Properties

Label 1560.1.cs.b
Level $1560$
Weight $1$
Character orbit 1560.cs
Analytic conductor $0.779$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -104
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.cs (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1521000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} + q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{5} - \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} + q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{5} - \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} - q^{10} + \zeta_{8}^{2} q^{12} - \zeta_{8} q^{13} - \zeta_{8}^{3} q^{15} - q^{16} + ( - \zeta_{8}^{2} - 1) q^{17} - \zeta_{8} q^{18} + \zeta_{8} q^{20} - \zeta_{8}^{3} q^{24} - \zeta_{8}^{2} q^{25} + \zeta_{8}^{2} q^{26} + q^{27} - q^{30} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{31} + \zeta_{8} q^{32} + (\zeta_{8}^{3} + \zeta_{8}) q^{34} + \zeta_{8}^{2} q^{36} + \zeta_{8}^{3} q^{37} - \zeta_{8} q^{39} - \zeta_{8}^{2} q^{40} + (\zeta_{8}^{2} + 1) q^{43} - \zeta_{8}^{3} q^{45} - q^{48} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{3} q^{50} + ( - \zeta_{8}^{2} - 1) q^{51} - \zeta_{8}^{3} q^{52} - \zeta_{8} q^{54} + \zeta_{8} q^{60} + ( - \zeta_{8}^{2} - 1) q^{62} - \zeta_{8}^{2} q^{64} - q^{65} + ( - \zeta_{8}^{2} + 1) q^{68} + (\zeta_{8}^{3} + \zeta_{8}) q^{71} - \zeta_{8}^{3} q^{72} + 2 q^{74} - \zeta_{8}^{2} q^{75} + \zeta_{8}^{2} q^{78} + \zeta_{8}^{3} q^{80} + q^{81} + (\zeta_{8}^{3} - \zeta_{8}) q^{85} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{86} - q^{90} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{93} + \zeta_{8} q^{96} - \zeta_{8}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 4 q^{10} - 4 q^{16} - 4 q^{17} + 4 q^{27} - 4 q^{30} + 4 q^{43} - 4 q^{48} - 4 q^{51} - 4 q^{62} - 4 q^{65} + 4 q^{68} + 8 q^{74} + 4 q^{81} - 4 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{8}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
467.1
0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i 1.00000 1.00000i 0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.707107 + 0.707107i 1.00000 −1.00000
467.2 0.707107 0.707107i 1.00000 1.00000i −0.707107 0.707107i 0.707107 0.707107i 0 −0.707107 0.707107i 1.00000 −1.00000
1403.1 −0.707107 0.707107i 1.00000 1.00000i 0.707107 0.707107i −0.707107 0.707107i 0 0.707107 0.707107i 1.00000 −1.00000
1403.2 0.707107 + 0.707107i 1.00000 1.00000i −0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.707107 + 0.707107i 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
104.h odd 2 1 CM by \(\Q(\sqrt{-26}) \)
8.d odd 2 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
120.q odd 4 1 inner
195.s even 4 1 inner
1560.cs odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.1.cs.b yes 4
3.b odd 2 1 1560.1.cs.a 4
5.c odd 4 1 1560.1.cs.a 4
8.d odd 2 1 inner 1560.1.cs.b yes 4
13.b even 2 1 inner 1560.1.cs.b yes 4
15.e even 4 1 inner 1560.1.cs.b yes 4
24.f even 2 1 1560.1.cs.a 4
39.d odd 2 1 1560.1.cs.a 4
40.k even 4 1 1560.1.cs.a 4
65.h odd 4 1 1560.1.cs.a 4
104.h odd 2 1 CM 1560.1.cs.b yes 4
120.q odd 4 1 inner 1560.1.cs.b yes 4
195.s even 4 1 inner 1560.1.cs.b yes 4
312.h even 2 1 1560.1.cs.a 4
520.bc even 4 1 1560.1.cs.a 4
1560.cs odd 4 1 inner 1560.1.cs.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.cs.a 4 3.b odd 2 1
1560.1.cs.a 4 5.c odd 4 1
1560.1.cs.a 4 24.f even 2 1
1560.1.cs.a 4 39.d odd 2 1
1560.1.cs.a 4 40.k even 4 1
1560.1.cs.a 4 65.h odd 4 1
1560.1.cs.a 4 312.h even 2 1
1560.1.cs.a 4 520.bc even 4 1
1560.1.cs.b yes 4 1.a even 1 1 trivial
1560.1.cs.b yes 4 8.d odd 2 1 inner
1560.1.cs.b yes 4 13.b even 2 1 inner
1560.1.cs.b yes 4 15.e even 4 1 inner
1560.1.cs.b yes 4 104.h odd 2 1 CM
1560.1.cs.b yes 4 120.q odd 4 1 inner
1560.1.cs.b yes 4 195.s even 4 1 inner
1560.1.cs.b yes 4 1560.cs odd 4 1 inner

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 2 \) Copy content Toggle raw display
\( T_{103} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 16 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) 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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