Properties

Label 4014.2.a.k
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( -2 \beta q^{5} \) \(+ q^{8}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( -2 \beta q^{5} \) \(+ q^{8}\) \( -2 \beta q^{10} \) \( + ( 3 - 3 \beta ) q^{11} \) \( -3 \beta q^{13} \) \(+ q^{16}\) \( -4 \beta q^{17} \) \( + ( 2 + \beta ) q^{19} \) \( -2 \beta q^{20} \) \( + ( 3 - 3 \beta ) q^{22} \) \( -2 q^{23} \) \( + ( -1 + 4 \beta ) q^{25} \) \( -3 \beta q^{26} \) \( + ( -1 + 5 \beta ) q^{29} \) \( + 6 q^{31} \) \(+ q^{32}\) \( -4 \beta q^{34} \) \( + 4 \beta q^{37} \) \( + ( 2 + \beta ) q^{38} \) \( -2 \beta q^{40} \) \( + 10 q^{41} \) \( + ( 7 + \beta ) q^{43} \) \( + ( 3 - 3 \beta ) q^{44} \) \( -2 q^{46} \) \( + ( -5 + \beta ) q^{47} \) \( -7 q^{49} \) \( + ( -1 + 4 \beta ) q^{50} \) \( -3 \beta q^{52} \) \( + 5 \beta q^{53} \) \( + 6 q^{55} \) \( + ( -1 + 5 \beta ) q^{58} \) \( + ( -2 - 3 \beta ) q^{59} \) \( + ( 7 - 11 \beta ) q^{61} \) \( + 6 q^{62} \) \(+ q^{64}\) \( + ( 6 + 6 \beta ) q^{65} \) \( + ( 2 + 8 \beta ) q^{67} \) \( -4 \beta q^{68} \) \( + ( -8 + 12 \beta ) q^{71} \) \( + ( 4 + 5 \beta ) q^{73} \) \( + 4 \beta q^{74} \) \( + ( 2 + \beta ) q^{76} \) \( + ( -3 - \beta ) q^{79} \) \( -2 \beta q^{80} \) \( + 10 q^{82} \) \( -12 q^{83} \) \( + ( 8 + 8 \beta ) q^{85} \) \( + ( 7 + \beta ) q^{86} \) \( + ( 3 - 3 \beta ) q^{88} \) \( + ( 6 - 2 \beta ) q^{89} \) \( -2 q^{92} \) \( + ( -5 + \beta ) q^{94} \) \( + ( -2 - 6 \beta ) q^{95} \) \( + ( 10 + 4 \beta ) q^{97} \) \( -7 q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 5q^{19} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut +\mathstrut 3q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut +\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 3q^{26} \) \(\mathstrut +\mathstrut 3q^{29} \) \(\mathstrut +\mathstrut 12q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut +\mathstrut 5q^{38} \) \(\mathstrut -\mathstrut 2q^{40} \) \(\mathstrut +\mathstrut 20q^{41} \) \(\mathstrut +\mathstrut 15q^{43} \) \(\mathstrut +\mathstrut 3q^{44} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 9q^{47} \) \(\mathstrut -\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 3q^{52} \) \(\mathstrut +\mathstrut 5q^{53} \) \(\mathstrut +\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 3q^{58} \) \(\mathstrut -\mathstrut 7q^{59} \) \(\mathstrut +\mathstrut 3q^{61} \) \(\mathstrut +\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 18q^{65} \) \(\mathstrut +\mathstrut 12q^{67} \) \(\mathstrut -\mathstrut 4q^{68} \) \(\mathstrut -\mathstrut 4q^{71} \) \(\mathstrut +\mathstrut 13q^{73} \) \(\mathstrut +\mathstrut 4q^{74} \) \(\mathstrut +\mathstrut 5q^{76} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut +\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 24q^{83} \) \(\mathstrut +\mathstrut 24q^{85} \) \(\mathstrut +\mathstrut 15q^{86} \) \(\mathstrut +\mathstrut 3q^{88} \) \(\mathstrut +\mathstrut 10q^{89} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut -\mathstrut 9q^{94} \) \(\mathstrut -\mathstrut 10q^{95} \) \(\mathstrut +\mathstrut 24q^{97} \) \(\mathstrut -\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.00000 0 1.00000 −3.23607 0 0 1.00000 0 −3.23607
1.2 1.00000 0 1.00000 1.23607 0 0 1.00000 0 1.23607
\(n\): 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
\(2\) \(-1\)
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\(T_{5}^{2} \) \(\mathstrut +\mathstrut 2 T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{7} \)
\(T_{11}^{2} \) \(\mathstrut -\mathstrut 3 T_{11} \) \(\mathstrut -\mathstrut 9 \)