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Properties

Label 928.1.bs.a
Level $928$
Weight $1$
Character orbit 928.bs
Analytic conductor $0.463$
Analytic rank $0$
Dimension $12$
Projective image $D_{28}$
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] = [928,1,Mod(97,928)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(928, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([0, 0, 23]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("928.97");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 928 = 2^{5} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 928.bs (of order \(28\), degree \(12\), 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: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{28}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{28}^{12} - \zeta_{28}^{4}) q^{5} - \zeta_{28}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{28}^{12} - \zeta_{28}^{4}) q^{5} - \zeta_{28}^{3} q^{9} + (\zeta_{28}^{13} + \zeta_{28}^{9}) q^{13} + ( - \zeta_{28}^{6} - \zeta_{28}) q^{17} + ( - \zeta_{28}^{10} + \cdots + \zeta_{28}^{2}) q^{25} + \cdots + (\zeta_{28}^{11} - 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{17} - 2 q^{25} + 2 q^{37} - 2 q^{41} + 2 q^{49} + 2 q^{61} + 2 q^{73} + 2 q^{81} - 2 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/928\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-\zeta_{28}^{9}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.974928 + 0.222521i
−0.974928 − 0.222521i
−0.781831 + 0.623490i
0.433884 + 0.900969i
0.433884 − 0.900969i
0.781831 + 0.623490i
−0.974928 + 0.222521i
0.974928 − 0.222521i
−0.781831 − 0.623490i
−0.433884 + 0.900969i
−0.433884 − 0.900969i
0.781831 − 0.623490i
0 0 0 −1.52446 − 0.347948i 0 0 0 −0.781831 − 0.623490i 0
193.1 0 0 0 −1.52446 − 0.347948i 0 0 0 0.781831 + 0.623490i 0
321.1 0 0 0 0.678448 − 0.541044i 0 0 0 −0.433884 − 0.900969i 0
385.1 0 0 0 0.846011 + 1.75676i 0 0 0 0.974928 + 0.222521i 0
417.1 0 0 0 0.846011 − 1.75676i 0 0 0 0.974928 − 0.222521i 0
449.1 0 0 0 0.678448 + 0.541044i 0 0 0 0.433884 − 0.900969i 0
577.1 0 0 0 −1.52446 + 0.347948i 0 0 0 0.781831 − 0.623490i 0
641.1 0 0 0 −1.52446 + 0.347948i 0 0 0 −0.781831 + 0.623490i 0
769.1 0 0 0 0.678448 + 0.541044i 0 0 0 −0.433884 + 0.900969i 0
801.1 0 0 0 0.846011 − 1.75676i 0 0 0 −0.974928 + 0.222521i 0
833.1 0 0 0 0.846011 + 1.75676i 0 0 0 −0.974928 − 0.222521i 0
897.1 0 0 0 0.678448 − 0.541044i 0 0 0 0.433884 + 0.900969i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
29.f odd 28 1 inner
116.l even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 928.1.bs.a ✓ 12
4.b odd 2 1 CM 928.1.bs.a ✓ 12
8.b even 2 1 1856.1.ca.a 12
8.d odd 2 1 1856.1.ca.a 12
29.f odd 28 1 inner 928.1.bs.a ✓ 12
116.l even 28 1 inner 928.1.bs.a ✓ 12
232.u odd 28 1 1856.1.ca.a 12
232.v even 28 1 1856.1.ca.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.1.bs.a ✓ 12 1.a even 1 1 trivial
928.1.bs.a ✓ 12 4.b odd 2 1 CM
928.1.bs.a ✓ 12 29.f odd 28 1 inner
928.1.bs.a ✓ 12 116.l even 28 1 inner
1856.1.ca.a 12 8.b even 2 1
1856.1.ca.a 12 8.d odd 2 1
1856.1.ca.a 12 232.u odd 28 1
1856.1.ca.a 12 232.v even 28 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(928, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 7 T^{3} - 7 T + 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} + 7 T^{10} + \cdots + 49 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} - 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{12} + 12 T^{11} + \cdots + 1 \) Copy content Toggle raw display
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