Properties

Label 672.2.a.j
Level 672
Weight 2
Character orbit 672.a
Self dual Yes
Analytic conductor 5.366
Analytic rank 0
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 672.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + 2 \beta q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + 2 \beta q^{5} - q^{7} + q^{9} + ( 2 - 2 \beta ) q^{11} + 2 q^{13} + 2 \beta q^{15} + ( 4 - 2 \beta ) q^{17} + 4 \beta q^{19} - q^{21} + ( 2 - 2 \beta ) q^{23} + 7 q^{25} + q^{27} + ( 2 - 4 \beta ) q^{29} + ( -4 - 4 \beta ) q^{31} + ( 2 - 2 \beta ) q^{33} -2 \beta q^{35} -2 q^{37} + 2 q^{39} + ( 8 + 2 \beta ) q^{41} -8 q^{43} + 2 \beta q^{45} + ( 4 + 4 \beta ) q^{47} + q^{49} + ( 4 - 2 \beta ) q^{51} -2 q^{53} + ( -12 + 4 \beta ) q^{55} + 4 \beta q^{57} + ( -8 + 4 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} - q^{63} + 4 \beta q^{65} + ( 4 - 4 \beta ) q^{67} + ( 2 - 2 \beta ) q^{69} + ( 6 + 2 \beta ) q^{71} + ( 6 + 4 \beta ) q^{73} + 7 q^{75} + ( -2 + 2 \beta ) q^{77} + ( -4 - 4 \beta ) q^{79} + q^{81} -4 q^{83} + ( -12 + 8 \beta ) q^{85} + ( 2 - 4 \beta ) q^{87} + 2 \beta q^{89} -2 q^{91} + ( -4 - 4 \beta ) q^{93} + 24 q^{95} + ( -2 - 4 \beta ) q^{97} + ( 2 - 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{7} + 2q^{9} + 4q^{11} + 4q^{13} + 8q^{17} - 2q^{21} + 4q^{23} + 14q^{25} + 2q^{27} + 4q^{29} - 8q^{31} + 4q^{33} - 4q^{37} + 4q^{39} + 16q^{41} - 16q^{43} + 8q^{47} + 2q^{49} + 8q^{51} - 4q^{53} - 24q^{55} - 16q^{59} - 4q^{61} - 2q^{63} + 8q^{67} + 4q^{69} + 12q^{71} + 12q^{73} + 14q^{75} - 4q^{77} - 8q^{79} + 2q^{81} - 8q^{83} - 24q^{85} + 4q^{87} - 4q^{91} - 8q^{93} + 48q^{95} - 4q^{97} + 4q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 1.00000 0 −3.46410 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 3.46410 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(672))\):

\( T_{5}^{2} - 12 \)
\( T_{11}^{2} - 4 T_{11} - 8 \)
\( T_{19}^{2} - 48 \)