Properties

Label 3800.2.a.g
Level $3800$
Weight $2$
Character orbit 3800.a
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} - 2 q^{7} + q^{9} + O(q^{10}) \) \( q + 2 q^{3} - 2 q^{7} + q^{9} + 4 q^{11} - 8 q^{17} - q^{19} - 4 q^{21} - 6 q^{23} - 4 q^{27} + 2 q^{29} - 8 q^{31} + 8 q^{33} - 6 q^{41} - 10 q^{43} + 6 q^{47} - 3 q^{49} - 16 q^{51} - 2 q^{57} - 4 q^{59} + 6 q^{61} - 2 q^{63} - 2 q^{67} - 12 q^{69} + 16 q^{71} + 16 q^{73} - 8 q^{77} + 8 q^{79} - 11 q^{81} - 10 q^{83} + 4 q^{87} - 10 q^{89} - 16 q^{93} - 4 q^{97} + 4 q^{99} + O(q^{100}) \)

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 2.00000 0 0 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 3800.2.a.g 1
4.b odd 2 1 7600.2.a.e 1
5.b even 2 1 3800.2.a.c 1
5.c odd 4 2 760.2.d.a 2
20.d odd 2 1 7600.2.a.q 1
20.e even 4 2 1520.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.d.a 2 5.c odd 4 2
1520.2.d.a 2 20.e even 4 2
3800.2.a.c 1 5.b even 2 1
3800.2.a.g 1 1.a even 1 1 trivial
7600.2.a.e 1 4.b odd 2 1
7600.2.a.q 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3800))\):

\( T_{3} - 2 \)
\( T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( 2 + T \)
$11$ \( -4 + T \)
$13$ \( T \)
$17$ \( 8 + T \)
$19$ \( 1 + T \)
$23$ \( 6 + T \)
$29$ \( -2 + T \)
$31$ \( 8 + T \)
$37$ \( T \)
$41$ \( 6 + T \)
$43$ \( 10 + T \)
$47$ \( -6 + T \)
$53$ \( T \)
$59$ \( 4 + T \)
$61$ \( -6 + T \)
$67$ \( 2 + T \)
$71$ \( -16 + T \)
$73$ \( -16 + T \)
$79$ \( -8 + T \)
$83$ \( 10 + T \)
$89$ \( 10 + T \)
$97$ \( 4 + T \)
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