Properties

Label 8400.2.a.h
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 2100)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} - q^{7} + q^{9} + q^{11} + 2q^{13} - 6q^{19} + q^{21} + q^{23} - q^{27} + q^{29} + 2q^{31} - q^{33} + 7q^{37} - 2q^{39} - 8q^{41} - q^{43} - 2q^{47} + q^{49} + 14q^{53} + 6q^{57} - 10q^{59} - q^{63} - 3q^{67} - q^{69} + 9q^{71} - q^{77} - q^{79} + q^{81} + 2q^{83} - q^{87} + 2q^{89} - 2q^{91} - 2q^{93} + 10q^{97} + q^{99} + O(q^{100}) \)

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.h 1
4.b odd 2 1 2100.2.a.p yes 1
5.b even 2 1 8400.2.a.cp 1
12.b even 2 1 6300.2.a.z 1
20.d odd 2 1 2100.2.a.c 1
20.e even 4 2 2100.2.k.d 2
60.h even 2 1 6300.2.a.k 1
60.l odd 4 2 6300.2.k.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.a.c 1 20.d odd 2 1
2100.2.a.p yes 1 4.b odd 2 1
2100.2.k.d 2 20.e even 4 2
6300.2.a.k 1 60.h even 2 1
6300.2.a.z 1 12.b even 2 1
6300.2.k.k 2 60.l odd 4 2
8400.2.a.h 1 1.a even 1 1 trivial
8400.2.a.cp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8400))\):

\( T_{11} - 1 \)
\( T_{13} - 2 \)
\( T_{17} \)
\( T_{19} + 6 \)
\( T_{23} - 1 \)