LMFDB - Embedded newform 8046.2.a.j.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.a (trivial)

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:	yes
Analytic conductor:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-2.72416$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.72416 q^{5} -2.55489 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72416 q^{5} -2.55489 q^{7} -1.00000 q^{8} -2.72416 q^{10} +0.632525 q^{11} -2.81471 q^{13} +2.55489 q^{14} +1.00000 q^{16} -1.96265 q^{17} -0.973194 q^{19} +2.72416 q^{20} -0.632525 q^{22} +7.05643 q^{23} +2.42105 q^{25} +2.81471 q^{26} -2.55489 q^{28} -6.39556 q^{29} +0.0203317 q^{31} -1.00000 q^{32} +1.96265 q^{34} -6.95994 q^{35} +5.11803 q^{37} +0.973194 q^{38} -2.72416 q^{40} +4.86133 q^{41} -3.52967 q^{43} +0.632525 q^{44} -7.05643 q^{46} -8.40734 q^{47} -0.472524 q^{49} -2.42105 q^{50} -2.81471 q^{52} +10.7975 q^{53} +1.72310 q^{55} +2.55489 q^{56} +6.39556 q^{58} +7.92337 q^{59} -10.7613 q^{61} -0.0203317 q^{62} +1.00000 q^{64} -7.66772 q^{65} -1.39314 q^{67} -1.96265 q^{68} +6.95994 q^{70} -2.93699 q^{71} +11.3462 q^{73} -5.11803 q^{74} -0.973194 q^{76} -1.61603 q^{77} +5.40421 q^{79} +2.72416 q^{80} -4.86133 q^{82} -5.25566 q^{83} -5.34658 q^{85} +3.52967 q^{86} -0.632525 q^{88} -1.53275 q^{89} +7.19128 q^{91} +7.05643 q^{92} +8.40734 q^{94} -2.65114 q^{95} -4.95227 q^{97} +0.472524 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8} + 3 q^{10} - 10 q^{11} + 5 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} + 2 q^{19} - 3 q^{20} + 10 q^{22} - 9 q^{23} + 7 q^{25} - 5 q^{26} + 6 q^{28} - 19 q^{29} + 10 q^{31} - 12 q^{32} + 8 q^{34} - 20 q^{35} + 11 q^{37} - 2 q^{38} + 3 q^{40} - 8 q^{41} + 13 q^{43} - 10 q^{44} + 9 q^{46} - 11 q^{47} + 2 q^{49} - 7 q^{50} + 5 q^{52} - 24 q^{53} + 3 q^{55} - 6 q^{56} + 19 q^{58} - 10 q^{59} - 10 q^{62} + 12 q^{64} - 28 q^{65} + 21 q^{67} - 8 q^{68} + 20 q^{70} - 37 q^{71} - 2 q^{73} - 11 q^{74} + 2 q^{76} - 2 q^{77} + 7 q^{79} - 3 q^{80} + 8 q^{82} - 22 q^{83} + 15 q^{85} - 13 q^{86} + 10 q^{88} - 40 q^{89} + q^{91} - 9 q^{92} + 11 q^{94} - 11 q^{95} + 7 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8 + 3 * q^10 - 10 * q^11 + 5 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 + 2 * q^19 - 3 * q^20 + 10 * q^22 - 9 * q^23 + 7 * q^25 - 5 * q^26 + 6 * q^28 - 19 * q^29 + 10 * q^31 - 12 * q^32 + 8 * q^34 - 20 * q^35 + 11 * q^37 - 2 * q^38 + 3 * q^40 - 8 * q^41 + 13 * q^43 - 10 * q^44 + 9 * q^46 - 11 * q^47 + 2 * q^49 - 7 * q^50 + 5 * q^52 - 24 * q^53 + 3 * q^55 - 6 * q^56 + 19 * q^58 - 10 * q^59 - 10 * q^62 + 12 * q^64 - 28 * q^65 + 21 * q^67 - 8 * q^68 + 20 * q^70 - 37 * q^71 - 2 * q^73 - 11 * q^74 + 2 * q^76 - 2 * q^77 + 7 * q^79 - 3 * q^80 + 8 * q^82 - 22 * q^83 + 15 * q^85 - 13 * q^86 + 10 * q^88 - 40 * q^89 + q^91 - 9 * q^92 + 11 * q^94 - 11 * q^95 + 7 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.72416	1.21828	0.609141	−	0.793062i	$-0.291514\pi$
$5$	2.72416	1.21828	0.609141	+	0.793062i	$0.291514\pi$
$6$	0	0
$7$	−2.55489	−0.965659	−0.482829	−	0.875714i	$-0.660391\pi$
$7$	−2.55489	−0.965659	−0.482829	+	0.875714i	$0.660391\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.72416	−0.861455
$11$	0.632525	0.190713	0.0953567	−	0.995443i	$-0.469601\pi$
$11$	0.632525	0.190713	0.0953567	+	0.995443i	$0.469601\pi$
$12$	0	0
$13$	−2.81471	−0.780660	−0.390330	−	0.920675i	$-0.627639\pi$
$13$	−2.81471	−0.780660	−0.390330	+	0.920675i	$0.627639\pi$
$14$	2.55489	0.682824
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.96265	−0.476013	−0.238007	−	0.971264i	$-0.576494\pi$
$17$	−1.96265	−0.476013	−0.238007	+	0.971264i	$0.576494\pi$
$18$	0	0
$19$	−0.973194	−0.223266	−0.111633	−	0.993750i	$-0.535608\pi$
$19$	−0.973194	−0.223266	−0.111633	+	0.993750i	$0.535608\pi$
$20$	2.72416	0.609141
$21$	0	0
$22$	−0.632525	−0.134855
$23$	7.05643	1.47137	0.735684	−	0.677325i	$-0.236861\pi$
$23$	7.05643	1.47137	0.735684	+	0.677325i	$0.236861\pi$
$24$	0	0
$25$	2.42105	0.484209
$26$	2.81471	0.552010
$27$	0	0
$28$	−2.55489	−0.482829
$29$	−6.39556	−1.18763	−0.593813	−	0.804603i	$-0.702378\pi$
$29$	−6.39556	−1.18763	−0.593813	+	0.804603i	$0.702378\pi$
$30$	0	0
$31$	0.0203317	0.00365167	0.00182584	−	0.999998i	$-0.499419\pi$
$31$	0.0203317	0.00365167	0.00182584	+	0.999998i	$0.499419\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.96265	0.336592
$35$	−6.95994	−1.17644
$36$	0	0
$37$	5.11803	0.841399	0.420700	−	0.907200i	$-0.361785\pi$
$37$	5.11803	0.841399	0.420700	+	0.907200i	$0.361785\pi$
$38$	0.973194	0.157873
$39$	0	0
$40$	−2.72416	−0.430728
$41$	4.86133	0.759212	0.379606	−	0.925148i	$-0.376060\pi$
$41$	4.86133	0.759212	0.379606	+	0.925148i	$0.376060\pi$
$42$	0	0
$43$	−3.52967	−0.538269	−0.269134	−	0.963103i	$-0.586738\pi$
$43$	−3.52967	−0.538269	−0.269134	+	0.963103i	$0.586738\pi$
$44$	0.632525	0.0953567
$45$	0	0
$46$	−7.05643	−1.04041
$47$	−8.40734	−1.22634	−0.613168	−	0.789953i	$-0.710105\pi$
$47$	−8.40734	−1.22634	−0.613168	+	0.789953i	$0.710105\pi$
$48$	0	0
$49$	−0.472524	−0.0675034
$50$	−2.42105	−0.342388
$51$	0	0
$52$	−2.81471	−0.390330
$53$	10.7975	1.48315	0.741574	−	0.670871i	$-0.234080\pi$
$53$	10.7975	1.48315	0.741574	+	0.670871i	$0.234080\pi$
$54$	0	0
$55$	1.72310	0.232343
$56$	2.55489	0.341412
$57$	0	0
$58$	6.39556	0.839778
$59$	7.92337	1.03153	0.515767	−	0.856729i	$-0.327507\pi$
$59$	7.92337	1.03153	0.515767	+	0.856729i	$0.327507\pi$
$60$	0	0
$61$	−10.7613	−1.37784	−0.688921	−	0.724836i	$-0.741915\pi$
$61$	−10.7613	−1.37784	−0.688921	+	0.724836i	$0.741915\pi$
$62$	−0.0203317	−0.00258212
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.66772	−0.951063
$66$	0	0
$67$	−1.39314	−0.170199	−0.0850995	−	0.996372i	$-0.527121\pi$
$67$	−1.39314	−0.170199	−0.0850995	+	0.996372i	$0.527121\pi$
$68$	−1.96265	−0.238007
$69$	0	0
$70$	6.95994	0.831871
$71$	−2.93699	−0.348556	−0.174278	−	0.984696i	$-0.555759\pi$
$71$	−2.93699	−0.348556	−0.174278	+	0.984696i	$0.555759\pi$
$72$	0	0
$73$	11.3462	1.32797	0.663984	−	0.747746i	$-0.268864\pi$
$73$	11.3462	1.32797	0.663984	+	0.747746i	$0.268864\pi$
$74$	−5.11803	−0.594959
$75$	0	0
$76$	−0.973194	−0.111633
$77$	−1.61603	−0.184164
$78$	0	0
$79$	5.40421	0.608021	0.304011	−	0.952669i	$-0.401674\pi$
$79$	5.40421	0.608021	0.304011	+	0.952669i	$0.401674\pi$
$80$	2.72416	0.304570
$81$	0	0
$82$	−4.86133	−0.536844
$83$	−5.25566	−0.576884	−0.288442	−	0.957497i	$-0.593137\pi$
$83$	−5.25566	−0.576884	−0.288442	+	0.957497i	$0.593137\pi$
$84$	0	0
$85$	−5.34658	−0.579918
$86$	3.52967	0.380614
$87$	0	0
$88$	−0.632525	−0.0674274
$89$	−1.53275	−0.162471	−0.0812357	−	0.996695i	$-0.525887\pi$
$89$	−1.53275	−0.162471	−0.0812357	+	0.996695i	$0.525887\pi$
$90$	0	0
$91$	7.19128	0.753851
$92$	7.05643	0.735684
$93$	0	0
$94$	8.40734	0.867150
$95$	−2.65114	−0.272001
$96$	0	0
$97$	−4.95227	−0.502827	−0.251414	−	0.967880i	$-0.580895\pi$
$97$	−4.95227	−0.502827	−0.251414	+	0.967880i	$0.580895\pi$
$98$	0.472524	0.0477321
$99$	0	0
$100$	2.42105	0.242105
$101$	−17.2893	−1.72035	−0.860173	−	0.510002i	$-0.829645\pi$
$101$	−17.2893	−1.72035	−0.860173	+	0.510002i	$0.829645\pi$
$102$	0	0
$103$	2.31423	0.228028	0.114014	−	0.993479i	$-0.463629\pi$
$103$	2.31423	0.228028	0.114014	+	0.993479i	$0.463629\pi$
$104$	2.81471	0.276005
$105$	0	0
$106$	−10.7975	−1.04874
$107$	−0.987568	−0.0954718	−0.0477359	−	0.998860i	$-0.515201\pi$
$107$	−0.987568	−0.0954718	−0.0477359	+	0.998860i	$0.515201\pi$
$108$	0	0
$109$	2.48978	0.238478	0.119239	−	0.992866i	$-0.461955\pi$
$109$	2.48978	0.238478	0.119239	+	0.992866i	$0.461955\pi$
$110$	−1.72310	−0.164291
$111$	0	0
$112$	−2.55489	−0.241415
$113$	−1.05663	−0.0993998	−0.0496999	−	0.998764i	$-0.515826\pi$
$113$	−1.05663	−0.0993998	−0.0496999	+	0.998764i	$0.515826\pi$
$114$	0	0
$115$	19.2229	1.79254
$116$	−6.39556	−0.593813
$117$	0	0
$118$	−7.92337	−0.729405
$119$	5.01437	0.459666
$120$	0	0
$121$	−10.5999	−0.963628
$122$	10.7613	0.974282
$123$	0	0
$124$	0.0203317	0.00182584
$125$	−7.02548	−0.628378
$126$	0	0
$127$	4.02592	0.357242	0.178621	−	0.983918i	$-0.442836\pi$
$127$	4.02592	0.357242	0.178621	+	0.983918i	$0.442836\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.66772	0.672503
$131$	−20.7829	−1.81581	−0.907904	−	0.419178i	$-0.862318\pi$
$131$	−20.7829	−1.81581	−0.907904	+	0.419178i	$0.862318\pi$
$132$	0	0
$133$	2.48641	0.215599
$134$	1.39314	0.120349
$135$	0	0
$136$	1.96265	0.168296
$137$	−13.2237	−1.12978	−0.564888	−	0.825168i	$-0.691081\pi$
$137$	−13.2237	−1.12978	−0.564888	+	0.825168i	$0.691081\pi$
$138$	0	0
$139$	21.6724	1.83823	0.919113	−	0.393993i	$-0.128907\pi$
$139$	21.6724	1.83823	0.919113	+	0.393993i	$0.128907\pi$
$140$	−6.95994	−0.588222
$141$	0	0
$142$	2.93699	0.246467
$143$	−1.78037	−0.148882
$144$	0	0
$145$	−17.4225	−1.44686
$146$	−11.3462	−0.939016
$147$	0	0
$148$	5.11803	0.420700
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	17.9676	1.46218	0.731092	−	0.682279i	$-0.239011\pi$
$151$	17.9676	1.46218	0.731092	+	0.682279i	$0.239011\pi$
$152$	0.973194	0.0789365
$153$	0	0
$154$	1.61603	0.130224
$155$	0.0553867	0.00444877
$156$	0	0
$157$	5.32120	0.424678	0.212339	−	0.977196i	$-0.431892\pi$
$157$	5.32120	0.424678	0.212339	+	0.977196i	$0.431892\pi$
$158$	−5.40421	−0.429936
$159$	0	0
$160$	−2.72416	−0.215364
$161$	−18.0284	−1.42084
$162$	0	0
$163$	−15.3370	−1.20129	−0.600645	−	0.799516i	$-0.705089\pi$
$163$	−15.3370	−1.20129	−0.600645	+	0.799516i	$0.705089\pi$
$164$	4.86133	0.379606
$165$	0	0
$166$	5.25566	0.407918
$167$	−15.5177	−1.20079	−0.600397	−	0.799702i	$-0.704991\pi$
$167$	−15.5177	−1.20079	−0.600397	+	0.799702i	$0.704991\pi$
$168$	0	0
$169$	−5.07742	−0.390570
$170$	5.34658	0.410064
$171$	0	0
$172$	−3.52967	−0.269134
$173$	−4.31289	−0.327903	−0.163952	−	0.986468i	$-0.552424\pi$
$173$	−4.31289	−0.327903	−0.163952	+	0.986468i	$0.552424\pi$
$174$	0	0
$175$	−6.18552	−0.467581
$176$	0.632525	0.0476784
$177$	0	0
$178$	1.53275	0.114885
$179$	−18.0202	−1.34690	−0.673448	−	0.739234i	$-0.735188\pi$
$179$	−18.0202	−1.34690	−0.673448	+	0.739234i	$0.735188\pi$
$180$	0	0
$181$	−3.78669	−0.281462	−0.140731	−	0.990048i	$-0.544945\pi$
$181$	−3.78669	−0.281462	−0.140731	+	0.990048i	$0.544945\pi$
$182$	−7.19128	−0.533053
$183$	0	0
$184$	−7.05643	−0.520207
$185$	13.9423	1.02506
$186$	0	0
$187$	−1.24143	−0.0907821
$188$	−8.40734	−0.613168
$189$	0	0
$190$	2.65114	0.192334
$191$	−21.3947	−1.54807	−0.774034	−	0.633144i	$-0.781764\pi$
$191$	−21.3947	−1.54807	−0.774034	+	0.633144i	$0.781764\pi$
$192$	0	0
$193$	−11.2761	−0.811675	−0.405837	−	0.913945i	$-0.633020\pi$
$193$	−11.2761	−0.811675	−0.405837	+	0.913945i	$0.633020\pi$
$194$	4.95227	0.355553
$195$	0	0
$196$	−0.472524	−0.0337517
$197$	−22.6373	−1.61284	−0.806419	−	0.591345i	$-0.798597\pi$
$197$	−22.6373	−1.61284	−0.806419	+	0.591345i	$0.798597\pi$
$198$	0	0
$199$	−3.22486	−0.228604	−0.114302	−	0.993446i	$-0.536463\pi$
$199$	−3.22486	−0.228604	−0.114302	+	0.993446i	$0.536463\pi$
$200$	−2.42105	−0.171194
$201$	0	0
$202$	17.2893	1.21647
$203$	16.3400	1.14684
$204$	0	0
$205$	13.2430	0.924934
$206$	−2.31423	−0.161240
$207$	0	0
$208$	−2.81471	−0.195165
$209$	−0.615569	−0.0425798
$210$	0	0
$211$	−3.11120	−0.214184	−0.107092	−	0.994249i	$-0.534154\pi$
$211$	−3.11120	−0.214184	−0.107092	+	0.994249i	$0.534154\pi$
$212$	10.7975	0.741574
$213$	0	0
$214$	0.987568	0.0675087
$215$	−9.61537	−0.655763
$216$	0	0
$217$	−0.0519452	−0.00352627
$218$	−2.48978	−0.168629
$219$	0	0
$220$	1.72310	0.116171
$221$	5.52429	0.371604
$222$	0	0
$223$	−11.8430	−0.793064	−0.396532	−	0.918021i	$-0.629786\pi$
$223$	−11.8430	−0.793064	−0.396532	+	0.918021i	$0.629786\pi$
$224$	2.55489	0.170706
$225$	0	0
$226$	1.05663	0.0702862
$227$	16.3151	1.08287	0.541436	−	0.840742i	$-0.317881\pi$
$227$	16.3151	1.08287	0.541436	+	0.840742i	$0.317881\pi$
$228$	0	0
$229$	12.4057	0.819788	0.409894	−	0.912133i	$-0.365566\pi$
$229$	12.4057	0.819788	0.409894	+	0.912133i	$0.365566\pi$
$230$	−19.2229	−1.26752
$231$	0	0
$232$	6.39556	0.419889
$233$	12.8942	0.844729	0.422365	−	0.906426i	$-0.361200\pi$
$233$	12.8942	0.844729	0.422365	+	0.906426i	$0.361200\pi$
$234$	0	0
$235$	−22.9029	−1.49402
$236$	7.92337	0.515767
$237$	0	0
$238$	−5.01437	−0.325033
$239$	−12.8100	−0.828612	−0.414306	−	0.910138i	$-0.635976\pi$
$239$	−12.8100	−0.828612	−0.414306	+	0.910138i	$0.635976\pi$
$240$	0	0
$241$	−29.8351	−1.92185	−0.960923	−	0.276816i	$-0.910721\pi$
$241$	−29.8351	−1.92185	−0.960923	+	0.276816i	$0.910721\pi$
$242$	10.5999	0.681388
$243$	0	0
$244$	−10.7613	−0.688921
$245$	−1.28723	−0.0822382
$246$	0	0
$247$	2.73926	0.174295
$248$	−0.0203317	−0.00129106
$249$	0	0
$250$	7.02548	0.444330
$251$	3.95644	0.249729	0.124864	−	0.992174i	$-0.460150\pi$
$251$	3.95644	0.249729	0.124864	+	0.992174i	$0.460150\pi$
$252$	0	0
$253$	4.46337	0.280610
$254$	−4.02592	−0.252608
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.32044	−0.581393	−0.290697	−	0.956815i	$-0.593887\pi$
$257$	−9.32044	−0.581393	−0.290697	+	0.956815i	$0.593887\pi$
$258$	0	0
$259$	−13.0760	−0.812504
$260$	−7.66772	−0.475532
$261$	0	0
$262$	20.7829	1.28397
$263$	−5.38249	−0.331899	−0.165949	−	0.986134i	$-0.553069\pi$
$263$	−5.38249	−0.331899	−0.165949	+	0.986134i	$0.553069\pi$
$264$	0	0
$265$	29.4141	1.80689
$266$	−2.48641	−0.152451
$267$	0	0
$268$	−1.39314	−0.0850995
$269$	9.77829	0.596193	0.298096	−	0.954536i	$-0.403648\pi$
$269$	9.77829	0.596193	0.298096	+	0.954536i	$0.403648\pi$
$270$	0	0
$271$	−9.67208	−0.587537	−0.293769	−	0.955877i	$-0.594910\pi$
$271$	−9.67208	−0.587537	−0.293769	+	0.955877i	$0.594910\pi$
$272$	−1.96265	−0.119003
$273$	0	0
$274$	13.2237	0.798872
$275$	1.53137	0.0923452
$276$	0	0
$277$	19.0125	1.14235	0.571174	−	0.820829i	$-0.306488\pi$
$277$	19.0125	1.14235	0.571174	+	0.820829i	$0.306488\pi$
$278$	−21.6724	−1.29982
$279$	0	0
$280$	6.95994	0.415936
$281$	19.6354	1.17135	0.585673	−	0.810547i	$-0.300830\pi$
$281$	19.6354	1.17135	0.585673	+	0.810547i	$0.300830\pi$
$282$	0	0
$283$	−7.55142	−0.448885	−0.224443	−	0.974487i	$-0.572056\pi$
$283$	−7.55142	−0.448885	−0.224443	+	0.974487i	$0.572056\pi$
$284$	−2.93699	−0.174278
$285$	0	0
$286$	1.78037	0.105276
$287$	−12.4202	−0.733140
$288$	0	0
$289$	−13.1480	−0.773411
$290$	17.4225	1.02309
$291$	0	0
$292$	11.3462	0.663984
$293$	−7.75677	−0.453155	−0.226578	−	0.973993i	$-0.572754\pi$
$293$	−7.75677	−0.453155	−0.226578	+	0.973993i	$0.572754\pi$
$294$	0	0
$295$	21.5845	1.25670
$296$	−5.11803	−0.297479
$297$	0	0
$298$	1.00000	0.0579284
$299$	−19.8618	−1.14864
$300$	0	0
$301$	9.01792	0.519784
$302$	−17.9676	−1.03392
$303$	0	0
$304$	−0.973194	−0.0558165
$305$	−29.3155	−1.67860
$306$	0	0
$307$	−9.37959	−0.535321	−0.267661	−	0.963513i	$-0.586251\pi$
$307$	−9.37959	−0.535321	−0.267661	+	0.963513i	$0.586251\pi$
$308$	−1.61603	−0.0920820
$309$	0	0
$310$	−0.0553867	−0.00314575
$311$	21.0155	1.19168	0.595839	−	0.803104i	$-0.296820\pi$
$311$	21.0155	1.19168	0.595839	+	0.803104i	$0.296820\pi$
$312$	0	0
$313$	4.59876	0.259937	0.129969	−	0.991518i	$-0.458512\pi$
$313$	4.59876	0.259937	0.129969	+	0.991518i	$0.458512\pi$
$314$	−5.32120	−0.300293
$315$	0	0
$316$	5.40421	0.304011
$317$	−15.4108	−0.865558	−0.432779	−	0.901500i	$-0.642467\pi$
$317$	−15.4108	−0.865558	−0.432779	+	0.901500i	$0.642467\pi$
$318$	0	0
$319$	−4.04535	−0.226496
$320$	2.72416	0.152285
$321$	0	0
$322$	18.0284	1.00469
$323$	1.91004	0.106278
$324$	0	0
$325$	−6.81454	−0.378003
$326$	15.3370	0.849440
$327$	0	0
$328$	−4.86133	−0.268422
$329$	21.4798	1.18422
$330$	0	0
$331$	30.2792	1.66429	0.832147	−	0.554555i	$-0.187111\pi$
$331$	30.2792	1.66429	0.832147	+	0.554555i	$0.187111\pi$
$332$	−5.25566	−0.288442
$333$	0	0
$334$	15.5177	0.849090
$335$	−3.79514	−0.207350
$336$	0	0
$337$	29.3354	1.59800	0.799001	−	0.601330i	$-0.205362\pi$
$337$	29.3354	1.59800	0.799001	+	0.601330i	$0.205362\pi$
$338$	5.07742	0.276175
$339$	0	0
$340$	−5.34658	−0.289959
$341$	0.0128603	0.000696423 0
$342$	0	0
$343$	19.0915	1.03084
$344$	3.52967	0.190307
$345$	0	0
$346$	4.31289	0.231862
$347$	12.7340	0.683597	0.341799	−	0.939773i	$-0.388964\pi$
$347$	12.7340	0.683597	0.341799	+	0.939773i	$0.388964\pi$
$348$	0	0
$349$	−14.5829	−0.780604	−0.390302	−	0.920687i	$-0.627629\pi$
$349$	−14.5829	−0.780604	−0.390302	+	0.920687i	$0.627629\pi$
$350$	6.18552	0.330630
$351$	0	0
$352$	−0.632525	−0.0337137
$353$	−18.5223	−0.985844	−0.492922	−	0.870073i	$-0.664071\pi$
$353$	−18.5223	−0.985844	−0.492922	+	0.870073i	$0.664071\pi$
$354$	0	0
$355$	−8.00083	−0.424640
$356$	−1.53275	−0.0812357
$357$	0	0
$358$	18.0202	0.952399
$359$	−19.9876	−1.05490	−0.527452	−	0.849585i	$-0.676853\pi$
$359$	−19.9876	−1.05490	−0.527452	+	0.849585i	$0.676853\pi$
$360$	0	0
$361$	−18.0529	−0.950152
$362$	3.78669	0.199024
$363$	0	0
$364$	7.19128	0.376925
$365$	30.9088	1.61784
$366$	0	0
$367$	9.76779	0.509874	0.254937	−	0.966958i	$-0.417945\pi$
$367$	9.76779	0.509874	0.254937	+	0.966958i	$0.417945\pi$
$368$	7.05643	0.367842
$369$	0	0
$370$	−13.9423	−0.724827
$371$	−27.5864	−1.43221
$372$	0	0
$373$	−28.0245	−1.45105	−0.725527	−	0.688194i	$-0.758404\pi$
$373$	−28.0245	−1.45105	−0.725527	+	0.688194i	$0.758404\pi$
$374$	1.24143	0.0641926
$375$	0	0
$376$	8.40734	0.433575
$377$	18.0016	0.927132
$378$	0	0
$379$	20.4183	1.04882	0.524409	−	0.851467i	$-0.324286\pi$
$379$	20.4183	1.04882	0.524409	+	0.851467i	$0.324286\pi$
$380$	−2.65114	−0.136000
$381$	0	0
$382$	21.3947	1.09465
$383$	−11.7192	−0.598824	−0.299412	−	0.954124i	$-0.596791\pi$
$383$	−11.7192	−0.598824	−0.299412	+	0.954124i	$0.596791\pi$
$384$	0	0
$385$	−4.40233	−0.224364
$386$	11.2761	0.573941
$387$	0	0
$388$	−4.95227	−0.251414
$389$	−30.9843	−1.57097	−0.785484	−	0.618882i	$-0.787586\pi$
$389$	−30.9843	−1.57097	−0.785484	+	0.618882i	$0.787586\pi$
$390$	0	0
$391$	−13.8493	−0.700391
$392$	0.472524	0.0238661
$393$	0	0
$394$	22.6373	1.14045
$395$	14.7219	0.740741
$396$	0	0
$397$	31.5773	1.58482	0.792411	−	0.609988i	$-0.208826\pi$
$397$	31.5773	1.58482	0.792411	+	0.609988i	$0.208826\pi$
$398$	3.22486	0.161648
$399$	0	0
$400$	2.42105	0.121052
$401$	4.27818	0.213642	0.106821	−	0.994278i	$-0.465933\pi$
$401$	4.27818	0.213642	0.106821	+	0.994278i	$0.465933\pi$
$402$	0	0
$403$	−0.0572277	−0.00285072
$404$	−17.2893	−0.860173
$405$	0	0
$406$	−16.3400	−0.810939
$407$	3.23728	0.160466
$408$	0	0
$409$	−24.8524	−1.22887	−0.614436	−	0.788967i	$-0.710616\pi$
$409$	−24.8524	−1.22887	−0.614436	+	0.788967i	$0.710616\pi$
$410$	−13.2430	−0.654027
$411$	0	0
$412$	2.31423	0.114014
$413$	−20.2434	−0.996110
$414$	0	0
$415$	−14.3173	−0.702807
$416$	2.81471	0.138002
$417$	0	0
$418$	0.615569	0.0301085
$419$	2.61800	0.127898	0.0639488	−	0.997953i	$-0.479631\pi$
$419$	2.61800	0.127898	0.0639488	+	0.997953i	$0.479631\pi$
$420$	0	0
$421$	9.68534	0.472034	0.236017	−	0.971749i	$-0.424158\pi$
$421$	9.68534	0.472034	0.236017	+	0.971749i	$0.424158\pi$
$422$	3.11120	0.151451
$423$	0	0
$424$	−10.7975	−0.524372
$425$	−4.75168	−0.230490
$426$	0	0
$427$	27.4939	1.33053
$428$	−0.987568	−0.0477359
$429$	0	0
$430$	9.61537	0.463694
$431$	−3.43813	−0.165609	−0.0828046	−	0.996566i	$-0.526388\pi$
$431$	−3.43813	−0.165609	−0.0828046	+	0.996566i	$0.526388\pi$
$432$	0	0
$433$	−10.1367	−0.487141	−0.243571	−	0.969883i	$-0.578319\pi$
$433$	−10.1367	−0.487141	−0.243571	+	0.969883i	$0.578319\pi$
$434$	0.0519452	0.00249345
$435$	0	0
$436$	2.48978	0.119239
$437$	−6.86728	−0.328507
$438$	0	0
$439$	4.61814	0.220412	0.110206	−	0.993909i	$-0.464849\pi$
$439$	4.61814	0.220412	0.110206	+	0.993909i	$0.464849\pi$
$440$	−1.72310	−0.0821455
$441$	0	0
$442$	−5.52429	−0.262764
$443$	16.8162	0.798960	0.399480	−	0.916742i	$-0.369191\pi$
$443$	16.8162	0.798960	0.399480	+	0.916742i	$0.369191\pi$
$444$	0	0
$445$	−4.17546	−0.197936
$446$	11.8430	0.560781
$447$	0	0
$448$	−2.55489	−0.120707
$449$	−21.1328	−0.997321	−0.498660	−	0.866797i	$-0.666175\pi$
$449$	−21.1328	−0.997321	−0.498660	+	0.866797i	$0.666175\pi$
$450$	0	0
$451$	3.07491	0.144792
$452$	−1.05663	−0.0496999
$453$	0	0
$454$	−16.3151	−0.765706
$455$	19.5902	0.918402
$456$	0	0
$457$	−16.4331	−0.768707	−0.384354	−	0.923186i	$-0.625576\pi$
$457$	−16.4331	−0.768707	−0.384354	+	0.923186i	$0.625576\pi$
$458$	−12.4057	−0.579678
$459$	0	0
$460$	19.2229	0.896270
$461$	30.7303	1.43125	0.715626	−	0.698483i	$-0.246141\pi$
$461$	30.7303	1.43125	0.715626	+	0.698483i	$0.246141\pi$
$462$	0	0
$463$	−4.10602	−0.190823	−0.0954115	−	0.995438i	$-0.530417\pi$
$463$	−4.10602	−0.190823	−0.0954115	+	0.995438i	$0.530417\pi$
$464$	−6.39556	−0.296907
$465$	0	0
$466$	−12.8942	−0.597314
$467$	19.3333	0.894638	0.447319	−	0.894375i	$-0.352379\pi$
$467$	19.3333	0.894638	0.447319	+	0.894375i	$0.352379\pi$
$468$	0	0
$469$	3.55932	0.164354
$470$	22.9029	1.05643
$471$	0	0
$472$	−7.92337	−0.364703
$473$	−2.23260	−0.102655
$474$	0	0
$475$	−2.35615	−0.108108
$476$	5.01437	0.229833
$477$	0	0
$478$	12.8100	0.585917
$479$	25.9840	1.18724	0.593619	−	0.804746i	$-0.297699\pi$
$479$	25.9840	1.18724	0.593619	+	0.804746i	$0.297699\pi$
$480$	0	0
$481$	−14.4058	−0.656846
$482$	29.8351	1.35895
$483$	0	0
$484$	−10.5999	−0.481814
$485$	−13.4908	−0.612585
$486$	0	0
$487$	0.511596	0.0231826	0.0115913	−	0.999933i	$-0.496310\pi$
$487$	0.511596	0.0231826	0.0115913	+	0.999933i	$0.496310\pi$
$488$	10.7613	0.487141
$489$	0	0
$490$	1.28723	0.0581512
$491$	−38.0649	−1.71784	−0.858922	−	0.512106i	$-0.828865\pi$
$491$	−38.0649	−1.71784	−0.858922	+	0.512106i	$0.828865\pi$
$492$	0	0
$493$	12.5523	0.565326
$494$	−2.73926	−0.123245
$495$	0	0
$496$	0.0203317	0.000912919 0
$497$	7.50369	0.336587
$498$	0	0
$499$	14.5486	0.651284	0.325642	−	0.945493i	$-0.394420\pi$
$499$	14.5486	0.651284	0.325642	+	0.945493i	$0.394420\pi$
$500$	−7.02548	−0.314189
$501$	0	0
$502$	−3.95644	−0.176585
$503$	43.4439	1.93707	0.968535	−	0.248879i	$-0.0800621\pi$
$503$	43.4439	1.93707	0.968535	+	0.248879i	$0.0800621\pi$
$504$	0	0
$505$	−47.0987	−2.09587
$506$	−4.46337	−0.198421
$507$	0	0
$508$	4.02592	0.178621
$509$	−1.18792	−0.0526534	−0.0263267	−	0.999653i	$-0.508381\pi$
$509$	−1.18792	−0.0526534	−0.0263267	+	0.999653i	$0.508381\pi$
$510$	0	0
$511$	−28.9882	−1.28236
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	9.32044	0.411107
$515$	6.30434	0.277802
$516$	0	0
$517$	−5.31785	−0.233879
$518$	13.0760	0.574527
$519$	0	0
$520$	7.66772	0.336252
$521$	−24.1359	−1.05741	−0.528706	−	0.848805i	$-0.677323\pi$
$521$	−24.1359	−1.05741	−0.528706	+	0.848805i	$0.677323\pi$
$522$	0	0
$523$	−24.2520	−1.06046	−0.530232	−	0.847852i	$-0.677895\pi$
$523$	−24.2520	−1.06046	−0.530232	+	0.847852i	$0.677895\pi$
$524$	−20.7829	−0.907904
$525$	0	0
$526$	5.38249	0.234688
$527$	−0.0399040	−0.00173825
$528$	0	0
$529$	26.7933	1.16492
$530$	−29.4141	−1.27766
$531$	0	0
$532$	2.48641	0.107799
$533$	−13.6832	−0.592686
$534$	0	0
$535$	−2.69029	−0.116311
$536$	1.39314	0.0601745
$537$	0	0
$538$	−9.77829	−0.421572
$539$	−0.298883	−0.0128738
$540$	0	0
$541$	−19.6530	−0.844950	−0.422475	−	0.906375i	$-0.638839\pi$
$541$	−19.6530	−0.844950	−0.422475	+	0.906375i	$0.638839\pi$
$542$	9.67208	0.415452
$543$	0	0
$544$	1.96265	0.0841480
$545$	6.78255	0.290533
$546$	0	0
$547$	4.04601	0.172995	0.0864975	−	0.996252i	$-0.472433\pi$
$547$	4.04601	0.172995	0.0864975	+	0.996252i	$0.472433\pi$
$548$	−13.2237	−0.564888
$549$	0	0
$550$	−1.53137	−0.0652979
$551$	6.22412	0.265157
$552$	0	0
$553$	−13.8072	−0.587141
$554$	−19.0125	−0.807763
$555$	0	0
$556$	21.6724	0.919113
$557$	−19.1385	−0.810923	−0.405461	−	0.914112i	$-0.632889\pi$
$557$	−19.1385	−0.810923	−0.405461	+	0.914112i	$0.632889\pi$
$558$	0	0
$559$	9.93498	0.420205
$560$	−6.95994	−0.294111
$561$	0	0
$562$	−19.6354	−0.828267
$563$	30.9841	1.30582	0.652911	−	0.757434i	$-0.273547\pi$
$563$	30.9841	1.30582	0.652911	+	0.757434i	$0.273547\pi$
$564$	0	0
$565$	−2.87844	−0.121097
$566$	7.55142	0.317410
$567$	0	0
$568$	2.93699	0.123233
$569$	−25.0139	−1.04864	−0.524319	−	0.851522i	$-0.675680\pi$
$569$	−25.0139	−1.04864	−0.524319	+	0.851522i	$0.675680\pi$
$570$	0	0
$571$	−0.392750	−0.0164361	−0.00821804	−	0.999966i	$-0.502616\pi$
$571$	−0.392750	−0.0164361	−0.00821804	+	0.999966i	$0.502616\pi$
$572$	−1.78037	−0.0744411
$573$	0	0
$574$	12.4202	0.518408
$575$	17.0840	0.712450
$576$	0	0
$577$	13.9820	0.582077	0.291039	−	0.956711i	$-0.405999\pi$
$577$	13.9820	0.582077	0.291039	+	0.956711i	$0.405999\pi$
$578$	13.1480	0.546884
$579$	0	0
$580$	−17.4225	−0.723431
$581$	13.4276	0.557073
$582$	0	0
$583$	6.82967	0.282856
$584$	−11.3462	−0.469508
$585$	0	0
$586$	7.75677	0.320429
$587$	−3.83546	−0.158306	−0.0791532	−	0.996862i	$-0.525222\pi$
$587$	−3.83546	−0.158306	−0.0791532	+	0.996862i	$0.525222\pi$
$588$	0	0
$589$	−0.0197867	−0.000815295 0
$590$	−21.5845	−0.888621
$591$	0	0
$592$	5.11803	0.210350
$593$	8.41016	0.345364	0.172682	−	0.984978i	$-0.444757\pi$
$593$	8.41016	0.345364	0.172682	+	0.984978i	$0.444757\pi$
$594$	0	0
$595$	13.6599	0.560003
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	19.8618	0.812210
$599$	6.25368	0.255518	0.127759	−	0.991805i	$-0.459222\pi$
$599$	6.25368	0.255518	0.127759	+	0.991805i	$0.459222\pi$
$600$	0	0
$601$	−12.3632	−0.504307	−0.252153	−	0.967687i	$-0.581139\pi$
$601$	−12.3632	−0.504307	−0.252153	+	0.967687i	$0.581139\pi$
$602$	−9.01792	−0.367543
$603$	0	0
$604$	17.9676	0.731092
$605$	−28.8759	−1.17397
$606$	0	0
$607$	−19.4820	−0.790749	−0.395375	−	0.918520i	$-0.629385\pi$
$607$	−19.4820	−0.790749	−0.395375	+	0.918520i	$0.629385\pi$
$608$	0.973194	0.0394682
$609$	0	0
$610$	29.3155	1.18695
$611$	23.6642	0.957351
$612$	0	0
$613$	−12.5738	−0.507849	−0.253925	−	0.967224i	$-0.581722\pi$
$613$	−12.5738	−0.507849	−0.253925	+	0.967224i	$0.581722\pi$
$614$	9.37959	0.378529
$615$	0	0
$616$	1.61603	0.0651118
$617$	43.0328	1.73243	0.866217	−	0.499668i	$-0.166545\pi$
$617$	43.0328	1.73243	0.866217	+	0.499668i	$0.166545\pi$
$618$	0	0
$619$	−15.3111	−0.615404	−0.307702	−	0.951483i	$-0.599560\pi$
$619$	−15.3111	−0.615404	−0.307702	+	0.951483i	$0.599560\pi$
$620$	0.0553867	0.00222438
$621$	0	0
$622$	−21.0155	−0.842644
$623$	3.91602	0.156892
$624$	0	0
$625$	−31.2438	−1.24975
$626$	−4.59876	−0.183804
$627$	0	0
$628$	5.32120	0.212339
$629$	−10.0449	−0.400517
$630$	0	0
$631$	13.8911	0.552995	0.276497	−	0.961015i	$-0.410826\pi$
$631$	13.8911	0.552995	0.276497	+	0.961015i	$0.410826\pi$
$632$	−5.40421	−0.214968
$633$	0	0
$634$	15.4108	0.612042
$635$	10.9672	0.435222
$636$	0	0
$637$	1.33002	0.0526972
$638$	4.04535	0.160157
$639$	0	0
$640$	−2.72416	−0.107682
$641$	−49.3131	−1.94775	−0.973876	−	0.227082i	$-0.927081\pi$
$641$	−49.3131	−1.94775	−0.973876	+	0.227082i	$0.927081\pi$
$642$	0	0
$643$	12.5417	0.494597	0.247298	−	0.968939i	$-0.420457\pi$
$643$	12.5417	0.494597	0.247298	+	0.968939i	$0.420457\pi$
$644$	−18.0284	−0.710420
$645$	0	0
$646$	−1.91004	−0.0751496
$647$	−9.18884	−0.361250	−0.180625	−	0.983552i	$-0.557812\pi$
$647$	−9.18884	−0.361250	−0.180625	+	0.983552i	$0.557812\pi$
$648$	0	0
$649$	5.01173	0.196727
$650$	6.81454	0.267288
$651$	0	0
$652$	−15.3370	−0.600645
$653$	1.20838	0.0472875	0.0236438	−	0.999720i	$-0.492473\pi$
$653$	1.20838	0.0472875	0.0236438	+	0.999720i	$0.492473\pi$
$654$	0	0
$655$	−56.6159	−2.21216
$656$	4.86133	0.189803
$657$	0	0
$658$	−21.4798	−0.837371
$659$	13.1368	0.511738	0.255869	−	0.966711i	$-0.417638\pi$
$659$	13.1368	0.511738	0.255869	+	0.966711i	$0.417638\pi$
$660$	0	0
$661$	−30.9600	−1.20420	−0.602102	−	0.798419i	$-0.705670\pi$
$661$	−30.9600	−1.20420	−0.602102	+	0.798419i	$0.705670\pi$
$662$	−30.2792	−1.17683
$663$	0	0
$664$	5.25566	0.203959
$665$	6.77337	0.262660
$666$	0	0
$667$	−45.1299	−1.74744
$668$	−15.5177	−0.600397
$669$	0	0
$670$	3.79514	0.146619
$671$	−6.80678	−0.262773
$672$	0	0
$673$	45.7351	1.76296	0.881479	−	0.472222i	$-0.156548\pi$
$673$	45.7351	1.76296	0.881479	+	0.472222i	$0.156548\pi$
$674$	−29.3354	−1.12996
$675$	0	0
$676$	−5.07742	−0.195285
$677$	33.3809	1.28293	0.641466	−	0.767151i	$-0.278326\pi$
$677$	33.3809	1.28293	0.641466	+	0.767151i	$0.278326\pi$
$678$	0	0
$679$	12.6525	0.485560
$680$	5.34658	0.205032
$681$	0	0
$682$	−0.0128603	−0.000492446 0
$683$	−37.9315	−1.45141	−0.725704	−	0.688007i	$-0.758486\pi$
$683$	−37.9315	−1.45141	−0.725704	+	0.688007i	$0.758486\pi$
$684$	0	0
$685$	−36.0234	−1.37638
$686$	−19.0915	−0.728917
$687$	0	0
$688$	−3.52967	−0.134567
$689$	−30.3918	−1.15783
$690$	0	0
$691$	6.65830	0.253294	0.126647	−	0.991948i	$-0.459579\pi$
$691$	6.65830	0.253294	0.126647	+	0.991948i	$0.459579\pi$
$692$	−4.31289	−0.163952
$693$	0	0
$694$	−12.7340	−0.483376
$695$	59.0390	2.23948
$696$	0	0
$697$	−9.54110	−0.361395
$698$	14.5829	0.551971
$699$	0	0
$700$	−6.18552	−0.233791
$701$	−39.0248	−1.47395	−0.736973	−	0.675922i	$-0.763746\pi$
$701$	−39.0248	−1.47395	−0.736973	+	0.675922i	$0.763746\pi$
$702$	0	0
$703$	−4.98084	−0.187856
$704$	0.632525	0.0238392
$705$	0	0
$706$	18.5223	0.697097
$707$	44.1722	1.66127
$708$	0	0
$709$	−1.10890	−0.0416457	−0.0208229	−	0.999783i	$-0.506629\pi$
$709$	−1.10890	−0.0416457	−0.0208229	+	0.999783i	$0.506629\pi$
$710$	8.00083	0.300266
$711$	0	0
$712$	1.53275	0.0574423
$713$	0.143469	0.00537296
$714$	0	0
$715$	−4.85002	−0.181380
$716$	−18.0202	−0.673448
$717$	0	0
$718$	19.9876	0.745930
$719$	1.34610	0.0502010	0.0251005	−	0.999685i	$-0.492009\pi$
$719$	1.34610	0.0502010	0.0251005	+	0.999685i	$0.492009\pi$
$720$	0	0
$721$	−5.91262	−0.220197
$722$	18.0529	0.671859
$723$	0	0
$724$	−3.78669	−0.140731
$725$	−15.4840	−0.575060
$726$	0	0
$727$	24.8410	0.921302	0.460651	−	0.887581i	$-0.347616\pi$
$727$	24.8410	0.921302	0.460651	+	0.887581i	$0.347616\pi$
$728$	−7.19128	−0.266526
$729$	0	0
$730$	−30.9088	−1.14399
$731$	6.92751	0.256223
$732$	0	0
$733$	−41.2648	−1.52415	−0.762075	−	0.647489i	$-0.775819\pi$
$733$	−41.2648	−1.52415	−0.762075	+	0.647489i	$0.775819\pi$
$734$	−9.76779	−0.360536
$735$	0	0
$736$	−7.05643	−0.260104
$737$	−0.881195	−0.0324593
$738$	0	0
$739$	19.2123	0.706737	0.353368	−	0.935484i	$-0.385036\pi$
$739$	19.2123	0.706737	0.353368	+	0.935484i	$0.385036\pi$
$740$	13.9423	0.512530
$741$	0	0
$742$	27.5864	1.01273
$743$	−42.4125	−1.55596	−0.777982	−	0.628287i	$-0.783756\pi$
$743$	−42.4125	−1.55596	−0.777982	+	0.628287i	$0.783756\pi$
$744$	0	0
$745$	−2.72416	−0.0998055
$746$	28.0245	1.02605
$747$	0	0
$748$	−1.24143	−0.0453910
$749$	2.52313	0.0921931
$750$	0	0
$751$	38.4936	1.40465	0.702325	−	0.711856i	$-0.252145\pi$
$751$	38.4936	1.40465	0.702325	+	0.711856i	$0.252145\pi$
$752$	−8.40734	−0.306584
$753$	0	0
$754$	−18.0016	−0.655581
$755$	48.9467	1.78135
$756$	0	0
$757$	−2.55661	−0.0929216	−0.0464608	−	0.998920i	$-0.514794\pi$
$757$	−2.55661	−0.0929216	−0.0464608	+	0.998920i	$0.514794\pi$
$758$	−20.4183	−0.741626
$759$	0	0
$760$	2.65114	0.0961668
$761$	12.2813	0.445197	0.222598	−	0.974910i	$-0.428546\pi$
$761$	12.2813	0.445197	0.222598	+	0.974910i	$0.428546\pi$
$762$	0	0
$763$	−6.36112	−0.230288
$764$	−21.3947	−0.774034
$765$	0	0
$766$	11.7192	0.423433
$767$	−22.3020	−0.805277
$768$	0	0
$769$	−21.8048	−0.786300	−0.393150	−	0.919474i	$-0.628615\pi$
$769$	−21.8048	−0.786300	−0.393150	+	0.919474i	$0.628615\pi$
$770$	4.40233	0.158649
$771$	0	0
$772$	−11.2761	−0.405837
$773$	5.99812	0.215738	0.107869	−	0.994165i	$-0.465597\pi$
$773$	5.99812	0.215738	0.107869	+	0.994165i	$0.465597\pi$
$774$	0	0
$775$	0.0492239	0.00176818
$776$	4.95227	0.177776
$777$	0	0
$778$	30.9843	1.11084
$779$	−4.73102	−0.169506
$780$	0	0
$781$	−1.85772	−0.0664744
$782$	13.8493	0.495251
$783$	0	0
$784$	−0.472524	−0.0168759
$785$	14.4958	0.517377
$786$	0	0
$787$	−26.1617	−0.932562	−0.466281	−	0.884637i	$-0.654406\pi$
$787$	−26.1617	−0.932562	−0.466281	+	0.884637i	$0.654406\pi$
$788$	−22.6373	−0.806419
$789$	0	0
$790$	−14.7219	−0.523783
$791$	2.69959	0.0959862
$792$	0	0
$793$	30.2899	1.07563
$794$	−31.5773	−1.12064
$795$	0	0
$796$	−3.22486	−0.114302
$797$	−1.06239	−0.0376319	−0.0188160	−	0.999823i	$-0.505990\pi$
$797$	−1.06239	−0.0376319	−0.0188160	+	0.999823i	$0.505990\pi$
$798$	0	0
$799$	16.5007	0.583752
$800$	−2.42105	−0.0855970
$801$	0	0
$802$	−4.27818	−0.151068
$803$	7.17674	0.253261
$804$	0	0
$805$	−49.1123	−1.73098
$806$	0.0572277	0.00201576
$807$	0	0
$808$	17.2893	0.608234
$809$	−41.1753	−1.44765	−0.723823	−	0.689986i	$-0.757617\pi$
$809$	−41.1753	−1.44765	−0.723823	+	0.689986i	$0.757617\pi$
$810$	0	0
$811$	−1.68038	−0.0590062	−0.0295031	−	0.999565i	$-0.509392\pi$
$811$	−1.68038	−0.0590062	−0.0295031	+	0.999565i	$0.509392\pi$
$812$	16.3400	0.573421
$813$	0	0
$814$	−3.23728	−0.113467
$815$	−41.7806	−1.46351
$816$	0	0
$817$	3.43505	0.120177
$818$	24.8524	0.868943
$819$	0	0
$820$	13.2430	0.462467
$821$	−47.1581	−1.64583	−0.822914	−	0.568166i	$-0.807653\pi$
$821$	−47.1581	−1.64583	−0.822914	+	0.568166i	$0.807653\pi$
$822$	0	0
$823$	24.8740	0.867052	0.433526	−	0.901141i	$-0.357269\pi$
$823$	24.8740	0.867052	0.433526	+	0.901141i	$0.357269\pi$
$824$	−2.31423	−0.0806201
$825$	0	0
$826$	20.2434	0.704356
$827$	−5.04048	−0.175275	−0.0876373	−	0.996152i	$-0.527932\pi$
$827$	−5.04048	−0.175275	−0.0876373	+	0.996152i	$0.527932\pi$
$828$	0	0
$829$	9.07901	0.315327	0.157663	−	0.987493i	$-0.449604\pi$
$829$	9.07901	0.315327	0.157663	+	0.987493i	$0.449604\pi$
$830$	14.3173	0.496959
$831$	0	0
$832$	−2.81471	−0.0975825
$833$	0.927400	0.0321325
$834$	0	0
$835$	−42.2726	−1.46290
$836$	−0.615569	−0.0212899
$837$	0	0
$838$	−2.61800	−0.0904372
$839$	36.4993	1.26010	0.630048	−	0.776556i	$-0.283035\pi$
$839$	36.4993	1.26010	0.630048	+	0.776556i	$0.283035\pi$
$840$	0	0
$841$	11.9032	0.410456
$842$	−9.68534	−0.333779
$843$	0	0
$844$	−3.11120	−0.107092
$845$	−13.8317	−0.475825
$846$	0	0
$847$	27.0816	0.930536
$848$	10.7975	0.370787
$849$	0	0
$850$	4.75168	0.162981
$851$	36.1150	1.23801
$852$	0	0
$853$	38.4430	1.31626	0.658131	−	0.752903i	$-0.271347\pi$
$853$	38.4430	1.31626	0.658131	+	0.752903i	$0.271347\pi$
$854$	−27.4939	−0.940823
$855$	0	0
$856$	0.987568	0.0337544
$857$	6.38063	0.217958	0.108979	−	0.994044i	$-0.465242\pi$
$857$	6.38063	0.217958	0.108979	+	0.994044i	$0.465242\pi$
$858$	0	0
$859$	−4.19177	−0.143021	−0.0715107	−	0.997440i	$-0.522782\pi$
$859$	−4.19177	−0.143021	−0.0715107	+	0.997440i	$0.522782\pi$
$860$	−9.61537	−0.327882
$861$	0	0
$862$	3.43813	0.117103
$863$	25.8085	0.878530	0.439265	−	0.898357i	$-0.355239\pi$
$863$	25.8085	0.878530	0.439265	+	0.898357i	$0.355239\pi$
$864$	0	0
$865$	−11.7490	−0.399478
$866$	10.1367	0.344461
$867$	0	0
$868$	−0.0519452	−0.00176314
$869$	3.41830	0.115958
$870$	0	0
$871$	3.92128	0.132868
$872$	−2.48978	−0.0843145
$873$	0	0
$874$	6.86728	0.232289
$875$	17.9493	0.606799
$876$	0	0
$877$	−47.9339	−1.61861	−0.809306	−	0.587388i	$-0.800156\pi$
$877$	−47.9339	−1.61861	−0.809306	+	0.587388i	$0.800156\pi$
$878$	−4.61814	−0.155855
$879$	0	0
$880$	1.72310	0.0580856
$881$	−24.2996	−0.818673	−0.409337	−	0.912383i	$-0.634240\pi$
$881$	−24.2996	−0.818673	−0.409337	+	0.912383i	$0.634240\pi$
$882$	0	0
$883$	12.0398	0.405170	0.202585	−	0.979265i	$-0.435066\pi$
$883$	12.0398	0.405170	0.202585	+	0.979265i	$0.435066\pi$
$884$	5.52429	0.185802
$885$	0	0
$886$	−16.8162	−0.564950
$887$	16.3841	0.550123	0.275061	−	0.961427i	$-0.411302\pi$
$887$	16.3841	0.550123	0.275061	+	0.961427i	$0.411302\pi$
$888$	0	0
$889$	−10.2858	−0.344974
$890$	4.17546	0.139962
$891$	0	0
$892$	−11.8430	−0.396532
$893$	8.18197	0.273799
$894$	0	0
$895$	−49.0900	−1.64090
$896$	2.55489	0.0853530
$897$	0	0
$898$	21.1328	0.705212
$899$	−0.130032	−0.00433682
$900$	0	0
$901$	−21.1917	−0.705998
$902$	−3.07491	−0.102383
$903$	0	0
$904$	1.05663	0.0351431
$905$	−10.3155	−0.342900
$906$	0	0
$907$	11.8652	0.393979	0.196989	−	0.980406i	$-0.436884\pi$
$907$	11.8652	0.393979	0.196989	+	0.980406i	$0.436884\pi$
$908$	16.3151	0.541436
$909$	0	0
$910$	−19.5902	−0.649408
$911$	−58.3128	−1.93199	−0.965995	−	0.258561i	$-0.916752\pi$
$911$	−58.3128	−1.93199	−0.965995	+	0.258561i	$0.916752\pi$
$912$	0	0
$913$	−3.32434	−0.110019
$914$	16.4331	0.543558
$915$	0	0
$916$	12.4057	0.409894
$917$	53.0980	1.75345
$918$	0	0
$919$	−38.8203	−1.28056	−0.640282	−	0.768140i	$-0.721183\pi$
$919$	−38.8203	−1.28056	−0.640282	+	0.768140i	$0.721183\pi$
$920$	−19.2229	−0.633759
$921$	0	0
$922$	−30.7303	−1.01205
$923$	8.26677	0.272104
$924$	0	0
$925$	12.3910	0.407413
$926$	4.10602	0.134932
$927$	0	0
$928$	6.39556	0.209945
$929$	16.5634	0.543426	0.271713	−	0.962378i	$-0.412410\pi$
$929$	16.5634	0.543426	0.271713	+	0.962378i	$0.412410\pi$
$930$	0	0
$931$	0.459858	0.0150712
$932$	12.8942	0.422365
$933$	0	0
$934$	−19.3333	−0.632604
$935$	−3.38184	−0.110598
$936$	0	0
$937$	3.59880	0.117568	0.0587838	−	0.998271i	$-0.481278\pi$
$937$	3.59880	0.117568	0.0587838	+	0.998271i	$0.481278\pi$
$938$	−3.55932	−0.116216
$939$	0	0
$940$	−22.9029	−0.747011
$941$	−50.1980	−1.63641	−0.818203	−	0.574929i	$-0.805030\pi$
$941$	−50.1980	−1.63641	−0.818203	+	0.574929i	$0.805030\pi$
$942$	0	0
$943$	34.3037	1.11708
$944$	7.92337	0.257884
$945$	0	0
$946$	2.23260	0.0725881
$947$	2.90731	0.0944750	0.0472375	−	0.998884i	$-0.484958\pi$
$947$	2.90731	0.0944750	0.0472375	+	0.998884i	$0.484958\pi$
$948$	0	0
$949$	−31.9362	−1.03669
$950$	2.35615	0.0764436
$951$	0	0
$952$	−5.01437	−0.162517
$953$	55.8194	1.80817	0.904084	−	0.427355i	$-0.140555\pi$
$953$	55.8194	1.80817	0.904084	+	0.427355i	$0.140555\pi$
$954$	0	0
$955$	−58.2827	−1.88598
$956$	−12.8100	−0.414306
$957$	0	0
$958$	−25.9840	−0.839504
$959$	33.7851	1.09098
$960$	0	0
$961$	−30.9996	−0.999987
$962$	14.4058	0.464460
$963$	0	0
$964$	−29.8351	−0.960923
$965$	−30.7180	−0.988848
$966$	0	0
$967$	−34.1191	−1.09720	−0.548599	−	0.836086i	$-0.684839\pi$
$967$	−34.1191	−1.09720	−0.548599	+	0.836086i	$0.684839\pi$
$968$	10.5999	0.340694
$969$	0	0
$970$	13.4908	0.433163
$971$	−7.07299	−0.226983	−0.113492	−	0.993539i	$-0.536203\pi$
$971$	−7.07299	−0.226983	−0.113492	+	0.993539i	$0.536203\pi$
$972$	0	0
$973$	−55.3706	−1.77510
$974$	−0.511596	−0.0163926
$975$	0	0
$976$	−10.7613	−0.344461
$977$	−1.38807	−0.0444084	−0.0222042	−	0.999753i	$-0.507068\pi$
$977$	−1.38807	−0.0444084	−0.0222042	+	0.999753i	$0.507068\pi$
$978$	0	0
$979$	−0.969504	−0.0309855
$980$	−1.28723	−0.0411191
$981$	0	0
$982$	38.0649	1.21470
$983$	−2.73500	−0.0872330	−0.0436165	−	0.999048i	$-0.513888\pi$
$983$	−2.73500	−0.0872330	−0.0436165	+	0.999048i	$0.513888\pi$
$984$	0	0
$985$	−61.6675	−1.96489
$986$	−12.5523	−0.399746
$987$	0	0
$988$	2.73926	0.0871474
$989$	−24.9069	−0.791992
$990$	0	0
$991$	26.2976	0.835372	0.417686	−	0.908591i	$-0.362841\pi$
$991$	26.2976	0.835372	0.417686	+	0.908591i	$0.362841\pi$
$992$	−0.0203317	−0.000645531 0
$993$	0	0
$994$	−7.50369	−0.238003
$995$	−8.78503	−0.278504
$996$	0	0
$997$	0.174863	0.00553795	0.00276898	−	0.999996i	$-0.499119\pi$
$997$	0.174863	0.00553795	0.00276898	+	0.999996i	$0.499119\pi$
$998$	−14.5486	−0.460527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.j.1.11	✓	12
3.2	odd	2		8046.2.a.o.1.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.11	✓	12	1.1	even	1	trivial
8046.2.a.o.1.2	yes	12	3.2	odd	2

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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72416 q^{5} -2.55489 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72416 q^{5} -2.55489 q^{7} -1.00000 q^{8} -2.72416 q^{10} +0.632525 q^{11} -2.81471 q^{13} +2.55489 q^{14} +1.00000 q^{16} -1.96265 q^{17} -0.973194 q^{19} +2.72416 q^{20} -0.632525 q^{22} +7.05643 q^{23} +2.42105 q^{25} +2.81471 q^{26} -2.55489 q^{28} -6.39556 q^{29} +0.0203317 q^{31} -1.00000 q^{32} +1.96265 q^{34} -6.95994 q^{35} +5.11803 q^{37} +0.973194 q^{38} -2.72416 q^{40} +4.86133 q^{41} -3.52967 q^{43} +0.632525 q^{44} -7.05643 q^{46} -8.40734 q^{47} -0.472524 q^{49} -2.42105 q^{50} -2.81471 q^{52} +10.7975 q^{53} +1.72310 q^{55} +2.55489 q^{56} +6.39556 q^{58} +7.92337 q^{59} -10.7613 q^{61} -0.0203317 q^{62} +1.00000 q^{64} -7.66772 q^{65} -1.39314 q^{67} -1.96265 q^{68} +6.95994 q^{70} -2.93699 q^{71} +11.3462 q^{73} -5.11803 q^{74} -0.973194 q^{76} -1.61603 q^{77} +5.40421 q^{79} +2.72416 q^{80} -4.86133 q^{82} -5.25566 q^{83} -5.34658 q^{85} +3.52967 q^{86} -0.632525 q^{88} -1.53275 q^{89} +7.19128 q^{91} +7.05643 q^{92} +8.40734 q^{94} -2.65114 q^{95} -4.95227 q^{97} +0.472524 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 3 q^{5} + 6 q^{7} - 12 q^{8} + 3 q^{10} - 10 q^{11} + 5 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} + 2 q^{19} - 3 q^{20} + 10 q^{22} - 9 q^{23} + 7 q^{25} - 5 q^{26} + 6 q^{28} - 19 q^{29} + 10 q^{31} - 12 q^{32} + 8 q^{34} - 20 q^{35} + 11 q^{37} - 2 q^{38} + 3 q^{40} - 8 q^{41} + 13 q^{43} - 10 q^{44} + 9 q^{46} - 11 q^{47} + 2 q^{49} - 7 q^{50} + 5 q^{52} - 24 q^{53} + 3 q^{55} - 6 q^{56} + 19 q^{58} - 10 q^{59} - 10 q^{62} + 12 q^{64} - 28 q^{65} + 21 q^{67} - 8 q^{68} + 20 q^{70} - 37 q^{71} - 2 q^{73} - 11 q^{74} + 2 q^{76} - 2 q^{77} + 7 q^{79} - 3 q^{80} + 8 q^{82} - 22 q^{83} + 15 q^{85} - 13 q^{86} + 10 q^{88} - 40 q^{89} + q^{91} - 9 q^{92} + 11 q^{94} - 11 q^{95} + 7 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 3 * q^5 + 6 * q^7 - 12 * q^8 + 3 * q^10 - 10 * q^11 + 5 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 + 2 * q^19 - 3 * q^20 + 10 * q^22 - 9 * q^23 + 7 * q^25 - 5 * q^26 + 6 * q^28 - 19 * q^29 + 10 * q^31 - 12 * q^32 + 8 * q^34 - 20 * q^35 + 11 * q^37 - 2 * q^38 + 3 * q^40 - 8 * q^41 + 13 * q^43 - 10 * q^44 + 9 * q^46 - 11 * q^47 + 2 * q^49 - 7 * q^50 + 5 * q^52 - 24 * q^53 + 3 * q^55 - 6 * q^56 + 19 * q^58 - 10 * q^59 - 10 * q^62 + 12 * q^64 - 28 * q^65 + 21 * q^67 - 8 * q^68 + 20 * q^70 - 37 * q^71 - 2 * q^73 - 11 * q^74 + 2 * q^76 - 2 * q^77 + 7 * q^79 - 3 * q^80 + 8 * q^82 - 22 * q^83 + 15 * q^85 - 13 * q^86 + 10 * q^88 - 40 * q^89 + q^91 - 9 * q^92 + 11 * q^94 - 11 * q^95 + 7 * q^97 - 2 * q^98

Introduction

L-functions

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