Properties

Label 9576.2.a.bf
Level $9576$
Weight $2$
Character orbit 9576.a
Self dual yes
Analytic conductor $76.465$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9576.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(76.4647449756\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{5} - q^{7} + 2 \beta q^{11} + 4 q^{13} + (\beta - 5) q^{17} + q^{19} + ( - 2 \beta + 4) q^{23} + ( - 2 \beta - 1) q^{25} + (\beta + 1) q^{29} - 2 \beta q^{31} + ( - \beta + 1) q^{35} + ( - 2 \beta - 4) q^{37} + ( - 2 \beta - 8) q^{41} + ( - 4 \beta - 2) q^{43} + ( - 5 \beta + 3) q^{47} + q^{49} + ( - 3 \beta - 3) q^{53} + ( - 2 \beta + 6) q^{55} - 8 q^{59} + ( - 2 \beta + 8) q^{61} + (4 \beta - 4) q^{65} - 4 q^{67} + (3 \beta + 1) q^{71} + ( - 2 \beta + 4) q^{73} - 2 \beta q^{77} + ( - 2 \beta - 2) q^{79} + (3 \beta - 5) q^{83} + ( - 6 \beta + 8) q^{85} + 6 \beta q^{89} - 4 q^{91} + (\beta - 1) q^{95} + 6 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61} - 8 q^{65} - 8 q^{67} + 2 q^{71} + 8 q^{73} - 4 q^{79} - 10 q^{83} + 16 q^{85} - 8 q^{91} - 2 q^{95} + 12 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 0 0 −2.73205 0 −1.00000 0 0 0
1.2 0 0 0 0.732051 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(19\) \(-1\)

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 9576.2.a.bf 2
3.b odd 2 1 9576.2.a.br yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9576.2.a.bf 2 1.a even 1 1 trivial
9576.2.a.br yes 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9576))\):

\( T_{5}^{2} + 2T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 10T_{17} + 22 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 10T + 22 \) Copy content Toggle raw display
$19$ \( (T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$31$ \( T^{2} - 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 52 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 66 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$59$ \( (T + 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 52 \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 26 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T - 2 \) Copy content Toggle raw display
$89$ \( T^{2} - 108 \) Copy content Toggle raw display
$97$ \( (T - 6)^{2} \) Copy content Toggle raw display
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