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Properties

Label 1044.1.ba.a
Level $1044$
Weight $1$
Character orbit 1044.ba
Analytic conductor $0.521$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1044,1,Mod(91,1044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1044, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1044.91");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1044 = 2^{2} \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1044.ba (of order \(14\), degree \(6\), 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: \(0.521023873189\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{28}^{3} q^{2} + \zeta_{28}^{6} q^{4} + (\zeta_{28}^{5} - \zeta_{28}) q^{5} + \zeta_{28}^{9} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{28}^{3} q^{2} + \zeta_{28}^{6} q^{4} + (\zeta_{28}^{5} - \zeta_{28}) q^{5} + \zeta_{28}^{9} q^{8} + (\zeta_{28}^{8} - \zeta_{28}^{4}) q^{10} + ( - \zeta_{28}^{6} + \zeta_{28}^{4}) q^{13} + \zeta_{28}^{12} q^{16} + (\zeta_{28}^{11} + \zeta_{28}^{3}) q^{17} + (\zeta_{28}^{11} - \zeta_{28}^{7}) q^{20} + (\zeta_{28}^{10} + \cdots + \zeta_{28}^{2}) q^{25}+ \cdots - \zeta_{28}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} - 4 q^{13} - 2 q^{16} + 2 q^{25} - 10 q^{34} - 14 q^{40} - 2 q^{49} + 4 q^{52} + 2 q^{58} + 2 q^{64} + 14 q^{73} - 4 q^{82} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1044\mathbb{Z}\right)^\times\).

\(n\) \(523\) \(901\) \(929\)
\(\chi(n)\) \(-1\) \(\zeta_{28}^{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
0.433884 + 0.900969i
−0.433884 − 0.900969i
0.781831 − 0.623490i
−0.781831 + 0.623490i
0.433884 − 0.900969i
−0.433884 + 0.900969i
−0.974928 − 0.222521i
0.974928 + 0.222521i
0.781831 + 0.623490i
−0.781831 − 0.623490i
−0.974928 + 0.222521i
0.974928 − 0.222521i
−0.974928 − 0.222521i 0 0.900969 + 0.433884i 0.347948 − 1.52446i 0 0 −0.781831 − 0.623490i 0 −0.678448 + 1.40881i
91.2 0.974928 + 0.222521i 0 0.900969 + 0.433884i −0.347948 + 1.52446i 0 0 0.781831 + 0.623490i 0 −0.678448 + 1.40881i
415.1 −0.433884 − 0.900969i 0 −0.623490 + 0.781831i −1.75676 + 0.846011i 0 0 0.974928 + 0.222521i 0 1.52446 + 1.21572i
415.2 0.433884 + 0.900969i 0 −0.623490 + 0.781831i 1.75676 − 0.846011i 0 0 −0.974928 − 0.222521i 0 1.52446 + 1.21572i
631.1 −0.974928 + 0.222521i 0 0.900969 − 0.433884i 0.347948 + 1.52446i 0 0 −0.781831 + 0.623490i 0 −0.678448 − 1.40881i
631.2 0.974928 − 0.222521i 0 0.900969 − 0.433884i −0.347948 − 1.52446i 0 0 0.781831 − 0.623490i 0 −0.678448 − 1.40881i
847.1 −0.781831 − 0.623490i 0 0.222521 + 0.974928i 0.541044 − 0.678448i 0 0 0.433884 − 0.900969i 0 −0.846011 + 0.193096i
847.2 0.781831 + 0.623490i 0 0.222521 + 0.974928i −0.541044 + 0.678448i 0 0 −0.433884 + 0.900969i 0 −0.846011 + 0.193096i
883.1 −0.433884 + 0.900969i 0 −0.623490 − 0.781831i −1.75676 − 0.846011i 0 0 0.974928 − 0.222521i 0 1.52446 − 1.21572i
883.2 0.433884 − 0.900969i 0 −0.623490 − 0.781831i 1.75676 + 0.846011i 0 0 −0.974928 + 0.222521i 0 1.52446 − 1.21572i
991.1 −0.781831 + 0.623490i 0 0.222521 − 0.974928i 0.541044 + 0.678448i 0 0 0.433884 + 0.900969i 0 −0.846011 − 0.193096i
991.2 0.781831 − 0.623490i 0 0.222521 − 0.974928i −0.541044 − 0.678448i 0 0 −0.433884 − 0.900969i 0 −0.846011 − 0.193096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner
29.e even 14 1 inner
87.h odd 14 1 inner
116.h odd 14 1 inner
348.t even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1044.1.ba.a ✓ 12
3.b odd 2 1 inner 1044.1.ba.a ✓ 12
4.b odd 2 1 CM 1044.1.ba.a ✓ 12
12.b even 2 1 inner 1044.1.ba.a ✓ 12
29.e even 14 1 inner 1044.1.ba.a ✓ 12
87.h odd 14 1 inner 1044.1.ba.a ✓ 12
116.h odd 14 1 inner 1044.1.ba.a ✓ 12
348.t even 14 1 inner 1044.1.ba.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1044.1.ba.a ✓ 12 1.a even 1 1 trivial
1044.1.ba.a ✓ 12 3.b odd 2 1 inner
1044.1.ba.a ✓ 12 4.b odd 2 1 CM
1044.1.ba.a ✓ 12 12.b even 2 1 inner
1044.1.ba.a ✓ 12 29.e even 14 1 inner
1044.1.ba.a ✓ 12 87.h odd 14 1 inner
1044.1.ba.a ✓ 12 116.h odd 14 1 inner
1044.1.ba.a ✓ 12 348.t even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1044, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 35 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) 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
$37$ \( (T^{6} + 7 T^{3} - 7 T + 7)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 5 T^{4} + 6 T^{2} + 1)^{2} \) 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
$61$ \( (T^{6} - 7 T^{3} + 7 T + 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{6} - 7 T^{5} + 21 T^{4} + \cdots + 7)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} - 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( (T^{6} + 7 T^{5} + 21 T^{4} + \cdots + 7)^{2} \) Copy content Toggle raw display
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