Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [36,18,Mod(1,36)] 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("36.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-130950] 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: \(65.9599514440\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 36.1

$q$-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)
 
\(f(q)\) \(=\) \(q-130950. q^{5} -1.48468e7 q^{7} +8.45470e8 q^{11} +1.75141e9 q^{13} +4.71479e7 q^{17} -5.69736e10 q^{19} +3.71395e11 q^{23} -7.45792e11 q^{25} +3.68117e12 q^{29} -5.47989e12 q^{31} +1.94419e12 q^{35} -5.44696e12 q^{37} -2.97733e13 q^{41} +9.84859e13 q^{43} -1.07862e14 q^{47} -1.22038e13 q^{49} -6.26473e14 q^{53} -1.10714e14 q^{55} +1.26097e15 q^{59} -9.56343e14 q^{61} -2.29348e14 q^{65} -5.51939e15 q^{67} -9.30305e15 q^{71} +3.69259e15 q^{73} -1.25525e16 q^{77} -2.59772e15 q^{79} -2.62667e16 q^{83} -6.17402e12 q^{85} -6.37172e16 q^{89} -2.60029e16 q^{91} +7.46069e15 q^{95} -7.85589e16 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −130950. −0.149920 −0.0749602 − 0.997187i \(-0.523883\pi\)
−0.0749602 + 0.997187i \(0.523883\pi\)
\(6\) 0 0
\(7\) −1.48468e7 −0.973417 −0.486708 − 0.873565i \(-0.661803\pi\)
−0.486708 + 0.873565i \(0.661803\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8.45470e8 1.18921 0.594607 − 0.804016i \(-0.297308\pi\)
0.594607 + 0.804016i \(0.297308\pi\)
\(12\) 0 0
\(13\) 1.75141e9 0.595485 0.297742 − 0.954646i \(-0.403766\pi\)
0.297742 + 0.954646i \(0.403766\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.71479e7 0.00163925 0.000819627 − 1.00000i \(-0.499739\pi\)
0.000819627 1.00000i \(0.499739\pi\)
\(18\) 0 0
\(19\) −5.69736e10 −0.769605 −0.384802 − 0.922999i \(-0.625730\pi\)
−0.384802 + 0.922999i \(0.625730\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.71395e11 0.988894 0.494447 − 0.869208i \(-0.335371\pi\)
0.494447 + 0.869208i \(0.335371\pi\)
\(24\) 0 0
\(25\) −7.45792e11 −0.977524
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.68117e12 1.36648 0.683239 − 0.730195i \(-0.260571\pi\)
0.683239 + 0.730195i \(0.260571\pi\)
\(30\) 0 0
\(31\) −5.47989e12 −1.15398 −0.576989 − 0.816752i \(-0.695772\pi\)
−0.576989 + 0.816752i \(0.695772\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) −2.97733e13 −0.582324 −0.291162 − 0.956674i \(-0.594042\pi\)
−0.291162 + 0.956674i \(0.594042\pi\)
\(42\) 0 0
\(43\) 9.84859e13 1.28497 0.642484 − 0.766299i \(-0.277904\pi\)
0.642484 + 0.766299i \(0.277904\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.07862e14 −0.660748 −0.330374 − 0.943850i \(-0.607175\pi\)
−0.330374 + 0.943850i \(0.607175\pi\)
\(48\) 0 0
\(49\) −1.22038e13 −0.0524598
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.26473e14 −1.38216 −0.691079 − 0.722780i \(-0.742864\pi\)
−0.691079 + 0.722780i \(0.742864\pi\)
\(54\) 0 0
\(55\) −1.10714e14 −0.178287
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.26097e15 1.11805 0.559027 − 0.829149i \(-0.311175\pi\)
0.559027 + 0.829149i \(0.311175\pi\)
\(60\) 0 0
\(61\) −9.56343e14 −0.638719 −0.319360 − 0.947634i \(-0.603468\pi\)
−0.319360 + 0.947634i \(0.603468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) −9.30305e15 −1.70974 −0.854869 − 0.518844i \(-0.826363\pi\)
−0.854869 + 0.518844i \(0.826363\pi\)
\(72\) 0 0
\(73\) 3.69259e15 0.535904 0.267952 − 0.963432i \(-0.413653\pi\)
0.267952 + 0.963432i \(0.413653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) −2.62667e16 −1.28010 −0.640048 − 0.768335i \(-0.721085\pi\)
−0.640048 + 0.768335i \(0.721085\pi\)
\(84\) 0 0
\(85\) −6.17402e12 −0.000245758 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.37172e16 −1.71570 −0.857850 − 0.513900i \(-0.828200\pi\)
−0.857850 + 0.513900i \(0.828200\pi\)
\(90\) 0 0
\(91\) −2.60029e16 −0.579655
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 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 36.18.a.a.1.1 1
3.2 odd 2 12.18.a.b.1.1 ✓ 1
12.11 even 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 3.2 odd 2
36.18.a.a.1.1 1 1.1 even 1 trivial
48.18.a.b.1.1 1 12.11 even 2