Properties

Label 5600.2.a.t
Level $5600$
Weight $2$
Character orbit 5600.a
Self dual yes
Analytic conductor $44.716$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5600 = 2^{5} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.7162251319\)
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

\(f(q)\) \(=\) \( q + 2q^{3} + q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + q^{7} + q^{9} - 4q^{11} + 4q^{13} + 2q^{17} - 6q^{19} + 2q^{21} - 8q^{23} - 4q^{27} + 2q^{29} - 4q^{31} - 8q^{33} - 10q^{37} + 8q^{39} - 10q^{41} - 4q^{43} - 4q^{47} + q^{49} + 4q^{51} + 2q^{53} - 12q^{57} + 10q^{59} - 8q^{61} + q^{63} + 8q^{67} - 16q^{69} + 6q^{73} - 4q^{77} - 16q^{79} - 11q^{81} - 2q^{83} + 4q^{87} + 18q^{89} + 4q^{91} - 8q^{93} + 2q^{97} - 4q^{99} + O(q^{100}) \)

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 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 5600.2.a.t 1
4.b odd 2 1 5600.2.a.c 1
5.b even 2 1 224.2.a.a 1
15.d odd 2 1 2016.2.a.e 1
20.d odd 2 1 224.2.a.b yes 1
35.c odd 2 1 1568.2.a.h 1
35.i odd 6 2 1568.2.i.c 2
35.j even 6 2 1568.2.i.k 2
40.e odd 2 1 448.2.a.b 1
40.f even 2 1 448.2.a.f 1
60.h even 2 1 2016.2.a.g 1
80.k odd 4 2 1792.2.b.b 2
80.q even 4 2 1792.2.b.f 2
120.i odd 2 1 4032.2.a.p 1
120.m even 2 1 4032.2.a.z 1
140.c even 2 1 1568.2.a.b 1
140.p odd 6 2 1568.2.i.b 2
140.s even 6 2 1568.2.i.j 2
280.c odd 2 1 3136.2.a.f 1
280.n even 2 1 3136.2.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 5.b even 2 1
224.2.a.b yes 1 20.d odd 2 1
448.2.a.b 1 40.e odd 2 1
448.2.a.f 1 40.f even 2 1
1568.2.a.b 1 140.c even 2 1
1568.2.a.h 1 35.c odd 2 1
1568.2.i.b 2 140.p odd 6 2
1568.2.i.c 2 35.i odd 6 2
1568.2.i.j 2 140.s even 6 2
1568.2.i.k 2 35.j even 6 2
1792.2.b.b 2 80.k odd 4 2
1792.2.b.f 2 80.q even 4 2
2016.2.a.e 1 15.d odd 2 1
2016.2.a.g 1 60.h even 2 1
3136.2.a.f 1 280.c odd 2 1
3136.2.a.y 1 280.n even 2 1
4032.2.a.p 1 120.i odd 2 1
4032.2.a.z 1 120.m even 2 1
5600.2.a.c 1 4.b odd 2 1
5600.2.a.t 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5600))\):

\( T_{3} - 2 \)
\( T_{11} + 4 \)
\( T_{13} - 4 \)
\( T_{19} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( 4 + T \)
$13$ \( -4 + T \)
$17$ \( -2 + T \)
$19$ \( 6 + T \)
$23$ \( 8 + T \)
$29$ \( -2 + T \)
$31$ \( 4 + T \)
$37$ \( 10 + T \)
$41$ \( 10 + T \)
$43$ \( 4 + T \)
$47$ \( 4 + T \)
$53$ \( -2 + T \)
$59$ \( -10 + T \)
$61$ \( 8 + T \)
$67$ \( -8 + T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( 16 + T \)
$83$ \( 2 + T \)
$89$ \( -18 + T \)
$97$ \( -2 + T \)
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