LMFDB - Embedded newform 2009.2.a.u.1.16

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:	$0$
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			1.16
Root			$-1.50500$ 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)

$f(q)$	$=$	$q+1.50500 q^{2} +0.350711 q^{3} +0.265025 q^{4} -3.19754 q^{5} +0.527821 q^{6} -2.61114 q^{8} -2.87700 q^{9} +O(q^{10})$	\(q+1.50500 q^{2} +0.350711 q^{3} +0.265025 q^{4} -3.19754 q^{5} +0.527821 q^{6} -2.61114 q^{8} -2.87700 q^{9} -4.81229 q^{10} +0.734523 q^{11} +0.0929472 q^{12} +6.80628 q^{13} -1.12141 q^{15} -4.45981 q^{16} +1.13300 q^{17} -4.32989 q^{18} +7.92756 q^{19} -0.847426 q^{20} +1.10546 q^{22} -4.32690 q^{23} -0.915756 q^{24} +5.22424 q^{25} +10.2435 q^{26} -2.06113 q^{27} +3.06561 q^{29} -1.68773 q^{30} +2.67042 q^{31} -1.48974 q^{32} +0.257606 q^{33} +1.70517 q^{34} -0.762477 q^{36} +1.08427 q^{37} +11.9310 q^{38} +2.38704 q^{39} +8.34921 q^{40} +1.00000 q^{41} +2.52496 q^{43} +0.194667 q^{44} +9.19932 q^{45} -6.51199 q^{46} -1.75609 q^{47} -1.56411 q^{48} +7.86248 q^{50} +0.397357 q^{51} +1.80383 q^{52} +3.77439 q^{53} -3.10200 q^{54} -2.34866 q^{55} +2.78029 q^{57} +4.61375 q^{58} +8.08503 q^{59} -0.297202 q^{60} +3.64075 q^{61} +4.01898 q^{62} +6.67756 q^{64} -21.7633 q^{65} +0.387696 q^{66} +13.1706 q^{67} +0.300274 q^{68} -1.51749 q^{69} +7.16682 q^{71} +7.51225 q^{72} -6.31108 q^{73} +1.63183 q^{74} +1.83220 q^{75} +2.10100 q^{76} +3.59250 q^{78} +4.34192 q^{79} +14.2604 q^{80} +7.90814 q^{81} +1.50500 q^{82} +2.74372 q^{83} -3.62282 q^{85} +3.80007 q^{86} +1.07515 q^{87} -1.91794 q^{88} -16.7897 q^{89} +13.8450 q^{90} -1.14674 q^{92} +0.936547 q^{93} -2.64292 q^{94} -25.3487 q^{95} -0.522469 q^{96} +14.9140 q^{97} -2.11322 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.50500	1.06420	0.532098	−	0.846683i	$-0.321404\pi$
$2$	1.50500	1.06420	0.532098	+	0.846683i	$0.321404\pi$
$3$	0.350711	0.202483	0.101242	−	0.994862i	$-0.467718\pi$
$3$	0.350711	0.202483	0.101242	+	0.994862i	$0.467718\pi$
$4$	0.265025	0.132512
$5$	−3.19754	−1.42998	−0.714991	−	0.699134i	$-0.753569\pi$
$5$	−3.19754	−1.42998	−0.714991	+	0.699134i	$0.753569\pi$
$6$	0.527821	0.215482
$7$	0	0
$7$	0	0
$8$	−2.61114	−0.923177
$9$	−2.87700	−0.959000
$10$	−4.81229	−1.52178
$11$	0.734523	0.221467	0.110733	−	0.993850i	$-0.464680\pi$
$11$	0.734523	0.221467	0.110733	+	0.993850i	$0.464680\pi$
$12$	0.0929472	0.0268316
$13$	6.80628	1.88772	0.943861	−	0.330342i	$-0.107164\pi$
$13$	6.80628	1.88772	0.943861	+	0.330342i	$0.107164\pi$
$14$	0	0
$15$	−1.12141	−0.289547
$16$	−4.45981	−1.11495
$17$	1.13300	0.274793	0.137397	−	0.990516i	$-0.456126\pi$
$17$	1.13300	0.274793	0.137397	+	0.990516i	$0.456126\pi$
$18$	−4.32989	−1.02056
$19$	7.92756	1.81871	0.909354	−	0.416024i	$-0.136577\pi$
$19$	7.92756	1.81871	0.909354	+	0.416024i	$0.136577\pi$
$20$	−0.847426	−0.189490
$21$	0	0
$22$	1.10546	0.235684
$23$	−4.32690	−0.902221	−0.451111	−	0.892468i	$-0.648972\pi$
$23$	−4.32690	−0.902221	−0.451111	+	0.892468i	$0.648972\pi$
$24$	−0.915756	−0.186928
$25$	5.22424	1.04485
$26$	10.2435	2.00891
$27$	−2.06113	−0.396665
$28$	0	0
$29$	3.06561	0.569270	0.284635	−	0.958636i	$-0.408128\pi$
$29$	3.06561	0.569270	0.284635	+	0.958636i	$0.408128\pi$
$30$	−1.68773	−0.308135
$31$	2.67042	0.479622	0.239811	−	0.970820i	$-0.422915\pi$
$31$	2.67042	0.479622	0.239811	+	0.970820i	$0.422915\pi$
$32$	−1.48974	−0.263351
$33$	0.257606	0.0448434
$34$	1.70517	0.292434
$35$	0	0
$36$	−0.762477	−0.127079
$37$	1.08427	0.178253	0.0891265	−	0.996020i	$-0.471592\pi$
$37$	1.08427	0.178253	0.0891265	+	0.996020i	$0.471592\pi$
$38$	11.9310	1.93546
$39$	2.38704	0.382232
$40$	8.34921	1.32013
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	2.52496	0.385053	0.192526	−	0.981292i	$-0.438332\pi$
$43$	2.52496	0.385053	0.192526	+	0.981292i	$0.438332\pi$
$44$	0.194667	0.0293471
$45$	9.19932	1.37135
$46$	−6.51199	−0.960140
$47$	−1.75609	−0.256153	−0.128076	−	0.991764i	$-0.540880\pi$
$47$	−1.75609	−0.256153	−0.128076	+	0.991764i	$0.540880\pi$
$48$	−1.56411	−0.225759
$49$	0	0
$50$	7.86248	1.11192
$51$	0.397357	0.0556411
$52$	1.80383	0.250147
$53$	3.77439	0.518452	0.259226	−	0.965817i	$-0.416533\pi$
$53$	3.77439	0.518452	0.259226	+	0.965817i	$0.416533\pi$
$54$	−3.10200	−0.422129
$55$	−2.34866	−0.316694
$56$	0	0
$57$	2.78029	0.368258
$58$	4.61375	0.605815
$59$	8.08503	1.05258	0.526291	−	0.850305i	$-0.323582\pi$
$59$	8.08503	1.05258	0.526291	+	0.850305i	$0.323582\pi$
$60$	−0.297202	−0.0383686
$61$	3.64075	0.466150	0.233075	−	0.972459i	$-0.425121\pi$
$61$	3.64075	0.466150	0.233075	+	0.972459i	$0.425121\pi$
$62$	4.01898	0.510411
$63$	0	0
$64$	6.67756	0.834695
$65$	−21.7633	−2.69941
$66$	0.387696	0.0477221
$67$	13.1706	1.60904	0.804522	−	0.593923i	$-0.202422\pi$
$67$	13.1706	1.60904	0.804522	+	0.593923i	$0.202422\pi$
$68$	0.300274	0.0364135
$69$	−1.51749	−0.182685
$70$	0	0
$71$	7.16682	0.850544	0.425272	−	0.905065i	$-0.360178\pi$
$71$	7.16682	0.850544	0.425272	+	0.905065i	$0.360178\pi$
$72$	7.51225	0.885327
$73$	−6.31108	−0.738657	−0.369328	−	0.929299i	$-0.620412\pi$
$73$	−6.31108	−0.738657	−0.369328	+	0.929299i	$0.620412\pi$
$74$	1.63183	0.189696
$75$	1.83220	0.211564
$76$	2.10100	0.241001
$77$	0	0
$78$	3.59250	0.406770
$79$	4.34192	0.488504	0.244252	−	0.969712i	$-0.421458\pi$
$79$	4.34192	0.488504	0.244252	+	0.969712i	$0.421458\pi$
$80$	14.2604	1.59436
$81$	7.90814	0.878682
$82$	1.50500	0.166199
$83$	2.74372	0.301163	0.150581	−	0.988598i	$-0.451885\pi$
$83$	2.74372	0.301163	0.150581	+	0.988598i	$0.451885\pi$
$84$	0	0
$85$	−3.62282	−0.392950
$86$	3.80007	0.409772
$87$	1.07515	0.115268
$88$	−1.91794	−0.204453
$89$	−16.7897	−1.77971	−0.889855	−	0.456244i	$-0.849194\pi$
$89$	−16.7897	−1.77971	−0.889855	+	0.456244i	$0.849194\pi$
$90$	13.8450	1.45939
$91$	0	0
$92$	−1.14674	−0.119556
$93$	0.936547	0.0971154
$94$	−2.64292	−0.272596
$95$	−25.3487	−2.60072
$96$	−0.522469	−0.0533243
$97$	14.9140	1.51429	0.757143	−	0.653249i	$-0.226595\pi$
$97$	14.9140	1.51429	0.757143	+	0.653249i	$0.226595\pi$
$98$	0	0
$99$	−2.11322	−0.212387
$100$	1.38455	0.138455
$101$	5.47492	0.544774	0.272387	−	0.962188i	$-0.412187\pi$
$101$	5.47492	0.544774	0.272387	+	0.962188i	$0.412187\pi$
$102$	0.598022	0.0592130
$103$	3.70848	0.365407	0.182704	−	0.983168i	$-0.441515\pi$
$103$	3.70848	0.365407	0.182704	+	0.983168i	$0.441515\pi$
$104$	−17.7721	−1.74270
$105$	0	0
$106$	5.68045	0.551734
$107$	−13.0408	−1.26070	−0.630352	−	0.776310i	$-0.717089\pi$
$107$	−13.0408	−1.26070	−0.630352	+	0.776310i	$0.717089\pi$
$108$	−0.546251	−0.0525630
$109$	12.9677	1.24208	0.621040	−	0.783779i	$-0.286710\pi$
$109$	12.9677	1.24208	0.621040	+	0.783779i	$0.286710\pi$
$110$	−3.53474	−0.337024
$111$	0.380266	0.0360933
$112$	0	0
$113$	−10.5277	−0.990367	−0.495184	−	0.868788i	$-0.664899\pi$
$113$	−10.5277	−0.990367	−0.495184	+	0.868788i	$0.664899\pi$
$114$	4.18433	0.391899
$115$	13.8354	1.29016
$116$	0.812463	0.0754353
$117$	−19.5817	−1.81033
$118$	12.1680	1.12015
$119$	0	0
$120$	2.92816	0.267303
$121$	−10.4605	−0.950952
$122$	5.47932	0.496075
$123$	0.350711	0.0316226
$124$	0.707728	0.0635558
$125$	−0.717006	−0.0641310
$126$	0	0
$127$	−16.9183	−1.50126	−0.750629	−	0.660724i	$-0.770249\pi$
$127$	−16.9183	−1.50126	−0.750629	+	0.660724i	$0.770249\pi$
$128$	13.0292	1.15163
$129$	0.885533	0.0779668
$130$	−32.7538	−2.87270
$131$	20.3424	1.77732	0.888661	−	0.458564i	$-0.151636\pi$
$131$	20.3424	1.77732	0.888661	+	0.458564i	$0.151636\pi$
$132$	0.0682719	0.00594230
$133$	0	0
$134$	19.8217	1.71234
$135$	6.59054	0.567224
$136$	−2.95843	−0.253683
$137$	−9.18145	−0.784425	−0.392212	−	0.919875i	$-0.628290\pi$
$137$	−9.18145	−0.784425	−0.392212	+	0.919875i	$0.628290\pi$
$138$	−2.28383	−0.194412
$139$	−15.5244	−1.31676	−0.658380	−	0.752685i	$-0.728758\pi$
$139$	−15.5244	−1.31676	−0.658380	+	0.752685i	$0.728758\pi$
$140$	0	0
$141$	−0.615882	−0.0518666
$142$	10.7861	0.905146
$143$	4.99937	0.418068
$144$	12.8309	1.06924
$145$	−9.80240	−0.814045
$146$	−9.49818	−0.786075
$147$	0	0
$148$	0.287359	0.0236207
$149$	−8.30757	−0.680582	−0.340291	−	0.940320i	$-0.610526\pi$
$149$	−8.30757	−0.680582	−0.340291	+	0.940320i	$0.610526\pi$
$150$	2.75746	0.225146
$151$	−8.34331	−0.678969	−0.339485	−	0.940612i	$-0.610253\pi$
$151$	−8.34331	−0.678969	−0.339485	+	0.940612i	$0.610253\pi$
$152$	−20.7000	−1.67899
$153$	−3.25965	−0.263527
$154$	0	0
$155$	−8.53877	−0.685850
$156$	0.632625	0.0506505
$157$	−7.44108	−0.593863	−0.296932	−	0.954899i	$-0.595963\pi$
$157$	−7.44108	−0.593863	−0.296932	+	0.954899i	$0.595963\pi$
$158$	6.53459	0.519864
$159$	1.32372	0.104978
$160$	4.76350	0.376588
$161$	0	0
$162$	11.9018	0.935090
$163$	−9.40425	−0.736598	−0.368299	−	0.929707i	$-0.620060\pi$
$163$	−9.40425	−0.736598	−0.368299	+	0.929707i	$0.620060\pi$
$164$	0.265025	0.0206950
$165$	−0.823703	−0.0641252
$166$	4.12930	0.320496
$167$	−17.5327	−1.35672	−0.678361	−	0.734729i	$-0.737310\pi$
$167$	−17.5327	−1.35672	−0.678361	+	0.734729i	$0.737310\pi$
$168$	0	0
$169$	33.3254	2.56350
$170$	−5.45234	−0.418175
$171$	−22.8076	−1.74414
$172$	0.669177	0.0510243
$173$	16.9827	1.29117	0.645584	−	0.763689i	$-0.276614\pi$
$173$	16.9827	1.29117	0.645584	+	0.763689i	$0.276614\pi$
$174$	1.61809	0.122667
$175$	0	0
$176$	−3.27583	−0.246925
$177$	2.83551	0.213130
$178$	−25.2686	−1.89396
$179$	−8.57603	−0.641003	−0.320501	−	0.947248i	$-0.603851\pi$
$179$	−8.57603	−0.641003	−0.320501	+	0.947248i	$0.603851\pi$
$180$	2.43805	0.181721
$181$	16.6966	1.24105	0.620523	−	0.784188i	$-0.286920\pi$
$181$	16.6966	1.24105	0.620523	+	0.784188i	$0.286920\pi$
$182$	0	0
$183$	1.27685	0.0943876
$184$	11.2981	0.832910
$185$	−3.46700	−0.254899
$186$	1.40950	0.103350
$187$	0.832216	0.0608577
$188$	−0.465408	−0.0339434
$189$	0	0
$190$	−38.1497	−2.76767
$191$	−20.8303	−1.50723	−0.753614	−	0.657318i	$-0.771691\pi$
$191$	−20.8303	−1.50723	−0.753614	+	0.657318i	$0.771691\pi$
$192$	2.34190	0.169012
$193$	−8.05805	−0.580031	−0.290016	−	0.957022i	$-0.593660\pi$
$193$	−8.05805	−0.580031	−0.290016	+	0.957022i	$0.593660\pi$
$194$	22.4455	1.61150
$195$	−7.63265	−0.546585
$196$	0	0
$197$	25.1312	1.79052	0.895261	−	0.445543i	$-0.146989\pi$
$197$	25.1312	1.79052	0.895261	+	0.445543i	$0.146989\pi$
$198$	−3.18040	−0.226021
$199$	4.26567	0.302385	0.151193	−	0.988504i	$-0.451689\pi$
$199$	4.26567	0.302385	0.151193	+	0.988504i	$0.451689\pi$
$200$	−13.6412	−0.964579
$201$	4.61908	0.325805
$202$	8.23975	0.579747
$203$	0	0
$204$	0.105309	0.00737314
$205$	−3.19754	−0.223326
$206$	5.58126	0.388865
$207$	12.4485	0.865231
$208$	−30.3547	−2.10472
$209$	5.82298	0.402784
$210$	0	0
$211$	20.1938	1.39020	0.695098	−	0.718915i	$-0.255361\pi$
$211$	20.1938	1.39020	0.695098	+	0.718915i	$0.255361\pi$
$212$	1.00031	0.0687013
$213$	2.51348	0.172221
$214$	−19.6264	−1.34164
$215$	−8.07365	−0.550619
$216$	5.38190	0.366192
$217$	0	0
$218$	19.5164	1.32182
$219$	−2.21337	−0.149566
$220$	−0.622454	−0.0419659
$221$	7.71153	0.518734
$222$	0.572301	0.0384103
$223$	2.18268	0.146163	0.0730815	−	0.997326i	$-0.476717\pi$
$223$	2.18268	0.146163	0.0730815	+	0.997326i	$0.476717\pi$
$224$	0	0
$225$	−15.0301	−1.00201
$226$	−15.8443	−1.05394
$227$	−19.9812	−1.32620	−0.663098	−	0.748532i	$-0.730759\pi$
$227$	−19.9812	−1.32620	−0.663098	+	0.748532i	$0.730759\pi$
$228$	0.736845	0.0487987
$229$	−1.76710	−0.116773	−0.0583867	−	0.998294i	$-0.518596\pi$
$229$	−1.76710	−0.116773	−0.0583867	+	0.998294i	$0.518596\pi$
$230$	20.8223	1.37298
$231$	0	0
$232$	−8.00473	−0.525537
$233$	22.9771	1.50528	0.752640	−	0.658432i	$-0.228780\pi$
$233$	22.9771	1.50528	0.752640	+	0.658432i	$0.228780\pi$
$234$	−29.4704	−1.92654
$235$	5.61517	0.366293
$236$	2.14273	0.139480
$237$	1.52276	0.0989140
$238$	0	0
$239$	0.233872	0.0151279	0.00756396	−	0.999971i	$-0.497592\pi$
$239$	0.233872	0.0151279	0.00756396	+	0.999971i	$0.497592\pi$
$240$	5.00129	0.322832
$241$	−2.52314	−0.162530	−0.0812648	−	0.996693i	$-0.525896\pi$
$241$	−2.52314	−0.162530	−0.0812648	+	0.996693i	$0.525896\pi$
$242$	−15.7430	−1.01200
$243$	8.95687	0.574584
$244$	0.964888	0.0617706
$245$	0	0
$246$	0.527821	0.0336526
$247$	53.9572	3.43321
$248$	−6.97284	−0.442775
$249$	0.962255	0.0609805
$250$	−1.07909	−0.0682479
$251$	21.5873	1.36258	0.681289	−	0.732015i	$-0.261420\pi$
$251$	21.5873	1.36258	0.681289	+	0.732015i	$0.261420\pi$
$252$	0	0
$253$	−3.17821	−0.199812
$254$	−25.4621	−1.59763
$255$	−1.27056	−0.0795658
$256$	6.25384	0.390865
$257$	12.3225	0.768654	0.384327	−	0.923197i	$-0.374433\pi$
$257$	12.3225	0.768654	0.384327	+	0.923197i	$0.374433\pi$
$258$	1.33273	0.0829719
$259$	0	0
$260$	−5.76782	−0.357705
$261$	−8.81977	−0.545930
$262$	30.6153	1.89142
$263$	−21.8177	−1.34533	−0.672667	−	0.739945i	$-0.734851\pi$
$263$	−21.8177	−1.34533	−0.672667	+	0.739945i	$0.734851\pi$
$264$	−0.672644	−0.0413984
$265$	−12.0687	−0.741377
$266$	0	0
$267$	−5.88836	−0.360362
$268$	3.49053	0.213218
$269$	29.3302	1.78829	0.894147	−	0.447774i	$-0.147783\pi$
$269$	29.3302	1.78829	0.894147	+	0.447774i	$0.147783\pi$
$270$	9.91877	0.603637
$271$	9.11838	0.553902	0.276951	−	0.960884i	$-0.410676\pi$
$271$	9.11838	0.553902	0.276951	+	0.960884i	$0.410676\pi$
$272$	−5.05298	−0.306382
$273$	0	0
$274$	−13.8181	−0.834781
$275$	3.83732	0.231399
$276$	−0.402174	−0.0242080
$277$	15.8915	0.954828	0.477414	−	0.878678i	$-0.341574\pi$
$277$	15.8915	0.954828	0.477414	+	0.878678i	$0.341574\pi$
$278$	−23.3642	−1.40129
$279$	−7.68280	−0.459957
$280$	0	0
$281$	6.78250	0.404610	0.202305	−	0.979323i	$-0.435157\pi$
$281$	6.78250	0.404610	0.202305	+	0.979323i	$0.435157\pi$
$282$	−0.926902	−0.0551962
$283$	−14.6704	−0.872064	−0.436032	−	0.899931i	$-0.643617\pi$
$283$	−14.6704	−0.872064	−0.436032	+	0.899931i	$0.643617\pi$
$284$	1.89938	0.112708
$285$	−8.89007	−0.526602
$286$	7.52405	0.444906
$287$	0	0
$288$	4.28599	0.252554
$289$	−15.7163	−0.924489
$290$	−14.7526	−0.866304
$291$	5.23050	0.306618
$292$	−1.67259	−0.0978812
$293$	−18.0683	−1.05556	−0.527781	−	0.849381i	$-0.676976\pi$
$293$	−18.0683	−1.05556	−0.527781	+	0.849381i	$0.676976\pi$
$294$	0	0
$295$	−25.8522	−1.50517
$296$	−2.83118	−0.164559
$297$	−1.51395	−0.0878482
$298$	−12.5029	−0.724273
$299$	−29.4501	−1.70314
$300$	0.485578	0.0280349
$301$	0	0
$302$	−12.5567	−0.722556
$303$	1.92012	0.110308
$304$	−35.3554	−2.02777
$305$	−11.6414	−0.666586
$306$	−4.90577	−0.280444
$307$	11.8602	0.676898	0.338449	−	0.940985i	$-0.390098\pi$
$307$	11.8602	0.676898	0.338449	+	0.940985i	$0.390098\pi$
$308$	0	0
$309$	1.30061	0.0739889
$310$	−12.8508	−0.729879
$311$	4.49238	0.254739	0.127370	−	0.991855i	$-0.459347\pi$
$311$	4.49238	0.254739	0.127370	+	0.991855i	$0.459347\pi$
$312$	−6.23289	−0.352868
$313$	25.5482	1.44407	0.722034	−	0.691858i	$-0.243208\pi$
$313$	25.5482	1.44407	0.722034	+	0.691858i	$0.243208\pi$
$314$	−11.1988	−0.631987
$315$	0	0
$316$	1.15072	0.0647329
$317$	−14.2798	−0.802031	−0.401015	−	0.916071i	$-0.631343\pi$
$317$	−14.2798	−0.802031	−0.401015	+	0.916071i	$0.631343\pi$
$318$	1.99220	0.111717
$319$	2.25176	0.126074
$320$	−21.3517	−1.19360
$321$	−4.57356	−0.255271
$322$	0	0
$323$	8.98195	0.499769
$324$	2.09585	0.116436
$325$	35.5576	1.97238
$326$	−14.1534	−0.783884
$327$	4.54792	0.251501
$328$	−2.61114	−0.144176
$329$	0	0
$330$	−1.23967	−0.0682418
$331$	20.4209	1.12243	0.561217	−	0.827669i	$-0.310333\pi$
$331$	20.4209	1.12243	0.561217	+	0.827669i	$0.310333\pi$
$332$	0.727155	0.0399078
$333$	−3.11945	−0.170945
$334$	−26.3867	−1.44382
$335$	−42.1134	−2.30090
$336$	0	0
$337$	−0.857111	−0.0466898	−0.0233449	−	0.999727i	$-0.507432\pi$
$337$	−0.857111	−0.0466898	−0.0233449	+	0.999727i	$0.507432\pi$
$338$	50.1548	2.72806
$339$	−3.69220	−0.200533
$340$	−0.960136	−0.0520707
$341$	1.96149	0.106220
$342$	−34.3254	−1.85611
$343$	0	0
$344$	−6.59302	−0.355472
$345$	4.85224	0.261236
$346$	25.5589	1.37406
$347$	18.0235	0.967551	0.483776	−	0.875192i	$-0.339265\pi$
$347$	18.0235	0.967551	0.483776	+	0.875192i	$0.339265\pi$
$348$	0.284940	0.0152744
$349$	−28.0324	−1.50054	−0.750270	−	0.661131i	$-0.770077\pi$
$349$	−28.0324	−1.50054	−0.750270	+	0.661131i	$0.770077\pi$
$350$	0	0
$351$	−14.0286	−0.748793
$352$	−1.09425	−0.0583237
$353$	−24.0153	−1.27820	−0.639102	−	0.769122i	$-0.720694\pi$
$353$	−24.0153	−1.27820	−0.639102	+	0.769122i	$0.720694\pi$
$354$	4.26745	0.226812
$355$	−22.9162	−1.21626
$356$	−4.44970	−0.235834
$357$	0	0
$358$	−12.9069	−0.682152
$359$	2.06332	0.108898	0.0544488	−	0.998517i	$-0.482660\pi$
$359$	2.06332	0.108898	0.0544488	+	0.998517i	$0.482660\pi$
$360$	−24.0207	−1.26600
$361$	43.8462	2.30770
$362$	25.1283	1.32072
$363$	−3.66861	−0.192552
$364$	0	0
$365$	20.1799	1.05627
$366$	1.92166	0.100447
$367$	25.5862	1.33559	0.667795	−	0.744345i	$-0.267238\pi$
$367$	25.5862	1.33559	0.667795	+	0.744345i	$0.267238\pi$
$368$	19.2972	1.00593
$369$	−2.87700	−0.149771
$370$	−5.21783	−0.271262
$371$	0	0
$372$	0.248208	0.0128690
$373$	18.3297	0.949076	0.474538	−	0.880235i	$-0.342615\pi$
$373$	18.3297	0.949076	0.474538	+	0.880235i	$0.342615\pi$
$374$	1.25249	0.0647645
$375$	−0.251462	−0.0129855
$376$	4.58540	0.236474
$377$	20.8654	1.07462
$378$	0	0
$379$	−31.8323	−1.63511	−0.817557	−	0.575848i	$-0.804672\pi$
$379$	−31.8323	−1.63511	−0.817557	+	0.575848i	$0.804672\pi$
$380$	−6.71803	−0.344627
$381$	−5.93345	−0.303980
$382$	−31.3496	−1.60398
$383$	24.0375	1.22826	0.614130	−	0.789205i	$-0.289507\pi$
$383$	24.0375	1.22826	0.614130	+	0.789205i	$0.289507\pi$
$384$	4.56949	0.233186
$385$	0	0
$386$	−12.1274	−0.617267
$387$	−7.26432	−0.369266
$388$	3.95258	0.200662
$389$	24.5682	1.24566	0.622828	−	0.782359i	$-0.285984\pi$
$389$	24.5682	1.24566	0.622828	+	0.782359i	$0.285984\pi$
$390$	−11.4871	−0.581674
$391$	−4.90239	−0.247925
$392$	0	0
$393$	7.13431	0.359878
$394$	37.8224	1.90547
$395$	−13.8835	−0.698552
$396$	−0.560057	−0.0281439
$397$	−3.14258	−0.157722	−0.0788608	−	0.996886i	$-0.525128\pi$
$397$	−3.14258	−0.157722	−0.0788608	+	0.996886i	$0.525128\pi$
$398$	6.41984	0.321797
$399$	0	0
$400$	−23.2991	−1.16496
$401$	12.5922	0.628826	0.314413	−	0.949286i	$-0.398192\pi$
$401$	12.5922	0.628826	0.314413	+	0.949286i	$0.398192\pi$
$402$	6.95171	0.346720
$403$	18.1756	0.905393
$404$	1.45099	0.0721894
$405$	−25.2866	−1.25650
$406$	0	0
$407$	0.796422	0.0394772
$408$	−1.03755	−0.0513666
$409$	−20.6417	−1.02066	−0.510332	−	0.859977i	$-0.670478\pi$
$409$	−20.6417	−1.02066	−0.510332	+	0.859977i	$0.670478\pi$
$410$	−4.81229	−0.237662
$411$	−3.22004	−0.158833
$412$	0.982839	0.0484210
$413$	0	0
$414$	18.7350	0.920775
$415$	−8.77316	−0.430657
$416$	−10.1396	−0.497134
$417$	−5.44458	−0.266622
$418$	8.76358	0.428641
$419$	27.5909	1.34790	0.673951	−	0.738776i	$-0.264596\pi$
$419$	27.5909	1.34790	0.673951	+	0.738776i	$0.264596\pi$
$420$	0	0
$421$	−26.3231	−1.28291	−0.641454	−	0.767161i	$-0.721669\pi$
$421$	−26.3231	−1.28291	−0.641454	+	0.767161i	$0.721669\pi$
$422$	30.3916	1.47944
$423$	5.05228	0.245650
$424$	−9.85544	−0.478623
$425$	5.91907	0.287117
$426$	3.78279	0.183277
$427$	0	0
$428$	−3.45614	−0.167059
$429$	1.75334	0.0846519
$430$	−12.1508	−0.585966
$431$	22.7390	1.09530	0.547650	−	0.836708i	$-0.315523\pi$
$431$	22.7390	1.09530	0.547650	+	0.836708i	$0.315523\pi$
$432$	9.19226	0.442263
$433$	3.07699	0.147871	0.0739354	−	0.997263i	$-0.476444\pi$
$433$	3.07699	0.147871	0.0739354	+	0.997263i	$0.476444\pi$
$434$	0	0
$435$	−3.43782	−0.164831
$436$	3.43676	0.164591
$437$	−34.3018	−1.64088
$438$	−3.33112	−0.159167
$439$	7.51743	0.358787	0.179394	−	0.983777i	$-0.442586\pi$
$439$	7.51743	0.358787	0.179394	+	0.983777i	$0.442586\pi$
$440$	6.13268	0.292364
$441$	0	0
$442$	11.6059	0.552034
$443$	−16.0864	−0.764286	−0.382143	−	0.924103i	$-0.624814\pi$
$443$	−16.0864	−0.764286	−0.382143	+	0.924103i	$0.624814\pi$
$444$	0.100780	0.00478281
$445$	53.6858	2.54495
$446$	3.28493	0.155546
$447$	−2.91356	−0.137807
$448$	0	0
$449$	−1.72694	−0.0814993	−0.0407497	−	0.999169i	$-0.512975\pi$
$449$	−1.72694	−0.0814993	−0.0407497	+	0.999169i	$0.512975\pi$
$450$	−22.6204	−1.06633
$451$	0.734523	0.0345873
$452$	−2.79011	−0.131236
$453$	−2.92610	−0.137480
$454$	−30.0716	−1.41133
$455$	0	0
$456$	−7.25971	−0.339967
$457$	10.8341	0.506798	0.253399	−	0.967362i	$-0.418451\pi$
$457$	10.8341	0.506798	0.253399	+	0.967362i	$0.418451\pi$
$458$	−2.65949	−0.124270
$459$	−2.33527	−0.109001
$460$	3.66673	0.170962
$461$	11.6300	0.541663	0.270831	−	0.962627i	$-0.412701\pi$
$461$	11.6300	0.541663	0.270831	+	0.962627i	$0.412701\pi$
$462$	0	0
$463$	35.5394	1.65166	0.825829	−	0.563921i	$-0.190708\pi$
$463$	35.5394	1.65166	0.825829	+	0.563921i	$0.190708\pi$
$464$	−13.6721	−0.634709
$465$	−2.99464	−0.138873
$466$	34.5805	1.60191
$467$	24.8054	1.14786	0.573929	−	0.818905i	$-0.305419\pi$
$467$	24.8054	1.14786	0.573929	+	0.818905i	$0.305419\pi$
$468$	−5.18963	−0.239891
$469$	0	0
$470$	8.45083	0.389808
$471$	−2.60967	−0.120247
$472$	−21.1111	−0.971718
$473$	1.85464	0.0852765
$474$	2.29176	0.105264
$475$	41.4155	1.90027
$476$	0	0
$477$	−10.8589	−0.497196
$478$	0.351977	0.0160991
$479$	13.9845	0.638967	0.319483	−	0.947592i	$-0.396491\pi$
$479$	13.9845	0.638967	0.319483	+	0.947592i	$0.396491\pi$
$480$	1.67061	0.0762528
$481$	7.37985	0.336492
$482$	−3.79732	−0.172963
$483$	0	0
$484$	−2.77229	−0.126013
$485$	−47.6880	−2.16540
$486$	13.4801	0.611469
$487$	25.9360	1.17527	0.587635	−	0.809126i	$-0.300059\pi$
$487$	25.9360	1.17527	0.587635	+	0.809126i	$0.300059\pi$
$488$	−9.50649	−0.430339
$489$	−3.29818	−0.149149
$490$	0	0
$491$	−20.7450	−0.936208	−0.468104	−	0.883673i	$-0.655063\pi$
$491$	−20.7450	−0.936208	−0.468104	+	0.883673i	$0.655063\pi$
$492$	0.0929472	0.00419039
$493$	3.47335	0.156432
$494$	81.2056	3.65361
$495$	6.75711	0.303709
$496$	−11.9096	−0.534756
$497$	0	0
$498$	1.44819	0.0648951
$499$	−41.5862	−1.86165	−0.930827	−	0.365460i	$-0.880912\pi$
$499$	−41.5862	−1.86165	−0.930827	+	0.365460i	$0.880912\pi$
$500$	−0.190024	−0.00849815
$501$	−6.14892	−0.274714
$502$	32.4889	1.45005
$503$	16.7465	0.746689	0.373345	−	0.927693i	$-0.378211\pi$
$503$	16.7465	0.746689	0.373345	+	0.927693i	$0.378211\pi$
$504$	0	0
$505$	−17.5062	−0.779017
$506$	−4.78320	−0.212639
$507$	11.6876	0.519065
$508$	−4.48377	−0.198935
$509$	−9.24958	−0.409980	−0.204990	−	0.978764i	$-0.565716\pi$
$509$	−9.24958	−0.409980	−0.204990	+	0.978764i	$0.565716\pi$
$510$	−1.91220	−0.0846735
$511$	0	0
$512$	−16.6464	−0.735674
$513$	−16.3397	−0.721418
$514$	18.5453	0.817999
$515$	−11.8580	−0.522526
$516$	0.234688	0.0103316
$517$	−1.28989	−0.0567293
$518$	0	0
$519$	5.95601	0.261440
$520$	56.8270	2.49203
$521$	2.13355	0.0934725	0.0467362	−	0.998907i	$-0.485118\pi$
$521$	2.13355	0.0934725	0.0467362	+	0.998907i	$0.485118\pi$
$522$	−13.2738	−0.580976
$523$	14.9578	0.654058	0.327029	−	0.945014i	$-0.393953\pi$
$523$	14.9578	0.654058	0.327029	+	0.945014i	$0.393953\pi$
$524$	5.39124	0.235517
$525$	0	0
$526$	−32.8356	−1.43170
$527$	3.02559	0.131797
$528$	−1.14887	−0.0499983
$529$	−4.27792	−0.185996
$530$	−18.1634	−0.788970
$531$	−23.2606	−1.00943
$532$	0	0
$533$	6.80628	0.294813
$534$	−8.86198	−0.383495
$535$	41.6985	1.80278
$536$	−34.3902	−1.48543
$537$	−3.00771	−0.129792
$538$	44.1419	1.90309
$539$	0	0
$540$	1.74666	0.0751642
$541$	34.7302	1.49317	0.746583	−	0.665292i	$-0.231693\pi$
$541$	34.7302	1.49317	0.746583	+	0.665292i	$0.231693\pi$
$542$	13.7232	0.589460
$543$	5.85568	0.251291
$544$	−1.68788	−0.0723673
$545$	−41.4647	−1.77615
$546$	0	0
$547$	−2.83477	−0.121206	−0.0606031	−	0.998162i	$-0.519302\pi$
$547$	−2.83477	−0.121206	−0.0606031	+	0.998162i	$0.519302\pi$
$548$	−2.43331	−0.103946
$549$	−10.4744	−0.447038
$550$	5.77517	0.246254
$551$	24.3028	1.03534
$552$	3.96239	0.168650
$553$	0	0
$554$	23.9167	1.01612
$555$	−1.21592	−0.0516127
$556$	−4.11435	−0.174487
$557$	−9.47763	−0.401580	−0.200790	−	0.979634i	$-0.564351\pi$
$557$	−9.47763	−0.401580	−0.200790	+	0.979634i	$0.564351\pi$
$558$	−11.5626	−0.489485
$559$	17.1856	0.726873
$560$	0	0
$561$	0.291868	0.0123227
$562$	10.2077	0.430584
$563$	20.5070	0.864267	0.432134	−	0.901810i	$-0.357761\pi$
$563$	20.5070	0.864267	0.432134	+	0.901810i	$0.357761\pi$
$564$	−0.163224	−0.00687297
$565$	33.6629	1.41621
$566$	−22.0789	−0.928047
$567$	0	0
$568$	−18.7135	−0.785203
$569$	−37.6081	−1.57661	−0.788306	−	0.615284i	$-0.789041\pi$
$569$	−37.6081	−1.57661	−0.788306	+	0.615284i	$0.789041\pi$
$570$	−13.3795	−0.560408
$571$	19.1163	0.799994	0.399997	−	0.916516i	$-0.369011\pi$
$571$	19.1163	0.799994	0.399997	+	0.916516i	$0.369011\pi$
$572$	1.32496	0.0553992
$573$	−7.30542	−0.305188
$574$	0	0
$575$	−22.6048	−0.942684
$576$	−19.2114	−0.800473
$577$	−39.5393	−1.64604	−0.823020	−	0.568012i	$-0.807713\pi$
$577$	−39.5393	−1.64604	−0.823020	+	0.568012i	$0.807713\pi$
$578$	−23.6530	−0.983837
$579$	−2.82605	−0.117447
$580$	−2.59788	−0.107871
$581$	0	0
$582$	7.87191	0.326301
$583$	2.77237	0.114820
$584$	16.4791	0.681910
$585$	62.6131	2.58873
$586$	−27.1928	−1.12332
$587$	37.9601	1.56678	0.783390	−	0.621530i	$-0.213489\pi$
$587$	37.9601	1.56678	0.783390	+	0.621530i	$0.213489\pi$
$588$	0	0
$589$	21.1699	0.872291
$590$	−38.9075	−1.60180
$591$	8.81379	0.362551
$592$	−4.83564	−0.198744
$593$	−32.3285	−1.32757	−0.663786	−	0.747922i	$-0.731052\pi$
$593$	−32.3285	−1.32757	−0.663786	+	0.747922i	$0.731052\pi$
$594$	−2.27849	−0.0934877
$595$	0	0
$596$	−2.20171	−0.0901856
$597$	1.49602	0.0612280
$598$	−44.3224	−1.81248
$599$	−36.6170	−1.49613	−0.748066	−	0.663625i	$-0.769017\pi$
$599$	−36.6170	−1.49613	−0.748066	+	0.663625i	$0.769017\pi$
$600$	−4.78413	−0.195311
$601$	−22.0254	−0.898436	−0.449218	−	0.893422i	$-0.648297\pi$
$601$	−22.0254	−0.898436	−0.449218	+	0.893422i	$0.648297\pi$
$602$	0	0
$603$	−37.8918	−1.54307
$604$	−2.21118	−0.0899718
$605$	33.4477	1.35984
$606$	2.88977	0.117389
$607$	−19.8990	−0.807677	−0.403838	−	0.914830i	$-0.632324\pi$
$607$	−19.8990	−0.807677	−0.403838	+	0.914830i	$0.632324\pi$
$608$	−11.8100	−0.478959
$609$	0	0
$610$	−17.5203	−0.709378
$611$	−11.9525	−0.483545
$612$	−0.863888	−0.0349206
$613$	−45.3077	−1.82996	−0.914980	−	0.403498i	$-0.867794\pi$
$613$	−45.3077	−1.82996	−0.914980	+	0.403498i	$0.867794\pi$
$614$	17.8496	0.720352
$615$	−1.12141	−0.0452197
$616$	0	0
$617$	−4.43517	−0.178553	−0.0892766	−	0.996007i	$-0.528456\pi$
$617$	−4.43517	−0.178553	−0.0892766	+	0.996007i	$0.528456\pi$
$618$	1.95741	0.0787387
$619$	−42.7889	−1.71983	−0.859916	−	0.510435i	$-0.829484\pi$
$619$	−42.7889	−1.71983	−0.859916	+	0.510435i	$0.829484\pi$
$620$	−2.26298	−0.0908837
$621$	8.91832	0.357880
$622$	6.76103	0.271093
$623$	0	0
$624$	−10.6457	−0.426171
$625$	−23.8285	−0.953141
$626$	38.4500	1.53677
$627$	2.04218	0.0815570
$628$	−1.97207	−0.0786942
$629$	1.22848	0.0489828
$630$	0	0
$631$	−37.1419	−1.47859	−0.739297	−	0.673379i	$-0.764842\pi$
$631$	−37.1419	−1.47859	−0.739297	+	0.673379i	$0.764842\pi$
$632$	−11.3374	−0.450976
$633$	7.08218	0.281491
$634$	−21.4910	−0.853518
$635$	54.0969	2.14677
$636$	0.350819	0.0139109
$637$	0	0
$638$	3.38890	0.134168
$639$	−20.6189	−0.815673
$640$	−41.6614	−1.64681
$641$	−17.4126	−0.687755	−0.343878	−	0.939015i	$-0.611741\pi$
$641$	−17.4126	−0.687755	−0.343878	+	0.939015i	$0.611741\pi$
$642$	−6.88321	−0.271659
$643$	−15.2166	−0.600084	−0.300042	−	0.953926i	$-0.597001\pi$
$643$	−15.2166	−0.600084	−0.300042	+	0.953926i	$0.597001\pi$
$644$	0	0
$645$	−2.83152	−0.111491
$646$	13.5178	0.531852
$647$	−29.9412	−1.17711	−0.588556	−	0.808457i	$-0.700303\pi$
$647$	−29.9412	−1.17711	−0.588556	+	0.808457i	$0.700303\pi$
$648$	−20.6492	−0.811179
$649$	5.93864	0.233112
$650$	53.5142	2.09900
$651$	0	0
$652$	−2.49236	−0.0976083
$653$	39.8140	1.55804	0.779021	−	0.626998i	$-0.215717\pi$
$653$	39.8140	1.55804	0.779021	+	0.626998i	$0.215717\pi$
$654$	6.84462	0.267646
$655$	−65.0455	−2.54154
$656$	−4.45981	−0.174126
$657$	18.1570	0.708372
$658$	0	0
$659$	−1.82438	−0.0710679	−0.0355340	−	0.999368i	$-0.511313\pi$
$659$	−1.82438	−0.0710679	−0.0355340	+	0.999368i	$0.511313\pi$
$660$	−0.218302	−0.00849739
$661$	−47.8128	−1.85970	−0.929851	−	0.367935i	$-0.880065\pi$
$661$	−47.8128	−1.85970	−0.929851	+	0.367935i	$0.880065\pi$
$662$	30.7334	1.19449
$663$	2.70452	0.105035
$664$	−7.16424	−0.278026
$665$	0	0
$666$	−4.69477	−0.181919
$667$	−13.2646	−0.513607
$668$	−4.64661	−0.179783
$669$	0.765491	0.0295956
$670$	−63.3807	−2.44861
$671$	2.67421	0.103237
$672$	0	0
$673$	15.8992	0.612870	0.306435	−	0.951892i	$-0.400864\pi$
$673$	15.8992	0.612870	0.306435	+	0.951892i	$0.400864\pi$
$674$	−1.28995	−0.0496871
$675$	−10.7678	−0.414454
$676$	8.83207	0.339695
$677$	14.4905	0.556915	0.278457	−	0.960449i	$-0.410177\pi$
$677$	14.4905	0.556915	0.278457	+	0.960449i	$0.410177\pi$
$678$	−5.55676	−0.213406
$679$	0	0
$680$	9.45967	0.362762
$681$	−7.00762	−0.268533
$682$	2.95204	0.113039
$683$	−17.4190	−0.666518	−0.333259	−	0.942835i	$-0.608148\pi$
$683$	−17.4190	−0.666518	−0.333259	+	0.942835i	$0.608148\pi$
$684$	−6.04458	−0.231120
$685$	29.3580	1.12171
$686$	0	0
$687$	−0.619743	−0.0236447
$688$	−11.2608	−0.429316
$689$	25.6895	0.978693
$690$	7.30262	0.278006
$691$	22.1064	0.840968	0.420484	−	0.907300i	$-0.361860\pi$
$691$	22.1064	0.840968	0.420484	+	0.907300i	$0.361860\pi$
$692$	4.50083	0.171096
$693$	0	0
$694$	27.1253	1.02966
$695$	49.6398	1.88294
$696$	−2.80735	−0.106412
$697$	1.13300	0.0429155
$698$	−42.1888	−1.59687
$699$	8.05834	0.304794
$700$	0	0
$701$	4.46122	0.168498	0.0842490	−	0.996445i	$-0.473151\pi$
$701$	4.46122	0.168498	0.0842490	+	0.996445i	$0.473151\pi$
$702$	−21.1131	−0.796863
$703$	8.59562	0.324190
$704$	4.90482	0.184857
$705$	1.96930	0.0741683
$706$	−36.1430	−1.36026
$707$	0	0
$708$	0.751481	0.0282424
$709$	7.76841	0.291749	0.145874	−	0.989303i	$-0.453400\pi$
$709$	7.76841	0.291749	0.145874	+	0.989303i	$0.453400\pi$
$710$	−34.4888	−1.29434
$711$	−12.4917	−0.468476
$712$	43.8403	1.64299
$713$	−11.5546	−0.432725
$714$	0	0
$715$	−15.9857	−0.597830
$716$	−2.27286	−0.0849408
$717$	0.0820216	0.00306315
$718$	3.10529	0.115888
$719$	−39.0425	−1.45604	−0.728019	−	0.685557i	$-0.759559\pi$
$719$	−39.0425	−1.45604	−0.728019	+	0.685557i	$0.759559\pi$
$720$	−41.0272	−1.52899
$721$	0	0
$722$	65.9886	2.45584
$723$	−0.884894	−0.0329095
$724$	4.42501	0.164454
$725$	16.0155	0.594800
$726$	−5.52126	−0.204913
$727$	−3.14097	−0.116492	−0.0582461	−	0.998302i	$-0.518551\pi$
$727$	−3.14097	−0.116492	−0.0582461	+	0.998302i	$0.518551\pi$
$728$	0	0
$729$	−20.5831	−0.762339
$730$	30.3708	1.12407
$731$	2.86079	0.105810
$732$	0.338397	0.0125075
$733$	30.4139	1.12336	0.561681	−	0.827354i	$-0.310155\pi$
$733$	30.4139	1.12336	0.561681	+	0.827354i	$0.310155\pi$
$734$	38.5073	1.42133
$735$	0	0
$736$	6.44596	0.237601
$737$	9.67410	0.356350
$738$	−4.32989	−0.159385
$739$	−38.3460	−1.41058	−0.705291	−	0.708918i	$-0.749184\pi$
$739$	−38.3460	−1.41058	−0.705291	+	0.708918i	$0.749184\pi$
$740$	−0.918840	−0.0337772
$741$	18.9234	0.695169
$742$	0	0
$743$	−4.88372	−0.179166	−0.0895832	−	0.995979i	$-0.528554\pi$
$743$	−4.88372	−0.179166	−0.0895832	+	0.995979i	$0.528554\pi$
$744$	−2.44545	−0.0896547
$745$	26.5637	0.973220
$746$	27.5862	1.01000
$747$	−7.89370	−0.288815
$748$	0.220558	0.00806440
$749$	0	0
$750$	−0.378451	−0.0138191
$751$	22.1942	0.809879	0.404940	−	0.914343i	$-0.367293\pi$
$751$	22.1942	0.809879	0.404940	+	0.914343i	$0.367293\pi$
$752$	7.83184	0.285598
$753$	7.57091	0.275899
$754$	31.4024	1.14361
$755$	26.6780	0.970913
$756$	0	0
$757$	−8.81952	−0.320551	−0.160275	−	0.987072i	$-0.551238\pi$
$757$	−8.81952	−0.320551	−0.160275	+	0.987072i	$0.551238\pi$
$758$	−47.9076	−1.74008
$759$	−1.11463	−0.0404587
$760$	66.1888	2.40092
$761$	7.54942	0.273666	0.136833	−	0.990594i	$-0.456308\pi$
$761$	7.54942	0.273666	0.136833	+	0.990594i	$0.456308\pi$
$762$	−8.92984	−0.323494
$763$	0	0
$764$	−5.52054	−0.199726
$765$	10.4228	0.376839
$766$	36.1765	1.30711
$767$	55.0290	1.98698
$768$	2.19329	0.0791437
$769$	−8.03405	−0.289715	−0.144858	−	0.989453i	$-0.546272\pi$
$769$	−8.03405	−0.289715	−0.144858	+	0.989453i	$0.546272\pi$
$770$	0	0
$771$	4.32163	0.155640
$772$	−2.13558	−0.0768613
$773$	13.5760	0.488297	0.244148	−	0.969738i	$-0.421492\pi$
$773$	13.5760	0.488297	0.244148	+	0.969738i	$0.421492\pi$
$774$	−10.9328	−0.392971
$775$	13.9509	0.501131
$776$	−38.9425	−1.39795
$777$	0	0
$778$	36.9751	1.32562
$779$	7.92756	0.284034
$780$	−2.02284	−0.0724293
$781$	5.26419	0.188368
$782$	−7.37810	−0.263840
$783$	−6.31863	−0.225809
$784$	0	0
$785$	23.7931	0.849213
$786$	10.7371	0.382981
$787$	−4.48717	−0.159950	−0.0799752	−	0.996797i	$-0.525484\pi$
$787$	−4.48717	−0.159950	−0.0799752	+	0.996797i	$0.525484\pi$
$788$	6.66038	0.237266
$789$	−7.65170	−0.272408
$790$	−20.8946	−0.743396
$791$	0	0
$792$	5.51792	0.196071
$793$	24.7799	0.879961
$794$	−4.72958	−0.167847
$795$	−4.23264	−0.150116
$796$	1.13051	0.0400698
$797$	−48.6900	−1.72469	−0.862344	−	0.506323i	$-0.831004\pi$
$797$	−48.6900	−1.72469	−0.862344	+	0.506323i	$0.831004\pi$
$798$	0	0
$799$	−1.98966	−0.0703890
$800$	−7.78276	−0.275162
$801$	48.3041	1.70674
$802$	18.9513	0.669194
$803$	−4.63564	−0.163588
$804$	1.22417	0.0431732
$805$	0	0
$806$	27.3543	0.963515
$807$	10.2864	0.362100
$808$	−14.2958	−0.502923
$809$	48.3228	1.69894	0.849469	−	0.527638i	$-0.176922\pi$
$809$	48.3228	1.69894	0.849469	+	0.527638i	$0.176922\pi$
$810$	−38.0563	−1.33716
$811$	43.8406	1.53945	0.769725	−	0.638375i	$-0.220393\pi$
$811$	43.8406	1.53945	0.769725	+	0.638375i	$0.220393\pi$
$812$	0	0
$813$	3.19792	0.112156
$814$	1.19861	0.0420114
$815$	30.0704	1.05332
$816$	−1.77214	−0.0620372
$817$	20.0168	0.700299
$818$	−31.0657	−1.08619
$819$	0	0
$820$	−0.847426	−0.0295934
$821$	44.6778	1.55927	0.779633	−	0.626237i	$-0.215406\pi$
$821$	44.6778	1.55927	0.779633	+	0.626237i	$0.215406\pi$
$822$	−4.84616	−0.169029
$823$	−48.8643	−1.70330	−0.851651	−	0.524109i	$-0.824398\pi$
$823$	−48.8643	−1.70330	−0.851651	+	0.524109i	$0.824398\pi$
$824$	−9.68335	−0.337336
$825$	1.34579	0.0468545
$826$	0	0
$827$	−11.1915	−0.389165	−0.194583	−	0.980886i	$-0.562335\pi$
$827$	−11.1915	−0.389165	−0.194583	+	0.980886i	$0.562335\pi$
$828$	3.29916	0.114654
$829$	−15.2289	−0.528922	−0.264461	−	0.964396i	$-0.585194\pi$
$829$	−15.2289	−0.528922	−0.264461	+	0.964396i	$0.585194\pi$
$830$	−13.2036	−0.458304
$831$	5.57333	0.193337
$832$	45.4494	1.57567
$833$	0	0
$834$	−8.19409	−0.283738
$835$	56.0615	1.94009
$836$	1.54323	0.0533738
$837$	−5.50409	−0.190249
$838$	41.5243	1.43443
$839$	14.1163	0.487348	0.243674	−	0.969857i	$-0.421647\pi$
$839$	14.1163	0.487348	0.243674	+	0.969857i	$0.421647\pi$
$840$	0	0
$841$	−19.6020	−0.675932
$842$	−39.6162	−1.36527
$843$	2.37870	0.0819268
$844$	5.35185	0.184218
$845$	−106.559	−3.66575
$846$	7.60369	0.261420
$847$	0	0
$848$	−16.8331	−0.578049
$849$	−5.14507	−0.176578
$850$	8.90821	0.305549
$851$	−4.69153	−0.160824
$852$	0.666136	0.0228214
$853$	9.68058	0.331457	0.165728	−	0.986171i	$-0.447003\pi$
$853$	9.68058	0.331457	0.165728	+	0.986171i	$0.447003\pi$
$854$	0	0
$855$	72.9281	2.49409
$856$	34.0514	1.16385
$857$	−20.4924	−0.700009	−0.350004	−	0.936748i	$-0.613820\pi$
$857$	−20.4924	−0.700009	−0.350004	+	0.936748i	$0.613820\pi$
$858$	2.63877	0.0900861
$859$	11.3722	0.388016	0.194008	−	0.981000i	$-0.437851\pi$
$859$	11.3722	0.388016	0.194008	+	0.981000i	$0.437851\pi$
$860$	−2.13972	−0.0729638
$861$	0	0
$862$	34.2222	1.16561
$863$	21.7167	0.739244	0.369622	−	0.929182i	$-0.379487\pi$
$863$	21.7167	0.739244	0.369622	+	0.929182i	$0.379487\pi$
$864$	3.07055	0.104462
$865$	−54.3027	−1.84635
$866$	4.63087	0.157363
$867$	−5.51189	−0.187194
$868$	0	0
$869$	3.18924	0.108188
$870$	−5.17391	−0.175412
$871$	89.6427	3.03743
$872$	−33.8604	−1.14666
$873$	−42.9076	−1.45220
$874$	−51.6242	−1.74621
$875$	0	0
$876$	−0.586598	−0.0198193
$877$	−30.0089	−1.01333	−0.506664	−	0.862143i	$-0.669122\pi$
$877$	−30.0089	−1.01333	−0.506664	+	0.862143i	$0.669122\pi$
$878$	11.3137	0.381820
$879$	−6.33676	−0.213734
$880$	10.4746	0.353099
$881$	41.1735	1.38717	0.693586	−	0.720374i	$-0.256030\pi$
$881$	41.1735	1.38717	0.693586	+	0.720374i	$0.256030\pi$
$882$	0	0
$883$	−43.7823	−1.47339	−0.736696	−	0.676224i	$-0.763615\pi$
$883$	−43.7823	−1.47339	−0.736696	+	0.676224i	$0.763615\pi$
$884$	2.04375	0.0687387
$885$	−9.06665	−0.304772
$886$	−24.2100	−0.813349
$887$	−0.447587	−0.0150285	−0.00751424	−	0.999972i	$-0.502392\pi$
$887$	−0.447587	−0.0150285	−0.00751424	+	0.999972i	$0.502392\pi$
$888$	−0.992927	−0.0333205
$889$	0	0
$890$	80.7971	2.70833
$891$	5.80871	0.194599
$892$	0.578464	0.0193684
$893$	−13.9215	−0.465866
$894$	−4.38491	−0.146653
$895$	27.4222	0.916622
$896$	0	0
$897$	−10.3285	−0.344858
$898$	−2.59904	−0.0867312
$899$	8.18647	0.273034
$900$	−3.98336	−0.132779
$901$	4.27639	0.142467
$902$	1.10546	0.0368077
$903$	0	0
$904$	27.4894	0.914284
$905$	−53.3879	−1.77467
$906$	−4.40377	−0.146306
$907$	−13.9211	−0.462243	−0.231121	−	0.972925i	$-0.574239\pi$
$907$	−13.9211	−0.462243	−0.231121	+	0.972925i	$0.574239\pi$
$908$	−5.29550	−0.175737
$909$	−15.7513	−0.522439
$910$	0	0
$911$	−15.9681	−0.529047	−0.264523	−	0.964379i	$-0.585215\pi$
$911$	−15.9681	−0.529047	−0.264523	+	0.964379i	$0.585215\pi$
$912$	−12.3996	−0.410590
$913$	2.01533	0.0666976
$914$	16.3053	0.539332
$915$	−4.08278	−0.134972
$916$	−0.468326	−0.0154739
$917$	0	0
$918$	−3.51458	−0.115998
$919$	−36.6336	−1.20843	−0.604215	−	0.796821i	$-0.706513\pi$
$919$	−36.6336	−1.20843	−0.604215	+	0.796821i	$0.706513\pi$
$920$	−36.1262	−1.19105
$921$	4.15951	0.137061
$922$	17.5031	0.576435
$923$	48.7793	1.60559
$924$	0	0
$925$	5.66449	0.186247
$926$	53.4868	1.75769
$927$	−10.6693	−0.350426
$928$	−4.56697	−0.149918
$929$	−34.1044	−1.11893	−0.559464	−	0.828855i	$-0.688993\pi$
$929$	−34.1044	−1.11893	−0.559464	+	0.828855i	$0.688993\pi$
$930$	−4.50694	−0.147788
$931$	0	0
$932$	6.08950	0.199468
$933$	1.57553	0.0515805
$934$	37.3321	1.22155
$935$	−2.66104	−0.0870254
$936$	51.1305	1.67125
$937$	−54.9105	−1.79385	−0.896925	−	0.442184i	$-0.854204\pi$
$937$	−54.9105	−1.79385	−0.896925	+	0.442184i	$0.854204\pi$
$938$	0	0
$939$	8.96003	0.292400
$940$	1.48816	0.0485384
$941$	−34.0556	−1.11018	−0.555090	−	0.831790i	$-0.687316\pi$
$941$	−34.0556	−1.11018	−0.555090	+	0.831790i	$0.687316\pi$
$942$	−3.92756	−0.127967
$943$	−4.32690	−0.140903
$944$	−36.0577	−1.17358
$945$	0	0
$946$	2.79124	0.0907509
$947$	3.79574	0.123345	0.0616726	−	0.998096i	$-0.480357\pi$
$947$	3.79574	0.123345	0.0616726	+	0.998096i	$0.480357\pi$
$948$	0.403570	0.0131073
$949$	−42.9550	−1.39438
$950$	62.3303	2.02226
$951$	−5.00807	−0.162398
$952$	0	0
$953$	13.8760	0.449489	0.224744	−	0.974418i	$-0.427845\pi$
$953$	13.8760	0.449489	0.224744	+	0.974418i	$0.427845\pi$
$954$	−16.3427	−0.529113
$955$	66.6056	2.15531
$956$	0.0619819	0.00200464
$957$	0.789719	0.0255280
$958$	21.0466	0.679986
$959$	0	0
$960$	−7.48830	−0.241684
$961$	−23.8689	−0.769963
$962$	11.1067	0.358094
$963$	37.5185	1.20902
$964$	−0.668694	−0.0215372
$965$	25.7659	0.829434
$966$	0	0
$967$	29.8546	0.960060	0.480030	−	0.877252i	$-0.340626\pi$
$967$	29.8546	0.960060	0.480030	+	0.877252i	$0.340626\pi$
$968$	27.3137	0.877897
$969$	3.15007	0.101195
$970$	−71.7704	−2.30441
$971$	−25.6455	−0.823003	−0.411502	−	0.911409i	$-0.634996\pi$
$971$	−25.6455	−0.823003	−0.411502	+	0.911409i	$0.634996\pi$
$972$	2.37379	0.0761395
$973$	0	0
$974$	39.0336	1.25072
$975$	12.4705	0.399374
$976$	−16.2370	−0.519735
$977$	−55.4206	−1.77306	−0.886531	−	0.462669i	$-0.846892\pi$
$977$	−55.4206	−1.77306	−0.886531	+	0.462669i	$0.846892\pi$
$978$	−4.96376	−0.158723
$979$	−12.3325	−0.394147
$980$	0	0
$981$	−37.3081	−1.19116
$982$	−31.2212	−0.996309
$983$	−31.8722	−1.01657	−0.508283	−	0.861190i	$-0.669720\pi$
$983$	−31.8722	−1.01657	−0.508283	+	0.861190i	$0.669720\pi$
$984$	−0.915756	−0.0291932
$985$	−80.3578	−2.56041
$986$	5.22739	0.166474
$987$	0	0
$988$	14.3000	0.454944
$989$	−10.9253	−0.347403
$990$	10.1694	0.323206
$991$	−49.6614	−1.57755	−0.788774	−	0.614684i	$-0.789284\pi$
$991$	−49.6614	−1.57755	−0.788774	+	0.614684i	$0.789284\pi$
$992$	−3.97823	−0.126309
$993$	7.16184	0.227274
$994$	0	0
$995$	−13.6396	−0.432406
$996$	0.255022	0.00808067
$997$	−13.4050	−0.424541	−0.212271	−	0.977211i	$-0.568086\pi$
$997$	−13.4050	−0.424541	−0.212271	+	0.977211i	$0.568086\pi$
$998$	−62.5872	−1.98116
$999$	−2.23483	−0.0707067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2009.2.a.u.1.16	yes	20
7.6	odd	2		2009.2.a.t.1.16	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2009.2.a.t.1.16	✓	20	7.6	odd	2
2009.2.a.u.1.16	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2009 = 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.50500 q^{2} +0.350711 q^{3} +0.265025 q^{4} -3.19754 q^{5} +0.527821 q^{6} -2.61114 q^{8} -2.87700 q^{9} +O(q^{10})\)	\(q+1.50500 q^{2} +0.350711 q^{3} +0.265025 q^{4} -3.19754 q^{5} +0.527821 q^{6} -2.61114 q^{8} -2.87700 q^{9} -4.81229 q^{10} +0.734523 q^{11} +0.0929472 q^{12} +6.80628 q^{13} -1.12141 q^{15} -4.45981 q^{16} +1.13300 q^{17} -4.32989 q^{18} +7.92756 q^{19} -0.847426 q^{20} +1.10546 q^{22} -4.32690 q^{23} -0.915756 q^{24} +5.22424 q^{25} +10.2435 q^{26} -2.06113 q^{27} +3.06561 q^{29} -1.68773 q^{30} +2.67042 q^{31} -1.48974 q^{32} +0.257606 q^{33} +1.70517 q^{34} -0.762477 q^{36} +1.08427 q^{37} +11.9310 q^{38} +2.38704 q^{39} +8.34921 q^{40} +1.00000 q^{41} +2.52496 q^{43} +0.194667 q^{44} +9.19932 q^{45} -6.51199 q^{46} -1.75609 q^{47} -1.56411 q^{48} +7.86248 q^{50} +0.397357 q^{51} +1.80383 q^{52} +3.77439 q^{53} -3.10200 q^{54} -2.34866 q^{55} +2.78029 q^{57} +4.61375 q^{58} +8.08503 q^{59} -0.297202 q^{60} +3.64075 q^{61} +4.01898 q^{62} +6.67756 q^{64} -21.7633 q^{65} +0.387696 q^{66} +13.1706 q^{67} +0.300274 q^{68} -1.51749 q^{69} +7.16682 q^{71} +7.51225 q^{72} -6.31108 q^{73} +1.63183 q^{74} +1.83220 q^{75} +2.10100 q^{76} +3.59250 q^{78} +4.34192 q^{79} +14.2604 q^{80} +7.90814 q^{81} +1.50500 q^{82} +2.74372 q^{83} -3.62282 q^{85} +3.80007 q^{86} +1.07515 q^{87} -1.91794 q^{88} -16.7897 q^{89} +13.8450 q^{90} -1.14674 q^{92} +0.936547 q^{93} -2.64292 q^{94} -25.3487 q^{95} -0.522469 q^{96} +14.9140 q^{97} -2.11322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 8 q^{3} + 18 q^{4} + 8 q^{5} + 12 q^{6} - 6 q^{8} + 16 q^{9}+O(q^{10}) \) 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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