LMFDB - Embedded newform 8019.2.a.i.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, 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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.724143 q^{2} -1.47562 q^{4} +2.15850 q^{5} -0.509330 q^{7} +2.51684 q^{8} +O(q^{10})$	\(q-0.724143 q^{2} -1.47562 q^{4} +2.15850 q^{5} -0.509330 q^{7} +2.51684 q^{8} -1.56306 q^{10} -1.00000 q^{11} -4.17016 q^{13} +0.368828 q^{14} +1.12868 q^{16} +1.93635 q^{17} -6.89214 q^{19} -3.18512 q^{20} +0.724143 q^{22} +1.75216 q^{23} -0.340885 q^{25} +3.01979 q^{26} +0.751576 q^{28} +7.66889 q^{29} -0.682974 q^{31} -5.85101 q^{32} -1.40220 q^{34} -1.09939 q^{35} +0.848056 q^{37} +4.99089 q^{38} +5.43260 q^{40} +8.95065 q^{41} +11.9425 q^{43} +1.47562 q^{44} -1.26881 q^{46} +0.714161 q^{47} -6.74058 q^{49} +0.246850 q^{50} +6.15356 q^{52} +6.72132 q^{53} -2.15850 q^{55} -1.28190 q^{56} -5.55337 q^{58} -11.8271 q^{59} -0.424113 q^{61} +0.494571 q^{62} +1.97961 q^{64} -9.00128 q^{65} -11.2701 q^{67} -2.85731 q^{68} +0.796114 q^{70} +12.4568 q^{71} -9.00610 q^{73} -0.614114 q^{74} +10.1702 q^{76} +0.509330 q^{77} +8.25308 q^{79} +2.43625 q^{80} -6.48155 q^{82} +2.56641 q^{83} +4.17961 q^{85} -8.64806 q^{86} -2.51684 q^{88} -6.86584 q^{89} +2.12399 q^{91} -2.58551 q^{92} -0.517155 q^{94} -14.8767 q^{95} -16.7142 q^{97} +4.88115 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * 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 $

$2$	−0.724143	−0.512046	−0.256023	−	0.966671i	$-0.582412\pi$
$2$	−0.724143	−0.512046	−0.256023	+	0.966671i	$0.582412\pi$
$3$	0	0
$3$	0	0
$4$	−1.47562	−0.737808
$5$	2.15850	0.965310	0.482655	−	0.875811i	$-0.339673\pi$
$5$	2.15850	0.965310	0.482655	+	0.875811i	$0.339673\pi$
$6$	0	0
$7$	−0.509330	−0.192509	−0.0962543	−	0.995357i	$-0.530686\pi$
$7$	−0.509330	−0.192509	−0.0962543	+	0.995357i	$0.530686\pi$
$8$	2.51684	0.889839
$9$	0	0
$10$	−1.56306	−0.494283
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.17016	−1.15659	−0.578297	−	0.815826i	$-0.696282\pi$
$13$	−4.17016	−1.15659	−0.578297	+	0.815826i	$0.696282\pi$
$14$	0.368828	0.0985734
$15$	0	0
$16$	1.12868	0.282170
$17$	1.93635	0.469634	0.234817	−	0.972040i	$-0.424551\pi$
$17$	1.93635	0.469634	0.234817	+	0.972040i	$0.424551\pi$
$18$	0	0
$19$	−6.89214	−1.58116	−0.790582	−	0.612356i	$-0.790222\pi$
$19$	−6.89214	−1.58116	−0.790582	+	0.612356i	$0.790222\pi$
$20$	−3.18512	−0.712214
$21$	0	0
$22$	0.724143	0.154388
$23$	1.75216	0.365350	0.182675	−	0.983173i	$-0.441524\pi$
$23$	1.75216	0.365350	0.182675	+	0.983173i	$0.441524\pi$
$24$	0	0
$25$	−0.340885	−0.0681771
$26$	3.01979	0.592230
$27$	0	0
$28$	0.751576	0.142034
$29$	7.66889	1.42408	0.712038	−	0.702141i	$-0.247772\pi$
$29$	7.66889	1.42408	0.712038	+	0.702141i	$0.247772\pi$
$30$	0	0
$31$	−0.682974	−0.122666	−0.0613329	−	0.998117i	$-0.519535\pi$
$31$	−0.682974	−0.122666	−0.0613329	+	0.998117i	$0.519535\pi$
$32$	−5.85101	−1.03432
$33$	0	0
$34$	−1.40220	−0.240475
$35$	−1.09939	−0.185830
$36$	0	0
$37$	0.848056	0.139420	0.0697098	−	0.997567i	$-0.477793\pi$
$37$	0.848056	0.139420	0.0697098	+	0.997567i	$0.477793\pi$
$38$	4.99089	0.809630
$39$	0	0
$40$	5.43260	0.858970
$41$	8.95065	1.39786	0.698928	−	0.715192i	$-0.253661\pi$
$41$	8.95065	1.39786	0.698928	+	0.715192i	$0.253661\pi$
$42$	0	0
$43$	11.9425	1.82121	0.910605	−	0.413279i	$-0.135616\pi$
$43$	11.9425	1.82121	0.910605	+	0.413279i	$0.135616\pi$
$44$	1.47562	0.222458
$45$	0	0
$46$	−1.26881	−0.187076
$47$	0.714161	0.104171	0.0520855	−	0.998643i	$-0.483413\pi$
$47$	0.714161	0.104171	0.0520855	+	0.998643i	$0.483413\pi$
$48$	0	0
$49$	−6.74058	−0.962940
$50$	0.246850	0.0349098
$51$	0	0
$52$	6.15356	0.853345
$53$	6.72132	0.923245	0.461622	−	0.887076i	$-0.347267\pi$
$53$	6.72132	0.923245	0.461622	+	0.887076i	$0.347267\pi$
$54$	0	0
$55$	−2.15850	−0.291052
$56$	−1.28190	−0.171302
$57$	0	0
$58$	−5.55337	−0.729193
$59$	−11.8271	−1.53976	−0.769881	−	0.638187i	$-0.779685\pi$
$59$	−11.8271	−1.53976	−0.769881	+	0.638187i	$0.779685\pi$
$60$	0	0
$61$	−0.424113	−0.0543020	−0.0271510	−	0.999631i	$-0.508644\pi$
$61$	−0.424113	−0.0543020	−0.0271510	+	0.999631i	$0.508644\pi$
$62$	0.494571	0.0628106
$63$	0	0
$64$	1.97961	0.247452
$65$	−9.00128	−1.11647
$66$	0	0
$67$	−11.2701	−1.37687	−0.688433	−	0.725300i	$-0.741701\pi$
$67$	−11.2701	−1.37687	−0.688433	+	0.725300i	$0.741701\pi$
$68$	−2.85731	−0.346500
$69$	0	0
$70$	0.796114	0.0951538
$71$	12.4568	1.47835	0.739173	−	0.673516i	$-0.235217\pi$
$71$	12.4568	1.47835	0.739173	+	0.673516i	$0.235217\pi$
$72$	0	0
$73$	−9.00610	−1.05408	−0.527042	−	0.849839i	$-0.676699\pi$
$73$	−9.00610	−1.05408	−0.527042	+	0.849839i	$0.676699\pi$
$74$	−0.614114	−0.0713893
$75$	0	0
$76$	10.1702	1.16660
$77$	0.509330	0.0580435
$78$	0	0
$79$	8.25308	0.928544	0.464272	−	0.885693i	$-0.346316\pi$
$79$	8.25308	0.928544	0.464272	+	0.885693i	$0.346316\pi$
$80$	2.43625	0.272381
$81$	0	0
$82$	−6.48155	−0.715767
$83$	2.56641	0.281700	0.140850	−	0.990031i	$-0.455016\pi$
$83$	2.56641	0.281700	0.140850	+	0.990031i	$0.455016\pi$
$84$	0	0
$85$	4.17961	0.453343
$86$	−8.64806	−0.932544
$87$	0	0
$88$	−2.51684	−0.268296
$89$	−6.86584	−0.727778	−0.363889	−	0.931442i	$-0.618551\pi$
$89$	−6.86584	−0.727778	−0.363889	+	0.931442i	$0.618551\pi$
$90$	0	0
$91$	2.12399	0.222654
$92$	−2.58551	−0.269558
$93$	0	0
$94$	−0.517155	−0.0533404
$95$	−14.8767	−1.52631
$96$	0	0
$97$	−16.7142	−1.69707	−0.848537	−	0.529137i	$-0.822516\pi$
$97$	−16.7142	−1.69707	−0.848537	+	0.529137i	$0.822516\pi$
$98$	4.88115	0.493070
$99$	0	0
$100$	0.503016	0.0503016
$101$	−4.83806	−0.481405	−0.240703	−	0.970599i	$-0.577378\pi$
$101$	−4.83806	−0.481405	−0.240703	+	0.970599i	$0.577378\pi$
$102$	0	0
$103$	9.18185	0.904715	0.452358	−	0.891837i	$-0.350583\pi$
$103$	9.18185	0.904715	0.452358	+	0.891837i	$0.350583\pi$
$104$	−10.4956	−1.02918
$105$	0	0
$106$	−4.86720	−0.472744
$107$	−4.68956	−0.453357	−0.226678	−	0.973970i	$-0.572787\pi$
$107$	−4.68956	−0.453357	−0.226678	+	0.973970i	$0.572787\pi$
$108$	0	0
$109$	−12.6401	−1.21071	−0.605353	−	0.795957i	$-0.706968\pi$
$109$	−12.6401	−1.21071	−0.605353	+	0.795957i	$0.706968\pi$
$110$	1.56306	0.149032
$111$	0	0
$112$	−0.574870	−0.0543201
$113$	−4.05635	−0.381590	−0.190795	−	0.981630i	$-0.561107\pi$
$113$	−4.05635	−0.381590	−0.190795	+	0.981630i	$0.561107\pi$
$114$	0	0
$115$	3.78203	0.352676
$116$	−11.3163	−1.05070
$117$	0	0
$118$	8.56454	0.788430
$119$	−0.986242	−0.0904087
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.307118	0.0278052
$123$	0	0
$124$	1.00781	0.0905039
$125$	−11.5283	−1.03112
$126$	0	0
$127$	−11.4012	−1.01169	−0.505847	−	0.862623i	$-0.668820\pi$
$127$	−11.4012	−1.01169	−0.505847	+	0.862623i	$0.668820\pi$
$128$	10.2685	0.907616
$129$	0	0
$130$	6.51821	0.571685
$131$	−2.92534	−0.255588	−0.127794	−	0.991801i	$-0.540790\pi$
$131$	−2.92534	−0.255588	−0.127794	+	0.991801i	$0.540790\pi$
$132$	0	0
$133$	3.51037	0.304388
$134$	8.16119	0.705019
$135$	0	0
$136$	4.87350	0.417899
$137$	8.39221	0.716995	0.358497	−	0.933531i	$-0.383289\pi$
$137$	8.39221	0.716995	0.358497	+	0.933531i	$0.383289\pi$
$138$	0	0
$139$	−5.18938	−0.440158	−0.220079	−	0.975482i	$-0.570631\pi$
$139$	−5.18938	−0.440158	−0.220079	+	0.975482i	$0.570631\pi$
$140$	1.62228	0.137107
$141$	0	0
$142$	−9.02048	−0.756982
$143$	4.17016	0.348726
$144$	0	0
$145$	16.5533	1.37468
$146$	6.52170	0.539740
$147$	0	0
$148$	−1.25141	−0.102865
$149$	−13.0673	−1.07051	−0.535256	−	0.844690i	$-0.679785\pi$
$149$	−13.0673	−1.07051	−0.535256	+	0.844690i	$0.679785\pi$
$150$	0	0
$151$	15.3073	1.24569	0.622847	−	0.782344i	$-0.285976\pi$
$151$	15.3073	1.24569	0.622847	+	0.782344i	$0.285976\pi$
$152$	−17.3464	−1.40698
$153$	0	0
$154$	−0.368828	−0.0297210
$155$	−1.47420	−0.118411
$156$	0	0
$157$	5.94889	0.474773	0.237387	−	0.971415i	$-0.423709\pi$
$157$	5.94889	0.474773	0.237387	+	0.971415i	$0.423709\pi$
$158$	−5.97641	−0.475458
$159$	0	0
$160$	−12.6294	−0.998442
$161$	−0.892426	−0.0703331
$162$	0	0
$163$	12.0236	0.941764	0.470882	−	0.882196i	$-0.343936\pi$
$163$	12.0236	0.941764	0.470882	+	0.882196i	$0.343936\pi$
$164$	−13.2077	−1.03135
$165$	0	0
$166$	−1.85845	−0.144244
$167$	−12.8937	−0.997741	−0.498871	−	0.866676i	$-0.666252\pi$
$167$	−12.8937	−0.997741	−0.498871	+	0.866676i	$0.666252\pi$
$168$	0	0
$169$	4.39022	0.337709
$170$	−3.02664	−0.232133
$171$	0	0
$172$	−17.6225	−1.34370
$173$	9.81386	0.746134	0.373067	−	0.927804i	$-0.378306\pi$
$173$	9.81386	0.746134	0.373067	+	0.927804i	$0.378306\pi$
$174$	0	0
$175$	0.173623	0.0131247
$176$	−1.12868	−0.0850774
$177$	0	0
$178$	4.97185	0.372656
$179$	22.3492	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$179$	22.3492	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$180$	0	0
$181$	−16.0287	−1.19140	−0.595701	−	0.803206i	$-0.703126\pi$
$181$	−16.0287	−1.19140	−0.595701	+	0.803206i	$0.703126\pi$
$182$	−1.53807	−0.114009
$183$	0	0
$184$	4.40991	0.325103
$185$	1.83053	0.134583
$186$	0	0
$187$	−1.93635	−0.141600
$188$	−1.05383	−0.0768583
$189$	0	0
$190$	10.7728	0.781543
$191$	−11.8737	−0.859148	−0.429574	−	0.903032i	$-0.641336\pi$
$191$	−11.8737	−0.859148	−0.429574	+	0.903032i	$0.641336\pi$
$192$	0	0
$193$	−15.4439	−1.11168	−0.555838	−	0.831291i	$-0.687602\pi$
$193$	−15.4439	−1.11168	−0.555838	+	0.831291i	$0.687602\pi$
$194$	12.1035	0.868980
$195$	0	0
$196$	9.94652	0.710466
$197$	3.32357	0.236795	0.118397	−	0.992966i	$-0.462224\pi$
$197$	3.32357	0.236795	0.118397	+	0.992966i	$0.462224\pi$
$198$	0	0
$199$	21.4203	1.51844	0.759222	−	0.650832i	$-0.225580\pi$
$199$	21.4203	1.51844	0.759222	+	0.650832i	$0.225580\pi$
$200$	−0.857955	−0.0606666
$201$	0	0
$202$	3.50345	0.246502
$203$	−3.90599	−0.274147
$204$	0	0
$205$	19.3200	1.34936
$206$	−6.64898	−0.463256
$207$	0	0
$208$	−4.70677	−0.326356
$209$	6.89214	0.476739
$210$	0	0
$211$	10.5251	0.724576	0.362288	−	0.932066i	$-0.381996\pi$
$211$	10.5251	0.724576	0.362288	+	0.932066i	$0.381996\pi$
$212$	−9.91810	−0.681178
$213$	0	0
$214$	3.39591	0.232140
$215$	25.7778	1.75803
$216$	0	0
$217$	0.347859	0.0236142
$218$	9.15327	0.619938
$219$	0	0
$220$	3.18512	0.214741
$221$	−8.07490	−0.543176
$222$	0	0
$223$	10.3522	0.693232	0.346616	−	0.938007i	$-0.387331\pi$
$223$	10.3522	0.693232	0.346616	+	0.938007i	$0.387331\pi$
$224$	2.98010	0.199116
$225$	0	0
$226$	2.93738	0.195392
$227$	−21.9953	−1.45988	−0.729941	−	0.683510i	$-0.760452\pi$
$227$	−21.9953	−1.45988	−0.729941	+	0.683510i	$0.760452\pi$
$228$	0	0
$229$	−9.50291	−0.627970	−0.313985	−	0.949428i	$-0.601664\pi$
$229$	−9.50291	−0.627970	−0.313985	+	0.949428i	$0.601664\pi$
$230$	−2.73873	−0.180587
$231$	0	0
$232$	19.3014	1.26720
$233$	−9.29529	−0.608955	−0.304477	−	0.952520i	$-0.598482\pi$
$233$	−9.29529	−0.608955	−0.304477	+	0.952520i	$0.598482\pi$
$234$	0	0
$235$	1.54151	0.100557
$236$	17.4523	1.13605
$237$	0	0
$238$	0.714181	0.0462935
$239$	10.8671	0.702932	0.351466	−	0.936201i	$-0.385683\pi$
$239$	10.8671	0.702932	0.351466	+	0.936201i	$0.385683\pi$
$240$	0	0
$241$	−15.5158	−0.999462	−0.499731	−	0.866181i	$-0.666568\pi$
$241$	−15.5158	−0.999462	−0.499731	+	0.866181i	$0.666568\pi$
$242$	−0.724143	−0.0465497
$243$	0	0
$244$	0.625828	0.0400645
$245$	−14.5495	−0.929536
$246$	0	0
$247$	28.7413	1.82876
$248$	−1.71894	−0.109153
$249$	0	0
$250$	8.34813	0.527982
$251$	−20.1726	−1.27328	−0.636640	−	0.771161i	$-0.719676\pi$
$251$	−20.1726	−1.27328	−0.636640	+	0.771161i	$0.719676\pi$
$252$	0	0
$253$	−1.75216	−0.110157
$254$	8.25610	0.518034
$255$	0	0
$256$	−11.3951	−0.712193
$257$	−13.8143	−0.861710	−0.430855	−	0.902421i	$-0.641788\pi$
$257$	−13.8143	−0.861710	−0.430855	+	0.902421i	$0.641788\pi$
$258$	0	0
$259$	−0.431940	−0.0268395
$260$	13.2824	0.823742
$261$	0	0
$262$	2.11837	0.130873
$263$	−5.00449	−0.308590	−0.154295	−	0.988025i	$-0.549311\pi$
$263$	−5.00449	−0.308590	−0.154295	+	0.988025i	$0.549311\pi$
$264$	0	0
$265$	14.5080	0.891217
$266$	−2.54201	−0.155861
$267$	0	0
$268$	16.6304	1.01586
$269$	−8.45546	−0.515539	−0.257769	−	0.966206i	$-0.582987\pi$
$269$	−8.45546	−0.515539	−0.257769	+	0.966206i	$0.582987\pi$
$270$	0	0
$271$	12.4012	0.753318	0.376659	−	0.926352i	$-0.377073\pi$
$271$	12.4012	0.753318	0.376659	+	0.926352i	$0.377073\pi$
$272$	2.18552	0.132517
$273$	0	0
$274$	−6.07716	−0.367135
$275$	0.340885	0.0205562
$276$	0	0
$277$	−11.1714	−0.671225	−0.335613	−	0.942000i	$-0.608943\pi$
$277$	−11.1714	−0.671225	−0.335613	+	0.942000i	$0.608943\pi$
$278$	3.75786	0.225381
$279$	0	0
$280$	−2.76699	−0.165359
$281$	−28.4479	−1.69706	−0.848529	−	0.529149i	$-0.822511\pi$
$281$	−28.4479	−1.69706	−0.848529	+	0.529149i	$0.822511\pi$
$282$	0	0
$283$	−14.9251	−0.887202	−0.443601	−	0.896224i	$-0.646299\pi$
$283$	−14.9251	−0.887202	−0.443601	+	0.896224i	$0.646299\pi$
$284$	−18.3814	−1.09074
$285$	0	0
$286$	−3.01979	−0.178564
$287$	−4.55883	−0.269099
$288$	0	0
$289$	−13.2505	−0.779443
$290$	−11.9869	−0.703898
$291$	0	0
$292$	13.2895	0.777712
$293$	−18.6181	−1.08768	−0.543840	−	0.839189i	$-0.683030\pi$
$293$	−18.6181	−1.08768	−0.543840	+	0.839189i	$0.683030\pi$
$294$	0	0
$295$	−25.5289	−1.48635
$296$	2.13442	0.124061
$297$	0	0
$298$	9.46256	0.548152
$299$	−7.30677	−0.422562
$300$	0	0
$301$	−6.08266	−0.350599
$302$	−11.0847	−0.637853
$303$	0	0
$304$	−7.77901	−0.446157
$305$	−0.915446	−0.0524183
$306$	0	0
$307$	−17.2804	−0.986246	−0.493123	−	0.869960i	$-0.664145\pi$
$307$	−17.2804	−0.986246	−0.493123	+	0.869960i	$0.664145\pi$
$308$	−0.751576	−0.0428250
$309$	0	0
$310$	1.06753	0.0606317
$311$	9.72891	0.551676	0.275838	−	0.961204i	$-0.411045\pi$
$311$	9.72891	0.551676	0.275838	+	0.961204i	$0.411045\pi$
$312$	0	0
$313$	30.7977	1.74079	0.870394	−	0.492356i	$-0.163864\pi$
$313$	30.7977	1.74079	0.870394	+	0.492356i	$0.163864\pi$
$314$	−4.30785	−0.243106
$315$	0	0
$316$	−12.1784	−0.685087
$317$	23.8528	1.33970	0.669852	−	0.742494i	$-0.266357\pi$
$317$	23.8528	1.33970	0.669852	+	0.742494i	$0.266357\pi$
$318$	0	0
$319$	−7.66889	−0.429375
$320$	4.27299	0.238867
$321$	0	0
$322$	0.646244	0.0360138
$323$	−13.3456	−0.742569
$324$	0	0
$325$	1.42155	0.0788532
$326$	−8.70683	−0.482227
$327$	0	0
$328$	22.5274	1.24387
$329$	−0.363743	−0.0200538
$330$	0	0
$331$	−5.83558	−0.320753	−0.160376	−	0.987056i	$-0.551271\pi$
$331$	−5.83558	−0.320753	−0.160376	+	0.987056i	$0.551271\pi$
$332$	−3.78704	−0.207841
$333$	0	0
$334$	9.33686	0.510890
$335$	−24.3266	−1.32910
$336$	0	0
$337$	−8.50521	−0.463308	−0.231654	−	0.972798i	$-0.574414\pi$
$337$	−8.50521	−0.463308	−0.231654	+	0.972798i	$0.574414\pi$
$338$	−3.17915	−0.172923
$339$	0	0
$340$	−6.16751	−0.334480
$341$	0.682974	0.0369851
$342$	0	0
$343$	6.99849	0.377883
$344$	30.0573	1.62058
$345$	0	0
$346$	−7.10664	−0.382055
$347$	−13.5442	−0.727093	−0.363546	−	0.931576i	$-0.618434\pi$
$347$	−13.5442	−0.727093	−0.363546	+	0.931576i	$0.618434\pi$
$348$	0	0
$349$	−7.77516	−0.416195	−0.208097	−	0.978108i	$-0.566727\pi$
$349$	−7.77516	−0.416195	−0.208097	+	0.978108i	$0.566727\pi$
$350$	−0.125728	−0.00672044
$351$	0	0
$352$	5.85101	0.311860
$353$	−8.84040	−0.470527	−0.235263	−	0.971932i	$-0.575595\pi$
$353$	−8.84040	−0.470527	−0.235263	+	0.971932i	$0.575595\pi$
$354$	0	0
$355$	26.8879	1.42706
$356$	10.1314	0.536961
$357$	0	0
$358$	−16.1840	−0.855352
$359$	27.5733	1.45526	0.727631	−	0.685969i	$-0.240621\pi$
$359$	27.5733	1.45526	0.727631	+	0.685969i	$0.240621\pi$
$360$	0	0
$361$	28.5015	1.50008
$362$	11.6070	0.610053
$363$	0	0
$364$	−3.13419	−0.164276
$365$	−19.4396	−1.01752
$366$	0	0
$367$	27.5711	1.43920	0.719599	−	0.694389i	$-0.244325\pi$
$367$	27.5711	1.43920	0.719599	+	0.694389i	$0.244325\pi$
$368$	1.97762	0.103091
$369$	0	0
$370$	−1.32556	−0.0689128
$371$	−3.42337	−0.177733
$372$	0	0
$373$	−20.4822	−1.06053	−0.530264	−	0.847832i	$-0.677907\pi$
$373$	−20.4822	−1.06053	−0.530264	+	0.847832i	$0.677907\pi$
$374$	1.40220	0.0725058
$375$	0	0
$376$	1.79743	0.0926954
$377$	−31.9805	−1.64708
$378$	0	0
$379$	23.3472	1.19926	0.599632	−	0.800276i	$-0.295314\pi$
$379$	23.3472	1.19926	0.599632	+	0.800276i	$0.295314\pi$
$380$	21.9523	1.12613
$381$	0	0
$382$	8.59823	0.439923
$383$	−31.4282	−1.60590	−0.802952	−	0.596043i	$-0.796739\pi$
$383$	−31.4282	−1.60590	−0.802952	+	0.596043i	$0.796739\pi$
$384$	0	0
$385$	1.09939	0.0560300
$386$	11.1836	0.569230
$387$	0	0
$388$	24.6638	1.25212
$389$	35.2565	1.78758	0.893788	−	0.448489i	$-0.148038\pi$
$389$	35.2565	1.78758	0.893788	+	0.448489i	$0.148038\pi$
$390$	0	0
$391$	3.39280	0.171581
$392$	−16.9650	−0.856862
$393$	0	0
$394$	−2.40674	−0.121250
$395$	17.8143	0.896332
$396$	0	0
$397$	34.9720	1.75520	0.877598	−	0.479397i	$-0.159145\pi$
$397$	34.9720	1.75520	0.877598	+	0.479397i	$0.159145\pi$
$398$	−15.5114	−0.777514
$399$	0	0
$400$	−0.384750	−0.0192375
$401$	−22.3173	−1.11447	−0.557236	−	0.830354i	$-0.688138\pi$
$401$	−22.3173	−1.11447	−0.557236	+	0.830354i	$0.688138\pi$
$402$	0	0
$403$	2.84811	0.141875
$404$	7.13913	0.355185
$405$	0	0
$406$	2.82850	0.140376
$407$	−0.848056	−0.0420366
$408$	0	0
$409$	−29.3429	−1.45091	−0.725457	−	0.688267i	$-0.758372\pi$
$409$	−29.3429	−1.45091	−0.725457	+	0.688267i	$0.758372\pi$
$410$	−13.9904	−0.690937
$411$	0	0
$412$	−13.5489	−0.667506
$413$	6.02392	0.296418
$414$	0	0
$415$	5.53959	0.271928
$416$	24.3996	1.19629
$417$	0	0
$418$	−4.99089	−0.244113
$419$	−17.5567	−0.857702	−0.428851	−	0.903375i	$-0.641081\pi$
$419$	−17.5567	−0.857702	−0.428851	+	0.903375i	$0.641081\pi$
$420$	0	0
$421$	19.6366	0.957031	0.478515	−	0.878079i	$-0.341175\pi$
$421$	19.6366	0.957031	0.478515	+	0.878079i	$0.341175\pi$
$422$	−7.62166	−0.371017
$423$	0	0
$424$	16.9165	0.821539
$425$	−0.660074	−0.0320183
$426$	0	0
$427$	0.216013	0.0104536
$428$	6.91999	0.334491
$429$	0	0
$430$	−18.6668	−0.900194
$431$	−15.6822	−0.755384	−0.377692	−	0.925931i	$-0.623282\pi$
$431$	−15.6822	−0.755384	−0.377692	+	0.925931i	$0.623282\pi$
$432$	0	0
$433$	−31.8457	−1.53041	−0.765203	−	0.643789i	$-0.777361\pi$
$433$	−31.8457	−1.53041	−0.765203	+	0.643789i	$0.777361\pi$
$434$	−0.251900	−0.0120916
$435$	0	0
$436$	18.6520	0.893269
$437$	−12.0761	−0.577679
$438$	0	0
$439$	−23.9357	−1.14239	−0.571194	−	0.820815i	$-0.693520\pi$
$439$	−23.9357	−1.14239	−0.571194	+	0.820815i	$0.693520\pi$
$440$	−5.43260	−0.258989
$441$	0	0
$442$	5.84738	0.278132
$443$	1.21644	0.0577949	0.0288974	−	0.999582i	$-0.490800\pi$
$443$	1.21644	0.0577949	0.0288974	+	0.999582i	$0.490800\pi$
$444$	0	0
$445$	−14.8199	−0.702531
$446$	−7.49645	−0.354967
$447$	0	0
$448$	−1.00828	−0.0476366
$449$	−3.66747	−0.173079	−0.0865394	−	0.996248i	$-0.527581\pi$
$449$	−3.66747	−0.173079	−0.0865394	+	0.996248i	$0.527581\pi$
$450$	0	0
$451$	−8.95065	−0.421470
$452$	5.98562	0.281540
$453$	0	0
$454$	15.9278	0.747527
$455$	4.58462	0.214930
$456$	0	0
$457$	−21.3178	−0.997204	−0.498602	−	0.866831i	$-0.666153\pi$
$457$	−21.3178	−0.997204	−0.498602	+	0.866831i	$0.666153\pi$
$458$	6.88147	0.321550
$459$	0	0
$460$	−5.58083	−0.260207
$461$	−22.3784	−1.04227	−0.521133	−	0.853475i	$-0.674491\pi$
$461$	−22.3784	−1.04227	−0.521133	+	0.853475i	$0.674491\pi$
$462$	0	0
$463$	−11.3768	−0.528724	−0.264362	−	0.964423i	$-0.585161\pi$
$463$	−11.3768	−0.528724	−0.264362	+	0.964423i	$0.585161\pi$
$464$	8.65571	0.401831
$465$	0	0
$466$	6.73112	0.311813
$467$	8.61284	0.398555	0.199277	−	0.979943i	$-0.436141\pi$
$467$	8.61284	0.398555	0.199277	+	0.979943i	$0.436141\pi$
$468$	0	0
$469$	5.74022	0.265059
$470$	−1.11628	−0.0514900
$471$	0	0
$472$	−29.7671	−1.37014
$473$	−11.9425	−0.549115
$474$	0	0
$475$	2.34943	0.107799
$476$	1.45532	0.0667043
$477$	0	0
$478$	−7.86931	−0.359934
$479$	−20.0089	−0.914228	−0.457114	−	0.889408i	$-0.651117\pi$
$479$	−20.0089	−0.914228	−0.457114	+	0.889408i	$0.651117\pi$
$480$	0	0
$481$	−3.53653	−0.161252
$482$	11.2357	0.511771
$483$	0	0
$484$	−1.47562	−0.0670735
$485$	−36.0776	−1.63820
$486$	0	0
$487$	−4.86040	−0.220246	−0.110123	−	0.993918i	$-0.535124\pi$
$487$	−4.86040	−0.220246	−0.110123	+	0.993918i	$0.535124\pi$
$488$	−1.06743	−0.0483201
$489$	0	0
$490$	10.5359	0.475966
$491$	14.8073	0.668246	0.334123	−	0.942529i	$-0.391560\pi$
$491$	14.8073	0.668246	0.334123	+	0.942529i	$0.391560\pi$
$492$	0	0
$493$	14.8497	0.668796
$494$	−20.8128	−0.936413
$495$	0	0
$496$	−0.770859	−0.0346126
$497$	−6.34460	−0.284594
$498$	0	0
$499$	−23.9242	−1.07099	−0.535497	−	0.844537i	$-0.679876\pi$
$499$	−23.9242	−1.07099	−0.535497	+	0.844537i	$0.679876\pi$
$500$	17.0113	0.760770
$501$	0	0
$502$	14.6078	0.651979
$503$	−0.419577	−0.0187080	−0.00935400	−	0.999956i	$-0.502978\pi$
$503$	−0.419577	−0.0187080	−0.00935400	+	0.999956i	$0.502978\pi$
$504$	0	0
$505$	−10.4430	−0.464705
$506$	1.26881	0.0564056
$507$	0	0
$508$	16.8238	0.746436
$509$	8.51681	0.377501	0.188750	−	0.982025i	$-0.439556\pi$
$509$	8.51681	0.377501	0.188750	+	0.982025i	$0.439556\pi$
$510$	0	0
$511$	4.58707	0.202920
$512$	−12.2853	−0.542940
$513$	0	0
$514$	10.0035	0.441236
$515$	19.8190	0.873330
$516$	0	0
$517$	−0.714161	−0.0314088
$518$	0.312787	0.0137431
$519$	0	0
$520$	−22.6548	−0.993479
$521$	−11.5681	−0.506806	−0.253403	−	0.967361i	$-0.581550\pi$
$521$	−11.5681	−0.506806	−0.253403	+	0.967361i	$0.581550\pi$
$522$	0	0
$523$	−30.2456	−1.32255	−0.661275	−	0.750144i	$-0.729984\pi$
$523$	−30.2456	−1.32255	−0.661275	+	0.750144i	$0.729984\pi$
$524$	4.31669	0.188575
$525$	0	0
$526$	3.62396	0.158012
$527$	−1.32248	−0.0576081
$528$	0	0
$529$	−19.9299	−0.866519
$530$	−10.5058	−0.456345
$531$	0	0
$532$	−5.17996	−0.224580
$533$	−37.3256	−1.61675
$534$	0	0
$535$	−10.1224	−0.437630
$536$	−28.3652	−1.22519
$537$	0	0
$538$	6.12296	0.263980
$539$	6.74058	0.290337
$540$	0	0
$541$	−20.3677	−0.875676	−0.437838	−	0.899054i	$-0.644256\pi$
$541$	−20.3677	−0.875676	−0.437838	+	0.899054i	$0.644256\pi$
$542$	−8.98022	−0.385734
$543$	0	0
$544$	−11.3296	−0.485754
$545$	−27.2837	−1.16871
$546$	0	0
$547$	39.3764	1.68361	0.841806	−	0.539780i	$-0.181493\pi$
$547$	39.3764	1.68361	0.841806	+	0.539780i	$0.181493\pi$
$548$	−12.3837	−0.529005
$549$	0	0
$550$	−0.246850	−0.0105257
$551$	−52.8550	−2.25170
$552$	0	0
$553$	−4.20354	−0.178753
$554$	8.08970	0.343699
$555$	0	0
$556$	7.65754	0.324752
$557$	−25.2840	−1.07132	−0.535659	−	0.844435i	$-0.679937\pi$
$557$	−25.2840	−1.07132	−0.535659	+	0.844435i	$0.679937\pi$
$558$	0	0
$559$	−49.8020	−2.10640
$560$	−1.24086	−0.0524357
$561$	0	0
$562$	20.6003	0.868973
$563$	−1.72267	−0.0726020	−0.0363010	−	0.999341i	$-0.511558\pi$
$563$	−1.72267	−0.0726020	−0.0363010	+	0.999341i	$0.511558\pi$
$564$	0	0
$565$	−8.75563	−0.368352
$566$	10.8079	0.454289
$567$	0	0
$568$	31.3517	1.31549
$569$	42.4559	1.77984	0.889922	−	0.456113i	$-0.150759\pi$
$569$	42.4559	1.77984	0.889922	+	0.456113i	$0.150759\pi$
$570$	0	0
$571$	−11.3508	−0.475018	−0.237509	−	0.971385i	$-0.576331\pi$
$571$	−11.3508	−0.475018	−0.237509	+	0.971385i	$0.576331\pi$
$572$	−6.15356	−0.257293
$573$	0	0
$574$	3.30125	0.137791
$575$	−0.597285	−0.0249085
$576$	0	0
$577$	−7.81696	−0.325424	−0.162712	−	0.986674i	$-0.552024\pi$
$577$	−7.81696	−0.325424	−0.162712	+	0.986674i	$0.552024\pi$
$578$	9.59529	0.399111
$579$	0	0
$580$	−24.4263	−1.01425
$581$	−1.30715	−0.0542297
$582$	0	0
$583$	−6.72132	−0.278369
$584$	−22.6669	−0.937964
$585$	0	0
$586$	13.4821	0.556942
$587$	14.7352	0.608185	0.304092	−	0.952643i	$-0.401647\pi$
$587$	14.7352	0.608185	0.304092	+	0.952643i	$0.401647\pi$
$588$	0	0
$589$	4.70715	0.193955
$590$	18.4865	0.761079
$591$	0	0
$592$	0.957183	0.0393400
$593$	−3.69214	−0.151618	−0.0758091	−	0.997122i	$-0.524154\pi$
$593$	−3.69214	−0.151618	−0.0758091	+	0.997122i	$0.524154\pi$
$594$	0	0
$595$	−2.12880	−0.0872724
$596$	19.2823	0.789832
$597$	0	0
$598$	5.29115	0.216371
$599$	3.45829	0.141302	0.0706509	−	0.997501i	$-0.477492\pi$
$599$	3.45829	0.141302	0.0706509	+	0.997501i	$0.477492\pi$
$600$	0	0
$601$	13.1702	0.537223	0.268611	−	0.963249i	$-0.413435\pi$
$601$	13.1702	0.537223	0.268611	+	0.963249i	$0.413435\pi$
$602$	4.40471	0.179523
$603$	0	0
$604$	−22.5878	−0.919083
$605$	2.15850	0.0877554
$606$	0	0
$607$	−0.178431	−0.00724230	−0.00362115	−	0.999993i	$-0.501153\pi$
$607$	−0.178431	−0.00724230	−0.00362115	+	0.999993i	$0.501153\pi$
$608$	40.3260	1.63543
$609$	0	0
$610$	0.662914	0.0268406
$611$	−2.97816	−0.120484
$612$	0	0
$613$	−47.1547	−1.90456	−0.952281	−	0.305224i	$-0.901269\pi$
$613$	−47.1547	−1.90456	−0.952281	+	0.305224i	$0.901269\pi$
$614$	12.5135	0.505004
$615$	0	0
$616$	1.28190	0.0516494
$617$	−19.3174	−0.777688	−0.388844	−	0.921304i	$-0.627126\pi$
$617$	−19.3174	−0.777688	−0.388844	+	0.921304i	$0.627126\pi$
$618$	0	0
$619$	−15.8572	−0.637353	−0.318677	−	0.947863i	$-0.603238\pi$
$619$	−15.8572	−0.637353	−0.318677	+	0.947863i	$0.603238\pi$
$620$	2.17535	0.0873643
$621$	0	0
$622$	−7.04512	−0.282484
$623$	3.49698	0.140104
$624$	0	0
$625$	−23.1794	−0.927175
$626$	−22.3019	−0.891364
$627$	0	0
$628$	−8.77828	−0.350292
$629$	1.64214	0.0654762
$630$	0	0
$631$	44.4044	1.76771	0.883855	−	0.467761i	$-0.154939\pi$
$631$	44.4044	1.76771	0.883855	+	0.467761i	$0.154939\pi$
$632$	20.7717	0.826254
$633$	0	0
$634$	−17.2728	−0.685991
$635$	−24.6095	−0.976597
$636$	0	0
$637$	28.1093	1.11373
$638$	5.55337	0.219860
$639$	0	0
$640$	22.1645	0.876131
$641$	−6.57176	−0.259569	−0.129784	−	0.991542i	$-0.541429\pi$
$641$	−6.57176	−0.259569	−0.129784	+	0.991542i	$0.541429\pi$
$642$	0	0
$643$	10.7622	0.424418	0.212209	−	0.977224i	$-0.431934\pi$
$643$	10.7622	0.424418	0.212209	+	0.977224i	$0.431934\pi$
$644$	1.31688	0.0518923
$645$	0	0
$646$	9.66413	0.380230
$647$	−16.3740	−0.643729	−0.321865	−	0.946786i	$-0.604310\pi$
$647$	−16.3740	−0.643729	−0.321865	+	0.946786i	$0.604310\pi$
$648$	0	0
$649$	11.8271	0.464256
$650$	−1.02940	−0.0403765
$651$	0	0
$652$	−17.7423	−0.694841
$653$	27.6482	1.08196	0.540978	−	0.841037i	$-0.318054\pi$
$653$	27.6482	1.08196	0.540978	+	0.841037i	$0.318054\pi$
$654$	0	0
$655$	−6.31435	−0.246722
$656$	10.1024	0.394433
$657$	0	0
$658$	0.263402	0.0102685
$659$	−1.65632	−0.0645209	−0.0322605	−	0.999479i	$-0.510271\pi$
$659$	−1.65632	−0.0645209	−0.0322605	+	0.999479i	$0.510271\pi$
$660$	0	0
$661$	22.5031	0.875271	0.437635	−	0.899152i	$-0.355816\pi$
$661$	22.5031	0.875271	0.437635	+	0.899152i	$0.355816\pi$
$662$	4.22580	0.164240
$663$	0	0
$664$	6.45925	0.250668
$665$	7.57713	0.293829
$666$	0	0
$667$	13.4371	0.520287
$668$	19.0261	0.736142
$669$	0	0
$670$	17.6159	0.680562
$671$	0.424113	0.0163727
$672$	0	0
$673$	−24.1240	−0.929913	−0.464956	−	0.885334i	$-0.653930\pi$
$673$	−24.1240	−0.929913	−0.464956	+	0.885334i	$0.653930\pi$
$674$	6.15899	0.237235
$675$	0	0
$676$	−6.47828	−0.249165
$677$	−18.3770	−0.706287	−0.353143	−	0.935569i	$-0.614887\pi$
$677$	−18.3770	−0.706287	−0.353143	+	0.935569i	$0.614887\pi$
$678$	0	0
$679$	8.51306	0.326701
$680$	10.5194	0.403402
$681$	0	0
$682$	−0.494571	−0.0189381
$683$	−2.65208	−0.101479	−0.0507395	−	0.998712i	$-0.516158\pi$
$683$	−2.65208	−0.101479	−0.0507395	+	0.998712i	$0.516158\pi$
$684$	0	0
$685$	18.1146	0.692122
$686$	−5.06791	−0.193494
$687$	0	0
$688$	13.4792	0.513890
$689$	−28.0290	−1.06782
$690$	0	0
$691$	−34.9604	−1.32995	−0.664977	−	0.746863i	$-0.731559\pi$
$691$	−34.9604	−1.32995	−0.664977	+	0.746863i	$0.731559\pi$
$692$	−14.4815	−0.550504
$693$	0	0
$694$	9.80796	0.372305
$695$	−11.2013	−0.424889
$696$	0	0
$697$	17.3316	0.656482
$698$	5.63033	0.213111
$699$	0	0
$700$	−0.256201	−0.00968349
$701$	−2.71694	−0.102617	−0.0513087	−	0.998683i	$-0.516339\pi$
$701$	−2.71694	−0.102617	−0.0513087	+	0.998683i	$0.516339\pi$
$702$	0	0
$703$	−5.84492	−0.220445
$704$	−1.97961	−0.0746095
$705$	0	0
$706$	6.40171	0.240932
$707$	2.46417	0.0926747
$708$	0	0
$709$	4.32021	0.162249	0.0811244	−	0.996704i	$-0.474149\pi$
$709$	4.32021	0.162249	0.0811244	+	0.996704i	$0.474149\pi$
$710$	−19.4707	−0.730722
$711$	0	0
$712$	−17.2803	−0.647605
$713$	−1.19668	−0.0448160
$714$	0	0
$715$	9.00128	0.336629
$716$	−32.9789	−1.23248
$717$	0	0
$718$	−19.9670	−0.745162
$719$	−0.0967893	−0.00360963	−0.00180482	−	0.999998i	$-0.500574\pi$
$719$	−0.0967893	−0.00360963	−0.00180482	+	0.999998i	$0.500574\pi$
$720$	0	0
$721$	−4.67659	−0.174165
$722$	−20.6392	−0.768111
$723$	0	0
$724$	23.6522	0.879026
$725$	−2.61421	−0.0970894
$726$	0	0
$727$	−15.0363	−0.557664	−0.278832	−	0.960340i	$-0.589947\pi$
$727$	−15.0363	−0.557664	−0.278832	+	0.960340i	$0.589947\pi$
$728$	5.34574	0.198126
$729$	0	0
$730$	14.0771	0.521016
$731$	23.1248	0.855303
$732$	0	0
$733$	26.0831	0.963401	0.481701	−	0.876336i	$-0.340019\pi$
$733$	26.0831	0.963401	0.481701	+	0.876336i	$0.340019\pi$
$734$	−19.9654	−0.736937
$735$	0	0
$736$	−10.2519	−0.377890
$737$	11.2701	0.415141
$738$	0	0
$739$	−37.9381	−1.39558	−0.697788	−	0.716305i	$-0.745832\pi$
$739$	−37.9381	−1.39558	−0.697788	+	0.716305i	$0.745832\pi$
$740$	−2.70116	−0.0992965
$741$	0	0
$742$	2.47901	0.0910074
$743$	38.4065	1.40900	0.704499	−	0.709705i	$-0.251172\pi$
$743$	38.4065	1.40900	0.704499	+	0.709705i	$0.251172\pi$
$744$	0	0
$745$	−28.2057	−1.03338
$746$	14.8321	0.543040
$747$	0	0
$748$	2.85731	0.104474
$749$	2.38853	0.0872751
$750$	0	0
$751$	−25.0720	−0.914889	−0.457445	−	0.889238i	$-0.651235\pi$
$751$	−25.0720	−0.914889	−0.457445	+	0.889238i	$0.651235\pi$
$752$	0.806058	0.0293939
$753$	0	0
$754$	23.1584	0.843381
$755$	33.0409	1.20248
$756$	0	0
$757$	0.840471	0.0305474	0.0152737	−	0.999883i	$-0.495138\pi$
$757$	0.840471	0.0305474	0.0152737	+	0.999883i	$0.495138\pi$
$758$	−16.9067	−0.614079
$759$	0	0
$760$	−37.4422	−1.35817
$761$	12.8481	0.465742	0.232871	−	0.972508i	$-0.425188\pi$
$761$	12.8481	0.465742	0.232871	+	0.972508i	$0.425188\pi$
$762$	0	0
$763$	6.43800	0.233071
$764$	17.5210	0.633886
$765$	0	0
$766$	22.7585	0.822298
$767$	49.3210	1.78088
$768$	0	0
$769$	39.9156	1.43939	0.719697	−	0.694288i	$-0.244281\pi$
$769$	39.9156	1.43939	0.719697	+	0.694288i	$0.244281\pi$
$770$	−0.796114	−0.0286900
$771$	0	0
$772$	22.7893	0.820204
$773$	2.55206	0.0917911	0.0458955	−	0.998946i	$-0.485386\pi$
$773$	2.55206	0.0917911	0.0458955	+	0.998946i	$0.485386\pi$
$774$	0	0
$775$	0.232816	0.00836300
$776$	−42.0671	−1.51012
$777$	0	0
$778$	−25.5308	−0.915322
$779$	−61.6891	−2.21024
$780$	0	0
$781$	−12.4568	−0.445738
$782$	−2.45687	−0.0878575
$783$	0	0
$784$	−7.60795	−0.271713
$785$	12.8407	0.458303
$786$	0	0
$787$	−36.3967	−1.29740	−0.648702	−	0.761043i	$-0.724688\pi$
$787$	−36.3967	−1.29740	−0.648702	+	0.761043i	$0.724688\pi$
$788$	−4.90432	−0.174709
$789$	0	0
$790$	−12.9001	−0.458964
$791$	2.06602	0.0734593
$792$	0	0
$793$	1.76862	0.0628054
$794$	−25.3248	−0.898742
$795$	0	0
$796$	−31.6081	−1.12032
$797$	46.1044	1.63310	0.816551	−	0.577273i	$-0.195883\pi$
$797$	46.1044	1.63310	0.816551	+	0.577273i	$0.195883\pi$
$798$	0	0
$799$	1.38287	0.0489223
$800$	1.99452	0.0705171
$801$	0	0
$802$	16.1609	0.570661
$803$	9.00610	0.317818
$804$	0	0
$805$	−1.92630	−0.0678932
$806$	−2.06244	−0.0726464
$807$	0	0
$808$	−12.1766	−0.428373
$809$	27.7521	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$809$	27.7521	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$810$	0	0
$811$	−29.6241	−1.04024	−0.520122	−	0.854092i	$-0.674114\pi$
$811$	−29.6241	−1.04024	−0.520122	+	0.854092i	$0.674114\pi$
$812$	5.76375	0.202268
$813$	0	0
$814$	0.614114	0.0215247
$815$	25.9530	0.909094
$816$	0	0
$817$	−82.3091	−2.87963
$818$	21.2485	0.742936
$819$	0	0
$820$	−28.5089	−0.995572
$821$	22.2924	0.778010	0.389005	−	0.921236i	$-0.372819\pi$
$821$	22.2924	0.778010	0.389005	+	0.921236i	$0.372819\pi$
$822$	0	0
$823$	−18.4210	−0.642117	−0.321059	−	0.947059i	$-0.604039\pi$
$823$	−18.4210	−0.642117	−0.321059	+	0.947059i	$0.604039\pi$
$824$	23.1093	0.805050
$825$	0	0
$826$	−4.36218	−0.151780
$827$	−15.8607	−0.551531	−0.275766	−	0.961225i	$-0.588931\pi$
$827$	−15.8607	−0.551531	−0.275766	+	0.961225i	$0.588931\pi$
$828$	0	0
$829$	−31.3842	−1.09002	−0.545009	−	0.838430i	$-0.683474\pi$
$829$	−31.3842	−1.09002	−0.545009	+	0.838430i	$0.683474\pi$
$830$	−4.01146	−0.139240
$831$	0	0
$832$	−8.25530	−0.286201
$833$	−13.0521	−0.452230
$834$	0	0
$835$	−27.8309	−0.963130
$836$	−10.1702	−0.351742
$837$	0	0
$838$	12.7136	0.439183
$839$	−41.8463	−1.44469	−0.722347	−	0.691531i	$-0.756937\pi$
$839$	−41.8463	−1.44469	−0.722347	+	0.691531i	$0.756937\pi$
$840$	0	0
$841$	29.8118	1.02799
$842$	−14.2197	−0.490044
$843$	0	0
$844$	−15.5310	−0.534598
$845$	9.47628	0.325994
$846$	0	0
$847$	−0.509330	−0.0175008
$848$	7.58622	0.260512
$849$	0	0
$850$	0.477988	0.0163949
$851$	1.48593	0.0509369
$852$	0	0
$853$	−0.921789	−0.0315614	−0.0157807	−	0.999875i	$-0.505023\pi$
$853$	−0.921789	−0.0315614	−0.0157807	+	0.999875i	$0.505023\pi$
$854$	−0.156424	−0.00535274
$855$	0	0
$856$	−11.8029	−0.403415
$857$	−34.0385	−1.16273	−0.581367	−	0.813641i	$-0.697482\pi$
$857$	−34.0385	−1.16273	−0.581367	+	0.813641i	$0.697482\pi$
$858$	0	0
$859$	−21.8869	−0.746772	−0.373386	−	0.927676i	$-0.621803\pi$
$859$	−21.8869	−0.746772	−0.373386	+	0.927676i	$0.621803\pi$
$860$	−38.0382	−1.29709
$861$	0	0
$862$	11.3561	0.386792
$863$	46.7103	1.59004	0.795018	−	0.606586i	$-0.207461\pi$
$863$	46.7103	1.59004	0.795018	+	0.606586i	$0.207461\pi$
$864$	0	0
$865$	21.1832	0.720250
$866$	23.0608	0.783639
$867$	0	0
$868$	−0.513307	−0.0174228
$869$	−8.25308	−0.279966
$870$	0	0
$871$	46.9982	1.59247
$872$	−31.8133	−1.07733
$873$	0	0
$874$	8.74483	0.295798
$875$	5.87170	0.198500
$876$	0	0
$877$	23.4354	0.791356	0.395678	−	0.918389i	$-0.370510\pi$
$877$	23.4354	0.791356	0.395678	+	0.918389i	$0.370510\pi$
$878$	17.3329	0.584956
$879$	0	0
$880$	−2.43625	−0.0821260
$881$	49.9423	1.68260	0.841300	−	0.540569i	$-0.181791\pi$
$881$	49.9423	1.68260	0.841300	+	0.540569i	$0.181791\pi$
$882$	0	0
$883$	−14.4253	−0.485451	−0.242725	−	0.970095i	$-0.578041\pi$
$883$	−14.4253	−0.485451	−0.242725	+	0.970095i	$0.578041\pi$
$884$	11.9155	0.400760
$885$	0	0
$886$	−0.880878	−0.0295937
$887$	−5.48153	−0.184052	−0.0920259	−	0.995757i	$-0.529334\pi$
$887$	−5.48153	−0.184052	−0.0920259	+	0.995757i	$0.529334\pi$
$888$	0	0
$889$	5.80698	0.194760
$890$	10.7317	0.359728
$891$	0	0
$892$	−15.2758	−0.511473
$893$	−4.92209	−0.164712
$894$	0	0
$895$	48.2407	1.61251
$896$	−5.23006	−0.174724
$897$	0	0
$898$	2.65577	0.0886243
$899$	−5.23765	−0.174686
$900$	0	0
$901$	13.0149	0.433588
$902$	6.48155	0.215812
$903$	0	0
$904$	−10.2092	−0.339553
$905$	−34.5979	−1.15007
$906$	0	0
$907$	−15.3153	−0.508536	−0.254268	−	0.967134i	$-0.581834\pi$
$907$	−15.3153	−0.508536	−0.254268	+	0.967134i	$0.581834\pi$
$908$	32.4567	1.07711
$909$	0	0
$910$	−3.31992	−0.110054
$911$	6.57828	0.217948	0.108974	−	0.994045i	$-0.465243\pi$
$911$	6.57828	0.217948	0.108974	+	0.994045i	$0.465243\pi$
$912$	0	0
$913$	−2.56641	−0.0849358
$914$	15.4371	0.510615
$915$	0	0
$916$	14.0227	0.463322
$917$	1.48997	0.0492030
$918$	0	0
$919$	−21.1917	−0.699049	−0.349524	−	0.936927i	$-0.613657\pi$
$919$	−21.1917	−0.699049	−0.349524	+	0.936927i	$0.613657\pi$
$920$	9.51878	0.313825
$921$	0	0
$922$	16.2052	0.533689
$923$	−51.9467	−1.70985
$924$	0	0
$925$	−0.289090	−0.00950521
$926$	8.23843	0.270732
$927$	0	0
$928$	−44.8708	−1.47295
$929$	55.1926	1.81081	0.905406	−	0.424547i	$-0.139567\pi$
$929$	55.1926	1.81081	0.905406	+	0.424547i	$0.139567\pi$
$930$	0	0
$931$	46.4570	1.52257
$932$	13.7163	0.449292
$933$	0	0
$934$	−6.23693	−0.204079
$935$	−4.17961	−0.136688
$936$	0	0
$937$	−33.5743	−1.09682	−0.548412	−	0.836209i	$-0.684767\pi$
$937$	−33.5743	−1.09682	−0.548412	+	0.836209i	$0.684767\pi$
$938$	−4.15674	−0.135722
$939$	0	0
$940$	−2.27469	−0.0741921
$941$	−60.3088	−1.96601	−0.983006	−	0.183575i	$-0.941233\pi$
$941$	−60.3088	−1.96601	−0.983006	+	0.183575i	$0.941233\pi$
$942$	0	0
$943$	15.6829	0.510707
$944$	−13.3490	−0.434474
$945$	0	0
$946$	8.64806	0.281173
$947$	−60.6168	−1.96978	−0.984890	−	0.173180i	$-0.944596\pi$
$947$	−60.6168	−1.96978	−0.984890	+	0.173180i	$0.944596\pi$
$948$	0	0
$949$	37.5568	1.21915
$950$	−1.70132	−0.0551982
$951$	0	0
$952$	−2.48222	−0.0804492
$953$	48.4040	1.56796	0.783980	−	0.620787i	$-0.213187\pi$
$953$	48.4040	1.56796	0.783980	+	0.620787i	$0.213187\pi$
$954$	0	0
$955$	−25.6293	−0.829344
$956$	−16.0356	−0.518629
$957$	0	0
$958$	14.4893	0.468127
$959$	−4.27440	−0.138028
$960$	0	0
$961$	−30.5335	−0.984953
$962$	2.56095	0.0825684
$963$	0	0
$964$	22.8954	0.737412
$965$	−33.3356	−1.07311
$966$	0	0
$967$	32.2073	1.03572	0.517859	−	0.855466i	$-0.326729\pi$
$967$	32.2073	1.03572	0.517859	+	0.855466i	$0.326729\pi$
$968$	2.51684	0.0808944
$969$	0	0
$970$	26.1254	0.838835
$971$	−5.30655	−0.170295	−0.0851477	−	0.996368i	$-0.527136\pi$
$971$	−5.30655	−0.170295	−0.0851477	+	0.996368i	$0.527136\pi$
$972$	0	0
$973$	2.64311	0.0847342
$974$	3.51963	0.112776
$975$	0	0
$976$	−0.478687	−0.0153224
$977$	43.5994	1.39487	0.697434	−	0.716649i	$-0.254325\pi$
$977$	43.5994	1.39487	0.697434	+	0.716649i	$0.254325\pi$
$978$	0	0
$979$	6.86584	0.219433
$980$	21.4695	0.685819
$981$	0	0
$982$	−10.7226	−0.342173
$983$	−57.5299	−1.83492	−0.917459	−	0.397830i	$-0.869763\pi$
$983$	−57.5299	−1.83492	−0.917459	+	0.397830i	$0.869763\pi$
$984$	0	0
$985$	7.17392	0.228580
$986$	−10.7533	−0.342454
$987$	0	0
$988$	−42.4111	−1.34928
$989$	20.9251	0.665379
$990$	0	0
$991$	6.10097	0.193804	0.0969018	−	0.995294i	$-0.469107\pi$
$991$	6.10097	0.193804	0.0969018	+	0.995294i	$0.469107\pi$
$992$	3.99609	0.126876
$993$	0	0
$994$	4.59440	0.145726
$995$	46.2357	1.46577
$996$	0	0
$997$	11.8169	0.374245	0.187122	−	0.982337i	$-0.440084\pi$
$997$	11.8169	0.374245	0.187122	+	0.982337i	$0.440084\pi$
$998$	17.3245	0.548398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.20	✓	48
3.2	odd	2		8019.2.a.j.1.29	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.20	✓	48	1.1	even	1	trivial
8019.2.a.j.1.29	yes	48	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-0.724143 q^{2} -1.47562 q^{4} +2.15850 q^{5} -0.509330 q^{7} +2.51684 q^{8} +O(q^{10})\)	\(q-0.724143 q^{2} -1.47562 q^{4} +2.15850 q^{5} -0.509330 q^{7} +2.51684 q^{8} -1.56306 q^{10} -1.00000 q^{11} -4.17016 q^{13} +0.368828 q^{14} +1.12868 q^{16} +1.93635 q^{17} -6.89214 q^{19} -3.18512 q^{20} +0.724143 q^{22} +1.75216 q^{23} -0.340885 q^{25} +3.01979 q^{26} +0.751576 q^{28} +7.66889 q^{29} -0.682974 q^{31} -5.85101 q^{32} -1.40220 q^{34} -1.09939 q^{35} +0.848056 q^{37} +4.99089 q^{38} +5.43260 q^{40} +8.95065 q^{41} +11.9425 q^{43} +1.47562 q^{44} -1.26881 q^{46} +0.714161 q^{47} -6.74058 q^{49} +0.246850 q^{50} +6.15356 q^{52} +6.72132 q^{53} -2.15850 q^{55} -1.28190 q^{56} -5.55337 q^{58} -11.8271 q^{59} -0.424113 q^{61} +0.494571 q^{62} +1.97961 q^{64} -9.00128 q^{65} -11.2701 q^{67} -2.85731 q^{68} +0.796114 q^{70} +12.4568 q^{71} -9.00610 q^{73} -0.614114 q^{74} +10.1702 q^{76} +0.509330 q^{77} +8.25308 q^{79} +2.43625 q^{80} -6.48155 q^{82} +2.56641 q^{83} +4.17961 q^{85} -8.64806 q^{86} -2.51684 q^{88} -6.86584 q^{89} +2.12399 q^{91} -2.58551 q^{92} -0.517155 q^{94} -14.8767 q^{95} -16.7142 q^{97} +4.88115 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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