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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Newspace parameters

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

Self dual: no
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
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 $C_9\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{18}^{8} q^{5} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{18}^{8} q^{5} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{4} q^{9} -\zeta_{18}^{6} q^{11} + \zeta_{18}^{5} q^{17} -2 \zeta_{18} q^{23} + \zeta_{18}^{2} q^{35} -\zeta_{18}^{8} q^{43} + \zeta_{18}^{3} q^{45} -\zeta_{18}^{4} q^{47} -\zeta_{18}^{5} q^{55} + \zeta_{18} q^{61} + \zeta_{18}^{7} q^{63} -\zeta_{18}^{2} q^{73} + q^{77} + \zeta_{18}^{8} q^{81} -2 \zeta_{18}^{3} q^{83} + \zeta_{18}^{4} q^{85} + \zeta_{18} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{7} + O(q^{10}) \) \( 6q + 3q^{7} + 3q^{11} + 3q^{45} + 6q^{77} - 6q^{83} + O(q^{100}) \)

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.

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 1345.1
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 \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

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