Properties

Label 1380.2.a.d
Level $1380$
Weight $2$
Character orbit 1380.a
Self dual yes
Analytic conductor $11.019$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0193554789\)
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

\(f(q)\) \(=\) \( q + q^{3} + q^{5} - 4q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} - 4q^{7} + q^{9} + 2q^{13} + q^{15} + 6q^{17} + 2q^{19} - 4q^{21} - q^{23} + q^{25} + q^{27} + 6q^{29} - 4q^{31} - 4q^{35} + 8q^{37} + 2q^{39} + 6q^{41} + 8q^{43} + q^{45} + 12q^{47} + 9q^{49} + 6q^{51} - 6q^{53} + 2q^{57} - 6q^{59} - 10q^{61} - 4q^{63} + 2q^{65} + 8q^{67} - q^{69} - 6q^{71} + 2q^{73} + q^{75} - 10q^{79} + q^{81} - 12q^{83} + 6q^{85} + 6q^{87} + 6q^{89} - 8q^{91} - 4q^{93} + 2q^{95} + 8q^{97} + O(q^{100}) \)

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 1.00000 0 1.00000 0 −4.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 1380.2.a.d 1
3.b odd 2 1 4140.2.a.a 1
4.b odd 2 1 5520.2.a.o 1
5.b even 2 1 6900.2.a.e 1
5.c odd 4 2 6900.2.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.a.d 1 1.a even 1 1 trivial
4140.2.a.a 1 3.b odd 2 1
5520.2.a.o 1 4.b odd 2 1
6900.2.a.e 1 5.b even 2 1
6900.2.f.d 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\).

Hecke characteristic polynomials

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