Properties

Label 4012.2.a.e
Level 4012
Weight 2
Character orbit 4012.a
Self dual Yes
Analytic conductor 32.036
Analytic rank 1
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{3} \) \(- q^{5}\) \( + \beta q^{7} \) \( + 2 q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta q^{3} \) \(- q^{5}\) \( + \beta q^{7} \) \( + 2 q^{9} \) \( -2 \beta q^{11} \) \( + ( -1 + \beta ) q^{13} \) \( + \beta q^{15} \) \(- q^{17}\) \( + 3 \beta q^{19} \) \( -5 q^{21} \) \( -2 \beta q^{23} \) \( -4 q^{25} \) \( + \beta q^{27} \) \(+ q^{29}\) \( + 6 q^{31} \) \( + 10 q^{33} \) \( -\beta q^{35} \) \( + ( -5 + 3 \beta ) q^{37} \) \( + ( -5 + \beta ) q^{39} \) \( + ( -3 + 2 \beta ) q^{41} \) \( + ( -4 - 2 \beta ) q^{43} \) \( -2 q^{45} \) \( -2 q^{47} \) \( -2 q^{49} \) \( + \beta q^{51} \) \( + ( 5 - 2 \beta ) q^{53} \) \( + 2 \beta q^{55} \) \( -15 q^{57} \) \(+ q^{59}\) \( + ( 7 + 3 \beta ) q^{61} \) \( + 2 \beta q^{63} \) \( + ( 1 - \beta ) q^{65} \) \( + ( -6 + 4 \beta ) q^{67} \) \( + 10 q^{69} \) \( + ( -2 - 2 \beta ) q^{71} \) \( + 6 \beta q^{73} \) \( + 4 \beta q^{75} \) \( -10 q^{77} \) \( + ( -2 - 3 \beta ) q^{79} \) \( -11 q^{81} \) \( + ( -1 - \beta ) q^{83} \) \(+ q^{85}\) \( -\beta q^{87} \) \( + ( 2 - 2 \beta ) q^{89} \) \( + ( 5 - \beta ) q^{91} \) \( -6 \beta q^{93} \) \( -3 \beta q^{95} \) \( + ( -1 + 3 \beta ) q^{97} \) \( -4 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut -\mathstrut 10q^{21} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 20q^{33} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 10q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 4q^{47} \) \(\mathstrut -\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 10q^{53} \) \(\mathstrut -\mathstrut 30q^{57} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 14q^{61} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 12q^{67} \) \(\mathstrut +\mathstrut 20q^{69} \) \(\mathstrut -\mathstrut 4q^{71} \) \(\mathstrut -\mathstrut 20q^{77} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut -\mathstrut 22q^{81} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut +\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 4q^{89} \) \(\mathstrut +\mathstrut 10q^{91} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.61803
−0.618034
0 −2.23607 0 −1.00000 0 2.23607 0 2.00000 0
1.2 0 2.23607 0 −1.00000 0 −2.23607 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

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