Properties

Label 1386.2.a.a
Level $1386$
Weight $2$
Character orbit 1386.a
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - 2q^{5} - q^{7} - q^{8} + O(q^{10}) \) \( q - q^{2} + q^{4} - 2q^{5} - q^{7} - q^{8} + 2q^{10} - q^{11} - 2q^{13} + q^{14} + q^{16} + 2q^{17} - 2q^{20} + q^{22} - q^{25} + 2q^{26} - q^{28} + 2q^{29} + 4q^{31} - q^{32} - 2q^{34} + 2q^{35} - 2q^{37} + 2q^{40} + 10q^{41} + 4q^{43} - q^{44} - 4q^{47} + q^{49} + q^{50} - 2q^{52} + 2q^{53} + 2q^{55} + q^{56} - 2q^{58} + 12q^{59} - 2q^{61} - 4q^{62} + q^{64} + 4q^{65} + 12q^{67} + 2q^{68} - 2q^{70} - 8q^{71} + 6q^{73} + 2q^{74} + q^{77} - 8q^{79} - 2q^{80} - 10q^{82} + 8q^{83} - 4q^{85} - 4q^{86} + q^{88} + 14q^{89} + 2q^{91} + 4q^{94} - 14q^{97} - q^{98} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(1\)

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 1386.2.a.a 1
3.b odd 2 1 462.2.a.g 1
7.b odd 2 1 9702.2.a.r 1
12.b even 2 1 3696.2.a.m 1
21.c even 2 1 3234.2.a.p 1
33.d even 2 1 5082.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.g 1 3.b odd 2 1
1386.2.a.a 1 1.a even 1 1 trivial
3234.2.a.p 1 21.c even 2 1
3696.2.a.m 1 12.b even 2 1
5082.2.a.n 1 33.d even 2 1
9702.2.a.r 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1386))\):

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

Hecke characteristic polynomials

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