Properties

Label 5408.2.a.g
Level $5408$
Weight $2$
Character orbit 5408.a
Self dual yes
Analytic conductor $43.183$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.1830974131\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{5} - 3q^{9} + O(q^{10}) \) \( q + 2q^{5} - 3q^{9} + 2q^{17} - q^{25} - 10q^{29} + 2q^{37} - 10q^{41} - 6q^{45} - 7q^{49} + 14q^{53} - 10q^{61} + 6q^{73} + 9q^{81} + 4q^{85} - 10q^{89} - 18q^{97} + O(q^{100}) \)

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 0 0 2.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(13\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5408.2.a.g 1
4.b odd 2 1 CM 5408.2.a.g 1
13.b even 2 1 32.2.a.a 1
39.d odd 2 1 288.2.a.d 1
52.b odd 2 1 32.2.a.a 1
65.d even 2 1 800.2.a.d 1
65.h odd 4 2 800.2.c.e 2
91.b odd 2 1 1568.2.a.e 1
91.r even 6 2 1568.2.i.g 2
91.s odd 6 2 1568.2.i.f 2
104.e even 2 1 64.2.a.a 1
104.h odd 2 1 64.2.a.a 1
117.n odd 6 2 2592.2.i.e 2
117.t even 6 2 2592.2.i.t 2
143.d odd 2 1 3872.2.a.f 1
156.h even 2 1 288.2.a.d 1
195.e odd 2 1 7200.2.a.v 1
195.s even 4 2 7200.2.f.m 2
208.o odd 4 2 256.2.b.b 2
208.p even 4 2 256.2.b.b 2
221.b even 2 1 9248.2.a.f 1
260.g odd 2 1 800.2.a.d 1
260.p even 4 2 800.2.c.e 2
312.b odd 2 1 576.2.a.c 1
312.h even 2 1 576.2.a.c 1
364.h even 2 1 1568.2.a.e 1
364.x even 6 2 1568.2.i.f 2
364.bl odd 6 2 1568.2.i.g 2
416.be odd 8 4 1024.2.e.j 4
416.bg even 8 4 1024.2.e.j 4
468.x even 6 2 2592.2.i.e 2
468.bg odd 6 2 2592.2.i.t 2
520.b odd 2 1 1600.2.a.n 1
520.p even 2 1 1600.2.a.n 1
520.bc even 4 2 1600.2.c.l 2
520.bg odd 4 2 1600.2.c.l 2
572.b even 2 1 3872.2.a.f 1
624.v even 4 2 2304.2.d.j 2
624.bi odd 4 2 2304.2.d.j 2
728.b even 2 1 3136.2.a.m 1
728.l odd 2 1 3136.2.a.m 1
780.d even 2 1 7200.2.a.v 1
780.w odd 4 2 7200.2.f.m 2
884.h odd 2 1 9248.2.a.f 1
1144.h odd 2 1 7744.2.a.v 1
1144.o even 2 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 13.b even 2 1
32.2.a.a 1 52.b odd 2 1
64.2.a.a 1 104.e even 2 1
64.2.a.a 1 104.h odd 2 1
256.2.b.b 2 208.o odd 4 2
256.2.b.b 2 208.p even 4 2
288.2.a.d 1 39.d odd 2 1
288.2.a.d 1 156.h even 2 1
576.2.a.c 1 312.b odd 2 1
576.2.a.c 1 312.h even 2 1
800.2.a.d 1 65.d even 2 1
800.2.a.d 1 260.g odd 2 1
800.2.c.e 2 65.h odd 4 2
800.2.c.e 2 260.p even 4 2
1024.2.e.j 4 416.be odd 8 4
1024.2.e.j 4 416.bg even 8 4
1568.2.a.e 1 91.b odd 2 1
1568.2.a.e 1 364.h even 2 1
1568.2.i.f 2 91.s odd 6 2
1568.2.i.f 2 364.x even 6 2
1568.2.i.g 2 91.r even 6 2
1568.2.i.g 2 364.bl odd 6 2
1600.2.a.n 1 520.b odd 2 1
1600.2.a.n 1 520.p even 2 1
1600.2.c.l 2 520.bc even 4 2
1600.2.c.l 2 520.bg odd 4 2
2304.2.d.j 2 624.v even 4 2
2304.2.d.j 2 624.bi odd 4 2
2592.2.i.e 2 117.n odd 6 2
2592.2.i.e 2 468.x even 6 2
2592.2.i.t 2 117.t even 6 2
2592.2.i.t 2 468.bg odd 6 2
3136.2.a.m 1 728.b even 2 1
3136.2.a.m 1 728.l odd 2 1
3872.2.a.f 1 143.d odd 2 1
3872.2.a.f 1 572.b even 2 1
5408.2.a.g 1 1.a even 1 1 trivial
5408.2.a.g 1 4.b odd 2 1 CM
7200.2.a.v 1 195.e odd 2 1
7200.2.a.v 1 780.d even 2 1
7200.2.f.m 2 195.s even 4 2
7200.2.f.m 2 780.w odd 4 2
7744.2.a.v 1 1144.h odd 2 1
7744.2.a.v 1 1144.o even 2 1
9248.2.a.f 1 221.b even 2 1
9248.2.a.f 1 884.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5408))\):

\( T_{3} \)
\( T_{5} - 2 \)
\( T_{7} \)
\( T_{37} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -2 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( T \)
$17$ \( -2 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( 10 + T \)
$31$ \( T \)
$37$ \( -2 + T \)
$41$ \( 10 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -14 + T \)
$59$ \( T \)
$61$ \( 10 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( 10 + T \)
$97$ \( 18 + T \)
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