Properties

Label 4008.2.a.e
Level 4008
Weight 2
Character orbit 4008.a
Self dual Yes
Analytic conductor 32.004
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + ( 2 - \beta_{2} ) q^{5} \) \( + ( -2 + 2 \beta_{1} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + ( 2 - \beta_{2} ) q^{5} \) \( + ( -2 + 2 \beta_{1} ) q^{7} \) \(+ q^{9}\) \( + ( -1 - \beta_{1} + \beta_{2} ) q^{11} \) \( + 2 q^{13} \) \( + ( -2 + \beta_{2} ) q^{15} \) \( + 3 \beta_{2} q^{17} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{19} \) \( + ( 2 - 2 \beta_{1} ) q^{21} \) \( -8 q^{23} \) \( + ( 2 - \beta_{1} - 5 \beta_{2} ) q^{25} \) \(- q^{27}\) \( + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{29} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} \) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{33} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{35} \) \( + ( -4 + 2 \beta_{1} ) q^{37} \) \( -2 q^{39} \) \( + ( 6 - 6 \beta_{1} - 3 \beta_{2} ) q^{41} \) \( + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{43} \) \( + ( 2 - \beta_{2} ) q^{45} \) \( + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{47} \) \( + ( 5 - 4 \beta_{1} + 4 \beta_{2} ) q^{49} \) \( -3 \beta_{2} q^{51} \) \( + ( -4 + 4 \beta_{1} + 3 \beta_{2} ) q^{53} \) \( + ( -6 + 4 \beta_{2} ) q^{55} \) \( + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{57} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{59} \) \( + ( -1 + 5 \beta_{1} + 5 \beta_{2} ) q^{61} \) \( + ( -2 + 2 \beta_{1} ) q^{63} \) \( + ( 4 - 2 \beta_{2} ) q^{65} \) \( + ( -8 - 2 \beta_{1} - \beta_{2} ) q^{67} \) \( + 8 q^{69} \) \( + ( -4 - 4 \beta_{1} - 6 \beta_{2} ) q^{71} \) \( + ( 8 - 2 \beta_{1} - 4 \beta_{2} ) q^{73} \) \( + ( -2 + \beta_{1} + 5 \beta_{2} ) q^{75} \) \( + ( -4 - 4 \beta_{2} ) q^{77} \) \( + ( 4 - 2 \beta_{1} + 5 \beta_{2} ) q^{79} \) \(+ q^{81}\) \( + ( -4 + 8 \beta_{1} + 6 \beta_{2} ) q^{83} \) \( + ( -9 + 3 \beta_{1} + 9 \beta_{2} ) q^{85} \) \( + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{87} \) \( + ( -6 \beta_{1} + 2 \beta_{2} ) q^{89} \) \( + ( -4 + 4 \beta_{1} ) q^{91} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{93} \) \( + ( 8 - 4 \beta_{1} - 8 \beta_{2} ) q^{95} \) \( + ( 2 - 6 \beta_{2} ) q^{97} \) \( + ( -1 - \beta_{1} + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut 24q^{23} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 3q^{27} \) \(\mathstrut -\mathstrut 14q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut +\mathstrut 12q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 11q^{49} \) \(\mathstrut -\mathstrut 8q^{53} \) \(\mathstrut -\mathstrut 18q^{55} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 12q^{65} \) \(\mathstrut -\mathstrut 26q^{67} \) \(\mathstrut +\mathstrut 24q^{69} \) \(\mathstrut -\mathstrut 16q^{71} \) \(\mathstrut +\mathstrut 22q^{73} \) \(\mathstrut -\mathstrut 5q^{75} \) \(\mathstrut -\mathstrut 12q^{77} \) \(\mathstrut +\mathstrut 10q^{79} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut 4q^{83} \) \(\mathstrut -\mathstrut 24q^{85} \) \(\mathstrut +\mathstrut 14q^{87} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut -\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 4q^{93} \) \(\mathstrut +\mathstrut 20q^{95} \) \(\mathstrut +\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

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

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
−1.48119
2.17009
0.311108
0 −1.00000 0 0.324869 0 −4.96239 0 1.00000 0
1.2 0 −1.00000 0 1.46081 0 2.34017 0 1.00000 0
1.3 0 −1.00000 0 4.21432 0 −1.37778 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
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{3} \) \(\mathstrut -\mathstrut 6 T_{5}^{2} \) \(\mathstrut +\mathstrut 8 T_{5} \) \(\mathstrut -\mathstrut 2 \)
\(T_{7}^{3} \) \(\mathstrut +\mathstrut 4 T_{7}^{2} \) \(\mathstrut -\mathstrut 8 T_{7} \) \(\mathstrut -\mathstrut 16 \)