Properties

Label 672.2.k.b
Level 672
Weight 2
Character orbit 672.k
Analytic conductor 5.366
Analytic rank 0
Dimension 8
CM disc. -84
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.k (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} -3 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} -3 q^{9} + \beta_{3} q^{11} + ( \beta_{3} + \beta_{6} ) q^{15} -\beta_{7} q^{17} + 2 \beta_{2} q^{19} + \beta_{4} q^{21} + ( \beta_{3} - \beta_{6} ) q^{23} + ( 5 - 2 \beta_{4} ) q^{25} + 3 \beta_{1} q^{27} + 2 \beta_{1} q^{31} + ( -2 \beta_{5} + \beta_{7} ) q^{33} + ( \beta_{3} + 2 \beta_{6} ) q^{35} + 2 \beta_{4} q^{37} + ( 2 \beta_{5} + \beta_{7} ) q^{41} -3 \beta_{5} q^{45} -7 q^{49} + ( \beta_{3} - 2 \beta_{6} ) q^{51} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{55} -2 \beta_{4} q^{57} + 3 \beta_{2} q^{63} + ( -\beta_{5} + 2 \beta_{7} ) q^{69} + ( \beta_{3} + 3 \beta_{6} ) q^{71} + ( -5 \beta_{1} - 6 \beta_{2} ) q^{75} + ( -\beta_{5} - 2 \beta_{7} ) q^{77} + 9 q^{81} + ( -2 + 2 \beta_{4} ) q^{85} + ( 4 \beta_{5} - \beta_{7} ) q^{89} + 6 q^{93} + ( -2 \beta_{3} - 4 \beta_{6} ) q^{95} -3 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 40q^{25} - 56q^{49} + 72q^{81} - 16q^{85} + 48q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 26 \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{6} - 9 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 20 \nu \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 3 \nu^{4} + \nu^{2} + 6 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 4 \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/10\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{7} - \nu^{5} - 7 \nu^{3} - 20 \nu \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{5} + \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + \beta_{2} + 3 \beta_{1} - 3\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 5 \beta_{6} + 3 \beta_{5}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{4} + 3 \beta_{2} - \beta_{1} - 1\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - 5 \beta_{6} + 7 \beta_{5} + 2 \beta_{3}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{2} - 9\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{6} + 2 \beta_{5} - 7 \beta_{3}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
1.09445 + 0.895644i
0.228425 + 1.39564i
−0.228425 1.39564i
−1.09445 0.895644i
1.09445 0.895644i
0.228425 1.39564i
−0.228425 + 1.39564i
−1.09445 + 0.895644i
0 1.73205i 0 −4.37780 0 2.64575i 0 −3.00000 0
545.2 0 1.73205i 0 −0.913701 0 2.64575i 0 −3.00000 0
545.3 0 1.73205i 0 0.913701 0 2.64575i 0 −3.00000 0
545.4 0 1.73205i 0 4.37780 0 2.64575i 0 −3.00000 0
545.5 0 1.73205i 0 −4.37780 0 2.64575i 0 −3.00000 0
545.6 0 1.73205i 0 −0.913701 0 2.64575i 0 −3.00000 0
545.7 0 1.73205i 0 0.913701 0 2.64575i 0 −3.00000 0
545.8 0 1.73205i 0 4.37780 0 2.64575i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
84.h Odd 1 CM by \(\Q(\sqrt{-21}) \) yes
3.b Odd 1 yes
4.b Odd 1 yes
7.b Odd 1 yes
12.b Even 1 yes
21.c Even 1 yes
28.d Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{4} - 20 T_{5}^{2} + 16 \)
\( T_{43} \)