Properties

Label 140.2.a.a
Level $140$
Weight $2$
Character orbit 140.a
Self dual yes
Analytic conductor $1.118$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} + q^{5} + q^{7} - 2q^{9} + 3q^{11} - q^{13} + q^{15} - 3q^{17} + 2q^{19} + q^{21} - 6q^{23} + q^{25} - 5q^{27} - 9q^{29} + 8q^{31} + 3q^{33} + q^{35} - 10q^{37} - q^{39} + 2q^{43} - 2q^{45} - 3q^{47} + q^{49} - 3q^{51} + 3q^{55} + 2q^{57} + 12q^{59} + 8q^{61} - 2q^{63} - q^{65} + 8q^{67} - 6q^{69} + 14q^{73} + q^{75} + 3q^{77} + 5q^{79} + q^{81} - 12q^{83} - 3q^{85} - 9q^{87} + 12q^{89} - q^{91} + 8q^{93} + 2q^{95} + 17q^{97} - 6q^{99} + O(q^{100}) \)

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 1.00000 0 1.00000 0 1.00000 0 −2.00000 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 140.2.a.a 1
3.b odd 2 1 1260.2.a.c 1
4.b odd 2 1 560.2.a.c 1
5.b even 2 1 700.2.a.d 1
5.c odd 4 2 700.2.e.c 2
7.b odd 2 1 980.2.a.c 1
7.c even 3 2 980.2.i.d 2
7.d odd 6 2 980.2.i.h 2
8.b even 2 1 2240.2.a.g 1
8.d odd 2 1 2240.2.a.r 1
12.b even 2 1 5040.2.a.h 1
15.d odd 2 1 6300.2.a.d 1
15.e even 4 2 6300.2.k.c 2
20.d odd 2 1 2800.2.a.y 1
20.e even 4 2 2800.2.g.j 2
21.c even 2 1 8820.2.a.r 1
28.d even 2 1 3920.2.a.u 1
35.c odd 2 1 4900.2.a.p 1
35.f even 4 2 4900.2.e.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 1.a even 1 1 trivial
560.2.a.c 1 4.b odd 2 1
700.2.a.d 1 5.b even 2 1
700.2.e.c 2 5.c odd 4 2
980.2.a.c 1 7.b odd 2 1
980.2.i.d 2 7.c even 3 2
980.2.i.h 2 7.d odd 6 2
1260.2.a.c 1 3.b odd 2 1
2240.2.a.g 1 8.b even 2 1
2240.2.a.r 1 8.d odd 2 1
2800.2.a.y 1 20.d odd 2 1
2800.2.g.j 2 20.e even 4 2
3920.2.a.u 1 28.d even 2 1
4900.2.a.p 1 35.c odd 2 1
4900.2.e.l 2 35.f even 4 2
5040.2.a.h 1 12.b even 2 1
6300.2.a.d 1 15.d odd 2 1
6300.2.k.c 2 15.e even 4 2
8820.2.a.r 1 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(140))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( -1 + T \)
$11$ \( -3 + T \)
$13$ \( 1 + T \)
$17$ \( 3 + T \)
$19$ \( -2 + T \)
$23$ \( 6 + T \)
$29$ \( 9 + T \)
$31$ \( -8 + T \)
$37$ \( 10 + T \)
$41$ \( T \)
$43$ \( -2 + T \)
$47$ \( 3 + T \)
$53$ \( T \)
$59$ \( -12 + T \)
$61$ \( -8 + T \)
$67$ \( -8 + T \)
$71$ \( T \)
$73$ \( -14 + T \)
$79$ \( -5 + T \)
$83$ \( 12 + T \)
$89$ \( -12 + T \)
$97$ \( -17 + T \)
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