Properties

Label 6336.2.a.n
Level $6336$
Weight $2$
Character orbit 6336.a
Self dual yes
Analytic conductor $50.593$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.5932147207\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - 4 q^{7} - q^{11} + 2 q^{13} + 2 q^{17} + 8 q^{23} - q^{25} - 6 q^{29} + 8 q^{31} + 8 q^{35} - 6 q^{37} + 2 q^{41} + 8 q^{47} + 9 q^{49} + 6 q^{53} + 2 q^{55} + 4 q^{59} - 6 q^{61} - 4 q^{65} - 4 q^{67} - 14 q^{73} + 4 q^{77} + 4 q^{79} - 12 q^{83} - 4 q^{85} + 6 q^{89} - 8 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0 0 0 −2.00000 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(11\) \(1\)

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 6336.2.a.n 1
3.b odd 2 1 2112.2.a.j 1
4.b odd 2 1 6336.2.a.x 1
8.b even 2 1 1584.2.a.o 1
8.d odd 2 1 99.2.a.b 1
12.b even 2 1 2112.2.a.bb 1
24.f even 2 1 33.2.a.a 1
24.h odd 2 1 528.2.a.g 1
40.e odd 2 1 2475.2.a.g 1
40.k even 4 2 2475.2.c.d 2
56.e even 2 1 4851.2.a.b 1
72.l even 6 2 891.2.e.e 2
72.p odd 6 2 891.2.e.g 2
88.g even 2 1 1089.2.a.j 1
120.m even 2 1 825.2.a.a 1
120.q odd 4 2 825.2.c.a 2
168.e odd 2 1 1617.2.a.j 1
264.m even 2 1 5808.2.a.t 1
264.p odd 2 1 363.2.a.b 1
264.r odd 10 4 363.2.e.g 4
264.w even 10 4 363.2.e.e 4
312.h even 2 1 5577.2.a.a 1
408.h even 2 1 9537.2.a.m 1
1320.b odd 2 1 9075.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 24.f even 2 1
99.2.a.b 1 8.d odd 2 1
363.2.a.b 1 264.p odd 2 1
363.2.e.e 4 264.w even 10 4
363.2.e.g 4 264.r odd 10 4
528.2.a.g 1 24.h odd 2 1
825.2.a.a 1 120.m even 2 1
825.2.c.a 2 120.q odd 4 2
891.2.e.e 2 72.l even 6 2
891.2.e.g 2 72.p odd 6 2
1089.2.a.j 1 88.g even 2 1
1584.2.a.o 1 8.b even 2 1
1617.2.a.j 1 168.e odd 2 1
2112.2.a.j 1 3.b odd 2 1
2112.2.a.bb 1 12.b even 2 1
2475.2.a.g 1 40.e odd 2 1
2475.2.c.d 2 40.k even 4 2
4851.2.a.b 1 56.e even 2 1
5577.2.a.a 1 312.h even 2 1
5808.2.a.t 1 264.m even 2 1
6336.2.a.n 1 1.a even 1 1 trivial
6336.2.a.x 1 4.b odd 2 1
9075.2.a.q 1 1320.b odd 2 1
9537.2.a.m 1 408.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6336))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23} - 8 \) Copy content Toggle raw display
\( T_{47} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

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