LMFDB - Embedded newform 3381.2.a.bi.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 13x^{8} + 41x^{7} + 47x^{6} - 165x^{5} - 45x^{4} + 207x^{3} + 12x^{2} - 76x - 2 $ x^10 - 3x^9 - 13x^8 + 41x^7 + 47x^6 - 165x^5 - 45x^4 + 207x^3 + 12x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0262565$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0262565 q^{2} -1.00000 q^{3} -1.99931 q^{4} +2.77828 q^{5} +0.0262565 q^{6} +0.105008 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.0262565 q^{2} -1.00000 q^{3} -1.99931 q^{4} +2.77828 q^{5} +0.0262565 q^{6} +0.105008 q^{8} +1.00000 q^{9} -0.0729480 q^{10} +0.660156 q^{11} +1.99931 q^{12} +4.12488 q^{13} -2.77828 q^{15} +3.99586 q^{16} +4.12381 q^{17} -0.0262565 q^{18} +4.76829 q^{19} -5.55465 q^{20} -0.0173334 q^{22} +1.00000 q^{23} -0.105008 q^{24} +2.71885 q^{25} -0.108305 q^{26} -1.00000 q^{27} -1.78025 q^{29} +0.0729480 q^{30} -3.62429 q^{31} -0.314933 q^{32} -0.660156 q^{33} -0.108277 q^{34} -1.99931 q^{36} -8.06211 q^{37} -0.125199 q^{38} -4.12488 q^{39} +0.291742 q^{40} -7.99969 q^{41} +12.5842 q^{43} -1.31986 q^{44} +2.77828 q^{45} -0.0262565 q^{46} +11.7622 q^{47} -3.99586 q^{48} -0.0713876 q^{50} -4.12381 q^{51} -8.24691 q^{52} -0.0608257 q^{53} +0.0262565 q^{54} +1.83410 q^{55} -4.76829 q^{57} +0.0467430 q^{58} -8.98717 q^{59} +5.55465 q^{60} +2.71267 q^{61} +0.0951611 q^{62} -7.98346 q^{64} +11.4601 q^{65} +0.0173334 q^{66} -2.68583 q^{67} -8.24477 q^{68} -1.00000 q^{69} +12.4462 q^{71} +0.105008 q^{72} -14.2871 q^{73} +0.211683 q^{74} -2.71885 q^{75} -9.53329 q^{76} +0.108305 q^{78} -6.94797 q^{79} +11.1016 q^{80} +1.00000 q^{81} +0.210044 q^{82} +9.40856 q^{83} +11.4571 q^{85} -0.330417 q^{86} +1.78025 q^{87} +0.0693216 q^{88} -5.92430 q^{89} -0.0729480 q^{90} -1.99931 q^{92} +3.62429 q^{93} -0.308834 q^{94} +13.2477 q^{95} +0.314933 q^{96} +2.53740 q^{97} +0.660156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 3 q^{2} - 10 q^{3} + 15 q^{4} - 5 q^{5} - 3 q^{6} + 9 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q + 3 * q^2 - 10 * q^3 + 15 * q^4 - 5 * q^5 - 3 * q^6 + 9 * q^8 + 10 * q^9	$ 10 q + 3 q^{2} - 10 q^{3} + 15 q^{4} - 5 q^{5} - 3 q^{6} + 9 q^{8} + 10 q^{9} + 11 q^{10} + 8 q^{11} - 15 q^{12} + 5 q^{15} + 37 q^{16} - 11 q^{17} + 3 q^{18} + q^{19} - 15 q^{20} + 6 q^{22} + 10 q^{23} - 9 q^{24} + 21 q^{25} + q^{26} - 10 q^{27} + 22 q^{29} - 11 q^{30} - 3 q^{31} + 11 q^{32} - 8 q^{33} - 3 q^{34} + 15 q^{36} - 3 q^{37} + 16 q^{38} + 39 q^{40} - 26 q^{41} + 27 q^{43} + 16 q^{44} - 5 q^{45} + 3 q^{46} + 11 q^{47} - 37 q^{48} + 2 q^{50} + 11 q^{51} + 29 q^{52} + 5 q^{53} - 3 q^{54} - 18 q^{55} - q^{57} + 16 q^{58} - 10 q^{59} + 15 q^{60} + 22 q^{61} - 32 q^{62} + 69 q^{64} - 11 q^{65} - 6 q^{66} - 2 q^{67} - 21 q^{68} - 10 q^{69} + 27 q^{71} + 9 q^{72} - 8 q^{73} + 14 q^{74} - 21 q^{75} - 22 q^{76} - q^{78} + 21 q^{79} - 53 q^{80} + 10 q^{81} + 36 q^{82} - 12 q^{83} + 23 q^{85} + 18 q^{86} - 22 q^{87} - 10 q^{88} + 6 q^{89} + 11 q^{90} + 15 q^{92} + 3 q^{93} + 35 q^{94} + 44 q^{95} - 11 q^{96} + 6 q^{97} + 8 q^{99}+O(q^{100}) $ 10 * q + 3 * q^2 - 10 * q^3 + 15 * q^4 - 5 * q^5 - 3 * q^6 + 9 * q^8 + 10 * q^9 + 11 * q^10 + 8 * q^11 - 15 * q^12 + 5 * q^15 + 37 * q^16 - 11 * q^17 + 3 * q^18 + q^19 - 15 * q^20 + 6 * q^22 + 10 * q^23 - 9 * q^24 + 21 * q^25 + q^26 - 10 * q^27 + 22 * q^29 - 11 * q^30 - 3 * q^31 + 11 * q^32 - 8 * q^33 - 3 * q^34 + 15 * q^36 - 3 * q^37 + 16 * q^38 + 39 * q^40 - 26 * q^41 + 27 * q^43 + 16 * q^44 - 5 * q^45 + 3 * q^46 + 11 * q^47 - 37 * q^48 + 2 * q^50 + 11 * q^51 + 29 * q^52 + 5 * q^53 - 3 * q^54 - 18 * q^55 - q^57 + 16 * q^58 - 10 * q^59 + 15 * q^60 + 22 * q^61 - 32 * q^62 + 69 * q^64 - 11 * q^65 - 6 * q^66 - 2 * q^67 - 21 * q^68 - 10 * q^69 + 27 * q^71 + 9 * q^72 - 8 * q^73 + 14 * q^74 - 21 * q^75 - 22 * q^76 - q^78 + 21 * q^79 - 53 * q^80 + 10 * q^81 + 36 * q^82 - 12 * q^83 + 23 * q^85 + 18 * q^86 - 22 * q^87 - 10 * q^88 + 6 * q^89 + 11 * q^90 + 15 * q^92 + 3 * q^93 + 35 * q^94 + 44 * q^95 - 11 * q^96 + 6 * q^97 + 8 * q^99

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 $

$2$	−0.0262565	−0.0185661	−0.00928307	−	0.999957i	$-0.502955\pi$
$2$	−0.0262565	−0.0185661	−0.00928307	+	0.999957i	$0.502955\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.99931	−0.999655
$5$	2.77828	1.24249	0.621243	−	0.783618i	$-0.286628\pi$
$5$	2.77828	1.24249	0.621243	+	0.783618i	$0.286628\pi$
$6$	0.0262565	0.0107192
$7$	0	0
$7$	0	0
$8$	0.105008	0.0371259
$9$	1.00000	0.333333
$10$	−0.0729480	−0.0230682
$11$	0.660156	0.199045	0.0995223	−	0.995035i	$-0.468269\pi$
$11$	0.660156	0.199045	0.0995223	+	0.995035i	$0.468269\pi$
$12$	1.99931	0.577151
$13$	4.12488	1.14404	0.572018	−	0.820241i	$-0.306161\pi$
$13$	4.12488	1.14404	0.572018	+	0.820241i	$0.306161\pi$
$14$	0	0
$15$	−2.77828	−0.717349
$16$	3.99586	0.998966
$17$	4.12381	1.00017	0.500085	−	0.865976i	$-0.333302\pi$
$17$	4.12381	1.00017	0.500085	+	0.865976i	$0.333302\pi$
$18$	−0.0262565	−0.00618871
$19$	4.76829	1.09392	0.546960	−	0.837159i	$-0.315785\pi$
$19$	4.76829	1.09392	0.546960	+	0.837159i	$0.315785\pi$
$20$	−5.55465	−1.24206
$21$	0	0
$22$	−0.0173334	−0.00369549
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−0.105008	−0.0214346
$25$	2.71885	0.543771
$26$	−0.108305	−0.0212403
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.78025	−0.330583	−0.165292	−	0.986245i	$-0.552857\pi$
$29$	−1.78025	−0.330583	−0.165292	+	0.986245i	$0.552857\pi$
$30$	0.0729480	0.0133184
$31$	−3.62429	−0.650941	−0.325471	−	0.945552i	$-0.605523\pi$
$31$	−3.62429	−0.650941	−0.325471	+	0.945552i	$0.605523\pi$
$32$	−0.314933	−0.0556728
$33$	−0.660156	−0.114918
$34$	−0.108277	−0.0185693
$35$	0	0
$36$	−1.99931	−0.333218
$37$	−8.06211	−1.32540	−0.662701	−	0.748884i	$-0.730590\pi$
$37$	−8.06211	−1.32540	−0.662701	+	0.748884i	$0.730590\pi$
$38$	−0.125199	−0.0203099
$39$	−4.12488	−0.660509
$40$	0.291742	0.0461284
$41$	−7.99969	−1.24934	−0.624671	−	0.780888i	$-0.714767\pi$
$41$	−7.99969	−1.24934	−0.624671	+	0.780888i	$0.714767\pi$
$42$	0	0
$43$	12.5842	1.91907	0.959536	−	0.281585i	$-0.0908601\pi$
$43$	12.5842	1.91907	0.959536	+	0.281585i	$0.0908601\pi$
$44$	−1.31986	−0.198976
$45$	2.77828	0.414162
$46$	−0.0262565	−0.00387131
$47$	11.7622	1.71569	0.857846	−	0.513908i	$-0.171803\pi$
$47$	11.7622	1.71569	0.857846	+	0.513908i	$0.171803\pi$
$48$	−3.99586	−0.576753
$49$	0	0
$50$	−0.0713876	−0.0100957
$51$	−4.12381	−0.577448
$52$	−8.24691	−1.14364
$53$	−0.0608257	−0.00835505	−0.00417752	−	0.999991i	$-0.501330\pi$
$53$	−0.0608257	−0.00835505	−0.00417752	+	0.999991i	$0.501330\pi$
$54$	0.0262565	0.00357306
$55$	1.83410	0.247310
$56$	0	0
$57$	−4.76829	−0.631575
$58$	0.0467430	0.00613766
$59$	−8.98717	−1.17003	−0.585015	−	0.811022i	$-0.698911\pi$
$59$	−8.98717	−1.17003	−0.585015	+	0.811022i	$0.698911\pi$
$60$	5.55465	0.717102
$61$	2.71267	0.347322	0.173661	−	0.984806i	$-0.444440\pi$
$61$	2.71267	0.347322	0.173661	+	0.984806i	$0.444440\pi$
$62$	0.0951611	0.0120855
$63$	0	0
$64$	−7.98346	−0.997932
$65$	11.4601	1.42145
$66$	0.0173334	0.00213359
$67$	−2.68583	−0.328126	−0.164063	−	0.986450i	$-0.552460\pi$
$67$	−2.68583	−0.328126	−0.164063	+	0.986450i	$0.552460\pi$
$68$	−8.24477	−0.999825
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.4462	1.47709	0.738545	−	0.674205i	$-0.235513\pi$
$71$	12.4462	1.47709	0.738545	+	0.674205i	$0.235513\pi$
$72$	0.105008	0.0123753
$73$	−14.2871	−1.67218	−0.836088	−	0.548596i	$-0.815163\pi$
$73$	−14.2871	−1.67218	−0.836088	+	0.548596i	$0.815163\pi$
$74$	0.211683	0.0246076
$75$	−2.71885	−0.313946
$76$	−9.53329	−1.09354
$77$	0	0
$78$	0.108305	0.0122631
$79$	−6.94797	−0.781708	−0.390854	−	0.920453i	$-0.627820\pi$
$79$	−6.94797	−0.781708	−0.390854	+	0.920453i	$0.627820\pi$
$80$	11.1016	1.24120
$81$	1.00000	0.111111
$82$	0.210044	0.0231955
$83$	9.40856	1.03272	0.516362	−	0.856371i	$-0.327286\pi$
$83$	9.40856	1.03272	0.516362	+	0.856371i	$0.327286\pi$
$84$	0	0
$85$	11.4571	1.24270
$86$	−0.330417	−0.0356298
$87$	1.78025	0.190862
$88$	0.0693216	0.00738971
$89$	−5.92430	−0.627975	−0.313987	−	0.949427i	$-0.601665\pi$
$89$	−5.92430	−0.627975	−0.313987	+	0.949427i	$0.601665\pi$
$90$	−0.0729480	−0.00768939
$91$	0	0
$92$	−1.99931	−0.208443
$93$	3.62429	0.375821
$94$	−0.308834	−0.0318538
$95$	13.2477	1.35918
$96$	0.314933	0.0321427
$97$	2.53740	0.257634	0.128817	−	0.991668i	$-0.458882\pi$
$97$	2.53740	0.257634	0.128817	+	0.991668i	$0.458882\pi$
$98$	0	0
$99$	0.660156	0.0663482
$100$	−5.43583	−0.543583
$101$	6.71178	0.667847	0.333923	−	0.942600i	$-0.391627\pi$
$101$	6.71178	0.667847	0.333923	+	0.942600i	$0.391627\pi$
$102$	0.108277	0.0107210
$103$	10.7478	1.05901	0.529507	−	0.848306i	$-0.322377\pi$
$103$	10.7478	1.05901	0.529507	+	0.848306i	$0.322377\pi$
$104$	0.433145	0.0424733
$105$	0	0
$106$	0.00159707	0.000155121 0
$107$	−0.484556	−0.0468438	−0.0234219	−	0.999726i	$-0.507456\pi$
$107$	−0.484556	−0.0468438	−0.0234219	+	0.999726i	$0.507456\pi$
$108$	1.99931	0.192384
$109$	12.4652	1.19395	0.596975	−	0.802260i	$-0.296369\pi$
$109$	12.4652	1.19395	0.596975	+	0.802260i	$0.296369\pi$
$110$	−0.0481570	−0.00459159
$111$	8.06211	0.765221
$112$	0	0
$113$	9.28875	0.873812	0.436906	−	0.899507i	$-0.356074\pi$
$113$	9.28875	0.873812	0.436906	+	0.899507i	$0.356074\pi$
$114$	0.125199	0.0117259
$115$	2.77828	0.259076
$116$	3.55926	0.330469
$117$	4.12488	0.381345
$118$	0.235972	0.0217229
$119$	0	0
$120$	−0.291742	−0.0266322
$121$	−10.5642	−0.960381
$122$	−0.0712252	−0.00644842
$123$	7.99969	0.721308
$124$	7.24608	0.650717
$125$	−6.33767	−0.566858
$126$	0	0
$127$	2.40074	0.213031	0.106515	−	0.994311i	$-0.466031\pi$
$127$	2.40074	0.213031	0.106515	+	0.994311i	$0.466031\pi$
$128$	0.839484	0.0742006
$129$	−12.5842	−1.10798
$130$	−0.300901	−0.0263908
$131$	−20.1505	−1.76056	−0.880278	−	0.474458i	$-0.842644\pi$
$131$	−20.1505	−1.76056	−0.880278	+	0.474458i	$0.842644\pi$
$132$	1.31986	0.114879
$133$	0	0
$134$	0.0705204	0.00609204
$135$	−2.77828	−0.239116
$136$	0.433032	0.0371322
$137$	7.00658	0.598613	0.299306	−	0.954157i	$-0.403245\pi$
$137$	7.00658	0.598613	0.299306	+	0.954157i	$0.403245\pi$
$138$	0.0262565	0.00223510
$139$	−2.88510	−0.244711	−0.122356	−	0.992486i	$-0.539045\pi$
$139$	−2.88510	−0.244711	−0.122356	+	0.992486i	$0.539045\pi$
$140$	0	0
$141$	−11.7622	−0.990555
$142$	−0.326793	−0.0274238
$143$	2.72306	0.227714
$144$	3.99586	0.332989
$145$	−4.94603	−0.410745
$146$	0.375128	0.0310458
$147$	0	0
$148$	16.1187	1.32495
$149$	19.0608	1.56152	0.780762	−	0.624829i	$-0.214831\pi$
$149$	19.0608	1.56152	0.780762	+	0.624829i	$0.214831\pi$
$150$	0.0713876	0.00582877
$151$	−5.06290	−0.412013	−0.206006	−	0.978551i	$-0.566047\pi$
$151$	−5.06290	−0.412013	−0.206006	+	0.978551i	$0.566047\pi$
$152$	0.500708	0.0406128
$153$	4.12381	0.333390
$154$	0	0
$155$	−10.0693	−0.808785
$156$	8.24691	0.660281
$157$	−2.74388	−0.218985	−0.109493	−	0.993988i	$-0.534923\pi$
$157$	−2.74388	−0.218985	−0.109493	+	0.993988i	$0.534923\pi$
$158$	0.182429	0.0145133
$159$	0.0608257	0.00482379
$160$	−0.874973	−0.0691727
$161$	0	0
$162$	−0.0262565	−0.00206290
$163$	15.3088	1.19908	0.599539	−	0.800346i	$-0.295351\pi$
$163$	15.3088	1.19908	0.599539	+	0.800346i	$0.295351\pi$
$164$	15.9939	1.24891
$165$	−1.83410	−0.142785
$166$	−0.247036	−0.0191737
$167$	13.0016	1.00610	0.503049	−	0.864258i	$-0.332212\pi$
$167$	13.0016	1.00610	0.503049	+	0.864258i	$0.332212\pi$
$168$	0	0
$169$	4.01461	0.308816
$170$	−0.300823	−0.0230721
$171$	4.76829	0.364640
$172$	−25.1597	−1.91841
$173$	−10.4220	−0.792372	−0.396186	−	0.918170i	$-0.629666\pi$
$173$	−10.4220	−0.792372	−0.396186	+	0.918170i	$0.629666\pi$
$174$	−0.0467430	−0.00354358
$175$	0	0
$176$	2.63789	0.198839
$177$	8.98717	0.675517
$178$	0.155551	0.0116591
$179$	19.2231	1.43680	0.718402	−	0.695628i	$-0.244874\pi$
$179$	19.2231	1.43680	0.718402	+	0.695628i	$0.244874\pi$
$180$	−5.55465	−0.414019
$181$	−11.1552	−0.829162	−0.414581	−	0.910012i	$-0.636072\pi$
$181$	−11.1552	−0.829162	−0.414581	+	0.910012i	$0.636072\pi$
$182$	0	0
$183$	−2.71267	−0.200526
$184$	0.105008	0.00774128
$185$	−22.3988	−1.64679
$186$	−0.0951611	−0.00697755
$187$	2.72236	0.199078
$188$	−23.5163	−1.71510
$189$	0	0
$190$	−0.347837	−0.0252347
$191$	19.2346	1.39177	0.695884	−	0.718154i	$-0.255013\pi$
$191$	19.2346	1.39177	0.695884	+	0.718154i	$0.255013\pi$
$192$	7.98346	0.576157
$193$	17.4527	1.25627	0.628136	−	0.778104i	$-0.283818\pi$
$193$	17.4527	1.25627	0.628136	+	0.778104i	$0.283818\pi$
$194$	−0.0666232	−0.00478326
$195$	−11.4601	−0.820673
$196$	0	0
$197$	−1.16239	−0.0828168	−0.0414084	−	0.999142i	$-0.513184\pi$
$197$	−1.16239	−0.0828168	−0.0414084	+	0.999142i	$0.513184\pi$
$198$	−0.0173334	−0.00123183
$199$	16.9768	1.20346	0.601728	−	0.798701i	$-0.294479\pi$
$199$	16.9768	1.20346	0.601728	+	0.798701i	$0.294479\pi$
$200$	0.285501	0.0201880
$201$	2.68583	0.189444
$202$	−0.176228	−0.0123993
$203$	0	0
$204$	8.24477	0.577249
$205$	−22.2254	−1.55229
$206$	−0.282200	−0.0196618
$207$	1.00000	0.0695048
$208$	16.4824	1.14285
$209$	3.14782	0.217739
$210$	0	0
$211$	11.6924	0.804938	0.402469	−	0.915434i	$-0.368152\pi$
$211$	11.6924	0.804938	0.402469	+	0.915434i	$0.368152\pi$
$212$	0.121609	0.00835217
$213$	−12.4462	−0.852798
$214$	0.0127227	0.000869708 0
$215$	34.9625	2.38442
$216$	−0.105008	−0.00714488
$217$	0	0
$218$	−0.327293	−0.0221671
$219$	14.2871	0.965431
$220$	−3.66694	−0.247225
$221$	17.0102	1.14423
$222$	−0.211683	−0.0142072
$223$	−15.5087	−1.03854	−0.519270	−	0.854610i	$-0.673796\pi$
$223$	−15.5087	−1.03854	−0.519270	+	0.854610i	$0.673796\pi$
$224$	0	0
$225$	2.71885	0.181257
$226$	−0.243890	−0.0162233
$227$	−15.3330	−1.01768	−0.508842	−	0.860860i	$-0.669926\pi$
$227$	−15.3330	−1.01768	−0.508842	+	0.860860i	$0.669926\pi$
$228$	9.53329	0.631358
$229$	−6.40484	−0.423243	−0.211622	−	0.977352i	$-0.567874\pi$
$229$	−6.40484	−0.423243	−0.211622	+	0.977352i	$0.567874\pi$
$230$	−0.0729480	−0.00481005
$231$	0	0
$232$	−0.186940	−0.0122732
$233$	−26.1430	−1.71269	−0.856344	−	0.516406i	$-0.827270\pi$
$233$	−26.1430	−1.71269	−0.856344	+	0.516406i	$0.827270\pi$
$234$	−0.108305	−0.00708011
$235$	32.6787	2.13172
$236$	17.9681	1.16963
$237$	6.94797	0.451319
$238$	0	0
$239$	17.6773	1.14345	0.571724	−	0.820446i	$-0.306275\pi$
$239$	17.6773	1.14345	0.571724	+	0.820446i	$0.306275\pi$
$240$	−11.1016	−0.716608
$241$	5.75969	0.371014	0.185507	−	0.982643i	$-0.440607\pi$
$241$	5.75969	0.371014	0.185507	+	0.982643i	$0.440607\pi$
$242$	0.277379	0.0178306
$243$	−1.00000	−0.0641500
$244$	−5.42347	−0.347202
$245$	0	0
$246$	−0.210044	−0.0133919
$247$	19.6686	1.25148
$248$	−0.380579	−0.0241668
$249$	−9.40856	−0.596243
$250$	0.166405	0.0105244
$251$	−10.3866	−0.655597	−0.327798	−	0.944748i	$-0.606307\pi$
$251$	−10.3866	−0.655597	−0.327798	+	0.944748i	$0.606307\pi$
$252$	0	0
$253$	0.660156	0.0415037
$254$	−0.0630349	−0.00395516
$255$	−11.4571	−0.717471
$256$	15.9449	0.996555
$257$	−2.69848	−0.168327	−0.0841633	−	0.996452i	$-0.526822\pi$
$257$	−2.69848	−0.168327	−0.0841633	+	0.996452i	$0.526822\pi$
$258$	0.330417	0.0205709
$259$	0	0
$260$	−22.9122	−1.42096
$261$	−1.78025	−0.110194
$262$	0.529081	0.0326867
$263$	−21.1490	−1.30411	−0.652053	−	0.758174i	$-0.726092\pi$
$263$	−21.1490	−1.30411	−0.652053	+	0.758174i	$0.726092\pi$
$264$	−0.0693216	−0.00426645
$265$	−0.168991	−0.0103810
$266$	0	0
$267$	5.92430	0.362561
$268$	5.36981	0.328013
$269$	−28.5411	−1.74018	−0.870090	−	0.492893i	$-0.835939\pi$
$269$	−28.5411	−1.74018	−0.870090	+	0.492893i	$0.835939\pi$
$270$	0.0729480	0.00443947
$271$	15.1472	0.920130	0.460065	−	0.887885i	$-0.347826\pi$
$271$	15.1472	0.920130	0.460065	+	0.887885i	$0.347826\pi$
$272$	16.4782	0.999136
$273$	0	0
$274$	−0.183968	−0.0111139
$275$	1.79487	0.108235
$276$	1.99931	0.120344
$277$	4.23717	0.254587	0.127294	−	0.991865i	$-0.459371\pi$
$277$	4.23717	0.254587	0.127294	+	0.991865i	$0.459371\pi$
$278$	0.0757527	0.00454335
$279$	−3.62429	−0.216980
$280$	0	0
$281$	0.762123	0.0454645	0.0227322	−	0.999742i	$-0.492763\pi$
$281$	0.762123	0.0454645	0.0227322	+	0.999742i	$0.492763\pi$
$282$	0.308834	0.0183908
$283$	17.1714	1.02073	0.510367	−	0.859957i	$-0.329510\pi$
$283$	17.1714	1.02073	0.510367	+	0.859957i	$0.329510\pi$
$284$	−24.8838	−1.47658
$285$	−13.2477	−0.784723
$286$	−0.0714981	−0.00422777
$287$	0	0
$288$	−0.314933	−0.0185576
$289$	0.00578278	0.000340164 0
$290$	0.129865	0.00762595
$291$	−2.53740	−0.148745
$292$	28.5643	1.67160
$293$	−1.15566	−0.0675144	−0.0337572	−	0.999430i	$-0.510747\pi$
$293$	−1.15566	−0.0675144	−0.0337572	+	0.999430i	$0.510747\pi$
$294$	0	0
$295$	−24.9689	−1.45375
$296$	−0.846585	−0.0492067
$297$	−0.660156	−0.0383061
$298$	−0.500470	−0.0289915
$299$	4.12488	0.238548
$300$	5.43583	0.313838
$301$	0	0
$302$	0.132934	0.00764949
$303$	−6.71178	−0.385581
$304$	19.0534	1.09279
$305$	7.53656	0.431542
$306$	−0.108277	−0.00618977
$307$	31.0365	1.77135	0.885673	−	0.464310i	$-0.153698\pi$
$307$	31.0365	1.77135	0.885673	+	0.464310i	$0.153698\pi$
$308$	0	0
$309$	−10.7478	−0.611422
$310$	0.264384	0.0150160
$311$	17.6141	0.998801	0.499401	−	0.866371i	$-0.333554\pi$
$311$	17.6141	0.998801	0.499401	+	0.866371i	$0.333554\pi$
$312$	−0.433145	−0.0245220
$313$	15.7406	0.889713	0.444856	−	0.895602i	$-0.353255\pi$
$313$	15.7406	0.889713	0.444856	+	0.895602i	$0.353255\pi$
$314$	0.0720446	0.00406571
$315$	0	0
$316$	13.8912	0.781438
$317$	−5.22139	−0.293263	−0.146631	−	0.989191i	$-0.546843\pi$
$317$	−5.22139	−0.293263	−0.146631	+	0.989191i	$0.546843\pi$
$318$	−0.00159707	−8.95592e−5 0
$319$	−1.17524	−0.0658008
$320$	−22.1803	−1.23992
$321$	0.484556	0.0270453
$322$	0	0
$323$	19.6635	1.09411
$324$	−1.99931	−0.111073
$325$	11.2149	0.622093
$326$	−0.401955	−0.0222623
$327$	−12.4652	−0.689328
$328$	−0.840030	−0.0463829
$329$	0	0
$330$	0.0481570	0.00265096
$331$	−13.9449	−0.766483	−0.383241	−	0.923648i	$-0.625192\pi$
$331$	−13.9449	−0.766483	−0.383241	+	0.923648i	$0.625192\pi$
$332$	−18.8106	−1.03237
$333$	−8.06211	−0.441801
$334$	−0.341378	−0.0186793
$335$	−7.46199	−0.407692
$336$	0	0
$337$	9.28645	0.505865	0.252933	−	0.967484i	$-0.418605\pi$
$337$	9.28645	0.505865	0.252933	+	0.967484i	$0.418605\pi$
$338$	−0.105410	−0.00573353
$339$	−9.28875	−0.504496
$340$	−22.9063	−1.24227
$341$	−2.39260	−0.129566
$342$	−0.125199	−0.00676996
$343$	0	0
$344$	1.32144	0.0712473
$345$	−2.77828	−0.149578
$346$	0.273646	0.0147113
$347$	7.67494	0.412013	0.206006	−	0.978551i	$-0.433953\pi$
$347$	7.67494	0.412013	0.206006	+	0.978551i	$0.433953\pi$
$348$	−3.55926	−0.190797
$349$	5.71924	0.306144	0.153072	−	0.988215i	$-0.451083\pi$
$349$	5.71924	0.306144	0.153072	+	0.988215i	$0.451083\pi$
$350$	0	0
$351$	−4.12488	−0.220170
$352$	−0.207905	−0.0110814
$353$	13.4196	0.714253	0.357127	−	0.934056i	$-0.383757\pi$
$353$	13.4196	0.714253	0.357127	+	0.934056i	$0.383757\pi$
$354$	−0.235972	−0.0125417
$355$	34.5790	1.83526
$356$	11.8445	0.627758
$357$	0	0
$358$	−0.504732	−0.0266759
$359$	17.8658	0.942922	0.471461	−	0.881887i	$-0.343727\pi$
$359$	17.8658	0.942922	0.471461	+	0.881887i	$0.343727\pi$
$360$	0.291742	0.0153761
$361$	3.73658	0.196662
$362$	0.292897	0.0153943
$363$	10.5642	0.554476
$364$	0	0
$365$	−39.6935	−2.07765
$366$	0.0712252	0.00372300
$367$	26.1522	1.36513	0.682566	−	0.730824i	$-0.260864\pi$
$367$	26.1522	1.36513	0.682566	+	0.730824i	$0.260864\pi$
$368$	3.99586	0.208299
$369$	−7.99969	−0.416447
$370$	0.588114	0.0305746
$371$	0	0
$372$	−7.24608	−0.375692
$373$	20.1300	1.04229	0.521147	−	0.853467i	$-0.325504\pi$
$373$	20.1300	1.04229	0.521147	+	0.853467i	$0.325504\pi$
$374$	−0.0714795	−0.00369612
$375$	6.33767	0.327276
$376$	1.23512	0.0636966
$377$	−7.34330	−0.378199
$378$	0	0
$379$	8.04896	0.413447	0.206724	−	0.978399i	$-0.433720\pi$
$379$	8.04896	0.413447	0.206724	+	0.978399i	$0.433720\pi$
$380$	−26.4862	−1.35871
$381$	−2.40074	−0.122993
$382$	−0.505034	−0.0258398
$383$	20.0694	1.02550	0.512749	−	0.858539i	$-0.328627\pi$
$383$	20.0694	1.02550	0.512749	+	0.858539i	$0.328627\pi$
$384$	−0.839484	−0.0428397
$385$	0	0
$386$	−0.458246	−0.0233241
$387$	12.5842	0.639691
$388$	−5.07305	−0.257545
$389$	12.4516	0.631322	0.315661	−	0.948872i	$-0.397774\pi$
$389$	12.4516	0.631322	0.315661	+	0.948872i	$0.397774\pi$
$390$	0.300901	0.0152367
$391$	4.12381	0.208550
$392$	0	0
$393$	20.1505	1.01646
$394$	0.0305203	0.00153759
$395$	−19.3034	−0.971261
$396$	−1.31986	−0.0663253
$397$	−6.93528	−0.348072	−0.174036	−	0.984739i	$-0.555681\pi$
$397$	−6.93528	−0.348072	−0.174036	+	0.984739i	$0.555681\pi$
$398$	−0.445752	−0.0223435
$399$	0	0
$400$	10.8642	0.543208
$401$	−14.4161	−0.719908	−0.359954	−	0.932970i	$-0.617208\pi$
$401$	−14.4161	−0.719908	−0.359954	+	0.932970i	$0.617208\pi$
$402$	−0.0705204	−0.00351724
$403$	−14.9497	−0.744700
$404$	−13.4189	−0.667617
$405$	2.77828	0.138054
$406$	0	0
$407$	−5.32225	−0.263814
$408$	−0.433032	−0.0214383
$409$	24.0265	1.18803	0.594016	−	0.804453i	$-0.297541\pi$
$409$	24.0265	1.18803	0.594016	+	0.804453i	$0.297541\pi$
$410$	0.583561	0.0288200
$411$	−7.00658	−0.345609
$412$	−21.4882	−1.05865
$413$	0	0
$414$	−0.0262565	−0.00129044
$415$	26.1396	1.28314
$416$	−1.29906	−0.0636917
$417$	2.88510	0.141284
$418$	−0.0826506	−0.00404257
$419$	31.8946	1.55815	0.779076	−	0.626930i	$-0.215689\pi$
$419$	31.8946	1.55815	0.779076	+	0.626930i	$0.215689\pi$
$420$	0	0
$421$	−18.8400	−0.918205	−0.459102	−	0.888383i	$-0.651829\pi$
$421$	−18.8400	−0.918205	−0.459102	+	0.888383i	$0.651829\pi$
$422$	−0.307002	−0.0149446
$423$	11.7622	0.571897
$424$	−0.00638717	−0.000310189 0
$425$	11.2120	0.543863
$426$	0.326793	0.0158332
$427$	0	0
$428$	0.968777	0.0468276
$429$	−2.72306	−0.131471
$430$	−0.917992	−0.0442695
$431$	−36.3431	−1.75058	−0.875292	−	0.483595i	$-0.839331\pi$
$431$	−36.3431	−1.75058	−0.875292	+	0.483595i	$0.839331\pi$
$432$	−3.99586	−0.192251
$433$	21.5486	1.03556	0.517780	−	0.855514i	$-0.326759\pi$
$433$	21.5486	1.03556	0.517780	+	0.855514i	$0.326759\pi$
$434$	0	0
$435$	4.94603	0.237144
$436$	−24.9218	−1.19354
$437$	4.76829	0.228098
$438$	−0.375128	−0.0179243
$439$	−10.7375	−0.512473	−0.256237	−	0.966614i	$-0.582483\pi$
$439$	−10.7375	−0.512473	−0.256237	+	0.966614i	$0.582483\pi$
$440$	0.192595	0.00918160
$441$	0	0
$442$	−0.446628	−0.0212439
$443$	−33.9531	−1.61316	−0.806580	−	0.591125i	$-0.798684\pi$
$443$	−33.9531	−1.61316	−0.806580	+	0.591125i	$0.798684\pi$
$444$	−16.1187	−0.764957
$445$	−16.4594	−0.780250
$446$	0.407204	0.0192817
$447$	−19.0608	−0.901546
$448$	0	0
$449$	6.94219	0.327622	0.163811	−	0.986492i	$-0.447621\pi$
$449$	6.94219	0.327622	0.163811	+	0.986492i	$0.447621\pi$
$450$	−0.0713876	−0.00336524
$451$	−5.28104	−0.248675
$452$	−18.5711	−0.873511
$453$	5.06290	0.237876
$454$	0.402590	0.0188945
$455$	0	0
$456$	−0.500708	−0.0234478
$457$	−39.5509	−1.85011	−0.925057	−	0.379829i	$-0.875983\pi$
$457$	−39.5509	−1.85011	−0.925057	+	0.379829i	$0.875983\pi$
$458$	0.168169	0.00785800
$459$	−4.12381	−0.192483
$460$	−5.55465	−0.258987
$461$	−4.90582	−0.228487	−0.114243	−	0.993453i	$-0.536444\pi$
$461$	−4.90582	−0.228487	−0.114243	+	0.993453i	$0.536444\pi$
$462$	0	0
$463$	−0.772106	−0.0358828	−0.0179414	−	0.999839i	$-0.505711\pi$
$463$	−0.772106	−0.0358828	−0.0179414	+	0.999839i	$0.505711\pi$
$464$	−7.11362	−0.330242
$465$	10.0693	0.466952
$466$	0.686425	0.0317980
$467$	−19.1977	−0.888365	−0.444182	−	0.895936i	$-0.646506\pi$
$467$	−19.1977	−0.888365	−0.444182	+	0.895936i	$0.646506\pi$
$468$	−8.24691	−0.381214
$469$	0	0
$470$	−0.858027	−0.0395778
$471$	2.74388	0.126431
$472$	−0.943724	−0.0434384
$473$	8.30754	0.381981
$474$	−0.182429	−0.00837926
$475$	12.9643	0.594842
$476$	0	0
$477$	−0.0608257	−0.00278502
$478$	−0.464143	−0.0212294
$479$	−18.2104	−0.832056	−0.416028	−	0.909352i	$-0.636578\pi$
$479$	−18.2104	−0.832056	−0.416028	+	0.909352i	$0.636578\pi$
$480$	0.874973	0.0399369
$481$	−33.2552	−1.51631
$482$	−0.151229	−0.00688830
$483$	0	0
$484$	21.1211	0.960050
$485$	7.04961	0.320106
$486$	0.0262565	0.00119102
$487$	−28.8813	−1.30873	−0.654367	−	0.756177i	$-0.727065\pi$
$487$	−28.8813	−1.30873	−0.654367	+	0.756177i	$0.727065\pi$
$488$	0.284852	0.0128946
$489$	−15.3088	−0.692288
$490$	0	0
$491$	−23.6817	−1.06874	−0.534369	−	0.845251i	$-0.679451\pi$
$491$	−23.6817	−1.06874	−0.534369	+	0.845251i	$0.679451\pi$
$492$	−15.9939	−0.721059
$493$	−7.34139	−0.330640
$494$	−0.516429	−0.0232352
$495$	1.83410	0.0824367
$496$	−14.4822	−0.650268
$497$	0	0
$498$	0.247036	0.0110699
$499$	25.3699	1.13571	0.567856	−	0.823128i	$-0.307773\pi$
$499$	25.3699	1.13571	0.567856	+	0.823128i	$0.307773\pi$
$500$	12.6710	0.566663
$501$	−13.0016	−0.580871
$502$	0.272716	0.0121719
$503$	−29.2636	−1.30480	−0.652400	−	0.757874i	$-0.726238\pi$
$503$	−29.2636	−1.30480	−0.652400	+	0.757874i	$0.726238\pi$
$504$	0	0
$505$	18.6472	0.829790
$506$	−0.0173334	−0.000770563 0
$507$	−4.01461	−0.178295
$508$	−4.79982	−0.212957
$509$	−40.6862	−1.80338	−0.901692	−	0.432379i	$-0.857674\pi$
$509$	−40.6862	−1.80338	−0.901692	+	0.432379i	$0.857674\pi$
$510$	0.300823	0.0133207
$511$	0	0
$512$	−2.09762	−0.0927028
$513$	−4.76829	−0.210525
$514$	0.0708526	0.00312518
$515$	29.8605	1.31581
$516$	25.1597	1.10760
$517$	7.76488	0.341499
$518$	0	0
$519$	10.4220	0.457476
$520$	1.20340	0.0527725
$521$	−33.7965	−1.48065	−0.740326	−	0.672248i	$-0.765329\pi$
$521$	−33.7965	−1.48065	−0.740326	+	0.672248i	$0.765329\pi$
$522$	0.0467430	0.00204589
$523$	12.8567	0.562186	0.281093	−	0.959681i	$-0.409303\pi$
$523$	12.8567	0.562186	0.281093	+	0.959681i	$0.409303\pi$
$524$	40.2871	1.75995
$525$	0	0
$526$	0.555300	0.0242122
$527$	−14.9459	−0.651052
$528$	−2.63789	−0.114800
$529$	1.00000	0.0434783
$530$	0.00443711	0.000192736 0
$531$	−8.98717	−0.390010
$532$	0	0
$533$	−32.9977	−1.42929
$534$	−0.155551	−0.00673137
$535$	−1.34623	−0.0582027
$536$	−0.282033	−0.0121820
$537$	−19.2231	−0.829539
$538$	0.749388	0.0323084
$539$	0	0
$540$	5.55465	0.239034
$541$	−11.5066	−0.494706	−0.247353	−	0.968925i	$-0.579561\pi$
$541$	−11.5066	−0.494706	−0.247353	+	0.968925i	$0.579561\pi$
$542$	−0.397714	−0.0170833
$543$	11.1552	0.478717
$544$	−1.29872	−0.0556823
$545$	34.6319	1.48347
$546$	0	0
$547$	8.26280	0.353292	0.176646	−	0.984274i	$-0.443475\pi$
$547$	8.26280	0.353292	0.176646	+	0.984274i	$0.443475\pi$
$548$	−14.0083	−0.598406
$549$	2.71267	0.115774
$550$	−0.0471269	−0.00200950
$551$	−8.48873	−0.361632
$552$	−0.105008	−0.00446943
$553$	0	0
$554$	−0.111253	−0.00472670
$555$	22.3988	0.950776
$556$	5.76822	0.244627
$557$	−22.6003	−0.957604	−0.478802	−	0.877923i	$-0.658929\pi$
$557$	−22.6003	−0.957604	−0.478802	+	0.877923i	$0.658929\pi$
$558$	0.0951611	0.00402849
$559$	51.9083	2.19549
$560$	0	0
$561$	−2.72236	−0.114938
$562$	−0.0200107	−0.000844100 0
$563$	−10.1504	−0.427786	−0.213893	−	0.976857i	$-0.568614\pi$
$563$	−10.1504	−0.427786	−0.213893	+	0.976857i	$0.568614\pi$
$564$	23.5163	0.990213
$565$	25.8068	1.08570
$566$	−0.450860	−0.0189511
$567$	0	0
$568$	1.30695	0.0548382
$569$	36.5753	1.53332	0.766659	−	0.642055i	$-0.221918\pi$
$569$	36.5753	1.53332	0.766659	+	0.642055i	$0.221918\pi$
$570$	0.347837	0.0145693
$571$	8.14486	0.340852	0.170426	−	0.985371i	$-0.445486\pi$
$571$	8.14486	0.340852	0.170426	+	0.985371i	$0.445486\pi$
$572$	−5.44425	−0.227635
$573$	−19.2346	−0.803538
$574$	0	0
$575$	2.71885	0.113384
$576$	−7.98346	−0.332644
$577$	−7.31887	−0.304689	−0.152344	−	0.988327i	$-0.548682\pi$
$577$	−7.31887	−0.304689	−0.152344	+	0.988327i	$0.548682\pi$
$578$	−0.000151836 0	−6.31553e−6 0
$579$	−17.4527	−0.725309
$580$	9.88864	0.410604
$581$	0	0
$582$	0.0666232	0.00276162
$583$	−0.0401544	−0.00166303
$584$	−1.50025	−0.0620810
$585$	11.4601	0.473816
$586$	0.0303436	0.00125348
$587$	−46.8386	−1.93324	−0.966618	−	0.256223i	$-0.917522\pi$
$587$	−46.8386	−1.93324	−0.966618	+	0.256223i	$0.917522\pi$
$588$	0	0
$589$	−17.2816	−0.712078
$590$	0.655596	0.0269905
$591$	1.16239	0.0478143
$592$	−32.2151	−1.32403
$593$	−16.0755	−0.660143	−0.330072	−	0.943956i	$-0.607073\pi$
$593$	−16.0755	−0.660143	−0.330072	+	0.943956i	$0.607073\pi$
$594$	0.0173334	0.000711197 0
$595$	0	0
$596$	−38.1085	−1.56099
$597$	−16.9768	−0.694816
$598$	−0.108305	−0.00442891
$599$	−27.5797	−1.12688	−0.563439	−	0.826158i	$-0.690522\pi$
$599$	−27.5797	−1.12688	−0.563439	+	0.826158i	$0.690522\pi$
$600$	−0.285501	−0.0116555
$601$	−32.9659	−1.34471	−0.672353	−	0.740231i	$-0.734716\pi$
$601$	−32.9659	−1.34471	−0.672353	+	0.740231i	$0.734716\pi$
$602$	0	0
$603$	−2.68583	−0.109375
$604$	10.1223	0.411871
$605$	−29.3503	−1.19326
$606$	0.176228	0.00715876
$607$	−36.8404	−1.49531	−0.747653	−	0.664090i	$-0.768819\pi$
$607$	−36.8404	−1.49531	−0.747653	+	0.664090i	$0.768819\pi$
$608$	−1.50169	−0.0609017
$609$	0	0
$610$	−0.197884	−0.00801208
$611$	48.5176	1.96281
$612$	−8.24477	−0.333275
$613$	22.1316	0.893887	0.446943	−	0.894562i	$-0.352513\pi$
$613$	22.1316	0.893887	0.446943	+	0.894562i	$0.352513\pi$
$614$	−0.814909	−0.0328871
$615$	22.2254	0.896214
$616$	0	0
$617$	25.5349	1.02800	0.513998	−	0.857792i	$-0.328164\pi$
$617$	25.5349	1.02800	0.513998	+	0.857792i	$0.328164\pi$
$618$	0.282200	0.0113517
$619$	30.3835	1.22121	0.610607	−	0.791933i	$-0.290925\pi$
$619$	30.3835	1.22121	0.610607	+	0.791933i	$0.290925\pi$
$620$	20.1316	0.808506
$621$	−1.00000	−0.0401286
$622$	−0.462483	−0.0185439
$623$	0	0
$624$	−16.4824	−0.659826
$625$	−31.2021	−1.24808
$626$	−0.413294	−0.0165185
$627$	−3.14782	−0.125712
$628$	5.48586	0.218910
$629$	−33.2466	−1.32563
$630$	0	0
$631$	−23.5164	−0.936175	−0.468087	−	0.883682i	$-0.655057\pi$
$631$	−23.5164	−0.936175	−0.468087	+	0.883682i	$0.655057\pi$
$632$	−0.729592	−0.0290216
$633$	−11.6924	−0.464731
$634$	0.137095	0.00544475
$635$	6.66992	0.264688
$636$	−0.121609	−0.00482213
$637$	0	0
$638$	0.0308577	0.00122167
$639$	12.4462	0.492363
$640$	2.33232	0.0921932
$641$	21.6111	0.853586	0.426793	−	0.904349i	$-0.359643\pi$
$641$	21.6111	0.853586	0.426793	+	0.904349i	$0.359643\pi$
$642$	−0.0127227	−0.000502126 0
$643$	−12.5132	−0.493472	−0.246736	−	0.969083i	$-0.579358\pi$
$643$	−12.5132	−0.493472	−0.246736	+	0.969083i	$0.579358\pi$
$644$	0	0
$645$	−34.9625	−1.37665
$646$	−0.516295	−0.0203133
$647$	18.8434	0.740811	0.370405	−	0.928870i	$-0.379219\pi$
$647$	18.8434	0.740811	0.370405	+	0.928870i	$0.379219\pi$
$648$	0.105008	0.00412510
$649$	−5.93294	−0.232888
$650$	−0.294465	−0.0115499
$651$	0	0
$652$	−30.6071	−1.19866
$653$	32.6635	1.27822	0.639111	−	0.769115i	$-0.279303\pi$
$653$	32.6635	1.27822	0.639111	+	0.769115i	$0.279303\pi$
$654$	0.327293	0.0127982
$655$	−55.9837	−2.18747
$656$	−31.9657	−1.24805
$657$	−14.2871	−0.557392
$658$	0	0
$659$	−31.3480	−1.22114	−0.610571	−	0.791961i	$-0.709060\pi$
$659$	−31.3480	−1.22114	−0.610571	+	0.791961i	$0.709060\pi$
$660$	3.66694	0.142735
$661$	37.9924	1.47773	0.738866	−	0.673852i	$-0.235362\pi$
$661$	37.9924	1.47773	0.738866	+	0.673852i	$0.235362\pi$
$662$	0.366145	0.0142306
$663$	−17.0102	−0.660621
$664$	0.987973	0.0383408
$665$	0	0
$666$	0.211683	0.00820253
$667$	−1.78025	−0.0689314
$668$	−25.9943	−1.00575
$669$	15.5087	0.599601
$670$	0.195926	0.00756927
$671$	1.79079	0.0691325
$672$	0	0
$673$	10.3170	0.397691	0.198845	−	0.980031i	$-0.436281\pi$
$673$	10.3170	0.397691	0.198845	+	0.980031i	$0.436281\pi$
$674$	−0.243830	−0.00939197
$675$	−2.71885	−0.104649
$676$	−8.02645	−0.308710
$677$	−27.3198	−1.04999	−0.524993	−	0.851106i	$-0.675932\pi$
$677$	−27.3198	−1.04999	−0.524993	+	0.851106i	$0.675932\pi$
$678$	0.243890	0.00936654
$679$	0	0
$680$	1.20309	0.0461362
$681$	15.3330	0.587560
$682$	0.0628212	0.00240555
$683$	−21.3082	−0.815337	−0.407669	−	0.913130i	$-0.633658\pi$
$683$	−21.3082	−0.815337	−0.407669	+	0.913130i	$0.633658\pi$
$684$	−9.53329	−0.364514
$685$	19.4663	0.743768
$686$	0	0
$687$	6.40484	0.244360
$688$	50.2848	1.91709
$689$	−0.250898	−0.00955847
$690$	0.0729480	0.00277708
$691$	−20.6702	−0.786332	−0.393166	−	0.919467i	$-0.628620\pi$
$691$	−20.6702	−0.786332	−0.393166	+	0.919467i	$0.628620\pi$
$692$	20.8369	0.792099
$693$	0	0
$694$	−0.201517	−0.00764949
$695$	−8.01563	−0.304050
$696$	0.186940	0.00708594
$697$	−32.9892	−1.24955
$698$	−0.150167	−0.00568391
$699$	26.1430	0.988821
$700$	0	0
$701$	−1.93401	−0.0730465	−0.0365232	−	0.999333i	$-0.511628\pi$
$701$	−1.93401	−0.0730465	−0.0365232	+	0.999333i	$0.511628\pi$
$702$	0.108305	0.00408770
$703$	−38.4425	−1.44988
$704$	−5.27033	−0.198633
$705$	−32.6787	−1.23075
$706$	−0.352352	−0.0132609
$707$	0	0
$708$	−17.9681	−0.675284
$709$	25.3329	0.951396	0.475698	−	0.879609i	$-0.342196\pi$
$709$	25.3329	0.951396	0.475698	+	0.879609i	$0.342196\pi$
$710$	−0.907923	−0.0340737
$711$	−6.94797	−0.260569
$712$	−0.622098	−0.0233141
$713$	−3.62429	−0.135731
$714$	0	0
$715$	7.56544	0.282931
$716$	−38.4330	−1.43631
$717$	−17.6773	−0.660170
$718$	−0.469094	−0.0175064
$719$	−11.7275	−0.437361	−0.218681	−	0.975796i	$-0.570175\pi$
$719$	−11.7275	−0.437361	−0.218681	+	0.975796i	$0.570175\pi$
$720$	11.1016	0.413734
$721$	0	0
$722$	−0.0981095	−0.00365126
$723$	−5.75969	−0.214205
$724$	22.3028	0.828876
$725$	−4.84023	−0.179762
$726$	−0.277379	−0.0102945
$727$	13.4547	0.499007	0.249503	−	0.968374i	$-0.419733\pi$
$727$	13.4547	0.499007	0.249503	+	0.968374i	$0.419733\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.04221	0.0385740
$731$	51.8948	1.91940
$732$	5.42347	0.200457
$733$	20.7942	0.768051	0.384025	−	0.923323i	$-0.374538\pi$
$733$	20.7942	0.768051	0.384025	+	0.923323i	$0.374538\pi$
$734$	−0.686664	−0.0253452
$735$	0	0
$736$	−0.314933	−0.0116086
$737$	−1.77307	−0.0653117
$738$	0.210044	0.00773182
$739$	−23.3524	−0.859034	−0.429517	−	0.903059i	$-0.641316\pi$
$739$	−23.3524	−0.859034	−0.429517	+	0.903059i	$0.641316\pi$
$740$	44.7822	1.64623
$741$	−19.6686	−0.722544
$742$	0	0
$743$	6.11578	0.224366	0.112183	−	0.993688i	$-0.464216\pi$
$743$	6.11578	0.224366	0.112183	+	0.993688i	$0.464216\pi$
$744$	0.380579	0.0139527
$745$	52.9564	1.94017
$746$	−0.528544	−0.0193514
$747$	9.40856	0.344241
$748$	−5.44284	−0.199010
$749$	0	0
$750$	−0.166405	−0.00607625
$751$	24.5032	0.894136	0.447068	−	0.894500i	$-0.352468\pi$
$751$	24.5032	0.894136	0.447068	+	0.894500i	$0.352468\pi$
$752$	47.0001	1.71392
$753$	10.3866	0.378509
$754$	0.192809	0.00702170
$755$	−14.0662	−0.511920
$756$	0	0
$757$	23.0058	0.836160	0.418080	−	0.908410i	$-0.362703\pi$
$757$	23.0058	0.836160	0.418080	+	0.908410i	$0.362703\pi$
$758$	−0.211337	−0.00767612
$759$	−0.660156	−0.0239621
$760$	1.39111	0.0504608
$761$	−10.6470	−0.385953	−0.192976	−	0.981203i	$-0.561814\pi$
$761$	−10.6470	−0.385953	−0.192976	+	0.981203i	$0.561814\pi$
$762$	0.0630349	0.00228351
$763$	0	0
$764$	−38.4560	−1.39129
$765$	11.4571	0.414232
$766$	−0.526952	−0.0190395
$767$	−37.0710	−1.33856
$768$	−15.9449	−0.575361
$769$	17.0235	0.613883	0.306942	−	0.951728i	$-0.400694\pi$
$769$	17.0235	0.613883	0.306942	+	0.951728i	$0.400694\pi$
$770$	0	0
$771$	2.69848	0.0971834
$772$	−34.8933	−1.25584
$773$	−7.84327	−0.282103	−0.141051	−	0.990002i	$-0.545048\pi$
$773$	−7.84327	−0.282103	−0.141051	+	0.990002i	$0.545048\pi$
$774$	−0.330417	−0.0118766
$775$	−9.85391	−0.353963
$776$	0.266447	0.00956488
$777$	0	0
$778$	−0.326936	−0.0117212
$779$	−38.1448	−1.36668
$780$	22.9122	0.820390
$781$	8.21642	0.294007
$782$	−0.108277	−0.00387197
$783$	1.78025	0.0636208
$784$	0	0
$785$	−7.62327	−0.272086
$786$	−0.529081	−0.0188717
$787$	−40.9580	−1.45999	−0.729997	−	0.683450i	$-0.760479\pi$
$787$	−40.9580	−1.45999	−0.729997	+	0.683450i	$0.760479\pi$
$788$	2.32398	0.0827883
$789$	21.1490	0.752926
$790$	0.506840	0.0180326
$791$	0	0
$792$	0.0693216	0.00246324
$793$	11.1894	0.397348
$794$	0.182096	0.00646235
$795$	0.168991	0.00599349
$796$	−33.9420	−1.20304
$797$	−11.8092	−0.418303	−0.209151	−	0.977883i	$-0.567070\pi$
$797$	−11.8092	−0.418303	−0.209151	+	0.977883i	$0.567070\pi$
$798$	0	0
$799$	48.5050	1.71598
$800$	−0.856257	−0.0302733
$801$	−5.92430	−0.209325
$802$	0.378517	0.0133659
$803$	−9.43170	−0.332837
$804$	−5.36981	−0.189378
$805$	0	0
$806$	0.392528	0.0138262
$807$	28.5411	1.00469
$808$	0.704789	0.0247944
$809$	13.0478	0.458736	0.229368	−	0.973340i	$-0.426334\pi$
$809$	13.0478	0.458736	0.229368	+	0.973340i	$0.426334\pi$
$810$	−0.0729480	−0.00256313
$811$	−51.3449	−1.80296	−0.901481	−	0.432818i	$-0.857519\pi$
$811$	−51.3449	−1.80296	−0.901481	+	0.432818i	$0.857519\pi$
$812$	0	0
$813$	−15.1472	−0.531237
$814$	0.139744	0.00489801
$815$	42.5322	1.48984
$816$	−16.4782	−0.576851
$817$	60.0051	2.09931
$818$	−0.630851	−0.0220572
$819$	0	0
$820$	44.4355	1.55175
$821$	56.3021	1.96496	0.982479	−	0.186374i	$-0.0596737\pi$
$821$	56.3021	1.96496	0.982479	+	0.186374i	$0.0596737\pi$
$822$	0.183968	0.00641663
$823$	−13.5565	−0.472551	−0.236276	−	0.971686i	$-0.575927\pi$
$823$	−13.5565	−0.472551	−0.236276	+	0.971686i	$0.575927\pi$
$824$	1.12860	0.0393168
$825$	−1.79487	−0.0624893
$826$	0	0
$827$	3.80574	0.132339	0.0661693	−	0.997808i	$-0.478922\pi$
$827$	3.80574	0.132339	0.0661693	+	0.997808i	$0.478922\pi$
$828$	−1.99931	−0.0694808
$829$	−38.8924	−1.35079	−0.675395	−	0.737457i	$-0.736027\pi$
$829$	−38.8924	−1.35079	−0.675395	+	0.737457i	$0.736027\pi$
$830$	−0.686335	−0.0238230
$831$	−4.23717	−0.146986
$832$	−32.9308	−1.14167
$833$	0	0
$834$	−0.0757527	−0.00262310
$835$	36.1222	1.25006
$836$	−6.29346	−0.217664
$837$	3.62429	0.125274
$838$	−0.837440	−0.0289289
$839$	5.77279	0.199299	0.0996495	−	0.995023i	$-0.468228\pi$
$839$	5.77279	0.199299	0.0996495	+	0.995023i	$0.468228\pi$
$840$	0	0
$841$	−25.8307	−0.890715
$842$	0.494672	0.0170475
$843$	−0.762123	−0.0262489
$844$	−23.3767	−0.804661
$845$	11.1537	0.383700
$846$	−0.308834	−0.0106179
$847$	0	0
$848$	−0.243051	−0.00834641
$849$	−17.1714	−0.589321
$850$	−0.294388	−0.0100974
$851$	−8.06211	−0.276365
$852$	24.8838	0.852504
$853$	6.63846	0.227296	0.113648	−	0.993521i	$-0.463746\pi$
$853$	6.63846	0.227296	0.113648	+	0.993521i	$0.463746\pi$
$854$	0	0
$855$	13.2477	0.453060
$856$	−0.0508822	−0.00173912
$857$	−25.7842	−0.880770	−0.440385	−	0.897809i	$-0.645158\pi$
$857$	−25.7842	−0.880770	−0.440385	+	0.897809i	$0.645158\pi$
$858$	0.0714981	0.00244090
$859$	21.1070	0.720162	0.360081	−	0.932921i	$-0.382749\pi$
$859$	21.1070	0.720162	0.360081	+	0.932921i	$0.382749\pi$
$860$	−69.9008	−2.38360
$861$	0	0
$862$	0.954242	0.0325016
$863$	12.4548	0.423965	0.211982	−	0.977273i	$-0.432008\pi$
$863$	12.4548	0.423965	0.211982	+	0.977273i	$0.432008\pi$
$864$	0.314933	0.0107142
$865$	−28.9553	−0.984511
$866$	−0.565791	−0.0192264
$867$	−0.00578278	−0.000196394 0
$868$	0	0
$869$	−4.58675	−0.155595
$870$	−0.129865	−0.00440285
$871$	−11.0787	−0.375388
$872$	1.30894	0.0443265
$873$	2.53740	0.0858779
$874$	−0.125199	−0.00423490
$875$	0	0
$876$	−28.5643	−0.965098
$877$	−45.9072	−1.55018	−0.775088	−	0.631854i	$-0.782294\pi$
$877$	−45.9072	−1.55018	−0.775088	+	0.631854i	$0.782294\pi$
$878$	0.281929	0.00951465
$879$	1.15566	0.0389794
$880$	7.32881	0.247054
$881$	17.7656	0.598539	0.299269	−	0.954169i	$-0.403257\pi$
$881$	17.7656	0.598539	0.299269	+	0.954169i	$0.403257\pi$
$882$	0	0
$883$	8.07698	0.271812	0.135906	−	0.990722i	$-0.456605\pi$
$883$	8.07698	0.271812	0.135906	+	0.990722i	$0.456605\pi$
$884$	−34.0087	−1.14384
$885$	24.9689	0.839320
$886$	0.891489	0.0299502
$887$	−51.6718	−1.73497	−0.867484	−	0.497464i	$-0.834265\pi$
$887$	−51.6718	−1.73497	−0.867484	+	0.497464i	$0.834265\pi$
$888$	0.846585	0.0284095
$889$	0	0
$890$	0.432166	0.0144862
$891$	0.660156	0.0221161
$892$	31.0067	1.03818
$893$	56.0855	1.87683
$894$	0.500470	0.0167382
$895$	53.4073	1.78521
$896$	0	0
$897$	−4.12488	−0.137726
$898$	−0.182278	−0.00608268
$899$	6.45212	0.215190
$900$	−5.43583	−0.181194
$901$	−0.250833	−0.00835647
$902$	0.138662	0.00461693
$903$	0	0
$904$	0.975392	0.0324411
$905$	−30.9924	−1.03022
$906$	−0.132934	−0.00441643
$907$	38.7795	1.28765	0.643826	−	0.765172i	$-0.277346\pi$
$907$	38.7795	1.28765	0.643826	+	0.765172i	$0.277346\pi$
$908$	30.6553	1.01733
$909$	6.71178	0.222616
$910$	0	0
$911$	9.13030	0.302500	0.151250	−	0.988496i	$-0.451670\pi$
$911$	9.13030	0.302500	0.151250	+	0.988496i	$0.451670\pi$
$912$	−19.0534	−0.630922
$913$	6.21112	0.205558
$914$	1.03847	0.0343495
$915$	−7.53656	−0.249151
$916$	12.8053	0.423098
$917$	0	0
$918$	0.108277	0.00357366
$919$	3.74922	0.123675	0.0618377	−	0.998086i	$-0.480304\pi$
$919$	3.74922	0.123675	0.0618377	+	0.998086i	$0.480304\pi$
$920$	0.291742	0.00961843
$921$	−31.0365	−1.02269
$922$	0.128810	0.00424212
$923$	51.3389	1.68984
$924$	0	0
$925$	−21.9197	−0.720715
$926$	0.0202728	0.000666206 0
$927$	10.7478	0.353004
$928$	0.560658	0.0184045
$929$	44.4320	1.45777	0.728883	−	0.684638i	$-0.240040\pi$
$929$	44.4320	1.45777	0.728883	+	0.684638i	$0.240040\pi$
$930$	−0.264384	−0.00866950
$931$	0	0
$932$	52.2681	1.71210
$933$	−17.6141	−0.576658
$934$	0.504065	0.0164935
$935$	7.56347	0.247352
$936$	0.433145	0.0141578
$937$	59.8482	1.95516	0.977579	−	0.210571i	$-0.0675324\pi$
$937$	59.8482	1.95516	0.977579	+	0.210571i	$0.0675324\pi$
$938$	0	0
$939$	−15.7406	−0.513676
$940$	−65.3348	−2.13099
$941$	21.7078	0.707654	0.353827	−	0.935311i	$-0.384880\pi$
$941$	21.7078	0.707654	0.353827	+	0.935311i	$0.384880\pi$
$942$	−0.0720446	−0.00234734
$943$	−7.99969	−0.260506
$944$	−35.9115	−1.16882
$945$	0	0
$946$	−0.218127	−0.00709191
$947$	16.5648	0.538285	0.269143	−	0.963100i	$-0.413260\pi$
$947$	16.5648	0.538285	0.269143	+	0.963100i	$0.413260\pi$
$948$	−13.8912	−0.451164
$949$	−58.9324	−1.91303
$950$	−0.340396	−0.0110439
$951$	5.22139	0.169315
$952$	0	0
$953$	51.8971	1.68111	0.840555	−	0.541726i	$-0.182229\pi$
$953$	51.8971	1.68111	0.840555	+	0.541726i	$0.182229\pi$
$954$	0.00159707	5.17070e−5 0
$955$	53.4392	1.72925
$956$	−35.3424	−1.14305
$957$	1.17524	0.0379901
$958$	0.478142	0.0154481
$959$	0	0
$960$	22.1803	0.715866
$961$	−17.8645	−0.576276
$962$	0.873165	0.0281520
$963$	−0.484556	−0.0156146
$964$	−11.5154	−0.370886
$965$	48.4885	1.56090
$966$	0	0
$967$	−1.28492	−0.0413202	−0.0206601	−	0.999787i	$-0.506577\pi$
$967$	−1.28492	−0.0413202	−0.0206601	+	0.999787i	$0.506577\pi$
$968$	−1.10932	−0.0356550
$969$	−19.6635	−0.631683
$970$	−0.185098	−0.00594314
$971$	−50.4972	−1.62053	−0.810266	−	0.586063i	$-0.800677\pi$
$971$	−50.4972	−1.62053	−0.810266	+	0.586063i	$0.800677\pi$
$972$	1.99931	0.0641279
$973$	0	0
$974$	0.758321	0.0242982
$975$	−11.2149	−0.359165
$976$	10.8395	0.346963
$977$	−5.77501	−0.184759	−0.0923794	−	0.995724i	$-0.529447\pi$
$977$	−5.77501	−0.184759	−0.0923794	+	0.995724i	$0.529447\pi$
$978$	0.401955	0.0128531
$979$	−3.91096	−0.124995
$980$	0	0
$981$	12.4652	0.397983
$982$	0.621797	0.0198423
$983$	44.3068	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$983$	44.3068	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$984$	0.840030	0.0267792
$985$	−3.22945	−0.102899
$986$	0.192759	0.00613870
$987$	0	0
$988$	−39.3237	−1.25105
$989$	12.5842	0.400154
$990$	−0.0481570	−0.00153053
$991$	9.45792	0.300441	0.150220	−	0.988653i	$-0.452002\pi$
$991$	9.45792	0.300441	0.150220	+	0.988653i	$0.452002\pi$
$992$	1.14141	0.0362397
$993$	13.9449	0.442529
$994$	0	0
$995$	47.1665	1.49528
$996$	18.8106	0.596038
$997$	−33.4467	−1.05927	−0.529635	−	0.848226i	$-0.677671\pi$
$997$	−33.4467	−1.05927	−0.529635	+	0.848226i	$0.677671\pi$
$998$	−0.666124	−0.0210858
$999$	8.06211	0.255074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bi.1.5		10
7.2	even	3		483.2.i.h.277.6	✓	20
7.4	even	3		483.2.i.h.415.6	yes	20
7.6	odd	2		3381.2.a.bj.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.h.277.6	✓	20	7.2	even	3
483.2.i.h.415.6	yes	20	7.4	even	3
3381.2.a.bi.1.5		10	1.1	even	1	trivial
3381.2.a.bj.1.5		10	7.6	odd	2

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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0262565 q^{2} -1.00000 q^{3} -1.99931 q^{4} +2.77828 q^{5} +0.0262565 q^{6} +0.105008 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.0262565 q^{2} -1.00000 q^{3} -1.99931 q^{4} +2.77828 q^{5} +0.0262565 q^{6} +0.105008 q^{8} +1.00000 q^{9} -0.0729480 q^{10} +0.660156 q^{11} +1.99931 q^{12} +4.12488 q^{13} -2.77828 q^{15} +3.99586 q^{16} +4.12381 q^{17} -0.0262565 q^{18} +4.76829 q^{19} -5.55465 q^{20} -0.0173334 q^{22} +1.00000 q^{23} -0.105008 q^{24} +2.71885 q^{25} -0.108305 q^{26} -1.00000 q^{27} -1.78025 q^{29} +0.0729480 q^{30} -3.62429 q^{31} -0.314933 q^{32} -0.660156 q^{33} -0.108277 q^{34} -1.99931 q^{36} -8.06211 q^{37} -0.125199 q^{38} -4.12488 q^{39} +0.291742 q^{40} -7.99969 q^{41} +12.5842 q^{43} -1.31986 q^{44} +2.77828 q^{45} -0.0262565 q^{46} +11.7622 q^{47} -3.99586 q^{48} -0.0713876 q^{50} -4.12381 q^{51} -8.24691 q^{52} -0.0608257 q^{53} +0.0262565 q^{54} +1.83410 q^{55} -4.76829 q^{57} +0.0467430 q^{58} -8.98717 q^{59} +5.55465 q^{60} +2.71267 q^{61} +0.0951611 q^{62} -7.98346 q^{64} +11.4601 q^{65} +0.0173334 q^{66} -2.68583 q^{67} -8.24477 q^{68} -1.00000 q^{69} +12.4462 q^{71} +0.105008 q^{72} -14.2871 q^{73} +0.211683 q^{74} -2.71885 q^{75} -9.53329 q^{76} +0.108305 q^{78} -6.94797 q^{79} +11.1016 q^{80} +1.00000 q^{81} +0.210044 q^{82} +9.40856 q^{83} +11.4571 q^{85} -0.330417 q^{86} +1.78025 q^{87} +0.0693216 q^{88} -5.92430 q^{89} -0.0729480 q^{90} -1.99931 q^{92} +3.62429 q^{93} -0.308834 q^{94} +13.2477 q^{95} +0.314933 q^{96} +2.53740 q^{97} +0.660156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 3 q^{2} - 10 q^{3} + 15 q^{4} - 5 q^{5} - 3 q^{6} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q + 3 * q^2 - 10 * q^3 + 15 * q^4 - 5 * q^5 - 3 * q^6 + 9 * q^8 + 10 * q^9	\( 10 q + 3 q^{2} - 10 q^{3} + 15 q^{4} - 5 q^{5} - 3 q^{6} + 9 q^{8} + 10 q^{9} + 11 q^{10} + 8 q^{11} - 15 q^{12} + 5 q^{15} + 37 q^{16} - 11 q^{17} + 3 q^{18} + q^{19} - 15 q^{20} + 6 q^{22} + 10 q^{23} - 9 q^{24} + 21 q^{25} + q^{26} - 10 q^{27} + 22 q^{29} - 11 q^{30} - 3 q^{31} + 11 q^{32} - 8 q^{33} - 3 q^{34} + 15 q^{36} - 3 q^{37} + 16 q^{38} + 39 q^{40} - 26 q^{41} + 27 q^{43} + 16 q^{44} - 5 q^{45} + 3 q^{46} + 11 q^{47} - 37 q^{48} + 2 q^{50} + 11 q^{51} + 29 q^{52} + 5 q^{53} - 3 q^{54} - 18 q^{55} - q^{57} + 16 q^{58} - 10 q^{59} + 15 q^{60} + 22 q^{61} - 32 q^{62} + 69 q^{64} - 11 q^{65} - 6 q^{66} - 2 q^{67} - 21 q^{68} - 10 q^{69} + 27 q^{71} + 9 q^{72} - 8 q^{73} + 14 q^{74} - 21 q^{75} - 22 q^{76} - q^{78} + 21 q^{79} - 53 q^{80} + 10 q^{81} + 36 q^{82} - 12 q^{83} + 23 q^{85} + 18 q^{86} - 22 q^{87} - 10 q^{88} + 6 q^{89} + 11 q^{90} + 15 q^{92} + 3 q^{93} + 35 q^{94} + 44 q^{95} - 11 q^{96} + 6 q^{97} + 8 q^{99}+O(q^{100}) \) 10 * q + 3 * q^2 - 10 * q^3 + 15 * q^4 - 5 * q^5 - 3 * q^6 + 9 * q^8 + 10 * q^9 + 11 * q^10 + 8 * q^11 - 15 * q^12 + 5 * q^15 + 37 * q^16 - 11 * q^17 + 3 * q^18 + q^19 - 15 * q^20 + 6 * q^22 + 10 * q^23 - 9 * q^24 + 21 * q^25 + q^26 - 10 * q^27 + 22 * q^29 - 11 * q^30 - 3 * q^31 + 11 * q^32 - 8 * q^33 - 3 * q^34 + 15 * q^36 - 3 * q^37 + 16 * q^38 + 39 * q^40 - 26 * q^41 + 27 * q^43 + 16 * q^44 - 5 * q^45 + 3 * q^46 + 11 * q^47 - 37 * q^48 + 2 * q^50 + 11 * q^51 + 29 * q^52 + 5 * q^53 - 3 * q^54 - 18 * q^55 - q^57 + 16 * q^58 - 10 * q^59 + 15 * q^60 + 22 * q^61 - 32 * q^62 + 69 * q^64 - 11 * q^65 - 6 * q^66 - 2 * q^67 - 21 * q^68 - 10 * q^69 + 27 * q^71 + 9 * q^72 - 8 * q^73 + 14 * q^74 - 21 * q^75 - 22 * q^76 - q^78 + 21 * q^79 - 53 * q^80 + 10 * q^81 + 36 * q^82 - 12 * q^83 + 23 * q^85 + 18 * q^86 - 22 * q^87 - 10 * q^88 + 6 * q^89 + 11 * q^90 + 15 * q^92 + 3 * q^93 + 35 * q^94 + 44 * q^95 - 11 * q^96 + 6 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

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