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Properties

Label 1560.1.cs.e
Level $1560$
Weight $1$
Character orbit 1560.cs
Analytic conductor $0.779$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -39
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: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.49353408000000.10

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{2} q^{3} + \zeta_{16}^{6} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{5} q^{6} - \zeta_{16} q^{8} + \zeta_{16}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{16}^{3} q^{2} + \zeta_{16}^{2} q^{3} + \zeta_{16}^{6} q^{4} - \zeta_{16}^{5} q^{5} + \zeta_{16}^{5} q^{6} - \zeta_{16} q^{8} + \zeta_{16}^{4} q^{9} + q^{10} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{11} - q^{12} + \zeta_{16}^{6} q^{13} - \zeta_{16}^{7} q^{15} - \zeta_{16}^{4} q^{16} + \zeta_{16}^{7} q^{18} + \zeta_{16}^{3} q^{20} + (\zeta_{16}^{4} + \zeta_{16}^{2}) q^{22} - \zeta_{16}^{3} q^{24} - \zeta_{16}^{2} q^{25} - \zeta_{16} q^{26} + \zeta_{16}^{6} q^{27} + \zeta_{16}^{2} q^{30} - \zeta_{16}^{7} q^{32} + (\zeta_{16}^{3} + \zeta_{16}) q^{33} - \zeta_{16}^{2} q^{36} - q^{39} + \zeta_{16}^{6} q^{40} + (\zeta_{16}^{5} - \zeta_{16}^{3}) q^{41} + ( - \zeta_{16}^{4} - 1) q^{43} + (\zeta_{16}^{7} + \zeta_{16}^{5}) q^{44} + \zeta_{16} q^{45} + (\zeta_{16}^{7} - \zeta_{16}^{5}) q^{47} - \zeta_{16}^{6} q^{48} + \zeta_{16}^{4} q^{49} - \zeta_{16}^{5} q^{50} - \zeta_{16}^{4} q^{52} - \zeta_{16} q^{54} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{55} + (\zeta_{16}^{7} + \zeta_{16}) q^{59} + \zeta_{16}^{5} q^{60} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{61} + \zeta_{16}^{2} q^{64} + \zeta_{16}^{3} q^{65} + (\zeta_{16}^{6} + \zeta_{16}^{4}) q^{66} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{71} - \zeta_{16}^{5} q^{72} - \zeta_{16}^{4} q^{75} - \zeta_{16}^{3} q^{78} + ( - \zeta_{16}^{6} + \zeta_{16}^{2}) q^{79} - \zeta_{16} q^{80} - q^{81} + ( - \zeta_{16}^{6} - 1) q^{82} + ( - \zeta_{16}^{3} - \zeta_{16}) q^{83} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{86} + ( - \zeta_{16}^{2} - 1) q^{88} + ( - \zeta_{16}^{5} - \zeta_{16}^{3}) q^{89} + \zeta_{16}^{4} q^{90} + ( - \zeta_{16}^{2} + 1) q^{94} + \zeta_{16} q^{96} + \zeta_{16}^{7} q^{98} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{10} - 8 q^{12} - 8 q^{39} - 8 q^{43} - 8 q^{81} - 8 q^{82} - 8 q^{88} + 8 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{16}^{4}\) \(-1\)

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
0.382683 + 0.923880i
−0.923880 + 0.382683i
0.923880 − 0.382683i
−0.382683 − 0.923880i
0.382683 − 0.923880i
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.382683 + 0.923880i
−0.923880 − 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.923880 + 0.382683i 0.923880 − 0.382683i 0 −0.382683 − 0.923880i − 1.00000i 1.00000
467.2 −0.382683 + 0.923880i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.382683 − 0.923880i 0.382683 + 0.923880i 0 0.923880 − 0.382683i − 1.00000i 1.00000
467.3 0.382683 − 0.923880i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.382683 + 0.923880i −0.382683 − 0.923880i 0 −0.923880 + 0.382683i − 1.00000i 1.00000
467.4 0.923880 + 0.382683i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.923880 − 0.382683i −0.923880 + 0.382683i 0 0.382683 + 0.923880i − 1.00000i 1.00000
1403.1 −0.923880 + 0.382683i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.923880 − 0.382683i 0.923880 + 0.382683i 0 −0.382683 + 0.923880i 1.00000i 1.00000
1403.2 −0.382683 − 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i −0.382683 + 0.923880i 0.382683 − 0.923880i 0 0.923880 + 0.382683i 1.00000i 1.00000
1403.3 0.382683 + 0.923880i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.382683 − 0.923880i −0.382683 + 0.923880i 0 −0.923880 − 0.382683i 1.00000i 1.00000
1403.4 0.923880 − 0.382683i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.923880 + 0.382683i −0.923880 − 0.382683i 0 0.382683 − 0.923880i 1.00000i 1.00000
\(n\): 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
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner
520.bc 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.e yes 8
3.b odd 2 1 inner 1560.1.cs.e yes 8
5.c odd 4 1 1560.1.cs.d ✓ 8
8.d odd 2 1 1560.1.cs.d ✓ 8
13.b even 2 1 inner 1560.1.cs.e yes 8
15.e even 4 1 1560.1.cs.d ✓ 8
24.f even 2 1 1560.1.cs.d ✓ 8
39.d odd 2 1 CM 1560.1.cs.e yes 8
40.k even 4 1 inner 1560.1.cs.e yes 8
65.h odd 4 1 1560.1.cs.d ✓ 8
104.h odd 2 1 1560.1.cs.d ✓ 8
120.q odd 4 1 inner 1560.1.cs.e yes 8
195.s even 4 1 1560.1.cs.d ✓ 8
312.h even 2 1 1560.1.cs.d ✓ 8
520.bc even 4 1 inner 1560.1.cs.e yes 8
1560.cs odd 4 1 inner 1560.1.cs.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.1.cs.d ✓ 8 5.c odd 4 1
1560.1.cs.d ✓ 8 8.d odd 2 1
1560.1.cs.d ✓ 8 15.e even 4 1
1560.1.cs.d ✓ 8 24.f even 2 1
1560.1.cs.d ✓ 8 65.h odd 4 1
1560.1.cs.d ✓ 8 104.h odd 2 1
1560.1.cs.d ✓ 8 195.s even 4 1
1560.1.cs.d ✓ 8 312.h even 2 1
1560.1.cs.e yes 8 1.a even 1 1 trivial
1560.1.cs.e yes 8 3.b odd 2 1 inner
1560.1.cs.e yes 8 13.b even 2 1 inner
1560.1.cs.e yes 8 39.d odd 2 1 CM
1560.1.cs.e yes 8 40.k even 4 1 inner
1560.1.cs.e yes 8 120.q odd 4 1 inner
1560.1.cs.e yes 8 520.bc even 4 1 inner
1560.1.cs.e 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 \(S_{1}^{\mathrm{new}}(1560, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) 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
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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