Properties

Label 1260.1.p.b
Level $1260$
Weight $1$
Character orbit 1260.p
Self dual yes
Analytic conductor $0.629$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -35
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1260.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.628821915918\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.140.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.2646000.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} - q^{7} + O(q^{10}) \) \( q + q^{5} - q^{7} + q^{11} + q^{13} - q^{17} + q^{25} + q^{29} - q^{35} - q^{47} + q^{49} + q^{55} + q^{65} - 2q^{71} - 2q^{73} - q^{77} - q^{79} + 2q^{83} - q^{85} - q^{91} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
0
0 0 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 1260.1.p.b 1
3.b odd 2 1 140.1.h.b yes 1
5.b even 2 1 1260.1.p.a 1
7.b odd 2 1 1260.1.p.a 1
12.b even 2 1 560.1.p.a 1
15.d odd 2 1 140.1.h.a 1
15.e even 4 2 700.1.d.a 2
21.c even 2 1 140.1.h.a 1
21.g even 6 2 980.1.n.b 2
21.h odd 6 2 980.1.n.a 2
24.f even 2 1 2240.1.p.d 1
24.h odd 2 1 2240.1.p.b 1
35.c odd 2 1 CM 1260.1.p.b 1
60.h even 2 1 560.1.p.b 1
60.l odd 4 2 2800.1.f.c 2
84.h odd 2 1 560.1.p.b 1
84.j odd 6 2 3920.1.br.a 2
84.n even 6 2 3920.1.br.b 2
105.g even 2 1 140.1.h.b yes 1
105.k odd 4 2 700.1.d.a 2
105.o odd 6 2 980.1.n.b 2
105.p even 6 2 980.1.n.a 2
120.i odd 2 1 2240.1.p.c 1
120.m even 2 1 2240.1.p.a 1
168.e odd 2 1 2240.1.p.a 1
168.i even 2 1 2240.1.p.c 1
420.o odd 2 1 560.1.p.a 1
420.w even 4 2 2800.1.f.c 2
420.ba even 6 2 3920.1.br.a 2
420.be odd 6 2 3920.1.br.b 2
840.b odd 2 1 2240.1.p.d 1
840.u even 2 1 2240.1.p.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 15.d odd 2 1
140.1.h.a 1 21.c even 2 1
140.1.h.b yes 1 3.b odd 2 1
140.1.h.b yes 1 105.g even 2 1
560.1.p.a 1 12.b even 2 1
560.1.p.a 1 420.o odd 2 1
560.1.p.b 1 60.h even 2 1
560.1.p.b 1 84.h odd 2 1
700.1.d.a 2 15.e even 4 2
700.1.d.a 2 105.k odd 4 2
980.1.n.a 2 21.h odd 6 2
980.1.n.a 2 105.p even 6 2
980.1.n.b 2 21.g even 6 2
980.1.n.b 2 105.o odd 6 2
1260.1.p.a 1 5.b even 2 1
1260.1.p.a 1 7.b odd 2 1
1260.1.p.b 1 1.a even 1 1 trivial
1260.1.p.b 1 35.c odd 2 1 CM
2240.1.p.a 1 120.m even 2 1
2240.1.p.a 1 168.e odd 2 1
2240.1.p.b 1 24.h odd 2 1
2240.1.p.b 1 840.u even 2 1
2240.1.p.c 1 120.i odd 2 1
2240.1.p.c 1 168.i even 2 1
2240.1.p.d 1 24.f even 2 1
2240.1.p.d 1 840.b odd 2 1
2800.1.f.c 2 60.l odd 4 2
2800.1.f.c 2 420.w even 4 2
3920.1.br.a 2 84.j odd 6 2
3920.1.br.a 2 420.ba even 6 2
3920.1.br.b 2 84.n even 6 2
3920.1.br.b 2 420.be odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1260, [\chi])\).

Hecke characteristic polynomials

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