Properties

Label 8372.2.a.e
Level $8372$
Weight $2$
Character orbit 8372.a
Self dual yes
Analytic conductor $66.851$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8372 = 2^{2} \cdot 7 \cdot 13 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8372.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.8507565722\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{7} - 3q^{9} + O(q^{10}) \) \( q - q^{7} - 3q^{9} - q^{13} + 6q^{17} - 4q^{19} - q^{23} - 5q^{25} - 2q^{29} - 10q^{31} + 4q^{37} + 10q^{41} - 4q^{43} + 6q^{47} + q^{49} - 2q^{53} + 2q^{59} + 10q^{61} + 3q^{63} - 16q^{67} + 10q^{71} - 2q^{73} - 8q^{79} + 9q^{81} + 16q^{83} - 8q^{89} + q^{91} - 4q^{97} + O(q^{100}) \)

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 0 0 0 0 −1.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)
\(23\) \(1\)

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 8372.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8372.2.a.e 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8372))\):

\( T_{3} \)
\( T_{5} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 1 + T \)
$11$ \( T \)
$13$ \( 1 + T \)
$17$ \( -6 + T \)
$19$ \( 4 + T \)
$23$ \( 1 + T \)
$29$ \( 2 + T \)
$31$ \( 10 + T \)
$37$ \( -4 + T \)
$41$ \( -10 + T \)
$43$ \( 4 + T \)
$47$ \( -6 + T \)
$53$ \( 2 + T \)
$59$ \( -2 + T \)
$61$ \( -10 + T \)
$67$ \( 16 + T \)
$71$ \( -10 + T \)
$73$ \( 2 + T \)
$79$ \( 8 + T \)
$83$ \( -16 + T \)
$89$ \( 8 + T \)
$97$ \( 4 + T \)
show more
show less