Properties

Label 1444.1.b.a
Level $1444$
Weight $1$
Character orbit 1444.b
Self dual yes
Analytic conductor $0.721$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -4
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1444 = 2^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1444.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
723.1
−0.618034
1.61803
−1.00000 0 1.00000 −1.61803 0 0 −1.00000 1.00000 1.61803
723.2 −1.00000 0 1.00000 0.618034 0 0 −1.00000 1.00000 −0.618034
\(n\): 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 \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.b.a 2
4.b odd 2 1 CM 1444.1.b.a 2
19.b odd 2 1 1444.1.b.b yes 2
19.c even 3 2 1444.1.g.b 4
19.d odd 6 2 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.b.b yes 2
76.f even 6 2 1444.1.g.a 4
76.g odd 6 2 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 1.a even 1 1 trivial
1444.1.b.a 2 4.b odd 2 1 CM
1444.1.b.b yes 2 19.b odd 2 1
1444.1.b.b yes 2 76.d even 2 1
1444.1.g.a 4 19.d odd 6 2
1444.1.g.a 4 76.f even 6 2
1444.1.g.b 4 19.c even 3 2
1444.1.g.b 4 76.g odd 6 2
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 \( T_{13}^{2} - T_{13} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1444, [\chi])\).

Hecke characteristic polynomials

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