Properties

Label 672.1.ba.a
Level 672
Weight 1
Character orbit 672.ba
Analytic conductor 0.335
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM disc. -24
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 672.ba (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.335371688489\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1176.1
Artin image size \(36\)
Artin image $C_6\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{3} + \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{11} + q^{15} -\zeta_{6}^{2} q^{21} + q^{27} + q^{29} -\zeta_{6} q^{31} -\zeta_{6}^{2} q^{33} - q^{35} -\zeta_{6} q^{45} + \zeta_{6}^{2} q^{49} -\zeta_{6} q^{53} - q^{55} + \zeta_{6} q^{59} - q^{63} -2 \zeta_{6} q^{73} + \zeta_{6}^{2} q^{77} + \zeta_{6}^{2} q^{79} -\zeta_{6} q^{81} - q^{83} -\zeta_{6} q^{87} + \zeta_{6}^{2} q^{93} - q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{5} + q^{7} - q^{9} + q^{11} + 2q^{15} + q^{21} + 2q^{27} + 2q^{29} - q^{31} + q^{33} - 2q^{35} - q^{45} - q^{49} - q^{53} - 2q^{55} + q^{59} - 2q^{63} - 2q^{73} - q^{77} - q^{79} - q^{81} - 2q^{83} - q^{87} - q^{93} - 2q^{97} - 2q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
401.1 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 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
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
7.c Even 1 yes
168.s Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(672, [\chi])\).