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Properties

Label 12.51.c.a.5.1
Level $12$
Weight $51$
Character 12.5
Self dual yes
Analytic conductor $190.003$
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] = [12,51,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, 51); 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, 51, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 51 \)
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: \(190.002544227\)
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)
 
\(f(q)\) \(=\) \(q-8.47289e11 q^{3} -1.14751e21 q^{7} +7.17898e23 q^{9} -1.20408e27 q^{13} -2.14825e30 q^{19} +9.72275e32 q^{21} +8.88178e34 q^{25} -6.08267e35 q^{27} -3.41461e37 q^{31} +1.39319e39 q^{37} +1.02020e39 q^{39} -2.51323e40 q^{43} -4.81678e41 q^{49} +1.82019e42 q^{57} -4.74733e44 q^{61} -8.23797e44 q^{63} -2.94809e45 q^{67} -7.56087e46 q^{73} -7.52543e46 q^{75} -1.90463e47 q^{79} +5.15378e47 q^{81} +1.38170e48 q^{91} +2.89316e49 q^{93} +2.38057e49 q^{97} +O(q^{100})\)

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
\(3\) −8.47289e11 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.14751e21 −0.855671 −0.427835 − 0.903857i \(-0.640724\pi\)
−0.427835 + 0.903857i \(0.640724\pi\)
\(8\) 0 0
\(9\) 7.17898e23 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.20408e27 −0.170636 −0.0853180 − 0.996354i \(-0.527191\pi\)
−0.0853180 + 0.996354i \(0.527191\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.14825e30 −0.0230805 −0.0115403 − 0.999933i \(-0.503673\pi\)
−0.0115403 + 0.999933i \(0.503673\pi\)
\(20\) 0 0
\(21\) 9.72275e32 0.855671
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 8.88178e34 1.00000
\(26\) 0 0
\(27\) −6.08267e35 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −3.41461e37 −1.77541 −0.887705 − 0.460413i \(-0.847701\pi\)
−0.887705 + 0.460413i \(0.847701\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.39319e39 0.868897 0.434448 − 0.900697i \(-0.356943\pi\)
0.434448 + 0.900697i \(0.356943\pi\)
\(38\) 0 0
\(39\) 1.02020e39 0.170636
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.51323e40 −0.366034 −0.183017 − 0.983110i \(-0.558586\pi\)
−0.183017 + 0.983110i \(0.558586\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −4.81678e41 −0.267827
\(50\) 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
\(57\) 1.82019e42 0.0230805
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −4.74733e44 −1.10460 −0.552298 − 0.833647i \(-0.686249\pi\)
−0.552298 + 0.833647i \(0.686249\pi\)
\(62\) 0 0
\(63\) −8.23797e44 −0.855671
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.94809e45 −0.657158 −0.328579 − 0.944476i \(-0.606570\pi\)
−0.328579 + 0.944476i \(0.606570\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −7.56087e46 −1.97470 −0.987348 − 0.158567i \(-0.949313\pi\)
−0.987348 + 0.158567i \(0.949313\pi\)
\(74\) 0 0
\(75\) −7.52543e46 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.90463e47 −0.690452 −0.345226 − 0.938520i \(-0.612198\pi\)
−0.345226 + 0.938520i \(0.612198\pi\)
\(80\) 0 0
\(81\) 5.15378e47 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
\(91\) 1.38170e48 0.146008
\(92\) 0 0
\(93\) 2.89316e49 1.77541
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.38057e49 0.509785 0.254892 − 0.966969i \(-0.417960\pi\)
0.254892 + 0.966969i \(0.417960\pi\)
\(98\) 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 12.51.c.a.5.1 ✓ 1
3.2 odd 2 CM 12.51.c.a.5.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.51.c.a.5.1 ✓ 1 1.1 even 1 trivial
12.51.c.a.5.1 ✓ 1 3.2 odd 2 CM