Properties

Label 96.3.h.b
Level $96$
Weight $3$
Character orbit 96.h
Self dual yes
Analytic conductor $2.616$
Analytic rank $0$
Dimension $1$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 96.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(2.61581053786\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} - 2q^{5} + 10q^{7} + 9q^{9} + O(q^{10}) \) \( q + 3q^{3} - 2q^{5} + 10q^{7} + 9q^{9} - 10q^{11} - 6q^{15} + 30q^{21} - 21q^{25} + 27q^{27} - 50q^{29} - 38q^{31} - 30q^{33} - 20q^{35} - 18q^{45} + 51q^{49} + 94q^{53} + 20q^{55} - 10q^{59} + 90q^{63} + 50q^{73} - 63q^{75} - 100q^{77} + 58q^{79} + 81q^{81} + 134q^{83} - 150q^{87} - 114q^{93} - 190q^{97} - 90q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0
0 3.00000 0 −2.00000 0 10.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.3.h.b 1
3.b odd 2 1 96.3.h.a 1
4.b odd 2 1 24.3.h.b yes 1
8.b even 2 1 96.3.h.a 1
8.d odd 2 1 24.3.h.a 1
12.b even 2 1 24.3.h.a 1
16.e even 4 2 768.3.e.c 2
16.f odd 4 2 768.3.e.d 2
24.f even 2 1 24.3.h.b yes 1
24.h odd 2 1 CM 96.3.h.b 1
48.i odd 4 2 768.3.e.c 2
48.k even 4 2 768.3.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.h.a 1 8.d odd 2 1
24.3.h.a 1 12.b even 2 1
24.3.h.b yes 1 4.b odd 2 1
24.3.h.b yes 1 24.f even 2 1
96.3.h.a 1 3.b odd 2 1
96.3.h.a 1 8.b even 2 1
96.3.h.b 1 1.a even 1 1 trivial
96.3.h.b 1 24.h odd 2 1 CM
768.3.e.c 2 16.e even 4 2
768.3.e.c 2 48.i odd 4 2
768.3.e.d 2 16.f odd 4 2
768.3.e.d 2 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{3}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 2 + T \)
$7$ \( -10 + T \)
$11$ \( 10 + T \)
$13$ \( T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 50 + T \)
$31$ \( 38 + T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -94 + T \)
$59$ \( 10 + T \)
$61$ \( T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -50 + T \)
$79$ \( -58 + T \)
$83$ \( -134 + T \)
$89$ \( T \)
$97$ \( 190 + T \)
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