Properties

Label 6776.2.a.g
Level $6776$
Weight $2$
Character orbit 6776.a
Self dual yes
Analytic conductor $54.107$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6776 = 2^{3} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6776.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.1066324096\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} + q^{7} - 3q^{9} + O(q^{10}) \) \( q + 2q^{5} + q^{7} - 3q^{9} - 2q^{13} + 6q^{17} - 8q^{19} - q^{25} - 6q^{29} + 8q^{31} + 2q^{35} - 2q^{37} - 2q^{41} + 4q^{43} - 6q^{45} - 8q^{47} + q^{49} + 6q^{53} + 6q^{61} - 3q^{63} - 4q^{65} - 4q^{67} - 8q^{71} - 10q^{73} - 16q^{79} + 9q^{81} - 8q^{83} + 12q^{85} - 6q^{89} - 2q^{91} - 16q^{95} - 6q^{97} + O(q^{100}) \)

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
0 0 0 2.00000 0 1.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 6776.2.a.g 1
11.b odd 2 1 56.2.a.a 1
33.d even 2 1 504.2.a.c 1
44.c even 2 1 112.2.a.b 1
55.d odd 2 1 1400.2.a.g 1
55.e even 4 2 1400.2.g.g 2
77.b even 2 1 392.2.a.d 1
77.h odd 6 2 392.2.i.c 2
77.i even 6 2 392.2.i.d 2
88.b odd 2 1 448.2.a.d 1
88.g even 2 1 448.2.a.e 1
132.d odd 2 1 1008.2.a.d 1
143.d odd 2 1 9464.2.a.c 1
176.i even 4 2 1792.2.b.d 2
176.l odd 4 2 1792.2.b.i 2
220.g even 2 1 2800.2.a.p 1
220.i odd 4 2 2800.2.g.p 2
231.h odd 2 1 3528.2.a.x 1
231.k odd 6 2 3528.2.s.e 2
231.l even 6 2 3528.2.s.t 2
264.m even 2 1 4032.2.a.bb 1
264.p odd 2 1 4032.2.a.bk 1
308.g odd 2 1 784.2.a.e 1
308.m odd 6 2 784.2.i.g 2
308.n even 6 2 784.2.i.e 2
385.h even 2 1 9800.2.a.u 1
616.g odd 2 1 3136.2.a.p 1
616.o even 2 1 3136.2.a.q 1
924.n even 2 1 7056.2.a.bo 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 11.b odd 2 1
112.2.a.b 1 44.c even 2 1
392.2.a.d 1 77.b even 2 1
392.2.i.c 2 77.h odd 6 2
392.2.i.d 2 77.i even 6 2
448.2.a.d 1 88.b odd 2 1
448.2.a.e 1 88.g even 2 1
504.2.a.c 1 33.d even 2 1
784.2.a.e 1 308.g odd 2 1
784.2.i.e 2 308.n even 6 2
784.2.i.g 2 308.m odd 6 2
1008.2.a.d 1 132.d odd 2 1
1400.2.a.g 1 55.d odd 2 1
1400.2.g.g 2 55.e even 4 2
1792.2.b.d 2 176.i even 4 2
1792.2.b.i 2 176.l odd 4 2
2800.2.a.p 1 220.g even 2 1
2800.2.g.p 2 220.i odd 4 2
3136.2.a.p 1 616.g odd 2 1
3136.2.a.q 1 616.o even 2 1
3528.2.a.x 1 231.h odd 2 1
3528.2.s.e 2 231.k odd 6 2
3528.2.s.t 2 231.l even 6 2
4032.2.a.bb 1 264.m even 2 1
4032.2.a.bk 1 264.p odd 2 1
6776.2.a.g 1 1.a even 1 1 trivial
7056.2.a.bo 1 924.n even 2 1
9464.2.a.c 1 143.d odd 2 1
9800.2.a.u 1 385.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6776))\):

\( T_{3} \)
\( T_{5} - 2 \)
\( T_{13} + 2 \)
\( T_{17} - 6 \)

Hecke characteristic polynomials

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