Properties

Label 672.2.c.a
Level 672
Weight 2
Character orbit 672.c
Analytic conductor 5.366
Analytic rank 0
Dimension 4
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.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{3} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} - q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{3} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{5} + q^{7} - q^{9} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{13} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{15} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{17} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} -\zeta_{12}^{3} q^{21} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{23} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{25} + \zeta_{12}^{3} q^{27} + ( 2 - 4 \zeta_{12}^{2} ) q^{29} -2 q^{31} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{33} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + ( 4 - 8 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{37} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{39} + ( -1 - 10 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{41} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{43} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{45} + q^{49} + ( -3 + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{51} + ( -6 + 12 \zeta_{12}^{2} ) q^{53} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{55} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( 4 - 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{59} + ( -4 + 8 \zeta_{12}^{2} ) q^{61} - q^{63} -4 q^{65} + 6 \zeta_{12}^{3} q^{67} + ( -1 + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{69} + ( -9 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{71} + ( 12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{73} + ( 2 - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{75} + ( -1 + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{77} + ( 2 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + q^{81} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{83} + ( 4 - 8 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{85} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 5 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{89} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{91} + 2 \zeta_{12}^{3} q^{93} + ( -10 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{95} + ( -8 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{97} + ( 1 - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{7} - 4q^{9} + 4q^{15} + 4q^{17} + 20q^{23} + 4q^{25} - 8q^{31} - 12q^{33} - 8q^{39} - 4q^{41} + 4q^{49} + 24q^{55} + 16q^{57} - 4q^{63} - 16q^{65} - 36q^{71} + 48q^{73} + 8q^{79} + 4q^{81} + 20q^{89} - 40q^{95} - 32q^{97} + O(q^{100}) \)

Character Values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
0 1.00000i 0 0.732051i 0 1.00000 0 −1.00000 0
337.2 0 1.00000i 0 2.73205i 0 1.00000 0 −1.00000 0
337.3 0 1.00000i 0 2.73205i 0 1.00000 0 −1.00000 0
337.4 0 1.00000i 0 0.732051i 0 1.00000 0 −1.00000 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
8.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8 T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).