Properties

Label 2704.2.a.j
Level $2704$
Weight $2$
Character orbit 2704.a
Self dual yes
Analytic conductor $21.592$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5915487066\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - 3 q^{5} + 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 3 q^{5} + 3 q^{7} - 2 q^{9} - 3 q^{15} - 3 q^{17} + 6 q^{19} + 3 q^{21} - 6 q^{23} + 4 q^{25} - 5 q^{27} - 9 q^{35} - 3 q^{37} - q^{43} + 6 q^{45} + 3 q^{47} + 2 q^{49} - 3 q^{51} - 6 q^{53} + 6 q^{57} - 6 q^{59} - 8 q^{61} - 6 q^{63} + 12 q^{67} - 6 q^{69} - 15 q^{71} - 6 q^{73} + 4 q^{75} - 10 q^{79} + q^{81} - 6 q^{83} + 9 q^{85} + 6 q^{89} - 18 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

Atkin-Lehner signs

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

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 2704.2.a.j 1
4.b odd 2 1 338.2.a.d 1
12.b even 2 1 3042.2.a.g 1
13.b even 2 1 2704.2.a.k 1
13.d odd 4 2 208.2.f.a 2
20.d odd 2 1 8450.2.a.h 1
39.f even 4 2 1872.2.c.f 2
52.b odd 2 1 338.2.a.b 1
52.f even 4 2 26.2.b.a 2
52.i odd 6 2 338.2.c.f 2
52.j odd 6 2 338.2.c.b 2
52.l even 12 4 338.2.e.c 4
104.j odd 4 2 832.2.f.b 2
104.m even 4 2 832.2.f.d 2
156.h even 2 1 3042.2.a.j 1
156.l odd 4 2 234.2.b.b 2
260.g odd 2 1 8450.2.a.u 1
260.l odd 4 2 650.2.c.a 2
260.s odd 4 2 650.2.c.d 2
260.u even 4 2 650.2.d.b 2
364.p odd 4 2 1274.2.d.c 2
364.bw odd 12 4 1274.2.n.c 4
364.ce even 12 4 1274.2.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 52.f even 4 2
208.2.f.a 2 13.d odd 4 2
234.2.b.b 2 156.l odd 4 2
338.2.a.b 1 52.b odd 2 1
338.2.a.d 1 4.b odd 2 1
338.2.c.b 2 52.j odd 6 2
338.2.c.f 2 52.i odd 6 2
338.2.e.c 4 52.l even 12 4
650.2.c.a 2 260.l odd 4 2
650.2.c.d 2 260.s odd 4 2
650.2.d.b 2 260.u even 4 2
832.2.f.b 2 104.j odd 4 2
832.2.f.d 2 104.m even 4 2
1274.2.d.c 2 364.p odd 4 2
1274.2.n.c 4 364.bw odd 12 4
1274.2.n.d 4 364.ce even 12 4
1872.2.c.f 2 39.f even 4 2
2704.2.a.j 1 1.a even 1 1 trivial
2704.2.a.k 1 13.b even 2 1
3042.2.a.g 1 12.b even 2 1
3042.2.a.j 1 156.h even 2 1
8450.2.a.h 1 20.d odd 2 1
8450.2.a.u 1 260.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T - 3 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T - 3 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T + 8 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 15 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 12 \) Copy content Toggle raw display
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