Properties

Label 380.2.a.b
Level $380$
Weight $2$
Character orbit 380.a
Self dual yes
Analytic conductor $3.034$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.03431527681\)
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

\(f(q)\) \(=\) \( q + 2q^{3} - q^{5} + 2q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} - q^{5} + 2q^{7} + q^{9} + 6q^{13} - 2q^{15} + 2q^{17} - q^{19} + 4q^{21} - 2q^{23} + q^{25} - 4q^{27} - 2q^{29} + 4q^{31} - 2q^{35} - 10q^{37} + 12q^{39} - 10q^{41} + 6q^{43} - q^{45} - 6q^{47} - 3q^{49} + 4q^{51} + 6q^{53} - 2q^{57} - 4q^{59} + 2q^{61} + 2q^{63} - 6q^{65} - 2q^{67} - 4q^{69} + 12q^{71} - 6q^{73} + 2q^{75} + 8q^{79} - 11q^{81} - 2q^{83} - 2q^{85} - 4q^{87} + 2q^{89} + 12q^{91} + 8q^{93} + q^{95} - 18q^{97} + O(q^{100}) \)

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 2.00000 0 −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
\(2\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 380.2.a.b 1
3.b odd 2 1 3420.2.a.f 1
4.b odd 2 1 1520.2.a.b 1
5.b even 2 1 1900.2.a.a 1
5.c odd 4 2 1900.2.c.a 2
8.b even 2 1 6080.2.a.f 1
8.d odd 2 1 6080.2.a.v 1
19.b odd 2 1 7220.2.a.b 1
20.d odd 2 1 7600.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.b 1 1.a even 1 1 trivial
1520.2.a.b 1 4.b odd 2 1
1900.2.a.a 1 5.b even 2 1
1900.2.c.a 2 5.c odd 4 2
3420.2.a.f 1 3.b odd 2 1
6080.2.a.f 1 8.b even 2 1
6080.2.a.v 1 8.d odd 2 1
7220.2.a.b 1 19.b odd 2 1
7600.2.a.r 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(380))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( 1 + T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( -2 + T \)
$19$ \( 1 + T \)
$23$ \( 2 + T \)
$29$ \( 2 + T \)
$31$ \( -4 + T \)
$37$ \( 10 + T \)
$41$ \( 10 + T \)
$43$ \( -6 + T \)
$47$ \( 6 + T \)
$53$ \( -6 + T \)
$59$ \( 4 + T \)
$61$ \( -2 + T \)
$67$ \( 2 + T \)
$71$ \( -12 + T \)
$73$ \( 6 + T \)
$79$ \( -8 + T \)
$83$ \( 2 + T \)
$89$ \( -2 + T \)
$97$ \( 18 + T \)
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