Properties

Label 8036.2.a.a
Level $8036$
Weight $2$
Character orbit 8036.a
Self dual yes
Analytic conductor $64.168$
Analytic rank $2$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{5} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - q^{5} - 2q^{9} - 5q^{11} - 2q^{13} + q^{15} - 3q^{17} - 3q^{19} - q^{23} - 4q^{25} + 5q^{27} - 10q^{29} - 11q^{31} + 5q^{33} + 7q^{37} + 2q^{39} - q^{41} + 8q^{43} + 2q^{45} - 7q^{47} + 3q^{51} - 11q^{53} + 5q^{55} + 3q^{57} - 7q^{59} - q^{61} + 2q^{65} + 7q^{67} + q^{69} - 12q^{71} - 5q^{73} + 4q^{75} + 9q^{79} + q^{81} + 3q^{85} + 10q^{87} - 3q^{89} + 11q^{93} + 3q^{95} + 2q^{97} + 10q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

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 8036.2.a.a 1
7.b odd 2 1 8036.2.a.e 1
7.c even 3 2 1148.2.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.b 2 7.c even 3 2
8036.2.a.a 1 1.a even 1 1 trivial
8036.2.a.e 1 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\( T_{3} + 1 \)
\( T_{5} + 1 \)
\( T_{11} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( 2 + T \)
$17$ \( 3 + T \)
$19$ \( 3 + T \)
$23$ \( 1 + T \)
$29$ \( 10 + T \)
$31$ \( 11 + T \)
$37$ \( -7 + T \)
$41$ \( 1 + T \)
$43$ \( -8 + T \)
$47$ \( 7 + T \)
$53$ \( 11 + T \)
$59$ \( 7 + T \)
$61$ \( 1 + T \)
$67$ \( -7 + T \)
$71$ \( 12 + T \)
$73$ \( 5 + T \)
$79$ \( -9 + T \)
$83$ \( T \)
$89$ \( 3 + T \)
$97$ \( -2 + T \)
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