Properties

Label 2.22.a.b.1.1
Level $2$
Weight $22$
Character 2.1
Self dual yes
Analytic conductor $5.590$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2,22,Mod(1,2)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1024] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.58954688574\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1024.00 q^{2} +59316.0 q^{3} +1.04858e6 q^{4} +4.97535e6 q^{5} +6.07396e7 q^{6} +1.42743e9 q^{7} +1.07374e9 q^{8} -6.94197e9 q^{9} +5.09476e9 q^{10} -1.06768e11 q^{11} +6.21973e10 q^{12} -1.50151e11 q^{13} +1.46168e12 q^{14} +2.95118e11 q^{15} +1.09951e12 q^{16} -1.12040e13 q^{17} -7.10857e12 q^{18} +1.10241e13 q^{19} +5.21703e12 q^{20} +8.46692e13 q^{21} -1.09330e14 q^{22} +1.29503e14 q^{23} +6.36901e13 q^{24} -4.52083e14 q^{25} -1.53754e14 q^{26} -1.03224e15 q^{27} +1.49676e15 q^{28} +2.38237e15 q^{29} +3.02201e14 q^{30} -8.78553e14 q^{31} +1.12590e15 q^{32} -6.33304e15 q^{33} -1.14729e16 q^{34} +7.10194e15 q^{35} -7.27918e15 q^{36} +3.11300e16 q^{37} +1.12886e16 q^{38} -8.90633e15 q^{39} +5.34224e15 q^{40} -2.46129e16 q^{41} +8.67013e16 q^{42} -1.33386e17 q^{43} -1.11954e17 q^{44} -3.45387e16 q^{45} +1.32611e17 q^{46} -1.92524e17 q^{47} +6.52186e16 q^{48} +1.47900e18 q^{49} -4.62933e17 q^{50} -6.64575e17 q^{51} -1.57444e17 q^{52} -5.94166e17 q^{53} -1.05701e18 q^{54} -5.31208e17 q^{55} +1.53269e18 q^{56} +6.53903e17 q^{57} +2.43955e18 q^{58} -2.95595e18 q^{59} +3.09454e17 q^{60} +7.98415e18 q^{61} -8.99638e17 q^{62} -9.90914e18 q^{63} +1.15292e18 q^{64} -7.47052e17 q^{65} -6.48504e18 q^{66} +4.83704e18 q^{67} -1.17482e19 q^{68} +7.68159e18 q^{69} +7.27239e18 q^{70} +8.84902e18 q^{71} -7.45388e18 q^{72} +3.66844e19 q^{73} +3.18771e19 q^{74} -2.68158e19 q^{75} +1.15596e19 q^{76} -1.52403e20 q^{77} -9.12008e18 q^{78} +3.38406e19 q^{79} +5.47046e18 q^{80} +1.13873e19 q^{81} -2.52036e19 q^{82} +2.04215e20 q^{83} +8.87821e19 q^{84} -5.57437e19 q^{85} -1.36587e20 q^{86} +1.41313e20 q^{87} -1.14641e20 q^{88} -4.10241e19 q^{89} -3.53676e19 q^{90} -2.14329e20 q^{91} +1.35794e20 q^{92} -5.21122e19 q^{93} -1.97145e20 q^{94} +5.48485e19 q^{95} +6.67839e19 q^{96} -7.27592e20 q^{97} +1.51449e21 q^{98} +7.41179e20 q^{99} +O(q^{100})\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1024.00 0.707107
\(3\) 59316.0 0.579961 0.289980 − 0.957033i \(-0.406351\pi\)
0.289980 + 0.957033i \(0.406351\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 4.97535e6 0.227845 0.113922 − 0.993490i \(-0.463659\pi\)
0.113922 + 0.993490i \(0.463659\pi\)
\(6\) 6.07396e7 0.410094
\(7\) 1.42743e9 1.90996 0.954980 − 0.296671i \(-0.0958765\pi\)
0.954980 + 0.296671i \(0.0958765\pi\)
\(8\) 1.07374e9 0.353553
\(9\) −6.94197e9 −0.663645
\(10\) 5.09476e9 0.161110
\(11\) −1.06768e11 −1.24113 −0.620565 − 0.784155i \(-0.713097\pi\)
−0.620565 + 0.784155i \(0.713097\pi\)
\(12\) 6.21973e10 0.289980
\(13\) −1.50151e11 −0.302080 −0.151040 − 0.988528i \(-0.548262\pi\)
−0.151040 + 0.988528i \(0.548262\pi\)
\(14\) 1.46168e12 1.35055
\(15\) 2.95118e11 0.132141
\(16\) 1.09951e12 0.250000
\(17\) −1.12040e13 −1.34790 −0.673952 − 0.738776i \(-0.735404\pi\)
−0.673952 + 0.738776i \(0.735404\pi\)
\(18\) −7.10857e12 −0.469268
\(19\) 1.10241e13 0.412504 0.206252 − 0.978499i \(-0.433873\pi\)
0.206252 + 0.978499i \(0.433873\pi\)
\(20\) 5.21703e12 0.113922
\(21\) 8.46692e13 1.10770
\(22\) −1.09330e14 −0.877612
\(23\) 1.29503e14 0.651834 0.325917 − 0.945398i \(-0.394327\pi\)
0.325917 + 0.945398i \(0.394327\pi\)
\(24\) 6.36901e13 0.205047
\(25\) −4.52083e14 −0.948087
\(26\) −1.53754e14 −0.213603
\(27\) −1.03224e15 −0.964849
\(28\) 1.49676e15 0.954980
\(29\) 2.38237e15 1.05155 0.525776 − 0.850623i \(-0.323775\pi\)
0.525776 + 0.850623i \(0.323775\pi\)
\(30\) 3.02201e14 0.0934377
\(31\) −8.78553e14 −0.192517 −0.0962587 − 0.995356i \(-0.530688\pi\)
−0.0962587 + 0.995356i \(0.530688\pi\)
\(32\) 1.12590e15 0.176777
\(33\) −6.33304e15 −0.719807
\(34\) −1.14729e16 −0.953111
\(35\) 7.10194e15 0.435174
\(36\) −7.27918e15 −0.331823
\(37\) 3.11300e16 1.06429 0.532146 − 0.846652i \(-0.321386\pi\)
0.532146 + 0.846652i \(0.321386\pi\)
\(38\) 1.12886e16 0.291685
\(39\) −8.90633e15 −0.175194
\(40\) 5.34224e15 0.0805552
\(41\) −2.46129e16 −0.286373 −0.143187 − 0.989696i \(-0.545735\pi\)
−0.143187 + 0.989696i \(0.545735\pi\)
\(42\) 8.67013e16 0.783263
\(43\) −1.33386e17 −0.941222 −0.470611 − 0.882341i \(-0.655966\pi\)
−0.470611 + 0.882341i \(0.655966\pi\)
\(44\) −1.11954e17 −0.620565
\(45\) −3.45387e16 −0.151208
\(46\) 1.32611e17 0.460916
\(47\) −1.92524e17 −0.533897 −0.266948 − 0.963711i \(-0.586015\pi\)
−0.266948 + 0.963711i \(0.586015\pi\)
\(48\) 6.52186e16 0.144990
\(49\) 1.47900e18 2.64794
\(50\) −4.62933e17 −0.670399
\(51\) −6.64575e17 −0.781731
\(52\) −1.57444e17 −0.151040
\(53\) −5.94166e17 −0.466672 −0.233336 − 0.972396i \(-0.574964\pi\)
−0.233336 + 0.972396i \(0.574964\pi\)
\(54\) −1.05701e18 −0.682251
\(55\) −5.31208e17 −0.282785
\(56\) 1.53269e18 0.675273
\(57\) 6.53903e17 0.239236
\(58\) 2.43955e18 0.743559
\(59\) −2.95595e18 −0.752925 −0.376462 − 0.926432i \(-0.622860\pi\)
−0.376462 + 0.926432i \(0.622860\pi\)
\(60\) 3.09454e17 0.0660705
\(61\) 7.98415e18 1.43306 0.716532 − 0.697554i \(-0.245728\pi\)
0.716532 + 0.697554i \(0.245728\pi\)
\(62\) −8.99638e17 −0.136130
\(63\) −9.90914e18 −1.26754
\(64\) 1.15292e18 0.125000
\(65\) −7.47052e17 −0.0688272
\(66\) −6.48504e18 −0.508981
\(67\) 4.83704e18 0.324186 0.162093 − 0.986775i \(-0.448176\pi\)
0.162093 + 0.986775i \(0.448176\pi\)
\(68\) −1.17482e19 −0.673952
\(69\) 7.68159e18 0.378038
\(70\) 7.27239e18 0.307714
\(71\) 8.84902e18 0.322613 0.161307 − 0.986904i \(-0.448429\pi\)
0.161307 + 0.986904i \(0.448429\pi\)
\(72\) −7.45388e18 −0.234634
\(73\) 3.66844e19 0.999060 0.499530 − 0.866297i \(-0.333506\pi\)
0.499530 + 0.866297i \(0.333506\pi\)
\(74\) 3.18771e19 0.752569
\(75\) −2.68158e19 −0.549853
\(76\) 1.15596e19 0.206252
\(77\) −1.52403e20 −2.37051
\(78\) −9.12008e18 −0.123881
\(79\) 3.38406e19 0.402118 0.201059 − 0.979579i \(-0.435562\pi\)
0.201059 + 0.979579i \(0.435562\pi\)
\(80\) 5.47046e18 0.0569611
\(81\) 1.13873e19 0.104071
\(82\) −2.52036e19 −0.202497
\(83\) 2.04215e20 1.44466 0.722332 − 0.691547i \(-0.243070\pi\)
0.722332 + 0.691547i \(0.243070\pi\)
\(84\) 8.87821e19 0.553851
\(85\) −5.57437e19 −0.307112
\(86\) −1.36587e20 −0.665544
\(87\) 1.41313e20 0.609858
\(88\) −1.14641e20 −0.438806
\(89\) −4.10241e19 −0.139458 −0.0697290 − 0.997566i \(-0.522213\pi\)
−0.0697290 + 0.997566i \(0.522213\pi\)
\(90\) −3.53676e19 −0.106920
\(91\) −2.14329e20 −0.576960
\(92\) 1.35794e20 0.325917
\(93\) −5.21122e19 −0.111653
\(94\) −1.97145e20 −0.377522
\(95\) 5.48485e19 0.0939869
\(96\) 6.67839e19 0.102524
\(97\) −7.27592e20 −1.00181 −0.500905 − 0.865503i \(-0.666999\pi\)
−0.500905 + 0.865503i \(0.666999\pi\)
\(98\) 1.51449e21 1.87238
\(99\) 7.41179e20 0.823671
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2.22.a.b.1.1 ✓ 1
3.2 odd 2 18.22.a.b.1.1 1
4.3 odd 2 16.22.a.b.1.1 1
5.2 odd 4 50.22.b.c.49.2 2
5.3 odd 4 50.22.b.c.49.1 2
5.4 even 2 50.22.a.a.1.1 1
8.3 odd 2 64.22.a.e.1.1 1
8.5 even 2 64.22.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.b.1.1 ✓ 1 1.1 even 1 trivial
16.22.a.b.1.1 1 4.3 odd 2
18.22.a.b.1.1 1 3.2 odd 2
50.22.a.a.1.1 1 5.4 even 2
50.22.b.c.49.1 2 5.3 odd 4
50.22.b.c.49.2 2 5.2 odd 4
64.22.a.c.1.1 1 8.5 even 2
64.22.a.e.1.1 1 8.3 odd 2