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Properties

Label 232.3.b.a.115.1
Level $232$
Weight $3$
Character 232.115
Self dual yes
Analytic conductor $6.322$
Analytic rank $0$
Dimension $1$
CM discriminant -232
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] = [232,3,Mod(115,232)] 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("232.115"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 232.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.32154213316\)
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 115.1
Character \(\chi\) \(=\) 232.115

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -8.00000 q^{8} +9.00000 q^{9} +16.0000 q^{16} -18.0000 q^{18} +25.0000 q^{25} +29.0000 q^{29} +54.0000 q^{31} -32.0000 q^{32} +36.0000 q^{36} +42.0000 q^{37} +22.0000 q^{47} +49.0000 q^{49} -50.0000 q^{50} -58.0000 q^{58} -114.000 q^{59} -6.00000 q^{61} -108.000 q^{62} +64.0000 q^{64} -98.0000 q^{67} -72.0000 q^{72} -84.0000 q^{74} -42.0000 q^{79} +81.0000 q^{81} -66.0000 q^{83} -44.0000 q^{94} -98.0000 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/232\mathbb{Z}\right)^\times\).

\(n\) \(89\) \(117\) \(175\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 4.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −8.00000 −1.00000
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −18.0000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 29.0000 1.00000
\(30\) 0 0
\(31\) 54.0000 1.74194 0.870968 − 0.491340i \(-0.163493\pi\)
0.870968 + 0.491340i \(0.163493\pi\)
\(32\) −32.0000 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 42.0000 1.13514 0.567568 − 0.823327i \(-0.307885\pi\)
0.567568 + 0.823327i \(0.307885\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 22.0000 0.468085 0.234043 − 0.972226i \(-0.424804\pi\)
0.234043 + 0.972226i \(0.424804\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) −50.0000 −1.00000
\(51\) 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
\(58\) −58.0000 −1.00000
\(59\) −114.000 −1.93220 −0.966102 − 0.258162i \(-0.916883\pi\)
−0.966102 + 0.258162i \(0.916883\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.0983607 −0.0491803 − 0.998790i \(-0.515661\pi\)
−0.0491803 + 0.998790i \(0.515661\pi\)
\(62\) −108.000 −1.74194
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −98.0000 −1.46269 −0.731343 − 0.682010i \(-0.761106\pi\)
−0.731343 + 0.682010i \(0.761106\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −72.0000 −1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −84.0000 −1.13514
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −42.0000 −0.531646 −0.265823 − 0.964022i \(-0.585644\pi\)
−0.265823 + 0.964022i \(0.585644\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −66.0000 −0.795181 −0.397590 − 0.917563i \(-0.630153\pi\)
−0.397590 + 0.917563i \(0.630153\pi\)
\(84\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −44.0000 −0.468085
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −98.0000 −1.00000
\(99\) 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 232.3.b.a.115.1 ✓ 1
4.3 odd 2 928.3.b.b.463.1 1
8.3 odd 2 232.3.b.b.115.1 yes 1
8.5 even 2 928.3.b.a.463.1 1
29.28 even 2 232.3.b.b.115.1 yes 1
116.115 odd 2 928.3.b.a.463.1 1
232.115 odd 2 CM 232.3.b.a.115.1 ✓ 1
232.173 even 2 928.3.b.b.463.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
232.3.b.a.115.1 ✓ 1 1.1 even 1 trivial
232.3.b.a.115.1 ✓ 1 232.115 odd 2 CM
232.3.b.b.115.1 yes 1 8.3 odd 2
232.3.b.b.115.1 yes 1 29.28 even 2
928.3.b.a.463.1 1 8.5 even 2
928.3.b.a.463.1 1 116.115 odd 2
928.3.b.b.463.1 1 4.3 odd 2
928.3.b.b.463.1 1 232.173 even 2