LMFDB - Embedded newform 3969.2.a.bi.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 $ x^12 - 4x^11 - 22x^10 + 92x^9 + 125x^8 - 620x^7 - 94x^6 + 1280x^5 - 234x^4 - 736x^3 + 96x^2 + 60*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.34584$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0683740 q^{2} -1.99532 q^{4} -2.66379 q^{5} +0.273176 q^{8} +O(q^{10})$	$q-0.0683740 q^{2} -1.99532 q^{4} -2.66379 q^{5} +0.273176 q^{8} +0.182134 q^{10} +1.59913 q^{11} -5.25380 q^{13} +3.97197 q^{16} -6.54721 q^{17} -1.90194 q^{19} +5.31513 q^{20} -0.109339 q^{22} +3.06837 q^{23} +2.09578 q^{25} +0.359223 q^{26} -6.38903 q^{29} -6.71923 q^{31} -0.817932 q^{32} +0.447659 q^{34} +4.22955 q^{37} +0.130043 q^{38} -0.727684 q^{40} +7.39296 q^{41} -11.2635 q^{43} -3.19078 q^{44} -0.209797 q^{46} +3.79918 q^{47} -0.143297 q^{50} +10.4830 q^{52} -8.89862 q^{53} -4.25974 q^{55} +0.436843 q^{58} -10.8928 q^{59} -2.71386 q^{61} +0.459420 q^{62} -7.88802 q^{64} +13.9950 q^{65} -3.32533 q^{67} +13.0638 q^{68} +12.3890 q^{71} -2.19863 q^{73} -0.289191 q^{74} +3.79498 q^{76} +0.813556 q^{79} -10.5805 q^{80} -0.505486 q^{82} +6.83684 q^{83} +17.4404 q^{85} +0.770132 q^{86} +0.436843 q^{88} +0.470572 q^{89} -6.12240 q^{92} -0.259765 q^{94} +5.06636 q^{95} +5.15246 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8	$ 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67} + 56 q^{71} + 68 q^{74} - 12 q^{79} - 12 q^{85} + 76 q^{86} + 16 q^{92} + 64 q^{95}+O(q^{100}) $ 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 20 * q^11 + 12 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^29 + 48 * q^32 + 12 * q^37 + 56 * q^44 - 24 * q^46 - 4 * q^50 + 32 * q^53 + 48 * q^64 + 60 * q^65 + 12 * q^67 + 56 * q^71 + 68 * q^74 - 12 * q^79 - 12 * q^85 + 76 * q^86 + 16 * q^92 + 64 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.0683740	−0.0483477	−0.0241739	−	0.999708i	$-0.507696\pi$
$2$	−0.0683740	−0.0483477	−0.0241739	+	0.999708i	$0.507696\pi$
$3$	0	0
$3$	0	0
$4$	−1.99532	−0.997662
$5$	−2.66379	−1.19128	−0.595642	−	0.803250i	$-0.703102\pi$
$5$	−2.66379	−1.19128	−0.595642	+	0.803250i	$0.703102\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0.273176	0.0965824
$9$	0	0
$10$	0.182134	0.0575958
$11$	1.59913	0.482155	0.241077	−	0.970506i	$-0.422499\pi$
$11$	1.59913	0.482155	0.241077	+	0.970506i	$0.422499\pi$
$12$	0	0
$13$	−5.25380	−1.45714	−0.728571	−	0.684970i	$-0.759815\pi$
$13$	−5.25380	−1.45714	−0.728571	+	0.684970i	$0.759815\pi$
$14$	0	0
$15$	0	0
$16$	3.97197	0.992993
$17$	−6.54721	−1.58793	−0.793966	−	0.607963i	$-0.791987\pi$
$17$	−6.54721	−1.58793	−0.793966	+	0.607963i	$0.791987\pi$
$18$	0	0
$19$	−1.90194	−0.436334	−0.218167	−	0.975911i	$-0.570008\pi$
$19$	−1.90194	−0.436334	−0.218167	+	0.975911i	$0.570008\pi$
$20$	5.31513	1.18850
$21$	0	0
$22$	−0.109339	−0.0233111
$23$	3.06837	0.639800	0.319900	−	0.947451i	$-0.396351\pi$
$23$	3.06837	0.639800	0.319900	+	0.947451i	$0.396351\pi$
$24$	0	0
$25$	2.09578	0.419157
$26$	0.359223	0.0704495
$27$	0	0
$28$	0	0
$29$	−6.38903	−1.18641	−0.593207	−	0.805050i	$-0.702138\pi$
$29$	−6.38903	−1.18641	−0.593207	+	0.805050i	$0.702138\pi$
$30$	0	0
$31$	−6.71923	−1.20681	−0.603405	−	0.797435i	$-0.706190\pi$
$31$	−6.71923	−1.20681	−0.603405	+	0.797435i	$0.706190\pi$
$32$	−0.817932	−0.144591
$33$	0	0
$34$	0.447659	0.0767728
$35$	0	0
$36$	0	0
$37$	4.22955	0.695333	0.347667	−	0.937618i	$-0.386974\pi$
$37$	4.22955	0.695333	0.347667	+	0.937618i	$0.386974\pi$
$38$	0.130043	0.0210958
$39$	0	0
$40$	−0.727684	−0.115057
$41$	7.39296	1.15459	0.577293	−	0.816537i	$-0.304109\pi$
$41$	7.39296	1.15459	0.577293	+	0.816537i	$0.304109\pi$
$42$	0	0
$43$	−11.2635	−1.71767	−0.858836	−	0.512251i	$-0.828812\pi$
$43$	−11.2635	−1.71767	−0.858836	+	0.512251i	$0.828812\pi$
$44$	−3.19078	−0.481028
$45$	0	0
$46$	−0.209797	−0.0309329
$47$	3.79918	0.554167	0.277083	−	0.960846i	$-0.410632\pi$
$47$	3.79918	0.554167	0.277083	+	0.960846i	$0.410632\pi$
$48$	0	0
$49$	0	0
$50$	−0.143297	−0.0202653
$51$	0	0
$52$	10.4830	1.45374
$53$	−8.89862	−1.22232	−0.611160	−	0.791507i	$-0.709297\pi$
$53$	−8.89862	−1.22232	−0.611160	+	0.791507i	$0.709297\pi$
$54$	0	0
$55$	−4.25974	−0.574383
$56$	0	0
$57$	0	0
$58$	0.436843	0.0573604
$59$	−10.8928	−1.41812	−0.709060	−	0.705148i	$-0.750880\pi$
$59$	−10.8928	−1.41812	−0.709060	+	0.705148i	$0.750880\pi$
$60$	0	0
$61$	−2.71386	−0.347475	−0.173737	−	0.984792i	$-0.555584\pi$
$61$	−2.71386	−0.347475	−0.173737	+	0.984792i	$0.555584\pi$
$62$	0.459420	0.0583465
$63$	0	0
$64$	−7.88802	−0.986002
$65$	13.9950	1.73587
$66$	0	0
$67$	−3.32533	−0.406254	−0.203127	−	0.979152i	$-0.565110\pi$
$67$	−3.32533	−0.406254	−0.203127	+	0.979152i	$0.565110\pi$
$68$	13.0638	1.58422
$69$	0	0
$70$	0	0
$71$	12.3890	1.47031	0.735154	−	0.677900i	$-0.237110\pi$
$71$	12.3890	1.47031	0.735154	+	0.677900i	$0.237110\pi$
$72$	0	0
$73$	−2.19863	−0.257331	−0.128665	−	0.991688i	$-0.541069\pi$
$73$	−2.19863	−0.257331	−0.128665	+	0.991688i	$0.541069\pi$
$74$	−0.289191	−0.0336178
$75$	0	0
$76$	3.79498	0.435314
$77$	0	0
$78$	0	0
$79$	0.813556	0.0915322	0.0457661	−	0.998952i	$-0.485427\pi$
$79$	0.813556	0.0915322	0.0457661	+	0.998952i	$0.485427\pi$
$80$	−10.5805	−1.18294
$81$	0	0
$82$	−0.505486	−0.0558216
$83$	6.83684	0.750441	0.375220	−	0.926936i	$-0.377567\pi$
$83$	6.83684	0.750441	0.375220	+	0.926936i	$0.377567\pi$
$84$	0	0
$85$	17.4404	1.89168
$86$	0.770132	0.0830455
$87$	0	0
$88$	0.436843	0.0465677
$89$	0.470572	0.0498805	0.0249403	−	0.999689i	$-0.492060\pi$
$89$	0.470572	0.0498805	0.0249403	+	0.999689i	$0.492060\pi$
$90$	0	0
$91$	0	0
$92$	−6.12240	−0.638305
$93$	0	0
$94$	−0.259765	−0.0267927
$95$	5.06636	0.519798
$96$	0	0
$97$	5.15246	0.523153	0.261576	−	0.965183i	$-0.415758\pi$
$97$	5.15246	0.523153	0.261576	+	0.965183i	$0.415758\pi$
$98$	0	0
$99$	0	0
$100$	−4.18177	−0.418177
$101$	−1.84488	−0.183572	−0.0917862	−	0.995779i	$-0.529258\pi$
$101$	−1.84488	−0.183572	−0.0917862	+	0.995779i	$0.529258\pi$
$102$	0	0
$103$	−5.17802	−0.510206	−0.255103	−	0.966914i	$-0.582109\pi$
$103$	−5.17802	−0.510206	−0.255103	+	0.966914i	$0.582109\pi$
$104$	−1.43521	−0.140734
$105$	0	0
$106$	0.608434	0.0590964
$107$	16.9489	1.63851	0.819256	−	0.573428i	$-0.194387\pi$
$107$	16.9489	1.63851	0.819256	+	0.573428i	$0.194387\pi$
$108$	0	0
$109$	−8.49992	−0.814145	−0.407073	−	0.913396i	$-0.633450\pi$
$109$	−8.49992	−0.814145	−0.407073	+	0.913396i	$0.633450\pi$
$110$	0.291255	0.0277701
$111$	0	0
$112$	0	0
$113$	−3.90392	−0.367250	−0.183625	−	0.982996i	$-0.558783\pi$
$113$	−3.90392	−0.367250	−0.183625	+	0.982996i	$0.558783\pi$
$114$	0	0
$115$	−8.17351	−0.762183
$116$	12.7482	1.18364
$117$	0	0
$118$	0.744783	0.0685628
$119$	0	0
$120$	0	0
$121$	−8.44279	−0.767527
$122$	0.185558	0.0167996
$123$	0	0
$124$	13.4070	1.20399
$125$	7.73623	0.691949
$126$	0	0
$127$	10.9533	0.971946	0.485973	−	0.873974i	$-0.338465\pi$
$127$	10.9533	0.971946	0.485973	+	0.873974i	$0.338465\pi$
$128$	2.17520	0.192262
$129$	0	0
$130$	−0.956896	−0.0839253
$131$	−4.45342	−0.389097	−0.194549	−	0.980893i	$-0.562324\pi$
$131$	−4.45342	−0.389097	−0.194549	+	0.980893i	$0.562324\pi$
$132$	0	0
$133$	0	0
$134$	0.227366	0.0196414
$135$	0	0
$136$	−1.78854	−0.153366
$137$	19.5360	1.66907	0.834537	−	0.550952i	$-0.185735\pi$
$137$	19.5360	1.66907	0.834537	+	0.550952i	$0.185735\pi$
$138$	0	0
$139$	−2.63079	−0.223141	−0.111570	−	0.993757i	$-0.535588\pi$
$139$	−2.63079	−0.223141	−0.111570	+	0.993757i	$0.535588\pi$
$140$	0	0
$141$	0	0
$142$	−0.847087	−0.0710860
$143$	−8.40149	−0.702568
$144$	0	0
$145$	17.0190	1.41335
$146$	0.150329	0.0124413
$147$	0	0
$148$	−8.43932	−0.693708
$149$	8.81281	0.721973	0.360987	−	0.932571i	$-0.382440\pi$
$149$	8.81281	0.721973	0.360987	+	0.932571i	$0.382440\pi$
$150$	0	0
$151$	4.66422	0.379569	0.189784	−	0.981826i	$-0.439221\pi$
$151$	4.66422	0.379569	0.189784	+	0.981826i	$0.439221\pi$
$152$	−0.519564	−0.0421422
$153$	0	0
$154$	0	0
$155$	17.8986	1.43765
$156$	0	0
$157$	4.07294	0.325056	0.162528	−	0.986704i	$-0.448035\pi$
$157$	4.07294	0.325056	0.162528	+	0.986704i	$0.448035\pi$
$158$	−0.0556261	−0.00442537
$159$	0	0
$160$	2.17880	0.172249
$161$	0	0
$162$	0	0
$163$	−12.1222	−0.949487	−0.474744	−	0.880124i	$-0.657459\pi$
$163$	−12.1222	−0.949487	−0.474744	+	0.880124i	$0.657459\pi$
$164$	−14.7514	−1.15189
$165$	0	0
$166$	−0.467462	−0.0362821
$167$	−4.79902	−0.371360	−0.185680	−	0.982610i	$-0.559449\pi$
$167$	−4.79902	−0.371360	−0.185680	+	0.982610i	$0.559449\pi$
$168$	0	0
$169$	14.6024	1.12326
$170$	−1.19247	−0.0914582
$171$	0	0
$172$	22.4744	1.71366
$173$	5.03171	0.382554	0.191277	−	0.981536i	$-0.438737\pi$
$173$	5.03171	0.382554	0.191277	+	0.981536i	$0.438737\pi$
$174$	0	0
$175$	0	0
$176$	6.35169	0.478776
$177$	0	0
$178$	−0.0321749	−0.00241161
$179$	16.3979	1.22564	0.612819	−	0.790224i	$-0.290036\pi$
$179$	16.3979	1.22564	0.612819	+	0.790224i	$0.290036\pi$
$180$	0	0
$181$	14.4345	1.07291	0.536454	−	0.843930i	$-0.319763\pi$
$181$	14.4345	1.07291	0.536454	+	0.843930i	$0.319763\pi$
$182$	0	0
$183$	0	0
$184$	0.838207	0.0617934
$185$	−11.2666	−0.828339
$186$	0	0
$187$	−10.4698	−0.765629
$188$	−7.58059	−0.552871
$189$	0	0
$190$	−0.346407	−0.0251310
$191$	−2.84131	−0.205590	−0.102795	−	0.994703i	$-0.532779\pi$
$191$	−2.84131	−0.205590	−0.102795	+	0.994703i	$0.532779\pi$
$192$	0	0
$193$	8.82886	0.635515	0.317758	−	0.948172i	$-0.397070\pi$
$193$	8.82886	0.635515	0.317758	+	0.948172i	$0.397070\pi$
$194$	−0.352294	−0.0252932
$195$	0	0
$196$	0	0
$197$	−5.72354	−0.407785	−0.203893	−	0.978993i	$-0.565359\pi$
$197$	−5.72354	−0.407785	−0.203893	+	0.978993i	$0.565359\pi$
$198$	0	0
$199$	11.4150	0.809191	0.404596	−	0.914496i	$-0.367412\pi$
$199$	11.4150	0.809191	0.404596	+	0.914496i	$0.367412\pi$
$200$	0.572518	0.0404832
$201$	0	0
$202$	0.126142	0.00887530
$203$	0	0
$204$	0	0
$205$	−19.6933	−1.37544
$206$	0.354042	0.0246673
$207$	0	0
$208$	−20.8679	−1.44693
$209$	−3.04144	−0.210381
$210$	0	0
$211$	−21.3837	−1.47212	−0.736059	−	0.676918i	$-0.763315\pi$
$211$	−21.3837	−1.47212	−0.736059	+	0.676918i	$0.763315\pi$
$212$	17.7556	1.21946
$213$	0	0
$214$	−1.15886	−0.0792183
$215$	30.0037	2.04623
$216$	0	0
$217$	0	0
$218$	0.581174	0.0393620
$219$	0	0
$220$	8.49956	0.573041
$221$	34.3977	2.31384
$222$	0	0
$223$	−7.16774	−0.479987	−0.239994	−	0.970774i	$-0.577145\pi$
$223$	−7.16774	−0.479987	−0.239994	+	0.970774i	$0.577145\pi$
$224$	0	0
$225$	0	0
$226$	0.266926	0.0177557
$227$	−13.7887	−0.915187	−0.457593	−	0.889162i	$-0.651288\pi$
$227$	−13.7887	−0.915187	−0.457593	+	0.889162i	$0.651288\pi$
$228$	0	0
$229$	26.3943	1.74418	0.872092	−	0.489341i	$-0.162763\pi$
$229$	26.3943	1.74418	0.872092	+	0.489341i	$0.162763\pi$
$230$	0.558855	0.0368498
$231$	0	0
$232$	−1.74533	−0.114587
$233$	12.6446	0.828375	0.414187	−	0.910192i	$-0.364066\pi$
$233$	12.6446	0.828375	0.414187	+	0.910192i	$0.364066\pi$
$234$	0	0
$235$	−10.1202	−0.660170
$236$	21.7346	1.41481
$237$	0	0
$238$	0	0
$239$	15.4328	0.998265	0.499133	−	0.866526i	$-0.333652\pi$
$239$	15.4328	0.998265	0.499133	+	0.866526i	$0.333652\pi$
$240$	0	0
$241$	−1.17988	−0.0760029	−0.0380015	−	0.999278i	$-0.512099\pi$
$241$	−1.17988	−0.0760029	−0.0380015	+	0.999278i	$0.512099\pi$
$242$	0.577267	0.0371082
$243$	0	0
$244$	5.41504	0.346662
$245$	0	0
$246$	0	0
$247$	9.99240	0.635801
$248$	−1.83553	−0.116557
$249$	0	0
$250$	−0.528957	−0.0334542
$251$	5.54970	0.350294	0.175147	−	0.984542i	$-0.443960\pi$
$251$	5.54970	0.350294	0.175147	+	0.984542i	$0.443960\pi$
$252$	0	0
$253$	4.90672	0.308483
$254$	−0.748919	−0.0469913
$255$	0	0
$256$	15.6273	0.976707
$257$	9.83076	0.613226	0.306613	−	0.951834i	$-0.400804\pi$
$257$	9.83076	0.613226	0.306613	+	0.951834i	$0.400804\pi$
$258$	0	0
$259$	0	0
$260$	−27.9246	−1.73181
$261$	0	0
$262$	0.304498	0.0188120
$263$	−11.9322	−0.735774	−0.367887	−	0.929870i	$-0.619919\pi$
$263$	−11.9322	−0.735774	−0.367887	+	0.929870i	$0.619919\pi$
$264$	0	0
$265$	23.7041	1.45613
$266$	0	0
$267$	0	0
$268$	6.63512	0.405304
$269$	29.9648	1.82699	0.913494	−	0.406853i	$-0.133374\pi$
$269$	29.9648	1.82699	0.913494	+	0.406853i	$0.133374\pi$
$270$	0	0
$271$	7.09650	0.431082	0.215541	−	0.976495i	$-0.430849\pi$
$271$	7.09650	0.431082	0.215541	+	0.976495i	$0.430849\pi$
$272$	−26.0053	−1.57680
$273$	0	0
$274$	−1.33575	−0.0806959
$275$	3.35142	0.202098
$276$	0	0
$277$	−9.82351	−0.590237	−0.295119	−	0.955461i	$-0.595359\pi$
$277$	−9.82351	−0.590237	−0.295119	+	0.955461i	$0.595359\pi$
$278$	0.179878	0.0107883
$279$	0	0
$280$	0	0
$281$	23.8777	1.42443	0.712213	−	0.701963i	$-0.247693\pi$
$281$	23.8777	1.42443	0.712213	+	0.701963i	$0.247693\pi$
$282$	0	0
$283$	3.01595	0.179280	0.0896399	−	0.995974i	$-0.471428\pi$
$283$	3.01595	0.179280	0.0896399	+	0.995974i	$0.471428\pi$
$284$	−24.7201	−1.46687
$285$	0	0
$286$	0.574443	0.0339675
$287$	0	0
$288$	0	0
$289$	25.8659	1.52153
$290$	−1.16366	−0.0683324
$291$	0	0
$292$	4.38699	0.256729
$293$	17.0583	0.996554	0.498277	−	0.867018i	$-0.333966\pi$
$293$	17.0583	0.996554	0.498277	+	0.867018i	$0.333966\pi$
$294$	0	0
$295$	29.0161	1.68938
$296$	1.15541	0.0671570
$297$	0	0
$298$	−0.602567	−0.0349058
$299$	−16.1206	−0.932280
$300$	0	0
$301$	0	0
$302$	−0.318911	−0.0183513
$303$	0	0
$304$	−7.55444	−0.433277
$305$	7.22917	0.413941
$306$	0	0
$307$	−23.2178	−1.32511	−0.662554	−	0.749014i	$-0.730527\pi$
$307$	−23.2178	−1.32511	−0.662554	+	0.749014i	$0.730527\pi$
$308$	0	0
$309$	0	0
$310$	−1.22380	−0.0695072
$311$	1.79093	0.101555	0.0507773	−	0.998710i	$-0.483830\pi$
$311$	1.79093	0.101555	0.0507773	+	0.998710i	$0.483830\pi$
$312$	0	0
$313$	−4.60917	−0.260526	−0.130263	−	0.991480i	$-0.541582\pi$
$313$	−4.60917	−0.260526	−0.130263	+	0.991480i	$0.541582\pi$
$314$	−0.278483	−0.0157157
$315$	0	0
$316$	−1.62331	−0.0913183
$317$	25.8841	1.45380	0.726898	−	0.686745i	$-0.240961\pi$
$317$	25.8841	1.45380	0.726898	+	0.686745i	$0.240961\pi$
$318$	0	0
$319$	−10.2169	−0.572035
$320$	21.0120	1.17461
$321$	0	0
$322$	0	0
$323$	12.4524	0.692869
$324$	0	0
$325$	−11.0108	−0.610771
$326$	0.828846	0.0459055
$327$	0	0
$328$	2.01958	0.111513
$329$	0	0
$330$	0	0
$331$	0.161323	0.00886714	0.00443357	−	0.999990i	$-0.498589\pi$
$331$	0.161323	0.00886714	0.00443357	+	0.999990i	$0.498589\pi$
$332$	−13.6417	−0.748687
$333$	0	0
$334$	0.328128	0.0179544
$335$	8.85799	0.483964
$336$	0	0
$337$	−9.05351	−0.493176	−0.246588	−	0.969120i	$-0.579309\pi$
$337$	−9.05351	−0.493176	−0.246588	+	0.969120i	$0.579309\pi$
$338$	−0.998425	−0.0543072
$339$	0	0
$340$	−34.7993	−1.88725
$341$	−10.7449	−0.581869
$342$	0	0
$343$	0	0
$344$	−3.07693	−0.165897
$345$	0	0
$346$	−0.344038	−0.0184956
$347$	5.81968	0.312417	0.156208	−	0.987724i	$-0.450073\pi$
$347$	5.81968	0.312417	0.156208	+	0.987724i	$0.450073\pi$
$348$	0	0
$349$	−27.2619	−1.45930	−0.729648	−	0.683823i	$-0.760316\pi$
$349$	−27.2619	−1.45930	−0.729648	+	0.683823i	$0.760316\pi$
$350$	0	0
$351$	0	0
$352$	−1.30798	−0.0697154
$353$	−24.1896	−1.28748	−0.643741	−	0.765244i	$-0.722618\pi$
$353$	−24.1896	−1.28748	−0.643741	+	0.765244i	$0.722618\pi$
$354$	0	0
$355$	−33.0018	−1.75155
$356$	−0.938944	−0.0497639
$357$	0	0
$358$	−1.12119	−0.0592568
$359$	21.0376	1.11032	0.555161	−	0.831743i	$-0.312657\pi$
$359$	21.0376	1.11032	0.555161	+	0.831743i	$0.312657\pi$
$360$	0	0
$361$	−15.3826	−0.809612
$362$	−0.986944	−0.0518726
$363$	0	0
$364$	0	0
$365$	5.85670	0.306554
$366$	0	0
$367$	−35.0380	−1.82897	−0.914485	−	0.404620i	$-0.867404\pi$
$367$	−35.0380	−1.82897	−0.914485	+	0.404620i	$0.867404\pi$
$368$	12.1875	0.635317
$369$	0	0
$370$	0.770345	0.0400483
$371$	0	0
$372$	0	0
$373$	1.12862	0.0584377	0.0292189	−	0.999573i	$-0.490698\pi$
$373$	1.12862	0.0584377	0.0292189	+	0.999573i	$0.490698\pi$
$374$	0.715863	0.0370164
$375$	0	0
$376$	1.03784	0.0535227
$377$	33.5667	1.72877
$378$	0	0
$379$	−21.9619	−1.12811	−0.564054	−	0.825738i	$-0.690759\pi$
$379$	−21.9619	−1.12811	−0.564054	+	0.825738i	$0.690759\pi$
$380$	−10.1090	−0.518583
$381$	0	0
$382$	0.194272	0.00993981
$383$	−23.0401	−1.17729	−0.588647	−	0.808390i	$-0.700339\pi$
$383$	−23.0401	−1.17729	−0.588647	+	0.808390i	$0.700339\pi$
$384$	0	0
$385$	0	0
$386$	−0.603664	−0.0307257
$387$	0	0
$388$	−10.2808	−0.521930
$389$	−15.7751	−0.799828	−0.399914	−	0.916553i	$-0.630960\pi$
$389$	−15.7751	−0.799828	−0.399914	+	0.916553i	$0.630960\pi$
$390$	0	0
$391$	−20.0893	−1.01596
$392$	0	0
$393$	0	0
$394$	0.391341	0.0197155
$395$	−2.16714	−0.109041
$396$	0	0
$397$	−16.5055	−0.828389	−0.414195	−	0.910188i	$-0.635937\pi$
$397$	−16.5055	−0.828389	−0.414195	+	0.910188i	$0.635937\pi$
$398$	−0.780492	−0.0391225
$399$	0	0
$400$	8.32439	0.416220
$401$	−21.6600	−1.08165	−0.540823	−	0.841136i	$-0.681887\pi$
$401$	−21.6600	−1.08165	−0.540823	+	0.841136i	$0.681887\pi$
$402$	0	0
$403$	35.3015	1.75849
$404$	3.68113	0.183143
$405$	0	0
$406$	0	0
$407$	6.76358	0.335258
$408$	0	0
$409$	30.5721	1.51169	0.755846	−	0.654750i	$-0.227226\pi$
$409$	30.5721	1.51169	0.755846	+	0.654750i	$0.227226\pi$
$410$	1.34651	0.0664993
$411$	0	0
$412$	10.3318	0.509013
$413$	0	0
$414$	0	0
$415$	−18.2119	−0.893988
$416$	4.29725	0.210690
$417$	0	0
$418$	0.207955	0.0101714
$419$	−21.6162	−1.05602	−0.528011	−	0.849237i	$-0.677062\pi$
$419$	−21.6162	−1.05602	−0.528011	+	0.849237i	$0.677062\pi$
$420$	0	0
$421$	−27.2434	−1.32776	−0.663881	−	0.747838i	$-0.731092\pi$
$421$	−27.2434	−1.32776	−0.663881	+	0.747838i	$0.731092\pi$
$422$	1.46209	0.0711735
$423$	0	0
$424$	−2.43089	−0.118055
$425$	−13.7215	−0.665592
$426$	0	0
$427$	0	0
$428$	−33.8186	−1.63468
$429$	0	0
$430$	−2.05147	−0.0989307
$431$	−8.19687	−0.394829	−0.197415	−	0.980320i	$-0.563255\pi$
$431$	−8.19687	−0.394829	−0.197415	+	0.980320i	$0.563255\pi$
$432$	0	0
$433$	−3.41468	−0.164099	−0.0820494	−	0.996628i	$-0.526147\pi$
$433$	−3.41468	−0.164099	−0.0820494	+	0.996628i	$0.526147\pi$
$434$	0	0
$435$	0	0
$436$	16.9601	0.812242
$437$	−5.83585	−0.279167
$438$	0	0
$439$	−6.58831	−0.314443	−0.157221	−	0.987563i	$-0.550254\pi$
$439$	−6.58831	−0.314443	−0.157221	+	0.987563i	$0.550254\pi$
$440$	−1.16366	−0.0554753
$441$	0	0
$442$	−2.35191	−0.111869
$443$	−28.6912	−1.36316	−0.681581	−	0.731743i	$-0.738707\pi$
$443$	−28.6912	−1.36316	−0.681581	+	0.731743i	$0.738707\pi$
$444$	0	0
$445$	−1.25350	−0.0594218
$446$	0.490087	0.0232063
$447$	0	0
$448$	0	0
$449$	−0.457724	−0.0216013	−0.0108007	−	0.999942i	$-0.503438\pi$
$449$	−0.457724	−0.0216013	−0.0108007	+	0.999942i	$0.503438\pi$
$450$	0	0
$451$	11.8223	0.556689
$452$	7.78958	0.366391
$453$	0	0
$454$	0.942787	0.0442472
$455$	0	0
$456$	0	0
$457$	20.2210	0.945900	0.472950	−	0.881089i	$-0.343189\pi$
$457$	20.2210	0.945900	0.472950	+	0.881089i	$0.343189\pi$
$458$	−1.80468	−0.0843273
$459$	0	0
$460$	16.3088	0.760402
$461$	24.2071	1.12744	0.563719	−	0.825967i	$-0.309370\pi$
$461$	24.2071	1.12744	0.563719	+	0.825967i	$0.309370\pi$
$462$	0	0
$463$	−4.80483	−0.223299	−0.111650	−	0.993748i	$-0.535613\pi$
$463$	−4.80483	−0.223299	−0.111650	+	0.993748i	$0.535613\pi$
$464$	−25.3770	−1.17810
$465$	0	0
$466$	−0.864561	−0.0400500
$467$	−27.2456	−1.26078	−0.630389	−	0.776279i	$-0.717105\pi$
$467$	−27.2456	−1.26078	−0.630389	+	0.776279i	$0.717105\pi$
$468$	0	0
$469$	0	0
$470$	0.691959	0.0319177
$471$	0	0
$472$	−2.97565	−0.136965
$473$	−18.0118	−0.828184
$474$	0	0
$475$	−3.98605	−0.182892
$476$	0	0
$477$	0	0
$478$	−1.05520	−0.0482638
$479$	20.5255	0.937834	0.468917	−	0.883242i	$-0.344644\pi$
$479$	20.5255	0.937834	0.468917	+	0.883242i	$0.344644\pi$
$480$	0	0
$481$	−22.2212	−1.01320
$482$	0.0806733	0.00367457
$483$	0	0
$484$	16.8461	0.765733
$485$	−13.7251	−0.623224
$486$	0	0
$487$	25.8449	1.17114	0.585571	−	0.810621i	$-0.300870\pi$
$487$	25.8449	1.17114	0.585571	+	0.810621i	$0.300870\pi$
$488$	−0.741363	−0.0335599
$489$	0	0
$490$	0	0
$491$	−15.6155	−0.704718	−0.352359	−	0.935865i	$-0.614620\pi$
$491$	−15.6155	−0.704718	−0.352359	+	0.935865i	$0.614620\pi$
$492$	0	0
$493$	41.8303	1.88394
$494$	−0.683220	−0.0307395
$495$	0	0
$496$	−26.6886	−1.19835
$497$	0	0
$498$	0	0
$499$	21.2690	0.952133	0.476066	−	0.879409i	$-0.342062\pi$
$499$	21.2690	0.952133	0.476066	+	0.879409i	$0.342062\pi$
$500$	−15.4363	−0.690332
$501$	0	0
$502$	−0.379455	−0.0169359
$503$	−16.3298	−0.728110	−0.364055	−	0.931377i	$-0.618608\pi$
$503$	−16.3298	−0.728110	−0.364055	+	0.931377i	$0.618608\pi$
$504$	0	0
$505$	4.91437	0.218687
$506$	−0.335492	−0.0149144
$507$	0	0
$508$	−21.8553	−0.969674
$509$	−13.4618	−0.596683	−0.298342	−	0.954459i	$-0.596433\pi$
$509$	−13.4618	−0.596683	−0.298342	+	0.954459i	$0.596433\pi$
$510$	0	0
$511$	0	0
$512$	−5.41890	−0.239484
$513$	0	0
$514$	−0.672168	−0.0296481
$515$	13.7932	0.607800
$516$	0	0
$517$	6.07536	0.267194
$518$	0	0
$519$	0	0
$520$	3.82311	0.167654
$521$	−1.42619	−0.0624826	−0.0312413	−	0.999512i	$-0.509946\pi$
$521$	−1.42619	−0.0624826	−0.0312413	+	0.999512i	$0.509946\pi$
$522$	0	0
$523$	7.71060	0.337161	0.168581	−	0.985688i	$-0.446082\pi$
$523$	7.71060	0.337161	0.168581	+	0.985688i	$0.446082\pi$
$524$	8.88603	0.388188
$525$	0	0
$526$	0.815855	0.0355730
$527$	43.9922	1.91633
$528$	0	0
$529$	−13.5851	−0.590656
$530$	−1.62074	−0.0704005
$531$	0	0
$532$	0	0
$533$	−38.8411	−1.68240
$534$	0	0
$535$	−45.1483	−1.95193
$536$	−0.908402	−0.0392370
$537$	0	0
$538$	−2.04881	−0.0883306
$539$	0	0
$540$	0	0
$541$	28.0456	1.20577	0.602886	−	0.797827i	$-0.294017\pi$
$541$	28.0456	1.20577	0.602886	+	0.797827i	$0.294017\pi$
$542$	−0.485216	−0.0208418
$543$	0	0
$544$	5.35517	0.229601
$545$	22.6420	0.969878
$546$	0	0
$547$	−35.4610	−1.51620	−0.758101	−	0.652137i	$-0.773873\pi$
$547$	−35.4610	−1.51620	−0.758101	+	0.652137i	$0.773873\pi$
$548$	−38.9807	−1.66517
$549$	0	0
$550$	−0.229150	−0.00977099
$551$	12.1515	0.517673
$552$	0	0
$553$	0	0
$554$	0.671672	0.0285366
$555$	0	0
$556$	5.24928	0.222619
$557$	−35.0419	−1.48477	−0.742386	−	0.669972i	$-0.766306\pi$
$557$	−35.0419	−1.48477	−0.742386	+	0.669972i	$0.766306\pi$
$558$	0	0
$559$	59.1763	2.50289
$560$	0	0
$561$	0	0
$562$	−1.63262	−0.0688677
$563$	16.0262	0.675425	0.337712	−	0.941249i	$-0.390347\pi$
$563$	16.0262	0.675425	0.337712	+	0.941249i	$0.390347\pi$
$564$	0	0
$565$	10.3992	0.437499
$566$	−0.206213	−0.00866776
$567$	0	0
$568$	3.38439	0.142006
$569$	−0.371302	−0.0155658	−0.00778290	−	0.999970i	$-0.502477\pi$
$569$	−0.371302	−0.0155658	−0.00778290	+	0.999970i	$0.502477\pi$
$570$	0	0
$571$	29.2304	1.22325	0.611626	−	0.791147i	$-0.290516\pi$
$571$	29.2304	1.22325	0.611626	+	0.791147i	$0.290516\pi$
$572$	16.7637	0.700926
$573$	0	0
$574$	0	0
$575$	6.43065	0.268177
$576$	0	0
$577$	15.0570	0.626833	0.313417	−	0.949616i	$-0.398526\pi$
$577$	15.0570	0.626833	0.313417	+	0.949616i	$0.398526\pi$
$578$	−1.76856	−0.0735623
$579$	0	0
$580$	−33.9585	−1.41005
$581$	0	0
$582$	0	0
$583$	−14.2300	−0.589347
$584$	−0.600615	−0.0248536
$585$	0	0
$586$	−1.16634	−0.0481811
$587$	1.67180	0.0690026	0.0345013	−	0.999405i	$-0.489016\pi$
$587$	1.67180	0.0690026	0.0345013	+	0.999405i	$0.489016\pi$
$588$	0	0
$589$	12.7795	0.526572
$590$	−1.98395	−0.0816778
$591$	0	0
$592$	16.7996	0.690461
$593$	−10.8174	−0.444218	−0.222109	−	0.975022i	$-0.571294\pi$
$593$	−10.8174	−0.444218	−0.222109	+	0.975022i	$0.571294\pi$
$594$	0	0
$595$	0	0
$596$	−17.5844	−0.720286
$597$	0	0
$598$	1.10223	0.0450736
$599$	−16.6401	−0.679898	−0.339949	−	0.940444i	$-0.610410\pi$
$599$	−16.6401	−0.679898	−0.339949	+	0.940444i	$0.610410\pi$
$600$	0	0
$601$	25.8022	1.05249	0.526246	−	0.850332i	$-0.323599\pi$
$601$	25.8022	1.05249	0.526246	+	0.850332i	$0.323599\pi$
$602$	0	0
$603$	0	0
$604$	−9.30663	−0.378681
$605$	22.4898	0.914342
$606$	0	0
$607$	37.8049	1.53445	0.767227	−	0.641376i	$-0.221636\pi$
$607$	37.8049	1.53445	0.767227	+	0.641376i	$0.221636\pi$
$608$	1.55566	0.0630901
$609$	0	0
$610$	−0.494287	−0.0200131
$611$	−19.9601	−0.807499
$612$	0	0
$613$	−12.9544	−0.523222	−0.261611	−	0.965173i	$-0.584254\pi$
$613$	−12.9544	−0.523222	−0.261611	+	0.965173i	$0.584254\pi$
$614$	1.58749	0.0640659
$615$	0	0
$616$	0	0
$617$	32.4403	1.30600	0.652999	−	0.757359i	$-0.273511\pi$
$617$	32.4403	1.30600	0.652999	+	0.757359i	$0.273511\pi$
$618$	0	0
$619$	33.1974	1.33431	0.667157	−	0.744917i	$-0.267511\pi$
$619$	33.1974	1.33431	0.667157	+	0.744917i	$0.267511\pi$
$620$	−35.7136	−1.43429
$621$	0	0
$622$	−0.122453	−0.00490993
$623$	0	0
$624$	0	0
$625$	−31.0866	−1.24346
$626$	0.315147	0.0125958
$627$	0	0
$628$	−8.12683	−0.324296
$629$	−27.6917	−1.10414
$630$	0	0
$631$	32.2773	1.28494	0.642470	−	0.766311i	$-0.277910\pi$
$631$	32.2773	1.28494	0.642470	+	0.766311i	$0.277910\pi$
$632$	0.222244	0.00884040
$633$	0	0
$634$	−1.76980	−0.0702877
$635$	−29.1772	−1.15786
$636$	0	0
$637$	0	0
$638$	0.698568	0.0276566
$639$	0	0
$640$	−5.79428	−0.229039
$641$	−43.0814	−1.70161	−0.850806	−	0.525480i	$-0.823886\pi$
$641$	−43.0814	−1.70161	−0.850806	+	0.525480i	$0.823886\pi$
$642$	0	0
$643$	6.40176	0.252461	0.126230	−	0.992001i	$-0.459712\pi$
$643$	6.40176	0.252461	0.126230	+	0.992001i	$0.459712\pi$
$644$	0	0
$645$	0	0
$646$	−0.851419	−0.0334986
$647$	−3.88807	−0.152856	−0.0764278	−	0.997075i	$-0.524351\pi$
$647$	−3.88807	−0.152856	−0.0764278	+	0.997075i	$0.524351\pi$
$648$	0	0
$649$	−17.4189	−0.683753
$650$	0.752854	0.0295294
$651$	0	0
$652$	24.1878	0.947268
$653$	−15.1035	−0.591044	−0.295522	−	0.955336i	$-0.595494\pi$
$653$	−15.1035	−0.591044	−0.295522	+	0.955336i	$0.595494\pi$
$654$	0	0
$655$	11.8630	0.463525
$656$	29.3646	1.14650
$657$	0	0
$658$	0	0
$659$	−14.2600	−0.555492	−0.277746	−	0.960655i	$-0.589587\pi$
$659$	−14.2600	−0.555492	−0.277746	+	0.960655i	$0.589587\pi$
$660$	0	0
$661$	−19.4193	−0.755323	−0.377662	−	0.925944i	$-0.623272\pi$
$661$	−19.4193	−0.755323	−0.377662	+	0.925944i	$0.623272\pi$
$662$	−0.0110303	−0.000428706 0
$663$	0	0
$664$	1.86766	0.0724794
$665$	0	0
$666$	0	0
$667$	−19.6039	−0.759067
$668$	9.57561	0.370492
$669$	0	0
$670$	−0.605656	−0.0233985
$671$	−4.33981	−0.167537
$672$	0	0
$673$	5.93126	0.228633	0.114317	−	0.993444i	$-0.463532\pi$
$673$	5.93126	0.228633	0.114317	+	0.993444i	$0.463532\pi$
$674$	0.619024	0.0238439
$675$	0	0
$676$	−29.1366	−1.12064
$677$	−36.9826	−1.42136	−0.710678	−	0.703518i	$-0.751612\pi$
$677$	−36.9826	−1.42136	−0.710678	+	0.703518i	$0.751612\pi$
$678$	0	0
$679$	0	0
$680$	4.76430	0.182703
$681$	0	0
$682$	0.734671	0.0281320
$683$	−13.1360	−0.502635	−0.251317	−	0.967905i	$-0.580864\pi$
$683$	−13.1360	−0.502635	−0.251317	+	0.967905i	$0.580864\pi$
$684$	0	0
$685$	−52.0398	−1.98834
$686$	0	0
$687$	0	0
$688$	−44.7384	−1.70564
$689$	46.7516	1.78109
$690$	0	0
$691$	14.7658	0.561719	0.280860	−	0.959749i	$-0.409380\pi$
$691$	14.7658	0.561719	0.280860	+	0.959749i	$0.409380\pi$
$692$	−10.0399	−0.381659
$693$	0	0
$694$	−0.397915	−0.0151046
$695$	7.00788	0.265824
$696$	0	0
$697$	−48.4032	−1.83340
$698$	1.86401	0.0705536
$699$	0	0
$700$	0	0
$701$	30.4627	1.15056	0.575281	−	0.817956i	$-0.304893\pi$
$701$	30.4627	1.15056	0.575281	+	0.817956i	$0.304893\pi$
$702$	0	0
$703$	−8.04433	−0.303398
$704$	−12.6139	−0.475406
$705$	0	0
$706$	1.65394	0.0622468
$707$	0	0
$708$	0	0
$709$	14.1030	0.529650	0.264825	−	0.964296i	$-0.414686\pi$
$709$	14.1030	0.529650	0.264825	+	0.964296i	$0.414686\pi$
$710$	2.25646	0.0846836
$711$	0	0
$712$	0.128549	0.00481758
$713$	−20.6171	−0.772117
$714$	0	0
$715$	22.3798	0.836958
$716$	−32.7192	−1.22277
$717$	0	0
$718$	−1.43842	−0.0536815
$719$	−14.9958	−0.559249	−0.279624	−	0.960109i	$-0.590210\pi$
$719$	−14.9958	−0.559249	−0.279624	+	0.960109i	$0.590210\pi$
$720$	0	0
$721$	0	0
$722$	1.05177	0.0391429
$723$	0	0
$724$	−28.8015	−1.07040
$725$	−13.3900	−0.497293
$726$	0	0
$727$	26.1055	0.968199	0.484099	−	0.875013i	$-0.339147\pi$
$727$	26.1055	0.968199	0.484099	+	0.875013i	$0.339147\pi$
$728$	0	0
$729$	0	0
$730$	−0.400446	−0.0148212
$731$	73.7447	2.72754
$732$	0	0
$733$	28.3821	1.04832	0.524159	−	0.851621i	$-0.324380\pi$
$733$	28.3821	1.04832	0.524159	+	0.851621i	$0.324380\pi$
$734$	2.39569	0.0884265
$735$	0	0
$736$	−2.50972	−0.0925096
$737$	−5.31763	−0.195877
$738$	0	0
$739$	46.5865	1.71371	0.856857	−	0.515555i	$-0.172414\pi$
$739$	46.5865	1.71371	0.856857	+	0.515555i	$0.172414\pi$
$740$	22.4806	0.826403
$741$	0	0
$742$	0	0
$743$	−0.339027	−0.0124377	−0.00621884	−	0.999981i	$-0.501980\pi$
$743$	−0.339027	−0.0124377	−0.00621884	+	0.999981i	$0.501980\pi$
$744$	0	0
$745$	−23.4755	−0.860075
$746$	−0.0771683	−0.00282533
$747$	0	0
$748$	20.8907	0.763839
$749$	0	0
$750$	0	0
$751$	−36.3662	−1.32702	−0.663510	−	0.748168i	$-0.730934\pi$
$751$	−36.3662	−1.32702	−0.663510	+	0.748168i	$0.730934\pi$
$752$	15.0902	0.550284
$753$	0	0
$754$	−2.29509	−0.0835822
$755$	−12.4245	−0.452174
$756$	0	0
$757$	−27.4703	−0.998424	−0.499212	−	0.866480i	$-0.666377\pi$
$757$	−27.4703	−0.998424	−0.499212	+	0.866480i	$0.666377\pi$
$758$	1.50162	0.0545415
$759$	0	0
$760$	1.38401	0.0502033
$761$	−33.0357	−1.19754	−0.598771	−	0.800920i	$-0.704344\pi$
$761$	−33.0357	−1.19754	−0.598771	+	0.800920i	$0.704344\pi$
$762$	0	0
$763$	0	0
$764$	5.66934	0.205110
$765$	0	0
$766$	1.57534	0.0569194
$767$	57.2285	2.06640
$768$	0	0
$769$	2.57751	0.0929475	0.0464738	−	0.998920i	$-0.485202\pi$
$769$	2.57751	0.0929475	0.0464738	+	0.998920i	$0.485202\pi$
$770$	0	0
$771$	0	0
$772$	−17.6164	−0.634030
$773$	−6.72973	−0.242051	−0.121026	−	0.992649i	$-0.538618\pi$
$773$	−6.72973	−0.242051	−0.121026	+	0.992649i	$0.538618\pi$
$774$	0	0
$775$	−14.0820	−0.505842
$776$	1.40753	0.0505274
$777$	0	0
$778$	1.07860	0.0386698
$779$	−14.0609	−0.503785
$780$	0	0
$781$	19.8116	0.708916
$782$	1.37358	0.0491193
$783$	0	0
$784$	0	0
$785$	−10.8495	−0.387234
$786$	0	0
$787$	28.6683	1.02191	0.510956	−	0.859607i	$-0.329291\pi$
$787$	28.6683	1.02191	0.510956	+	0.859607i	$0.329291\pi$
$788$	11.4203	0.406832
$789$	0	0
$790$	0.148176	0.00527187
$791$	0	0
$792$	0	0
$793$	14.2581	0.506320
$794$	1.12855	0.0400507
$795$	0	0
$796$	−22.7767	−0.807300
$797$	22.9825	0.814084	0.407042	−	0.913410i	$-0.366560\pi$
$797$	22.9825	0.814084	0.407042	+	0.913410i	$0.366560\pi$
$798$	0	0
$799$	−24.8740	−0.879979
$800$	−1.71421	−0.0606064
$801$	0	0
$802$	1.48098	0.0522951
$803$	−3.51589	−0.124073
$804$	0	0
$805$	0	0
$806$	−2.41370	−0.0850191
$807$	0	0
$808$	−0.503977	−0.0177299
$809$	16.4779	0.579332	0.289666	−	0.957128i	$-0.406456\pi$
$809$	16.4779	0.579332	0.289666	+	0.957128i	$0.406456\pi$
$810$	0	0
$811$	40.4318	1.41975	0.709876	−	0.704326i	$-0.248751\pi$
$811$	40.4318	1.41975	0.709876	+	0.704326i	$0.248751\pi$
$812$	0	0
$813$	0	0
$814$	−0.462453	−0.0162090
$815$	32.2911	1.13111
$816$	0	0
$817$	21.4225	0.749479
$818$	−2.09033	−0.0730868
$819$	0	0
$820$	39.2945	1.37222
$821$	28.1086	0.980998	0.490499	−	0.871442i	$-0.336814\pi$
$821$	28.1086	0.980998	0.490499	+	0.871442i	$0.336814\pi$
$822$	0	0
$823$	25.9058	0.903019	0.451510	−	0.892266i	$-0.350886\pi$
$823$	25.9058	0.903019	0.451510	+	0.892266i	$0.350886\pi$
$824$	−1.41451	−0.0492769
$825$	0	0
$826$	0	0
$827$	17.7998	0.618961	0.309480	−	0.950906i	$-0.399845\pi$
$827$	17.7998	0.618961	0.309480	+	0.950906i	$0.399845\pi$
$828$	0	0
$829$	−15.7069	−0.545523	−0.272761	−	0.962082i	$-0.587937\pi$
$829$	−15.7069	−0.545523	−0.272761	+	0.962082i	$0.587937\pi$
$830$	1.24522	0.0432223
$831$	0	0
$832$	41.4421	1.43675
$833$	0	0
$834$	0	0
$835$	12.7836	0.442395
$836$	6.06866	0.209889
$837$	0	0
$838$	1.47799	0.0510563
$839$	−7.39643	−0.255353	−0.127677	−	0.991816i	$-0.540752\pi$
$839$	−7.39643	−0.255353	−0.127677	+	0.991816i	$0.540752\pi$
$840$	0	0
$841$	11.8197	0.407576
$842$	1.86274	0.0641942
$843$	0	0
$844$	42.6675	1.46868
$845$	−38.8978	−1.33812
$846$	0	0
$847$	0	0
$848$	−35.3451	−1.21376
$849$	0	0
$850$	0.938196	0.0321798
$851$	12.9778	0.444874
$852$	0	0
$853$	53.1262	1.81901	0.909503	−	0.415698i	$-0.136463\pi$
$853$	53.1262	1.81901	0.909503	+	0.415698i	$0.136463\pi$
$854$	0	0
$855$	0	0
$856$	4.63004	0.158251
$857$	−3.81530	−0.130328	−0.0651640	−	0.997875i	$-0.520757\pi$
$857$	−3.81530	−0.130328	−0.0651640	+	0.997875i	$0.520757\pi$
$858$	0	0
$859$	−38.9768	−1.32987	−0.664936	−	0.746901i	$-0.731541\pi$
$859$	−38.9768	−1.32987	−0.664936	+	0.746901i	$0.731541\pi$
$860$	−59.8671	−2.04145
$861$	0	0
$862$	0.560452	0.0190891
$863$	−26.6736	−0.907978	−0.453989	−	0.891007i	$-0.650000\pi$
$863$	−26.6736	−0.907978	−0.453989	+	0.891007i	$0.650000\pi$
$864$	0	0
$865$	−13.4034	−0.455730
$866$	0.233475	0.00793380
$867$	0	0
$868$	0	0
$869$	1.30098	0.0441327
$870$	0	0
$871$	17.4706	0.591970
$872$	−2.32198	−0.0786321
$873$	0	0
$874$	0.399020	0.0134971
$875$	0	0
$876$	0	0
$877$	24.0135	0.810879	0.405440	−	0.914122i	$-0.367118\pi$
$877$	24.0135	0.810879	0.405440	+	0.914122i	$0.367118\pi$
$878$	0.450469	0.0152026
$879$	0	0
$880$	−16.9196	−0.570358
$881$	−4.67326	−0.157446	−0.0787231	−	0.996897i	$-0.525084\pi$
$881$	−4.67326	−0.157446	−0.0787231	+	0.996897i	$0.525084\pi$
$882$	0	0
$883$	−35.6948	−1.20122	−0.600612	−	0.799541i	$-0.705076\pi$
$883$	−35.6948	−1.20122	−0.600612	+	0.799541i	$0.705076\pi$
$884$	−68.6346	−2.30843
$885$	0	0
$886$	1.96173	0.0659058
$887$	29.1032	0.977191	0.488596	−	0.872510i	$-0.337509\pi$
$887$	29.1032	0.977191	0.488596	+	0.872510i	$0.337509\pi$
$888$	0	0
$889$	0	0
$890$	0.0857071	0.00287291
$891$	0	0
$892$	14.3020	0.478865
$893$	−7.22579	−0.241802
$894$	0	0
$895$	−43.6806	−1.46008
$896$	0	0
$897$	0	0
$898$	0.0312964	0.00104437
$899$	42.9294	1.43177
$900$	0	0
$901$	58.2611	1.94096
$902$	−0.808336	−0.0269146
$903$	0	0
$904$	−1.06646	−0.0354699
$905$	−38.4505	−1.27814
$906$	0	0
$907$	−41.2142	−1.36849	−0.684247	−	0.729250i	$-0.739869\pi$
$907$	−41.2142	−1.36849	−0.684247	+	0.729250i	$0.739869\pi$
$908$	27.5129	0.913047
$909$	0	0
$910$	0	0
$911$	57.7239	1.91248	0.956239	−	0.292588i	$-0.0945163\pi$
$911$	57.7239	1.91248	0.956239	+	0.292588i	$0.0945163\pi$
$912$	0	0
$913$	10.9330	0.361829
$914$	−1.38259	−0.0457321
$915$	0	0
$916$	−52.6652	−1.74011
$917$	0	0
$918$	0	0
$919$	−51.5598	−1.70080	−0.850400	−	0.526137i	$-0.823640\pi$
$919$	−51.5598	−1.70080	−0.850400	+	0.526137i	$0.823640\pi$
$920$	−2.23281	−0.0736135
$921$	0	0
$922$	−1.65514	−0.0545090
$923$	−65.0895	−2.14245
$924$	0	0
$925$	8.86422	0.291454
$926$	0.328525	0.0107960
$927$	0	0
$928$	5.22579	0.171545
$929$	−50.2824	−1.64971	−0.824856	−	0.565343i	$-0.808744\pi$
$929$	−50.2824	−1.64971	−0.824856	+	0.565343i	$0.808744\pi$
$930$	0	0
$931$	0	0
$932$	−25.2301	−0.826439
$933$	0	0
$934$	1.86289	0.0609557
$935$	27.8894	0.912081
$936$	0	0
$937$	−18.1400	−0.592607	−0.296303	−	0.955094i	$-0.595754\pi$
$937$	−18.1400	−0.592607	−0.296303	+	0.955094i	$0.595754\pi$
$938$	0	0
$939$	0	0
$940$	20.1931	0.658627
$941$	−17.0332	−0.555267	−0.277633	−	0.960687i	$-0.589550\pi$
$941$	−17.0332	−0.555267	−0.277633	+	0.960687i	$0.589550\pi$
$942$	0	0
$943$	22.6844	0.738704
$944$	−43.2658	−1.40818
$945$	0	0
$946$	1.23154	0.0400408
$947$	−31.5059	−1.02381	−0.511903	−	0.859044i	$-0.671059\pi$
$947$	−31.5059	−1.02381	−0.511903	+	0.859044i	$0.671059\pi$
$948$	0	0
$949$	11.5512	0.374967
$950$	0.272542	0.00884243
$951$	0	0
$952$	0	0
$953$	−16.0677	−0.520485	−0.260242	−	0.965543i	$-0.583803\pi$
$953$	−16.0677	−0.520485	−0.260242	+	0.965543i	$0.583803\pi$
$954$	0	0
$955$	7.56866	0.244916
$956$	−30.7935	−0.995932
$957$	0	0
$958$	−1.40341	−0.0453421
$959$	0	0
$960$	0	0
$961$	14.1480	0.456388
$962$	1.51935	0.0489859
$963$	0	0
$964$	2.35425	0.0758253
$965$	−23.5182	−0.757079
$966$	0	0
$967$	−26.6098	−0.855713	−0.427857	−	0.903847i	$-0.640731\pi$
$967$	−26.6098	−0.855713	−0.427857	+	0.903847i	$0.640731\pi$
$968$	−2.30637	−0.0741296
$969$	0	0
$970$	0.938438	0.0301314
$971$	−56.5678	−1.81535	−0.907674	−	0.419676i	$-0.862144\pi$
$971$	−56.5678	−1.81535	−0.907674	+	0.419676i	$0.862144\pi$
$972$	0	0
$973$	0	0
$974$	−1.76712	−0.0566221
$975$	0	0
$976$	−10.7794	−0.345040
$977$	−53.7560	−1.71981	−0.859904	−	0.510456i	$-0.829477\pi$
$977$	−53.7560	−1.71981	−0.859904	+	0.510456i	$0.829477\pi$
$978$	0	0
$979$	0.752504	0.0240501
$980$	0	0
$981$	0	0
$982$	1.06769	0.0340715
$983$	27.4102	0.874249	0.437125	−	0.899401i	$-0.355997\pi$
$983$	27.4102	0.874249	0.437125	+	0.899401i	$0.355997\pi$
$984$	0	0
$985$	15.2463	0.485788
$986$	−2.86011	−0.0910843
$987$	0	0
$988$	−19.9381	−0.634315
$989$	−34.5607	−1.09897
$990$	0	0
$991$	−17.3374	−0.550740	−0.275370	−	0.961338i	$-0.588800\pi$
$991$	−17.3374	−0.550740	−0.275370	+	0.961338i	$0.588800\pi$
$992$	5.49587	0.174494
$993$	0	0
$994$	0	0
$995$	−30.4073	−0.963976
$996$	0	0
$997$	35.6638	1.12948	0.564742	−	0.825268i	$-0.308976\pi$
$997$	35.6638	1.12948	0.564742	+	0.825268i	$0.308976\pi$
$998$	−1.45425	−0.0460334
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bi.1.5		12
3.2	odd	2		3969.2.a.bh.1.8		12
7.6	odd	2	inner	3969.2.a.bi.1.6		12
9.2	odd	6		441.2.f.h.148.5	✓	24
9.4	even	3		1323.2.f.h.883.8		24
9.5	odd	6		441.2.f.h.295.5	yes	24
9.7	even	3		1323.2.f.h.442.8		24
21.20	even	2		3969.2.a.bh.1.7		12
63.2	odd	6		441.2.g.h.67.6		24
63.4	even	3		1323.2.g.h.667.7		24
63.5	even	6		441.2.h.h.214.7		24
63.11	odd	6		441.2.h.h.373.8		24
63.13	odd	6		1323.2.f.h.883.7		24
63.16	even	3		1323.2.g.h.361.7		24
63.20	even	6		441.2.f.h.148.6	yes	24
63.23	odd	6		441.2.h.h.214.8		24
63.25	even	3		1323.2.h.h.226.5		24
63.31	odd	6		1323.2.g.h.667.8		24
63.32	odd	6		441.2.g.h.79.6		24
63.34	odd	6		1323.2.f.h.442.7		24
63.38	even	6		441.2.h.h.373.7		24
63.40	odd	6		1323.2.h.h.802.6		24
63.41	even	6		441.2.f.h.295.6	yes	24
63.47	even	6		441.2.g.h.67.5		24
63.52	odd	6		1323.2.h.h.226.6		24
63.58	even	3		1323.2.h.h.802.5		24
63.59	even	6		441.2.g.h.79.5		24
63.61	odd	6		1323.2.g.h.361.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.h.148.5	✓	24	9.2	odd	6
441.2.f.h.148.6	yes	24	63.20	even	6
441.2.f.h.295.5	yes	24	9.5	odd	6
441.2.f.h.295.6	yes	24	63.41	even	6
441.2.g.h.67.5		24	63.47	even	6
441.2.g.h.67.6		24	63.2	odd	6
441.2.g.h.79.5		24	63.59	even	6
441.2.g.h.79.6		24	63.32	odd	6
441.2.h.h.214.7		24	63.5	even	6
441.2.h.h.214.8		24	63.23	odd	6
441.2.h.h.373.7		24	63.38	even	6
441.2.h.h.373.8		24	63.11	odd	6
1323.2.f.h.442.7		24	63.34	odd	6
1323.2.f.h.442.8		24	9.7	even	3
1323.2.f.h.883.7		24	63.13	odd	6
1323.2.f.h.883.8		24	9.4	even	3
1323.2.g.h.361.7		24	63.16	even	3
1323.2.g.h.361.8		24	63.61	odd	6
1323.2.g.h.667.7		24	63.4	even	3
1323.2.g.h.667.8		24	63.31	odd	6
1323.2.h.h.226.5		24	63.25	even	3
1323.2.h.h.226.6		24	63.52	odd	6
1323.2.h.h.802.5		24	63.58	even	3
1323.2.h.h.802.6		24	63.40	odd	6
3969.2.a.bh.1.7		12	21.20	even	2
3969.2.a.bh.1.8		12	3.2	odd	2
3969.2.a.bi.1.5		12	1.1	even	1	trivial
3969.2.a.bi.1.6		12	7.6	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0683740 q^{2} -1.99532 q^{4} -2.66379 q^{5} +0.273176 q^{8} +O(q^{10})\)	\(q-0.0683740 q^{2} -1.99532 q^{4} -2.66379 q^{5} +0.273176 q^{8} +0.182134 q^{10} +1.59913 q^{11} -5.25380 q^{13} +3.97197 q^{16} -6.54721 q^{17} -1.90194 q^{19} +5.31513 q^{20} -0.109339 q^{22} +3.06837 q^{23} +2.09578 q^{25} +0.359223 q^{26} -6.38903 q^{29} -6.71923 q^{31} -0.817932 q^{32} +0.447659 q^{34} +4.22955 q^{37} +0.130043 q^{38} -0.727684 q^{40} +7.39296 q^{41} -11.2635 q^{43} -3.19078 q^{44} -0.209797 q^{46} +3.79918 q^{47} -0.143297 q^{50} +10.4830 q^{52} -8.89862 q^{53} -4.25974 q^{55} +0.436843 q^{58} -10.8928 q^{59} -2.71386 q^{61} +0.459420 q^{62} -7.88802 q^{64} +13.9950 q^{65} -3.32533 q^{67} +13.0638 q^{68} +12.3890 q^{71} -2.19863 q^{73} -0.289191 q^{74} +3.79498 q^{76} +0.813556 q^{79} -10.5805 q^{80} -0.505486 q^{82} +6.83684 q^{83} +17.4404 q^{85} +0.770132 q^{86} +0.436843 q^{88} +0.470572 q^{89} -6.12240 q^{92} -0.259765 q^{94} +5.06636 q^{95} +5.15246 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8	\( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 20 q^{11} + 12 q^{16} + 32 q^{23} + 12 q^{25} + 16 q^{29} + 48 q^{32} + 12 q^{37} + 56 q^{44} - 24 q^{46} - 4 q^{50} + 32 q^{53} + 48 q^{64} + 60 q^{65} + 12 q^{67} + 56 q^{71} + 68 q^{74} - 12 q^{79} - 12 q^{85} + 76 q^{86} + 16 q^{92} + 64 q^{95}+O(q^{100}) \) 12 * q + 4 * q^2 + 12 * q^4 + 12 * q^8 + 20 * q^11 + 12 * q^16 + 32 * q^23 + 12 * q^25 + 16 * q^29 + 48 * q^32 + 12 * q^37 + 56 * q^44 - 24 * q^46 - 4 * q^50 + 32 * q^53 + 48 * q^64 + 60 * q^65 + 12 * q^67 + 56 * q^71 + 68 * q^74 - 12 * q^79 - 12 * q^85 + 76 * q^86 + 16 * q^92 + 64 * q^95

Introduction

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