Properties

Label 3136.2.a.d
Level $3136$
Weight $2$
Character orbit 3136.a
Self dual yes
Analytic conductor $25.041$
Analytic rank $2$
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: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1568)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{3} - 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} - 2 q^{5} + q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{15} - 4 q^{17} - 6 q^{19} - 4 q^{23} - q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 8 q^{33} + 6 q^{37} + 12 q^{39} + 4 q^{41} - 12 q^{43} - 2 q^{45} - 12 q^{47} + 8 q^{51} - 6 q^{53} + 8 q^{55} + 12 q^{57} - 6 q^{59} + 6 q^{61} + 12 q^{65} - 12 q^{67} + 8 q^{69} + 8 q^{71} + 2 q^{75} - 11 q^{81} - 6 q^{83} + 8 q^{85} - 12 q^{87} - 16 q^{89} + 8 q^{93} + 12 q^{95} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) 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 −2.00000 0 −2.00000 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.d 1
4.b odd 2 1 3136.2.a.x 1
7.b odd 2 1 3136.2.a.ba 1
8.b even 2 1 1568.2.a.i yes 1
8.d odd 2 1 1568.2.a.c yes 1
28.d even 2 1 3136.2.a.g 1
56.e even 2 1 1568.2.a.g yes 1
56.h odd 2 1 1568.2.a.a 1
56.j odd 6 2 1568.2.i.l 2
56.k odd 6 2 1568.2.i.i 2
56.m even 6 2 1568.2.i.d 2
56.p even 6 2 1568.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.a 1 56.h odd 2 1
1568.2.a.c yes 1 8.d odd 2 1
1568.2.a.g yes 1 56.e even 2 1
1568.2.a.i yes 1 8.b even 2 1
1568.2.i.a 2 56.p even 6 2
1568.2.i.d 2 56.m even 6 2
1568.2.i.i 2 56.k odd 6 2
1568.2.i.l 2 56.j odd 6 2
3136.2.a.d 1 1.a even 1 1 trivial
3136.2.a.g 1 28.d even 2 1
3136.2.a.x 1 4.b odd 2 1
3136.2.a.ba 1 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 4 \) Copy content Toggle raw display
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T - 6 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 6 \) Copy content Toggle raw display
$89$ \( T + 16 \) Copy content Toggle raw display
$97$ \( T + 12 \) Copy content Toggle raw display
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