LMFDB - Embedded newform 2883.1.z.a.419.1

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			419.1
Root			$0.796938 - 0.604061i$ of defining polynomial
Character	$\chi$	$=$	2883.419
Dual form			2883.1.z.a.1211.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.960010 - 0.279964i) q^{3} +(-0.643650 - 0.765320i) q^{4} +(0.415421 - 1.92244i) q^{7} +(0.843240 - 0.537537i) q^{9} +(-0.832173 - 0.554517i) q^{12} +(1.23016 - 1.01348i) q^{13} +(-0.171430 + 0.985196i) q^{16} +(-1.02247 + 1.71697i) q^{19} +(-0.139407 - 1.96187i) q^{21} +(-0.0506492 + 0.998717i) q^{25} +(0.659028 - 0.752118i) q^{27} +(-1.73867 + 0.919451i) q^{28} +(0.771804 - 0.635861i) q^{31} +(-0.954139 - 0.299363i) q^{36} +(0.0658532 - 0.254340i) q^{37} +(0.897227 - 1.31736i) q^{39} +(-1.10461 + 1.48841i) q^{43} +(0.111245 + 0.993793i) q^{48} +(-2.61244 - 1.18435i) q^{49} +(-1.56743 - 0.289137i) q^{52} +(-0.500893 + 1.93456i) q^{57} +(-0.280378 + 0.757044i) q^{61} +(-0.683086 - 1.84439i) q^{63} +(0.864331 - 0.502923i) q^{64} +(-0.0198531 + 0.00407992i) q^{67} +(0.190162 + 0.628311i) q^{73} +(0.230981 + 0.972958i) q^{75} +(1.97214 - 0.322609i) q^{76} +(-0.381813 + 1.26154i) q^{79} +(0.422108 - 0.906546i) q^{81} +(-1.41173 + 1.36945i) q^{84} +(-1.43733 - 2.78594i) q^{91} +(0.562921 - 0.826511i) q^{93} +(0.215942 - 0.0535821i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2883\mathbb{Z}\right)^\times$.

$n$	$962$	$964$
$\chi(n)$	$-1$	$e\left(\frac{141}{155}\right)$

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		0.422108	−	0.906546i	$-0.361290\pi$
$2$		0			0		−0.422108	+	0.906546i	$0.638710\pi$
$3$	0.960010	−	0.279964i	0.960010	−	0.279964i
$3$	0.960010	−	0.279964i	0.960010	−	0.279964i
$4$	−0.643650	−	0.765320i	−0.643650	−	0.765320i
$5$		0			0		−0.688967	−	0.724793i	$-0.741935\pi$
$5$		0			0		0.688967	+	0.724793i	$0.258065\pi$
$6$		0			0
$7$	0.415421	−	1.92244i	0.415421	−	1.92244i	0.0303978	−	0.999538i	$-0.490323\pi$
$7$	0.415421	−	1.92244i	0.415421	−	1.92244i	0.385023	−	0.922907i	$-0.374194\pi$
$8$		0			0
$9$	0.843240	−	0.537537i	0.843240	−	0.537537i
$10$		0			0
$11$		0			0		0.853961	−	0.520337i	$-0.174194\pi$
$11$		0			0		−0.853961	+	0.520337i	$0.825806\pi$
$12$	−0.832173	−	0.554517i	−0.832173	−	0.554517i
$13$	1.23016	−	1.01348i	1.23016	−	1.01348i	0.230981	−	0.972958i	$-0.425806\pi$
$13$	1.23016	−	1.01348i	1.23016	−	1.01348i	0.999179	−	0.0405256i	$-0.0129032\pi$
$14$		0			0
$15$		0			0
$16$	−0.171430	+	0.985196i	−0.171430	+	0.985196i
$17$		0			0		−0.494138	−	0.869384i	$-0.664516\pi$
$17$		0			0		0.494138	+	0.869384i	$0.335484\pi$
$18$		0			0
$19$	−1.02247	+	1.71697i	−1.02247	+	1.71697i	−0.476416	+	0.879220i	$0.658065\pi$
$19$	−1.02247	+	1.71697i	−1.02247	+	1.71697i	−0.546055	+	0.837749i	$0.683871\pi$
$20$		0			0
$21$	−0.139407	−	1.96187i	−0.139407	−	1.96187i
$22$		0			0
$23$		0			0		−0.902221	−	0.431273i	$-0.858065\pi$
$23$		0			0		0.902221	+	0.431273i	$0.141935\pi$
$24$		0			0
$25$	−0.0506492	+	0.998717i	−0.0506492	+	0.998717i
$26$		0			0
$27$	0.659028	−	0.752118i	0.659028	−	0.752118i
$28$	−1.73867	+	0.919451i	−1.73867	+	0.919451i
$29$		0			0		0.131363	−	0.991334i	$-0.458065\pi$
$29$		0			0		−0.131363	+	0.991334i	$0.541935\pi$
$30$		0			0
$31$	0.771804	−	0.635861i	0.771804	−	0.635861i
$31$	0.771804	−	0.635861i	0.771804	−	0.635861i
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.954139	−	0.299363i	−0.954139	−	0.299363i
$37$	0.0658532	−	0.254340i	0.0658532	−	0.254340i	−0.926761	−	0.375650i	$-0.877419\pi$
$37$	0.0658532	−	0.254340i	0.0658532	−	0.254340i	0.992615	+	0.121311i	$0.0387097\pi$
$38$		0			0
$39$	0.897227	−	1.31736i	0.897227	−	1.31736i
$40$		0			0
$41$		0			0		−0.546055	−	0.837749i	$-0.683871\pi$
$41$		0			0		0.546055	+	0.837749i	$0.316129\pi$
$42$		0			0
$43$	−1.10461	+	1.48841i	−1.10461	+	1.48841i	−0.250653	+	0.968077i	$0.580645\pi$
$43$	−1.10461	+	1.48841i	−1.10461	+	1.48841i	−0.853961	+	0.520337i	$0.825806\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.983408	−	0.181405i	$-0.0580645\pi$
$47$		0			0		−0.983408	+	0.181405i	$0.941935\pi$
$48$	0.111245	+	0.993793i	0.111245	+	0.993793i
$49$	−2.61244	−	1.18435i	−2.61244	−	1.18435i
$50$		0			0
$51$		0			0
$52$	−1.56743	−	0.289137i	−1.56743	−	0.289137i
$53$		0			0		−0.941224	−	0.337784i	$-0.890323\pi$
$53$		0			0		0.941224	+	0.337784i	$0.109677\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.500893	+	1.93456i	−0.500893	+	1.93456i
$58$		0			0
$59$		0			0		0.771804	−	0.635861i	$-0.219355\pi$
$59$		0			0		−0.771804	+	0.635861i	$0.780645\pi$
$60$		0			0
$61$	−0.280378	+	0.757044i	−0.280378	+	0.757044i	0.717774	+	0.696277i	$0.245161\pi$
$61$	−0.280378	+	0.757044i	−0.280378	+	0.757044i	−0.998152	+	0.0607676i	$0.980645\pi$
$62$		0			0
$63$	−0.683086	−	1.84439i	−0.683086	−	1.84439i
$64$	0.864331	−	0.502923i	0.864331	−	0.502923i
$65$		0			0
$66$		0			0
$67$	−0.0198531	+	0.00407992i	−0.0198531	+	0.00407992i	−0.211215	−	0.977440i	$-0.567742\pi$
$67$	−0.0198531	+	0.00407992i	−0.0198531	+	0.00407992i	0.191362	+	0.981520i	$0.438710\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.0910811	−	0.995843i	$-0.470968\pi$
$71$		0			0		−0.0910811	+	0.995843i	$0.529032\pi$
$72$		0			0
$73$	0.190162	+	0.628311i	0.190162	+	0.628311i	0.999179	+	0.0405256i	$0.0129032\pi$
$73$	0.190162	+	0.628311i	0.190162	+	0.628311i	−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$	0.230981	+	0.972958i	0.230981	+	0.972958i
$76$	1.97214	−	0.322609i	1.97214	−	0.322609i
$77$		0			0
$78$		0			0
$79$	−0.381813	+	1.26154i	−0.381813	+	1.26154i	0.528964	+	0.848644i	$0.322581\pi$
$79$	−0.381813	+	1.26154i	−0.381813	+	1.26154i	−0.910777	+	0.412899i	$0.864516\pi$
$80$		0			0
$81$	0.422108	−	0.906546i	0.422108	−	0.906546i
$82$		0			0
$83$		0			0		0.422108	−	0.906546i	$-0.361290\pi$
$83$		0			0		−0.422108	+	0.906546i	$0.638710\pi$
$84$	−1.41173	+	1.36945i	−1.41173	+	1.36945i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.403648	−	0.914914i	$-0.367742\pi$
$89$		0			0		−0.403648	+	0.914914i	$0.632258\pi$
$90$		0			0
$91$	−1.43733	−	2.78594i	−1.43733	−	2.78594i
$92$		0			0
$93$	0.562921	−	0.826511i	0.562921	−	0.826511i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.215942	−	0.0535821i	0.215942	−	0.0535821i	−0.131363	−	0.991334i	$-0.541935\pi$
$97$	0.215942	−	0.0535821i	0.215942	−	0.0535821i	0.347305	+	0.937752i	$0.387097\pi$
$98$		0			0
$99$		0			0
$100$	0.796938	−	0.604061i	0.796938	−	0.604061i
$101$		0			0		−0.731738	−	0.681586i	$-0.761290\pi$
$101$		0			0		0.731738	+	0.681586i	$0.238710\pi$
$102$		0			0
$103$	−0.965428	−	0.510542i	−0.965428	−	0.510542i	−0.0910811	−	0.995843i	$-0.529032\pi$
$103$	−0.965428	−	0.510542i	−0.965428	−	0.510542i	−0.874347	+	0.485302i	$0.838710\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.843240	−	0.537537i	$-0.180645\pi$
$107$		0			0		−0.843240	+	0.537537i	$0.819355\pi$
$108$	−0.999795	−	0.0202670i	−0.999795	−	0.0202670i
$109$	1.24674	−	1.54492i	1.24674	−	1.54492i	0.528964	−	0.848644i	$-0.322581\pi$
$109$	1.24674	−	1.54492i	1.24674	−	1.54492i	0.717774	−	0.696277i	$-0.245161\pi$
$110$		0			0
$111$	−0.00798633	−	0.262605i	−0.00798633	−	0.262605i
$112$	1.82277	+	0.738836i	1.82277	+	0.738836i
$113$		0			0		0.494138	−	0.869384i	$-0.335484\pi$
$113$		0			0		−0.494138	+	0.869384i	$0.664516\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.492535	−	1.51587i	0.492535	−	1.51587i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.458499	−	0.888695i	0.458499	−	0.888695i
$122$		0			0
$123$		0			0
$124$	−0.983408	−	0.181405i	−0.983408	−	0.181405i
$125$		0			0
$126$		0			0
$127$	1.70599	+	0.992652i	1.70599	+	0.992652i	0.934184	+	0.356791i	$0.116129\pi$
$127$	1.70599	+	0.992652i	1.70599	+	0.992652i	0.771804	+	0.635861i	$0.219355\pi$
$128$		0			0
$129$	−0.643738	+	1.73815i	−0.643738	+	1.73815i
$130$		0			0
$131$		0			0		0.111245	−	0.993793i	$-0.464516\pi$
$131$		0			0		−0.111245	+	0.993793i	$0.535484\pi$
$132$		0			0
$133$	2.87602	+	2.67891i	2.87602	+	2.67891i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.975249	−	0.221109i	$-0.929032\pi$
$137$		0			0		0.975249	+	0.221109i	$0.0709677\pi$
$138$		0			0
$139$	−0.335841	+	0.853286i	−0.335841	+	0.853286i	0.659028	+	0.752118i	$0.270968\pi$
$139$	−0.335841	+	0.853286i	−0.335841	+	0.853286i	−0.994869	+	0.101168i	$0.967742\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.385023	+	0.922907i	0.385023	+	0.922907i
$145$		0			0
$146$		0			0
$147$	−2.83955	−	0.405596i	−2.83955	−	0.405596i
$148$	−0.237038	+	0.113307i	−0.237038	+	0.113307i
$149$		0			0		−0.347305	−	0.937752i	$-0.612903\pi$
$149$		0			0		0.347305	+	0.937752i	$0.387097\pi$
$150$		0			0
$151$	0.165844	−	1.48155i	0.165844	−	1.48155i	−0.579556	−	0.814932i	$-0.696774\pi$
$151$	0.165844	−	1.48155i	0.165844	−	1.48155i	0.745401	−	0.666616i	$-0.232258\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.58570	+	0.161250i	−1.58570	+	0.161250i
$157$	1.35629	+	0.549755i	1.35629	+	0.549755i	0.934184	−	0.356791i	$-0.116129\pi$
$157$	1.35629	+	0.549755i	1.35629	+	0.549755i	0.422108	+	0.906546i	$0.361290\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.35619	+	1.07192i	−1.35619	+	1.07192i	−0.366239	+	0.930521i	$0.619355\pi$
$163$	−1.35619	+	1.07192i	−1.35619	+	1.07192i	−0.989952	+	0.141403i	$0.954839\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.745401	−	0.666616i	$-0.232258\pi$
$167$		0			0		−0.745401	+	0.666616i	$0.767742\pi$
$168$		0			0
$169$	0.294782	−	1.51197i	0.294782	−	1.51197i
$170$		0			0
$171$	0.0607457	+	1.99743i	0.0607457	+	1.99743i
$172$	1.85010	−	0.112634i	1.85010	−	0.112634i
$173$		0			0		0.884003	−	0.467482i	$-0.154839\pi$
$173$		0			0		−0.884003	+	0.467482i	$0.845161\pi$
$174$		0			0
$175$	1.89894	+	0.512258i	1.89894	+	0.512258i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.934184	−	0.356791i	$-0.883871\pi$
$179$		0			0		0.934184	+	0.356791i	$0.116129\pi$
$180$		0			0
$181$	−1.34149	−	1.15163i	−1.34149	−	1.15163i	−0.975249	−	0.221109i	$-0.929032\pi$
$181$	−1.34149	−	1.15163i	−1.34149	−	1.15163i	−0.366239	−	0.930521i	$-0.619355\pi$
$182$		0			0
$183$	−0.0572209	+	0.805266i	−0.0572209	+	0.805266i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.17213	−	1.57939i	−1.17213	−	1.57939i
$190$		0			0
$191$		0			0		−0.612106	−	0.790776i	$-0.709677\pi$
$191$		0			0		0.612106	+	0.790776i	$0.290323\pi$
$192$	0.688967	−	0.724793i	0.688967	−	0.724793i
$193$	0.421649	+	0.0256700i	0.421649	+	0.0256700i	0.270221	−	0.962798i	$-0.412903\pi$
$193$	0.421649	+	0.0256700i	0.421649	+	0.0256700i	0.151428	+	0.988468i	$0.451613\pi$
$194$		0			0
$195$		0			0
$196$	0.775094	+	2.76166i	0.775094	+	2.76166i
$197$		0			0		0.211215	−	0.977440i	$-0.432258\pi$
$197$		0			0		−0.211215	+	0.977440i	$0.567742\pi$
$198$		0			0
$199$	0.0381801	−	0.0473117i	0.0381801	−	0.0473117i	−0.758758	−	0.651372i	$-0.774194\pi$
$199$	0.0381801	−	0.0473117i	0.0381801	−	0.0473117i	0.796938	+	0.604061i	$0.206452\pi$
$200$		0			0
$201$	−0.0179170	+	0.00947492i	−0.0179170	+	0.00947492i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.787594	+	1.38569i	0.787594	+	1.38569i
$209$		0			0
$210$		0			0
$211$	−0.387455	−	0.621614i	−0.387455	−	0.621614i	0.595954	−	0.803019i	$-0.296774\pi$
$211$	−0.387455	−	0.621614i	−0.387455	−	0.621614i	−0.983408	+	0.181405i	$0.941935\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.901784	−	1.74790i	−0.901784	−	1.74790i
$218$		0			0
$219$	0.358462	+	0.549947i	0.358462	+	0.549947i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.36914	+	1.17536i	−1.36914	+	1.17536i	−0.403648	+	0.914914i	$0.632258\pi$
$223$	−1.36914	+	1.17536i	−1.36914	+	1.17536i	−0.965487	+	0.260450i	$0.916129\pi$
$224$		0			0
$225$	0.494138	+	0.869384i	0.494138	+	0.869384i
$226$		0			0
$227$		0			0		0.864331	−	0.502923i	$-0.167742\pi$
$227$		0			0		−0.864331	+	0.502923i	$0.832258\pi$
$228$	1.80296	−	0.861838i	1.80296	−	0.861838i
$229$	−1.57041	+	1.14097i	−1.57041	+	1.14097i	−0.643650	+	0.765320i	$0.722581\pi$
$229$	−1.57041	+	1.14097i	−1.57041	+	1.14097i	−0.926761	+	0.375650i	$0.877419\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.674136	−	0.738607i	$-0.735484\pi$
$233$		0			0		0.674136	+	0.738607i	$0.264516\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.0133572	+	1.31799i	−0.0133572	+	1.31799i
$238$		0			0
$239$		0			0		0.998152	−	0.0607676i	$-0.0193548\pi$
$239$		0			0		−0.998152	+	0.0607676i	$0.980645\pi$
$240$		0			0
$241$	0.202351	−	0.458651i	0.202351	−	0.458651i	−0.784532	−	0.620088i	$-0.787097\pi$
$241$	0.202351	−	0.458651i	0.202351	−	0.458651i	0.986883	+	0.161437i	$0.0516129\pi$
$242$		0			0
$243$	0.151428	−	0.988468i	0.151428	−	0.988468i
$244$	0.759847	−	0.272692i	0.759847	−	0.272692i
$245$		0			0
$246$		0			0
$247$	0.482318	+	3.14840i	0.482318	+	3.14840i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.230981	−	0.972958i	$-0.574194\pi$
$251$		0			0		0.230981	+	0.972958i	$0.425806\pi$
$252$	−0.971878	+	1.70992i	−0.971878	+	1.70992i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.941224	−	0.337784i	−0.941224	−	0.337784i
$257$		0			0		0.476416	−	0.879220i	$-0.341935\pi$
$257$		0			0		−0.476416	+	0.879220i	$0.658065\pi$
$258$		0			0
$259$	−0.461598	−	0.232257i	−0.461598	−	0.232257i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.975249	−	0.221109i	$-0.929032\pi$
$263$		0			0		0.975249	+	0.221109i	$0.0709677\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.0159009	+	0.0125679i	0.0159009	+	0.0125679i
$269$		0			0		−0.926761	−	0.375650i	$-0.877419\pi$
$269$		0			0		0.926761	+	0.375650i	$0.122581\pi$
$270$		0			0
$271$	1.87089	−	0.306045i	1.87089	−	0.306045i	0.884003	−	0.467482i	$-0.154839\pi$
$271$	1.87089	−	0.306045i	1.87089	−	0.306045i	0.986883	+	0.161437i	$0.0516129\pi$
$272$		0			0
$273$	−2.15982	−	2.27213i	−2.15982	−	2.27213i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.13116	+	1.01160i	−1.13116	+	1.01160i	−0.131363	+	0.991334i	$0.541935\pi$
$277$	−1.13116	+	1.01160i	−1.13116	+	1.01160i	−0.999795	+	0.0202670i	$0.993548\pi$
$278$		0			0
$279$	0.309017	−	0.951057i	0.309017	−	0.951057i
$280$		0			0
$281$		0			0		0.0303978	−	0.999538i	$-0.490323\pi$
$281$		0			0		−0.0303978	+	0.999538i	$0.509677\pi$
$282$		0			0
$283$	1.42907	+	1.12952i	1.42907	+	1.12952i	0.970568	+	0.240829i	$0.0774194\pi$
$283$	1.42907	+	1.12952i	1.42907	+	1.12952i	0.458499	+	0.888695i	$0.348387\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.511656	+	0.859190i	−0.511656	+	0.859190i
$290$		0			0
$291$	0.192305	−	0.111895i	0.192305	−	0.111895i
$292$	0.358462	−	0.549947i	0.358462	−	0.549947i
$293$		0			0		−0.989952	−	0.141403i	$-0.954839\pi$
$293$		0			0		0.989952	+	0.141403i	$0.0451613\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.595954	−	0.803019i	0.595954	−	0.803019i
$301$	2.40251	+	2.74188i	2.40251	+	2.74188i
$302$		0			0
$303$		0			0
$304$	−1.51627	−	1.30167i	−1.51627	−	1.30167i
$305$		0			0
$306$		0			0
$307$	0.642339	+	0.703769i	0.642339	+	0.703769i	0.970568	−	0.240829i	$-0.0774194\pi$
$307$	0.642339	+	0.703769i	0.642339	+	0.703769i	−0.328229	+	0.944598i	$0.606452\pi$
$308$		0			0
$309$	−1.06975	−	0.219840i	−1.06975	−	0.219840i
$310$		0			0
$311$		0			0		0.979530	−	0.201299i	$-0.0645161\pi$
$311$		0			0		−0.979530	+	0.201299i	$0.935484\pi$
$312$		0			0
$313$	0.285319	−	1.63971i	0.285319	−	1.63971i	−0.403648	−	0.914914i	$-0.632258\pi$
$313$	0.285319	−	1.63971i	0.285319	−	1.63971i	0.688967	−	0.724793i	$-0.258065\pi$
$314$		0			0
$315$		0			0
$316$	1.21124	−	0.519783i	1.21124	−	0.519783i
$317$		0			0		0.999179	−	0.0405256i	$-0.0129032\pi$
$317$		0			0		−0.999179	+	0.0405256i	$0.987097\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.965487	+	0.260450i	−0.965487	+	0.260450i
$325$	0.949876	+	1.27991i	0.949876	+	1.27991i
$326$		0			0
$327$	0.764359	−	1.83218i	0.764359	−	1.83218i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.15422	+	1.43028i	1.15422	+	1.43028i	0.884003	+	0.467482i	$0.154839\pi$
$331$	1.15422	+	1.43028i	1.15422	+	1.43028i	0.270221	+	0.962798i	$0.412903\pi$
$332$		0			0
$333$	−0.0811871	−	0.249868i	−0.0811871	−	0.249868i
$334$		0			0
$335$		0			0
$336$	1.95673	+	0.198980i	1.95673	+	0.198980i
$337$	0.552719	+	0.536165i	0.552719	+	0.536165i	0.918958	−	0.394356i	$-0.129032\pi$
$337$	0.552719	+	0.536165i	0.552719	+	0.536165i	−0.366239	+	0.930521i	$0.619355\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.18997	+	2.95088i	−2.18997	+	2.95088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.954139	−	0.299363i	$-0.0967742\pi$
$347$		0			0		−0.954139	+	0.299363i	$0.903226\pi$
$348$		0			0
$349$	−0.265289	−	1.22768i	−0.265289	−	1.22768i	−0.893295	−	0.449470i	$-0.851613\pi$
$349$	−0.265289	−	1.22768i	−0.265289	−	1.22768i	0.628007	−	0.778208i	$-0.283871\pi$
$350$		0			0
$351$	0.0484504	−	1.59314i	0.0484504	−	1.59314i
$352$		0			0
$353$		0			0		0.910777	−	0.412899i	$-0.135484\pi$
$353$		0			0		−0.910777	+	0.412899i	$0.864516\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.960010	−	0.279964i	$-0.0903226\pi$
$359$		0			0		−0.960010	+	0.279964i	$0.909677\pi$
$360$		0			0
$361$	−1.42612	−	2.63188i	−1.42612	−	2.63188i
$362$		0			0
$363$	0.191362	−	0.981520i	0.191362	−	0.981520i
$364$	−1.20699	+	2.89319i	−1.20699	+	2.89319i
$365$		0			0
$366$		0			0
$367$	−1.63310	−	1.13667i	−1.63310	−	1.13667i	−0.874347	−	0.485302i	$-0.838710\pi$
$367$	−1.63310	−	1.13667i	−1.63310	−	1.13667i	−0.758758	−	0.651372i	$-0.774194\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.994869	+	0.101168i	−0.994869	+	0.101168i
$373$	−0.680935	−	1.09246i	−0.680935	−	1.09246i	−0.989952	−	0.141403i	$-0.954839\pi$
$373$	−0.680935	−	1.09246i	−0.680935	−	1.09246i	0.309017	−	0.951057i	$-0.400000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.105573	+	1.15430i	0.105573	+	1.15430i	0.864331	+	0.502923i	$0.167742\pi$
$379$	0.105573	+	1.15430i	0.105573	+	1.15430i	−0.758758	+	0.651372i	$0.774194\pi$
$380$		0			0
$381$	1.91567	+	0.475340i	1.91567	+	0.475340i
$382$		0			0
$383$		0			0		−0.562921	−	0.826511i	$-0.690323\pi$
$383$		0			0		0.562921	+	0.826511i	$0.309677\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.131377	+	1.84886i	−0.131377	+	1.84886i
$388$	−0.179998	−	0.130776i	−0.179998	−	0.130776i
$389$		0			0		0.934184	−	0.356791i	$-0.116129\pi$
$389$		0			0		−0.934184	+	0.356791i	$0.883871\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.31921	+	0.566116i	1.31921	+	0.566116i	0.934184	−	0.356791i	$-0.116129\pi$
$397$	1.31921	+	0.566116i	1.31921	+	0.566116i	0.385023	+	0.922907i	$0.374194\pi$
$398$		0			0
$399$	3.51101	+	1.76660i	3.51101	+	1.76660i
$400$	−0.975249	−	0.221109i	−0.975249	−	0.221109i
$401$		0			0		−0.366239	−	0.930521i	$-0.619355\pi$
$401$		0			0		0.366239	+	0.930521i	$0.380645\pi$
$402$		0			0
$403$	0.305007	−	1.56442i	0.305007	−	1.56442i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.73972	−	0.176912i	1.73972	−	0.176912i	0.820763	−	0.571268i	$-0.193548\pi$
$409$	1.73972	−	0.176912i	1.73972	−	0.176912i	0.918958	+	0.394356i	$0.129032\pi$
$410$		0			0
$411$		0			0
$412$	0.230670	+	1.06747i	0.230670	+	1.06747i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.0835213	+	0.913187i	−0.0835213	+	0.913187i
$418$		0			0
$419$		0			0		0.270221	−	0.962798i	$-0.412903\pi$
$419$		0			0		−0.270221	+	0.962798i	$0.587097\pi$
$420$		0			0
$421$	−1.29436	−	1.41814i	−1.29436	−	1.41814i	−0.853961	−	0.520337i	$-0.825806\pi$
$421$	−1.29436	−	1.41814i	−1.29436	−	1.41814i	−0.440394	−	0.897805i	$-0.645161\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.33890	+	0.853504i	1.33890	+	0.853504i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.999795	−	0.0202670i	$-0.993548\pi$
$431$		0			0		0.999795	+	0.0202670i	$0.00645161\pi$
$432$	0.628007	+	0.778208i	0.628007	+	0.778208i
$433$	−0.516750	−	0.667585i	−0.516750	−	0.667585i	0.458499	−	0.888695i	$-0.348387\pi$
$433$	−0.516750	−	0.667585i	−0.516750	−	0.667585i	−0.975249	+	0.221109i	$0.929032\pi$
$434$		0			0
$435$		0			0
$436$	−1.98482	+	0.0402345i	−1.98482	+	0.0402345i
$437$		0			0
$438$		0			0
$439$	−1.99959			−1.99959			−0.999795	−	0.0202670i	$-0.993548\pi$
$439$	−1.99959			−1.99959			−0.999795	+	0.0202670i	$0.993548\pi$
$440$		0			0
$441$	−2.83955	+	0.405596i	−2.83955	+	0.405596i
$442$		0			0
$443$		0			0		0.784532	−	0.620088i	$-0.212903\pi$
$443$		0			0		−0.784532	+	0.620088i	$0.787097\pi$
$444$	−0.195837	+	0.175138i	−0.195837	+	0.175138i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.607780	−	1.87055i	−0.607780	−	1.87055i
$449$		0			0		0.703515	−	0.710681i	$-0.251613\pi$
$449$		0			0		−0.703515	+	0.710681i	$0.748387\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.255568	−	1.46873i	−0.255568	−	1.46873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.402566	−	0.566060i	−0.402566	−	0.566060i	0.562921	−	0.826511i	$-0.309677\pi$
$457$	−0.402566	−	0.566060i	−0.402566	−	0.566060i	−0.965487	+	0.260450i	$0.916129\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.986883	−	0.161437i	$-0.948387\pi$
$461$		0			0		0.986883	+	0.161437i	$0.0516129\pi$
$462$		0			0
$463$	1.60690	+	1.07076i	1.60690	+	1.07076i	0.947876	+	0.318639i	$0.103226\pi$
$463$	1.60690	+	1.07076i	1.60690	+	1.07076i	0.659028	+	0.752118i	$0.270968\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.960010	−	0.279964i	$-0.909677\pi$
$467$		0			0		0.960010	+	0.279964i	$0.0903226\pi$
$468$	−1.47714	+	0.598740i	−1.47714	+	0.598740i
$469$	−0.000403976		0.0398614i	−0.000403976		0.0398614i
$470$		0			0
$471$	1.45597	+	0.148057i	1.45597	+	0.148057i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.66298	−	1.10812i	−1.66298	−	1.10812i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.659028	−	0.752118i	$-0.729032\pi$
$479$		0			0		0.659028	+	0.752118i	$0.270968\pi$
$480$		0			0
$481$	−0.176759	−	0.379620i	−0.176759	−	0.379620i
$482$		0			0
$483$		0			0
$484$	−0.975249	+	0.221109i	−0.975249	+	0.221109i
$485$		0			0
$486$		0			0
$487$	−1.26945	−	0.606814i	−1.26945	−	0.606814i	−0.328229	−	0.944598i	$-0.606452\pi$
$487$	−1.26945	−	0.606814i	−1.26945	−	0.606814i	−0.941224	+	0.337784i	$0.890323\pi$
$488$		0			0
$489$	−1.00186	+	1.40874i	−1.00186	+	1.40874i
$490$		0			0
$491$		0			0		0.874347	−	0.485302i	$-0.161290\pi$
$491$		0			0		−0.874347	+	0.485302i	$0.838710\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.494138	+	0.869384i	0.494138	+	0.869384i
$497$		0			0
$498$		0			0
$499$	1.96318	−	0.362140i	1.96318	−	0.362140i	0.970568	−	0.240829i	$-0.0774194\pi$
$499$	1.96318	−	0.362140i	1.96318	−	0.362140i	0.992615	−	0.121311i	$-0.0387097\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.191362	−	0.981520i	$-0.561290\pi$
$503$		0			0		0.191362	+	0.981520i	$0.438710\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.140305	−	1.53404i	−0.140305	−	1.53404i
$508$	−0.338363	−	1.94455i	−0.338363	−	1.94455i
$509$		0			0		0.983408	−	0.181405i	$-0.0580645\pi$
$509$		0			0		−0.983408	+	0.181405i	$0.941935\pi$
$510$		0			0
$511$	1.28689	−	0.104562i	1.28689	−	0.104562i
$512$		0			0
$513$	0.617526	+	1.90055i	0.617526	+	1.90055i
$514$		0			0
$515$		0			0
$516$	1.74458	−	0.626091i	1.74458	−	0.626091i
$517$		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
$523$	−0.140335	+	0.0200452i	−0.140335	+	0.0200452i	−0.211215	−	0.977440i	$-0.567742\pi$
$523$	−0.140335	+	0.0200452i	−0.140335	+	0.0200452i	0.0708797	+	0.997485i	$0.477419\pi$
$524$		0			0
$525$	1.96641	−	0.0398614i	1.96641	−	0.0398614i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.628007	+	0.778208i	0.628007	+	0.778208i
$530$		0			0
$531$		0			0
$532$	0.199072	−	3.92536i	0.199072	−	3.92536i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.12492	+	0.0456254i	1.12492	+	0.0456254i	0.595954	−	0.803019i	$-0.296774\pi$
$541$	1.12492	+	0.0456254i	1.12492	+	0.0456254i	0.528964	+	0.848644i	$0.322581\pi$
$542$		0			0
$543$	−1.61026	−	0.730008i	−1.61026	−	0.730008i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.484867	+	1.49227i	−0.484867	+	1.49227i	0.347305	+	0.937752i	$0.387097\pi$
$547$	−0.484867	+	1.49227i	−0.484867	+	1.49227i	−0.832173	+	0.554517i	$0.812903\pi$
$548$		0			0
$549$	0.170513	+	0.789084i	0.170513	+	0.789084i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.26664	+	1.25809i	2.26664	+	1.25809i
$554$		0			0
$555$		0			0
$556$	0.869201	−	0.292192i	0.869201	−	0.292192i
$557$		0			0		0.0506492	−	0.998717i	$-0.483871\pi$
$557$		0			0		−0.0506492	+	0.998717i	$0.516129\pi$
$558$		0			0
$559$	0.149632	+	2.95049i	0.149632	+	2.95049i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.440394	−	0.897805i	$-0.645161\pi$
$563$		0			0		0.440394	+	0.897805i	$0.354839\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.56743	−	1.18808i	−1.56743	−	1.18808i
$568$		0			0
$569$		0			0		−0.986883	−	0.161437i	$-0.948387\pi$
$569$		0			0		0.986883	+	0.161437i	$0.0516129\pi$
$570$		0			0
$571$	0.372347	−	1.32667i	0.372347	−	1.32667i	−0.511656	−	0.859190i	$-0.670968\pi$
$571$	0.372347	−	1.32667i	0.372347	−	1.32667i	0.884003	−	0.467482i	$-0.154839\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.458499	−	0.888695i	0.458499	−	0.888695i
$577$	−0.309185	−	0.785560i	−0.309185	−	0.785560i	−0.998152	−	0.0607676i	$-0.980645\pi$
$577$	−0.309185	−	0.785560i	−0.309185	−	0.785560i	0.688967	−	0.724793i	$-0.258065\pi$
$578$		0			0
$579$	0.411974	−	0.0934031i	0.411974	−	0.0934031i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$	1.51726	+	2.43422i	1.51726	+	2.43422i
$589$	0.302607	+	1.97531i	0.302607	+	1.97531i
$590$		0			0
$591$		0			0
$592$	0.239286	+	0.108480i	0.239286	+	0.108480i
$593$		0			0		0.579556	−	0.814932i	$-0.303226\pi$
$593$		0			0		−0.579556	+	0.814932i	$0.696774\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.0234077	−	0.0561087i	0.0234077	−	0.0561087i
$598$		0			0
$599$		0			0		0.910777	−	0.412899i	$-0.135484\pi$
$599$		0			0		−0.910777	+	0.412899i	$0.864516\pi$
$600$		0			0
$601$	1.61770	−	0.0327926i	1.61770	−	0.0327926i	0.796938	−	0.604061i	$-0.206452\pi$
$601$	1.61770	−	0.0327926i	1.61770	−	0.0327926i	0.820763	+	0.571268i	$0.193548\pi$
$602$		0			0
$603$	−0.0145478	+	0.0141121i	−0.0145478	+	0.0141121i
$604$	−1.24060	+	0.826674i	−1.24060	+	0.826674i
$605$		0			0
$606$		0			0
$607$	0.0932044	+	0.156512i	0.0932044	+	0.156512i	0.902221	−	0.431273i	$-0.141935\pi$
$607$	0.0932044	+	0.156512i	0.0932044	+	0.156512i	−0.809017	+	0.587785i	$0.800000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.946066	−	1.07970i	0.946066	−	1.07970i	−0.0506492	−	0.998717i	$-0.516129\pi$
$613$	0.946066	−	1.07970i	0.946066	−	1.07970i	0.996715	−	0.0809846i	$-0.0258065\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.366239	−	0.930521i	$-0.380645\pi$
$617$		0			0		−0.366239	+	0.930521i	$0.619355\pi$
$618$		0			0
$619$	−1.18297	−	0.507654i	−1.18297	−	0.507654i	−0.289679	−	0.957124i	$-0.593548\pi$
$619$	−1.18297	−	0.507654i	−1.18297	−	0.507654i	−0.893295	+	0.449470i	$0.851613\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.14404	+	1.10978i	1.14404	+	1.10978i
$625$	−0.994869	−	0.101168i	−0.994869	−	0.101168i
$626$		0			0
$627$		0			0
$628$	−0.452239	−	1.39185i	−0.452239	−	1.39185i
$629$		0			0
$630$		0			0
$631$	0.467874	−	1.06049i	0.467874	−	1.06049i	−0.511656	−	0.859190i	$-0.670968\pi$
$631$	0.467874	−	1.06049i	0.467874	−	1.06049i	0.979530	−	0.201299i	$-0.0645161\pi$
$632$		0			0
$633$	−0.545990	−	0.488282i	−0.545990	−	0.488282i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−4.41404	+	1.19073i	−4.41404	+	1.19073i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.960010	−	0.279964i	$-0.909677\pi$
$641$		0			0		0.960010	+	0.279964i	$0.0903226\pi$
$642$		0			0
$643$	−0.182336	+	0.336498i	−0.182336	+	0.336498i	−0.954139	−	0.299363i	$-0.903226\pi$
$643$	−0.182336	+	0.336498i	−0.182336	+	0.336498i	0.771804	+	0.635861i	$0.219355\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.731738	−	0.681586i	$-0.238710\pi$
$647$		0			0		−0.731738	+	0.681586i	$0.761290\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.35507	−	1.42553i	−1.35507	−	1.42553i
$652$	1.69328	+	0.347977i	1.69328	+	0.347977i
$653$		0			0		0.511656	−	0.859190i	$-0.329032\pi$
$653$		0			0		−0.511656	+	0.859190i	$0.670968\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.498092	+	0.427598i	0.498092	+	0.427598i
$658$		0			0
$659$		0			0		−0.853961	−	0.520337i	$-0.825806\pi$
$659$		0			0		0.853961	+	0.520337i	$0.174194\pi$
$660$		0			0
$661$	−0.964273	+	1.29931i	−0.964273	+	1.29931i	−0.0101340	+	0.999949i	$0.503226\pi$
$661$	−0.964273	+	1.29931i	−0.964273	+	1.29931i	−0.954139	+	0.299363i	$0.903226\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.985325	+	1.51167i	−0.985325	+	1.51167i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.25498	+	0.0509007i	1.25498	+	0.0509007i	0.659028	−	0.752118i	$-0.270968\pi$
$673$	1.25498	+	0.0509007i	1.25498	+	0.0509007i	0.595954	+	0.803019i	$0.296774\pi$
$674$		0			0
$675$	0.717774	+	0.696277i	0.717774	+	0.696277i
$676$	−1.34688	+	0.747580i	−1.34688	+	0.747580i
$677$		0			0		0.820763	−	0.571268i	$-0.193548\pi$
$677$		0			0		−0.820763	+	0.571268i	$0.806452\pi$
$678$		0			0
$679$	−0.0133020	−	0.437395i	−0.0133020	−	0.437395i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.688967	−	0.724793i	$-0.741935\pi$
$683$		0			0		0.688967	+	0.724793i	$0.258065\pi$
$684$	1.48958	−	1.33214i	1.48958	−	1.33214i
$685$		0			0
$686$		0			0
$687$	−1.18818	+	1.53500i	−1.18818	+	1.53500i
$688$	−1.27702	−	1.34342i	−1.27702	−	1.34342i
$689$		0			0
$690$		0			0
$691$	−0.724785	+	1.73732i	−0.724785	+	1.73732i	−0.0506492	+	0.998717i	$0.516129\pi$
$691$	−0.724785	+	1.73732i	−0.724785	+	1.73732i	−0.674136	+	0.738607i	$0.735484\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.830209	−	1.78301i	−0.830209	−	1.78301i
$701$		0			0		−0.546055	−	0.837749i	$-0.683871\pi$
$701$		0			0		0.546055	+	0.837749i	$0.316129\pi$
$702$		0			0
$703$	0.369361	+	0.373123i	0.369361	+	0.373123i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.942952	+	1.65903i	−0.942952	+	1.65903i	−0.211215	+	0.977440i	$0.567742\pi$
$709$	−0.942952	+	1.65903i	−0.942952	+	1.65903i	−0.731738	+	0.681586i	$0.761290\pi$
$710$		0			0
$711$	0.356167	+	1.26902i	0.356167	+	1.26902i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.612106	−	0.790776i	$-0.709677\pi$
$719$		0			0		0.612106	+	0.790776i	$0.290323\pi$
$720$		0			0
$721$	−1.38255	+	1.64389i	−1.38255	+	1.64389i
$722$		0			0
$723$	0.0658532	−	0.496961i	0.0658532	−	0.496961i
$724$	−0.0179170	+	1.76791i	−0.0179170	+	1.76791i
$725$		0			0
$726$		0			0
$727$	−0.221611	−	0.206422i	−0.221611	−	0.206422i	0.562921	−	0.826511i	$-0.309677\pi$
$727$	−0.221611	−	0.206422i	−0.221611	−	0.206422i	−0.784532	+	0.620088i	$0.787097\pi$
$728$		0			0
$729$	−0.131363	−	0.991334i	−0.131363	−	0.991334i
$730$		0			0
$731$		0			0
$732$	0.653117	−	0.474517i	0.653117	−	0.474517i
$733$	0.954485	−	0.456256i	0.954485	−	0.456256i	0.111245	−	0.993793i	$-0.464516\pi$
$733$	0.954485	−	0.456256i	0.954485	−	0.456256i	0.843240	+	0.537537i	$0.180645\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.532405	−	0.228473i	−0.532405	−	0.228473i	0.111245	−	0.993793i	$-0.464516\pi$
$739$	−0.532405	−	0.228473i	−0.532405	−	0.228473i	−0.643650	+	0.765320i	$0.722581\pi$
$740$		0			0
$741$	1.34447	+	2.88747i	1.34447	+	2.88747i
$742$		0			0
$743$		0			0		−0.440394	−	0.897805i	$-0.645161\pi$
$743$		0			0		0.440394	+	0.897805i	$0.354839\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.267441	+	0.641061i	0.267441	+	0.641061i	0.999179	−	0.0405256i	$-0.0129032\pi$
$751$	0.267441	+	0.641061i	0.267441	+	0.641061i	−0.731738	+	0.681586i	$0.761290\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.454297	+	1.91363i	−0.454297	+	1.91363i
$757$	−0.0675365	+	0.950436i	−0.0675365	+	0.950436i	0.843240	+	0.537537i	$0.180645\pi$
$757$	−0.0675365	+	0.950436i	−0.0675365	+	0.950436i	−0.910777	+	0.412899i	$0.864516\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.211215	−	0.977440i	$-0.567742\pi$
$761$		0			0		0.211215	+	0.977440i	$0.432258\pi$
$762$		0			0
$763$	−2.45210	−	3.03858i	−2.45210	−	3.03858i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.998152	−	0.0607676i	−0.998152	−	0.0607676i
$769$	0.821185	−	0.863886i	0.821185	−	0.863886i	−0.171430	−	0.985196i	$-0.554839\pi$
$769$	0.821185	−	0.863886i	0.821185	−	0.863886i	0.992615	+	0.121311i	$0.0387097\pi$
$770$		0			0
$771$		0			0
$772$	−0.251748	−	0.339219i	−0.251748	−	0.339219i
$773$		0			0		−0.771804	−	0.635861i	$-0.780645\pi$
$773$		0			0		0.771804	+	0.635861i	$0.219355\pi$
$774$		0			0
$775$	0.595954	+	0.803019i	0.595954	+	0.803019i
$776$		0			0
$777$	−0.508162	−	0.0937385i	−0.508162	−	0.0937385i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.61467	−	2.37074i	1.61467	−	2.37074i
$785$		0			0
$786$		0			0
$787$	−0.845566	−	0.246589i	−0.845566	−	0.246589i	−0.171430	−	0.985196i	$-0.554839\pi$
$787$	−0.845566	−	0.246589i	−0.845566	−	0.246589i	−0.674136	+	0.738607i	$0.735484\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.422342	+	1.21544i	0.422342	+	1.21544i
$794$		0			0
$795$		0			0
$796$	−0.0607832	+	0.00123214i	−0.0607832	+	0.00123214i
$797$		0			0		0.289679	−	0.957124i	$-0.406452\pi$
$797$		0			0		−0.289679	+	0.957124i	$0.593548\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.0187836	+	0.00761368i	0.0187836	+	0.00761368i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.270221	−	0.962798i	$-0.412903\pi$
$809$		0			0		−0.270221	+	0.962798i	$0.587097\pi$
$810$		0			0
$811$	−0.862759	−	0.177301i	−0.862759	−	0.177301i	−0.250653	−	0.968077i	$-0.580645\pi$
$811$	−0.862759	−	0.177301i	−0.862759	−	0.177301i	−0.612106	+	0.790776i	$0.709677\pi$
$812$		0			0
$813$	1.71039	−	0.817587i	1.71039	−	0.817587i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.42612	−	3.41845i	−1.42612	−	3.41845i
$818$		0			0
$819$	−2.70956	−	1.57659i	−2.70956	−	1.57659i
$820$		0			0
$821$		0			0		−0.947876	−	0.318639i	$-0.896774\pi$
$821$		0			0		0.947876	+	0.318639i	$0.103226\pi$
$822$		0			0
$823$	−1.12561	+	1.65268i	−1.12561	+	1.65268i	−0.546055	+	0.837749i	$0.683871\pi$
$823$	−1.12561	+	1.65268i	−1.12561	+	1.65268i	−0.579556	+	0.814932i	$0.696774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.230981	−	0.972958i	$-0.425806\pi$
$827$		0			0		−0.230981	+	0.972958i	$0.574194\pi$
$828$		0			0
$829$	−1.70652	−	0.0692145i	−1.70652	−	0.0692145i	−0.832173	−	0.554517i	$-0.812903\pi$
$829$	−1.70652	−	0.0692145i	−1.70652	−	0.0692145i	−0.874347	+	0.485302i	$0.838710\pi$
$830$		0			0
$831$	−0.802711	+	1.28783i	−0.802711	+	1.28783i
$832$	0.553562	−	1.49466i	0.553562	−	1.49466i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.0303978	−	0.999538i	0.0303978	−	0.999538i
$838$		0			0
$839$		0			0		−0.703515	−	0.710681i	$-0.748387\pi$
$839$		0			0		0.703515	+	0.710681i	$0.251613\pi$
$840$		0			0
$841$	−0.965487	−	0.260450i	−0.965487	−	0.260450i
$842$		0			0
$843$		0			0
$844$	−0.226348	+	0.696629i	−0.226348	+	0.696629i
$845$		0			0
$846$		0			0
$847$	−1.51800	−	1.25062i	−1.51800	−	1.25062i
$848$		0			0
$849$	1.68815	+	0.684267i	1.68815	+	0.684267i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.98933	+	0.0403259i	1.98933	+	0.0403259i	0.996715	−	0.0809846i	$-0.0258065\pi$
$853$	1.98933	+	0.0403259i	1.98933	+	0.0403259i	0.992615	+	0.121311i	$0.0387097\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.579556	−	0.814932i	$-0.303226\pi$
$857$		0			0		−0.579556	+	0.814932i	$0.696774\pi$
$858$		0			0
$859$	1.86223	−	0.711237i	1.86223	−	0.711237i	0.902221	−	0.431273i	$-0.141935\pi$
$859$	1.86223	−	0.711237i	1.86223	−	0.711237i	0.960010	−	0.279964i	$-0.0903226\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.758758	−	0.651372i	$-0.225806\pi$
$863$		0			0		−0.758758	+	0.651372i	$0.774194\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.250653	+	0.968077i	−0.250653	+	0.968077i
$868$	−0.757270	+	1.81519i	−0.757270	+	1.81519i
$869$		0			0
$870$		0			0
$871$	−0.0202876	+	0.0251398i	−0.0202876	+	0.0251398i
$872$		0			0
$873$	0.153288	−	0.161259i	0.153288	−	0.161259i
$874$		0			0
$875$		0			0
$876$	0.190162	−	0.628311i	0.190162	−	0.628311i
$877$	−0.878707	+	0.852390i	−0.878707	+	0.852390i	−0.989952	−	0.141403i	$-0.954839\pi$
$877$	−0.878707	+	0.852390i	−0.878707	+	0.852390i	0.111245	+	0.993793i	$0.464516\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.832173	−	0.554517i	$-0.187097\pi$
$881$		0			0		−0.832173	+	0.554517i	$0.812903\pi$
$882$		0			0
$883$	−0.609846	−	1.02408i	−0.609846	−	1.02408i	−0.994869	−	0.101168i	$-0.967742\pi$
$883$	−0.609846	−	1.02408i	−0.609846	−	1.02408i	0.385023	−	0.922907i	$-0.374194\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.771804	−	0.635861i	$-0.780645\pi$
$887$		0			0		0.771804	+	0.635861i	$0.219355\pi$
$888$		0			0
$889$	2.61702	−	2.86730i	2.61702	−	2.86730i
$890$		0			0
$891$		0			0
$892$	1.78077	+	0.291304i	1.78077	+	0.291304i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.347305	−	0.937752i	0.347305	−	0.937752i
$901$		0			0
$902$		0			0
$903$	3.07407	+	1.95961i	3.07407	+	1.95961i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.680745	−	1.54298i	−0.680745	−	1.54298i	−0.832173	−	0.554517i	$-0.812903\pi$
$907$	−0.680745	−	1.54298i	−0.680745	−	1.54298i	0.151428	−	0.988468i	$-0.451613\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.884003	−	0.467482i	$-0.845161\pi$
$911$		0			0		0.884003	+	0.467482i	$0.154839\pi$
$912$	−1.82006	−	0.825120i	−1.82006	−	0.825120i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.88400	+	0.467482i	1.88400	+	0.467482i
$917$		0			0
$918$		0			0
$919$	−1.86492	−	0.712262i	−1.86492	−	0.712262i	−0.954139	−	0.299363i	$-0.903226\pi$
$919$	−1.86492	−	0.712262i	−1.86492	−	0.712262i	−0.910777	−	0.412899i	$-0.864516\pi$
$920$		0			0
$921$	0.813682	+	0.495794i	0.813682	+	0.495794i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.250678	+	0.0786507i	0.250678	+	0.0786507i
$926$		0			0
$927$	−1.08852	+	0.0884440i	−1.08852	+	0.0884440i
$928$		0			0
$929$		0			0		−0.820763	−	0.571268i	$-0.806452\pi$
$929$		0			0		0.820763	+	0.571268i	$0.193548\pi$
$930$		0			0
$931$	4.70464	−	3.27452i	4.70464	−	3.27452i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.394498	+	0.240376i	−0.394498	+	0.240376i	−0.703515	−	0.710681i	$-0.748387\pi$
$937$	−0.394498	+	0.240376i	−0.394498	+	0.240376i	0.309017	+	0.951057i	$0.400000\pi$
$938$		0			0
$939$	−0.185150	−	1.65401i	−0.185150	−	1.65401i
$940$		0			0
$941$		0			0		0.111245	−	0.993793i	$-0.464516\pi$
$941$		0			0		−0.111245	+	0.993793i	$0.535484\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.796938	−	0.604061i	$-0.793548\pi$
$947$		0			0		0.796938	+	0.604061i	$0.206452\pi$
$948$	1.01728	−	0.838101i	1.01728	−	0.838101i
$949$	0.870712	+	0.580197i	0.870712	+	0.580197i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.965487	−	0.260450i	$-0.0838710\pi$
$953$		0			0		−0.965487	+	0.260450i	$0.916129\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.191362	−	0.981520i	0.191362	−	0.981520i
$962$		0			0
$963$		0			0
$964$	−0.481258	+	0.140347i	−0.481258	+	0.140347i
$965$		0			0
$966$		0			0
$967$	0.301861	−	1.97044i	0.301861	−	1.97044i	0.0708797	−	0.997485i	$-0.477419\pi$
$967$	0.301861	−	1.97044i	0.301861	−	1.97044i	0.230981	−	0.972958i	$-0.425806\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.947876	−	0.318639i	$-0.103226\pi$
$971$		0			0		−0.947876	+	0.318639i	$0.896774\pi$
$972$	−0.853961	+	0.520337i	−0.853961	+	0.520337i
$973$	1.50088	+	1.00011i	1.50088	+	1.00011i
$974$		0			0
$975$	1.27022	+	0.962798i	1.27022	+	0.962798i
$976$	−0.697772	−	0.406008i	−0.697772	−	0.406008i
$977$		0			0		0.171430	−	0.985196i	$-0.445161\pi$
$977$		0			0		−0.171430	+	0.985196i	$0.554839\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.220847	−	1.97291i	0.220847	−	1.97291i
$982$		0			0
$983$		0			0		−0.111245	−	0.993793i	$-0.535484\pi$
$983$		0			0		0.111245	+	0.993793i	$0.464516\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.09909	−	2.39560i	2.09909	−	2.39560i
$989$		0			0
$990$		0			0
$991$	−0.149512	+	0.104064i	−0.149512	+	0.104064i	−0.643650	−	0.765320i	$-0.722581\pi$
$991$	−0.149512	+	0.104064i	−0.149512	+	0.104064i	0.494138	+	0.869384i	$0.335484\pi$
$992$		0			0
$993$	1.50849	+	1.04994i	1.50849	+	1.04994i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.58802	+	0.498244i	1.58802	+	0.498244i	0.960010	−	0.279964i	$-0.0903226\pi$
$997$	1.58802	+	0.498244i	1.58802	+	0.498244i	0.628007	+	0.778208i	$0.283871\pi$
$998$		0			0
$999$	−0.147895	−	0.217147i	−0.147895	−	0.217147i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2883.1.z.a.419.1	✓	120
3.2	odd	2	CM	2883.1.z.a.419.1	✓	120
961.250	even	155	inner	2883.1.z.a.1211.1	yes	120
2883.1211	odd	310	inner	2883.1.z.a.1211.1	yes	120

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2883.1.z.a.419.1	✓	120	1.1	even	1	trivial
2883.1.z.a.419.1	✓	120	3.2	odd	2	CM
2883.1.z.a.1211.1	yes	120	961.250	even	155	inner
2883.1.z.a.1211.1	yes	120	2883.1211	odd	310	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:	\( N \)	\(=\)	\( 2883 = 3 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2883.z (of order \(310\), degree \(120\), minimal)

\(f(q)\)	\(=\)	\(q+(0.960010 - 0.279964i) q^{3} +(-0.643650 - 0.765320i) q^{4} +(0.415421 - 1.92244i) q^{7} +(0.843240 - 0.537537i) q^{9} +(-0.832173 - 0.554517i) q^{12} +(1.23016 - 1.01348i) q^{13} +(-0.171430 + 0.985196i) q^{16} +(-1.02247 + 1.71697i) q^{19} +(-0.139407 - 1.96187i) q^{21} +(-0.0506492 + 0.998717i) q^{25} +(0.659028 - 0.752118i) q^{27} +(-1.73867 + 0.919451i) q^{28} +(0.771804 - 0.635861i) q^{31} +(-0.954139 - 0.299363i) q^{36} +(0.0658532 - 0.254340i) q^{37} +(0.897227 - 1.31736i) q^{39} +(-1.10461 + 1.48841i) q^{43} +(0.111245 + 0.993793i) q^{48} +(-2.61244 - 1.18435i) q^{49} +(-1.56743 - 0.289137i) q^{52} +(-0.500893 + 1.93456i) q^{57} +(-0.280378 + 0.757044i) q^{61} +(-0.683086 - 1.84439i) q^{63} +(0.864331 - 0.502923i) q^{64} +(-0.0198531 + 0.00407992i) q^{67} +(0.190162 + 0.628311i) q^{73} +(0.230981 + 0.972958i) q^{75} +(1.97214 - 0.322609i) q^{76} +(-0.381813 + 1.26154i) q^{79} +(0.422108 - 0.906546i) q^{81} +(-1.41173 + 1.36945i) q^{84} +(-1.43733 - 2.78594i) q^{91} +(0.562921 - 0.826511i) q^{93} +(0.215942 - 0.0535821i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + q^{3} + q^{4} + 2 q^{7} + q^{9} + q^{12} + 2 q^{13} + q^{16} + 2 q^{19} - 3 q^{21} - 4 q^{25} + q^{27} - 3 q^{28} + q^{31} - 4 q^{36} + 2 q^{37} - 3 q^{39} - 3 q^{43} + q^{48} + 3 q^{49}+ \cdots - 3 q^{97}+O(q^{100}) \) 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

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