Properties

Label 2704.2.a.g
Level $2704$
Weight $2$
Character orbit 2704.a
Self dual yes
Analytic conductor $21.592$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - 2 q^{7} - 3 q^{9} - 2 q^{11} + 6 q^{17} - 6 q^{19} - 8 q^{23} - q^{25} + 2 q^{29} + 10 q^{31} + 4 q^{35} + 6 q^{37} + 6 q^{41} - 4 q^{43} + 6 q^{45} - 2 q^{47} - 3 q^{49} + 6 q^{53} + 4 q^{55} - 10 q^{59} - 2 q^{61} + 6 q^{63} + 10 q^{67} + 10 q^{71} - 2 q^{73} + 4 q^{77} + 4 q^{79} + 9 q^{81} - 6 q^{83} - 12 q^{85} + 6 q^{89} + 12 q^{95} - 2 q^{97} + 6 q^{99}+O(q^{100}) \) 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 0 0 −2.00000 0 −2.00000 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

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.g 1
4.b odd 2 1 676.2.a.c 1
12.b even 2 1 6084.2.a.m 1
13.b even 2 1 208.2.a.c 1
13.d odd 4 2 2704.2.f.f 2
39.d odd 2 1 1872.2.a.f 1
52.b odd 2 1 52.2.a.a 1
52.f even 4 2 676.2.d.c 2
52.i odd 6 2 676.2.e.c 2
52.j odd 6 2 676.2.e.b 2
52.l even 12 4 676.2.h.c 4
65.d even 2 1 5200.2.a.q 1
104.e even 2 1 832.2.a.f 1
104.h odd 2 1 832.2.a.e 1
156.h even 2 1 468.2.a.b 1
156.l odd 4 2 6084.2.b.m 2
208.o odd 4 2 3328.2.b.q 2
208.p even 4 2 3328.2.b.e 2
260.g odd 2 1 1300.2.a.d 1
260.p even 4 2 1300.2.c.c 2
312.b odd 2 1 7488.2.a.bw 1
312.h even 2 1 7488.2.a.bn 1
364.h even 2 1 2548.2.a.e 1
364.x even 6 2 2548.2.j.f 2
364.bl odd 6 2 2548.2.j.e 2
468.x even 6 2 4212.2.i.i 2
468.bg odd 6 2 4212.2.i.d 2
572.b even 2 1 6292.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 52.b odd 2 1
208.2.a.c 1 13.b even 2 1
468.2.a.b 1 156.h even 2 1
676.2.a.c 1 4.b odd 2 1
676.2.d.c 2 52.f even 4 2
676.2.e.b 2 52.j odd 6 2
676.2.e.c 2 52.i odd 6 2
676.2.h.c 4 52.l even 12 4
832.2.a.e 1 104.h odd 2 1
832.2.a.f 1 104.e even 2 1
1300.2.a.d 1 260.g odd 2 1
1300.2.c.c 2 260.p even 4 2
1872.2.a.f 1 39.d odd 2 1
2548.2.a.e 1 364.h even 2 1
2548.2.j.e 2 364.bl odd 6 2
2548.2.j.f 2 364.x even 6 2
2704.2.a.g 1 1.a even 1 1 trivial
2704.2.f.f 2 13.d odd 4 2
3328.2.b.e 2 208.p even 4 2
3328.2.b.q 2 208.o odd 4 2
4212.2.i.d 2 468.bg odd 6 2
4212.2.i.i 2 468.x even 6 2
5200.2.a.q 1 65.d even 2 1
6084.2.a.m 1 12.b even 2 1
6084.2.b.m 2 156.l odd 4 2
6292.2.a.g 1 572.b even 2 1
7488.2.a.bn 1 312.h even 2 1
7488.2.a.bw 1 312.b odd 2 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 2 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 2 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 10 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 10 \) Copy content Toggle raw display
$71$ \( T - 10 \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T - 4 \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
show more
show less