Properties

Label 3136.2.a.y
Level $3136$
Weight $2$
Character orbit 3136.a
Self dual yes
Analytic conductor $25.041$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + q^{9} - 4q^{11} - 4q^{13} + 2q^{17} + 6q^{19} - 8q^{23} - 5q^{25} - 4q^{27} - 2q^{29} - 4q^{31} - 8q^{33} - 10q^{37} - 8q^{39} + 10q^{41} + 4q^{43} + 4q^{47} + 4q^{51} + 2q^{53} + 12q^{57} - 10q^{59} - 8q^{61} - 8q^{67} - 16q^{69} + 6q^{73} - 10q^{75} + 16q^{79} - 11q^{81} - 2q^{83} - 4q^{87} - 18q^{89} - 8q^{93} + 2q^{97} - 4q^{99} + O(q^{100}) \)

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 2.00000 0 0 0 0 0 1.00000 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.y 1
4.b odd 2 1 3136.2.a.f 1
7.b odd 2 1 448.2.a.b 1
8.b even 2 1 1568.2.a.b 1
8.d odd 2 1 1568.2.a.h 1
21.c even 2 1 4032.2.a.z 1
28.d even 2 1 448.2.a.f 1
56.e even 2 1 224.2.a.a 1
56.h odd 2 1 224.2.a.b yes 1
56.j odd 6 2 1568.2.i.b 2
56.k odd 6 2 1568.2.i.c 2
56.m even 6 2 1568.2.i.k 2
56.p even 6 2 1568.2.i.j 2
84.h odd 2 1 4032.2.a.p 1
112.j even 4 2 1792.2.b.f 2
112.l odd 4 2 1792.2.b.b 2
168.e odd 2 1 2016.2.a.e 1
168.i even 2 1 2016.2.a.g 1
280.c odd 2 1 5600.2.a.c 1
280.n even 2 1 5600.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 56.e even 2 1
224.2.a.b yes 1 56.h odd 2 1
448.2.a.b 1 7.b odd 2 1
448.2.a.f 1 28.d even 2 1
1568.2.a.b 1 8.b even 2 1
1568.2.a.h 1 8.d odd 2 1
1568.2.i.b 2 56.j odd 6 2
1568.2.i.c 2 56.k odd 6 2
1568.2.i.j 2 56.p even 6 2
1568.2.i.k 2 56.m even 6 2
1792.2.b.b 2 112.l odd 4 2
1792.2.b.f 2 112.j even 4 2
2016.2.a.e 1 168.e odd 2 1
2016.2.a.g 1 168.i even 2 1
3136.2.a.f 1 4.b odd 2 1
3136.2.a.y 1 1.a even 1 1 trivial
4032.2.a.p 1 84.h odd 2 1
4032.2.a.z 1 21.c even 2 1
5600.2.a.c 1 280.c odd 2 1
5600.2.a.t 1 280.n even 2 1

Hecke kernels

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

\( T_{3} - 2 \)
\( T_{5} \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 4 + T \)
$17$ \( -2 + T \)
$19$ \( -6 + T \)
$23$ \( 8 + T \)
$29$ \( 2 + T \)
$31$ \( 4 + T \)
$37$ \( 10 + T \)
$41$ \( -10 + T \)
$43$ \( -4 + T \)
$47$ \( -4 + T \)
$53$ \( -2 + T \)
$59$ \( 10 + T \)
$61$ \( 8 + T \)
$67$ \( 8 + T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( -16 + T \)
$83$ \( 2 + T \)
$89$ \( 18 + T \)
$97$ \( -2 + T \)
show more
show less