Properties

Label 672.2.c.b
Level 672
Weight 2
Character orbit 672.c
Analytic conductor 5.366
Analytic rank 0
Dimension 8
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: \(8\)
Coefficient field: 8.0.386672896.3
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} - q^{7} - q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} - q^{7} - q^{9} + ( -2 \beta_{2} - \beta_{7} ) q^{11} + ( \beta_{2} - \beta_{6} ) q^{13} + \beta_{5} q^{15} + ( 1 + \beta_{3} - \beta_{5} ) q^{17} + ( \beta_{1} + 2 \beta_{2} - \beta_{7} ) q^{19} -\beta_{2} q^{21} + ( -1 + \beta_{3} - \beta_{5} ) q^{23} + ( -3 + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{25} -\beta_{2} q^{27} + ( \beta_{1} + 2 \beta_{6} - \beta_{7} ) q^{29} + ( -2 + \beta_{4} + \beta_{5} ) q^{31} + ( 2 - \beta_{4} ) q^{33} -\beta_{1} q^{35} + ( -\beta_{1} - 2 \beta_{2} - \beta_{7} ) q^{37} + ( -1 + \beta_{3} ) q^{39} + ( -1 - \beta_{3} + \beta_{5} ) q^{41} + ( \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} ) q^{43} -\beta_{1} q^{45} + ( -2 \beta_{4} + 2 \beta_{5} ) q^{47} + q^{49} + ( \beta_{1} + \beta_{2} + \beta_{6} ) q^{51} + ( \beta_{1} + 2 \beta_{2} - \beta_{7} ) q^{53} + ( 2 + \beta_{4} - 3 \beta_{5} ) q^{55} + ( -2 - \beta_{4} + \beta_{5} ) q^{57} -4 \beta_{2} q^{59} + ( 2 \beta_{1} + \beta_{2} - \beta_{6} ) q^{61} + q^{63} + ( -4 + 4 \beta_{5} ) q^{65} + ( \beta_{1} + 3 \beta_{2} - \beta_{6} - \beta_{7} ) q^{67} + ( \beta_{1} - \beta_{2} + \beta_{6} ) q^{69} + ( 5 - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{71} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{73} + ( \beta_{1} - 3 \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{75} + ( 2 \beta_{2} + \beta_{7} ) q^{77} + ( 6 + 2 \beta_{3} - 2 \beta_{5} ) q^{79} + q^{81} + ( 8 \beta_{2} + 2 \beta_{7} ) q^{83} + ( -3 \beta_{1} + 4 \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{85} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{87} + ( 1 + \beta_{3} + 3 \beta_{5} ) q^{89} + ( -\beta_{2} + \beta_{6} ) q^{91} + ( -\beta_{1} - 2 \beta_{2} - \beta_{7} ) q^{93} + ( -6 + 2 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 4 - 2 \beta_{3} + 2 \beta_{4} ) q^{97} + ( 2 \beta_{2} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q - 8q^{7} - 8q^{9} + 4q^{15} + 4q^{17} - 12q^{23} - 24q^{25} - 8q^{31} + 12q^{33} - 8q^{39} - 4q^{41} + 8q^{49} + 8q^{55} - 16q^{57} + 8q^{63} - 16q^{65} + 28q^{71} - 8q^{73} + 40q^{79} + 8q^{81} + 20q^{89} - 40q^{95} + 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} - 4 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + \nu^{3} - 2 \nu^{2} - 4 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} - 3 \nu^{5} + 4 \nu^{4} - 2 \nu^{3} + 4 \nu + 24 \)\()/16\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} + 6 \nu^{3} + 8 \nu^{2} - 12 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} + \nu^{5} + 4 \nu^{4} + 2 \nu^{3} + 8 \nu^{2} + 20 \nu + 8 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} - \nu^{5} + 6 \nu^{3} - 4 \nu + 8 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 2 \nu^{6} - 7 \nu^{5} - 4 \nu^{4} + 6 \nu^{3} - 16 \nu^{2} + 20 \nu + 24 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{7} + 6 \nu^{6} + 3 \nu^{5} + 6 \nu^{3} - 16 \nu^{2} - 36 \nu - 40 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} + 2\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} + 9 \beta_{2} - 4\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_{1} + 14\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 16\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(4 \beta_{7} + \beta_{6} + 5 \beta_{5} + 5 \beta_{4} - 6 \beta_{3} - 9 \beta_{2} - 10 \beta_{1} + 2\)\()/4\)

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
0.621372 + 1.27039i
−0.835949 1.14070i
1.40961 + 0.114062i
−1.19503 + 0.756243i
−1.19503 0.756243i
1.40961 0.114062i
−0.835949 + 1.14070i
0.621372 1.27039i
0 1.00000i 0 3.69833i 0 −1.00000 0 −1.00000 0
337.2 0 1.00000i 0 0.467138i 0 −1.00000 0 −1.00000 0
337.3 0 1.00000i 0 1.12875i 0 −1.00000 0 −1.00000 0
337.4 0 1.00000i 0 4.10245i 0 −1.00000 0 −1.00000 0
337.5 0 1.00000i 0 4.10245i 0 −1.00000 0 −1.00000 0
337.6 0 1.00000i 0 1.12875i 0 −1.00000 0 −1.00000 0
337.7 0 1.00000i 0 0.467138i 0 −1.00000 0 −1.00000 0
337.8 0 1.00000i 0 3.69833i 0 −1.00000 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
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 \( T_{5}^{8} + 32 T_{5}^{6} + 276 T_{5}^{4} + 352 T_{5}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).