Properties

Label 1560.1.dk.b
Level $1560$
Weight $1$
Character orbit 1560.dk
Analytic conductor $0.779$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -120
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.dk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.16039857600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} - \zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} - \zeta_{12}^{4} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + \zeta_{12}^{5} q^{6} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} - \zeta_{12}^{4} q^{10} + \zeta_{12} q^{11} + q^{12} + q^{13} + \zeta_{12} q^{15} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{3} q^{18} + \zeta_{12}^{5} q^{20} - \zeta_{12}^{2} q^{22} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{23} - \zeta_{12} q^{24} - q^{25} - \zeta_{12} q^{26} - q^{27} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{29} - \zeta_{12}^{2} q^{30} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{31} - \zeta_{12}^{5} q^{32} - \zeta_{12}^{5} q^{33} - \zeta_{12}^{4} q^{36} + (\zeta_{12}^{2} + 1) q^{37} - \zeta_{12}^{4} q^{39} + q^{40} - \zeta_{12}^{2} q^{43} + \zeta_{12}^{3} q^{44} - \zeta_{12}^{5} q^{45} + ( - \zeta_{12}^{4} + 1) q^{46} - \zeta_{12}^{3} q^{47} + \zeta_{12}^{2} q^{48} - \zeta_{12}^{4} q^{49} + \zeta_{12} q^{50} + \zeta_{12}^{2} q^{52} + \zeta_{12} q^{54} + \zeta_{12}^{4} q^{55} + (\zeta_{12}^{4} - 1) q^{58} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{3} q^{60} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{62} - q^{64} + \zeta_{12}^{3} q^{65} - q^{66} + (\zeta_{12}^{3} + \zeta_{12}) q^{69} + \zeta_{12}^{5} q^{72} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{74} + \zeta_{12}^{4} q^{75} + \zeta_{12}^{5} q^{78} - q^{79} - \zeta_{12} q^{80} + \zeta_{12}^{4} q^{81} + \zeta_{12}^{3} q^{86} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{87} - \zeta_{12}^{4} q^{88} - q^{90} + (\zeta_{12}^{5} - \zeta_{12}) q^{92} + (\zeta_{12}^{2} + 1) q^{93} + \zeta_{12}^{4} q^{94} - \zeta_{12}^{3} q^{96} + \zeta_{12}^{5} q^{98} - \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{4} - 2 q^{9} + 2 q^{10} + 4 q^{12} + 4 q^{13} - 2 q^{16} - 2 q^{22} - 4 q^{25} - 4 q^{27} - 2 q^{30} + 2 q^{36} + 6 q^{37} + 2 q^{39} + 4 q^{40} - 2 q^{43} + 6 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{52} - 2 q^{55} - 6 q^{58} - 4 q^{64} - 4 q^{66} - 2 q^{75} - 4 q^{79} - 2 q^{81} + 2 q^{88} - 4 q^{90} + 6 q^{93} - 2 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1109.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i 0 1.00000i −0.500000 0.866025i 0.500000 0.866025i
1109.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i 0 1.00000i −0.500000 0.866025i 0.500000 0.866025i
1349.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i 0 1.00000i −0.500000 + 0.866025i 0.500000 + 0.866025i
1349.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i 0 1.00000i −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)
3.b odd 2 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
40.f even 2 1 inner
520.bp even 6 1 inner
1560.dk odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.1.dk.b yes 4
3.b odd 2 1 inner 1560.1.dk.b yes 4
5.b even 2 1 1560.1.dk.a 4
8.b even 2 1 1560.1.dk.a 4
13.e even 6 1 inner 1560.1.dk.b yes 4
15.d odd 2 1 1560.1.dk.a 4
24.h odd 2 1 1560.1.dk.a 4
39.h odd 6 1 inner 1560.1.dk.b yes 4
40.f even 2 1 inner 1560.1.dk.b yes 4
65.l even 6 1 1560.1.dk.a 4
104.s even 6 1 1560.1.dk.a 4
120.i odd 2 1 CM 1560.1.dk.b yes 4
195.y odd 6 1 1560.1.dk.a 4
312.bg odd 6 1 1560.1.dk.a 4
520.bp even 6 1 inner 1560.1.dk.b yes 4
1560.dk odd 6 1 inner 1560.1.dk.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.dk.a 4 5.b even 2 1
1560.1.dk.a 4 8.b even 2 1
1560.1.dk.a 4 15.d odd 2 1
1560.1.dk.a 4 24.h odd 2 1
1560.1.dk.a 4 65.l even 6 1
1560.1.dk.a 4 104.s even 6 1
1560.1.dk.a 4 195.y odd 6 1
1560.1.dk.a 4 312.bg odd 6 1
1560.1.dk.b yes 4 1.a even 1 1 trivial
1560.1.dk.b yes 4 3.b odd 2 1 inner
1560.1.dk.b yes 4 13.e even 6 1 inner
1560.1.dk.b yes 4 39.h odd 6 1 inner
1560.1.dk.b yes 4 40.f even 2 1 inner
1560.1.dk.b yes 4 120.i odd 2 1 CM
1560.1.dk.b yes 4 520.bp even 6 1 inner
1560.1.dk.b yes 4 1560.dk odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{37}^{2} - 3T_{37} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1560, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$29$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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