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Properties

Label 1444.1.j.a
Level $1444$
Weight $1$
Character orbit 1444.j
Analytic conductor $0.721$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -19
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1444,1,Mod(333,1444)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1444, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1444.333");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.j (of order \(18\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.76.1
Artin image: $S_3\times C_9$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{18}^{8} q^{5} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{4} q^{9} - \zeta_{18}^{6} q^{11} + \zeta_{18}^{5} q^{17} - 2 \zeta_{18} q^{23} + \zeta_{18}^{2} q^{35} - \zeta_{18}^{8} q^{43} + \zeta_{18}^{3} q^{45} + \cdots + \zeta_{18} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{7} + 3 q^{11} + 3 q^{45} + 6 q^{77} - 6 q^{83}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(1\) \(\zeta_{18}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
333.1
0.939693 − 0.342020i
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
−0.173648 − 0.984808i
−0.766044 − 0.642788i
0 0 0 0.939693 + 0.342020i 0 0.500000 − 0.866025i 0 0.173648 − 0.984808i 0
477.1 0 0 0 0.939693 − 0.342020i 0 0.500000 + 0.866025i 0 0.173648 + 0.984808i 0
849.1 0 0 0 −0.173648 − 0.984808i 0 0.500000 − 0.866025i 0 0.766044 + 0.642788i 0
1021.1 0 0 0 −0.766044 − 0.642788i 0 0.500000 + 0.866025i 0 −0.939693 − 0.342020i 0
1029.1 0 0 0 −0.173648 + 0.984808i 0 0.500000 + 0.866025i 0 0.766044 − 0.642788i 0
1345.1 0 0 0 −0.766044 + 0.642788i 0 0.500000 − 0.866025i 0 −0.939693 + 0.342020i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 333.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
19.c even 3 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.j.a 6
19.b odd 2 1 CM 1444.1.j.a 6
19.c even 3 2 inner 1444.1.j.a 6
19.d odd 6 2 inner 1444.1.j.a 6
19.e even 9 1 76.1.c.a ✓ 1
19.e even 9 2 1444.1.h.a 2
19.e even 9 3 inner 1444.1.j.a 6
19.f odd 18 1 76.1.c.a ✓ 1
19.f odd 18 2 1444.1.h.a 2
19.f odd 18 3 inner 1444.1.j.a 6
57.j even 18 1 684.1.h.a 1
57.l odd 18 1 684.1.h.a 1
76.k even 18 1 304.1.e.a 1
76.l odd 18 1 304.1.e.a 1
95.o odd 18 1 1900.1.e.a 1
95.p even 18 1 1900.1.e.a 1
95.q odd 36 2 1900.1.g.a 2
95.r even 36 2 1900.1.g.a 2
133.u even 9 1 3724.1.bc.c 2
133.w even 9 1 3724.1.bc.c 2
133.x odd 18 1 3724.1.bc.b 2
133.y odd 18 1 3724.1.e.c 1
133.z odd 18 1 3724.1.bc.b 2
133.ba even 18 1 3724.1.e.c 1
133.bb even 18 1 3724.1.bc.b 2
133.bd odd 18 1 3724.1.bc.c 2
133.be odd 18 1 3724.1.bc.c 2
133.bf even 18 1 3724.1.bc.b 2
152.s odd 18 1 1216.1.e.a 1
152.t even 18 1 1216.1.e.a 1
152.u odd 18 1 1216.1.e.b 1
152.v even 18 1 1216.1.e.b 1
228.u odd 18 1 2736.1.o.b 1
228.v even 18 1 2736.1.o.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a ✓ 1 19.e even 9 1
76.1.c.a ✓ 1 19.f odd 18 1
304.1.e.a 1 76.k even 18 1
304.1.e.a 1 76.l odd 18 1
684.1.h.a 1 57.j even 18 1
684.1.h.a 1 57.l odd 18 1
1216.1.e.a 1 152.s odd 18 1
1216.1.e.a 1 152.t even 18 1
1216.1.e.b 1 152.u odd 18 1
1216.1.e.b 1 152.v even 18 1
1444.1.h.a 2 19.e even 9 2
1444.1.h.a 2 19.f odd 18 2
1444.1.j.a 6 1.a even 1 1 trivial
1444.1.j.a 6 19.b odd 2 1 CM
1444.1.j.a 6 19.c even 3 2 inner
1444.1.j.a 6 19.d odd 6 2 inner
1444.1.j.a 6 19.e even 9 3 inner
1444.1.j.a 6 19.f odd 18 3 inner
1900.1.e.a 1 95.o odd 18 1
1900.1.e.a 1 95.p even 18 1
1900.1.g.a 2 95.q odd 36 2
1900.1.g.a 2 95.r even 36 2
2736.1.o.b 1 228.u odd 18 1
2736.1.o.b 1 228.v even 18 1
3724.1.e.c 1 133.y odd 18 1
3724.1.e.c 1 133.ba even 18 1
3724.1.bc.b 2 133.x odd 18 1
3724.1.bc.b 2 133.z odd 18 1
3724.1.bc.b 2 133.bb even 18 1
3724.1.bc.b 2 133.bf even 18 1
3724.1.bc.c 2 133.u even 9 1
3724.1.bc.c 2 133.w even 9 1
3724.1.bc.c 2 133.bd odd 18 1
3724.1.bc.c 2 133.be odd 18 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 8T^{3} + 64 \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} \) Copy content Toggle raw display
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