Properties

Label 2003.1.b.b
Level 2003
Weight 1
Character orbit 2003.b
Self dual Yes
Analytic conductor 1.000
Analytic rank 0
Dimension 3
Projective image \(D_{9}\)
CM disc. -2003
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 2003 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2003.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.999627220304\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{9}\)
Projective field Galois closure of 9.1.16096216216081.1
Artin image size \(18\)
Artin image $D_9$
Artin field Galois closure of 9.1.16096216216081.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{4} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{4} + ( 1 + \beta_{2} ) q^{9} -\beta_{1} q^{12} + ( \beta_{1} - \beta_{2} ) q^{13} + q^{16} - q^{19} + q^{25} + ( -1 - \beta_{1} ) q^{27} + ( 1 + \beta_{2} ) q^{36} + ( -1 + \beta_{1} - \beta_{2} ) q^{39} + ( \beta_{1} - \beta_{2} ) q^{47} -\beta_{1} q^{48} + q^{49} + ( \beta_{1} - \beta_{2} ) q^{52} - q^{53} + \beta_{1} q^{57} + \beta_{2} q^{59} + q^{64} + \beta_{2} q^{73} -\beta_{1} q^{75} - q^{76} + \beta_{2} q^{79} + ( 1 + \beta_{1} ) q^{81} - q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{4} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{4} + 3q^{9} + 3q^{16} - 3q^{19} + 3q^{25} - 3q^{27} + 3q^{36} - 3q^{39} + 3q^{49} - 3q^{53} + 3q^{64} - 3q^{76} + 3q^{81} - 3q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2003\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2002.1
1.87939
−0.347296
−1.53209
0 −1.87939 1.00000 0 0 0 0 2.53209 0
2002.2 0 0.347296 1.00000 0 0 0 0 −0.879385 0
2002.3 0 1.53209 1.00000 0 0 0 0 1.34730 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
2003.b Odd 1 CM by \(\Q(\sqrt{-2003}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3}^{3} - 3 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2003, [\chi])\).