Properties

Label 8400.2.a.q
Level $8400$
Weight $2$
Character orbit 8400.a
Self dual yes
Analytic conductor $67.074$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{3} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{7} + q^{9} - 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} - q^{21} + 8q^{23} - q^{27} - 2q^{29} + 4q^{33} + 2q^{37} - 2q^{39} + 10q^{41} + 4q^{43} + q^{49} - 6q^{51} - 14q^{53} + 4q^{57} - 12q^{59} - 2q^{61} + q^{63} - 4q^{67} - 8q^{69} - 2q^{73} - 4q^{77} + 8q^{79} + q^{81} - 4q^{83} + 2q^{87} - 6q^{89} + 2q^{91} + 6q^{97} - 4q^{99} + O(q^{100}) \)

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.q 1
4.b odd 2 1 4200.2.a.z 1
5.b even 2 1 1680.2.a.p 1
15.d odd 2 1 5040.2.a.j 1
20.d odd 2 1 840.2.a.f 1
20.e even 4 2 4200.2.t.q 2
40.e odd 2 1 6720.2.a.bn 1
40.f even 2 1 6720.2.a.h 1
60.h even 2 1 2520.2.a.g 1
140.c even 2 1 5880.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.a.f 1 20.d odd 2 1
1680.2.a.p 1 5.b even 2 1
2520.2.a.g 1 60.h even 2 1
4200.2.a.z 1 4.b odd 2 1
4200.2.t.q 2 20.e even 4 2
5040.2.a.j 1 15.d odd 2 1
5880.2.a.z 1 140.c even 2 1
6720.2.a.h 1 40.f even 2 1
6720.2.a.bn 1 40.e odd 2 1
8400.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):

\( T_{11} + 4 \)
\( T_{13} - 2 \)
\( T_{17} - 6 \)
\( T_{19} + 4 \)
\( T_{23} - 8 \)

Hecke characteristic polynomials

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