Properties

Label 1248.2.a.c
Level $1248$
Weight $2$
Character orbit 1248.a
Self dual yes
Analytic conductor $9.965$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.96533017226\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - 2 q^{7} + q^{9} + q^{13} + 2 q^{17} + 2 q^{19} + 2 q^{21} - 8 q^{23} - 5 q^{25} - q^{27} + 6 q^{29} - 2 q^{31} - 6 q^{37} - q^{39} - 4 q^{43} - 8 q^{47} - 3 q^{49} - 2 q^{51} - 6 q^{53} - 2 q^{57} - 4 q^{59} + 2 q^{61} - 2 q^{63} - 2 q^{67} + 8 q^{69} - 4 q^{71} - 2 q^{73} + 5 q^{75} - 12 q^{79} + q^{81} + 12 q^{83} - 6 q^{87} + 12 q^{89} - 2 q^{91} + 2 q^{93} - 18 q^{97}+O(q^{100}) \) 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 −1.00000 0 0 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(-1\)

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 1248.2.a.c 1
3.b odd 2 1 3744.2.a.g 1
4.b odd 2 1 1248.2.a.i yes 1
8.b even 2 1 2496.2.a.w 1
8.d odd 2 1 2496.2.a.i 1
12.b even 2 1 3744.2.a.j 1
24.f even 2 1 7488.2.a.be 1
24.h odd 2 1 7488.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1248.2.a.c 1 1.a even 1 1 trivial
1248.2.a.i yes 1 4.b odd 2 1
2496.2.a.i 1 8.d odd 2 1
2496.2.a.w 1 8.b even 2 1
3744.2.a.g 1 3.b odd 2 1
3744.2.a.j 1 12.b even 2 1
7488.2.a.ba 1 24.h odd 2 1
7488.2.a.be 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1248))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T + 4 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T + 12 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T + 18 \) Copy content Toggle raw display
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