Properties

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

Related objects

Downloads

Learn more

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 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + 2 q^{7} - q^{11} - 4 q^{13} + 2 q^{17} - q^{23} - 4 q^{25} - 7 q^{31} + 2 q^{35} - 3 q^{37} + 8 q^{41} - 6 q^{43} + 8 q^{47} - 3 q^{49} - 6 q^{53} - q^{55} - 5 q^{59} - 12 q^{61} - 4 q^{65} - 7 q^{67} - 3 q^{71} + 4 q^{73} - 2 q^{77} + 10 q^{79} + 6 q^{83} + 2 q^{85} - 15 q^{89} - 8 q^{91} - 7 q^{97}+O(q^{100}) \) 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 1.00000 0 2.00000 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.bu 1
3.b odd 2 1 704.2.a.c 1
4.b odd 2 1 6336.2.a.br 1
8.b even 2 1 1584.2.a.g 1
8.d odd 2 1 99.2.a.d 1
12.b even 2 1 704.2.a.h 1
24.f even 2 1 11.2.a.a 1
24.h odd 2 1 176.2.a.b 1
33.d even 2 1 7744.2.a.k 1
40.e odd 2 1 2475.2.a.a 1
40.k even 4 2 2475.2.c.a 2
48.i odd 4 2 2816.2.c.f 2
48.k even 4 2 2816.2.c.j 2
56.e even 2 1 4851.2.a.t 1
72.l even 6 2 891.2.e.k 2
72.p odd 6 2 891.2.e.b 2
88.g even 2 1 1089.2.a.b 1
120.i odd 2 1 4400.2.a.i 1
120.m even 2 1 275.2.a.b 1
120.q odd 4 2 275.2.b.a 2
120.w even 4 2 4400.2.b.h 2
132.d odd 2 1 7744.2.a.x 1
168.e odd 2 1 539.2.a.a 1
168.i even 2 1 8624.2.a.j 1
168.v even 6 2 539.2.e.h 2
168.be odd 6 2 539.2.e.g 2
264.m even 2 1 1936.2.a.i 1
264.p odd 2 1 121.2.a.d 1
264.r odd 10 4 121.2.c.a 4
264.w even 10 4 121.2.c.e 4
312.h even 2 1 1859.2.a.b 1
408.h even 2 1 3179.2.a.a 1
456.l odd 2 1 3971.2.a.b 1
552.h odd 2 1 5819.2.a.a 1
696.l even 2 1 9251.2.a.d 1
1320.b odd 2 1 3025.2.a.a 1
1848.e even 2 1 5929.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 24.f even 2 1
99.2.a.d 1 8.d odd 2 1
121.2.a.d 1 264.p odd 2 1
121.2.c.a 4 264.r odd 10 4
121.2.c.e 4 264.w even 10 4
176.2.a.b 1 24.h odd 2 1
275.2.a.b 1 120.m even 2 1
275.2.b.a 2 120.q odd 4 2
539.2.a.a 1 168.e odd 2 1
539.2.e.g 2 168.be odd 6 2
539.2.e.h 2 168.v even 6 2
704.2.a.c 1 3.b odd 2 1
704.2.a.h 1 12.b even 2 1
891.2.e.b 2 72.p odd 6 2
891.2.e.k 2 72.l even 6 2
1089.2.a.b 1 88.g even 2 1
1584.2.a.g 1 8.b even 2 1
1859.2.a.b 1 312.h even 2 1
1936.2.a.i 1 264.m even 2 1
2475.2.a.a 1 40.e odd 2 1
2475.2.c.a 2 40.k even 4 2
2816.2.c.f 2 48.i odd 4 2
2816.2.c.j 2 48.k even 4 2
3025.2.a.a 1 1320.b odd 2 1
3179.2.a.a 1 408.h even 2 1
3971.2.a.b 1 456.l odd 2 1
4400.2.a.i 1 120.i odd 2 1
4400.2.b.h 2 120.w even 4 2
4851.2.a.t 1 56.e even 2 1
5819.2.a.a 1 552.h odd 2 1
5929.2.a.h 1 1848.e even 2 1
6336.2.a.br 1 4.b odd 2 1
6336.2.a.bu 1 1.a even 1 1 trivial
7744.2.a.k 1 33.d even 2 1
7744.2.a.x 1 132.d odd 2 1
8624.2.a.j 1 168.i even 2 1
9251.2.a.d 1 696.l even 2 1

Hecke kernels

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

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23} + 1 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 4 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T - 8 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 5 \) Copy content Toggle raw display
$61$ \( T + 12 \) Copy content Toggle raw display
$67$ \( T + 7 \) Copy content Toggle raw display
$71$ \( T + 3 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
show more
show less