Properties

Label 672.2.i.e
Level 672
Weight 2
Character orbit 672.i
Analytic conductor 5.366
Analytic rank 0
Dimension 8
CM disc. -56
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.i (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10070523904.11
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{6} - \beta_{7} ) q^{5} + \beta_{3} q^{7} + ( \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{6} - \beta_{7} ) q^{5} + \beta_{3} q^{7} + ( \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{1} - \beta_{6} - \beta_{7} ) q^{13} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{15} + ( -\beta_{1} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{5} + \beta_{6} ) q^{21} -2 \beta_{4} q^{23} + ( -5 - 2 \beta_{3} ) q^{25} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{27} + ( 3 \beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{35} + ( -5 - 4 \beta_{2} - \beta_{3} + \beta_{4} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{45} + 7 q^{49} + ( -1 - 5 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{57} + ( -3 \beta_{1} + \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{59} + ( 4 \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{61} + ( 7 + \beta_{4} ) q^{63} + ( -2 \beta_{2} - 4 \beta_{4} ) q^{65} + ( -2 \beta_{1} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{69} -4 \beta_{2} q^{71} + ( -5 \beta_{1} - 2 \beta_{5} - 2 \beta_{6} ) q^{75} + 2 \beta_{3} q^{79} + ( 5 + 2 \beta_{4} ) q^{81} + ( -3 \beta_{1} + \beta_{5} - 3 \beta_{6} + 5 \beta_{7} ) q^{83} + ( -\beta_{1} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} + ( 8 \beta_{2} - 2 \beta_{4} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{15} - 40q^{25} - 40q^{39} + 56q^{49} - 8q^{57} + 56q^{63} + 40q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 10 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - \nu^{2} \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 19 \nu^{2} \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 5 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 9 \nu^{5} + 37 \nu^{3} + 63 \nu \)\()/54\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} + 9 \nu^{5} + 20 \nu^{3} - 63 \nu \)\()/54\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + \nu^{3} \)\()/18\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{5}\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} + 5\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{3} + 19 \beta_{2}\)
\(\nu^{7}\)\(=\)\(-19 \beta_{7} + \beta_{6} + \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−1.68014 0.420861i
−1.68014 + 0.420861i
−0.420861 1.68014i
−0.420861 + 1.68014i
0.420861 1.68014i
0.420861 + 1.68014i
1.68014 0.420861i
1.68014 + 0.420861i
0 −1.68014 0.420861i 0 3.91044i 0 2.64575 0 2.64575 + 1.41421i 0
209.2 0 −1.68014 + 0.420861i 0 3.91044i 0 2.64575 0 2.64575 1.41421i 0
209.3 0 −0.420861 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 + 1.41421i 0
209.4 0 −0.420861 + 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 1.41421i 0
209.5 0 0.420861 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 1.41421i 0
209.6 0 0.420861 + 1.68014i 0 2.16991i 0 −2.64575 0 −2.64575 + 1.41421i 0
209.7 0 1.68014 0.420861i 0 3.91044i 0 2.64575 0 2.64575 1.41421i 0
209.8 0 1.68014 + 0.420861i 0 3.91044i 0 2.64575 0 2.64575 + 1.41421i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
56.h Odd 1 CM by \(\Q(\sqrt{-14}) \) yes
3.b Odd 1 yes
7.b Odd 1 yes
8.b Even 1 yes
21.c Even 1 yes
24.h Odd 1 yes
168.i Even 1 yes

Hecke kernels

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

\( T_{5}^{4} + 20 T_{5}^{2} + 72 \)
\( T_{11} \)
\( T_{13}^{4} - 52 T_{13}^{2} + 648 \)