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Properties

Label 1444.1.g.b
Level $1444$
Weight $1$
Character orbit 1444.g
Analytic conductor $0.721$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
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] = [1444,1,Mod(1151,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.1151");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.g (of order \(6\), degree \(2\), not 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: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.2085136.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} + (\beta_{3} - \beta_1 + 1) q^{5} - q^{8} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} + (\beta_{3} - \beta_1 + 1) q^{5} - q^{8} + \beta_{3} q^{9} + (\beta_{3} - \beta_{2} - \beta_1) q^{10} + (\beta_{2} + \beta_1) q^{13} + ( - \beta_{3} - 1) q^{16} + \beta_1 q^{17} - q^{18} + ( - \beta_{2} - 1) q^{20} + (\beta_{3} - \beta_{2} - \beta_1) q^{25} + \beta_{2} q^{26} + (\beta_{2} + \beta_1) q^{29} - \beta_{3} q^{32} + (\beta_{2} + \beta_1) q^{34} + ( - \beta_{3} - 1) q^{36} + (\beta_{2} + 1) q^{37} + ( - \beta_{3} + \beta_1 - 1) q^{40} + ( - \beta_{3} + \beta_1 - 1) q^{41} + ( - \beta_{2} - 1) q^{45} + q^{49} + ( - \beta_{2} - 1) q^{50} - \beta_1 q^{52} + (\beta_{3} - \beta_{2} - \beta_1) q^{53} + \beta_{2} q^{58} + ( - \beta_{2} - \beta_1) q^{61} + q^{64} + q^{65} + \beta_{2} q^{68} - \beta_{3} q^{72} + (\beta_{3} - \beta_1 + 1) q^{73} + (\beta_{3} - \beta_1 + 1) q^{74} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{80} + ( - \beta_{3} - 1) q^{81} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{82} - \beta_{3} q^{85} + (\beta_{3} - \beta_{2} - \beta_1) q^{89} + ( - \beta_{3} + \beta_1 - 1) q^{90} - \beta_1 q^{97} + (\beta_{3} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + q^{5} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + q^{5} - 4 q^{8} - 2 q^{9} - q^{10} - q^{13} - 2 q^{16} + q^{17} - 4 q^{18} - 2 q^{20} - q^{25} - 2 q^{26} - q^{29} + 2 q^{32} - q^{34} - 2 q^{36} + 2 q^{37} - q^{40} - q^{41} - 2 q^{45} + 4 q^{49} - 2 q^{50} - q^{52} - q^{53} - 2 q^{58} + q^{61} + 4 q^{64} + 4 q^{65} - 2 q^{68} + 2 q^{72} + q^{73} + q^{74} + q^{80} - 2 q^{81} + q^{82} + 2 q^{85} - q^{89} - q^{90} - q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times\).

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.809017 − 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 − 0.535233i
0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.309017 + 0.535233i 0 0 −1.00000 −0.500000 − 0.866025i 0.309017 + 0.535233i
1151.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.809017 − 1.40126i 0 0 −1.00000 −0.500000 − 0.866025i −0.809017 − 1.40126i
1375.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.309017 − 0.535233i 0 0 −1.00000 −0.500000 + 0.866025i 0.309017 − 0.535233i
1375.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.809017 + 1.40126i 0 0 −1.00000 −0.500000 + 0.866025i −0.809017 + 1.40126i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
19.c even 3 1 inner
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.g.b 4
4.b odd 2 1 CM 1444.1.g.b 4
19.b odd 2 1 1444.1.g.a 4
19.c even 3 1 1444.1.b.a ✓ 2
19.c even 3 1 inner 1444.1.g.b 4
19.d odd 6 1 1444.1.b.b yes 2
19.d odd 6 1 1444.1.g.a 4
19.e even 9 6 1444.1.l.b 12
19.f odd 18 6 1444.1.l.a 12
76.d even 2 1 1444.1.g.a 4
76.f even 6 1 1444.1.b.b yes 2
76.f even 6 1 1444.1.g.a 4
76.g odd 6 1 1444.1.b.a ✓ 2
76.g odd 6 1 inner 1444.1.g.b 4
76.k even 18 6 1444.1.l.a 12
76.l odd 18 6 1444.1.l.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.b.a ✓ 2 19.c even 3 1
1444.1.b.a ✓ 2 76.g odd 6 1
1444.1.b.b yes 2 19.d odd 6 1
1444.1.b.b yes 2 76.f even 6 1
1444.1.g.a 4 19.b odd 2 1
1444.1.g.a 4 19.d odd 6 1
1444.1.g.a 4 76.d even 2 1
1444.1.g.a 4 76.f even 6 1
1444.1.g.b 4 1.a even 1 1 trivial
1444.1.g.b 4 4.b odd 2 1 CM
1444.1.g.b 4 19.c even 3 1 inner
1444.1.g.b 4 76.g odd 6 1 inner
1444.1.l.a 12 19.f odd 18 6
1444.1.l.a 12 76.k even 18 6
1444.1.l.b 12 19.e even 9 6
1444.1.l.b 12 76.l odd 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} + T_{13}^{3} + 2T_{13}^{2} - T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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