Properties

Label 1560.1.y.c
Level $1560$
Weight $1$
Character orbit 1560.y
Self dual yes
Analytic conductor $0.779$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -39, -1560, 40
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1560.y (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{10}, \sqrt{-39})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.60840.4

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{12} + q^{13} + q^{15} + q^{16} + q^{18} - q^{20} - q^{24} + q^{25} + q^{26} - q^{27} + q^{30} + q^{32} + q^{36} - q^{39} - q^{40} + 2 q^{41} + 2 q^{43} - q^{45} - q^{48} - q^{49} + q^{50} + q^{52} - q^{54} + q^{60} + q^{64} - q^{65} - 2 q^{71} + q^{72} - q^{75} - q^{78} - 2 q^{79} - q^{80} + q^{81} + 2 q^{82} - 2 q^{83} + 2 q^{86} - 2 q^{89} - q^{90} - q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
389.1
0
1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
40.f even 2 1 RM by \(\Q(\sqrt{10}) \)
1560.y odd 2 1 CM by \(\Q(\sqrt{-390}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.1.y.c yes 1
3.b odd 2 1 1560.1.y.a 1
5.b even 2 1 1560.1.y.b yes 1
8.b even 2 1 1560.1.y.b yes 1
13.b even 2 1 1560.1.y.a 1
15.d odd 2 1 1560.1.y.d yes 1
24.h odd 2 1 1560.1.y.d yes 1
39.d odd 2 1 CM 1560.1.y.c yes 1
40.f even 2 1 RM 1560.1.y.c yes 1
65.d even 2 1 1560.1.y.d yes 1
104.e even 2 1 1560.1.y.d yes 1
120.i odd 2 1 1560.1.y.a 1
195.e odd 2 1 1560.1.y.b yes 1
312.b odd 2 1 1560.1.y.b yes 1
520.p even 2 1 1560.1.y.a 1
1560.y odd 2 1 CM 1560.1.y.c yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.y.a 1 3.b odd 2 1
1560.1.y.a 1 13.b even 2 1
1560.1.y.a 1 120.i odd 2 1
1560.1.y.a 1 520.p even 2 1
1560.1.y.b yes 1 5.b even 2 1
1560.1.y.b yes 1 8.b even 2 1
1560.1.y.b yes 1 195.e odd 2 1
1560.1.y.b yes 1 312.b odd 2 1
1560.1.y.c yes 1 1.a even 1 1 trivial
1560.1.y.c yes 1 39.d odd 2 1 CM
1560.1.y.c yes 1 40.f even 2 1 RM
1560.1.y.c yes 1 1560.y odd 2 1 CM
1560.1.y.d yes 1 15.d odd 2 1
1560.1.y.d yes 1 24.h odd 2 1
1560.1.y.d yes 1 65.d even 2 1
1560.1.y.d yes 1 104.e even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1560, [\chi])\):

\( T_{11} \) Copy content Toggle raw display
\( T_{41} - 2 \) Copy content Toggle raw display
\( T_{43} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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