Properties

Label 672.2.q.f
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( 1 - 3 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} -4 q^{13} -3 q^{15} + ( -4 + 4 \zeta_{6} ) q^{17} + ( -2 - \zeta_{6} ) q^{21} + 8 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} - q^{27} -7 q^{29} + ( 11 - 11 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} + ( -9 + 6 \zeta_{6} ) q^{35} -4 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} -4 q^{41} + 2 q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} -2 \zeta_{6} q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + 4 \zeta_{6} q^{51} + ( 11 - 11 \zeta_{6} ) q^{53} + 3 q^{55} + ( 7 - 7 \zeta_{6} ) q^{59} -10 \zeta_{6} q^{61} + ( -3 + 2 \zeta_{6} ) q^{63} + 12 \zeta_{6} q^{65} + ( 10 - 10 \zeta_{6} ) q^{67} + 8 q^{69} -6 q^{71} + ( 6 - 6 \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{75} + ( 2 + \zeta_{6} ) q^{77} + 11 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -11 q^{83} + 12 q^{85} + ( -7 + 7 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{89} + ( -4 + 12 \zeta_{6} ) q^{91} -11 \zeta_{6} q^{93} + 7 q^{97} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - 3q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - 3q^{5} - q^{7} - q^{9} - q^{11} - 8q^{13} - 6q^{15} - 4q^{17} - 5q^{21} + 8q^{23} - 4q^{25} - 2q^{27} - 14q^{29} + 11q^{31} + q^{33} - 12q^{35} - 4q^{37} - 4q^{39} - 8q^{41} + 4q^{43} - 3q^{45} - 2q^{47} - 13q^{49} + 4q^{51} + 11q^{53} + 6q^{55} + 7q^{59} - 10q^{61} - 4q^{63} + 12q^{65} + 10q^{67} + 16q^{69} - 12q^{71} + 6q^{73} + 4q^{75} + 5q^{77} + 11q^{79} - q^{81} - 22q^{83} + 24q^{85} - 7q^{87} - 6q^{89} + 4q^{91} - 11q^{93} + 14q^{97} + 2q^{99} + O(q^{100}) \)

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 −1.50000 2.59808i 0 −0.500000 2.59808i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −1.50000 + 2.59808i 0 −0.500000 + 2.59808i 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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} + T_{11} + 1 \)
\( T_{13} + 4 \)