Properties

Label 4012.2.a.f
Level 4012
Weight 2
Character orbit 4012.a
Self dual Yes
Analytic conductor 32.036
Analytic rank 1
Dimension 3
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{1} q^{5} + q^{7} -2 q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{1} q^{5} + q^{7} -2 q^{9} -2 q^{11} + ( 1 + \beta_{1} ) q^{13} + \beta_{1} q^{15} + q^{17} + ( -2 + \beta_{1} ) q^{19} - q^{21} + ( -\beta_{1} - \beta_{2} ) q^{23} + ( 6 + 2 \beta_{2} ) q^{25} + 5 q^{27} + ( -2 - \beta_{1} ) q^{29} + ( 2 + \beta_{1} - \beta_{2} ) q^{31} + 2 q^{33} -\beta_{1} q^{35} + ( 3 + 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 - \beta_{1} ) q^{39} + ( -2 + \beta_{1} ) q^{41} + ( -2 + \beta_{1} - \beta_{2} ) q^{43} + 2 \beta_{1} q^{45} + ( -4 + \beta_{1} + \beta_{2} ) q^{47} -6 q^{49} - q^{51} + ( 5 + 2 \beta_{2} ) q^{53} + 2 \beta_{1} q^{55} + ( 2 - \beta_{1} ) q^{57} - q^{59} + ( -3 - \beta_{1} ) q^{61} -2 q^{63} + ( -11 - \beta_{1} - 2 \beta_{2} ) q^{65} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( \beta_{1} + \beta_{2} ) q^{69} + ( -2 + 2 \beta_{2} ) q^{71} + 2 q^{73} + ( -6 - 2 \beta_{2} ) q^{75} -2 q^{77} -5 q^{79} + q^{81} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{83} -\beta_{1} q^{85} + ( 2 + \beta_{1} ) q^{87} + ( 4 + \beta_{1} - \beta_{2} ) q^{89} + ( 1 + \beta_{1} ) q^{91} + ( -2 - \beta_{1} + \beta_{2} ) q^{93} + ( -11 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -3 + \beta_{1} - 2 \beta_{2} ) q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + q^{5} + 3q^{7} - 6q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + q^{5} + 3q^{7} - 6q^{9} - 6q^{11} + 2q^{13} - q^{15} + 3q^{17} - 7q^{19} - 3q^{21} + 20q^{25} + 15q^{27} - 5q^{29} + 4q^{31} + 6q^{33} + q^{35} + 6q^{37} - 2q^{39} - 7q^{41} - 8q^{43} - 2q^{45} - 12q^{47} - 18q^{49} - 3q^{51} + 17q^{53} - 2q^{55} + 7q^{57} - 3q^{59} - 8q^{61} - 6q^{63} - 34q^{65} - 6q^{67} - 4q^{71} + 6q^{73} - 20q^{75} - 6q^{77} - 15q^{79} + 3q^{81} - 6q^{83} + q^{85} + 5q^{87} + 10q^{89} + 2q^{91} - 4q^{93} - 37q^{95} - 12q^{97} + 12q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 6\)\()/2\)

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
2.46050
0.239123
−1.69963
0 −1.00000 0 −3.92101 0 1.00000 0 −2.00000 0
1.2 0 −1.00000 0 0.521753 0 1.00000 0 −2.00000 0
1.3 0 −1.00000 0 4.39926 0 1.00000 0 −2.00000 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
\(2\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\( T_{3} + 1 \)
\( T_{5}^{3} - T_{5}^{2} - 17 T_{5} + 9 \)