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

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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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

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