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Properties

Label 1320.1.dk.a
Level $1320$
Weight $1$
Character orbit 1320.dk
Analytic conductor $0.659$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1320,1,Mod(173,1320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1320, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 10, 10, 15, 6]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1320.173");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1320.dk (of order \(20\), degree \(8\), 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: \(0.658765816676\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{40}^{17} q^{2} + \zeta_{40} q^{3} - \zeta_{40}^{14} q^{4} + \zeta_{40}^{19} q^{5} - \zeta_{40}^{18} q^{6} + ( - \zeta_{40}^{12} - \zeta_{40}^{6}) q^{7} - \zeta_{40}^{11} q^{8} + \zeta_{40}^{2} q^{9} + \cdots + \zeta_{40}^{17} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} - 4 q^{10} - 16 q^{15} + 4 q^{16} + 4 q^{22} - 4 q^{24} - 16 q^{28} - 4 q^{33} + 4 q^{36} - 4 q^{42} + 16 q^{54} + 16 q^{58} + 4 q^{63} - 4 q^{70} - 4 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87}+ \cdots - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(881\) \(991\) \(1057\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{40}^{10}\) \(\zeta_{40}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1
0.453990 − 0.891007i
−0.453990 + 0.891007i
−0.891007 + 0.453990i
0.891007 − 0.453990i
0.156434 + 0.987688i
−0.156434 − 0.987688i
−0.891007 − 0.453990i
0.891007 + 0.453990i
0.453990 + 0.891007i
−0.453990 − 0.891007i
−0.987688 + 0.156434i
0.987688 − 0.156434i
−0.987688 − 0.156434i
0.987688 + 0.156434i
0.156434 − 0.987688i
−0.156434 + 0.987688i
−0.987688 − 0.156434i 0.453990 − 0.891007i 0.951057 + 0.309017i −0.453990 − 0.891007i −0.587785 + 0.809017i −1.76007 + 0.896802i −0.891007 − 0.453990i −0.587785 − 0.809017i 0.309017 + 0.951057i
173.2 0.987688 + 0.156434i −0.453990 + 0.891007i 0.951057 + 0.309017i 0.453990 + 0.891007i −0.587785 + 0.809017i −1.76007 + 0.896802i 0.891007 + 0.453990i −0.587785 − 0.809017i 0.309017 + 0.951057i
293.1 −0.156434 − 0.987688i −0.891007 + 0.453990i −0.951057 + 0.309017i 0.891007 + 0.453990i 0.587785 + 0.809017i 0.142040 − 0.278768i 0.453990 + 0.891007i 0.587785 − 0.809017i 0.309017 − 0.951057i
293.2 0.156434 + 0.987688i 0.891007 − 0.453990i −0.951057 + 0.309017i −0.891007 − 0.453990i 0.587785 + 0.809017i 0.142040 − 0.278768i −0.453990 − 0.891007i 0.587785 − 0.809017i 0.309017 − 0.951057i
413.1 −0.453990 + 0.891007i 0.156434 + 0.987688i −0.587785 − 0.809017i −0.156434 + 0.987688i −0.951057 − 0.309017i 0.896802 + 0.142040i 0.987688 − 0.156434i −0.951057 + 0.309017i −0.809017 − 0.587785i
413.2 0.453990 − 0.891007i −0.156434 − 0.987688i −0.587785 − 0.809017i 0.156434 − 0.987688i −0.951057 − 0.309017i 0.896802 + 0.142040i −0.987688 + 0.156434i −0.951057 + 0.309017i −0.809017 − 0.587785i
437.1 −0.156434 + 0.987688i −0.891007 − 0.453990i −0.951057 − 0.309017i 0.891007 − 0.453990i 0.587785 − 0.809017i 0.142040 + 0.278768i 0.453990 − 0.891007i 0.587785 + 0.809017i 0.309017 + 0.951057i
437.2 0.156434 − 0.987688i 0.891007 + 0.453990i −0.951057 − 0.309017i −0.891007 + 0.453990i 0.587785 − 0.809017i 0.142040 + 0.278768i −0.453990 + 0.891007i 0.587785 + 0.809017i 0.309017 + 0.951057i
557.1 −0.987688 + 0.156434i 0.453990 + 0.891007i 0.951057 − 0.309017i −0.453990 + 0.891007i −0.587785 − 0.809017i −1.76007 − 0.896802i −0.891007 + 0.453990i −0.587785 + 0.809017i 0.309017 − 0.951057i
557.2 0.987688 − 0.156434i −0.453990 − 0.891007i 0.951057 − 0.309017i 0.453990 − 0.891007i −0.587785 − 0.809017i −1.76007 − 0.896802i 0.891007 − 0.453990i −0.587785 + 0.809017i 0.309017 − 0.951057i
677.1 −0.891007 − 0.453990i −0.987688 + 0.156434i 0.587785 + 0.809017i 0.987688 + 0.156434i 0.951057 + 0.309017i −0.278768 + 1.76007i −0.156434 − 0.987688i 0.951057 − 0.309017i −0.809017 − 0.587785i
677.2 0.891007 + 0.453990i 0.987688 − 0.156434i 0.587785 + 0.809017i −0.987688 − 0.156434i 0.951057 + 0.309017i −0.278768 + 1.76007i 0.156434 + 0.987688i 0.951057 − 0.309017i −0.809017 − 0.587785i
893.1 −0.891007 + 0.453990i −0.987688 − 0.156434i 0.587785 − 0.809017i 0.987688 − 0.156434i 0.951057 − 0.309017i −0.278768 − 1.76007i −0.156434 + 0.987688i 0.951057 + 0.309017i −0.809017 + 0.587785i
893.2 0.891007 − 0.453990i 0.987688 + 0.156434i 0.587785 − 0.809017i −0.987688 + 0.156434i 0.951057 − 0.309017i −0.278768 − 1.76007i 0.156434 − 0.987688i 0.951057 + 0.309017i −0.809017 + 0.587785i
1157.1 −0.453990 − 0.891007i 0.156434 − 0.987688i −0.587785 + 0.809017i −0.156434 − 0.987688i −0.951057 + 0.309017i 0.896802 − 0.142040i 0.987688 + 0.156434i −0.951057 − 0.309017i −0.809017 + 0.587785i
1157.2 0.453990 + 0.891007i −0.156434 + 0.987688i −0.587785 + 0.809017i 0.156434 + 0.987688i −0.951057 + 0.309017i 0.896802 − 0.142040i −0.987688 − 0.156434i −0.951057 − 0.309017i −0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
55.l even 20 1 inner
165.u odd 20 1 inner
440.bu even 20 1 inner
1320.dk odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1320.1.dk.a ✓ 16
3.b odd 2 1 inner 1320.1.dk.a ✓ 16
5.c odd 4 1 1320.1.dk.b yes 16
8.b even 2 1 inner 1320.1.dk.a ✓ 16
11.d odd 10 1 1320.1.dk.b yes 16
15.e even 4 1 1320.1.dk.b yes 16
24.h odd 2 1 CM 1320.1.dk.a ✓ 16
33.f even 10 1 1320.1.dk.b yes 16
40.i odd 4 1 1320.1.dk.b yes 16
55.l even 20 1 inner 1320.1.dk.a ✓ 16
88.p odd 10 1 1320.1.dk.b yes 16
120.w even 4 1 1320.1.dk.b yes 16
165.u odd 20 1 inner 1320.1.dk.a ✓ 16
264.u even 10 1 1320.1.dk.b yes 16
440.bu even 20 1 inner 1320.1.dk.a ✓ 16
1320.dk odd 20 1 inner 1320.1.dk.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.1.dk.a ✓ 16 1.a even 1 1 trivial
1320.1.dk.a ✓ 16 3.b odd 2 1 inner
1320.1.dk.a ✓ 16 8.b even 2 1 inner
1320.1.dk.a ✓ 16 24.h odd 2 1 CM
1320.1.dk.a ✓ 16 55.l even 20 1 inner
1320.1.dk.a ✓ 16 165.u odd 20 1 inner
1320.1.dk.a ✓ 16 440.bu even 20 1 inner
1320.1.dk.a ✓ 16 1320.dk odd 20 1 inner
1320.1.dk.b yes 16 5.c odd 4 1
1320.1.dk.b yes 16 11.d odd 10 1
1320.1.dk.b yes 16 15.e even 4 1
1320.1.dk.b yes 16 33.f even 10 1
1320.1.dk.b yes 16 40.i odd 4 1
1320.1.dk.b yes 16 88.p odd 10 1
1320.1.dk.b yes 16 120.w even 4 1
1320.1.dk.b yes 16 264.u even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 2T_{7}^{7} + 2T_{7}^{6} - 4T_{7}^{4} - 10T_{7}^{3} + 13T_{7}^{2} - 4T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} - 11 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 2 T^{7} + 7 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + 4 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
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