LMFDB - Embedded newform 2299.1.t.a.892.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,1,Mod(56,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2299.56"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([2, 11])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2299.t (of order $22$, degree $10$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.14735046404$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} + \cdots)$

Embedding invariants

Embedding label			892.1
Root			$-0.841254 + 0.540641i$ of defining polynomial
Character	$\chi$	$=$	2299.892
Dual form			2299.1.t.a.683.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.142315 + 0.989821i) q^{4} +(-0.239446 - 0.153882i) q^{5} +(1.25667 - 0.368991i) q^{7} +1.00000 q^{9} +1.00000 q^{11} +(-0.959493 - 0.281733i) q^{16} +(-0.118239 - 0.258908i) q^{17} +(0.415415 - 0.909632i) q^{19} +(0.186393 - 0.215109i) q^{20} +(-0.797176 - 0.234072i) q^{23} +(-0.381761 - 0.835939i) q^{25} +(0.186393 + 1.29639i) q^{28} +(-0.357685 - 0.105026i) q^{35} +(-0.142315 + 0.989821i) q^{36} +(0.698939 - 0.449181i) q^{43} +(-0.142315 + 0.989821i) q^{44} +(-0.239446 - 0.153882i) q^{45} +(-1.10181 + 1.27155i) q^{47} +(0.601808 - 0.386758i) q^{49} +(-0.239446 - 0.153882i) q^{55} +(-1.30972 + 1.51150i) q^{61} +(1.25667 - 0.368991i) q^{63} +(0.415415 - 0.909632i) q^{64} +(0.273100 - 0.0801894i) q^{68} +(1.25667 + 0.368991i) q^{73} +(0.841254 + 0.540641i) q^{76} +(1.25667 - 0.368991i) q^{77} +(0.186393 + 0.215109i) q^{80} +1.00000 q^{81} +(-0.797176 + 0.234072i) q^{83} +(-0.0115295 + 0.0801894i) q^{85} +(0.345139 - 0.755750i) q^{92} +(-0.239446 + 0.153882i) q^{95} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{4} - 2 q^{5} - 2 q^{7} + 10 q^{9} + 10 q^{11} - q^{16} - 2 q^{17} - q^{19} - 2 q^{20} - 2 q^{23} - 3 q^{25} - 2 q^{28} - 4 q^{35} - q^{36} - 2 q^{43} - q^{44} - 2 q^{45} - 2 q^{47} - 3 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) $ 10 * q - q^4 - 2 * q^5 - 2 * q^7 + 10 * q^9 + 10 * q^11 - q^16 - 2 * q^17 - q^19 - 2 * q^20 - 2 * q^23 - 3 * q^25 - 2 * q^28 - 4 * q^35 - q^36 - 2 * q^43 - q^44 - 2 * q^45 - 2 * q^47 - 3 * q^49 - 2 * q^55 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^68 - 2 * q^73 - q^76 - 2 * q^77 - 2 * q^80 + 10 * q^81 - 2 * q^83 - 4 * q^85 + 9 * q^92 - 2 * q^95 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2299\mathbb{Z}\right)^\times$.

$n$	$970$	$1332$
$\chi(n)$	$e\left(\frac{3}{11}\right)$	$-1$

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			0		−0.654861	−	0.755750i	$-0.727273\pi$
$2$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$5$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$5$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$6$		0			0
$7$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$7$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$8$		0			0
$9$	1.00000			1.00000
$10$		0			0
$11$	1.00000			1.00000
$11$	1.00000			1.00000
$12$		0			0
$13$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$13$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$14$		0			0
$15$		0			0
$16$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$17$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$17$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$18$		0			0
$19$	0.415415	−	0.909632i	0.415415	−	0.909632i
$19$	0.415415	−	0.909632i	0.415415	−	0.909632i
$20$	0.186393	−	0.215109i	0.186393	−	0.215109i
$21$		0			0
$22$		0			0
$23$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$23$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$24$		0			0
$25$	−0.381761	−	0.835939i	−0.381761	−	0.835939i
$26$		0			0
$27$		0			0
$28$	0.186393	+	1.29639i	0.186393	+	1.29639i
$29$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$29$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$30$		0			0
$31$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$31$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.357685	−	0.105026i	−0.357685	−	0.105026i
$36$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$37$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$37$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$41$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$42$		0			0
$43$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$43$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$44$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$45$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$46$		0			0
$47$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$47$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$48$		0			0
$49$	0.601808	−	0.386758i	0.601808	−	0.386758i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$53$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$54$		0			0
$55$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$59$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$60$		0			0
$61$	−1.30972	+	1.51150i	−1.30972	+	1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$61$	−1.30972	+	1.51150i	−1.30972	+	1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$62$		0			0
$63$	1.25667	−	0.368991i	1.25667	−	0.368991i
$64$	0.415415	−	0.909632i	0.415415	−	0.909632i
$65$		0			0
$66$		0			0
$67$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$67$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$68$	0.273100	−	0.0801894i	0.273100	−	0.0801894i
$69$		0			0
$70$		0			0
$71$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$71$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$72$		0			0
$73$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$73$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$74$		0			0
$75$		0			0
$76$	0.841254	+	0.540641i	0.841254	+	0.540641i
$77$	1.25667	−	0.368991i	1.25667	−	0.368991i
$78$		0			0
$79$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$79$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$80$	0.186393	+	0.215109i	0.186393	+	0.215109i
$81$	1.00000			1.00000
$82$		0			0
$83$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$83$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$84$		0			0
$85$	−0.0115295	+	0.0801894i	−0.0115295	+	0.0801894i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$89$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$90$		0			0
$91$		0			0
$92$	0.345139	−	0.755750i	0.345139	−	0.755750i
$93$		0			0
$94$		0			0
$95$	−0.239446	+	0.153882i	−0.239446	+	0.153882i
$96$		0			0
$97$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$97$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$98$		0			0
$99$	1.00000			1.00000
$100$	0.881761	−	0.258908i	0.881761	−	0.258908i
$101$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$101$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$102$		0			0
$103$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$103$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$107$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$108$		0			0
$109$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$109$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$110$		0			0
$111$		0			0
$112$	−1.30972			−1.30972
$113$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$113$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$114$		0			0
$115$	0.154861	+	0.178719i	0.154861	+	0.178719i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.244123	−	0.281733i	−0.244123	−	0.281733i
$120$		0			0
$121$	1.00000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.0777324	+	0.540641i	−0.0777324	+	0.540641i
$126$		0			0
$127$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$127$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$131$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$132$		0			0
$133$	0.186393	−	1.29639i	0.186393	−	1.29639i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.91899	−	0.563465i	−1.91899	−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$137$	−1.91899	−	0.563465i	−1.91899	−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$138$		0			0
$139$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$139$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$140$	0.154861	−	0.339098i	0.154861	−	0.339098i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$149$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$150$		0			0
$151$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$151$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$152$		0			0
$153$	−0.118239	−	0.258908i	−0.118239	−	0.258908i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$157$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.08816			−1.08816
$162$		0			0
$163$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$163$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$167$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$168$		0			0
$169$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$170$		0			0
$171$	0.415415	−	0.909632i	0.415415	−	0.909632i
$172$	0.345139	+	0.755750i	0.345139	+	0.755750i
$173$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$173$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$174$		0			0
$175$	−0.788201	−	0.909632i	−0.788201	−	0.909632i
$176$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$177$		0			0
$178$		0			0
$179$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$179$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$180$	0.186393	−	0.215109i	0.186393	−	0.215109i
$181$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$181$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.118239	−	0.258908i	−0.118239	−	0.258908i
$188$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$189$		0			0
$190$		0			0
$191$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$191$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$192$		0			0
$193$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$193$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$194$		0			0
$195$		0			0
$196$	0.297176	+	0.650724i	0.297176	+	0.650724i
$197$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$197$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$198$		0			0
$199$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$199$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$208$		0			0
$209$	0.415415	−	0.909632i	0.415415	−	0.909632i
$210$		0			0
$211$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$211$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.236479			−0.236479
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.186393	−	0.215109i	0.186393	−	0.215109i
$221$		0			0
$222$		0			0
$223$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$223$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$224$		0			0
$225$	−0.381761	−	0.835939i	−0.381761	−	0.835939i
$226$		0			0
$227$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$227$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$228$		0			0
$229$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$229$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$233$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$234$		0			0
$235$	0.459493	−	0.134919i	0.459493	−	0.134919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$239$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	−1.30972	−	1.51150i	−1.30972	−	1.51150i
$245$	−0.203616			−0.203616
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$251$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$252$	0.186393	+	1.29639i	0.186393	+	1.29639i
$253$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$254$		0			0
$255$		0			0
$256$	0.841254	+	0.540641i	0.841254	+	0.540641i
$257$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$257$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$263$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$271$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$272$	0.0405070	+	0.281733i	0.0405070	+	0.281733i
$273$		0			0
$274$		0			0
$275$	−0.381761	−	0.835939i	−0.381761	−	0.835939i
$276$		0			0
$277$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$277$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$281$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$282$		0			0
$283$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$283$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.601808	−	0.694523i	0.601808	−	0.694523i
$290$		0			0
$291$		0			0
$292$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$293$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$293$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.712591	−	0.822373i	0.712591	−	0.822373i
$302$		0			0
$303$		0			0
$304$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$305$	0.546200	−	0.160379i	0.546200	−	0.160379i
$306$		0			0
$307$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$307$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$308$	0.186393	+	1.29639i	0.186393	+	1.29639i
$309$		0			0
$310$		0			0
$311$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$311$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$312$		0			0
$313$	0.830830	−	1.81926i	0.830830	−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$313$	0.830830	−	1.81926i	0.830830	−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$314$		0			0
$315$	−0.357685	−	0.105026i	−0.357685	−	0.105026i
$316$		0			0
$317$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$317$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$318$		0			0
$319$		0			0
$320$	−0.239446	+	0.153882i	−0.239446	+	0.153882i
$321$		0			0
$322$		0			0
$323$	−0.284630			−0.284630
$324$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.915415	+	2.00448i	−0.915415	+	2.00448i
$330$		0			0
$331$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$331$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$332$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$337$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$338$		0			0
$339$		0			0
$340$	−0.0777324	−	0.0228243i	−0.0777324	−	0.0228243i
$341$		0			0
$342$		0			0
$343$	−0.244123	+	0.281733i	−0.244123	+	0.281733i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$347$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$348$		0			0
$349$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$349$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$353$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$359$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$360$		0			0
$361$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.244123	−	0.281733i	−0.244123	−	0.281733i
$366$		0			0
$367$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$367$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$368$	0.698939	+	0.449181i	0.698939	+	0.449181i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$373$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$379$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$380$	−0.118239	−	0.258908i	−0.118239	−	0.258908i
$381$		0			0
$382$		0			0
$383$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$383$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$384$		0			0
$385$	−0.357685	−	0.105026i	−0.357685	−	0.105026i
$386$		0			0
$387$	0.698939	−	0.449181i	0.698939	−	0.449181i
$388$		0			0
$389$	1.68251	−	1.08128i	1.68251	−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$389$	1.68251	−	1.08128i	1.68251	−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$390$		0			0
$391$	0.0336545	+	0.234072i	0.0336545	+	0.234072i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$397$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$397$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$398$		0			0
$399$		0			0
$400$	0.130785	+	0.909632i	0.130785	+	0.909632i
$401$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$401$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$402$		0			0
$403$		0			0
$404$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$405$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$409$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.226900	+	0.0666238i	0.226900	+	0.0666238i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.186393	−	0.215109i	0.186393	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$419$	0.186393	−	0.215109i	0.186393	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$420$		0			0
$421$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$421$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$422$		0			0
$423$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$424$		0			0
$425$	−0.171292	+	0.197682i	−0.171292	+	0.197682i
$426$		0			0
$427$	−1.08816	+	2.38273i	−1.08816	+	2.38273i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$431$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$432$		0			0
$433$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$433$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.544078	+	0.627899i	−0.544078	+	0.627899i
$438$		0			0
$439$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$439$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$440$		0			0
$441$	0.601808	−	0.386758i	0.601808	−	0.386758i
$442$		0			0
$443$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$443$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.186393	−	1.29639i	0.186393	−	1.29639i
$449$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$449$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$457$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$458$		0			0
$459$		0			0
$460$	−0.198939	+	0.127850i	−0.198939	+	0.127850i
$461$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$461$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$463$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$467$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.698939	−	0.449181i	0.698939	−	0.449181i
$474$		0			0
$475$	−0.918986			−0.918986
$476$	0.313607	−	0.201543i	0.313607	−	0.201543i
$477$		0			0
$478$		0			0
$479$	1.41542	+	0.909632i	1.41542	+	0.909632i	1.00000			$0$
$479$	1.41542	+	0.909632i	1.41542	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$491$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$499$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$500$	−0.524075	−	0.153882i	−0.524075	−	0.153882i
$501$		0			0
$502$		0			0
$503$	0.345139	−	0.755750i	0.345139	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$503$	0.345139	−	0.755750i	0.345139	−	0.755750i	1.00000			$0$
$504$		0			0
$505$	0.459493	−	0.134919i	0.459493	−	0.134919i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$	1.71537			1.71537
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$521$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$522$		0			0
$523$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$523$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$524$	−1.91899			−1.91899
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.260554	−	0.167448i	−0.260554	−	0.167448i
$530$		0			0
$531$		0			0
$532$	1.25667	+	0.368991i	1.25667	+	0.368991i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.601808	−	0.386758i	0.601808	−	0.386758i
$540$		0			0
$541$	0.345139	−	0.755750i	0.345139	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$541$	0.345139	−	0.755750i	0.345139	−	0.755750i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$547$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$548$	0.830830	−	1.81926i	0.830830	−	1.81926i
$549$	−1.30972	+	1.51150i	−1.30972	+	1.51150i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.10181	+	0.708089i	−1.10181	+	0.708089i
$557$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$557$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$558$		0			0
$559$		0			0
$560$	0.313607	+	0.201543i	0.313607	+	0.201543i
$561$		0			0
$562$		0			0
$563$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$563$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.25667	−	0.368991i	1.25667	−	0.368991i
$568$		0			0
$569$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$569$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$570$		0			0
$571$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$571$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.108660	+	0.755750i	0.108660	+	0.755750i
$576$	0.415415	−	0.909632i	0.415415	−	0.909632i
$577$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	−0.654861	−	0.755750i	$-0.727273\pi$
$577$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	0.415415	−	0.909632i	$-0.363636\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.915415	+	0.588302i	−0.915415	+	0.588302i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$587$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$593$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$594$		0			0
$595$	0.0151004	+	0.105026i	0.0151004	+	0.105026i
$596$	0.830830			0.830830
$597$		0			0
$598$		0			0
$599$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$599$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$600$		0			0
$601$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$601$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$606$		0			0
$607$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$607$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.273100	−	0.0801894i	0.273100	−	0.0801894i
$613$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$613$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$617$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$618$		0			0
$619$	1.41542	+	0.909632i	1.41542	+	0.909632i	1.00000			$0$
$619$	1.41542	+	0.909632i	1.41542	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.577031i	−0.500000	+	0.577031i
$626$		0			0
$627$		0			0
$628$	−1.61435	−	0.474017i	−1.61435	−	0.474017i
$629$		0			0
$630$		0			0
$631$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$631$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$641$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$642$		0			0
$643$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$643$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$644$	0.154861	−	1.07708i	0.154861	−	1.07708i
$645$		0			0
$646$		0			0
$647$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$647$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.857685	−	0.989821i	0.857685	−	0.989821i
$653$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$653$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$654$		0			0
$655$	0.226900	−	0.496841i	0.226900	−	0.496841i
$656$		0			0
$657$	1.25667	+	0.368991i	1.25667	+	0.368991i
$658$		0			0
$659$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$659$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$660$		0			0
$661$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$661$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.244123	+	0.281733i	−0.244123	+	0.281733i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.30972	+	1.51150i	−1.30972	+	1.51150i
$672$		0			0
$673$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$673$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$674$		0			0
$675$		0			0
$676$	0.415415	−	0.909632i	0.415415	−	0.909632i
$677$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$677$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$683$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$684$	0.841254	+	0.540641i	0.841254	+	0.540641i
$685$	0.372786	+	0.430218i	0.372786	+	0.430218i
$686$		0			0
$687$		0			0
$688$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$689$		0			0
$690$		0			0
$691$	−1.91899	−	0.563465i	−1.91899	−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$691$	−1.91899	−	0.563465i	−1.91899	−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$692$		0			0
$693$	1.25667	−	0.368991i	1.25667	−	0.368991i
$694$		0			0
$695$	−0.0530529	−	0.368991i	−0.0530529	−	0.368991i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.01255	−	0.650724i	1.01255	−	0.650724i
$701$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$701$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$702$		0			0
$703$		0			0
$704$	0.415415	−	0.909632i	0.415415	−	0.909632i
$705$		0			0
$706$		0			0
$707$	−0.915415	+	2.00448i	−0.915415	+	2.00448i
$708$		0			0
$709$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$709$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$719$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$720$	0.186393	+	0.215109i	0.186393	+	0.215109i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.00000			2.00000			1.00000			$0$
$727$	2.00000			2.00000			1.00000			$0$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$	−0.198939	−	0.127850i	−0.198939	−	0.127850i
$732$		0			0
$733$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$733$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$739$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$743$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$744$		0			0
$745$	−0.0982369	+	0.215109i	−0.0982369	+	0.215109i
$746$		0			0
$747$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$748$	0.273100	−	0.0801894i	0.273100	−	0.0801894i
$749$		0			0
$750$		0			0
$751$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$751$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$752$	1.41542	−	0.909632i	1.41542	−	0.909632i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$757$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$761$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$		0			0
$764$	0.273100	−	0.0801894i	0.273100	−	0.0801894i
$765$	−0.0115295	+	0.0801894i	−0.0115295	+	0.0801894i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$769$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$773$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.686393	+	0.201543i	−0.686393	+	0.201543i
$785$	0.313607	−	0.361922i	0.313607	−	0.361922i
$786$		0			0
$787$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$787$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$788$	1.25667	−	1.45027i	1.25667	−	1.45027i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.345139	+	0.755750i	0.345139	+	0.755750i
$797$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$797$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$798$		0			0
$799$	0.459493	+	0.134919i	0.459493	+	0.134919i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.25667	+	0.368991i	1.25667	+	0.368991i
$804$		0			0
$805$	0.260554	+	0.167448i	0.260554	+	0.167448i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$809$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$810$		0			0
$811$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$811$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.154861	+	0.339098i	0.154861	+	0.339098i
$816$		0			0
$817$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$821$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$822$		0			0
$823$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$823$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$827$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$828$	0.345139	−	0.755750i	0.345139	−	0.755750i
$829$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$829$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.171292	−	0.110083i	−0.171292	−	0.110083i
$834$		0			0
$835$		0			0
$836$	0.841254	+	0.540641i	0.841254	+	0.540641i
$837$		0			0
$838$		0			0
$839$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$839$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$840$		0			0
$841$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.186393	+	0.215109i	0.186393	+	0.215109i
$846$		0			0
$847$	1.25667	−	0.368991i	1.25667	−	0.368991i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$853$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$854$		0			0
$855$	−0.239446	+	0.153882i	−0.239446	+	0.153882i
$856$		0			0
$857$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$857$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$858$		0			0
$859$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$859$	−0.118239	+	0.822373i	−0.118239	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$860$	0.0336545	−	0.234072i	0.0336545	−	0.234072i
$861$		0			0
$862$		0			0
$863$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$863$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.101808	+	0.708089i	0.101808	+	0.708089i
$876$		0			0
$877$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$877$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$878$		0			0
$879$		0			0
$880$	0.186393	+	0.215109i	0.186393	+	0.215109i
$881$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$881$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$882$		0			0
$883$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$883$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.00000			1.00000
$892$		0			0
$893$	0.698939	+	1.53046i	0.698939	+	1.53046i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.881761	−	0.258908i	0.881761	−	0.258908i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$907$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$910$		0			0
$911$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$911$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$912$		0			0
$913$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$914$		0			0
$915$		0			0
$916$	0.273100	+	0.0801894i	0.273100	+	0.0801894i
$917$	1.04408	+	2.28621i	1.04408	+	2.28621i
$918$		0			0
$919$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$919$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$929$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$930$		0			0
$931$	−0.101808	−	0.708089i	−0.101808	−	0.708089i
$932$	0.186393	−	1.29639i	0.186393	−	1.29639i
$933$		0			0
$934$		0			0
$935$	−0.0115295	+	0.0801894i	−0.0115295	+	0.0801894i
$936$		0			0
$937$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$937$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$938$		0			0
$939$		0			0
$940$	0.0681534	+	0.474017i	0.0681534	+	0.474017i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$947$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$953$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$	−0.0115295	+	0.0801894i	−0.0115295	+	0.0801894i
$956$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$957$		0			0
$958$		0			0
$959$	−2.61944			−2.61944
$960$		0			0
$961$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$967$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	1.44306	+	0.927399i	1.44306	+	0.927399i
$974$		0			0
$975$		0			0
$976$	1.68251	−	1.08128i	1.68251	−	1.08128i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	0.0289775	−	0.201543i	0.0289775	−	0.201543i
$981$		0			0
$982$		0			0
$983$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$983$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$984$		0			0
$985$	0.226900	+	0.496841i	0.226900	+	0.496841i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.662317	+	0.194474i	−0.662317	+	0.194474i
$990$		0			0
$991$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$991$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.236479			−0.236479
$996$		0			0
$997$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$997$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.1.t.a.892.1	yes	10
19.18	odd	2	CM	2299.1.t.a.892.1	yes	10
121.78	even	11	inner	2299.1.t.a.683.1	✓	10
2299.683	odd	22	inner	2299.1.t.a.683.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2299.1.t.a.683.1	✓	10	121.78	even	11	inner
2299.1.t.a.683.1	✓	10	2299.683	odd	22	inner
2299.1.t.a.892.1	yes	10	1.1	even	1	trivial
2299.1.t.a.892.1	yes	10	19.18	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2299.t (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.142315 + 0.989821i) q^{4} +(-0.239446 - 0.153882i) q^{5} +(1.25667 - 0.368991i) q^{7} +1.00000 q^{9} +1.00000 q^{11} +(-0.959493 - 0.281733i) q^{16} +(-0.118239 - 0.258908i) q^{17} +(0.415415 - 0.909632i) q^{19} +(0.186393 - 0.215109i) q^{20} +(-0.797176 - 0.234072i) q^{23} +(-0.381761 - 0.835939i) q^{25} +(0.186393 + 1.29639i) q^{28} +(-0.357685 - 0.105026i) q^{35} +(-0.142315 + 0.989821i) q^{36} +(0.698939 - 0.449181i) q^{43} +(-0.142315 + 0.989821i) q^{44} +(-0.239446 - 0.153882i) q^{45} +(-1.10181 + 1.27155i) q^{47} +(0.601808 - 0.386758i) q^{49} +(-0.239446 - 0.153882i) q^{55} +(-1.30972 + 1.51150i) q^{61} +(1.25667 - 0.368991i) q^{63} +(0.415415 - 0.909632i) q^{64} +(0.273100 - 0.0801894i) q^{68} +(1.25667 + 0.368991i) q^{73} +(0.841254 + 0.540641i) q^{76} +(1.25667 - 0.368991i) q^{77} +(0.186393 + 0.215109i) q^{80} +1.00000 q^{81} +(-0.797176 + 0.234072i) q^{83} +(-0.0115295 + 0.0801894i) q^{85} +(0.345139 - 0.755750i) q^{92} +(-0.239446 + 0.153882i) q^{95} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{4} - 2 q^{5} - 2 q^{7} + 10 q^{9} + 10 q^{11} - q^{16} - 2 q^{17} - q^{19} - 2 q^{20} - 2 q^{23} - 3 q^{25} - 2 q^{28} - 4 q^{35} - q^{36} - 2 q^{43} - q^{44} - 2 q^{45} - 2 q^{47} - 3 q^{49}+ \cdots + 10 q^{99}+O(q^{100}) \) 10 * q - q^4 - 2 * q^5 - 2 * q^7 + 10 * q^9 + 10 * q^11 - q^16 - 2 * q^17 - q^19 - 2 * q^20 - 2 * q^23 - 3 * q^25 - 2 * q^28 - 4 * q^35 - q^36 - 2 * q^43 - q^44 - 2 * q^45 - 2 * q^47 - 3 * q^49 - 2 * q^55 - 2 * q^61 - 2 * q^63 - q^64 - 2 * q^68 - 2 * q^73 - q^76 - 2 * q^77 - 2 * q^80 + 10 * q^81 - 2 * q^83 - 4 * q^85 + 9 * q^92 - 2 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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