Properties

Label 3648.2.a.r
Level $3648$
Weight $2$
Character orbit 3648.a
Self dual yes
Analytic conductor $29.129$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 3q^{5} + 5q^{7} + q^{9} + O(q^{10}) \) \( q - q^{3} + 3q^{5} + 5q^{7} + q^{9} + q^{11} - 2q^{13} - 3q^{15} - q^{17} - q^{19} - 5q^{21} + 4q^{23} + 4q^{25} - q^{27} + 2q^{29} + 6q^{31} - q^{33} + 15q^{35} + 2q^{39} - q^{43} + 3q^{45} + 9q^{47} + 18q^{49} + q^{51} - 10q^{53} + 3q^{55} + q^{57} - 8q^{59} + q^{61} + 5q^{63} - 6q^{65} + 8q^{67} - 4q^{69} + 12q^{71} - 11q^{73} - 4q^{75} + 5q^{77} - 16q^{79} + q^{81} + 12q^{83} - 3q^{85} - 2q^{87} - 6q^{89} - 10q^{91} - 6q^{93} - 3q^{95} - 10q^{97} + q^{99} + O(q^{100}) \)

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 −1.00000 0 3.00000 0 5.00000 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.r 1
4.b odd 2 1 3648.2.a.bh 1
8.b even 2 1 912.2.a.g 1
8.d odd 2 1 57.2.a.a 1
24.f even 2 1 171.2.a.d 1
24.h odd 2 1 2736.2.a.v 1
40.e odd 2 1 1425.2.a.j 1
40.k even 4 2 1425.2.c.b 2
56.e even 2 1 2793.2.a.b 1
88.g even 2 1 6897.2.a.f 1
104.h odd 2 1 9633.2.a.o 1
120.m even 2 1 4275.2.a.b 1
152.b even 2 1 1083.2.a.e 1
168.e odd 2 1 8379.2.a.p 1
456.l odd 2 1 3249.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.a 1 8.d odd 2 1
171.2.a.d 1 24.f even 2 1
912.2.a.g 1 8.b even 2 1
1083.2.a.e 1 152.b even 2 1
1425.2.a.j 1 40.e odd 2 1
1425.2.c.b 2 40.k even 4 2
2736.2.a.v 1 24.h odd 2 1
2793.2.a.b 1 56.e even 2 1
3249.2.a.b 1 456.l odd 2 1
3648.2.a.r 1 1.a even 1 1 trivial
3648.2.a.bh 1 4.b odd 2 1
4275.2.a.b 1 120.m even 2 1
6897.2.a.f 1 88.g even 2 1
8379.2.a.p 1 168.e odd 2 1
9633.2.a.o 1 104.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3648))\):

\( T_{5} - 3 \)
\( T_{7} - 5 \)
\( T_{11} - 1 \)
\( T_{23} - 4 \)
\( T_{31} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( -3 + T \)
$7$ \( -5 + T \)
$11$ \( -1 + T \)
$13$ \( 2 + T \)
$17$ \( 1 + T \)
$19$ \( 1 + T \)
$23$ \( -4 + T \)
$29$ \( -2 + T \)
$31$ \( -6 + T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( 1 + T \)
$47$ \( -9 + T \)
$53$ \( 10 + T \)
$59$ \( 8 + T \)
$61$ \( -1 + T \)
$67$ \( -8 + T \)
$71$ \( -12 + T \)
$73$ \( 11 + T \)
$79$ \( 16 + T \)
$83$ \( -12 + T \)
$89$ \( 6 + T \)
$97$ \( 10 + T \)
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