LMFDB - Embedded newform 2009.2.a.t.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2009,2,Mod(1,2009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2009, 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("2009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2009 = 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2009.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:	$16.0419457661$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 27 x^{18} + 52 x^{17} + 302 x^{16} - 552 x^{15} - 1820 x^{14} + 3090 x^{13} + \cdots + 49 $ x^20 - 2x^19 - 27x^18 + 52x^17 + 302x^16 - 552x^15 - 1820x^14 + 3090x^13 + 6449x^12 - 9852x^11 - 13797x^10 + 18080x^9 + 17721x^8 - 18446x^7 - 13352x^6 + 9524x^5 + 5690x^4 - 1972x^3 - 1179x^2 + 42*x + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Root			$-2.51950$ of defining polynomial
Character	$\chi$	$=$	2009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.51950 q^{2} -2.78497 q^{3} +4.34790 q^{4} -1.47804 q^{5} -7.01674 q^{6} +5.91555 q^{8} +4.75604 q^{9} +O(q^{10})$	\(q+2.51950 q^{2} -2.78497 q^{3} +4.34790 q^{4} -1.47804 q^{5} -7.01674 q^{6} +5.91555 q^{8} +4.75604 q^{9} -3.72392 q^{10} -4.83604 q^{11} -12.1088 q^{12} +4.02077 q^{13} +4.11628 q^{15} +6.20845 q^{16} +2.82150 q^{17} +11.9829 q^{18} -4.32444 q^{19} -6.42636 q^{20} -12.1844 q^{22} -7.27952 q^{23} -16.4746 q^{24} -2.81541 q^{25} +10.1304 q^{26} -4.89052 q^{27} +7.18920 q^{29} +10.3710 q^{30} -7.03562 q^{31} +3.81112 q^{32} +13.4682 q^{33} +7.10877 q^{34} +20.6788 q^{36} -8.74950 q^{37} -10.8955 q^{38} -11.1977 q^{39} -8.74340 q^{40} -1.00000 q^{41} +0.174038 q^{43} -21.0266 q^{44} -7.02960 q^{45} -18.3408 q^{46} -5.63943 q^{47} -17.2903 q^{48} -7.09344 q^{50} -7.85778 q^{51} +17.4819 q^{52} +7.57218 q^{53} -12.3217 q^{54} +7.14784 q^{55} +12.0434 q^{57} +18.1132 q^{58} -3.13471 q^{59} +17.8972 q^{60} -4.21867 q^{61} -17.7263 q^{62} -2.81477 q^{64} -5.94285 q^{65} +33.9332 q^{66} -8.79107 q^{67} +12.2676 q^{68} +20.2732 q^{69} -13.6811 q^{71} +28.1346 q^{72} -2.46761 q^{73} -22.0444 q^{74} +7.84083 q^{75} -18.8023 q^{76} -28.2127 q^{78} +16.3988 q^{79} -9.17631 q^{80} -0.648197 q^{81} -2.51950 q^{82} -6.42502 q^{83} -4.17027 q^{85} +0.438490 q^{86} -20.0217 q^{87} -28.6078 q^{88} -11.4841 q^{89} -17.7111 q^{90} -31.6506 q^{92} +19.5940 q^{93} -14.2086 q^{94} +6.39168 q^{95} -10.6138 q^{96} -15.7940 q^{97} -23.0004 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) $ 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9	$ 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9} - 16 q^{10} - 6 q^{12} - 12 q^{13} + 14 q^{16} - 8 q^{17} - 18 q^{18} - 36 q^{19} - 24 q^{20} - 8 q^{22} - 12 q^{23} - 36 q^{24} + 20 q^{25} + 22 q^{26} - 32 q^{27} + 4 q^{29} + 28 q^{30} - 80 q^{31} + 6 q^{32} + 12 q^{33} - 48 q^{34} + 26 q^{36} + 4 q^{37} - 12 q^{38} - 28 q^{39} + 4 q^{40} - 20 q^{41} - 20 q^{44} - 40 q^{45} + 8 q^{46} - 32 q^{47} - 16 q^{48} + 6 q^{50} - 20 q^{51} - 36 q^{52} + 4 q^{53} + 50 q^{54} - 64 q^{55} - 4 q^{57} - 32 q^{59} + 20 q^{60} - 44 q^{61} + 8 q^{62} - 30 q^{64} - 8 q^{65} - 32 q^{66} - 4 q^{67} + 48 q^{68} + 24 q^{69} + 8 q^{71} - 8 q^{72} - 48 q^{73} - 38 q^{74} - 24 q^{75} - 84 q^{76} + 30 q^{78} - 4 q^{79} - 56 q^{80} + 2 q^{82} - 8 q^{83} - 12 q^{85} - 24 q^{86} - 40 q^{87} - 48 q^{88} - 20 q^{89} - 48 q^{90} - 50 q^{92} + 48 q^{93} - 26 q^{94} + 20 q^{95} - 70 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) $ 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9 - 16 * q^10 - 6 * q^12 - 12 * q^13 + 14 * q^16 - 8 * q^17 - 18 * q^18 - 36 * q^19 - 24 * q^20 - 8 * q^22 - 12 * q^23 - 36 * q^24 + 20 * q^25 + 22 * q^26 - 32 * q^27 + 4 * q^29 + 28 * q^30 - 80 * q^31 + 6 * q^32 + 12 * q^33 - 48 * q^34 + 26 * q^36 + 4 * q^37 - 12 * q^38 - 28 * q^39 + 4 * q^40 - 20 * q^41 - 20 * q^44 - 40 * q^45 + 8 * q^46 - 32 * q^47 - 16 * q^48 + 6 * q^50 - 20 * q^51 - 36 * q^52 + 4 * q^53 + 50 * q^54 - 64 * q^55 - 4 * q^57 - 32 * q^59 + 20 * q^60 - 44 * q^61 + 8 * q^62 - 30 * q^64 - 8 * q^65 - 32 * q^66 - 4 * q^67 + 48 * q^68 + 24 * q^69 + 8 * q^71 - 8 * q^72 - 48 * q^73 - 38 * q^74 - 24 * q^75 - 84 * q^76 + 30 * q^78 - 4 * q^79 - 56 * q^80 + 2 * q^82 - 8 * q^83 - 12 * q^85 - 24 * q^86 - 40 * q^87 - 48 * q^88 - 20 * q^89 - 48 * q^90 - 50 * q^92 + 48 * q^93 - 26 * q^94 + 20 * q^95 - 70 * q^96 - 8 * 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 $

$2$	2.51950	1.78156	0.890779	−	0.454436i	$-0.150159\pi$
$2$	2.51950	1.78156	0.890779	+	0.454436i	$0.150159\pi$
$3$	−2.78497	−1.60790	−0.803951	−	0.594696i	$-0.797272\pi$
$3$	−2.78497	−1.60790	−0.803951	+	0.594696i	$0.797272\pi$
$4$	4.34790	2.17395
$5$	−1.47804	−0.660998	−0.330499	−	0.943806i	$-0.607217\pi$
$5$	−1.47804	−0.660998	−0.330499	+	0.943806i	$0.607217\pi$
$6$	−7.01674	−2.86457
$7$	0	0
$7$	0	0
$8$	5.91555	2.09146
$9$	4.75604	1.58535
$10$	−3.72392	−1.17761
$11$	−4.83604	−1.45812	−0.729060	−	0.684450i	$-0.760043\pi$
$11$	−4.83604	−1.45812	−0.729060	+	0.684450i	$0.760043\pi$
$12$	−12.1088	−3.49550
$13$	4.02077	1.11516	0.557581	−	0.830123i	$-0.311730\pi$
$13$	4.02077	1.11516	0.557581	+	0.830123i	$0.311730\pi$
$14$	0	0
$15$	4.11628	1.06282
$16$	6.20845	1.55211
$17$	2.82150	0.684314	0.342157	−	0.939643i	$-0.388843\pi$
$17$	2.82150	0.684314	0.342157	+	0.939643i	$0.388843\pi$
$18$	11.9829	2.82439
$19$	−4.32444	−0.992096	−0.496048	−	0.868295i	$-0.665216\pi$
$19$	−4.32444	−0.992096	−0.496048	+	0.868295i	$0.665216\pi$
$20$	−6.42636	−1.43698
$21$	0	0
$22$	−12.1844	−2.59773
$23$	−7.27952	−1.51788	−0.758942	−	0.651158i	$-0.774283\pi$
$23$	−7.27952	−1.51788	−0.758942	+	0.651158i	$0.774283\pi$
$24$	−16.4746	−3.36287
$25$	−2.81541	−0.563082
$26$	10.1304	1.98673
$27$	−4.89052	−0.941180
$28$	0	0
$29$	7.18920	1.33500	0.667500	−	0.744609i	$-0.267364\pi$
$29$	7.18920	1.33500	0.667500	+	0.744609i	$0.267364\pi$
$30$	10.3710	1.89347
$31$	−7.03562	−1.26363	−0.631817	−	0.775118i	$-0.717691\pi$
$31$	−7.03562	−1.26363	−0.631817	+	0.775118i	$0.717691\pi$
$32$	3.81112	0.673718
$33$	13.4682	2.34451
$34$	7.10877	1.21914
$35$	0	0
$36$	20.6788	3.44647
$37$	−8.74950	−1.43841	−0.719204	−	0.694799i	$-0.755493\pi$
$37$	−8.74950	−1.43841	−0.719204	+	0.694799i	$0.755493\pi$
$38$	−10.8955	−1.76748
$39$	−11.1977	−1.79307
$40$	−8.74340	−1.38245
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	0.174038	0.0265406	0.0132703	−	0.999912i	$-0.495776\pi$
$43$	0.174038	0.0265406	0.0132703	+	0.999912i	$0.495776\pi$
$44$	−21.0266	−3.16988
$45$	−7.02960	−1.04791
$46$	−18.3408	−2.70420
$47$	−5.63943	−0.822595	−0.411298	−	0.911501i	$-0.634924\pi$
$47$	−5.63943	−0.822595	−0.411298	+	0.911501i	$0.634924\pi$
$48$	−17.2903	−2.49565
$49$	0	0
$50$	−7.09344	−1.00316
$51$	−7.85778	−1.10031
$52$	17.4819	2.42431
$53$	7.57218	1.04012	0.520059	−	0.854130i	$-0.325910\pi$
$53$	7.57218	1.04012	0.520059	+	0.854130i	$0.325910\pi$
$54$	−12.3217	−1.67677
$55$	7.14784	0.963814
$56$	0	0
$57$	12.0434	1.59519
$58$	18.1132	2.37838
$59$	−3.13471	−0.408105	−0.204052	−	0.978960i	$-0.565411\pi$
$59$	−3.13471	−0.408105	−0.204052	+	0.978960i	$0.565411\pi$
$60$	17.8972	2.31052
$61$	−4.21867	−0.540145	−0.270073	−	0.962840i	$-0.587048\pi$
$61$	−4.21867	−0.540145	−0.270073	+	0.962840i	$0.587048\pi$
$62$	−17.7263	−2.25124
$63$	0	0
$64$	−2.81477	−0.351846
$65$	−5.94285	−0.737119
$66$	33.9332	4.17689
$67$	−8.79107	−1.07400	−0.537000	−	0.843582i	$-0.680442\pi$
$67$	−8.79107	−1.07400	−0.537000	+	0.843582i	$0.680442\pi$
$68$	12.2676	1.48766
$69$	20.2732	2.44061
$70$	0	0
$71$	−13.6811	−1.62365	−0.811824	−	0.583902i	$-0.801525\pi$
$71$	−13.6811	−1.62365	−0.811824	+	0.583902i	$0.801525\pi$
$72$	28.1346	3.31570
$73$	−2.46761	−0.288811	−0.144406	−	0.989519i	$-0.546127\pi$
$73$	−2.46761	−0.288811	−0.144406	+	0.989519i	$0.546127\pi$
$74$	−22.0444	−2.56261
$75$	7.84083	0.905381
$76$	−18.8023	−2.15677
$77$	0	0
$78$	−28.2127	−3.19446
$79$	16.3988	1.84500	0.922502	−	0.385992i	$-0.126141\pi$
$79$	16.3988	1.84500	0.922502	+	0.385992i	$0.126141\pi$
$80$	−9.17631	−1.02594
$81$	−0.648197	−0.0720219
$82$	−2.51950	−0.278233
$83$	−6.42502	−0.705238	−0.352619	−	0.935767i	$-0.614709\pi$
$83$	−6.42502	−0.705238	−0.352619	+	0.935767i	$0.614709\pi$
$84$	0	0
$85$	−4.17027	−0.452330
$86$	0.438490	0.0472836
$87$	−20.0217	−2.14655
$88$	−28.6078	−3.04960
$89$	−11.4841	−1.21731	−0.608655	−	0.793435i	$-0.708291\pi$
$89$	−11.4841	−1.21731	−0.608655	+	0.793435i	$0.708291\pi$
$90$	−17.7111	−1.86691
$91$	0	0
$92$	−31.6506	−3.29981
$93$	19.5940	2.03180
$94$	−14.2086	−1.46550
$95$	6.39168	0.655773
$96$	−10.6138	−1.08327
$97$	−15.7940	−1.60364	−0.801820	−	0.597566i	$-0.796135\pi$
$97$	−15.7940	−1.60364	−0.801820	+	0.597566i	$0.796135\pi$
$98$	0	0
$99$	−23.0004	−2.31163
$100$	−12.2411	−1.22411
$101$	18.0266	1.79371	0.896856	−	0.442322i	$-0.145845\pi$
$101$	18.0266	1.79371	0.896856	+	0.442322i	$0.145845\pi$
$102$	−19.7977	−1.96026
$103$	6.37036	0.627691	0.313845	−	0.949474i	$-0.398383\pi$
$103$	6.37036	0.627691	0.313845	+	0.949474i	$0.398383\pi$
$104$	23.7851	2.33232
$105$	0	0
$106$	19.0781	1.85303
$107$	4.47397	0.432515	0.216257	−	0.976336i	$-0.430615\pi$
$107$	4.47397	0.432515	0.216257	+	0.976336i	$0.430615\pi$
$108$	−21.2635	−2.04608
$109$	5.49312	0.526145	0.263073	−	0.964776i	$-0.415264\pi$
$109$	5.49312	0.526145	0.263073	+	0.964776i	$0.415264\pi$
$110$	18.0090	1.71709
$111$	24.3671	2.31282
$112$	0	0
$113$	−4.20891	−0.395941	−0.197970	−	0.980208i	$-0.563435\pi$
$113$	−4.20891	−0.395941	−0.197970	+	0.980208i	$0.563435\pi$
$114$	30.3435	2.84193
$115$	10.7594	1.00332
$116$	31.2579	2.90223
$117$	19.1230	1.76792
$118$	−7.89792	−0.727063
$119$	0	0
$120$	24.3501	2.22285
$121$	12.3873	1.12611
$122$	−10.6290	−0.962301
$123$	2.78497	0.251112
$124$	−30.5902	−2.74708
$125$	11.5515	1.03319
$126$	0	0
$127$	−6.78035	−0.601658	−0.300829	−	0.953678i	$-0.597263\pi$
$127$	−6.78035	−0.601658	−0.300829	+	0.953678i	$0.597263\pi$
$128$	−14.7141	−1.30055
$129$	−0.484691	−0.0426747
$130$	−14.9730	−1.31322
$131$	10.3439	0.903752	0.451876	−	0.892081i	$-0.350755\pi$
$131$	10.3439	0.903752	0.451876	+	0.892081i	$0.350755\pi$
$132$	58.5584	5.09686
$133$	0	0
$134$	−22.1491	−1.91339
$135$	7.22836	0.622118
$136$	16.6907	1.43122
$137$	5.30605	0.453327	0.226663	−	0.973973i	$-0.427218\pi$
$137$	5.30605	0.453327	0.226663	+	0.973973i	$0.427218\pi$
$138$	51.0784	4.34809
$139$	0.682588	0.0578963	0.0289482	−	0.999581i	$-0.490784\pi$
$139$	0.682588	0.0578963	0.0289482	+	0.999581i	$0.490784\pi$
$140$	0	0
$141$	15.7056	1.32265
$142$	−34.4696	−2.89263
$143$	−19.4446	−1.62604
$144$	29.5277	2.46064
$145$	−10.6259	−0.882432
$146$	−6.21714	−0.514535
$147$	0	0
$148$	−38.0420	−3.12703
$149$	12.3279	1.00994	0.504971	−	0.863137i	$-0.331503\pi$
$149$	12.3279	1.00994	0.504971	+	0.863137i	$0.331503\pi$
$150$	19.7550	1.61299
$151$	14.6635	1.19330	0.596648	−	0.802503i	$-0.296499\pi$
$151$	14.6635	1.19330	0.596648	+	0.802503i	$0.296499\pi$
$152$	−25.5815	−2.07493
$153$	13.4192	1.08487
$154$	0	0
$155$	10.3989	0.835259
$156$	−48.6866	−3.89805
$157$	−7.47508	−0.596577	−0.298288	−	0.954476i	$-0.596416\pi$
$157$	−7.47508	−0.596577	−0.298288	+	0.954476i	$0.596416\pi$
$158$	41.3167	3.28698
$159$	−21.0883	−1.67241
$160$	−5.63297	−0.445326
$161$	0	0
$162$	−1.63314	−0.128311
$163$	15.2168	1.19187	0.595935	−	0.803033i	$-0.296782\pi$
$163$	15.2168	1.19187	0.595935	+	0.803033i	$0.296782\pi$
$164$	−4.34790	−0.339514
$165$	−19.9065	−1.54972
$166$	−16.1879	−1.25642
$167$	19.9981	1.54750	0.773749	−	0.633492i	$-0.218379\pi$
$167$	19.9981	1.54750	0.773749	+	0.633492i	$0.218379\pi$
$168$	0	0
$169$	3.16662	0.243586
$170$	−10.5070	−0.805852
$171$	−20.5672	−1.57282
$172$	0.756702	0.0576980
$173$	−7.22448	−0.549267	−0.274633	−	0.961549i	$-0.588557\pi$
$173$	−7.22448	−0.549267	−0.274633	+	0.961549i	$0.588557\pi$
$174$	−50.4447	−3.82420
$175$	0	0
$176$	−30.0243	−2.26317
$177$	8.73007	0.656192
$178$	−28.9342	−2.16871
$179$	18.2822	1.36648	0.683239	−	0.730194i	$-0.260571\pi$
$179$	18.2822	1.36648	0.683239	+	0.730194i	$0.260571\pi$
$180$	−30.5640	−2.27811
$181$	−6.10150	−0.453521	−0.226761	−	0.973951i	$-0.572814\pi$
$181$	−6.10150	−0.453521	−0.226761	+	0.973951i	$0.572814\pi$
$182$	0	0
$183$	11.7489	0.868500
$184$	−43.0624	−3.17460
$185$	12.9321	0.950785
$186$	49.3671	3.61977
$187$	−13.6449	−0.997812
$188$	−24.5197	−1.78828
$189$	0	0
$190$	16.1039	1.16830
$191$	7.22078	0.522477	0.261239	−	0.965274i	$-0.415869\pi$
$191$	7.22078	0.522477	0.261239	+	0.965274i	$0.415869\pi$
$192$	7.83903	0.565734
$193$	−18.4620	−1.32892	−0.664461	−	0.747323i	$-0.731339\pi$
$193$	−18.4620	−1.32892	−0.664461	+	0.747323i	$0.731339\pi$
$194$	−39.7931	−2.85698
$195$	16.5506	1.18522
$196$	0	0
$197$	−8.35764	−0.595457	−0.297728	−	0.954651i	$-0.596229\pi$
$197$	−8.35764	−0.595457	−0.297728	+	0.954651i	$0.596229\pi$
$198$	−57.9496	−4.11830
$199$	−27.6030	−1.95673	−0.978363	−	0.206894i	$-0.933664\pi$
$199$	−27.6030	−1.95673	−0.978363	+	0.206894i	$0.933664\pi$
$200$	−16.6547	−1.17767
$201$	24.4828	1.72689
$202$	45.4181	3.19560
$203$	0	0
$204$	−34.1648	−2.39202
$205$	1.47804	0.103230
$206$	16.0502	1.11827
$207$	−34.6217	−2.40637
$208$	24.9628	1.73086
$209$	20.9132	1.44659
$210$	0	0
$211$	4.43107	0.305047	0.152524	−	0.988300i	$-0.451260\pi$
$211$	4.43107	0.305047	0.152524	+	0.988300i	$0.451260\pi$
$212$	32.9231	2.26117
$213$	38.1014	2.61067
$214$	11.2722	0.770550
$215$	−0.257235	−0.0175433
$216$	−28.9301	−1.96844
$217$	0	0
$218$	13.8399	0.937358
$219$	6.87220	0.464380
$220$	31.0781	2.09528
$221$	11.3446	0.763121
$222$	61.3929	4.12042
$223$	27.9961	1.87476	0.937378	−	0.348315i	$-0.113246\pi$
$223$	27.9961	1.87476	0.937378	+	0.348315i	$0.113246\pi$
$224$	0	0
$225$	−13.3902	−0.892681
$226$	−10.6044	−0.705392
$227$	10.7502	0.713515	0.356757	−	0.934197i	$-0.383882\pi$
$227$	10.7502	0.713515	0.356757	+	0.934197i	$0.383882\pi$
$228$	52.3637	3.46787
$229$	−14.0857	−0.930812	−0.465406	−	0.885097i	$-0.654092\pi$
$229$	−14.0857	−0.930812	−0.465406	+	0.885097i	$0.654092\pi$
$230$	27.1083	1.78747
$231$	0	0
$232$	42.5281	2.79211
$233$	6.86735	0.449895	0.224948	−	0.974371i	$-0.427779\pi$
$233$	6.86735	0.449895	0.224948	+	0.974371i	$0.427779\pi$
$234$	48.1804	3.14965
$235$	8.33528	0.543733
$236$	−13.6294	−0.887200
$237$	−45.6700	−2.96658
$238$	0	0
$239$	0.467300	0.0302271	0.0151136	−	0.999886i	$-0.495189\pi$
$239$	0.467300	0.0302271	0.0151136	+	0.999886i	$0.495189\pi$
$240$	25.5557	1.64962
$241$	13.9496	0.898574	0.449287	−	0.893387i	$-0.351678\pi$
$241$	13.9496	0.898574	0.449287	+	0.893387i	$0.351678\pi$
$242$	31.2098	2.00624
$243$	16.4768	1.05698
$244$	−18.3424	−1.17425
$245$	0	0
$246$	7.01674	0.447371
$247$	−17.3876	−1.10635
$248$	−41.6196	−2.64284
$249$	17.8935	1.13395
$250$	29.1039	1.84070
$251$	−19.8529	−1.25311	−0.626553	−	0.779379i	$-0.715535\pi$
$251$	−19.8529	−1.25311	−0.626553	+	0.779379i	$0.715535\pi$
$252$	0	0
$253$	35.2040	2.21326
$254$	−17.0831	−1.07189
$255$	11.6141	0.727302
$256$	−31.4426	−1.96516
$257$	24.3571	1.51935	0.759677	−	0.650301i	$-0.225357\pi$
$257$	24.3571	1.51935	0.759677	+	0.650301i	$0.225357\pi$
$258$	−1.22118	−0.0760274
$259$	0	0
$260$	−25.8389	−1.60246
$261$	34.1921	2.11644
$262$	26.0615	1.61009
$263$	−16.4878	−1.01668	−0.508341	−	0.861156i	$-0.669741\pi$
$263$	−16.4878	−1.01668	−0.508341	+	0.861156i	$0.669741\pi$
$264$	79.6719	4.90346
$265$	−11.1919	−0.687516
$266$	0	0
$267$	31.9828	1.95732
$268$	−38.2227	−2.33482
$269$	−9.69457	−0.591088	−0.295544	−	0.955329i	$-0.595501\pi$
$269$	−9.69457	−0.591088	−0.295544	+	0.955329i	$0.595501\pi$
$270$	18.2119	1.10834
$271$	−14.2386	−0.864933	−0.432467	−	0.901650i	$-0.642357\pi$
$271$	−14.2386	−0.864933	−0.432467	+	0.901650i	$0.642357\pi$
$272$	17.5171	1.06213
$273$	0	0
$274$	13.3686	0.807628
$275$	13.6154	0.821041
$276$	88.1460	5.30576
$277$	−15.4775	−0.929950	−0.464975	−	0.885324i	$-0.653937\pi$
$277$	−15.4775	−0.929950	−0.464975	+	0.885324i	$0.653937\pi$
$278$	1.71978	0.103146
$279$	−33.4617	−2.00330
$280$	0	0
$281$	−28.5644	−1.70401	−0.852005	−	0.523534i	$-0.824613\pi$
$281$	−28.5644	−1.70401	−0.852005	+	0.523534i	$0.824613\pi$
$282$	39.5704	2.35638
$283$	21.5025	1.27819	0.639094	−	0.769128i	$-0.279309\pi$
$283$	21.5025	1.27819	0.639094	+	0.769128i	$0.279309\pi$
$284$	−59.4841	−3.52973
$285$	−17.8006	−1.05442
$286$	−48.9908	−2.89689
$287$	0	0
$288$	18.1259	1.06808
$289$	−9.03915	−0.531715
$290$	−26.7720	−1.57210
$291$	43.9858	2.57849
$292$	−10.7289	−0.627862
$293$	2.33534	0.136432	0.0682159	−	0.997671i	$-0.478269\pi$
$293$	2.33534	0.136432	0.0682159	+	0.997671i	$0.478269\pi$
$294$	0	0
$295$	4.63322	0.269756
$296$	−51.7581	−3.00838
$297$	23.6507	1.37235
$298$	31.0602	1.79927
$299$	−29.2693	−1.69269
$300$	34.0911	1.96825
$301$	0	0
$302$	36.9447	2.12593
$303$	−50.2034	−2.88411
$304$	−26.8481	−1.53984
$305$	6.23534	0.357035
$306$	33.8096	1.93277
$307$	16.6878	0.952422	0.476211	−	0.879331i	$-0.342010\pi$
$307$	16.6878	0.952422	0.476211	+	0.879331i	$0.342010\pi$
$308$	0	0
$309$	−17.7413	−1.00926
$310$	26.2001	1.48806
$311$	2.12955	0.120756	0.0603779	−	0.998176i	$-0.480769\pi$
$311$	2.12955	0.120756	0.0603779	+	0.998176i	$0.480769\pi$
$312$	−66.2407	−3.75014
$313$	−31.5460	−1.78309	−0.891544	−	0.452935i	$-0.850377\pi$
$313$	−31.5460	−1.78309	−0.891544	+	0.452935i	$0.850377\pi$
$314$	−18.8335	−1.06284
$315$	0	0
$316$	71.3002	4.01095
$317$	−9.04710	−0.508136	−0.254068	−	0.967186i	$-0.581769\pi$
$317$	−9.04710	−0.508136	−0.254068	+	0.967186i	$0.581769\pi$
$318$	−53.1320	−2.97949
$319$	−34.7672	−1.94659
$320$	4.16033	0.232569
$321$	−12.4598	−0.695441
$322$	0	0
$323$	−12.2014	−0.678904
$324$	−2.81830	−0.156572
$325$	−11.3201	−0.627928
$326$	38.3387	2.12339
$327$	−15.2981	−0.845989
$328$	−5.91555	−0.326632
$329$	0	0
$330$	−50.1545	−2.76091
$331$	6.04330	0.332170	0.166085	−	0.986111i	$-0.446887\pi$
$331$	6.04330	0.332170	0.166085	+	0.986111i	$0.446887\pi$
$332$	−27.9354	−1.53315
$333$	−41.6130	−2.28038
$334$	50.3853	2.75696
$335$	12.9935	0.709911
$336$	0	0
$337$	−11.4122	−0.621661	−0.310830	−	0.950465i	$-0.600607\pi$
$337$	−11.4122	−0.621661	−0.310830	+	0.950465i	$0.600607\pi$
$338$	7.97832	0.433963
$339$	11.7217	0.636634
$340$	−18.1319	−0.983343
$341$	34.0245	1.84253
$342$	−51.8192	−2.80206
$343$	0	0
$344$	1.02953	0.0555087
$345$	−29.9645	−1.61324
$346$	−18.2021	−0.978551
$347$	25.2610	1.35608	0.678040	−	0.735025i	$-0.262829\pi$
$347$	25.2610	1.35608	0.678040	+	0.735025i	$0.262829\pi$
$348$	−87.0523	−4.66649
$349$	22.0231	1.17887	0.589434	−	0.807816i	$-0.299351\pi$
$349$	22.0231	1.17887	0.589434	+	0.807816i	$0.299351\pi$
$350$	0	0
$351$	−19.6637	−1.04957
$352$	−18.4307	−0.982361
$353$	−0.376661	−0.0200476	−0.0100238	−	0.999950i	$-0.503191\pi$
$353$	−0.376661	−0.0200476	−0.0100238	+	0.999950i	$0.503191\pi$
$354$	21.9954	1.16904
$355$	20.2212	1.07323
$356$	−49.9317	−2.64637
$357$	0	0
$358$	46.0622	2.43446
$359$	27.7277	1.46341	0.731705	−	0.681622i	$-0.238725\pi$
$359$	27.7277	1.46341	0.731705	+	0.681622i	$0.238725\pi$
$360$	−41.5839	−2.19167
$361$	−0.299182	−0.0157464
$362$	−15.3728	−0.807975
$363$	−34.4981	−1.81068
$364$	0	0
$365$	3.64721	0.190904
$366$	29.6013	1.54728
$367$	−15.1846	−0.792627	−0.396314	−	0.918115i	$-0.629711\pi$
$367$	−15.1846	−0.792627	−0.396314	+	0.918115i	$0.629711\pi$
$368$	−45.1945	−2.35593
$369$	−4.75604	−0.247590
$370$	32.5824	1.69388
$371$	0	0
$372$	85.1926	4.41703
$373$	9.65905	0.500127	0.250063	−	0.968229i	$-0.419549\pi$
$373$	9.65905	0.500127	0.250063	+	0.968229i	$0.419549\pi$
$374$	−34.3783	−1.77766
$375$	−32.1704	−1.66127
$376$	−33.3603	−1.72043
$377$	28.9061	1.48874
$378$	0	0
$379$	−17.8484	−0.916811	−0.458406	−	0.888743i	$-0.651579\pi$
$379$	−17.8484	−0.916811	−0.458406	+	0.888743i	$0.651579\pi$
$380$	27.7904	1.42562
$381$	18.8830	0.967407
$382$	18.1928	0.930824
$383$	3.32996	0.170153	0.0850765	−	0.996374i	$-0.472887\pi$
$383$	3.32996	0.170153	0.0850765	+	0.996374i	$0.472887\pi$
$384$	40.9782	2.09116
$385$	0	0
$386$	−46.5150	−2.36755
$387$	0.827733	0.0420761
$388$	−68.6709	−3.48624
$389$	15.6783	0.794920	0.397460	−	0.917620i	$-0.369892\pi$
$389$	15.6783	0.794920	0.397460	+	0.917620i	$0.369892\pi$
$390$	41.6994	2.11153
$391$	−20.5391	−1.03871
$392$	0	0
$393$	−28.8075	−1.45314
$394$	−21.0571	−1.06084
$395$	−24.2379	−1.21954
$396$	−100.003	−5.02536
$397$	3.32922	0.167089	0.0835444	−	0.996504i	$-0.473376\pi$
$397$	3.32922	0.167089	0.0835444	+	0.996504i	$0.473376\pi$
$398$	−69.5459	−3.48602
$399$	0	0
$400$	−17.4793	−0.873967
$401$	7.17799	0.358451	0.179226	−	0.983808i	$-0.442641\pi$
$401$	7.17799	0.358451	0.179226	+	0.983808i	$0.442641\pi$
$402$	61.6846	3.07655
$403$	−28.2886	−1.40916
$404$	78.3778	3.89944
$405$	0.958058	0.0476063
$406$	0	0
$407$	42.3129	2.09737
$408$	−46.4831	−2.30126
$409$	−9.00790	−0.445412	−0.222706	−	0.974886i	$-0.571489\pi$
$409$	−9.00790	−0.445412	−0.222706	+	0.974886i	$0.571489\pi$
$410$	3.72392	0.183911
$411$	−14.7772	−0.728905
$412$	27.6977	1.36457
$413$	0	0
$414$	−87.2295	−4.28709
$415$	9.49641	0.466161
$416$	15.3237	0.751304
$417$	−1.90098	−0.0930916
$418$	52.6908	2.57719
$419$	3.35984	0.164139	0.0820694	−	0.996627i	$-0.473847\pi$
$419$	3.35984	0.164139	0.0820694	+	0.996627i	$0.473847\pi$
$420$	0	0
$421$	−37.6716	−1.83600	−0.918001	−	0.396578i	$-0.870198\pi$
$421$	−37.6716	−1.83600	−0.918001	+	0.396578i	$0.870198\pi$
$422$	11.1641	0.543460
$423$	−26.8214	−1.30410
$424$	44.7936	2.17537
$425$	−7.94367	−0.385325
$426$	95.9967	4.65106
$427$	0	0
$428$	19.4524	0.940266
$429$	54.1526	2.61451
$430$	−0.648104	−0.0312544
$431$	12.2998	0.592458	0.296229	−	0.955117i	$-0.404271\pi$
$431$	12.2998	0.592458	0.296229	+	0.955117i	$0.404271\pi$
$432$	−30.3625	−1.46082
$433$	−0.961159	−0.0461904	−0.0230952	−	0.999733i	$-0.507352\pi$
$433$	−0.961159	−0.0461904	−0.0230952	+	0.999733i	$0.507352\pi$
$434$	0	0
$435$	29.5928	1.41886
$436$	23.8835	1.14381
$437$	31.4799	1.50589
$438$	17.3145	0.827321
$439$	−17.2056	−0.821180	−0.410590	−	0.911820i	$-0.634677\pi$
$439$	−17.2056	−0.821180	−0.410590	+	0.911820i	$0.634677\pi$
$440$	42.2834	2.01578
$441$	0	0
$442$	28.5828	1.35954
$443$	−33.4729	−1.59035	−0.795173	−	0.606383i	$-0.792620\pi$
$443$	−33.4729	−1.59035	−0.795173	+	0.606383i	$0.792620\pi$
$444$	105.946	5.02796
$445$	16.9739	0.804640
$446$	70.5362	3.33999
$447$	−34.3328	−1.62389
$448$	0	0
$449$	−4.12640	−0.194737	−0.0973685	−	0.995248i	$-0.531043\pi$
$449$	−4.12640	−0.194737	−0.0973685	+	0.995248i	$0.531043\pi$
$450$	−33.7367	−1.59036
$451$	4.83604	0.227720
$452$	−18.2999	−0.860756
$453$	−40.8373	−1.91870
$454$	27.0851	1.27117
$455$	0	0
$456$	71.2436	3.33629
$457$	16.2146	0.758486	0.379243	−	0.925297i	$-0.376184\pi$
$457$	16.2146	0.758486	0.379243	+	0.925297i	$0.376184\pi$
$458$	−35.4891	−1.65830
$459$	−13.7986	−0.644062
$460$	46.7808	2.18116
$461$	6.50231	0.302843	0.151421	−	0.988469i	$-0.451615\pi$
$461$	6.50231	0.302843	0.151421	+	0.988469i	$0.451615\pi$
$462$	0	0
$463$	6.28120	0.291912	0.145956	−	0.989291i	$-0.453374\pi$
$463$	6.28120	0.291912	0.145956	+	0.989291i	$0.453374\pi$
$464$	44.6338	2.07207
$465$	−28.9606	−1.34301
$466$	17.3023	0.801515
$467$	32.3281	1.49597	0.747984	−	0.663717i	$-0.231022\pi$
$467$	32.3281	1.49597	0.747984	+	0.663717i	$0.231022\pi$
$468$	83.1448	3.84337
$469$	0	0
$470$	21.0008	0.968693
$471$	20.8179	0.959236
$472$	−18.5436	−0.853536
$473$	−0.841656	−0.0386994
$474$	−115.066	−5.28514
$475$	12.1751	0.558631
$476$	0	0
$477$	36.0136	1.64895
$478$	1.17736	0.0538514
$479$	6.20951	0.283720	0.141860	−	0.989887i	$-0.454692\pi$
$479$	6.20951	0.283720	0.141860	+	0.989887i	$0.454692\pi$
$480$	15.6876	0.716040
$481$	−35.1798	−1.60406
$482$	35.1461	1.60086
$483$	0	0
$484$	53.8586	2.44812
$485$	23.3441	1.06000
$486$	41.5133	1.88308
$487$	−4.60588	−0.208712	−0.104356	−	0.994540i	$-0.533278\pi$
$487$	−4.60588	−0.208712	−0.104356	+	0.994540i	$0.533278\pi$
$488$	−24.9558	−1.12969
$489$	−42.3782	−1.91641
$490$	0	0
$491$	−27.1445	−1.22501	−0.612507	−	0.790465i	$-0.709839\pi$
$491$	−27.1445	−1.22501	−0.612507	+	0.790465i	$0.709839\pi$
$492$	12.1088	0.545905
$493$	20.2843	0.913559
$494$	−43.8082	−1.97102
$495$	33.9954	1.52798
$496$	−43.6803	−1.96130
$497$	0	0
$498$	45.0827	2.02020
$499$	−31.1462	−1.39430	−0.697148	−	0.716927i	$-0.745548\pi$
$499$	−31.1462	−1.39430	−0.697148	+	0.716927i	$0.745548\pi$
$500$	50.2246	2.24611
$501$	−55.6940	−2.48822
$502$	−50.0195	−2.23248
$503$	−2.38608	−0.106390	−0.0531949	−	0.998584i	$-0.516940\pi$
$503$	−2.38608	−0.106390	−0.0531949	+	0.998584i	$0.516940\pi$
$504$	0	0
$505$	−26.6439	−1.18564
$506$	88.6967	3.94305
$507$	−8.81894	−0.391663
$508$	−29.4803	−1.30798
$509$	−5.11092	−0.226538	−0.113269	−	0.993564i	$-0.536132\pi$
$509$	−5.11092	−0.226538	−0.113269	+	0.993564i	$0.536132\pi$
$510$	29.2617	1.29573
$511$	0	0
$512$	−49.7917	−2.20050
$513$	21.1488	0.933741
$514$	61.3678	2.70682
$515$	−9.41562	−0.414902
$516$	−2.10739	−0.0927726
$517$	27.2725	1.19944
$518$	0	0
$519$	20.1199	0.883167
$520$	−35.1552	−1.54166
$521$	11.9410	0.523145	0.261573	−	0.965184i	$-0.415759\pi$
$521$	11.9410	0.523145	0.261573	+	0.965184i	$0.415759\pi$
$522$	86.1472	3.77056
$523$	14.1037	0.616710	0.308355	−	0.951271i	$-0.400222\pi$
$523$	14.1037	0.616710	0.308355	+	0.951271i	$0.400222\pi$
$524$	44.9743	1.96471
$525$	0	0
$526$	−41.5411	−1.81128
$527$	−19.8510	−0.864722
$528$	83.6167	3.63895
$529$	29.9914	1.30397
$530$	−28.1982	−1.22485
$531$	−14.9088	−0.646988
$532$	0	0
$533$	−4.02077	−0.174159
$534$	80.5808	3.48707
$535$	−6.61268	−0.285891
$536$	−52.0040	−2.24623
$537$	−50.9154	−2.19716
$538$	−24.4255	−1.05306
$539$	0	0
$540$	31.4282	1.35245
$541$	2.06663	0.0888512	0.0444256	−	0.999013i	$-0.485854\pi$
$541$	2.06663	0.0888512	0.0444256	+	0.999013i	$0.485854\pi$
$542$	−35.8742	−1.54093
$543$	16.9925	0.729217
$544$	10.7531	0.461034
$545$	−8.11902	−0.347781
$546$	0	0
$547$	16.6533	0.712043	0.356022	−	0.934478i	$-0.384133\pi$
$547$	16.6533	0.712043	0.356022	+	0.934478i	$0.384133\pi$
$548$	23.0702	0.985510
$549$	−20.0642	−0.856318
$550$	34.3041	1.46273
$551$	−31.0893	−1.32445
$552$	119.927	5.10444
$553$	0	0
$554$	−38.9955	−1.65676
$555$	−36.0154	−1.52877
$556$	2.96782	0.125864
$557$	−12.1868	−0.516373	−0.258187	−	0.966095i	$-0.583125\pi$
$557$	−12.1868	−0.516373	−0.258187	+	0.966095i	$0.583125\pi$
$558$	−84.3069	−3.56899
$559$	0.699769	0.0295971
$560$	0	0
$561$	38.0005	1.60438
$562$	−71.9681	−3.03579
$563$	−22.8469	−0.962882	−0.481441	−	0.876479i	$-0.659886\pi$
$563$	−22.8469	−0.962882	−0.481441	+	0.876479i	$0.659886\pi$
$564$	68.2865	2.87538
$565$	6.22092	0.261716
$566$	54.1755	2.27717
$567$	0	0
$568$	−80.9313	−3.39580
$569$	−22.9419	−0.961773	−0.480886	−	0.876783i	$-0.659685\pi$
$569$	−22.9419	−0.961773	−0.480886	+	0.876783i	$0.659685\pi$
$570$	−44.8488	−1.87851
$571$	−6.21311	−0.260010	−0.130005	−	0.991513i	$-0.541499\pi$
$571$	−6.21311	−0.260010	−0.130005	+	0.991513i	$0.541499\pi$
$572$	−84.5433	−3.53493
$573$	−20.1096	−0.840092
$574$	0	0
$575$	20.4948	0.854693
$576$	−13.3871	−0.557798
$577$	−9.46737	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$577$	−9.46737	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$578$	−22.7742	−0.947281
$579$	51.4160	2.13678
$580$	−46.2003	−1.91836
$581$	0	0
$582$	110.822	4.59374
$583$	−36.6193	−1.51662
$584$	−14.5972	−0.604038
$585$	−28.2644	−1.16859
$586$	5.88389	0.243061
$587$	−5.16741	−0.213282	−0.106641	−	0.994298i	$-0.534010\pi$
$587$	−5.16741	−0.213282	−0.106641	+	0.994298i	$0.534010\pi$
$588$	0	0
$589$	30.4251	1.25365
$590$	11.6734	0.480587
$591$	23.2757	0.957436
$592$	−54.3209	−2.23257
$593$	45.0942	1.85180	0.925898	−	0.377774i	$-0.123310\pi$
$593$	45.0942	1.85180	0.925898	+	0.377774i	$0.123310\pi$
$594$	59.5881	2.44493
$595$	0	0
$596$	53.6005	2.19556
$597$	76.8735	3.14622
$598$	−73.7441	−3.01562
$599$	26.7653	1.09360	0.546801	−	0.837263i	$-0.315846\pi$
$599$	26.7653	1.09360	0.546801	+	0.837263i	$0.315846\pi$
$600$	46.3828	1.89357
$601$	−30.9277	−1.26157	−0.630784	−	0.775958i	$-0.717267\pi$
$601$	−30.9277	−1.26157	−0.630784	+	0.775958i	$0.717267\pi$
$602$	0	0
$603$	−41.8107	−1.70266
$604$	63.7553	2.59417
$605$	−18.3088	−0.744359
$606$	−126.488	−5.13822
$607$	0.361880	0.0146883	0.00734413	−	0.999973i	$-0.497662\pi$
$607$	0.361880	0.0146883	0.00734413	+	0.999973i	$0.497662\pi$
$608$	−16.4810	−0.668392
$609$	0	0
$610$	15.7100	0.636078
$611$	−22.6749	−0.917327
$612$	58.3452	2.35846
$613$	23.4574	0.947436	0.473718	−	0.880677i	$-0.342912\pi$
$613$	23.4574	0.947436	0.473718	+	0.880677i	$0.342912\pi$
$614$	42.0449	1.69680
$615$	−4.11628	−0.165984
$616$	0	0
$617$	20.7266	0.834421	0.417211	−	0.908810i	$-0.363008\pi$
$617$	20.7266	0.834421	0.417211	+	0.908810i	$0.363008\pi$
$618$	−44.6992	−1.79806
$619$	−15.5058	−0.623230	−0.311615	−	0.950209i	$-0.600870\pi$
$619$	−15.5058	−0.623230	−0.311615	+	0.950209i	$0.600870\pi$
$620$	45.2134	1.81581
$621$	35.6006	1.42860
$622$	5.36542	0.215134
$623$	0	0
$624$	−69.5205	−2.78305
$625$	−2.99641	−0.119856
$626$	−79.4804	−3.17667
$627$	−58.2425	−2.32598
$628$	−32.5009	−1.29693
$629$	−24.6867	−0.984323
$630$	0	0
$631$	−9.91913	−0.394874	−0.197437	−	0.980316i	$-0.563262\pi$
$631$	−9.91913	−0.394874	−0.197437	+	0.980316i	$0.563262\pi$
$632$	97.0077	3.85876
$633$	−12.3404	−0.490486
$634$	−22.7942	−0.905274
$635$	10.0216	0.397695
$636$	−91.6897	−3.63573
$637$	0	0
$638$	−87.5962	−3.46797
$639$	−65.0679	−2.57405
$640$	21.7479	0.859662
$641$	−29.2989	−1.15724	−0.578619	−	0.815598i	$-0.696408\pi$
$641$	−29.2989	−1.15724	−0.578619	+	0.815598i	$0.696408\pi$
$642$	−31.3926	−1.23897
$643$	15.7181	0.619860	0.309930	−	0.950759i	$-0.399694\pi$
$643$	15.7181	0.619860	0.309930	+	0.950759i	$0.399694\pi$
$644$	0	0
$645$	0.716390	0.0282078
$646$	−30.7415	−1.20951
$647$	−10.7783	−0.423738	−0.211869	−	0.977298i	$-0.567955\pi$
$647$	−10.7783	−0.423738	−0.211869	+	0.977298i	$0.567955\pi$
$648$	−3.83444	−0.150631
$649$	15.1596	0.595066
$650$	−28.5211	−1.11869
$651$	0	0
$652$	66.1611	2.59107
$653$	−6.82326	−0.267015	−0.133507	−	0.991048i	$-0.542624\pi$
$653$	−6.82326	−0.267015	−0.133507	+	0.991048i	$0.542624\pi$
$654$	−38.5437	−1.50718
$655$	−15.2887	−0.597378
$656$	−6.20845	−0.242399
$657$	−11.7360	−0.457866
$658$	0	0
$659$	31.9112	1.24309	0.621543	−	0.783380i	$-0.286506\pi$
$659$	31.9112	1.24309	0.621543	+	0.783380i	$0.286506\pi$
$660$	−86.5515	−3.36901
$661$	−11.6275	−0.452256	−0.226128	−	0.974098i	$-0.572607\pi$
$661$	−11.6275	−0.452256	−0.226128	+	0.974098i	$0.572607\pi$
$662$	15.2261	0.591780
$663$	−31.5943	−1.22702
$664$	−38.0076	−1.47498
$665$	0	0
$666$	−104.844	−4.06263
$667$	−52.3339	−2.02638
$668$	86.9497	3.36419
$669$	−77.9681	−3.01442
$670$	32.7372	1.26475
$671$	20.4016	0.787597
$672$	0	0
$673$	−39.0645	−1.50582	−0.752912	−	0.658121i	$-0.771352\pi$
$673$	−39.0645	−1.50582	−0.752912	+	0.658121i	$0.771352\pi$
$674$	−28.7530	−1.10753
$675$	13.7688	0.529962
$676$	13.7682	0.529545
$677$	8.82385	0.339128	0.169564	−	0.985519i	$-0.445764\pi$
$677$	8.82385	0.339128	0.169564	+	0.985519i	$0.445764\pi$
$678$	29.5328	1.13420
$679$	0	0
$680$	−24.6695	−0.946031
$681$	−29.9389	−1.14726
$682$	85.7249	3.28258
$683$	−12.3490	−0.472522	−0.236261	−	0.971690i	$-0.575922\pi$
$683$	−12.3490	−0.472522	−0.236261	+	0.971690i	$0.575922\pi$
$684$	−89.4243	−3.41922
$685$	−7.84254	−0.299648
$686$	0	0
$687$	39.2283	1.49665
$688$	1.08051	0.0411940
$689$	30.4460	1.15990
$690$	−75.4958	−2.87407
$691$	−29.1443	−1.10870	−0.554351	−	0.832283i	$-0.687033\pi$
$691$	−29.1443	−1.10870	−0.554351	+	0.832283i	$0.687033\pi$
$692$	−31.4113	−1.19408
$693$	0	0
$694$	63.6452	2.41594
$695$	−1.00889	−0.0382693
$696$	−118.439	−4.48943
$697$	−2.82150	−0.106872
$698$	55.4873	2.10022
$699$	−19.1254	−0.723387
$700$	0	0
$701$	−16.8243	−0.635447	−0.317723	−	0.948183i	$-0.602918\pi$
$701$	−16.8243	−0.635447	−0.317723	+	0.948183i	$0.602918\pi$
$702$	−49.5427	−1.86987
$703$	37.8367	1.42704
$704$	13.6123	0.513034
$705$	−23.2135	−0.874270
$706$	−0.948998	−0.0357160
$707$	0	0
$708$	37.9575	1.42653
$709$	23.6356	0.887655	0.443827	−	0.896112i	$-0.353620\pi$
$709$	23.6356	0.887655	0.443827	+	0.896112i	$0.353620\pi$
$710$	50.9473	1.91202
$711$	77.9932	2.92497
$712$	−67.9347	−2.54596
$713$	51.2159	1.91805
$714$	0	0
$715$	28.7398	1.07481
$716$	79.4894	2.97066
$717$	−1.30141	−0.0486022
$718$	69.8600	2.60715
$719$	−14.5753	−0.543567	−0.271784	−	0.962358i	$-0.587614\pi$
$719$	−14.5753	−0.543567	−0.271784	+	0.962358i	$0.587614\pi$
$720$	−43.6429	−1.62648
$721$	0	0
$722$	−0.753790	−0.0280532
$723$	−38.8492	−1.44482
$724$	−26.5287	−0.985933
$725$	−20.2405	−0.751715
$726$	−86.9182	−3.22584
$727$	−6.71319	−0.248978	−0.124489	−	0.992221i	$-0.539729\pi$
$727$	−6.71319	−0.248978	−0.124489	+	0.992221i	$0.539729\pi$
$728$	0	0
$729$	−43.9426	−1.62750
$730$	9.18916	0.340106
$731$	0.491049	0.0181621
$732$	51.0829	1.88808
$733$	−7.34544	−0.271310	−0.135655	−	0.990756i	$-0.543314\pi$
$733$	−7.34544	−0.271310	−0.135655	+	0.990756i	$0.543314\pi$
$734$	−38.2575	−1.41211
$735$	0	0
$736$	−27.7431	−1.02263
$737$	42.5139	1.56602
$738$	−11.9829	−0.441095
$739$	−41.6009	−1.53031	−0.765157	−	0.643844i	$-0.777339\pi$
$739$	−41.6009	−1.53031	−0.765157	+	0.643844i	$0.777339\pi$
$740$	56.2274	2.06696
$741$	48.4239	1.77890
$742$	0	0
$743$	−45.8763	−1.68304	−0.841519	−	0.540228i	$-0.818338\pi$
$743$	−45.8763	−1.68304	−0.841519	+	0.540228i	$0.818338\pi$
$744$	115.909	4.24943
$745$	−18.2211	−0.667569
$746$	24.3360	0.891005
$747$	−30.5577	−1.11805
$748$	−59.3266	−2.16919
$749$	0	0
$750$	−81.0535	−2.95966
$751$	−4.94985	−0.180623	−0.0903113	−	0.995914i	$-0.528786\pi$
$751$	−4.94985	−0.180623	−0.0903113	+	0.995914i	$0.528786\pi$
$752$	−35.0121	−1.27676
$753$	55.2898	2.01487
$754$	72.8292	2.65228
$755$	−21.6731	−0.788766
$756$	0	0
$757$	−5.55579	−0.201929	−0.100964	−	0.994890i	$-0.532193\pi$
$757$	−5.55579	−0.201929	−0.100964	+	0.994890i	$0.532193\pi$
$758$	−44.9692	−1.63335
$759$	−98.0420	−3.55870
$760$	37.8103	1.37152
$761$	−43.1027	−1.56247	−0.781237	−	0.624235i	$-0.785411\pi$
$761$	−43.1027	−1.56247	−0.781237	+	0.624235i	$0.785411\pi$
$762$	47.5759	1.72349
$763$	0	0
$764$	31.3953	1.13584
$765$	−19.8340	−0.717099
$766$	8.38985	0.303138
$767$	−12.6040	−0.455103
$768$	87.5666	3.15979
$769$	−40.9910	−1.47817	−0.739087	−	0.673610i	$-0.764743\pi$
$769$	−40.9910	−1.47817	−0.739087	+	0.673610i	$0.764743\pi$
$770$	0	0
$771$	−67.8337	−2.44297
$772$	−80.2709	−2.88901
$773$	16.9255	0.608769	0.304385	−	0.952549i	$-0.401549\pi$
$773$	16.9255	0.608769	0.304385	+	0.952549i	$0.401549\pi$
$774$	2.08548	0.0749610
$775$	19.8082	0.711530
$776$	−93.4303	−3.35395
$777$	0	0
$778$	39.5015	1.41620
$779$	4.32444	0.154939
$780$	71.9605	2.57660
$781$	66.1624	2.36748
$782$	−51.7484	−1.85052
$783$	−35.1589	−1.25648
$784$	0	0
$785$	11.0484	0.394336
$786$	−72.5805	−2.58886
$787$	3.43935	0.122600	0.0612999	−	0.998119i	$-0.480475\pi$
$787$	3.43935	0.122600	0.0612999	+	0.998119i	$0.480475\pi$
$788$	−36.3382	−1.29449
$789$	45.9180	1.63472
$790$	−61.0676	−2.17269
$791$	0	0
$792$	−136.060	−4.83468
$793$	−16.9623	−0.602350
$794$	8.38799	0.297679
$795$	31.1692	1.10546
$796$	−120.015	−4.25383
$797$	−48.0233	−1.70107	−0.850536	−	0.525917i	$-0.823722\pi$
$797$	−48.0233	−1.70107	−0.850536	+	0.525917i	$0.823722\pi$
$798$	0	0
$799$	−15.9116	−0.562913
$800$	−10.7299	−0.379358
$801$	−54.6188	−1.92986
$802$	18.0850	0.638602
$803$	11.9334	0.421122
$804$	106.449	3.75417
$805$	0	0
$806$	−71.2733	−2.51050
$807$	26.9991	0.950412
$808$	106.637	3.75148
$809$	−28.9663	−1.01840	−0.509200	−	0.860648i	$-0.670059\pi$
$809$	−28.9663	−1.01840	−0.509200	+	0.860648i	$0.670059\pi$
$810$	2.41383	0.0848134
$811$	−15.4521	−0.542595	−0.271298	−	0.962495i	$-0.587453\pi$
$811$	−15.4521	−0.542595	−0.271298	+	0.962495i	$0.587453\pi$
$812$	0	0
$813$	39.6540	1.39073
$814$	106.608	3.73659
$815$	−22.4909	−0.787823
$816$	−48.7846	−1.70780
$817$	−0.752619	−0.0263308
$818$	−22.6954	−0.793527
$819$	0	0
$820$	6.42636	0.224418
$821$	1.92930	0.0673329	0.0336664	−	0.999433i	$-0.489282\pi$
$821$	1.92930	0.0673329	0.0336664	+	0.999433i	$0.489282\pi$
$822$	−37.2312	−1.29859
$823$	16.1295	0.562239	0.281120	−	0.959673i	$-0.409294\pi$
$823$	16.1295	0.562239	0.281120	+	0.959673i	$0.409294\pi$
$824$	37.6842	1.31279
$825$	−37.9185	−1.32015
$826$	0	0
$827$	−20.3885	−0.708976	−0.354488	−	0.935060i	$-0.615345\pi$
$827$	−20.3885	−0.708976	−0.354488	+	0.935060i	$0.615345\pi$
$828$	−150.532	−5.23134
$829$	22.6093	0.785254	0.392627	−	0.919698i	$-0.371566\pi$
$829$	22.6093	0.785254	0.392627	+	0.919698i	$0.371566\pi$
$830$	23.9263	0.830492
$831$	43.1042	1.49527
$832$	−11.3175	−0.392365
$833$	0	0
$834$	−4.78954	−0.165848
$835$	−29.5579	−1.02289
$836$	90.9285	3.14483
$837$	34.4078	1.18931
$838$	8.46513	0.292423
$839$	−21.3958	−0.738665	−0.369333	−	0.929297i	$-0.620414\pi$
$839$	−21.3958	−0.738665	−0.369333	+	0.929297i	$0.620414\pi$
$840$	0	0
$841$	22.6846	0.782227
$842$	−94.9138	−3.27095
$843$	79.5509	2.73988
$844$	19.2658	0.663158
$845$	−4.68038	−0.161010
$846$	−67.5765	−2.32333
$847$	0	0
$848$	47.0115	1.61438
$849$	−59.8836	−2.05520
$850$	−20.0141	−0.686479
$851$	63.6921	2.18334
$852$	165.661	5.67546
$853$	−18.6430	−0.638325	−0.319163	−	0.947700i	$-0.603402\pi$
$853$	−18.6430	−0.638325	−0.319163	+	0.947700i	$0.603402\pi$
$854$	0	0
$855$	30.3991	1.03963
$856$	26.4660	0.904588
$857$	−13.6233	−0.465362	−0.232681	−	0.972553i	$-0.574750\pi$
$857$	−13.6233	−0.465362	−0.232681	+	0.972553i	$0.574750\pi$
$858$	136.438	4.65791
$859$	−0.968694	−0.0330514	−0.0165257	−	0.999863i	$-0.505261\pi$
$859$	−0.968694	−0.0330514	−0.0165257	+	0.999863i	$0.505261\pi$
$860$	−1.11843	−0.0381382
$861$	0	0
$862$	30.9893	1.05550
$863$	5.80688	0.197668	0.0988342	−	0.995104i	$-0.468489\pi$
$863$	5.80688	0.197668	0.0988342	+	0.995104i	$0.468489\pi$
$864$	−18.6384	−0.634090
$865$	10.6780	0.363064
$866$	−2.42165	−0.0822909
$867$	25.1737	0.854945
$868$	0	0
$869$	−79.3050	−2.69024
$870$	74.5591	2.52779
$871$	−35.3469	−1.19768
$872$	32.4948	1.10041
$873$	−75.1170	−2.54233
$874$	79.3137	2.68282
$875$	0	0
$876$	29.8797	1.00954
$877$	33.1986	1.12104	0.560518	−	0.828142i	$-0.310602\pi$
$877$	33.1986	1.12104	0.560518	+	0.828142i	$0.310602\pi$
$878$	−43.3497	−1.46298
$879$	−6.50383	−0.219369
$880$	44.3770	1.49595
$881$	43.0324	1.44980	0.724899	−	0.688855i	$-0.241886\pi$
$881$	43.0324	1.44980	0.724899	+	0.688855i	$0.241886\pi$
$882$	0	0
$883$	25.8894	0.871249	0.435624	−	0.900129i	$-0.356528\pi$
$883$	25.8894	0.871249	0.435624	+	0.900129i	$0.356528\pi$
$884$	49.3252	1.65899
$885$	−12.9034	−0.433741
$886$	−84.3352	−2.83330
$887$	41.8678	1.40578	0.702891	−	0.711297i	$-0.251892\pi$
$887$	41.8678	1.40578	0.702891	+	0.711297i	$0.251892\pi$
$888$	144.145	4.83718
$889$	0	0
$890$	42.7658	1.43351
$891$	3.13471	0.105017
$892$	121.724	4.07563
$893$	24.3874	0.816093
$894$	−86.5016	−2.89305
$895$	−27.0218	−0.903239
$896$	0	0
$897$	81.5140	2.72167
$898$	−10.3965	−0.346935
$899$	−50.5804	−1.68695
$900$	−58.2193	−1.94064
$901$	21.3649	0.711767
$902$	12.1844	0.405697
$903$	0	0
$904$	−24.8980	−0.828096
$905$	9.01824	0.299776
$906$	−102.890	−3.41828
$907$	−17.7115	−0.588099	−0.294050	−	0.955790i	$-0.595003\pi$
$907$	−17.7115	−0.588099	−0.294050	+	0.955790i	$0.595003\pi$
$908$	46.7408	1.55115
$909$	85.7352	2.84366
$910$	0	0
$911$	18.4955	0.612783	0.306392	−	0.951906i	$-0.400878\pi$
$911$	18.4955	0.612783	0.306392	+	0.951906i	$0.400878\pi$
$912$	74.7711	2.47592
$913$	31.0717	1.02832
$914$	40.8527	1.35129
$915$	−17.3652	−0.574077
$916$	−61.2434	−2.02354
$917$	0	0
$918$	−34.7656	−1.14744
$919$	−36.9355	−1.21839	−0.609196	−	0.793020i	$-0.708508\pi$
$919$	−36.9355	−1.21839	−0.609196	+	0.793020i	$0.708508\pi$
$920$	63.6477	2.09840
$921$	−46.4749	−1.53140
$922$	16.3826	0.539532
$923$	−55.0086	−1.81063
$924$	0	0
$925$	24.6334	0.809942
$926$	15.8255	0.520059
$927$	30.2977	0.995107
$928$	27.3989	0.899413
$929$	−23.0313	−0.755632	−0.377816	−	0.925881i	$-0.623325\pi$
$929$	−23.0313	−0.755632	−0.377816	+	0.925881i	$0.623325\pi$
$930$	−72.9663	−2.39266
$931$	0	0
$932$	29.8586	0.978050
$933$	−5.93074	−0.194164
$934$	81.4509	2.66515
$935$	20.1676	0.659551
$936$	113.123	3.69754
$937$	41.7706	1.36459	0.682293	−	0.731078i	$-0.260983\pi$
$937$	41.7706	1.36459	0.682293	+	0.731078i	$0.260983\pi$
$938$	0	0
$939$	87.8547	2.86703
$940$	36.2410	1.18205
$941$	−17.1269	−0.558320	−0.279160	−	0.960245i	$-0.590056\pi$
$941$	−17.1269	−0.558320	−0.279160	+	0.960245i	$0.590056\pi$
$942$	52.4507	1.70894
$943$	7.27952	0.237054
$944$	−19.4617	−0.633425
$945$	0	0
$946$	−2.12056	−0.0689452
$947$	6.15920	0.200147	0.100073	−	0.994980i	$-0.468092\pi$
$947$	6.15920	0.200147	0.100073	+	0.994980i	$0.468092\pi$
$948$	−198.569	−6.44921
$949$	−9.92169	−0.322072
$950$	30.6752	0.995234
$951$	25.1959	0.817032
$952$	0	0
$953$	−2.98600	−0.0967261	−0.0483630	−	0.998830i	$-0.515400\pi$
$953$	−2.98600	−0.0967261	−0.0483630	+	0.998830i	$0.515400\pi$
$954$	90.7364	2.93770
$955$	−10.6726	−0.345356
$956$	2.03177	0.0657123
$957$	96.8256	3.12993
$958$	15.6449	0.505463
$959$	0	0
$960$	−11.5864	−0.373949
$961$	18.4999	0.596771
$962$	−88.6356	−2.85772
$963$	21.2784	0.685686
$964$	60.6516	1.95346
$965$	27.2875	0.878414
$966$	0	0
$967$	2.66622	0.0857399	0.0428699	−	0.999081i	$-0.486350\pi$
$967$	2.66622	0.0857399	0.0428699	+	0.999081i	$0.486350\pi$
$968$	73.2775	2.35523
$969$	33.9805	1.09161
$970$	58.8156	1.88846
$971$	−6.45629	−0.207192	−0.103596	−	0.994619i	$-0.533035\pi$
$971$	−6.45629	−0.207192	−0.103596	+	0.994619i	$0.533035\pi$
$972$	71.6393	2.29783
$973$	0	0
$974$	−11.6045	−0.371833
$975$	31.5262	1.00965
$976$	−26.1914	−0.838367
$977$	−21.0479	−0.673382	−0.336691	−	0.941615i	$-0.609308\pi$
$977$	−21.0479	−0.673382	−0.336691	+	0.941615i	$0.609308\pi$
$978$	−106.772	−3.41420
$979$	55.5375	1.77499
$980$	0	0
$981$	26.1255	0.834123
$982$	−68.3907	−2.18243
$983$	−33.1038	−1.05585	−0.527924	−	0.849292i	$-0.677029\pi$
$983$	−33.1038	−1.05585	−0.527924	+	0.849292i	$0.677029\pi$
$984$	16.4746	0.525192
$985$	12.3529	0.393596
$986$	51.1064	1.62756
$987$	0	0
$988$	−75.5996	−2.40515
$989$	−1.26691	−0.0402855
$990$	85.6516	2.72219
$991$	33.0725	1.05058	0.525292	−	0.850922i	$-0.323956\pi$
$991$	33.0725	1.05058	0.525292	+	0.850922i	$0.323956\pi$
$992$	−26.8136	−0.851332
$993$	−16.8304	−0.534097
$994$	0	0
$995$	40.7983	1.29339
$996$	77.7991	2.46516
$997$	−11.9187	−0.377468	−0.188734	−	0.982028i	$-0.560438\pi$
$997$	−11.9187	−0.377468	−0.188734	+	0.982028i	$0.560438\pi$
$998$	−78.4730	−2.48402
$999$	42.7896	1.35380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.t.1.19	✓	20
7.6	odd	2		2009.2.a.u.1.19	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.19	✓	20	1.1	even	1	trivial
2009.2.a.u.1.19	yes	20	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:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+2.51950 q^{2} -2.78497 q^{3} +4.34790 q^{4} -1.47804 q^{5} -7.01674 q^{6} +5.91555 q^{8} +4.75604 q^{9} +O(q^{10})\)	\(q+2.51950 q^{2} -2.78497 q^{3} +4.34790 q^{4} -1.47804 q^{5} -7.01674 q^{6} +5.91555 q^{8} +4.75604 q^{9} -3.72392 q^{10} -4.83604 q^{11} -12.1088 q^{12} +4.02077 q^{13} +4.11628 q^{15} +6.20845 q^{16} +2.82150 q^{17} +11.9829 q^{18} -4.32444 q^{19} -6.42636 q^{20} -12.1844 q^{22} -7.27952 q^{23} -16.4746 q^{24} -2.81541 q^{25} +10.1304 q^{26} -4.89052 q^{27} +7.18920 q^{29} +10.3710 q^{30} -7.03562 q^{31} +3.81112 q^{32} +13.4682 q^{33} +7.10877 q^{34} +20.6788 q^{36} -8.74950 q^{37} -10.8955 q^{38} -11.1977 q^{39} -8.74340 q^{40} -1.00000 q^{41} +0.174038 q^{43} -21.0266 q^{44} -7.02960 q^{45} -18.3408 q^{46} -5.63943 q^{47} -17.2903 q^{48} -7.09344 q^{50} -7.85778 q^{51} +17.4819 q^{52} +7.57218 q^{53} -12.3217 q^{54} +7.14784 q^{55} +12.0434 q^{57} +18.1132 q^{58} -3.13471 q^{59} +17.8972 q^{60} -4.21867 q^{61} -17.7263 q^{62} -2.81477 q^{64} -5.94285 q^{65} +33.9332 q^{66} -8.79107 q^{67} +12.2676 q^{68} +20.2732 q^{69} -13.6811 q^{71} +28.1346 q^{72} -2.46761 q^{73} -22.0444 q^{74} +7.84083 q^{75} -18.8023 q^{76} -28.2127 q^{78} +16.3988 q^{79} -9.17631 q^{80} -0.648197 q^{81} -2.51950 q^{82} -6.42502 q^{83} -4.17027 q^{85} +0.438490 q^{86} -20.0217 q^{87} -28.6078 q^{88} -11.4841 q^{89} -17.7111 q^{90} -31.6506 q^{92} +19.5940 q^{93} -14.2086 q^{94} +6.39168 q^{95} -10.6138 q^{96} -15.7940 q^{97} -23.0004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9	\( 20 q - 2 q^{2} - 8 q^{3} + 18 q^{4} - 8 q^{5} - 12 q^{6} - 6 q^{8} + 16 q^{9} - 16 q^{10} - 6 q^{12} - 12 q^{13} + 14 q^{16} - 8 q^{17} - 18 q^{18} - 36 q^{19} - 24 q^{20} - 8 q^{22} - 12 q^{23} - 36 q^{24} + 20 q^{25} + 22 q^{26} - 32 q^{27} + 4 q^{29} + 28 q^{30} - 80 q^{31} + 6 q^{32} + 12 q^{33} - 48 q^{34} + 26 q^{36} + 4 q^{37} - 12 q^{38} - 28 q^{39} + 4 q^{40} - 20 q^{41} - 20 q^{44} - 40 q^{45} + 8 q^{46} - 32 q^{47} - 16 q^{48} + 6 q^{50} - 20 q^{51} - 36 q^{52} + 4 q^{53} + 50 q^{54} - 64 q^{55} - 4 q^{57} - 32 q^{59} + 20 q^{60} - 44 q^{61} + 8 q^{62} - 30 q^{64} - 8 q^{65} - 32 q^{66} - 4 q^{67} + 48 q^{68} + 24 q^{69} + 8 q^{71} - 8 q^{72} - 48 q^{73} - 38 q^{74} - 24 q^{75} - 84 q^{76} + 30 q^{78} - 4 q^{79} - 56 q^{80} + 2 q^{82} - 8 q^{83} - 12 q^{85} - 24 q^{86} - 40 q^{87} - 48 q^{88} - 20 q^{89} - 48 q^{90} - 50 q^{92} + 48 q^{93} - 26 q^{94} + 20 q^{95} - 70 q^{96} - 8 q^{97} - 8 q^{99}+O(q^{100}) \) 20 * q - 2 * q^2 - 8 * q^3 + 18 * q^4 - 8 * q^5 - 12 * q^6 - 6 * q^8 + 16 * q^9 - 16 * q^10 - 6 * q^12 - 12 * q^13 + 14 * q^16 - 8 * q^17 - 18 * q^18 - 36 * q^19 - 24 * q^20 - 8 * q^22 - 12 * q^23 - 36 * q^24 + 20 * q^25 + 22 * q^26 - 32 * q^27 + 4 * q^29 + 28 * q^30 - 80 * q^31 + 6 * q^32 + 12 * q^33 - 48 * q^34 + 26 * q^36 + 4 * q^37 - 12 * q^38 - 28 * q^39 + 4 * q^40 - 20 * q^41 - 20 * q^44 - 40 * q^45 + 8 * q^46 - 32 * q^47 - 16 * q^48 + 6 * q^50 - 20 * q^51 - 36 * q^52 + 4 * q^53 + 50 * q^54 - 64 * q^55 - 4 * q^57 - 32 * q^59 + 20 * q^60 - 44 * q^61 + 8 * q^62 - 30 * q^64 - 8 * q^65 - 32 * q^66 - 4 * q^67 + 48 * q^68 + 24 * q^69 + 8 * q^71 - 8 * q^72 - 48 * q^73 - 38 * q^74 - 24 * q^75 - 84 * q^76 + 30 * q^78 - 4 * q^79 - 56 * q^80 + 2 * q^82 - 8 * q^83 - 12 * q^85 - 24 * q^86 - 40 * q^87 - 48 * q^88 - 20 * q^89 - 48 * q^90 - 50 * q^92 + 48 * q^93 - 26 * q^94 + 20 * q^95 - 70 * q^96 - 8 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

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Database

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