LMFDB - Embedded newform 1323.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 7 $ x^4 - 6*x^2 + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.25928$ 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)

$f(q)$	$=$	$q-1.25928 q^{2} -0.414214 q^{4} -1.25928 q^{5} +3.04017 q^{8} +1.58579 q^{10} -4.82106 q^{11} +1.17157 q^{13} -3.00000 q^{16} -4.29945 q^{17} +1.58579 q^{19} +0.521611 q^{20} +6.07107 q^{22} +6.60195 q^{23} -3.41421 q^{25} -1.47534 q^{26} -5.03712 q^{29} +10.6569 q^{31} -2.30250 q^{32} +5.41421 q^{34} -5.24264 q^{37} -1.99695 q^{38} -3.82843 q^{40} -0.521611 q^{41} +7.07107 q^{43} +1.99695 q^{44} -8.31371 q^{46} -11.4230 q^{47} +4.29945 q^{50} -0.485281 q^{52} -1.04322 q^{53} +6.07107 q^{55} +6.34315 q^{58} +1.78089 q^{59} +10.4853 q^{61} -13.4200 q^{62} +8.89949 q^{64} -1.47534 q^{65} +5.65685 q^{67} +1.78089 q^{68} +8.81496 q^{71} +8.58579 q^{73} +6.60195 q^{74} -0.656854 q^{76} -13.8995 q^{79} +3.77784 q^{80} +0.656854 q^{82} +14.6792 q^{83} +5.41421 q^{85} -8.90446 q^{86} -14.6569 q^{88} +17.7194 q^{89} -2.73462 q^{92} +14.3848 q^{94} -1.99695 q^{95} +9.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 12 q^{10} + 16 q^{13} - 12 q^{16} + 12 q^{19} - 4 q^{22} - 8 q^{25} + 20 q^{31} + 16 q^{34} - 4 q^{37} - 4 q^{40} + 12 q^{46} + 32 q^{52} - 4 q^{55} + 48 q^{58} + 8 q^{61} - 4 q^{64} + 40 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 12 * q^10 + 16 * q^13 - 12 * q^16 + 12 * q^19 - 4 * q^22 - 8 * q^25 + 20 * q^31 + 16 * q^34 - 4 * q^37 - 4 * q^40 + 12 * q^46 + 32 * q^52 - 4 * q^55 + 48 * q^58 + 8 * q^61 - 4 * q^64 + 40 * q^73 + 20 * q^76 - 16 * q^79 - 20 * q^82 + 16 * q^85 - 36 * q^88 - 16 * q^94 + 16 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.25928	−0.890446	−0.445223	−	0.895420i	$-0.646876\pi$
$2$	−1.25928	−0.890446	−0.445223	+	0.895420i	$0.646876\pi$
$3$	0	0
$3$	0	0
$4$	−0.414214	−0.207107
$5$	−1.25928	−0.563167	−0.281584	−	0.959537i	$-0.590860\pi$
$5$	−1.25928	−0.563167	−0.281584	+	0.959537i	$0.590860\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.04017	1.07486
$9$	0	0
$10$	1.58579	0.501470
$11$	−4.82106	−1.45360	−0.726802	−	0.686847i	$-0.758994\pi$
$11$	−4.82106	−1.45360	−0.726802	+	0.686847i	$0.758994\pi$
$12$	0	0
$13$	1.17157	0.324936	0.162468	−	0.986714i	$-0.448055\pi$
$13$	1.17157	0.324936	0.162468	+	0.986714i	$0.448055\pi$
$14$	0	0
$15$	0	0
$16$	−3.00000	−0.750000
$17$	−4.29945	−1.04277	−0.521385	−	0.853322i	$-0.674584\pi$
$17$	−4.29945	−1.04277	−0.521385	+	0.853322i	$0.674584\pi$
$18$	0	0
$19$	1.58579	0.363804	0.181902	−	0.983317i	$-0.441775\pi$
$19$	1.58579	0.363804	0.181902	+	0.983317i	$0.441775\pi$
$20$	0.521611	0.116636
$21$	0	0
$22$	6.07107	1.29436
$23$	6.60195	1.37660	0.688301	−	0.725425i	$-0.258357\pi$
$23$	6.60195	1.37660	0.688301	+	0.725425i	$0.258357\pi$
$24$	0	0
$25$	−3.41421	−0.682843
$26$	−1.47534	−0.289338
$27$	0	0
$28$	0	0
$29$	−5.03712	−0.935370	−0.467685	−	0.883895i	$-0.654912\pi$
$29$	−5.03712	−0.935370	−0.467685	+	0.883895i	$0.654912\pi$
$30$	0	0
$31$	10.6569	1.91403	0.957014	−	0.290043i	$-0.0936695\pi$
$31$	10.6569	1.91403	0.957014	+	0.290043i	$0.0936695\pi$
$32$	−2.30250	−0.407029
$33$	0	0
$34$	5.41421	0.928530
$35$	0	0
$36$	0	0
$37$	−5.24264	−0.861885	−0.430942	−	0.902379i	$-0.641819\pi$
$37$	−5.24264	−0.861885	−0.430942	+	0.902379i	$0.641819\pi$
$38$	−1.99695	−0.323948
$39$	0	0
$40$	−3.82843	−0.605327
$41$	−0.521611	−0.0814619	−0.0407310	−	0.999170i	$-0.512969\pi$
$41$	−0.521611	−0.0814619	−0.0407310	+	0.999170i	$0.512969\pi$
$42$	0	0
$43$	7.07107	1.07833	0.539164	−	0.842201i	$-0.318740\pi$
$43$	7.07107	1.07833	0.539164	+	0.842201i	$0.318740\pi$
$44$	1.99695	0.301051
$45$	0	0
$46$	−8.31371	−1.22579
$47$	−11.4230	−1.66622	−0.833109	−	0.553109i	$-0.813441\pi$
$47$	−11.4230	−1.66622	−0.833109	+	0.553109i	$0.813441\pi$
$48$	0	0
$49$	0	0
$50$	4.29945	0.608034
$51$	0	0
$52$	−0.485281	−0.0672964
$53$	−1.04322	−0.143298	−0.0716488	−	0.997430i	$-0.522826\pi$
$53$	−1.04322	−0.143298	−0.0716488	+	0.997430i	$0.522826\pi$
$54$	0	0
$55$	6.07107	0.818623
$56$	0	0
$57$	0	0
$58$	6.34315	0.832896
$59$	1.78089	0.231852	0.115926	−	0.993258i	$-0.463016\pi$
$59$	1.78089	0.231852	0.115926	+	0.993258i	$0.463016\pi$
$60$	0	0
$61$	10.4853	1.34250	0.671251	−	0.741230i	$-0.265757\pi$
$61$	10.4853	1.34250	0.671251	+	0.741230i	$0.265757\pi$
$62$	−13.4200	−1.70434
$63$	0	0
$64$	8.89949	1.11244
$65$	−1.47534	−0.182993
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	1.78089	0.215965
$69$	0	0
$70$	0	0
$71$	8.81496	1.04614	0.523072	−	0.852289i	$-0.324786\pi$
$71$	8.81496	1.04614	0.523072	+	0.852289i	$0.324786\pi$
$72$	0	0
$73$	8.58579	1.00489	0.502445	−	0.864609i	$-0.332434\pi$
$73$	8.58579	1.00489	0.502445	+	0.864609i	$0.332434\pi$
$74$	6.60195	0.767461
$75$	0	0
$76$	−0.656854	−0.0753463
$77$	0	0
$78$	0	0
$79$	−13.8995	−1.56382	−0.781908	−	0.623394i	$-0.785753\pi$
$79$	−13.8995	−1.56382	−0.781908	+	0.623394i	$0.785753\pi$
$80$	3.77784	0.422375
$81$	0	0
$82$	0.656854	0.0725374
$83$	14.6792	1.61126	0.805628	−	0.592421i	$-0.201828\pi$
$83$	14.6792	1.61126	0.805628	+	0.592421i	$0.201828\pi$
$84$	0	0
$85$	5.41421	0.587254
$86$	−8.90446	−0.960192
$87$	0	0
$88$	−14.6569	−1.56243
$89$	17.7194	1.87825	0.939127	−	0.343570i	$-0.111636\pi$
$89$	17.7194	1.87825	0.939127	+	0.343570i	$0.111636\pi$
$90$	0	0
$91$	0	0
$92$	−2.73462	−0.285104
$93$	0	0
$94$	14.3848	1.48368
$95$	−1.99695	−0.204883
$96$	0	0
$97$	9.65685	0.980505	0.490252	−	0.871580i	$-0.336905\pi$
$97$	9.65685	0.980505	0.490252	+	0.871580i	$0.336905\pi$
$98$	0	0
$99$	0	0
$100$	1.41421	0.141421
$101$	16.4601	1.63784	0.818922	−	0.573904i	$-0.194572\pi$
$101$	16.4601	1.63784	0.818922	+	0.573904i	$0.194572\pi$
$102$	0	0
$103$	−3.48528	−0.343415	−0.171707	−	0.985148i	$-0.554928\pi$
$103$	−3.48528	−0.343415	−0.171707	+	0.985148i	$0.554928\pi$
$104$	3.56178	0.349261
$105$	0	0
$106$	1.31371	0.127599
$107$	6.81801	0.659122	0.329561	−	0.944134i	$-0.393099\pi$
$107$	6.81801	0.659122	0.329561	+	0.944134i	$0.393099\pi$
$108$	0	0
$109$	2.65685	0.254480	0.127240	−	0.991872i	$-0.459388\pi$
$109$	2.65685	0.254480	0.127240	+	0.991872i	$0.459388\pi$
$110$	−7.64518	−0.728939
$111$	0	0
$112$	0	0
$113$	−11.4230	−1.07459	−0.537293	−	0.843395i	$-0.680553\pi$
$113$	−11.4230	−1.07459	−0.537293	+	0.843395i	$0.680553\pi$
$114$	0	0
$115$	−8.31371	−0.775257
$116$	2.08644	0.193721
$117$	0	0
$118$	−2.24264	−0.206452
$119$	0	0
$120$	0	0
$121$	12.2426	1.11297
$122$	−13.2039	−1.19543
$123$	0	0
$124$	−4.41421	−0.396408
$125$	10.5959	0.947722
$126$	0	0
$127$	−4.82843	−0.428454	−0.214227	−	0.976784i	$-0.568723\pi$
$127$	−4.82843	−0.428454	−0.214227	+	0.976784i	$0.568723\pi$
$128$	−6.60195	−0.583536
$129$	0	0
$130$	1.85786	0.162945
$131$	15.4169	1.34698	0.673491	−	0.739195i	$-0.264794\pi$
$131$	15.4169	1.34698	0.673491	+	0.739195i	$0.264794\pi$
$132$	0	0
$133$	0	0
$134$	−7.12356	−0.615382
$135$	0	0
$136$	−13.0711	−1.12083
$137$	4.60500	0.393432	0.196716	−	0.980461i	$-0.436972\pi$
$137$	4.60500	0.393432	0.196716	+	0.980461i	$0.436972\pi$
$138$	0	0
$139$	−8.82843	−0.748817	−0.374409	−	0.927264i	$-0.622154\pi$
$139$	−8.82843	−0.748817	−0.374409	+	0.927264i	$0.622154\pi$
$140$	0	0
$141$	0	0
$142$	−11.1005	−0.931534
$143$	−5.64823	−0.472328
$144$	0	0
$145$	6.34315	0.526770
$146$	−10.8119	−0.894800
$147$	0	0
$148$	2.17157	0.178502
$149$	−6.81801	−0.558553	−0.279277	−	0.960211i	$-0.590095\pi$
$149$	−6.81801	−0.558553	−0.279277	+	0.960211i	$0.590095\pi$
$150$	0	0
$151$	4.24264	0.345261	0.172631	−	0.984987i	$-0.444773\pi$
$151$	4.24264	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$152$	4.82106	0.391040
$153$	0	0
$154$	0	0
$155$	−13.4200	−1.07792
$156$	0	0
$157$	18.4853	1.47529	0.737643	−	0.675191i	$-0.235939\pi$
$157$	18.4853	1.47529	0.737643	+	0.675191i	$0.235939\pi$
$158$	17.5034	1.39249
$159$	0	0
$160$	2.89949	0.229225
$161$	0	0
$162$	0	0
$163$	−11.6569	−0.913035	−0.456518	−	0.889714i	$-0.650903\pi$
$163$	−11.6569	−0.913035	−0.456518	+	0.889714i	$0.650903\pi$
$164$	0.216058	0.0168713
$165$	0	0
$166$	−18.4853	−1.43474
$167$	−11.4230	−0.883939	−0.441970	−	0.897030i	$-0.645720\pi$
$167$	−11.4230	−0.883939	−0.441970	+	0.897030i	$0.645720\pi$
$168$	0	0
$169$	−11.6274	−0.894417
$170$	−6.81801	−0.522918
$171$	0	0
$172$	−2.92893	−0.223329
$173$	13.1144	0.997070	0.498535	−	0.866869i	$-0.333871\pi$
$173$	13.1144	0.997070	0.498535	+	0.866869i	$0.333871\pi$
$174$	0	0
$175$	0	0
$176$	14.4632	1.09020
$177$	0	0
$178$	−22.3137	−1.67248
$179$	17.9355	1.34056	0.670280	−	0.742108i	$-0.266174\pi$
$179$	17.9355	1.34056	0.670280	+	0.742108i	$0.266174\pi$
$180$	0	0
$181$	4.10051	0.304788	0.152394	−	0.988320i	$-0.451302\pi$
$181$	4.10051	0.304788	0.152394	+	0.988320i	$0.451302\pi$
$182$	0	0
$183$	0	0
$184$	20.0711	1.47966
$185$	6.60195	0.485385
$186$	0	0
$187$	20.7279	1.51578
$188$	4.73157	0.345085
$189$	0	0
$190$	2.51472	0.182437
$191$	−9.12051	−0.659937	−0.329969	−	0.943992i	$-0.607038\pi$
$191$	−9.12051	−0.659937	−0.329969	+	0.943992i	$0.607038\pi$
$192$	0	0
$193$	23.6569	1.70286	0.851429	−	0.524470i	$-0.175737\pi$
$193$	23.6569	1.70286	0.851429	+	0.524470i	$0.175737\pi$
$194$	−12.1607	−0.873086
$195$	0	0
$196$	0	0
$197$	3.25623	0.231997	0.115998	−	0.993249i	$-0.462993\pi$
$197$	3.25623	0.231997	0.115998	+	0.993249i	$0.462993\pi$
$198$	0	0
$199$	−3.92893	−0.278515	−0.139257	−	0.990256i	$-0.544472\pi$
$199$	−3.92893	−0.278515	−0.139257	+	0.990256i	$0.544472\pi$
$200$	−10.3798	−0.733962
$201$	0	0
$202$	−20.7279	−1.45841
$203$	0	0
$204$	0	0
$205$	0.656854	0.0458767
$206$	4.38895	0.305792
$207$	0	0
$208$	−3.51472	−0.243702
$209$	−7.64518	−0.528828
$210$	0	0
$211$	−8.24264	−0.567447	−0.283723	−	0.958906i	$-0.591570\pi$
$211$	−8.24264	−0.567447	−0.283723	+	0.958906i	$0.591570\pi$
$212$	0.432117	0.0296779
$213$	0	0
$214$	−8.58579	−0.586912
$215$	−8.90446	−0.607279
$216$	0	0
$217$	0	0
$218$	−3.34572	−0.226601
$219$	0	0
$220$	−2.51472	−0.169542
$221$	−5.03712	−0.338833
$222$	0	0
$223$	−13.7279	−0.919290	−0.459645	−	0.888103i	$-0.652023\pi$
$223$	−13.7279	−0.919290	−0.459645	+	0.888103i	$0.652023\pi$
$224$	0	0
$225$	0	0
$226$	14.3848	0.956861
$227$	−7.86123	−0.521768	−0.260884	−	0.965370i	$-0.584014\pi$
$227$	−7.86123	−0.521768	−0.260884	+	0.965370i	$0.584014\pi$
$228$	0	0
$229$	1.51472	0.100095	0.0500477	−	0.998747i	$-0.484063\pi$
$229$	1.51472	0.100095	0.0500477	+	0.998747i	$0.484063\pi$
$230$	10.4693	0.690324
$231$	0	0
$232$	−15.3137	−1.00539
$233$	20.0219	1.31168	0.655840	−	0.754900i	$-0.272315\pi$
$233$	20.0219	1.31168	0.655840	+	0.754900i	$0.272315\pi$
$234$	0	0
$235$	14.3848	0.938359
$236$	−0.737669	−0.0480182
$237$	0	0
$238$	0	0
$239$	1.78089	0.115196	0.0575981	−	0.998340i	$-0.481656\pi$
$239$	1.78089	0.115196	0.0575981	+	0.998340i	$0.481656\pi$
$240$	0	0
$241$	−24.7279	−1.59287	−0.796433	−	0.604727i	$-0.793282\pi$
$241$	−24.7279	−1.59287	−0.796433	+	0.604727i	$0.793282\pi$
$242$	−15.4169	−0.991037
$243$	0	0
$244$	−4.34315	−0.278041
$245$	0	0
$246$	0	0
$247$	1.85786	0.118213
$248$	32.3987	2.05732
$249$	0	0
$250$	−13.3431	−0.843895
$251$	11.7286	0.740301	0.370150	−	0.928972i	$-0.379306\pi$
$251$	11.7286	0.740301	0.370150	+	0.928972i	$0.379306\pi$
$252$	0	0
$253$	−31.8284	−2.00104
$254$	6.08034	0.381515
$255$	0	0
$256$	−9.48528	−0.592830
$257$	−10.2903	−0.641891	−0.320946	−	0.947098i	$-0.604001\pi$
$257$	−10.2903	−0.641891	−0.320946	+	0.947098i	$0.604001\pi$
$258$	0	0
$259$	0	0
$260$	0.611105	0.0378991
$261$	0	0
$262$	−19.4142	−1.19941
$263$	−6.16984	−0.380448	−0.190224	−	0.981741i	$-0.560921\pi$
$263$	−6.16984	−0.380448	−0.190224	+	0.981741i	$0.560921\pi$
$264$	0	0
$265$	1.31371	0.0807005
$266$	0	0
$267$	0	0
$268$	−2.34315	−0.143130
$269$	0.521611	0.0318032	0.0159016	−	0.999874i	$-0.494938\pi$
$269$	0.521611	0.0318032	0.0159016	+	0.999874i	$0.494938\pi$
$270$	0	0
$271$	−16.8284	−1.02225	−0.511127	−	0.859505i	$-0.670772\pi$
$271$	−16.8284	−1.02225	−0.511127	+	0.859505i	$0.670772\pi$
$272$	12.8984	0.782078
$273$	0	0
$274$	−5.79899	−0.350330
$275$	16.4601	0.992584
$276$	0	0
$277$	8.55635	0.514101	0.257051	−	0.966398i	$-0.417249\pi$
$277$	8.55635	0.514101	0.257051	+	0.966398i	$0.417249\pi$
$278$	11.1175	0.666781
$279$	0	0
$280$	0	0
$281$	22.5405	1.34465	0.672326	−	0.740255i	$-0.265295\pi$
$281$	22.5405	1.34465	0.672326	+	0.740255i	$0.265295\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	−3.65128	−0.216663
$285$	0	0
$286$	7.11270	0.420583
$287$	0	0
$288$	0	0
$289$	1.48528	0.0873695
$290$	−7.98780	−0.469060
$291$	0	0
$292$	−3.55635	−0.208120
$293$	−9.94768	−0.581149	−0.290575	−	0.956852i	$-0.593846\pi$
$293$	−9.94768	−0.581149	−0.290575	+	0.956852i	$0.593846\pi$
$294$	0	0
$295$	−2.24264	−0.130572
$296$	−15.9385	−0.926408
$297$	0	0
$298$	8.58579	0.497361
$299$	7.73467	0.447307
$300$	0	0
$301$	0	0
$302$	−5.34267	−0.307436
$303$	0	0
$304$	−4.75736	−0.272853
$305$	−13.2039	−0.756053
$306$	0	0
$307$	−9.92893	−0.566674	−0.283337	−	0.959020i	$-0.591442\pi$
$307$	−9.92893	−0.566674	−0.283337	+	0.959020i	$0.591442\pi$
$308$	0	0
$309$	0	0
$310$	16.8995	0.959827
$311$	−22.2349	−1.26083	−0.630413	−	0.776260i	$-0.717115\pi$
$311$	−22.2349	−1.26083	−0.630413	+	0.776260i	$0.717115\pi$
$312$	0	0
$313$	24.8284	1.40339	0.701693	−	0.712480i	$-0.252428\pi$
$313$	24.8284	1.40339	0.701693	+	0.712480i	$0.252428\pi$
$314$	−23.2781	−1.31366
$315$	0	0
$316$	5.75736	0.323877
$317$	19.4108	1.09022	0.545110	−	0.838365i	$-0.316488\pi$
$317$	19.4108	1.09022	0.545110	+	0.838365i	$0.316488\pi$
$318$	0	0
$319$	24.2843	1.35966
$320$	−11.2070	−0.626488
$321$	0	0
$322$	0	0
$323$	−6.81801	−0.379364
$324$	0	0
$325$	−4.00000	−0.221880
$326$	14.6792	0.813008
$327$	0	0
$328$	−1.58579	−0.0875604
$329$	0	0
$330$	0	0
$331$	−2.97056	−0.163277	−0.0816384	−	0.996662i	$-0.526015\pi$
$331$	−2.97056	−0.163277	−0.0816384	+	0.996662i	$0.526015\pi$
$332$	−6.08034	−0.333702
$333$	0	0
$334$	14.3848	0.787100
$335$	−7.12356	−0.389202
$336$	0	0
$337$	−28.7990	−1.56878	−0.784390	−	0.620267i	$-0.787024\pi$
$337$	−28.7990	−1.56878	−0.784390	+	0.620267i	$0.787024\pi$
$338$	14.6422	0.796429
$339$	0	0
$340$	−2.24264	−0.121624
$341$	−51.3774	−2.78224
$342$	0	0
$343$	0	0
$344$	21.4973	1.15905
$345$	0	0
$346$	−16.5147	−0.887837
$347$	−11.6391	−0.624818	−0.312409	−	0.949948i	$-0.601136\pi$
$347$	−11.6391	−0.624818	−0.312409	+	0.949948i	$0.601136\pi$
$348$	0	0
$349$	−1.07107	−0.0573329	−0.0286665	−	0.999589i	$-0.509126\pi$
$349$	−1.07107	−0.0573329	−0.0286665	+	0.999589i	$0.509126\pi$
$350$	0	0
$351$	0	0
$352$	11.1005	0.591659
$353$	−28.4048	−1.51183	−0.755916	−	0.654668i	$-0.772808\pi$
$353$	−28.4048	−1.51183	−0.755916	+	0.654668i	$0.772808\pi$
$354$	0	0
$355$	−11.1005	−0.589154
$356$	−7.33962	−0.388999
$357$	0	0
$358$	−22.5858	−1.19370
$359$	−26.5344	−1.40043	−0.700215	−	0.713932i	$-0.746913\pi$
$359$	−26.5344	−1.40043	−0.700215	+	0.713932i	$0.746913\pi$
$360$	0	0
$361$	−16.4853	−0.867646
$362$	−5.16368	−0.271397
$363$	0	0
$364$	0	0
$365$	−10.8119	−0.565921
$366$	0	0
$367$	−7.38478	−0.385482	−0.192741	−	0.981250i	$-0.561738\pi$
$367$	−7.38478	−0.385482	−0.192741	+	0.981250i	$0.561738\pi$
$368$	−19.8059	−1.03245
$369$	0	0
$370$	−8.31371	−0.432209
$371$	0	0
$372$	0	0
$373$	13.1421	0.680474	0.340237	−	0.940340i	$-0.389493\pi$
$373$	13.1421	0.680474	0.340237	+	0.940340i	$0.389493\pi$
$374$	−26.1023	−1.34972
$375$	0	0
$376$	−34.7279	−1.79096
$377$	−5.90135	−0.303935
$378$	0	0
$379$	−1.89949	−0.0975705	−0.0487853	−	0.998809i	$-0.515535\pi$
$379$	−1.89949	−0.0975705	−0.0487853	+	0.998809i	$0.515535\pi$
$380$	0.827164	0.0424326
$381$	0	0
$382$	11.4853	0.587638
$383$	6.81801	0.348384	0.174192	−	0.984712i	$-0.444269\pi$
$383$	6.81801	0.348384	0.174192	+	0.984712i	$0.444269\pi$
$384$	0	0
$385$	0	0
$386$	−29.7906	−1.51630
$387$	0	0
$388$	−4.00000	−0.203069
$389$	−11.1175	−0.563678	−0.281839	−	0.959462i	$-0.590944\pi$
$389$	−11.1175	−0.563678	−0.281839	+	0.959462i	$0.590944\pi$
$390$	0	0
$391$	−28.3848	−1.43548
$392$	0	0
$393$	0	0
$394$	−4.10051	−0.206580
$395$	17.5034	0.880690
$396$	0	0
$397$	8.92893	0.448130	0.224065	−	0.974574i	$-0.428067\pi$
$397$	8.92893	0.448130	0.224065	+	0.974574i	$0.428067\pi$
$398$	4.94763	0.248002
$399$	0	0
$400$	10.2426	0.512132
$401$	−21.8028	−1.08878	−0.544390	−	0.838832i	$-0.683239\pi$
$401$	−21.8028	−1.08878	−0.544390	+	0.838832i	$0.683239\pi$
$402$	0	0
$403$	12.4853	0.621936
$404$	−6.81801	−0.339209
$405$	0	0
$406$	0	0
$407$	25.2751	1.25284
$408$	0	0
$409$	−9.07107	−0.448535	−0.224268	−	0.974528i	$-0.571999\pi$
$409$	−9.07107	−0.448535	−0.224268	+	0.974528i	$0.571999\pi$
$410$	−0.827164	−0.0408507
$411$	0	0
$412$	1.44365	0.0711236
$413$	0	0
$414$	0	0
$415$	−18.4853	−0.907407
$416$	−2.69755	−0.132258
$417$	0	0
$418$	9.62742	0.470892
$419$	17.5034	0.855095	0.427547	−	0.903993i	$-0.359378\pi$
$419$	17.5034	0.855095	0.427547	+	0.903993i	$0.359378\pi$
$420$	0	0
$421$	22.0711	1.07568	0.537839	−	0.843048i	$-0.319241\pi$
$421$	22.0711	1.07568	0.537839	+	0.843048i	$0.319241\pi$
$422$	10.3798	0.505280
$423$	0	0
$424$	−3.17157	−0.154025
$425$	14.6792	0.712048
$426$	0	0
$427$	0	0
$428$	−2.82411	−0.136509
$429$	0	0
$430$	11.2132	0.540749
$431$	17.7194	0.853514	0.426757	−	0.904366i	$-0.359656\pi$
$431$	17.7194	0.853514	0.426757	+	0.904366i	$0.359656\pi$
$432$	0	0
$433$	24.9706	1.20001	0.600004	−	0.799997i	$-0.295166\pi$
$433$	24.9706	1.20001	0.600004	+	0.799997i	$0.295166\pi$
$434$	0	0
$435$	0	0
$436$	−1.10051	−0.0527046
$437$	10.4693	0.500814
$438$	0	0
$439$	19.7990	0.944954	0.472477	−	0.881343i	$-0.343360\pi$
$439$	19.7990	0.944954	0.472477	+	0.881343i	$0.343360\pi$
$440$	18.4571	0.879907
$441$	0	0
$442$	6.34315	0.301713
$443$	14.4632	0.687167	0.343583	−	0.939122i	$-0.388359\pi$
$443$	14.4632	0.687167	0.343583	+	0.939122i	$0.388359\pi$
$444$	0	0
$445$	−22.3137	−1.05777
$446$	17.2873	0.818577
$447$	0	0
$448$	0	0
$449$	9.94768	0.469460	0.234730	−	0.972061i	$-0.424579\pi$
$449$	9.94768	0.469460	0.234730	+	0.972061i	$0.424579\pi$
$450$	0	0
$451$	2.51472	0.118413
$452$	4.73157	0.222554
$453$	0	0
$454$	9.89949	0.464606
$455$	0	0
$456$	0	0
$457$	19.9706	0.934184	0.467092	−	0.884209i	$-0.345302\pi$
$457$	19.9706	0.934184	0.467092	+	0.884209i	$0.345302\pi$
$458$	−1.90746	−0.0891295
$459$	0	0
$460$	3.44365	0.160561
$461$	−30.3122	−1.41178	−0.705890	−	0.708321i	$-0.749453\pi$
$461$	−30.3122	−1.41178	−0.705890	+	0.708321i	$0.749453\pi$
$462$	0	0
$463$	11.8995	0.553016	0.276508	−	0.961012i	$-0.410823\pi$
$463$	11.8995	0.553016	0.276508	+	0.961012i	$0.410823\pi$
$464$	15.1114	0.701527
$465$	0	0
$466$	−25.2132	−1.16798
$467$	−2.82411	−0.130684	−0.0653422	−	0.997863i	$-0.520814\pi$
$467$	−2.82411	−0.130684	−0.0653422	+	0.997863i	$0.520814\pi$
$468$	0	0
$469$	0	0
$470$	−18.1145	−0.835558
$471$	0	0
$472$	5.41421	0.249209
$473$	−34.0901	−1.56746
$474$	0	0
$475$	−5.41421	−0.248421
$476$	0	0
$477$	0	0
$478$	−2.24264	−0.102576
$479$	26.4078	1.20660	0.603302	−	0.797513i	$-0.293851\pi$
$479$	26.4078	1.20660	0.603302	+	0.797513i	$0.293851\pi$
$480$	0	0
$481$	−6.14214	−0.280057
$482$	31.1394	1.41836
$483$	0	0
$484$	−5.07107	−0.230503
$485$	−12.1607	−0.552188
$486$	0	0
$487$	32.5858	1.47660	0.738301	−	0.674471i	$-0.235628\pi$
$487$	32.5858	1.47660	0.738301	+	0.674471i	$0.235628\pi$
$488$	31.8771	1.44301
$489$	0	0
$490$	0	0
$491$	11.2070	0.505763	0.252881	−	0.967497i	$-0.418622\pi$
$491$	11.2070	0.505763	0.252881	+	0.967497i	$0.418622\pi$
$492$	0	0
$493$	21.6569	0.975376
$494$	−2.33957	−0.105262
$495$	0	0
$496$	−31.9706	−1.43552
$497$	0	0
$498$	0	0
$499$	−41.5563	−1.86032	−0.930159	−	0.367157i	$-0.880331\pi$
$499$	−41.5563	−1.86032	−0.930159	+	0.367157i	$0.880331\pi$
$500$	−4.38895	−0.196280
$501$	0	0
$502$	−14.7696	−0.659197
$503$	18.2410	0.813327	0.406664	−	0.913578i	$-0.366692\pi$
$503$	18.2410	0.813327	0.406664	+	0.913578i	$0.366692\pi$
$504$	0	0
$505$	−20.7279	−0.922380
$506$	40.0809	1.78181
$507$	0	0
$508$	2.00000	0.0887357
$509$	−8.90446	−0.394683	−0.197342	−	0.980335i	$-0.563231\pi$
$509$	−8.90446	−0.394683	−0.197342	+	0.980335i	$0.563231\pi$
$510$	0	0
$511$	0	0
$512$	25.1485	1.11142
$513$	0	0
$514$	12.9584	0.571569
$515$	4.38895	0.193400
$516$	0	0
$517$	55.0711	2.42202
$518$	0	0
$519$	0	0
$520$	−4.48528	−0.196693
$521$	−10.4693	−0.458668	−0.229334	−	0.973348i	$-0.573655\pi$
$521$	−10.4693	−0.458668	−0.229334	+	0.973348i	$0.573655\pi$
$522$	0	0
$523$	−29.7696	−1.30173	−0.650866	−	0.759193i	$-0.725594\pi$
$523$	−29.7696	−1.30173	−0.650866	+	0.759193i	$0.725594\pi$
$524$	−6.38589	−0.278969
$525$	0	0
$526$	7.76955	0.338769
$527$	−45.8186	−1.99589
$528$	0	0
$529$	20.5858	0.895034
$530$	−1.65433	−0.0718594
$531$	0	0
$532$	0	0
$533$	−0.611105	−0.0264699
$534$	0	0
$535$	−8.58579	−0.371196
$536$	17.1978	0.742832
$537$	0	0
$538$	−0.656854	−0.0283190
$539$	0	0
$540$	0	0
$541$	15.3848	0.661443	0.330722	−	0.943728i	$-0.392708\pi$
$541$	15.3848	0.661443	0.330722	+	0.943728i	$0.392708\pi$
$542$	21.1917	0.910262
$543$	0	0
$544$	9.89949	0.424437
$545$	−3.34572	−0.143315
$546$	0	0
$547$	24.4853	1.04692	0.523458	−	0.852052i	$-0.324642\pi$
$547$	24.4853	1.04692	0.523458	+	0.852052i	$0.324642\pi$
$548$	−1.90746	−0.0814824
$549$	0	0
$550$	−20.7279	−0.883842
$551$	−7.98780	−0.340292
$552$	0	0
$553$	0	0
$554$	−10.7748	−0.457779
$555$	0	0
$556$	3.65685	0.155085
$557$	39.7383	1.68377	0.841883	−	0.539661i	$-0.181448\pi$
$557$	39.7383	1.68377	0.841883	+	0.539661i	$0.181448\pi$
$558$	0	0
$559$	8.28427	0.350387
$560$	0	0
$561$	0	0
$562$	−28.3848	−1.19734
$563$	40.4760	1.70586	0.852929	−	0.522027i	$-0.174824\pi$
$563$	40.4760	1.70586	0.852929	+	0.522027i	$0.174824\pi$
$564$	0	0
$565$	14.3848	0.605172
$566$	−10.6853	−0.449139
$567$	0	0
$568$	26.7990	1.12446
$569$	−24.6269	−1.03241	−0.516207	−	0.856464i	$-0.672657\pi$
$569$	−24.6269	−1.03241	−0.516207	+	0.856464i	$0.672657\pi$
$570$	0	0
$571$	−32.0416	−1.34090	−0.670450	−	0.741954i	$-0.733899\pi$
$571$	−32.0416	−1.34090	−0.670450	+	0.741954i	$0.733899\pi$
$572$	2.33957	0.0978224
$573$	0	0
$574$	0	0
$575$	−22.5405	−0.940003
$576$	0	0
$577$	41.3553	1.72165	0.860823	−	0.508905i	$-0.169950\pi$
$577$	41.3553	1.72165	0.860823	+	0.508905i	$0.169950\pi$
$578$	−1.87039	−0.0777978
$579$	0	0
$580$	−2.62742	−0.109098
$581$	0	0
$582$	0	0
$583$	5.02944	0.208298
$584$	26.1023	1.08012
$585$	0	0
$586$	12.5269	0.517482
$587$	−35.1333	−1.45011	−0.725053	−	0.688693i	$-0.758185\pi$
$587$	−35.1333	−1.45011	−0.725053	+	0.688693i	$0.758185\pi$
$588$	0	0
$589$	16.8995	0.696332
$590$	2.82411	0.116267
$591$	0	0
$592$	15.7279	0.646414
$593$	−1.87039	−0.0768075	−0.0384038	−	0.999262i	$-0.512227\pi$
$593$	−1.87039	−0.0768075	−0.0384038	+	0.999262i	$0.512227\pi$
$594$	0	0
$595$	0	0
$596$	2.82411	0.115680
$597$	0	0
$598$	−9.74012	−0.398303
$599$	44.5593	1.82065	0.910323	−	0.413899i	$-0.135833\pi$
$599$	44.5593	1.82065	0.910323	+	0.413899i	$0.135833\pi$
$600$	0	0
$601$	−18.6274	−0.759828	−0.379914	−	0.925022i	$-0.624046\pi$
$601$	−18.6274	−0.759828	−0.379914	+	0.925022i	$0.624046\pi$
$602$	0	0
$603$	0	0
$604$	−1.75736	−0.0715059
$605$	−15.4169	−0.626787
$606$	0	0
$607$	29.3137	1.18981	0.594903	−	0.803797i	$-0.297190\pi$
$607$	29.3137	1.18981	0.594903	+	0.803797i	$0.297190\pi$
$608$	−3.65128	−0.148079
$609$	0	0
$610$	16.6274	0.673224
$611$	−13.3829	−0.541414
$612$	0	0
$613$	−39.6274	−1.60054	−0.800268	−	0.599642i	$-0.795310\pi$
$613$	−39.6274	−1.60054	−0.800268	+	0.599642i	$0.795310\pi$
$614$	12.5033	0.504592
$615$	0	0
$616$	0	0
$617$	−11.8551	−0.477270	−0.238635	−	0.971109i	$-0.576700\pi$
$617$	−11.8551	−0.477270	−0.238635	+	0.971109i	$0.576700\pi$
$618$	0	0
$619$	16.4558	0.661416	0.330708	−	0.943733i	$-0.392712\pi$
$619$	16.4558	0.661416	0.330708	+	0.943733i	$0.392712\pi$
$620$	5.55873	0.223244
$621$	0	0
$622$	28.0000	1.12270
$623$	0	0
$624$	0	0
$625$	3.72792	0.149117
$626$	−31.2659	−1.24964
$627$	0	0
$628$	−7.65685	−0.305542
$629$	22.5405	0.898748
$630$	0	0
$631$	29.3553	1.16862	0.584309	−	0.811531i	$-0.301366\pi$
$631$	29.3553	1.16862	0.584309	+	0.811531i	$0.301366\pi$
$632$	−42.2568	−1.68089
$633$	0	0
$634$	−24.4437	−0.970781
$635$	6.08034	0.241291
$636$	0	0
$637$	0	0
$638$	−30.5807	−1.21070
$639$	0	0
$640$	8.31371	0.328628
$641$	45.3865	1.79266	0.896330	−	0.443388i	$-0.146224\pi$
$641$	45.3865	1.79266	0.896330	+	0.443388i	$0.146224\pi$
$642$	0	0
$643$	−2.75736	−0.108740	−0.0543698	−	0.998521i	$-0.517315\pi$
$643$	−2.75736	−0.108740	−0.0543698	+	0.998521i	$0.517315\pi$
$644$	0	0
$645$	0	0
$646$	8.58579	0.337803
$647$	−6.81801	−0.268044	−0.134022	−	0.990978i	$-0.542789\pi$
$647$	−6.81801	−0.268044	−0.134022	+	0.990978i	$0.542789\pi$
$648$	0	0
$649$	−8.58579	−0.337022
$650$	5.03712	0.197572
$651$	0	0
$652$	4.82843	0.189096
$653$	37.6518	1.47343	0.736715	−	0.676203i	$-0.236376\pi$
$653$	37.6518	1.47343	0.736715	+	0.676203i	$0.236376\pi$
$654$	0	0
$655$	−19.4142	−0.758576
$656$	1.56483	0.0610965
$657$	0	0
$658$	0	0
$659$	−13.8521	−0.539600	−0.269800	−	0.962916i	$-0.586958\pi$
$659$	−13.8521	−0.539600	−0.269800	+	0.962916i	$0.586958\pi$
$660$	0	0
$661$	1.85786	0.0722625	0.0361313	−	0.999347i	$-0.488497\pi$
$661$	1.85786	0.0722625	0.0361313	+	0.999347i	$0.488497\pi$
$662$	3.74077	0.145389
$663$	0	0
$664$	44.6274	1.73188
$665$	0	0
$666$	0	0
$667$	−33.2548	−1.28763
$668$	4.73157	0.183070
$669$	0	0
$670$	8.97056	0.346563
$671$	−50.5502	−1.95147
$672$	0	0
$673$	15.1716	0.584821	0.292411	−	0.956293i	$-0.405543\pi$
$673$	15.1716	0.584821	0.292411	+	0.956293i	$0.405543\pi$
$674$	36.2660	1.39691
$675$	0	0
$676$	4.81623	0.185240
$677$	23.4942	0.902956	0.451478	−	0.892282i	$-0.350897\pi$
$677$	23.4942	0.902956	0.451478	+	0.892282i	$0.350897\pi$
$678$	0	0
$679$	0	0
$680$	16.4601	0.631217
$681$	0	0
$682$	64.6985	2.47743
$683$	−13.8521	−0.530035	−0.265018	−	0.964244i	$-0.585378\pi$
$683$	−13.8521	−0.530035	−0.265018	+	0.964244i	$0.585378\pi$
$684$	0	0
$685$	−5.79899	−0.221568
$686$	0	0
$687$	0	0
$688$	−21.2132	−0.808746
$689$	−1.22221	−0.0465625
$690$	0	0
$691$	−44.8284	−1.70535	−0.852677	−	0.522439i	$-0.825022\pi$
$691$	−44.8284	−1.70535	−0.852677	+	0.522439i	$0.825022\pi$
$692$	−5.43217	−0.206500
$693$	0	0
$694$	14.6569	0.556367
$695$	11.1175	0.421709
$696$	0	0
$697$	2.24264	0.0849461
$698$	1.34877	0.0510519
$699$	0	0
$700$	0	0
$701$	−2.51856	−0.0951247	−0.0475624	−	0.998868i	$-0.515145\pi$
$701$	−2.51856	−0.0951247	−0.0475624	+	0.998868i	$0.515145\pi$
$702$	0	0
$703$	−8.31371	−0.313557
$704$	−42.9050	−1.61704
$705$	0	0
$706$	35.7696	1.34620
$707$	0	0
$708$	0	0
$709$	31.3431	1.17712	0.588558	−	0.808455i	$-0.299696\pi$
$709$	31.3431	1.17712	0.588558	+	0.808455i	$0.299696\pi$
$710$	13.9786	0.524609
$711$	0	0
$712$	53.8701	2.01887
$713$	70.3561	2.63485
$714$	0	0
$715$	7.11270	0.266000
$716$	−7.42912	−0.277639
$717$	0	0
$718$	33.4142	1.24701
$719$	29.0529	1.08349	0.541746	−	0.840542i	$-0.317764\pi$
$719$	29.0529	1.08349	0.541746	+	0.840542i	$0.317764\pi$
$720$	0	0
$721$	0	0
$722$	20.7596	0.772592
$723$	0	0
$724$	−1.69848	−0.0631237
$725$	17.1978	0.638710
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	0	0
$730$	13.6152	0.503922
$731$	−30.4017	−1.12445
$732$	0	0
$733$	24.2426	0.895422	0.447711	−	0.894178i	$-0.352239\pi$
$733$	24.2426	0.895422	0.447711	+	0.894178i	$0.352239\pi$
$734$	9.29950	0.343251
$735$	0	0
$736$	−15.2010	−0.560317
$737$	−27.2720	−1.00458
$738$	0	0
$739$	1.55635	0.0572512	0.0286256	−	0.999590i	$-0.490887\pi$
$739$	1.55635	0.0572512	0.0286256	+	0.999590i	$0.490887\pi$
$740$	−2.73462	−0.100527
$741$	0	0
$742$	0	0
$743$	−10.2903	−0.377514	−0.188757	−	0.982024i	$-0.560446\pi$
$743$	−10.2903	−0.377514	−0.188757	+	0.982024i	$0.560446\pi$
$744$	0	0
$745$	8.58579	0.314559
$746$	−16.5496	−0.605925
$747$	0	0
$748$	−8.58579	−0.313927
$749$	0	0
$750$	0	0
$751$	−31.8995	−1.16403	−0.582015	−	0.813178i	$-0.697735\pi$
$751$	−31.8995	−1.16403	−0.582015	+	0.813178i	$0.697735\pi$
$752$	34.2690	1.24966
$753$	0	0
$754$	7.43146	0.270638
$755$	−5.34267	−0.194440
$756$	0	0
$757$	1.65685	0.0602194	0.0301097	−	0.999547i	$-0.490414\pi$
$757$	1.65685	0.0602194	0.0301097	+	0.999547i	$0.490414\pi$
$758$	2.39200	0.0868812
$759$	0	0
$760$	−6.07107	−0.220221
$761$	0.305553	0.0110763	0.00553814	−	0.999985i	$-0.498237\pi$
$761$	0.305553	0.0110763	0.00553814	+	0.999985i	$0.498237\pi$
$762$	0	0
$763$	0	0
$764$	3.77784	0.136677
$765$	0	0
$766$	−8.58579	−0.310217
$767$	2.08644	0.0753371
$768$	0	0
$769$	22.9706	0.828340	0.414170	−	0.910200i	$-0.364072\pi$
$769$	22.9706	0.828340	0.414170	+	0.910200i	$0.364072\pi$
$770$	0	0
$771$	0	0
$772$	−9.79899	−0.352673
$773$	−6.72852	−0.242008	−0.121004	−	0.992652i	$-0.538611\pi$
$773$	−6.72852	−0.242008	−0.121004	+	0.992652i	$0.538611\pi$
$774$	0	0
$775$	−36.3848	−1.30698
$776$	29.3585	1.05391
$777$	0	0
$778$	14.0000	0.501924
$779$	−0.827164	−0.0296362
$780$	0	0
$781$	−42.4975	−1.52068
$782$	35.7444	1.27822
$783$	0	0
$784$	0	0
$785$	−23.2781	−0.830833
$786$	0	0
$787$	−17.6569	−0.629399	−0.314699	−	0.949191i	$-0.601904\pi$
$787$	−17.6569	−0.629399	−0.314699	+	0.949191i	$0.601904\pi$
$788$	−1.34877	−0.0480481
$789$	0	0
$790$	−22.0416	−0.784206
$791$	0	0
$792$	0	0
$793$	12.2843	0.436227
$794$	−11.2440	−0.399036
$795$	0	0
$796$	1.62742	0.0576823
$797$	−3.16674	−0.112172	−0.0560858	−	0.998426i	$-0.517862\pi$
$797$	−3.16674	−0.112172	−0.0560858	+	0.998426i	$0.517862\pi$
$798$	0	0
$799$	49.1127	1.73748
$800$	7.86123	0.277937
$801$	0	0
$802$	27.4558	0.969500
$803$	−41.3926	−1.46071
$804$	0	0
$805$	0	0
$806$	−15.7225	−0.553800
$807$	0	0
$808$	50.0416	1.76046
$809$	−29.9696	−1.05367	−0.526837	−	0.849966i	$-0.676622\pi$
$809$	−29.9696	−1.05367	−0.526837	+	0.849966i	$0.676622\pi$
$810$	0	0
$811$	37.6274	1.32128	0.660639	−	0.750704i	$-0.270286\pi$
$811$	37.6274	1.32128	0.660639	+	0.750704i	$0.270286\pi$
$812$	0	0
$813$	0	0
$814$	−31.8284	−1.11559
$815$	14.6792	0.514192
$816$	0	0
$817$	11.2132	0.392300
$818$	11.4230	0.399396
$819$	0	0
$820$	−0.272078	−0.00950137
$821$	−42.2568	−1.47477	−0.737387	−	0.675471i	$-0.763941\pi$
$821$	−42.2568	−1.47477	−0.737387	+	0.675471i	$0.763941\pi$
$822$	0	0
$823$	−50.7279	−1.76826	−0.884132	−	0.467237i	$-0.845249\pi$
$823$	−50.7279	−1.76826	−0.884132	+	0.467237i	$0.845249\pi$
$824$	−10.5959	−0.369124
$825$	0	0
$826$	0	0
$827$	18.1515	0.631191	0.315595	−	0.948894i	$-0.397796\pi$
$827$	18.1515	0.631191	0.315595	+	0.948894i	$0.397796\pi$
$828$	0	0
$829$	39.2132	1.36193	0.680965	−	0.732316i	$-0.261560\pi$
$829$	39.2132	1.36193	0.680965	+	0.732316i	$0.261560\pi$
$830$	23.2781	0.807996
$831$	0	0
$832$	10.4264	0.361471
$833$	0	0
$834$	0	0
$835$	14.3848	0.497806
$836$	3.16674	0.109524
$837$	0	0
$838$	−22.0416	−0.761415
$839$	−12.1607	−0.419833	−0.209917	−	0.977719i	$-0.567319\pi$
$839$	−12.1607	−0.419833	−0.209917	+	0.977719i	$0.567319\pi$
$840$	0	0
$841$	−3.62742	−0.125083
$842$	−27.7937	−0.957833
$843$	0	0
$844$	3.41421	0.117522
$845$	14.6422	0.503706
$846$	0	0
$847$	0	0
$848$	3.12967	0.107473
$849$	0	0
$850$	−18.4853	−0.634040
$851$	−34.6117	−1.18647
$852$	0	0
$853$	38.8284	1.32946	0.664730	−	0.747084i	$-0.268547\pi$
$853$	38.8284	1.32946	0.664730	+	0.747084i	$0.268547\pi$
$854$	0	0
$855$	0	0
$856$	20.7279	0.708466
$857$	33.0098	1.12759	0.563796	−	0.825914i	$-0.309340\pi$
$857$	33.0098	1.12759	0.563796	+	0.825914i	$0.309340\pi$
$858$	0	0
$859$	−6.27208	−0.214001	−0.107000	−	0.994259i	$-0.534125\pi$
$859$	−6.27208	−0.214001	−0.107000	+	0.994259i	$0.534125\pi$
$860$	3.68835	0.125772
$861$	0	0
$862$	−22.3137	−0.760008
$863$	40.3494	1.37351	0.686755	−	0.726889i	$-0.259035\pi$
$863$	40.3494	1.37351	0.686755	+	0.726889i	$0.259035\pi$
$864$	0	0
$865$	−16.5147	−0.561517
$866$	−31.4449	−1.06854
$867$	0	0
$868$	0	0
$869$	67.0103	2.27317
$870$	0	0
$871$	6.62742	0.224561
$872$	8.07729	0.273532
$873$	0	0
$874$	−13.1838	−0.445948
$875$	0	0
$876$	0	0
$877$	−21.1716	−0.714913	−0.357457	−	0.933930i	$-0.616356\pi$
$877$	−21.1716	−0.714913	−0.357457	+	0.933930i	$0.616356\pi$
$878$	−24.9325	−0.841430
$879$	0	0
$880$	−18.2132	−0.613967
$881$	−52.1150	−1.75580	−0.877900	−	0.478844i	$-0.841056\pi$
$881$	−52.1150	−1.75580	−0.877900	+	0.478844i	$0.841056\pi$
$882$	0	0
$883$	0.627417	0.0211143	0.0105571	−	0.999944i	$-0.496639\pi$
$883$	0.627417	0.0211143	0.0105571	+	0.999944i	$0.496639\pi$
$884$	2.08644	0.0701747
$885$	0	0
$886$	−18.2132	−0.611885
$887$	−36.3031	−1.21894	−0.609469	−	0.792810i	$-0.708617\pi$
$887$	−36.3031	−1.21894	−0.609469	+	0.792810i	$0.708617\pi$
$888$	0	0
$889$	0	0
$890$	28.0992	0.941888
$891$	0	0
$892$	5.68629	0.190391
$893$	−18.1145	−0.606177
$894$	0	0
$895$	−22.5858	−0.754960
$896$	0	0
$897$	0	0
$898$	−12.5269	−0.418028
$899$	−53.6799	−1.79032
$900$	0	0
$901$	4.48528	0.149426
$902$	−3.16674	−0.105441
$903$	0	0
$904$	−34.7279	−1.15503
$905$	−5.16368	−0.171647
$906$	0	0
$907$	50.7279	1.68439	0.842197	−	0.539171i	$-0.181262\pi$
$907$	50.7279	1.68439	0.842197	+	0.539171i	$0.181262\pi$
$908$	3.25623	0.108062
$909$	0	0
$910$	0	0
$911$	0.916658	0.0303702	0.0151851	−	0.999885i	$-0.495166\pi$
$911$	0.916658	0.0303702	0.0151851	+	0.999885i	$0.495166\pi$
$912$	0	0
$913$	−70.7696	−2.34213
$914$	−25.1485	−0.831840
$915$	0	0
$916$	−0.627417	−0.0207304
$917$	0	0
$918$	0	0
$919$	−13.1716	−0.434490	−0.217245	−	0.976117i	$-0.569707\pi$
$919$	−13.1716	−0.434490	−0.217245	+	0.976117i	$0.569707\pi$
$920$	−25.2751	−0.833295
$921$	0	0
$922$	38.1716	1.25711
$923$	10.3274	0.339929
$924$	0	0
$925$	17.8995	0.588532
$926$	−14.9848	−0.492431
$927$	0	0
$928$	11.5980	0.380722
$929$	−38.6951	−1.26954	−0.634772	−	0.772700i	$-0.718906\pi$
$929$	−38.6951	−1.26954	−0.634772	+	0.772700i	$0.718906\pi$
$930$	0	0
$931$	0	0
$932$	−8.29335	−0.271658
$933$	0	0
$934$	3.55635	0.116367
$935$	−26.1023	−0.853635
$936$	0	0
$937$	−31.2548	−1.02105	−0.510525	−	0.859863i	$-0.670549\pi$
$937$	−31.2548	−1.02105	−0.510525	+	0.859863i	$0.670549\pi$
$938$	0	0
$939$	0	0
$940$	−5.95837	−0.194341
$941$	−10.2903	−0.335454	−0.167727	−	0.985833i	$-0.553643\pi$
$941$	−10.2903	−0.335454	−0.167727	+	0.985833i	$0.553643\pi$
$942$	0	0
$943$	−3.44365	−0.112141
$944$	−5.34267	−0.173889
$945$	0	0
$946$	42.9289	1.39574
$947$	6.16984	0.200493	0.100246	−	0.994963i	$-0.468037\pi$
$947$	6.16984	0.200493	0.100246	+	0.994963i	$0.468037\pi$
$948$	0	0
$949$	10.0589	0.326525
$950$	6.81801	0.221206
$951$	0	0
$952$	0	0
$953$	31.2659	1.01280	0.506402	−	0.862298i	$-0.330975\pi$
$953$	31.2659	1.01280	0.506402	+	0.862298i	$0.330975\pi$
$954$	0	0
$955$	11.4853	0.371655
$956$	−0.737669	−0.0238579
$957$	0	0
$958$	−33.2548	−1.07441
$959$	0	0
$960$	0	0
$961$	82.5685	2.66350
$962$	7.73467	0.249376
$963$	0	0
$964$	10.2426	0.329893
$965$	−29.7906	−0.958994
$966$	0	0
$967$	−17.8995	−0.575609	−0.287804	−	0.957689i	$-0.592925\pi$
$967$	−17.8995	−0.575609	−0.287804	+	0.957689i	$0.592925\pi$
$968$	37.2197	1.19629
$969$	0	0
$970$	15.3137	0.491694
$971$	−0.432117	−0.0138673	−0.00693364	−	0.999976i	$-0.502207\pi$
$971$	−0.432117	−0.0138673	−0.00693364	+	0.999976i	$0.502207\pi$
$972$	0	0
$973$	0	0
$974$	−41.0346	−1.31483
$975$	0	0
$976$	−31.4558	−1.00688
$977$	18.0620	0.577856	0.288928	−	0.957351i	$-0.406701\pi$
$977$	18.0620	0.577856	0.288928	+	0.957351i	$0.406701\pi$
$978$	0	0
$979$	−85.4264	−2.73024
$980$	0	0
$981$	0	0
$982$	−14.1127	−0.450354
$983$	5.77479	0.184187	0.0920936	−	0.995750i	$-0.470644\pi$
$983$	5.77479	0.184187	0.0920936	+	0.995750i	$0.470644\pi$
$984$	0	0
$985$	−4.10051	−0.130653
$986$	−27.2720	−0.868519
$987$	0	0
$988$	−0.769553	−0.0244827
$989$	46.6829	1.48443
$990$	0	0
$991$	−6.97056	−0.221427	−0.110714	−	0.993852i	$-0.535314\pi$
$991$	−6.97056	−0.221427	−0.110714	+	0.993852i	$0.535314\pi$
$992$	−24.5374	−0.779064
$993$	0	0
$994$	0	0
$995$	4.94763	0.156850
$996$	0	0
$997$	−0.870058	−0.0275550	−0.0137775	−	0.999905i	$-0.504386\pi$
$997$	−0.870058	−0.0275550	−0.0137775	+	0.999905i	$0.504386\pi$
$998$	52.3311	1.65651
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.bd.1.2	yes	4
3.2	odd	2	inner	1323.2.a.bd.1.3	yes	4
7.6	odd	2		1323.2.a.bc.1.2	✓	4
21.20	even	2		1323.2.a.bc.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.a.bc.1.2	✓	4	7.6	odd	2
1323.2.a.bc.1.3	yes	4	21.20	even	2
1323.2.a.bd.1.2	yes	4	1.1	even	1	trivial
1323.2.a.bd.1.3	yes	4	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-1.25928 q^{2} -0.414214 q^{4} -1.25928 q^{5} +3.04017 q^{8} +1.58579 q^{10} -4.82106 q^{11} +1.17157 q^{13} -3.00000 q^{16} -4.29945 q^{17} +1.58579 q^{19} +0.521611 q^{20} +6.07107 q^{22} +6.60195 q^{23} -3.41421 q^{25} -1.47534 q^{26} -5.03712 q^{29} +10.6569 q^{31} -2.30250 q^{32} +5.41421 q^{34} -5.24264 q^{37} -1.99695 q^{38} -3.82843 q^{40} -0.521611 q^{41} +7.07107 q^{43} +1.99695 q^{44} -8.31371 q^{46} -11.4230 q^{47} +4.29945 q^{50} -0.485281 q^{52} -1.04322 q^{53} +6.07107 q^{55} +6.34315 q^{58} +1.78089 q^{59} +10.4853 q^{61} -13.4200 q^{62} +8.89949 q^{64} -1.47534 q^{65} +5.65685 q^{67} +1.78089 q^{68} +8.81496 q^{71} +8.58579 q^{73} +6.60195 q^{74} -0.656854 q^{76} -13.8995 q^{79} +3.77784 q^{80} +0.656854 q^{82} +14.6792 q^{83} +5.41421 q^{85} -8.90446 q^{86} -14.6569 q^{88} +17.7194 q^{89} -2.73462 q^{92} +14.3848 q^{94} -1.99695 q^{95} +9.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 12 q^{10} + 16 q^{13} - 12 q^{16} + 12 q^{19} - 4 q^{22} - 8 q^{25} + 20 q^{31} + 16 q^{34} - 4 q^{37} - 4 q^{40} + 12 q^{46} + 32 q^{52} - 4 q^{55} + 48 q^{58} + 8 q^{61} - 4 q^{64} + 40 q^{73}+ \cdots + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 12 * q^10 + 16 * q^13 - 12 * q^16 + 12 * q^19 - 4 * q^22 - 8 * q^25 + 20 * q^31 + 16 * q^34 - 4 * q^37 - 4 * q^40 + 12 * q^46 + 32 * q^52 - 4 * q^55 + 48 * q^58 + 8 * q^61 - 4 * q^64 + 40 * q^73 + 20 * q^76 - 16 * q^79 - 20 * q^82 + 16 * q^85 - 36 * q^88 - 16 * q^94 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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