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Properties

Label 140.1.h.b
Level 140
Weight 1
Character orbit 140.h
Self dual yes
Analytic conductor 0.070
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -35
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 140.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0698691017686\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.140.1
Artin image $D_6$
Artin field Galois closure of 6.2.98000.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{5} - q^{7} + O(q^{10}) \) \( q + q^{3} - q^{5} - q^{7} - q^{11} + q^{13} - q^{15} + q^{17} - q^{21} + q^{25} - q^{27} - q^{29} - q^{33} + q^{35} + q^{39} + q^{47} + q^{49} + q^{51} + q^{55} - q^{65} + 2q^{71} - 2q^{73} + q^{75} + q^{77} - q^{79} - q^{81} - 2q^{83} - q^{85} - q^{87} - q^{91} + q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0
0 1.00000 0 −1.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.1.h.b yes 1
3.b odd 2 1 1260.1.p.b 1
4.b odd 2 1 560.1.p.a 1
5.b even 2 1 140.1.h.a ✓ 1
5.c odd 4 2 700.1.d.a 2
7.b odd 2 1 140.1.h.a ✓ 1
7.c even 3 2 980.1.n.a 2
7.d odd 6 2 980.1.n.b 2
8.b even 2 1 2240.1.p.b 1
8.d odd 2 1 2240.1.p.d 1
15.d odd 2 1 1260.1.p.a 1
20.d odd 2 1 560.1.p.b 1
20.e even 4 2 2800.1.f.c 2
21.c even 2 1 1260.1.p.a 1
28.d even 2 1 560.1.p.b 1
28.f even 6 2 3920.1.br.a 2
28.g odd 6 2 3920.1.br.b 2
35.c odd 2 1 CM 140.1.h.b yes 1
35.f even 4 2 700.1.d.a 2
35.i odd 6 2 980.1.n.a 2
35.j even 6 2 980.1.n.b 2
40.e odd 2 1 2240.1.p.a 1
40.f even 2 1 2240.1.p.c 1
56.e even 2 1 2240.1.p.a 1
56.h odd 2 1 2240.1.p.c 1
105.g even 2 1 1260.1.p.b 1
140.c even 2 1 560.1.p.a 1
140.j odd 4 2 2800.1.f.c 2
140.p odd 6 2 3920.1.br.a 2
140.s even 6 2 3920.1.br.b 2
280.c odd 2 1 2240.1.p.b 1
280.n even 2 1 2240.1.p.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a ✓ 1 5.b even 2 1
140.1.h.a ✓ 1 7.b odd 2 1
140.1.h.b yes 1 1.a even 1 1 trivial
140.1.h.b yes 1 35.c odd 2 1 CM
560.1.p.a 1 4.b odd 2 1
560.1.p.a 1 140.c even 2 1
560.1.p.b 1 20.d odd 2 1
560.1.p.b 1 28.d even 2 1
700.1.d.a 2 5.c odd 4 2
700.1.d.a 2 35.f even 4 2
980.1.n.a 2 7.c even 3 2
980.1.n.a 2 35.i odd 6 2
980.1.n.b 2 7.d odd 6 2
980.1.n.b 2 35.j even 6 2
1260.1.p.a 1 15.d odd 2 1
1260.1.p.a 1 21.c even 2 1
1260.1.p.b 1 3.b odd 2 1
1260.1.p.b 1 105.g even 2 1
2240.1.p.a 1 40.e odd 2 1
2240.1.p.a 1 56.e even 2 1
2240.1.p.b 1 8.b even 2 1
2240.1.p.b 1 280.c odd 2 1
2240.1.p.c 1 40.f even 2 1
2240.1.p.c 1 56.h odd 2 1
2240.1.p.d 1 8.d odd 2 1
2240.1.p.d 1 280.n even 2 1
2800.1.f.c 2 20.e even 4 2
2800.1.f.c 2 140.j odd 4 2
3920.1.br.a 2 28.f even 6 2
3920.1.br.a 2 140.p odd 6 2
3920.1.br.b 2 28.g odd 6 2
3920.1.br.b 2 140.s even 6 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( 1 + T + T^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( 1 - T + T^{2} \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( ( 1 - T )( 1 + T ) \)
$29$ \( 1 + T + T^{2} \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( ( 1 - T )( 1 + T ) \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( 1 - T + T^{2} \)
$53$ \( ( 1 - T )( 1 + T ) \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( ( 1 - T )( 1 + T ) \)
$71$ \( ( 1 - T )^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( 1 + T )^{2} \)
$89$ \( ( 1 - T )( 1 + T ) \)
$97$ \( 1 - T + T^{2} \)
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