Properties

Label 6776.2.a.d
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 616)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{5} - q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} - q^{7} - 2 q^{9} + q^{15} + 2 q^{17} + 2 q^{19} + q^{21} - 7 q^{23} - 4 q^{25} + 5 q^{27} + 10 q^{29} + 7 q^{31} + q^{35} - 9 q^{37} + 2 q^{41} + 4 q^{43} + 2 q^{45} + 8 q^{47} + q^{49} - 2 q^{51} + 2 q^{53} - 2 q^{57} - 15 q^{59} + 14 q^{61} + 2 q^{63} + 3 q^{67} + 7 q^{69} + 3 q^{71} - 10 q^{73} + 4 q^{75} - 10 q^{79} + q^{81} - 2 q^{85} - 10 q^{87} - 11 q^{89} - 7 q^{93} - 2 q^{95} + 7 q^{97}+O(q^{100}) \) 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 −1.00000 0 −1.00000 0 −1.00000 0 −2.00000 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.d 1
11.b odd 2 1 616.2.a.b 1
33.d even 2 1 5544.2.a.o 1
44.c even 2 1 1232.2.a.i 1
77.b even 2 1 4312.2.a.h 1
88.b odd 2 1 4928.2.a.ba 1
88.g even 2 1 4928.2.a.k 1
308.g odd 2 1 8624.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.a.b 1 11.b odd 2 1
1232.2.a.i 1 44.c even 2 1
4312.2.a.h 1 77.b even 2 1
4928.2.a.k 1 88.g even 2 1
4928.2.a.ba 1 88.b odd 2 1
5544.2.a.o 1 33.d even 2 1
6776.2.a.d 1 1.a even 1 1 trivial
8624.2.a.m 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6776))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T + 7 \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T - 7 \) Copy content Toggle raw display
$37$ \( T + 9 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T + 15 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T - 3 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 11 \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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