Properties

Label 6422.2.a.d
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + 2 q^{3} + q^{4} - 3 q^{5} - 2 q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + 2 q^{3} + q^{4} - 3 q^{5} - 2 q^{6} - q^{8} + q^{9} + 3 q^{10} - 6 q^{11} + 2 q^{12} - 6 q^{15} + q^{16} + 7 q^{17} - q^{18} + q^{19} - 3 q^{20} + 6 q^{22} - 4 q^{23} - 2 q^{24} + 4 q^{25} - 4 q^{27} - 9 q^{29} + 6 q^{30} - q^{32} - 12 q^{33} - 7 q^{34} + q^{36} + 3 q^{37} - q^{38} + 3 q^{40} + 5 q^{41} - 6 q^{44} - 3 q^{45} + 4 q^{46} - 2 q^{47} + 2 q^{48} - 7 q^{49} - 4 q^{50} + 14 q^{51} + 11 q^{53} + 4 q^{54} + 18 q^{55} + 2 q^{57} + 9 q^{58} - 10 q^{59} - 6 q^{60} + 13 q^{61} + q^{64} + 12 q^{66} + 10 q^{67} + 7 q^{68} - 8 q^{69} + 6 q^{71} - q^{72} - q^{73} - 3 q^{74} + 8 q^{75} + q^{76} - 10 q^{79} - 3 q^{80} - 11 q^{81} - 5 q^{82} - 6 q^{83} - 21 q^{85} - 18 q^{87} + 6 q^{88} + 2 q^{89} + 3 q^{90} - 4 q^{92} + 2 q^{94} - 3 q^{95} - 2 q^{96} + 18 q^{97} + 7 q^{98} - 6 q^{99} + O(q^{100}) \)

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
−1.00000 2.00000 1.00000 −3.00000 −2.00000 0 −1.00000 1.00000 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(1\)
\(19\) \(-1\)

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 6422.2.a.d 1
13.b even 2 1 6422.2.a.i 1
13.c even 3 2 494.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.g.a 2 13.c even 3 2
6422.2.a.d 1 1.a even 1 1 trivial
6422.2.a.i 1 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3} - 2 \)
\( T_{5} + 3 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -2 + T \)
$5$ \( 3 + T \)
$7$ \( T \)
$11$ \( 6 + T \)
$13$ \( T \)
$17$ \( -7 + T \)
$19$ \( -1 + T \)
$23$ \( 4 + T \)
$29$ \( 9 + T \)
$31$ \( T \)
$37$ \( -3 + T \)
$41$ \( -5 + T \)
$43$ \( T \)
$47$ \( 2 + T \)
$53$ \( -11 + T \)
$59$ \( 10 + T \)
$61$ \( -13 + T \)
$67$ \( -10 + T \)
$71$ \( -6 + T \)
$73$ \( 1 + T \)
$79$ \( 10 + T \)
$83$ \( 6 + T \)
$89$ \( -2 + T \)
$97$ \( -18 + T \)
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