Properties

Label 1152.2.a.d
Level $1152$
Weight $2$
Character orbit 1152.a
Self dual yes
Analytic conductor $9.199$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - 2 q^{7} + 4 q^{11} - 2 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} - q^{25} - 6 q^{29} + 6 q^{31} + 4 q^{35} + 2 q^{37} + 12 q^{41} + 12 q^{43} + 8 q^{47} - 3 q^{49} - 6 q^{53} - 8 q^{55} + 8 q^{59} + 10 q^{61} + 4 q^{65} - 8 q^{67} + 2 q^{73} - 8 q^{77} + 14 q^{79} + 12 q^{83} - 8 q^{85} - 8 q^{89} + 4 q^{91} + 8 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −2.00000 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.a.d 1
3.b odd 2 1 1152.2.a.n yes 1
4.b odd 2 1 1152.2.a.f yes 1
8.b even 2 1 1152.2.a.o yes 1
8.d odd 2 1 1152.2.a.q yes 1
12.b even 2 1 1152.2.a.p yes 1
16.e even 4 2 2304.2.d.p 2
16.f odd 4 2 2304.2.d.g 2
24.f even 2 1 1152.2.a.g yes 1
24.h odd 2 1 1152.2.a.e yes 1
48.i odd 4 2 2304.2.d.n 2
48.k even 4 2 2304.2.d.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.a.d 1 1.a even 1 1 trivial
1152.2.a.e yes 1 24.h odd 2 1
1152.2.a.f yes 1 4.b odd 2 1
1152.2.a.g yes 1 24.f even 2 1
1152.2.a.n yes 1 3.b odd 2 1
1152.2.a.o yes 1 8.b even 2 1
1152.2.a.p yes 1 12.b even 2 1
1152.2.a.q yes 1 8.d odd 2 1
2304.2.d.e 2 48.k even 4 2
2304.2.d.g 2 16.f odd 4 2
2304.2.d.n 2 48.i odd 4 2
2304.2.d.p 2 16.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T - 4 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 6 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 12 \) Copy content Toggle raw display
$43$ \( T - 12 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 14 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 8 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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