Properties

Label 672.2.bc.b
Level 672
Weight 2
Character orbit 672.bc
Analytic conductor 5.366
Analytic rank 0
Dimension 4
CM No
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + 5 \zeta_{12} q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} + ( 1 - \zeta_{12}^{2} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{19} + ( -4 - \zeta_{12}^{2} ) q^{21} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{23} + ( 4 - 4 \zeta_{12}^{2} ) q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -2 + 4 \zeta_{12}^{2} ) q^{29} -9 \zeta_{12} q^{31} + ( 5 - 10 \zeta_{12}^{2} ) q^{33} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{35} -3 \zeta_{12}^{2} q^{37} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{39} + 8 q^{41} -3 q^{45} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{51} + ( 7 + 7 \zeta_{12}^{2} ) q^{53} + 5 \zeta_{12}^{3} q^{55} + ( -1 - \zeta_{12}^{2} ) q^{57} + ( 5 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{59} + ( -14 + 7 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{63} + ( -4 + 2 \zeta_{12}^{2} ) q^{65} + ( 7 \zeta_{12} - 14 \zeta_{12}^{3} ) q^{67} + ( -5 - 5 \zeta_{12}^{2} ) q^{69} -2 \zeta_{12}^{3} q^{71} + ( 5 + 5 \zeta_{12}^{2} ) q^{73} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{75} + ( 15 - 5 \zeta_{12}^{2} ) q^{77} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83} + q^{85} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{87} -13 \zeta_{12}^{2} q^{89} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} + ( -9 + 18 \zeta_{12}^{2} ) q^{93} + \zeta_{12} q^{95} + ( 2 - 4 \zeta_{12}^{2} ) q^{97} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{5} - 6q^{9} + 2q^{17} - 18q^{21} + 8q^{25} - 6q^{37} + 32q^{41} - 12q^{45} - 4q^{49} + 42q^{53} - 6q^{57} - 42q^{61} - 12q^{65} - 30q^{69} + 30q^{73} + 50q^{77} - 18q^{81} + 4q^{85} - 26q^{89} + O(q^{100}) \)

Character Values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0 1.73205 + 2.00000i 0 −1.50000 2.59808i 0
257.2 0 0.866025 1.50000i 0 0.500000 0.866025i 0 −1.73205 2.00000i 0 −1.50000 2.59808i 0
353.1 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0 1.73205 2.00000i 0 −1.50000 + 2.59808i 0
353.2 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 −1.73205 + 2.00000i 0 −1.50000 + 2.59808i 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
4.b Odd 1 yes
21.g Even 1 yes
84.j 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 \(S_{2}^{\mathrm{new}}(672, [\chi])\).