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Properties

Label 3888.1.q.a.161.1
Level $3888$
Weight $1$
Character 3888.161
Analytic conductor $1.940$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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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] = [3888,1,Mod(161,3888)] 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("3888.161"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3888 = 2^{4} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3888.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-1,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.94036476912\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 972)
Projective image: \(D_{3}\)
Projective field: Galois closure of \(\Q(\sqrt[3]{12})\)
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.2056589122535424.23

Embedding invariants

Embedding label 161.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3888.161
Dual form 3888.1.q.a.2753.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{7} +(-1.00000 + 1.73205i) q^{13} -2.00000 q^{19} +(-0.500000 - 0.866025i) q^{25} +(-0.500000 + 0.866025i) q^{31} -1.00000 q^{37} +(-0.500000 - 0.866025i) q^{43} +(0.500000 + 0.866025i) q^{61} +(-0.500000 + 0.866025i) q^{67} -1.00000 q^{73} +(1.00000 + 1.73205i) q^{79} +2.00000 q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{7} - 2 q^{13} - 4 q^{19} - q^{25} - q^{31} - 2 q^{37} - q^{43} + q^{61} - q^{67} - 2 q^{73} + 2 q^{79} + 4 q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3888\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2431\) \(2917\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 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
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 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
\(91\) 2.00000 2.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(98\) 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 3888.1.q.a.161.1 2
3.2 odd 2 CM 3888.1.q.a.161.1 2
4.3 odd 2 972.1.g.b.161.1 2
9.2 odd 6 inner 3888.1.q.a.2753.1 2
9.4 even 3 3888.1.e.c.1457.1 1
9.5 odd 6 3888.1.e.c.1457.1 1
9.7 even 3 inner 3888.1.q.a.2753.1 2
12.11 even 2 972.1.g.b.161.1 2
36.7 odd 6 972.1.g.b.809.1 2
36.11 even 6 972.1.g.b.809.1 2
36.23 even 6 972.1.c.a.485.1 ✓ 1
36.31 odd 6 972.1.c.a.485.1 ✓ 1
108.7 odd 18 2916.1.k.a.161.1 6
108.11 even 18 2916.1.k.a.1133.1 6
108.23 even 18 2916.1.k.a.809.1 6
108.31 odd 18 2916.1.k.a.809.1 6
108.43 odd 18 2916.1.k.a.1133.1 6
108.47 even 18 2916.1.k.a.161.1 6
108.59 even 18 2916.1.k.a.2753.1 6
108.67 odd 18 2916.1.k.a.1781.1 6
108.79 odd 18 2916.1.k.a.2105.1 6
108.83 even 18 2916.1.k.a.2105.1 6
108.95 even 18 2916.1.k.a.1781.1 6
108.103 odd 18 2916.1.k.a.2753.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
972.1.c.a.485.1 ✓ 1 36.23 even 6
972.1.c.a.485.1 ✓ 1 36.31 odd 6
972.1.g.b.161.1 2 4.3 odd 2
972.1.g.b.161.1 2 12.11 even 2
972.1.g.b.809.1 2 36.7 odd 6
972.1.g.b.809.1 2 36.11 even 6
2916.1.k.a.161.1 6 108.7 odd 18
2916.1.k.a.161.1 6 108.47 even 18
2916.1.k.a.809.1 6 108.23 even 18
2916.1.k.a.809.1 6 108.31 odd 18
2916.1.k.a.1133.1 6 108.11 even 18
2916.1.k.a.1133.1 6 108.43 odd 18
2916.1.k.a.1781.1 6 108.67 odd 18
2916.1.k.a.1781.1 6 108.95 even 18
2916.1.k.a.2105.1 6 108.79 odd 18
2916.1.k.a.2105.1 6 108.83 even 18
2916.1.k.a.2753.1 6 108.59 even 18
2916.1.k.a.2753.1 6 108.103 odd 18
3888.1.e.c.1457.1 1 9.4 even 3
3888.1.e.c.1457.1 1 9.5 odd 6
3888.1.q.a.161.1 2 1.1 even 1 trivial
3888.1.q.a.161.1 2 3.2 odd 2 CM
3888.1.q.a.2753.1 2 9.2 odd 6 inner
3888.1.q.a.2753.1 2 9.7 even 3 inner