Properties

Label 1444.1.l.a
Level $1444$
Weight $1$
Character orbit 1444.l
Analytic conductor $0.721$
Analytic rank $0$
Dimension $12$
Projective image $D_{5}$
CM discriminant -4
Inner twists $12$

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Newspace parameters

Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.l (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: 12.0.6053445140625.1
Defining polynomial: \(x^{12} - 4 x^{9} + 17 x^{6} + 4 x^{3} + 1\)
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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{9} ) q^{2} + ( -\beta_{5} + \beta_{11} ) q^{4} + ( \beta_{1} - \beta_{7} + \beta_{9} ) q^{5} -\beta_{8} q^{8} + \beta_{5} q^{9} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{9} ) q^{2} + ( -\beta_{5} + \beta_{11} ) q^{4} + ( \beta_{1} - \beta_{7} + \beta_{9} ) q^{5} -\beta_{8} q^{8} + \beta_{5} q^{9} + ( \beta_{4} - \beta_{5} ) q^{10} + ( \beta_{4} + \beta_{10} ) q^{13} + \beta_{3} q^{16} -\beta_{7} q^{17} + q^{18} + ( -1 - \beta_{2} ) q^{20} + ( \beta_{4} + \beta_{10} - \beta_{11} ) q^{25} + ( -\beta_{2} + \beta_{6} ) q^{26} -\beta_{4} q^{29} -\beta_{11} q^{32} -\beta_{10} q^{34} + ( -\beta_{3} + \beta_{9} ) q^{36} + ( -1 - \beta_{2} ) q^{37} + ( \beta_{3} + \beta_{7} - \beta_{9} ) q^{40} + ( -\beta_{1} - \beta_{3} ) q^{41} + ( 1 + \beta_{2} - \beta_{6} - \beta_{8} ) q^{45} -\beta_{8} q^{49} + ( -1 - \beta_{2} + \beta_{6} + \beta_{8} ) q^{50} + ( -\beta_{1} + \beta_{7} ) q^{52} + ( \beta_{5} + \beta_{10} - \beta_{11} ) q^{53} + \beta_{2} q^{58} -\beta_{10} q^{61} + ( -1 + \beta_{8} ) q^{64} + \beta_{8} q^{65} -\beta_{6} q^{68} + ( -\beta_{5} + \beta_{11} ) q^{72} + ( -\beta_{1} - \beta_{3} ) q^{73} + ( \beta_{3} + \beta_{7} - \beta_{9} ) q^{74} + ( \beta_{5} + \beta_{10} - \beta_{11} ) q^{80} -\beta_{9} q^{81} + ( -\beta_{4} - \beta_{10} + \beta_{11} ) q^{82} -\beta_{5} q^{85} + ( -\beta_{4} - \beta_{10} + \beta_{11} ) q^{89} + ( \beta_{1} - \beta_{7} + \beta_{9} ) q^{90} -\beta_{7} q^{97} + \beta_{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{8} + O(q^{10}) \) \( 12q - 6q^{8} + 12q^{18} - 6q^{20} + 3q^{26} - 6q^{37} + 3q^{45} - 6q^{49} - 3q^{50} - 6q^{58} - 6q^{64} + 6q^{65} + 3q^{68} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{9} + 17 x^{6} + 4 x^{3} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} + 21 \)\()/34\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{10} + 55 \nu \)\()/34\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{11} - 55 \nu^{2} \)\()/34\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} + 89 \nu^{2} \)\()/34\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{9} + 17 \nu^{6} - 85 \nu^{3} + 1 \)\()/34\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{10} - 17 \nu^{7} + 68 \nu^{4} + 16 \nu \)\()/17\)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{9} + 17 \nu^{6} - 68 \nu^{3} + 1 \)\()/17\)
\(\beta_{9}\)\(=\)\((\)\( -12 \nu^{10} + 51 \nu^{7} - 221 \nu^{4} + 3 \nu \)\()/34\)
\(\beta_{10}\)\(=\)\((\)\( -12 \nu^{11} + 51 \nu^{8} - 221 \nu^{5} + 3 \nu^{2} \)\()/34\)
\(\beta_{11}\)\(=\)\((\)\( 21 \nu^{11} - 85 \nu^{8} + 357 \nu^{5} + 84 \nu^{2} \)\()/34\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4}\)
\(\nu^{3}\)\(=\)\(\beta_{8} - 2 \beta_{6}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{9} - 3 \beta_{7} + 3 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-3 \beta_{11} - 5 \beta_{10} + 3 \beta_{5}\)
\(\nu^{6}\)\(=\)\(5 \beta_{8} - 8 \beta_{6} + 8 \beta_{2} - 5\)
\(\nu^{7}\)\(=\)\(-8 \beta_{9} - 13 \beta_{7} + 8 \beta_{3}\)
\(\nu^{8}\)\(=\)\(-13 \beta_{11} - 21 \beta_{10} - 21 \beta_{4}\)
\(\nu^{9}\)\(=\)\(34 \beta_{2} - 21\)
\(\nu^{10}\)\(=\)\(34 \beta_{3} - 55 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-55 \beta_{5} - 89 \beta_{4}\)

Character values

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

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(-1\) \(-\beta_{5} + \beta_{11}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.107320 + 0.608645i
−0.280969 1.59345i
−0.580762 + 0.211380i
1.52045 0.553400i
0.107320 0.608645i
−0.280969 + 1.59345i
−0.580762 0.211380i
1.52045 + 0.553400i
−1.23949 1.04005i
0.473442 + 0.397265i
−1.23949 + 1.04005i
0.473442 0.397265i
−0.939693 0.342020i 0 0.766044 + 0.642788i −1.23949 + 1.04005i 0 0 −0.500000 0.866025i −0.939693 + 0.342020i 1.52045 0.553400i
99.2 −0.939693 0.342020i 0 0.766044 + 0.642788i 0.473442 0.397265i 0 0 −0.500000 0.866025i −0.939693 + 0.342020i −0.580762 + 0.211380i
415.1 0.766044 + 0.642788i 0 0.173648 + 0.984808i −0.280969 + 1.59345i 0 0 −0.500000 + 0.866025i 0.766044 0.642788i −1.23949 + 1.04005i
415.2 0.766044 + 0.642788i 0 0.173648 + 0.984808i 0.107320 0.608645i 0 0 −0.500000 + 0.866025i 0.766044 0.642788i 0.473442 0.397265i
423.1 −0.939693 + 0.342020i 0 0.766044 0.642788i −1.23949 1.04005i 0 0 −0.500000 + 0.866025i −0.939693 0.342020i 1.52045 + 0.553400i
423.2 −0.939693 + 0.342020i 0 0.766044 0.642788i 0.473442 + 0.397265i 0 0 −0.500000 + 0.866025i −0.939693 0.342020i −0.580762 0.211380i
595.1 0.766044 0.642788i 0 0.173648 0.984808i −0.280969 1.59345i 0 0 −0.500000 0.866025i 0.766044 + 0.642788i −1.23949 1.04005i
595.2 0.766044 0.642788i 0 0.173648 0.984808i 0.107320 + 0.608645i 0 0 −0.500000 0.866025i 0.766044 + 0.642788i 0.473442 + 0.397265i
967.1 0.173648 0.984808i 0 −0.939693 0.342020i −0.580762 + 0.211380i 0 0 −0.500000 + 0.866025i 0.173648 + 0.984808i 0.107320 + 0.608645i
967.2 0.173648 0.984808i 0 −0.939693 0.342020i 1.52045 0.553400i 0 0 −0.500000 + 0.866025i 0.173648 + 0.984808i −0.280969 1.59345i
1111.1 0.173648 + 0.984808i 0 −0.939693 + 0.342020i −0.580762 0.211380i 0 0 −0.500000 0.866025i 0.173648 0.984808i 0.107320 0.608645i
1111.2 0.173648 + 0.984808i 0 −0.939693 + 0.342020i 1.52045 + 0.553400i 0 0 −0.500000 0.866025i 0.173648 0.984808i −0.280969 + 1.59345i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1111.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
19.c even 3 2 inner
19.e even 9 3 inner
76.g odd 6 2 inner
76.l 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.l.a 12
4.b odd 2 1 CM 1444.1.l.a 12
19.b odd 2 1 1444.1.l.b 12
19.c even 3 2 inner 1444.1.l.a 12
19.d odd 6 2 1444.1.l.b 12
19.e even 9 1 1444.1.b.b yes 2
19.e even 9 2 1444.1.g.a 4
19.e even 9 3 inner 1444.1.l.a 12
19.f odd 18 1 1444.1.b.a 2
19.f odd 18 2 1444.1.g.b 4
19.f odd 18 3 1444.1.l.b 12
76.d even 2 1 1444.1.l.b 12
76.f even 6 2 1444.1.l.b 12
76.g odd 6 2 inner 1444.1.l.a 12
76.k even 18 1 1444.1.b.a 2
76.k even 18 2 1444.1.g.b 4
76.k even 18 3 1444.1.l.b 12
76.l odd 18 1 1444.1.b.b yes 2
76.l odd 18 2 1444.1.g.a 4
76.l odd 18 3 inner 1444.1.l.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.b.a 2 19.f odd 18 1
1444.1.b.a 2 76.k even 18 1
1444.1.b.b yes 2 19.e even 9 1
1444.1.b.b yes 2 76.l odd 18 1
1444.1.g.a 4 19.e even 9 2
1444.1.g.a 4 76.l odd 18 2
1444.1.g.b 4 19.f odd 18 2
1444.1.g.b 4 76.k even 18 2
1444.1.l.a 12 1.a even 1 1 trivial
1444.1.l.a 12 4.b odd 2 1 CM
1444.1.l.a 12 19.c even 3 2 inner
1444.1.l.a 12 19.e even 9 3 inner
1444.1.l.a 12 76.g odd 6 2 inner
1444.1.l.a 12 76.l odd 18 3 inner
1444.1.l.b 12 19.b odd 2 1
1444.1.l.b 12 19.d odd 6 2
1444.1.l.b 12 19.f odd 18 3
1444.1.l.b 12 76.d even 2 1
1444.1.l.b 12 76.f even 6 2
1444.1.l.b 12 76.k even 18 3

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{12} - 4 T_{13}^{9} + 17 T_{13}^{6} + 4 T_{13}^{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$3$ \( T^{12} \)
$5$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$17$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$31$ \( T^{12} \)
$37$ \( ( -1 + T + T^{2} )^{6} \)
$41$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$79$ \( T^{12} \)
$83$ \( T^{12} \)
$89$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
$97$ \( 1 + 4 T^{3} + 17 T^{6} - 4 T^{9} + T^{12} \)
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