pi$
$809$	0.236777	−	0.376828i	0.236777	−	0.376828i	0.943883	+	0.330279i	$0.107143\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.411851	−	0.516445i	0.411851	−	0.516445i
$819$		0			0
$820$	0.277479	−	0.792990i	0.277479	−	0.792990i
$821$	−1.79061	+	0.862311i	−1.79061	+	0.862311i	−0.846724	+	0.532032i	$0.821429\pi$
$821$	−1.79061	+	0.862311i	−1.79061	+	0.862311i	−0.943883	+	0.330279i	$0.892857\pi$
$822$		0			0
$823$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$823$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$827$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$828$		0			0
$829$	−0.752407	−	0.752407i	−0.752407	−	0.752407i	0.222521	−	0.974928i	$-0.428571\pi$
$829$	−0.752407	−	0.752407i	−0.752407	−	0.752407i	−0.974928	+	0.222521i	$0.928571\pi$
$830$		0			0
$831$		0			0
$832$	1.75676	−	0.846011i	1.75676	−	0.846011i
$833$	−0.411851	+	1.17700i	−0.411851	+	1.17700i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$839$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$840$		0			0
$841$	0.781831	−	0.623490i	0.781831	−	0.623490i
$842$		−	1.41421i		−	1.41421i
$843$		0			0
$844$		0			0
$845$	1.17700	−	5.15679i	1.17700	−	5.15679i
$846$		0			0
$847$		0			0
$848$	−0.218315	+	0.0498289i	−0.218315	+	0.0498289i
$849$		0			0
$850$	−1.38705	−	2.88025i	−1.38705	−	2.88025i
$851$		0			0
$852$		0			0
$853$	1.33485	−	1.33485i	1.33485	−	1.33485i	0.433884	−	0.900969i	$-0.357143\pi$
$853$	1.33485	−	1.33485i	1.33485	−	1.33485i	0.900969	−	0.433884i	$-0.142857\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.613604	−	1.27416i	0.613604	−	1.27416i	−0.330279	−	0.943883i	$-0.607143\pi$
$857$	0.613604	−	1.27416i	0.613604	−	1.27416i	0.943883	−	0.330279i	$-0.107143\pi$
$858$		0			0
$859$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$859$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$863$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$864$		0			0
$865$	0.263565	+	0.330500i	0.263565	+	0.330500i
$866$	1.37876	+	0.314692i	1.37876	+	0.314692i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.442244	+	0.0498289i	0.442244	+	0.0498289i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$877$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.47592	−	0.516445i	1.47592	−	0.516445i	0.532032	−	0.846724i	$-0.321429\pi$
$881$	1.47592	−	0.516445i	1.47592	−	0.516445i	0.943883	+	0.330279i	$0.107143\pi$
$882$		0			0
$883$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$883$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$884$	−0.272234	−	2.41614i	−0.272234	−	2.41614i
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	0.710763	+	1.47592i	0.710763	+	1.47592i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		−	1.24698i		−	1.24698i
$899$		0			0
$900$		0			0
$901$	−0.0312644	+	0.277479i	−0.0312644	+	0.277479i
$902$		0			0
$903$		0			0
$904$	−1.40881	+	1.12349i	−1.40881	+	1.12349i
$905$	1.02137	−	1.28075i	1.02137	−	1.28075i
$906$		0			0
$907$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$907$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$	0.862311	−	0.0971591i	0.862311	−	0.0971591i
$915$		0			0
$916$	1.40532	+	1.40532i	1.40532	+	1.40532i
$917$		0			0
$918$		0			0
$919$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$919$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.30978	−	1.45133i	−2.30978	−	1.45133i
$926$		0			0
$927$		0			0
$928$	0.974928	+	0.222521i	0.974928	+	0.222521i
$929$		−	1.06406i		−	1.06406i	−0.846724	−	0.532032i	$-0.821429\pi$
$929$		−	1.06406i		−	1.06406i	0.846724	−	0.532032i	$-0.178571\pi$
$930$		0			0
$931$		0			0
$932$	0.146988	−	0.643997i	0.146988	−	0.643997i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.781831	+	1.62349i	0.781831	+	1.62349i	0.781831	+	0.623490i	$0.214286\pi$
$937$	0.781831	+	1.62349i	0.781831	+	1.62349i			1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.27416	+	0.613604i	1.27416	+	0.613604i	0.943883	−	0.330279i	$-0.107143\pi$
$941$	1.27416	+	0.613604i	1.27416	+	0.613604i	0.330279	+	0.943883i	$0.392857\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$947$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$948$		0			0
$949$	−0.412127	−	0.144209i	−0.412127	−	0.144209i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.03739	+	0.236777i	1.03739	+	0.236777i	0.707107	−	0.707107i	$-0.250000\pi$
$953$	1.03739	+	0.236777i	1.03739	+	0.236777i	0.330279	+	0.943883i	$0.392857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$962$	−1.29360	−	1.62212i	−1.29360	−	1.62212i
$963$		0			0
$964$	0.277479	+	1.21572i	0.277479	+	1.21572i
$965$	−1.17700	−	0.411851i	−1.17700	−	0.411851i
$966$		0			0
$967$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$967$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$968$	−0.943883	+	0.330279i	−0.943883	+	0.330279i
$969$		0			0
$970$	−1.38705	+	2.88025i	−1.38705	+	2.88025i
$971$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$971$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.68280	+	1.05737i	−1.68280	+	1.05737i
$977$	0.286605	+	0.595142i	0.286605	+	0.595142i	0.993712	−	0.111964i	$-0.0357143\pi$
$977$	0.286605	+	0.595142i	0.286605	+	0.595142i	−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$	−1.47592	−	1.17700i	−1.47592	−	1.17700i
$981$		0			0
$982$		0			0
$983$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$983$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$984$		0			0
$985$			1.24698i			1.24698i
$986$	0.663433	−	1.05585i	0.663433	−	1.05585i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$991$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.623490	+	1.78183i	0.623490	+	1.78183i	0.623490	+	0.781831i	$0.285714\pi$
$997$	0.623490	+	1.78183i	0.623490	+	1.78183i			1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1044.1.bi.a.143.1	✓	24
3.2	odd	2	inner	1044.1.bi.a.143.2	yes	24
4.3	odd	2	CM	1044.1.bi.a.143.1	✓	24
12.11	even	2	inner	1044.1.bi.a.143.2	yes	24
29.14	odd	28	inner	1044.1.bi.a.971.2	yes	24
87.14	even	28	inner	1044.1.bi.a.971.1	yes	24
116.43	even	28	inner	1044.1.bi.a.971.2	yes	24
348.275	odd	28	inner	1044.1.bi.a.971.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1044.1.bi.a.143.1	✓	24	1.1	even	1	trivial
1044.1.bi.a.143.1	✓	24	4.3	odd	2	CM
1044.1.bi.a.143.2	yes	24	3.2	odd	2	inner
1044.1.bi.a.143.2	yes	24	12.11	even	2	inner
1044.1.bi.a.971.1	yes	24	87.14	even	28	inner
1044.1.bi.a.971.1	yes	24	348.275	odd	28	inner
1044.1.bi.a.971.2	yes	24	29.14	odd	28	inner
1044.1.bi.a.971.2	yes	24	116.43	even	28	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:	\( N \)	\(=\)	\( 1044 = 2^{2} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1044.bi (of order \(28\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.993712 - 0.111964i) q^{2} +(0.974928 + 0.222521i) q^{4} +(1.17700 + 1.47592i) q^{5} +(-0.943883 - 0.330279i) q^{8} +O(q^{10})\)	\(q+(-0.993712 - 0.111964i) q^{2} +(0.974928 + 0.222521i) q^{4} +(1.17700 + 1.47592i) q^{5} +(-0.943883 - 0.330279i) q^{8} +(-1.00435 - 1.59842i) q^{10} +(0.846011 - 1.75676i) q^{13} +(0.900969 + 0.433884i) q^{16} +(0.881748 - 0.881748i) q^{17} +(0.819071 + 1.70082i) q^{20} +(-0.570469 + 2.49939i) q^{25} +(-1.03739 + 1.65099i) q^{26} +(-0.943883 + 0.330279i) q^{29} +(-0.846724 - 0.532032i) q^{32} +(-0.974928 + 0.777479i) q^{34} +(-0.351438 + 1.00435i) q^{37} +(-0.623490 - 1.78183i) q^{40} +(-0.314692 - 0.314692i) q^{41} +(-0.900969 + 0.433884i) q^{49} +(0.846724 - 2.41980i) q^{50} +(1.21572 - 1.52446i) q^{52} +(-0.175075 + 0.139617i) q^{53} +(0.974928 - 0.222521i) q^{58} +(-1.05737 + 1.68280i) q^{61} +(0.781831 + 0.623490i) q^{64} +(3.58859 - 0.819071i) q^{65} +(1.05585 - 0.663433i) q^{68} +(-0.0250721 - 0.222521i) q^{73} +(0.461680 - 0.958689i) q^{74} +(0.420068 + 1.84044i) q^{80} +(0.277479 + 0.347948i) q^{82} +(2.33921 + 0.263565i) q^{85} +(-0.862311 - 0.0971591i) q^{89} +(-0.900969 - 1.43388i) q^{97} +(0.943883 - 0.330279i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 4 q^{10} + 4 q^{16} - 4 q^{25} - 4 q^{37} + 4 q^{40} - 4 q^{49} + 4 q^{61} - 24 q^{73} + 8 q^{82} + 8 q^{85} - 4 q^{97}+O(q^{100}) \) 24 * q - 4 * q^10 + 4 * q^16 - 4 * q^25 - 4 * q^37 + 4 * q^40 - 4 * q^49 + 4 * q^61 - 24 * q^73 + 8 * q^82 + 8 * q^85 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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