Properties

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

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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: \(0\)
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

\(f(q)\) \(=\) \( q + 3 q^{3} - q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - q^{5} + 6 q^{9} + q^{11} + 2 q^{13} - 3 q^{15} - 3 q^{17} + 5 q^{19} - 3 q^{23} - 4 q^{25} + 9 q^{27} + 6 q^{29} + q^{31} + 3 q^{33} + 5 q^{37} + 6 q^{39} + 10 q^{41} + 4 q^{43} - 6 q^{45} - q^{47} - 9 q^{51} + 9 q^{53} - q^{55} + 15 q^{57} + 3 q^{59} + 3 q^{61} - 2 q^{65} - 11 q^{67} - 9 q^{69} + 16 q^{71} - 7 q^{73} - 12 q^{75} - 11 q^{79} + 9 q^{81} - 4 q^{83} + 3 q^{85} + 18 q^{87} + 9 q^{89} + 3 q^{93} - 5 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) 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 3.00000 0 −1.00000 0 0 0 6.00000 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.bb 1
4.b odd 2 1 3136.2.a.a 1
7.b odd 2 1 3136.2.a.b 1
7.d odd 6 2 448.2.i.f 2
8.b even 2 1 392.2.a.a 1
8.d odd 2 1 784.2.a.j 1
24.f even 2 1 7056.2.a.s 1
24.h odd 2 1 3528.2.a.k 1
28.d even 2 1 3136.2.a.bc 1
28.f even 6 2 448.2.i.a 2
40.f even 2 1 9800.2.a.bp 1
56.e even 2 1 784.2.a.a 1
56.h odd 2 1 392.2.a.f 1
56.j odd 6 2 56.2.i.a 2
56.k odd 6 2 784.2.i.a 2
56.m even 6 2 112.2.i.c 2
56.p even 6 2 392.2.i.f 2
168.e odd 2 1 7056.2.a.bi 1
168.i even 2 1 3528.2.a.r 1
168.s odd 6 2 3528.2.s.o 2
168.ba even 6 2 504.2.s.e 2
168.be odd 6 2 1008.2.s.e 2
280.c odd 2 1 9800.2.a.b 1
280.bk odd 6 2 1400.2.q.g 2
280.bv even 12 4 1400.2.bh.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.i.a 2 56.j odd 6 2
112.2.i.c 2 56.m even 6 2
392.2.a.a 1 8.b even 2 1
392.2.a.f 1 56.h odd 2 1
392.2.i.f 2 56.p even 6 2
448.2.i.a 2 28.f even 6 2
448.2.i.f 2 7.d odd 6 2
504.2.s.e 2 168.ba even 6 2
784.2.a.a 1 56.e even 2 1
784.2.a.j 1 8.d odd 2 1
784.2.i.a 2 56.k odd 6 2
1008.2.s.e 2 168.be odd 6 2
1400.2.q.g 2 280.bk odd 6 2
1400.2.bh.f 4 280.bv even 12 4
3136.2.a.a 1 4.b odd 2 1
3136.2.a.b 1 7.b odd 2 1
3136.2.a.bb 1 1.a even 1 1 trivial
3136.2.a.bc 1 28.d even 2 1
3528.2.a.k 1 24.h odd 2 1
3528.2.a.r 1 168.i even 2 1
3528.2.s.o 2 168.s odd 6 2
7056.2.a.s 1 24.f even 2 1
7056.2.a.bi 1 168.e odd 2 1
9800.2.a.b 1 280.c odd 2 1
9800.2.a.bp 1 40.f even 2 1

Hecke kernels

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

\( T_{3} - 3 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 3 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T + 3 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T - 5 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 1 \) Copy content Toggle raw display
$53$ \( T - 9 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T - 3 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T + 7 \) Copy content Toggle raw display
$79$ \( T + 11 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 9 \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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