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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Newspace parameters

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

Self dual: no
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} - 1\)

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.

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 \( T_{13}^{4} + T_{13}^{3} + 2 T_{13}^{2} - T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

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