Properties

Label 3.32.a.a
Level $3$
Weight $32$
Character orbit 3.a
Self dual yes
Analytic conductor $18.263$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.2631398457\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \( x^{2} - x - 2875320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{11501281}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 19764) q^{2} + 14348907 q^{3} + (39528 \beta - 100683488) q^{4} + ( - 1179920 \beta - 3965258610) q^{5} + ( - 14348907 \beta - 283591797948) q^{6} + (140643216 \beta - 5244437118088) q^{7} + (1466935744 \beta - 21032884217088) q^{8} + 205891132094649 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 19764) q^{2} + 14348907 q^{3} + (39528 \beta - 100683488) q^{4} + ( - 1179920 \beta - 3965258610) q^{5} + ( - 14348907 \beta - 283591797948) q^{6} + (140643216 \beta - 5244437118088) q^{7} + (1466935744 \beta - 21032884217088) q^{8} + 205891132094649 q^{9} + (27285197490 \beta + 20\!\cdots\!20) q^{10}+ \cdots + (88\!\cdots\!96 \beta + 11\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 39528 q^{2} + 28697814 q^{3} - 201366976 q^{4} - 7930517220 q^{5} - 567183595896 q^{6} - 10488874236176 q^{7} - 42065768434176 q^{8} + 411782264189298 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 39528 q^{2} + 28697814 q^{3} - 201366976 q^{4} - 7930517220 q^{5} - 567183595896 q^{6} - 10488874236176 q^{7} - 42065768434176 q^{8} + 411782264189298 q^{9} + 40\!\cdots\!40 q^{10}+ \cdots + 22\!\cdots\!92 q^{99}+O(q^{100}) \) 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
1696.18
−1695.18
−60460.2 1.43489e7 1.50796e9 −5.19836e10 −8.67538e11 4.79214e11 3.86659e13 2.05891e14 3.14294e15
1.2 20932.2 1.43489e7 −1.70932e9 4.40531e10 3.00355e11 −1.09681e13 −8.07317e13 2.05891e14 9.22129e14
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3.32.a.a 2
3.b odd 2 1 9.32.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.32.a.a 2 1.a even 1 1 trivial
9.32.a.b 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 39528T_{2} - 1265568768 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 39528 T - 1265568768 \) Copy content Toggle raw display
$3$ \( (T - 14348907)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 7930517220 T - 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + 10488874236176 T - 52\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 27\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 19\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 27\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 18\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 23\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 19\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 89\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 35\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 49\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
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