Properties

Label 1560.1.y.e
Level $1560$
Weight $1$
Character orbit 1560.y
Analytic conductor $0.779$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -39, -120, 520
Inner twists $8$

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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: no
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-30}, \sqrt{-39})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.3701505600.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - i q^{2} - q^{3} - q^{4} - i q^{5} + i q^{6} + i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} - q^{3} - q^{4} - i q^{5} + i q^{6} + i q^{8} + q^{9} - q^{10} - i q^{11} + q^{12} + q^{13} + i q^{15} + q^{16} - i q^{18} + i q^{20} - 2 q^{22} - i q^{24} - q^{25} - i q^{26} - q^{27} + q^{30} - i q^{32} + 2 i q^{33} - q^{36} - q^{39} + q^{40} - q^{43} + 2 i q^{44} - i q^{45} - i q^{47} - q^{48} - q^{49} + i q^{50} - q^{52} + i q^{54} - 2 q^{55} + i q^{59} - i q^{60} - q^{64} - i q^{65} + 2 q^{66} + i q^{72} + q^{75} + i q^{78} + q^{79} - i q^{80} + q^{81} + 2 i q^{86} + 2 q^{88} - q^{90} - 2 q^{94} + i q^{96} + i q^{98} - 2 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} + 2 q^{13} + 2 q^{16} - 4 q^{22} - 2 q^{25} - 2 q^{27} + 2 q^{30} - 2 q^{36} - 2 q^{39} + 2 q^{40} - 4 q^{43} - 2 q^{48} - 2 q^{49} - 2 q^{52} - 4 q^{55} - 2 q^{64} + 4 q^{66} + 2 q^{75} + 4 q^{79} + 2 q^{81} + 4 q^{88} - 2 q^{90} - 4 q^{94}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
389.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 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 \(\Q(\sqrt{-39}) \)
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)
520.p even 2 1 RM by \(\Q(\sqrt{130}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
40.f even 2 1 inner
1560.y odd 2 1 inner

Twists

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

Hecke kernels

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

\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{41} \) Copy content Toggle raw display
\( T_{43} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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