Properties

Label 6422.2.a.c
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} - q^{8} - 2q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} - q^{8} - 2q^{9} - 3q^{10} - 2q^{11} + q^{12} + q^{14} + 3q^{15} + q^{16} + q^{17} + 2q^{18} + q^{19} + 3q^{20} - q^{21} + 2q^{22} - 4q^{23} - q^{24} + 4q^{25} - 5q^{27} - q^{28} + 2q^{29} - 3q^{30} + 4q^{31} - q^{32} - 2q^{33} - q^{34} - 3q^{35} - 2q^{36} - 3q^{37} - q^{38} - 3q^{40} - 10q^{41} + q^{42} - 11q^{43} - 2q^{44} - 6q^{45} + 4q^{46} + 3q^{47} + q^{48} - 6q^{49} - 4q^{50} + q^{51} + 5q^{54} - 6q^{55} + q^{56} + q^{57} - 2q^{58} + 6q^{59} + 3q^{60} + 6q^{61} - 4q^{62} + 2q^{63} + q^{64} + 2q^{66} + 2q^{67} + q^{68} - 4q^{69} + 3q^{70} + 3q^{71} + 2q^{72} + 4q^{73} + 3q^{74} + 4q^{75} + q^{76} + 2q^{77} - 6q^{79} + 3q^{80} + q^{81} + 10q^{82} + 14q^{83} - q^{84} + 3q^{85} + 11q^{86} + 2q^{87} + 2q^{88} - 8q^{89} + 6q^{90} - 4q^{92} + 4q^{93} - 3q^{94} + 3q^{95} - q^{96} - 16q^{97} + 6q^{98} + 4q^{99} + O(q^{100}) \)

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
−1.00000 1.00000 1.00000 3.00000 −1.00000 −1.00000 −1.00000 −2.00000 −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.c 1
13.b even 2 1 6422.2.a.g 1
13.d odd 4 2 494.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.d.a 2 13.d odd 4 2
6422.2.a.c 1 1.a even 1 1 trivial
6422.2.a.g 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

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

Hecke characteristic polynomials

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