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Properties

Label 12.57.c.a.5.1
Level $12$
Weight $57$
Character 12.5
Self dual yes
Analytic conductor $238.333$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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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] = [12,57,Mod(5,12)] 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("12.5"); S:= CuspForms(chi, 57); 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, 1])) N = Newforms(chi, 57, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 57 \)
Character orbit: \([\chi]\) \(=\) 12.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(238.332749918\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 5.1
Character \(\chi\) \(=\) 12.5

$q$-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)
 
\(f(q)\) \(=\) \(q+2.28768e13 q^{3} -5.91717e23 q^{7} +5.23348e26 q^{9} -3.09919e31 q^{13} -8.01094e35 q^{19} -1.35366e37 q^{21} +1.38778e39 q^{25} +1.19725e40 q^{27} -1.77219e41 q^{31} -1.31479e44 q^{37} -7.08996e44 q^{39} +6.26064e45 q^{43} +1.38542e47 q^{49} -1.83265e49 q^{57} +1.60882e50 q^{61} -3.09674e50 q^{63} +2.18634e51 q^{67} -2.68247e52 q^{73} +3.17479e52 q^{75} +2.69218e53 q^{79} +2.73893e53 q^{81} +1.83385e55 q^{91} -4.05421e54 q^{93} +8.17123e55 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 2.28768e13 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −5.91717e23 −1.28638 −0.643190 − 0.765707i \(-0.722389\pi\)
−0.643190 + 0.765707i \(0.722389\pi\)
\(8\) 0 0
\(9\) 5.23348e26 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −3.09919e31 −1.99910 −0.999550 − 0.0299867i \(-0.990453\pi\)
−0.999550 + 0.0299867i \(0.990453\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −8.01094e35 −1.25482 −0.627412 − 0.778688i \(-0.715886\pi\)
−0.627412 + 0.778688i \(0.715886\pi\)
\(20\) 0 0
\(21\) −1.35366e37 −1.28638
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.38778e39 1.00000
\(26\) 0 0
\(27\) 1.19725e40 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.77219e41 −0.309302 −0.154651 − 0.987969i \(-0.549425\pi\)
−0.154651 + 0.987969i \(0.549425\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.31479e44 −1.61886 −0.809429 − 0.587217i \(-0.800223\pi\)
−0.809429 + 0.587217i \(0.800223\pi\)
\(38\) 0 0
\(39\) −7.08996e44 −1.99910
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 6.26064e45 1.14684 0.573419 − 0.819262i \(-0.305617\pi\)
0.573419 + 0.819262i \(0.305617\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.38542e47 0.654773
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.83265e49 −1.25482
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.60882e50 1.64920 0.824598 − 0.565718i \(-0.191401\pi\)
0.824598 + 0.565718i \(0.191401\pi\)
\(62\) 0 0
\(63\) −3.09674e50 −1.28638
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.18634e51 1.62040 0.810201 − 0.586153i \(-0.199358\pi\)
0.810201 + 0.586153i \(0.199358\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −2.68247e52 −1.80092 −0.900460 − 0.434939i \(-0.856770\pi\)
−0.900460 + 0.434939i \(0.856770\pi\)
\(74\) 0 0
\(75\) 3.17479e52 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.69218e53 1.97946 0.989729 − 0.142958i \(-0.0456615\pi\)
0.989729 + 0.142958i \(0.0456615\pi\)
\(80\) 0 0
\(81\) 2.73893e53 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.83385e55 2.57160
\(92\) 0 0
\(93\) −4.05421e54 −0.309302
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.17123e55 1.91725 0.958625 − 0.284673i \(-0.0918849\pi\)
0.958625 + 0.284673i \(0.0918849\pi\)
\(98\) 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 12.57.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.57.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.57.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.57.c.a.5.1 ✓ 1 3.2 odd 2 CM