LMFDB - Embedded newform 2919.1.cp.a.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2919,1,Mod(26,2919)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2919, base_ring=CyclotomicField(138))

chi = DirichletCharacter(H, H._module([69, 115, 65]))

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

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

chi := DirichletCharacter("2919.26");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2919 = 3 \cdot 7 \cdot 139 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2919.cp (of order $138$, degree $44$, 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:	$1.45677077188$
Analytic rank:	$0$
Dimension:	$44$
Coefficient field:	$\Q(\zeta_{69})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{44} - x^{43} + x^{41} - x^{40} + x^{38} - x^{37} + x^{35} - x^{34} + x^{32} - x^{31} + x^{29} + \cdots + 1 $ x^44 - x^43 + x^41 - x^40 + x^38 - x^37 + x^35 - x^34 + x^32 - x^31 + x^29 - x^28 + x^26 - x^25 + x^23 - x^22 + x^21 - x^19 + x^18 - x^16 + x^15 - x^13 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{138}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{138} - \cdots)$

Embedding invariants

Embedding label			269.1
Root			$-0.829885 - 0.557934i$ of defining polynomial
Character	$\chi$	$=$	2919.269
Dual form			2919.1.cp.a.803.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.995857 + 0.0909349i) q^{3} +(-0.898128 + 0.439735i) q^{4} +(-0.538899 - 0.842371i) q^{7} +(0.983462 + 0.181116i) q^{9} +O(q^{10})$	\(q+(0.995857 + 0.0909349i) q^{3} +(-0.898128 + 0.439735i) q^{4} +(-0.538899 - 0.842371i) q^{7} +(0.983462 + 0.181116i) q^{9} +(-0.934394 + 0.356242i) q^{12} +(-0.127129 + 1.39223i) q^{13} +(0.613267 - 0.789876i) q^{16} +(-0.267087 + 1.04418i) q^{19} +(-0.460065 - 0.887885i) q^{21} +(0.917211 - 0.398401i) q^{25} +(0.962917 + 0.269797i) q^{27} +(0.854419 + 0.519584i) q^{28} +(0.359888 + 1.95420i) q^{31} +(-0.962917 + 0.269797i) q^{36} +(1.81734 - 0.789381i) q^{37} +(-0.253205 + 1.37490i) q^{39} +(0.899946 - 0.519584i) q^{43} +(0.682553 - 0.730836i) q^{48} +(-0.419177 + 0.907905i) q^{49} +(-0.498035 - 1.30631i) q^{52} +(-0.360932 + 1.01557i) q^{57} +(0.621544 + 0.971557i) q^{61} +(-0.377419 - 0.926043i) q^{63} +(-0.203456 + 0.979084i) q^{64} +(-1.62278 - 0.986835i) q^{67} +(-0.315646 + 0.256797i) q^{73} +(0.949640 - 0.313344i) q^{75} +(-0.219284 - 1.05525i) q^{76} +(-0.582687 + 0.0265483i) q^{79} +(0.934394 + 0.356242i) q^{81} +(0.803631 + 0.595128i) q^{84} +(1.24129 - 0.643182i) q^{91} +(0.180693 + 1.97882i) q^{93} +(0.898128 - 1.55560i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + q^{3} - q^{4} - q^{7} + q^{9}+O(q^{10}) $ 44 * q + q^3 - q^4 - q^7 + q^9	$ 44 q + q^{3} - q^{4} - q^{7} + q^{9} - q^{12} - 3 q^{13} + q^{16} - q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 2 q^{28} + 3 q^{31} + 2 q^{36} - 4 q^{37} + 3 q^{39} - 2 q^{48} + q^{49} - 3 q^{52} + 2 q^{57} - 2 q^{61} - q^{63} + 2 q^{64} + 2 q^{67} - 4 q^{73} - q^{75} - 2 q^{76} - q^{79} + q^{81} + q^{84} - 3 q^{93} + q^{97}+O(q^{100}) $ 44 * q + q^3 - q^4 - q^7 + q^9 - q^12 - 3 * q^13 + q^16 - q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 2 * q^28 + 3 * q^31 + 2 * q^36 - 4 * q^37 + 3 * q^39 - 2 * q^48 + q^49 - 3 * q^52 + 2 * q^57 - 2 * q^61 - q^63 + 2 * q^64 + 2 * q^67 - 4 * q^73 - q^75 - 2 * q^76 - q^79 + q^81 + q^84 - 3 * q^93 + q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2919\mathbb{Z}\right)^\times$.

$n$	$974$	$1114$	$1669$
$\chi(n)$	$-1$	$e\left(\frac{13}{138}\right)$	$e\left(\frac{1}{6}\right)$

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		−0.225690	−	0.974199i	$-0.572464\pi$
$2$		0			0		0.225690	+	0.974199i	$0.427536\pi$
$3$	0.995857	+	0.0909349i	0.995857	+	0.0909349i
$3$	0.995857	+	0.0909349i	0.995857	+	0.0909349i
$4$	−0.898128	+	0.439735i	−0.898128	+	0.439735i
$5$		0			0		0.979084	−	0.203456i	$-0.0652174\pi$
$5$		0			0		−0.979084	+	0.203456i	$0.934783\pi$
$6$		0			0
$7$	−0.538899	−	0.842371i	−0.538899	−	0.842371i
$7$	−0.538899	−	0.842371i	−0.538899	−	0.842371i
$8$		0			0
$9$	0.983462	+	0.181116i	0.983462	+	0.181116i
$10$		0			0
$11$		0			0		0.854419	−	0.519584i	$-0.173913\pi$
$11$		0			0		−0.854419	+	0.519584i	$0.826087\pi$
$12$	−0.934394	+	0.356242i	−0.934394	+	0.356242i
$13$	−0.127129	+	1.39223i	−0.127129	+	1.39223i	0.648582	+	0.761145i	$0.275362\pi$
$13$	−0.127129	+	1.39223i	−0.127129	+	1.39223i	−0.775711	+	0.631088i	$0.782609\pi$
$14$		0			0
$15$		0			0
$16$	0.613267	−	0.789876i	0.613267	−	0.789876i
$17$		0			0		−0.715110	−	0.699012i	$-0.753623\pi$
$17$		0			0		0.715110	+	0.699012i	$0.246377\pi$
$18$		0			0
$19$	−0.267087	+	1.04418i	−0.267087	+	1.04418i	0.682553	+	0.730836i	$0.260870\pi$
$19$	−0.267087	+	1.04418i	−0.267087	+	1.04418i	−0.949640	+	0.313344i	$0.898551\pi$
$20$		0			0
$21$	−0.460065	−	0.887885i	−0.460065	−	0.887885i
$22$		0			0
$23$		0			0		0.926043	−	0.377419i	$-0.123188\pi$
$23$		0			0		−0.926043	+	0.377419i	$0.876812\pi$
$24$		0			0
$25$	0.917211	−	0.398401i	0.917211	−	0.398401i
$26$		0			0
$27$	0.962917	+	0.269797i	0.962917	+	0.269797i
$28$	0.854419	+	0.519584i	0.854419	+	0.519584i
$29$		0			0		0.898128	−	0.439735i	$-0.144928\pi$
$29$		0			0		−0.898128	+	0.439735i	$0.855072\pi$
$30$		0			0
$31$	0.359888	+	1.95420i	0.359888	+	1.95420i	0.291646	+	0.956526i	$0.405797\pi$
$31$	0.359888	+	1.95420i	0.359888	+	1.95420i	0.0682424	+	0.997669i	$0.478261\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.962917	+	0.269797i	−0.962917	+	0.269797i
$37$	1.81734	−	0.789381i	1.81734	−	0.789381i	0.854419	−	0.519584i	$-0.173913\pi$
$37$	1.81734	−	0.789381i	1.81734	−	0.789381i	0.962917	−	0.269797i	$-0.0869565\pi$
$38$		0			0
$39$	−0.253205	+	1.37490i	−0.253205	+	1.37490i
$40$		0			0
$41$		0			0		0.0455146	−	0.998964i	$-0.485507\pi$
$41$		0			0		−0.0455146	+	0.998964i	$0.514493\pi$
$42$		0			0
$43$	0.899946	−	0.519584i	0.899946	−	0.519584i	0.0227632	−	0.999741i	$-0.492754\pi$
$43$	0.899946	−	0.519584i	0.899946	−	0.519584i	0.877183	+	0.480157i	$0.159420\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.979084	−	0.203456i	$-0.934783\pi$
$47$		0			0		0.979084	+	0.203456i	$0.0652174\pi$
$48$	0.682553	−	0.730836i	0.682553	−	0.730836i
$49$	−0.419177	+	0.907905i	−0.419177	+	0.907905i
$50$		0			0
$51$		0			0
$52$	−0.498035	−	1.30631i	−0.498035	−	1.30631i
$53$		0			0		−0.956526	−	0.291646i	$-0.905797\pi$
$53$		0			0		0.956526	+	0.291646i	$0.0942029\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.360932	+	1.01557i	−0.360932	+	1.01557i
$58$		0			0
$59$		0			0		−0.898128	−	0.439735i	$-0.855072\pi$
$59$		0			0		0.898128	+	0.439735i	$0.144928\pi$
$60$		0			0
$61$	0.621544	+	0.971557i	0.621544	+	0.971557i	0.998964	+	0.0455146i	$0.0144928\pi$
$61$	0.621544	+	0.971557i	0.621544	+	0.971557i	−0.377419	+	0.926043i	$0.623188\pi$
$62$		0			0
$63$	−0.377419	−	0.926043i	−0.377419	−	0.926043i
$64$	−0.203456	+	0.979084i	−0.203456	+	0.979084i
$65$		0			0
$66$		0			0
$67$	−1.62278	−	0.986835i	−1.62278	−	0.986835i	−0.974199	−	0.225690i	$-0.927536\pi$
$67$	−1.62278	−	0.986835i	−1.62278	−	0.986835i	−0.648582	−	0.761145i	$-0.724638\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.746184	−	0.665740i	$-0.231884\pi$
$71$		0			0		−0.746184	+	0.665740i	$0.768116\pi$
$72$		0			0
$73$	−0.315646	+	0.256797i	−0.315646	+	0.256797i	−0.775711	−	0.631088i	$-0.782609\pi$
$73$	−0.315646	+	0.256797i	−0.315646	+	0.256797i	0.460065	+	0.887885i	$0.347826\pi$
$74$		0			0
$75$	0.949640	−	0.313344i	0.949640	−	0.313344i
$76$	−0.219284	−	1.05525i	−0.219284	−	1.05525i
$77$		0			0
$78$		0			0
$79$	−0.582687	+	0.0265483i	−0.582687	+	0.0265483i	−0.334880	−	0.942261i	$-0.608696\pi$
$79$	−0.582687	+	0.0265483i	−0.582687	+	0.0265483i	−0.247808	+	0.968809i	$0.579710\pi$
$80$		0			0
$81$	0.934394	+	0.356242i	0.934394	+	0.356242i
$82$		0			0
$83$		0			0		0.999741	−	0.0227632i	$-0.00724638\pi$
$83$		0			0		−0.999741	+	0.0227632i	$0.992754\pi$
$84$	0.803631	+	0.595128i	0.803631	+	0.595128i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.398401	−	0.917211i	$-0.369565\pi$
$89$		0			0		−0.398401	+	0.917211i	$0.630435\pi$
$90$		0			0
$91$	1.24129	−	0.643182i	1.24129	−	0.643182i
$92$		0			0
$93$	0.180693	+	1.97882i	0.180693	+	1.97882i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.898128	−	1.55560i	0.898128	−	1.55560i	0.0682424	−	0.997669i	$-0.478261\pi$
$97$	0.898128	−	1.55560i	0.898128	−	1.55560i	0.829885	−	0.557934i	$-0.188406\pi$
$98$		0			0
$99$		0			0
$100$	−0.648582	+	0.761145i	−0.648582	+	0.761145i
$101$		0			0		0.0227632	−	0.999741i	$-0.492754\pi$
$101$		0			0		−0.0227632	+	0.999741i	$0.507246\pi$
$102$		0			0
$103$	−0.226951	−	0.221842i	−0.226951	−	0.221842i	0.576680	−	0.816970i	$-0.304348\pi$
$103$	−0.226951	−	0.221842i	−0.226951	−	0.221842i	−0.803631	+	0.595128i	$0.797101\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.576680	−	0.816970i	$-0.695652\pi$
$107$		0			0		0.576680	+	0.816970i	$0.304348\pi$
$108$	−0.983462	+	0.181116i	−0.983462	+	0.181116i
$109$	−1.11254	−	0.948014i	−1.11254	−	0.948014i	−0.113580	−	0.993529i	$-0.536232\pi$
$109$	−1.11254	−	0.948014i	−1.11254	−	0.948014i	−0.998964	+	0.0455146i	$0.985507\pi$
$110$		0			0
$111$	1.88159	−	0.620851i	1.88159	−	0.620851i
$112$	−0.995857	−	0.0909349i	−0.995857	−	0.0909349i
$113$		0			0		0.377419	−	0.926043i	$-0.376812\pi$
$113$		0			0		−0.377419	+	0.926043i	$0.623188\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.377183	+	1.34618i	−0.377183	+	1.34618i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.460065	−	0.887885i	0.460065	−	0.887885i
$122$		0			0
$123$		0			0
$124$	−1.18255	−	1.59686i	−1.18255	−	1.59686i
$125$		0			0
$126$		0			0
$127$	−0.278387	+	1.73214i	−0.278387	+	1.73214i	0.334880	+	0.942261i	$0.391304\pi$
$127$	−0.278387	+	1.73214i	−0.278387	+	1.73214i	−0.613267	+	0.789876i	$0.710145\pi$
$128$		0			0
$129$	0.943465	−	0.435595i	0.943465	−	0.435595i
$130$		0			0
$131$		0			0		0.356242	−	0.934394i	$-0.384058\pi$
$131$		0			0		−0.356242	+	0.934394i	$0.615942\pi$
$132$		0			0
$133$	1.02352	−	0.337721i	1.02352	−	0.337721i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.682553	−	0.730836i	$-0.739130\pi$
$137$		0			0		0.682553	+	0.730836i	$0.260870\pi$
$138$		0			0
$139$	0.247808	+	0.968809i	0.247808	+	0.968809i
$139$	0.247808	+	0.968809i	0.247808	+	0.968809i
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.746184	−	0.665740i	0.746184	−	0.665740i
$145$		0			0
$146$		0			0
$147$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$148$	−1.28508	+	1.50811i	−1.28508	+	1.50811i
$149$		0			0		0.0909349	−	0.995857i	$-0.471014\pi$
$149$		0			0		−0.0909349	+	0.995857i	$0.528986\pi$
$150$		0			0
$151$	−1.25923	+	1.54781i	−1.25923	+	1.54781i	−0.576680	+	0.816970i	$0.695652\pi$
$151$	−1.25923	+	1.54781i	−1.25923	+	1.54781i	−0.682553	+	0.730836i	$0.739130\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.377183	−	1.34618i	−0.377183	−	1.34618i
$157$	−0.226925	+	0.0103391i	−0.226925	+	0.0103391i	−0.158683	−	0.987330i	$-0.550725\pi$
$157$	−0.226925	+	0.0103391i	−0.226925	+	0.0103391i	−0.0682424	+	0.997669i	$0.521739\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.00931405	−	0.136167i	−0.00931405	−	0.136167i	0.990686	−	0.136167i	$-0.0434783\pi$
$163$	−0.00931405	−	0.136167i	−0.00931405	−	0.136167i	−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.313344	−	0.949640i	$-0.601449\pi$
$167$		0			0		0.313344	+	0.949640i	$0.398551\pi$
$168$		0			0
$169$	−0.938688	−	0.172871i	−0.938688	−	0.172871i
$170$		0			0
$171$	−0.451787	+	0.978537i	−0.451787	+	0.978537i
$172$	−0.579787	+	0.862390i	−0.579787	+	0.862390i
$173$		0			0		−0.999741	−	0.0227632i	$-0.992754\pi$
$173$		0			0		0.999741	+	0.0227632i	$0.00724638\pi$
$174$		0			0
$175$	−0.829885	−	0.557934i	−0.829885	−	0.557934i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.987330	−	0.158683i	$-0.949275\pi$
$179$		0			0		0.987330	+	0.158683i	$0.0507246\pi$
$180$		0			0
$181$	−1.41503	−	0.816970i	−1.41503	−	0.816970i	−0.419177	−	0.907905i	$-0.637681\pi$
$181$	−1.41503	−	0.816970i	−1.41503	−	0.816970i	−0.995857	+	0.0909349i	$0.971014\pi$
$182$		0			0
$183$	0.530621	+	1.02405i	0.530621	+	1.02405i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.291646	−	0.956526i	−0.291646	−	0.956526i
$190$		0			0
$191$		0			0		0.158683	−	0.987330i	$-0.449275\pi$
$191$		0			0		−0.158683	+	0.987330i	$0.550725\pi$
$192$	−0.291646	+	0.956526i	−0.291646	+	0.956526i
$193$	−0.628038	−	1.21206i	−0.628038	−	1.21206i	−0.962917	−	0.269797i	$-0.913043\pi$
$193$	−0.628038	−	1.21206i	−0.628038	−	1.21206i	0.334880	−	0.942261i	$-0.391304\pi$
$194$		0			0
$195$		0			0
$196$	−0.0227632	−	0.999741i	−0.0227632	−	0.999741i
$197$		0			0		0.987330	−	0.158683i	$-0.0507246\pi$
$197$		0			0		−0.987330	+	0.158683i	$0.949275\pi$
$198$		0			0
$199$	−1.00709	−	1.57421i	−1.00709	−	1.57421i	−0.803631	−	0.595128i	$-0.797101\pi$
$199$	−1.00709	−	1.57421i	−1.00709	−	1.57421i	−0.203456	−	0.979084i	$-0.565217\pi$
$200$		0			0
$201$	−1.52632	−	1.13031i	−1.52632	−	1.13031i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.02173	+	0.954226i	1.02173	+	0.954226i
$209$		0			0
$210$		0			0
$211$	0.272263	+	0.00619918i	0.272263	+	0.00619918i	0.158683	−	0.987330i	$-0.449275\pi$
$211$	0.272263	+	0.00619918i	0.272263	+	0.00619918i	0.113580	+	0.993529i	$0.463768\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.45221	−	1.35627i	1.45221	−	1.35627i
$218$		0			0
$219$	−0.337690	+	0.227030i	−0.337690	+	0.227030i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.13288	+	1.21302i	1.13288	+	1.21302i	0.974199	+	0.225690i	$0.0724638\pi$
$223$	1.13288	+	1.21302i	1.13288	+	1.21302i	0.158683	+	0.987330i	$0.449275\pi$
$224$		0			0
$225$	0.974199	−	0.225690i	0.974199	−	0.225690i
$226$		0			0
$227$		0			0		0.334880	−	0.942261i	$-0.391304\pi$
$227$		0			0		−0.334880	+	0.942261i	$0.608696\pi$
$228$	−0.122416	−	1.07082i	−0.122416	−	1.07082i
$229$	1.31181	−	1.28228i	1.31181	−	1.28228i	0.377419	−	0.926043i	$-0.376812\pi$
$229$	1.31181	−	1.28228i	1.31181	−	1.28228i	0.934394	−	0.356242i	$-0.115942\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.480157	−	0.877183i	$-0.340580\pi$
$233$		0			0		−0.480157	+	0.877183i	$0.659420\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.582687	−	0.0265483i	−0.582687	−	0.0265483i
$238$		0			0
$239$		0			0		−0.334880	−	0.942261i	$-0.608696\pi$
$239$		0			0		0.334880	+	0.942261i	$0.391304\pi$
$240$		0			0
$241$	−1.60060	−	1.18532i	−1.60060	−	1.18532i	−0.854419	−	0.519584i	$-0.826087\pi$
$241$	−1.60060	−	1.18532i	−1.60060	−	1.18532i	−0.746184	−	0.665740i	$-0.768116\pi$
$242$		0			0
$243$	0.898128	+	0.439735i	0.898128	+	0.439735i
$244$	−0.985454	−	0.599268i	−0.985454	−	0.599268i
$245$		0			0
$246$		0			0
$247$	−1.41979	−	0.504592i	−1.41979	−	0.504592i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.269797	−	0.962917i	$-0.413043\pi$
$251$		0			0		−0.269797	+	0.962917i	$0.586957\pi$
$252$	0.746184	+	0.665740i	0.746184	+	0.665740i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.247808	−	0.968809i	−0.247808	−	0.968809i
$257$		0			0		0.956526	−	0.291646i	$-0.0942029\pi$
$257$		0			0		−0.956526	+	0.291646i	$0.905797\pi$
$258$		0			0
$259$	−1.64431	−	1.10547i	−1.64431	−	1.10547i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.917211	−	0.398401i	$-0.869565\pi$
$263$		0			0		0.917211	+	0.398401i	$0.130435\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.89141	+	0.172711i	1.89141	+	0.172711i
$269$		0			0		−0.334880	−	0.942261i	$-0.608696\pi$
$269$		0			0		0.334880	+	0.942261i	$0.391304\pi$
$270$		0			0
$271$	1.59229	+	0.218856i	1.59229	+	0.218856i	0.877183	−	0.480157i	$-0.159420\pi$
$271$	1.59229	+	0.218856i	1.59229	+	0.218856i	0.715110	+	0.699012i	$0.246377\pi$
$272$		0			0
$273$	1.29463	−	0.527641i	1.29463	−	0.527641i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.60395	−	0.926043i	−1.60395	−	0.926043i	−0.990686	−	0.136167i	$-0.956522\pi$
$277$	−1.60395	−	0.926043i	−1.60395	−	0.926043i	−0.613267	−	0.789876i	$-0.710145\pi$
$278$		0			0
$279$			1.98706i			1.98706i
$280$		0			0
$281$		0			0		0.907905	−	0.419177i	$-0.137681\pi$
$281$		0			0		−0.907905	+	0.419177i	$0.862319\pi$
$282$		0			0
$283$	0.0606018	+	0.0679245i	0.0606018	+	0.0679245i	0.775711	−	0.631088i	$-0.217391\pi$
$283$	0.0606018	+	0.0679245i	0.0606018	+	0.0679245i	−0.715110	+	0.699012i	$0.753623\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.0227632	+	0.999741i	0.0227632	+	0.999741i
$290$		0			0
$291$	1.03587	−	1.46749i	1.03587	−	1.46749i
$292$	0.170568	−	0.369437i	0.170568	−	0.369437i
$293$		0			0		−0.113580	−	0.993529i	$-0.536232\pi$
$293$		0			0		0.113580	+	0.993529i	$0.463768\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.715110	+	0.699012i	−0.715110	+	0.699012i
$301$	−0.922662	−	0.478085i	−0.922662	−	0.478085i
$302$		0			0
$303$		0			0
$304$	0.660977	+	0.851326i	0.660977	+	0.851326i
$305$		0			0
$306$		0			0
$307$	−0.868326	−	1.77350i	−0.868326	−	1.77350i	−0.576680	−	0.816970i	$-0.695652\pi$
$307$	−0.868326	−	1.77350i	−0.868326	−	1.77350i	−0.291646	−	0.956526i	$-0.594203\pi$
$308$		0			0
$309$	−0.205837	−	0.241561i	−0.205837	−	0.241561i
$310$		0			0
$311$		0			0		−0.829885	−	0.557934i	$-0.811594\pi$
$311$		0			0		0.829885	+	0.557934i	$0.188406\pi$
$312$		0			0
$313$	0.275576	−	0.562846i	0.275576	−	0.562846i	−0.715110	−	0.699012i	$-0.753623\pi$
$313$	0.275576	−	0.562846i	0.275576	−	0.562846i	0.990686	+	0.136167i	$0.0434783\pi$
$314$		0			0
$315$		0			0
$316$	0.511653	−	0.280072i	0.511653	−	0.280072i
$317$		0			0		−0.761145	−	0.648582i	$-0.775362\pi$
$317$		0			0		0.761145	+	0.648582i	$0.224638\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.995857	+	0.0909349i	−0.995857	+	0.0909349i
$325$	0.438063	+	1.32762i	0.438063	+	1.32762i
$326$		0			0
$327$	−1.02173	−	1.04526i	−1.02173	−	1.04526i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.132917	−	0.124136i	0.132917	−	0.124136i	−0.613267	−	0.789876i	$-0.710145\pi$
$331$	0.132917	−	0.124136i	0.132917	−	0.124136i	0.746184	+	0.665740i	$0.231884\pi$
$332$		0			0
$333$	1.93025	−	0.447177i	1.93025	−	0.447177i
$334$		0			0
$335$		0			0
$336$	−0.983462	−	0.181116i	−0.983462	−	0.181116i
$337$	1.99482	−	0.136449i	1.99482	−	0.136449i	0.995857	−	0.0909349i	$-0.0289855\pi$
$337$	1.99482	−	0.136449i	1.99482	−	0.136449i	0.998964	−	0.0455146i	$-0.0144928\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.990686	−	0.136167i	0.990686	−	0.136167i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.974199	−	0.225690i	$-0.927536\pi$
$347$		0			0		0.974199	+	0.225690i	$0.0724638\pi$
$348$		0			0
$349$	−1.12160	−	1.25713i	−1.12160	−	1.25713i	−0.962917	−	0.269797i	$-0.913043\pi$
$349$	−1.12160	−	1.25713i	−1.12160	−	1.25713i	−0.158683	−	0.987330i	$-0.550725\pi$
$350$		0			0
$351$	−0.498035	+	1.30631i	−0.498035	+	1.30631i
$352$		0			0
$353$		0			0		−0.746184	−	0.665740i	$-0.768116\pi$
$353$		0			0		0.746184	+	0.665740i	$0.231884\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.775711	−	0.631088i	$-0.217391\pi$
$359$		0			0		−0.775711	+	0.631088i	$0.782609\pi$
$360$		0			0
$361$	−0.141794	−	0.0776159i	−0.141794	−	0.0776159i
$362$		0			0
$363$	0.538899	−	0.842371i	0.538899	−	0.842371i
$364$	−0.832003	+	1.12350i	−0.832003	+	1.12350i
$365$		0			0
$366$		0			0
$367$	−1.80406	+	0.206240i	−1.80406	+	0.206240i	−0.949640	−	0.313344i	$-0.898551\pi$
$367$	−1.80406	+	0.206240i	−1.80406	+	0.206240i	−0.854419	+	0.519584i	$0.826087\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.03244	−	1.69778i	−1.03244	−	1.69778i
$373$	0.454538	+	0.290786i	0.454538	+	0.290786i	0.746184	−	0.665740i	$-0.231884\pi$
$373$	0.454538	+	0.290786i	0.454538	+	0.290786i	−0.291646	+	0.956526i	$0.594203\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.461101	−	0.842371i	−0.461101	−	0.842371i	0.538899	−	0.842371i	$-0.318841\pi$
$379$	−0.461101	−	0.842371i	−0.461101	−	0.842371i	−1.00000			$\pi$
$380$		0			0
$381$	−0.434745	+	1.69965i	−0.434745	+	1.69965i
$382$		0			0
$383$		0			0		0.0227632	−	0.999741i	$-0.492754\pi$
$383$		0			0		−0.0227632	+	0.999741i	$0.507246\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.979167	−	0.347996i	0.979167	−	0.347996i
$388$	−0.122581	+	1.79207i	−0.122581	+	1.79207i
$389$		0			0		−0.979084	−	0.203456i	$-0.934783\pi$
$389$		0			0		0.979084	+	0.203456i	$0.0652174\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.539792	+	1.32444i	−0.539792	+	1.32444i	0.377419	+	0.926043i	$0.376812\pi$
$397$	−0.539792	+	1.32444i	−0.539792	+	1.32444i	−0.917211	+	0.398401i	$0.869565\pi$
$398$		0			0
$399$	1.04999	−	0.243248i	1.04999	−	0.243248i
$400$	0.247808	−	0.968809i	0.247808	−	0.968809i
$401$		0			0		−0.789876	−	0.613267i	$-0.789855\pi$
$401$		0			0		0.789876	+	0.613267i	$0.210145\pi$
$402$		0			0
$403$	−2.76645	+	0.252613i	−2.76645	+	0.252613i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.24618	−	0.200285i	1.24618	−	0.200285i	0.500000	−	0.866025i	$-0.333333\pi$
$409$	1.24618	−	0.200285i	1.24618	−	0.200285i	0.746184	+	0.665740i	$0.231884\pi$
$410$		0			0
$411$		0			0
$412$	0.301382	+	0.0994444i	0.301382	+	0.0994444i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.158683	+	0.987330i	0.158683	+	0.987330i
$418$		0			0
$419$		0			0		0.291646	−	0.956526i	$-0.405797\pi$
$419$		0			0		−0.291646	+	0.956526i	$0.594203\pi$
$420$		0			0
$421$	−0.763602	−	0.513372i	−0.763602	−	0.513372i	0.113580	−	0.993529i	$-0.463768\pi$
$421$	−0.763602	−	0.513372i	−0.763602	−	0.513372i	−0.877183	+	0.480157i	$0.840580\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.483462	−	1.04714i	0.483462	−	1.04714i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.0909349	−	0.995857i	$-0.528986\pi$
$431$		0			0		0.0909349	+	0.995857i	$0.471014\pi$
$432$	0.803631	−	0.595128i	0.803631	−	0.595128i
$433$	1.23301	−	0.502526i	1.23301	−	0.502526i	0.334880	−	0.942261i	$-0.391304\pi$
$433$	1.23301	−	0.502526i	1.23301	−	0.502526i	0.898128	+	0.439735i	$0.144928\pi$
$434$		0			0
$435$		0			0
$436$	1.41608	+	0.362214i	1.41608	+	0.362214i
$437$		0			0
$438$		0			0
$439$	1.43703	+	0.402636i	1.43703	+	0.402636i	0.898128	−	0.439735i	$-0.144928\pi$
$439$	1.43703	+	0.402636i	1.43703	+	0.402636i	0.538899	+	0.842371i	$0.318841\pi$
$440$		0			0
$441$	−0.576680	+	0.816970i	−0.576680	+	0.816970i
$442$		0			0
$443$		0			0		−0.136167	−	0.990686i	$-0.543478\pi$
$443$		0			0		0.136167	+	0.990686i	$0.456522\pi$
$444$	−1.41690	+	1.38500i	−1.41690	+	1.38500i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.934394	−	0.356242i	0.934394	−	0.356242i
$449$		0			0		−0.0909349	−	0.995857i	$-0.528986\pi$
$449$		0			0		0.0909349	+	0.995857i	$0.471014\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.39477	+	1.42688i	−1.39477	+	1.42688i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.449640	+	0.552681i	−0.449640	+	0.552681i	−0.949640	−	0.313344i	$-0.898551\pi$
$457$	−0.449640	+	0.552681i	−0.449640	+	0.552681i	0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.887885	−	0.460065i	$-0.152174\pi$
$461$		0			0		−0.887885	+	0.460065i	$0.847826\pi$
$462$		0			0
$463$	0.716305	−	1.55146i	0.716305	−	1.55146i	−0.113580	−	0.993529i	$-0.536232\pi$
$463$	0.716305	−	1.55146i	0.716305	−	1.55146i	0.829885	−	0.557934i	$-0.188406\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.917211	−	0.398401i	$-0.869565\pi$
$467$		0			0		0.917211	+	0.398401i	$0.130435\pi$
$468$	−0.253205	−	1.37490i	−0.253205	−	1.37490i
$469$	0.0432337	+	1.89879i	0.0432337	+	1.89879i
$470$		0			0
$471$	−0.226925	−	0.0103391i	−0.226925	−	0.0103391i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.171028	+	1.06414i	0.171028	+	1.06414i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.113580	−	0.993529i	$-0.463768\pi$
$479$		0			0		−0.113580	+	0.993529i	$0.536232\pi$
$480$		0			0
$481$	0.867965	+	2.63051i	0.867965	+	2.63051i
$482$		0			0
$483$		0			0
$484$	−0.0227632	+	0.999741i	−0.0227632	+	0.999741i
$485$		0			0
$486$		0			0
$487$	−0.179831	+	0.776244i	−0.179831	+	0.776244i	0.803631	+	0.595128i	$0.202899\pi$
$487$	−0.179831	+	0.776244i	−0.179831	+	0.776244i	−0.983462	+	0.181116i	$0.942029\pi$
$488$		0			0
$489$	0.00310683	−	0.136449i	0.00310683	−	0.136449i
$490$		0			0
$491$		0			0		0.398401	−	0.917211i	$-0.369565\pi$
$491$		0			0		−0.398401	+	0.917211i	$0.630435\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.76428	+	0.914176i	1.76428	+	0.914176i
$497$		0			0
$498$		0			0
$499$	−0.350934	−	1.90557i	−0.350934	−	1.90557i	−0.419177	−	0.907905i	$-0.637681\pi$
$499$	−0.350934	−	1.90557i	−0.350934	−	1.90557i	0.0682424	−	0.997669i	$-0.478261\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.225690	−	0.974199i	$-0.572464\pi$
$503$		0			0		0.225690	+	0.974199i	$0.427536\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.919079	−	0.257514i	−0.919079	−	0.257514i
$508$	−0.511653	−	1.67810i	−0.511653	−	1.67810i
$509$		0			0		0.203456	−	0.979084i	$-0.434783\pi$
$509$		0			0		−0.203456	+	0.979084i	$0.565217\pi$
$510$		0			0
$511$	0.386420	+	0.127503i	0.386420	+	0.127503i
$512$		0			0
$513$	−0.538899	+	0.933400i	−0.538899	+	0.933400i
$514$		0			0
$515$		0			0
$516$	−0.655806	+	0.806094i	−0.655806	+	0.806094i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.0682424	−	0.997669i	$-0.521739\pi$
$521$		0			0		0.0682424	+	0.997669i	$0.478261\pi$
$522$		0			0
$523$	−0.350382	−	0.715630i	−0.350382	−	0.715630i	0.648582	−	0.761145i	$-0.275362\pi$
$523$	−0.350382	−	0.715630i	−0.350382	−	0.715630i	−0.998964	+	0.0455146i	$0.985507\pi$
$524$		0			0
$525$	−0.775711	−	0.631088i	−0.775711	−	0.631088i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.715110	−	0.699012i	0.715110	−	0.699012i
$530$		0			0
$531$		0			0
$532$	−0.770743	+	0.753394i	−0.770743	+	0.753394i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.19931	+	0.888149i	−1.19931	+	0.888149i	−0.995857	−	0.0909349i	$-0.971014\pi$
$541$	−1.19931	+	0.888149i	−1.19931	+	0.888149i	−0.203456	+	0.979084i	$0.565217\pi$
$542$		0			0
$543$	−1.33488	−	0.942261i	−1.33488	−	0.942261i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.01345	−	0.863574i	−1.01345	−	0.863574i	−0.0227632	−	0.999741i	$-0.507246\pi$
$547$	−1.01345	−	0.863574i	−1.01345	−	0.863574i	−0.990686	+	0.136167i	$0.956522\pi$
$548$		0			0
$549$	0.435300	+	1.06806i	0.435300	+	1.06806i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.336373	+	0.476532i	0.336373	+	0.476532i
$554$		0			0
$555$		0			0
$556$	−0.648582	−	0.761145i	−0.648582	−	0.761145i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	0.608972	+	1.31899i	0.608972	+	1.31899i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.926043	−	0.377419i	$-0.123188\pi$
$563$		0			0		−0.926043	+	0.377419i	$0.876812\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.203456	−	0.979084i	−0.203456	−	0.979084i
$568$		0			0
$569$		0			0		−0.419177	−	0.907905i	$-0.637681\pi$
$569$		0			0		0.419177	+	0.907905i	$0.362319\pi$
$570$		0			0
$571$	−0.474199	+	1.09172i	−0.474199	+	1.09172i	0.500000	+	0.866025i	$0.333333\pi$
$571$	−0.474199	+	1.09172i	−0.474199	+	1.09172i	−0.974199	+	0.225690i	$0.927536\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.377419	+	0.926043i	−0.377419	+	0.926043i
$577$	−0.515217	+	0.551663i	−0.515217	+	0.551663i	−0.934394	−	0.356242i	$-0.884058\pi$
$577$	−0.515217	+	0.551663i	−0.515217	+	0.551663i	0.419177	+	0.907905i	$0.362319\pi$
$578$		0			0
$579$	−0.515217	−	1.26415i	−0.515217	−	1.26415i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.761145	−	0.648582i	$-0.224638\pi$
$587$		0			0		−0.761145	+	0.648582i	$0.775362\pi$
$588$	0.0682424	−	0.997669i	0.0682424	−	0.997669i
$589$	−2.13665	−	0.146151i	−2.13665	−	0.146151i
$590$		0			0
$591$		0			0
$592$	0.490999	−	1.91957i	0.490999	−	1.91957i
$593$		0			0		−0.398401	−	0.917211i	$-0.630435\pi$
$593$		0			0		0.398401	+	0.917211i	$0.369565\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.859764	−	1.65927i	−0.859764	−	1.65927i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	0.579787	+	0.953419i	0.579787	+	0.953419i	0.998964	+	0.0455146i	$0.0144928\pi$
$601$	0.579787	+	0.953419i	0.579787	+	0.953419i	−0.419177	+	0.907905i	$0.637681\pi$
$602$		0			0
$603$	−1.41721	−	1.26443i	−1.41721	−	1.26443i
$604$	0.450328	−	1.94386i	0.450328	−	1.94386i
$605$		0			0
$606$		0			0
$607$	0.542303	−	0.462104i	0.542303	−	0.462104i	−0.334880	−	0.942261i	$-0.608696\pi$
$607$	0.542303	−	0.462104i	0.542303	−	0.462104i	0.877183	+	0.480157i	$0.159420\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.489575	+	0.574541i	0.489575	+	0.574541i	0.949640	−	0.313344i	$-0.101449\pi$
$613$	0.489575	+	0.574541i	0.489575	+	0.574541i	−0.460065	+	0.887885i	$0.652174\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.0455146	−	0.998964i	$-0.514493\pi$
$617$		0			0		0.0455146	+	0.998964i	$0.485507\pi$
$618$		0			0
$619$	0.599444	−	1.81671i	0.599444	−	1.81671i	0.0227632	−	0.999741i	$-0.492754\pi$
$619$	0.599444	−	1.81671i	0.599444	−	1.81671i	0.576680	−	0.816970i	$-0.304348\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.930721	+	1.04318i	0.930721	+	1.04318i
$625$	0.682553	−	0.730836i	0.682553	−	0.730836i
$626$		0			0
$627$		0			0
$628$	0.199261	−	0.109073i	0.199261	−	0.109073i
$629$		0			0
$630$		0			0
$631$	−1.87304	−	0.389222i	−1.87304	−	0.389222i	−0.877183	−	0.480157i	$-0.840580\pi$
$631$	−1.87304	−	0.389222i	−1.87304	−	0.389222i	−0.995857	+	0.0909349i	$0.971014\pi$
$632$		0			0
$633$	0.270571	+	0.0309317i	0.270571	+	0.0309317i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.21073	−	0.699012i	−1.21073	−	0.699012i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.887885	−	0.460065i	$-0.152174\pi$
$641$		0			0		−0.887885	+	0.460065i	$0.847826\pi$
$642$		0			0
$643$	0.951436	+	0.774051i	0.951436	+	0.774051i	0.974199	−	0.225690i	$-0.0724638\pi$
$643$	0.951436	+	0.774051i	0.951436	+	0.774051i	−0.0227632	+	0.999741i	$0.507246\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.956526	−	0.291646i	$-0.0942029\pi$
$647$		0			0		−0.956526	+	0.291646i	$0.905797\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.56953	−	1.21860i	1.56953	−	1.21860i
$652$	0.0682424	+	0.118199i	0.0682424	+	0.118199i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.356936	+	0.195382i	−0.356936	+	0.195382i
$658$		0			0
$659$		0			0		−0.269797	−	0.962917i	$-0.586957\pi$
$659$		0			0		0.269797	+	0.962917i	$0.413043\pi$
$660$		0			0
$661$	−1.36088	+	0.744926i	−1.36088	+	0.744926i	−0.983462	−	0.181116i	$-0.942029\pi$
$661$	−1.36088	+	0.744926i	−1.36088	+	0.744926i	−0.377419	+	0.926043i	$0.623188\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.01788	+	1.31101i	1.01788	+	1.31101i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.42874	−	1.39658i	1.42874	−	1.39658i	0.682553	−	0.730836i	$-0.260870\pi$
$673$	1.42874	−	1.39658i	1.42874	−	1.39658i	0.746184	−	0.665740i	$-0.231884\pi$
$674$		0			0
$675$	0.990686	−	0.136167i	0.990686	−	0.136167i
$676$	0.919079	−	0.257514i	0.919079	−	0.257514i
$677$		0			0		−0.887885	−	0.460065i	$-0.847826\pi$
$677$		0			0		0.887885	+	0.460065i	$0.152174\pi$
$678$		0			0
$679$	−1.79439	+	0.0817558i	−1.79439	+	0.0817558i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.775711	−	0.631088i	$-0.782609\pi$
$683$		0			0		0.775711	+	0.631088i	$0.217391\pi$
$684$	−0.0245341	−	1.07752i	−0.0245341	−	1.07752i
$685$		0			0
$686$		0			0
$687$	1.42298	−	1.15768i	1.42298	−	1.15768i
$688$	0.141500	−	1.02949i	0.141500	−	1.02949i
$689$		0			0
$690$		0			0
$691$	−0.134228	+	1.96234i	−0.134228	+	1.96234i	0.113580	+	0.993529i	$0.463768\pi$
$691$	−0.134228	+	1.96234i	−0.134228	+	1.96234i	−0.247808	+	0.968809i	$0.579710\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.990686	+	0.136167i	0.990686	+	0.136167i
$701$		0			0		0.990686	−	0.136167i	$-0.0434783\pi$
$701$		0			0		−0.990686	+	0.136167i	$0.956522\pi$
$702$		0			0
$703$	0.338869	+	2.10846i	0.338869	+	2.10846i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.43426	−	0.662195i	−1.43426	−	0.662195i	−0.460065	−	0.887885i	$-0.652174\pi$
$709$	−1.43426	−	0.662195i	−1.43426	−	0.662195i	−0.974199	+	0.225690i	$0.927536\pi$
$710$		0			0
$711$	−0.577859	−	0.0794249i	−0.577859	−	0.0794249i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.997669	−	0.0682424i	$-0.0217391\pi$
$719$		0			0		−0.997669	+	0.0682424i	$0.978261\pi$
$720$		0			0
$721$	−0.0645698	+	0.310727i	−0.0645698	+	0.310727i
$722$		0			0
$723$	−1.48618	−	1.32596i	−1.48618	−	1.32596i
$724$	1.63013	+	0.111504i	1.63013	+	0.111504i
$725$		0			0
$726$		0			0
$727$	−1.70176	+	0.155393i	−1.70176	+	0.155393i	−0.898128	−	0.439735i	$-0.855072\pi$
$727$	−1.70176	+	0.155393i	−1.70176	+	0.155393i	−0.803631	+	0.595128i	$0.797101\pi$
$728$		0			0
$729$	0.854419	+	0.519584i	0.854419	+	0.519584i
$730$		0			0
$731$		0			0
$732$	−0.926877	−	0.686397i	−0.926877	−	0.686397i
$733$	−0.0944966	+	0.155393i	−0.0944966	+	0.155393i	−0.898128	−	0.439735i	$-0.855072\pi$
$733$	−0.0944966	+	0.155393i	−0.0944966	+	0.155393i	0.803631	+	0.595128i	$0.202899\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.94638	+	0.0886806i	1.94638	+	0.0886806i	0.983462	−	0.181116i	$-0.0579710\pi$
$739$	1.94638	+	0.0886806i	1.94638	+	0.0886806i	0.962917	+	0.269797i	$0.0869565\pi$
$740$		0			0
$741$	−1.36802	−	0.631610i	−1.36802	−	0.631610i
$742$		0			0
$743$		0			0		−0.730836	−	0.682553i	$-0.760870\pi$
$743$		0			0		0.730836	+	0.682553i	$0.239130\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.579787	−	0.953419i	−0.579787	−	0.953419i	−0.998964	−	0.0455146i	$-0.985507\pi$
$751$	−0.579787	−	0.953419i	−0.579787	−	0.953419i	0.419177	−	0.907905i	$-0.362319\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.682553	+	0.730836i	0.682553	+	0.730836i
$757$	0.750796	+	1.72850i	0.750796	+	1.72850i	0.682553	+	0.730836i	$0.260870\pi$
$757$	0.750796	+	1.72850i	0.750796	+	1.72850i	0.0682424	+	0.997669i	$0.478261\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.987330	−	0.158683i	$-0.0507246\pi$
$761$		0			0		−0.987330	+	0.158683i	$0.949275\pi$
$762$		0			0
$763$	−0.199031	+	1.44806i	−0.199031	+	1.44806i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.158683	−	0.987330i	−0.158683	−	0.987330i
$769$	1.76640	+	0.767255i	1.76640	+	0.767255i	0.990686	+	0.136167i	$0.0434783\pi$
$769$	1.76640	+	0.767255i	1.76640	+	0.767255i	0.775711	+	0.631088i	$0.217391\pi$
$770$		0			0
$771$		0			0
$772$	1.09704	+	0.812413i	1.09704	+	0.812413i
$773$		0			0		−0.887885	−	0.460065i	$-0.847826\pi$
$773$		0			0		0.887885	+	0.460065i	$0.152174\pi$
$774$		0			0
$775$	1.10865	+	1.64903i	1.10865	+	1.64903i
$776$		0			0
$777$	−1.53697	−	1.25042i	−1.53697	−	1.25042i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.460065	+	0.887885i	0.460065	+	0.887885i
$785$		0			0
$786$		0			0
$787$	−1.89731	+	0.626038i	−1.89731	+	0.626038i	−0.934394	+	0.356242i	$0.884058\pi$
$787$	−1.89731	+	0.626038i	−1.89731	+	0.626038i	−0.962917	+	0.269797i	$0.913043\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.43165	+	0.741821i	−1.43165	+	0.741821i
$794$		0			0
$795$		0			0
$796$	1.59673	+	0.970992i	1.59673	+	0.970992i
$797$		0			0		0.113580	−	0.993529i	$-0.463768\pi$
$797$		0			0		−0.113580	+	0.993529i	$0.536232\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.86787	+	0.343990i	1.86787	+	0.343990i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.398401	−	0.917211i	$-0.630435\pi$
$809$		0			0		0.398401	+	0.917211i	$0.369565\pi$
$810$		0			0
$811$	−0.130763	+	0.951369i	−0.130763	+	0.951369i	0.803631	+	0.595128i	$0.202899\pi$
$811$	−0.130763	+	0.951369i	−0.130763	+	0.951369i	−0.934394	+	0.356242i	$0.884058\pi$
$812$		0			0
$813$	1.56579	+	0.362744i	1.56579	+	0.362744i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.302176	+	1.07848i	0.302176	+	1.07848i
$818$		0			0
$819$	1.33725	−	0.407728i	1.33725	−	0.407728i
$820$		0			0
$821$		0			0		0.907905	−	0.419177i	$-0.137681\pi$
$821$		0			0		−0.907905	+	0.419177i	$0.862319\pi$
$822$		0			0
$823$	0.325617	+	0.535454i	0.325617	+	0.535454i	0.974199	−	0.225690i	$-0.0724638\pi$
$823$	0.325617	+	0.535454i	0.325617	+	0.535454i	−0.648582	+	0.761145i	$0.724638\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.789876	−	0.613267i	$-0.789855\pi$
$827$		0			0		0.789876	+	0.613267i	$0.210145\pi$
$828$		0			0
$829$	0.686586	−	0.612568i	0.686586	−	0.612568i	−0.247808	−	0.968809i	$-0.579710\pi$
$829$	0.686586	−	0.612568i	0.686586	−	0.612568i	0.934394	+	0.356242i	$0.115942\pi$
$830$		0			0
$831$	−1.51310	−	1.06806i	−1.51310	−	1.06806i
$832$	−1.33725	−	0.407728i	−1.33725	−	0.407728i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.180693	+	1.97882i	−0.180693	+	1.97882i
$838$		0			0
$839$		0			0		0.313344	−	0.949640i	$-0.398551\pi$
$839$		0			0		−0.313344	+	0.949640i	$0.601449\pi$
$840$		0			0
$841$	0.613267	−	0.789876i	0.613267	−	0.789876i
$842$		0			0
$843$		0			0
$844$	−0.247253	+	0.114156i	−0.247253	+	0.114156i
$845$		0			0
$846$		0			0
$847$	−0.995857	+	0.0909349i	−0.995857	+	0.0909349i
$848$		0			0
$849$	0.0541740	+	0.0731539i	0.0541740	+	0.0731539i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.478951	−	0.468170i	−0.478951	−	0.468170i	0.419177	−	0.907905i	$-0.362319\pi$
$853$	−0.478951	−	0.468170i	−0.478951	−	0.468170i	−0.898128	+	0.439735i	$0.855072\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.613267	−	0.789876i	$-0.289855\pi$
$857$		0			0		−0.613267	+	0.789876i	$0.710145\pi$
$858$		0			0
$859$	−1.84010	−	0.210360i	−1.84010	−	0.210360i	−0.877183	−	0.480157i	$-0.840580\pi$
$859$	−1.84010	−	0.210360i	−1.84010	−	0.210360i	−0.962917	+	0.269797i	$0.913043\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.829885	−	0.557934i	$-0.811594\pi$
$863$		0			0		0.829885	+	0.557934i	$0.188406\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.0682424	+	0.997669i	−0.0682424	+	0.997669i
$868$	−0.707873	+	1.85669i	−0.707873	+	1.85669i
$869$		0			0
$870$		0			0
$871$	1.58021	−	2.13383i	1.58021	−	2.13383i
$872$		0			0
$873$	1.16502	−	1.36721i	1.16502	−	1.36721i
$874$		0			0
$875$		0			0
$876$	0.203456	−	0.352396i	0.203456	−	0.352396i
$877$	1.57254	+	0.907905i	1.57254	+	0.907905i	0.995857	+	0.0909349i	$0.0289855\pi$
$877$	1.57254	+	0.907905i	1.57254	+	0.907905i	0.576680	+	0.816970i	$0.304348\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.665740	−	0.746184i	$-0.268116\pi$
$881$		0			0		−0.665740	+	0.746184i	$0.731884\pi$
$882$		0			0
$883$	−0.398292	−	1.55713i	−0.398292	−	1.55713i	−0.775711	−	0.631088i	$-0.782609\pi$
$883$	−0.398292	−	1.55713i	−0.398292	−	1.55713i	0.377419	−	0.926043i	$-0.376812\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.682553	−	0.730836i	$-0.739130\pi$
$887$		0			0		0.682553	+	0.730836i	$0.260870\pi$
$888$		0			0
$889$	1.60912	−	0.698941i	1.60912	−	0.698941i
$890$		0			0
$891$		0			0
$892$	−1.55088	−	0.591280i	−1.55088	−	0.591280i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.775711	+	0.631088i	−0.775711	+	0.631088i
$901$		0			0
$902$		0			0
$903$	−0.875365	−	0.560006i	−0.875365	−	0.560006i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.960699	+	1.18086i	0.960699	+	1.18086i	0.983462	+	0.181116i	$0.0579710\pi$
$907$	0.960699	+	1.18086i	0.960699	+	1.18086i	−0.0227632	+	0.999741i	$0.507246\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.917211	−	0.398401i	$-0.130435\pi$
$911$		0			0		−0.917211	+	0.398401i	$0.869565\pi$
$912$	0.580823	+	0.907905i	0.580823	+	0.907905i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.614311	+	1.72850i	−0.614311	+	1.72850i
$917$		0			0
$918$		0			0
$919$	−0.0808235	+	1.77393i	−0.0808235	+	1.77393i	0.419177	+	0.907905i	$0.362319\pi$
$919$	−0.0808235	+	1.77393i	−0.0808235	+	1.77393i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−0.703456	−	1.84511i	−0.703456	−	1.84511i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.35239	−	1.44806i	1.35239	−	1.44806i
$926$		0			0
$927$	−0.183018	−	0.259278i	−0.183018	−	0.259278i
$928$		0			0
$929$		0			0		0.538899	−	0.842371i	$-0.318841\pi$
$929$		0			0		−0.538899	+	0.842371i	$0.681159\pi$
$930$		0			0
$931$	−0.836060	−	0.680185i	−0.836060	−	0.680185i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.85442	−	0.519584i	1.85442	−	0.519584i	0.854419	−	0.519584i	$-0.173913\pi$
$937$	1.85442	−	0.519584i	1.85442	−	0.519584i	1.00000			$0$
$938$		0			0
$939$	0.325617	−	0.535454i	0.325617	−	0.535454i
$940$		0			0
$941$		0			0		−0.907905	−	0.419177i	$-0.862319\pi$
$941$		0			0		0.907905	+	0.419177i	$0.137681\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.990686	−	0.136167i	$-0.0434783\pi$
$947$		0			0		−0.990686	+	0.136167i	$0.956522\pi$
$948$	0.535002	−	0.232384i	0.535002	−	0.232384i
$949$	−0.317394	−	0.472099i	−0.317394	−	0.472099i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.789876	−	0.613267i	$-0.210145\pi$
$953$		0			0		−0.789876	+	0.613267i	$0.789855\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.75497	+	1.05034i	−2.75497	+	1.05034i
$962$		0			0
$963$		0			0
$964$	1.95877	+	0.360732i	1.95877	+	0.360732i
$965$		0			0
$966$		0			0
$967$	−0.247253	−	1.79890i	−0.247253	−	1.79890i	−0.538899	−	0.842371i	$-0.681159\pi$
$967$	−0.247253	−	1.79890i	−0.247253	−	1.79890i	0.291646	−	0.956526i	$-0.405797\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.225690	−	0.974199i	$-0.572464\pi$
$971$		0			0		0.225690	+	0.974199i	$0.427536\pi$
$972$	−1.00000			−1.00000
$973$	0.682553	−	0.730836i	0.682553	−	0.730836i
$974$		0			0
$975$	0.315521	+	1.36195i	0.315521	+	1.36195i
$976$	1.14858	+	0.104881i	1.14858	+	0.104881i
$977$		0			0		0.898128	−	0.439735i	$-0.144928\pi$
$977$		0			0		−0.898128	+	0.439735i	$0.855072\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.922444	−	1.13384i	−0.922444	−	1.13384i
$982$		0			0
$983$		0			0		−0.419177	−	0.907905i	$-0.637681\pi$
$983$		0			0		0.419177	+	0.907905i	$0.362319\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.49704	−	0.171141i	1.49704	−	0.171141i
$989$		0			0
$990$		0			0
$991$	0.0692864	+	1.52071i	0.0692864	+	1.52071i	0.682553	+	0.730836i	$0.260870\pi$
$991$	0.0692864	+	1.52071i	0.0692864	+	1.52071i	−0.613267	+	0.789876i	$0.710145\pi$
$992$		0			0
$993$	0.143655	−	0.111535i	0.143655	−	0.111535i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.09253	−	1.62505i	−1.09253	−	1.62505i	−0.715110	−	0.699012i	$-0.753623\pi$
$997$	−1.09253	−	1.62505i	−1.09253	−	1.62505i	−0.377419	−	0.926043i	$-0.623188\pi$
$998$		0			0
$999$	1.96292	−	0.269797i	1.96292	−	0.269797i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2919.1.cp.a.269.1	yes	44
3.2	odd	2	CM	2919.1.cp.a.269.1	yes	44
7.5	odd	6		2919.1.cc.a.1937.1	✓	44
21.5	even	6		2919.1.cc.a.1937.1	✓	44
139.108	odd	138		2919.1.cc.a.2054.1	yes	44
417.386	even	138		2919.1.cc.a.2054.1	yes	44
973.803	even	138	inner	2919.1.cp.a.803.1	yes	44
2919.803	odd	138	inner	2919.1.cp.a.803.1	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2919.1.cc.a.1937.1	✓	44	7.5	odd	6
2919.1.cc.a.1937.1	✓	44	21.5	even	6
2919.1.cc.a.2054.1	yes	44	139.108	odd	138
2919.1.cc.a.2054.1	yes	44	417.386	even	138
2919.1.cp.a.269.1	yes	44	1.1	even	1	trivial
2919.1.cp.a.269.1	yes	44	3.2	odd	2	CM
2919.1.cp.a.803.1	yes	44	973.803	even	138	inner
2919.1.cp.a.803.1	yes	44	2919.803	odd	138	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:	\( N \)	\(=\)	\( 2919 = 3 \cdot 7 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2919.cp (of order \(138\), degree \(44\), minimal)

\(f(q)\)	\(=\)	\(q+(0.995857 + 0.0909349i) q^{3} +(-0.898128 + 0.439735i) q^{4} +(-0.538899 - 0.842371i) q^{7} +(0.983462 + 0.181116i) q^{9} +O(q^{10})\)	\(q+(0.995857 + 0.0909349i) q^{3} +(-0.898128 + 0.439735i) q^{4} +(-0.538899 - 0.842371i) q^{7} +(0.983462 + 0.181116i) q^{9} +(-0.934394 + 0.356242i) q^{12} +(-0.127129 + 1.39223i) q^{13} +(0.613267 - 0.789876i) q^{16} +(-0.267087 + 1.04418i) q^{19} +(-0.460065 - 0.887885i) q^{21} +(0.917211 - 0.398401i) q^{25} +(0.962917 + 0.269797i) q^{27} +(0.854419 + 0.519584i) q^{28} +(0.359888 + 1.95420i) q^{31} +(-0.962917 + 0.269797i) q^{36} +(1.81734 - 0.789381i) q^{37} +(-0.253205 + 1.37490i) q^{39} +(0.899946 - 0.519584i) q^{43} +(0.682553 - 0.730836i) q^{48} +(-0.419177 + 0.907905i) q^{49} +(-0.498035 - 1.30631i) q^{52} +(-0.360932 + 1.01557i) q^{57} +(0.621544 + 0.971557i) q^{61} +(-0.377419 - 0.926043i) q^{63} +(-0.203456 + 0.979084i) q^{64} +(-1.62278 - 0.986835i) q^{67} +(-0.315646 + 0.256797i) q^{73} +(0.949640 - 0.313344i) q^{75} +(-0.219284 - 1.05525i) q^{76} +(-0.582687 + 0.0265483i) q^{79} +(0.934394 + 0.356242i) q^{81} +(0.803631 + 0.595128i) q^{84} +(1.24129 - 0.643182i) q^{91} +(0.180693 + 1.97882i) q^{93} +(0.898128 - 1.55560i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + q^{3} - q^{4} - q^{7} + q^{9}+O(q^{10}) \) 44 * q + q^3 - q^4 - q^7 + q^9	\( 44 q + q^{3} - q^{4} - q^{7} + q^{9} - q^{12} - 3 q^{13} + q^{16} - q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 2 q^{28} + 3 q^{31} + 2 q^{36} - 4 q^{37} + 3 q^{39} - 2 q^{48} + q^{49} - 3 q^{52} + 2 q^{57} - 2 q^{61} - q^{63} + 2 q^{64} + 2 q^{67} - 4 q^{73} - q^{75} - 2 q^{76} - q^{79} + q^{81} + q^{84} - 3 q^{93} + q^{97}+O(q^{100}) \) 44 * q + q^3 - q^4 - q^7 + q^9 - q^12 - 3 * q^13 + q^16 - q^19 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 2 * q^28 + 3 * q^31 + 2 * q^36 - 4 * q^37 + 3 * q^39 - 2 * q^48 + q^49 - 3 * q^52 + 2 * q^57 - 2 * q^61 - q^63 + 2 * q^64 + 2 * q^67 - 4 * q^73 - q^75 - 2 * q^76 - q^79 + q^81 + q^84 - 3 * q^93 + q^97

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