Properties

Label 6422.2.a.j
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

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

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 3.00000 1.00000 3.00000 3.00000 −3.00000 1.00000 6.00000 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.j 1
13.b even 2 1 494.2.a.c 1
39.d odd 2 1 4446.2.a.u 1
52.b odd 2 1 3952.2.a.a 1
247.d odd 2 1 9386.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.a.c 1 13.b even 2 1
3952.2.a.a 1 52.b odd 2 1
4446.2.a.u 1 39.d odd 2 1
6422.2.a.j 1 1.a even 1 1 trivial
9386.2.a.h 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

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

Hecke characteristic polynomials

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