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Properties

Label 2132.1.cs.a
Level $2132$
Weight $1$
Character orbit 2132.cs
Analytic conductor $1.064$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2132,1,Mod(167,2132)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2132, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([12, 14, 9]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2132.167");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2132 = 2^{2} \cdot 13 \cdot 41 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2132.cs (of order \(24\), 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: \(1.06400660693\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{4} q^{2} + \zeta_{24}^{8} q^{4} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{5} + q^{8} + \zeta_{24}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{4} q^{2} + \zeta_{24}^{8} q^{4} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{5} + q^{8} + \zeta_{24}^{5} q^{9} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{10} + \zeta_{24}^{4} q^{13} - \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{10} + \zeta_{24}^{9}) q^{17} - \zeta_{24}^{9} q^{18} + ( - \zeta_{24}^{3} - \zeta_{24}) q^{20} + (\zeta_{24}^{10} - \zeta_{24}^{2} - 1) q^{25} - \zeta_{24}^{8} q^{26} + (\zeta_{24}^{11} - \zeta_{24}^{6}) q^{29} + \zeta_{24}^{8} q^{32} + ( - \zeta_{24}^{2} + \zeta_{24}) q^{34} - \zeta_{24} q^{36} - \zeta_{24}^{7} q^{37} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{40} - q^{41} + (\zeta_{24}^{10} - 1) q^{45} - \zeta_{24}^{7} q^{49} + (\zeta_{24}^{6} + \cdots + \zeta_{24}^{2}) q^{50} + \cdots + \zeta_{24}^{11} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + 8 q^{8} + 4 q^{13} - 4 q^{16} - 8 q^{25} + 4 q^{26} - 4 q^{32} - 8 q^{41} - 8 q^{45} + 4 q^{50} - 8 q^{52} - 4 q^{53} + 8 q^{61} + 8 q^{64} + 4 q^{82} - 4 q^{85} + 4 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2132\mathbb{Z}\right)^\times\).

\(n\) \(821\) \(1067\) \(1769\)
\(\chi(n)\) \(\zeta_{24}^{2}\) \(-1\) \(\zeta_{24}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
167.1
0.258819 − 0.965926i
0.258819 + 0.965926i
0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.500000 − 0.866025i 0 −0.500000 + 0.866025i − 0.517638i 0 0 1.00000 0.965926 − 0.258819i −0.448288 + 0.258819i
383.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.517638i 0 0 1.00000 0.965926 + 0.258819i −0.448288 − 0.258819i
735.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i − 1.93185i 0 0 1.00000 0.258819 − 0.965926i 1.67303 + 0.965926i
847.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.93185i 0 0 1.00000 0.258819 + 0.965926i 1.67303 − 0.965926i
1151.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.93185i 0 0 1.00000 −0.258819 + 0.965926i −1.67303 − 0.965926i
1367.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i − 1.93185i 0 0 1.00000 −0.258819 − 0.965926i −1.67303 + 0.965926i
1883.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.517638i 0 0 1.00000 −0.965926 + 0.258819i 0.448288 − 0.258819i
1995.1 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i − 0.517638i 0 0 1.00000 −0.965926 − 0.258819i 0.448288 + 0.258819i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 167.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
533.bj even 24 1 inner
2132.cs odd 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2132.1.cs.a ✓ 8
4.b odd 2 1 CM 2132.1.cs.a ✓ 8
13.f odd 12 1 2132.1.cy.a yes 8
41.e odd 8 1 2132.1.cy.a yes 8
52.l even 12 1 2132.1.cy.a yes 8
164.i even 8 1 2132.1.cy.a yes 8
533.bj even 24 1 inner 2132.1.cs.a ✓ 8
2132.cs odd 24 1 inner 2132.1.cs.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2132.1.cs.a ✓ 8 1.a even 1 1 trivial
2132.1.cs.a ✓ 8 4.b odd 2 1 CM
2132.1.cs.a ✓ 8 533.bj even 24 1 inner
2132.1.cs.a ✓ 8 2132.cs odd 24 1 inner
2132.1.cy.a yes 8 13.f odd 12 1
2132.1.cy.a yes 8 41.e odd 8 1
2132.1.cy.a yes 8 52.l even 12 1
2132.1.cy.a yes 8 164.i even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2132, [\chi])\).

Hecke characteristic polynomials

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