Properties

Label 672.2.b.a
Level 672
Weight 2
Character orbit 672.b
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.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{5} + \beta_{4} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{5} + \beta_{4} q^{7} + q^{9} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{11} -\beta_{5} q^{13} -\beta_{1} q^{15} + \beta_{1} q^{17} + ( -2 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{19} -\beta_{4} q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{23} + ( -2 + \beta_{3} + \beta_{6} ) q^{25} - q^{27} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{29} + ( 2 + \beta_{6} ) q^{31} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{33} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} ) q^{35} + ( 1 - \beta_{3} + \beta_{6} ) q^{37} + \beta_{5} q^{39} + ( \beta_{1} - 2 \beta_{5} ) q^{41} + ( \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{43} + \beta_{1} q^{45} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{49} -\beta_{1} q^{51} + ( 4 - 2 \beta_{2} + 2 \beta_{4} - \beta_{6} ) q^{53} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{55} + ( 2 - \beta_{2} + \beta_{4} - \beta_{6} ) q^{57} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{59} + ( -\beta_{5} - 2 \beta_{7} ) q^{61} + \beta_{4} q^{63} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{65} + ( 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{67} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{69} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{5} ) q^{73} + ( 2 - \beta_{3} - \beta_{6} ) q^{75} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{77} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{79} + q^{81} + ( -5 + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} ) q^{83} + ( -7 + \beta_{3} + \beta_{6} ) q^{85} + ( 1 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{87} + ( -\beta_{1} - 4 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -2 + \beta_{1} + \beta_{6} + \beta_{7} ) q^{91} + ( -2 - \beta_{6} ) q^{93} + ( -4 \beta_{1} + 2 \beta_{5} + 2 \beta_{7} ) q^{95} + ( -2 \beta_{5} - 2 \beta_{7} ) q^{97} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} - 4q^{7} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{3} - 4q^{7} + 8q^{9} - 8q^{19} + 4q^{21} - 16q^{25} - 8q^{27} + 16q^{31} + 8q^{35} + 8q^{37} - 16q^{47} + 16q^{53} + 8q^{55} + 8q^{57} - 32q^{59} - 4q^{63} + 16q^{65} + 16q^{75} + 32q^{77} + 8q^{81} - 16q^{83} - 56q^{85} - 16q^{91} - 16q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 14 x^{6} + 61 x^{4} + 84 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 9 \nu^{3} + 16 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 11 \nu^{4} + 30 \nu^{2} + 4 \nu + 12 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 9 \nu^{4} - 16 \nu^{2} + 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} - 11 \nu^{4} - 30 \nu^{2} + 4 \nu - 12 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + 11 \nu^{3} + 26 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 11 \nu^{4} - 26 \nu^{2} + 4 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 13 \nu^{5} + 50 \nu^{3} + 54 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - \beta_{4} + \beta_{2} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{5} - 5 \beta_{4} - 5 \beta_{2} - 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 42\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-18 \beta_{5} + 29 \beta_{4} + 29 \beta_{2} + 22 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(47 \beta_{6} - 29 \beta_{4} - 22 \beta_{3} + 29 \beta_{2} - 246\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(4 \beta_{7} + 134 \beta_{5} - 181 \beta_{4} - 181 \beta_{2} - 186 \beta_{1}\)\()/2\)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
2.06644i
2.63640i
0.222191i
1.65222i
1.65222i
0.222191i
2.63640i
2.06644i
0 −1.00000 0 4.33660i 0 −1.65222 + 2.06644i 0 1.00000 0
223.2 0 −1.00000 0 2.31423i 0 0.222191 2.63640i 0 1.00000 0
223.3 0 −1.00000 0 1.72844i 0 −2.63640 0.222191i 0 1.00000 0
223.4 0 −1.00000 0 0.922382i 0 2.06644 + 1.65222i 0 1.00000 0
223.5 0 −1.00000 0 0.922382i 0 2.06644 1.65222i 0 1.00000 0
223.6 0 −1.00000 0 1.72844i 0 −2.63640 + 0.222191i 0 1.00000 0
223.7 0 −1.00000 0 2.31423i 0 0.222191 + 2.63640i 0 1.00000 0
223.8 0 −1.00000 0 4.33660i 0 −1.65222 2.06644i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
28.d Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{19}^{4} + 4 T_{19}^{3} - 44 T_{19}^{2} - 96 T_{19} + 64 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).