LMFDB - Embedded newform 1323.4.a.bg.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
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} - 42x^{4} + 369x^{2} - 112 $ x^6 - 42x^4 + 369x^2 - 112
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.560992$ 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)

$f(q)$	$=$	$q+0.560992 q^{2} -7.68529 q^{4} +12.9561 q^{5} -8.79932 q^{8} +O(q^{10})$	$q+0.560992 q^{2} -7.68529 q^{4} +12.9561 q^{5} -8.79932 q^{8} +7.26827 q^{10} +48.2286 q^{11} +36.3241 q^{13} +56.5460 q^{16} +83.2713 q^{17} -67.4359 q^{19} -99.5714 q^{20} +27.0559 q^{22} -30.6300 q^{23} +42.8607 q^{25} +20.3775 q^{26} +294.244 q^{29} -270.844 q^{31} +102.116 q^{32} +46.7145 q^{34} -204.315 q^{37} -37.8310 q^{38} -114.005 q^{40} +287.529 q^{41} -55.4901 q^{43} -370.651 q^{44} -17.1832 q^{46} -191.301 q^{47} +24.0445 q^{50} -279.161 q^{52} +521.107 q^{53} +624.855 q^{55} +165.069 q^{58} -381.074 q^{59} -155.465 q^{61} -151.941 q^{62} -395.081 q^{64} +470.619 q^{65} -65.1959 q^{67} -639.964 q^{68} -256.370 q^{71} -318.625 q^{73} -114.619 q^{74} +518.264 q^{76} +77.7532 q^{79} +732.615 q^{80} +161.301 q^{82} +836.549 q^{83} +1078.87 q^{85} -31.1295 q^{86} -424.379 q^{88} +1590.06 q^{89} +235.400 q^{92} -107.319 q^{94} -873.706 q^{95} +1189.88 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 36 q^{4}+O(q^{10}) $ 6 * q + 36 * q^4	$ 6 q + 36 q^{4} + 180 q^{10} + 108 q^{13} + 420 q^{16} + 198 q^{19} - 84 q^{22} + 420 q^{25} - 90 q^{31} - 648 q^{34} - 402 q^{37} + 2844 q^{40} - 660 q^{43} - 1332 q^{46} + 1224 q^{52} + 846 q^{55} - 1800 q^{58} + 1152 q^{61} + 2964 q^{64} + 924 q^{67} + 1260 q^{73} + 5868 q^{76} - 1500 q^{79} + 4500 q^{82} - 2232 q^{85} - 2460 q^{88} - 4968 q^{94} + 3312 q^{97}+O(q^{100}) $ 6 * q + 36 * q^4 + 180 * q^10 + 108 * q^13 + 420 * q^16 + 198 * q^19 - 84 * q^22 + 420 * q^25 - 90 * q^31 - 648 * q^34 - 402 * q^37 + 2844 * q^40 - 660 * q^43 - 1332 * q^46 + 1224 * q^52 + 846 * q^55 - 1800 * q^58 + 1152 * q^61 + 2964 * q^64 + 924 * q^67 + 1260 * q^73 + 5868 * q^76 - 1500 * q^79 + 4500 * q^82 - 2232 * q^85 - 2460 * q^88 - 4968 * q^94 + 3312 * q^97

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 $

$2$	0.560992	0.198341	0.0991703	−	0.995070i	$-0.468381\pi$
$2$	0.560992	0.198341	0.0991703	+	0.995070i	$0.468381\pi$
$3$	0	0
$3$	0	0
$4$	−7.68529	−0.960661
$5$	12.9561	1.15883	0.579415	−	0.815033i	$-0.303281\pi$
$5$	12.9561	1.15883	0.579415	+	0.815033i	$0.303281\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−8.79932	−0.388879
$9$	0	0
$10$	7.26827	0.229843
$11$	48.2286	1.32195	0.660976	−	0.750407i	$-0.270142\pi$
$11$	48.2286	1.32195	0.660976	+	0.750407i	$0.270142\pi$
$12$	0	0
$13$	36.3241	0.774962	0.387481	−	0.921878i	$-0.373345\pi$
$13$	36.3241	0.774962	0.387481	+	0.921878i	$0.373345\pi$
$14$	0	0
$15$	0	0
$16$	56.5460	0.883531
$17$	83.2713	1.18802	0.594008	−	0.804459i	$-0.297545\pi$
$17$	83.2713	1.18802	0.594008	+	0.804459i	$0.297545\pi$
$18$	0	0
$19$	−67.4359	−0.814255	−0.407128	−	0.913371i	$-0.633470\pi$
$19$	−67.4359	−0.814255	−0.407128	+	0.913371i	$0.633470\pi$
$20$	−99.5714	−1.11324
$21$	0	0
$22$	27.0559	0.262197
$23$	−30.6300	−0.277687	−0.138843	−	0.990314i	$-0.544338\pi$
$23$	−30.6300	−0.277687	−0.138843	+	0.990314i	$0.544338\pi$
$24$	0	0
$25$	42.8607	0.342885
$26$	20.3775	0.153706
$27$	0	0
$28$	0	0
$29$	294.244	1.88413	0.942065	−	0.335431i	$-0.108882\pi$
$29$	294.244	1.88413	0.942065	+	0.335431i	$0.108882\pi$
$30$	0	0
$31$	−270.844	−1.56919	−0.784597	−	0.620006i	$-0.787130\pi$
$31$	−270.844	−1.56919	−0.784597	+	0.620006i	$0.787130\pi$
$32$	102.116	0.564119
$33$	0	0
$34$	46.7145	0.235632
$35$	0	0
$36$	0	0
$37$	−204.315	−0.907817	−0.453909	−	0.891048i	$-0.649971\pi$
$37$	−204.315	−0.907817	−0.453909	+	0.891048i	$0.649971\pi$
$38$	−37.8310	−0.161500
$39$	0	0
$40$	−114.005	−0.450644
$41$	287.529	1.09523	0.547615	−	0.836730i	$-0.315536\pi$
$41$	287.529	1.09523	0.547615	+	0.836730i	$0.315536\pi$
$42$	0	0
$43$	−55.4901	−0.196794	−0.0983972	−	0.995147i	$-0.531372\pi$
$43$	−55.4901	−0.196794	−0.0983972	+	0.995147i	$0.531372\pi$
$44$	−370.651	−1.26995
$45$	0	0
$46$	−17.1832	−0.0550765
$47$	−191.301	−0.593706	−0.296853	−	0.954923i	$-0.595937\pi$
$47$	−191.301	−0.593706	−0.296853	+	0.954923i	$0.595937\pi$
$48$	0	0
$49$	0	0
$50$	24.0445	0.0680081
$51$	0	0
$52$	−279.161	−0.744475
$53$	521.107	1.35056	0.675278	−	0.737563i	$-0.264024\pi$
$53$	521.107	1.35056	0.675278	+	0.737563i	$0.264024\pi$
$54$	0	0
$55$	624.855	1.53192
$56$	0	0
$57$	0	0
$58$	165.069	0.373699
$59$	−381.074	−0.840874	−0.420437	−	0.907322i	$-0.638123\pi$
$59$	−381.074	−0.840874	−0.420437	+	0.907322i	$0.638123\pi$
$60$	0	0
$61$	−155.465	−0.326315	−0.163158	−	0.986600i	$-0.552168\pi$
$61$	−155.465	−0.326315	−0.163158	+	0.986600i	$0.552168\pi$
$62$	−151.941	−0.311235
$63$	0	0
$64$	−395.081	−0.771643
$65$	470.619	0.898048
$66$	0	0
$67$	−65.1959	−0.118880	−0.0594399	−	0.998232i	$-0.518931\pi$
$67$	−65.1959	−0.118880	−0.0594399	+	0.998232i	$0.518931\pi$
$68$	−639.964	−1.14128
$69$	0	0
$70$	0	0
$71$	−256.370	−0.428529	−0.214265	−	0.976776i	$-0.568735\pi$
$71$	−256.370	−0.428529	−0.214265	+	0.976776i	$0.568735\pi$
$72$	0	0
$73$	−318.625	−0.510852	−0.255426	−	0.966829i	$-0.582216\pi$
$73$	−318.625	−0.510852	−0.255426	+	0.966829i	$0.582216\pi$
$74$	−114.619	−0.180057
$75$	0	0
$76$	518.264	0.782223
$77$	0	0
$78$	0	0
$79$	77.7532	0.110733	0.0553666	−	0.998466i	$-0.482367\pi$
$79$	77.7532	0.110733	0.0553666	+	0.998466i	$0.482367\pi$
$80$	732.615	1.02386
$81$	0	0
$82$	161.301	0.217229
$83$	836.549	1.10630	0.553152	−	0.833081i	$-0.313425\pi$
$83$	836.549	1.10630	0.553152	+	0.833081i	$0.313425\pi$
$84$	0	0
$85$	1078.87	1.37671
$86$	−31.1295	−0.0390323
$87$	0	0
$88$	−424.379	−0.514079
$89$	1590.06	1.89377	0.946887	−	0.321565i	$-0.104209\pi$
$89$	1590.06	1.89377	0.946887	+	0.321565i	$0.104209\pi$
$90$	0	0
$91$	0	0
$92$	235.400	0.266763
$93$	0	0
$94$	−107.319	−0.117756
$95$	−873.706	−0.943583
$96$	0	0
$97$	1189.88	1.24550	0.622751	−	0.782420i	$-0.286015\pi$
$97$	1189.88	1.24550	0.622751	+	0.782420i	$0.286015\pi$
$98$	0	0
$99$	0	0
$100$	−329.397	−0.329397
$101$	507.845	0.500321	0.250161	−	0.968204i	$-0.419517\pi$
$101$	507.845	0.500321	0.250161	+	0.968204i	$0.419517\pi$
$102$	0	0
$103$	1852.78	1.77242	0.886212	−	0.463280i	$-0.153328\pi$
$103$	1852.78	1.77242	0.886212	+	0.463280i	$0.153328\pi$
$104$	−319.628	−0.301366
$105$	0	0
$106$	292.337	0.267870
$107$	614.904	0.555561	0.277781	−	0.960645i	$-0.410401\pi$
$107$	614.904	0.555561	0.277781	+	0.960645i	$0.410401\pi$
$108$	0	0
$109$	1017.92	0.894487	0.447243	−	0.894412i	$-0.352406\pi$
$109$	1017.92	0.894487	0.447243	+	0.894412i	$0.352406\pi$
$110$	350.539	0.303841
$111$	0	0
$112$	0	0
$113$	−1204.42	−1.00267	−0.501337	−	0.865252i	$-0.667158\pi$
$113$	−1204.42	−1.00267	−0.501337	+	0.865252i	$0.667158\pi$
$114$	0	0
$115$	−396.845	−0.321792
$116$	−2261.35	−1.81001
$117$	0	0
$118$	−213.779	−0.166779
$119$	0	0
$120$	0	0
$121$	995.000	0.747559
$122$	−87.2145	−0.0647215
$123$	0	0
$124$	2081.52	1.50746
$125$	−1064.21	−0.761484
$126$	0	0
$127$	−2515.55	−1.75763	−0.878813	−	0.477165i	$-0.841664\pi$
$127$	−2515.55	−1.75763	−0.878813	+	0.477165i	$0.841664\pi$
$128$	−1038.57	−0.717167
$129$	0	0
$130$	264.014	0.178119
$131$	−2165.37	−1.44419	−0.722096	−	0.691793i	$-0.756821\pi$
$131$	−2165.37	−1.44419	−0.722096	+	0.691793i	$0.756821\pi$
$132$	0	0
$133$	0	0
$134$	−36.5743	−0.0235787
$135$	0	0
$136$	−732.731	−0.461994
$137$	795.668	0.496194	0.248097	−	0.968735i	$-0.420195\pi$
$137$	795.668	0.496194	0.248097	+	0.968735i	$0.420195\pi$
$138$	0	0
$139$	2400.60	1.46487	0.732433	−	0.680839i	$-0.238385\pi$
$139$	2400.60	1.46487	0.732433	+	0.680839i	$0.238385\pi$
$140$	0	0
$141$	0	0
$142$	−143.822	−0.0849947
$143$	1751.86	1.02446
$144$	0	0
$145$	3812.26	2.18338
$146$	−178.746	−0.101323
$147$	0	0
$148$	1570.22	0.872105
$149$	1404.26	0.772091	0.386046	−	0.922480i	$-0.373841\pi$
$149$	1404.26	0.772091	0.386046	+	0.922480i	$0.373841\pi$
$150$	0	0
$151$	−1761.48	−0.949322	−0.474661	−	0.880169i	$-0.657429\pi$
$151$	−1761.48	−0.949322	−0.474661	+	0.880169i	$0.657429\pi$
$152$	593.390	0.316646
$153$	0	0
$154$	0	0
$155$	−3509.09	−1.81843
$156$	0	0
$157$	1082.03	0.550034	0.275017	−	0.961439i	$-0.411317\pi$
$157$	1082.03	0.550034	0.275017	+	0.961439i	$0.411317\pi$
$158$	43.6189	0.0219629
$159$	0	0
$160$	1323.03	0.653717
$161$	0	0
$162$	0	0
$163$	−2473.27	−1.18847	−0.594237	−	0.804290i	$-0.702546\pi$
$163$	−2473.27	−1.18847	−0.594237	+	0.804290i	$0.702546\pi$
$164$	−2209.74	−1.05215
$165$	0	0
$166$	469.297	0.219425
$167$	2240.74	1.03828	0.519142	−	0.854688i	$-0.326252\pi$
$167$	2240.74	1.03828	0.519142	+	0.854688i	$0.326252\pi$
$168$	0	0
$169$	−877.557	−0.399434
$170$	605.238	0.273057
$171$	0	0
$172$	426.457	0.189053
$173$	−2121.28	−0.932242	−0.466121	−	0.884721i	$-0.654349\pi$
$173$	−2121.28	−0.932242	−0.466121	+	0.884721i	$0.654349\pi$
$174$	0	0
$175$	0	0
$176$	2727.13	1.16799
$177$	0	0
$178$	892.010	0.375612
$179$	1578.84	0.659262	0.329631	−	0.944110i	$-0.393076\pi$
$179$	1578.84	0.659262	0.329631	+	0.944110i	$0.393076\pi$
$180$	0	0
$181$	4609.64	1.89299	0.946496	−	0.322716i	$-0.104596\pi$
$181$	4609.64	1.89299	0.946496	+	0.322716i	$0.104596\pi$
$182$	0	0
$183$	0	0
$184$	269.523	0.107986
$185$	−2647.13	−1.05201
$186$	0	0
$187$	4016.06	1.57050
$188$	1470.21	0.570350
$189$	0	0
$190$	−490.142	−0.187151
$191$	1291.72	0.489350	0.244675	−	0.969605i	$-0.421319\pi$
$191$	1291.72	0.489350	0.244675	+	0.969605i	$0.421319\pi$
$192$	0	0
$193$	1176.01	0.438605	0.219302	−	0.975657i	$-0.429622\pi$
$193$	1176.01	0.438605	0.219302	+	0.975657i	$0.429622\pi$
$194$	667.511	0.247033
$195$	0	0
$196$	0	0
$197$	4560.71	1.64943	0.824713	−	0.565552i	$-0.191337\pi$
$197$	4560.71	1.64943	0.824713	+	0.565552i	$0.191337\pi$
$198$	0	0
$199$	−986.014	−0.351240	−0.175620	−	0.984458i	$-0.556193\pi$
$199$	−986.014	−0.351240	−0.175620	+	0.984458i	$0.556193\pi$
$200$	−377.145	−0.133341
$201$	0	0
$202$	284.897	0.0992340
$203$	0	0
$204$	0	0
$205$	3725.25	1.26919
$206$	1039.39	0.351543
$207$	0	0
$208$	2053.98	0.684702
$209$	−3252.34	−1.07641
$210$	0	0
$211$	5459.28	1.78120	0.890598	−	0.454792i	$-0.150286\pi$
$211$	5459.28	1.78120	0.890598	+	0.454792i	$0.150286\pi$
$212$	−4004.85	−1.29743
$213$	0	0
$214$	344.956	0.110190
$215$	−718.936	−0.228051
$216$	0	0
$217$	0	0
$218$	571.045	0.177413
$219$	0	0
$220$	−4802.19	−1.47165
$221$	3024.76	0.920666
$222$	0	0
$223$	4626.41	1.38927	0.694636	−	0.719362i	$-0.255566\pi$
$223$	4626.41	1.38927	0.694636	+	0.719362i	$0.255566\pi$
$224$	0	0
$225$	0	0
$226$	−675.669	−0.198871
$227$	2437.77	0.712776	0.356388	−	0.934338i	$-0.384008\pi$
$227$	2437.77	0.712776	0.356388	+	0.934338i	$0.384008\pi$
$228$	0	0
$229$	2946.50	0.850264	0.425132	−	0.905131i	$-0.360228\pi$
$229$	2946.50	0.850264	0.425132	+	0.905131i	$0.360228\pi$
$230$	−222.627	−0.0638243
$231$	0	0
$232$	−2589.15	−0.732698
$233$	−3441.49	−0.967638	−0.483819	−	0.875168i	$-0.660751\pi$
$233$	−3441.49	−0.967638	−0.483819	+	0.875168i	$0.660751\pi$
$234$	0	0
$235$	−2478.52	−0.688004
$236$	2928.66	0.807795
$237$	0	0
$238$	0	0
$239$	744.021	0.201367	0.100684	−	0.994918i	$-0.467897\pi$
$239$	744.021	0.201367	0.100684	+	0.994918i	$0.467897\pi$
$240$	0	0
$241$	−4714.37	−1.26008	−0.630040	−	0.776562i	$-0.716962\pi$
$241$	−4714.37	−1.26008	−0.630040	+	0.776562i	$0.716962\pi$
$242$	558.187	0.148271
$243$	0	0
$244$	1194.79	0.313478
$245$	0	0
$246$	0	0
$247$	−2449.55	−0.631017
$248$	2383.24	0.610226
$249$	0	0
$250$	−597.011	−0.151033
$251$	−2248.41	−0.565411	−0.282706	−	0.959207i	$-0.591232\pi$
$251$	−2248.41	−0.565411	−0.282706	+	0.959207i	$0.591232\pi$
$252$	0	0
$253$	−1477.24	−0.367089
$254$	−1411.20	−0.348609
$255$	0	0
$256$	2578.02	0.629400
$257$	6099.67	1.48049	0.740247	−	0.672335i	$-0.234709\pi$
$257$	6099.67	1.48049	0.740247	+	0.672335i	$0.234709\pi$
$258$	0	0
$259$	0	0
$260$	−3616.85	−0.862720
$261$	0	0
$262$	−1214.75	−0.286442
$263$	3493.26	0.819026	0.409513	−	0.912304i	$-0.365699\pi$
$263$	3493.26	0.819026	0.409513	+	0.912304i	$0.365699\pi$
$264$	0	0
$265$	6751.51	1.56506
$266$	0	0
$267$	0	0
$268$	501.049	0.114203
$269$	−6393.14	−1.44906	−0.724529	−	0.689244i	$-0.757943\pi$
$269$	−6393.14	−1.44906	−0.724529	+	0.689244i	$0.757943\pi$
$270$	0	0
$271$	−4222.81	−0.946559	−0.473279	−	0.880912i	$-0.656930\pi$
$271$	−4222.81	−0.946559	−0.473279	+	0.880912i	$0.656930\pi$
$272$	4708.66	1.04965
$273$	0	0
$274$	446.363	0.0984153
$275$	2067.11	0.453278
$276$	0	0
$277$	−5240.37	−1.13669	−0.568345	−	0.822790i	$-0.692416\pi$
$277$	−5240.37	−1.13669	−0.568345	+	0.822790i	$0.692416\pi$
$278$	1346.72	0.290542
$279$	0	0
$280$	0	0
$281$	5924.79	1.25781	0.628903	−	0.777484i	$-0.283504\pi$
$281$	5924.79	1.25781	0.628903	+	0.777484i	$0.283504\pi$
$282$	0	0
$283$	−1885.61	−0.396070	−0.198035	−	0.980195i	$-0.563456\pi$
$283$	−1885.61	−0.396070	−0.198035	+	0.980195i	$0.563456\pi$
$284$	1970.28	0.411671
$285$	0	0
$286$	982.781	0.203192
$287$	0	0
$288$	0	0
$289$	2021.11	0.411380
$290$	2138.65	0.433054
$291$	0	0
$292$	2448.72	0.490756
$293$	−3173.12	−0.632680	−0.316340	−	0.948646i	$-0.602454\pi$
$293$	−3173.12	−0.632680	−0.316340	+	0.948646i	$0.602454\pi$
$294$	0	0
$295$	−4937.23	−0.974430
$296$	1797.84	0.353031
$297$	0	0
$298$	787.779	0.153137
$299$	−1112.61	−0.215197
$300$	0	0
$301$	0	0
$302$	−988.179	−0.188289
$303$	0	0
$304$	−3813.23	−0.719419
$305$	−2014.22	−0.378144
$306$	0	0
$307$	6188.30	1.15044	0.575220	−	0.817999i	$-0.304916\pi$
$307$	6188.30	1.15044	0.575220	+	0.817999i	$0.304916\pi$
$308$	0	0
$309$	0	0
$310$	−1968.57	−0.360668
$311$	3750.93	0.683909	0.341954	−	0.939717i	$-0.388911\pi$
$311$	3750.93	0.683909	0.341954	+	0.939717i	$0.388911\pi$
$312$	0	0
$313$	−4172.29	−0.753456	−0.376728	−	0.926324i	$-0.622951\pi$
$313$	−4172.29	−0.753456	−0.376728	+	0.926324i	$0.622951\pi$
$314$	607.009	0.109094
$315$	0	0
$316$	−597.556	−0.106377
$317$	8410.38	1.49014	0.745069	−	0.666987i	$-0.232416\pi$
$317$	8410.38	1.49014	0.745069	+	0.666987i	$0.232416\pi$
$318$	0	0
$319$	14191.0	2.49073
$320$	−5118.71	−0.894203
$321$	0	0
$322$	0	0
$323$	−5615.47	−0.967347
$324$	0	0
$325$	1556.88	0.265723
$326$	−1387.48	−0.235723
$327$	0	0
$328$	−2530.06	−0.425912
$329$	0	0
$330$	0	0
$331$	−1217.72	−0.202211	−0.101105	−	0.994876i	$-0.532238\pi$
$331$	−1217.72	−0.202211	−0.101105	+	0.994876i	$0.532238\pi$
$332$	−6429.12	−1.06278
$333$	0	0
$334$	1257.03	0.205934
$335$	−844.684	−0.137761
$336$	0	0
$337$	3880.01	0.627175	0.313587	−	0.949559i	$-0.398469\pi$
$337$	3880.01	0.627175	0.313587	+	0.949559i	$0.398469\pi$
$338$	−492.302	−0.0792240
$339$	0	0
$340$	−8291.44	−1.32255
$341$	−13062.4	−2.07440
$342$	0	0
$343$	0	0
$344$	488.275	0.0765291
$345$	0	0
$346$	−1190.02	−0.184901
$347$	−9672.45	−1.49638	−0.748191	−	0.663484i	$-0.769077\pi$
$347$	−9672.45	−1.49638	−0.748191	+	0.663484i	$0.769077\pi$
$348$	0	0
$349$	3960.64	0.607473	0.303737	−	0.952756i	$-0.401766\pi$
$349$	3960.64	0.607473	0.303737	+	0.952756i	$0.401766\pi$
$350$	0	0
$351$	0	0
$352$	4924.93	0.745738
$353$	−7639.67	−1.15189	−0.575947	−	0.817487i	$-0.695367\pi$
$353$	−7639.67	−1.15189	−0.575947	+	0.817487i	$0.695367\pi$
$354$	0	0
$355$	−3321.56	−0.496592
$356$	−12220.1	−1.81928
$357$	0	0
$358$	885.715	0.130758
$359$	−9411.16	−1.38357	−0.691785	−	0.722103i	$-0.743176\pi$
$359$	−9411.16	−1.38357	−0.691785	+	0.722103i	$0.743176\pi$
$360$	0	0
$361$	−2311.40	−0.336989
$362$	2585.97	0.375457
$363$	0	0
$364$	0	0
$365$	−4128.14	−0.591991
$366$	0	0
$367$	5707.80	0.811838	0.405919	−	0.913909i	$-0.366952\pi$
$367$	5707.80	0.811838	0.405919	+	0.913909i	$0.366952\pi$
$368$	−1732.00	−0.245345
$369$	0	0
$370$	−1485.02	−0.208655
$371$	0	0
$372$	0	0
$373$	−9579.55	−1.32979	−0.664893	−	0.746938i	$-0.731523\pi$
$373$	−9579.55	−1.32979	−0.664893	+	0.746938i	$0.731523\pi$
$374$	2252.98	0.311494
$375$	0	0
$376$	1683.32	0.230880
$377$	10688.2	1.46013
$378$	0	0
$379$	8730.92	1.18332	0.591658	−	0.806189i	$-0.298473\pi$
$379$	8730.92	1.18332	0.591658	+	0.806189i	$0.298473\pi$
$380$	6714.68	0.906463
$381$	0	0
$382$	724.646	0.0970579
$383$	9623.21	1.28387	0.641936	−	0.766758i	$-0.278131\pi$
$383$	9623.21	1.28387	0.641936	+	0.766758i	$0.278131\pi$
$384$	0	0
$385$	0	0
$386$	659.730	0.0869931
$387$	0	0
$388$	−9144.54	−1.19650
$389$	−11319.7	−1.47541	−0.737704	−	0.675125i	$-0.764090\pi$
$389$	−11319.7	−1.47541	−0.737704	+	0.675125i	$0.764090\pi$
$390$	0	0
$391$	−2550.60	−0.329896
$392$	0	0
$393$	0	0
$394$	2558.52	0.327148
$395$	1007.38	0.128321
$396$	0	0
$397$	2855.87	0.361038	0.180519	−	0.983572i	$-0.442222\pi$
$397$	2855.87	0.361038	0.180519	+	0.983572i	$0.442222\pi$
$398$	−553.146	−0.0696651
$399$	0	0
$400$	2423.60	0.302950
$401$	−1975.97	−0.246073	−0.123036	−	0.992402i	$-0.539263\pi$
$401$	−1975.97	−0.246073	−0.123036	+	0.992402i	$0.539263\pi$
$402$	0	0
$403$	−9838.18	−1.21607
$404$	−3902.94	−0.480639
$405$	0	0
$406$	0	0
$407$	−9853.85	−1.20009
$408$	0	0
$409$	8244.62	0.996749	0.498375	−	0.866962i	$-0.333930\pi$
$409$	8244.62	0.996749	0.498375	+	0.866962i	$0.333930\pi$
$410$	2089.84	0.251731
$411$	0	0
$412$	−14239.1	−1.70270
$413$	0	0
$414$	0	0
$415$	10838.4	1.28202
$416$	3709.29	0.437170
$417$	0	0
$418$	−1824.54	−0.213495
$419$	12861.0	1.49952	0.749761	−	0.661709i	$-0.230169\pi$
$419$	12861.0	1.49952	0.749761	+	0.661709i	$0.230169\pi$
$420$	0	0
$421$	−16467.7	−1.90638	−0.953188	−	0.302377i	$-0.902220\pi$
$421$	−16467.7	−1.90638	−0.953188	+	0.302377i	$0.902220\pi$
$422$	3062.61	0.353283
$423$	0	0
$424$	−4585.38	−0.525203
$425$	3569.06	0.407353
$426$	0	0
$427$	0	0
$428$	−4725.72	−0.533706
$429$	0	0
$430$	−403.317	−0.0452318
$431$	273.346	0.0305490	0.0152745	−	0.999883i	$-0.495138\pi$
$431$	273.346	0.0305490	0.0152745	+	0.999883i	$0.495138\pi$
$432$	0	0
$433$	−13765.3	−1.52776	−0.763878	−	0.645361i	$-0.776707\pi$
$433$	−13765.3	−1.52776	−0.763878	+	0.645361i	$0.776707\pi$
$434$	0	0
$435$	0	0
$436$	−7823.01	−0.859298
$437$	2065.56	0.226108
$438$	0	0
$439$	2965.97	0.322456	0.161228	−	0.986917i	$-0.448455\pi$
$439$	2965.97	0.322456	0.161228	+	0.986917i	$0.448455\pi$
$440$	−5498.30	−0.595730
$441$	0	0
$442$	1696.86	0.182605
$443$	−14038.4	−1.50561	−0.752806	−	0.658242i	$-0.771300\pi$
$443$	−14038.4	−1.50561	−0.752806	+	0.658242i	$0.771300\pi$
$444$	0	0
$445$	20601.0	2.19456
$446$	2595.38	0.275549
$447$	0	0
$448$	0	0
$449$	6338.13	0.666180	0.333090	−	0.942895i	$-0.391909\pi$
$449$	6338.13	0.666180	0.333090	+	0.942895i	$0.391909\pi$
$450$	0	0
$451$	13867.1	1.44784
$452$	9256.30	0.963230
$453$	0	0
$454$	1367.57	0.141372
$455$	0	0
$456$	0	0
$457$	9120.26	0.933540	0.466770	−	0.884379i	$-0.345418\pi$
$457$	9120.26	0.933540	0.466770	+	0.884379i	$0.345418\pi$
$458$	1652.97	0.168642
$459$	0	0
$460$	3049.87	0.309133
$461$	−15109.6	−1.52651	−0.763256	−	0.646096i	$-0.776400\pi$
$461$	−15109.6	−1.52651	−0.763256	+	0.646096i	$0.776400\pi$
$462$	0	0
$463$	6418.72	0.644283	0.322142	−	0.946692i	$-0.395597\pi$
$463$	6418.72	0.644283	0.322142	+	0.946692i	$0.395597\pi$
$464$	16638.3	1.66469
$465$	0	0
$466$	−1930.65	−0.191922
$467$	4304.81	0.426559	0.213279	−	0.976991i	$-0.431586\pi$
$467$	4304.81	0.426559	0.213279	+	0.976991i	$0.431586\pi$
$468$	0	0
$469$	0	0
$470$	−1390.43	−0.136459
$471$	0	0
$472$	3353.19	0.326998
$473$	−2676.21	−0.260153
$474$	0	0
$475$	−2890.35	−0.279196
$476$	0	0
$477$	0	0
$478$	417.390	0.0399393
$479$	5030.65	0.479867	0.239934	−	0.970789i	$-0.422874\pi$
$479$	5030.65	0.479867	0.239934	+	0.970789i	$0.422874\pi$
$480$	0	0
$481$	−7421.58	−0.703524
$482$	−2644.72	−0.249925
$483$	0	0
$484$	−7646.87	−0.718150
$485$	15416.2	1.44332
$486$	0	0
$487$	−13300.6	−1.23759	−0.618794	−	0.785553i	$-0.712379\pi$
$487$	−13300.6	−1.23759	−0.618794	+	0.785553i	$0.712379\pi$
$488$	1367.98	0.126897
$489$	0	0
$490$	0	0
$491$	9246.30	0.849857	0.424928	−	0.905227i	$-0.360299\pi$
$491$	9246.30	0.849857	0.424928	+	0.905227i	$0.360299\pi$
$492$	0	0
$493$	24502.1	2.23837
$494$	−1374.18	−0.125156
$495$	0	0
$496$	−15315.1	−1.38643
$497$	0	0
$498$	0	0
$499$	−16233.3	−1.45632	−0.728160	−	0.685407i	$-0.759624\pi$
$499$	−16233.3	−1.45632	−0.728160	+	0.685407i	$0.759624\pi$
$500$	8178.73	0.731528
$501$	0	0
$502$	−1261.34	−0.112144
$503$	1664.13	0.147515	0.0737575	−	0.997276i	$-0.476501\pi$
$503$	1664.13	0.147515	0.0737575	+	0.997276i	$0.476501\pi$
$504$	0	0
$505$	6579.69	0.579787
$506$	−828.721	−0.0728086
$507$	0	0
$508$	19332.7	1.68848
$509$	8757.17	0.762583	0.381291	−	0.924455i	$-0.375479\pi$
$509$	8757.17	0.762583	0.381291	+	0.924455i	$0.375479\pi$
$510$	0	0
$511$	0	0
$512$	9754.80	0.842002
$513$	0	0
$514$	3421.86	0.293642
$515$	24004.8	2.05394
$516$	0	0
$517$	−9226.21	−0.784851
$518$	0	0
$519$	0	0
$520$	−4141.13	−0.349232
$521$	−1641.01	−0.137993	−0.0689963	−	0.997617i	$-0.521980\pi$
$521$	−1641.01	−0.137993	−0.0689963	+	0.997617i	$0.521980\pi$
$522$	0	0
$523$	−16530.6	−1.38209	−0.691043	−	0.722813i	$-0.742849\pi$
$523$	−16530.6	−1.38209	−0.691043	+	0.722813i	$0.742849\pi$
$524$	16641.5	1.38738
$525$	0	0
$526$	1959.69	0.162446
$527$	−22553.5	−1.86423
$528$	0	0
$529$	−11228.8	−0.922890
$530$	3787.54	0.310416
$531$	0	0
$532$	0	0
$533$	10444.2	0.848762
$534$	0	0
$535$	7966.77	0.643801
$536$	573.679	0.0462298
$537$	0	0
$538$	−3586.50	−0.287407
$539$	0	0
$540$	0	0
$541$	−3418.03	−0.271631	−0.135816	−	0.990734i	$-0.543365\pi$
$541$	−3418.03	−0.271631	−0.135816	+	0.990734i	$0.543365\pi$
$542$	−2368.96	−0.187741
$543$	0	0
$544$	8503.36	0.670181
$545$	13188.3	1.03656
$546$	0	0
$547$	−15983.1	−1.24934	−0.624668	−	0.780890i	$-0.714766\pi$
$547$	−15983.1	−1.24934	−0.624668	+	0.780890i	$0.714766\pi$
$548$	−6114.94	−0.476674
$549$	0	0
$550$	1159.63	0.0899034
$551$	−19842.6	−1.53416
$552$	0	0
$553$	0	0
$554$	−2939.80	−0.225452
$555$	0	0
$556$	−18449.3	−1.40724
$557$	−8736.81	−0.664615	−0.332307	−	0.943171i	$-0.607827\pi$
$557$	−8736.81	−0.664615	−0.332307	+	0.943171i	$0.607827\pi$
$558$	0	0
$559$	−2015.63	−0.152508
$560$	0	0
$561$	0	0
$562$	3323.76	0.249474
$563$	−24107.2	−1.80461	−0.902306	−	0.431095i	$-0.858127\pi$
$563$	−24107.2	−1.80461	−0.902306	+	0.431095i	$0.858127\pi$
$564$	0	0
$565$	−15604.6	−1.16193
$566$	−1057.81	−0.0785567
$567$	0	0
$568$	2255.89	0.166646
$569$	−13291.7	−0.979292	−0.489646	−	0.871921i	$-0.662874\pi$
$569$	−13291.7	−0.979292	−0.489646	+	0.871921i	$0.662874\pi$
$570$	0	0
$571$	15208.2	1.11461	0.557305	−	0.830308i	$-0.311836\pi$
$571$	15208.2	1.11461	0.557305	+	0.830308i	$0.311836\pi$
$572$	−13463.6	−0.984161
$573$	0	0
$574$	0	0
$575$	−1312.82	−0.0952147
$576$	0	0
$577$	10809.5	0.779904	0.389952	−	0.920835i	$-0.372492\pi$
$577$	10809.5	0.779904	0.389952	+	0.920835i	$0.372492\pi$
$578$	1133.83	0.0815933
$579$	0	0
$580$	−29298.3	−2.09749
$581$	0	0
$582$	0	0
$583$	25132.3	1.78537
$584$	2803.68	0.198660
$585$	0	0
$586$	−1780.09	−0.125486
$587$	3843.26	0.270236	0.135118	−	0.990830i	$-0.456859\pi$
$587$	3843.26	0.270236	0.135118	+	0.990830i	$0.456859\pi$
$588$	0	0
$589$	18264.6	1.27773
$590$	−2769.75	−0.193269
$591$	0	0
$592$	−11553.2	−0.802084
$593$	−1402.87	−0.0971484	−0.0485742	−	0.998820i	$-0.515468\pi$
$593$	−1402.87	−0.0971484	−0.0485742	+	0.998820i	$0.515468\pi$
$594$	0	0
$595$	0	0
$596$	−10792.2	−0.741718
$597$	0	0
$598$	−624.164	−0.0426822
$599$	12738.8	0.868937	0.434469	−	0.900687i	$-0.356936\pi$
$599$	12738.8	0.868937	0.434469	+	0.900687i	$0.356936\pi$
$600$	0	0
$601$	−17292.7	−1.17368	−0.586841	−	0.809702i	$-0.699629\pi$
$601$	−17292.7	−1.17368	−0.586841	+	0.809702i	$0.699629\pi$
$602$	0	0
$603$	0	0
$604$	13537.5	0.911977
$605$	12891.3	0.866293
$606$	0	0
$607$	−27165.6	−1.81650	−0.908251	−	0.418426i	$-0.862582\pi$
$607$	−27165.6	−1.81650	−0.908251	+	0.418426i	$0.862582\pi$
$608$	−6886.31	−0.459336
$609$	0	0
$610$	−1129.96	−0.0750012
$611$	−6948.86	−0.460100
$612$	0	0
$613$	−10405.5	−0.685599	−0.342800	−	0.939409i	$-0.611375\pi$
$613$	−10405.5	−0.685599	−0.342800	+	0.939409i	$0.611375\pi$
$614$	3471.59	0.228179
$615$	0	0
$616$	0	0
$617$	18326.4	1.19578	0.597889	−	0.801579i	$-0.296006\pi$
$617$	18326.4	1.19578	0.597889	+	0.801579i	$0.296006\pi$
$618$	0	0
$619$	3575.13	0.232143	0.116072	−	0.993241i	$-0.462970\pi$
$619$	3575.13	0.232143	0.116072	+	0.993241i	$0.462970\pi$
$620$	26968.3	1.74689
$621$	0	0
$622$	2104.24	0.135647
$623$	0	0
$624$	0	0
$625$	−19145.5	−1.22531
$626$	−2340.62	−0.149441
$627$	0	0
$628$	−8315.70	−0.528396
$629$	−17013.6	−1.07850
$630$	0	0
$631$	−11722.1	−0.739538	−0.369769	−	0.929124i	$-0.620563\pi$
$631$	−11722.1	−0.739538	−0.369769	+	0.929124i	$0.620563\pi$
$632$	−684.175	−0.0430617
$633$	0	0
$634$	4718.15	0.295555
$635$	−32591.7	−2.03679
$636$	0	0
$637$	0	0
$638$	7961.03	0.494013
$639$	0	0
$640$	−13455.8	−0.831074
$641$	−24327.5	−1.49903	−0.749514	−	0.661988i	$-0.769713\pi$
$641$	−24327.5	−1.49903	−0.749514	+	0.661988i	$0.769713\pi$
$642$	0	0
$643$	−10684.8	−0.655316	−0.327658	−	0.944796i	$-0.606259\pi$
$643$	−10684.8	−0.655316	−0.327658	+	0.944796i	$0.606259\pi$
$644$	0	0
$645$	0	0
$646$	−3150.23	−0.191864
$647$	21203.0	1.28837	0.644184	−	0.764870i	$-0.277197\pi$
$647$	21203.0	1.28837	0.644184	+	0.764870i	$0.277197\pi$
$648$	0	0
$649$	−18378.7	−1.11160
$650$	873.395	0.0527037
$651$	0	0
$652$	19007.8	1.14172
$653$	27562.2	1.65175	0.825874	−	0.563854i	$-0.190682\pi$
$653$	27562.2	1.65175	0.825874	+	0.563854i	$0.190682\pi$
$654$	0	0
$655$	−28054.7	−1.67357
$656$	16258.6	0.967670
$657$	0	0
$658$	0	0
$659$	1389.06	0.0821091	0.0410546	−	0.999157i	$-0.486928\pi$
$659$	1389.06	0.0821091	0.0410546	+	0.999157i	$0.486928\pi$
$660$	0	0
$661$	12490.6	0.734989	0.367494	−	0.930026i	$-0.380216\pi$
$661$	12490.6	0.734989	0.367494	+	0.930026i	$0.380216\pi$
$662$	−683.130	−0.0401066
$663$	0	0
$664$	−7361.06	−0.430218
$665$	0	0
$666$	0	0
$667$	−9012.70	−0.523198
$668$	−17220.7	−0.997438
$669$	0	0
$670$	−473.861	−0.0273237
$671$	−7497.85	−0.431373
$672$	0	0
$673$	−13234.5	−0.758027	−0.379014	−	0.925391i	$-0.623737\pi$
$673$	−13234.5	−0.758027	−0.379014	+	0.925391i	$0.623737\pi$
$674$	2176.66	0.124394
$675$	0	0
$676$	6744.28	0.383721
$677$	4788.89	0.271864	0.135932	−	0.990718i	$-0.456597\pi$
$677$	4788.89	0.271864	0.135932	+	0.990718i	$0.456597\pi$
$678$	0	0
$679$	0	0
$680$	−9493.34	−0.535372
$681$	0	0
$682$	−7327.92	−0.411438
$683$	−7543.01	−0.422584	−0.211292	−	0.977423i	$-0.567767\pi$
$683$	−7543.01	−0.422584	−0.211292	+	0.977423i	$0.567767\pi$
$684$	0	0
$685$	10308.8	0.575004
$686$	0	0
$687$	0	0
$688$	−3137.74	−0.173874
$689$	18928.7	1.04663
$690$	0	0
$691$	−27685.3	−1.52416	−0.762082	−	0.647481i	$-0.775823\pi$
$691$	−27685.3	−1.52416	−0.762082	+	0.647481i	$0.775823\pi$
$692$	16302.6	0.895568
$693$	0	0
$694$	−5426.16	−0.296793
$695$	31102.4	1.69753
$696$	0	0
$697$	23942.9	1.30115
$698$	2221.89	0.120487
$699$	0	0
$700$	0	0
$701$	−19866.5	−1.07040	−0.535198	−	0.844726i	$-0.679763\pi$
$701$	−19866.5	−1.07040	−0.535198	+	0.844726i	$0.679763\pi$
$702$	0	0
$703$	13778.2	0.739195
$704$	−19054.2	−1.02008
$705$	0	0
$706$	−4285.79	−0.228467
$707$	0	0
$708$	0	0
$709$	−3572.12	−0.189216	−0.0946078	−	0.995515i	$-0.530160\pi$
$709$	−3572.12	−0.189216	−0.0946078	+	0.995515i	$0.530160\pi$
$710$	−1863.37	−0.0984944
$711$	0	0
$712$	−13991.4	−0.736448
$713$	8295.95	0.435745
$714$	0	0
$715$	22697.3	1.18718
$716$	−12133.8	−0.633327
$717$	0	0
$718$	−5279.58	−0.274418
$719$	32955.5	1.70936	0.854681	−	0.519154i	$-0.173753\pi$
$719$	32955.5	1.70936	0.854681	+	0.519154i	$0.173753\pi$
$720$	0	0
$721$	0	0
$722$	−1296.68	−0.0668385
$723$	0	0
$724$	−35426.4	−1.81852
$725$	12611.5	0.646041
$726$	0	0
$727$	6347.31	0.323808	0.161904	−	0.986806i	$-0.448236\pi$
$727$	6347.31	0.323808	0.161904	+	0.986806i	$0.448236\pi$
$728$	0	0
$729$	0	0
$730$	−2315.85	−0.117416
$731$	−4620.73	−0.233795
$732$	0	0
$733$	−22488.7	−1.13321	−0.566603	−	0.823991i	$-0.691743\pi$
$733$	−22488.7	−1.13321	−0.566603	+	0.823991i	$0.691743\pi$
$734$	3202.03	0.161020
$735$	0	0
$736$	−3127.82	−0.156648
$737$	−3144.31	−0.157153
$738$	0	0
$739$	19859.5	0.988559	0.494279	−	0.869303i	$-0.335432\pi$
$739$	19859.5	0.988559	0.494279	+	0.869303i	$0.335432\pi$
$740$	20344.0	1.01062
$741$	0	0
$742$	0	0
$743$	−16900.0	−0.834455	−0.417228	−	0.908802i	$-0.636998\pi$
$743$	−16900.0	−0.834455	−0.417228	+	0.908802i	$0.636998\pi$
$744$	0	0
$745$	18193.8	0.894722
$746$	−5374.05	−0.263751
$747$	0	0
$748$	−30864.6	−1.50872
$749$	0	0
$750$	0	0
$751$	19441.1	0.944627	0.472314	−	0.881431i	$-0.343419\pi$
$751$	19441.1	0.944627	0.472314	+	0.881431i	$0.343419\pi$
$752$	−10817.3	−0.524558
$753$	0	0
$754$	5995.97	0.289603
$755$	−22822.0	−1.10010
$756$	0	0
$757$	−17481.7	−0.839343	−0.419671	−	0.907676i	$-0.637855\pi$
$757$	−17481.7	−0.839343	−0.419671	+	0.907676i	$0.637855\pi$
$758$	4897.97	0.234700
$759$	0	0
$760$	7688.02	0.366939
$761$	−4160.98	−0.198207	−0.0991035	−	0.995077i	$-0.531597\pi$
$761$	−4160.98	−0.198207	−0.0991035	+	0.995077i	$0.531597\pi$
$762$	0	0
$763$	0	0
$764$	−9927.26	−0.470099
$765$	0	0
$766$	5398.54	0.254644
$767$	−13842.2	−0.651645
$768$	0	0
$769$	33004.0	1.54766	0.773832	−	0.633391i	$-0.218338\pi$
$769$	33004.0	1.54766	0.773832	+	0.633391i	$0.218338\pi$
$770$	0	0
$771$	0	0
$772$	−9037.94	−0.421351
$773$	29291.1	1.36291	0.681455	−	0.731860i	$-0.261348\pi$
$773$	29291.1	1.36291	0.681455	+	0.731860i	$0.261348\pi$
$774$	0	0
$775$	−11608.6	−0.538054
$776$	−10470.1	−0.484349
$777$	0	0
$778$	−6350.28	−0.292633
$779$	−19389.8	−0.891797
$780$	0	0
$781$	−12364.4	−0.566495
$782$	−1430.87	−0.0654318
$783$	0	0
$784$	0	0
$785$	14018.9	0.637395
$786$	0	0
$787$	7935.32	0.359420	0.179710	−	0.983720i	$-0.442484\pi$
$787$	7935.32	0.359420	0.179710	+	0.983720i	$0.442484\pi$
$788$	−35050.3	−1.58454
$789$	0	0
$790$	565.131	0.0254512
$791$	0	0
$792$	0	0
$793$	−5647.12	−0.252882
$794$	1602.12	0.0716084
$795$	0	0
$796$	7577.80	0.337422
$797$	20316.5	0.902947	0.451473	−	0.892285i	$-0.350899\pi$
$797$	20316.5	0.902947	0.451473	+	0.892285i	$0.350899\pi$
$798$	0	0
$799$	−15929.9	−0.705332
$800$	4376.78	0.193428
$801$	0	0
$802$	−1108.50	−0.0488062
$803$	−15366.8	−0.675323
$804$	0	0
$805$	0	0
$806$	−5519.14	−0.241195
$807$	0	0
$808$	−4468.69	−0.194564
$809$	−12447.0	−0.540931	−0.270465	−	0.962730i	$-0.587178\pi$
$809$	−12447.0	−0.540931	−0.270465	+	0.962730i	$0.587178\pi$
$810$	0	0
$811$	−9478.03	−0.410380	−0.205190	−	0.978722i	$-0.565781\pi$
$811$	−9478.03	−0.410380	−0.205190	+	0.978722i	$0.565781\pi$
$812$	0	0
$813$	0	0
$814$	−5527.93	−0.238027
$815$	−32043.9	−1.37724
$816$	0	0
$817$	3742.02	0.160241
$818$	4625.17	0.197696
$819$	0	0
$820$	−28629.7	−1.21926
$821$	−3763.09	−0.159967	−0.0799835	−	0.996796i	$-0.525487\pi$
$821$	−3763.09	−0.159967	−0.0799835	+	0.996796i	$0.525487\pi$
$822$	0	0
$823$	21474.8	0.909554	0.454777	−	0.890605i	$-0.349719\pi$
$823$	21474.8	0.909554	0.454777	+	0.890605i	$0.349719\pi$
$824$	−16303.2	−0.689258
$825$	0	0
$826$	0	0
$827$	−45304.9	−1.90496	−0.952481	−	0.304597i	$-0.901478\pi$
$827$	−45304.9	−1.90496	−0.952481	+	0.304597i	$0.901478\pi$
$828$	0	0
$829$	−4614.78	−0.193339	−0.0966694	−	0.995317i	$-0.530819\pi$
$829$	−4614.78	−0.193339	−0.0966694	+	0.995317i	$0.530819\pi$
$830$	6080.26	0.254276
$831$	0	0
$832$	−14351.0	−0.597994
$833$	0	0
$834$	0	0
$835$	29031.2	1.20319
$836$	24995.2	1.03406
$837$	0	0
$838$	7214.90	0.297416
$839$	−18308.4	−0.753368	−0.376684	−	0.926342i	$-0.622936\pi$
$839$	−18308.4	−0.753368	−0.376684	+	0.926342i	$0.622936\pi$
$840$	0	0
$841$	62190.6	2.54995
$842$	−9238.22	−0.378112
$843$	0	0
$844$	−41956.1	−1.71112
$845$	−11369.7	−0.462876
$846$	0	0
$847$	0	0
$848$	29466.5	1.19326
$849$	0	0
$850$	2002.22	0.0807946
$851$	6258.18	0.252089
$852$	0	0
$853$	−13373.0	−0.536790	−0.268395	−	0.963309i	$-0.586493\pi$
$853$	−13373.0	−0.536790	−0.268395	+	0.963309i	$0.586493\pi$
$854$	0	0
$855$	0	0
$856$	−5410.74	−0.216046
$857$	−17658.0	−0.703835	−0.351917	−	0.936031i	$-0.614470\pi$
$857$	−17658.0	−0.703835	−0.351917	+	0.936031i	$0.614470\pi$
$858$	0	0
$859$	12413.0	0.493044	0.246522	−	0.969137i	$-0.420712\pi$
$859$	12413.0	0.493044	0.246522	+	0.969137i	$0.420712\pi$
$860$	5525.23	0.219080
$861$	0	0
$862$	153.345	0.00605910
$863$	10219.2	0.403087	0.201544	−	0.979480i	$-0.435404\pi$
$863$	10219.2	0.403087	0.201544	+	0.979480i	$0.435404\pi$
$864$	0	0
$865$	−27483.5	−1.08031
$866$	−7722.22	−0.303016
$867$	0	0
$868$	0	0
$869$	3749.93	0.146384
$870$	0	0
$871$	−2368.18	−0.0921272
$872$	−8957.00	−0.347847
$873$	0	0
$874$	1158.76	0.0448464
$875$	0	0
$876$	0	0
$877$	952.207	0.0366633	0.0183317	−	0.999832i	$-0.494165\pi$
$877$	952.207	0.0366633	0.0183317	+	0.999832i	$0.494165\pi$
$878$	1663.89	0.0639561
$879$	0	0
$880$	35333.0	1.35350
$881$	25182.5	0.963021	0.481510	−	0.876440i	$-0.340088\pi$
$881$	25182.5	0.963021	0.481510	+	0.876440i	$0.340088\pi$
$882$	0	0
$883$	−3941.95	−0.150234	−0.0751172	−	0.997175i	$-0.523933\pi$
$883$	−3941.95	−0.150234	−0.0751172	+	0.997175i	$0.523933\pi$
$884$	−23246.1	−0.884448
$885$	0	0
$886$	−7875.45	−0.298624
$887$	41630.5	1.57589	0.787946	−	0.615744i	$-0.211144\pi$
$887$	41630.5	1.57589	0.787946	+	0.615744i	$0.211144\pi$
$888$	0	0
$889$	0	0
$890$	11557.0	0.435271
$891$	0	0
$892$	−35555.3	−1.33462
$893$	12900.6	0.483428
$894$	0	0
$895$	20455.6	0.763972
$896$	0	0
$897$	0	0
$898$	3555.64	0.132130
$899$	−79694.3	−2.95657
$900$	0	0
$901$	43393.2	1.60448
$902$	7779.34	0.287166
$903$	0	0
$904$	10598.1	0.389918
$905$	59722.9	2.19365
$906$	0	0
$907$	34713.1	1.27081	0.635407	−	0.772177i	$-0.280832\pi$
$907$	34713.1	1.27081	0.635407	+	0.772177i	$0.280832\pi$
$908$	−18734.9	−0.684737
$909$	0	0
$910$	0	0
$911$	−22126.6	−0.804705	−0.402352	−	0.915485i	$-0.631807\pi$
$911$	−22126.6	−0.804705	−0.402352	+	0.915485i	$0.631807\pi$
$912$	0	0
$913$	40345.6	1.46248
$914$	5116.39	0.185159
$915$	0	0
$916$	−22644.7	−0.816816
$917$	0	0
$918$	0	0
$919$	−38198.8	−1.37112	−0.685561	−	0.728015i	$-0.740443\pi$
$919$	−38198.8	−1.37112	−0.685561	+	0.728015i	$0.740443\pi$
$920$	3491.97	0.125138
$921$	0	0
$922$	−8476.33	−0.302769
$923$	−9312.43	−0.332094
$924$	0	0
$925$	−8757.09	−0.311277
$926$	3600.85	0.127787
$927$	0	0
$928$	30047.1	1.06287
$929$	−13387.9	−0.472813	−0.236407	−	0.971654i	$-0.575970\pi$
$929$	−13387.9	−0.472813	−0.236407	+	0.971654i	$0.575970\pi$
$930$	0	0
$931$	0	0
$932$	26448.8	0.929572
$933$	0	0
$934$	2414.96	0.0846039
$935$	52032.5	1.81994
$936$	0	0
$937$	34574.1	1.20543	0.602715	−	0.797957i	$-0.294086\pi$
$937$	34574.1	1.20543	0.602715	+	0.797957i	$0.294086\pi$
$938$	0	0
$939$	0	0
$940$	19048.2	0.660939
$941$	−2922.81	−0.101255	−0.0506274	−	0.998718i	$-0.516122\pi$
$941$	−2922.81	−0.101255	−0.0506274	+	0.998718i	$0.516122\pi$
$942$	0	0
$943$	−8807.01	−0.304131
$944$	−21548.2	−0.742938
$945$	0	0
$946$	−1501.33	−0.0515989
$947$	−9913.84	−0.340186	−0.170093	−	0.985428i	$-0.554407\pi$
$947$	−9913.84	−0.340186	−0.170093	+	0.985428i	$0.554407\pi$
$948$	0	0
$949$	−11573.8	−0.395891
$950$	−1621.46	−0.0553759
$951$	0	0
$952$	0	0
$953$	51173.4	1.73942	0.869710	−	0.493562i	$-0.164306\pi$
$953$	51173.4	1.73942	0.869710	+	0.493562i	$0.164306\pi$
$954$	0	0
$955$	16735.7	0.567073
$956$	−5718.02	−0.193446
$957$	0	0
$958$	2822.16	0.0951772
$959$	0	0
$960$	0	0
$961$	43565.5	1.46237
$962$	−4163.45	−0.139537
$963$	0	0
$964$	36231.3	1.21051
$965$	15236.5	0.508268
$966$	0	0
$967$	8758.72	0.291273	0.145637	−	0.989338i	$-0.453477\pi$
$967$	8758.72	0.291273	0.145637	+	0.989338i	$0.453477\pi$
$968$	−8755.33	−0.290710
$969$	0	0
$970$	8648.34	0.286270
$971$	8293.18	0.274089	0.137045	−	0.990565i	$-0.456240\pi$
$971$	8293.18	0.274089	0.137045	+	0.990565i	$0.456240\pi$
$972$	0	0
$973$	0	0
$974$	−7461.50	−0.245464
$975$	0	0
$976$	−8790.91	−0.288309
$977$	18969.7	0.621181	0.310590	−	0.950544i	$-0.399473\pi$
$977$	18969.7	0.621181	0.310590	+	0.950544i	$0.399473\pi$
$978$	0	0
$979$	76686.4	2.50348
$980$	0	0
$981$	0	0
$982$	5187.10	0.168561
$983$	56313.2	1.82718	0.913588	−	0.406642i	$-0.133300\pi$
$983$	56313.2	1.82718	0.913588	+	0.406642i	$0.133300\pi$
$984$	0	0
$985$	59089.0	1.91140
$986$	13745.5	0.443960
$987$	0	0
$988$	18825.5	0.606193
$989$	1699.66	0.0546472
$990$	0	0
$991$	−8558.30	−0.274332	−0.137166	−	0.990548i	$-0.543799\pi$
$991$	−8558.30	−0.274332	−0.137166	+	0.990548i	$0.543799\pi$
$992$	−27657.6	−0.885212
$993$	0	0
$994$	0	0
$995$	−12774.9	−0.407027
$996$	0	0
$997$	36163.8	1.14876	0.574382	−	0.818587i	$-0.305242\pi$
$997$	36163.8	1.14876	0.574382	+	0.818587i	$0.305242\pi$
$998$	−9106.77	−0.288847
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bg.1.4	yes	6
3.2	odd	2	inner	1323.4.a.bg.1.3	yes	6
7.6	odd	2		1323.4.a.bf.1.4	yes	6
21.20	even	2		1323.4.a.bf.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bf.1.3	✓	6	21.20	even	2
1323.4.a.bf.1.4	yes	6	7.6	odd	2
1323.4.a.bg.1.3	yes	6	3.2	odd	2	inner
1323.4.a.bg.1.4	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.560992 q^{2} -7.68529 q^{4} +12.9561 q^{5} -8.79932 q^{8} +O(q^{10})\)	\(q+0.560992 q^{2} -7.68529 q^{4} +12.9561 q^{5} -8.79932 q^{8} +7.26827 q^{10} +48.2286 q^{11} +36.3241 q^{13} +56.5460 q^{16} +83.2713 q^{17} -67.4359 q^{19} -99.5714 q^{20} +27.0559 q^{22} -30.6300 q^{23} +42.8607 q^{25} +20.3775 q^{26} +294.244 q^{29} -270.844 q^{31} +102.116 q^{32} +46.7145 q^{34} -204.315 q^{37} -37.8310 q^{38} -114.005 q^{40} +287.529 q^{41} -55.4901 q^{43} -370.651 q^{44} -17.1832 q^{46} -191.301 q^{47} +24.0445 q^{50} -279.161 q^{52} +521.107 q^{53} +624.855 q^{55} +165.069 q^{58} -381.074 q^{59} -155.465 q^{61} -151.941 q^{62} -395.081 q^{64} +470.619 q^{65} -65.1959 q^{67} -639.964 q^{68} -256.370 q^{71} -318.625 q^{73} -114.619 q^{74} +518.264 q^{76} +77.7532 q^{79} +732.615 q^{80} +161.301 q^{82} +836.549 q^{83} +1078.87 q^{85} -31.1295 q^{86} -424.379 q^{88} +1590.06 q^{89} +235.400 q^{92} -107.319 q^{94} -873.706 q^{95} +1189.88 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 36 q^{4}+O(q^{10}) \) 6 * q + 36 * q^4	\( 6 q + 36 q^{4} + 180 q^{10} + 108 q^{13} + 420 q^{16} + 198 q^{19} - 84 q^{22} + 420 q^{25} - 90 q^{31} - 648 q^{34} - 402 q^{37} + 2844 q^{40} - 660 q^{43} - 1332 q^{46} + 1224 q^{52} + 846 q^{55} - 1800 q^{58} + 1152 q^{61} + 2964 q^{64} + 924 q^{67} + 1260 q^{73} + 5868 q^{76} - 1500 q^{79} + 4500 q^{82} - 2232 q^{85} - 2460 q^{88} - 4968 q^{94} + 3312 q^{97}+O(q^{100}) \) 6 * q + 36 * q^4 + 180 * q^10 + 108 * q^13 + 420 * q^16 + 198 * q^19 - 84 * q^22 + 420 * q^25 - 90 * q^31 - 648 * q^34 - 402 * q^37 + 2844 * q^40 - 660 * q^43 - 1332 * q^46 + 1224 * q^52 + 846 * q^55 - 1800 * q^58 + 1152 * q^61 + 2964 * q^64 + 924 * q^67 + 1260 * q^73 + 5868 * q^76 - 1500 * q^79 + 4500 * q^82 - 2232 * q^85 - 2460 * q^88 - 4968 * q^94 + 3312 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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