Properties

Label 18.22.a.b.1.1
Level $18$
Weight $22$
Character 18.1
Self dual yes
Analytic conductor $50.306$
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] = [18,22,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1024,0,1048576,-4975350] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(50.3059219717\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 18.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +1.04858e6 q^{4} -4.97535e6 q^{5} +1.42743e9 q^{7} -1.07374e9 q^{8} +5.09476e9 q^{10} +1.06768e11 q^{11} -1.50151e11 q^{13} -1.46168e12 q^{14} +1.09951e12 q^{16} +1.12040e13 q^{17} +1.10241e13 q^{19} -5.21703e12 q^{20} -1.09330e14 q^{22} -1.29503e14 q^{23} -4.52083e14 q^{25} +1.53754e14 q^{26} +1.49676e15 q^{28} -2.38237e15 q^{29} -8.78553e14 q^{31} -1.12590e15 q^{32} -1.14729e16 q^{34} -7.10194e15 q^{35} +3.11300e16 q^{37} -1.12886e16 q^{38} +5.34224e15 q^{40} +2.46129e16 q^{41} -1.33386e17 q^{43} +1.11954e17 q^{44} +1.32611e17 q^{46} +1.92524e17 q^{47} +1.47900e18 q^{49} +4.62933e17 q^{50} -1.57444e17 q^{52} +5.94166e17 q^{53} -5.31208e17 q^{55} -1.53269e18 q^{56} +2.43955e18 q^{58} +2.95595e18 q^{59} +7.98415e18 q^{61} +8.99638e17 q^{62} +1.15292e18 q^{64} +7.47052e17 q^{65} +4.83704e18 q^{67} +1.17482e19 q^{68} +7.27239e18 q^{70} -8.84902e18 q^{71} +3.66844e19 q^{73} -3.18771e19 q^{74} +1.15596e19 q^{76} +1.52403e20 q^{77} +3.38406e19 q^{79} -5.47046e18 q^{80} -2.52036e19 q^{82} -2.04215e20 q^{83} -5.57437e19 q^{85} +1.36587e20 q^{86} -1.14641e20 q^{88} +4.10241e19 q^{89} -2.14329e20 q^{91} -1.35794e20 q^{92} -1.97145e20 q^{94} -5.48485e19 q^{95} -7.27592e20 q^{97} -1.51449e21 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.04858e6 0.500000
\(5\) −4.97535e6 −0.227845 −0.113922 − 0.993490i \(-0.536341\pi\)
−0.113922 + 0.993490i \(0.536341\pi\)
\(6\) 0 0
\(7\) 1.42743e9 1.90996 0.954980 − 0.296671i \(-0.0958765\pi\)
0.954980 + 0.296671i \(0.0958765\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 0 0
\(10\) 5.09476e9 0.161110
\(11\) 1.06768e11 1.24113 0.620565 − 0.784155i \(-0.286903\pi\)
0.620565 + 0.784155i \(0.286903\pi\)
\(12\) 0 0
\(13\) −1.50151e11 −0.302080 −0.151040 − 0.988528i \(-0.548262\pi\)
−0.151040 + 0.988528i \(0.548262\pi\)
\(14\) −1.46168e12 −1.35055
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.12040e13 1.34790 0.673952 − 0.738776i \(-0.264596\pi\)
0.673952 + 0.738776i \(0.264596\pi\)
\(18\) 0 0
\(19\) 1.10241e13 0.412504 0.206252 − 0.978499i \(-0.433873\pi\)
0.206252 + 0.978499i \(0.433873\pi\)
\(20\) −5.21703e12 −0.113922
\(21\) 0 0
\(22\) −1.09330e14 −0.877612
\(23\) −1.29503e14 −0.651834 −0.325917 − 0.945398i \(-0.605673\pi\)
−0.325917 + 0.945398i \(0.605673\pi\)
\(24\) 0 0
\(25\) −4.52083e14 −0.948087
\(26\) 1.53754e14 0.213603
\(27\) 0 0
\(28\) 1.49676e15 0.954980
\(29\) −2.38237e15 −1.05155 −0.525776 − 0.850623i \(-0.676225\pi\)
−0.525776 + 0.850623i \(0.676225\pi\)
\(30\) 0 0
\(31\) −8.78553e14 −0.192517 −0.0962587 − 0.995356i \(-0.530688\pi\)
−0.0962587 + 0.995356i \(0.530688\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 0 0
\(34\) −1.14729e16 −0.953111
\(35\) −7.10194e15 −0.435174
\(36\) 0 0
\(37\) 3.11300e16 1.06429 0.532146 − 0.846652i \(-0.321386\pi\)
0.532146 + 0.846652i \(0.321386\pi\)
\(38\) −1.12886e16 −0.291685
\(39\) 0 0
\(40\) 5.34224e15 0.0805552
\(41\) 2.46129e16 0.286373 0.143187 − 0.989696i \(-0.454265\pi\)
0.143187 + 0.989696i \(0.454265\pi\)
\(42\) 0 0
\(43\) −1.33386e17 −0.941222 −0.470611 − 0.882341i \(-0.655966\pi\)
−0.470611 + 0.882341i \(0.655966\pi\)
\(44\) 1.11954e17 0.620565
\(45\) 0 0
\(46\) 1.32611e17 0.460916
\(47\) 1.92524e17 0.533897 0.266948 − 0.963711i \(-0.413985\pi\)
0.266948 + 0.963711i \(0.413985\pi\)
\(48\) 0 0
\(49\) 1.47900e18 2.64794
\(50\) 4.62933e17 0.670399
\(51\) 0 0
\(52\) −1.57444e17 −0.151040
\(53\) 5.94166e17 0.466672 0.233336 − 0.972396i \(-0.425036\pi\)
0.233336 + 0.972396i \(0.425036\pi\)
\(54\) 0 0
\(55\) −5.31208e17 −0.282785
\(56\) −1.53269e18 −0.675273
\(57\) 0 0
\(58\) 2.43955e18 0.743559
\(59\) 2.95595e18 0.752925 0.376462 − 0.926432i \(-0.377140\pi\)
0.376462 + 0.926432i \(0.377140\pi\)
\(60\) 0 0
\(61\) 7.98415e18 1.43306 0.716532 − 0.697554i \(-0.245728\pi\)
0.716532 + 0.697554i \(0.245728\pi\)
\(62\) 8.99638e17 0.136130
\(63\) 0 0
\(64\) 1.15292e18 0.125000
\(65\) 7.47052e17 0.0688272
\(66\) 0 0
\(67\) 4.83704e18 0.324186 0.162093 − 0.986775i \(-0.448176\pi\)
0.162093 + 0.986775i \(0.448176\pi\)
\(68\) 1.17482e19 0.673952
\(69\) 0 0
\(70\) 7.27239e18 0.307714
\(71\) −8.84902e18 −0.322613 −0.161307 − 0.986904i \(-0.551571\pi\)
−0.161307 + 0.986904i \(0.551571\pi\)
\(72\) 0 0
\(73\) 3.66844e19 0.999060 0.499530 − 0.866297i \(-0.333506\pi\)
0.499530 + 0.866297i \(0.333506\pi\)
\(74\) −3.18771e19 −0.752569
\(75\) 0 0
\(76\) 1.15596e19 0.206252
\(77\) 1.52403e20 2.37051
\(78\) 0 0
\(79\) 3.38406e19 0.402118 0.201059 − 0.979579i \(-0.435562\pi\)
0.201059 + 0.979579i \(0.435562\pi\)
\(80\) −5.47046e18 −0.0569611
\(81\) 0 0
\(82\) −2.52036e19 −0.202497
\(83\) −2.04215e20 −1.44466 −0.722332 − 0.691547i \(-0.756930\pi\)
−0.722332 + 0.691547i \(0.756930\pi\)
\(84\) 0 0
\(85\) −5.57437e19 −0.307112
\(86\) 1.36587e20 0.665544
\(87\) 0 0
\(88\) −1.14641e20 −0.438806
\(89\) 4.10241e19 0.139458 0.0697290 − 0.997566i \(-0.477787\pi\)
0.0697290 + 0.997566i \(0.477787\pi\)
\(90\) 0 0
\(91\) −2.14329e20 −0.576960
\(92\) −1.35794e20 −0.325917
\(93\) 0 0
\(94\) −1.97145e20 −0.377522
\(95\) −5.48485e19 −0.0939869
\(96\) 0 0
\(97\) −7.27592e20 −1.00181 −0.500905 − 0.865503i \(-0.666999\pi\)
−0.500905 + 0.865503i \(0.666999\pi\)
\(98\) −1.51449e21 −1.87238
\(99\) 0 0
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 18.22.a.b.1.1 1
3.2 odd 2 2.22.a.b.1.1 ✓ 1
12.11 even 2 16.22.a.b.1.1 1
15.2 even 4 50.22.b.c.49.2 2
15.8 even 4 50.22.b.c.49.1 2
15.14 odd 2 50.22.a.a.1.1 1
24.5 odd 2 64.22.a.c.1.1 1
24.11 even 2 64.22.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.b.1.1 ✓ 1 3.2 odd 2
16.22.a.b.1.1 1 12.11 even 2
18.22.a.b.1.1 1 1.1 even 1 trivial
50.22.a.a.1.1 1 15.14 odd 2
50.22.b.c.49.1 2 15.8 even 4
50.22.b.c.49.2 2 15.2 even 4
64.22.a.c.1.1 1 24.5 odd 2
64.22.a.e.1.1 1 24.11 even 2