Properties

Label 381.2.a.a
Level $381$
Weight $2$
Character orbit 381.a
Self dual yes
Analytic conductor $3.042$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.04230031701\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 2q^{4} - q^{5} - 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} - 2q^{4} - q^{5} - 2q^{7} + q^{9} - 4q^{11} - 2q^{12} - 3q^{13} - q^{15} + 4q^{16} - 4q^{19} + 2q^{20} - 2q^{21} - 3q^{23} - 4q^{25} + q^{27} + 4q^{28} + 5q^{29} - 5q^{31} - 4q^{33} + 2q^{35} - 2q^{36} + 5q^{37} - 3q^{39} + 4q^{41} - 4q^{43} + 8q^{44} - q^{45} + 12q^{47} + 4q^{48} - 3q^{49} + 6q^{52} - q^{53} + 4q^{55} - 4q^{57} + 5q^{59} + 2q^{60} - 5q^{61} - 2q^{63} - 8q^{64} + 3q^{65} - 8q^{67} - 3q^{69} - 6q^{71} - q^{73} - 4q^{75} + 8q^{76} + 8q^{77} + 8q^{79} - 4q^{80} + q^{81} - 3q^{83} + 4q^{84} + 5q^{87} + 7q^{89} + 6q^{91} + 6q^{92} - 5q^{93} + 4q^{95} + 4q^{97} - 4q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(127\) \(-1\)

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 381.2.a.a 1
3.b odd 2 1 1143.2.a.c 1
4.b odd 2 1 6096.2.a.e 1
5.b even 2 1 9525.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.a 1 1.a even 1 1 trivial
1143.2.a.c 1 3.b odd 2 1
6096.2.a.e 1 4.b odd 2 1
9525.2.a.d 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(381))\).

Hecke characteristic polynomials

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