Properties

Label 4012.2.a.d
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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 −3.23607 0 0.381966 0 −3.00000 0 7.47214 0
1.2 0 1.23607 0 2.61803 0 −3.00000 0 −1.47214 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\(T_{3}^{2} \) \(\mathstrut +\mathstrut 2 T_{3} \) \(\mathstrut -\mathstrut 4 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 3 T_{5} \) \(\mathstrut +\mathstrut 1 \)