Properties

Label 672.2.q.l
Level 672
Weight 2
Character orbit 672.q
Analytic conductor 5.366
Analytic rank 0
Dimension 6
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} + ( -1 - \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -\beta_{1} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{7} + ( -1 - \beta_{2} ) q^{9} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{11} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} ) q^{13} -\beta_{1} q^{15} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{17} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{19} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{23} + ( -1 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} - q^{27} + ( 4 - \beta_{1} ) q^{29} + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{31} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{33} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{35} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{37} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{39} + ( -4 + 2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( -4 - \beta_{3} + \beta_{5} ) q^{43} -\beta_{4} q^{45} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{47} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{5} ) q^{49} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{51} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{53} + ( 4 - 3 \beta_{1} ) q^{55} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{57} + ( 1 + \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( 6 + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -1 + \beta_{3} ) q^{63} + ( -8 - 6 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{65} + ( -1 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{67} + ( -2 - 2 \beta_{1} ) q^{69} + 2 \beta_{1} q^{71} + ( -1 - \beta_{1} + 10 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{75} + ( -8 + \beta_{1} - 12 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{77} + ( -9 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{79} + \beta_{2} q^{81} + ( -5 + 4 \beta_{1} - \beta_{3} + \beta_{5} ) q^{83} + ( 14 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( -4 \beta_{2} - \beta_{4} ) q^{87} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{89} + ( 7 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{91} + ( -1 + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{93} + ( -2 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{95} + ( 1 - \beta_{3} + \beta_{5} ) q^{97} + ( -1 + \beta_{3} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{3} + 3q^{7} - 3q^{9} + O(q^{10}) \) \( 6q + 3q^{3} + 3q^{7} - 3q^{9} - 6q^{13} - 6q^{17} + 3q^{19} - 6q^{23} - 3q^{25} - 6q^{27} + 24q^{29} + 3q^{31} + 12q^{35} - 3q^{37} - 3q^{39} - 12q^{41} - 30q^{43} + 12q^{47} + 9q^{49} + 6q^{51} - 6q^{53} + 24q^{55} + 6q^{57} + 12q^{59} + 18q^{61} - 3q^{63} - 24q^{65} + 9q^{67} - 12q^{69} - 33q^{73} + 3q^{75} - 12q^{77} - 27q^{79} - 3q^{81} - 36q^{83} + 72q^{85} + 12q^{87} + 12q^{89} + 51q^{91} - 3q^{93} - 24q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 27 x^{2} - 18 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 3 \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 22 \nu^{3} - 28 \nu^{2} + 43 \nu - 18 \)\()/2\)
\(\beta_{3}\)\(=\)\( -3 \nu^{5} + 7 \nu^{4} - 31 \nu^{3} + 37 \nu^{2} - 56 \nu + 22 \)
\(\beta_{4}\)\(=\)\((\)\( -6 \nu^{5} + 15 \nu^{4} - 64 \nu^{3} + 82 \nu^{2} - 121 \nu + 50 \)\()/2\)
\(\beta_{5}\)\(=\)\( -3 \nu^{5} + 8 \nu^{4} - 33 \nu^{3} + 44 \nu^{2} - 62 \nu + 24 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{1} - 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{5} + 8 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-5 \beta_{5} + 18 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} - 17 \beta_{1} + 27\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(13 \beta_{5} - 28 \beta_{4} + 3 \beta_{3} - 34 \beta_{2} - 11 \beta_{1} + 49\)\()/2\)

Character Values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
0.500000 + 0.0585812i
0.500000 + 1.51496i
0.500000 2.43956i
0.500000 0.0585812i
0.500000 1.51496i
0.500000 + 2.43956i
0 0.500000 0.866025i 0 −1.37328 2.37860i 0 2.64510 + 0.0585812i 0 −0.500000 0.866025i 0
193.2 0 0.500000 0.866025i 0 −0.227452 0.393958i 0 −2.16908 + 1.51496i 0 −0.500000 0.866025i 0
193.3 0 0.500000 0.866025i 0 1.60074 + 2.77256i 0 1.02398 2.43956i 0 −0.500000 0.866025i 0
289.1 0 0.500000 + 0.866025i 0 −1.37328 + 2.37860i 0 2.64510 0.0585812i 0 −0.500000 + 0.866025i 0
289.2 0 0.500000 + 0.866025i 0 −0.227452 + 0.393958i 0 −2.16908 1.51496i 0 −0.500000 + 0.866025i 0
289.3 0 0.500000 + 0.866025i 0 1.60074 2.77256i 0 1.02398 + 2.43956i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{6} + 9 T_{5}^{4} + 8 T_{5}^{3} + 81 T_{5}^{2} + 36 T_{5} + 16 \)
\( T_{11}^{6} + 27 T_{11}^{4} - 76 T_{11}^{3} + 729 T_{11}^{2} - 1026 T_{11} + 1444 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 36 T_{13} - 112 \)