Properties

Label 3136.2.a.e
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 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.e 1
4.b odd 2 1 3136.2.a.z 1
7.b odd 2 1 448.2.a.g 1
8.b even 2 1 98.2.a.a 1
8.d odd 2 1 784.2.a.b 1
21.c even 2 1 4032.2.a.w 1
24.f even 2 1 7056.2.a.bd 1
24.h odd 2 1 882.2.a.i 1
28.d even 2 1 448.2.a.a 1
40.f even 2 1 2450.2.a.t 1
40.i odd 4 2 2450.2.c.c 2
56.e even 2 1 112.2.a.c 1
56.h odd 2 1 14.2.a.a 1
56.j odd 6 2 98.2.c.b 2
56.k odd 6 2 784.2.i.i 2
56.m even 6 2 784.2.i.c 2
56.p even 6 2 98.2.c.a 2
84.h odd 2 1 4032.2.a.r 1
112.j even 4 2 1792.2.b.g 2
112.l odd 4 2 1792.2.b.c 2
168.e odd 2 1 1008.2.a.h 1
168.i even 2 1 126.2.a.b 1
168.s odd 6 2 882.2.g.d 2
168.ba even 6 2 882.2.g.c 2
280.c odd 2 1 350.2.a.f 1
280.n even 2 1 2800.2.a.g 1
280.s even 4 2 350.2.c.d 2
280.y odd 4 2 2800.2.g.h 2
504.bn odd 6 2 1134.2.f.l 2
504.cc even 6 2 1134.2.f.f 2
616.o even 2 1 1694.2.a.e 1
728.l odd 2 1 2366.2.a.j 1
728.ba even 4 2 2366.2.d.b 2
840.u even 2 1 3150.2.a.i 1
840.bp odd 4 2 3150.2.g.j 2
952.e odd 2 1 4046.2.a.f 1
1064.f even 2 1 5054.2.a.c 1
1288.m even 2 1 7406.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 56.h odd 2 1
98.2.a.a 1 8.b even 2 1
98.2.c.a 2 56.p even 6 2
98.2.c.b 2 56.j odd 6 2
112.2.a.c 1 56.e even 2 1
126.2.a.b 1 168.i even 2 1
350.2.a.f 1 280.c odd 2 1
350.2.c.d 2 280.s even 4 2
448.2.a.a 1 28.d even 2 1
448.2.a.g 1 7.b odd 2 1
784.2.a.b 1 8.d odd 2 1
784.2.i.c 2 56.m even 6 2
784.2.i.i 2 56.k odd 6 2
882.2.a.i 1 24.h odd 2 1
882.2.g.c 2 168.ba even 6 2
882.2.g.d 2 168.s odd 6 2
1008.2.a.h 1 168.e odd 2 1
1134.2.f.f 2 504.cc even 6 2
1134.2.f.l 2 504.bn odd 6 2
1694.2.a.e 1 616.o even 2 1
1792.2.b.c 2 112.l odd 4 2
1792.2.b.g 2 112.j even 4 2
2366.2.a.j 1 728.l odd 2 1
2366.2.d.b 2 728.ba even 4 2
2450.2.a.t 1 40.f even 2 1
2450.2.c.c 2 40.i odd 4 2
2800.2.a.g 1 280.n even 2 1
2800.2.g.h 2 280.y odd 4 2
3136.2.a.e 1 1.a even 1 1 trivial
3136.2.a.z 1 4.b odd 2 1
3150.2.a.i 1 840.u even 2 1
3150.2.g.j 2 840.bp odd 4 2
4032.2.a.r 1 84.h odd 2 1
4032.2.a.w 1 21.c even 2 1
4046.2.a.f 1 952.e odd 2 1
5054.2.a.c 1 1064.f even 2 1
7056.2.a.bd 1 24.f even 2 1
7406.2.a.a 1 1288.m even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( 4 + T \)
$17$ \( 6 + T \)
$19$ \( -2 + T \)
$23$ \( T \)
$29$ \( -6 + T \)
$31$ \( -4 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( 8 + T \)
$47$ \( -12 + T \)
$53$ \( 6 + T \)
$59$ \( 6 + T \)
$61$ \( -8 + T \)
$67$ \( -4 + T \)
$71$ \( T \)
$73$ \( 2 + T \)
$79$ \( -8 + T \)
$83$ \( 6 + T \)
$89$ \( -6 + T \)
$97$ \( -10 + T \)
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