Properties

Label 4056.2.a.g
Level $4056$
Weight $2$
Character orbit 4056.a
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 2 q^{5} - 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 2 q^{5} - 4 q^{7} + q^{9} - 2 q^{15} + 2 q^{17} - 8 q^{19} + 4 q^{21} + 8 q^{23} - q^{25} - q^{27} - 2 q^{29} - 4 q^{31} - 8 q^{35} + 10 q^{37} - 2 q^{41} - 4 q^{43} + 2 q^{45} + 12 q^{47} + 9 q^{49} - 2 q^{51} + 6 q^{53} + 8 q^{57} - 2 q^{61} - 4 q^{63} - 8 q^{67} - 8 q^{69} + 12 q^{71} - 10 q^{73} + q^{75} - 8 q^{79} + q^{81} + 4 q^{85} + 2 q^{87} + 14 q^{89} + 4 q^{93} - 16 q^{95} - 2 q^{97}+O(q^{100}) \) 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 −1.00000 0 2.00000 0 −4.00000 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 4056.2.a.g 1
4.b odd 2 1 8112.2.a.bg 1
13.b even 2 1 312.2.a.a 1
13.d odd 4 2 4056.2.c.c 2
39.d odd 2 1 936.2.a.h 1
52.b odd 2 1 624.2.a.f 1
65.d even 2 1 7800.2.a.n 1
104.e even 2 1 2496.2.a.bb 1
104.h odd 2 1 2496.2.a.k 1
156.h even 2 1 1872.2.a.n 1
312.b odd 2 1 7488.2.a.t 1
312.h even 2 1 7488.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.a.a 1 13.b even 2 1
624.2.a.f 1 52.b odd 2 1
936.2.a.h 1 39.d odd 2 1
1872.2.a.n 1 156.h even 2 1
2496.2.a.k 1 104.h odd 2 1
2496.2.a.bb 1 104.e even 2 1
4056.2.a.g 1 1.a even 1 1 trivial
4056.2.c.c 2 13.d odd 4 2
7488.2.a.i 1 312.h even 2 1
7488.2.a.t 1 312.b odd 2 1
7800.2.a.n 1 65.d even 2 1
8112.2.a.bg 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4056))\):

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