Properties

Label 8400.2.a.v
Level $8400$
Weight $2$
Character orbit 8400.a
Self dual yes
Analytic conductor $67.074$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.0743376979\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{7} + q^{9} - 2 q^{11} + 2 q^{13} + 6 q^{19} - q^{21} - q^{27} - 6 q^{29} - 10 q^{31} + 2 q^{33} - 2 q^{39} + 6 q^{41} - 8 q^{43} - 12 q^{47} + q^{49} + 6 q^{53} - 6 q^{57} - 6 q^{61} + q^{63} + 4 q^{67} - 6 q^{71} + 14 q^{73} - 2 q^{77} - 4 q^{79} + q^{81} + 6 q^{87} - 6 q^{89} + 2 q^{91} + 10 q^{93} + 2 q^{97} - 2 q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 8400.2.a.v 1
4.b odd 2 1 4200.2.a.w 1
5.b even 2 1 8400.2.a.bs 1
5.c odd 4 2 1680.2.t.b 2
15.e even 4 2 5040.2.t.l 2
20.d odd 2 1 4200.2.a.j 1
20.e even 4 2 840.2.t.b 2
60.l odd 4 2 2520.2.t.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.b 2 20.e even 4 2
1680.2.t.b 2 5.c odd 4 2
2520.2.t.e 2 60.l odd 4 2
4200.2.a.j 1 20.d odd 2 1
4200.2.a.w 1 4.b odd 2 1
5040.2.t.l 2 15.e even 4 2
8400.2.a.v 1 1.a even 1 1 trivial
8400.2.a.bs 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):

\( T_{11} + 2 \)
\( T_{13} - 2 \)
\( T_{17} \)
\( T_{19} - 6 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( 2 + T \)
$13$ \( -2 + T \)
$17$ \( T \)
$19$ \( -6 + T \)
$23$ \( T \)
$29$ \( 6 + T \)
$31$ \( 10 + T \)
$37$ \( T \)
$41$ \( -6 + T \)
$43$ \( 8 + T \)
$47$ \( 12 + T \)
$53$ \( -6 + T \)
$59$ \( T \)
$61$ \( 6 + T \)
$67$ \( -4 + T \)
$71$ \( 6 + T \)
$73$ \( -14 + T \)
$79$ \( 4 + T \)
$83$ \( T \)
$89$ \( 6 + T \)
$97$ \( -2 + T \)
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