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Properties

Label 36.18.a.c.1.1
Level $36$
Weight $18$
Character 36.1
Self dual yes
Analytic conductor $65.960$
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] = [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 = [1,0,0,0,1608930] 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)
 
\(f(q)\) \(=\) \(q+1.60893e6 q^{5} -9.41718e6 q^{7} +1.86911e8 q^{11} -2.62544e9 q^{13} -4.37823e10 q^{17} -9.65950e10 q^{19} -2.90868e11 q^{23} +1.82572e12 q^{25} -1.39862e12 q^{29} +7.64790e12 q^{31} -1.51516e13 q^{35} -3.33695e13 q^{37} +1.20327e13 q^{41} -7.55092e11 q^{43} +2.80540e14 q^{47} -1.43947e14 q^{49} -4.60570e14 q^{53} +3.00726e14 q^{55} -1.07847e15 q^{59} -1.98078e15 q^{61} -4.22415e15 q^{65} +4.85019e15 q^{67} -2.70757e15 q^{71} -5.00226e15 q^{73} -1.76017e15 q^{77} -9.77448e15 q^{79} -1.71129e16 q^{83} -7.04427e16 q^{85} -3.46982e16 q^{89} +2.47243e16 q^{91} -1.55415e17 q^{95} +6.86169e16 q^{97} +O(q^{100})\)

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
\(5\) 1.60893e6 1.84201 0.921005 − 0.389550i \(-0.127369\pi\)
0.921005 + 0.389550i \(0.127369\pi\)
\(6\) 0 0
\(7\) −9.41718e6 −0.617430 −0.308715 − 0.951155i \(-0.599899\pi\)
−0.308715 + 0.951155i \(0.599899\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.86911e8 0.262903 0.131452 − 0.991323i \(-0.458036\pi\)
0.131452 + 0.991323i \(0.458036\pi\)
\(12\) 0 0
\(13\) −2.62544e9 −0.892656 −0.446328 − 0.894869i \(-0.647268\pi\)
−0.446328 + 0.894869i \(0.647268\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.37823e10 −1.52224 −0.761120 − 0.648612i \(-0.775350\pi\)
−0.761120 + 0.648612i \(0.775350\pi\)
\(18\) 0 0
\(19\) −9.65950e10 −1.30482 −0.652408 − 0.757868i \(-0.726241\pi\)
−0.652408 + 0.757868i \(0.726241\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.90868e11 −0.774478 −0.387239 − 0.921979i \(-0.626571\pi\)
−0.387239 + 0.921979i \(0.626571\pi\)
\(24\) 0 0
\(25\) 1.82572e12 2.39300
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.39862e12 −0.519178 −0.259589 − 0.965719i \(-0.583587\pi\)
−0.259589 + 0.965719i \(0.583587\pi\)
\(30\) 0 0
\(31\) 7.64790e12 1.61053 0.805263 − 0.592918i \(-0.202024\pi\)
0.805263 + 0.592918i \(0.202024\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.51516e13 −1.13731
\(36\) 0 0
\(37\) −3.33695e13 −1.56184 −0.780918 − 0.624634i \(-0.785248\pi\)
−0.780918 + 0.624634i \(0.785248\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.20327e13 0.235343 0.117672 − 0.993053i \(-0.462457\pi\)
0.117672 + 0.993053i \(0.462457\pi\)
\(42\) 0 0
\(43\) −7.55092e11 −0.00985186 −0.00492593 − 0.999988i \(-0.501568\pi\)
−0.00492593 + 0.999988i \(0.501568\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.80540e14 1.71856 0.859278 − 0.511509i \(-0.170913\pi\)
0.859278 + 0.511509i \(0.170913\pi\)
\(48\) 0 0
\(49\) −1.43947e14 −0.618780
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.60570e14 −1.01613 −0.508067 − 0.861318i \(-0.669640\pi\)
−0.508067 + 0.861318i \(0.669640\pi\)
\(54\) 0 0
\(55\) 3.00726e14 0.484271
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.07847e15 −0.956236 −0.478118 − 0.878296i \(-0.658681\pi\)
−0.478118 + 0.878296i \(0.658681\pi\)
\(60\) 0 0
\(61\) −1.98078e15 −1.32292 −0.661458 − 0.749982i \(-0.730062\pi\)
−0.661458 + 0.749982i \(0.730062\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.22415e15 −1.64428
\(66\) 0 0
\(67\) 4.85019e15 1.45923 0.729614 − 0.683860i \(-0.239700\pi\)
0.729614 + 0.683860i \(0.239700\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.70757e15 −0.497605 −0.248802 − 0.968554i \(-0.580037\pi\)
−0.248802 + 0.968554i \(0.580037\pi\)
\(72\) 0 0
\(73\) −5.00226e15 −0.725976 −0.362988 − 0.931794i \(-0.618243\pi\)
−0.362988 + 0.931794i \(0.618243\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.76017e15 −0.162324
\(78\) 0 0
\(79\) −9.77448e15 −0.724875 −0.362438 − 0.932008i \(-0.618055\pi\)
−0.362438 + 0.932008i \(0.618055\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.71129e16 −0.833989 −0.416994 − 0.908909i \(-0.636917\pi\)
−0.416994 + 0.908909i \(0.636917\pi\)
\(84\) 0 0
\(85\) −7.04427e16 −2.80398
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.46982e16 −0.934311 −0.467156 − 0.884175i \(-0.654721\pi\)
−0.467156 + 0.884175i \(0.654721\pi\)
\(90\) 0 0
\(91\) 2.47243e16 0.551152
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.55415e17 −2.40348
\(96\) 0 0
\(97\) 6.86169e16 0.888938 0.444469 − 0.895794i \(-0.353392\pi\)
0.444469 + 0.895794i \(0.353392\pi\)
\(98\) 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.c.1.1 1
3.2 odd 2 12.18.a.a.1.1 ✓ 1
12.11 even 2 48.18.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.18.a.a.1.1 ✓ 1 3.2 odd 2
36.18.a.c.1.1 1 1.1 even 1 trivial
48.18.a.c.1.1 1 12.11 even 2