Properties

Label 672.2.i.b
Level 672
Weight 2
Character orbit 672.i
Analytic conductor 5.366
Analytic rank 0
Dimension 4
CM disc. -24
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + 2 \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + 2 \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} -3 q^{9} -4 \beta_{1} q^{11} -6 q^{15} + ( 3 \beta_{1} - \beta_{2} ) q^{21} -7 q^{25} -3 \beta_{2} q^{27} + 2 \beta_{1} q^{29} -2 \beta_{3} q^{31} + 4 \beta_{3} q^{33} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{35} -6 \beta_{2} q^{45} + ( -5 - 2 \beta_{3} ) q^{49} -10 \beta_{1} q^{53} + 8 \beta_{3} q^{55} + 6 \beta_{2} q^{59} + ( 3 - 3 \beta_{3} ) q^{63} -4 \beta_{3} q^{73} -7 \beta_{2} q^{75} + ( 4 \beta_{1} + 8 \beta_{2} ) q^{77} -10 q^{79} + 9 q^{81} + 10 \beta_{2} q^{83} -2 \beta_{3} q^{87} -6 \beta_{1} q^{93} + 8 \beta_{3} q^{97} + 12 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{7} - 12q^{9} - 24q^{15} - 28q^{25} - 20q^{49} + 12q^{63} - 40q^{79} + 36q^{81} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

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
0.707107 1.22474i
−0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 1.73205i 0 3.46410i 0 −1.00000 2.44949i 0 −3.00000 0
209.2 0 1.73205i 0 3.46410i 0 −1.00000 + 2.44949i 0 −3.00000 0
209.3 0 1.73205i 0 3.46410i 0 −1.00000 2.44949i 0 −3.00000 0
209.4 0 1.73205i 0 3.46410i 0 −1.00000 + 2.44949i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
24.h Odd 1 CM by \(\Q(\sqrt{-6}) \) yes
3.b Odd 1 yes
7.b Odd 1 yes
8.b Even 1 yes
21.c Even 1 yes
56.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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 12 \)
\( T_{11}^{2} - 32 \)
\( T_{13} \)