LMFDB - Embedded newform 1323.4.a.bm.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 $ x^8 - 54x^6 + 887x^4 - 4176*x^2 + 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$5.00098$ 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)

$f(q)$	$=$	$q+5.00098 q^{2} +17.0098 q^{4} +3.98256 q^{5} +45.0578 q^{8} +O(q^{10})$	$q+5.00098 q^{2} +17.0098 q^{4} +3.98256 q^{5} +45.0578 q^{8} +19.9167 q^{10} -5.53706 q^{11} +33.8302 q^{13} +89.2547 q^{16} +103.352 q^{17} +124.036 q^{19} +67.7424 q^{20} -27.6907 q^{22} -151.087 q^{23} -109.139 q^{25} +169.184 q^{26} +103.634 q^{29} -60.2612 q^{31} +85.8986 q^{32} +516.859 q^{34} +413.638 q^{37} +620.303 q^{38} +179.445 q^{40} +153.584 q^{41} -261.492 q^{43} -94.1843 q^{44} -755.582 q^{46} -104.807 q^{47} -545.803 q^{50} +575.444 q^{52} +716.150 q^{53} -22.0517 q^{55} +518.272 q^{58} -647.384 q^{59} +269.514 q^{61} -301.365 q^{62} -284.461 q^{64} +134.731 q^{65} +483.808 q^{67} +1757.99 q^{68} -362.885 q^{71} +908.060 q^{73} +2068.60 q^{74} +2109.83 q^{76} +162.439 q^{79} +355.462 q^{80} +768.069 q^{82} -419.276 q^{83} +411.603 q^{85} -1307.72 q^{86} -249.488 q^{88} +1019.47 q^{89} -2569.96 q^{92} -524.140 q^{94} +493.981 q^{95} -346.785 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 44 q^{4}+O(q^{10}) $ 8 * q + 44 * q^4	$ 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55} - 1200 q^{58} + 2400 q^{61} - 1388 q^{64} + 1824 q^{73} + 2844 q^{76} + 2368 q^{79} + 2436 q^{82} + 3512 q^{85} + 3780 q^{88} + 4368 q^{94} + 5712 q^{97}+O(q^{100}) $ 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * 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 $

$2$	5.00098	1.76811	0.884057	−	0.467380i	$-0.154802\pi$
$2$	5.00098	1.76811	0.884057	+	0.467380i	$0.154802\pi$
$3$	0	0
$3$	0	0
$4$	17.0098	2.12622
$5$	3.98256	0.356211	0.178105	−	0.984011i	$-0.443003\pi$
$5$	3.98256	0.356211	0.178105	+	0.984011i	$0.443003\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	45.0578	1.99129
$9$	0	0
$10$	19.9167	0.629821
$11$	−5.53706	−0.151772	−0.0758858	−	0.997117i	$-0.524178\pi$
$11$	−5.53706	−0.151772	−0.0758858	+	0.997117i	$0.524178\pi$
$12$	0	0
$13$	33.8302	0.721754	0.360877	−	0.932613i	$-0.382477\pi$
$13$	33.8302	0.721754	0.360877	+	0.932613i	$0.382477\pi$
$14$	0	0
$15$	0	0
$16$	89.2547	1.39460
$17$	103.352	1.47450	0.737248	−	0.675623i	$-0.236125\pi$
$17$	103.352	1.47450	0.737248	+	0.675623i	$0.236125\pi$
$18$	0	0
$19$	124.036	1.49768	0.748839	−	0.662752i	$-0.230612\pi$
$19$	124.036	1.49768	0.748839	+	0.662752i	$0.230612\pi$
$20$	67.7424	0.757384
$21$	0	0
$22$	−27.6907	−0.268349
$23$	−151.087	−1.36973	−0.684865	−	0.728670i	$-0.740139\pi$
$23$	−151.087	−1.36973	−0.684865	+	0.728670i	$0.740139\pi$
$24$	0	0
$25$	−109.139	−0.873114
$26$	169.184	1.27614
$27$	0	0
$28$	0	0
$29$	103.634	0.663599	0.331799	−	0.943350i	$-0.392344\pi$
$29$	103.634	0.663599	0.331799	+	0.943350i	$0.392344\pi$
$30$	0	0
$31$	−60.2612	−0.349137	−0.174568	−	0.984645i	$-0.555853\pi$
$31$	−60.2612	−0.349137	−0.174568	+	0.984645i	$0.555853\pi$
$32$	85.8986	0.474527
$33$	0	0
$34$	516.859	2.60707
$35$	0	0
$36$	0	0
$37$	413.638	1.83788	0.918942	−	0.394392i	$-0.129045\pi$
$37$	413.638	1.83788	0.918942	+	0.394392i	$0.129045\pi$
$38$	620.303	2.64806
$39$	0	0
$40$	179.445	0.709319
$41$	153.584	0.585018	0.292509	−	0.956263i	$-0.405510\pi$
$41$	153.584	0.585018	0.292509	+	0.956263i	$0.405510\pi$
$42$	0	0
$43$	−261.492	−0.927377	−0.463689	−	0.885998i	$-0.653474\pi$
$43$	−261.492	−0.927377	−0.463689	+	0.885998i	$0.653474\pi$
$44$	−94.1843	−0.322700
$45$	0	0
$46$	−755.582	−2.42184
$47$	−104.807	−0.325271	−0.162636	−	0.986686i	$-0.551999\pi$
$47$	−104.807	−0.325271	−0.162636	+	0.986686i	$0.551999\pi$
$48$	0	0
$49$	0	0
$50$	−545.803	−1.54376
$51$	0	0
$52$	575.444	1.53461
$53$	716.150	1.85605	0.928027	−	0.372514i	$-0.121504\pi$
$53$	716.150	1.85605	0.928027	+	0.372514i	$0.121504\pi$
$54$	0	0
$55$	−22.0517	−0.0540626
$56$	0	0
$57$	0	0
$58$	518.272	1.17332
$59$	−647.384	−1.42851	−0.714256	−	0.699884i	$-0.753235\pi$
$59$	−647.384	−1.42851	−0.714256	+	0.699884i	$0.753235\pi$
$60$	0	0
$61$	269.514	0.565700	0.282850	−	0.959164i	$-0.408720\pi$
$61$	269.514	0.565700	0.282850	+	0.959164i	$0.408720\pi$
$62$	−301.365	−0.617313
$63$	0	0
$64$	−284.461	−0.555587
$65$	134.731	0.257096
$66$	0	0
$67$	483.808	0.882187	0.441094	−	0.897461i	$-0.354591\pi$
$67$	483.808	0.882187	0.441094	+	0.897461i	$0.354591\pi$
$68$	1757.99	3.13511
$69$	0	0
$70$	0	0
$71$	−362.885	−0.606572	−0.303286	−	0.952900i	$-0.598084\pi$
$71$	−362.885	−0.606572	−0.303286	+	0.952900i	$0.598084\pi$
$72$	0	0
$73$	908.060	1.45590	0.727948	−	0.685633i	$-0.240474\pi$
$73$	908.060	1.45590	0.727948	+	0.685633i	$0.240474\pi$
$74$	2068.60	3.24959
$75$	0	0
$76$	2109.83	3.18440
$77$	0	0
$78$	0	0
$79$	162.439	0.231339	0.115669	−	0.993288i	$-0.463099\pi$
$79$	162.439	0.231339	0.115669	+	0.993288i	$0.463099\pi$
$80$	355.462	0.496773
$81$	0	0
$82$	768.069	1.03438
$83$	−419.276	−0.554476	−0.277238	−	0.960801i	$-0.589419\pi$
$83$	−419.276	−0.554476	−0.277238	+	0.960801i	$0.589419\pi$
$84$	0	0
$85$	411.603	0.525231
$86$	−1307.72	−1.63971
$87$	0	0
$88$	−249.488	−0.302221
$89$	1019.47	1.21420	0.607101	−	0.794625i	$-0.292332\pi$
$89$	1019.47	1.21420	0.607101	+	0.794625i	$0.292332\pi$
$90$	0	0
$91$	0	0
$92$	−2569.96	−2.91235
$93$	0	0
$94$	−524.140	−0.575116
$95$	493.981	0.533489
$96$	0	0
$97$	−346.785	−0.362997	−0.181499	−	0.983391i	$-0.558095\pi$
$97$	−346.785	−0.362997	−0.181499	+	0.983391i	$0.558095\pi$
$98$	0	0
$99$	0	0
$100$	−1856.44	−1.85644
$101$	1427.73	1.40658	0.703291	−	0.710902i	$-0.251713\pi$
$101$	1427.73	1.40658	0.703291	+	0.710902i	$0.251713\pi$
$102$	0	0
$103$	−853.502	−0.816486	−0.408243	−	0.912873i	$-0.633858\pi$
$103$	−853.502	−0.816486	−0.408243	+	0.912873i	$0.633858\pi$
$104$	1524.31	1.43722
$105$	0	0
$106$	3581.45	3.28171
$107$	−730.508	−0.660008	−0.330004	−	0.943980i	$-0.607050\pi$
$107$	−730.508	−0.660008	−0.330004	+	0.943980i	$0.607050\pi$
$108$	0	0
$109$	−1812.87	−1.59304	−0.796521	−	0.604611i	$-0.793329\pi$
$109$	−1812.87	−1.59304	−0.796521	+	0.604611i	$0.793329\pi$
$110$	−110.280	−0.0955889
$111$	0	0
$112$	0	0
$113$	472.343	0.393224	0.196612	−	0.980481i	$-0.437006\pi$
$113$	472.343	0.393224	0.196612	+	0.980481i	$0.437006\pi$
$114$	0	0
$115$	−601.712	−0.487912
$116$	1762.79	1.41096
$117$	0	0
$118$	−3237.55	−2.52577
$119$	0	0
$120$	0	0
$121$	−1300.34	−0.976965
$122$	1347.83	1.00022
$123$	0	0
$124$	−1025.03	−0.742342
$125$	−932.473	−0.667223
$126$	0	0
$127$	−660.594	−0.461561	−0.230781	−	0.973006i	$-0.574128\pi$
$127$	−660.594	−0.461561	−0.230781	+	0.973006i	$0.574128\pi$
$128$	−2109.77	−1.45687
$129$	0	0
$130$	673.785	0.454575
$131$	1294.58	0.863417	0.431709	−	0.902013i	$-0.357911\pi$
$131$	1294.58	0.863417	0.431709	+	0.902013i	$0.357911\pi$
$132$	0	0
$133$	0	0
$134$	2419.51	1.55981
$135$	0	0
$136$	4656.79	2.93615
$137$	−798.203	−0.497774	−0.248887	−	0.968532i	$-0.580065\pi$
$137$	−798.203	−0.497774	−0.248887	+	0.968532i	$0.580065\pi$
$138$	0	0
$139$	956.430	0.583621	0.291811	−	0.956476i	$-0.405742\pi$
$139$	956.430	0.583621	0.291811	+	0.956476i	$0.405742\pi$
$140$	0	0
$141$	0	0
$142$	−1814.78	−1.07249
$143$	−187.320	−0.109542
$144$	0	0
$145$	412.729	0.236381
$146$	4541.19	2.57419
$147$	0	0
$148$	7035.90	3.90775
$149$	−1212.30	−0.666546	−0.333273	−	0.942830i	$-0.608153\pi$
$149$	−1212.30	−0.666546	−0.333273	+	0.942830i	$0.608153\pi$
$150$	0	0
$151$	2686.26	1.44771	0.723857	−	0.689950i	$-0.242368\pi$
$151$	2686.26	1.44771	0.723857	+	0.689950i	$0.242368\pi$
$152$	5588.80	2.98231
$153$	0	0
$154$	0	0
$155$	−239.994	−0.124366
$156$	0	0
$157$	−1441.07	−0.732549	−0.366275	−	0.930507i	$-0.619367\pi$
$157$	−1441.07	−0.732549	−0.366275	+	0.930507i	$0.619367\pi$
$158$	812.352	0.409033
$159$	0	0
$160$	342.096	0.169032
$161$	0	0
$162$	0	0
$163$	−3093.82	−1.48667	−0.743334	−	0.668921i	$-0.766757\pi$
$163$	−3093.82	−1.48667	−0.743334	+	0.668921i	$0.766757\pi$
$164$	2612.43	1.24388
$165$	0	0
$166$	−2096.79	−0.980377
$167$	−1905.39	−0.882896	−0.441448	−	0.897287i	$-0.645535\pi$
$167$	−1905.39	−0.882896	−0.441448	+	0.897287i	$0.645535\pi$
$168$	0	0
$169$	−1052.52	−0.479071
$170$	2058.42	0.928668
$171$	0	0
$172$	−4447.93	−1.97181
$173$	−2901.01	−1.27491	−0.637456	−	0.770487i	$-0.720013\pi$
$173$	−2901.01	−1.27491	−0.637456	+	0.770487i	$0.720013\pi$
$174$	0	0
$175$	0	0
$176$	−494.209	−0.211661
$177$	0	0
$178$	5098.36	2.14685
$179$	1375.68	0.574430	0.287215	−	0.957866i	$-0.407271\pi$
$179$	1375.68	0.574430	0.287215	+	0.957866i	$0.407271\pi$
$180$	0	0
$181$	−4558.15	−1.87185	−0.935924	−	0.352203i	$-0.885433\pi$
$181$	−4558.15	−1.87185	−0.935924	+	0.352203i	$0.885433\pi$
$182$	0	0
$183$	0	0
$184$	−6807.64	−2.72753
$185$	1647.34	0.654674
$186$	0	0
$187$	−572.264	−0.223786
$188$	−1782.75	−0.691599
$189$	0	0
$190$	2470.39	0.943268
$191$	−4740.73	−1.79595	−0.897976	−	0.440044i	$-0.854963\pi$
$191$	−4740.73	−1.79595	−0.897976	+	0.440044i	$0.854963\pi$
$192$	0	0
$193$	2427.21	0.905255	0.452628	−	0.891700i	$-0.350487\pi$
$193$	2427.21	0.905255	0.452628	+	0.891700i	$0.350487\pi$
$194$	−1734.27	−0.641820
$195$	0	0
$196$	0	0
$197$	4374.12	1.58194	0.790972	−	0.611852i	$-0.209575\pi$
$197$	4374.12	1.58194	0.790972	+	0.611852i	$0.209575\pi$
$198$	0	0
$199$	367.938	0.131068	0.0655338	−	0.997850i	$-0.479125\pi$
$199$	367.938	0.131068	0.0655338	+	0.997850i	$0.479125\pi$
$200$	−4917.57	−1.73862
$201$	0	0
$202$	7140.07	2.48700
$203$	0	0
$204$	0	0
$205$	611.656	0.208390
$206$	−4268.34	−1.44364
$207$	0	0
$208$	3019.50	1.00656
$209$	−686.797	−0.227305
$210$	0	0
$211$	2224.16	0.725677	0.362838	−	0.931852i	$-0.381808\pi$
$211$	2224.16	0.725677	0.362838	+	0.931852i	$0.381808\pi$
$212$	12181.6	3.94638
$213$	0	0
$214$	−3653.25	−1.16697
$215$	−1041.41	−0.330342
$216$	0	0
$217$	0	0
$218$	−9066.13	−2.81668
$219$	0	0
$220$	−375.094	−0.114949
$221$	3496.40	1.06422
$222$	0	0
$223$	3675.92	1.10385	0.551923	−	0.833895i	$-0.313894\pi$
$223$	3675.92	1.10385	0.551923	+	0.833895i	$0.313894\pi$
$224$	0	0
$225$	0	0
$226$	2362.18	0.695264
$227$	−5870.47	−1.71646	−0.858230	−	0.513265i	$-0.828436\pi$
$227$	−5870.47	−1.71646	−0.858230	+	0.513265i	$0.828436\pi$
$228$	0	0
$229$	1505.34	0.434393	0.217196	−	0.976128i	$-0.430309\pi$
$229$	1505.34	0.434393	0.217196	+	0.976128i	$0.430309\pi$
$230$	−3009.15	−0.862684
$231$	0	0
$232$	4669.52	1.32142
$233$	713.257	0.200545	0.100273	−	0.994960i	$-0.468028\pi$
$233$	713.257	0.200545	0.100273	+	0.994960i	$0.468028\pi$
$234$	0	0
$235$	−417.402	−0.115865
$236$	−11011.9	−3.03734
$237$	0	0
$238$	0	0
$239$	195.054	0.0527908	0.0263954	−	0.999652i	$-0.491597\pi$
$239$	195.054	0.0527908	0.0263954	+	0.999652i	$0.491597\pi$
$240$	0	0
$241$	968.228	0.258793	0.129396	−	0.991593i	$-0.458696\pi$
$241$	968.228	0.258793	0.129396	+	0.991593i	$0.458696\pi$
$242$	−6502.98	−1.72739
$243$	0	0
$244$	4584.37	1.20280
$245$	0	0
$246$	0	0
$247$	4196.17	1.08095
$248$	−2715.24	−0.695233
$249$	0	0
$250$	−4663.28	−1.17973
$251$	−3482.07	−0.875642	−0.437821	−	0.899062i	$-0.644250\pi$
$251$	−3482.07	−0.875642	−0.437821	+	0.899062i	$0.644250\pi$
$252$	0	0
$253$	836.577	0.207886
$254$	−3303.62	−0.816092
$255$	0	0
$256$	−8275.23	−2.02032
$257$	−5337.90	−1.29560	−0.647800	−	0.761811i	$-0.724311\pi$
$257$	−5337.90	−1.29560	−0.647800	+	0.761811i	$0.724311\pi$
$258$	0	0
$259$	0	0
$260$	2291.74	0.546645
$261$	0	0
$262$	6474.15	1.52662
$263$	−1182.24	−0.277186	−0.138593	−	0.990349i	$-0.544258\pi$
$263$	−1182.24	−0.277186	−0.138593	+	0.990349i	$0.544258\pi$
$264$	0	0
$265$	2852.11	0.661146
$266$	0	0
$267$	0	0
$268$	8229.47	1.87573
$269$	−5038.16	−1.14194	−0.570971	−	0.820971i	$-0.693433\pi$
$269$	−5038.16	−1.14194	−0.570971	+	0.820971i	$0.693433\pi$
$270$	0	0
$271$	3004.05	0.673370	0.336685	−	0.941617i	$-0.390694\pi$
$271$	3004.05	0.673370	0.336685	+	0.941617i	$0.390694\pi$
$272$	9224.61	2.05634
$273$	0	0
$274$	−3991.80	−0.880121
$275$	604.311	0.132514
$276$	0	0
$277$	−322.006	−0.0698465	−0.0349233	−	0.999390i	$-0.511119\pi$
$277$	−322.006	−0.0698465	−0.0349233	+	0.999390i	$0.511119\pi$
$278$	4783.09	1.03191
$279$	0	0
$280$	0	0
$281$	249.602	0.0529894	0.0264947	−	0.999649i	$-0.491565\pi$
$281$	249.602	0.0529894	0.0264947	+	0.999649i	$0.491565\pi$
$282$	0	0
$283$	2539.25	0.533366	0.266683	−	0.963784i	$-0.414072\pi$
$283$	2539.25	0.533366	0.266683	+	0.963784i	$0.414072\pi$
$284$	−6172.61	−1.28971
$285$	0	0
$286$	−936.782	−0.193682
$287$	0	0
$288$	0	0
$289$	5768.53	1.17414
$290$	2064.05	0.417948
$291$	0	0
$292$	15445.9	3.09556
$293$	−8928.79	−1.78029	−0.890146	−	0.455675i	$-0.849398\pi$
$293$	−8928.79	−1.78029	−0.890146	+	0.455675i	$0.849398\pi$
$294$	0	0
$295$	−2578.24	−0.508851
$296$	18637.6	3.65976
$297$	0	0
$298$	−6062.68	−1.17853
$299$	−5111.29	−0.988608
$300$	0	0
$301$	0	0
$302$	13433.9	2.55972
$303$	0	0
$304$	11070.8	2.08867
$305$	1073.35	0.201508
$306$	0	0
$307$	6792.76	1.26281	0.631406	−	0.775452i	$-0.282478\pi$
$307$	6792.76	1.26281	0.631406	+	0.775452i	$0.282478\pi$
$308$	0	0
$309$	0	0
$310$	−1200.20	−0.219893
$311$	699.942	0.127621	0.0638104	−	0.997962i	$-0.479675\pi$
$311$	699.942	0.127621	0.0638104	+	0.997962i	$0.479675\pi$
$312$	0	0
$313$	9620.06	1.73725	0.868623	−	0.495474i	$-0.165005\pi$
$313$	9620.06	1.73725	0.868623	+	0.495474i	$0.165005\pi$
$314$	−7206.78	−1.29523
$315$	0	0
$316$	2763.05	0.491878
$317$	−3863.90	−0.684601	−0.342300	−	0.939591i	$-0.611206\pi$
$317$	−3863.90	−0.684601	−0.342300	+	0.939591i	$0.611206\pi$
$318$	0	0
$319$	−573.828	−0.100715
$320$	−1132.88	−0.197906
$321$	0	0
$322$	0	0
$323$	12819.3	2.20832
$324$	0	0
$325$	−3692.20	−0.630173
$326$	−15472.1	−2.62860
$327$	0	0
$328$	6920.14	1.16494
$329$	0	0
$330$	0	0
$331$	−5030.07	−0.835280	−0.417640	−	0.908613i	$-0.637143\pi$
$331$	−5030.07	−0.835280	−0.417640	+	0.908613i	$0.637143\pi$
$332$	−7131.80	−1.17894
$333$	0	0
$334$	−9528.83	−1.56106
$335$	1926.79	0.314244
$336$	0	0
$337$	−7351.34	−1.18829	−0.594144	−	0.804359i	$-0.702509\pi$
$337$	−7351.34	−1.18829	−0.594144	+	0.804359i	$0.702509\pi$
$338$	−5263.63	−0.847052
$339$	0	0
$340$	7001.28	1.11676
$341$	333.670	0.0529890
$342$	0	0
$343$	0	0
$344$	−11782.3	−1.84668
$345$	0	0
$346$	−14507.9	−2.25419
$347$	−12562.2	−1.94344	−0.971719	−	0.236141i	$-0.924117\pi$
$347$	−12562.2	−1.94344	−0.971719	+	0.236141i	$0.924117\pi$
$348$	0	0
$349$	1050.54	0.161129	0.0805647	−	0.996749i	$-0.474328\pi$
$349$	1050.54	0.161129	0.0805647	+	0.996749i	$0.474328\pi$
$350$	0	0
$351$	0	0
$352$	−475.626	−0.0720197
$353$	7770.20	1.17157	0.585787	−	0.810465i	$-0.300785\pi$
$353$	7770.20	1.17157	0.585787	+	0.810465i	$0.300785\pi$
$354$	0	0
$355$	−1445.21	−0.216067
$356$	17341.0	2.58166
$357$	0	0
$358$	6879.74	1.01566
$359$	−5712.35	−0.839794	−0.419897	−	0.907572i	$-0.637934\pi$
$359$	−5712.35	−0.839794	−0.419897	+	0.907572i	$0.637934\pi$
$360$	0	0
$361$	8526.00	1.24304
$362$	−22795.2	−3.30964
$363$	0	0
$364$	0	0
$365$	3616.40	0.518605
$366$	0	0
$367$	6489.45	0.923016	0.461508	−	0.887136i	$-0.347309\pi$
$367$	6489.45	0.923016	0.461508	+	0.887136i	$0.347309\pi$
$368$	−13485.2	−1.91023
$369$	0	0
$370$	8238.30	1.15754
$371$	0	0
$372$	0	0
$373$	−6133.45	−0.851415	−0.425707	−	0.904861i	$-0.639975\pi$
$373$	−6133.45	−0.851415	−0.425707	+	0.904861i	$0.639975\pi$
$374$	−2861.88	−0.395680
$375$	0	0
$376$	−4722.39	−0.647709
$377$	3505.96	0.478955
$378$	0	0
$379$	−3611.83	−0.489518	−0.244759	−	0.969584i	$-0.578709\pi$
$379$	−3611.83	−0.489518	−0.244759	+	0.969584i	$0.578709\pi$
$380$	8402.52	1.13432
$381$	0	0
$382$	−23708.3	−3.17545
$383$	−800.143	−0.106750	−0.0533752	−	0.998575i	$-0.516998\pi$
$383$	−800.143	−0.106750	−0.0533752	+	0.998575i	$0.516998\pi$
$384$	0	0
$385$	0	0
$386$	12138.4	1.60059
$387$	0	0
$388$	−5898.75	−0.771813
$389$	10775.2	1.40443	0.702214	−	0.711966i	$-0.252195\pi$
$389$	10775.2	1.40443	0.702214	+	0.711966i	$0.252195\pi$
$390$	0	0
$391$	−15615.1	−2.01966
$392$	0	0
$393$	0	0
$394$	21874.9	2.79706
$395$	646.921	0.0824053
$396$	0	0
$397$	−7359.38	−0.930369	−0.465185	−	0.885214i	$-0.654012\pi$
$397$	−7359.38	−0.930369	−0.465185	+	0.885214i	$0.654012\pi$
$398$	1840.05	0.231742
$399$	0	0
$400$	−9741.19	−1.21765
$401$	−13789.0	−1.71718	−0.858589	−	0.512664i	$-0.828659\pi$
$401$	−13789.0	−1.71718	−0.858589	+	0.512664i	$0.828659\pi$
$402$	0	0
$403$	−2038.65	−0.251991
$404$	24285.5	2.99071
$405$	0	0
$406$	0	0
$407$	−2290.34	−0.278939
$408$	0	0
$409$	7668.11	0.927050	0.463525	−	0.886084i	$-0.346584\pi$
$409$	7668.11	0.927050	0.463525	+	0.886084i	$0.346584\pi$
$410$	3058.88	0.368456
$411$	0	0
$412$	−14517.9	−1.73603
$413$	0	0
$414$	0	0
$415$	−1669.79	−0.197510
$416$	2905.96	0.342492
$417$	0	0
$418$	−3434.66	−0.401901
$419$	−10488.8	−1.22293	−0.611467	−	0.791270i	$-0.709420\pi$
$419$	−10488.8	−1.22293	−0.611467	+	0.791270i	$0.709420\pi$
$420$	0	0
$421$	5841.77	0.676272	0.338136	−	0.941097i	$-0.390204\pi$
$421$	5841.77	0.676272	0.338136	+	0.941097i	$0.390204\pi$
$422$	11123.0	1.28308
$423$	0	0
$424$	32268.1	3.69594
$425$	−11279.7	−1.28740
$426$	0	0
$427$	0	0
$428$	−12425.8	−1.40333
$429$	0	0
$430$	−5208.06	−0.584081
$431$	13825.6	1.54515	0.772573	−	0.634926i	$-0.218969\pi$
$431$	13825.6	1.54515	0.772573	+	0.634926i	$0.218969\pi$
$432$	0	0
$433$	9226.76	1.02404	0.512021	−	0.858973i	$-0.328897\pi$
$433$	9226.76	1.02404	0.512021	+	0.858973i	$0.328897\pi$
$434$	0	0
$435$	0	0
$436$	−30836.6	−3.38716
$437$	−18740.3	−2.05141
$438$	0	0
$439$	−10674.4	−1.16051	−0.580255	−	0.814435i	$-0.697047\pi$
$439$	−10674.4	−1.16051	−0.580255	+	0.814435i	$0.697047\pi$
$440$	−993.599	−0.107654
$441$	0	0
$442$	17485.4	1.88167
$443$	−7788.79	−0.835342	−0.417671	−	0.908598i	$-0.637153\pi$
$443$	−7788.79	−0.835342	−0.417671	+	0.908598i	$0.637153\pi$
$444$	0	0
$445$	4060.11	0.432511
$446$	18383.2	1.95172
$447$	0	0
$448$	0	0
$449$	−6445.65	−0.677481	−0.338741	−	0.940880i	$-0.610001\pi$
$449$	−6445.65	−0.677481	−0.338741	+	0.940880i	$0.610001\pi$
$450$	0	0
$451$	−850.402	−0.0887891
$452$	8034.45	0.836081
$453$	0	0
$454$	−29358.1	−3.03490
$455$	0	0
$456$	0	0
$457$	−8375.21	−0.857278	−0.428639	−	0.903476i	$-0.641007\pi$
$457$	−8375.21	−0.857278	−0.428639	+	0.903476i	$0.641007\pi$
$458$	7528.19	0.768055
$459$	0	0
$460$	−10235.0	−1.03741
$461$	−14339.5	−1.44871	−0.724356	−	0.689426i	$-0.757863\pi$
$461$	−14339.5	−1.44871	−0.724356	+	0.689426i	$0.757863\pi$
$462$	0	0
$463$	285.260	0.0286332	0.0143166	−	0.999898i	$-0.495443\pi$
$463$	285.260	0.0286332	0.0143166	+	0.999898i	$0.495443\pi$
$464$	9249.83	0.925458
$465$	0	0
$466$	3566.99	0.354587
$467$	−15511.4	−1.53700	−0.768501	−	0.639849i	$-0.778997\pi$
$467$	−15511.4	−1.53700	−0.768501	+	0.639849i	$0.778997\pi$
$468$	0	0
$469$	0	0
$470$	−2087.42	−0.204862
$471$	0	0
$472$	−29169.7	−2.84458
$473$	1447.90	0.140749
$474$	0	0
$475$	−13537.2	−1.30764
$476$	0	0
$477$	0	0
$478$	975.461	0.0933400
$479$	16229.9	1.54815	0.774074	−	0.633095i	$-0.218216\pi$
$479$	16229.9	1.54815	0.774074	+	0.633095i	$0.218216\pi$
$480$	0	0
$481$	13993.5	1.32650
$482$	4842.09	0.457575
$483$	0	0
$484$	−22118.5	−2.07725
$485$	−1381.09	−0.129303
$486$	0	0
$487$	17014.8	1.58319	0.791596	−	0.611044i	$-0.209250\pi$
$487$	17014.8	1.58319	0.791596	+	0.611044i	$0.209250\pi$
$488$	12143.7	1.12647
$489$	0	0
$490$	0	0
$491$	−9821.77	−0.902750	−0.451375	−	0.892334i	$-0.649066\pi$
$491$	−9821.77	−0.902750	−0.451375	+	0.892334i	$0.649066\pi$
$492$	0	0
$493$	10710.7	0.978473
$494$	20985.0	1.91125
$495$	0	0
$496$	−5378.59	−0.486907
$497$	0	0
$498$	0	0
$499$	−8967.25	−0.804468	−0.402234	−	0.915537i	$-0.631766\pi$
$499$	−8967.25	−0.804468	−0.402234	+	0.915537i	$0.631766\pi$
$500$	−15861.2	−1.41867
$501$	0	0
$502$	−17413.7	−1.54823
$503$	−16119.0	−1.42884	−0.714422	−	0.699715i	$-0.753310\pi$
$503$	−16119.0	−1.42884	−0.714422	+	0.699715i	$0.753310\pi$
$504$	0	0
$505$	5686.03	0.501040
$506$	4183.71	0.367566
$507$	0	0
$508$	−11236.6	−0.981382
$509$	10807.7	0.941141	0.470570	−	0.882362i	$-0.344048\pi$
$509$	10807.7	0.941141	0.470570	+	0.882362i	$0.344048\pi$
$510$	0	0
$511$	0	0
$512$	−24506.1	−2.11529
$513$	0	0
$514$	−26694.7	−2.29077
$515$	−3399.12	−0.290841
$516$	0	0
$517$	580.325	0.0493669
$518$	0	0
$519$	0	0
$520$	6070.66	0.511954
$521$	−13837.5	−1.16359	−0.581797	−	0.813334i	$-0.697650\pi$
$521$	−13837.5	−1.16359	−0.581797	+	0.813334i	$0.697650\pi$
$522$	0	0
$523$	20126.9	1.68276	0.841382	−	0.540440i	$-0.181742\pi$
$523$	20126.9	1.68276	0.841382	+	0.540440i	$0.181742\pi$
$524$	22020.5	1.83582
$525$	0	0
$526$	−5912.35	−0.490097
$527$	−6228.09	−0.514800
$528$	0	0
$529$	10660.2	0.876160
$530$	14263.3	1.16898
$531$	0	0
$532$	0	0
$533$	5195.76	0.422239
$534$	0	0
$535$	−2909.29	−0.235102
$536$	21799.3	1.75669
$537$	0	0
$538$	−25195.7	−2.01908
$539$	0	0
$540$	0	0
$541$	−3879.16	−0.308277	−0.154139	−	0.988049i	$-0.549260\pi$
$541$	−3879.16	−0.308277	−0.154139	+	0.988049i	$0.549260\pi$
$542$	15023.2	1.19059
$543$	0	0
$544$	8877.75	0.699688
$545$	−7219.86	−0.567459
$546$	0	0
$547$	−14131.4	−1.10460	−0.552298	−	0.833647i	$-0.686249\pi$
$547$	−14131.4	−1.10460	−0.552298	+	0.833647i	$0.686249\pi$
$548$	−13577.3	−1.05838
$549$	0	0
$550$	3022.15	0.234300
$551$	12854.4	0.993857
$552$	0	0
$553$	0	0
$554$	−1610.35	−0.123497
$555$	0	0
$556$	16268.7	1.24091
$557$	−2501.36	−0.190280	−0.0951402	−	0.995464i	$-0.530330\pi$
$557$	−2501.36	−0.190280	−0.0951402	+	0.995464i	$0.530330\pi$
$558$	0	0
$559$	−8846.33	−0.669338
$560$	0	0
$561$	0	0
$562$	1248.25	0.0936912
$563$	23407.2	1.75221	0.876106	−	0.482118i	$-0.160132\pi$
$563$	23407.2	1.75221	0.876106	+	0.482118i	$0.160132\pi$
$564$	0	0
$565$	1881.13	0.140070
$566$	12698.7	0.943051
$567$	0	0
$568$	−16350.8	−1.20786
$569$	−18623.5	−1.37213	−0.686063	−	0.727542i	$-0.740663\pi$
$569$	−18623.5	−1.37213	−0.686063	+	0.727542i	$0.740663\pi$
$570$	0	0
$571$	−9576.20	−0.701841	−0.350921	−	0.936405i	$-0.614131\pi$
$571$	−9576.20	−0.701841	−0.350921	+	0.936405i	$0.614131\pi$
$572$	−3186.27	−0.232910
$573$	0	0
$574$	0	0
$575$	16489.5	1.19593
$576$	0	0
$577$	−8469.72	−0.611090	−0.305545	−	0.952178i	$-0.598839\pi$
$577$	−8469.72	−0.611090	−0.305545	+	0.952178i	$0.598839\pi$
$578$	28848.3	2.07601
$579$	0	0
$580$	7020.43	0.502599
$581$	0	0
$582$	0	0
$583$	−3965.37	−0.281696
$584$	40915.2	2.89911
$585$	0	0
$586$	−44652.7	−3.14776
$587$	−445.290	−0.0313102	−0.0156551	−	0.999877i	$-0.504983\pi$
$587$	−445.290	−0.0313102	−0.0156551	+	0.999877i	$0.504983\pi$
$588$	0	0
$589$	−7474.58	−0.522894
$590$	−12893.7	−0.899707
$591$	0	0
$592$	36919.2	2.56312
$593$	−5518.68	−0.382167	−0.191083	−	0.981574i	$-0.561200\pi$
$593$	−5518.68	−0.382167	−0.191083	+	0.981574i	$0.561200\pi$
$594$	0	0
$595$	0	0
$596$	−20620.9	−1.41723
$597$	0	0
$598$	−25561.5	−1.74797
$599$	24007.1	1.63757	0.818784	−	0.574101i	$-0.194649\pi$
$599$	24007.1	1.63757	0.818784	+	0.574101i	$0.194649\pi$
$600$	0	0
$601$	19449.9	1.32009	0.660047	−	0.751224i	$-0.270536\pi$
$601$	19449.9	1.32009	0.660047	+	0.751224i	$0.270536\pi$
$602$	0	0
$603$	0	0
$604$	45692.7	3.07816
$605$	−5178.68	−0.348005
$606$	0	0
$607$	17741.4	1.18633	0.593165	−	0.805081i	$-0.297878\pi$
$607$	17741.4	1.18633	0.593165	+	0.805081i	$0.297878\pi$
$608$	10654.5	0.710689
$609$	0	0
$610$	5367.82	0.356289
$611$	−3545.65	−0.234766
$612$	0	0
$613$	−5438.44	−0.358331	−0.179165	−	0.983819i	$-0.557340\pi$
$613$	−5438.44	−0.358331	−0.179165	+	0.983819i	$0.557340\pi$
$614$	33970.5	2.23280
$615$	0	0
$616$	0	0
$617$	3557.73	0.232138	0.116069	−	0.993241i	$-0.462971\pi$
$617$	3557.73	0.232138	0.116069	+	0.993241i	$0.462971\pi$
$618$	0	0
$619$	−17458.2	−1.13361	−0.566806	−	0.823851i	$-0.691821\pi$
$619$	−17458.2	−1.13361	−0.566806	+	0.823851i	$0.691821\pi$
$620$	−4082.24	−0.264430
$621$	0	0
$622$	3500.40	0.225648
$623$	0	0
$624$	0	0
$625$	9928.78	0.635442
$626$	48109.7	3.07165
$627$	0	0
$628$	−24512.4	−1.55756
$629$	42750.1	2.70995
$630$	0	0
$631$	10359.5	0.653573	0.326787	−	0.945098i	$-0.394034\pi$
$631$	10359.5	0.653573	0.326787	+	0.945098i	$0.394034\pi$
$632$	7319.12	0.460663
$633$	0	0
$634$	−19323.3	−1.21045
$635$	−2630.85	−0.164413
$636$	0	0
$637$	0	0
$638$	−2869.70	−0.178076
$639$	0	0
$640$	−8402.28	−0.518952
$641$	−1433.28	−0.0883172	−0.0441586	−	0.999025i	$-0.514061\pi$
$641$	−1433.28	−0.0883172	−0.0441586	+	0.999025i	$0.514061\pi$
$642$	0	0
$643$	25175.8	1.54407	0.772034	−	0.635581i	$-0.219239\pi$
$643$	25175.8	1.54407	0.772034	+	0.635581i	$0.219239\pi$
$644$	0	0
$645$	0	0
$646$	64109.2	3.90456
$647$	−7551.11	−0.458832	−0.229416	−	0.973328i	$-0.573682\pi$
$647$	−7551.11	−0.458832	−0.229416	+	0.973328i	$0.573682\pi$
$648$	0	0
$649$	3584.61	0.216808
$650$	−18464.6	−1.11422
$651$	0	0
$652$	−52625.3	−3.16099
$653$	2302.17	0.137965	0.0689824	−	0.997618i	$-0.478025\pi$
$653$	2302.17	0.137965	0.0689824	+	0.997618i	$0.478025\pi$
$654$	0	0
$655$	5155.72	0.307558
$656$	13708.1	0.815869
$657$	0	0
$658$	0	0
$659$	10145.8	0.599733	0.299866	−	0.953981i	$-0.403058\pi$
$659$	10145.8	0.599733	0.299866	+	0.953981i	$0.403058\pi$
$660$	0	0
$661$	18080.7	1.06393	0.531965	−	0.846766i	$-0.321454\pi$
$661$	18080.7	1.06393	0.531965	+	0.846766i	$0.321454\pi$
$662$	−25155.3	−1.47687
$663$	0	0
$664$	−18891.7	−1.10412
$665$	0	0
$666$	0	0
$667$	−15657.7	−0.908951
$668$	−32410.3	−1.87724
$669$	0	0
$670$	9635.85	0.555620
$671$	−1492.31	−0.0858572
$672$	0	0
$673$	−1488.08	−0.0852324	−0.0426162	−	0.999092i	$-0.513569\pi$
$673$	−1488.08	−0.0852324	−0.0426162	+	0.999092i	$0.513569\pi$
$674$	−36763.9	−2.10103
$675$	0	0
$676$	−17903.1	−1.01861
$677$	19326.7	1.09717	0.548587	−	0.836094i	$-0.315166\pi$
$677$	19326.7	1.09717	0.548587	+	0.836094i	$0.315166\pi$
$678$	0	0
$679$	0	0
$680$	18545.9	1.04589
$681$	0	0
$682$	1668.68	0.0936905
$683$	26176.2	1.46648	0.733238	−	0.679972i	$-0.238008\pi$
$683$	26176.2	1.46648	0.733238	+	0.679972i	$0.238008\pi$
$684$	0	0
$685$	−3178.89	−0.177312
$686$	0	0
$687$	0	0
$688$	−23339.4	−1.29332
$689$	24227.5	1.33961
$690$	0	0
$691$	25360.2	1.39616	0.698082	−	0.716018i	$-0.254037\pi$
$691$	25360.2	1.39616	0.698082	+	0.716018i	$0.254037\pi$
$692$	−49345.6	−2.71075
$693$	0	0
$694$	−62823.2	−3.43622
$695$	3809.04	0.207892
$696$	0	0
$697$	15873.1	0.862607
$698$	5253.73	0.284895
$699$	0	0
$700$	0	0
$701$	6628.89	0.357161	0.178580	−	0.983925i	$-0.442850\pi$
$701$	6628.89	0.357161	0.178580	+	0.983925i	$0.442850\pi$
$702$	0	0
$703$	51306.2	2.75256
$704$	1575.08	0.0843223
$705$	0	0
$706$	38858.6	2.07148
$707$	0	0
$708$	0	0
$709$	3173.90	0.168122	0.0840608	−	0.996461i	$-0.473211\pi$
$709$	3173.90	0.168122	0.0840608	+	0.996461i	$0.473211\pi$
$710$	−7227.47	−0.382031
$711$	0	0
$712$	45935.2	2.41783
$713$	9104.68	0.478223
$714$	0	0
$715$	−746.011	−0.0390199
$716$	23400.0	1.22137
$717$	0	0
$718$	−28567.3	−1.48485
$719$	−2683.55	−0.139193	−0.0695963	−	0.997575i	$-0.522171\pi$
$719$	−2683.55	−0.139193	−0.0695963	+	0.997575i	$0.522171\pi$
$720$	0	0
$721$	0	0
$722$	42638.3	2.19783
$723$	0	0
$724$	−77533.1	−3.97997
$725$	−11310.5	−0.579397
$726$	0	0
$727$	6401.88	0.326592	0.163296	−	0.986577i	$-0.447787\pi$
$727$	6401.88	0.326592	0.163296	+	0.986577i	$0.447787\pi$
$728$	0	0
$729$	0	0
$730$	18085.5	0.916953
$731$	−27025.6	−1.36741
$732$	0	0
$733$	−5888.94	−0.296744	−0.148372	−	0.988932i	$-0.547403\pi$
$733$	−5888.94	−0.296744	−0.148372	+	0.988932i	$0.547403\pi$
$734$	32453.6	1.63200
$735$	0	0
$736$	−12978.1	−0.649974
$737$	−2678.87	−0.133891
$738$	0	0
$739$	18593.3	0.925530	0.462765	−	0.886481i	$-0.346857\pi$
$739$	18593.3	0.925530	0.462765	+	0.886481i	$0.346857\pi$
$740$	28020.9	1.39198
$741$	0	0
$742$	0	0
$743$	−27522.3	−1.35894	−0.679471	−	0.733702i	$-0.737791\pi$
$743$	−27522.3	−1.35894	−0.679471	+	0.733702i	$0.737791\pi$
$744$	0	0
$745$	−4828.05	−0.237431
$746$	−30673.2	−1.50540
$747$	0	0
$748$	−9734.09	−0.475820
$749$	0	0
$750$	0	0
$751$	16609.7	0.807053	0.403526	−	0.914968i	$-0.367784\pi$
$751$	16609.7	0.807053	0.403526	+	0.914968i	$0.367784\pi$
$752$	−9354.56	−0.453624
$753$	0	0
$754$	17533.2	0.846847
$755$	10698.2	0.515691
$756$	0	0
$757$	−23386.2	−1.12284	−0.561418	−	0.827532i	$-0.689744\pi$
$757$	−23386.2	−1.12284	−0.561418	+	0.827532i	$0.689744\pi$
$758$	−18062.7	−0.865522
$759$	0	0
$760$	22257.7	1.06233
$761$	−20245.9	−0.964409	−0.482204	−	0.876059i	$-0.660164\pi$
$761$	−20245.9	−0.964409	−0.482204	+	0.876059i	$0.660164\pi$
$762$	0	0
$763$	0	0
$764$	−80638.8	−3.81860
$765$	0	0
$766$	−4001.50	−0.188747
$767$	−21901.1	−1.03103
$768$	0	0
$769$	8846.54	0.414843	0.207422	−	0.978252i	$-0.433493\pi$
$769$	8846.54	0.414843	0.207422	+	0.978252i	$0.433493\pi$
$770$	0	0
$771$	0	0
$772$	41286.3	1.92478
$773$	9788.82	0.455471	0.227736	−	0.973723i	$-0.426868\pi$
$773$	9788.82	0.455471	0.227736	+	0.973723i	$0.426868\pi$
$774$	0	0
$775$	6576.86	0.304836
$776$	−15625.4	−0.722833
$777$	0	0
$778$	53886.4	2.48319
$779$	19049.9	0.876169
$780$	0	0
$781$	2009.32	0.0920603
$782$	−78090.6	−3.57099
$783$	0	0
$784$	0	0
$785$	−5739.16	−0.260942
$786$	0	0
$787$	−6582.93	−0.298165	−0.149083	−	0.988825i	$-0.547632\pi$
$787$	−6582.93	−0.298165	−0.149083	+	0.988825i	$0.547632\pi$
$788$	74402.8	3.36357
$789$	0	0
$790$	3235.24	0.145702
$791$	0	0
$792$	0	0
$793$	9117.69	0.408296
$794$	−36804.1	−1.64500
$795$	0	0
$796$	6258.55	0.278679
$797$	−37369.9	−1.66087	−0.830434	−	0.557117i	$-0.811907\pi$
$797$	−37369.9	−1.66087	−0.830434	+	0.557117i	$0.811907\pi$
$798$	0	0
$799$	−10832.0	−0.479611
$800$	−9374.91	−0.414316
$801$	0	0
$802$	−68958.4	−3.03617
$803$	−5027.99	−0.220964
$804$	0	0
$805$	0	0
$806$	−10195.2	−0.445548
$807$	0	0
$808$	64330.5	2.80092
$809$	28089.7	1.22074	0.610371	−	0.792116i	$-0.291020\pi$
$809$	28089.7	1.22074	0.610371	+	0.792116i	$0.291020\pi$
$810$	0	0
$811$	−9732.34	−0.421392	−0.210696	−	0.977552i	$-0.567573\pi$
$811$	−9732.34	−0.421392	−0.210696	+	0.977552i	$0.567573\pi$
$812$	0	0
$813$	0	0
$814$	−11453.9	−0.493195
$815$	−12321.3	−0.529567
$816$	0	0
$817$	−32434.5	−1.38891
$818$	38348.0	1.63913
$819$	0	0
$820$	10404.1	0.443083
$821$	31157.5	1.32449	0.662244	−	0.749288i	$-0.269604\pi$
$821$	31157.5	1.32449	0.662244	+	0.749288i	$0.269604\pi$
$822$	0	0
$823$	36684.6	1.55376	0.776880	−	0.629649i	$-0.216801\pi$
$823$	36684.6	1.55376	0.776880	+	0.629649i	$0.216801\pi$
$824$	−38456.9	−1.62586
$825$	0	0
$826$	0	0
$827$	−22589.9	−0.949853	−0.474927	−	0.880025i	$-0.657525\pi$
$827$	−22589.9	−0.949853	−0.474927	+	0.880025i	$0.657525\pi$
$828$	0	0
$829$	−9519.37	−0.398820	−0.199410	−	0.979916i	$-0.563902\pi$
$829$	−9519.37	−0.398820	−0.199410	+	0.979916i	$0.563902\pi$
$830$	−8350.59	−0.349221
$831$	0	0
$832$	−9623.35	−0.400997
$833$	0	0
$834$	0	0
$835$	−7588.33	−0.314497
$836$	−11682.3	−0.483301
$837$	0	0
$838$	−52454.1	−2.16229
$839$	−38805.6	−1.59680	−0.798402	−	0.602124i	$-0.794321\pi$
$839$	−38805.6	−1.59680	−0.798402	+	0.602124i	$0.794321\pi$
$840$	0	0
$841$	−13649.0	−0.559637
$842$	29214.6	1.19573
$843$	0	0
$844$	37832.6	1.54295
$845$	−4191.72	−0.170650
$846$	0	0
$847$	0	0
$848$	63919.8	2.58846
$849$	0	0
$850$	−56409.6	−2.27627
$851$	−62495.3	−2.51741
$852$	0	0
$853$	203.914	0.00818507	0.00409254	−	0.999992i	$-0.498697\pi$
$853$	203.914	0.00818507	0.00409254	+	0.999992i	$0.498697\pi$
$854$	0	0
$855$	0	0
$856$	−32915.1	−1.31427
$857$	−5606.57	−0.223473	−0.111737	−	0.993738i	$-0.535641\pi$
$857$	−5606.57	−0.223473	−0.111737	+	0.993738i	$0.535641\pi$
$858$	0	0
$859$	−13425.5	−0.533263	−0.266632	−	0.963799i	$-0.585911\pi$
$859$	−13425.5	−0.533263	−0.266632	+	0.963799i	$0.585911\pi$
$860$	−17714.1	−0.702380
$861$	0	0
$862$	69141.8	2.73199
$863$	−10490.2	−0.413778	−0.206889	−	0.978364i	$-0.566334\pi$
$863$	−10490.2	−0.413778	−0.206889	+	0.978364i	$0.566334\pi$
$864$	0	0
$865$	−11553.4	−0.454137
$866$	46142.9	1.81062
$867$	0	0
$868$	0	0
$869$	−899.432	−0.0351106
$870$	0	0
$871$	16367.3	0.636722
$872$	−81684.0	−3.17221
$873$	0	0
$874$	−93719.6	−3.62713
$875$	0	0
$876$	0	0
$877$	−20405.9	−0.785699	−0.392850	−	0.919603i	$-0.628511\pi$
$877$	−20405.9	−0.785699	−0.392850	+	0.919603i	$0.628511\pi$
$878$	−53382.7	−2.05191
$879$	0	0
$880$	−1968.21	−0.0753960
$881$	29977.0	1.14637	0.573184	−	0.819427i	$-0.305708\pi$
$881$	29977.0	1.14637	0.573184	+	0.819427i	$0.305708\pi$
$882$	0	0
$883$	18717.7	0.713365	0.356682	−	0.934226i	$-0.383908\pi$
$883$	18717.7	0.713365	0.356682	+	0.934226i	$0.383908\pi$
$884$	59473.0	2.26278
$885$	0	0
$886$	−38951.6	−1.47698
$887$	28343.6	1.07293	0.536463	−	0.843924i	$-0.319760\pi$
$887$	28343.6	1.07293	0.536463	+	0.843924i	$0.319760\pi$
$888$	0	0
$889$	0	0
$890$	20304.5	0.764729
$891$	0	0
$892$	62526.6	2.34702
$893$	−12999.9	−0.487151
$894$	0	0
$895$	5478.71	0.204618
$896$	0	0
$897$	0	0
$898$	−32234.6	−1.19786
$899$	−6245.11	−0.231687
$900$	0	0
$901$	74015.2	2.73674
$902$	−4252.85	−0.156989
$903$	0	0
$904$	21282.7	0.783023
$905$	−18153.1	−0.666772
$906$	0	0
$907$	29101.8	1.06539	0.532695	−	0.846307i	$-0.321179\pi$
$907$	29101.8	1.06539	0.532695	+	0.846307i	$0.321179\pi$
$908$	−99855.4	−3.64958
$909$	0	0
$910$	0	0
$911$	7099.57	0.258199	0.129099	−	0.991632i	$-0.458791\pi$
$911$	7099.57	0.258199	0.129099	+	0.991632i	$0.458791\pi$
$912$	0	0
$913$	2321.56	0.0841538
$914$	−41884.3	−1.51576
$915$	0	0
$916$	25605.6	0.923616
$917$	0	0
$918$	0	0
$919$	−17499.6	−0.628138	−0.314069	−	0.949400i	$-0.601692\pi$
$919$	−17499.6	−0.628138	−0.314069	+	0.949400i	$0.601692\pi$
$920$	−27111.8	−0.971575
$921$	0	0
$922$	−71711.5	−2.56149
$923$	−12276.5	−0.437795
$924$	0	0
$925$	−45144.2	−1.60468
$926$	1426.58	0.0506267
$927$	0	0
$928$	8902.02	0.314896
$929$	638.171	0.0225379	0.0112690	−	0.999937i	$-0.496413\pi$
$929$	638.171	0.0225379	0.0112690	+	0.999937i	$0.496413\pi$
$930$	0	0
$931$	0	0
$932$	12132.4	0.426404
$933$	0	0
$934$	−77571.9	−2.71759
$935$	−2279.07	−0.0797151
$936$	0	0
$937$	−3768.72	−0.131397	−0.0656984	−	0.997840i	$-0.520928\pi$
$937$	−3768.72	−0.131397	−0.0656984	+	0.997840i	$0.520928\pi$
$938$	0	0
$939$	0	0
$940$	−7099.91	−0.246355
$941$	14086.8	0.488009	0.244004	−	0.969774i	$-0.421539\pi$
$941$	14086.8	0.488009	0.244004	+	0.969774i	$0.421539\pi$
$942$	0	0
$943$	−23204.5	−0.801317
$944$	−57782.1	−1.99221
$945$	0	0
$946$	7240.92	0.248861
$947$	40861.3	1.40213	0.701064	−	0.713099i	$-0.252709\pi$
$947$	40861.3	1.40213	0.701064	+	0.713099i	$0.252709\pi$
$948$	0	0
$949$	30719.8	1.05080
$950$	−67699.4	−2.31206
$951$	0	0
$952$	0	0
$953$	34318.7	1.16652	0.583259	−	0.812286i	$-0.301777\pi$
$953$	34318.7	1.16652	0.583259	+	0.812286i	$0.301777\pi$
$954$	0	0
$955$	−18880.2	−0.639737
$956$	3317.83	0.112245
$957$	0	0
$958$	81165.4	2.73730
$959$	0	0
$960$	0	0
$961$	−26159.6	−0.878104
$962$	69981.0	2.34540
$963$	0	0
$964$	16469.3	0.550251
$965$	9666.49	0.322461
$966$	0	0
$967$	56456.3	1.87747	0.938735	−	0.344639i	$-0.111999\pi$
$967$	56456.3	1.87747	0.938735	+	0.344639i	$0.111999\pi$
$968$	−58590.5	−1.94542
$969$	0	0
$970$	−6906.81	−0.228623
$971$	−4430.47	−0.146427	−0.0732134	−	0.997316i	$-0.523325\pi$
$971$	−4430.47	−0.146427	−0.0732134	+	0.997316i	$0.523325\pi$
$972$	0	0
$973$	0	0
$974$	85090.7	2.79926
$975$	0	0
$976$	24055.4	0.788928
$977$	−44537.3	−1.45842	−0.729209	−	0.684291i	$-0.760112\pi$
$977$	−44537.3	−1.45842	−0.729209	+	0.684291i	$0.760112\pi$
$978$	0	0
$979$	−5644.89	−0.184281
$980$	0	0
$981$	0	0
$982$	−49118.4	−1.59616
$983$	18190.6	0.590225	0.295112	−	0.955463i	$-0.404643\pi$
$983$	18190.6	0.590225	0.295112	+	0.955463i	$0.404643\pi$
$984$	0	0
$985$	17420.2	0.563505
$986$	53564.2	1.73005
$987$	0	0
$988$	71376.0	2.29835
$989$	39508.1	1.27026
$990$	0	0
$991$	−5077.83	−0.162768	−0.0813838	−	0.996683i	$-0.525934\pi$
$991$	−5077.83	−0.162768	−0.0813838	+	0.996683i	$0.525934\pi$
$992$	−5176.35	−0.165675
$993$	0	0
$994$	0	0
$995$	1465.33	0.0466877
$996$	0	0
$997$	54583.4	1.73387	0.866937	−	0.498418i	$-0.166085\pi$
$997$	54583.4	1.73387	0.866937	+	0.498418i	$0.166085\pi$
$998$	−44845.1	−1.42239
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bm.1.8	yes	8
3.2	odd	2	inner	1323.4.a.bm.1.1	yes	8
7.6	odd	2		1323.4.a.bl.1.8	yes	8
21.20	even	2		1323.4.a.bl.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.4.a.bl.1.1	✓	8	21.20	even	2
1323.4.a.bl.1.8	yes	8	7.6	odd	2
1323.4.a.bm.1.1	yes	8	3.2	odd	2	inner
1323.4.a.bm.1.8	yes	8	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)

\(f(q)\)	\(=\)	\(q+5.00098 q^{2} +17.0098 q^{4} +3.98256 q^{5} +45.0578 q^{8} +O(q^{10})\)	\(q+5.00098 q^{2} +17.0098 q^{4} +3.98256 q^{5} +45.0578 q^{8} +19.9167 q^{10} -5.53706 q^{11} +33.8302 q^{13} +89.2547 q^{16} +103.352 q^{17} +124.036 q^{19} +67.7424 q^{20} -27.6907 q^{22} -151.087 q^{23} -109.139 q^{25} +169.184 q^{26} +103.634 q^{29} -60.2612 q^{31} +85.8986 q^{32} +516.859 q^{34} +413.638 q^{37} +620.303 q^{38} +179.445 q^{40} +153.584 q^{41} -261.492 q^{43} -94.1843 q^{44} -755.582 q^{46} -104.807 q^{47} -545.803 q^{50} +575.444 q^{52} +716.150 q^{53} -22.0517 q^{55} +518.272 q^{58} -647.384 q^{59} +269.514 q^{61} -301.365 q^{62} -284.461 q^{64} +134.731 q^{65} +483.808 q^{67} +1757.99 q^{68} -362.885 q^{71} +908.060 q^{73} +2068.60 q^{74} +2109.83 q^{76} +162.439 q^{79} +355.462 q^{80} +768.069 q^{82} -419.276 q^{83} +411.603 q^{85} -1307.72 q^{86} -249.488 q^{88} +1019.47 q^{89} -2569.96 q^{92} -524.140 q^{94} +493.981 q^{95} -346.785 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 44 q^{4}+O(q^{10}) \) 8 * q + 44 * q^4	\( 8 q + 44 q^{4} + 132 q^{10} + 336 q^{13} + 204 q^{16} - 288 q^{19} + 484 q^{22} + 152 q^{25} + 120 q^{31} + 1008 q^{34} + 592 q^{37} + 1620 q^{40} - 1872 q^{43} - 1644 q^{46} + 2400 q^{52} + 1344 q^{55} - 1200 q^{58} + 2400 q^{61} - 1388 q^{64} + 1824 q^{73} + 2844 q^{76} + 2368 q^{79} + 2436 q^{82} + 3512 q^{85} + 3780 q^{88} + 4368 q^{94} + 5712 q^{97}+O(q^{100}) \) 8 * q + 44 * q^4 + 132 * q^10 + 336 * q^13 + 204 * q^16 - 288 * q^19 + 484 * q^22 + 152 * q^25 + 120 * q^31 + 1008 * q^34 + 592 * q^37 + 1620 * q^40 - 1872 * q^43 - 1644 * q^46 + 2400 * q^52 + 1344 * q^55 - 1200 * q^58 + 2400 * q^61 - 1388 * q^64 + 1824 * q^73 + 2844 * q^76 + 2368 * q^79 + 2436 * q^82 + 3512 * q^85 + 3780 * q^88 + 4368 * q^94 + 5712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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