Properties

Label 6422.2.a.e
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} - 2 q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} - 2 q^{9} - q^{10} - q^{12} + q^{14} + q^{15} + q^{16} - 3 q^{17} - 2 q^{18} + q^{19} - q^{20} - q^{21} + 6 q^{23} - q^{24} - 4 q^{25} + 5 q^{27} + q^{28} - 8 q^{29} + q^{30} + 8 q^{31} + q^{32} - 3 q^{34} - q^{35} - 2 q^{36} + 5 q^{37} + q^{38} - q^{40} + 2 q^{41} - q^{42} - q^{43} + 2 q^{45} + 6 q^{46} - 3 q^{47} - q^{48} - 6 q^{49} - 4 q^{50} + 3 q^{51} - 2 q^{53} + 5 q^{54} + q^{56} - q^{57} - 8 q^{58} + 10 q^{59} + q^{60} - 14 q^{61} + 8 q^{62} - 2 q^{63} + q^{64} + 4 q^{67} - 3 q^{68} - 6 q^{69} - q^{70} - 3 q^{71} - 2 q^{72} - 16 q^{73} + 5 q^{74} + 4 q^{75} + q^{76} + 4 q^{79} - q^{80} + q^{81} + 2 q^{82} - 16 q^{83} - q^{84} + 3 q^{85} - q^{86} + 8 q^{87} - 8 q^{89} + 2 q^{90} + 6 q^{92} - 8 q^{93} - 3 q^{94} - q^{95} - q^{96} + 10 q^{97} - 6 q^{98} + O(q^{100}) \)

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 −1.00000 1.00000 −1.00000 −1.00000 1.00000 1.00000 −2.00000 −1.00000
\(n\): 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.e 1
13.b even 2 1 494.2.a.a 1
39.d odd 2 1 4446.2.a.n 1
52.b odd 2 1 3952.2.a.i 1
247.d odd 2 1 9386.2.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.a.a 1 13.b even 2 1
3952.2.a.i 1 52.b odd 2 1
4446.2.a.n 1 39.d odd 2 1
6422.2.a.e 1 1.a even 1 1 trivial
9386.2.a.l 1 247.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3} + 1 \)
\( T_{5} + 1 \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( 3 + T \)
$19$ \( -1 + T \)
$23$ \( -6 + T \)
$29$ \( 8 + T \)
$31$ \( -8 + T \)
$37$ \( -5 + T \)
$41$ \( -2 + T \)
$43$ \( 1 + T \)
$47$ \( 3 + T \)
$53$ \( 2 + T \)
$59$ \( -10 + T \)
$61$ \( 14 + T \)
$67$ \( -4 + T \)
$71$ \( 3 + T \)
$73$ \( 16 + T \)
$79$ \( -4 + T \)
$83$ \( 16 + T \)
$89$ \( 8 + T \)
$97$ \( -10 + T \)
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