Properties

Label 1044.1.bb.a
Level $1044$
Weight $1$
Character orbit 1044.bb
Analytic conductor $0.521$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
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] = [1044,1,Mod(199,1044)]
 
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, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1044.199");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1044 = 2^{2} \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1044.bb (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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.38068692544.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{14} q^{2} + \zeta_{14}^{2} q^{4} + (\zeta_{14}^{5} - \zeta_{14}^{4}) q^{5} + \zeta_{14}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{14} q^{2} + \zeta_{14}^{2} q^{4} + (\zeta_{14}^{5} - \zeta_{14}^{4}) q^{5} + \zeta_{14}^{3} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{5}) q^{10} + (\zeta_{14}^{6} + \zeta_{14}^{2}) q^{13} + \zeta_{14}^{4} q^{16} + ( - \zeta_{14}^{6} + \zeta_{14}) q^{17} + ( - \zeta_{14}^{6} - 1) q^{20} + ( - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14}) q^{25} + (\zeta_{14}^{3} - 1) q^{26} - \zeta_{14}^{4} q^{29} + \zeta_{14}^{5} q^{32} + (\zeta_{14}^{2} + 1) q^{34} + ( - \zeta_{14}^{5} - \zeta_{14}) q^{37} + ( - \zeta_{14} + 1) q^{40} + (\zeta_{14}^{5} - \zeta_{14}^{2}) q^{41} - \zeta_{14}^{3} q^{49} + ( - \zeta_{14}^{4} + \cdots - \zeta_{14}^{2}) q^{50} + \cdots - \zeta_{14}^{4} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{4} + 2 q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{4} + 2 q^{5} + q^{8} - 2 q^{10} - 2 q^{13} - q^{16} + 2 q^{17} - 5 q^{20} - 3 q^{25} - 5 q^{26} + q^{29} + q^{32} + 5 q^{34} - 2 q^{37} + 5 q^{40} + 2 q^{41} - q^{49} + 3 q^{50} - 2 q^{52} - 5 q^{53} - q^{58} - 2 q^{61} - q^{64} - 3 q^{65} + 2 q^{68} + 5 q^{73} + 2 q^{74} + 2 q^{80} - 2 q^{82} - 4 q^{85} + 2 q^{89} + 5 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(523\) \(901\) \(929\)
\(\chi(n)\) \(-1\) \(\zeta_{14}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.222521 0.974928i
−0.623490 0.781831i
0.900969 + 0.433884i
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.900969 0.433884i
0.222521 0.974928i 0 −0.900969 0.433884i 0.277479 1.21572i 0 0 −0.623490 + 0.781831i 0 −1.12349 0.541044i
343.1 −0.623490 0.781831i 0 −0.222521 + 0.974928i 1.12349 + 1.40881i 0 0 0.900969 0.433884i 0 0.400969 1.75676i
451.1 0.900969 + 0.433884i 0 0.623490 + 0.781831i −0.400969 0.193096i 0 0 0.222521 + 0.974928i 0 −0.277479 0.347948i
487.1 −0.623490 + 0.781831i 0 −0.222521 0.974928i 1.12349 1.40881i 0 0 0.900969 + 0.433884i 0 0.400969 + 1.75676i
703.1 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 0.277479 + 1.21572i 0 0 −0.623490 0.781831i 0 −1.12349 + 0.541044i
919.1 0.900969 0.433884i 0 0.623490 0.781831i −0.400969 + 0.193096i 0 0 0.222521 0.974928i 0 −0.277479 + 0.347948i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.1
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.d even 7 1 inner
116.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1044.1.bb.a 6
3.b odd 2 1 116.1.j.a 6
4.b odd 2 1 CM 1044.1.bb.a 6
12.b even 2 1 116.1.j.a 6
15.d odd 2 1 2900.1.bj.a 6
15.e even 4 2 2900.1.bd.a 12
24.f even 2 1 1856.1.bh.a 6
24.h odd 2 1 1856.1.bh.a 6
29.d even 7 1 inner 1044.1.bb.a 6
60.h even 2 1 2900.1.bj.a 6
60.l odd 4 2 2900.1.bd.a 12
87.d odd 2 1 3364.1.j.d 6
87.f even 4 2 3364.1.h.e 12
87.h odd 14 1 3364.1.b.b 3
87.h odd 14 2 3364.1.j.c 6
87.h odd 14 1 3364.1.j.d 6
87.h odd 14 2 3364.1.j.e 6
87.j odd 14 1 116.1.j.a 6
87.j odd 14 1 3364.1.b.c 3
87.j odd 14 2 3364.1.j.a 6
87.j odd 14 2 3364.1.j.b 6
87.k even 28 2 3364.1.d.a 6
87.k even 28 4 3364.1.h.c 12
87.k even 28 4 3364.1.h.d 12
87.k even 28 2 3364.1.h.e 12
116.j odd 14 1 inner 1044.1.bb.a 6
348.b even 2 1 3364.1.j.d 6
348.k odd 4 2 3364.1.h.e 12
348.s even 14 1 116.1.j.a 6
348.s even 14 1 3364.1.b.c 3
348.s even 14 2 3364.1.j.a 6
348.s even 14 2 3364.1.j.b 6
348.t even 14 1 3364.1.b.b 3
348.t even 14 2 3364.1.j.c 6
348.t even 14 1 3364.1.j.d 6
348.t even 14 2 3364.1.j.e 6
348.v odd 28 2 3364.1.d.a 6
348.v odd 28 4 3364.1.h.c 12
348.v odd 28 4 3364.1.h.d 12
348.v odd 28 2 3364.1.h.e 12
435.w odd 14 1 2900.1.bj.a 6
435.bj even 28 2 2900.1.bd.a 12
696.z odd 14 1 1856.1.bh.a 6
696.bf even 14 1 1856.1.bh.a 6
1740.br even 14 1 2900.1.bj.a 6
1740.ci odd 28 2 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 3.b odd 2 1
116.1.j.a 6 12.b even 2 1
116.1.j.a 6 87.j odd 14 1
116.1.j.a 6 348.s even 14 1
1044.1.bb.a 6 1.a even 1 1 trivial
1044.1.bb.a 6 4.b odd 2 1 CM
1044.1.bb.a 6 29.d even 7 1 inner
1044.1.bb.a 6 116.j odd 14 1 inner
1856.1.bh.a 6 24.f even 2 1
1856.1.bh.a 6 24.h odd 2 1
1856.1.bh.a 6 696.z odd 14 1
1856.1.bh.a 6 696.bf even 14 1
2900.1.bd.a 12 15.e even 4 2
2900.1.bd.a 12 60.l odd 4 2
2900.1.bd.a 12 435.bj even 28 2
2900.1.bd.a 12 1740.ci odd 28 2
2900.1.bj.a 6 15.d odd 2 1
2900.1.bj.a 6 60.h even 2 1
2900.1.bj.a 6 435.w odd 14 1
2900.1.bj.a 6 1740.br even 14 1
3364.1.b.b 3 87.h odd 14 1
3364.1.b.b 3 348.t even 14 1
3364.1.b.c 3 87.j odd 14 1
3364.1.b.c 3 348.s even 14 1
3364.1.d.a 6 87.k even 28 2
3364.1.d.a 6 348.v odd 28 2
3364.1.h.c 12 87.k even 28 4
3364.1.h.c 12 348.v odd 28 4
3364.1.h.d 12 87.k even 28 4
3364.1.h.d 12 348.v odd 28 4
3364.1.h.e 12 87.f even 4 2
3364.1.h.e 12 87.k even 28 2
3364.1.h.e 12 348.k odd 4 2
3364.1.h.e 12 348.v odd 28 2
3364.1.j.a 6 87.j odd 14 2
3364.1.j.a 6 348.s even 14 2
3364.1.j.b 6 87.j odd 14 2
3364.1.j.b 6 348.s even 14 2
3364.1.j.c 6 87.h odd 14 2
3364.1.j.c 6 348.t even 14 2
3364.1.j.d 6 87.d odd 2 1
3364.1.j.d 6 87.h odd 14 1
3364.1.j.d 6 348.b even 2 1
3364.1.j.d 6 348.t even 14 1
3364.1.j.e 6 87.h odd 14 2
3364.1.j.e 6 348.t even 14 2

Hecke kernels

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

Hecke characteristic polynomials

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