Properties

Label 1444.1.h.a
Level $1444$
Weight $1$
Character orbit 1444.h
Analytic conductor $0.721$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -19
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.720649878242\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
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 $C_3\times S_3$
Artin field Galois closure of 6.0.39617584.1

$q$-expansion

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

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

Character values

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

\(n\) \(723\) \(1085\)
\(\chi(n)\) \(1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0
293.1 0 0 0 0.500000 0.866025i 0 −1.00000 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
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 1 inner
19.d 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.h.a 2
19.b odd 2 1 CM 1444.1.h.a 2
19.c even 3 1 76.1.c.a 1
19.c even 3 1 inner 1444.1.h.a 2
19.d odd 6 1 76.1.c.a 1
19.d odd 6 1 inner 1444.1.h.a 2
19.e even 9 6 1444.1.j.a 6
19.f odd 18 6 1444.1.j.a 6
57.f even 6 1 684.1.h.a 1
57.h odd 6 1 684.1.h.a 1
76.f even 6 1 304.1.e.a 1
76.g odd 6 1 304.1.e.a 1
95.h odd 6 1 1900.1.e.a 1
95.i even 6 1 1900.1.e.a 1
95.l even 12 2 1900.1.g.a 2
95.m odd 12 2 1900.1.g.a 2
133.g even 3 1 3724.1.bc.c 2
133.h even 3 1 3724.1.bc.c 2
133.i even 6 1 3724.1.bc.b 2
133.j odd 6 1 3724.1.bc.c 2
133.k odd 6 1 3724.1.bc.b 2
133.m odd 6 1 3724.1.e.c 1
133.n odd 6 1 3724.1.bc.c 2
133.p even 6 1 3724.1.e.c 1
133.s even 6 1 3724.1.bc.b 2
133.t odd 6 1 3724.1.bc.b 2
152.k odd 6 1 1216.1.e.b 1
152.l odd 6 1 1216.1.e.a 1
152.o even 6 1 1216.1.e.b 1
152.p even 6 1 1216.1.e.a 1
228.m even 6 1 2736.1.o.b 1
228.n odd 6 1 2736.1.o.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 19.c even 3 1
76.1.c.a 1 19.d odd 6 1
304.1.e.a 1 76.f even 6 1
304.1.e.a 1 76.g odd 6 1
684.1.h.a 1 57.f even 6 1
684.1.h.a 1 57.h odd 6 1
1216.1.e.a 1 152.l odd 6 1
1216.1.e.a 1 152.p even 6 1
1216.1.e.b 1 152.k odd 6 1
1216.1.e.b 1 152.o even 6 1
1444.1.h.a 2 1.a even 1 1 trivial
1444.1.h.a 2 19.b odd 2 1 CM
1444.1.h.a 2 19.c even 3 1 inner
1444.1.h.a 2 19.d odd 6 1 inner
1444.1.j.a 6 19.e even 9 6
1444.1.j.a 6 19.f odd 18 6
1900.1.e.a 1 95.h odd 6 1
1900.1.e.a 1 95.i even 6 1
1900.1.g.a 2 95.l even 12 2
1900.1.g.a 2 95.m odd 12 2
2736.1.o.b 1 228.m even 6 1
2736.1.o.b 1 228.n odd 6 1
3724.1.e.c 1 133.m odd 6 1
3724.1.e.c 1 133.p even 6 1
3724.1.bc.b 2 133.i even 6 1
3724.1.bc.b 2 133.k odd 6 1
3724.1.bc.b 2 133.s even 6 1
3724.1.bc.b 2 133.t odd 6 1
3724.1.bc.c 2 133.g even 3 1
3724.1.bc.c 2 133.h even 3 1
3724.1.bc.c 2 133.j odd 6 1
3724.1.bc.c 2 133.n odd 6 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

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