Properties

Label 672.2.i.a
Level 672
Weight 2
Character orbit 672.i
Analytic conductor 5.366
Analytic rank 0
Dimension 4
CM No
Inner twists 4

Related objects

Downloads

Learn more about

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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{3} + \beta_{1} q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{3} + \beta_{1} q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 + 2 \beta_{1} ) q^{9} + \beta_{3} q^{11} -2 q^{13} + ( 2 - \beta_{1} ) q^{15} + 3 \beta_{3} q^{17} -4 q^{19} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + \beta_{1} q^{23} + 3 q^{25} + ( 5 - \beta_{1} ) q^{27} + 2 \beta_{3} q^{29} -4 \beta_{2} q^{31} + ( -2 \beta_{2} - \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{3} ) q^{35} -6 \beta_{2} q^{37} + ( 2 + 2 \beta_{1} ) q^{39} -\beta_{3} q^{41} + 2 \beta_{2} q^{43} + ( -4 - \beta_{1} ) q^{45} + 2 \beta_{3} q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + ( -6 \beta_{2} - 3 \beta_{3} ) q^{51} + 2 \beta_{2} q^{55} + ( 4 + 4 \beta_{1} ) q^{57} -4 \beta_{1} q^{59} + 10 q^{61} + ( -2 + 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{63} -2 \beta_{1} q^{65} -2 \beta_{2} q^{67} + ( 2 - \beta_{1} ) q^{69} + \beta_{1} q^{71} + ( -3 - 3 \beta_{1} ) q^{75} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{77} -8 q^{79} + ( -7 - 4 \beta_{1} ) q^{81} + 8 \beta_{1} q^{83} + 6 \beta_{2} q^{85} + ( -4 \beta_{2} - 2 \beta_{3} ) q^{87} -3 \beta_{3} q^{89} + ( -4 + 2 \beta_{2} ) q^{91} + ( 4 \beta_{2} - 4 \beta_{3} ) q^{93} -4 \beta_{1} q^{95} + 8 \beta_{2} q^{97} + ( 4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 8q^{7} - 4q^{9} - 8q^{13} + 8q^{15} - 16q^{19} - 8q^{21} + 12q^{25} + 20q^{27} + 8q^{39} - 16q^{45} + 4q^{49} + 16q^{57} + 40q^{61} - 8q^{63} + 8q^{69} - 12q^{75} - 32q^{79} - 28q^{81} - 16q^{91} + O(q^{100}) \)

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
1.22474 + 0.707107i
−1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
0 −1.00000 1.41421i 0 1.41421i 0 2.00000 1.73205i 0 −1.00000 + 2.82843i 0
209.2 0 −1.00000 1.41421i 0 1.41421i 0 2.00000 + 1.73205i 0 −1.00000 + 2.82843i 0
209.3 0 −1.00000 + 1.41421i 0 1.41421i 0 2.00000 1.73205i 0 −1.00000 2.82843i 0
209.4 0 −1.00000 + 1.41421i 0 1.41421i 0 2.00000 + 1.73205i 0 −1.00000 2.82843i 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
3.b Odd 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 \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 6 \)
\( T_{13} + 2 \)