LMFDB - Embedded newform 1323.4.a.ba.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:	$4$
Coefficient field:	$\Q(\sqrt{5}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 4 $ x^4 - 9*x^2 + 4
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			$0.684742$ 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)

$f(q)$	$=$	$q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} +O(q^{10})$	$q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} -87.7802 q^{10} -30.3531 q^{11} +89.5603 q^{13} +132.967 q^{16} -13.4164 q^{17} +5.37355 q^{19} -316.496 q^{20} -156.527 q^{22} +167.449 q^{23} +164.747 q^{25} +461.852 q^{26} -135.085 q^{29} +18.9339 q^{31} +248.664 q^{32} -69.1868 q^{34} +402.560 q^{37} +27.7107 q^{38} -929.889 q^{40} +434.345 q^{41} +53.1868 q^{43} -564.367 q^{44} +863.516 q^{46} +155.218 q^{47} +849.581 q^{50} +1665.23 q^{52} -301.707 q^{53} +516.669 q^{55} -696.615 q^{58} -412.635 q^{59} +571.253 q^{61} +97.6396 q^{62} +218.593 q^{64} -1524.49 q^{65} +820.549 q^{67} -249.456 q^{68} -8.95825 q^{71} -21.9339 q^{73} +2075.95 q^{74} +99.9124 q^{76} +619.428 q^{79} -2263.36 q^{80} +2239.87 q^{82} +259.532 q^{83} +228.374 q^{85} +274.278 q^{86} -1658.15 q^{88} +484.329 q^{89} +3113.45 q^{92} +800.440 q^{94} -91.4683 q^{95} +252.747 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 26 q^{4}+O(q^{10}) $ 4 * q + 26 * q^4	$ 4 q + 26 q^{4} - 206 q^{10} + 68 q^{13} + 290 q^{16} - 172 q^{19} - 94 q^{22} + 272 q^{25} - 408 q^{31} - 180 q^{34} + 1320 q^{37} - 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 3952 q^{52} - 352 q^{55} - 1432 q^{58} + 2672 q^{61} + 826 q^{64} + 1444 q^{67} + 396 q^{73} + 1222 q^{76} + 1220 q^{79} + 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 3492 q^{94} + 624 q^{97}+O(q^{100}) $ 4 * q + 26 * q^4 - 206 * q^10 + 68 * q^13 + 290 * q^16 - 172 * q^19 - 94 * q^22 + 272 * q^25 - 408 * q^31 - 180 * q^34 + 1320 * q^37 - 1446 * q^40 + 116 * q^43 + 1374 * q^46 + 3952 * q^52 - 352 * q^55 - 1432 * q^58 + 2672 * q^61 + 826 * q^64 + 1444 * q^67 + 396 * q^73 + 1222 * q^76 + 1220 * q^79 + 3590 * q^82 + 720 * q^85 - 6294 * q^88 + 3492 * q^94 + 624 * 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 $

$2$	5.15688	1.82323	0.911616	−	0.411043i	$-0.134836\pi$
$2$	5.15688	1.82323	0.911616	+	0.411043i	$0.134836\pi$
$3$	0	0
$3$	0	0
$4$	18.5934	2.32417
$5$	−17.0220	−1.52249	−0.761245	−	0.648464i	$-0.775412\pi$
$5$	−17.0220	−1.52249	−0.761245	+	0.648464i	$0.775412\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	54.6288	2.41427
$9$	0	0
$10$	−87.7802	−2.77585
$11$	−30.3531	−0.831982	−0.415991	−	0.909369i	$-0.636565\pi$
$11$	−30.3531	−0.831982	−0.415991	+	0.909369i	$0.636565\pi$
$12$	0	0
$13$	89.5603	1.91074	0.955368	−	0.295419i	$-0.0954592\pi$
$13$	89.5603	1.91074	0.955368	+	0.295419i	$0.0954592\pi$
$14$	0	0
$15$	0	0
$16$	132.967	2.07761
$17$	−13.4164	−0.191409	−0.0957046	−	0.995410i	$-0.530510\pi$
$17$	−13.4164	−0.191409	−0.0957046	+	0.995410i	$0.530510\pi$
$18$	0	0
$19$	5.37355	0.0648830	0.0324415	−	0.999474i	$-0.489672\pi$
$19$	5.37355	0.0648830	0.0324415	+	0.999474i	$0.489672\pi$
$20$	−316.496	−3.53853
$21$	0	0
$22$	−156.527	−1.51690
$23$	167.449	1.51807	0.759035	−	0.651050i	$-0.225671\pi$
$23$	167.449	1.51807	0.759035	+	0.651050i	$0.225671\pi$
$24$	0	0
$25$	164.747	1.31798
$26$	461.852	3.48371
$27$	0	0
$28$	0	0
$29$	−135.085	−0.864986	−0.432493	−	0.901637i	$-0.642366\pi$
$29$	−135.085	−0.864986	−0.432493	+	0.901637i	$0.642366\pi$
$30$	0	0
$31$	18.9339	0.109698	0.0548488	−	0.998495i	$-0.482532\pi$
$31$	18.9339	0.109698	0.0548488	+	0.998495i	$0.482532\pi$
$32$	248.664	1.37369
$33$	0	0
$34$	−69.1868	−0.348983
$35$	0	0
$36$	0	0
$37$	402.560	1.78866	0.894331	−	0.447406i	$-0.147652\pi$
$37$	402.560	1.78866	0.894331	+	0.447406i	$0.147652\pi$
$38$	27.7107	0.118297
$39$	0	0
$40$	−929.889	−3.67571
$41$	434.345	1.65447	0.827236	−	0.561855i	$-0.189912\pi$
$41$	434.345	1.65447	0.827236	+	0.561855i	$0.189912\pi$
$42$	0	0
$43$	53.1868	0.188626	0.0943129	−	0.995543i	$-0.469935\pi$
$43$	53.1868	0.188626	0.0943129	+	0.995543i	$0.469935\pi$
$44$	−564.367	−1.93367
$45$	0	0
$46$	863.516	2.76779
$47$	155.218	0.481720	0.240860	−	0.970560i	$-0.422570\pi$
$47$	155.218	0.481720	0.240860	+	0.970560i	$0.422570\pi$
$48$	0	0
$49$	0	0
$50$	849.581	2.40298
$51$	0	0
$52$	1665.23	4.44088
$53$	−301.707	−0.781938	−0.390969	−	0.920404i	$-0.627860\pi$
$53$	−301.707	−0.781938	−0.390969	+	0.920404i	$0.627860\pi$
$54$	0	0
$55$	516.669	1.26669
$56$	0	0
$57$	0	0
$58$	−696.615	−1.57707
$59$	−412.635	−0.910518	−0.455259	−	0.890359i	$-0.650453\pi$
$59$	−412.635	−0.910518	−0.455259	+	0.890359i	$0.650453\pi$
$60$	0	0
$61$	571.253	1.19904	0.599520	−	0.800360i	$-0.295358\pi$
$61$	571.253	1.19904	0.599520	+	0.800360i	$0.295358\pi$
$62$	97.6396	0.200004
$63$	0	0
$64$	218.593	0.426940
$65$	−1524.49	−2.90908
$66$	0	0
$67$	820.549	1.49621	0.748104	−	0.663581i	$-0.230964\pi$
$67$	820.549	1.49621	0.748104	+	0.663581i	$0.230964\pi$
$68$	−249.456	−0.444868
$69$	0	0
$70$	0	0
$71$	−8.95825	−0.0149739	−0.00748696	−	0.999972i	$-0.502383\pi$
$71$	−8.95825	−0.0149739	−0.00748696	+	0.999972i	$0.502383\pi$
$72$	0	0
$73$	−21.9339	−0.0351666	−0.0175833	−	0.999845i	$-0.505597\pi$
$73$	−21.9339	−0.0351666	−0.0175833	+	0.999845i	$0.505597\pi$
$74$	2075.95	3.26115
$75$	0	0
$76$	99.9124	0.150799
$77$	0	0
$78$	0	0
$79$	619.428	0.882166	0.441083	−	0.897466i	$-0.354594\pi$
$79$	619.428	0.882166	0.441083	+	0.897466i	$0.354594\pi$
$80$	−2263.36	−3.16314
$81$	0	0
$82$	2239.87	3.01649
$83$	259.532	0.343221	0.171610	−	0.985165i	$-0.445103\pi$
$83$	259.532	0.343221	0.171610	+	0.985165i	$0.445103\pi$
$84$	0	0
$85$	228.374	0.291419
$86$	274.278	0.343908
$87$	0	0
$88$	−1658.15	−2.00863
$89$	484.329	0.576840	0.288420	−	0.957504i	$-0.406870\pi$
$89$	484.329	0.576840	0.288420	+	0.957504i	$0.406870\pi$
$90$	0	0
$91$	0	0
$92$	3113.45	3.52826
$93$	0	0
$94$	800.440	0.878288
$95$	−91.4683	−0.0987837
$96$	0	0
$97$	252.747	0.264563	0.132281	−	0.991212i	$-0.457770\pi$
$97$	252.747	0.264563	0.132281	+	0.991212i	$0.457770\pi$
$98$	0	0
$99$	0	0
$100$	3063.21	3.06321
$101$	−629.378	−0.620054	−0.310027	−	0.950728i	$-0.600338\pi$
$101$	−629.378	−0.620054	−0.310027	+	0.950728i	$0.600338\pi$
$102$	0	0
$103$	−1172.54	−1.12168	−0.560842	−	0.827923i	$-0.689523\pi$
$103$	−1172.54	−1.12168	−0.560842	+	0.827923i	$0.689523\pi$
$104$	4892.57	4.61304
$105$	0	0
$106$	−1555.87	−1.42565
$107$	1002.87	0.906087	0.453044	−	0.891488i	$-0.350338\pi$
$107$	1002.87	0.906087	0.453044	+	0.891488i	$0.350338\pi$
$108$	0	0
$109$	542.802	0.476981	0.238491	−	0.971145i	$-0.423347\pi$
$109$	542.802	0.476981	0.238491	+	0.971145i	$0.423347\pi$
$110$	2664.40	2.30946
$111$	0	0
$112$	0	0
$113$	1388.74	1.15612	0.578062	−	0.815993i	$-0.303809\pi$
$113$	1388.74	1.15612	0.578062	+	0.815993i	$0.303809\pi$
$114$	0	0
$115$	−2850.32	−2.31125
$116$	−2511.68	−2.01038
$117$	0	0
$118$	−2127.91	−1.66009
$119$	0	0
$120$	0	0
$121$	−409.689	−0.307805
$122$	2945.88	2.18613
$123$	0	0
$124$	352.045	0.254956
$125$	−676.573	−0.484117
$126$	0	0
$127$	−2330.00	−1.62798	−0.813991	−	0.580877i	$-0.802710\pi$
$127$	−2330.00	−1.62798	−0.813991	+	0.580877i	$0.802710\pi$
$128$	−862.051	−0.595276
$129$	0	0
$130$	−7861.62	−5.30392
$131$	226.817	0.151276	0.0756379	−	0.997135i	$-0.475901\pi$
$131$	226.817	0.151276	0.0756379	+	0.997135i	$0.475901\pi$
$132$	0	0
$133$	0	0
$134$	4231.47	2.72793
$135$	0	0
$136$	−732.922	−0.462114
$137$	−1900.83	−1.18540	−0.592698	−	0.805425i	$-0.701937\pi$
$137$	−1900.83	−1.18540	−0.592698	+	0.805425i	$0.701937\pi$
$138$	0	0
$139$	−1958.99	−1.19539	−0.597695	−	0.801724i	$-0.703917\pi$
$139$	−1958.99	−1.19539	−0.597695	+	0.801724i	$0.703917\pi$
$140$	0	0
$141$	0	0
$142$	−46.1966	−0.0273009
$143$	−2718.43	−1.58970
$144$	0	0
$145$	2299.40	1.31693
$146$	−113.110	−0.0641169
$147$	0	0
$148$	7484.96	4.15716
$149$	1518.37	0.834831	0.417416	−	0.908716i	$-0.362936\pi$
$149$	1518.37	0.834831	0.417416	+	0.908716i	$0.362936\pi$
$150$	0	0
$151$	−1815.18	−0.978262	−0.489131	−	0.872210i	$-0.662686\pi$
$151$	−1815.18	−0.978262	−0.489131	+	0.872210i	$0.662686\pi$
$152$	293.550	0.156645
$153$	0	0
$154$	0	0
$155$	−322.291	−0.167013
$156$	0	0
$157$	1858.79	0.944889	0.472445	−	0.881360i	$-0.343372\pi$
$157$	1858.79	0.944889	0.472445	+	0.881360i	$0.343372\pi$
$158$	3194.31	1.60839
$159$	0	0
$160$	−4232.75	−2.09142
$161$	0	0
$162$	0	0
$163$	870.089	0.418102	0.209051	−	0.977905i	$-0.432962\pi$
$163$	870.089	0.418102	0.209051	+	0.977905i	$0.432962\pi$
$164$	8075.95	3.84528
$165$	0	0
$166$	1338.37	0.625771
$167$	−1749.13	−0.810488	−0.405244	−	0.914209i	$-0.632813\pi$
$167$	−1749.13	−0.810488	−0.405244	+	0.914209i	$0.632813\pi$
$168$	0	0
$169$	5824.05	2.65091
$170$	1177.69	0.531324
$171$	0	0
$172$	988.922	0.438399
$173$	−39.3374	−0.0172877	−0.00864383	−	0.999963i	$-0.502751\pi$
$173$	−39.3374	−0.0172877	−0.00864383	+	0.999963i	$0.502751\pi$
$174$	0	0
$175$	0	0
$176$	−4035.96	−1.72853
$177$	0	0
$178$	2497.62	1.05171
$179$	−2650.58	−1.10678	−0.553391	−	0.832922i	$-0.686666\pi$
$179$	−2650.58	−1.10678	−0.553391	+	0.832922i	$0.686666\pi$
$180$	0	0
$181$	−2719.71	−1.11688	−0.558438	−	0.829546i	$-0.688599\pi$
$181$	−2719.71	−1.11688	−0.558438	+	0.829546i	$0.688599\pi$
$182$	0	0
$183$	0	0
$184$	9147.55	3.66504
$185$	−6852.37	−2.72322
$186$	0	0
$187$	407.230	0.159249
$188$	2886.03	1.11960
$189$	0	0
$190$	−471.691	−0.180106
$191$	4339.00	1.64376	0.821881	−	0.569659i	$-0.192925\pi$
$191$	4339.00	1.64376	0.821881	+	0.569659i	$0.192925\pi$
$192$	0	0
$193$	977.689	0.364640	0.182320	−	0.983239i	$-0.441639\pi$
$193$	977.689	0.364640	0.182320	+	0.983239i	$0.441639\pi$
$194$	1303.39	0.482359
$195$	0	0
$196$	0	0
$197$	−1556.32	−0.562859	−0.281430	−	0.959582i	$-0.590809\pi$
$197$	−1556.32	−0.562859	−0.281430	+	0.959582i	$0.590809\pi$
$198$	0	0
$199$	−1723.21	−0.613846	−0.306923	−	0.951734i	$-0.599299\pi$
$199$	−1723.21	−0.613846	−0.306923	+	0.951734i	$0.599299\pi$
$200$	8999.94	3.18196
$201$	0	0
$202$	−3245.62	−1.13050
$203$	0	0
$204$	0	0
$205$	−7393.41	−2.51892
$206$	−6046.63	−2.04509
$207$	0	0
$208$	11908.6	3.96976
$209$	−163.104	−0.0539815
$210$	0	0
$211$	1295.19	0.422581	0.211290	−	0.977423i	$-0.432233\pi$
$211$	1295.19	0.422581	0.211290	+	0.977423i	$0.432233\pi$
$212$	−5609.76	−1.81736
$213$	0	0
$214$	5171.69	1.65201
$215$	−905.343	−0.287181
$216$	0	0
$217$	0	0
$218$	2799.16	0.869647
$219$	0	0
$220$	9606.63	2.94400
$221$	−1201.58	−0.365732
$222$	0	0
$223$	483.611	0.145224	0.0726121	−	0.997360i	$-0.476866\pi$
$223$	483.611	0.145224	0.0726121	+	0.997360i	$0.476866\pi$
$224$	0	0
$225$	0	0
$226$	7161.58	2.10788
$227$	4019.86	1.17536	0.587681	−	0.809092i	$-0.300041\pi$
$227$	4019.86	1.17536	0.587681	+	0.809092i	$0.300041\pi$
$228$	0	0
$229$	−5070.37	−1.46314	−0.731570	−	0.681766i	$-0.761212\pi$
$229$	−5070.37	−1.46314	−0.731570	+	0.681766i	$0.761212\pi$
$230$	−14698.7	−4.21394
$231$	0	0
$232$	−7379.51	−2.08831
$233$	−4418.26	−1.24227	−0.621137	−	0.783702i	$-0.713329\pi$
$233$	−4418.26	−1.24227	−0.621137	+	0.783702i	$0.713329\pi$
$234$	0	0
$235$	−2642.11	−0.733415
$236$	−7672.29	−2.11620
$237$	0	0
$238$	0	0
$239$	−3228.15	−0.873690	−0.436845	−	0.899537i	$-0.643904\pi$
$239$	−3228.15	−0.873690	−0.436845	+	0.899537i	$0.643904\pi$
$240$	0	0
$241$	1537.14	0.410854	0.205427	−	0.978672i	$-0.434142\pi$
$241$	1537.14	0.410854	0.205427	+	0.978672i	$0.434142\pi$
$242$	−2112.72	−0.561200
$243$	0	0
$244$	10621.5	2.78678
$245$	0	0
$246$	0	0
$247$	481.257	0.123974
$248$	1034.33	0.264840
$249$	0	0
$250$	−3489.01	−0.882657
$251$	−178.384	−0.0448587	−0.0224293	−	0.999748i	$-0.507140\pi$
$251$	−178.384	−0.0448587	−0.0224293	+	0.999748i	$0.507140\pi$
$252$	0	0
$253$	−5082.61	−1.26301
$254$	−12015.5	−2.96819
$255$	0	0
$256$	−6194.24	−1.51227
$257$	−3271.92	−0.794152	−0.397076	−	0.917786i	$-0.629975\pi$
$257$	−3271.92	−0.794152	−0.397076	+	0.917786i	$0.629975\pi$
$258$	0	0
$259$	0	0
$260$	−28345.5	−6.76120
$261$	0	0
$262$	1169.67	0.275811
$263$	−4738.45	−1.11097	−0.555486	−	0.831526i	$-0.687468\pi$
$263$	−4738.45	−1.11097	−0.555486	+	0.831526i	$0.687468\pi$
$264$	0	0
$265$	5135.65	1.19049
$266$	0	0
$267$	0	0
$268$	15256.8	3.47745
$269$	−418.456	−0.0948466	−0.0474233	−	0.998875i	$-0.515101\pi$
$269$	−418.456	−0.0948466	−0.0474233	+	0.998875i	$0.515101\pi$
$270$	0	0
$271$	−5263.95	−1.17993	−0.589967	−	0.807428i	$-0.700859\pi$
$271$	−5263.95	−1.17993	−0.589967	+	0.807428i	$0.700859\pi$
$272$	−1783.94	−0.397673
$273$	0	0
$274$	−9802.37	−2.16125
$275$	−5000.59	−1.09653
$276$	0	0
$277$	2855.61	0.619412	0.309706	−	0.950832i	$-0.399769\pi$
$277$	2855.61	0.619412	0.309706	+	0.950832i	$0.399769\pi$
$278$	−10102.3	−2.17947
$279$	0	0
$280$	0	0
$281$	8589.72	1.82356	0.911779	−	0.410682i	$-0.134709\pi$
$281$	8589.72	1.82356	0.911779	+	0.410682i	$0.134709\pi$
$282$	0	0
$283$	−3681.18	−0.773228	−0.386614	−	0.922242i	$-0.626355\pi$
$283$	−3681.18	−0.773228	−0.386614	+	0.922242i	$0.626355\pi$
$284$	−166.564	−0.0348020
$285$	0	0
$286$	−14018.6	−2.89839
$287$	0	0
$288$	0	0
$289$	−4733.00	−0.963363
$290$	11857.7	2.40107
$291$	0	0
$292$	−407.825	−0.0817334
$293$	−1958.68	−0.390536	−0.195268	−	0.980750i	$-0.562558\pi$
$293$	−1958.68	−0.390536	−0.195268	+	0.980750i	$0.562558\pi$
$294$	0	0
$295$	7023.86	1.38625
$296$	21991.4	4.31832
$297$	0	0
$298$	7830.06	1.52209
$299$	14996.8	2.90063
$300$	0	0
$301$	0	0
$302$	−9360.68	−1.78360
$303$	0	0
$304$	714.504	0.134801
$305$	−9723.84	−1.82553
$306$	0	0
$307$	5672.32	1.05452	0.527258	−	0.849706i	$-0.323220\pi$
$307$	5672.32	1.05452	0.527258	+	0.849706i	$0.323220\pi$
$308$	0	0
$309$	0	0
$310$	−1662.02	−0.304504
$311$	−1271.71	−0.231872	−0.115936	−	0.993257i	$-0.536987\pi$
$311$	−1271.71	−0.231872	−0.115936	+	0.993257i	$0.536987\pi$
$312$	0	0
$313$	4868.38	0.879160	0.439580	−	0.898204i	$-0.355127\pi$
$313$	4868.38	0.879160	0.439580	+	0.898204i	$0.355127\pi$
$314$	9585.55	1.72275
$315$	0	0
$316$	11517.3	2.05031
$317$	3224.18	0.571256	0.285628	−	0.958341i	$-0.407798\pi$
$317$	3224.18	0.571256	0.285628	+	0.958341i	$0.407798\pi$
$318$	0	0
$319$	4100.24	0.719653
$320$	−3720.89	−0.650012
$321$	0	0
$322$	0	0
$323$	−72.0937	−0.0124192
$324$	0	0
$325$	14754.8	2.51831
$326$	4486.94	0.762297
$327$	0	0
$328$	23727.8	3.99435
$329$	0	0
$330$	0	0
$331$	5309.27	0.881643	0.440821	−	0.897595i	$-0.354687\pi$
$331$	5309.27	0.881643	0.440821	+	0.897595i	$0.354687\pi$
$332$	4825.57	0.797704
$333$	0	0
$334$	−9020.04	−1.47771
$335$	−13967.3	−2.27796
$336$	0	0
$337$	−2304.33	−0.372478	−0.186239	−	0.982505i	$-0.559630\pi$
$337$	−2304.33	−0.372478	−0.186239	+	0.982505i	$0.559630\pi$
$338$	30033.9	4.83322
$339$	0	0
$340$	4246.24	0.677308
$341$	−574.702	−0.0912664
$342$	0	0
$343$	0	0
$344$	2905.53	0.455394
$345$	0	0
$346$	−202.858	−0.0315194
$347$	−5196.20	−0.803881	−0.401940	−	0.915666i	$-0.631664\pi$
$347$	−5196.20	−0.803881	−0.401940	+	0.915666i	$0.631664\pi$
$348$	0	0
$349$	9030.60	1.38509	0.692546	−	0.721374i	$-0.256489\pi$
$349$	9030.60	1.38509	0.692546	+	0.721374i	$0.256489\pi$
$350$	0	0
$351$	0	0
$352$	−7547.72	−1.14288
$353$	8885.51	1.33974	0.669869	−	0.742479i	$-0.266350\pi$
$353$	8885.51	1.33974	0.669869	+	0.742479i	$0.266350\pi$
$354$	0	0
$355$	152.487	0.0227976
$356$	9005.31	1.34068
$357$	0	0
$358$	−13668.7	−2.01792
$359$	−37.2983	−0.00548337	−0.00274169	−	0.999996i	$-0.500873\pi$
$359$	−37.2983	−0.00548337	−0.00274169	+	0.999996i	$0.500873\pi$
$360$	0	0
$361$	−6830.12	−0.995790
$362$	−14025.2	−2.03632
$363$	0	0
$364$	0	0
$365$	373.357	0.0535409
$366$	0	0
$367$	−3335.79	−0.474460	−0.237230	−	0.971454i	$-0.576240\pi$
$367$	−3335.79	−0.474460	−0.237230	+	0.971454i	$0.576240\pi$
$368$	22265.2	3.15395
$369$	0	0
$370$	−35336.8	−4.96506
$371$	0	0
$372$	0	0
$373$	3935.81	0.546349	0.273175	−	0.961964i	$-0.411926\pi$
$373$	3935.81	0.546349	0.273175	+	0.961964i	$0.411926\pi$
$374$	2100.03	0.290348
$375$	0	0
$376$	8479.37	1.16301
$377$	−12098.2	−1.65276
$378$	0	0
$379$	2308.00	0.312808	0.156404	−	0.987693i	$-0.450010\pi$
$379$	2308.00	0.312808	0.156404	+	0.987693i	$0.450010\pi$
$380$	−1700.71	−0.229590
$381$	0	0
$382$	22375.7	2.99696
$383$	2931.62	0.391119	0.195559	−	0.980692i	$-0.437348\pi$
$383$	2931.62	0.391119	0.195559	+	0.980692i	$0.437348\pi$
$384$	0	0
$385$	0	0
$386$	5041.82	0.664824
$387$	0	0
$388$	4699.42	0.614890
$389$	−7610.12	−0.991898	−0.495949	−	0.868352i	$-0.665180\pi$
$389$	−7610.12	−0.991898	−0.495949	+	0.868352i	$0.665180\pi$
$390$	0	0
$391$	−2246.57	−0.290572
$392$	0	0
$393$	0	0
$394$	−8025.76	−1.02622
$395$	−10543.9	−1.34309
$396$	0	0
$397$	4269.74	0.539778	0.269889	−	0.962891i	$-0.413013\pi$
$397$	4269.74	0.539778	0.269889	+	0.962891i	$0.413013\pi$
$398$	−8886.41	−1.11918
$399$	0	0
$400$	21905.9	2.73824
$401$	−11128.8	−1.38590	−0.692948	−	0.720988i	$-0.743688\pi$
$401$	−11128.8	−1.38590	−0.692948	+	0.720988i	$0.743688\pi$
$402$	0	0
$403$	1695.72	0.209603
$404$	−11702.3	−1.44111
$405$	0	0
$406$	0	0
$407$	−12219.0	−1.48814
$408$	0	0
$409$	−7220.39	−0.872922	−0.436461	−	0.899723i	$-0.643768\pi$
$409$	−7220.39	−0.872922	−0.436461	+	0.899723i	$0.643768\pi$
$410$	−38126.9	−4.59257
$411$	0	0
$412$	−21801.4	−2.60699
$413$	0	0
$414$	0	0
$415$	−4417.74	−0.522550
$416$	22270.4	2.62475
$417$	0	0
$418$	−841.106	−0.0984207
$419$	16631.2	1.93911	0.969556	−	0.244870i	$-0.0787454\pi$
$419$	16631.2	1.93911	0.969556	+	0.244870i	$0.0787454\pi$
$420$	0	0
$421$	−654.498	−0.0757679	−0.0378839	−	0.999282i	$-0.512062\pi$
$421$	−654.498	−0.0757679	−0.0378839	+	0.999282i	$0.512062\pi$
$422$	6679.14	0.770463
$423$	0	0
$424$	−16481.9	−1.88781
$425$	−2210.31	−0.252273
$426$	0	0
$427$	0	0
$428$	18646.8	2.10590
$429$	0	0
$430$	−4668.74	−0.523597
$431$	−3049.91	−0.340856	−0.170428	−	0.985370i	$-0.554515\pi$
$431$	−3049.91	−0.340856	−0.170428	+	0.985370i	$0.554515\pi$
$432$	0	0
$433$	1599.41	0.177512	0.0887562	−	0.996053i	$-0.471711\pi$
$433$	1599.41	0.177512	0.0887562	+	0.996053i	$0.471711\pi$
$434$	0	0
$435$	0	0
$436$	10092.5	1.10859
$437$	899.797	0.0984968
$438$	0	0
$439$	15893.9	1.72796	0.863979	−	0.503528i	$-0.167965\pi$
$439$	15893.9	1.72796	0.863979	+	0.503528i	$0.167965\pi$
$440$	28225.0	3.05813
$441$	0	0
$442$	−6196.39	−0.666815
$443$	−6160.22	−0.660679	−0.330339	−	0.943862i	$-0.607163\pi$
$443$	−6160.22	−0.660679	−0.330339	+	0.943862i	$0.607163\pi$
$444$	0	0
$445$	−8244.23	−0.878233
$446$	2493.92	0.264777
$447$	0	0
$448$	0	0
$449$	8742.47	0.918892	0.459446	−	0.888206i	$-0.348048\pi$
$449$	8742.47	0.918892	0.459446	+	0.888206i	$0.348048\pi$
$450$	0	0
$451$	−13183.7	−1.37649
$452$	25821.4	2.68703
$453$	0	0
$454$	20729.9	2.14296
$455$	0	0
$456$	0	0
$457$	−580.112	−0.0593796	−0.0296898	−	0.999559i	$-0.509452\pi$
$457$	−580.112	−0.0593796	−0.0296898	+	0.999559i	$0.509452\pi$
$458$	−26147.3	−2.66764
$459$	0	0
$460$	−52997.0	−5.37174
$461$	−1957.05	−0.197720	−0.0988599	−	0.995101i	$-0.531520\pi$
$461$	−1957.05	−0.197720	−0.0988599	+	0.995101i	$0.531520\pi$
$462$	0	0
$463$	8111.80	0.814227	0.407114	−	0.913378i	$-0.366535\pi$
$463$	8111.80	0.814227	0.407114	+	0.913378i	$0.366535\pi$
$464$	−17961.8	−1.79710
$465$	0	0
$466$	−22784.4	−2.26495
$467$	−1240.29	−0.122899	−0.0614493	−	0.998110i	$-0.519572\pi$
$467$	−1240.29	−0.122899	−0.0614493	+	0.998110i	$0.519572\pi$
$468$	0	0
$469$	0	0
$470$	−13625.1	−1.33718
$471$	0	0
$472$	−22541.8	−2.19824
$473$	−1614.38	−0.156933
$474$	0	0
$475$	885.276	0.0855142
$476$	0	0
$477$	0	0
$478$	−16647.2	−1.59294
$479$	−18605.0	−1.77470	−0.887352	−	0.461092i	$-0.847458\pi$
$479$	−18605.0	−1.77470	−0.887352	+	0.461092i	$0.847458\pi$
$480$	0	0
$481$	36053.4	3.41766
$482$	7926.84	0.749083
$483$	0	0
$484$	−7617.51	−0.715393
$485$	−4302.25	−0.402794
$486$	0	0
$487$	874.431	0.0813640	0.0406820	−	0.999172i	$-0.487047\pi$
$487$	874.431	0.0813640	0.0406820	+	0.999172i	$0.487047\pi$
$488$	31206.9	2.89481
$489$	0	0
$490$	0	0
$491$	5849.95	0.537688	0.268844	−	0.963184i	$-0.413358\pi$
$491$	5849.95	0.537688	0.268844	+	0.963184i	$0.413358\pi$
$492$	0	0
$493$	1812.35	0.165566
$494$	2481.78	0.226034
$495$	0	0
$496$	2517.58	0.227908
$497$	0	0
$498$	0	0
$499$	−1095.76	−0.0983029	−0.0491514	−	0.998791i	$-0.515652\pi$
$499$	−1095.76	−0.0983029	−0.0491514	+	0.998791i	$0.515652\pi$
$500$	−12579.8	−1.12517
$501$	0	0
$502$	−919.907	−0.0817878
$503$	−4988.25	−0.442177	−0.221088	−	0.975254i	$-0.570961\pi$
$503$	−4988.25	−0.442177	−0.221088	+	0.975254i	$0.570961\pi$
$504$	0	0
$505$	10713.2	0.944025
$506$	−26210.4	−2.30275
$507$	0	0
$508$	−43322.5	−3.78371
$509$	16831.7	1.46572	0.732859	−	0.680381i	$-0.238186\pi$
$509$	16831.7	1.46572	0.732859	+	0.680381i	$0.238186\pi$
$510$	0	0
$511$	0	0
$512$	−25046.5	−2.16193
$513$	0	0
$514$	−16872.9	−1.44792
$515$	19958.9	1.70775
$516$	0	0
$517$	−4711.35	−0.400783
$518$	0	0
$519$	0	0
$520$	−83281.2	−7.02331
$521$	12826.2	1.07855	0.539277	−	0.842129i	$-0.318698\pi$
$521$	12826.2	1.07855	0.539277	+	0.842129i	$0.318698\pi$
$522$	0	0
$523$	3377.82	0.282412	0.141206	−	0.989980i	$-0.454902\pi$
$523$	3377.82	0.282412	0.141206	+	0.989980i	$0.454902\pi$
$524$	4217.30	0.351591
$525$	0	0
$526$	−24435.6	−2.02556
$527$	−254.024	−0.0209971
$528$	0	0
$529$	15872.3	1.30453
$530$	26483.9	2.17054
$531$	0	0
$532$	0	0
$533$	38900.1	3.16126
$534$	0	0
$535$	−17070.8	−1.37951
$536$	44825.6	3.61226
$537$	0	0
$538$	−2157.93	−0.172927
$539$	0	0
$540$	0	0
$541$	4866.90	0.386774	0.193387	−	0.981123i	$-0.438053\pi$
$541$	4866.90	0.386774	0.193387	+	0.981123i	$0.438053\pi$
$542$	−27145.5	−2.15129
$543$	0	0
$544$	−3336.18	−0.262936
$545$	−9239.55	−0.726199
$546$	0	0
$547$	6870.41	0.537034	0.268517	−	0.963275i	$-0.413467\pi$
$547$	6870.41	0.537034	0.268517	+	0.963275i	$0.413467\pi$
$548$	−35342.9	−2.75506
$549$	0	0
$550$	−25787.4	−1.99923
$551$	−725.883	−0.0561228
$552$	0	0
$553$	0	0
$554$	14726.0	1.12933
$555$	0	0
$556$	−36424.2	−2.77829
$557$	−19795.9	−1.50589	−0.752943	−	0.658086i	$-0.771366\pi$
$557$	−19795.9	−1.50589	−0.752943	+	0.658086i	$0.771366\pi$
$558$	0	0
$559$	4763.42	0.360414
$560$	0	0
$561$	0	0
$562$	44296.1	3.32477
$563$	−6981.58	−0.522626	−0.261313	−	0.965254i	$-0.584156\pi$
$563$	−6981.58	−0.522626	−0.261313	+	0.965254i	$0.584156\pi$
$564$	0	0
$565$	−23639.1	−1.76019
$566$	−18983.4	−1.40977
$567$	0	0
$568$	−489.378	−0.0361512
$569$	−3412.25	−0.251404	−0.125702	−	0.992068i	$-0.540118\pi$
$569$	−3412.25	−0.251404	−0.125702	+	0.992068i	$0.540118\pi$
$570$	0	0
$571$	20647.3	1.51325	0.756624	−	0.653850i	$-0.226847\pi$
$571$	20647.3	1.51325	0.756624	+	0.653850i	$0.226847\pi$
$572$	−50544.9	−3.69473
$573$	0	0
$574$	0	0
$575$	27586.8	2.00078
$576$	0	0
$577$	−12938.7	−0.933530	−0.466765	−	0.884381i	$-0.654581\pi$
$577$	−12938.7	−0.933530	−0.466765	+	0.884381i	$0.654581\pi$
$578$	−24407.5	−1.75643
$579$	0	0
$580$	42753.7	3.06078
$581$	0	0
$582$	0	0
$583$	9157.75	0.650558
$584$	−1198.22	−0.0849019
$585$	0	0
$586$	−10100.7	−0.712038
$587$	−14032.4	−0.986676	−0.493338	−	0.869838i	$-0.664223\pi$
$587$	−14032.4	−0.986676	−0.493338	+	0.869838i	$0.664223\pi$
$588$	0	0
$589$	101.742	0.00711750
$590$	36221.2	2.52746
$591$	0	0
$592$	53527.2	3.71614
$593$	−23344.9	−1.61663	−0.808313	−	0.588752i	$-0.799619\pi$
$593$	−23344.9	−1.61663	−0.808313	+	0.588752i	$0.799619\pi$
$594$	0	0
$595$	0	0
$596$	28231.7	1.94029
$597$	0	0
$598$	77336.7	5.28852
$599$	21813.4	1.48793	0.743966	−	0.668217i	$-0.232942\pi$
$599$	21813.4	1.48793	0.743966	+	0.668217i	$0.232942\pi$
$600$	0	0
$601$	−1010.28	−0.0685690	−0.0342845	−	0.999412i	$-0.510915\pi$
$601$	−1010.28	−0.0685690	−0.0342845	+	0.999412i	$0.510915\pi$
$602$	0	0
$603$	0	0
$604$	−33750.4	−2.27365
$605$	6973.71	0.468631
$606$	0	0
$607$	1763.19	0.117901	0.0589504	−	0.998261i	$-0.481225\pi$
$607$	1763.19	0.117901	0.0589504	+	0.998261i	$0.481225\pi$
$608$	1336.21	0.0891288
$609$	0	0
$610$	−50144.7	−3.32836
$611$	13901.4	0.920440
$612$	0	0
$613$	264.802	0.0174474	0.00872368	−	0.999962i	$-0.497223\pi$
$613$	264.802	0.0174474	0.00872368	+	0.999962i	$0.497223\pi$
$614$	29251.4	1.92263
$615$	0	0
$616$	0	0
$617$	−17930.8	−1.16996	−0.584980	−	0.811048i	$-0.698898\pi$
$617$	−17930.8	−1.16996	−0.584980	+	0.811048i	$0.698898\pi$
$618$	0	0
$619$	19581.1	1.27146	0.635729	−	0.771912i	$-0.280700\pi$
$619$	19581.1	1.27146	0.635729	+	0.771912i	$0.280700\pi$
$620$	−5992.49	−0.388168
$621$	0	0
$622$	−6558.07	−0.422757
$623$	0	0
$624$	0	0
$625$	−9076.78	−0.580914
$626$	25105.6	1.60291
$627$	0	0
$628$	34561.2	2.19609
$629$	−5400.91	−0.342366
$630$	0	0
$631$	13595.9	0.857754	0.428877	−	0.903363i	$-0.358909\pi$
$631$	13595.9	0.857754	0.428877	+	0.903363i	$0.358909\pi$
$632$	33838.6	2.12979
$633$	0	0
$634$	16626.7	1.04153
$635$	39661.1	2.47859
$636$	0	0
$637$	0	0
$638$	21144.4	1.31209
$639$	0	0
$640$	14673.8	0.906301
$641$	−9626.58	−0.593178	−0.296589	−	0.955005i	$-0.595849\pi$
$641$	−9626.58	−0.593178	−0.296589	+	0.955005i	$0.595849\pi$
$642$	0	0
$643$	−26571.5	−1.62967	−0.814834	−	0.579695i	$-0.803172\pi$
$643$	−26571.5	−1.62967	−0.814834	+	0.579695i	$0.803172\pi$
$644$	0	0
$645$	0	0
$646$	−371.778	−0.0226431
$647$	29729.0	1.80644	0.903220	−	0.429179i	$-0.141197\pi$
$647$	29729.0	1.80644	0.903220	+	0.429179i	$0.141197\pi$
$648$	0	0
$649$	12524.8	0.757535
$650$	76088.7	4.59145
$651$	0	0
$652$	16177.9	0.971742
$653$	−16796.1	−1.00656	−0.503278	−	0.864125i	$-0.667873\pi$
$653$	−16796.1	−1.00656	−0.503278	+	0.864125i	$0.667873\pi$
$654$	0	0
$655$	−3860.88	−0.230316
$656$	57753.6	3.43734
$657$	0	0
$658$	0	0
$659$	−2078.96	−0.122891	−0.0614453	−	0.998110i	$-0.519571\pi$
$659$	−2078.96	−0.122891	−0.0614453	+	0.998110i	$0.519571\pi$
$660$	0	0
$661$	6600.40	0.388390	0.194195	−	0.980963i	$-0.437791\pi$
$661$	6600.40	0.388390	0.194195	+	0.980963i	$0.437791\pi$
$662$	27379.2	1.60744
$663$	0	0
$664$	14177.9	0.828629
$665$	0	0
$666$	0	0
$667$	−22619.8	−1.31311
$668$	−32522.2	−1.88372
$669$	0	0
$670$	−72027.9	−4.15325
$671$	−17339.3	−0.997580
$672$	0	0
$673$	−21644.8	−1.23974	−0.619871	−	0.784704i	$-0.712815\pi$
$673$	−21644.8	−1.23974	−0.619871	+	0.784704i	$0.712815\pi$
$674$	−11883.2	−0.679113
$675$	0	0
$676$	108289.	6.16118
$677$	−349.464	−0.0198390	−0.00991949	−	0.999951i	$-0.503158\pi$
$677$	−349.464	−0.0198390	−0.00991949	+	0.999951i	$0.503158\pi$
$678$	0	0
$679$	0	0
$680$	12475.8	0.703565
$681$	0	0
$682$	−2963.67	−0.166400
$683$	−14807.9	−0.829589	−0.414795	−	0.909915i	$-0.636147\pi$
$683$	−14807.9	−0.829589	−0.414795	+	0.909915i	$0.636147\pi$
$684$	0	0
$685$	32355.9	1.80475
$686$	0	0
$687$	0	0
$688$	7072.08	0.391890
$689$	−27021.0	−1.49408
$690$	0	0
$691$	−21679.4	−1.19352	−0.596760	−	0.802420i	$-0.703546\pi$
$691$	−21679.4	−1.19352	−0.596760	+	0.802420i	$0.703546\pi$
$692$	−731.415	−0.0401795
$693$	0	0
$694$	−26796.2	−1.46566
$695$	33345.8	1.81997
$696$	0	0
$697$	−5827.35	−0.316681
$698$	46569.7	2.52534
$699$	0	0
$700$	0	0
$701$	5065.92	0.272949	0.136474	−	0.990644i	$-0.456423\pi$
$701$	5065.92	0.272949	0.136474	+	0.990644i	$0.456423\pi$
$702$	0	0
$703$	2163.18	0.116054
$704$	−6634.99	−0.355207
$705$	0	0
$706$	45821.5	2.44265
$707$	0	0
$708$	0	0
$709$	−16981.8	−0.899527	−0.449764	−	0.893148i	$-0.648492\pi$
$709$	−16981.8	−0.899527	−0.449764	+	0.893148i	$0.648492\pi$
$710$	786.356	0.0415654
$711$	0	0
$712$	26458.3	1.39265
$713$	3170.46	0.166528
$714$	0	0
$715$	46273.1	2.42030
$716$	−49283.3	−2.57235
$717$	0	0
$718$	−192.343	−0.00999746
$719$	31450.6	1.63130	0.815652	−	0.578543i	$-0.196378\pi$
$719$	31450.6	1.63130	0.815652	+	0.578543i	$0.196378\pi$
$720$	0	0
$721$	0	0
$722$	−35222.1	−1.81556
$723$	0	0
$724$	−50568.7	−2.59581
$725$	−22254.8	−1.14003
$726$	0	0
$727$	−14930.2	−0.761665	−0.380833	−	0.924644i	$-0.624363\pi$
$727$	−14930.2	−0.761665	−0.380833	+	0.924644i	$0.624363\pi$
$728$	0	0
$729$	0	0
$730$	1925.36	0.0976174
$731$	−713.575	−0.0361047
$732$	0	0
$733$	−26799.5	−1.35043	−0.675213	−	0.737623i	$-0.735948\pi$
$733$	−26799.5	−1.35043	−0.675213	+	0.737623i	$0.735948\pi$
$734$	−17202.3	−0.865050
$735$	0	0
$736$	41638.6	2.08535
$737$	−24906.2	−1.24482
$738$	0	0
$739$	22511.2	1.12055	0.560275	−	0.828307i	$-0.310695\pi$
$739$	22511.2	1.12055	0.560275	+	0.828307i	$0.310695\pi$
$740$	−127409.	−6.32924
$741$	0	0
$742$	0	0
$743$	−23783.0	−1.17431	−0.587155	−	0.809474i	$-0.699752\pi$
$743$	−23783.0	−1.17431	−0.587155	+	0.809474i	$0.699752\pi$
$744$	0	0
$745$	−25845.7	−1.27102
$746$	20296.5	0.996121
$747$	0	0
$748$	7571.78	0.370123
$749$	0	0
$750$	0	0
$751$	−6729.39	−0.326976	−0.163488	−	0.986545i	$-0.552274\pi$
$751$	−6729.39	−0.326976	−0.163488	+	0.986545i	$0.552274\pi$
$752$	20638.8	1.00083
$753$	0	0
$754$	−62389.0	−3.01336
$755$	30898.0	1.48939
$756$	0	0
$757$	34646.1	1.66345	0.831726	−	0.555187i	$-0.187353\pi$
$757$	34646.1	1.66345	0.831726	+	0.555187i	$0.187353\pi$
$758$	11902.1	0.570321
$759$	0	0
$760$	−4996.80	−0.238491
$761$	−24096.3	−1.14782	−0.573908	−	0.818919i	$-0.694573\pi$
$761$	−24096.3	−1.14782	−0.573908	+	0.818919i	$0.694573\pi$
$762$	0	0
$763$	0	0
$764$	80676.6	3.82039
$765$	0	0
$766$	15118.0	0.713100
$767$	−36955.8	−1.73976
$768$	0	0
$769$	28099.8	1.31769	0.658846	−	0.752278i	$-0.271045\pi$
$769$	28099.8	1.31769	0.658846	+	0.752278i	$0.271045\pi$
$770$	0	0
$771$	0	0
$772$	18178.5	0.847487
$773$	−9455.98	−0.439984	−0.219992	−	0.975502i	$-0.570603\pi$
$773$	−9455.98	−0.439984	−0.219992	+	0.975502i	$0.570603\pi$
$774$	0	0
$775$	3119.30	0.144579
$776$	13807.3	0.638727
$777$	0	0
$778$	−39244.5	−1.80846
$779$	2333.98	0.107347
$780$	0	0
$781$	271.911	0.0124580
$782$	−11585.3	−0.529781
$783$	0	0
$784$	0	0
$785$	−31640.2	−1.43858
$786$	0	0
$787$	16846.0	0.763017	0.381509	−	0.924365i	$-0.375405\pi$
$787$	16846.0	0.763017	0.381509	+	0.924365i	$0.375405\pi$
$788$	−28937.3	−1.30818
$789$	0	0
$790$	−54373.5	−2.44876
$791$	0	0
$792$	0	0
$793$	51161.6	2.29105
$794$	22018.5	0.984140
$795$	0	0
$796$	−32040.4	−1.42669
$797$	35955.5	1.59801	0.799003	−	0.601327i	$-0.205361\pi$
$797$	35955.5	1.59801	0.799003	+	0.601327i	$0.205361\pi$
$798$	0	0
$799$	−2082.47	−0.0922057
$800$	40966.6	1.81049
$801$	0	0
$802$	−57389.7	−2.52681
$803$	665.761	0.0292580
$804$	0	0
$805$	0	0
$806$	8744.64	0.382155
$807$	0	0
$808$	−34382.1	−1.49698
$809$	13680.6	0.594543	0.297272	−	0.954793i	$-0.403923\pi$
$809$	13680.6	0.594543	0.297272	+	0.954793i	$0.403923\pi$
$810$	0	0
$811$	−10215.1	−0.442294	−0.221147	−	0.975241i	$-0.570980\pi$
$811$	−10215.1	−0.442294	−0.221147	+	0.975241i	$0.570980\pi$
$812$	0	0
$813$	0	0
$814$	−63011.7	−2.71322
$815$	−14810.6	−0.636557
$816$	0	0
$817$	285.802	0.0122386
$818$	−37234.6	−1.59154
$819$	0	0
$820$	−137469.	−5.85440
$821$	14255.6	0.605996	0.302998	−	0.952991i	$-0.402012\pi$
$821$	14255.6	0.605996	0.302998	+	0.952991i	$0.402012\pi$
$822$	0	0
$823$	−31349.9	−1.32781	−0.663905	−	0.747817i	$-0.731102\pi$
$823$	−31349.9	−1.32781	−0.663905	+	0.747817i	$0.731102\pi$
$824$	−64054.3	−2.70805
$825$	0	0
$826$	0	0
$827$	−6114.87	−0.257116	−0.128558	−	0.991702i	$-0.541035\pi$
$827$	−6114.87	−0.257116	−0.128558	+	0.991702i	$0.541035\pi$
$828$	0	0
$829$	−32418.4	−1.35819	−0.679093	−	0.734052i	$-0.737627\pi$
$829$	−32418.4	−1.35819	−0.679093	+	0.734052i	$0.737627\pi$
$830$	−22781.7	−0.952730
$831$	0	0
$832$	19577.3	0.815770
$833$	0	0
$834$	0	0
$835$	29773.6	1.23396
$836$	−3032.65	−0.125462
$837$	0	0
$838$	85765.1	3.53545
$839$	44683.5	1.83867	0.919336	−	0.393474i	$-0.128727\pi$
$839$	44683.5	1.83867	0.919336	+	0.393474i	$0.128727\pi$
$840$	0	0
$841$	−6141.15	−0.251800
$842$	−3375.17	−0.138142
$843$	0	0
$844$	24082.0	0.982151
$845$	−99136.8	−4.03599
$846$	0	0
$847$	0	0
$848$	−40117.1	−1.62456
$849$	0	0
$850$	−11398.3	−0.459952
$851$	67408.4	2.71531
$852$	0	0
$853$	9965.17	0.400001	0.200001	−	0.979796i	$-0.435906\pi$
$853$	9965.17	0.400001	0.200001	+	0.979796i	$0.435906\pi$
$854$	0	0
$855$	0	0
$856$	54785.7	2.18754
$857$	−8639.89	−0.344379	−0.172190	−	0.985064i	$-0.555084\pi$
$857$	−8639.89	−0.344379	−0.172190	+	0.985064i	$0.555084\pi$
$858$	0	0
$859$	−15551.5	−0.617707	−0.308853	−	0.951110i	$-0.599945\pi$
$859$	−15551.5	−0.617707	−0.308853	+	0.951110i	$0.599945\pi$
$860$	−16833.4	−0.667458
$861$	0	0
$862$	−15728.0	−0.621460
$863$	−47746.4	−1.88332	−0.941661	−	0.336563i	$-0.890735\pi$
$863$	−47746.4	−1.88332	−0.941661	+	0.336563i	$0.890735\pi$
$864$	0	0
$865$	669.599	0.0263203
$866$	8247.97	0.323646
$867$	0	0
$868$	0	0
$869$	−18801.6	−0.733946
$870$	0	0
$871$	73488.6	2.85886
$872$	29652.6	1.15156
$873$	0	0
$874$	4640.14	0.179583
$875$	0	0
$876$	0	0
$877$	−37958.8	−1.46155	−0.730773	−	0.682620i	$-0.760840\pi$
$877$	−37958.8	−1.46155	−0.730773	+	0.682620i	$0.760840\pi$
$878$	81962.8	3.15047
$879$	0	0
$880$	68699.9	2.63168
$881$	−32323.6	−1.23611	−0.618053	−	0.786136i	$-0.712078\pi$
$881$	−32323.6	−1.23611	−0.618053	+	0.786136i	$0.712078\pi$
$882$	0	0
$883$	27760.0	1.05798	0.528991	−	0.848628i	$-0.322571\pi$
$883$	27760.0	1.05798	0.528991	+	0.848628i	$0.322571\pi$
$884$	−22341.4	−0.850026
$885$	0	0
$886$	−31767.5	−1.20457
$887$	32759.6	1.24009	0.620044	−	0.784567i	$-0.287115\pi$
$887$	32759.6	1.24009	0.620044	+	0.784567i	$0.287115\pi$
$888$	0	0
$889$	0	0
$890$	−42514.5	−1.60122
$891$	0	0
$892$	8991.97	0.337526
$893$	834.071	0.0312554
$894$	0	0
$895$	45118.1	1.68506
$896$	0	0
$897$	0	0
$898$	45083.8	1.67535
$899$	−2557.67	−0.0948868
$900$	0	0
$901$	4047.83	0.149670
$902$	−67986.9	−2.50966
$903$	0	0
$904$	75865.3	2.79120
$905$	46294.8	1.70043
$906$	0	0
$907$	−42682.6	−1.56257	−0.781286	−	0.624174i	$-0.785436\pi$
$907$	−42682.6	−1.56257	−0.781286	+	0.624174i	$0.785436\pi$
$908$	74742.8	2.73175
$909$	0	0
$910$	0	0
$911$	−50130.0	−1.82314	−0.911570	−	0.411144i	$-0.865129\pi$
$911$	−50130.0	−1.82314	−0.911570	+	0.411144i	$0.865129\pi$
$912$	0	0
$913$	−7877.60	−0.285554
$914$	−2991.57	−0.108263
$915$	0	0
$916$	−94275.3	−3.40059
$917$	0	0
$918$	0	0
$919$	−33784.7	−1.21268	−0.606341	−	0.795205i	$-0.707363\pi$
$919$	−33784.7	−1.21268	−0.606341	+	0.795205i	$0.707363\pi$
$920$	−155709.	−5.57998
$921$	0	0
$922$	−10092.3	−0.360489
$923$	−802.303	−0.0286112
$924$	0	0
$925$	66320.6	2.35742
$926$	41831.5	1.48452
$927$	0	0
$928$	−33590.7	−1.18822
$929$	50854.8	1.79601	0.898004	−	0.439987i	$-0.145017\pi$
$929$	50854.8	1.79601	0.898004	+	0.439987i	$0.145017\pi$
$930$	0	0
$931$	0	0
$932$	−82150.4	−2.88726
$933$	0	0
$934$	−6396.01	−0.224073
$935$	−6931.85	−0.242455
$936$	0	0
$937$	3684.00	0.128443	0.0642215	−	0.997936i	$-0.479544\pi$
$937$	3684.00	0.128443	0.0642215	+	0.997936i	$0.479544\pi$
$938$	0	0
$939$	0	0
$940$	−49125.8	−1.70458
$941$	28660.5	0.992886	0.496443	−	0.868069i	$-0.334639\pi$
$941$	28660.5	0.992886	0.496443	+	0.868069i	$0.334639\pi$
$942$	0	0
$943$	72730.8	2.51160
$944$	−54866.9	−1.89170
$945$	0	0
$946$	−8325.18	−0.286126
$947$	31873.0	1.09370	0.546849	−	0.837231i	$-0.315827\pi$
$947$	31873.0	1.09370	0.546849	+	0.837231i	$0.315827\pi$
$948$	0	0
$949$	−1964.40	−0.0671942
$950$	4565.26	0.155912
$951$	0	0
$952$	0	0
$953$	−3310.08	−0.112512	−0.0562561	−	0.998416i	$-0.517916\pi$
$953$	−3310.08	−0.112512	−0.0562561	+	0.998416i	$0.517916\pi$
$954$	0	0
$955$	−73858.2	−2.50261
$956$	−60022.3	−2.03061
$957$	0	0
$958$	−95943.6	−3.23570
$959$	0	0
$960$	0	0
$961$	−29432.5	−0.987966
$962$	185923.	6.23119
$963$	0	0
$964$	28580.6	0.954897
$965$	−16642.2	−0.555161
$966$	0	0
$967$	−168.413	−0.00560061	−0.00280031	−	0.999996i	$-0.500891\pi$
$967$	−168.413	−0.00560061	−0.00280031	+	0.999996i	$0.500891\pi$
$968$	−22380.8	−0.743127
$969$	0	0
$970$	−22186.2	−0.734387
$971$	−37669.9	−1.24499	−0.622495	−	0.782624i	$-0.713881\pi$
$971$	−37669.9	−1.24499	−0.622495	+	0.782624i	$0.713881\pi$
$972$	0	0
$973$	0	0
$974$	4509.33	0.148345
$975$	0	0
$976$	75957.7	2.49114
$977$	26688.7	0.873948	0.436974	−	0.899474i	$-0.356050\pi$
$977$	26688.7	0.873948	0.436974	+	0.899474i	$0.356050\pi$
$978$	0	0
$979$	−14700.9	−0.479921
$980$	0	0
$981$	0	0
$982$	30167.5	0.980329
$983$	1102.29	0.0357655	0.0178828	−	0.999840i	$-0.494307\pi$
$983$	1102.29	0.0357655	0.0178828	+	0.999840i	$0.494307\pi$
$984$	0	0
$985$	26491.6	0.856948
$986$	9346.07	0.301866
$987$	0	0
$988$	8948.19	0.288137
$989$	8906.09	0.286347
$990$	0	0
$991$	16523.6	0.529656	0.264828	−	0.964296i	$-0.414685\pi$
$991$	16523.6	0.529656	0.264828	+	0.964296i	$0.414685\pi$
$992$	4708.17	0.150690
$993$	0	0
$994$	0	0
$995$	29332.5	0.934575
$996$	0	0
$997$	−17994.5	−0.571605	−0.285802	−	0.958289i	$-0.592260\pi$
$997$	−17994.5	−0.571605	−0.285802	+	0.958289i	$0.592260\pi$
$998$	−5650.72	−0.179229
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.ba.1.4		4
3.2	odd	2	inner	1323.4.a.ba.1.1		4
7.6	odd	2		189.4.a.l.1.4	yes	4
21.20	even	2		189.4.a.l.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.a.l.1.1	✓	4	21.20	even	2
189.4.a.l.1.4	yes	4	7.6	odd	2
1323.4.a.ba.1.1		4	3.2	odd	2	inner
1323.4.a.ba.1.4		4	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)

\(f(q)\)	\(=\)	\(q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} +O(q^{10})\)	\(q+5.15688 q^{2} +18.5934 q^{4} -17.0220 q^{5} +54.6288 q^{8} -87.7802 q^{10} -30.3531 q^{11} +89.5603 q^{13} +132.967 q^{16} -13.4164 q^{17} +5.37355 q^{19} -316.496 q^{20} -156.527 q^{22} +167.449 q^{23} +164.747 q^{25} +461.852 q^{26} -135.085 q^{29} +18.9339 q^{31} +248.664 q^{32} -69.1868 q^{34} +402.560 q^{37} +27.7107 q^{38} -929.889 q^{40} +434.345 q^{41} +53.1868 q^{43} -564.367 q^{44} +863.516 q^{46} +155.218 q^{47} +849.581 q^{50} +1665.23 q^{52} -301.707 q^{53} +516.669 q^{55} -696.615 q^{58} -412.635 q^{59} +571.253 q^{61} +97.6396 q^{62} +218.593 q^{64} -1524.49 q^{65} +820.549 q^{67} -249.456 q^{68} -8.95825 q^{71} -21.9339 q^{73} +2075.95 q^{74} +99.9124 q^{76} +619.428 q^{79} -2263.36 q^{80} +2239.87 q^{82} +259.532 q^{83} +228.374 q^{85} +274.278 q^{86} -1658.15 q^{88} +484.329 q^{89} +3113.45 q^{92} +800.440 q^{94} -91.4683 q^{95} +252.747 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 26 q^{4}+O(q^{10}) \) 4 * q + 26 * q^4	\( 4 q + 26 q^{4} - 206 q^{10} + 68 q^{13} + 290 q^{16} - 172 q^{19} - 94 q^{22} + 272 q^{25} - 408 q^{31} - 180 q^{34} + 1320 q^{37} - 1446 q^{40} + 116 q^{43} + 1374 q^{46} + 3952 q^{52} - 352 q^{55} - 1432 q^{58} + 2672 q^{61} + 826 q^{64} + 1444 q^{67} + 396 q^{73} + 1222 q^{76} + 1220 q^{79} + 3590 q^{82} + 720 q^{85} - 6294 q^{88} + 3492 q^{94} + 624 q^{97}+O(q^{100}) \) 4 * q + 26 * q^4 - 206 * q^10 + 68 * q^13 + 290 * q^16 - 172 * q^19 - 94 * q^22 + 272 * q^25 - 408 * q^31 - 180 * q^34 + 1320 * q^37 - 1446 * q^40 + 116 * q^43 + 1374 * q^46 + 3952 * q^52 - 352 * q^55 - 1432 * q^58 + 2672 * q^61 + 826 * q^64 + 1444 * q^67 + 396 * q^73 + 1222 * q^76 + 1220 * q^79 + 3590 * q^82 + 720 * q^85 - 6294 * q^88 + 3492 * q^94 + 624 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more