Properties

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

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - 2q^{5} - 4q^{7} + q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} + q^{4} - 2q^{5} - 4q^{7} + q^{8} - 3q^{9} - 2q^{10} - 4q^{11} - 4q^{14} + q^{16} + 2q^{17} - 3q^{18} - q^{19} - 2q^{20} - 4q^{22} - 8q^{23} - q^{25} - 4q^{28} + 2q^{29} + q^{32} + 2q^{34} + 8q^{35} - 3q^{36} - 10q^{37} - q^{38} - 2q^{40} + 2q^{41} + 12q^{43} - 4q^{44} + 6q^{45} - 8q^{46} - 4q^{47} + 9q^{49} - q^{50} - 6q^{53} + 8q^{55} - 4q^{56} + 2q^{58} + 12q^{59} + 14q^{61} + 12q^{63} + q^{64} + 12q^{67} + 2q^{68} + 8q^{70} - 8q^{71} - 3q^{72} - 2q^{73} - 10q^{74} - q^{76} + 16q^{77} - 16q^{79} - 2q^{80} + 9q^{81} + 2q^{82} + 12q^{83} - 4q^{85} + 12q^{86} - 4q^{88} - 6q^{89} + 6q^{90} - 8q^{92} - 4q^{94} + 2q^{95} + 2q^{97} + 9q^{98} + 12q^{99} + O(q^{100}) \)

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 0 1.00000 −2.00000 0 −4.00000 1.00000 −3.00000 −2.00000
\(n\): 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.f 1
13.b even 2 1 494.2.a.b 1
39.d odd 2 1 4446.2.a.m 1
52.b odd 2 1 3952.2.a.e 1
247.d odd 2 1 9386.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.a.b 1 13.b even 2 1
3952.2.a.e 1 52.b odd 2 1
4446.2.a.m 1 39.d odd 2 1
6422.2.a.f 1 1.a even 1 1 trivial
9386.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\( T_{3} \)
\( T_{5} + 2 \)
\( T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( 2 + T \)
$7$ \( 4 + T \)
$11$ \( 4 + T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( 8 + T \)
$29$ \( -2 + T \)
$31$ \( T \)
$37$ \( 10 + T \)
$41$ \( -2 + T \)
$43$ \( -12 + T \)
$47$ \( 4 + T \)
$53$ \( 6 + T \)
$59$ \( -12 + T \)
$61$ \( -14 + T \)
$67$ \( -12 + T \)
$71$ \( 8 + T \)
$73$ \( 2 + T \)
$79$ \( 16 + T \)
$83$ \( -12 + T \)
$89$ \( 6 + T \)
$97$ \( -2 + T \)
show more
show less