LMFDB - Embedded newform 8379.2.a.bv.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,8,0,0,-6,0,-10,6,0,-2,0,0,2,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 931)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.92022$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.92022 q^{2} +1.68725 q^{4} -2.00692 q^{5} -0.600553 q^{8} -3.85372 q^{10} +0.645701 q^{11} +0.232973 q^{13} -4.52769 q^{16} +5.71912 q^{17} +1.00000 q^{19} -3.38617 q^{20} +1.23989 q^{22} -4.21494 q^{23} -0.972286 q^{25} +0.447359 q^{26} -3.65262 q^{29} -1.75375 q^{31} -7.49306 q^{32} +10.9820 q^{34} +3.06650 q^{37} +1.92022 q^{38} +1.20526 q^{40} +6.88559 q^{41} +3.13184 q^{43} +1.08946 q^{44} -8.09361 q^{46} +12.5133 q^{47} -1.86700 q^{50} +0.393083 q^{52} +0.719116 q^{53} -1.29587 q^{55} -7.01383 q^{58} -3.78086 q^{59} -3.12493 q^{61} -3.36758 q^{62} -5.33295 q^{64} -0.467557 q^{65} +3.07702 q^{67} +9.64957 q^{68} +6.14844 q^{71} +3.95485 q^{73} +5.88835 q^{74} +1.68725 q^{76} -14.1872 q^{79} +9.08670 q^{80} +13.2219 q^{82} +15.6761 q^{83} -11.4778 q^{85} +6.01383 q^{86} -0.387778 q^{88} +8.41604 q^{89} -7.11165 q^{92} +24.0282 q^{94} -2.00692 q^{95} -15.7044 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{5} - 6 q^{8} - 10 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} - 14 q^{22} + 8 q^{23} + 20 q^{25} + 16 q^{26} - 2 q^{29} - 2 q^{32} + 22 q^{34} + 10 q^{37} - 22 q^{40}+ \cdots - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^5 - 6 * q^8 - 10 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 - 14 * q^22 + 8 * q^23 + 20 * q^25 + 16 * q^26 - 2 * q^29 - 2 * q^32 + 22 * q^34 + 10 * q^37 - 22 * q^40 + 12 * q^41 + 4 * q^43 + 14 * q^44 - 8 * q^46 + 16 * q^47 - 12 * q^50 - 28 * q^52 - 12 * q^53 + 8 * q^55 + 4 * q^58 + 14 * q^59 - 20 * q^61 - 20 * q^62 - 20 * q^64 - 10 * q^65 + 2 * q^67 - 24 * q^68 + 2 * q^71 + 16 * q^73 + 26 * q^74 + 2 * q^76 - 8 * q^79 + 28 * q^80 + 12 * q^82 + 20 * q^83 - 14 * q^85 - 8 * q^86 - 2 * q^88 + 16 * q^89 - 26 * q^92 - 10 * q^94 + 8 * q^95 - 2 * q^97

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 $

$2$	1.92022	1.35780	0.678901	−	0.734230i	$-0.262457\pi$
$2$	1.92022	1.35780	0.678901	+	0.734230i	$0.262457\pi$
$3$	0	0
$3$	0	0
$4$	1.68725	0.843624
$5$	−2.00692	−0.897520	−0.448760	−	0.893652i	$-0.648134\pi$
$5$	−2.00692	−0.897520	−0.448760	+	0.893652i	$0.648134\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−0.600553	−0.212327
$9$	0	0
$10$	−3.85372	−1.21865
$11$	0.645701	0.194686	0.0973431	−	0.995251i	$-0.468966\pi$
$11$	0.645701	0.194686	0.0973431	+	0.995251i	$0.468966\pi$
$12$	0	0
$13$	0.232973	0.0646150	0.0323075	−	0.999478i	$-0.489714\pi$
$13$	0.232973	0.0646150	0.0323075	+	0.999478i	$0.489714\pi$
$14$	0	0
$15$	0	0
$16$	−4.52769	−1.13192
$17$	5.71912	1.38709	0.693545	−	0.720414i	$-0.256048\pi$
$17$	5.71912	1.38709	0.693545	+	0.720414i	$0.256048\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.38617	−0.757170
$21$	0	0
$22$	1.23989	0.264345
$23$	−4.21494	−0.878875	−0.439438	−	0.898273i	$-0.644822\pi$
$23$	−4.21494	−0.878875	−0.439438	+	0.898273i	$0.644822\pi$
$24$	0	0
$25$	−0.972286	−0.194457
$26$	0.447359	0.0877343
$27$	0	0
$28$	0	0
$29$	−3.65262	−0.678274	−0.339137	−	0.940737i	$-0.610135\pi$
$29$	−3.65262	−0.678274	−0.339137	+	0.940737i	$0.610135\pi$
$30$	0	0
$31$	−1.75375	−0.314982	−0.157491	−	0.987520i	$-0.550341\pi$
$31$	−1.75375	−0.314982	−0.157491	+	0.987520i	$0.550341\pi$
$32$	−7.49306	−1.32460
$33$	0	0
$34$	10.9820	1.88339
$35$	0	0
$36$	0	0
$37$	3.06650	0.504129	0.252065	−	0.967710i	$-0.418890\pi$
$37$	3.06650	0.504129	0.252065	+	0.967710i	$0.418890\pi$
$38$	1.92022	0.311501
$39$	0	0
$40$	1.20526	0.190568
$41$	6.88559	1.07535	0.537674	−	0.843153i	$-0.319303\pi$
$41$	6.88559	1.07535	0.537674	+	0.843153i	$0.319303\pi$
$42$	0	0
$43$	3.13184	0.477602	0.238801	−	0.971069i	$-0.423246\pi$
$43$	3.13184	0.477602	0.238801	+	0.971069i	$0.423246\pi$
$44$	1.08946	0.164242
$45$	0	0
$46$	−8.09361	−1.19334
$47$	12.5133	1.82525	0.912623	−	0.408802i	$-0.134053\pi$
$47$	12.5133	1.82525	0.912623	+	0.408802i	$0.134053\pi$
$48$	0	0
$49$	0	0
$50$	−1.86700	−0.264034
$51$	0	0
$52$	0.393083	0.0545108
$53$	0.719116	0.0987781	0.0493891	−	0.998780i	$-0.484273\pi$
$53$	0.719116	0.0987781	0.0493891	+	0.998780i	$0.484273\pi$
$54$	0	0
$55$	−1.29587	−0.174735
$56$	0	0
$57$	0	0
$58$	−7.01383	−0.920961
$59$	−3.78086	−0.492226	−0.246113	−	0.969241i	$-0.579153\pi$
$59$	−3.78086	−0.492226	−0.246113	+	0.969241i	$0.579153\pi$
$60$	0	0
$61$	−3.12493	−0.400106	−0.200053	−	0.979785i	$-0.564111\pi$
$61$	−3.12493	−0.400106	−0.200053	+	0.979785i	$0.564111\pi$
$62$	−3.36758	−0.427683
$63$	0	0
$64$	−5.33295	−0.666619
$65$	−0.467557	−0.0579933
$66$	0	0
$67$	3.07702	0.375917	0.187959	−	0.982177i	$-0.439813\pi$
$67$	3.07702	0.375917	0.187959	+	0.982177i	$0.439813\pi$
$68$	9.64957	1.17018
$69$	0	0
$70$	0	0
$71$	6.14844	0.729686	0.364843	−	0.931069i	$-0.381123\pi$
$71$	6.14844	0.729686	0.364843	+	0.931069i	$0.381123\pi$
$72$	0	0
$73$	3.95485	0.462880	0.231440	−	0.972849i	$-0.425656\pi$
$73$	3.95485	0.462880	0.231440	+	0.972849i	$0.425656\pi$
$74$	5.88835	0.684507
$75$	0	0
$76$	1.68725	0.193541
$77$	0	0
$78$	0	0
$79$	−14.1872	−1.59619	−0.798094	−	0.602533i	$-0.794158\pi$
$79$	−14.1872	−1.59619	−0.798094	+	0.602533i	$0.794158\pi$
$80$	9.08670	1.01592
$81$	0	0
$82$	13.2219	1.46011
$83$	15.6761	1.72068	0.860339	−	0.509722i	$-0.170252\pi$
$83$	15.6761	1.72068	0.860339	+	0.509722i	$0.170252\pi$
$84$	0	0
$85$	−11.4778	−1.24494
$86$	6.01383	0.648488
$87$	0	0
$88$	−0.387778	−0.0413372
$89$	8.41604	0.892099	0.446049	−	0.895008i	$-0.352831\pi$
$89$	8.41604	0.892099	0.446049	+	0.895008i	$0.352831\pi$
$90$	0	0
$91$	0	0
$92$	−7.11165	−0.741440
$93$	0	0
$94$	24.0282	2.47832
$95$	−2.00692	−0.205905
$96$	0	0
$97$	−15.7044	−1.59454	−0.797270	−	0.603623i	$-0.793723\pi$
$97$	−15.7044	−1.59454	−0.797270	+	0.603623i	$0.793723\pi$
$98$	0	0
$99$	0	0
$100$	−1.64049	−0.164049
$101$	18.0900	1.80002	0.900012	−	0.435866i	$-0.143558\pi$
$101$	18.0900	1.80002	0.900012	+	0.435866i	$0.143558\pi$
$102$	0	0
$103$	4.02351	0.396448	0.198224	−	0.980157i	$-0.436483\pi$
$103$	4.02351	0.396448	0.198224	+	0.980157i	$0.436483\pi$
$104$	−0.139912	−0.0137195
$105$	0	0
$106$	1.38086	0.134121
$107$	−15.4133	−1.49006	−0.745029	−	0.667032i	$-0.767564\pi$
$107$	−15.4133	−1.49006	−0.745029	+	0.667032i	$0.767564\pi$
$108$	0	0
$109$	13.4133	1.28476	0.642380	−	0.766387i	$-0.277947\pi$
$109$	13.4133	1.28476	0.642380	+	0.766387i	$0.277947\pi$
$110$	−2.48835	−0.237255
$111$	0	0
$112$	0	0
$113$	14.8022	1.39247	0.696237	−	0.717812i	$-0.254856\pi$
$113$	14.8022	1.39247	0.696237	+	0.717812i	$0.254856\pi$
$114$	0	0
$115$	8.45903	0.788809
$116$	−6.16287	−0.572208
$117$	0	0
$118$	−7.26009	−0.668345
$119$	0	0
$120$	0	0
$121$	−10.5831	−0.962097
$122$	−6.00055	−0.543264
$123$	0	0
$124$	−2.95900	−0.265726
$125$	11.9859	1.07205
$126$	0	0
$127$	17.9639	1.59404	0.797021	−	0.603952i	$-0.206408\pi$
$127$	17.9639	1.59404	0.797021	+	0.603952i	$0.206408\pi$
$128$	4.74568	0.419463
$129$	0	0
$130$	−0.897812	−0.0787433
$131$	10.2183	0.892773	0.446386	−	0.894840i	$-0.352711\pi$
$131$	10.2183	0.892773	0.446386	+	0.894840i	$0.352711\pi$
$132$	0	0
$133$	0	0
$134$	5.90855	0.510421
$135$	0	0
$136$	−3.43463	−0.294517
$137$	2.79889	0.239126	0.119563	−	0.992827i	$-0.461851\pi$
$137$	2.79889	0.239126	0.119563	+	0.992827i	$0.461851\pi$
$138$	0	0
$139$	−16.2440	−1.37780	−0.688901	−	0.724856i	$-0.741906\pi$
$139$	−16.2440	−1.37780	−0.688901	+	0.724856i	$0.741906\pi$
$140$	0	0
$141$	0	0
$142$	11.8064	0.990768
$143$	0.150431	0.0125797
$144$	0	0
$145$	7.33050	0.608765
$146$	7.59419	0.628499
$147$	0	0
$148$	5.17394	0.425295
$149$	23.8055	1.95022	0.975112	−	0.221712i	$-0.0711644\pi$
$149$	23.8055	1.95022	0.975112	+	0.221712i	$0.0711644\pi$
$150$	0	0
$151$	13.0836	1.06473	0.532366	−	0.846514i	$-0.321303\pi$
$151$	13.0836	1.06473	0.532366	+	0.846514i	$0.321303\pi$
$152$	−0.600553	−0.0487112
$153$	0	0
$154$	0	0
$155$	3.51962	0.282703
$156$	0	0
$157$	15.5734	1.24289	0.621446	−	0.783457i	$-0.286546\pi$
$157$	15.5734	1.24289	0.621446	+	0.783457i	$0.286546\pi$
$158$	−27.2426	−2.16731
$159$	0	0
$160$	15.0379	1.18885
$161$	0	0
$162$	0	0
$163$	5.26285	0.412218	0.206109	−	0.978529i	$-0.433920\pi$
$163$	5.26285	0.412218	0.206109	+	0.978529i	$0.433920\pi$
$164$	11.6177	0.907190
$165$	0	0
$166$	30.1016	2.33634
$167$	−17.3853	−1.34531	−0.672657	−	0.739955i	$-0.734847\pi$
$167$	−17.3853	−1.34531	−0.672657	+	0.739955i	$0.734847\pi$
$168$	0	0
$169$	−12.9457	−0.995825
$170$	−22.0399	−1.69038
$171$	0	0
$172$	5.28420	0.402916
$173$	−5.63819	−0.428663	−0.214332	−	0.976761i	$-0.568757\pi$
$173$	−5.63819	−0.428663	−0.214332	+	0.976761i	$0.568757\pi$
$174$	0	0
$175$	0	0
$176$	−2.92354	−0.220370
$177$	0	0
$178$	16.1607	1.21129
$179$	11.1795	0.835593	0.417796	−	0.908541i	$-0.362803\pi$
$179$	11.1795	0.835593	0.417796	+	0.908541i	$0.362803\pi$
$180$	0	0
$181$	−9.64409	−0.716840	−0.358420	−	0.933561i	$-0.616684\pi$
$181$	−9.64409	−0.716840	−0.358420	+	0.933561i	$0.616684\pi$
$182$	0	0
$183$	0	0
$184$	2.53129	0.186609
$185$	−6.15421	−0.452466
$186$	0	0
$187$	3.69284	0.270047
$188$	21.1130	1.53982
$189$	0	0
$190$	−3.85372	−0.279578
$191$	−0.298871	−0.0216255	−0.0108128	−	0.999942i	$-0.503442\pi$
$191$	−0.298871	−0.0216255	−0.0108128	+	0.999942i	$0.503442\pi$
$192$	0	0
$193$	−13.4100	−0.965270	−0.482635	−	0.875821i	$-0.660320\pi$
$193$	−13.4100	−0.965270	−0.482635	+	0.875821i	$0.660320\pi$
$194$	−30.1559	−2.16507
$195$	0	0
$196$	0	0
$197$	−9.80636	−0.698674	−0.349337	−	0.936997i	$-0.613593\pi$
$197$	−9.80636	−0.698674	−0.349337	+	0.936997i	$0.613593\pi$
$198$	0	0
$199$	14.9515	1.05989	0.529943	−	0.848033i	$-0.322213\pi$
$199$	14.9515	1.05989	0.529943	+	0.848033i	$0.322213\pi$
$200$	0.583909	0.0412886
$201$	0	0
$202$	34.7368	2.44407
$203$	0	0
$204$	0	0
$205$	−13.8188	−0.965147
$206$	7.72603	0.538298
$207$	0	0
$208$	−1.05483	−0.0731392
$209$	0.645701	0.0446641
$210$	0	0
$211$	−6.50966	−0.448143	−0.224072	−	0.974573i	$-0.571935\pi$
$211$	−6.50966	−0.448143	−0.224072	+	0.974573i	$0.571935\pi$
$212$	1.21333	0.0833316
$213$	0	0
$214$	−29.5969	−2.02320
$215$	−6.28535	−0.428657
$216$	0	0
$217$	0	0
$218$	25.7565	1.74445
$219$	0	0
$220$	−2.18645	−0.147411
$221$	1.33240	0.0896268
$222$	0	0
$223$	5.10113	0.341597	0.170798	−	0.985306i	$-0.445365\pi$
$223$	5.10113	0.341597	0.170798	+	0.985306i	$0.445365\pi$
$224$	0	0
$225$	0	0
$226$	28.4235	1.89070
$227$	−2.70500	−0.179537	−0.0897684	−	0.995963i	$-0.528613\pi$
$227$	−2.70500	−0.179537	−0.0897684	+	0.995963i	$0.528613\pi$
$228$	0	0
$229$	−2.55624	−0.168921	−0.0844606	−	0.996427i	$-0.526917\pi$
$229$	−2.55624	−0.168921	−0.0844606	+	0.996427i	$0.526917\pi$
$230$	16.2432	1.07105
$231$	0	0
$232$	2.19359	0.144016
$233$	23.5692	1.54407	0.772034	−	0.635581i	$-0.219239\pi$
$233$	23.5692	1.54407	0.772034	+	0.635581i	$0.219239\pi$
$234$	0	0
$235$	−25.1131	−1.63820
$236$	−6.37925	−0.415254
$237$	0	0
$238$	0	0
$239$	−19.7767	−1.27925	−0.639623	−	0.768689i	$-0.720909\pi$
$239$	−19.7767	−1.27925	−0.639623	+	0.768689i	$0.720909\pi$
$240$	0	0
$241$	−9.14568	−0.589125	−0.294562	−	0.955632i	$-0.595174\pi$
$241$	−9.14568	−0.589125	−0.294562	+	0.955632i	$0.595174\pi$
$242$	−20.3218	−1.30634
$243$	0	0
$244$	−5.27253	−0.337539
$245$	0	0
$246$	0	0
$247$	0.232973	0.0148237
$248$	1.05322	0.0668793
$249$	0	0
$250$	23.0155	1.45563
$251$	−8.53737	−0.538874	−0.269437	−	0.963018i	$-0.586838\pi$
$251$	−8.53737	−0.538874	−0.269437	+	0.963018i	$0.586838\pi$
$252$	0	0
$253$	−2.72159	−0.171105
$254$	34.4947	2.16439
$255$	0	0
$256$	19.7786	1.23617
$257$	−0.529129	−0.0330061	−0.0165031	−	0.999864i	$-0.505253\pi$
$257$	−0.529129	−0.0330061	−0.0165031	+	0.999864i	$0.505253\pi$
$258$	0	0
$259$	0	0
$260$	−0.788884	−0.0489245
$261$	0	0
$262$	19.6213	1.21221
$263$	19.4235	1.19771	0.598853	−	0.800859i	$-0.295624\pi$
$263$	19.4235	1.19771	0.598853	+	0.800859i	$0.295624\pi$
$264$	0	0
$265$	−1.44321	−0.0886554
$266$	0	0
$267$	0	0
$268$	5.19169	0.317133
$269$	25.3323	1.54454	0.772270	−	0.635294i	$-0.219121\pi$
$269$	25.3323	1.54454	0.772270	+	0.635294i	$0.219121\pi$
$270$	0	0
$271$	8.39164	0.509756	0.254878	−	0.966973i	$-0.417965\pi$
$271$	8.39164	0.509756	0.254878	+	0.966973i	$0.417965\pi$
$272$	−25.8944	−1.57008
$273$	0	0
$274$	5.37450	0.324685
$275$	−0.627806	−0.0378582
$276$	0	0
$277$	−6.32575	−0.380077	−0.190039	−	0.981777i	$-0.560861\pi$
$277$	−6.32575	−0.380077	−0.190039	+	0.981777i	$0.560861\pi$
$278$	−31.1922	−1.87078
$279$	0	0
$280$	0	0
$281$	16.6210	0.991527	0.495763	−	0.868458i	$-0.334888\pi$
$281$	16.6210	0.991527	0.495763	+	0.868458i	$0.334888\pi$
$282$	0	0
$283$	23.8495	1.41771	0.708853	−	0.705356i	$-0.249213\pi$
$283$	23.8495	1.41771	0.708853	+	0.705356i	$0.249213\pi$
$284$	10.3739	0.615580
$285$	0	0
$286$	0.288860	0.0170807
$287$	0	0
$288$	0	0
$289$	15.7083	0.924017
$290$	14.0762	0.826582
$291$	0	0
$292$	6.67282	0.390497
$293$	2.00305	0.117019	0.0585097	−	0.998287i	$-0.481365\pi$
$293$	2.00305	0.117019	0.0585097	+	0.998287i	$0.481365\pi$
$294$	0	0
$295$	7.58787	0.441783
$296$	−1.84159	−0.107040
$297$	0	0
$298$	45.7119	2.64802
$299$	−0.981965	−0.0567885
$300$	0	0
$301$	0	0
$302$	25.1235	1.44569
$303$	0	0
$304$	−4.52769	−0.259681
$305$	6.27147	0.359103
$306$	0	0
$307$	7.55379	0.431118	0.215559	−	0.976491i	$-0.430843\pi$
$307$	7.55379	0.431118	0.215559	+	0.976491i	$0.430843\pi$
$308$	0	0
$309$	0	0
$310$	6.75845	0.383854
$311$	14.1352	0.801531	0.400766	−	0.916181i	$-0.368744\pi$
$311$	14.1352	0.801531	0.400766	+	0.916181i	$0.368744\pi$
$312$	0	0
$313$	28.2914	1.59912	0.799561	−	0.600585i	$-0.205066\pi$
$313$	28.2914	1.59912	0.799561	+	0.600585i	$0.205066\pi$
$314$	29.9043	1.68760
$315$	0	0
$316$	−23.9374	−1.34658
$317$	−23.9013	−1.34243	−0.671215	−	0.741262i	$-0.734227\pi$
$317$	−23.9013	−1.34243	−0.671215	+	0.741262i	$0.734227\pi$
$318$	0	0
$319$	−2.35850	−0.132051
$320$	10.7028	0.598304
$321$	0	0
$322$	0	0
$323$	5.71912	0.318220
$324$	0	0
$325$	−0.226516	−0.0125649
$326$	10.1058	0.559710
$327$	0	0
$328$	−4.13516	−0.228326
$329$	0	0
$330$	0	0
$331$	−4.23381	−0.232711	−0.116356	−	0.993208i	$-0.537121\pi$
$331$	−4.23381	−0.232711	−0.116356	+	0.993208i	$0.537121\pi$
$332$	26.4495	1.45161
$333$	0	0
$334$	−33.3836	−1.82667
$335$	−6.17531	−0.337393
$336$	0	0
$337$	−12.4792	−0.679784	−0.339892	−	0.940464i	$-0.610391\pi$
$337$	−12.4792	−0.679784	−0.339892	+	0.940464i	$0.610391\pi$
$338$	−24.8586	−1.35213
$339$	0	0
$340$	−19.3659	−1.05026
$341$	−1.13240	−0.0613227
$342$	0	0
$343$	0	0
$344$	−1.88084	−0.101408
$345$	0	0
$346$	−10.8266	−0.582040
$347$	34.2362	1.83790	0.918949	−	0.394377i	$-0.129040\pi$
$347$	34.2362	1.83790	0.918949	+	0.394377i	$0.129040\pi$
$348$	0	0
$349$	2.57805	0.138000	0.0690000	−	0.997617i	$-0.478019\pi$
$349$	2.57805	0.138000	0.0690000	+	0.997617i	$0.478019\pi$
$350$	0	0
$351$	0	0
$352$	−4.83828	−0.257881
$353$	−3.24044	−0.172471	−0.0862356	−	0.996275i	$-0.527484\pi$
$353$	−3.24044	−0.172471	−0.0862356	+	0.996275i	$0.527484\pi$
$354$	0	0
$355$	−12.3394	−0.654908
$356$	14.2000	0.752596
$357$	0	0
$358$	21.4670	1.13457
$359$	−11.3254	−0.597734	−0.298867	−	0.954295i	$-0.596609\pi$
$359$	−11.3254	−0.597734	−0.298867	+	0.954295i	$0.596609\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−18.5188	−0.973326
$363$	0	0
$364$	0	0
$365$	−7.93706	−0.415445
$366$	0	0
$367$	20.2579	1.05745	0.528726	−	0.848792i	$-0.322670\pi$
$367$	20.2579	1.05745	0.528726	+	0.848792i	$0.322670\pi$
$368$	19.0839	0.994819
$369$	0	0
$370$	−11.8174	−0.614359
$371$	0	0
$372$	0	0
$373$	33.9822	1.75953	0.879765	−	0.475408i	$-0.157700\pi$
$373$	33.9822	1.75953	0.879765	+	0.475408i	$0.157700\pi$
$374$	7.09107	0.366670
$375$	0	0
$376$	−7.51487	−0.387550
$377$	−0.850960	−0.0438267
$378$	0	0
$379$	−20.2122	−1.03823	−0.519115	−	0.854704i	$-0.673738\pi$
$379$	−20.2122	−1.03823	−0.519115	+	0.854704i	$0.673738\pi$
$380$	−3.38617	−0.173707
$381$	0	0
$382$	−0.573898	−0.0293632
$383$	32.4978	1.66056	0.830279	−	0.557348i	$-0.188181\pi$
$383$	32.4978	1.66056	0.830279	+	0.557348i	$0.188181\pi$
$384$	0	0
$385$	0	0
$386$	−25.7501	−1.31065
$387$	0	0
$388$	−26.4972	−1.34519
$389$	6.65953	0.337652	0.168826	−	0.985646i	$-0.446002\pi$
$389$	6.65953	0.337652	0.168826	+	0.985646i	$0.446002\pi$
$390$	0	0
$391$	−24.1057	−1.21908
$392$	0	0
$393$	0	0
$394$	−18.8304	−0.948661
$395$	28.4726	1.43261
$396$	0	0
$397$	−27.3950	−1.37491	−0.687457	−	0.726225i	$-0.741273\pi$
$397$	−27.3950	−1.37491	−0.687457	+	0.726225i	$0.741273\pi$
$398$	28.7103	1.43911
$399$	0	0
$400$	4.40221	0.220111
$401$	−11.5291	−0.575737	−0.287869	−	0.957670i	$-0.592947\pi$
$401$	−11.5291	−0.575737	−0.287869	+	0.957670i	$0.592947\pi$
$402$	0	0
$403$	−0.408575	−0.0203526
$404$	30.5223	1.51854
$405$	0	0
$406$	0	0
$407$	1.98004	0.0981470
$408$	0	0
$409$	39.7423	1.96513	0.982565	−	0.185920i	$-0.0595264\pi$
$409$	39.7423	1.96513	0.982565	+	0.185920i	$0.0595264\pi$
$410$	−26.5352	−1.31048
$411$	0	0
$412$	6.78866	0.334453
$413$	0	0
$414$	0	0
$415$	−31.4607	−1.54434
$416$	−1.74568	−0.0855889
$417$	0	0
$418$	1.23989	0.0606450
$419$	−22.8335	−1.11549	−0.557744	−	0.830013i	$-0.688333\pi$
$419$	−22.8335	−1.11549	−0.557744	+	0.830013i	$0.688333\pi$
$420$	0	0
$421$	18.8016	0.916334	0.458167	−	0.888866i	$-0.348506\pi$
$421$	18.8016	0.916334	0.458167	+	0.888866i	$0.348506\pi$
$422$	−12.5000	−0.608489
$423$	0	0
$424$	−0.431867	−0.0209733
$425$	−5.56062	−0.269730
$426$	0	0
$427$	0	0
$428$	−26.0060	−1.25705
$429$	0	0
$430$	−12.0693	−0.582031
$431$	−26.7224	−1.28717	−0.643587	−	0.765373i	$-0.722555\pi$
$431$	−26.7224	−1.28717	−0.643587	+	0.765373i	$0.722555\pi$
$432$	0	0
$433$	−20.7393	−0.996668	−0.498334	−	0.866985i	$-0.666055\pi$
$433$	−20.7393	−0.996668	−0.498334	+	0.866985i	$0.666055\pi$
$434$	0	0
$435$	0	0
$436$	22.6315	1.08385
$437$	−4.21494	−0.201628
$438$	0	0
$439$	−28.9789	−1.38309	−0.691544	−	0.722334i	$-0.743069\pi$
$439$	−28.9789	−1.38309	−0.691544	+	0.722334i	$0.743069\pi$
$440$	0.778237	0.0371010
$441$	0	0
$442$	2.55850	0.121695
$443$	9.13936	0.434224	0.217112	−	0.976147i	$-0.430336\pi$
$443$	9.13936	0.434224	0.217112	+	0.976147i	$0.430336\pi$
$444$	0	0
$445$	−16.8903	−0.800677
$446$	9.79529	0.463821
$447$	0	0
$448$	0	0
$449$	−7.36398	−0.347528	−0.173764	−	0.984787i	$-0.555593\pi$
$449$	−7.36398	−0.347528	−0.173764	+	0.984787i	$0.555593\pi$
$450$	0	0
$451$	4.44604	0.209356
$452$	24.9750	1.17473
$453$	0	0
$454$	−5.19419	−0.243775
$455$	0	0
$456$	0	0
$457$	−18.8052	−0.879671	−0.439835	−	0.898078i	$-0.644963\pi$
$457$	−18.8052	−0.879671	−0.439835	+	0.898078i	$0.644963\pi$
$458$	−4.90855	−0.229362
$459$	0	0
$460$	14.2725	0.665458
$461$	−6.42512	−0.299248	−0.149624	−	0.988743i	$-0.547806\pi$
$461$	−6.42512	−0.299248	−0.149624	+	0.988743i	$0.547806\pi$
$462$	0	0
$463$	2.03434	0.0945439	0.0472720	−	0.998882i	$-0.484947\pi$
$463$	2.03434	0.0945439	0.0472720	+	0.998882i	$0.484947\pi$
$464$	16.5379	0.767754
$465$	0	0
$466$	45.2580	2.09654
$467$	−11.0506	−0.511362	−0.255681	−	0.966761i	$-0.582300\pi$
$467$	−11.0506	−0.511362	−0.255681	+	0.966761i	$0.582300\pi$
$468$	0	0
$469$	0	0
$470$	−48.2226	−2.22434
$471$	0	0
$472$	2.27061	0.104513
$473$	2.02224	0.0929825
$474$	0	0
$475$	−0.972286	−0.0446115
$476$	0	0
$477$	0	0
$478$	−37.9756	−1.73696
$479$	−17.0714	−0.780011	−0.390006	−	0.920813i	$-0.627527\pi$
$479$	−17.0714	−0.780011	−0.390006	+	0.920813i	$0.627527\pi$
$480$	0	0
$481$	0.714410	0.0325743
$482$	−17.5617	−0.799914
$483$	0	0
$484$	−17.8563	−0.811648
$485$	31.5174	1.43113
$486$	0	0
$487$	5.59220	0.253407	0.126703	−	0.991941i	$-0.459560\pi$
$487$	5.59220	0.253407	0.126703	+	0.991941i	$0.459560\pi$
$488$	1.87668	0.0849535
$489$	0	0
$490$	0	0
$491$	−30.2011	−1.36295	−0.681477	−	0.731839i	$-0.738662\pi$
$491$	−30.2011	−1.36295	−0.681477	+	0.731839i	$0.738662\pi$
$492$	0	0
$493$	−20.8897	−0.940827
$494$	0.447359	0.0201276
$495$	0	0
$496$	7.94042	0.356535
$497$	0	0
$498$	0	0
$499$	−0.903338	−0.0404390	−0.0202195	−	0.999796i	$-0.506436\pi$
$499$	−0.903338	−0.0404390	−0.0202195	+	0.999796i	$0.506436\pi$
$500$	20.2232	0.904407
$501$	0	0
$502$	−16.3936	−0.731684
$503$	0.668202	0.0297937	0.0148968	−	0.999889i	$-0.495258\pi$
$503$	0.668202	0.0297937	0.0148968	+	0.999889i	$0.495258\pi$
$504$	0	0
$505$	−36.3051	−1.61556
$506$	−5.22606	−0.232327
$507$	0	0
$508$	30.3096	1.34477
$509$	−7.41600	−0.328708	−0.164354	−	0.986401i	$-0.552554\pi$
$509$	−7.41600	−0.328708	−0.164354	+	0.986401i	$0.552554\pi$
$510$	0	0
$511$	0	0
$512$	28.4880	1.25900
$513$	0	0
$514$	−1.01604	−0.0448158
$515$	−8.07485	−0.355821
$516$	0	0
$517$	8.07983	0.355350
$518$	0	0
$519$	0	0
$520$	0.280792	0.0123136
$521$	39.6778	1.73832	0.869159	−	0.494533i	$-0.164661\pi$
$521$	39.6778	1.73832	0.869159	+	0.494533i	$0.164661\pi$
$522$	0	0
$523$	−31.7681	−1.38912	−0.694562	−	0.719433i	$-0.744402\pi$
$523$	−31.7681	−1.38912	−0.694562	+	0.719433i	$0.744402\pi$
$524$	17.2407	0.753165
$525$	0	0
$526$	37.2974	1.62625
$527$	−10.0299	−0.436908
$528$	0	0
$529$	−5.23430	−0.227578
$530$	−2.77127	−0.120376
$531$	0	0
$532$	0	0
$533$	1.60415	0.0694836
$534$	0	0
$535$	30.9332	1.33736
$536$	−1.84791	−0.0798176
$537$	0	0
$538$	48.6437	2.09718
$539$	0	0
$540$	0	0
$541$	34.2022	1.47047	0.735233	−	0.677815i	$-0.237073\pi$
$541$	34.2022	1.47047	0.735233	+	0.677815i	$0.237073\pi$
$542$	16.1138	0.692147
$543$	0	0
$544$	−42.8537	−1.83734
$545$	−26.9193	−1.15310
$546$	0	0
$547$	−8.74798	−0.374037	−0.187018	−	0.982356i	$-0.559882\pi$
$547$	−8.74798	−0.374037	−0.187018	+	0.982356i	$0.559882\pi$
$548$	4.72243	0.201732
$549$	0	0
$550$	−1.20553	−0.0514038
$551$	−3.65262	−0.155607
$552$	0	0
$553$	0	0
$554$	−12.1468	−0.516069
$555$	0	0
$556$	−27.4077	−1.16235
$557$	−24.2097	−1.02580	−0.512899	−	0.858449i	$-0.671429\pi$
$557$	−24.2097	−1.02580	−0.512899	+	0.858449i	$0.671429\pi$
$558$	0	0
$559$	0.729634	0.0308602
$560$	0	0
$561$	0	0
$562$	31.9160	1.34630
$563$	−23.5382	−0.992017	−0.496009	−	0.868318i	$-0.665202\pi$
$563$	−23.5382	−0.992017	−0.496009	+	0.868318i	$0.665202\pi$
$564$	0	0
$565$	−29.7068	−1.24977
$566$	45.7963	1.92496
$567$	0	0
$568$	−3.69246	−0.154932
$569$	−40.8720	−1.71344	−0.856721	−	0.515780i	$-0.827502\pi$
$569$	−40.8720	−1.71344	−0.856721	+	0.515780i	$0.827502\pi$
$570$	0	0
$571$	−8.95010	−0.374550	−0.187275	−	0.982308i	$-0.559966\pi$
$571$	−8.95010	−0.374550	−0.187275	+	0.982308i	$0.559966\pi$
$572$	0.253814	0.0106125
$573$	0	0
$574$	0	0
$575$	4.09813	0.170904
$576$	0	0
$577$	−10.9327	−0.455133	−0.227566	−	0.973763i	$-0.573077\pi$
$577$	−10.9327	−0.455133	−0.227566	+	0.973763i	$0.573077\pi$
$578$	30.1634	1.25463
$579$	0	0
$580$	12.3684	0.513569
$581$	0	0
$582$	0	0
$583$	0.464334	0.0192307
$584$	−2.37510	−0.0982822
$585$	0	0
$586$	3.84630	0.158889
$587$	−37.5213	−1.54867	−0.774335	−	0.632775i	$-0.781916\pi$
$587$	−37.5213	−1.54867	−0.774335	+	0.632775i	$0.781916\pi$
$588$	0	0
$589$	−1.75375	−0.0722618
$590$	14.5704	0.599853
$591$	0	0
$592$	−13.8842	−0.570635
$593$	12.1249	0.497909	0.248955	−	0.968515i	$-0.419913\pi$
$593$	12.1249	0.497909	0.248955	+	0.968515i	$0.419913\pi$
$594$	0	0
$595$	0	0
$596$	40.1658	1.64526
$597$	0	0
$598$	−1.88559	−0.0771075
$599$	−21.8083	−0.891061	−0.445531	−	0.895267i	$-0.646985\pi$
$599$	−21.8083	−0.891061	−0.445531	+	0.895267i	$0.646985\pi$
$600$	0	0
$601$	18.1878	0.741895	0.370947	−	0.928654i	$-0.379033\pi$
$601$	18.1878	0.741895	0.370947	+	0.928654i	$0.379033\pi$
$602$	0	0
$603$	0	0
$604$	22.0754	0.898234
$605$	21.2393	0.863502
$606$	0	0
$607$	27.4341	1.11351	0.556757	−	0.830675i	$-0.312045\pi$
$607$	27.4341	1.11351	0.556757	+	0.830675i	$0.312045\pi$
$608$	−7.49306	−0.303884
$609$	0	0
$610$	12.0426	0.487591
$611$	2.91525	0.117938
$612$	0	0
$613$	24.8456	1.00350	0.501752	−	0.865011i	$-0.332689\pi$
$613$	24.8456	1.00350	0.501752	+	0.865011i	$0.332689\pi$
$614$	14.5049	0.585372
$615$	0	0
$616$	0	0
$617$	−22.9991	−0.925909	−0.462955	−	0.886382i	$-0.653211\pi$
$617$	−22.9991	−0.925909	−0.462955	+	0.886382i	$0.653211\pi$
$618$	0	0
$619$	1.64294	0.0660353	0.0330176	−	0.999455i	$-0.489488\pi$
$619$	1.64294	0.0660353	0.0330176	+	0.999455i	$0.489488\pi$
$620$	5.93848	0.238495
$621$	0	0
$622$	27.1426	1.08832
$623$	0	0
$624$	0	0
$625$	−19.1932	−0.767729
$626$	54.3256	2.17129
$627$	0	0
$628$	26.2762	1.04853
$629$	17.5377	0.699272
$630$	0	0
$631$	−5.63823	−0.224455	−0.112227	−	0.993683i	$-0.535798\pi$
$631$	−5.63823	−0.224455	−0.112227	+	0.993683i	$0.535798\pi$
$632$	8.52017	0.338914
$633$	0	0
$634$	−45.8958	−1.82275
$635$	−36.0521	−1.43068
$636$	0	0
$637$	0	0
$638$	−4.52884	−0.179299
$639$	0	0
$640$	−9.52418	−0.376476
$641$	−34.9878	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$641$	−34.9878	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$642$	0	0
$643$	9.93598	0.391837	0.195918	−	0.980620i	$-0.437231\pi$
$643$	9.93598	0.391837	0.195918	+	0.980620i	$0.437231\pi$
$644$	0	0
$645$	0	0
$646$	10.9820	0.432080
$647$	−0.220703	−0.00867674	−0.00433837	−	0.999991i	$-0.501381\pi$
$647$	−0.220703	−0.00867674	−0.00433837	+	0.999991i	$0.501381\pi$
$648$	0	0
$649$	−2.44131	−0.0958297
$650$	−0.434961	−0.0170606
$651$	0	0
$652$	8.87973	0.347757
$653$	−26.3684	−1.03188	−0.515938	−	0.856626i	$-0.672556\pi$
$653$	−26.3684	−1.03188	−0.515938	+	0.856626i	$0.672556\pi$
$654$	0	0
$655$	−20.5072	−0.801282
$656$	−31.1758	−1.21721
$657$	0	0
$658$	0	0
$659$	19.0242	0.741079	0.370540	−	0.928817i	$-0.379173\pi$
$659$	19.0242	0.741079	0.370540	+	0.928817i	$0.379173\pi$
$660$	0	0
$661$	−33.1950	−1.29114	−0.645568	−	0.763703i	$-0.723379\pi$
$661$	−33.1950	−1.29114	−0.645568	+	0.763703i	$0.723379\pi$
$662$	−8.12985	−0.315976
$663$	0	0
$664$	−9.41434	−0.365347
$665$	0	0
$666$	0	0
$667$	15.3956	0.596118
$668$	−29.3333	−1.13494
$669$	0	0
$670$	−11.8580	−0.458113
$671$	−2.01777	−0.0778952
$672$	0	0
$673$	−31.3013	−1.20658	−0.603289	−	0.797523i	$-0.706143\pi$
$673$	−31.3013	−1.20658	−0.603289	+	0.797523i	$0.706143\pi$
$674$	−23.9628	−0.923012
$675$	0	0
$676$	−21.8426	−0.840102
$677$	−12.0271	−0.462240	−0.231120	−	0.972925i	$-0.574239\pi$
$677$	−12.0271	−0.462240	−0.231120	+	0.972925i	$0.574239\pi$
$678$	0	0
$679$	0	0
$680$	6.89301	0.264335
$681$	0	0
$682$	−2.17445	−0.0832640
$683$	7.00387	0.267995	0.133998	−	0.990982i	$-0.457219\pi$
$683$	7.00387	0.267995	0.133998	+	0.990982i	$0.457219\pi$
$684$	0	0
$685$	−5.61715	−0.214620
$686$	0	0
$687$	0	0
$688$	−14.1800	−0.540608
$689$	0.167534	0.00638255
$690$	0	0
$691$	−38.5256	−1.46558	−0.732792	−	0.680453i	$-0.761783\pi$
$691$	−38.5256	−1.46558	−0.732792	+	0.680453i	$0.761783\pi$
$692$	−9.51302	−0.361631
$693$	0	0
$694$	65.7411	2.49550
$695$	32.6004	1.23661
$696$	0	0
$697$	39.3795	1.49160
$698$	4.95043	0.187377
$699$	0	0
$700$	0	0
$701$	32.9668	1.24514	0.622569	−	0.782565i	$-0.286089\pi$
$701$	32.9668	1.24514	0.622569	+	0.782565i	$0.286089\pi$
$702$	0	0
$703$	3.06650	0.115655
$704$	−3.44349	−0.129782
$705$	0	0
$706$	−6.22236	−0.234182
$707$	0	0
$708$	0	0
$709$	32.2564	1.21142	0.605708	−	0.795687i	$-0.292890\pi$
$709$	32.2564	1.21142	0.605708	+	0.795687i	$0.292890\pi$
$710$	−23.6944	−0.889234
$711$	0	0
$712$	−5.05428	−0.189417
$713$	7.39193	0.276830
$714$	0	0
$715$	−0.301902	−0.0112905
$716$	18.8625	0.704926
$717$	0	0
$718$	−21.7473	−0.811603
$719$	−37.9595	−1.41565	−0.707825	−	0.706387i	$-0.750324\pi$
$719$	−37.9595	−1.41565	−0.707825	+	0.706387i	$0.750324\pi$
$720$	0	0
$721$	0	0
$722$	1.92022	0.0714632
$723$	0	0
$724$	−16.2720	−0.604743
$725$	3.55139	0.131895
$726$	0	0
$727$	−5.40908	−0.200612	−0.100306	−	0.994957i	$-0.531982\pi$
$727$	−5.40908	−0.200612	−0.100306	+	0.994957i	$0.531982\pi$
$728$	0	0
$729$	0	0
$730$	−15.2409	−0.564091
$731$	17.9114	0.662476
$732$	0	0
$733$	40.3312	1.48967	0.744833	−	0.667251i	$-0.232529\pi$
$733$	40.3312	1.48967	0.744833	+	0.667251i	$0.232529\pi$
$734$	38.8996	1.43581
$735$	0	0
$736$	31.5828	1.16416
$737$	1.98683	0.0731860
$738$	0	0
$739$	−17.8947	−0.658268	−0.329134	−	0.944283i	$-0.606757\pi$
$739$	−17.8947	−0.658268	−0.329134	+	0.944283i	$0.606757\pi$
$740$	−10.3837	−0.381711
$741$	0	0
$742$	0	0
$743$	0.0581426	0.00213305	0.00106652	−	0.999999i	$-0.499661\pi$
$743$	0.0581426	0.00213305	0.00106652	+	0.999999i	$0.499661\pi$
$744$	0	0
$745$	−47.7757	−1.75037
$746$	65.2533	2.38909
$747$	0	0
$748$	6.23074	0.227818
$749$	0	0
$750$	0	0
$751$	−46.2886	−1.68909	−0.844547	−	0.535482i	$-0.820130\pi$
$751$	−46.2886	−1.68909	−0.844547	+	0.535482i	$0.820130\pi$
$752$	−56.6561	−2.06604
$753$	0	0
$754$	−1.63403	−0.0595079
$755$	−26.2578	−0.955619
$756$	0	0
$757$	28.3733	1.03125	0.515623	−	0.856815i	$-0.327560\pi$
$757$	28.3733	1.03125	0.515623	+	0.856815i	$0.327560\pi$
$758$	−38.8118	−1.40971
$759$	0	0
$760$	1.20526	0.0437193
$761$	−39.2965	−1.42450	−0.712249	−	0.701927i	$-0.752323\pi$
$761$	−39.2965	−1.42450	−0.712249	+	0.701927i	$0.752323\pi$
$762$	0	0
$763$	0	0
$764$	−0.504270	−0.0182438
$765$	0	0
$766$	62.4029	2.25471
$767$	−0.880837	−0.0318052
$768$	0	0
$769$	−41.1096	−1.48245	−0.741225	−	0.671256i	$-0.765755\pi$
$769$	−41.1096	−1.48245	−0.741225	+	0.671256i	$0.765755\pi$
$770$	0	0
$771$	0	0
$772$	−22.6259	−0.814325
$773$	44.3906	1.59662	0.798309	−	0.602248i	$-0.205728\pi$
$773$	44.3906	1.59662	0.798309	+	0.602248i	$0.205728\pi$
$774$	0	0
$775$	1.70514	0.0612505
$776$	9.43132	0.338564
$777$	0	0
$778$	12.7878	0.458464
$779$	6.88559	0.246702
$780$	0	0
$781$	3.97006	0.142060
$782$	−46.2883	−1.65527
$783$	0	0
$784$	0	0
$785$	−31.2545	−1.11552
$786$	0	0
$787$	−39.4575	−1.40651	−0.703254	−	0.710939i	$-0.748270\pi$
$787$	−39.4575	−1.40651	−0.703254	+	0.710939i	$0.748270\pi$
$788$	−16.5458	−0.589419
$789$	0	0
$790$	54.6736	1.94520
$791$	0	0
$792$	0	0
$793$	−0.728023	−0.0258528
$794$	−52.6044	−1.86686
$795$	0	0
$796$	25.2270	0.894146
$797$	−23.3858	−0.828369	−0.414184	−	0.910193i	$-0.635933\pi$
$797$	−23.3858	−0.828369	−0.414184	+	0.910193i	$0.635933\pi$
$798$	0	0
$799$	71.5648	2.53178
$800$	7.28540	0.257578
$801$	0	0
$802$	−22.1385	−0.781737
$803$	2.55365	0.0901164
$804$	0	0
$805$	0	0
$806$	−0.784554	−0.0276347
$807$	0	0
$808$	−10.8640	−0.382194
$809$	12.5745	0.442094	0.221047	−	0.975263i	$-0.429053\pi$
$809$	12.5745	0.442094	0.221047	+	0.975263i	$0.429053\pi$
$810$	0	0
$811$	−7.91138	−0.277806	−0.138903	−	0.990306i	$-0.544358\pi$
$811$	−7.91138	−0.277806	−0.138903	+	0.990306i	$0.544358\pi$
$812$	0	0
$813$	0	0
$814$	3.80212	0.133264
$815$	−10.5621	−0.369974
$816$	0	0
$817$	3.13184	0.109569
$818$	76.3140	2.66826
$819$	0	0
$820$	−23.3158	−0.814221
$821$	−31.2830	−1.09178	−0.545892	−	0.837855i	$-0.683809\pi$
$821$	−31.2830	−1.09178	−0.545892	+	0.837855i	$0.683809\pi$
$822$	0	0
$823$	−6.04702	−0.210786	−0.105393	−	0.994431i	$-0.533610\pi$
$823$	−6.04702	−0.210786	−0.105393	+	0.994431i	$0.533610\pi$
$824$	−2.41633	−0.0841769
$825$	0	0
$826$	0	0
$827$	30.6559	1.06601	0.533005	−	0.846112i	$-0.321063\pi$
$827$	30.6559	1.06601	0.533005	+	0.846112i	$0.321063\pi$
$828$	0	0
$829$	42.1300	1.46324	0.731618	−	0.681715i	$-0.238765\pi$
$829$	42.1300	1.46324	0.731618	+	0.681715i	$0.238765\pi$
$830$	−60.4115	−2.09691
$831$	0	0
$832$	−1.24243	−0.0430736
$833$	0	0
$834$	0	0
$835$	34.8908	1.20745
$836$	1.08946	0.0376797
$837$	0	0
$838$	−43.8453	−1.51461
$839$	−14.4798	−0.499897	−0.249949	−	0.968259i	$-0.580414\pi$
$839$	−14.4798	−0.499897	−0.249949	+	0.968259i	$0.580414\pi$
$840$	0	0
$841$	−15.6584	−0.539944
$842$	36.1032	1.24420
$843$	0	0
$844$	−10.9834	−0.378064
$845$	25.9810	0.893773
$846$	0	0
$847$	0	0
$848$	−3.25593	−0.111809
$849$	0	0
$850$	−10.6776	−0.366239
$851$	−12.9251	−0.443067
$852$	0	0
$853$	31.7188	1.08603	0.543016	−	0.839722i	$-0.317282\pi$
$853$	31.7188	1.08603	0.543016	+	0.839722i	$0.317282\pi$
$854$	0	0
$855$	0	0
$856$	9.25648	0.316380
$857$	−33.3132	−1.13796	−0.568978	−	0.822353i	$-0.692661\pi$
$857$	−33.3132	−1.13796	−0.568978	+	0.822353i	$0.692661\pi$
$858$	0	0
$859$	−5.57339	−0.190162	−0.0950808	−	0.995470i	$-0.530311\pi$
$859$	−5.57339	−0.190162	−0.0950808	+	0.995470i	$0.530311\pi$
$860$	−10.6049	−0.361626
$861$	0	0
$862$	−51.3130	−1.74773
$863$	30.8841	1.05131	0.525654	−	0.850699i	$-0.323821\pi$
$863$	30.8841	1.05131	0.525654	+	0.850699i	$0.323821\pi$
$864$	0	0
$865$	11.3154	0.384734
$866$	−39.8241	−1.35328
$867$	0	0
$868$	0	0
$869$	−9.16071	−0.310756
$870$	0	0
$871$	0.716861	0.0242899
$872$	−8.05538	−0.272790
$873$	0	0
$874$	−8.09361	−0.273771
$875$	0	0
$876$	0	0
$877$	8.83319	0.298276	0.149138	−	0.988816i	$-0.452350\pi$
$877$	8.83319	0.298276	0.149138	+	0.988816i	$0.452350\pi$
$878$	−55.6459	−1.87796
$879$	0	0
$880$	5.86729	0.197786
$881$	36.3004	1.22299	0.611496	−	0.791247i	$-0.290568\pi$
$881$	36.3004	1.22299	0.611496	+	0.791247i	$0.290568\pi$
$882$	0	0
$883$	39.9694	1.34508	0.672539	−	0.740062i	$-0.265204\pi$
$883$	39.9694	1.34508	0.672539	+	0.740062i	$0.265204\pi$
$884$	2.24809	0.0756113
$885$	0	0
$886$	17.5496	0.589590
$887$	3.13202	0.105163	0.0525814	−	0.998617i	$-0.483255\pi$
$887$	3.13202	0.105163	0.0525814	+	0.998617i	$0.483255\pi$
$888$	0	0
$889$	0	0
$890$	−32.4331	−1.08716
$891$	0	0
$892$	8.60687	0.288179
$893$	12.5133	0.418740
$894$	0	0
$895$	−22.4363	−0.749961
$896$	0	0
$897$	0	0
$898$	−14.1405	−0.471873
$899$	6.40576	0.213644
$900$	0	0
$901$	4.11271	0.137014
$902$	8.53737	0.284263
$903$	0	0
$904$	−8.88950	−0.295661
$905$	19.3549	0.643378
$906$	0	0
$907$	10.5535	0.350424	0.175212	−	0.984531i	$-0.443939\pi$
$907$	10.5535	0.350424	0.175212	+	0.984531i	$0.443939\pi$
$908$	−4.56400	−0.151462
$909$	0	0
$910$	0	0
$911$	20.6476	0.684087	0.342043	−	0.939684i	$-0.388881\pi$
$911$	20.6476	0.684087	0.342043	+	0.939684i	$0.388881\pi$
$912$	0	0
$913$	10.1221	0.334993
$914$	−36.1102	−1.19442
$915$	0	0
$916$	−4.31302	−0.142506
$917$	0	0
$918$	0	0
$919$	29.5158	0.973638	0.486819	−	0.873503i	$-0.338157\pi$
$919$	29.5158	0.973638	0.486819	+	0.873503i	$0.338157\pi$
$920$	−5.08009	−0.167486
$921$	0	0
$922$	−12.3377	−0.406319
$923$	1.43242	0.0471486
$924$	0	0
$925$	−2.98151	−0.0980316
$926$	3.90639	0.128372
$927$	0	0
$928$	27.3693	0.898441
$929$	−1.17531	−0.0385608	−0.0192804	−	0.999814i	$-0.506138\pi$
$929$	−1.17531	−0.0385608	−0.0192804	+	0.999814i	$0.506138\pi$
$930$	0	0
$931$	0	0
$932$	39.7671	1.30261
$933$	0	0
$934$	−21.2196	−0.694328
$935$	−7.41122	−0.242373
$936$	0	0
$937$	−6.87262	−0.224519	−0.112259	−	0.993679i	$-0.535809\pi$
$937$	−6.87262	−0.224519	−0.112259	+	0.993679i	$0.535809\pi$
$938$	0	0
$939$	0	0
$940$	−42.3720	−1.38202
$941$	−28.0438	−0.914202	−0.457101	−	0.889415i	$-0.651112\pi$
$941$	−28.0438	−0.914202	−0.457101	+	0.889415i	$0.651112\pi$
$942$	0	0
$943$	−29.0223	−0.945097
$944$	17.1186	0.557162
$945$	0	0
$946$	3.88314	0.126252
$947$	9.09378	0.295508	0.147754	−	0.989024i	$-0.452796\pi$
$947$	9.09378	0.295508	0.147754	+	0.989024i	$0.452796\pi$
$948$	0	0
$949$	0.921372	0.0299090
$950$	−1.86700	−0.0605736
$951$	0	0
$952$	0	0
$953$	−7.70805	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$953$	−7.70805	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$954$	0	0
$955$	0.599809	0.0194094
$956$	−33.3681	−1.07920
$957$	0	0
$958$	−32.7808	−1.05910
$959$	0	0
$960$	0	0
$961$	−27.9244	−0.900786
$962$	1.37183	0.0442294
$963$	0	0
$964$	−15.4310	−0.497000
$965$	26.9127	0.866350
$966$	0	0
$967$	−16.2862	−0.523729	−0.261864	−	0.965105i	$-0.584337\pi$
$967$	−16.2862	−0.523729	−0.261864	+	0.965105i	$0.584337\pi$
$968$	6.35569	0.204280
$969$	0	0
$970$	60.5204	1.94319
$971$	15.2756	0.490218	0.245109	−	0.969496i	$-0.421176\pi$
$971$	15.2756	0.490218	0.245109	+	0.969496i	$0.421176\pi$
$972$	0	0
$973$	0	0
$974$	10.7383	0.344076
$975$	0	0
$976$	14.1487	0.452889
$977$	−51.3383	−1.64246	−0.821229	−	0.570598i	$-0.806711\pi$
$977$	−51.3383	−1.64246	−0.821229	+	0.570598i	$0.806711\pi$
$978$	0	0
$979$	5.43425	0.173679
$980$	0	0
$981$	0	0
$982$	−57.9927	−1.85062
$983$	26.2107	0.835993	0.417996	−	0.908449i	$-0.362732\pi$
$983$	26.2107	0.835993	0.417996	+	0.908449i	$0.362732\pi$
$984$	0	0
$985$	19.6806	0.627075
$986$	−40.1129	−1.27746
$987$	0	0
$988$	0.393083	0.0125056
$989$	−13.2005	−0.419752
$990$	0	0
$991$	27.9607	0.888201	0.444101	−	0.895977i	$-0.353523\pi$
$991$	27.9607	0.888201	0.444101	+	0.895977i	$0.353523\pi$
$992$	13.1409	0.417225
$993$	0	0
$994$	0	0
$995$	−30.0065	−0.951270
$996$	0	0
$997$	−26.4041	−0.836226	−0.418113	−	0.908395i	$-0.637308\pi$
$997$	−26.4041	−0.836226	−0.418113	+	0.908395i	$0.637308\pi$
$998$	−1.73461	−0.0549081
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bv.1.4		4
3.2	odd	2		931.2.a.l.1.1	✓	4
7.6	odd	2		8379.2.a.bu.1.4		4
21.2	odd	6		931.2.f.o.704.4		8
21.5	even	6		931.2.f.n.704.4		8
21.11	odd	6		931.2.f.o.324.4		8
21.17	even	6		931.2.f.n.324.4		8
21.20	even	2		931.2.a.m.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.l.1.1	✓	4	3.2	odd	2
931.2.a.m.1.1	yes	4	21.20	even	2
931.2.f.n.324.4		8	21.17	even	6
931.2.f.n.704.4		8	21.5	even	6
931.2.f.o.324.4		8	21.11	odd	6
931.2.f.o.704.4		8	21.2	odd	6
8379.2.a.bu.1.4		4	7.6	odd	2
8379.2.a.bv.1.4		4	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+1.92022 q^{2} +1.68725 q^{4} -2.00692 q^{5} -0.600553 q^{8} -3.85372 q^{10} +0.645701 q^{11} +0.232973 q^{13} -4.52769 q^{16} +5.71912 q^{17} +1.00000 q^{19} -3.38617 q^{20} +1.23989 q^{22} -4.21494 q^{23} -0.972286 q^{25} +0.447359 q^{26} -3.65262 q^{29} -1.75375 q^{31} -7.49306 q^{32} +10.9820 q^{34} +3.06650 q^{37} +1.92022 q^{38} +1.20526 q^{40} +6.88559 q^{41} +3.13184 q^{43} +1.08946 q^{44} -8.09361 q^{46} +12.5133 q^{47} -1.86700 q^{50} +0.393083 q^{52} +0.719116 q^{53} -1.29587 q^{55} -7.01383 q^{58} -3.78086 q^{59} -3.12493 q^{61} -3.36758 q^{62} -5.33295 q^{64} -0.467557 q^{65} +3.07702 q^{67} +9.64957 q^{68} +6.14844 q^{71} +3.95485 q^{73} +5.88835 q^{74} +1.68725 q^{76} -14.1872 q^{79} +9.08670 q^{80} +13.2219 q^{82} +15.6761 q^{83} -11.4778 q^{85} +6.01383 q^{86} -0.387778 q^{88} +8.41604 q^{89} -7.11165 q^{92} +24.0282 q^{94} -2.00692 q^{95} -15.7044 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{5} - 6 q^{8} - 10 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} - 14 q^{22} + 8 q^{23} + 20 q^{25} + 16 q^{26} - 2 q^{29} - 2 q^{32} + 22 q^{34} + 10 q^{37} - 22 q^{40}+ \cdots - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^5 - 6 * q^8 - 10 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 - 14 * q^22 + 8 * q^23 + 20 * q^25 + 16 * q^26 - 2 * q^29 - 2 * q^32 + 22 * q^34 + 10 * q^37 - 22 * q^40 + 12 * q^41 + 4 * q^43 + 14 * q^44 - 8 * q^46 + 16 * q^47 - 12 * q^50 - 28 * q^52 - 12 * q^53 + 8 * q^55 + 4 * q^58 + 14 * q^59 - 20 * q^61 - 20 * q^62 - 20 * q^64 - 10 * q^65 + 2 * q^67 - 24 * q^68 + 2 * q^71 + 16 * q^73 + 26 * q^74 + 2 * q^76 - 8 * q^79 + 28 * q^80 + 12 * q^82 + 20 * q^83 - 14 * q^85 - 8 * q^86 - 2 * q^88 + 16 * q^89 - 26 * q^92 - 10 * q^94 + 8 * q^95 - 2 * q^97

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