LMFDB - Embedded newform 8046.2.a.l.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:	$12$
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} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
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			$-1.12518$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.12518 q^{5} -0.686663 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.12518 q^{5} -0.686663 q^{7} -1.00000 q^{8} +1.12518 q^{10} +3.05448 q^{11} -3.99040 q^{13} +0.686663 q^{14} +1.00000 q^{16} +5.80043 q^{17} +8.32168 q^{19} -1.12518 q^{20} -3.05448 q^{22} -2.20462 q^{23} -3.73397 q^{25} +3.99040 q^{26} -0.686663 q^{28} +6.02040 q^{29} +8.16044 q^{31} -1.00000 q^{32} -5.80043 q^{34} +0.772621 q^{35} +4.59174 q^{37} -8.32168 q^{38} +1.12518 q^{40} +5.82690 q^{41} +0.944768 q^{43} +3.05448 q^{44} +2.20462 q^{46} -6.19531 q^{47} -6.52849 q^{49} +3.73397 q^{50} -3.99040 q^{52} -8.18515 q^{53} -3.43684 q^{55} +0.686663 q^{56} -6.02040 q^{58} +8.39608 q^{59} -14.1320 q^{61} -8.16044 q^{62} +1.00000 q^{64} +4.48992 q^{65} +6.33902 q^{67} +5.80043 q^{68} -0.772621 q^{70} +2.82097 q^{71} -3.33968 q^{73} -4.59174 q^{74} +8.32168 q^{76} -2.09740 q^{77} -8.49208 q^{79} -1.12518 q^{80} -5.82690 q^{82} +6.34293 q^{83} -6.52653 q^{85} -0.944768 q^{86} -3.05448 q^{88} +6.07267 q^{89} +2.74006 q^{91} -2.20462 q^{92} +6.19531 q^{94} -9.36339 q^{95} -11.5459 q^{97} +6.52849 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * 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
$5$	−1.12518	−0.503196	−0.251598	−	0.967832i	$-0.580956\pi$
$5$	−1.12518	−0.503196	−0.251598	+	0.967832i	$0.580956\pi$
$6$	0	0
$7$	−0.686663	−0.259534	−0.129767	−	0.991544i	$-0.541423\pi$
$7$	−0.686663	−0.259534	−0.129767	+	0.991544i	$0.541423\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.12518	0.355814
$11$	3.05448	0.920960	0.460480	−	0.887670i	$-0.347677\pi$
$11$	3.05448	0.920960	0.460480	+	0.887670i	$0.347677\pi$
$12$	0	0
$13$	−3.99040	−1.10674	−0.553369	−	0.832936i	$-0.686658\pi$
$13$	−3.99040	−1.10674	−0.553369	+	0.832936i	$0.686658\pi$
$14$	0.686663	0.183519
$15$	0	0
$16$	1.00000	0.250000
$17$	5.80043	1.40681	0.703405	−	0.710789i	$-0.251662\pi$
$17$	5.80043	1.40681	0.703405	+	0.710789i	$0.251662\pi$
$18$	0	0
$19$	8.32168	1.90912	0.954562	−	0.298013i	$-0.0963240\pi$
$19$	8.32168	1.90912	0.954562	+	0.298013i	$0.0963240\pi$
$20$	−1.12518	−0.251598
$21$	0	0
$22$	−3.05448	−0.651217
$23$	−2.20462	−0.459696	−0.229848	−	0.973227i	$-0.573823\pi$
$23$	−2.20462	−0.459696	−0.229848	+	0.973227i	$0.573823\pi$
$24$	0	0
$25$	−3.73397	−0.746793
$26$	3.99040	0.782582
$27$	0	0
$28$	−0.686663	−0.129767
$29$	6.02040	1.11796	0.558980	−	0.829181i	$-0.311193\pi$
$29$	6.02040	1.11796	0.558980	+	0.829181i	$0.311193\pi$
$30$	0	0
$31$	8.16044	1.46566	0.732829	−	0.680413i	$-0.238199\pi$
$31$	8.16044	1.46566	0.732829	+	0.680413i	$0.238199\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.80043	−0.994765
$35$	0.772621	0.130597
$36$	0	0
$37$	4.59174	0.754877	0.377438	−	0.926035i	$-0.376805\pi$
$37$	4.59174	0.754877	0.377438	+	0.926035i	$0.376805\pi$
$38$	−8.32168	−1.34995
$39$	0	0
$40$	1.12518	0.177907
$41$	5.82690	0.910009	0.455004	−	0.890489i	$-0.349638\pi$
$41$	5.82690	0.910009	0.455004	+	0.890489i	$0.349638\pi$
$42$	0	0
$43$	0.944768	0.144076	0.0720379	−	0.997402i	$-0.477050\pi$
$43$	0.944768	0.144076	0.0720379	+	0.997402i	$0.477050\pi$
$44$	3.05448	0.460480
$45$	0	0
$46$	2.20462	0.325054
$47$	−6.19531	−0.903679	−0.451839	−	0.892099i	$-0.649232\pi$
$47$	−6.19531	−0.903679	−0.451839	+	0.892099i	$0.649232\pi$
$48$	0	0
$49$	−6.52849	−0.932642
$50$	3.73397	0.528063
$51$	0	0
$52$	−3.99040	−0.553369
$53$	−8.18515	−1.12432	−0.562159	−	0.827029i	$-0.690029\pi$
$53$	−8.18515	−1.12432	−0.562159	+	0.827029i	$0.690029\pi$
$54$	0	0
$55$	−3.43684	−0.463424
$56$	0.686663	0.0917593
$57$	0	0
$58$	−6.02040	−0.790517
$59$	8.39608	1.09308	0.546538	−	0.837434i	$-0.315945\pi$
$59$	8.39608	1.09308	0.546538	+	0.837434i	$0.315945\pi$
$60$	0	0
$61$	−14.1320	−1.80942	−0.904712	−	0.426024i	$-0.859914\pi$
$61$	−14.1320	−1.80942	−0.904712	+	0.426024i	$0.859914\pi$
$62$	−8.16044	−1.03638
$63$	0	0
$64$	1.00000	0.125000
$65$	4.48992	0.556906
$66$	0	0
$67$	6.33902	0.774434	0.387217	−	0.921989i	$-0.373436\pi$
$67$	6.33902	0.774434	0.387217	+	0.921989i	$0.373436\pi$
$68$	5.80043	0.703405
$69$	0	0
$70$	−0.772621	−0.0923459
$71$	2.82097	0.334787	0.167393	−	0.985890i	$-0.446465\pi$
$71$	2.82097	0.334787	0.167393	+	0.985890i	$0.446465\pi$
$72$	0	0
$73$	−3.33968	−0.390880	−0.195440	−	0.980716i	$-0.562613\pi$
$73$	−3.33968	−0.390880	−0.195440	+	0.980716i	$0.562613\pi$
$74$	−4.59174	−0.533779
$75$	0	0
$76$	8.32168	0.954562
$77$	−2.09740	−0.239021
$78$	0	0
$79$	−8.49208	−0.955433	−0.477717	−	0.878514i	$-0.658535\pi$
$79$	−8.49208	−0.955433	−0.477717	+	0.878514i	$0.658535\pi$
$80$	−1.12518	−0.125799
$81$	0	0
$82$	−5.82690	−0.643473
$83$	6.34293	0.696227	0.348113	−	0.937452i	$-0.386822\pi$
$83$	6.34293	0.696227	0.348113	+	0.937452i	$0.386822\pi$
$84$	0	0
$85$	−6.52653	−0.707902
$86$	−0.944768	−0.101877
$87$	0	0
$88$	−3.05448	−0.325609
$89$	6.07267	0.643702	0.321851	−	0.946790i	$-0.395695\pi$
$89$	6.07267	0.643702	0.321851	+	0.946790i	$0.395695\pi$
$90$	0	0
$91$	2.74006	0.287236
$92$	−2.20462	−0.229848
$93$	0	0
$94$	6.19531	0.638997
$95$	−9.36339	−0.960664
$96$	0	0
$97$	−11.5459	−1.17231	−0.586153	−	0.810200i	$-0.699358\pi$
$97$	−11.5459	−1.17231	−0.586153	+	0.810200i	$0.699358\pi$
$98$	6.52849	0.659477
$99$	0	0
$100$	−3.73397	−0.373397
$101$	−13.1430	−1.30778	−0.653891	−	0.756589i	$-0.726864\pi$
$101$	−13.1430	−1.30778	−0.653891	+	0.756589i	$0.726864\pi$
$102$	0	0
$103$	−11.8728	−1.16986	−0.584932	−	0.811082i	$-0.698879\pi$
$103$	−11.8728	−1.16986	−0.584932	+	0.811082i	$0.698879\pi$
$104$	3.99040	0.391291
$105$	0	0
$106$	8.18515	0.795012
$107$	10.9151	1.05521	0.527603	−	0.849491i	$-0.323091\pi$
$107$	10.9151	1.05521	0.527603	+	0.849491i	$0.323091\pi$
$108$	0	0
$109$	2.92308	0.279980	0.139990	−	0.990153i	$-0.455293\pi$
$109$	2.92308	0.279980	0.139990	+	0.990153i	$0.455293\pi$
$110$	3.43684	0.327690
$111$	0	0
$112$	−0.686663	−0.0648836
$113$	5.24148	0.493077	0.246539	−	0.969133i	$-0.420707\pi$
$113$	5.24148	0.493077	0.246539	+	0.969133i	$0.420707\pi$
$114$	0	0
$115$	2.48060	0.231317
$116$	6.02040	0.558980
$117$	0	0
$118$	−8.39608	−0.772922
$119$	−3.98294	−0.365116
$120$	0	0
$121$	−1.67015	−0.151832
$122$	14.1320	1.27946
$123$	0	0
$124$	8.16044	0.732829
$125$	9.82730	0.878980
$126$	0	0
$127$	−13.7317	−1.21849	−0.609247	−	0.792981i	$-0.708528\pi$
$127$	−13.7317	−1.21849	−0.609247	+	0.792981i	$0.708528\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.48992	−0.393792
$131$	17.6560	1.54261	0.771307	−	0.636463i	$-0.219603\pi$
$131$	17.6560	1.54261	0.771307	+	0.636463i	$0.219603\pi$
$132$	0	0
$133$	−5.71419	−0.495483
$134$	−6.33902	−0.547608
$135$	0	0
$136$	−5.80043	−0.497383
$137$	18.9694	1.62066	0.810332	−	0.585971i	$-0.199287\pi$
$137$	18.9694	1.62066	0.810332	+	0.585971i	$0.199287\pi$
$138$	0	0
$139$	8.88482	0.753601	0.376800	−	0.926294i	$-0.377024\pi$
$139$	8.88482	0.753601	0.376800	+	0.926294i	$0.377024\pi$
$140$	0.772621	0.0652984
$141$	0	0
$142$	−2.82097	−0.236730
$143$	−12.1886	−1.01926
$144$	0	0
$145$	−6.77404	−0.562553
$146$	3.33968	0.276394
$147$	0	0
$148$	4.59174	0.377438
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−15.2920	−1.24444	−0.622222	−	0.782841i	$-0.713770\pi$
$151$	−15.2920	−1.24444	−0.622222	+	0.782841i	$0.713770\pi$
$152$	−8.32168	−0.674977
$153$	0	0
$154$	2.09740	0.169013
$155$	−9.18198	−0.737514
$156$	0	0
$157$	−6.18546	−0.493654	−0.246827	−	0.969060i	$-0.579388\pi$
$157$	−6.18546	−0.493654	−0.246827	+	0.969060i	$0.579388\pi$
$158$	8.49208	0.675593
$159$	0	0
$160$	1.12518	0.0889534
$161$	1.51383	0.119307
$162$	0	0
$163$	16.8203	1.31747	0.658733	−	0.752376i	$-0.271093\pi$
$163$	16.8203	1.31747	0.658733	+	0.752376i	$0.271093\pi$
$164$	5.82690	0.455004
$165$	0	0
$166$	−6.34293	−0.492307
$167$	−9.10633	−0.704669	−0.352334	−	0.935874i	$-0.614612\pi$
$167$	−9.10633	−0.704669	−0.352334	+	0.935874i	$0.614612\pi$
$168$	0	0
$169$	2.92329	0.224868
$170$	6.52653	0.500562
$171$	0	0
$172$	0.944768	0.0720379
$173$	19.5509	1.48643	0.743213	−	0.669055i	$-0.233301\pi$
$173$	19.5509	1.48643	0.743213	+	0.669055i	$0.233301\pi$
$174$	0	0
$175$	2.56398	0.193819
$176$	3.05448	0.230240
$177$	0	0
$178$	−6.07267	−0.455166
$179$	11.9617	0.894063	0.447031	−	0.894518i	$-0.352481\pi$
$179$	11.9617	0.894063	0.447031	+	0.894518i	$0.352481\pi$
$180$	0	0
$181$	0.902790	0.0671039	0.0335519	−	0.999437i	$-0.489318\pi$
$181$	0.902790	0.0671039	0.0335519	+	0.999437i	$0.489318\pi$
$182$	−2.74006	−0.203107
$183$	0	0
$184$	2.20462	0.162527
$185$	−5.16654	−0.379851
$186$	0	0
$187$	17.7173	1.29562
$188$	−6.19531	−0.451839
$189$	0	0
$190$	9.36339	0.679292
$191$	−10.3662	−0.750070	−0.375035	−	0.927011i	$-0.622369\pi$
$191$	−10.3662	−0.750070	−0.375035	+	0.927011i	$0.622369\pi$
$192$	0	0
$193$	−17.1187	−1.23223	−0.616116	−	0.787656i	$-0.711295\pi$
$193$	−17.1187	−1.23223	−0.616116	+	0.787656i	$0.711295\pi$
$194$	11.5459	0.828945
$195$	0	0
$196$	−6.52849	−0.466321
$197$	−11.7114	−0.834405	−0.417202	−	0.908814i	$-0.636989\pi$
$197$	−11.7114	−0.834405	−0.417202	+	0.908814i	$0.636989\pi$
$198$	0	0
$199$	23.9115	1.69504	0.847520	−	0.530763i	$-0.178094\pi$
$199$	23.9115	1.69504	0.847520	+	0.530763i	$0.178094\pi$
$200$	3.73397	0.264031
$201$	0	0
$202$	13.1430	0.924741
$203$	−4.13398	−0.290149
$204$	0	0
$205$	−6.55632	−0.457913
$206$	11.8728	0.827219
$207$	0	0
$208$	−3.99040	−0.276684
$209$	25.4184	1.75823
$210$	0	0
$211$	−14.8387	−1.02154	−0.510769	−	0.859718i	$-0.670639\pi$
$211$	−14.8387	−1.02154	−0.510769	+	0.859718i	$0.670639\pi$
$212$	−8.18515	−0.562159
$213$	0	0
$214$	−10.9151	−0.746143
$215$	−1.06304	−0.0724984
$216$	0	0
$217$	−5.60348	−0.380389
$218$	−2.92308	−0.197976
$219$	0	0
$220$	−3.43684	−0.231712
$221$	−23.1460	−1.55697
$222$	0	0
$223$	15.8699	1.06273	0.531365	−	0.847143i	$-0.321679\pi$
$223$	15.8699	1.06273	0.531365	+	0.847143i	$0.321679\pi$
$224$	0.686663	0.0458796
$225$	0	0
$226$	−5.24148	−0.348658
$227$	15.2519	1.01230	0.506151	−	0.862445i	$-0.331068\pi$
$227$	15.2519	1.01230	0.506151	+	0.862445i	$0.331068\pi$
$228$	0	0
$229$	24.3514	1.60919	0.804594	−	0.593826i	$-0.202383\pi$
$229$	24.3514	1.60919	0.804594	+	0.593826i	$0.202383\pi$
$230$	−2.48060	−0.163566
$231$	0	0
$232$	−6.02040	−0.395258
$233$	7.46438	0.489008	0.244504	−	0.969648i	$-0.421375\pi$
$233$	7.46438	0.489008	0.244504	+	0.969648i	$0.421375\pi$
$234$	0	0
$235$	6.97085	0.454728
$236$	8.39608	0.546538
$237$	0	0
$238$	3.98294	0.258176
$239$	19.9977	1.29354	0.646772	−	0.762684i	$-0.276119\pi$
$239$	19.9977	1.29354	0.646772	+	0.762684i	$0.276119\pi$
$240$	0	0
$241$	12.9019	0.831081	0.415541	−	0.909575i	$-0.363592\pi$
$241$	12.9019	0.831081	0.415541	+	0.909575i	$0.363592\pi$
$242$	1.67015	0.107361
$243$	0	0
$244$	−14.1320	−0.904712
$245$	7.34574	0.469302
$246$	0	0
$247$	−33.2068	−2.11290
$248$	−8.16044	−0.518189
$249$	0	0
$250$	−9.82730	−0.621533
$251$	−13.6343	−0.860589	−0.430295	−	0.902688i	$-0.641590\pi$
$251$	−13.6343	−0.860589	−0.430295	+	0.902688i	$0.641590\pi$
$252$	0	0
$253$	−6.73398	−0.423361
$254$	13.7317	0.861605
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.4900	−1.65240	−0.826200	−	0.563377i	$-0.809502\pi$
$257$	−26.4900	−1.65240	−0.826200	+	0.563377i	$0.809502\pi$
$258$	0	0
$259$	−3.15298	−0.195916
$260$	4.48992	0.278453
$261$	0	0
$262$	−17.6560	−1.09079
$263$	4.98189	0.307196	0.153598	−	0.988133i	$-0.450914\pi$
$263$	4.98189	0.307196	0.153598	+	0.988133i	$0.450914\pi$
$264$	0	0
$265$	9.20978	0.565752
$266$	5.71419	0.350359
$267$	0	0
$268$	6.33902	0.387217
$269$	7.74980	0.472514	0.236257	−	0.971691i	$-0.424079\pi$
$269$	7.74980	0.472514	0.236257	+	0.971691i	$0.424079\pi$
$270$	0	0
$271$	20.0627	1.21872	0.609361	−	0.792893i	$-0.291426\pi$
$271$	20.0627	1.21872	0.609361	+	0.792893i	$0.291426\pi$
$272$	5.80043	0.351703
$273$	0	0
$274$	−18.9694	−1.14598
$275$	−11.4053	−0.687767
$276$	0	0
$277$	18.1869	1.09275	0.546373	−	0.837542i	$-0.316008\pi$
$277$	18.1869	1.09275	0.546373	+	0.837542i	$0.316008\pi$
$278$	−8.88482	−0.532876
$279$	0	0
$280$	−0.772621	−0.0461729
$281$	15.9505	0.951525	0.475763	−	0.879574i	$-0.342172\pi$
$281$	15.9505	0.951525	0.475763	+	0.879574i	$0.342172\pi$
$282$	0	0
$283$	1.04252	0.0619712	0.0309856	−	0.999520i	$-0.490135\pi$
$283$	1.04252	0.0619712	0.0309856	+	0.999520i	$0.490135\pi$
$284$	2.82097	0.167393
$285$	0	0
$286$	12.1886	0.720727
$287$	−4.00112	−0.236179
$288$	0	0
$289$	16.6450	0.979115
$290$	6.77404	0.397785
$291$	0	0
$292$	−3.33968	−0.195440
$293$	−1.65871	−0.0969029	−0.0484515	−	0.998826i	$-0.515429\pi$
$293$	−1.65871	−0.0969029	−0.0484515	+	0.998826i	$0.515429\pi$
$294$	0	0
$295$	−9.44711	−0.550032
$296$	−4.59174	−0.266889
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	8.79733	0.508762
$300$	0	0
$301$	−0.648738	−0.0373926
$302$	15.2920	0.879955
$303$	0	0
$304$	8.32168	0.477281
$305$	15.9011	0.910495
$306$	0	0
$307$	−30.3028	−1.72947	−0.864737	−	0.502225i	$-0.832515\pi$
$307$	−30.3028	−1.72947	−0.864737	+	0.502225i	$0.832515\pi$
$308$	−2.09740	−0.119510
$309$	0	0
$310$	9.18198	0.521501
$311$	6.82658	0.387100	0.193550	−	0.981090i	$-0.438000\pi$
$311$	6.82658	0.387100	0.193550	+	0.981090i	$0.438000\pi$
$312$	0	0
$313$	−3.07494	−0.173806	−0.0869030	−	0.996217i	$-0.527697\pi$
$313$	−3.07494	−0.173806	−0.0869030	+	0.996217i	$0.527697\pi$
$314$	6.18546	0.349066
$315$	0	0
$316$	−8.49208	−0.477717
$317$	−14.5491	−0.817161	−0.408581	−	0.912722i	$-0.633976\pi$
$317$	−14.5491	−0.817161	−0.408581	+	0.912722i	$0.633976\pi$
$318$	0	0
$319$	18.3892	1.02960
$320$	−1.12518	−0.0628996
$321$	0	0
$322$	−1.51383	−0.0843626
$323$	48.2693	2.68577
$324$	0	0
$325$	14.9000	0.826504
$326$	−16.8203	−0.931590
$327$	0	0
$328$	−5.82690	−0.321737
$329$	4.25409	0.234536
$330$	0	0
$331$	18.2660	1.00399	0.501994	−	0.864871i	$-0.332600\pi$
$331$	18.2660	1.00399	0.501994	+	0.864871i	$0.332600\pi$
$332$	6.34293	0.348113
$333$	0	0
$334$	9.10633	0.498276
$335$	−7.13254	−0.389693
$336$	0	0
$337$	23.9834	1.30646	0.653228	−	0.757161i	$-0.273414\pi$
$337$	23.9834	1.30646	0.653228	+	0.757161i	$0.273414\pi$
$338$	−2.92329	−0.159006
$339$	0	0
$340$	−6.52653	−0.353951
$341$	24.9259	1.34981
$342$	0	0
$343$	9.28952	0.501587
$344$	−0.944768	−0.0509385
$345$	0	0
$346$	−19.5509	−1.05106
$347$	−21.2095	−1.13859	−0.569293	−	0.822135i	$-0.692783\pi$
$347$	−21.2095	−1.13859	−0.569293	+	0.822135i	$0.692783\pi$
$348$	0	0
$349$	13.4739	0.721241	0.360620	−	0.932713i	$-0.382565\pi$
$349$	13.4739	0.721241	0.360620	+	0.932713i	$0.382565\pi$
$350$	−2.56398	−0.137050
$351$	0	0
$352$	−3.05448	−0.162804
$353$	−17.1231	−0.911369	−0.455685	−	0.890141i	$-0.650605\pi$
$353$	−17.1231	−0.911369	−0.455685	+	0.890141i	$0.650605\pi$
$354$	0	0
$355$	−3.17410	−0.168464
$356$	6.07267	0.321851
$357$	0	0
$358$	−11.9617	−0.632198
$359$	33.0700	1.74537	0.872683	−	0.488288i	$-0.162378\pi$
$359$	33.0700	1.74537	0.872683	+	0.488288i	$0.162378\pi$
$360$	0	0
$361$	50.2503	2.64475
$362$	−0.902790	−0.0474496
$363$	0	0
$364$	2.74006	0.143618
$365$	3.75775	0.196689
$366$	0	0
$367$	0.517529	0.0270148	0.0135074	−	0.999909i	$-0.495700\pi$
$367$	0.517529	0.0270148	0.0135074	+	0.999909i	$0.495700\pi$
$368$	−2.20462	−0.114924
$369$	0	0
$370$	5.16654	0.268595
$371$	5.62045	0.291799
$372$	0	0
$373$	−24.3848	−1.26260	−0.631299	−	0.775540i	$-0.717478\pi$
$373$	−24.3848	−1.26260	−0.631299	+	0.775540i	$0.717478\pi$
$374$	−17.7173	−0.916139
$375$	0	0
$376$	6.19531	0.319499
$377$	−24.0238	−1.23729
$378$	0	0
$379$	10.9892	0.564479	0.282239	−	0.959344i	$-0.408923\pi$
$379$	10.9892	0.564479	0.282239	+	0.959344i	$0.408923\pi$
$380$	−9.36339	−0.480332
$381$	0	0
$382$	10.3662	0.530380
$383$	25.8159	1.31913	0.659565	−	0.751648i	$-0.270741\pi$
$383$	25.8159	1.31913	0.659565	+	0.751648i	$0.270741\pi$
$384$	0	0
$385$	2.35995	0.120274
$386$	17.1187	0.871319
$387$	0	0
$388$	−11.5459	−0.586153
$389$	−37.0725	−1.87965	−0.939825	−	0.341656i	$-0.889012\pi$
$389$	−37.0725	−1.87965	−0.939825	+	0.341656i	$0.889012\pi$
$390$	0	0
$391$	−12.7878	−0.646704
$392$	6.52849	0.329739
$393$	0	0
$394$	11.7114	0.590013
$395$	9.55513	0.480771
$396$	0	0
$397$	−19.1725	−0.962242	−0.481121	−	0.876654i	$-0.659770\pi$
$397$	−19.1725	−0.962242	−0.481121	+	0.876654i	$0.659770\pi$
$398$	−23.9115	−1.19857
$399$	0	0
$400$	−3.73397	−0.186698
$401$	9.85673	0.492222	0.246111	−	0.969242i	$-0.420847\pi$
$401$	9.85673	0.492222	0.246111	+	0.969242i	$0.420847\pi$
$402$	0	0
$403$	−32.5634	−1.62210
$404$	−13.1430	−0.653891
$405$	0	0
$406$	4.13398	0.205166
$407$	14.0254	0.695212
$408$	0	0
$409$	−8.27230	−0.409039	−0.204519	−	0.978863i	$-0.565563\pi$
$409$	−8.27230	−0.409039	−0.204519	+	0.978863i	$0.565563\pi$
$410$	6.55632	0.323794
$411$	0	0
$412$	−11.8728	−0.584932
$413$	−5.76528	−0.283691
$414$	0	0
$415$	−7.13694	−0.350339
$416$	3.99040	0.195645
$417$	0	0
$418$	−25.4184	−1.24325
$419$	−5.77203	−0.281982	−0.140991	−	0.990011i	$-0.545029\pi$
$419$	−5.77203	−0.281982	−0.140991	+	0.990011i	$0.545029\pi$
$420$	0	0
$421$	8.50728	0.414619	0.207310	−	0.978275i	$-0.433529\pi$
$421$	8.50728	0.414619	0.207310	+	0.978275i	$0.433529\pi$
$422$	14.8387	0.722336
$423$	0	0
$424$	8.18515	0.397506
$425$	−21.6586	−1.05060
$426$	0	0
$427$	9.70396	0.469608
$428$	10.9151	0.527603
$429$	0	0
$430$	1.06304	0.0512641
$431$	−29.8564	−1.43813	−0.719067	−	0.694941i	$-0.755431\pi$
$431$	−29.8564	−1.43813	−0.719067	+	0.694941i	$0.755431\pi$
$432$	0	0
$433$	15.7016	0.754571	0.377286	−	0.926097i	$-0.376858\pi$
$433$	15.7016	0.754571	0.377286	+	0.926097i	$0.376858\pi$
$434$	5.60348	0.268975
$435$	0	0
$436$	2.92308	0.139990
$437$	−18.3462	−0.877616
$438$	0	0
$439$	15.4229	0.736097	0.368048	−	0.929807i	$-0.380026\pi$
$439$	15.4229	0.736097	0.368048	+	0.929807i	$0.380026\pi$
$440$	3.43684	0.163845
$441$	0	0
$442$	23.1460	1.10094
$443$	−5.88701	−0.279700	−0.139850	−	0.990173i	$-0.544662\pi$
$443$	−5.88701	−0.279700	−0.139850	+	0.990173i	$0.544662\pi$
$444$	0	0
$445$	−6.83285	−0.323908
$446$	−15.8699	−0.751463
$447$	0	0
$448$	−0.686663	−0.0324418
$449$	−23.1092	−1.09059	−0.545295	−	0.838244i	$-0.683582\pi$
$449$	−23.1092	−1.09059	−0.545295	+	0.838244i	$0.683582\pi$
$450$	0	0
$451$	17.7982	0.838082
$452$	5.24148	0.246539
$453$	0	0
$454$	−15.2519	−0.715806
$455$	−3.08307	−0.144536
$456$	0	0
$457$	16.5419	0.773798	0.386899	−	0.922122i	$-0.373546\pi$
$457$	16.5419	0.773798	0.386899	+	0.922122i	$0.373546\pi$
$458$	−24.3514	−1.13787
$459$	0	0
$460$	2.48060	0.115659
$461$	−3.97700	−0.185227	−0.0926137	−	0.995702i	$-0.529522\pi$
$461$	−3.97700	−0.185227	−0.0926137	+	0.995702i	$0.529522\pi$
$462$	0	0
$463$	18.7496	0.871366	0.435683	−	0.900100i	$-0.356507\pi$
$463$	18.7496	0.871366	0.435683	+	0.900100i	$0.356507\pi$
$464$	6.02040	0.279490
$465$	0	0
$466$	−7.46438	−0.345781
$467$	9.33972	0.432191	0.216095	−	0.976372i	$-0.430668\pi$
$467$	9.33972	0.432191	0.216095	+	0.976372i	$0.430668\pi$
$468$	0	0
$469$	−4.35277	−0.200992
$470$	−6.97085	−0.321541
$471$	0	0
$472$	−8.39608	−0.386461
$473$	2.88577	0.132688
$474$	0	0
$475$	−31.0729	−1.42572
$476$	−3.98294	−0.182558
$477$	0	0
$478$	−19.9977	−0.914673
$479$	−6.50916	−0.297411	−0.148705	−	0.988882i	$-0.547511\pi$
$479$	−6.50916	−0.297411	−0.148705	+	0.988882i	$0.547511\pi$
$480$	0	0
$481$	−18.3229	−0.835451
$482$	−12.9019	−0.587663
$483$	0	0
$484$	−1.67015	−0.0759160
$485$	12.9912	0.589900
$486$	0	0
$487$	17.0058	0.770608	0.385304	−	0.922790i	$-0.374097\pi$
$487$	17.0058	0.770608	0.385304	+	0.922790i	$0.374097\pi$
$488$	14.1320	0.639728
$489$	0	0
$490$	−7.34574	−0.331847
$491$	26.8032	1.20961	0.604807	−	0.796372i	$-0.293250\pi$
$491$	26.8032	1.20961	0.604807	+	0.796372i	$0.293250\pi$
$492$	0	0
$493$	34.9209	1.57276
$494$	33.2068	1.49405
$495$	0	0
$496$	8.16044	0.366415
$497$	−1.93705	−0.0868887
$498$	0	0
$499$	21.4285	0.959272	0.479636	−	0.877467i	$-0.340769\pi$
$499$	21.4285	0.959272	0.479636	+	0.877467i	$0.340769\pi$
$500$	9.82730	0.439490
$501$	0	0
$502$	13.6343	0.608529
$503$	−14.9635	−0.667189	−0.333594	−	0.942717i	$-0.608262\pi$
$503$	−14.9635	−0.667189	−0.333594	+	0.942717i	$0.608262\pi$
$504$	0	0
$505$	14.7883	0.658071
$506$	6.73398	0.299362
$507$	0	0
$508$	−13.7317	−0.609247
$509$	−5.63400	−0.249723	−0.124861	−	0.992174i	$-0.539849\pi$
$509$	−5.63400	−0.249723	−0.124861	+	0.992174i	$0.539849\pi$
$510$	0	0
$511$	2.29324	0.101447
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.4900	1.16842
$515$	13.3591	0.588671
$516$	0	0
$517$	−18.9234	−0.832252
$518$	3.15298	0.138534
$519$	0	0
$520$	−4.48992	−0.196896
$521$	4.52985	0.198456	0.0992281	−	0.995065i	$-0.468363\pi$
$521$	4.52985	0.198456	0.0992281	+	0.995065i	$0.468363\pi$
$522$	0	0
$523$	−15.0316	−0.657285	−0.328643	−	0.944454i	$-0.606591\pi$
$523$	−15.0316	−0.657285	−0.328643	+	0.944454i	$0.606591\pi$
$524$	17.6560	0.771307
$525$	0	0
$526$	−4.98189	−0.217221
$527$	47.3340	2.06190
$528$	0	0
$529$	−18.1396	−0.788680
$530$	−9.20978	−0.400047
$531$	0	0
$532$	−5.71419	−0.247742
$533$	−23.2517	−1.00714
$534$	0	0
$535$	−12.2815	−0.530975
$536$	−6.33902	−0.273804
$537$	0	0
$538$	−7.74980	−0.334118
$539$	−19.9412	−0.858926
$540$	0	0
$541$	−13.2437	−0.569391	−0.284696	−	0.958618i	$-0.591893\pi$
$541$	−13.2437	−0.569391	−0.284696	+	0.958618i	$0.591893\pi$
$542$	−20.0627	−0.861767
$543$	0	0
$544$	−5.80043	−0.248691
$545$	−3.28900	−0.140885
$546$	0	0
$547$	−0.855180	−0.0365649	−0.0182824	−	0.999833i	$-0.505820\pi$
$547$	−0.855180	−0.0365649	−0.0182824	+	0.999833i	$0.505820\pi$
$548$	18.9694	0.810332
$549$	0	0
$550$	11.4053	0.486325
$551$	50.0998	2.13432
$552$	0	0
$553$	5.83120	0.247968
$554$	−18.1869	−0.772689
$555$	0	0
$556$	8.88482	0.376800
$557$	27.0002	1.14403	0.572017	−	0.820242i	$-0.306161\pi$
$557$	27.0002	1.14403	0.572017	+	0.820242i	$0.306161\pi$
$558$	0	0
$559$	−3.77000	−0.159454
$560$	0.772621	0.0326492
$561$	0	0
$562$	−15.9505	−0.672830
$563$	−13.5674	−0.571799	−0.285900	−	0.958260i	$-0.592292\pi$
$563$	−13.5674	−0.571799	−0.285900	+	0.958260i	$0.592292\pi$
$564$	0	0
$565$	−5.89762	−0.248115
$566$	−1.04252	−0.0438202
$567$	0	0
$568$	−2.82097	−0.118365
$569$	3.74436	0.156972	0.0784858	−	0.996915i	$-0.474991\pi$
$569$	3.74436	0.156972	0.0784858	+	0.996915i	$0.474991\pi$
$570$	0	0
$571$	−17.6994	−0.740697	−0.370349	−	0.928893i	$-0.620762\pi$
$571$	−17.6994	−0.740697	−0.370349	+	0.928893i	$0.620762\pi$
$572$	−12.1886	−0.509631
$573$	0	0
$574$	4.00112	0.167003
$575$	8.23199	0.343298
$576$	0	0
$577$	−25.8371	−1.07561	−0.537805	−	0.843069i	$-0.680746\pi$
$577$	−25.8371	−1.07561	−0.537805	+	0.843069i	$0.680746\pi$
$578$	−16.6450	−0.692339
$579$	0	0
$580$	−6.77404	−0.281277
$581$	−4.35546	−0.180695
$582$	0	0
$583$	−25.0014	−1.03545
$584$	3.33968	0.138197
$585$	0	0
$586$	1.65871	0.0685207
$587$	18.4980	0.763495	0.381748	−	0.924267i	$-0.375322\pi$
$587$	18.4980	0.763495	0.381748	+	0.924267i	$0.375322\pi$
$588$	0	0
$589$	67.9086	2.79812
$590$	9.44711	0.388931
$591$	0	0
$592$	4.59174	0.188719
$593$	33.8650	1.39067	0.695335	−	0.718686i	$-0.255256\pi$
$593$	33.8650	1.39067	0.695335	+	0.718686i	$0.255256\pi$
$594$	0	0
$595$	4.48153	0.183725
$596$	1.00000	0.0409616
$597$	0	0
$598$	−8.79733	−0.359749
$599$	3.73165	0.152471	0.0762354	−	0.997090i	$-0.475710\pi$
$599$	3.73165	0.152471	0.0762354	+	0.997090i	$0.475710\pi$
$600$	0	0
$601$	−23.3136	−0.950980	−0.475490	−	0.879721i	$-0.657729\pi$
$601$	−23.3136	−0.950980	−0.475490	+	0.879721i	$0.657729\pi$
$602$	0.648738	0.0264406
$603$	0	0
$604$	−15.2920	−0.622222
$605$	1.87922	0.0764013
$606$	0	0
$607$	6.50372	0.263978	0.131989	−	0.991251i	$-0.457864\pi$
$607$	6.50372	0.263978	0.131989	+	0.991251i	$0.457864\pi$
$608$	−8.32168	−0.337489
$609$	0	0
$610$	−15.9011	−0.643817
$611$	24.7218	1.00014
$612$	0	0
$613$	21.0805	0.851434	0.425717	−	0.904856i	$-0.360022\pi$
$613$	21.0805	0.851434	0.425717	+	0.904856i	$0.360022\pi$
$614$	30.3028	1.22292
$615$	0	0
$616$	2.09740	0.0845066
$617$	3.92176	0.157884	0.0789422	−	0.996879i	$-0.474846\pi$
$617$	3.92176	0.157884	0.0789422	+	0.996879i	$0.474846\pi$
$618$	0	0
$619$	−23.6404	−0.950188	−0.475094	−	0.879935i	$-0.657586\pi$
$619$	−23.6404	−0.950188	−0.475094	+	0.879935i	$0.657586\pi$
$620$	−9.18198	−0.368757
$621$	0	0
$622$	−6.82658	−0.273721
$623$	−4.16988	−0.167063
$624$	0	0
$625$	7.61234	0.304494
$626$	3.07494	0.122899
$627$	0	0
$628$	−6.18546	−0.246827
$629$	26.6340	1.06197
$630$	0	0
$631$	3.61784	0.144024	0.0720120	−	0.997404i	$-0.477058\pi$
$631$	3.61784	0.144024	0.0720120	+	0.997404i	$0.477058\pi$
$632$	8.49208	0.337797
$633$	0	0
$634$	14.5491	0.577820
$635$	15.4507	0.613141
$636$	0	0
$637$	26.0513	1.03219
$638$	−18.3892	−0.728034
$639$	0	0
$640$	1.12518	0.0444767
$641$	25.6454	1.01293	0.506467	−	0.862259i	$-0.330951\pi$
$641$	25.6454	1.01293	0.506467	+	0.862259i	$0.330951\pi$
$642$	0	0
$643$	41.2134	1.62530	0.812649	−	0.582754i	$-0.198025\pi$
$643$	41.2134	1.62530	0.812649	+	0.582754i	$0.198025\pi$
$644$	1.51383	0.0596534
$645$	0	0
$646$	−48.2693	−1.89913
$647$	35.3255	1.38879	0.694394	−	0.719595i	$-0.255673\pi$
$647$	35.3255	1.38879	0.694394	+	0.719595i	$0.255673\pi$
$648$	0	0
$649$	25.6457	1.00668
$650$	−14.9000	−0.584427
$651$	0	0
$652$	16.8203	0.658733
$653$	−23.0974	−0.903872	−0.451936	−	0.892050i	$-0.649267\pi$
$653$	−23.0974	−0.903872	−0.451936	+	0.892050i	$0.649267\pi$
$654$	0	0
$655$	−19.8662	−0.776238
$656$	5.82690	0.227502
$657$	0	0
$658$	−4.25409	−0.165842
$659$	−20.6520	−0.804487	−0.402244	−	0.915533i	$-0.631770\pi$
$659$	−20.6520	−0.804487	−0.402244	+	0.915533i	$0.631770\pi$
$660$	0	0
$661$	20.9399	0.814466	0.407233	−	0.913324i	$-0.366494\pi$
$661$	20.9399	0.814466	0.407233	+	0.913324i	$0.366494\pi$
$662$	−18.2660	−0.709927
$663$	0	0
$664$	−6.34293	−0.246153
$665$	6.42950	0.249325
$666$	0	0
$667$	−13.2727	−0.513921
$668$	−9.10633	−0.352334
$669$	0	0
$670$	7.13254	0.275554
$671$	−43.1661	−1.66641
$672$	0	0
$673$	9.84995	0.379687	0.189844	−	0.981814i	$-0.439202\pi$
$673$	9.84995	0.379687	0.189844	+	0.981814i	$0.439202\pi$
$674$	−23.9834	−0.923804
$675$	0	0
$676$	2.92329	0.112434
$677$	28.1520	1.08197	0.540985	−	0.841032i	$-0.318052\pi$
$677$	28.1520	1.08197	0.540985	+	0.841032i	$0.318052\pi$
$678$	0	0
$679$	7.92813	0.304254
$680$	6.52653	0.250281
$681$	0	0
$682$	−24.9259	−0.954462
$683$	−9.42277	−0.360552	−0.180276	−	0.983616i	$-0.557699\pi$
$683$	−9.42277	−0.360552	−0.180276	+	0.983616i	$0.557699\pi$
$684$	0	0
$685$	−21.3440	−0.815512
$686$	−9.28952	−0.354676
$687$	0	0
$688$	0.944768	0.0360189
$689$	32.6620	1.24432
$690$	0	0
$691$	−51.7355	−1.96811	−0.984056	−	0.177856i	$-0.943084\pi$
$691$	−51.7355	−1.96811	−0.984056	+	0.177856i	$0.943084\pi$
$692$	19.5509	0.743213
$693$	0	0
$694$	21.2095	0.805101
$695$	−9.99704	−0.379209
$696$	0	0
$697$	33.7985	1.28021
$698$	−13.4739	−0.509994
$699$	0	0
$700$	2.56398	0.0969093
$701$	17.0599	0.644345	0.322172	−	0.946681i	$-0.395587\pi$
$701$	17.0599	0.644345	0.322172	+	0.946681i	$0.395587\pi$
$702$	0	0
$703$	38.2109	1.44115
$704$	3.05448	0.115120
$705$	0	0
$706$	17.1231	0.644435
$707$	9.02485	0.339414
$708$	0	0
$709$	−37.2820	−1.40016	−0.700078	−	0.714066i	$-0.746851\pi$
$709$	−37.2820	−1.40016	−0.700078	+	0.714066i	$0.746851\pi$
$710$	3.17410	0.119122
$711$	0	0
$712$	−6.07267	−0.227583
$713$	−17.9907	−0.673757
$714$	0	0
$715$	13.7144	0.512889
$716$	11.9617	0.447031
$717$	0	0
$718$	−33.0700	−1.23416
$719$	28.8178	1.07472	0.537362	−	0.843352i	$-0.319421\pi$
$719$	28.8178	1.07472	0.537362	+	0.843352i	$0.319421\pi$
$720$	0	0
$721$	8.15263	0.303620
$722$	−50.2503	−1.87012
$723$	0	0
$724$	0.902790	0.0335519
$725$	−22.4800	−0.834885
$726$	0	0
$727$	6.62094	0.245557	0.122779	−	0.992434i	$-0.460820\pi$
$727$	6.62094	0.245557	0.122779	+	0.992434i	$0.460820\pi$
$728$	−2.74006	−0.101553
$729$	0	0
$730$	−3.75775	−0.139080
$731$	5.48006	0.202687
$732$	0	0
$733$	27.7091	1.02346	0.511730	−	0.859146i	$-0.329005\pi$
$733$	27.7091	1.02346	0.511730	+	0.859146i	$0.329005\pi$
$734$	−0.517529	−0.0191023
$735$	0	0
$736$	2.20462	0.0812635
$737$	19.3624	0.713223
$738$	0	0
$739$	−22.1604	−0.815183	−0.407591	−	0.913164i	$-0.633631\pi$
$739$	−22.1604	−0.815183	−0.407591	+	0.913164i	$0.633631\pi$
$740$	−5.16654	−0.189926
$741$	0	0
$742$	−5.62045	−0.206333
$743$	16.1890	0.593917	0.296959	−	0.954890i	$-0.404028\pi$
$743$	16.1890	0.593917	0.296959	+	0.954890i	$0.404028\pi$
$744$	0	0
$745$	−1.12518	−0.0412235
$746$	24.3848	0.892791
$747$	0	0
$748$	17.7173	0.647808
$749$	−7.49502	−0.273862
$750$	0	0
$751$	−46.6646	−1.70282	−0.851408	−	0.524503i	$-0.824251\pi$
$751$	−46.6646	−1.70282	−0.851408	+	0.524503i	$0.824251\pi$
$752$	−6.19531	−0.225920
$753$	0	0
$754$	24.0238	0.874895
$755$	17.2063	0.626200
$756$	0	0
$757$	39.3659	1.43078	0.715389	−	0.698727i	$-0.246250\pi$
$757$	39.3659	1.43078	0.715389	+	0.698727i	$0.246250\pi$
$758$	−10.9892	−0.399147
$759$	0	0
$760$	9.36339	0.339646
$761$	−25.2105	−0.913879	−0.456940	−	0.889498i	$-0.651054\pi$
$761$	−25.2105	−0.913879	−0.456940	+	0.889498i	$0.651054\pi$
$762$	0	0
$763$	−2.00717	−0.0726645
$764$	−10.3662	−0.375035
$765$	0	0
$766$	−25.8159	−0.932766
$767$	−33.5037	−1.20975
$768$	0	0
$769$	23.2413	0.838103	0.419051	−	0.907963i	$-0.362363\pi$
$769$	23.2413	0.838103	0.419051	+	0.907963i	$0.362363\pi$
$770$	−2.35995	−0.0850469
$771$	0	0
$772$	−17.1187	−0.616116
$773$	17.1288	0.616080	0.308040	−	0.951373i	$-0.400327\pi$
$773$	17.1288	0.616080	0.308040	+	0.951373i	$0.400327\pi$
$774$	0	0
$775$	−30.4708	−1.09454
$776$	11.5459	0.414473
$777$	0	0
$778$	37.0725	1.32911
$779$	48.4896	1.73732
$780$	0	0
$781$	8.61658	0.308326
$782$	12.7878	0.457289
$783$	0	0
$784$	−6.52849	−0.233160
$785$	6.95977	0.248405
$786$	0	0
$787$	−22.7947	−0.812542	−0.406271	−	0.913753i	$-0.633171\pi$
$787$	−22.7947	−0.812542	−0.406271	+	0.913753i	$0.633171\pi$
$788$	−11.7114	−0.417202
$789$	0	0
$790$	−9.55513	−0.339956
$791$	−3.59914	−0.127971
$792$	0	0
$793$	56.3925	2.00256
$794$	19.1725	0.680408
$795$	0	0
$796$	23.9115	0.847520
$797$	−24.3325	−0.861902	−0.430951	−	0.902375i	$-0.641822\pi$
$797$	−24.3325	−0.861902	−0.430951	+	0.902375i	$0.641822\pi$
$798$	0	0
$799$	−35.9354	−1.27130
$800$	3.73397	0.132016
$801$	0	0
$802$	−9.85673	−0.348053
$803$	−10.2010	−0.359985
$804$	0	0
$805$	−1.70334	−0.0600348
$806$	32.5634	1.14700
$807$	0	0
$808$	13.1430	0.462371
$809$	−29.9622	−1.05341	−0.526707	−	0.850047i	$-0.676574\pi$
$809$	−29.9622	−1.05341	−0.526707	+	0.850047i	$0.676574\pi$
$810$	0	0
$811$	37.8887	1.33045	0.665226	−	0.746642i	$-0.268335\pi$
$811$	37.8887	1.33045	0.665226	+	0.746642i	$0.268335\pi$
$812$	−4.13398	−0.145074
$813$	0	0
$814$	−14.0254	−0.491589
$815$	−18.9259	−0.662945
$816$	0	0
$817$	7.86205	0.275058
$818$	8.27230	0.289234
$819$	0	0
$820$	−6.55632	−0.228957
$821$	42.2822	1.47566	0.737829	−	0.674988i	$-0.235851\pi$
$821$	42.2822	1.47566	0.737829	+	0.674988i	$0.235851\pi$
$822$	0	0
$823$	−28.6134	−0.997399	−0.498699	−	0.866775i	$-0.666189\pi$
$823$	−28.6134	−0.997399	−0.498699	+	0.866775i	$0.666189\pi$
$824$	11.8728	0.413609
$825$	0	0
$826$	5.76528	0.200600
$827$	−21.9207	−0.762259	−0.381130	−	0.924522i	$-0.624465\pi$
$827$	−21.9207	−0.762259	−0.381130	+	0.924522i	$0.624465\pi$
$828$	0	0
$829$	11.3236	0.393284	0.196642	−	0.980475i	$-0.436996\pi$
$829$	11.3236	0.393284	0.196642	+	0.980475i	$0.436996\pi$
$830$	7.13694	0.247727
$831$	0	0
$832$	−3.99040	−0.138342
$833$	−37.8681	−1.31205
$834$	0	0
$835$	10.2463	0.354587
$836$	25.4184	0.879114
$837$	0	0
$838$	5.77203	0.199392
$839$	43.3273	1.49583	0.747913	−	0.663797i	$-0.231056\pi$
$839$	43.3273	1.49583	0.747913	+	0.663797i	$0.231056\pi$
$840$	0	0
$841$	7.24516	0.249833
$842$	−8.50728	−0.293180
$843$	0	0
$844$	−14.8387	−0.510769
$845$	−3.28923	−0.113153
$846$	0	0
$847$	1.14683	0.0394056
$848$	−8.18515	−0.281079
$849$	0	0
$850$	21.6586	0.742884
$851$	−10.1230	−0.347014
$852$	0	0
$853$	16.1588	0.553266	0.276633	−	0.960976i	$-0.410781\pi$
$853$	16.1588	0.553266	0.276633	+	0.960976i	$0.410781\pi$
$854$	−9.70396	−0.332063
$855$	0	0
$856$	−10.9151	−0.373071
$857$	11.2150	0.383098	0.191549	−	0.981483i	$-0.438649\pi$
$857$	11.2150	0.383098	0.191549	+	0.981483i	$0.438649\pi$
$858$	0	0
$859$	46.2975	1.57965	0.789825	−	0.613333i	$-0.210171\pi$
$859$	46.2975	1.57965	0.789825	+	0.613333i	$0.210171\pi$
$860$	−1.06304	−0.0362492
$861$	0	0
$862$	29.8564	1.01691
$863$	−12.4665	−0.424365	−0.212182	−	0.977230i	$-0.568057\pi$
$863$	−12.4665	−0.424365	−0.212182	+	0.977230i	$0.568057\pi$
$864$	0	0
$865$	−21.9983	−0.747964
$866$	−15.7016	−0.533562
$867$	0	0
$868$	−5.60348	−0.190194
$869$	−25.9389	−0.879916
$870$	0	0
$871$	−25.2952	−0.857096
$872$	−2.92308	−0.0989880
$873$	0	0
$874$	18.3462	0.620568
$875$	−6.74804	−0.228126
$876$	0	0
$877$	58.2751	1.96781	0.983905	−	0.178695i	$-0.0571875\pi$
$877$	58.2751	1.96781	0.983905	+	0.178695i	$0.0571875\pi$
$878$	−15.4229	−0.520499
$879$	0	0
$880$	−3.43684	−0.115856
$881$	−26.0826	−0.878745	−0.439372	−	0.898305i	$-0.644799\pi$
$881$	−26.0826	−0.878745	−0.439372	+	0.898305i	$0.644799\pi$
$882$	0	0
$883$	48.1927	1.62182	0.810908	−	0.585174i	$-0.198974\pi$
$883$	48.1927	1.62182	0.810908	+	0.585174i	$0.198974\pi$
$884$	−23.1460	−0.778485
$885$	0	0
$886$	5.88701	0.197778
$887$	−29.2074	−0.980689	−0.490345	−	0.871529i	$-0.663129\pi$
$887$	−29.2074	−0.980689	−0.490345	+	0.871529i	$0.663129\pi$
$888$	0	0
$889$	9.42907	0.316241
$890$	6.83285	0.229038
$891$	0	0
$892$	15.8699	0.531365
$893$	−51.5554	−1.72523
$894$	0	0
$895$	−13.4591	−0.449889
$896$	0.686663	0.0229398
$897$	0	0
$898$	23.1092	0.771163
$899$	49.1291	1.63855
$900$	0	0
$901$	−47.4774	−1.58170
$902$	−17.7982	−0.592614
$903$	0	0
$904$	−5.24148	−0.174329
$905$	−1.01580	−0.0337664
$906$	0	0
$907$	−50.0879	−1.66314	−0.831571	−	0.555419i	$-0.812558\pi$
$907$	−50.0879	−1.66314	−0.831571	+	0.555419i	$0.812558\pi$
$908$	15.2519	0.506151
$909$	0	0
$910$	3.08307	0.102203
$911$	12.0650	0.399731	0.199866	−	0.979823i	$-0.435949\pi$
$911$	12.0650	0.399731	0.199866	+	0.979823i	$0.435949\pi$
$912$	0	0
$913$	19.3743	0.641197
$914$	−16.5419	−0.547158
$915$	0	0
$916$	24.3514	0.804594
$917$	−12.1237	−0.400361
$918$	0	0
$919$	18.0290	0.594721	0.297360	−	0.954765i	$-0.403894\pi$
$919$	18.0290	0.594721	0.297360	+	0.954765i	$0.403894\pi$
$920$	−2.48060	−0.0817830
$921$	0	0
$922$	3.97700	0.130975
$923$	−11.2568	−0.370521
$924$	0	0
$925$	−17.1454	−0.563737
$926$	−18.7496	−0.616149
$927$	0	0
$928$	−6.02040	−0.197629
$929$	26.6010	0.872749	0.436375	−	0.899765i	$-0.356262\pi$
$929$	26.6010	0.872749	0.436375	+	0.899765i	$0.356262\pi$
$930$	0	0
$931$	−54.3280	−1.78053
$932$	7.46438	0.244504
$933$	0	0
$934$	−9.33972	−0.305605
$935$	−19.9352	−0.651950
$936$	0	0
$937$	12.3300	0.402804	0.201402	−	0.979509i	$-0.435450\pi$
$937$	12.3300	0.402804	0.201402	+	0.979509i	$0.435450\pi$
$938$	4.35277	0.142123
$939$	0	0
$940$	6.97085	0.227364
$941$	47.7282	1.55589	0.777947	−	0.628330i	$-0.216261\pi$
$941$	47.7282	1.55589	0.777947	+	0.628330i	$0.216261\pi$
$942$	0	0
$943$	−12.8461	−0.418327
$944$	8.39608	0.273269
$945$	0	0
$946$	−2.88577	−0.0938246
$947$	20.7544	0.674427	0.337214	−	0.941428i	$-0.390516\pi$
$947$	20.7544	0.674427	0.337214	+	0.941428i	$0.390516\pi$
$948$	0	0
$949$	13.3267	0.432602
$950$	31.0729	1.00814
$951$	0	0
$952$	3.98294	0.129088
$953$	−33.0722	−1.07131	−0.535657	−	0.844436i	$-0.679936\pi$
$953$	−33.0722	−1.07131	−0.535657	+	0.844436i	$0.679936\pi$
$954$	0	0
$955$	11.6638	0.377432
$956$	19.9977	0.646772
$957$	0	0
$958$	6.50916	0.210301
$959$	−13.0256	−0.420618
$960$	0	0
$961$	35.5928	1.14816
$962$	18.3229	0.590753
$963$	0	0
$964$	12.9019	0.415541
$965$	19.2616	0.620054
$966$	0	0
$967$	18.9226	0.608511	0.304255	−	0.952590i	$-0.401592\pi$
$967$	18.9226	0.608511	0.304255	+	0.952590i	$0.401592\pi$
$968$	1.67015	0.0536807
$969$	0	0
$970$	−12.9912	−0.417122
$971$	13.8653	0.444959	0.222480	−	0.974937i	$-0.428585\pi$
$971$	13.8653	0.444959	0.222480	+	0.974937i	$0.428585\pi$
$972$	0	0
$973$	−6.10088	−0.195585
$974$	−17.0058	−0.544902
$975$	0	0
$976$	−14.1320	−0.452356
$977$	16.4738	0.527042	0.263521	−	0.964654i	$-0.415116\pi$
$977$	16.4738	0.527042	0.263521	+	0.964654i	$0.415116\pi$
$978$	0	0
$979$	18.5488	0.592824
$980$	7.34574	0.234651
$981$	0	0
$982$	−26.8032	−0.855326
$983$	−0.130489	−0.00416194	−0.00208097	−	0.999998i	$-0.500662\pi$
$983$	−0.130489	−0.00416194	−0.00208097	+	0.999998i	$0.500662\pi$
$984$	0	0
$985$	13.1775	0.419870
$986$	−34.9209	−1.11211
$987$	0	0
$988$	−33.2068	−1.05645
$989$	−2.08286	−0.0662310
$990$	0	0
$991$	9.54294	0.303142	0.151571	−	0.988446i	$-0.451567\pi$
$991$	9.54294	0.303142	0.151571	+	0.988446i	$0.451567\pi$
$992$	−8.16044	−0.259094
$993$	0	0
$994$	1.93705	0.0614396
$995$	−26.9048	−0.852938
$996$	0	0
$997$	−35.1838	−1.11428	−0.557140	−	0.830418i	$-0.688101\pi$
$997$	−35.1838	−1.11428	−0.557140	+	0.830418i	$0.688101\pi$
$998$	−21.4285	−0.678308
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.l.1.5	✓	12
3.2	odd	2		8046.2.a.m.1.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.5	✓	12	1.1	even	1	trivial
8046.2.a.m.1.8	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.12518 q^{5} -0.686663 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.12518 q^{5} -0.686663 q^{7} -1.00000 q^{8} +1.12518 q^{10} +3.05448 q^{11} -3.99040 q^{13} +0.686663 q^{14} +1.00000 q^{16} +5.80043 q^{17} +8.32168 q^{19} -1.12518 q^{20} -3.05448 q^{22} -2.20462 q^{23} -3.73397 q^{25} +3.99040 q^{26} -0.686663 q^{28} +6.02040 q^{29} +8.16044 q^{31} -1.00000 q^{32} -5.80043 q^{34} +0.772621 q^{35} +4.59174 q^{37} -8.32168 q^{38} +1.12518 q^{40} +5.82690 q^{41} +0.944768 q^{43} +3.05448 q^{44} +2.20462 q^{46} -6.19531 q^{47} -6.52849 q^{49} +3.73397 q^{50} -3.99040 q^{52} -8.18515 q^{53} -3.43684 q^{55} +0.686663 q^{56} -6.02040 q^{58} +8.39608 q^{59} -14.1320 q^{61} -8.16044 q^{62} +1.00000 q^{64} +4.48992 q^{65} +6.33902 q^{67} +5.80043 q^{68} -0.772621 q^{70} +2.82097 q^{71} -3.33968 q^{73} -4.59174 q^{74} +8.32168 q^{76} -2.09740 q^{77} -8.49208 q^{79} -1.12518 q^{80} -5.82690 q^{82} +6.34293 q^{83} -6.52653 q^{85} -0.944768 q^{86} -3.05448 q^{88} +6.07267 q^{89} +2.74006 q^{91} -2.20462 q^{92} +6.19531 q^{94} -9.36339 q^{95} -11.5459 q^{97} +6.52849 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * q^98

Introduction

L-functions

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