Properties

Label 3822.2.a.bf
Level $3822$
Weight $2$
Character orbit 3822.a
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 5q^{11} + q^{12} + q^{13} + q^{16} - 7q^{17} + q^{18} - 7q^{19} - 5q^{22} + 2q^{23} + q^{24} - 5q^{25} + q^{26} + q^{27} - 9q^{29} + q^{32} - 5q^{33} - 7q^{34} + q^{36} + 4q^{37} - 7q^{38} + q^{39} - 4q^{41} + 2q^{43} - 5q^{44} + 2q^{46} + 3q^{47} + q^{48} - 5q^{50} - 7q^{51} + q^{52} + q^{53} + q^{54} - 7q^{57} - 9q^{58} - 7q^{59} - 13q^{61} + q^{64} - 5q^{66} + 3q^{67} - 7q^{68} + 2q^{69} + 9q^{71} + q^{72} + 10q^{73} + 4q^{74} - 5q^{75} - 7q^{76} + q^{78} + 14q^{79} + q^{81} - 4q^{82} - 16q^{83} + 2q^{86} - 9q^{87} - 5q^{88} + 12q^{89} + 2q^{92} + 3q^{94} + q^{96} - 6q^{97} - 5q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(13\) \(-1\)

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 3822.2.a.bf 1
7.b odd 2 1 3822.2.a.u 1
7.d odd 6 2 546.2.i.d 2
21.g even 6 2 1638.2.j.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.i.d 2 7.d odd 6 2
1638.2.j.i 2 21.g even 6 2
3822.2.a.u 1 7.b odd 2 1
3822.2.a.bf 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5} \)
\( T_{11} + 5 \)
\( T_{17} + 7 \)
\( T_{29} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 5 + T \)
$13$ \( -1 + T \)
$17$ \( 7 + T \)
$19$ \( 7 + T \)
$23$ \( -2 + T \)
$29$ \( 9 + T \)
$31$ \( T \)
$37$ \( -4 + T \)
$41$ \( 4 + T \)
$43$ \( -2 + T \)
$47$ \( -3 + T \)
$53$ \( -1 + T \)
$59$ \( 7 + T \)
$61$ \( 13 + T \)
$67$ \( -3 + T \)
$71$ \( -9 + T \)
$73$ \( -10 + T \)
$79$ \( -14 + T \)
$83$ \( 16 + T \)
$89$ \( -12 + T \)
$97$ \( 6 + T \)
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