Properties

Label 672.2.j.b
Level 672
Weight 2
Character orbit 672.j
Analytic conductor 5.366
Analytic rank 0
Dimension 4
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.j (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} ) q^{3} + ( \beta_{1} - \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{9} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{13} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} -2 \beta_{2} q^{17} + ( -4 + \beta_{1} - \beta_{3} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{25} + ( 4 + 3 \beta_{2} - \beta_{3} ) q^{27} + ( -4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{31} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{33} + ( \beta_{1} + \beta_{3} ) q^{35} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{39} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -6 - 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( -4 + 3 \beta_{1} + \beta_{3} ) q^{45} + ( -4 - 2 \beta_{1} + 2 \beta_{3} ) q^{47} - q^{49} + ( -2 + 2 \beta_{3} ) q^{51} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -2 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{57} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{59} + ( 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{61} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{63} -4 \beta_{2} q^{65} + ( -2 - 4 \beta_{1} + 4 \beta_{3} ) q^{67} + ( -4 - 4 \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{69} + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{71} + ( 6 + 2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( 4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{75} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{77} + ( 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{79} + ( 1 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{81} + ( \beta_{1} - 10 \beta_{2} + \beta_{3} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{85} -6 \beta_{2} q^{89} + ( 2 + \beta_{1} - \beta_{3} ) q^{91} + ( -12 - 4 \beta_{1} + 8 \beta_{2} ) q^{93} + ( 4 - 6 \beta_{1} + 6 \beta_{3} ) q^{95} + ( 10 + 2 \beta_{1} - 2 \beta_{3} ) q^{97} + ( 4 + 4 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 4q^{5} + O(q^{10}) \) \( 4q + 2q^{3} - 4q^{5} - 12q^{15} - 20q^{19} + 2q^{21} + 16q^{23} + 4q^{25} + 14q^{27} - 20q^{33} - 12q^{39} - 16q^{43} - 20q^{45} - 8q^{47} - 4q^{49} - 4q^{51} + 24q^{53} - 20q^{57} - 8q^{63} + 8q^{67} - 12q^{69} + 8q^{71} + 16q^{73} + 22q^{75} + 4q^{81} + 4q^{91} - 40q^{93} + 40q^{95} + 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.618034i
0.618034i
1.61803i
1.61803i
0 −0.618034 1.61803i 0 1.23607 0 1.00000i 0 −2.23607 + 2.00000i 0
239.2 0 −0.618034 + 1.61803i 0 1.23607 0 1.00000i 0 −2.23607 2.00000i 0
239.3 0 1.61803 0.618034i 0 −3.23607 0 1.00000i 0 2.23607 2.00000i 0
239.4 0 1.61803 + 0.618034i 0 −3.23607 0 1.00000i 0 2.23607 + 2.00000i 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.f Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2 T_{5} - 4 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\).