Properties

Label 3136.2.a.j
Level $3136$
Weight $2$
Character orbit 3136.a
Self dual yes
Analytic conductor $25.041$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - 2 q^{9} + 3 q^{11} + 6 q^{13} - q^{15} - 5 q^{17} + q^{19} + 7 q^{23} - 4 q^{25} + 5 q^{27} - 2 q^{29} + 5 q^{31} - 3 q^{33} - 3 q^{37} - 6 q^{39} - 2 q^{41} - 4 q^{43} - 2 q^{45} - 5 q^{47} + 5 q^{51} + q^{53} + 3 q^{55} - q^{57} + 15 q^{59} + 5 q^{61} + 6 q^{65} - 9 q^{67} - 7 q^{69} + 7 q^{73} + 4 q^{75} - q^{79} + q^{81} + 12 q^{83} - 5 q^{85} + 2 q^{87} + 7 q^{89} - 5 q^{93} + q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) \) 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 −1.00000 0 1.00000 0 0 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

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 3136.2.a.j 1
4.b odd 2 1 3136.2.a.u 1
7.b odd 2 1 3136.2.a.t 1
7.c even 3 2 448.2.i.d 2
8.b even 2 1 784.2.a.h 1
8.d odd 2 1 392.2.a.c 1
24.f even 2 1 3528.2.a.p 1
24.h odd 2 1 7056.2.a.bj 1
28.d even 2 1 3136.2.a.i 1
28.g odd 6 2 448.2.i.b 2
40.e odd 2 1 9800.2.a.be 1
56.e even 2 1 392.2.a.e 1
56.h odd 2 1 784.2.a.c 1
56.j odd 6 2 784.2.i.h 2
56.k odd 6 2 56.2.i.b 2
56.m even 6 2 392.2.i.b 2
56.p even 6 2 112.2.i.a 2
168.e odd 2 1 3528.2.a.j 1
168.i even 2 1 7056.2.a.u 1
168.s odd 6 2 1008.2.s.g 2
168.v even 6 2 504.2.s.c 2
168.be odd 6 2 3528.2.s.q 2
280.n even 2 1 9800.2.a.s 1
280.bi odd 6 2 1400.2.q.d 2
280.br even 12 4 1400.2.bh.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.b 2 56.k odd 6 2
112.2.i.a 2 56.p even 6 2
392.2.a.c 1 8.d odd 2 1
392.2.a.e 1 56.e even 2 1
392.2.i.b 2 56.m even 6 2
448.2.i.b 2 28.g odd 6 2
448.2.i.d 2 7.c even 3 2
504.2.s.c 2 168.v even 6 2
784.2.a.c 1 56.h odd 2 1
784.2.a.h 1 8.b even 2 1
784.2.i.h 2 56.j odd 6 2
1008.2.s.g 2 168.s odd 6 2
1400.2.q.d 2 280.bi odd 6 2
1400.2.bh.a 4 280.br even 12 4
3136.2.a.i 1 28.d even 2 1
3136.2.a.j 1 1.a even 1 1 trivial
3136.2.a.t 1 7.b odd 2 1
3136.2.a.u 1 4.b odd 2 1
3528.2.a.j 1 168.e odd 2 1
3528.2.a.p 1 24.f even 2 1
3528.2.s.q 2 168.be odd 6 2
7056.2.a.u 1 168.i even 2 1
7056.2.a.bj 1 24.h odd 2 1
9800.2.a.s 1 280.n even 2 1
9800.2.a.be 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{11} - 3 \) 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
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T + 5 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T - 7 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T + 3 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 5 \) Copy content Toggle raw display
$53$ \( T - 1 \) Copy content Toggle raw display
$59$ \( T - 15 \) Copy content Toggle raw display
$61$ \( T - 5 \) Copy content Toggle raw display
$67$ \( T + 9 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 7 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 7 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
show more
show less