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Properties

Label 1560.1.cs.f
Level $1560$
Weight $1$
Character orbit 1560.cs
Analytic conductor $0.779$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -104
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1560,1,Mod(467,1560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1560, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1560.467");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q - \zeta_{24}^{9} q^{2} + \zeta_{24}^{2} q^{3} - \zeta_{24}^{6} q^{4} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{11} q^{6} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{9} q^{2} + \zeta_{24}^{2} q^{3} - \zeta_{24}^{6} q^{4} - \zeta_{24}^{11} q^{5} - \zeta_{24}^{11} q^{6} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{3} q^{8} + \zeta_{24}^{4} q^{9} - \zeta_{24}^{8} q^{10} - \zeta_{24}^{8} q^{12} + \zeta_{24}^{9} q^{13} + (\zeta_{24}^{10} - \zeta_{24}^{2}) q^{14} + \zeta_{24} q^{15} - q^{16} + ( - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{17} + \zeta_{24} q^{18} - \zeta_{24}^{5} q^{20} + ( - \zeta_{24}^{7} - \zeta_{24}^{3}) q^{21} - \zeta_{24}^{5} q^{24} - \zeta_{24}^{10} q^{25} + \zeta_{24}^{6} q^{26} + \zeta_{24}^{6} q^{27} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{28} - \zeta_{24}^{10} q^{30} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{31} + \zeta_{24}^{9} q^{32} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{34} + ( - \zeta_{24}^{4} - 1) q^{35} - \zeta_{24}^{10} q^{36} + \zeta_{24}^{3} q^{37} + \zeta_{24}^{11} q^{39} - \zeta_{24}^{2} q^{40} + ( - \zeta_{24}^{4} - 1) q^{42} + (\zeta_{24}^{10} + \zeta_{24}^{8}) q^{43} + \zeta_{24}^{3} q^{45} + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{47} - \zeta_{24}^{2} q^{48} + (\zeta_{24}^{10} + \cdots + \zeta_{24}^{2}) q^{49} + \cdots + ( - \zeta_{24}^{11} + \cdots + \zeta_{24}^{3}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} + 4 q^{10} + 4 q^{12} - 8 q^{16} - 4 q^{17} - 12 q^{35} - 12 q^{42} - 4 q^{43} + 4 q^{51} + 8 q^{62} - 4 q^{65} + 4 q^{68} + 8 q^{74} + 8 q^{75} - 4 q^{78} - 4 q^{81} + 8 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(\zeta_{24}^{6}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
467.1
0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.258819 − 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.707107 + 0.707107i −0.866025 + 0.500000i − 1.00000i 0.258819 − 0.965926i 0.258819 − 0.965926i −1.22474 − 1.22474i 0.707107 + 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
467.2 −0.707107 + 0.707107i 0.866025 + 0.500000i − 1.00000i −0.965926 + 0.258819i −0.965926 + 0.258819i 1.22474 + 1.22474i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
467.3 0.707107 − 0.707107i −0.866025 + 0.500000i − 1.00000i −0.258819 + 0.965926i −0.258819 + 0.965926i 1.22474 + 1.22474i −0.707107 − 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
467.4 0.707107 − 0.707107i 0.866025 + 0.500000i − 1.00000i 0.965926 − 0.258819i 0.965926 − 0.258819i −1.22474 − 1.22474i −0.707107 − 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
1403.1 −0.707107 − 0.707107i −0.866025 − 0.500000i 1.00000i 0.258819 + 0.965926i 0.258819 + 0.965926i −1.22474 + 1.22474i 0.707107 − 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
1403.2 −0.707107 − 0.707107i 0.866025 − 0.500000i 1.00000i −0.965926 − 0.258819i −0.965926 − 0.258819i 1.22474 − 1.22474i 0.707107 − 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
1403.3 0.707107 + 0.707107i −0.866025 − 0.500000i 1.00000i −0.258819 − 0.965926i −0.258819 − 0.965926i 1.22474 − 1.22474i −0.707107 + 0.707107i 0.500000 + 0.866025i 0.500000 − 0.866025i
1403.4 0.707107 + 0.707107i 0.866025 − 0.500000i 1.00000i 0.965926 + 0.258819i 0.965926 + 0.258819i −1.22474 + 1.22474i −0.707107 + 0.707107i 0.500000 − 0.866025i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 467.4
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.f yes 8
3.b odd 2 1 1560.1.cs.c ✓ 8
5.c odd 4 1 1560.1.cs.c ✓ 8
8.d odd 2 1 inner 1560.1.cs.f yes 8
13.b even 2 1 inner 1560.1.cs.f yes 8
15.e even 4 1 inner 1560.1.cs.f yes 8
24.f even 2 1 1560.1.cs.c ✓ 8
39.d odd 2 1 1560.1.cs.c ✓ 8
40.k even 4 1 1560.1.cs.c ✓ 8
65.h odd 4 1 1560.1.cs.c ✓ 8
104.h odd 2 1 CM 1560.1.cs.f yes 8
120.q odd 4 1 inner 1560.1.cs.f yes 8
195.s even 4 1 inner 1560.1.cs.f yes 8
312.h even 2 1 1560.1.cs.c ✓ 8
520.bc even 4 1 1560.1.cs.c ✓ 8
1560.cs odd 4 1 inner 1560.1.cs.f yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.cs.c ✓ 8 3.b odd 2 1
1560.1.cs.c ✓ 8 5.c odd 4 1
1560.1.cs.c ✓ 8 24.f even 2 1
1560.1.cs.c ✓ 8 39.d odd 2 1
1560.1.cs.c ✓ 8 40.k even 4 1
1560.1.cs.c ✓ 8 65.h odd 4 1
1560.1.cs.c ✓ 8 312.h even 2 1
1560.1.cs.c ✓ 8 520.bc even 4 1
1560.1.cs.f yes 8 1.a even 1 1 trivial
1560.1.cs.f yes 8 8.d odd 2 1 inner
1560.1.cs.f yes 8 13.b even 2 1 inner
1560.1.cs.f yes 8 15.e even 4 1 inner
1560.1.cs.f yes 8 104.h odd 2 1 CM
1560.1.cs.f yes 8 120.q odd 4 1 inner
1560.1.cs.f yes 8 195.s even 4 1 inner
1560.1.cs.f yes 8 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 \(S_{1}^{\mathrm{new}}(1560, [\chi])\):

\( T_{7}^{4} + 9 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} + 2T_{17}^{2} - 2T_{17} + 1 \) Copy content Toggle raw display
\( T_{103} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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