Properties

Label 3648.2.a.m
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3648.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.1294266574\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1824)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{5} + q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + q^{5} + q^{7} + q^{9} + 3q^{11} - q^{15} - 7q^{17} + q^{19} - q^{21} - 8q^{23} - 4q^{25} - q^{27} + 2q^{31} - 3q^{33} + q^{35} - 4q^{37} - 4q^{41} - q^{43} + q^{45} + 3q^{47} - 6q^{49} + 7q^{51} - 6q^{53} + 3q^{55} - q^{57} - 6q^{59} + 5q^{61} + q^{63} - 2q^{67} + 8q^{69} + 2q^{71} - 11q^{73} + 4q^{75} + 3q^{77} + 10q^{79} + q^{81} + 16q^{83} - 7q^{85} - 14q^{89} - 2q^{93} + q^{95} - 8q^{97} + 3q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)

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 3648.2.a.m 1
4.b odd 2 1 3648.2.a.bd 1
8.b even 2 1 1824.2.a.i yes 1
8.d odd 2 1 1824.2.a.b 1
24.f even 2 1 5472.2.a.n 1
24.h odd 2 1 5472.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1824.2.a.b 1 8.d odd 2 1
1824.2.a.i yes 1 8.b even 2 1
3648.2.a.m 1 1.a even 1 1 trivial
3648.2.a.bd 1 4.b odd 2 1
5472.2.a.n 1 24.f even 2 1
5472.2.a.r 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5} - 1 \)
\( T_{7} - 1 \)
\( T_{11} - 3 \)
\( T_{23} + 8 \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( -3 + T \)
$13$ \( T \)
$17$ \( 7 + T \)
$19$ \( -1 + T \)
$23$ \( 8 + T \)
$29$ \( T \)
$31$ \( -2 + T \)
$37$ \( 4 + T \)
$41$ \( 4 + T \)
$43$ \( 1 + T \)
$47$ \( -3 + T \)
$53$ \( 6 + T \)
$59$ \( 6 + T \)
$61$ \( -5 + T \)
$67$ \( 2 + T \)
$71$ \( -2 + T \)
$73$ \( 11 + T \)
$79$ \( -10 + T \)
$83$ \( -16 + T \)
$89$ \( 14 + T \)
$97$ \( 8 + T \)
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