Properties

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

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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 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + 3 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 3 q^{5} - 2 q^{9} - 3 q^{11} + 2 q^{13} - 3 q^{15} - 3 q^{17} + q^{19} - 3 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 7 q^{31} + 3 q^{33} + q^{37} - 2 q^{39} - 6 q^{41} - 4 q^{43} - 6 q^{45} - 9 q^{47} + 3 q^{51} - 3 q^{53} - 9 q^{55} - q^{57} - 9 q^{59} - q^{61} + 6 q^{65} - 7 q^{67} + 3 q^{69} + q^{73} - 4 q^{75} + 13 q^{79} + q^{81} - 12 q^{83} - 9 q^{85} - 6 q^{87} - 15 q^{89} + 7 q^{93} + 3 q^{95} + 10 q^{97} + 6 q^{99}+O(q^{100}) \) 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 3.00000 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.k 1
4.b odd 2 1 3136.2.a.v 1
7.b odd 2 1 3136.2.a.s 1
7.d odd 6 2 448.2.i.c 2
8.b even 2 1 784.2.a.g 1
8.d odd 2 1 196.2.a.a 1
24.f even 2 1 1764.2.a.j 1
24.h odd 2 1 7056.2.a.bw 1
28.d even 2 1 3136.2.a.h 1
28.f even 6 2 448.2.i.e 2
40.e odd 2 1 4900.2.a.n 1
40.k even 4 2 4900.2.e.h 2
56.e even 2 1 196.2.a.b 1
56.h odd 2 1 784.2.a.d 1
56.j odd 6 2 112.2.i.b 2
56.k odd 6 2 196.2.e.a 2
56.m even 6 2 28.2.e.a 2
56.p even 6 2 784.2.i.d 2
168.e odd 2 1 1764.2.a.a 1
168.i even 2 1 7056.2.a.f 1
168.v even 6 2 1764.2.k.b 2
168.ba even 6 2 1008.2.s.p 2
168.be odd 6 2 252.2.k.c 2
280.n even 2 1 4900.2.a.g 1
280.y odd 4 2 4900.2.e.i 2
280.ba even 6 2 700.2.i.c 2
280.bp odd 12 4 700.2.r.b 4
504.u odd 6 2 2268.2.l.a 2
504.bf even 6 2 2268.2.i.a 2
504.cm odd 6 2 2268.2.i.h 2
504.cz even 6 2 2268.2.l.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 56.m even 6 2
112.2.i.b 2 56.j odd 6 2
196.2.a.a 1 8.d odd 2 1
196.2.a.b 1 56.e even 2 1
196.2.e.a 2 56.k odd 6 2
252.2.k.c 2 168.be odd 6 2
448.2.i.c 2 7.d odd 6 2
448.2.i.e 2 28.f even 6 2
700.2.i.c 2 280.ba even 6 2
700.2.r.b 4 280.bp odd 12 4
784.2.a.d 1 56.h odd 2 1
784.2.a.g 1 8.b even 2 1
784.2.i.d 2 56.p even 6 2
1008.2.s.p 2 168.ba even 6 2
1764.2.a.a 1 168.e odd 2 1
1764.2.a.j 1 24.f even 2 1
1764.2.k.b 2 168.v even 6 2
2268.2.i.a 2 504.bf even 6 2
2268.2.i.h 2 504.cm odd 6 2
2268.2.l.a 2 504.u odd 6 2
2268.2.l.h 2 504.cz even 6 2
3136.2.a.h 1 28.d even 2 1
3136.2.a.k 1 1.a even 1 1 trivial
3136.2.a.s 1 7.b odd 2 1
3136.2.a.v 1 4.b odd 2 1
4900.2.a.g 1 280.n even 2 1
4900.2.a.n 1 40.e odd 2 1
4900.2.e.h 2 40.k even 4 2
4900.2.e.i 2 280.y odd 4 2
7056.2.a.f 1 168.i even 2 1
7056.2.a.bw 1 24.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} - 3 \) 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
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 3 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T - 1 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 9 \) Copy content Toggle raw display
$53$ \( T + 3 \) Copy content Toggle raw display
$59$ \( T + 9 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T + 7 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 1 \) Copy content Toggle raw display
$79$ \( T - 13 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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