LMFDB - Embedded newform 1323.4.a.bd.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1323,4,Mod(1,1323)] 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("1323.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,32,0,0,0,0,0,-20,0,0,52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$78.0595269376$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 40x^{4} + 453x^{2} - 1278 $ x^6 - 40x^4 + 453x^2 - 1278
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.05936$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05936 q^{2} -3.75905 q^{4} +14.5041 q^{5} -24.2161 q^{8} +29.8691 q^{10} +29.8094 q^{11} -13.3320 q^{13} -19.7971 q^{16} -64.2830 q^{17} -110.350 q^{19} -54.5216 q^{20} +61.3881 q^{22} +19.3456 q^{23} +85.3686 q^{25} -27.4552 q^{26} -111.113 q^{29} +192.938 q^{31} +152.959 q^{32} -132.382 q^{34} -71.5214 q^{37} -227.249 q^{38} -351.232 q^{40} -277.562 q^{41} -178.522 q^{43} -112.055 q^{44} +39.8395 q^{46} -531.874 q^{47} +175.804 q^{50} +50.1155 q^{52} -310.832 q^{53} +432.357 q^{55} -228.821 q^{58} +722.023 q^{59} +663.070 q^{61} +397.328 q^{62} +473.375 q^{64} -193.368 q^{65} -608.559 q^{67} +241.643 q^{68} -976.305 q^{71} +261.148 q^{73} -147.288 q^{74} +414.810 q^{76} +1236.15 q^{79} -287.139 q^{80} -571.600 q^{82} -1225.79 q^{83} -932.367 q^{85} -367.640 q^{86} -721.866 q^{88} +791.982 q^{89} -72.7211 q^{92} -1095.32 q^{94} -1600.52 q^{95} -935.253 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 32 q^{4} - 20 q^{10} + 52 q^{13} - 148 q^{16} - 62 q^{19} - 356 q^{22} - 46 q^{25} - 82 q^{31} + 420 q^{34} - 1132 q^{37} - 444 q^{40} - 1566 q^{43} - 888 q^{46} - 72 q^{52} + 224 q^{55} + 4 q^{58}+ \cdots - 682 q^{97}+O(q^{100}) $ 6 * q + 32 * q^4 - 20 * q^10 + 52 * q^13 - 148 * q^16 - 62 * q^19 - 356 * q^22 - 46 * q^25 - 82 * q^31 + 420 * q^34 - 1132 * q^37 - 444 * q^40 - 1566 * q^43 - 888 * q^46 - 72 * q^52 + 224 * q^55 + 4 * q^58 + 886 * q^61 - 924 * q^64 - 2084 * q^67 - 2398 * q^73 + 3204 * q^76 + 984 * q^79 - 3892 * q^82 - 3600 * q^85 - 5796 * q^88 + 2772 * q^94 - 682 * 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 $

$2$	2.05936	0.728092	0.364046	−	0.931381i	$-0.381395\pi$
$2$	2.05936	0.728092	0.364046	+	0.931381i	$0.381395\pi$
$3$	0	0
$3$	0	0
$4$	−3.75905	−0.469882
$5$	14.5041	1.29729	0.648643	−	0.761093i	$-0.275337\pi$
$5$	14.5041	1.29729	0.648643	+	0.761093i	$0.275337\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−24.2161	−1.07021
$9$	0	0
$10$	29.8691	0.944543
$11$	29.8094	0.817078	0.408539	−	0.912741i	$-0.366038\pi$
$11$	29.8094	0.817078	0.408539	+	0.912741i	$0.366038\pi$
$12$	0	0
$13$	−13.3320	−0.284432	−0.142216	−	0.989836i	$-0.545423\pi$
$13$	−13.3320	−0.284432	−0.142216	+	0.989836i	$0.545423\pi$
$14$	0	0
$15$	0	0
$16$	−19.7971	−0.309330
$17$	−64.2830	−0.917113	−0.458557	−	0.888665i	$-0.651633\pi$
$17$	−64.2830	−0.917113	−0.458557	+	0.888665i	$0.651633\pi$
$18$	0	0
$19$	−110.350	−1.33242	−0.666209	−	0.745765i	$-0.732084\pi$
$19$	−110.350	−1.33242	−0.666209	+	0.745765i	$0.732084\pi$
$20$	−54.5216	−0.609570
$21$	0	0
$22$	61.3881	0.594908
$23$	19.3456	0.175384	0.0876921	−	0.996148i	$-0.472051\pi$
$23$	19.3456	0.175384	0.0876921	+	0.996148i	$0.472051\pi$
$24$	0	0
$25$	85.3686	0.682949
$26$	−27.4552	−0.207093
$27$	0	0
$28$	0	0
$29$	−111.113	−0.711487	−0.355744	−	0.934584i	$-0.615772\pi$
$29$	−111.113	−0.711487	−0.355744	+	0.934584i	$0.615772\pi$
$30$	0	0
$31$	192.938	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$31$	192.938	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$32$	152.959	0.844989
$33$	0	0
$34$	−132.382	−0.667743
$35$	0	0
$36$	0	0
$37$	−71.5214	−0.317785	−0.158892	−	0.987296i	$-0.550792\pi$
$37$	−71.5214	−0.317785	−0.158892	+	0.987296i	$0.550792\pi$
$38$	−227.249	−0.970123
$39$	0	0
$40$	−351.232	−1.38837
$41$	−277.562	−1.05727	−0.528634	−	0.848850i	$-0.677296\pi$
$41$	−277.562	−1.05727	−0.528634	+	0.848850i	$0.677296\pi$
$42$	0	0
$43$	−178.522	−0.633124	−0.316562	−	0.948572i	$-0.602528\pi$
$43$	−178.522	−0.633124	−0.316562	+	0.948572i	$0.602528\pi$
$44$	−112.055	−0.383930
$45$	0	0
$46$	39.8395	0.127696
$47$	−531.874	−1.65068	−0.825338	−	0.564638i	$-0.809016\pi$
$47$	−531.874	−1.65068	−0.825338	+	0.564638i	$0.809016\pi$
$48$	0	0
$49$	0	0
$50$	175.804	0.497250
$51$	0	0
$52$	50.1155	0.133649
$53$	−310.832	−0.805586	−0.402793	−	0.915291i	$-0.631961\pi$
$53$	−310.832	−0.805586	−0.402793	+	0.915291i	$0.631961\pi$
$54$	0	0
$55$	432.357	1.05998
$56$	0	0
$57$	0	0
$58$	−228.821	−0.518029
$59$	722.023	1.59321	0.796605	−	0.604500i	$-0.206627\pi$
$59$	722.023	1.59321	0.796605	+	0.604500i	$0.206627\pi$
$60$	0	0
$61$	663.070	1.39176	0.695880	−	0.718158i	$-0.255014\pi$
$61$	663.070	1.39176	0.695880	+	0.718158i	$0.255014\pi$
$62$	397.328	0.813882
$63$	0	0
$64$	473.375	0.924560
$65$	−193.368	−0.368990
$66$	0	0
$67$	−608.559	−1.10966	−0.554830	−	0.831964i	$-0.687217\pi$
$67$	−608.559	−1.10966	−0.554830	+	0.831964i	$0.687217\pi$
$68$	241.643	0.430934
$69$	0	0
$70$	0	0
$71$	−976.305	−1.63192	−0.815958	−	0.578111i	$-0.803790\pi$
$71$	−976.305	−1.63192	−0.815958	+	0.578111i	$0.803790\pi$
$72$	0	0
$73$	261.148	0.418700	0.209350	−	0.977841i	$-0.432865\pi$
$73$	261.148	0.418700	0.209350	+	0.977841i	$0.432865\pi$
$74$	−147.288	−0.231377
$75$	0	0
$76$	414.810	0.626078
$77$	0	0
$78$	0	0
$79$	1236.15	1.76048	0.880239	−	0.474531i	$-0.157382\pi$
$79$	1236.15	1.76048	0.880239	+	0.474531i	$0.157382\pi$
$80$	−287.139	−0.401289
$81$	0	0
$82$	−571.600	−0.769789
$83$	−1225.79	−1.62106	−0.810529	−	0.585699i	$-0.800820\pi$
$83$	−1225.79	−1.62106	−0.810529	+	0.585699i	$0.800820\pi$
$84$	0	0
$85$	−932.367	−1.18976
$86$	−367.640	−0.460973
$87$	0	0
$88$	−721.866	−0.874445
$89$	791.982	0.943257	0.471629	−	0.881797i	$-0.343666\pi$
$89$	791.982	0.943257	0.471629	+	0.881797i	$0.343666\pi$
$90$	0	0
$91$	0	0
$92$	−72.7211	−0.0824098
$93$	0	0
$94$	−1095.32	−1.20185
$95$	−1600.52	−1.72853
$96$	0	0
$97$	−935.253	−0.978975	−0.489488	−	0.872010i	$-0.662816\pi$
$97$	−935.253	−0.978975	−0.489488	+	0.872010i	$0.662816\pi$
$98$	0	0
$99$	0	0
$100$	−320.905	−0.320905
$101$	442.969	0.436407	0.218203	−	0.975903i	$-0.429980\pi$
$101$	442.969	0.436407	0.218203	+	0.975903i	$0.429980\pi$
$102$	0	0
$103$	−699.401	−0.669068	−0.334534	−	0.942384i	$-0.608579\pi$
$103$	−699.401	−0.669068	−0.334534	+	0.942384i	$0.608579\pi$
$104$	322.848	0.304402
$105$	0	0
$106$	−640.114	−0.586541
$107$	−1911.18	−1.72674	−0.863369	−	0.504573i	$-0.831650\pi$
$107$	−1911.18	−1.72674	−0.863369	+	0.504573i	$0.831650\pi$
$108$	0	0
$109$	−2071.42	−1.82024	−0.910118	−	0.414349i	$-0.864009\pi$
$109$	−2071.42	−1.82024	−0.910118	+	0.414349i	$0.864009\pi$
$110$	890.378	0.771766
$111$	0	0
$112$	0	0
$113$	2313.34	1.92585	0.962926	−	0.269767i	$-0.0869467\pi$
$113$	2313.34	1.92585	0.962926	+	0.269767i	$0.0869467\pi$
$114$	0	0
$115$	280.590	0.227523
$116$	417.679	0.334315
$117$	0	0
$118$	1486.90	1.16000
$119$	0	0
$120$	0	0
$121$	−442.403	−0.332384
$122$	1365.50	1.01333
$123$	0	0
$124$	−725.263	−0.525246
$125$	−574.817	−0.411306
$126$	0	0
$127$	−1317.50	−0.920542	−0.460271	−	0.887778i	$-0.652248\pi$
$127$	−1317.50	−0.920542	−0.460271	+	0.887778i	$0.652248\pi$
$128$	−248.828	−0.171824
$129$	0	0
$130$	−398.213	−0.268659
$131$	458.841	0.306024	0.153012	−	0.988224i	$-0.451103\pi$
$131$	458.841	0.306024	0.153012	+	0.988224i	$0.451103\pi$
$132$	0	0
$133$	0	0
$134$	−1253.24	−0.807935
$135$	0	0
$136$	1556.68	0.981503
$137$	−1332.76	−0.831132	−0.415566	−	0.909563i	$-0.636416\pi$
$137$	−1332.76	−0.831132	−0.415566	+	0.909563i	$0.636416\pi$
$138$	0	0
$139$	−406.835	−0.248254	−0.124127	−	0.992266i	$-0.539613\pi$
$139$	−406.835	−0.248254	−0.124127	+	0.992266i	$0.539613\pi$
$140$	0	0
$141$	0	0
$142$	−2010.56	−1.18819
$143$	−397.417	−0.232403
$144$	0	0
$145$	−1611.59	−0.923002
$146$	537.797	0.304852
$147$	0	0
$148$	268.853	0.149321
$149$	435.523	0.239459	0.119730	−	0.992807i	$-0.461797\pi$
$149$	435.523	0.239459	0.119730	+	0.992807i	$0.461797\pi$
$150$	0	0
$151$	−1965.34	−1.05918	−0.529592	−	0.848253i	$-0.677655\pi$
$151$	−1965.34	−1.05918	−0.529592	+	0.848253i	$0.677655\pi$
$152$	2672.23	1.42597
$153$	0	0
$154$	0	0
$155$	2798.39	1.45014
$156$	0	0
$157$	831.483	0.422672	0.211336	−	0.977413i	$-0.432219\pi$
$157$	831.483	0.422672	0.211336	+	0.977413i	$0.432219\pi$
$158$	2545.67	1.28179
$159$	0	0
$160$	2218.54	1.09619
$161$	0	0
$162$	0	0
$163$	−1176.12	−0.565159	−0.282579	−	0.959244i	$-0.591190\pi$
$163$	−1176.12	−0.565159	−0.282579	+	0.959244i	$0.591190\pi$
$164$	1043.37	0.496791
$165$	0	0
$166$	−2524.34	−1.18028
$167$	−1674.29	−0.775809	−0.387905	−	0.921700i	$-0.626801\pi$
$167$	−1674.29	−0.775809	−0.387905	+	0.921700i	$0.626801\pi$
$168$	0	0
$169$	−2019.26	−0.919098
$170$	−1920.08	−0.866253
$171$	0	0
$172$	671.073	0.297493
$173$	−1432.92	−0.629728	−0.314864	−	0.949137i	$-0.601959\pi$
$173$	−1432.92	−0.629728	−0.314864	+	0.949137i	$0.601959\pi$
$174$	0	0
$175$	0	0
$176$	−590.139	−0.252747
$177$	0	0
$178$	1630.97	0.686778
$179$	1194.53	0.498790	0.249395	−	0.968402i	$-0.419768\pi$
$179$	1194.53	0.498790	0.249395	+	0.968402i	$0.419768\pi$
$180$	0	0
$181$	741.424	0.304473	0.152236	−	0.988344i	$-0.451352\pi$
$181$	741.424	0.304473	0.152236	+	0.988344i	$0.451352\pi$
$182$	0	0
$183$	0	0
$184$	−468.474	−0.187698
$185$	−1037.35	−0.412258
$186$	0	0
$187$	−1916.24	−0.749353
$188$	1999.34	0.775623
$189$	0	0
$190$	−3296.04	−1.25853
$191$	3903.98	1.47897	0.739483	−	0.673175i	$-0.235070\pi$
$191$	3903.98	1.47897	0.739483	+	0.673175i	$0.235070\pi$
$192$	0	0
$193$	1771.25	0.660609	0.330305	−	0.943874i	$-0.392849\pi$
$193$	1771.25	0.660609	0.330305	+	0.943874i	$0.392849\pi$
$194$	−1926.02	−0.712784
$195$	0	0
$196$	0	0
$197$	−830.366	−0.300310	−0.150155	−	0.988662i	$-0.547977\pi$
$197$	−830.366	−0.300310	−0.150155	+	0.988662i	$0.547977\pi$
$198$	0	0
$199$	1062.51	0.378487	0.189244	−	0.981930i	$-0.439396\pi$
$199$	1062.51	0.378487	0.189244	+	0.981930i	$0.439396\pi$
$200$	−2067.29	−0.730898
$201$	0	0
$202$	912.232	0.317745
$203$	0	0
$204$	0	0
$205$	−4025.79	−1.37158
$206$	−1440.32	−0.487143
$207$	0	0
$208$	263.934	0.0879834
$209$	−3289.45	−1.08869
$210$	0	0
$211$	−3848.67	−1.25570	−0.627851	−	0.778333i	$-0.716065\pi$
$211$	−3848.67	−1.25570	−0.627851	+	0.778333i	$0.716065\pi$
$212$	1168.43	0.378530
$213$	0	0
$214$	−3935.81	−1.25722
$215$	−2589.30	−0.821342
$216$	0	0
$217$	0	0
$218$	−4265.79	−1.32530
$219$	0	0
$220$	−1625.25	−0.498066
$221$	857.018	0.260856
$222$	0	0
$223$	5482.75	1.64642	0.823211	−	0.567736i	$-0.192181\pi$
$223$	5482.75	1.64642	0.823211	+	0.567736i	$0.192181\pi$
$224$	0	0
$225$	0	0
$226$	4764.00	1.40220
$227$	−394.837	−0.115446	−0.0577230	−	0.998333i	$-0.518384\pi$
$227$	−394.837	−0.115446	−0.0577230	+	0.998333i	$0.518384\pi$
$228$	0	0
$229$	−3823.57	−1.10336	−0.551678	−	0.834057i	$-0.686012\pi$
$229$	−3823.57	−1.10336	−0.551678	+	0.834057i	$0.686012\pi$
$230$	577.835	0.165658
$231$	0	0
$232$	2690.72	0.761441
$233$	2537.54	0.713475	0.356738	−	0.934205i	$-0.383889\pi$
$233$	2537.54	0.713475	0.356738	+	0.934205i	$0.383889\pi$
$234$	0	0
$235$	−7714.35	−2.14140
$236$	−2714.12	−0.748620
$237$	0	0
$238$	0	0
$239$	−5368.81	−1.45305	−0.726526	−	0.687139i	$-0.758866\pi$
$239$	−5368.81	−1.45305	−0.726526	+	0.687139i	$0.758866\pi$
$240$	0	0
$241$	2108.02	0.563442	0.281721	−	0.959496i	$-0.409095\pi$
$241$	2108.02	0.563442	0.281721	+	0.959496i	$0.409095\pi$
$242$	−911.064	−0.242006
$243$	0	0
$244$	−2492.51	−0.653962
$245$	0	0
$246$	0	0
$247$	1471.18	0.378982
$248$	−4672.20	−1.19631
$249$	0	0
$250$	−1183.75	−0.299469
$251$	5717.94	1.43790	0.718951	−	0.695061i	$-0.244623\pi$
$251$	5717.94	1.43790	0.718951	+	0.695061i	$0.244623\pi$
$252$	0	0
$253$	576.680	0.143303
$254$	−2713.19	−0.670240
$255$	0	0
$256$	−4299.42	−1.04966
$257$	−4987.47	−1.21054	−0.605272	−	0.796019i	$-0.706936\pi$
$257$	−4987.47	−1.21054	−0.605272	+	0.796019i	$0.706936\pi$
$258$	0	0
$259$	0	0
$260$	726.880	0.173381
$261$	0	0
$262$	944.917	0.222814
$263$	−615.273	−0.144256	−0.0721281	−	0.997395i	$-0.522979\pi$
$263$	−615.273	−0.144256	−0.0721281	+	0.997395i	$0.522979\pi$
$264$	0	0
$265$	−4508.34	−1.04508
$266$	0	0
$267$	0	0
$268$	2287.60	0.521409
$269$	793.962	0.179958	0.0899790	−	0.995944i	$-0.471320\pi$
$269$	793.962	0.179958	0.0899790	+	0.995944i	$0.471320\pi$
$270$	0	0
$271$	3764.95	0.843927	0.421964	−	0.906613i	$-0.361341\pi$
$271$	3764.95	0.843927	0.421964	+	0.906613i	$0.361341\pi$
$272$	1272.62	0.283690
$273$	0	0
$274$	−2744.62	−0.605141
$275$	2544.78	0.558022
$276$	0	0
$277$	−3196.54	−0.693364	−0.346682	−	0.937983i	$-0.612692\pi$
$277$	−3196.54	−0.693364	−0.346682	+	0.937983i	$0.612692\pi$
$278$	−837.817	−0.180752
$279$	0	0
$280$	0	0
$281$	6225.15	1.32157	0.660785	−	0.750575i	$-0.270223\pi$
$281$	6225.15	1.32157	0.660785	+	0.750575i	$0.270223\pi$
$282$	0	0
$283$	−3717.50	−0.780857	−0.390429	−	0.920633i	$-0.627673\pi$
$283$	−3717.50	−0.780857	−0.390429	+	0.920633i	$0.627673\pi$
$284$	3669.98	0.766807
$285$	0	0
$286$	−818.423	−0.169211
$287$	0	0
$288$	0	0
$289$	−780.693	−0.158904
$290$	−3318.84	−0.672031
$291$	0	0
$292$	−981.669	−0.196739
$293$	6392.56	1.27460	0.637299	−	0.770616i	$-0.280052\pi$
$293$	6392.56	1.27460	0.637299	+	0.770616i	$0.280052\pi$
$294$	0	0
$295$	10472.3	2.06685
$296$	1731.97	0.340096
$297$	0	0
$298$	896.897	0.174348
$299$	−257.915	−0.0498849
$300$	0	0
$301$	0	0
$302$	−4047.33	−0.771183
$303$	0	0
$304$	2184.60	0.412156
$305$	9617.22	1.80551
$306$	0	0
$307$	939.255	0.174613	0.0873063	−	0.996182i	$-0.472174\pi$
$307$	939.255	0.174613	0.0873063	+	0.996182i	$0.472174\pi$
$308$	0	0
$309$	0	0
$310$	5762.88	1.05584
$311$	2751.61	0.501703	0.250851	−	0.968026i	$-0.419289\pi$
$311$	2751.61	0.501703	0.250851	+	0.968026i	$0.419289\pi$
$312$	0	0
$313$	−637.099	−0.115051	−0.0575255	−	0.998344i	$-0.518321\pi$
$313$	−637.099	−0.115051	−0.0575255	+	0.998344i	$0.518321\pi$
$314$	1712.32	0.307744
$315$	0	0
$316$	−4646.75	−0.827216
$317$	2872.19	0.508891	0.254445	−	0.967087i	$-0.418107\pi$
$317$	2872.19	0.508891	0.254445	+	0.967087i	$0.418107\pi$
$318$	0	0
$319$	−3312.20	−0.581341
$320$	6865.87	1.19942
$321$	0	0
$322$	0	0
$323$	7093.60	1.22198
$324$	0	0
$325$	−1138.13	−0.194253
$326$	−2422.05	−0.411488
$327$	0	0
$328$	6721.47	1.13150
$329$	0	0
$330$	0	0
$331$	−6393.60	−1.06170	−0.530852	−	0.847464i	$-0.678128\pi$
$331$	−6393.60	−1.06170	−0.530852	+	0.847464i	$0.678128\pi$
$332$	4607.80	0.761705
$333$	0	0
$334$	−3447.95	−0.564861
$335$	−8826.59	−1.43955
$336$	0	0
$337$	−12107.6	−1.95710	−0.978549	−	0.206016i	$-0.933950\pi$
$337$	−12107.6	−1.95710	−0.978549	+	0.206016i	$0.933950\pi$
$338$	−4158.37	−0.669188
$339$	0	0
$340$	3504.81	0.559045
$341$	5751.35	0.913352
$342$	0	0
$343$	0	0
$344$	4323.10	0.677575
$345$	0	0
$346$	−2950.89	−0.458500
$347$	−2710.14	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$347$	−2710.14	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$348$	0	0
$349$	−1175.22	−0.180252	−0.0901261	−	0.995930i	$-0.528727\pi$
$349$	−1175.22	−0.180252	−0.0901261	+	0.995930i	$0.528727\pi$
$350$	0	0
$351$	0	0
$352$	4559.62	0.690422
$353$	−5276.41	−0.795566	−0.397783	−	0.917480i	$-0.630220\pi$
$353$	−5276.41	−0.795566	−0.397783	+	0.917480i	$0.630220\pi$
$354$	0	0
$355$	−14160.4	−2.11706
$356$	−2977.10	−0.443219
$357$	0	0
$358$	2459.96	0.363165
$359$	11286.5	1.65928	0.829639	−	0.558300i	$-0.188546\pi$
$359$	11286.5	1.65928	0.829639	+	0.558300i	$0.188546\pi$
$360$	0	0
$361$	5318.03	0.775336
$362$	1526.86	0.221684
$363$	0	0
$364$	0	0
$365$	3787.72	0.543173
$366$	0	0
$367$	7567.59	1.07636	0.538181	−	0.842829i	$-0.319112\pi$
$367$	7567.59	1.07636	0.538181	+	0.842829i	$0.319112\pi$
$368$	−382.987	−0.0542516
$369$	0	0
$370$	−2136.28	−0.300162
$371$	0	0
$372$	0	0
$373$	−6503.97	−0.902850	−0.451425	−	0.892309i	$-0.649084\pi$
$373$	−6503.97	−0.902850	−0.451425	+	0.892309i	$0.649084\pi$
$374$	−3946.21	−0.545598
$375$	0	0
$376$	12879.9	1.76657
$377$	1481.35	0.202370
$378$	0	0
$379$	10773.8	1.46019	0.730097	−	0.683344i	$-0.239475\pi$
$379$	10773.8	1.46019	0.730097	+	0.683344i	$0.239475\pi$
$380$	6016.44	0.812202
$381$	0	0
$382$	8039.70	1.07682
$383$	6366.13	0.849331	0.424666	−	0.905350i	$-0.360392\pi$
$383$	6366.13	0.849331	0.424666	+	0.905350i	$0.360392\pi$
$384$	0	0
$385$	0	0
$386$	3647.64	0.480985
$387$	0	0
$388$	3515.67	0.460002
$389$	5903.44	0.769450	0.384725	−	0.923031i	$-0.374296\pi$
$389$	5903.44	0.769450	0.384725	+	0.923031i	$0.374296\pi$
$390$	0	0
$391$	−1243.59	−0.160847
$392$	0	0
$393$	0	0
$394$	−1710.02	−0.218654
$395$	17929.2	2.28384
$396$	0	0
$397$	3766.78	0.476194	0.238097	−	0.971241i	$-0.423476\pi$
$397$	3766.78	0.476194	0.238097	+	0.971241i	$0.423476\pi$
$398$	2188.08	0.275574
$399$	0	0
$400$	−1690.05	−0.211256
$401$	−7252.00	−0.903112	−0.451556	−	0.892243i	$-0.649131\pi$
$401$	−7252.00	−0.903112	−0.451556	+	0.892243i	$0.649131\pi$
$402$	0	0
$403$	−2572.24	−0.317946
$404$	−1665.15	−0.205060
$405$	0	0
$406$	0	0
$407$	−2132.01	−0.259655
$408$	0	0
$409$	12547.4	1.51694	0.758471	−	0.651707i	$-0.225947\pi$
$409$	12547.4	1.51694	0.758471	+	0.651707i	$0.225947\pi$
$410$	−8290.54	−0.998635
$411$	0	0
$412$	2629.08	0.314383
$413$	0	0
$414$	0	0
$415$	−17778.9	−2.10297
$416$	−2039.25	−0.240342
$417$	0	0
$418$	−6774.15	−0.792666
$419$	12631.5	1.47276	0.736381	−	0.676567i	$-0.236533\pi$
$419$	12631.5	1.47276	0.736381	+	0.676567i	$0.236533\pi$
$420$	0	0
$421$	−3495.94	−0.404707	−0.202354	−	0.979312i	$-0.564859\pi$
$421$	−3495.94	−0.404707	−0.202354	+	0.979312i	$0.564859\pi$
$422$	−7925.78	−0.914267
$423$	0	0
$424$	7527.13	0.862146
$425$	−5487.75	−0.626341
$426$	0	0
$427$	0	0
$428$	7184.23	0.811362
$429$	0	0
$430$	−5332.28	−0.598013
$431$	2345.18	0.262096	0.131048	−	0.991376i	$-0.458166\pi$
$431$	2345.18	0.262096	0.131048	+	0.991376i	$0.458166\pi$
$432$	0	0
$433$	3250.63	0.360774	0.180387	−	0.983596i	$-0.442265\pi$
$433$	3250.63	0.360774	0.180387	+	0.983596i	$0.442265\pi$
$434$	0	0
$435$	0	0
$436$	7786.56	0.855295
$437$	−2134.78	−0.233685
$438$	0	0
$439$	16240.5	1.76564	0.882820	−	0.469711i	$-0.155642\pi$
$439$	16240.5	1.76564	0.882820	+	0.469711i	$0.155642\pi$
$440$	−10470.0	−1.13440
$441$	0	0
$442$	1764.91	0.189928
$443$	−4849.04	−0.520056	−0.260028	−	0.965601i	$-0.583732\pi$
$443$	−4849.04	−0.520056	−0.260028	+	0.965601i	$0.583732\pi$
$444$	0	0
$445$	11487.0	1.22367
$446$	11290.9	1.19875
$447$	0	0
$448$	0	0
$449$	−4139.93	−0.435134	−0.217567	−	0.976045i	$-0.569812\pi$
$449$	−4139.93	−0.435134	−0.217567	+	0.976045i	$0.569812\pi$
$450$	0	0
$451$	−8273.96	−0.863870
$452$	−8695.98	−0.904922
$453$	0	0
$454$	−813.110	−0.0840553
$455$	0	0
$456$	0	0
$457$	9232.48	0.945027	0.472513	−	0.881323i	$-0.343347\pi$
$457$	9232.48	0.945027	0.472513	+	0.881323i	$0.343347\pi$
$458$	−7874.08	−0.803344
$459$	0	0
$460$	−1054.75	−0.106909
$461$	1274.90	0.128803	0.0644013	−	0.997924i	$-0.479486\pi$
$461$	1274.90	0.128803	0.0644013	+	0.997924i	$0.479486\pi$
$462$	0	0
$463$	2061.80	0.206955	0.103477	−	0.994632i	$-0.467003\pi$
$463$	2061.80	0.206955	0.103477	+	0.994632i	$0.467003\pi$
$464$	2199.71	0.220084
$465$	0	0
$466$	5225.70	0.519476
$467$	−968.041	−0.0959221	−0.0479610	−	0.998849i	$-0.515272\pi$
$467$	−968.041	−0.0959221	−0.0479610	+	0.998849i	$0.515272\pi$
$468$	0	0
$469$	0	0
$470$	−15886.6	−1.55914
$471$	0	0
$472$	−17484.6	−1.70507
$473$	−5321.62	−0.517311
$474$	0	0
$475$	−9420.39	−0.909973
$476$	0	0
$477$	0	0
$478$	−11056.3	−1.05796
$479$	6200.05	0.591414	0.295707	−	0.955279i	$-0.404445\pi$
$479$	6200.05	0.591414	0.295707	+	0.955279i	$0.404445\pi$
$480$	0	0
$481$	953.519	0.0903882
$482$	4341.17	0.410238
$483$	0	0
$484$	1663.01	0.156181
$485$	−13565.0	−1.27001
$486$	0	0
$487$	−9829.66	−0.914629	−0.457315	−	0.889305i	$-0.651189\pi$
$487$	−9829.66	−0.914629	−0.457315	+	0.889305i	$0.651189\pi$
$488$	−16056.9	−1.48948
$489$	0	0
$490$	0	0
$491$	−5053.57	−0.464489	−0.232245	−	0.972657i	$-0.574607\pi$
$491$	−5053.57	−0.464489	−0.232245	+	0.972657i	$0.574607\pi$
$492$	0	0
$493$	7142.67	0.652514
$494$	3029.67	0.275934
$495$	0	0
$496$	−3819.61	−0.345777
$497$	0	0
$498$	0	0
$499$	14061.6	1.26149	0.630747	−	0.775988i	$-0.282749\pi$
$499$	14061.6	1.26149	0.630747	+	0.775988i	$0.282749\pi$
$500$	2160.77	0.193265
$501$	0	0
$502$	11775.3	1.04693
$503$	8001.67	0.709298	0.354649	−	0.934999i	$-0.384600\pi$
$503$	8001.67	0.709298	0.354649	+	0.934999i	$0.384600\pi$
$504$	0	0
$505$	6424.87	0.566144
$506$	1187.59	0.104337
$507$	0	0
$508$	4952.53	0.432546
$509$	−10913.8	−0.950386	−0.475193	−	0.879881i	$-0.657622\pi$
$509$	−10913.8	−0.950386	−0.475193	+	0.879881i	$0.657622\pi$
$510$	0	0
$511$	0	0
$512$	−6863.42	−0.592428
$513$	0	0
$514$	−10271.0	−0.881388
$515$	−10144.2	−0.867972
$516$	0	0
$517$	−15854.8	−1.34873
$518$	0	0
$519$	0	0
$520$	4682.61	0.394896
$521$	−15803.4	−1.32890	−0.664452	−	0.747331i	$-0.731335\pi$
$521$	−15803.4	−1.32890	−0.664452	+	0.747331i	$0.731335\pi$
$522$	0	0
$523$	−646.094	−0.0540186	−0.0270093	−	0.999635i	$-0.508598\pi$
$523$	−646.094	−0.0540186	−0.0270093	+	0.999635i	$0.508598\pi$
$524$	−1724.81	−0.143795
$525$	0	0
$526$	−1267.07	−0.105032
$527$	−12402.6	−1.02517
$528$	0	0
$529$	−11792.7	−0.969240
$530$	−9284.27	−0.760911
$531$	0	0
$532$	0	0
$533$	3700.45	0.300721
$534$	0	0
$535$	−27720.0	−2.24007
$536$	14736.9	1.18757
$537$	0	0
$538$	1635.05	0.131026
$539$	0	0
$540$	0	0
$541$	10246.7	0.814303	0.407152	−	0.913361i	$-0.366522\pi$
$541$	10246.7	0.814303	0.407152	+	0.913361i	$0.366522\pi$
$542$	7753.37	0.614457
$543$	0	0
$544$	−9832.69	−0.774950
$545$	−30044.0	−2.36137
$546$	0	0
$547$	4368.98	0.341506	0.170753	−	0.985314i	$-0.445380\pi$
$547$	4368.98	0.341506	0.170753	+	0.985314i	$0.445380\pi$
$548$	5009.90	0.390533
$549$	0	0
$550$	5240.61	0.406292
$551$	12261.3	0.947998
$552$	0	0
$553$	0	0
$554$	−6582.82	−0.504833
$555$	0	0
$556$	1529.31	0.116650
$557$	−8396.43	−0.638722	−0.319361	−	0.947633i	$-0.603468\pi$
$557$	−8396.43	−0.638722	−0.319361	+	0.947633i	$0.603468\pi$
$558$	0	0
$559$	2380.04	0.180081
$560$	0	0
$561$	0	0
$562$	12819.8	0.962226
$563$	12142.3	0.908944	0.454472	−	0.890761i	$-0.349828\pi$
$563$	12142.3	0.908944	0.454472	+	0.890761i	$0.349828\pi$
$564$	0	0
$565$	33553.0	2.49838
$566$	−7655.66	−0.568536
$567$	0	0
$568$	23642.3	1.74649
$569$	11196.1	0.824896	0.412448	−	0.910981i	$-0.364674\pi$
$569$	11196.1	0.824896	0.412448	+	0.910981i	$0.364674\pi$
$570$	0	0
$571$	19659.6	1.44085	0.720427	−	0.693531i	$-0.243946\pi$
$571$	19659.6	1.44085	0.720427	+	0.693531i	$0.243946\pi$
$572$	1493.91	0.109202
$573$	0	0
$574$	0	0
$575$	1651.51	0.119778
$576$	0	0
$577$	8516.45	0.614462	0.307231	−	0.951635i	$-0.400598\pi$
$577$	8516.45	0.614462	0.307231	+	0.951635i	$0.400598\pi$
$578$	−1607.73	−0.115696
$579$	0	0
$580$	6058.05	0.433702
$581$	0	0
$582$	0	0
$583$	−9265.70	−0.658227
$584$	−6323.98	−0.448096
$585$	0	0
$586$	13164.6	0.928025
$587$	−7115.53	−0.500323	−0.250161	−	0.968204i	$-0.580484\pi$
$587$	−7115.53	−0.500323	−0.250161	+	0.968204i	$0.580484\pi$
$588$	0	0
$589$	−21290.6	−1.48941
$590$	21566.2	1.50486
$591$	0	0
$592$	1415.92	0.0983003
$593$	6144.48	0.425504	0.212752	−	0.977106i	$-0.431757\pi$
$593$	6144.48	0.425504	0.212752	+	0.977106i	$0.431757\pi$
$594$	0	0
$595$	0	0
$596$	−1637.15	−0.112517
$597$	0	0
$598$	−531.138	−0.0363208
$599$	9578.39	0.653360	0.326680	−	0.945135i	$-0.394070\pi$
$599$	9578.39	0.653360	0.326680	+	0.945135i	$0.394070\pi$
$600$	0	0
$601$	−19113.8	−1.29728	−0.648642	−	0.761094i	$-0.724663\pi$
$601$	−19113.8	−1.29728	−0.648642	+	0.761094i	$0.724663\pi$
$602$	0	0
$603$	0	0
$604$	7387.80	0.497691
$605$	−6416.65	−0.431196
$606$	0	0
$607$	20442.6	1.36695	0.683477	−	0.729972i	$-0.260467\pi$
$607$	20442.6	1.36695	0.683477	+	0.729972i	$0.260467\pi$
$608$	−16879.0	−1.12588
$609$	0	0
$610$	19805.3	1.31458
$611$	7090.92	0.469506
$612$	0	0
$613$	17422.7	1.14796	0.573978	−	0.818870i	$-0.305399\pi$
$613$	17422.7	1.14796	0.573978	+	0.818870i	$0.305399\pi$
$614$	1934.26	0.127134
$615$	0	0
$616$	0	0
$617$	−1805.17	−0.117785	−0.0588926	−	0.998264i	$-0.518757\pi$
$617$	−1805.17	−0.117785	−0.0588926	+	0.998264i	$0.518757\pi$
$618$	0	0
$619$	8142.23	0.528698	0.264349	−	0.964427i	$-0.414843\pi$
$619$	8142.23	0.528698	0.264349	+	0.964427i	$0.414843\pi$
$620$	−10519.3	−0.681394
$621$	0	0
$622$	5666.55	0.365286
$623$	0	0
$624$	0	0
$625$	−19008.3	−1.21653
$626$	−1312.01	−0.0837678
$627$	0	0
$628$	−3125.59	−0.198606
$629$	4597.61	0.291445
$630$	0	0
$631$	3630.07	0.229019	0.114509	−	0.993422i	$-0.463470\pi$
$631$	3630.07	0.229019	0.114509	+	0.993422i	$0.463470\pi$
$632$	−29934.7	−1.88408
$633$	0	0
$634$	5914.87	0.370519
$635$	−19109.1	−1.19421
$636$	0	0
$637$	0	0
$638$	−6821.00	−0.423270
$639$	0	0
$640$	−3609.02	−0.222905
$641$	22223.6	1.36939	0.684696	−	0.728829i	$-0.259935\pi$
$641$	22223.6	1.36939	0.684696	+	0.728829i	$0.259935\pi$
$642$	0	0
$643$	−5013.00	−0.307455	−0.153727	−	0.988113i	$-0.549128\pi$
$643$	−5013.00	−0.307455	−0.153727	+	0.988113i	$0.549128\pi$
$644$	0	0
$645$	0	0
$646$	14608.3	0.889712
$647$	7716.16	0.468862	0.234431	−	0.972133i	$-0.424677\pi$
$647$	7716.16	0.468862	0.234431	+	0.972133i	$0.424677\pi$
$648$	0	0
$649$	21523.0	1.30178
$650$	−2343.82	−0.141434
$651$	0	0
$652$	4421.10	0.265558
$653$	−5644.40	−0.338258	−0.169129	−	0.985594i	$-0.554095\pi$
$653$	−5644.40	−0.338258	−0.169129	+	0.985594i	$0.554095\pi$
$654$	0	0
$655$	6655.07	0.397000
$656$	5494.94	0.327044
$657$	0	0
$658$	0	0
$659$	8812.27	0.520906	0.260453	−	0.965487i	$-0.416128\pi$
$659$	8812.27	0.520906	0.260453	+	0.965487i	$0.416128\pi$
$660$	0	0
$661$	−6191.02	−0.364300	−0.182150	−	0.983271i	$-0.558306\pi$
$661$	−6191.02	−0.364300	−0.182150	+	0.983271i	$0.558306\pi$
$662$	−13166.7	−0.773019
$663$	0	0
$664$	29683.8	1.73487
$665$	0	0
$666$	0	0
$667$	−2149.54	−0.124784
$668$	6293.73	0.364538
$669$	0	0
$670$	−18177.1	−1.04812
$671$	19765.7	1.13718
$672$	0	0
$673$	29580.7	1.69428	0.847141	−	0.531368i	$-0.178322\pi$
$673$	29580.7	1.69428	0.847141	+	0.531368i	$0.178322\pi$
$674$	−24933.8	−1.42495
$675$	0	0
$676$	7590.50	0.431867
$677$	26290.0	1.49248	0.746239	−	0.665678i	$-0.231857\pi$
$677$	26290.0	1.49248	0.746239	+	0.665678i	$0.231857\pi$
$678$	0	0
$679$	0	0
$680$	22578.3	1.27329
$681$	0	0
$682$	11844.1	0.665005
$683$	−810.684	−0.0454172	−0.0227086	−	0.999742i	$-0.507229\pi$
$683$	−810.684	−0.0454172	−0.0227086	+	0.999742i	$0.507229\pi$
$684$	0	0
$685$	−19330.4	−1.07821
$686$	0	0
$687$	0	0
$688$	3534.22	0.195844
$689$	4144.00	0.229135
$690$	0	0
$691$	1171.38	0.0644883	0.0322442	−	0.999480i	$-0.489735\pi$
$691$	1171.38	0.0644883	0.0322442	+	0.999480i	$0.489735\pi$
$692$	5386.42	0.295897
$693$	0	0
$694$	−5581.14	−0.305269
$695$	−5900.76	−0.322056
$696$	0	0
$697$	17842.6	0.969634
$698$	−2420.19	−0.131240
$699$	0	0
$700$	0	0
$701$	−1057.70	−0.0569885	−0.0284943	−	0.999594i	$-0.509071\pi$
$701$	−1057.70	−0.0569885	−0.0284943	+	0.999594i	$0.509071\pi$
$702$	0	0
$703$	7892.35	0.423422
$704$	14111.0	0.755437
$705$	0	0
$706$	−10866.0	−0.579245
$707$	0	0
$708$	0	0
$709$	−29243.7	−1.54904	−0.774521	−	0.632548i	$-0.782009\pi$
$709$	−29243.7	−1.54904	−0.774521	+	0.632548i	$0.782009\pi$
$710$	−29161.3	−1.54142
$711$	0	0
$712$	−19178.7	−1.00948
$713$	3732.50	0.196049
$714$	0	0
$715$	−5764.17	−0.301493
$716$	−4490.30	−0.234372
$717$	0	0
$718$	23243.0	1.20811
$719$	24557.7	1.27378	0.636890	−	0.770955i	$-0.280221\pi$
$719$	24557.7	1.27378	0.636890	+	0.770955i	$0.280221\pi$
$720$	0	0
$721$	0	0
$722$	10951.7	0.564516
$723$	0	0
$724$	−2787.05	−0.143066
$725$	−9485.55	−0.485909
$726$	0	0
$727$	−23444.2	−1.19601	−0.598003	−	0.801494i	$-0.704039\pi$
$727$	−23444.2	−1.19601	−0.598003	+	0.801494i	$0.704039\pi$
$728$	0	0
$729$	0	0
$730$	7800.25	0.395480
$731$	11475.9	0.580646
$732$	0	0
$733$	−35971.3	−1.81259	−0.906296	−	0.422644i	$-0.861102\pi$
$733$	−35971.3	−1.81259	−0.906296	+	0.422644i	$0.861102\pi$
$734$	15584.4	0.783692
$735$	0	0
$736$	2959.09	0.148198
$737$	−18140.7	−0.906679
$738$	0	0
$739$	12068.6	0.600746	0.300373	−	0.953822i	$-0.402889\pi$
$739$	12068.6	0.600746	0.300373	+	0.953822i	$0.402889\pi$
$740$	3899.46	0.193712
$741$	0	0
$742$	0	0
$743$	−5475.09	−0.270339	−0.135169	−	0.990823i	$-0.543158\pi$
$743$	−5475.09	−0.270339	−0.135169	+	0.990823i	$0.543158\pi$
$744$	0	0
$745$	6316.86	0.310647
$746$	−13394.0	−0.657358
$747$	0	0
$748$	7203.23	0.352107
$749$	0	0
$750$	0	0
$751$	25274.6	1.22807	0.614036	−	0.789278i	$-0.289545\pi$
$751$	25274.6	1.22807	0.614036	+	0.789278i	$0.289545\pi$
$752$	10529.6	0.510604
$753$	0	0
$754$	3050.63	0.147344
$755$	−28505.4	−1.37406
$756$	0	0
$757$	−23583.7	−1.13232	−0.566159	−	0.824296i	$-0.691571\pi$
$757$	−23583.7	−1.13232	−0.566159	+	0.824296i	$0.691571\pi$
$758$	22187.1	1.06316
$759$	0	0
$760$	38758.3	1.84988
$761$	−16202.7	−0.771810	−0.385905	−	0.922539i	$-0.626111\pi$
$761$	−16202.7	−0.771810	−0.385905	+	0.922539i	$0.626111\pi$
$762$	0	0
$763$	0	0
$764$	−14675.3	−0.694939
$765$	0	0
$766$	13110.1	0.618392
$767$	−9625.98	−0.453160
$768$	0	0
$769$	−18603.1	−0.872362	−0.436181	−	0.899859i	$-0.643669\pi$
$769$	−18603.1	−0.872362	−0.436181	+	0.899859i	$0.643669\pi$
$770$	0	0
$771$	0	0
$772$	−6658.23	−0.310408
$773$	27281.3	1.26939	0.634697	−	0.772761i	$-0.281125\pi$
$773$	27281.3	1.26939	0.634697	+	0.772761i	$0.281125\pi$
$774$	0	0
$775$	16470.8	0.763419
$776$	22648.2	1.04771
$777$	0	0
$778$	12157.3	0.560231
$779$	30628.9	1.40872
$780$	0	0
$781$	−29103.0	−1.33340
$782$	−2561.00	−0.117112
$783$	0	0
$784$	0	0
$785$	12059.9	0.548326
$786$	0	0
$787$	571.719	0.0258953	0.0129476	−	0.999916i	$-0.495879\pi$
$787$	571.719	0.0258953	0.0129476	+	0.999916i	$0.495879\pi$
$788$	3121.39	0.141110
$789$	0	0
$790$	36922.7	1.66285
$791$	0	0
$792$	0	0
$793$	−8840.01	−0.395861
$794$	7757.14	0.346714
$795$	0	0
$796$	−3994.01	−0.177844
$797$	−33080.5	−1.47023	−0.735113	−	0.677944i	$-0.762871\pi$
$797$	−33080.5	−1.47023	−0.735113	+	0.677944i	$0.762871\pi$
$798$	0	0
$799$	34190.5	1.51386
$800$	13057.9	0.577084
$801$	0	0
$802$	−14934.5	−0.657549
$803$	7784.66	0.342110
$804$	0	0
$805$	0	0
$806$	−5297.15	−0.231494
$807$	0	0
$808$	−10727.0	−0.467047
$809$	−32838.2	−1.42711	−0.713554	−	0.700601i	$-0.752915\pi$
$809$	−32838.2	−1.42711	−0.713554	+	0.700601i	$0.752915\pi$
$810$	0	0
$811$	−20453.4	−0.885592	−0.442796	−	0.896622i	$-0.646013\pi$
$811$	−20453.4	−0.885592	−0.442796	+	0.896622i	$0.646013\pi$
$812$	0	0
$813$	0	0
$814$	−4390.56	−0.189053
$815$	−17058.6	−0.733172
$816$	0	0
$817$	19699.8	0.843585
$818$	25839.6	1.10447
$819$	0	0
$820$	15133.2	0.644479
$821$	−32462.4	−1.37996	−0.689980	−	0.723828i	$-0.742381\pi$
$821$	−32462.4	−1.37996	−0.689980	+	0.723828i	$0.742381\pi$
$822$	0	0
$823$	−47088.3	−1.99440	−0.997202	−	0.0747505i	$-0.976184\pi$
$823$	−47088.3	−1.99440	−0.997202	+	0.0747505i	$0.976184\pi$
$824$	16936.7	0.716043
$825$	0	0
$826$	0	0
$827$	−17641.7	−0.741792	−0.370896	−	0.928674i	$-0.620949\pi$
$827$	−17641.7	−0.741792	−0.370896	+	0.928674i	$0.620949\pi$
$828$	0	0
$829$	−1583.53	−0.0663427	−0.0331714	−	0.999450i	$-0.510561\pi$
$829$	−1583.53	−0.0663427	−0.0331714	+	0.999450i	$0.510561\pi$
$830$	−36613.2	−1.53116
$831$	0	0
$832$	−6311.01	−0.262975
$833$	0	0
$834$	0	0
$835$	−24284.0	−1.00645
$836$	12365.2	0.511555
$837$	0	0
$838$	26012.7	1.07231
$839$	39612.3	1.63000	0.815000	−	0.579461i	$-0.196737\pi$
$839$	39612.3	1.63000	0.815000	+	0.579461i	$0.196737\pi$
$840$	0	0
$841$	−12042.9	−0.493786
$842$	−7199.39	−0.294664
$843$	0	0
$844$	14467.3	0.590031
$845$	−29287.5	−1.19233
$846$	0	0
$847$	0	0
$848$	6153.58	0.249192
$849$	0	0
$850$	−11301.2	−0.456034
$851$	−1383.62	−0.0557344
$852$	0	0
$853$	−9443.21	−0.379049	−0.189525	−	0.981876i	$-0.560695\pi$
$853$	−9443.21	−0.379049	−0.189525	+	0.981876i	$0.560695\pi$
$854$	0	0
$855$	0	0
$856$	46281.3	1.84797
$857$	−20138.7	−0.802713	−0.401356	−	0.915922i	$-0.631461\pi$
$857$	−20138.7	−0.802713	−0.401356	+	0.915922i	$0.631461\pi$
$858$	0	0
$859$	−18457.4	−0.733129	−0.366564	−	0.930393i	$-0.619466\pi$
$859$	−18457.4	−0.733129	−0.366564	+	0.930393i	$0.619466\pi$
$860$	9733.30	0.385933
$861$	0	0
$862$	4829.57	0.190830
$863$	48840.2	1.92646	0.963232	−	0.268670i	$-0.0865840\pi$
$863$	48840.2	1.92646	0.963232	+	0.268670i	$0.0865840\pi$
$864$	0	0
$865$	−20783.2	−0.816936
$866$	6694.20	0.262677
$867$	0	0
$868$	0	0
$869$	36848.8	1.43845
$870$	0	0
$871$	8113.27	0.315623
$872$	50161.6	1.94803
$873$	0	0
$874$	−4396.27	−0.170144
$875$	0	0
$876$	0	0
$877$	−5584.39	−0.215019	−0.107509	−	0.994204i	$-0.534288\pi$
$877$	−5584.39	−0.215019	−0.107509	+	0.994204i	$0.534288\pi$
$878$	33445.0	1.28555
$879$	0	0
$880$	−8559.43	−0.327884
$881$	−24753.3	−0.946606	−0.473303	−	0.880900i	$-0.656938\pi$
$881$	−24753.3	−0.946606	−0.473303	+	0.880900i	$0.656938\pi$
$882$	0	0
$883$	−12368.7	−0.471394	−0.235697	−	0.971827i	$-0.575737\pi$
$883$	−12368.7	−0.471394	−0.235697	+	0.971827i	$0.575737\pi$
$884$	−3221.58	−0.122572
$885$	0	0
$886$	−9985.90	−0.378649
$887$	738.713	0.0279634	0.0139817	−	0.999902i	$-0.495549\pi$
$887$	738.713	0.0279634	0.0139817	+	0.999902i	$0.495549\pi$
$888$	0	0
$889$	0	0
$890$	23655.8	0.890947
$891$	0	0
$892$	−20609.9	−0.773623
$893$	58692.1	2.19939
$894$	0	0
$895$	17325.6	0.647073
$896$	0	0
$897$	0	0
$898$	−8525.58	−0.316818
$899$	−21437.9	−0.795320
$900$	0	0
$901$	19981.2	0.738814
$902$	−17039.0	−0.628977
$903$	0	0
$904$	−56020.1	−2.06106
$905$	10753.7	0.394988
$906$	0	0
$907$	−776.937	−0.0284430	−0.0142215	−	0.999899i	$-0.504527\pi$
$907$	−776.937	−0.0284430	−0.0142215	+	0.999899i	$0.504527\pi$
$908$	1484.21	0.0542459
$909$	0	0
$910$	0	0
$911$	−31326.5	−1.13929	−0.569645	−	0.821891i	$-0.692919\pi$
$911$	−31326.5	−1.13929	−0.569645	+	0.821891i	$0.692919\pi$
$912$	0	0
$913$	−36540.0	−1.32453
$914$	19013.0	0.688067
$915$	0	0
$916$	14373.0	0.518446
$917$	0	0
$918$	0	0
$919$	6149.72	0.220741	0.110370	−	0.993891i	$-0.464796\pi$
$919$	6149.72	0.220741	0.110370	+	0.993891i	$0.464796\pi$
$920$	−6794.80	−0.243498
$921$	0	0
$922$	2625.47	0.0937802
$923$	13016.1	0.464170
$924$	0	0
$925$	−6105.68	−0.217031
$926$	4245.98	0.150682
$927$	0	0
$928$	−16995.7	−0.601199
$929$	−31050.1	−1.09658	−0.548288	−	0.836289i	$-0.684720\pi$
$929$	−31050.1	−1.09658	−0.548288	+	0.836289i	$0.684720\pi$
$930$	0	0
$931$	0	0
$932$	−9538.74	−0.335249
$933$	0	0
$934$	−1993.54	−0.0698401
$935$	−27793.2	−0.972124
$936$	0	0
$937$	22802.9	0.795023	0.397512	−	0.917597i	$-0.369874\pi$
$937$	22802.9	0.795023	0.397512	+	0.917597i	$0.369874\pi$
$938$	0	0
$939$	0	0
$940$	28998.6	1.00620
$941$	−46442.5	−1.60891	−0.804455	−	0.594014i	$-0.797542\pi$
$941$	−46442.5	−1.60891	−0.804455	+	0.594014i	$0.797542\pi$
$942$	0	0
$943$	−5369.61	−0.185428
$944$	−14294.0	−0.492828
$945$	0	0
$946$	−10959.1	−0.376651
$947$	29207.7	1.00224	0.501121	−	0.865377i	$-0.332921\pi$
$947$	29207.7	1.00224	0.501121	+	0.865377i	$0.332921\pi$
$948$	0	0
$949$	−3481.61	−0.119092
$950$	−19399.9	−0.662544
$951$	0	0
$952$	0	0
$953$	6829.15	0.232128	0.116064	−	0.993242i	$-0.462972\pi$
$953$	6829.15	0.232128	0.116064	+	0.993242i	$0.462972\pi$
$954$	0	0
$955$	56623.7	1.91864
$956$	20181.6	0.682762
$957$	0	0
$958$	12768.1	0.430604
$959$	0	0
$960$	0	0
$961$	7433.99	0.249538
$962$	1963.64	0.0658110
$963$	0	0
$964$	−7924.16	−0.264751
$965$	25690.4	0.856999
$966$	0	0
$967$	−2583.84	−0.0859263	−0.0429632	−	0.999077i	$-0.513680\pi$
$967$	−2583.84	−0.0859263	−0.0429632	+	0.999077i	$0.513680\pi$
$968$	10713.3	0.355720
$969$	0	0
$970$	−27935.2	−0.924684
$971$	−32515.6	−1.07464	−0.537321	−	0.843378i	$-0.680563\pi$
$971$	−32515.6	−1.07464	−0.537321	+	0.843378i	$0.680563\pi$
$972$	0	0
$973$	0	0
$974$	−20242.8	−0.665935
$975$	0	0
$976$	−13126.9	−0.430513
$977$	−222.043	−0.00727102	−0.00363551	−	0.999993i	$-0.501157\pi$
$977$	−222.043	−0.00727102	−0.00363551	+	0.999993i	$0.501157\pi$
$978$	0	0
$979$	23608.5	0.770715
$980$	0	0
$981$	0	0
$982$	−10407.1	−0.338191
$983$	10916.7	0.354210	0.177105	−	0.984192i	$-0.443327\pi$
$983$	10916.7	0.354210	0.177105	+	0.984192i	$0.443327\pi$
$984$	0	0
$985$	−12043.7	−0.389588
$986$	14709.3	0.475091
$987$	0	0
$988$	−5530.22	−0.178077
$989$	−3453.61	−0.111040
$990$	0	0
$991$	8015.59	0.256936	0.128468	−	0.991714i	$-0.458994\pi$
$991$	8015.59	0.256936	0.128468	+	0.991714i	$0.458994\pi$
$992$	29511.6	0.944552
$993$	0	0
$994$	0	0
$995$	15410.7	0.491006
$996$	0	0
$997$	−21833.7	−0.693559	−0.346780	−	0.937947i	$-0.612725\pi$
$997$	−21833.7	−0.693559	−0.346780	+	0.937947i	$0.612725\pi$
$998$	28957.9	0.918484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bd.1.4		6
3.2	odd	2	inner	1323.4.a.bd.1.3		6
7.2	even	3		189.4.e.e.109.3	✓	12
7.4	even	3		189.4.e.e.163.3	yes	12
7.6	odd	2		1323.4.a.be.1.4		6
21.2	odd	6		189.4.e.e.109.4	yes	12
21.11	odd	6		189.4.e.e.163.4	yes	12
21.20	even	2		1323.4.a.be.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.e.109.3	✓	12	7.2	even	3
189.4.e.e.109.4	yes	12	21.2	odd	6
189.4.e.e.163.3	yes	12	7.4	even	3
189.4.e.e.163.4	yes	12	21.11	odd	6
1323.4.a.bd.1.3		6	3.2	odd	2	inner
1323.4.a.bd.1.4		6	1.1	even	1	trivial
1323.4.a.be.1.3		6	21.20	even	2
1323.4.a.be.1.4		6	7.6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05936 q^{2} -3.75905 q^{4} +14.5041 q^{5} -24.2161 q^{8} +29.8691 q^{10} +29.8094 q^{11} -13.3320 q^{13} -19.7971 q^{16} -64.2830 q^{17} -110.350 q^{19} -54.5216 q^{20} +61.3881 q^{22} +19.3456 q^{23} +85.3686 q^{25} -27.4552 q^{26} -111.113 q^{29} +192.938 q^{31} +152.959 q^{32} -132.382 q^{34} -71.5214 q^{37} -227.249 q^{38} -351.232 q^{40} -277.562 q^{41} -178.522 q^{43} -112.055 q^{44} +39.8395 q^{46} -531.874 q^{47} +175.804 q^{50} +50.1155 q^{52} -310.832 q^{53} +432.357 q^{55} -228.821 q^{58} +722.023 q^{59} +663.070 q^{61} +397.328 q^{62} +473.375 q^{64} -193.368 q^{65} -608.559 q^{67} +241.643 q^{68} -976.305 q^{71} +261.148 q^{73} -147.288 q^{74} +414.810 q^{76} +1236.15 q^{79} -287.139 q^{80} -571.600 q^{82} -1225.79 q^{83} -932.367 q^{85} -367.640 q^{86} -721.866 q^{88} +791.982 q^{89} -72.7211 q^{92} -1095.32 q^{94} -1600.52 q^{95} -935.253 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 32 q^{4} - 20 q^{10} + 52 q^{13} - 148 q^{16} - 62 q^{19} - 356 q^{22} - 46 q^{25} - 82 q^{31} + 420 q^{34} - 1132 q^{37} - 444 q^{40} - 1566 q^{43} - 888 q^{46} - 72 q^{52} + 224 q^{55} + 4 q^{58}+ \cdots - 682 q^{97}+O(q^{100}) \) 6 * q + 32 * q^4 - 20 * q^10 + 52 * q^13 - 148 * q^16 - 62 * q^19 - 356 * q^22 - 46 * q^25 - 82 * q^31 + 420 * q^34 - 1132 * q^37 - 444 * q^40 - 1566 * q^43 - 888 * q^46 - 72 * q^52 + 224 * q^55 + 4 * q^58 + 886 * q^61 - 924 * q^64 - 2084 * q^67 - 2398 * q^73 + 3204 * q^76 + 984 * q^79 - 3892 * q^82 - 3600 * q^85 - 5796 * q^88 + 2772 * q^94 - 682 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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