Properties

Label 381.2.a.b
Level $381$
Weight $2$
Character orbit 381.a
Self dual yes
Analytic conductor $3.042$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 4q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{2} + q^{3} + 2q^{4} + 3q^{5} + 2q^{6} - 4q^{7} + q^{9} + 6q^{10} + 6q^{11} + 2q^{12} - 7q^{13} - 8q^{14} + 3q^{15} - 4q^{16} - 2q^{17} + 2q^{18} + 6q^{20} - 4q^{21} + 12q^{22} + q^{23} + 4q^{25} - 14q^{26} + q^{27} - 8q^{28} + 9q^{29} + 6q^{30} - 5q^{31} - 8q^{32} + 6q^{33} - 4q^{34} - 12q^{35} + 2q^{36} - 3q^{37} - 7q^{39} - 6q^{41} - 8q^{42} + 4q^{43} + 12q^{44} + 3q^{45} + 2q^{46} + 2q^{47} - 4q^{48} + 9q^{49} + 8q^{50} - 2q^{51} - 14q^{52} - q^{53} + 2q^{54} + 18q^{55} + 18q^{58} + 13q^{59} + 6q^{60} - 5q^{61} - 10q^{62} - 4q^{63} - 8q^{64} - 21q^{65} + 12q^{66} - 2q^{67} - 4q^{68} + q^{69} - 24q^{70} + 6q^{71} - q^{73} - 6q^{74} + 4q^{75} - 24q^{77} - 14q^{78} - 12q^{80} + q^{81} - 12q^{82} - 7q^{83} - 8q^{84} - 6q^{85} + 8q^{86} + 9q^{87} + 15q^{89} + 6q^{90} + 28q^{91} + 2q^{92} - 5q^{93} + 4q^{94} - 8q^{96} + 2q^{97} + 18q^{98} + 6q^{99} + O(q^{100}) \)

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
2.00000 1.00000 2.00000 3.00000 2.00000 −4.00000 0 1.00000 6.00000
\(n\): 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.b 1
3.b odd 2 1 1143.2.a.a 1
4.b odd 2 1 6096.2.a.j 1
5.b even 2 1 9525.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.b 1 1.a even 1 1 trivial
1143.2.a.a 1 3.b odd 2 1
6096.2.a.j 1 4.b odd 2 1
9525.2.a.c 1 5.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T \)
$3$ \( -1 + T \)
$5$ \( -3 + T \)
$7$ \( 4 + T \)
$11$ \( -6 + T \)
$13$ \( 7 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( -1 + T \)
$29$ \( -9 + T \)
$31$ \( 5 + T \)
$37$ \( 3 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( -2 + T \)
$53$ \( 1 + T \)
$59$ \( -13 + T \)
$61$ \( 5 + T \)
$67$ \( 2 + T \)
$71$ \( -6 + T \)
$73$ \( 1 + T \)
$79$ \( T \)
$83$ \( 7 + T \)
$89$ \( -15 + T \)
$97$ \( -2 + T \)
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