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Properties

Label 36.14.a.d.1.1
Level $36$
Weight $14$
Character 36.1
Self dual yes
Analytic conductor $38.603$
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] = [36,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,24570] 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: \(38.6031344227\)
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)
 
\(f(q)\) \(=\) \(q+24570.0 q^{5} -173704. q^{7} +970164. q^{11} -2.41494e7 q^{13} +1.57098e8 q^{17} -1.19525e8 q^{19} +9.49750e7 q^{23} -6.17018e8 q^{25} -4.97957e9 q^{29} +5.63827e9 q^{31} -4.26791e9 q^{35} -5.88141e9 q^{37} -2.57538e10 q^{41} -6.84564e10 q^{43} -2.96176e9 q^{47} -6.67159e10 q^{49} -3.12743e11 q^{53} +2.38369e10 q^{55} -4.61474e11 q^{59} +2.83119e11 q^{61} -5.93351e11 q^{65} -1.30344e12 q^{67} +1.26398e12 q^{71} +5.94014e11 q^{73} -1.68521e11 q^{77} -1.15379e12 q^{79} +4.82038e12 q^{83} +3.85990e12 q^{85} -7.28549e11 q^{89} +4.19485e12 q^{91} -2.93672e12 q^{95} +2.58874e12 q^{97} +O(q^{100})\)

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
\(5\) 24570.0 0.703234 0.351617 − 0.936144i \(-0.385632\pi\)
0.351617 + 0.936144i \(0.385632\pi\)
\(6\) 0 0
\(7\) −173704. −0.558049 −0.279025 − 0.960284i \(-0.590011\pi\)
−0.279025 + 0.960284i \(0.590011\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 970164. 0.165117 0.0825587 − 0.996586i \(-0.473691\pi\)
0.0825587 + 0.996586i \(0.473691\pi\)
\(12\) 0 0
\(13\) −2.41494e7 −1.38763 −0.693817 − 0.720152i \(-0.744072\pi\)
−0.693817 + 0.720152i \(0.744072\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.57098e8 1.57853 0.789264 − 0.614054i \(-0.210462\pi\)
0.789264 + 0.614054i \(0.210462\pi\)
\(18\) 0 0
\(19\) −1.19525e8 −0.582854 −0.291427 − 0.956593i \(-0.594130\pi\)
−0.291427 + 0.956593i \(0.594130\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.49750e7 0.133776 0.0668880 − 0.997760i \(-0.478693\pi\)
0.0668880 + 0.997760i \(0.478693\pi\)
\(24\) 0 0
\(25\) −6.17018e8 −0.505461
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.97957e9 −1.55455 −0.777276 − 0.629160i \(-0.783399\pi\)
−0.777276 + 0.629160i \(0.783399\pi\)
\(30\) 0 0
\(31\) 5.63827e9 1.14103 0.570513 − 0.821289i \(-0.306745\pi\)
0.570513 + 0.821289i \(0.306745\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.26791e9 −0.392439
\(36\) 0 0
\(37\) −5.88141e9 −0.376852 −0.188426 − 0.982087i \(-0.560338\pi\)
−0.188426 + 0.982087i \(0.560338\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.57538e10 −0.846734 −0.423367 − 0.905958i \(-0.639152\pi\)
−0.423367 + 0.905958i \(0.639152\pi\)
\(42\) 0 0
\(43\) −6.84564e10 −1.65146 −0.825732 − 0.564063i \(-0.809237\pi\)
−0.825732 + 0.564063i \(0.809237\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.96176e9 −0.0400787 −0.0200394 − 0.999799i \(-0.506379\pi\)
−0.0200394 + 0.999799i \(0.506379\pi\)
\(48\) 0 0
\(49\) −6.67159e10 −0.688581
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.12743e11 −1.93818 −0.969090 − 0.246708i \(-0.920651\pi\)
−0.969090 + 0.246708i \(0.920651\pi\)
\(54\) 0 0
\(55\) 2.38369e10 0.116116
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.61474e11 −1.42433 −0.712163 − 0.702014i \(-0.752284\pi\)
−0.712163 + 0.702014i \(0.752284\pi\)
\(60\) 0 0
\(61\) 2.83119e11 0.703599 0.351800 − 0.936075i \(-0.385570\pi\)
0.351800 + 0.936075i \(0.385570\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.93351e11 −0.975832
\(66\) 0 0
\(67\) −1.30344e12 −1.76037 −0.880186 − 0.474629i \(-0.842582\pi\)
−0.880186 + 0.474629i \(0.842582\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.26398e12 1.17101 0.585507 − 0.810667i \(-0.300895\pi\)
0.585507 + 0.810667i \(0.300895\pi\)
\(72\) 0 0
\(73\) 5.94014e11 0.459408 0.229704 − 0.973261i \(-0.426224\pi\)
0.229704 + 0.973261i \(0.426224\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.68521e11 −0.0921436
\(78\) 0 0
\(79\) −1.15379e12 −0.534013 −0.267007 − 0.963695i \(-0.586035\pi\)
−0.267007 + 0.963695i \(0.586035\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.82038e12 1.61835 0.809177 − 0.587565i \(-0.199913\pi\)
0.809177 + 0.587565i \(0.199913\pi\)
\(84\) 0 0
\(85\) 3.85990e12 1.11008
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.28549e11 −0.155390 −0.0776951 − 0.996977i \(-0.524756\pi\)
−0.0776951 + 0.996977i \(0.524756\pi\)
\(90\) 0 0
\(91\) 4.19485e12 0.774368
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.93672e12 −0.409883
\(96\) 0 0
\(97\) 2.58874e12 0.315552 0.157776 − 0.987475i \(-0.449568\pi\)
0.157776 + 0.987475i \(0.449568\pi\)
\(98\) 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 36.14.a.d.1.1 1
3.2 odd 2 12.14.a.b.1.1 ✓ 1
4.3 odd 2 144.14.a.j.1.1 1
12.11 even 2 48.14.a.a.1.1 1
24.5 odd 2 192.14.a.d.1.1 1
24.11 even 2 192.14.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.14.a.b.1.1 ✓ 1 3.2 odd 2
36.14.a.d.1.1 1 1.1 even 1 trivial
48.14.a.a.1.1 1 12.11 even 2
144.14.a.j.1.1 1 4.3 odd 2
192.14.a.d.1.1 1 24.5 odd 2
192.14.a.i.1.1 1 24.11 even 2