Properties

Label 672.2.q.b
Level 672
Weight 2
Character orbit 672.q
Analytic conductor 5.366
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -3 + \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{11} + q^{15} + ( 8 - 8 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + ( 2 - 3 \zeta_{6} ) q^{21} + 4 \zeta_{6} q^{23} + ( 4 - 4 \zeta_{6} ) q^{25} + q^{27} -5 q^{29} + ( 7 - 7 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} + ( 1 + 2 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + 4 q^{41} -10 q^{43} + ( -1 + \zeta_{6} ) q^{45} + 6 \zeta_{6} q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + 8 \zeta_{6} q^{51} + ( 1 - \zeta_{6} ) q^{53} - q^{55} + 4 q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + 2 \zeta_{6} q^{61} + ( 1 + 2 \zeta_{6} ) q^{63} + ( 2 - 2 \zeta_{6} ) q^{67} -4 q^{69} -6 q^{71} + ( -2 + 2 \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{75} + ( -2 + 3 \zeta_{6} ) q^{77} -9 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 3 q^{83} -8 q^{85} + ( 5 - 5 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + 7 \zeta_{6} q^{93} + ( -4 + 4 \zeta_{6} ) q^{95} - q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - q^{5} - 5q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{5} - 5q^{7} - q^{9} + q^{11} + 2q^{15} + 8q^{17} - 4q^{19} + q^{21} + 4q^{23} + 4q^{25} + 2q^{27} - 10q^{29} + 7q^{31} + q^{33} + 4q^{35} - 8q^{37} + 8q^{41} - 20q^{43} - q^{45} + 6q^{47} + 11q^{49} + 8q^{51} + q^{53} - 2q^{55} + 8q^{57} + 9q^{59} + 2q^{61} + 4q^{63} + 2q^{67} - 8q^{69} - 12q^{71} - 2q^{73} + 4q^{75} - q^{77} - 9q^{79} - q^{81} + 6q^{83} - 16q^{85} + 5q^{87} + 6q^{89} + 7q^{93} - 4q^{95} - 2q^{97} - 2q^{99} + O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −2.50000 + 0.866025i 0 −0.500000 0.866025i 0
289.1 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −2.50000 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
7.c Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \)
\( T_{11}^{2} - T_{11} + 1 \)
\( T_{13} \)