LMFDB - Embedded newform 2883.1.z.a.314.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2883,1,Mod(2,2883)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2883, base_ring=CyclotomicField(310))

chi = DirichletCharacter(H, H._module([155, 208]))

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

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

chi := DirichletCharacter("2883.2");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2883 = 3 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2883.z (of order $310$, degree $120$, 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:	$1.43880443142$
Analytic rank:	$0$
Dimension:	$120$
Coefficient field:	$\Q(\zeta_{155})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{120} - x^{119} + x^{115} - x^{114} + x^{110} - x^{109} + x^{105} - x^{104} + x^{100} - x^{99} + \cdots + 1 $ x^120 - x^119 + x^115 - x^114 + x^110 - x^109 + x^105 - x^104 + x^100 - x^99 + x^95 - x^94 + x^90 - x^88 + x^85 - x^83 + x^80 - x^78 + x^75 - x^73 + x^70 - x^68 + x^65 - x^63 + x^60 - x^57 + x^55 - x^52 + x^50 - x^47 + x^45 - x^42 + x^40 - x^37 + x^35 - x^32 + x^30 - x^26 + x^25 - x^21 + x^20 - x^16 + x^15 - x^11 + x^10 - x^6 + x^5 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{155}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{155} - \cdots)$

Embedding invariants

Embedding label			314.1
Root			$-0.366239 - 0.930521i$ of defining polynomial
Character	$\chi$	$=$	2883.314
Dual form			2883.1.z.a.101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.659028 - 0.752118i) q^{3} +(0.864331 - 0.502923i) q^{4} +(-1.01784 + 1.37149i) q^{7} +(-0.131363 - 0.991334i) q^{9} +(0.191362 - 0.981520i) q^{12} +(0.348965 + 0.644010i) q^{13} +(0.494138 - 0.869384i) q^{16} +(1.98360 + 0.0804526i) q^{19} +(0.360738 + 1.66939i) q^{21} +(0.151428 - 0.988468i) q^{25} +(-0.832173 - 0.554517i) q^{27} +(-0.189998 + 1.69732i) q^{28} +(-0.476416 - 0.879220i) q^{31} +(-0.612106 - 0.790776i) q^{36} +(0.530536 - 0.558124i) q^{37} +(0.714349 + 0.161958i) q^{39} +(0.759847 + 0.272692i) q^{43} +(-0.328229 - 0.944598i) q^{48} +(-0.555313 - 1.83481i) q^{49} +(0.625508 + 0.381135i) q^{52} +(1.36776 - 1.43888i) q^{57} +(-1.65754 + 0.920012i) q^{61} +(1.49332 + 0.828858i) q^{63} +(-0.0101340 - 0.999949i) q^{64} +(0.0498988 - 0.0347306i) q^{67} +(1.30163 + 1.07237i) q^{73} +(-0.643650 - 0.765320i) q^{75} +(1.75495 - 0.928059i) q^{76} +(-1.28455 + 1.05829i) q^{79} +(-0.965487 + 0.260450i) q^{81} +(1.15137 + 1.26148i) q^{84} +(-1.23845 - 0.176897i) q^{91} +(-0.975249 - 0.221109i) q^{93} +(-0.489324 + 0.437605i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 120 q + q^{3} + q^{4} + 2 q^{7} + q^{9} + q^{12} + 2 q^{13} + q^{16} + 2 q^{19} - 3 q^{21} - 4 q^{25} + q^{27} - 3 q^{28} + q^{31} - 4 q^{36} + 2 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + 3 q^{49}+ \cdots - 3 q^{97}+O(q^{100}) $ 120 * q + q^3 + q^4 + 2 * q^7 + q^9 + q^12 + 2 * q^13 + q^16 + 2 * q^19 - 3 * q^21 - 4 * q^25 + q^27 - 3 * q^28 + q^31 - 4 * q^36 + 2 * q^37 - 3 * q^39 - 3 * q^43 + q^48 + 3 * q^49 + 2 * q^52 + 2 * q^57 + 2 * q^61 + 2 * q^63 + q^64 + 2 * q^67 - 29 * q^73 + q^75 - 3 * q^76 - 3 * q^79 + q^81 + 2 * q^84 - 32 * q^91 + q^93 - 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2883\mathbb{Z}\right)^\times$.

$n$	$962$	$964$
$\chi(n)$	$-1$	$e\left(\frac{113}{155}\right)$

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		0.965487	−	0.260450i	$-0.0838710\pi$
$2$		0			0		−0.965487	+	0.260450i	$0.916129\pi$
$3$	0.659028	−	0.752118i	0.659028	−	0.752118i
$3$	0.659028	−	0.752118i	0.659028	−	0.752118i
$4$	0.864331	−	0.502923i	0.864331	−	0.502923i
$5$		0			0		0.758758	−	0.651372i	$-0.225806\pi$
$5$		0			0		−0.758758	+	0.651372i	$0.774194\pi$
$6$		0			0
$7$	−1.01784	+	1.37149i	−1.01784	+	1.37149i	−0.0910811	+	0.995843i	$0.529032\pi$
$7$	−1.01784	+	1.37149i	−1.01784	+	1.37149i	−0.926761	+	0.375650i	$0.877419\pi$
$8$		0			0
$9$	−0.131363	−	0.991334i	−0.131363	−	0.991334i
$10$		0			0
$11$		0			0		−0.0708797	−	0.997485i	$-0.522581\pi$
$11$		0			0		0.0708797	+	0.997485i	$0.477419\pi$
$12$	0.191362	−	0.981520i	0.191362	−	0.981520i
$13$	0.348965	+	0.644010i	0.348965	+	0.644010i	0.992615	−	0.121311i	$-0.0387097\pi$
$13$	0.348965	+	0.644010i	0.348965	+	0.644010i	−0.643650	+	0.765320i	$0.722581\pi$
$14$		0			0
$15$		0			0
$16$	0.494138	−	0.869384i	0.494138	−	0.869384i
$17$		0			0		−0.999795	−	0.0202670i	$-0.993548\pi$
$17$		0			0		0.999795	+	0.0202670i	$0.00645161\pi$
$18$		0			0
$19$	1.98360	+	0.0804526i	1.98360	+	0.0804526i	0.996715	−	0.0809846i	$-0.0258065\pi$
$19$	1.98360	+	0.0804526i	1.98360	+	0.0804526i	0.986883	+	0.161437i	$0.0516129\pi$
$20$		0			0
$21$	0.360738	+	1.66939i	0.360738	+	1.66939i
$22$		0			0
$23$		0			0		−0.230981	−	0.972958i	$-0.574194\pi$
$23$		0			0		0.230981	+	0.972958i	$0.425806\pi$
$24$		0			0
$25$	0.151428	−	0.988468i	0.151428	−	0.988468i
$26$		0			0
$27$	−0.832173	−	0.554517i	−0.832173	−	0.554517i
$28$	−0.189998	+	1.69732i	−0.189998	+	1.69732i
$29$		0			0		0.385023	−	0.922907i	$-0.374194\pi$
$29$		0			0		−0.385023	+	0.922907i	$0.625806\pi$
$30$		0			0
$31$	−0.476416	−	0.879220i	−0.476416	−	0.879220i
$31$	−0.476416	−	0.879220i	−0.476416	−	0.879220i
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.612106	−	0.790776i	−0.612106	−	0.790776i
$37$	0.530536	−	0.558124i	0.530536	−	0.558124i	−0.403648	−	0.914914i	$-0.632258\pi$
$37$	0.530536	−	0.558124i	0.530536	−	0.558124i	0.934184	+	0.356791i	$0.116129\pi$
$38$		0			0
$39$	0.714349	+	0.161958i	0.714349	+	0.161958i
$40$		0			0
$41$		0			0		0.986883	−	0.161437i	$-0.0516129\pi$
$41$		0			0		−0.986883	+	0.161437i	$0.948387\pi$
$42$		0			0
$43$	0.759847	+	0.272692i	0.759847	+	0.272692i	0.688967	−	0.724793i	$-0.258065\pi$
$43$	0.759847	+	0.272692i	0.759847	+	0.272692i	0.0708797	+	0.997485i	$0.477419\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.853961	−	0.520337i	$-0.174194\pi$
$47$		0			0		−0.853961	+	0.520337i	$0.825806\pi$
$48$	−0.328229	−	0.944598i	−0.328229	−	0.944598i
$49$	−0.555313	−	1.83481i	−0.555313	−	1.83481i
$50$		0			0
$51$		0			0
$52$	0.625508	+	0.381135i	0.625508	+	0.381135i
$53$		0			0		−0.511656	−	0.859190i	$-0.670968\pi$
$53$		0			0		0.511656	+	0.859190i	$0.329032\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.36776	−	1.43888i	1.36776	−	1.43888i
$58$		0			0
$59$		0			0		−0.476416	−	0.879220i	$-0.658065\pi$
$59$		0			0		0.476416	+	0.879220i	$0.341935\pi$
$60$		0			0
$61$	−1.65754	+	0.920012i	−1.65754	+	0.920012i	−0.674136	+	0.738607i	$0.735484\pi$
$61$	−1.65754	+	0.920012i	−1.65754	+	0.920012i	−0.983408	+	0.181405i	$0.941935\pi$
$62$		0			0
$63$	1.49332	+	0.828858i	1.49332	+	0.828858i
$64$	−0.0101340	−	0.999949i	−0.0101340	−	0.999949i
$65$		0			0
$66$		0			0
$67$	0.0498988	−	0.0347306i	0.0498988	−	0.0347306i	−0.546055	−	0.837749i	$-0.683871\pi$
$67$	0.0498988	−	0.0347306i	0.0498988	−	0.0347306i	0.595954	+	0.803019i	$0.296774\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.270221	−	0.962798i	$-0.412903\pi$
$71$		0			0		−0.270221	+	0.962798i	$0.587097\pi$
$72$		0			0
$73$	1.30163	+	1.07237i	1.30163	+	1.07237i	0.992615	+	0.121311i	$0.0387097\pi$
$73$	1.30163	+	1.07237i	1.30163	+	1.07237i	0.309017	+	0.951057i	$0.400000\pi$
$74$		0			0
$75$	−0.643650	−	0.765320i	−0.643650	−	0.765320i
$76$	1.75495	−	0.928059i	1.75495	−	0.928059i
$77$		0			0
$78$		0			0
$79$	−1.28455	+	1.05829i	−1.28455	+	1.05829i	−0.289679	+	0.957124i	$0.593548\pi$
$79$	−1.28455	+	1.05829i	−1.28455	+	1.05829i	−0.994869	+	0.101168i	$0.967742\pi$
$80$		0			0
$81$	−0.965487	+	0.260450i	−0.965487	+	0.260450i
$82$		0			0
$83$		0			0		0.965487	−	0.260450i	$-0.0838710\pi$
$83$		0			0		−0.965487	+	0.260450i	$0.916129\pi$
$84$	1.15137	+	1.26148i	1.15137	+	1.26148i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.947876	−	0.318639i	$-0.103226\pi$
$89$		0			0		−0.947876	+	0.318639i	$0.896774\pi$
$90$		0			0
$91$	−1.23845	−	0.176897i	−1.23845	−	0.176897i
$92$		0			0
$93$	−0.975249	−	0.221109i	−0.975249	−	0.221109i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.489324	+	0.437605i	−0.489324	+	0.437605i	−0.874347	−	0.485302i	$-0.838710\pi$
$97$	−0.489324	+	0.437605i	−0.489324	+	0.437605i	0.385023	+	0.922907i	$0.374194\pi$
$98$		0			0
$99$		0			0
$100$	−0.366239	−	0.930521i	−0.366239	−	0.930521i
$101$		0			0		0.628007	−	0.778208i	$-0.283871\pi$
$101$		0			0		−0.628007	+	0.778208i	$0.716129\pi$
$102$		0			0
$103$	0.219572	+	1.96151i	0.219572	+	1.96151i	0.270221	+	0.962798i	$0.412903\pi$
$103$	0.219572	+	1.96151i	0.219572	+	1.96151i	−0.0506492	+	0.998717i	$0.516129\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.131363	−	0.991334i	$-0.541935\pi$
$107$		0			0		0.131363	+	0.991334i	$0.458065\pi$
$108$	−0.998152	−	0.0607676i	−0.998152	−	0.0607676i
$109$	−1.66901	−	0.839775i	−1.66901	−	0.839775i	−0.994869	−	0.101168i	$-0.967742\pi$
$109$	−1.66901	−	0.839775i	−1.66901	−	0.839775i	−0.674136	−	0.738607i	$-0.735484\pi$
$110$		0			0
$111$	−0.0701366	−	0.766845i	−0.0701366	−	0.766845i
$112$	0.689400	+	1.56260i	0.689400	+	1.56260i
$113$		0			0		0.999795	−	0.0202670i	$-0.00645161\pi$
$113$		0			0		−0.999795	+	0.0202670i	$0.993548\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.592588	−	0.430540i	0.592588	−	0.430540i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.989952	+	0.141403i	−0.989952	+	0.141403i
$122$		0			0
$123$		0			0
$124$	−0.853961	−	0.520337i	−0.853961	−	0.520337i
$125$		0			0
$126$		0			0
$127$	−0.0179170	+	1.76791i	−0.0179170	+	1.76791i	0.458499	+	0.888695i	$0.348387\pi$
$127$	−0.0179170	+	1.76791i	−0.0179170	+	1.76791i	−0.476416	+	0.879220i	$0.658065\pi$
$128$		0			0
$129$	0.705857	−	0.391783i	0.705857	−	0.391783i
$130$		0			0
$131$		0			0		0.328229	−	0.944598i	$-0.393548\pi$
$131$		0			0		−0.328229	+	0.944598i	$0.606452\pi$
$132$		0			0
$133$	−2.12933	+	2.63860i	−2.12933	+	2.63860i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.784532	−	0.620088i	$-0.787097\pi$
$137$		0			0		0.784532	+	0.620088i	$0.212903\pi$
$138$		0			0
$139$	−1.78631	+	0.853880i	−1.78631	+	0.853880i	−0.832173	+	0.554517i	$0.812903\pi$
$139$	−1.78631	+	0.853880i	−1.78631	+	0.853880i	−0.954139	+	0.299363i	$0.903226\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.926761	−	0.375650i	−0.926761	−	0.375650i
$145$		0			0
$146$		0			0
$147$	−1.74596	−	0.791528i	−1.74596	−	0.791528i
$148$	0.177866	−	0.749222i	0.177866	−	0.749222i
$149$		0			0		−0.874347	−	0.485302i	$-0.838710\pi$
$149$		0			0		0.874347	+	0.485302i	$0.161290\pi$
$150$		0			0
$151$	0.380454	−	1.09490i	0.380454	−	1.09490i	−0.579556	−	0.814932i	$-0.696774\pi$
$151$	0.380454	−	1.09490i	0.380454	−	1.09490i	0.960010	−	0.279964i	$-0.0903226\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.698887	−	0.219277i	0.698887	−	0.219277i
$157$	−0.506988	−	1.14914i	−0.506988	−	1.14914i	−0.965487	−	0.260450i	$-0.916129\pi$
$157$	−0.506988	−	1.14914i	−0.506988	−	1.14914i	0.458499	−	0.888695i	$-0.348387\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.00855528	−	0.0183739i	−0.00855528	−	0.0183739i	0.902221	−	0.431273i	$-0.141935\pi$
$163$	−0.00855528	−	0.0183739i	−0.00855528	−	0.0183739i	−0.910777	+	0.412899i	$0.864516\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.579556	−	0.814932i	$-0.696774\pi$
$167$		0			0		0.579556	+	0.814932i	$0.303226\pi$
$168$		0			0
$169$	0.253083	−	0.388276i	0.253083	−	0.388276i
$170$		0			0
$171$	−0.180817	−	1.97698i	−0.180817	−	1.97698i
$172$	0.793902	−	0.146448i	0.793902	−	0.146448i
$173$		0			0		0.111245	−	0.993793i	$-0.464516\pi$
$173$		0			0		−0.111245	+	0.993793i	$0.535484\pi$
$174$		0			0
$175$	1.20155	+	1.21379i	1.20155	+	1.21379i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.458499	−	0.888695i	$-0.651613\pi$
$179$		0			0		0.458499	+	0.888695i	$0.348387\pi$
$180$		0			0
$181$	0.117689	−	0.188815i	0.117689	−	0.188815i	−0.784532	−	0.620088i	$-0.787097\pi$
$181$	0.117689	−	0.188815i	0.117689	−	0.188815i	0.902221	+	0.431273i	$0.141935\pi$
$182$		0			0
$183$	−0.400411	+	1.85298i	−0.400411	+	1.85298i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.60754	−	0.576909i	1.60754	−	0.576909i
$190$		0			0
$191$		0			0		0.918958	−	0.394356i	$-0.129032\pi$
$191$		0			0		−0.918958	+	0.394356i	$0.870968\pi$
$192$	−0.758758	−	0.651372i	−0.758758	−	0.651372i
$193$	−1.17213	−	0.216218i	−1.17213	−	0.216218i	−0.440394	−	0.897805i	$-0.645161\pi$
$193$	−1.17213	−	0.216218i	−1.17213	−	0.216218i	−0.731738	+	0.681586i	$0.761290\pi$
$194$		0			0
$195$		0			0
$196$	−1.40274	−	1.30660i	−1.40274	−	1.30660i
$197$		0			0		0.595954	−	0.803019i	$-0.296774\pi$
$197$		0			0		−0.595954	+	0.803019i	$0.703226\pi$
$198$		0			0
$199$	0.162725	+	0.0818764i	0.162725	+	0.0818764i	0.528964	−	0.848644i	$-0.322581\pi$
$199$	0.162725	+	0.0818764i	0.162725	+	0.0818764i	−0.366239	+	0.930521i	$0.619355\pi$
$200$		0			0
$201$	0.00676322	−	0.0604183i	0.00676322	−	0.0604183i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.732328	+	0.0148451i	0.732328	+	0.0148451i
$209$		0			0
$210$		0			0
$211$	−1.79518	+	0.182552i	−1.79518	+	0.182552i	−0.941224	−	0.337784i	$-0.890323\pi$
$211$	−1.79518	+	0.182552i	−1.79518	+	0.182552i	−0.853961	+	0.520337i	$0.825806\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.69076	+	0.241505i	1.69076	+	0.241505i
$218$		0			0
$219$	1.66436	−	0.272261i	1.66436	−	0.272261i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.244361	+	0.392042i	0.244361	+	0.392042i	0.947876	−	0.318639i	$-0.103226\pi$
$223$	0.244361	+	0.392042i	0.244361	+	0.392042i	−0.703515	+	0.710681i	$0.748387\pi$
$224$		0			0
$225$	−0.999795	−	0.0202670i	−0.999795	−	0.0202670i
$226$		0			0
$227$		0			0		−0.0101340	−	0.999949i	$-0.503226\pi$
$227$		0			0		0.0101340	+	0.999949i	$0.496774\pi$
$228$	0.458550	−	1.93155i	0.458550	−	1.93155i
$229$	0.460683	+	1.41784i	0.460683	+	1.41784i	0.864331	+	0.502923i	$0.167742\pi$
$229$	0.460683	+	1.41784i	0.460683	+	1.41784i	−0.403648	+	0.914914i	$0.632258\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.796938	−	0.604061i	$-0.206452\pi$
$233$		0			0		−0.796938	+	0.604061i	$0.793548\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.0505925	+	1.66358i	−0.0505925	+	1.66358i
$238$		0			0
$239$		0			0		0.983408	−	0.181405i	$-0.0580645\pi$
$239$		0			0		−0.983408	+	0.181405i	$0.941935\pi$
$240$		0			0
$241$	1.30611	−	0.439064i	1.30611	−	0.439064i	0.422108	−	0.906546i	$-0.361290\pi$
$241$	1.30611	−	0.439064i	1.30611	−	0.439064i	0.884003	+	0.467482i	$0.154839\pi$
$242$		0			0
$243$	−0.440394	+	0.897805i	−0.440394	+	0.897805i
$244$	−0.969973	+	1.62881i	−0.969973	+	1.62881i
$245$		0			0
$246$		0			0
$247$	0.640394	+	1.30553i	0.640394	+	1.30553i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.643650	−	0.765320i	$-0.722581\pi$
$251$		0			0		0.643650	+	0.765320i	$0.277419\pi$
$252$	1.70757	−	0.0346144i	1.70757	−	0.0346144i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.511656	−	0.859190i	−0.511656	−	0.859190i
$257$		0			0		0.996715	−	0.0809846i	$-0.0258065\pi$
$257$		0			0		−0.996715	+	0.0809846i	$0.974194\pi$
$258$		0			0
$259$	0.225461	+	1.29571i	0.225461	+	1.29571i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.784532	−	0.620088i	$-0.787097\pi$
$263$		0			0		0.784532	+	0.620088i	$0.212903\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.0256623	−	0.0551140i	0.0256623	−	0.0551140i
$269$		0			0		−0.403648	−	0.914914i	$-0.632258\pi$
$269$		0			0		0.403648	+	0.914914i	$0.367742\pi$
$270$		0			0
$271$	0.995248	−	0.526311i	0.995248	−	0.526311i	0.111245	−	0.993793i	$-0.464516\pi$
$271$	0.995248	−	0.526311i	0.995248	−	0.526311i	0.884003	+	0.467482i	$0.154839\pi$
$272$		0			0
$273$	−0.949219	+	0.814878i	−0.949219	+	0.814878i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.613129	−	0.862139i	−0.613129	−	0.862139i	0.385023	−	0.922907i	$-0.374194\pi$
$277$	−0.613129	−	0.862139i	−0.613129	−	0.862139i	−0.998152	+	0.0607676i	$0.980645\pi$
$278$		0			0
$279$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$280$		0			0
$281$		0			0		0.0910811	−	0.995843i	$-0.470968\pi$
$281$		0			0		−0.0910811	+	0.995843i	$0.529032\pi$
$282$		0			0
$283$	−0.244551	+	0.525214i	−0.244551	+	0.525214i	−0.989952	−	0.141403i	$-0.954839\pi$
$283$	−0.244551	+	0.525214i	−0.244551	+	0.525214i	0.745401	+	0.666616i	$0.232258\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.999179	+	0.0405256i	0.999179	+	0.0405256i
$290$		0			0
$291$	0.00665253	+	0.656423i	0.00665253	+	0.656423i
$292$	1.66436	+	0.272261i	1.66436	+	0.272261i
$293$		0			0		−0.910777	−	0.412899i	$-0.864516\pi$
$293$		0			0		0.910777	+	0.412899i	$0.135484\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.941224	−	0.337784i	−0.941224	−	0.337784i
$301$	−1.14740	+	0.764567i	−1.14740	+	0.764567i
$302$		0			0
$303$		0			0
$304$	1.05011	−	1.68475i	1.05011	−	1.68475i
$305$		0			0
$306$		0			0
$307$	1.58864	−	1.20415i	1.58864	−	1.20415i	0.745401	−	0.666616i	$-0.232258\pi$
$307$	1.58864	−	1.20415i	1.58864	−	1.20415i	0.843240	−	0.537537i	$-0.180645\pi$
$308$		0			0
$309$	1.61999	+	1.12755i	1.61999	+	1.12755i
$310$		0			0
$311$		0			0		0.820763	−	0.571268i	$-0.193548\pi$
$311$		0			0		−0.820763	+	0.571268i	$0.806452\pi$
$312$		0			0
$313$	0.189118	−	0.332733i	0.189118	−	0.332733i	−0.758758	−	0.651372i	$-0.774194\pi$
$313$	0.189118	−	0.332733i	0.189118	−	0.332733i	0.947876	+	0.318639i	$0.103226\pi$
$314$		0			0
$315$		0			0
$316$	−0.578036	+	1.56074i	−0.578036	+	1.56074i
$317$		0			0		0.992615	−	0.121311i	$-0.0387097\pi$
$317$		0			0		−0.992615	+	0.121311i	$0.961290\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.703515	+	0.710681i	−0.703515	+	0.710681i
$325$	0.689426	−	0.247420i	0.689426	−	0.247420i
$326$		0			0
$327$	−1.73153	+	0.701853i	−1.73153	+	0.701853i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.620492	+	0.312207i	−0.620492	+	0.312207i	−0.731738	−	0.681586i	$-0.761290\pi$
$331$	−0.620492	+	0.312207i	−0.620492	+	0.312207i	0.111245	+	0.993793i	$0.464516\pi$
$332$		0			0
$333$	−0.622980	−	0.452622i	−0.622980	−	0.452622i
$334$		0			0
$335$		0			0
$336$	1.62960	+	0.511289i	1.62960	+	0.511289i
$337$	1.24953	−	1.36903i	1.24953	−	1.36903i	0.347305	−	0.937752i	$-0.387097\pi$
$337$	1.24953	−	1.36903i	1.24953	−	1.36903i	0.902221	−	0.431273i	$-0.141935\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.47411	+	0.529025i	1.47411	+	0.529025i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.612106	−	0.790776i	$-0.290323\pi$
$347$		0			0		−0.612106	+	0.790776i	$0.709677\pi$
$348$		0			0
$349$	−1.06473	−	1.43467i	−1.06473	−	1.43467i	−0.893295	−	0.449470i	$-0.851613\pi$
$349$	−1.06473	−	1.43467i	−1.06473	−	1.43467i	−0.171430	−	0.985196i	$-0.554839\pi$
$350$		0			0
$351$	0.0667150	−	0.729434i	0.0667150	−	0.729434i
$352$		0			0
$353$		0			0		0.289679	−	0.957124i	$-0.406452\pi$
$353$		0			0		−0.289679	+	0.957124i	$0.593548\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.659028	−	0.752118i	$-0.270968\pi$
$359$		0			0		−0.659028	+	0.752118i	$0.729032\pi$
$360$		0			0
$361$	2.93147	+	0.238187i	2.93147	+	0.238187i
$362$		0			0
$363$	−0.546055	+	0.837749i	−0.546055	+	0.837749i
$364$	−1.15939	+	0.469945i	−1.15939	+	0.469945i
$365$		0			0
$366$		0			0
$367$	0.478315	−	1.84736i	0.478315	−	1.84736i	−0.0506492	−	0.998717i	$-0.516129\pi$
$367$	0.478315	−	1.84736i	0.478315	−	1.84736i	0.528964	−	0.848644i	$-0.322581\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.954139	+	0.299363i	−0.954139	+	0.299363i
$373$	−1.71979	+	0.174886i	−1.71979	+	0.174886i	−0.910777	−	0.412899i	$-0.864516\pi$
$373$	−1.71979	+	0.174886i	−1.71979	+	0.174886i	−0.809017	+	0.587785i	$0.800000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.518830	+	1.84859i	0.518830	+	1.84859i	0.528964	+	0.848644i	$0.322581\pi$
$379$	0.518830	+	1.84859i	0.518830	+	1.84859i	−0.0101340	+	0.999949i	$0.503226\pi$
$380$		0			0
$381$	1.31787	+	1.17858i	1.31787	+	1.17858i
$382$		0			0
$383$		0			0		0.975249	−	0.221109i	$-0.0709677\pi$
$383$		0			0		−0.975249	+	0.221109i	$0.929032\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	0.170513	−	0.789084i	0.170513	−	0.789084i
$388$	−0.202856	+	0.624328i	−0.202856	+	0.624328i
$389$		0			0		0.458499	−	0.888695i	$-0.348387\pi$
$389$		0			0		−0.458499	+	0.888695i	$0.651613\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.468262	−	1.26435i	−0.468262	−	1.26435i	−0.926761	−	0.375650i	$-0.877419\pi$
$397$	−0.468262	−	1.26435i	−0.468262	−	1.26435i	0.458499	−	0.888695i	$-0.348387\pi$
$398$		0			0
$399$	0.581253	+	3.34042i	0.581253	+	3.34042i
$400$	−0.784532	−	0.620088i	−0.784532	−	0.620088i
$401$		0			0		−0.902221	−	0.431273i	$-0.858065\pi$
$401$		0			0		0.902221	+	0.431273i	$0.141935\pi$
$402$		0			0
$403$	0.399973	−	0.613634i	0.399973	−	0.613634i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.0966527	−	0.0303250i	0.0966527	−	0.0303250i	−0.250653	−	0.968077i	$-0.580645\pi$
$409$	0.0966527	−	0.0303250i	0.0966527	−	0.0303250i	0.347305	+	0.937752i	$0.387097\pi$
$410$		0			0
$411$		0			0
$412$	1.17627	+	1.58497i	1.17627	+	1.58497i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.535012	+	1.90625i	−0.535012	+	1.90625i
$418$		0			0
$419$		0			0		0.731738	−	0.681586i	$-0.238710\pi$
$419$		0			0		−0.731738	+	0.681586i	$0.761290\pi$
$420$		0			0
$421$	1.05041	−	0.796186i	1.05041	−	0.796186i	0.0708797	−	0.997485i	$-0.477419\pi$
$421$	1.05041	−	0.796186i	1.05041	−	0.796186i	0.979530	+	0.201299i	$0.0645161\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.425328	−	3.20974i	0.425328	−	3.20974i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.998152	−	0.0607676i	$-0.980645\pi$
$431$		0			0		0.998152	+	0.0607676i	$0.0193548\pi$
$432$	−0.893295	+	0.449470i	−0.893295	+	0.449470i
$433$	−1.77448	+	0.761491i	−1.77448	+	0.761491i	−0.784532	+	0.620088i	$0.787097\pi$
$433$	−1.77448	+	0.761491i	−1.77448	+	0.761491i	−0.989952	+	0.141403i	$0.954839\pi$
$434$		0			0
$435$		0			0
$436$	−1.86492	+	0.113536i	−1.86492	+	0.113536i
$437$		0			0
$438$		0			0
$439$	−1.99630			−1.99630			−0.998152	−	0.0607676i	$-0.980645\pi$
$439$	−1.99630			−1.99630			−0.998152	+	0.0607676i	$0.980645\pi$
$440$		0			0
$441$	−1.74596	+	0.791528i	−1.74596	+	0.791528i
$442$		0			0
$443$		0			0		−0.422108	−	0.906546i	$-0.638710\pi$
$443$		0			0		0.422108	+	0.906546i	$0.361290\pi$
$444$	−0.446285	−	0.627535i	−0.446285	−	0.627535i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.38174	+	1.00389i	1.38174	+	1.00389i
$449$		0			0		−0.717774	−	0.696277i	$-0.754839\pi$
$449$		0			0		0.717774	+	0.696277i	$0.245161\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.572761	−	1.00771i	−0.572761	−	1.00771i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.67876	+	0.489571i	−1.67876	+	0.489571i	−0.975249	−	0.221109i	$-0.929032\pi$
$457$	−1.67876	+	0.489571i	−1.67876	+	0.489571i	−0.703515	+	0.710681i	$0.748387\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.884003	−	0.467482i	$-0.845161\pi$
$461$		0			0		0.884003	+	0.467482i	$0.154839\pi$
$462$		0			0
$463$	−0.269251	+	1.38103i	−0.269251	+	1.38103i	0.562921	+	0.826511i	$0.309677\pi$
$463$	−0.269251	+	1.38103i	−0.269251	+	1.38103i	−0.832173	+	0.554517i	$0.812903\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.659028	−	0.752118i	$-0.729032\pi$
$467$		0			0		0.659028	+	0.752118i	$0.270968\pi$
$468$	0.295664	−	0.670155i	0.295664	−	0.670155i
$469$	−0.00315634	+	0.103786i	−0.00315634	+	0.103786i
$470$		0			0
$471$	−1.19841	−	0.376004i	−1.19841	−	0.376004i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.379897	−	1.94854i	0.379897	−	1.94854i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.832173	−	0.554517i	$-0.187097\pi$
$479$		0			0		−0.832173	+	0.554517i	$0.812903\pi$
$480$		0			0
$481$	0.544575	+	0.146905i	0.544575	+	0.146905i
$482$		0			0
$483$		0			0
$484$	−0.784532	+	0.620088i	−0.784532	+	0.620088i
$485$		0			0
$486$		0			0
$487$	0.331584	+	1.39673i	0.331584	+	1.39673i	0.843240	+	0.537537i	$0.180645\pi$
$487$	0.331584	+	1.39673i	0.331584	+	1.39673i	−0.511656	+	0.859190i	$0.670968\pi$
$488$		0			0
$489$	−0.0194575	−	0.00567431i	−0.0194575	−	0.00567431i
$490$		0			0
$491$		0			0		0.0506492	−	0.998717i	$-0.483871\pi$
$491$		0			0		−0.0506492	+	0.998717i	$0.516129\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.999795	−	0.0202670i	−0.999795	−	0.0202670i
$497$		0			0
$498$		0			0
$499$	1.67959	−	1.02341i	1.67959	−	1.02341i	0.745401	−	0.666616i	$-0.232258\pi$
$499$	1.67959	−	1.02341i	1.67959	−	1.02341i	0.934184	−	0.356791i	$-0.116129\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.546055	−	0.837749i	$-0.683871\pi$
$503$		0			0		0.546055	+	0.837749i	$0.316129\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.125241	−	0.446233i	−0.125241	−	0.446233i
$508$	0.873638	+	1.53707i	0.873638	+	1.53707i
$509$		0			0		0.853961	−	0.520337i	$-0.174194\pi$
$509$		0			0		−0.853961	+	0.520337i	$0.825806\pi$
$510$		0			0
$511$	−2.79560	+	0.693679i	−2.79560	+	0.693679i
$512$		0			0
$513$	−1.60608	−	1.16689i	−1.60608	−	1.16689i
$514$		0			0
$515$		0			0
$516$	0.413058	−	0.693622i	0.413058	−	0.693622i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	0.384739	−	0.174421i	0.384739	−	0.174421i	−0.211215	−	0.977440i	$-0.567742\pi$
$523$	0.384739	−	0.174421i	0.384739	−	0.174421i	0.595954	+	0.803019i	$0.296774\pi$
$524$		0			0
$525$	1.70477	−	0.103786i	1.70477	−	0.103786i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.893295	+	0.449470i	−0.893295	+	0.449470i
$530$		0			0
$531$		0			0
$532$	−0.513434	+	3.35152i	−0.513434	+	3.35152i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.93609	−	0.236616i	−1.93609	−	0.236616i	−0.941224	−	0.337784i	$-0.890323\pi$
$541$	−1.93609	−	0.236616i	−1.93609	−	0.236616i	−0.994869	+	0.101168i	$0.967742\pi$
$542$		0			0
$543$	−0.0644506	−	0.212951i	−0.0644506	−	0.212951i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.682985	+	0.496218i	−0.682985	+	0.496218i	−0.874347	−	0.485302i	$-0.838710\pi$
$547$	−0.682985	+	0.496218i	−0.682985	+	0.496218i	0.191362	+	0.981520i	$0.438710\pi$
$548$		0			0
$549$	1.12978	+	1.52232i	1.12978	+	1.52232i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.143974	−	2.83892i	−0.143974	−	2.83892i
$554$		0			0
$555$		0			0
$556$	−1.11453	+	1.63641i	−1.11453	+	1.63641i
$557$		0			0		0.151428	−	0.988468i	$-0.451613\pi$
$557$		0			0		−0.151428	+	0.988468i	$0.548387\pi$
$558$		0			0
$559$	0.0895434	+	0.584509i	0.0895434	+	0.584509i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.979530	−	0.201299i	$-0.935484\pi$
$563$		0			0		0.979530	+	0.201299i	$0.0645161\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.625508	−	1.58926i	0.625508	−	1.58926i
$568$		0			0
$569$		0			0		−0.884003	−	0.467482i	$-0.845161\pi$
$569$		0			0		0.884003	+	0.467482i	$0.154839\pi$
$570$		0			0
$571$	1.11042	−	1.03432i	1.11042	−	1.03432i	0.111245	−	0.993793i	$-0.464516\pi$
$571$	1.11042	−	1.03432i	1.11042	−	1.03432i	0.999179	−	0.0405256i	$-0.0129032\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.989952	+	0.141403i	−0.989952	+	0.141403i
$577$	−1.74217	−	0.832778i	−1.74217	−	0.832778i	−0.983408	−	0.181405i	$-0.941935\pi$
$577$	−1.74217	−	0.832778i	−1.74217	−	0.832778i	−0.758758	−	0.651372i	$-0.774194\pi$
$578$		0			0
$579$	−0.935089	+	0.739088i	−0.935089	+	0.739088i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$587$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$588$	−1.90716	+	0.193940i	−1.90716	+	0.193940i
$589$	−0.874283	−	1.78235i	−0.874283	−	1.78235i
$590$		0			0
$591$		0			0
$592$	−0.223066	−	0.737029i	−0.223066	−	0.737029i
$593$		0			0		−0.960010	−	0.279964i	$-0.909677\pi$
$593$		0			0		0.960010	+	0.279964i	$0.0903226\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.168821	−	0.0684293i	0.168821	−	0.0684293i
$598$		0			0
$599$		0			0		0.289679	−	0.957124i	$-0.406452\pi$
$599$		0			0		−0.289679	+	0.957124i	$0.593548\pi$
$600$		0			0
$601$	−0.616892	+	0.0375564i	−0.616892	+	0.0375564i	−0.366239	−	0.930521i	$-0.619355\pi$
$601$	−0.616892	+	0.0375564i	−0.616892	+	0.0375564i	−0.250653	+	0.968077i	$0.580645\pi$
$602$		0			0
$603$	−0.0409845	−	0.0449041i	−0.0409845	−	0.0449041i
$604$	−0.221810	−	1.13769i	−0.221810	−	1.13769i
$605$		0			0
$606$		0			0
$607$	0.539998	−	0.0219017i	0.539998	−	0.0219017i	0.230981	−	0.972958i	$-0.425806\pi$
$607$	0.539998	−	0.0219017i	0.539998	−	0.0219017i	0.309017	+	0.951057i	$0.400000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.12200	+	0.747639i	1.12200	+	0.747639i	0.970568	−	0.240829i	$-0.0774194\pi$
$613$	1.12200	+	0.747639i	1.12200	+	0.747639i	0.151428	+	0.988468i	$0.451613\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.902221	−	0.431273i	$-0.141935\pi$
$617$		0			0		−0.902221	+	0.431273i	$0.858065\pi$
$618$		0			0
$619$	0.600374	+	1.62106i	0.600374	+	1.62106i	0.771804	+	0.635861i	$0.219355\pi$
$619$	0.600374	+	1.62106i	0.600374	+	1.62106i	−0.171430	+	0.985196i	$0.554839\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.493790	−	0.541014i	0.493790	−	0.541014i
$625$	−0.954139	−	0.299363i	−0.954139	−	0.299363i
$626$		0			0
$627$		0			0
$628$	−1.01614	−	0.738266i	−1.01614	−	0.738266i
$629$		0			0
$630$		0			0
$631$	1.81994	−	0.611794i	1.81994	−	0.611794i	0.820763	−	0.571268i	$-0.193548\pi$
$631$	1.81994	−	0.611794i	1.81994	−	0.611794i	0.999179	−	0.0405256i	$-0.0129032\pi$
$632$		0			0
$633$	−1.04578	+	1.47050i	−1.04578	+	1.47050i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.987848	−	0.997910i	0.987848	−	0.997910i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.659028	−	0.752118i	$-0.729032\pi$
$641$		0			0		0.659028	+	0.752118i	$0.270968\pi$
$642$		0			0
$643$	−1.08852	+	0.0884440i	−1.08852	+	0.0884440i	−0.612106	−	0.790776i	$-0.709677\pi$
$643$	−1.08852	+	0.0884440i	−1.08852	+	0.0884440i	−0.476416	+	0.879220i	$0.658065\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.628007	−	0.778208i	$-0.716129\pi$
$647$		0			0		0.628007	+	0.778208i	$0.283871\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.29590	−	1.11249i	1.29590	−	1.11249i
$652$	−0.0166352	−	0.0115785i	−0.0166352	−	0.0115785i
$653$		0			0		−0.999179	−	0.0405256i	$-0.987097\pi$
$653$		0			0		0.999179	+	0.0405256i	$0.0129032\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.892087	−	1.43122i	0.892087	−	1.43122i
$658$		0			0
$659$		0			0		0.0708797	−	0.997485i	$-0.477419\pi$
$659$		0			0		−0.0708797	+	0.997485i	$0.522581\pi$
$660$		0			0
$661$	−0.581708	−	0.208762i	−0.581708	−	0.208762i	0.0303978	−	0.999538i	$-0.490323\pi$
$661$	−0.581708	−	0.208762i	−0.581708	−	0.208762i	−0.612106	+	0.790776i	$0.709677\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.455903	+	0.0745778i	0.455903	+	0.0745778i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.77340	−	0.216732i	−1.77340	−	0.216732i	−0.832173	−	0.554517i	$-0.812903\pi$
$673$	−1.77340	−	0.216732i	−1.77340	−	0.216732i	−0.941224	+	0.337784i	$0.890323\pi$
$674$		0			0
$675$	−0.674136	+	0.738607i	−0.674136	+	0.738607i
$676$	0.0234746	−	0.462880i	0.0234746	−	0.462880i
$677$		0			0		−0.250653	−	0.968077i	$-0.580645\pi$
$677$		0			0		0.250653	+	0.968077i	$0.419355\pi$
$678$		0			0
$679$	−0.102118	−	1.11652i	−0.102118	−	1.11652i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.758758	−	0.651372i	$-0.225806\pi$
$683$		0			0		−0.758758	+	0.651372i	$0.774194\pi$
$684$	−1.15055	−	1.61783i	−1.15055	−	1.61783i
$685$		0			0
$686$		0			0
$687$	1.36998	+	0.587906i	1.36998	+	0.587906i
$688$	0.612543	−	0.525851i	0.612543	−	0.525851i
$689$		0			0
$690$		0			0
$691$	0.948366	−	0.384407i	0.948366	−	0.384407i	0.151428	−	0.988468i	$-0.451613\pi$
$691$	0.948366	−	0.384407i	0.948366	−	0.384407i	0.796938	+	0.604061i	$0.206452\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.64898	+	0.444829i	1.64898	+	0.444829i
$701$		0			0		0.986883	−	0.161437i	$-0.0516129\pi$
$701$		0			0		−0.986883	+	0.161437i	$0.948387\pi$
$702$		0			0
$703$	1.09727	−	1.06441i	1.09727	−	1.06441i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.22396	−	0.0248110i	1.22396	−	0.0248110i	0.595954	−	0.803019i	$-0.296774\pi$
$709$	1.22396	−	0.0248110i	1.22396	−	0.0248110i	0.628007	+	0.778208i	$0.283871\pi$
$710$		0			0
$711$	1.21786	+	1.13440i	1.21786	+	1.13440i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.918958	−	0.394356i	$-0.129032\pi$
$719$		0			0		−0.918958	+	0.394356i	$0.870968\pi$
$720$		0			0
$721$	−2.91370	−	1.69537i	−2.91370	−	1.69537i
$722$		0			0
$723$	0.530536	−	1.27170i	0.530536	−	1.27170i
$724$	0.00676322	−	0.222387i	0.00676322	−	0.222387i
$725$		0			0
$726$		0			0
$727$	−0.553141	+	0.685436i	−0.553141	+	0.685436i	−0.975249	−	0.221109i	$-0.929032\pi$
$727$	−0.553141	+	0.685436i	−0.553141	+	0.685436i	0.422108	+	0.906546i	$0.361290\pi$
$728$		0			0
$729$	0.385023	+	0.922907i	0.385023	+	0.922907i
$730$		0			0
$731$		0			0
$732$	0.585820	+	1.80297i	0.585820	+	1.80297i
$733$	−0.459592	+	1.93593i	−0.459592	+	1.93593i	−0.131363	+	0.991334i	$0.541935\pi$
$733$	−0.459592	+	1.93593i	−0.459592	+	1.93593i	−0.328229	+	0.944598i	$0.606452\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.536103	+	1.44752i	0.536103	+	1.44752i	0.864331	+	0.502923i	$0.167742\pi$
$739$	0.536103	+	1.44752i	0.536103	+	1.44752i	−0.328229	+	0.944598i	$0.606452\pi$
$740$		0			0
$741$	1.40395	+	0.378730i	1.40395	+	0.378730i
$742$		0			0
$743$		0			0		−0.979530	−	0.201299i	$-0.935484\pi$
$743$		0			0		0.979530	+	0.201299i	$0.0645161\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.62062	+	0.656897i	1.62062	+	0.656897i	0.992615	−	0.121311i	$-0.0387097\pi$
$751$	1.62062	+	0.656897i	1.62062	+	0.656897i	0.628007	+	0.778208i	$0.283871\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.09930	−	1.30711i	1.09930	−	1.30711i
$757$	−0.421042	+	1.94846i	−0.421042	+	1.94846i	−0.131363	+	0.991334i	$0.541935\pi$
$757$	−0.421042	+	1.94846i	−0.421042	+	1.94846i	−0.289679	+	0.957124i	$0.593548\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.595954	−	0.803019i	$-0.703226\pi$
$761$		0			0		0.595954	+	0.803019i	$0.296774\pi$
$762$		0			0
$763$	2.85053	−	1.43427i	2.85053	−	1.43427i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.983408	−	0.181405i	−0.983408	−	0.181405i
$769$	1.42832	+	1.22617i	1.42832	+	1.22617i	0.934184	+	0.356791i	$0.116129\pi$
$769$	1.42832	+	1.22617i	1.42832	+	1.22617i	0.494138	+	0.869384i	$0.335484\pi$
$770$		0			0
$771$		0			0
$772$	−1.12185	+	0.402608i	−1.12185	+	0.402608i
$773$		0			0		0.476416	−	0.879220i	$-0.341935\pi$
$773$		0			0		−0.476416	+	0.879220i	$0.658065\pi$
$774$		0			0
$775$	−0.941224	+	0.337784i	−0.941224	+	0.337784i
$776$		0			0
$777$	1.12311	+	0.684336i	1.12311	+	0.684336i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.86955	−	0.423866i	−1.86955	−	0.423866i
$785$		0			0
$786$		0			0
$787$	1.29108	+	1.47344i	1.29108	+	1.47344i	0.796938	+	0.604061i	$0.206452\pi$
$787$	1.29108	+	1.47344i	1.29108	+	1.47344i	0.494138	+	0.869384i	$0.335484\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.17092	−	0.746423i	−1.17092	−	0.746423i
$794$		0			0
$795$		0			0
$796$	0.181826	−	0.0110696i	0.181826	−	0.0110696i
$797$		0			0		0.771804	−	0.635861i	$-0.219355\pi$
$797$		0			0		−0.771804	+	0.635861i	$0.780645\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.0245401	−	0.0556228i	−0.0245401	−	0.0556228i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.731738	−	0.681586i	$-0.238710\pi$
$809$		0			0		−0.731738	+	0.681586i	$0.761290\pi$
$810$		0			0
$811$	1.60792	+	1.11915i	1.60792	+	1.11915i	0.918958	+	0.394356i	$0.129032\pi$
$811$	1.60792	+	1.11915i	1.60792	+	1.11915i	0.688967	+	0.724793i	$0.258065\pi$
$812$		0			0
$813$	0.260048	−	1.09540i	0.260048	−	1.09540i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.48529	+	0.602043i	1.48529	+	0.602043i
$818$		0			0
$819$	−0.0126778	+	1.25095i	−0.0126778	+	1.25095i
$820$		0			0
$821$		0			0		−0.562921	−	0.826511i	$-0.690323\pi$
$821$		0			0		0.562921	+	0.826511i	$0.309677\pi$
$822$		0			0
$823$	1.94689	+	0.441401i	1.94689	+	0.441401i	0.986883	+	0.161437i	$0.0516129\pi$
$823$	1.94689	+	0.441401i	1.94689	+	0.441401i	0.960010	+	0.279964i	$0.0903226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.643650	−	0.765320i	$-0.277419\pi$
$827$		0			0		−0.643650	+	0.765320i	$0.722581\pi$
$828$		0			0
$829$	0.140712	+	0.0171969i	0.140712	+	0.0171969i	0.191362	−	0.981520i	$-0.438710\pi$
$829$	0.140712	+	0.0171969i	0.140712	+	0.0171969i	−0.0506492	+	0.998717i	$0.516129\pi$
$830$		0			0
$831$	−1.05250	−	0.107029i	−1.05250	−	0.107029i
$832$	0.640440	−	0.355473i	0.640440	−	0.355473i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.0910811	+	0.995843i	−0.0910811	+	0.995843i
$838$		0			0
$839$		0			0		0.717774	−	0.696277i	$-0.245161\pi$
$839$		0			0		−0.717774	+	0.696277i	$0.754839\pi$
$840$		0			0
$841$	−0.703515	−	0.710681i	−0.703515	−	0.710681i
$842$		0			0
$843$		0			0
$844$	−1.45982	+	1.06062i	−1.45982	+	1.06062i
$845$		0			0
$846$		0			0
$847$	0.813682	−	1.50164i	0.813682	−	1.50164i
$848$		0			0
$849$	0.233856	+	0.530062i	0.233856	+	0.530062i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.90475	+	0.115961i	1.90475	+	0.115961i	0.970568	−	0.240829i	$-0.0774194\pi$
$853$	1.90475	+	0.115961i	1.90475	+	0.115961i	0.934184	+	0.356791i	$0.116129\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.960010	−	0.279964i	$-0.909677\pi$
$857$		0			0		0.960010	+	0.279964i	$0.0903226\pi$
$858$		0			0
$859$	0.890009	−	1.72508i	0.890009	−	1.72508i	0.230981	−	0.972958i	$-0.425806\pi$
$859$	0.890009	−	1.72508i	0.890009	−	1.72508i	0.659028	−	0.752118i	$-0.270968\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.528964	−	0.848644i	$-0.677419\pi$
$863$		0			0		0.528964	+	0.848644i	$0.322581\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.688967	−	0.724793i	0.688967	−	0.724793i
$868$	1.58284	−	0.641582i	1.58284	−	0.641582i
$869$		0			0
$870$		0			0
$871$	0.0397798	+	0.0200156i	0.0397798	+	0.0200156i
$872$		0			0
$873$	0.498092	+	0.427598i	0.498092	+	0.427598i
$874$		0			0
$875$		0			0
$876$	1.30163	−	1.07237i	1.30163	−	1.07237i
$877$	−1.23901	−	1.35750i	−1.23901	−	1.35750i	−0.910777	−	0.412899i	$-0.864516\pi$
$877$	−1.23901	−	1.35750i	−1.23901	−	1.35750i	−0.328229	−	0.944598i	$-0.606452\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.191362	−	0.981520i	$-0.561290\pi$
$881$		0			0		0.191362	+	0.981520i	$0.438710\pi$
$882$		0			0
$883$	−1.88090	+	0.0762873i	−1.88090	+	0.0762873i	−0.954139	−	0.299363i	$-0.903226\pi$
$883$	−1.88090	+	0.0762873i	−1.88090	+	0.0762873i	−0.926761	+	0.375650i	$0.877419\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.476416	−	0.879220i	$-0.341935\pi$
$887$		0			0		−0.476416	+	0.879220i	$0.658065\pi$
$888$		0			0
$889$	−2.40645	−	1.82403i	−2.40645	−	1.82403i
$890$		0			0
$891$		0			0
$892$	0.408376	+	0.215959i	0.408376	+	0.215959i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.874347	+	0.485302i	−0.874347	+	0.485302i
$901$		0			0
$902$		0			0
$903$	−0.181124	+	1.36685i	−0.181124	+	1.36685i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.249033	−	0.0837151i	−0.249033	−	0.0837151i	0.191362	−	0.981520i	$-0.438710\pi$
$907$	−0.249033	−	0.0837151i	−0.249033	−	0.0837151i	−0.440394	+	0.897805i	$0.645161\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.111245	−	0.993793i	$-0.535484\pi$
$911$		0			0		0.111245	+	0.993793i	$0.464516\pi$
$912$	−0.575078	−	1.90011i	−0.575078	−	1.90011i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.11125	+	0.993793i	1.11125	+	0.993793i
$917$		0			0
$918$		0			0
$919$	−0.901784	−	1.74790i	−0.901784	−	1.74790i	−0.612106	−	0.790776i	$-0.709677\pi$
$919$	−0.901784	−	1.74790i	−0.901784	−	1.74790i	−0.289679	−	0.957124i	$-0.593548\pi$
$920$		0			0
$921$	0.141294	−	1.98842i	0.141294	−	1.98842i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.471350	−	0.608934i	−0.471350	−	0.608934i
$926$		0			0
$927$	1.91567	−	0.475340i	1.91567	−	0.475340i
$928$		0			0
$929$		0			0		0.250653	−	0.968077i	$-0.419355\pi$
$929$		0			0		−0.250653	+	0.968077i	$0.580645\pi$
$930$		0			0
$931$	−0.953904	−	3.68419i	−0.953904	−	3.68419i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.0912434	−	1.28406i	−0.0912434	−	1.28406i	−0.809017	−	0.587785i	$-0.800000\pi$
$937$	−0.0912434	−	1.28406i	−0.0912434	−	1.28406i	0.717774	−	0.696277i	$-0.245161\pi$
$938$		0			0
$939$	−0.125621	−	0.361520i	−0.125621	−	0.361520i
$940$		0			0
$941$		0			0		0.328229	−	0.944598i	$-0.393548\pi$
$941$		0			0		−0.328229	+	0.944598i	$0.606452\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.366239	−	0.930521i	$-0.380645\pi$
$947$		0			0		−0.366239	+	0.930521i	$0.619355\pi$
$948$	0.792921	+	1.46333i	0.792921	+	1.46333i
$949$	−0.236391	+	1.21248i	−0.236391	+	1.21248i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.703515	−	0.710681i	$-0.251613\pi$
$953$		0			0		−0.703515	+	0.710681i	$0.748387\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.546055	+	0.837749i	−0.546055	+	0.837749i
$962$		0			0
$963$		0			0
$964$	0.908097	−	1.03637i	0.908097	−	1.03637i
$965$		0			0
$966$		0			0
$967$	−0.854865	+	1.74276i	−0.854865	+	1.74276i	−0.211215	+	0.977440i	$0.567742\pi$
$967$	−0.854865	+	1.74276i	−0.854865	+	1.74276i	−0.643650	+	0.765320i	$0.722581\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.562921	−	0.826511i	$-0.309677\pi$
$971$		0			0		−0.562921	+	0.826511i	$0.690323\pi$
$972$	0.0708797	+	0.997485i	0.0708797	+	0.997485i
$973$	0.647094	−	3.31903i	0.647094	−	3.31903i
$974$		0			0
$975$	0.268262	−	0.681586i	0.268262	−	0.681586i
$976$	−0.0192115	+	1.89565i	−0.0192115	+	1.89565i
$977$		0			0		0.494138	−	0.869384i	$-0.335484\pi$
$977$		0			0		−0.494138	+	0.869384i	$0.664516\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.613252	+	1.76486i	−0.613252	+	1.76486i
$982$		0			0
$983$		0			0		−0.328229	−	0.944598i	$-0.606452\pi$
$983$		0			0		0.328229	+	0.944598i	$0.393548\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.21009	+	0.806343i	1.21009	+	0.806343i
$989$		0			0
$990$		0			0
$991$	−0.135463	−	0.523190i	−0.135463	−	0.523190i	−0.999795	−	0.0202670i	$-0.993548\pi$
$991$	−0.135463	−	0.523190i	−0.135463	−	0.523190i	0.864331	−	0.502923i	$-0.167742\pi$
$992$		0			0
$993$	−0.174106	+	0.672437i	−0.174106	+	0.672437i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.234267	−	0.302648i	−0.234267	−	0.302648i	0.659028	−	0.752118i	$-0.270968\pi$
$997$	−0.234267	−	0.302648i	−0.234267	−	0.302648i	−0.893295	+	0.449470i	$0.851613\pi$
$998$		0			0
$999$	−0.750986	+	0.170264i	−0.750986	+	0.170264i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2883.1.z.a.314.1	yes	120
3.2	odd	2	CM	2883.1.z.a.314.1	yes	120
961.101	even	155	inner	2883.1.z.a.101.1	✓	120
2883.101	odd	310	inner	2883.1.z.a.101.1	✓	120

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.1.z.a.101.1	✓	120	961.101	even	155	inner
2883.1.z.a.101.1	✓	120	2883.101	odd	310	inner
2883.1.z.a.314.1	yes	120	1.1	even	1	trivial
2883.1.z.a.314.1	yes	120	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2883 = 3 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2883.z (of order \(310\), degree \(120\), minimal)

\(f(q)\)	\(=\)	\(q+(0.659028 - 0.752118i) q^{3} +(0.864331 - 0.502923i) q^{4} +(-1.01784 + 1.37149i) q^{7} +(-0.131363 - 0.991334i) q^{9} +(0.191362 - 0.981520i) q^{12} +(0.348965 + 0.644010i) q^{13} +(0.494138 - 0.869384i) q^{16} +(1.98360 + 0.0804526i) q^{19} +(0.360738 + 1.66939i) q^{21} +(0.151428 - 0.988468i) q^{25} +(-0.832173 - 0.554517i) q^{27} +(-0.189998 + 1.69732i) q^{28} +(-0.476416 - 0.879220i) q^{31} +(-0.612106 - 0.790776i) q^{36} +(0.530536 - 0.558124i) q^{37} +(0.714349 + 0.161958i) q^{39} +(0.759847 + 0.272692i) q^{43} +(-0.328229 - 0.944598i) q^{48} +(-0.555313 - 1.83481i) q^{49} +(0.625508 + 0.381135i) q^{52} +(1.36776 - 1.43888i) q^{57} +(-1.65754 + 0.920012i) q^{61} +(1.49332 + 0.828858i) q^{63} +(-0.0101340 - 0.999949i) q^{64} +(0.0498988 - 0.0347306i) q^{67} +(1.30163 + 1.07237i) q^{73} +(-0.643650 - 0.765320i) q^{75} +(1.75495 - 0.928059i) q^{76} +(-1.28455 + 1.05829i) q^{79} +(-0.965487 + 0.260450i) q^{81} +(1.15137 + 1.26148i) q^{84} +(-1.23845 - 0.176897i) q^{91} +(-0.975249 - 0.221109i) q^{93} +(-0.489324 + 0.437605i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + q^{3} + q^{4} + 2 q^{7} + q^{9} + q^{12} + 2 q^{13} + q^{16} + 2 q^{19} - 3 q^{21} - 4 q^{25} + q^{27} - 3 q^{28} + q^{31} - 4 q^{36} + 2 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + 3 q^{49}+ \cdots - 3 q^{97}+O(q^{100}) \) 120 * q + q^3 + q^4 + 2 * q^7 + q^9 + q^12 + 2 * q^13 + q^16 + 2 * q^19 - 3 * q^21 - 4 * q^25 + q^27 - 3 * q^28 + q^31 - 4 * q^36 + 2 * q^37 - 3 * q^39 - 3 * q^43 + q^48 + 3 * q^49 + 2 * q^52 + 2 * q^57 + 2 * q^61 + 2 * q^63 + q^64 + 2 * q^67 - 29 * q^73 + q^75 - 3 * q^76 - 3 * q^79 + q^81 + 2 * q^84 - 32 * q^91 + q^93 - 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more