Properties

Label 9576.2.a.bt
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - q^{7} + (3 \beta - 1) q^{11} + ( - 4 \beta + 3) q^{13} + ( - \beta + 3) q^{17} - q^{19} + (2 \beta - 1) q^{23} - 4 q^{25} + ( - 5 \beta + 3) q^{29} + (5 \beta - 7) q^{31} - q^{35} + (4 \beta - 7) q^{37} + (\beta + 1) q^{41} - 2 q^{43} + ( - 6 \beta - 3) q^{47} + q^{49} + ( - 5 \beta + 8) q^{53} + (3 \beta - 1) q^{55} + (8 \beta - 3) q^{59} - 5 q^{61} + ( - 4 \beta + 3) q^{65} + (7 \beta - 11) q^{67} - 9 q^{71} + (3 \beta - 4) q^{73} + ( - 3 \beta + 1) q^{77} + ( - 8 \beta + 10) q^{79} + ( - \beta - 6) q^{83} + ( - \beta + 3) q^{85} + ( - 2 \beta + 6) q^{89} + (4 \beta - 3) q^{91} - q^{95} + ( - 4 \beta - 7) q^{97}+O(q^{100}) \) 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 \( 2 q + 2 q^{5} - 2 q^{7} + q^{11} + 2 q^{13} + 5 q^{17} - 2 q^{19} - 8 q^{25} + q^{29} - 9 q^{31} - 2 q^{35} - 10 q^{37} + 3 q^{41} - 4 q^{43} - 12 q^{47} + 2 q^{49} + 11 q^{53} + q^{55} + 2 q^{59} - 10 q^{61} + 2 q^{65} - 15 q^{67} - 18 q^{71} - 5 q^{73} - q^{77} + 12 q^{79} - 13 q^{83} + 5 q^{85} + 10 q^{89} - 2 q^{91} - 2 q^{95} - 18 q^{97}+O(q^{100}) \) 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
−0.618034
1.61803
0 0 0 1.00000 0 −1.00000 0 0 0
1.2 0 0 0 1.00000 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.bt 2
3.b odd 2 1 1064.2.a.d 2
12.b even 2 1 2128.2.a.f 2
21.c even 2 1 7448.2.a.z 2
24.f even 2 1 8512.2.a.y 2
24.h odd 2 1 8512.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.d 2 3.b odd 2 1
2128.2.a.f 2 12.b even 2 1
7448.2.a.z 2 21.c even 2 1
8512.2.a.q 2 24.h odd 2 1
8512.2.a.y 2 24.f even 2 1
9576.2.a.bt 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9576))\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 11 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 19 \) Copy content Toggle raw display
\( T_{17}^{2} - 5T_{17} + 5 \) 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
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 5 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$31$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 5 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12T - 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T - 1 \) Copy content Toggle raw display
$59$ \( T^{2} - 2T - 79 \) Copy content Toggle raw display
$61$ \( (T + 5)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 15T - 5 \) Copy content Toggle raw display
$71$ \( (T + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} + 13T + 41 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$97$ \( T^{2} + 18T + 61 \) Copy content Toggle raw display
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