LMFDB - Embedded newform 8046.2.a.q.1.5

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
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			1.5
Root			$2.04034$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.04034 q^{5} +2.45177 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.04034 q^{5} +2.45177 q^{7} -1.00000 q^{8} +2.04034 q^{10} -0.203477 q^{11} +3.95618 q^{13} -2.45177 q^{14} +1.00000 q^{16} -2.98757 q^{17} +6.08465 q^{19} -2.04034 q^{20} +0.203477 q^{22} +0.149808 q^{23} -0.837025 q^{25} -3.95618 q^{26} +2.45177 q^{28} -1.18201 q^{29} +1.40369 q^{31} -1.00000 q^{32} +2.98757 q^{34} -5.00245 q^{35} -3.56672 q^{37} -6.08465 q^{38} +2.04034 q^{40} +4.57686 q^{41} +10.1987 q^{43} -0.203477 q^{44} -0.149808 q^{46} +8.79796 q^{47} -0.988805 q^{49} +0.837025 q^{50} +3.95618 q^{52} -8.68416 q^{53} +0.415161 q^{55} -2.45177 q^{56} +1.18201 q^{58} +5.65615 q^{59} -8.79078 q^{61} -1.40369 q^{62} +1.00000 q^{64} -8.07193 q^{65} -1.11318 q^{67} -2.98757 q^{68} +5.00245 q^{70} +6.56830 q^{71} +16.7120 q^{73} +3.56672 q^{74} +6.08465 q^{76} -0.498879 q^{77} -4.54460 q^{79} -2.04034 q^{80} -4.57686 q^{82} +2.21378 q^{83} +6.09564 q^{85} -10.1987 q^{86} +0.203477 q^{88} +10.6761 q^{89} +9.69965 q^{91} +0.149808 q^{92} -8.79796 q^{94} -12.4147 q^{95} -8.22556 q^{97} +0.988805 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * 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
$5$	−2.04034	−0.912466	−0.456233	−	0.889860i	$-0.650802\pi$
$5$	−2.04034	−0.912466	−0.456233	+	0.889860i	$0.650802\pi$
$6$	0	0
$7$	2.45177	0.926683	0.463342	−	0.886180i	$-0.346650\pi$
$7$	2.45177	0.926683	0.463342	+	0.886180i	$0.346650\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.04034	0.645211
$11$	−0.203477	−0.0613506	−0.0306753	−	0.999529i	$-0.509766\pi$
$11$	−0.203477	−0.0613506	−0.0306753	+	0.999529i	$0.509766\pi$
$12$	0	0
$13$	3.95618	1.09725	0.548623	−	0.836070i	$-0.315152\pi$
$13$	3.95618	1.09725	0.548623	+	0.836070i	$0.315152\pi$
$14$	−2.45177	−0.655264
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.98757	−0.724591	−0.362296	−	0.932063i	$-0.618007\pi$
$17$	−2.98757	−0.724591	−0.362296	+	0.932063i	$0.618007\pi$
$18$	0	0
$19$	6.08465	1.39591	0.697957	−	0.716140i	$-0.254093\pi$
$19$	6.08465	1.39591	0.697957	+	0.716140i	$0.254093\pi$
$20$	−2.04034	−0.456233
$21$	0	0
$22$	0.203477	0.0433814
$23$	0.149808	0.0312372	0.0156186	−	0.999878i	$-0.495028\pi$
$23$	0.149808	0.0312372	0.0156186	+	0.999878i	$0.495028\pi$
$24$	0	0
$25$	−0.837025	−0.167405
$26$	−3.95618	−0.775870
$27$	0	0
$28$	2.45177	0.463342
$29$	−1.18201	−0.219494	−0.109747	−	0.993960i	$-0.535004\pi$
$29$	−1.18201	−0.219494	−0.109747	+	0.993960i	$0.535004\pi$
$30$	0	0
$31$	1.40369	0.252111	0.126055	−	0.992023i	$-0.459768\pi$
$31$	1.40369	0.252111	0.126055	+	0.992023i	$0.459768\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.98757	0.512363
$35$	−5.00245	−0.845568
$36$	0	0
$37$	−3.56672	−0.586365	−0.293182	−	0.956057i	$-0.594714\pi$
$37$	−3.56672	−0.586365	−0.293182	+	0.956057i	$0.594714\pi$
$38$	−6.08465	−0.987060
$39$	0	0
$40$	2.04034	0.322606
$41$	4.57686	0.714785	0.357393	−	0.933954i	$-0.383666\pi$
$41$	4.57686	0.714785	0.357393	+	0.933954i	$0.383666\pi$
$42$	0	0
$43$	10.1987	1.55528	0.777640	−	0.628709i	$-0.216417\pi$
$43$	10.1987	1.55528	0.777640	+	0.628709i	$0.216417\pi$
$44$	−0.203477	−0.0306753
$45$	0	0
$46$	−0.149808	−0.0220880
$47$	8.79796	1.28331	0.641657	−	0.766991i	$-0.278247\pi$
$47$	8.79796	1.28331	0.641657	+	0.766991i	$0.278247\pi$
$48$	0	0
$49$	−0.988805	−0.141258
$50$	0.837025	0.118373
$51$	0	0
$52$	3.95618	0.548623
$53$	−8.68416	−1.19286	−0.596431	−	0.802664i	$-0.703415\pi$
$53$	−8.68416	−1.19286	−0.596431	+	0.802664i	$0.703415\pi$
$54$	0	0
$55$	0.415161	0.0559804
$56$	−2.45177	−0.327632
$57$	0	0
$58$	1.18201	0.155205
$59$	5.65615	0.736367	0.368184	−	0.929753i	$-0.379980\pi$
$59$	5.65615	0.736367	0.368184	+	0.929753i	$0.379980\pi$
$60$	0	0
$61$	−8.79078	−1.12554	−0.562772	−	0.826612i	$-0.690265\pi$
$61$	−8.79078	−1.12554	−0.562772	+	0.826612i	$0.690265\pi$
$62$	−1.40369	−0.178269
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.07193	−1.00120
$66$	0	0
$67$	−1.11318	−0.135996	−0.0679980	−	0.997685i	$-0.521661\pi$
$67$	−1.11318	−0.135996	−0.0679980	+	0.997685i	$0.521661\pi$
$68$	−2.98757	−0.362296
$69$	0	0
$70$	5.00245	0.597907
$71$	6.56830	0.779513	0.389757	−	0.920918i	$-0.372559\pi$
$71$	6.56830	0.779513	0.389757	+	0.920918i	$0.372559\pi$
$72$	0	0
$73$	16.7120	1.95599	0.977995	−	0.208629i	$-0.0669000\pi$
$73$	16.7120	1.95599	0.977995	+	0.208629i	$0.0669000\pi$
$74$	3.56672	0.414622
$75$	0	0
$76$	6.08465	0.697957
$77$	−0.498879	−0.0568526
$78$	0	0
$79$	−4.54460	−0.511307	−0.255653	−	0.966768i	$-0.582291\pi$
$79$	−4.54460	−0.511307	−0.255653	+	0.966768i	$0.582291\pi$
$80$	−2.04034	−0.228117
$81$	0	0
$82$	−4.57686	−0.505429
$83$	2.21378	0.242994	0.121497	−	0.992592i	$-0.461231\pi$
$83$	2.21378	0.242994	0.121497	+	0.992592i	$0.461231\pi$
$84$	0	0
$85$	6.09564	0.661165
$86$	−10.1987	−1.09975
$87$	0	0
$88$	0.203477	0.0216907
$89$	10.6761	1.13166	0.565830	−	0.824522i	$-0.308556\pi$
$89$	10.6761	1.13166	0.565830	+	0.824522i	$0.308556\pi$
$90$	0	0
$91$	9.69965	1.01680
$92$	0.149808	0.0156186
$93$	0	0
$94$	−8.79796	−0.907440
$95$	−12.4147	−1.27372
$96$	0	0
$97$	−8.22556	−0.835179	−0.417590	−	0.908636i	$-0.637125\pi$
$97$	−8.22556	−0.835179	−0.417590	+	0.908636i	$0.637125\pi$
$98$	0.988805	0.0998844
$99$	0	0
$100$	−0.837025	−0.0837025
$101$	1.19588	0.118994	0.0594972	−	0.998228i	$-0.481050\pi$
$101$	1.19588	0.118994	0.0594972	+	0.998228i	$0.481050\pi$
$102$	0	0
$103$	−3.95892	−0.390084	−0.195042	−	0.980795i	$-0.562484\pi$
$103$	−3.95892	−0.390084	−0.195042	+	0.980795i	$0.562484\pi$
$104$	−3.95618	−0.387935
$105$	0	0
$106$	8.68416	0.843481
$107$	5.27886	0.510326	0.255163	−	0.966898i	$-0.417871\pi$
$107$	5.27886	0.510326	0.255163	+	0.966898i	$0.417871\pi$
$108$	0	0
$109$	−12.5318	−1.20033	−0.600163	−	0.799878i	$-0.704898\pi$
$109$	−12.5318	−1.20033	−0.600163	+	0.799878i	$0.704898\pi$
$110$	−0.415161	−0.0395841
$111$	0	0
$112$	2.45177	0.231671
$113$	−5.30005	−0.498587	−0.249293	−	0.968428i	$-0.580198\pi$
$113$	−5.30005	−0.498587	−0.249293	+	0.968428i	$0.580198\pi$
$114$	0	0
$115$	−0.305659	−0.0285029
$116$	−1.18201	−0.109747
$117$	0	0
$118$	−5.65615	−0.520690
$119$	−7.32484	−0.671467
$120$	0	0
$121$	−10.9586	−0.996236
$122$	8.79078	0.795880
$123$	0	0
$124$	1.40369	0.126055
$125$	11.9095	1.06522
$126$	0	0
$127$	4.45202	0.395053	0.197526	−	0.980298i	$-0.436709\pi$
$127$	4.45202	0.395053	0.197526	+	0.980298i	$0.436709\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	8.07193	0.707955
$131$	12.2984	1.07451	0.537256	−	0.843419i	$-0.319461\pi$
$131$	12.2984	1.07451	0.537256	+	0.843419i	$0.319461\pi$
$132$	0	0
$133$	14.9182	1.29357
$134$	1.11318	0.0961637
$135$	0	0
$136$	2.98757	0.256182
$137$	−18.7145	−1.59889	−0.799445	−	0.600740i	$-0.794873\pi$
$137$	−18.7145	−1.59889	−0.799445	+	0.600740i	$0.794873\pi$
$138$	0	0
$139$	−7.98547	−0.677319	−0.338659	−	0.940909i	$-0.609973\pi$
$139$	−7.98547	−0.677319	−0.338659	+	0.940909i	$0.609973\pi$
$140$	−5.00245	−0.422784
$141$	0	0
$142$	−6.56830	−0.551199
$143$	−0.804990	−0.0673167
$144$	0	0
$145$	2.41170	0.200281
$146$	−16.7120	−1.38309
$147$	0	0
$148$	−3.56672	−0.293182
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	12.5737	1.02324	0.511619	−	0.859213i	$-0.329046\pi$
$151$	12.5737	1.02324	0.511619	+	0.859213i	$0.329046\pi$
$152$	−6.08465	−0.493530
$153$	0	0
$154$	0.498879	0.0402008
$155$	−2.86401	−0.230043
$156$	0	0
$157$	7.13867	0.569728	0.284864	−	0.958568i	$-0.408052\pi$
$157$	7.13867	0.569728	0.284864	+	0.958568i	$0.408052\pi$
$158$	4.54460	0.361549
$159$	0	0
$160$	2.04034	0.161303
$161$	0.367296	0.0289470
$162$	0	0
$163$	1.08015	0.0846039	0.0423019	−	0.999105i	$-0.486531\pi$
$163$	1.08015	0.0846039	0.0423019	+	0.999105i	$0.486531\pi$
$164$	4.57686	0.357393
$165$	0	0
$166$	−2.21378	−0.171823
$167$	−17.8060	−1.37787	−0.688936	−	0.724822i	$-0.741922\pi$
$167$	−17.8060	−1.37787	−0.688936	+	0.724822i	$0.741922\pi$
$168$	0	0
$169$	2.65133	0.203948
$170$	−6.09564	−0.467514
$171$	0	0
$172$	10.1987	0.777640
$173$	−18.3462	−1.39484	−0.697419	−	0.716664i	$-0.745668\pi$
$173$	−18.3462	−1.39484	−0.697419	+	0.716664i	$0.745668\pi$
$174$	0	0
$175$	−2.05219	−0.155131
$176$	−0.203477	−0.0153376
$177$	0	0
$178$	−10.6761	−0.800205
$179$	16.2742	1.21639	0.608194	−	0.793788i	$-0.291894\pi$
$179$	16.2742	1.21639	0.608194	+	0.793788i	$0.291894\pi$
$180$	0	0
$181$	6.27919	0.466729	0.233364	−	0.972389i	$-0.425027\pi$
$181$	6.27919	0.466729	0.233364	+	0.972389i	$0.425027\pi$
$182$	−9.69965	−0.718986
$183$	0	0
$184$	−0.149808	−0.0110440
$185$	7.27730	0.535038
$186$	0	0
$187$	0.607901	0.0444541
$188$	8.79796	0.641657
$189$	0	0
$190$	12.4147	0.900660
$191$	15.9069	1.15099	0.575493	−	0.817807i	$-0.304810\pi$
$191$	15.9069	1.15099	0.575493	+	0.817807i	$0.304810\pi$
$192$	0	0
$193$	−14.9415	−1.07552	−0.537758	−	0.843099i	$-0.680729\pi$
$193$	−14.9415	−1.07552	−0.537758	+	0.843099i	$0.680729\pi$
$194$	8.22556	0.590561
$195$	0	0
$196$	−0.988805	−0.0706289
$197$	6.88235	0.490347	0.245174	−	0.969479i	$-0.421155\pi$
$197$	6.88235	0.490347	0.245174	+	0.969479i	$0.421155\pi$
$198$	0	0
$199$	−8.62978	−0.611749	−0.305875	−	0.952072i	$-0.598949\pi$
$199$	−8.62978	−0.611749	−0.305875	+	0.952072i	$0.598949\pi$
$200$	0.837025	0.0591866
$201$	0	0
$202$	−1.19588	−0.0841418
$203$	−2.89802	−0.203401
$204$	0	0
$205$	−9.33833	−0.652217
$206$	3.95892	0.275831
$207$	0	0
$208$	3.95618	0.274311
$209$	−1.23809	−0.0856402
$210$	0	0
$211$	11.9484	0.822564	0.411282	−	0.911508i	$-0.365081\pi$
$211$	11.9484	0.822564	0.411282	+	0.911508i	$0.365081\pi$
$212$	−8.68416	−0.596431
$213$	0	0
$214$	−5.27886	−0.360855
$215$	−20.8087	−1.41914
$216$	0	0
$217$	3.44154	0.233627
$218$	12.5318	0.848759
$219$	0	0
$220$	0.415161	0.0279902
$221$	−11.8193	−0.795055
$222$	0	0
$223$	16.3096	1.09217	0.546087	−	0.837729i	$-0.316117\pi$
$223$	16.3096	1.09217	0.546087	+	0.837729i	$0.316117\pi$
$224$	−2.45177	−0.163816
$225$	0	0
$226$	5.30005	0.352554
$227$	25.9668	1.72348	0.861739	−	0.507352i	$-0.169376\pi$
$227$	25.9668	1.72348	0.861739	+	0.507352i	$0.169376\pi$
$228$	0	0
$229$	−17.1880	−1.13581	−0.567907	−	0.823093i	$-0.692247\pi$
$229$	−17.1880	−1.13581	−0.567907	+	0.823093i	$0.692247\pi$
$230$	0.305659	0.0201546
$231$	0	0
$232$	1.18201	0.0776027
$233$	−11.7241	−0.768070	−0.384035	−	0.923319i	$-0.625466\pi$
$233$	−11.7241	−0.768070	−0.384035	+	0.923319i	$0.625466\pi$
$234$	0	0
$235$	−17.9508	−1.17098
$236$	5.65615	0.368184
$237$	0	0
$238$	7.32484	0.474799
$239$	−11.3448	−0.733833	−0.366917	−	0.930254i	$-0.619587\pi$
$239$	−11.3448	−0.733833	−0.366917	+	0.930254i	$0.619587\pi$
$240$	0	0
$241$	12.7772	0.823054	0.411527	−	0.911398i	$-0.364996\pi$
$241$	12.7772	0.823054	0.411527	+	0.911398i	$0.364996\pi$
$242$	10.9586	0.704445
$243$	0	0
$244$	−8.79078	−0.562772
$245$	2.01749	0.128893
$246$	0	0
$247$	24.0719	1.53166
$248$	−1.40369	−0.0891346
$249$	0	0
$250$	−11.9095	−0.753223
$251$	−20.9749	−1.32392	−0.661961	−	0.749538i	$-0.730276\pi$
$251$	−20.9749	−1.32392	−0.661961	+	0.749538i	$0.730276\pi$
$252$	0	0
$253$	−0.0304825	−0.00191642
$254$	−4.45202	−0.279344
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.1377	0.632373	0.316186	−	0.948697i	$-0.397598\pi$
$257$	10.1377	0.632373	0.316186	+	0.948697i	$0.397598\pi$
$258$	0	0
$259$	−8.74478	−0.543374
$260$	−8.07193	−0.500600
$261$	0	0
$262$	−12.2984	−0.759795
$263$	−2.00396	−0.123569	−0.0617847	−	0.998090i	$-0.519679\pi$
$263$	−2.00396	−0.123569	−0.0617847	+	0.998090i	$0.519679\pi$
$264$	0	0
$265$	17.7186	1.08845
$266$	−14.9182	−0.914693
$267$	0	0
$268$	−1.11318	−0.0679980
$269$	1.93175	0.117781	0.0588904	−	0.998264i	$-0.481244\pi$
$269$	1.93175	0.117781	0.0588904	+	0.998264i	$0.481244\pi$
$270$	0	0
$271$	24.0451	1.46064	0.730318	−	0.683108i	$-0.239372\pi$
$271$	24.0451	1.46064	0.730318	+	0.683108i	$0.239372\pi$
$272$	−2.98757	−0.181148
$273$	0	0
$274$	18.7145	1.13059
$275$	0.170315	0.0102704
$276$	0	0
$277$	11.0962	0.666706	0.333353	−	0.942802i	$-0.391820\pi$
$277$	11.0962	0.666706	0.333353	+	0.942802i	$0.391820\pi$
$278$	7.98547	0.478937
$279$	0	0
$280$	5.00245	0.298953
$281$	16.2571	0.969819	0.484909	−	0.874564i	$-0.338853\pi$
$281$	16.2571	0.969819	0.484909	+	0.874564i	$0.338853\pi$
$282$	0	0
$283$	0.215180	0.0127911	0.00639555	−	0.999980i	$-0.497964\pi$
$283$	0.215180	0.0127911	0.00639555	+	0.999980i	$0.497964\pi$
$284$	6.56830	0.389757
$285$	0	0
$286$	0.804990	0.0476001
$287$	11.2214	0.662379
$288$	0	0
$289$	−8.07445	−0.474967
$290$	−2.41170	−0.141620
$291$	0	0
$292$	16.7120	0.977995
$293$	17.9241	1.04714	0.523570	−	0.851983i	$-0.324600\pi$
$293$	17.9241	1.04714	0.523570	+	0.851983i	$0.324600\pi$
$294$	0	0
$295$	−11.5404	−0.671911
$296$	3.56672	0.207311
$297$	0	0
$298$	1.00000	0.0579284
$299$	0.592667	0.0342748
$300$	0	0
$301$	25.0048	1.44125
$302$	−12.5737	−0.723538
$303$	0	0
$304$	6.08465	0.348979
$305$	17.9362	1.02702
$306$	0	0
$307$	8.70157	0.496625	0.248312	−	0.968680i	$-0.420124\pi$
$307$	8.70157	0.496625	0.248312	+	0.968680i	$0.420124\pi$
$308$	−0.498879	−0.0284263
$309$	0	0
$310$	2.86401	0.162665
$311$	−5.96807	−0.338418	−0.169209	−	0.985580i	$-0.554121\pi$
$311$	−5.96807	−0.338418	−0.169209	+	0.985580i	$0.554121\pi$
$312$	0	0
$313$	12.0096	0.678821	0.339410	−	0.940638i	$-0.389772\pi$
$313$	12.0096	0.678821	0.339410	+	0.940638i	$0.389772\pi$
$314$	−7.13867	−0.402858
$315$	0	0
$316$	−4.54460	−0.255653
$317$	9.00626	0.505842	0.252921	−	0.967487i	$-0.418609\pi$
$317$	9.00626	0.505842	0.252921	+	0.967487i	$0.418609\pi$
$318$	0	0
$319$	0.240512	0.0134661
$320$	−2.04034	−0.114058
$321$	0	0
$322$	−0.367296	−0.0204686
$323$	−18.1783	−1.01147
$324$	0	0
$325$	−3.31142	−0.183684
$326$	−1.08015	−0.0598240
$327$	0	0
$328$	−4.57686	−0.252715
$329$	21.5706	1.18923
$330$	0	0
$331$	−21.9340	−1.20560	−0.602802	−	0.797891i	$-0.705949\pi$
$331$	−21.9340	−1.20560	−0.602802	+	0.797891i	$0.705949\pi$
$332$	2.21378	0.121497
$333$	0	0
$334$	17.8060	0.974303
$335$	2.27125	0.124092
$336$	0	0
$337$	6.33234	0.344944	0.172472	−	0.985014i	$-0.444824\pi$
$337$	6.33234	0.344944	0.172472	+	0.985014i	$0.444824\pi$
$338$	−2.65133	−0.144213
$339$	0	0
$340$	6.09564	0.330583
$341$	−0.285619	−0.0154671
$342$	0	0
$343$	−19.5867	−1.05758
$344$	−10.1987	−0.549875
$345$	0	0
$346$	18.3462	0.986299
$347$	17.8450	0.957969	0.478984	−	0.877823i	$-0.341005\pi$
$347$	17.8450	0.957969	0.478984	+	0.877823i	$0.341005\pi$
$348$	0	0
$349$	4.18716	0.224134	0.112067	−	0.993701i	$-0.464253\pi$
$349$	4.18716	0.224134	0.112067	+	0.993701i	$0.464253\pi$
$350$	2.05219	0.109694
$351$	0	0
$352$	0.203477	0.0108454
$353$	13.9825	0.744214	0.372107	−	0.928190i	$-0.378635\pi$
$353$	13.9825	0.744214	0.372107	+	0.928190i	$0.378635\pi$
$354$	0	0
$355$	−13.4015	−0.711280
$356$	10.6761	0.565830
$357$	0	0
$358$	−16.2742	−0.860116
$359$	3.55787	0.187777	0.0938886	−	0.995583i	$-0.470070\pi$
$359$	3.55787	0.187777	0.0938886	+	0.995583i	$0.470070\pi$
$360$	0	0
$361$	18.0230	0.948576
$362$	−6.27919	−0.330027
$363$	0	0
$364$	9.69965	0.508400
$365$	−34.0981	−1.78478
$366$	0	0
$367$	28.0206	1.46267	0.731333	−	0.682021i	$-0.238899\pi$
$367$	28.0206	1.46267	0.731333	+	0.682021i	$0.238899\pi$
$368$	0.149808	0.00780929
$369$	0	0
$370$	−7.27730	−0.378329
$371$	−21.2916	−1.10541
$372$	0	0
$373$	−25.6903	−1.33020	−0.665098	−	0.746756i	$-0.731610\pi$
$373$	−25.6903	−1.33020	−0.665098	+	0.746756i	$0.731610\pi$
$374$	−0.607901	−0.0314338
$375$	0	0
$376$	−8.79796	−0.453720
$377$	−4.67624	−0.240838
$378$	0	0
$379$	27.3594	1.40536	0.702679	−	0.711507i	$-0.251987\pi$
$379$	27.3594	1.40536	0.702679	+	0.711507i	$0.251987\pi$
$380$	−12.4147	−0.636862
$381$	0	0
$382$	−15.9069	−0.813870
$383$	−23.6995	−1.21099	−0.605493	−	0.795851i	$-0.707024\pi$
$383$	−23.6995	−1.21099	−0.605493	+	0.795851i	$0.707024\pi$
$384$	0	0
$385$	1.01788	0.0518761
$386$	14.9415	0.760504
$387$	0	0
$388$	−8.22556	−0.417590
$389$	−12.0397	−0.610435	−0.305217	−	0.952283i	$-0.598729\pi$
$389$	−12.0397	−0.610435	−0.305217	+	0.952283i	$0.598729\pi$
$390$	0	0
$391$	−0.447562	−0.0226342
$392$	0.988805	0.0499422
$393$	0	0
$394$	−6.88235	−0.346728
$395$	9.27251	0.466550
$396$	0	0
$397$	−7.93861	−0.398428	−0.199214	−	0.979956i	$-0.563839\pi$
$397$	−7.93861	−0.398428	−0.199214	+	0.979956i	$0.563839\pi$
$398$	8.62978	0.432572
$399$	0	0
$400$	−0.837025	−0.0418512
$401$	−20.3167	−1.01457	−0.507285	−	0.861779i	$-0.669351\pi$
$401$	−20.3167	−1.01457	−0.507285	+	0.861779i	$0.669351\pi$
$402$	0	0
$403$	5.55326	0.276628
$404$	1.19588	0.0594972
$405$	0	0
$406$	2.89802	0.143826
$407$	0.725744	0.0359738
$408$	0	0
$409$	−1.10081	−0.0544318	−0.0272159	−	0.999630i	$-0.508664\pi$
$409$	−1.10081	−0.0544318	−0.0272159	+	0.999630i	$0.508664\pi$
$410$	9.33833	0.461187
$411$	0	0
$412$	−3.95892	−0.195042
$413$	13.8676	0.682380
$414$	0	0
$415$	−4.51685	−0.221724
$416$	−3.95618	−0.193967
$417$	0	0
$418$	1.23809	0.0605567
$419$	10.2288	0.499710	0.249855	−	0.968283i	$-0.419617\pi$
$419$	10.2288	0.499710	0.249855	+	0.968283i	$0.419617\pi$
$420$	0	0
$421$	−16.7435	−0.816028	−0.408014	−	0.912976i	$-0.633779\pi$
$421$	−16.7435	−0.816028	−0.408014	+	0.912976i	$0.633779\pi$
$422$	−11.9484	−0.581640
$423$	0	0
$424$	8.68416	0.421740
$425$	2.50067	0.121300
$426$	0	0
$427$	−21.5530	−1.04302
$428$	5.27886	0.255163
$429$	0	0
$430$	20.8087	1.00348
$431$	24.2430	1.16774	0.583871	−	0.811846i	$-0.301537\pi$
$431$	24.2430	1.16774	0.583871	+	0.811846i	$0.301537\pi$
$432$	0	0
$433$	27.4417	1.31877	0.659383	−	0.751807i	$-0.270818\pi$
$433$	27.4417	1.31877	0.659383	+	0.751807i	$0.270818\pi$
$434$	−3.44154	−0.165199
$435$	0	0
$436$	−12.5318	−0.600163
$437$	0.911530	0.0436044
$438$	0	0
$439$	21.7342	1.03732	0.518658	−	0.854982i	$-0.326432\pi$
$439$	21.7342	1.03732	0.518658	+	0.854982i	$0.326432\pi$
$440$	−0.415161	−0.0197920
$441$	0	0
$442$	11.8193	0.562189
$443$	12.5322	0.595424	0.297712	−	0.954656i	$-0.403776\pi$
$443$	12.5322	0.595424	0.297712	+	0.954656i	$0.403776\pi$
$444$	0	0
$445$	−21.7828	−1.03260
$446$	−16.3096	−0.772283
$447$	0	0
$448$	2.45177	0.115835
$449$	18.9161	0.892704	0.446352	−	0.894857i	$-0.352723\pi$
$449$	18.9161	0.892704	0.446352	+	0.894857i	$0.352723\pi$
$450$	0	0
$451$	−0.931285	−0.0438525
$452$	−5.30005	−0.249293
$453$	0	0
$454$	−25.9668	−1.21868
$455$	−19.7906	−0.927795
$456$	0	0
$457$	−8.15610	−0.381526	−0.190763	−	0.981636i	$-0.561096\pi$
$457$	−8.15610	−0.381526	−0.190763	+	0.981636i	$0.561096\pi$
$458$	17.1880	0.803142
$459$	0	0
$460$	−0.305659	−0.0142514
$461$	16.2395	0.756351	0.378176	−	0.925734i	$-0.376552\pi$
$461$	16.2395	0.756351	0.378176	+	0.925734i	$0.376552\pi$
$462$	0	0
$463$	14.8238	0.688922	0.344461	−	0.938801i	$-0.388062\pi$
$463$	14.8238	0.688922	0.344461	+	0.938801i	$0.388062\pi$
$464$	−1.18201	−0.0548734
$465$	0	0
$466$	11.7241	0.543107
$467$	−15.7271	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$467$	−15.7271	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$468$	0	0
$469$	−2.72926	−0.126025
$470$	17.9508	0.828009
$471$	0	0
$472$	−5.65615	−0.260345
$473$	−2.07519	−0.0954174
$474$	0	0
$475$	−5.09300	−0.233683
$476$	−7.32484	−0.335733
$477$	0	0
$478$	11.3448	0.518899
$479$	23.6260	1.07950	0.539749	−	0.841826i	$-0.318519\pi$
$479$	23.6260	1.07950	0.539749	+	0.841826i	$0.318519\pi$
$480$	0	0
$481$	−14.1106	−0.643386
$482$	−12.7772	−0.581987
$483$	0	0
$484$	−10.9586	−0.498118
$485$	16.7829	0.762073
$486$	0	0
$487$	31.8503	1.44328	0.721638	−	0.692271i	$-0.243389\pi$
$487$	31.8503	1.44328	0.721638	+	0.692271i	$0.243389\pi$
$488$	8.79078	0.397940
$489$	0	0
$490$	−2.01749	−0.0911411
$491$	8.22229	0.371067	0.185533	−	0.982638i	$-0.440599\pi$
$491$	8.22229	0.371067	0.185533	+	0.982638i	$0.440599\pi$
$492$	0	0
$493$	3.53133	0.159043
$494$	−24.0719	−1.08305
$495$	0	0
$496$	1.40369	0.0630277
$497$	16.1040	0.722362
$498$	0	0
$499$	34.4260	1.54112	0.770560	−	0.637367i	$-0.219976\pi$
$499$	34.4260	1.54112	0.770560	+	0.637367i	$0.219976\pi$
$500$	11.9095	0.532609
$501$	0	0
$502$	20.9749	0.936155
$503$	7.44402	0.331913	0.165956	−	0.986133i	$-0.446929\pi$
$503$	7.44402	0.331913	0.165956	+	0.986133i	$0.446929\pi$
$504$	0	0
$505$	−2.44000	−0.108578
$506$	0.0304825	0.00135511
$507$	0	0
$508$	4.45202	0.197526
$509$	23.1214	1.02484	0.512418	−	0.858736i	$-0.328750\pi$
$509$	23.1214	1.02484	0.512418	+	0.858736i	$0.328750\pi$
$510$	0	0
$511$	40.9740	1.81258
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.1377	−0.447155
$515$	8.07753	0.355939
$516$	0	0
$517$	−1.79018	−0.0787321
$518$	8.74478	0.384224
$519$	0	0
$520$	8.07193	0.353978
$521$	8.90428	0.390104	0.195052	−	0.980793i	$-0.437513\pi$
$521$	8.90428	0.390104	0.195052	+	0.980793i	$0.437513\pi$
$522$	0	0
$523$	17.7942	0.778084	0.389042	−	0.921220i	$-0.372806\pi$
$523$	17.7942	0.778084	0.389042	+	0.921220i	$0.372806\pi$
$524$	12.2984	0.537256
$525$	0	0
$526$	2.00396	0.0873767
$527$	−4.19363	−0.182677
$528$	0	0
$529$	−22.9776	−0.999024
$530$	−17.7186	−0.769648
$531$	0	0
$532$	14.9182	0.646785
$533$	18.1069	0.784295
$534$	0	0
$535$	−10.7706	−0.465656
$536$	1.11318	0.0480819
$537$	0	0
$538$	−1.93175	−0.0832836
$539$	0.201199	0.00866625
$540$	0	0
$541$	−0.335476	−0.0144232	−0.00721161	−	0.999974i	$-0.502296\pi$
$541$	−0.335476	−0.0144232	−0.00721161	+	0.999974i	$0.502296\pi$
$542$	−24.0451	−1.03283
$543$	0	0
$544$	2.98757	0.128091
$545$	25.5690	1.09526
$546$	0	0
$547$	−2.81996	−0.120573	−0.0602864	−	0.998181i	$-0.519201\pi$
$547$	−2.81996	−0.120573	−0.0602864	+	0.998181i	$0.519201\pi$
$548$	−18.7145	−0.799445
$549$	0	0
$550$	−0.170315	−0.00726226
$551$	−7.19211	−0.306394
$552$	0	0
$553$	−11.1423	−0.473820
$554$	−11.0962	−0.471432
$555$	0	0
$556$	−7.98547	−0.338659
$557$	17.2759	0.732004	0.366002	−	0.930614i	$-0.380726\pi$
$557$	17.2759	0.732004	0.366002	+	0.930614i	$0.380726\pi$
$558$	0	0
$559$	40.3477	1.70653
$560$	−5.00245	−0.211392
$561$	0	0
$562$	−16.2571	−0.685766
$563$	−23.2936	−0.981709	−0.490854	−	0.871242i	$-0.663315\pi$
$563$	−23.2936	−0.981709	−0.490854	+	0.871242i	$0.663315\pi$
$564$	0	0
$565$	10.8139	0.454944
$566$	−0.215180	−0.00904468
$567$	0	0
$568$	−6.56830	−0.275600
$569$	−5.17932	−0.217129	−0.108564	−	0.994089i	$-0.534625\pi$
$569$	−5.17932	−0.217129	−0.108564	+	0.994089i	$0.534625\pi$
$570$	0	0
$571$	−40.3418	−1.68825	−0.844126	−	0.536145i	$-0.819880\pi$
$571$	−40.3418	−1.68825	−0.844126	+	0.536145i	$0.819880\pi$
$572$	−0.804990	−0.0336583
$573$	0	0
$574$	−11.2214	−0.468373
$575$	−0.125393	−0.00522925
$576$	0	0
$577$	43.0244	1.79113	0.895564	−	0.444932i	$-0.146772\pi$
$577$	43.0244	1.79113	0.895564	+	0.444932i	$0.146772\pi$
$578$	8.07445	0.335853
$579$	0	0
$580$	2.41170	0.100140
$581$	5.42768	0.225178
$582$	0	0
$583$	1.76703	0.0731828
$584$	−16.7120	−0.691547
$585$	0	0
$586$	−17.9241	−0.740439
$587$	35.9418	1.48348	0.741738	−	0.670689i	$-0.234002\pi$
$587$	35.9418	1.48348	0.741738	+	0.670689i	$0.234002\pi$
$588$	0	0
$589$	8.54098	0.351925
$590$	11.5404	0.475113
$591$	0	0
$592$	−3.56672	−0.146591
$593$	8.28724	0.340316	0.170158	−	0.985417i	$-0.445572\pi$
$593$	8.28724	0.340316	0.170158	+	0.985417i	$0.445572\pi$
$594$	0	0
$595$	14.9451	0.612691
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−0.592667	−0.0242360
$599$	−38.7266	−1.58233	−0.791164	−	0.611605i	$-0.790524\pi$
$599$	−38.7266	−1.58233	−0.791164	+	0.611605i	$0.790524\pi$
$600$	0	0
$601$	21.0216	0.857490	0.428745	−	0.903426i	$-0.358956\pi$
$601$	21.0216	0.857490	0.428745	+	0.903426i	$0.358956\pi$
$602$	−25.0048	−1.01912
$603$	0	0
$604$	12.5737	0.511619
$605$	22.3592	0.909032
$606$	0	0
$607$	−6.91848	−0.280812	−0.140406	−	0.990094i	$-0.544841\pi$
$607$	−6.91848	−0.280812	−0.140406	+	0.990094i	$0.544841\pi$
$608$	−6.08465	−0.246765
$609$	0	0
$610$	−17.9362	−0.726214
$611$	34.8063	1.40811
$612$	0	0
$613$	−17.1798	−0.693885	−0.346943	−	0.937886i	$-0.612780\pi$
$613$	−17.1798	−0.693885	−0.346943	+	0.937886i	$0.612780\pi$
$614$	−8.70157	−0.351167
$615$	0	0
$616$	0.498879	0.0201004
$617$	−29.4917	−1.18729	−0.593646	−	0.804726i	$-0.702312\pi$
$617$	−29.4917	−1.18729	−0.593646	+	0.804726i	$0.702312\pi$
$618$	0	0
$619$	19.8981	0.799772	0.399886	−	0.916565i	$-0.369050\pi$
$619$	19.8981	0.799772	0.399886	+	0.916565i	$0.369050\pi$
$620$	−2.86401	−0.115021
$621$	0	0
$622$	5.96807	0.239298
$623$	26.1753	1.04869
$624$	0	0
$625$	−20.1143	−0.804571
$626$	−12.0096	−0.479999
$627$	0	0
$628$	7.13867	0.284864
$629$	10.6558	0.424875
$630$	0	0
$631$	35.8054	1.42539	0.712696	−	0.701473i	$-0.247474\pi$
$631$	35.8054	1.42539	0.712696	+	0.701473i	$0.247474\pi$
$632$	4.54460	0.180774
$633$	0	0
$634$	−9.00626	−0.357684
$635$	−9.08362	−0.360472
$636$	0	0
$637$	−3.91189	−0.154995
$638$	−0.240512	−0.00952194
$639$	0	0
$640$	2.04034	0.0806514
$641$	23.1561	0.914611	0.457305	−	0.889310i	$-0.348815\pi$
$641$	23.1561	0.914611	0.457305	+	0.889310i	$0.348815\pi$
$642$	0	0
$643$	11.9119	0.469761	0.234880	−	0.972024i	$-0.424530\pi$
$643$	11.9119	0.469761	0.234880	+	0.972024i	$0.424530\pi$
$644$	0.367296	0.0144735
$645$	0	0
$646$	18.1783	0.715215
$647$	0.801674	0.0315171	0.0157585	−	0.999876i	$-0.494984\pi$
$647$	0.801674	0.0315171	0.0157585	+	0.999876i	$0.494984\pi$
$648$	0	0
$649$	−1.15089	−0.0451766
$650$	3.31142	0.129884
$651$	0	0
$652$	1.08015	0.0423019
$653$	14.6278	0.572431	0.286215	−	0.958165i	$-0.407603\pi$
$653$	14.6278	0.572431	0.286215	+	0.958165i	$0.407603\pi$
$654$	0	0
$655$	−25.0928	−0.980456
$656$	4.57686	0.178696
$657$	0	0
$658$	−21.5706	−0.840910
$659$	−36.8855	−1.43686	−0.718428	−	0.695601i	$-0.755138\pi$
$659$	−36.8855	−1.43686	−0.718428	+	0.695601i	$0.755138\pi$
$660$	0	0
$661$	−4.83682	−0.188130	−0.0940652	−	0.995566i	$-0.529986\pi$
$661$	−4.83682	−0.188130	−0.0940652	+	0.995566i	$0.529986\pi$
$662$	21.9340	0.852491
$663$	0	0
$664$	−2.21378	−0.0859113
$665$	−30.4381	−1.18034
$666$	0	0
$667$	−0.177075	−0.00685636
$668$	−17.8060	−0.688936
$669$	0	0
$670$	−2.27125	−0.0877462
$671$	1.78872	0.0690528
$672$	0	0
$673$	−1.43783	−0.0554245	−0.0277122	−	0.999616i	$-0.508822\pi$
$673$	−1.43783	−0.0554245	−0.0277122	+	0.999616i	$0.508822\pi$
$674$	−6.33234	−0.243913
$675$	0	0
$676$	2.65133	0.101974
$677$	−51.4167	−1.97610	−0.988052	−	0.154123i	$-0.950745\pi$
$677$	−51.4167	−1.97610	−0.988052	+	0.154123i	$0.950745\pi$
$678$	0	0
$679$	−20.1672	−0.773947
$680$	−6.09564	−0.233757
$681$	0	0
$682$	0.285619	0.0109369
$683$	−36.2412	−1.38673	−0.693365	−	0.720586i	$-0.743873\pi$
$683$	−36.2412	−1.38673	−0.693365	+	0.720586i	$0.743873\pi$
$684$	0	0
$685$	38.1839	1.45893
$686$	19.5867	0.747825
$687$	0	0
$688$	10.1987	0.388820
$689$	−34.3561	−1.30886
$690$	0	0
$691$	11.2663	0.428589	0.214294	−	0.976769i	$-0.431255\pi$
$691$	11.2663	0.428589	0.214294	+	0.976769i	$0.431255\pi$
$692$	−18.3462	−0.697419
$693$	0	0
$694$	−17.8450	−0.677386
$695$	16.2931	0.618031
$696$	0	0
$697$	−13.6737	−0.517927
$698$	−4.18716	−0.158486
$699$	0	0
$700$	−2.05219	−0.0775657
$701$	22.6248	0.854526	0.427263	−	0.904127i	$-0.359478\pi$
$701$	22.6248	0.854526	0.427263	+	0.904127i	$0.359478\pi$
$702$	0	0
$703$	−21.7022	−0.818515
$704$	−0.203477	−0.00766882
$705$	0	0
$706$	−13.9825	−0.526239
$707$	2.93203	0.110270
$708$	0	0
$709$	23.7184	0.890763	0.445381	−	0.895341i	$-0.353068\pi$
$709$	23.7184	0.890763	0.445381	+	0.895341i	$0.353068\pi$
$710$	13.4015	0.502951
$711$	0	0
$712$	−10.6761	−0.400102
$713$	0.210285	0.00787523
$714$	0	0
$715$	1.64245	0.0614242
$716$	16.2742	0.608194
$717$	0	0
$718$	−3.55787	−0.132778
$719$	7.15270	0.266751	0.133375	−	0.991066i	$-0.457418\pi$
$719$	7.15270	0.266751	0.133375	+	0.991066i	$0.457418\pi$
$720$	0	0
$721$	−9.70638	−0.361485
$722$	−18.0230	−0.670745
$723$	0	0
$724$	6.27919	0.233364
$725$	0.989371	0.0367443
$726$	0	0
$727$	−9.45384	−0.350623	−0.175312	−	0.984513i	$-0.556093\pi$
$727$	−9.45384	−0.350623	−0.175312	+	0.984513i	$0.556093\pi$
$728$	−9.69965	−0.359493
$729$	0	0
$730$	34.0981	1.26203
$731$	−30.4692	−1.12694
$732$	0	0
$733$	−28.1192	−1.03860	−0.519302	−	0.854591i	$-0.673808\pi$
$733$	−28.1192	−1.03860	−0.519302	+	0.854591i	$0.673808\pi$
$734$	−28.0206	−1.03426
$735$	0	0
$736$	−0.149808	−0.00552200
$737$	0.226506	0.00834344
$738$	0	0
$739$	43.7960	1.61106	0.805532	−	0.592553i	$-0.201880\pi$
$739$	43.7960	1.61106	0.805532	+	0.592553i	$0.201880\pi$
$740$	7.27730	0.267519
$741$	0	0
$742$	21.2916	0.781639
$743$	37.3780	1.37126	0.685632	−	0.727948i	$-0.259526\pi$
$743$	37.3780	1.37126	0.685632	+	0.727948i	$0.259526\pi$
$744$	0	0
$745$	2.04034	0.0747522
$746$	25.6903	0.940590
$747$	0	0
$748$	0.607901	0.0222270
$749$	12.9426	0.472911
$750$	0	0
$751$	−12.4139	−0.452989	−0.226495	−	0.974012i	$-0.572727\pi$
$751$	−12.4139	−0.452989	−0.226495	+	0.974012i	$0.572727\pi$
$752$	8.79796	0.320829
$753$	0	0
$754$	4.67624	0.170298
$755$	−25.6547	−0.933670
$756$	0	0
$757$	−22.7545	−0.827028	−0.413514	−	0.910498i	$-0.635699\pi$
$757$	−22.7545	−0.827028	−0.413514	+	0.910498i	$0.635699\pi$
$758$	−27.3594	−0.993738
$759$	0	0
$760$	12.4147	0.450330
$761$	32.3320	1.17204	0.586018	−	0.810298i	$-0.300695\pi$
$761$	32.3320	1.17204	0.586018	+	0.810298i	$0.300695\pi$
$762$	0	0
$763$	−30.7251	−1.11232
$764$	15.9069	0.575493
$765$	0	0
$766$	23.6995	0.856296
$767$	22.3767	0.807976
$768$	0	0
$769$	−17.9719	−0.648083	−0.324041	−	0.946043i	$-0.605042\pi$
$769$	−17.9719	−0.648083	−0.324041	+	0.946043i	$0.605042\pi$
$770$	−1.01788	−0.0366819
$771$	0	0
$772$	−14.9415	−0.537758
$773$	8.22939	0.295990	0.147995	−	0.988988i	$-0.452718\pi$
$773$	8.22939	0.295990	0.147995	+	0.988988i	$0.452718\pi$
$774$	0	0
$775$	−1.17493	−0.0422046
$776$	8.22556	0.295281
$777$	0	0
$778$	12.0397	0.431642
$779$	27.8486	0.997779
$780$	0	0
$781$	−1.33650	−0.0478236
$782$	0.447562	0.0160048
$783$	0	0
$784$	−0.988805	−0.0353145
$785$	−14.5653	−0.519857
$786$	0	0
$787$	36.8243	1.31264	0.656322	−	0.754481i	$-0.272111\pi$
$787$	36.8243	1.31264	0.656322	+	0.754481i	$0.272111\pi$
$788$	6.88235	0.245174
$789$	0	0
$790$	−9.27251	−0.329901
$791$	−12.9945	−0.462032
$792$	0	0
$793$	−34.7779	−1.23500
$794$	7.93861	0.281731
$795$	0	0
$796$	−8.62978	−0.305875
$797$	−45.4961	−1.61156	−0.805778	−	0.592217i	$-0.798253\pi$
$797$	−45.4961	−1.61156	−0.805778	+	0.592217i	$0.798253\pi$
$798$	0	0
$799$	−26.2845	−0.929879
$800$	0.837025	0.0295933
$801$	0	0
$802$	20.3167	0.717409
$803$	−3.40050	−0.120001
$804$	0	0
$805$	−0.749407	−0.0264131
$806$	−5.55326	−0.195605
$807$	0	0
$808$	−1.19588	−0.0420709
$809$	−42.0253	−1.47753	−0.738766	−	0.673962i	$-0.764591\pi$
$809$	−42.0253	−1.47753	−0.738766	+	0.673962i	$0.764591\pi$
$810$	0	0
$811$	33.5255	1.17724	0.588619	−	0.808411i	$-0.299672\pi$
$811$	33.5255	1.17724	0.588619	+	0.808411i	$0.299672\pi$
$812$	−2.89802	−0.101701
$813$	0	0
$814$	−0.725744	−0.0254373
$815$	−2.20387	−0.0771982
$816$	0	0
$817$	62.0552	2.17104
$818$	1.10081	0.0384891
$819$	0	0
$820$	−9.33833	−0.326109
$821$	10.1280	0.353470	0.176735	−	0.984258i	$-0.443446\pi$
$821$	10.1280	0.353470	0.176735	+	0.984258i	$0.443446\pi$
$822$	0	0
$823$	34.5274	1.20355	0.601775	−	0.798666i	$-0.294461\pi$
$823$	34.5274	1.20355	0.601775	+	0.798666i	$0.294461\pi$
$824$	3.95892	0.137916
$825$	0	0
$826$	−13.8676	−0.482515
$827$	−12.0445	−0.418827	−0.209414	−	0.977827i	$-0.567155\pi$
$827$	−12.0445	−0.418827	−0.209414	+	0.977827i	$0.567155\pi$
$828$	0	0
$829$	22.4078	0.778257	0.389128	−	0.921184i	$-0.372776\pi$
$829$	22.4078	0.778257	0.389128	+	0.921184i	$0.372776\pi$
$830$	4.51685	0.156782
$831$	0	0
$832$	3.95618	0.137156
$833$	2.95412	0.102354
$834$	0	0
$835$	36.3303	1.25726
$836$	−1.23809	−0.0428201
$837$	0	0
$838$	−10.2288	−0.353348
$839$	34.9694	1.20728	0.603640	−	0.797257i	$-0.293717\pi$
$839$	34.9694	1.20728	0.603640	+	0.797257i	$0.293717\pi$
$840$	0	0
$841$	−27.6029	−0.951823
$842$	16.7435	0.577019
$843$	0	0
$844$	11.9484	0.411282
$845$	−5.40961	−0.186096
$846$	0	0
$847$	−26.8680	−0.923195
$848$	−8.68416	−0.298215
$849$	0	0
$850$	−2.50067	−0.0857721
$851$	−0.534323	−0.0183164
$852$	0	0
$853$	43.7366	1.49751	0.748756	−	0.662846i	$-0.230652\pi$
$853$	43.7366	1.49751	0.748756	+	0.662846i	$0.230652\pi$
$854$	21.5530	0.737529
$855$	0	0
$856$	−5.27886	−0.180428
$857$	−47.8442	−1.63433	−0.817163	−	0.576407i	$-0.804454\pi$
$857$	−47.8442	−1.63433	−0.817163	+	0.576407i	$0.804454\pi$
$858$	0	0
$859$	8.60137	0.293475	0.146737	−	0.989175i	$-0.453123\pi$
$859$	8.60137	0.293475	0.146737	+	0.989175i	$0.453123\pi$
$860$	−20.8087	−0.709571
$861$	0	0
$862$	−24.2430	−0.825719
$863$	19.0673	0.649059	0.324529	−	0.945876i	$-0.394794\pi$
$863$	19.0673	0.649059	0.324529	+	0.945876i	$0.394794\pi$
$864$	0	0
$865$	37.4325	1.27274
$866$	−27.4417	−0.932509
$867$	0	0
$868$	3.44154	0.116813
$869$	0.924720	0.0313690
$870$	0	0
$871$	−4.40392	−0.149221
$872$	12.5318	0.424379
$873$	0	0
$874$	−0.911530	−0.0308330
$875$	29.1994	0.987120
$876$	0	0
$877$	−29.2798	−0.988708	−0.494354	−	0.869261i	$-0.664595\pi$
$877$	−29.2798	−0.988708	−0.494354	+	0.869261i	$0.664595\pi$
$878$	−21.7342	−0.733494
$879$	0	0
$880$	0.415161	0.0139951
$881$	15.2528	0.513881	0.256940	−	0.966427i	$-0.417286\pi$
$881$	15.2528	0.513881	0.256940	+	0.966427i	$0.417286\pi$
$882$	0	0
$883$	−22.4516	−0.755557	−0.377778	−	0.925896i	$-0.623312\pi$
$883$	−22.4516	−0.755557	−0.377778	+	0.925896i	$0.623312\pi$
$884$	−11.8193	−0.397527
$885$	0	0
$886$	−12.5322	−0.421028
$887$	19.7620	0.663544	0.331772	−	0.943360i	$-0.392353\pi$
$887$	19.7620	0.663544	0.331772	+	0.943360i	$0.392353\pi$
$888$	0	0
$889$	10.9153	0.366089
$890$	21.7828	0.730160
$891$	0	0
$892$	16.3096	0.546087
$893$	53.5325	1.79140
$894$	0	0
$895$	−33.2048	−1.10991
$896$	−2.45177	−0.0819080
$897$	0	0
$898$	−18.9161	−0.631237
$899$	−1.65918	−0.0553367
$900$	0	0
$901$	25.9445	0.864337
$902$	0.931285	0.0310084
$903$	0	0
$904$	5.30005	0.176277
$905$	−12.8117	−0.425874
$906$	0	0
$907$	−46.9011	−1.55733	−0.778663	−	0.627443i	$-0.784102\pi$
$907$	−46.9011	−1.55733	−0.778663	+	0.627443i	$0.784102\pi$
$908$	25.9668	0.861739
$909$	0	0
$910$	19.7906	0.656050
$911$	32.7025	1.08348	0.541742	−	0.840545i	$-0.317765\pi$
$911$	32.7025	1.08348	0.541742	+	0.840545i	$0.317765\pi$
$912$	0	0
$913$	−0.450453	−0.0149078
$914$	8.15610	0.269780
$915$	0	0
$916$	−17.1880	−0.567907
$917$	30.1528	0.995732
$918$	0	0
$919$	5.93251	0.195696	0.0978478	−	0.995201i	$-0.468804\pi$
$919$	5.93251	0.195696	0.0978478	+	0.995201i	$0.468804\pi$
$920$	0.305659	0.0100773
$921$	0	0
$922$	−16.2395	−0.534821
$923$	25.9853	0.855318
$924$	0	0
$925$	2.98543	0.0981603
$926$	−14.8238	−0.487141
$927$	0	0
$928$	1.18201	0.0388013
$929$	4.38637	0.143912	0.0719560	−	0.997408i	$-0.477076\pi$
$929$	4.38637	0.143912	0.0719560	+	0.997408i	$0.477076\pi$
$930$	0	0
$931$	−6.01653	−0.197184
$932$	−11.7241	−0.384035
$933$	0	0
$934$	15.7271	0.514608
$935$	−1.24032	−0.0405629
$936$	0	0
$937$	−22.8270	−0.745726	−0.372863	−	0.927886i	$-0.621624\pi$
$937$	−22.8270	−0.745726	−0.372863	+	0.927886i	$0.621624\pi$
$938$	2.72926	0.0891133
$939$	0	0
$940$	−17.9508	−0.585491
$941$	−13.3308	−0.434571	−0.217285	−	0.976108i	$-0.569720\pi$
$941$	−13.3308	−0.434571	−0.217285	+	0.976108i	$0.569720\pi$
$942$	0	0
$943$	0.685651	0.0223279
$944$	5.65615	0.184092
$945$	0	0
$946$	2.07519	0.0674703
$947$	51.2535	1.66552	0.832758	−	0.553637i	$-0.186761\pi$
$947$	51.2535	1.66552	0.832758	+	0.553637i	$0.186761\pi$
$948$	0	0
$949$	66.1156	2.14620
$950$	5.09300	0.165239
$951$	0	0
$952$	7.32484	0.237399
$953$	1.57768	0.0511061	0.0255531	−	0.999673i	$-0.491865\pi$
$953$	1.57768	0.0511061	0.0255531	+	0.999673i	$0.491865\pi$
$954$	0	0
$955$	−32.4555	−1.05024
$956$	−11.3448	−0.366917
$957$	0	0
$958$	−23.6260	−0.763320
$959$	−45.8838	−1.48166
$960$	0	0
$961$	−29.0296	−0.936440
$962$	14.1106	0.454943
$963$	0	0
$964$	12.7772	0.411527
$965$	30.4858	0.981372
$966$	0	0
$967$	−30.8568	−0.992287	−0.496143	−	0.868241i	$-0.665251\pi$
$967$	−30.8568	−0.992287	−0.496143	+	0.868241i	$0.665251\pi$
$968$	10.9586	0.352223
$969$	0	0
$970$	−16.7829	−0.538867
$971$	−30.6296	−0.982952	−0.491476	−	0.870891i	$-0.663542\pi$
$971$	−30.6296	−0.982952	−0.491476	+	0.870891i	$0.663542\pi$
$972$	0	0
$973$	−19.5786	−0.627660
$974$	−31.8503	−1.02055
$975$	0	0
$976$	−8.79078	−0.281386
$977$	−19.6525	−0.628738	−0.314369	−	0.949301i	$-0.601793\pi$
$977$	−19.6525	−0.628738	−0.314369	+	0.949301i	$0.601793\pi$
$978$	0	0
$979$	−2.17233	−0.0694280
$980$	2.01749	0.0644465
$981$	0	0
$982$	−8.22229	−0.262384
$983$	27.1718	0.866646	0.433323	−	0.901239i	$-0.357341\pi$
$983$	27.1718	0.866646	0.433323	+	0.901239i	$0.357341\pi$
$984$	0	0
$985$	−14.0423	−0.447425
$986$	−3.53133	−0.112460
$987$	0	0
$988$	24.0719	0.765831
$989$	1.52784	0.0485825
$990$	0	0
$991$	39.8356	1.26542	0.632711	−	0.774388i	$-0.281942\pi$
$991$	39.8356	1.26542	0.632711	+	0.774388i	$0.281942\pi$
$992$	−1.40369	−0.0445673
$993$	0	0
$994$	−16.1040	−0.510787
$995$	17.6077	0.558201
$996$	0	0
$997$	20.3388	0.644137	0.322068	−	0.946716i	$-0.395622\pi$
$997$	20.3388	0.644137	0.322068	+	0.946716i	$0.395622\pi$
$998$	−34.4260	−1.08974
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.q.1.5	✓	14
3.2	odd	2		8046.2.a.r.1.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.5	✓	14	1.1	even	1	trivial
8046.2.a.r.1.10	yes	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.04034 q^{5} +2.45177 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.04034 q^{5} +2.45177 q^{7} -1.00000 q^{8} +2.04034 q^{10} -0.203477 q^{11} +3.95618 q^{13} -2.45177 q^{14} +1.00000 q^{16} -2.98757 q^{17} +6.08465 q^{19} -2.04034 q^{20} +0.203477 q^{22} +0.149808 q^{23} -0.837025 q^{25} -3.95618 q^{26} +2.45177 q^{28} -1.18201 q^{29} +1.40369 q^{31} -1.00000 q^{32} +2.98757 q^{34} -5.00245 q^{35} -3.56672 q^{37} -6.08465 q^{38} +2.04034 q^{40} +4.57686 q^{41} +10.1987 q^{43} -0.203477 q^{44} -0.149808 q^{46} +8.79796 q^{47} -0.988805 q^{49} +0.837025 q^{50} +3.95618 q^{52} -8.68416 q^{53} +0.415161 q^{55} -2.45177 q^{56} +1.18201 q^{58} +5.65615 q^{59} -8.79078 q^{61} -1.40369 q^{62} +1.00000 q^{64} -8.07193 q^{65} -1.11318 q^{67} -2.98757 q^{68} +5.00245 q^{70} +6.56830 q^{71} +16.7120 q^{73} +3.56672 q^{74} +6.08465 q^{76} -0.498879 q^{77} -4.54460 q^{79} -2.04034 q^{80} -4.57686 q^{82} +2.21378 q^{83} +6.09564 q^{85} -10.1987 q^{86} +0.203477 q^{88} +10.6761 q^{89} +9.69965 q^{91} +0.149808 q^{92} -8.79796 q^{94} -12.4147 q^{95} -8.22556 q^{97} +0.988805 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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