Properties

Label 9576.2.a.bi
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: \(\Q(\sqrt{5}) \)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). 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 \( q + ( - \beta - 1) q^{5} + q^{7} - 2 q^{11} + 2 \beta q^{13} + (\beta + 1) q^{17} - q^{19} + 2 q^{23} + (2 \beta + 1) q^{25} + ( - 3 \beta - 1) q^{29} + ( - 2 \beta - 6) q^{31} + ( - \beta - 1) q^{35} + ( - 2 \beta + 4) q^{37} + (2 \beta - 4) q^{41} + 8 q^{43} + ( - 3 \beta - 5) q^{47} + q^{49} + ( - \beta + 9) q^{53} + (2 \beta + 2) q^{55} + 4 \beta q^{59} + 6 \beta q^{61} + ( - 2 \beta - 10) q^{65} + ( - 2 \beta - 6) q^{67} + ( - 3 \beta + 1) q^{71} + ( - 2 \beta + 8) q^{73} - 2 q^{77} + 4 q^{79} + (3 \beta - 7) q^{83} + ( - 2 \beta - 6) q^{85} - 2 \beta q^{89} + 2 \beta q^{91} + (\beta + 1) q^{95} + (6 \beta - 4) q^{97} +O(q^{100}) \) 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 \( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 2 q^{35} + 8 q^{37} - 8 q^{41} + 16 q^{43} - 10 q^{47} + 2 q^{49} + 18 q^{53} + 4 q^{55} - 20 q^{65} - 12 q^{67} + 2 q^{71} + 16 q^{73} - 4 q^{77} + 8 q^{79} - 14 q^{83} - 12 q^{85} + 2 q^{95} - 8 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 0 0 −3.23607 0 1.00000 0 0 0
1.2 0 0 0 1.23607 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.bi 2
3.b odd 2 1 9576.2.a.bv yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9576.2.a.bi 2 1.a even 1 1 trivial
9576.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9576))\):

\( T_{5}^{2} + 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 4 \) 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
$5$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 10T - 20 \) Copy content Toggle raw display
$53$ \( T^{2} - 18T + 76 \) Copy content Toggle raw display
$59$ \( T^{2} - 80 \) Copy content Toggle raw display
$61$ \( T^{2} - 180 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 14T + 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 20 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 164 \) Copy content Toggle raw display
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