Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
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

\(f(q)\) \(=\) \( q + q^{3} + q^{5} - 2q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} - 2q^{9} + 3q^{11} - 2q^{13} + q^{15} - q^{17} + 3q^{19} - 5q^{23} - 4q^{25} - 5q^{27} - 2q^{29} - 5q^{31} + 3q^{33} + 7q^{37} - 2q^{39} - q^{41} + 4q^{43} - 2q^{45} + 3q^{47} - q^{51} - 3q^{53} + 3q^{55} + 3q^{57} - 5q^{59} - 3q^{61} - 2q^{65} - 13q^{67} - 5q^{69} + q^{73} - 4q^{75} - 11q^{79} + q^{81} - 4q^{83} - q^{85} - 2q^{87} - 5q^{89} - 5q^{93} + 3q^{95} - 2q^{97} - 6q^{99} + O(q^{100}) \)

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.f 1
7.b odd 2 1 8036.2.a.b 1
7.d odd 6 2 1148.2.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.c 2 7.d odd 6 2
8036.2.a.b 1 7.b odd 2 1
8036.2.a.f 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( 2 + T \)
$17$ \( 1 + T \)
$19$ \( -3 + T \)
$23$ \( 5 + T \)
$29$ \( 2 + T \)
$31$ \( 5 + T \)
$37$ \( -7 + T \)
$41$ \( 1 + T \)
$43$ \( -4 + T \)
$47$ \( -3 + T \)
$53$ \( 3 + T \)
$59$ \( 5 + T \)
$61$ \( 3 + T \)
$67$ \( 13 + T \)
$71$ \( T \)
$73$ \( -1 + T \)
$79$ \( 11 + T \)
$83$ \( 4 + T \)
$89$ \( 5 + T \)
$97$ \( 2 + T \)
show more
show less