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Properties

Label 48.14.a.a.1.1
Level $48$
Weight $14$
Character 48.1
Self dual yes
Analytic conductor $51.471$
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] = [48,14,Mod(1,48)] 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("48.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-729,0,-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: \(51.4708458969\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 48.1

$q$-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)
 
\(f(q)\) \(=\) \(q-729.000 q^{3} -24570.0 q^{5} +173704. q^{7} +531441. q^{9} +970164. q^{11} -2.41494e7 q^{13} +1.79115e7 q^{15} -1.57098e8 q^{17} +1.19525e8 q^{19} -1.26630e8 q^{21} +9.49750e7 q^{23} -6.17018e8 q^{25} -3.87420e8 q^{27} +4.97957e9 q^{29} -5.63827e9 q^{31} -7.07250e8 q^{33} -4.26791e9 q^{35} -5.88141e9 q^{37} +1.76049e10 q^{39} +2.57538e10 q^{41} +6.84564e10 q^{43} -1.30575e10 q^{45} -2.96176e9 q^{47} -6.67159e10 q^{49} +1.14524e11 q^{51} +3.12743e11 q^{53} -2.38369e10 q^{55} -8.71336e10 q^{57} -4.61474e11 q^{59} +2.83119e11 q^{61} +9.23134e10 q^{63} +5.93351e11 q^{65} +1.30344e12 q^{67} -6.92368e10 q^{69} +1.26398e12 q^{71} +5.94014e11 q^{73} +4.49806e11 q^{75} +1.68521e11 q^{77} +1.15379e12 q^{79} +2.82430e11 q^{81} +4.82038e12 q^{83} +3.85990e12 q^{85} -3.63011e12 q^{87} +7.28549e11 q^{89} -4.19485e12 q^{91} +4.11030e12 q^{93} -2.93672e12 q^{95} +2.58874e12 q^{97} +5.15585e11 q^{99} +O(q^{100})\)

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
\(3\) −729.000 −0.577350
\(4\) 0 0
\(5\) −24570.0 −0.703234 −0.351617 − 0.936144i \(-0.614368\pi\)
−0.351617 + 0.936144i \(0.614368\pi\)
\(6\) 0 0
\(7\) 173704. 0.558049 0.279025 − 0.960284i \(-0.409989\pi\)
0.279025 + 0.960284i \(0.409989\pi\)
\(8\) 0 0
\(9\) 531441. 0.333333
\(10\) 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
\(15\) 1.79115e7 0.406013
\(16\) 0 0
\(17\) −1.57098e8 −1.57853 −0.789264 − 0.614054i \(-0.789538\pi\)
−0.789264 + 0.614054i \(0.789538\pi\)
\(18\) 0 0
\(19\) 1.19525e8 0.582854 0.291427 − 0.956593i \(-0.405870\pi\)
0.291427 + 0.956593i \(0.405870\pi\)
\(20\) 0 0
\(21\) −1.26630e8 −0.322190
\(22\) 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
\(27\) −3.87420e8 −0.192450
\(28\) 0 0
\(29\) 4.97957e9 1.55455 0.777276 − 0.629160i \(-0.216601\pi\)
0.777276 + 0.629160i \(0.216601\pi\)
\(30\) 0 0
\(31\) −5.63827e9 −1.14103 −0.570513 − 0.821289i \(-0.693255\pi\)
−0.570513 + 0.821289i \(0.693255\pi\)
\(32\) 0 0
\(33\) −7.07250e8 −0.0953305
\(34\) 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
\(39\) 1.76049e10 0.801150
\(40\) 0 0
\(41\) 2.57538e10 0.846734 0.423367 − 0.905958i \(-0.360848\pi\)
0.423367 + 0.905958i \(0.360848\pi\)
\(42\) 0 0
\(43\) 6.84564e10 1.65146 0.825732 − 0.564063i \(-0.190763\pi\)
0.825732 + 0.564063i \(0.190763\pi\)
\(44\) 0 0
\(45\) −1.30575e10 −0.234411
\(46\) 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
\(51\) 1.14524e11 0.911364
\(52\) 0 0
\(53\) 3.12743e11 1.93818 0.969090 − 0.246708i \(-0.0793488\pi\)
0.969090 + 0.246708i \(0.0793488\pi\)
\(54\) 0 0
\(55\) −2.38369e10 −0.116116
\(56\) 0 0
\(57\) −8.71336e10 −0.336511
\(58\) 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
\(63\) 9.23134e10 0.186016
\(64\) 0 0
\(65\) 5.93351e11 0.975832
\(66\) 0 0
\(67\) 1.30344e12 1.76037 0.880186 − 0.474629i \(-0.157418\pi\)
0.880186 + 0.474629i \(0.157418\pi\)
\(68\) 0 0
\(69\) −6.92368e10 −0.0772356
\(70\) 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
\(75\) 4.49806e11 0.291828
\(76\) 0 0
\(77\) 1.68521e11 0.0921436
\(78\) 0 0
\(79\) 1.15379e12 0.534013 0.267007 − 0.963695i \(-0.413965\pi\)
0.267007 + 0.963695i \(0.413965\pi\)
\(80\) 0 0
\(81\) 2.82430e11 0.111111
\(82\) 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
\(87\) −3.63011e12 −0.897521
\(88\) 0 0
\(89\) 7.28549e11 0.155390 0.0776951 − 0.996977i \(-0.475244\pi\)
0.0776951 + 0.996977i \(0.475244\pi\)
\(90\) 0 0
\(91\) −4.19485e12 −0.774368
\(92\) 0 0
\(93\) 4.11030e12 0.658771
\(94\) 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
\(99\) 5.15585e11 0.0550391
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 48.14.a.a.1.1 1
3.2 odd 2 144.14.a.j.1.1 1
4.3 odd 2 12.14.a.b.1.1 ✓ 1
8.3 odd 2 192.14.a.d.1.1 1
8.5 even 2 192.14.a.i.1.1 1
12.11 even 2 36.14.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.14.a.b.1.1 ✓ 1 4.3 odd 2
36.14.a.d.1.1 1 12.11 even 2
48.14.a.a.1.1 1 1.1 even 1 trivial
144.14.a.j.1.1 1 3.2 odd 2
192.14.a.d.1.1 1 8.3 odd 2
192.14.a.i.1.1 1 8.5 even 2