LMFDB - Embedded newform 8046.2.a.t.1.8

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
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			1.8
Root			$0.0881948$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.0881948 q^{5} +4.92686 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.0881948 q^{5} +4.92686 q^{7} +1.00000 q^{8} +0.0881948 q^{10} +0.970875 q^{11} +4.90725 q^{13} +4.92686 q^{14} +1.00000 q^{16} -3.81432 q^{17} +2.32609 q^{19} +0.0881948 q^{20} +0.970875 q^{22} +0.0904339 q^{23} -4.99222 q^{25} +4.90725 q^{26} +4.92686 q^{28} -6.73087 q^{29} +4.73318 q^{31} +1.00000 q^{32} -3.81432 q^{34} +0.434524 q^{35} +5.77847 q^{37} +2.32609 q^{38} +0.0881948 q^{40} -6.48650 q^{41} +2.46680 q^{43} +0.970875 q^{44} +0.0904339 q^{46} +12.0519 q^{47} +17.2740 q^{49} -4.99222 q^{50} +4.90725 q^{52} +6.91596 q^{53} +0.0856261 q^{55} +4.92686 q^{56} -6.73087 q^{58} +7.13615 q^{59} +10.1951 q^{61} +4.73318 q^{62} +1.00000 q^{64} +0.432793 q^{65} -12.9524 q^{67} -3.81432 q^{68} +0.434524 q^{70} -11.3961 q^{71} +2.68616 q^{73} +5.77847 q^{74} +2.32609 q^{76} +4.78337 q^{77} -10.3924 q^{79} +0.0881948 q^{80} -6.48650 q^{82} -14.3068 q^{83} -0.336403 q^{85} +2.46680 q^{86} +0.970875 q^{88} +14.1825 q^{89} +24.1773 q^{91} +0.0904339 q^{92} +12.0519 q^{94} +0.205149 q^{95} +9.41121 q^{97} +17.2740 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * 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
$5$	0.0881948	0.0394419	0.0197209	−	0.999806i	$-0.493722\pi$
$5$	0.0881948	0.0394419	0.0197209	+	0.999806i	$0.493722\pi$
$6$	0	0
$7$	4.92686	1.86218	0.931090	−	0.364790i	$-0.118859\pi$
$7$	4.92686	1.86218	0.931090	+	0.364790i	$0.118859\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.0881948	0.0278896
$11$	0.970875	0.292730	0.146365	−	0.989231i	$-0.453243\pi$
$11$	0.970875	0.292730	0.146365	+	0.989231i	$0.453243\pi$
$12$	0	0
$13$	4.90725	1.36103	0.680513	−	0.732736i	$-0.261757\pi$
$13$	4.90725	1.36103	0.680513	+	0.732736i	$0.261757\pi$
$14$	4.92686	1.31676
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.81432	−0.925109	−0.462555	−	0.886591i	$-0.653067\pi$
$17$	−3.81432	−0.925109	−0.462555	+	0.886591i	$0.653067\pi$
$18$	0	0
$19$	2.32609	0.533641	0.266820	−	0.963746i	$-0.414027\pi$
$19$	2.32609	0.533641	0.266820	+	0.963746i	$0.414027\pi$
$20$	0.0881948	0.0197209
$21$	0	0
$22$	0.970875	0.206991
$23$	0.0904339	0.0188568	0.00942838	−	0.999956i	$-0.496999\pi$
$23$	0.0904339	0.0188568	0.00942838	+	0.999956i	$0.496999\pi$
$24$	0	0
$25$	−4.99222	−0.998444
$26$	4.90725	0.962390
$27$	0	0
$28$	4.92686	0.931090
$29$	−6.73087	−1.24989	−0.624946	−	0.780668i	$-0.714879\pi$
$29$	−6.73087	−1.24989	−0.624946	+	0.780668i	$0.714879\pi$
$30$	0	0
$31$	4.73318	0.850103	0.425052	−	0.905169i	$-0.360256\pi$
$31$	4.73318	0.850103	0.425052	+	0.905169i	$0.360256\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.81432	−0.654151
$35$	0.434524	0.0734479
$36$	0	0
$37$	5.77847	0.949974	0.474987	−	0.879993i	$-0.342453\pi$
$37$	5.77847	0.949974	0.474987	+	0.879993i	$0.342453\pi$
$38$	2.32609	0.377341
$39$	0	0
$40$	0.0881948	0.0139448
$41$	−6.48650	−1.01302	−0.506511	−	0.862234i	$-0.669065\pi$
$41$	−6.48650	−1.01302	−0.506511	+	0.862234i	$0.669065\pi$
$42$	0	0
$43$	2.46680	0.376184	0.188092	−	0.982151i	$-0.439770\pi$
$43$	2.46680	0.376184	0.188092	+	0.982151i	$0.439770\pi$
$44$	0.970875	0.146365
$45$	0	0
$46$	0.0904339	0.0133337
$47$	12.0519	1.75795	0.878974	−	0.476869i	$-0.158228\pi$
$47$	12.0519	1.75795	0.878974	+	0.476869i	$0.158228\pi$
$48$	0	0
$49$	17.2740	2.46771
$50$	−4.99222	−0.706007
$51$	0	0
$52$	4.90725	0.680513
$53$	6.91596	0.949981	0.474990	−	0.879991i	$-0.342452\pi$
$53$	6.91596	0.949981	0.474990	+	0.879991i	$0.342452\pi$
$54$	0	0
$55$	0.0856261	0.0115458
$56$	4.92686	0.658380
$57$	0	0
$58$	−6.73087	−0.883807
$59$	7.13615	0.929047	0.464523	−	0.885561i	$-0.346226\pi$
$59$	7.13615	0.929047	0.464523	+	0.885561i	$0.346226\pi$
$60$	0	0
$61$	10.1951	1.30534	0.652671	−	0.757641i	$-0.273648\pi$
$61$	10.1951	1.30534	0.652671	+	0.757641i	$0.273648\pi$
$62$	4.73318	0.601114
$63$	0	0
$64$	1.00000	0.125000
$65$	0.432793	0.0536814
$66$	0	0
$67$	−12.9524	−1.58239	−0.791197	−	0.611562i	$-0.790542\pi$
$67$	−12.9524	−1.58239	−0.791197	+	0.611562i	$0.790542\pi$
$68$	−3.81432	−0.462555
$69$	0	0
$70$	0.434524	0.0519355
$71$	−11.3961	−1.35247	−0.676235	−	0.736686i	$-0.736390\pi$
$71$	−11.3961	−1.35247	−0.676235	+	0.736686i	$0.736390\pi$
$72$	0	0
$73$	2.68616	0.314392	0.157196	−	0.987567i	$-0.449755\pi$
$73$	2.68616	0.314392	0.157196	+	0.987567i	$0.449755\pi$
$74$	5.77847	0.671733
$75$	0	0
$76$	2.32609	0.266820
$77$	4.78337	0.545115
$78$	0	0
$79$	−10.3924	−1.16924	−0.584620	−	0.811307i	$-0.698757\pi$
$79$	−10.3924	−1.16924	−0.584620	+	0.811307i	$0.698757\pi$
$80$	0.0881948	0.00986047
$81$	0	0
$82$	−6.48650	−0.716315
$83$	−14.3068	−1.57037	−0.785186	−	0.619261i	$-0.787432\pi$
$83$	−14.3068	−1.57037	−0.785186	+	0.619261i	$0.787432\pi$
$84$	0	0
$85$	−0.336403	−0.0364881
$86$	2.46680	0.266002
$87$	0	0
$88$	0.970875	0.103496
$89$	14.1825	1.50335	0.751674	−	0.659535i	$-0.229247\pi$
$89$	14.1825	1.50335	0.751674	+	0.659535i	$0.229247\pi$
$90$	0	0
$91$	24.1773	2.53447
$92$	0.0904339	0.00942838
$93$	0	0
$94$	12.0519	1.24306
$95$	0.205149	0.0210478
$96$	0	0
$97$	9.41121	0.955563	0.477782	−	0.878479i	$-0.341441\pi$
$97$	9.41121	0.955563	0.477782	+	0.878479i	$0.341441\pi$
$98$	17.2740	1.74494
$99$	0	0
$100$	−4.99222	−0.499222
$101$	−13.3267	−1.32605	−0.663026	−	0.748596i	$-0.730728\pi$
$101$	−13.3267	−1.32605	−0.663026	+	0.748596i	$0.730728\pi$
$102$	0	0
$103$	5.53443	0.545323	0.272662	−	0.962110i	$-0.412096\pi$
$103$	5.53443	0.545323	0.272662	+	0.962110i	$0.412096\pi$
$104$	4.90725	0.481195
$105$	0	0
$106$	6.91596	0.671738
$107$	3.57082	0.345204	0.172602	−	0.984992i	$-0.444783\pi$
$107$	3.57082	0.345204	0.172602	+	0.984992i	$0.444783\pi$
$108$	0	0
$109$	0.913614	0.0875083	0.0437542	−	0.999042i	$-0.486068\pi$
$109$	0.913614	0.0875083	0.0437542	+	0.999042i	$0.486068\pi$
$110$	0.0856261	0.00816413
$111$	0	0
$112$	4.92686	0.465545
$113$	−0.782261	−0.0735889	−0.0367944	−	0.999323i	$-0.511715\pi$
$113$	−0.782261	−0.0735889	−0.0367944	+	0.999323i	$0.511715\pi$
$114$	0	0
$115$	0.00797580	0.000743747 0
$116$	−6.73087	−0.624946
$117$	0	0
$118$	7.13615	0.656935
$119$	−18.7927	−1.72272
$120$	0	0
$121$	−10.0574	−0.914309
$122$	10.1951	0.923017
$123$	0	0
$124$	4.73318	0.425052
$125$	−0.881262	−0.0788224
$126$	0	0
$127$	−0.107606	−0.00954844	−0.00477422	−	0.999989i	$-0.501520\pi$
$127$	−0.107606	−0.00954844	−0.00477422	+	0.999989i	$0.501520\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.432793	0.0379585
$131$	3.75057	0.327689	0.163844	−	0.986486i	$-0.447610\pi$
$131$	3.75057	0.327689	0.163844	+	0.986486i	$0.447610\pi$
$132$	0	0
$133$	11.4603	0.993735
$134$	−12.9524	−1.11892
$135$	0	0
$136$	−3.81432	−0.327075
$137$	2.39376	0.204512	0.102256	−	0.994758i	$-0.467394\pi$
$137$	2.39376	0.204512	0.102256	+	0.994758i	$0.467394\pi$
$138$	0	0
$139$	−19.2425	−1.63213	−0.816066	−	0.577959i	$-0.803849\pi$
$139$	−19.2425	−1.63213	−0.816066	+	0.577959i	$0.803849\pi$
$140$	0.434524	0.0367240
$141$	0	0
$142$	−11.3961	−0.956341
$143$	4.76432	0.398413
$144$	0	0
$145$	−0.593628	−0.0492981
$146$	2.68616	0.222308
$147$	0	0
$148$	5.77847	0.474987
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−14.7457	−1.19999	−0.599995	−	0.800004i	$-0.704831\pi$
$151$	−14.7457	−1.19999	−0.599995	+	0.800004i	$0.704831\pi$
$152$	2.32609	0.188670
$153$	0	0
$154$	4.78337	0.385455
$155$	0.417441	0.0335297
$156$	0	0
$157$	3.98919	0.318372	0.159186	−	0.987249i	$-0.449113\pi$
$157$	3.98919	0.318372	0.159186	+	0.987249i	$0.449113\pi$
$158$	−10.3924	−0.826778
$159$	0	0
$160$	0.0881948	0.00697241
$161$	0.445556	0.0351147
$162$	0	0
$163$	−16.6118	−1.30114	−0.650569	−	0.759447i	$-0.725469\pi$
$163$	−16.6118	−1.30114	−0.650569	+	0.759447i	$0.725469\pi$
$164$	−6.48650	−0.506511
$165$	0	0
$166$	−14.3068	−1.11042
$167$	−19.3057	−1.49392	−0.746959	−	0.664870i	$-0.768487\pi$
$167$	−19.3057	−1.49392	−0.746959	+	0.664870i	$0.768487\pi$
$168$	0	0
$169$	11.0811	0.852389
$170$	−0.336403	−0.0258010
$171$	0	0
$172$	2.46680	0.188092
$173$	1.91358	0.145487	0.0727436	−	0.997351i	$-0.476825\pi$
$173$	1.91358	0.145487	0.0727436	+	0.997351i	$0.476825\pi$
$174$	0	0
$175$	−24.5960	−1.85928
$176$	0.970875	0.0731824
$177$	0	0
$178$	14.1825	1.06303
$179$	−20.7029	−1.54741	−0.773704	−	0.633547i	$-0.781598\pi$
$179$	−20.7029	−1.54741	−0.773704	+	0.633547i	$0.781598\pi$
$180$	0	0
$181$	−13.0756	−0.971901	−0.485951	−	0.873986i	$-0.661527\pi$
$181$	−13.0756	−0.971901	−0.485951	+	0.873986i	$0.661527\pi$
$182$	24.1773	1.79214
$183$	0	0
$184$	0.0904339	0.00666687
$185$	0.509631	0.0374688
$186$	0	0
$187$	−3.70323	−0.270807
$188$	12.0519	0.878974
$189$	0	0
$190$	0.205149	0.0148830
$191$	22.7732	1.64781	0.823904	−	0.566729i	$-0.191791\pi$
$191$	22.7732	1.64781	0.823904	+	0.566729i	$0.191791\pi$
$192$	0	0
$193$	−6.80693	−0.489974	−0.244987	−	0.969526i	$-0.578784\pi$
$193$	−6.80693	−0.489974	−0.244987	+	0.969526i	$0.578784\pi$
$194$	9.41121	0.675685
$195$	0	0
$196$	17.2740	1.23386
$197$	3.11279	0.221777	0.110888	−	0.993833i	$-0.464630\pi$
$197$	3.11279	0.221777	0.110888	+	0.993833i	$0.464630\pi$
$198$	0	0
$199$	−2.78337	−0.197308	−0.0986539	−	0.995122i	$-0.531454\pi$
$199$	−2.78337	−0.197308	−0.0986539	+	0.995122i	$0.531454\pi$
$200$	−4.99222	−0.353003
$201$	0	0
$202$	−13.3267	−0.937661
$203$	−33.1621	−2.32752
$204$	0	0
$205$	−0.572076	−0.0399555
$206$	5.53443	0.385602
$207$	0	0
$208$	4.90725	0.340256
$209$	2.25834	0.156213
$210$	0	0
$211$	11.3351	0.780339	0.390169	−	0.920743i	$-0.372416\pi$
$211$	11.3351	0.780339	0.390169	+	0.920743i	$0.372416\pi$
$212$	6.91596	0.474990
$213$	0	0
$214$	3.57082	0.244096
$215$	0.217559	0.0148374
$216$	0	0
$217$	23.3197	1.58305
$218$	0.913614	0.0618777
$219$	0	0
$220$	0.0856261	0.00577291
$221$	−18.7178	−1.25910
$222$	0	0
$223$	23.9825	1.60599	0.802994	−	0.595987i	$-0.203239\pi$
$223$	23.9825	1.60599	0.802994	+	0.595987i	$0.203239\pi$
$224$	4.92686	0.329190
$225$	0	0
$226$	−0.782261	−0.0520352
$227$	27.7114	1.83927	0.919635	−	0.392774i	$-0.128484\pi$
$227$	27.7114	1.83927	0.919635	+	0.392774i	$0.128484\pi$
$228$	0	0
$229$	7.16447	0.473441	0.236721	−	0.971578i	$-0.423927\pi$
$229$	7.16447	0.473441	0.236721	+	0.971578i	$0.423927\pi$
$230$	0.00797580	0.000525908 0
$231$	0	0
$232$	−6.73087	−0.441903
$233$	9.86828	0.646492	0.323246	−	0.946315i	$-0.395226\pi$
$233$	9.86828	0.646492	0.323246	+	0.946315i	$0.395226\pi$
$234$	0	0
$235$	1.06291	0.0693368
$236$	7.13615	0.464523
$237$	0	0
$238$	−18.7927	−1.21815
$239$	−5.01270	−0.324244	−0.162122	−	0.986771i	$-0.551834\pi$
$239$	−5.01270	−0.324244	−0.162122	+	0.986771i	$0.551834\pi$
$240$	0	0
$241$	−5.18151	−0.333770	−0.166885	−	0.985976i	$-0.553371\pi$
$241$	−5.18151	−0.333770	−0.166885	+	0.985976i	$0.553371\pi$
$242$	−10.0574	−0.646514
$243$	0	0
$244$	10.1951	0.652671
$245$	1.52348	0.0973313
$246$	0	0
$247$	11.4147	0.726298
$248$	4.73318	0.300557
$249$	0	0
$250$	−0.881262	−0.0557359
$251$	22.1063	1.39534	0.697669	−	0.716420i	$-0.254220\pi$
$251$	22.1063	1.39534	0.697669	+	0.716420i	$0.254220\pi$
$252$	0	0
$253$	0.0878000	0.00551994
$254$	−0.107606	−0.00675177
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.64178	0.476681	0.238341	−	0.971182i	$-0.423397\pi$
$257$	7.64178	0.476681	0.238341	+	0.971182i	$0.423397\pi$
$258$	0	0
$259$	28.4697	1.76902
$260$	0.432793	0.0268407
$261$	0	0
$262$	3.75057	0.231711
$263$	18.5855	1.14603	0.573014	−	0.819545i	$-0.305774\pi$
$263$	18.5855	1.14603	0.573014	+	0.819545i	$0.305774\pi$
$264$	0	0
$265$	0.609952	0.0374690
$266$	11.4603	0.702677
$267$	0	0
$268$	−12.9524	−0.791197
$269$	7.65749	0.466886	0.233443	−	0.972371i	$-0.425001\pi$
$269$	7.65749	0.466886	0.233443	+	0.972371i	$0.425001\pi$
$270$	0	0
$271$	14.1832	0.861568	0.430784	−	0.902455i	$-0.358237\pi$
$271$	14.1832	0.861568	0.430784	+	0.902455i	$0.358237\pi$
$272$	−3.81432	−0.231277
$273$	0	0
$274$	2.39376	0.144612
$275$	−4.84682	−0.292274
$276$	0	0
$277$	−4.38297	−0.263347	−0.131673	−	0.991293i	$-0.542035\pi$
$277$	−4.38297	−0.263347	−0.131673	+	0.991293i	$0.542035\pi$
$278$	−19.2425	−1.15409
$279$	0	0
$280$	0.434524	0.0259678
$281$	6.79238	0.405200	0.202600	−	0.979262i	$-0.435061\pi$
$281$	6.79238	0.405200	0.202600	+	0.979262i	$0.435061\pi$
$282$	0	0
$283$	13.3655	0.794495	0.397247	−	0.917712i	$-0.369965\pi$
$283$	13.3655	0.794495	0.397247	+	0.917712i	$0.369965\pi$
$284$	−11.3961	−0.676235
$285$	0	0
$286$	4.76432	0.281720
$287$	−31.9581	−1.88643
$288$	0	0
$289$	−2.45094	−0.144173
$290$	−0.593628	−0.0348590
$291$	0	0
$292$	2.68616	0.157196
$293$	0.611700	0.0357359	0.0178680	−	0.999840i	$-0.494312\pi$
$293$	0.611700	0.0357359	0.0178680	+	0.999840i	$0.494312\pi$
$294$	0	0
$295$	0.629371	0.0366434
$296$	5.77847	0.335867
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	0.443781	0.0256645
$300$	0	0
$301$	12.1536	0.700522
$302$	−14.7457	−0.848521
$303$	0	0
$304$	2.32609	0.133410
$305$	0.899150	0.0514852
$306$	0	0
$307$	−24.7080	−1.41016	−0.705080	−	0.709128i	$-0.749089\pi$
$307$	−24.7080	−1.41016	−0.705080	+	0.709128i	$0.749089\pi$
$308$	4.78337	0.272558
$309$	0	0
$310$	0.417441	0.0237091
$311$	−18.0461	−1.02330	−0.511651	−	0.859193i	$-0.670966\pi$
$311$	−18.0461	−1.02330	−0.511651	+	0.859193i	$0.670966\pi$
$312$	0	0
$313$	−9.22092	−0.521197	−0.260598	−	0.965447i	$-0.583920\pi$
$313$	−9.22092	−0.521197	−0.260598	+	0.965447i	$0.583920\pi$
$314$	3.98919	0.225123
$315$	0	0
$316$	−10.3924	−0.584620
$317$	21.7478	1.22148	0.610740	−	0.791831i	$-0.290872\pi$
$317$	21.7478	1.22148	0.610740	+	0.791831i	$0.290872\pi$
$318$	0	0
$319$	−6.53483	−0.365880
$320$	0.0881948	0.00493024
$321$	0	0
$322$	0.445556	0.0248298
$323$	−8.87244	−0.493676
$324$	0	0
$325$	−24.4981	−1.35891
$326$	−16.6118	−0.920043
$327$	0	0
$328$	−6.48650	−0.358157
$329$	59.3780	3.27362
$330$	0	0
$331$	1.40235	0.0770799	0.0385399	−	0.999257i	$-0.487729\pi$
$331$	1.40235	0.0770799	0.0385399	+	0.999257i	$0.487729\pi$
$332$	−14.3068	−0.785186
$333$	0	0
$334$	−19.3057	−1.05636
$335$	−1.14234	−0.0624126
$336$	0	0
$337$	24.1287	1.31438	0.657188	−	0.753727i	$-0.271746\pi$
$337$	24.1287	1.31438	0.657188	+	0.753727i	$0.271746\pi$
$338$	11.0811	0.602730
$339$	0	0
$340$	−0.336403	−0.0182440
$341$	4.59532	0.248851
$342$	0	0
$343$	50.6186	2.73315
$344$	2.46680	0.133001
$345$	0	0
$346$	1.91358	0.102875
$347$	−15.7437	−0.845166	−0.422583	−	0.906324i	$-0.638877\pi$
$347$	−15.7437	−0.845166	−0.422583	+	0.906324i	$0.638877\pi$
$348$	0	0
$349$	−10.6347	−0.569261	−0.284630	−	0.958637i	$-0.591871\pi$
$349$	−10.6347	−0.569261	−0.284630	+	0.958637i	$0.591871\pi$
$350$	−24.5960	−1.31471
$351$	0	0
$352$	0.970875	0.0517478
$353$	−20.4552	−1.08872	−0.544359	−	0.838852i	$-0.683227\pi$
$353$	−20.4552	−1.08872	−0.544359	+	0.838852i	$0.683227\pi$
$354$	0	0
$355$	−1.00508	−0.0533440
$356$	14.1825	0.751674
$357$	0	0
$358$	−20.7029	−1.09418
$359$	13.8320	0.730025	0.365013	−	0.931003i	$-0.381065\pi$
$359$	13.8320	0.730025	0.365013	+	0.931003i	$0.381065\pi$
$360$	0	0
$361$	−13.5893	−0.715228
$362$	−13.0756	−0.687238
$363$	0	0
$364$	24.1773	1.26724
$365$	0.236906	0.0124002
$366$	0	0
$367$	−6.17541	−0.322354	−0.161177	−	0.986926i	$-0.551529\pi$
$367$	−6.17541	−0.322354	−0.161177	+	0.986926i	$0.551529\pi$
$368$	0.0904339	0.00471419
$369$	0	0
$370$	0.509631	0.0264944
$371$	34.0740	1.76903
$372$	0	0
$373$	−20.9319	−1.08381	−0.541907	−	0.840438i	$-0.682297\pi$
$373$	−20.9319	−1.08381	−0.541907	+	0.840438i	$0.682297\pi$
$374$	−3.70323	−0.191489
$375$	0	0
$376$	12.0519	0.621529
$377$	−33.0300	−1.70113
$378$	0	0
$379$	−20.4879	−1.05239	−0.526196	−	0.850363i	$-0.676382\pi$
$379$	−20.4879	−1.05239	−0.526196	+	0.850363i	$0.676382\pi$
$380$	0.205149	0.0105239
$381$	0	0
$382$	22.7732	1.16518
$383$	−36.0988	−1.84456	−0.922281	−	0.386520i	$-0.873677\pi$
$383$	−36.0988	−1.84456	−0.922281	+	0.386520i	$0.873677\pi$
$384$	0	0
$385$	0.421868	0.0215004
$386$	−6.80693	−0.346464
$387$	0	0
$388$	9.41121	0.477782
$389$	−2.12718	−0.107852	−0.0539262	−	0.998545i	$-0.517174\pi$
$389$	−2.12718	−0.107852	−0.0539262	+	0.998545i	$0.517174\pi$
$390$	0	0
$391$	−0.344944	−0.0174446
$392$	17.2740	0.872468
$393$	0	0
$394$	3.11279	0.156820
$395$	−0.916559	−0.0461171
$396$	0	0
$397$	32.9751	1.65497	0.827487	−	0.561484i	$-0.189770\pi$
$397$	32.9751	1.65497	0.827487	+	0.561484i	$0.189770\pi$
$398$	−2.78337	−0.139518
$399$	0	0
$400$	−4.99222	−0.249611
$401$	33.8024	1.68801	0.844006	−	0.536333i	$-0.180191\pi$
$401$	33.8024	1.68801	0.844006	+	0.536333i	$0.180191\pi$
$402$	0	0
$403$	23.2269	1.15701
$404$	−13.3267	−0.663026
$405$	0	0
$406$	−33.1621	−1.64581
$407$	5.61017	0.278086
$408$	0	0
$409$	34.0554	1.68393	0.841965	−	0.539533i	$-0.181399\pi$
$409$	34.0554	1.68393	0.841965	+	0.539533i	$0.181399\pi$
$410$	−0.572076	−0.0282528
$411$	0	0
$412$	5.53443	0.272662
$413$	35.1588	1.73005
$414$	0	0
$415$	−1.26178	−0.0619384
$416$	4.90725	0.240598
$417$	0	0
$418$	2.25834	0.110459
$419$	10.4507	0.510552	0.255276	−	0.966868i	$-0.417834\pi$
$419$	10.4507	0.510552	0.255276	+	0.966868i	$0.417834\pi$
$420$	0	0
$421$	−17.4742	−0.851641	−0.425821	−	0.904808i	$-0.640015\pi$
$421$	−17.4742	−0.851641	−0.425821	+	0.904808i	$0.640015\pi$
$422$	11.3351	0.551783
$423$	0	0
$424$	6.91596	0.335869
$425$	19.0419	0.923670
$426$	0	0
$427$	50.2296	2.43078
$428$	3.57082	0.172602
$429$	0	0
$430$	0.217559	0.0104916
$431$	20.0952	0.967953	0.483977	−	0.875081i	$-0.339192\pi$
$431$	20.0952	0.967953	0.483977	+	0.875081i	$0.339192\pi$
$432$	0	0
$433$	25.0729	1.20493	0.602463	−	0.798147i	$-0.294186\pi$
$433$	25.0729	1.20493	0.602463	+	0.798147i	$0.294186\pi$
$434$	23.3197	1.11938
$435$	0	0
$436$	0.913614	0.0437542
$437$	0.210357	0.0100627
$438$	0	0
$439$	6.01704	0.287178	0.143589	−	0.989637i	$-0.454136\pi$
$439$	6.01704	0.287178	0.143589	+	0.989637i	$0.454136\pi$
$440$	0.0856261	0.00408206
$441$	0	0
$442$	−18.7178	−0.890316
$443$	15.2612	0.725080	0.362540	−	0.931968i	$-0.381910\pi$
$443$	15.2612	0.725080	0.362540	+	0.931968i	$0.381910\pi$
$444$	0	0
$445$	1.25083	0.0592949
$446$	23.9825	1.13560
$447$	0	0
$448$	4.92686	0.232772
$449$	−25.2828	−1.19317	−0.596585	−	0.802550i	$-0.703476\pi$
$449$	−25.2828	−1.19317	−0.596585	+	0.802550i	$0.703476\pi$
$450$	0	0
$451$	−6.29758	−0.296542
$452$	−0.782261	−0.0367944
$453$	0	0
$454$	27.7114	1.30056
$455$	2.13231	0.0999644
$456$	0	0
$457$	−23.9133	−1.11862	−0.559309	−	0.828959i	$-0.688933\pi$
$457$	−23.9133	−1.11862	−0.559309	+	0.828959i	$0.688933\pi$
$458$	7.16447	0.334774
$459$	0	0
$460$	0.00797580	0.000371873 0
$461$	−20.8889	−0.972895	−0.486447	−	0.873710i	$-0.661707\pi$
$461$	−20.8889	−0.972895	−0.486447	+	0.873710i	$0.661707\pi$
$462$	0	0
$463$	−6.09906	−0.283447	−0.141724	−	0.989906i	$-0.545264\pi$
$463$	−6.09906	−0.283447	−0.141724	+	0.989906i	$0.545264\pi$
$464$	−6.73087	−0.312473
$465$	0	0
$466$	9.86828	0.457139
$467$	24.8312	1.14905	0.574525	−	0.818487i	$-0.305187\pi$
$467$	24.8312	1.14905	0.574525	+	0.818487i	$0.305187\pi$
$468$	0	0
$469$	−63.8149	−2.94670
$470$	1.06291	0.0490285
$471$	0	0
$472$	7.13615	0.328468
$473$	2.39496	0.110120
$474$	0	0
$475$	−11.6123	−0.532811
$476$	−18.7927	−0.861360
$477$	0	0
$478$	−5.01270	−0.229275
$479$	−17.2202	−0.786813	−0.393406	−	0.919365i	$-0.628703\pi$
$479$	−17.2202	−0.786813	−0.393406	+	0.919365i	$0.628703\pi$
$480$	0	0
$481$	28.3564	1.29294
$482$	−5.18151	−0.236011
$483$	0	0
$484$	−10.0574	−0.457155
$485$	0.830019	0.0376892
$486$	0	0
$487$	−24.7651	−1.12221	−0.561106	−	0.827744i	$-0.689624\pi$
$487$	−24.7651	−1.12221	−0.561106	+	0.827744i	$0.689624\pi$
$488$	10.1951	0.461508
$489$	0	0
$490$	1.52348	0.0688236
$491$	−30.4005	−1.37195	−0.685977	−	0.727624i	$-0.740625\pi$
$491$	−30.4005	−1.37195	−0.685977	+	0.727624i	$0.740625\pi$
$492$	0	0
$493$	25.6737	1.15629
$494$	11.4147	0.513571
$495$	0	0
$496$	4.73318	0.212526
$497$	−56.1471	−2.51854
$498$	0	0
$499$	34.5454	1.54647	0.773233	−	0.634122i	$-0.218638\pi$
$499$	34.5454	1.54647	0.773233	+	0.634122i	$0.218638\pi$
$500$	−0.881262	−0.0394112
$501$	0	0
$502$	22.1063	0.986654
$503$	6.32713	0.282113	0.141056	−	0.990002i	$-0.454950\pi$
$503$	6.32713	0.282113	0.141056	+	0.990002i	$0.454950\pi$
$504$	0	0
$505$	−1.17534	−0.0523020
$506$	0.0878000	0.00390318
$507$	0	0
$508$	−0.107606	−0.00477422
$509$	−24.6816	−1.09399	−0.546996	−	0.837135i	$-0.684229\pi$
$509$	−24.6816	−1.09399	−0.546996	+	0.837135i	$0.684229\pi$
$510$	0	0
$511$	13.2344	0.585454
$512$	1.00000	0.0441942
$513$	0	0
$514$	7.64178	0.337065
$515$	0.488107	0.0215086
$516$	0	0
$517$	11.7009	0.514604
$518$	28.4697	1.25089
$519$	0	0
$520$	0.432793	0.0189792
$521$	−29.9820	−1.31354	−0.656769	−	0.754092i	$-0.728077\pi$
$521$	−29.9820	−1.31354	−0.656769	+	0.754092i	$0.728077\pi$
$522$	0	0
$523$	−20.1006	−0.878939	−0.439470	−	0.898257i	$-0.644834\pi$
$523$	−20.1006	−0.878939	−0.439470	+	0.898257i	$0.644834\pi$
$524$	3.75057	0.163844
$525$	0	0
$526$	18.5855	0.810365
$527$	−18.0539	−0.786438
$528$	0	0
$529$	−22.9918	−0.999644
$530$	0.609952	0.0264946
$531$	0	0
$532$	11.4603	0.496867
$533$	−31.8309	−1.37875
$534$	0	0
$535$	0.314928	0.0136155
$536$	−12.9524	−0.559460
$537$	0	0
$538$	7.65749	0.330138
$539$	16.7709	0.722373
$540$	0	0
$541$	−3.77343	−0.162232	−0.0811161	−	0.996705i	$-0.525848\pi$
$541$	−3.77343	−0.162232	−0.0811161	+	0.996705i	$0.525848\pi$
$542$	14.1832	0.609220
$543$	0	0
$544$	−3.81432	−0.163538
$545$	0.0805760	0.00345149
$546$	0	0
$547$	31.4264	1.34370	0.671848	−	0.740689i	$-0.265501\pi$
$547$	31.4264	1.34370	0.671848	+	0.740689i	$0.265501\pi$
$548$	2.39376	0.102256
$549$	0	0
$550$	−4.84682	−0.206669
$551$	−15.6566	−0.666993
$552$	0	0
$553$	−51.2021	−2.17734
$554$	−4.38297	−0.186214
$555$	0	0
$556$	−19.2425	−0.816066
$557$	−9.33409	−0.395498	−0.197749	−	0.980253i	$-0.563363\pi$
$557$	−9.33409	−0.395498	−0.197749	+	0.980253i	$0.563363\pi$
$558$	0	0
$559$	12.1052	0.511996
$560$	0.434524	0.0183620
$561$	0	0
$562$	6.79238	0.286519
$563$	30.6418	1.29140	0.645699	−	0.763592i	$-0.276566\pi$
$563$	30.6418	1.29140	0.645699	+	0.763592i	$0.276566\pi$
$564$	0	0
$565$	−0.0689913	−0.00290249
$566$	13.3655	0.561793
$567$	0	0
$568$	−11.3961	−0.478171
$569$	16.4797	0.690865	0.345433	−	0.938444i	$-0.387732\pi$
$569$	16.4797	0.690865	0.345433	+	0.938444i	$0.387732\pi$
$570$	0	0
$571$	14.8993	0.623515	0.311757	−	0.950162i	$-0.399082\pi$
$571$	14.8993	0.623515	0.311757	+	0.950162i	$0.399082\pi$
$572$	4.76432	0.199206
$573$	0	0
$574$	−31.9581	−1.33391
$575$	−0.451466	−0.0188274
$576$	0	0
$577$	−11.7369	−0.488614	−0.244307	−	0.969698i	$-0.578561\pi$
$577$	−11.7369	−0.488614	−0.244307	+	0.969698i	$0.578561\pi$
$578$	−2.45094	−0.101946
$579$	0	0
$580$	−0.593628	−0.0246490
$581$	−70.4875	−2.92431
$582$	0	0
$583$	6.71453	0.278088
$584$	2.68616	0.111154
$585$	0	0
$586$	0.611700	0.0252691
$587$	1.99541	0.0823592	0.0411796	−	0.999152i	$-0.486888\pi$
$587$	1.99541	0.0823592	0.0411796	+	0.999152i	$0.486888\pi$
$588$	0	0
$589$	11.0098	0.453650
$590$	0.629371	0.0259108
$591$	0	0
$592$	5.77847	0.237494
$593$	2.57712	0.105830	0.0529148	−	0.998599i	$-0.483149\pi$
$593$	2.57712	0.105830	0.0529148	+	0.998599i	$0.483149\pi$
$594$	0	0
$595$	−1.65741	−0.0679473
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	0.443781	0.0181476
$599$	−34.7954	−1.42170	−0.710851	−	0.703343i	$-0.751690\pi$
$599$	−34.7954	−1.42170	−0.710851	+	0.703343i	$0.751690\pi$
$600$	0	0
$601$	−0.827051	−0.0337361	−0.0168680	−	0.999858i	$-0.505370\pi$
$601$	−0.827051	−0.0337361	−0.0168680	+	0.999858i	$0.505370\pi$
$602$	12.1536	0.495344
$603$	0	0
$604$	−14.7457	−0.599995
$605$	−0.887010	−0.0360621
$606$	0	0
$607$	−2.40715	−0.0977032	−0.0488516	−	0.998806i	$-0.515556\pi$
$607$	−2.40715	−0.0977032	−0.0488516	+	0.998806i	$0.515556\pi$
$608$	2.32609	0.0943352
$609$	0	0
$610$	0.899150	0.0364055
$611$	59.1416	2.39261
$612$	0	0
$613$	16.4454	0.664225	0.332113	−	0.943240i	$-0.392239\pi$
$613$	16.4454	0.664225	0.332113	+	0.943240i	$0.392239\pi$
$614$	−24.7080	−0.997133
$615$	0	0
$616$	4.78337	0.192727
$617$	17.2514	0.694516	0.347258	−	0.937770i	$-0.387113\pi$
$617$	17.2514	0.694516	0.347258	+	0.937770i	$0.387113\pi$
$618$	0	0
$619$	19.5623	0.786274	0.393137	−	0.919480i	$-0.371390\pi$
$619$	19.5623	0.786274	0.393137	+	0.919480i	$0.371390\pi$
$620$	0.417441	0.0167648
$621$	0	0
$622$	−18.0461	−0.723584
$623$	69.8755	2.79950
$624$	0	0
$625$	24.8834	0.995335
$626$	−9.22092	−0.368542
$627$	0	0
$628$	3.98919	0.159186
$629$	−22.0409	−0.878830
$630$	0	0
$631$	−24.2848	−0.966761	−0.483381	−	0.875410i	$-0.660591\pi$
$631$	−24.2848	−0.966761	−0.483381	+	0.875410i	$0.660591\pi$
$632$	−10.3924	−0.413389
$633$	0	0
$634$	21.7478	0.863717
$635$	−0.00949024	−0.000376609 0
$636$	0	0
$637$	84.7677	3.35862
$638$	−6.53483	−0.258716
$639$	0	0
$640$	0.0881948	0.00348620
$641$	−18.9752	−0.749474	−0.374737	−	0.927131i	$-0.622267\pi$
$641$	−18.9752	−0.749474	−0.374737	+	0.927131i	$0.622267\pi$
$642$	0	0
$643$	−8.78985	−0.346638	−0.173319	−	0.984866i	$-0.555449\pi$
$643$	−8.78985	−0.346638	−0.173319	+	0.984866i	$0.555449\pi$
$644$	0.445556	0.0175573
$645$	0	0
$646$	−8.87244	−0.349082
$647$	−48.1946	−1.89472	−0.947362	−	0.320164i	$-0.896262\pi$
$647$	−48.1946	−1.89472	−0.947362	+	0.320164i	$0.896262\pi$
$648$	0	0
$649$	6.92830	0.271960
$650$	−24.4981	−0.960893
$651$	0	0
$652$	−16.6118	−0.650569
$653$	−49.4358	−1.93457	−0.967287	−	0.253685i	$-0.918357\pi$
$653$	−49.4358	−1.93457	−0.967287	+	0.253685i	$0.918357\pi$
$654$	0	0
$655$	0.330781	0.0129247
$656$	−6.48650	−0.253255
$657$	0	0
$658$	59.3780	2.31480
$659$	2.25528	0.0878534	0.0439267	−	0.999035i	$-0.486013\pi$
$659$	2.25528	0.0878534	0.0439267	+	0.999035i	$0.486013\pi$
$660$	0	0
$661$	−47.9401	−1.86465	−0.932327	−	0.361616i	$-0.882225\pi$
$661$	−47.9401	−1.86465	−0.932327	+	0.361616i	$0.882225\pi$
$662$	1.40235	0.0545037
$663$	0	0
$664$	−14.3068	−0.555210
$665$	1.01074	0.0391948
$666$	0	0
$667$	−0.608699	−0.0235689
$668$	−19.3057	−0.746959
$669$	0	0
$670$	−1.14234	−0.0441324
$671$	9.89812	0.382113
$672$	0	0
$673$	−1.24697	−0.0480670	−0.0240335	−	0.999711i	$-0.507651\pi$
$673$	−1.24697	−0.0480670	−0.0240335	+	0.999711i	$0.507651\pi$
$674$	24.1287	0.929404
$675$	0	0
$676$	11.0811	0.426195
$677$	−16.7283	−0.642923	−0.321461	−	0.946923i	$-0.604174\pi$
$677$	−16.7283	−0.642923	−0.321461	+	0.946923i	$0.604174\pi$
$678$	0	0
$679$	46.3677	1.77943
$680$	−0.336403	−0.0129005
$681$	0	0
$682$	4.59532	0.175964
$683$	17.4482	0.667636	0.333818	−	0.942638i	$-0.391663\pi$
$683$	17.4482	0.667636	0.333818	+	0.942638i	$0.391663\pi$
$684$	0	0
$685$	0.211117	0.00806635
$686$	50.6186	1.93263
$687$	0	0
$688$	2.46680	0.0940459
$689$	33.9383	1.29295
$690$	0	0
$691$	5.55660	0.211383	0.105691	−	0.994399i	$-0.466294\pi$
$691$	5.55660	0.211383	0.105691	+	0.994399i	$0.466294\pi$
$692$	1.91358	0.0727436
$693$	0	0
$694$	−15.7437	−0.597623
$695$	−1.69709	−0.0643743
$696$	0	0
$697$	24.7416	0.937156
$698$	−10.6347	−0.402528
$699$	0	0
$700$	−24.5960	−0.929641
$701$	−22.0085	−0.831250	−0.415625	−	0.909536i	$-0.636437\pi$
$701$	−22.0085	−0.831250	−0.415625	+	0.909536i	$0.636437\pi$
$702$	0	0
$703$	13.4412	0.506945
$704$	0.970875	0.0365912
$705$	0	0
$706$	−20.4552	−0.769840
$707$	−65.6587	−2.46935
$708$	0	0
$709$	−38.9532	−1.46292	−0.731459	−	0.681885i	$-0.761160\pi$
$709$	−38.9532	−1.46292	−0.731459	+	0.681885i	$0.761160\pi$
$710$	−1.00508	−0.0377199
$711$	0	0
$712$	14.1825	0.531513
$713$	0.428039	0.0160302
$714$	0	0
$715$	0.420188	0.0157141
$716$	−20.7029	−0.773704
$717$	0	0
$718$	13.8320	0.516206
$719$	−29.8868	−1.11459	−0.557296	−	0.830314i	$-0.688161\pi$
$719$	−29.8868	−1.11459	−0.557296	+	0.830314i	$0.688161\pi$
$720$	0	0
$721$	27.2674	1.01549
$722$	−13.5893	−0.505742
$723$	0	0
$724$	−13.0756	−0.485951
$725$	33.6020	1.24795
$726$	0	0
$727$	−11.2047	−0.415561	−0.207780	−	0.978176i	$-0.566624\pi$
$727$	−11.2047	−0.415561	−0.207780	+	0.978176i	$0.566624\pi$
$728$	24.1773	0.896072
$729$	0	0
$730$	0.236906	0.00876827
$731$	−9.40918	−0.348011
$732$	0	0
$733$	39.3537	1.45356	0.726780	−	0.686870i	$-0.241016\pi$
$733$	39.3537	1.45356	0.726780	+	0.686870i	$0.241016\pi$
$734$	−6.17541	−0.227938
$735$	0	0
$736$	0.0904339	0.00333344
$737$	−12.5752	−0.463214
$738$	0	0
$739$	−40.7948	−1.50066	−0.750331	−	0.661063i	$-0.770106\pi$
$739$	−40.7948	−1.50066	−0.750331	+	0.661063i	$0.770106\pi$
$740$	0.509631	0.0187344
$741$	0	0
$742$	34.0740	1.25090
$743$	18.6384	0.683775	0.341888	−	0.939741i	$-0.388934\pi$
$743$	18.6384	0.683775	0.341888	+	0.939741i	$0.388934\pi$
$744$	0	0
$745$	−0.0881948	−0.00323121
$746$	−20.9319	−0.766372
$747$	0	0
$748$	−3.70323	−0.135403
$749$	17.5929	0.642832
$750$	0	0
$751$	30.4976	1.11287	0.556437	−	0.830890i	$-0.312168\pi$
$751$	30.4976	1.11287	0.556437	+	0.830890i	$0.312168\pi$
$752$	12.0519	0.439487
$753$	0	0
$754$	−33.0300	−1.20288
$755$	−1.30050	−0.0473299
$756$	0	0
$757$	28.2758	1.02770	0.513850	−	0.857880i	$-0.328219\pi$
$757$	28.2758	1.02770	0.513850	+	0.857880i	$0.328219\pi$
$758$	−20.4879	−0.744153
$759$	0	0
$760$	0.205149	0.00744152
$761$	−13.3321	−0.483288	−0.241644	−	0.970365i	$-0.577687\pi$
$761$	−13.3321	−0.483288	−0.241644	+	0.970365i	$0.577687\pi$
$762$	0	0
$763$	4.50125	0.162956
$764$	22.7732	0.823904
$765$	0	0
$766$	−36.0988	−1.30430
$767$	35.0188	1.26446
$768$	0	0
$769$	10.1745	0.366901	0.183450	−	0.983029i	$-0.441273\pi$
$769$	10.1745	0.366901	0.183450	+	0.983029i	$0.441273\pi$
$770$	0.421868	0.0152031
$771$	0	0
$772$	−6.80693	−0.244987
$773$	−21.8027	−0.784188	−0.392094	−	0.919925i	$-0.628249\pi$
$773$	−21.8027	−0.784188	−0.392094	+	0.919925i	$0.628249\pi$
$774$	0	0
$775$	−23.6291	−0.848781
$776$	9.41121	0.337843
$777$	0	0
$778$	−2.12718	−0.0762631
$779$	−15.0882	−0.540590
$780$	0	0
$781$	−11.0642	−0.395908
$782$	−0.344944	−0.0123352
$783$	0	0
$784$	17.2740	0.616928
$785$	0.351826	0.0125572
$786$	0	0
$787$	15.5986	0.556030	0.278015	−	0.960577i	$-0.410324\pi$
$787$	15.5986	0.556030	0.278015	+	0.960577i	$0.410324\pi$
$788$	3.11279	0.110888
$789$	0	0
$790$	−0.916559	−0.0326097
$791$	−3.85409	−0.137036
$792$	0	0
$793$	50.0296	1.77660
$794$	32.9751	1.17024
$795$	0	0
$796$	−2.78337	−0.0986539
$797$	−15.1466	−0.536520	−0.268260	−	0.963347i	$-0.586449\pi$
$797$	−15.1466	−0.536520	−0.268260	+	0.963347i	$0.586449\pi$
$798$	0	0
$799$	−45.9698	−1.62629
$800$	−4.99222	−0.176502
$801$	0	0
$802$	33.8024	1.19361
$803$	2.60793	0.0920318
$804$	0	0
$805$	0.0392957	0.00138499
$806$	23.2269	0.818131
$807$	0	0
$808$	−13.3267	−0.468831
$809$	−14.7874	−0.519899	−0.259949	−	0.965622i	$-0.583706\pi$
$809$	−14.7874	−0.519899	−0.259949	+	0.965622i	$0.583706\pi$
$810$	0	0
$811$	−44.2078	−1.55235	−0.776174	−	0.630519i	$-0.782842\pi$
$811$	−44.2078	−1.55235	−0.776174	+	0.630519i	$0.782842\pi$
$812$	−33.1621	−1.16376
$813$	0	0
$814$	5.61017	0.196636
$815$	−1.46507	−0.0513193
$816$	0	0
$817$	5.73799	0.200747
$818$	34.0554	1.19072
$819$	0	0
$820$	−0.572076	−0.0199778
$821$	33.7451	1.17771	0.588856	−	0.808238i	$-0.299579\pi$
$821$	33.7451	1.17771	0.588856	+	0.808238i	$0.299579\pi$
$822$	0	0
$823$	46.2281	1.61141	0.805706	−	0.592316i	$-0.201786\pi$
$823$	46.2281	1.61141	0.805706	+	0.592316i	$0.201786\pi$
$824$	5.53443	0.192801
$825$	0	0
$826$	35.1588	1.22333
$827$	−10.4540	−0.363522	−0.181761	−	0.983343i	$-0.558180\pi$
$827$	−10.4540	−0.363522	−0.181761	+	0.983343i	$0.558180\pi$
$828$	0	0
$829$	15.5590	0.540386	0.270193	−	0.962806i	$-0.412912\pi$
$829$	15.5590	0.540386	0.270193	+	0.962806i	$0.412912\pi$
$830$	−1.26178	−0.0437971
$831$	0	0
$832$	4.90725	0.170128
$833$	−65.8886	−2.28290
$834$	0	0
$835$	−1.70266	−0.0589230
$836$	2.25834	0.0781063
$837$	0	0
$838$	10.4507	0.361015
$839$	−45.2713	−1.56294	−0.781469	−	0.623944i	$-0.785529\pi$
$839$	−45.2713	−1.56294	−0.781469	+	0.623944i	$0.785529\pi$
$840$	0	0
$841$	16.3046	0.562228
$842$	−17.4742	−0.602201
$843$	0	0
$844$	11.3351	0.390169
$845$	0.977292	0.0336199
$846$	0	0
$847$	−49.5515	−1.70261
$848$	6.91596	0.237495
$849$	0	0
$850$	19.0419	0.653133
$851$	0.522569	0.0179134
$852$	0	0
$853$	9.79856	0.335496	0.167748	−	0.985830i	$-0.446350\pi$
$853$	9.79856	0.335496	0.167748	+	0.985830i	$0.446350\pi$
$854$	50.2296	1.71882
$855$	0	0
$856$	3.57082	0.122048
$857$	7.50138	0.256242	0.128121	−	0.991759i	$-0.459105\pi$
$857$	7.50138	0.256242	0.128121	+	0.991759i	$0.459105\pi$
$858$	0	0
$859$	−38.7068	−1.32066	−0.660330	−	0.750976i	$-0.729583\pi$
$859$	−38.7068	−1.32066	−0.660330	+	0.750976i	$0.729583\pi$
$860$	0.217559	0.00741870
$861$	0	0
$862$	20.0952	0.684446
$863$	−56.6219	−1.92743	−0.963717	−	0.266927i	$-0.913992\pi$
$863$	−56.6219	−1.92743	−0.963717	+	0.266927i	$0.913992\pi$
$864$	0	0
$865$	0.168768	0.00573829
$866$	25.0729	0.852011
$867$	0	0
$868$	23.3197	0.791523
$869$	−10.0898	−0.342272
$870$	0	0
$871$	−63.5608	−2.15368
$872$	0.913614	0.0309389
$873$	0	0
$874$	0.210357	0.00711543
$875$	−4.34186	−0.146782
$876$	0	0
$877$	34.6454	1.16989	0.584946	−	0.811072i	$-0.301116\pi$
$877$	34.6454	1.16989	0.584946	+	0.811072i	$0.301116\pi$
$878$	6.01704	0.203065
$879$	0	0
$880$	0.0856261	0.00288645
$881$	46.4275	1.56418	0.782092	−	0.623163i	$-0.214153\pi$
$881$	46.4275	1.56418	0.782092	+	0.623163i	$0.214153\pi$
$882$	0	0
$883$	25.3416	0.852813	0.426407	−	0.904532i	$-0.359779\pi$
$883$	25.3416	0.852813	0.426407	+	0.904532i	$0.359779\pi$
$884$	−18.7178	−0.629548
$885$	0	0
$886$	15.2612	0.512709
$887$	−27.5977	−0.926639	−0.463320	−	0.886191i	$-0.653342\pi$
$887$	−27.5977	−0.926639	−0.463320	+	0.886191i	$0.653342\pi$
$888$	0	0
$889$	−0.530158	−0.0177809
$890$	1.25083	0.0419278
$891$	0	0
$892$	23.9825	0.802994
$893$	28.0337	0.938113
$894$	0	0
$895$	−1.82589	−0.0610327
$896$	4.92686	0.164595
$897$	0	0
$898$	−25.2828	−0.843698
$899$	−31.8584	−1.06254
$900$	0	0
$901$	−26.3797	−0.878836
$902$	−6.29758	−0.209687
$903$	0	0
$904$	−0.782261	−0.0260176
$905$	−1.15320	−0.0383336
$906$	0	0
$907$	20.1531	0.669173	0.334587	−	0.942365i	$-0.391403\pi$
$907$	20.1531	0.669173	0.334587	+	0.942365i	$0.391403\pi$
$908$	27.7114	0.919635
$909$	0	0
$910$	2.13231	0.0706855
$911$	13.5053	0.447452	0.223726	−	0.974652i	$-0.428178\pi$
$911$	13.5053	0.447452	0.223726	+	0.974652i	$0.428178\pi$
$912$	0	0
$913$	−13.8901	−0.459694
$914$	−23.9133	−0.790982
$915$	0	0
$916$	7.16447	0.236721
$917$	18.4786	0.610216
$918$	0	0
$919$	2.91103	0.0960260	0.0480130	−	0.998847i	$-0.484711\pi$
$919$	2.91103	0.0960260	0.0480130	+	0.998847i	$0.484711\pi$
$920$	0.00797580	0.000262954 0
$921$	0	0
$922$	−20.8889	−0.687940
$923$	−55.9236	−1.84075
$924$	0	0
$925$	−28.8474	−0.948496
$926$	−6.09906	−0.200428
$927$	0	0
$928$	−6.73087	−0.220952
$929$	1.46556	0.0480833	0.0240417	−	0.999711i	$-0.492347\pi$
$929$	1.46556	0.0480833	0.0240417	+	0.999711i	$0.492347\pi$
$930$	0	0
$931$	40.1808	1.31687
$932$	9.86828	0.323246
$933$	0	0
$934$	24.8312	0.812501
$935$	−0.326605	−0.0106811
$936$	0	0
$937$	−46.3585	−1.51447	−0.757233	−	0.653145i	$-0.773449\pi$
$937$	−46.3585	−1.51447	−0.757233	+	0.653145i	$0.773449\pi$
$938$	−63.8149	−2.08363
$939$	0	0
$940$	1.06291	0.0346684
$941$	11.0132	0.359019	0.179509	−	0.983756i	$-0.442549\pi$
$941$	11.0132	0.359019	0.179509	+	0.983756i	$0.442549\pi$
$942$	0	0
$943$	−0.586600	−0.0191023
$944$	7.13615	0.232262
$945$	0	0
$946$	2.39496	0.0778667
$947$	23.2761	0.756370	0.378185	−	0.925730i	$-0.376548\pi$
$947$	23.2761	0.756370	0.378185	+	0.925730i	$0.376548\pi$
$948$	0	0
$949$	13.1817	0.427895
$950$	−11.6123	−0.376754
$951$	0	0
$952$	−18.7927	−0.609073
$953$	52.7100	1.70744	0.853722	−	0.520728i	$-0.174340\pi$
$953$	52.7100	1.70744	0.853722	+	0.520728i	$0.174340\pi$
$954$	0	0
$955$	2.00847	0.0649927
$956$	−5.01270	−0.162122
$957$	0	0
$958$	−17.2202	−0.556361
$959$	11.7937	0.380839
$960$	0	0
$961$	−8.59705	−0.277324
$962$	28.3564	0.914246
$963$	0	0
$964$	−5.18151	−0.166885
$965$	−0.600336	−0.0193255
$966$	0	0
$967$	34.6587	1.11455	0.557274	−	0.830329i	$-0.311847\pi$
$967$	34.6587	1.11455	0.557274	+	0.830329i	$0.311847\pi$
$968$	−10.0574	−0.323257
$969$	0	0
$970$	0.830019	0.0266503
$971$	43.9859	1.41158	0.705788	−	0.708423i	$-0.250593\pi$
$971$	43.9859	1.41158	0.705788	+	0.708423i	$0.250593\pi$
$972$	0	0
$973$	−94.8054	−3.03932
$974$	−24.7651	−0.793524
$975$	0	0
$976$	10.1951	0.326336
$977$	34.0033	1.08786	0.543931	−	0.839130i	$-0.316935\pi$
$977$	34.0033	1.08786	0.543931	+	0.839130i	$0.316935\pi$
$978$	0	0
$979$	13.7695	0.440074
$980$	1.52348	0.0486657
$981$	0	0
$982$	−30.4005	−0.970117
$983$	−23.1598	−0.738683	−0.369341	−	0.929294i	$-0.620417\pi$
$983$	−23.1598	−0.738683	−0.369341	+	0.929294i	$0.620417\pi$
$984$	0	0
$985$	0.274531	0.00874730
$986$	25.6737	0.817618
$987$	0	0
$988$	11.4147	0.363149
$989$	0.223082	0.00709361
$990$	0	0
$991$	−23.9435	−0.760589	−0.380295	−	0.924865i	$-0.624177\pi$
$991$	−23.9435	−0.760589	−0.380295	+	0.924865i	$0.624177\pi$
$992$	4.73318	0.150278
$993$	0	0
$994$	−56.1471	−1.78088
$995$	−0.245479	−0.00778219
$996$	0	0
$997$	−4.08595	−0.129403	−0.0647016	−	0.997905i	$-0.520610\pi$
$997$	−4.08595	−0.129403	−0.0647016	+	0.997905i	$0.520610\pi$
$998$	34.5454	1.09352
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.8	yes	16
3.2	odd	2		8046.2.a.s.1.9	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.9	✓	16	3.2	odd	2
8046.2.a.t.1.8	yes	16	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.0881948 q^{5} +4.92686 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.0881948 q^{5} +4.92686 q^{7} +1.00000 q^{8} +0.0881948 q^{10} +0.970875 q^{11} +4.90725 q^{13} +4.92686 q^{14} +1.00000 q^{16} -3.81432 q^{17} +2.32609 q^{19} +0.0881948 q^{20} +0.970875 q^{22} +0.0904339 q^{23} -4.99222 q^{25} +4.90725 q^{26} +4.92686 q^{28} -6.73087 q^{29} +4.73318 q^{31} +1.00000 q^{32} -3.81432 q^{34} +0.434524 q^{35} +5.77847 q^{37} +2.32609 q^{38} +0.0881948 q^{40} -6.48650 q^{41} +2.46680 q^{43} +0.970875 q^{44} +0.0904339 q^{46} +12.0519 q^{47} +17.2740 q^{49} -4.99222 q^{50} +4.90725 q^{52} +6.91596 q^{53} +0.0856261 q^{55} +4.92686 q^{56} -6.73087 q^{58} +7.13615 q^{59} +10.1951 q^{61} +4.73318 q^{62} +1.00000 q^{64} +0.432793 q^{65} -12.9524 q^{67} -3.81432 q^{68} +0.434524 q^{70} -11.3961 q^{71} +2.68616 q^{73} +5.77847 q^{74} +2.32609 q^{76} +4.78337 q^{77} -10.3924 q^{79} +0.0881948 q^{80} -6.48650 q^{82} -14.3068 q^{83} -0.336403 q^{85} +2.46680 q^{86} +0.970875 q^{88} +14.1825 q^{89} +24.1773 q^{91} +0.0904339 q^{92} +12.0519 q^{94} +0.205149 q^{95} +9.41121 q^{97} +17.2740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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