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Properties

Label 12.18.a.b.1.1
Level $12$
Weight $18$
Character 12.1
Self dual yes
Analytic conductor $21.987$
Analytic rank $1$
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] = [12,18,Mod(1,12)] 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("12.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,6561] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(21.9866504813\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 12.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6561.00 q^{3} +130950. q^{5} -1.48468e7 q^{7} +4.30467e7 q^{9} -8.45470e8 q^{11} +1.75141e9 q^{13} +8.59163e8 q^{15} -4.71479e7 q^{17} -5.69736e10 q^{19} -9.74097e10 q^{21} -3.71395e11 q^{23} -7.45792e11 q^{25} +2.82430e11 q^{27} -3.68117e12 q^{29} -5.47989e12 q^{31} -5.54713e12 q^{33} -1.94419e12 q^{35} -5.44696e12 q^{37} +1.14910e13 q^{39} +2.97733e13 q^{41} +9.84859e13 q^{43} +5.63697e12 q^{45} +1.07862e14 q^{47} -1.22038e13 q^{49} -3.09337e11 q^{51} +6.26473e14 q^{53} -1.10714e14 q^{55} -3.73804e14 q^{57} -1.26097e15 q^{59} -9.56343e14 q^{61} -6.39105e14 q^{63} +2.29348e14 q^{65} -5.51939e15 q^{67} -2.43673e15 q^{69} +9.30305e15 q^{71} +3.69259e15 q^{73} -4.89314e15 q^{75} +1.25525e16 q^{77} -2.59772e15 q^{79} +1.85302e15 q^{81} +2.62667e16 q^{83} -6.17402e12 q^{85} -2.41521e16 q^{87} +6.37172e16 q^{89} -2.60029e16 q^{91} -3.59536e16 q^{93} -7.46069e15 q^{95} -7.85589e16 q^{97} -3.63947e16 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 6561.00 0.577350
\(4\) 0 0
\(5\) 130950. 0.149920 0.0749602 − 0.997187i \(-0.476117\pi\)
0.0749602 + 0.997187i \(0.476117\pi\)
\(6\) 0 0
\(7\) −1.48468e7 −0.973417 −0.486708 − 0.873565i \(-0.661803\pi\)
−0.486708 + 0.873565i \(0.661803\pi\)
\(8\) 0 0
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −8.45470e8 −1.18921 −0.594607 − 0.804016i \(-0.702692\pi\)
−0.594607 + 0.804016i \(0.702692\pi\)
\(12\) 0 0
\(13\) 1.75141e9 0.595485 0.297742 − 0.954646i \(-0.403766\pi\)
0.297742 + 0.954646i \(0.403766\pi\)
\(14\) 0 0
\(15\) 8.59163e8 0.0865565
\(16\) 0 0
\(17\) −4.71479e7 −0.00163925 −0.000819627 − 1.00000i \(-0.500261\pi\)
−0.000819627 1.00000i \(0.500261\pi\)
\(18\) 0 0
\(19\) −5.69736e10 −0.769605 −0.384802 − 0.922999i \(-0.625730\pi\)
−0.384802 + 0.922999i \(0.625730\pi\)
\(20\) 0 0
\(21\) −9.74097e10 −0.562002
\(22\) 0 0
\(23\) −3.71395e11 −0.988894 −0.494447 − 0.869208i \(-0.664629\pi\)
−0.494447 + 0.869208i \(0.664629\pi\)
\(24\) 0 0
\(25\) −7.45792e11 −0.977524
\(26\) 0 0
\(27\) 2.82430e11 0.192450
\(28\) 0 0
\(29\) −3.68117e12 −1.36648 −0.683239 − 0.730195i \(-0.739429\pi\)
−0.683239 + 0.730195i \(0.739429\pi\)
\(30\) 0 0
\(31\) −5.47989e12 −1.15398 −0.576989 − 0.816752i \(-0.695772\pi\)
−0.576989 + 0.816752i \(0.695772\pi\)
\(32\) 0 0
\(33\) −5.54713e12 −0.686593
\(34\) 0 0
\(35\) −1.94419e12 −0.145935
\(36\) 0 0
\(37\) −5.44696e12 −0.254941 −0.127470 − 0.991842i \(-0.540686\pi\)
−0.127470 + 0.991842i \(0.540686\pi\)
\(38\) 0 0
\(39\) 1.14910e13 0.343803
\(40\) 0 0
\(41\) 2.97733e13 0.582324 0.291162 − 0.956674i \(-0.405958\pi\)
0.291162 + 0.956674i \(0.405958\pi\)
\(42\) 0 0
\(43\) 9.84859e13 1.28497 0.642484 − 0.766299i \(-0.277904\pi\)
0.642484 + 0.766299i \(0.277904\pi\)
\(44\) 0 0
\(45\) 5.63697e12 0.0499734
\(46\) 0 0
\(47\) 1.07862e14 0.660748 0.330374 − 0.943850i \(-0.392825\pi\)
0.330374 + 0.943850i \(0.392825\pi\)
\(48\) 0 0
\(49\) −1.22038e13 −0.0524598
\(50\) 0 0
\(51\) −3.09337e11 −0.000946424 0
\(52\) 0 0
\(53\) 6.26473e14 1.38216 0.691079 − 0.722780i \(-0.257136\pi\)
0.691079 + 0.722780i \(0.257136\pi\)
\(54\) 0 0
\(55\) −1.10714e14 −0.178287
\(56\) 0 0
\(57\) −3.73804e14 −0.444332
\(58\) 0 0
\(59\) −1.26097e15 −1.11805 −0.559027 − 0.829149i \(-0.688825\pi\)
−0.559027 + 0.829149i \(0.688825\pi\)
\(60\) 0 0
\(61\) −9.56343e14 −0.638719 −0.319360 − 0.947634i \(-0.603468\pi\)
−0.319360 + 0.947634i \(0.603468\pi\)
\(62\) 0 0
\(63\) −6.39105e14 −0.324472
\(64\) 0 0
\(65\) 2.29348e14 0.0892752
\(66\) 0 0
\(67\) −5.51939e15 −1.66056 −0.830281 − 0.557345i \(-0.811820\pi\)
−0.830281 + 0.557345i \(0.811820\pi\)
\(68\) 0 0
\(69\) −2.43673e15 −0.570938
\(70\) 0 0
\(71\) 9.30305e15 1.70974 0.854869 − 0.518844i \(-0.173637\pi\)
0.854869 + 0.518844i \(0.173637\pi\)
\(72\) 0 0
\(73\) 3.69259e15 0.535904 0.267952 − 0.963432i \(-0.413653\pi\)
0.267952 + 0.963432i \(0.413653\pi\)
\(74\) 0 0
\(75\) −4.89314e15 −0.564374
\(76\) 0 0
\(77\) 1.25525e16 1.15760
\(78\) 0 0
\(79\) −2.59772e15 −0.192647 −0.0963235 − 0.995350i \(-0.530708\pi\)
−0.0963235 + 0.995350i \(0.530708\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 0 0
\(83\) 2.62667e16 1.28010 0.640048 − 0.768335i \(-0.278915\pi\)
0.640048 + 0.768335i \(0.278915\pi\)
\(84\) 0 0
\(85\) −6.17402e12 −0.000245758 0
\(86\) 0 0
\(87\) −2.41521e16 −0.788937
\(88\) 0 0
\(89\) 6.37172e16 1.71570 0.857850 − 0.513900i \(-0.171800\pi\)
0.857850 + 0.513900i \(0.171800\pi\)
\(90\) 0 0
\(91\) −2.60029e16 −0.579655
\(92\) 0 0
\(93\) −3.59536e16 −0.666249
\(94\) 0 0
\(95\) −7.46069e15 −0.115379
\(96\) 0 0
\(97\) −7.85589e16 −1.01774 −0.508869 − 0.860844i \(-0.669936\pi\)
−0.508869 + 0.860844i \(0.669936\pi\)
\(98\) 0 0
\(99\) −3.63947e16 −0.396405
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 12.18.a.b.1.1 ✓ 1
3.2 odd 2 36.18.a.a.1.1 1
4.3 odd 2 48.18.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.18.a.b.1.1 ✓ 1 1.1 even 1 trivial
36.18.a.a.1.1 1 3.2 odd 2
48.18.a.b.1.1 1 4.3 odd 2