Properties

Label 4018.2.a.x
Level 4018
Weight 2
Character orbit 4018.a
Self dual Yes
Analytic conductor 32.084
Analytic rank 1
Dimension 2
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4018.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0838915322\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( -2 q^{3} \) \(+ q^{4}\) \( + ( -1 - \beta ) q^{5} \) \( -2 q^{6} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( -2 q^{3} \) \(+ q^{4}\) \( + ( -1 - \beta ) q^{5} \) \( -2 q^{6} \) \(+ q^{8}\) \(+ q^{9}\) \( + ( -1 - \beta ) q^{10} \) \( + ( -1 - \beta ) q^{11} \) \( -2 q^{12} \) \( + ( 2 + 2 \beta ) q^{13} \) \( + ( 2 + 2 \beta ) q^{15} \) \(+ q^{16}\) \(+ q^{18}\) \( + 2 \beta q^{19} \) \( + ( -1 - \beta ) q^{20} \) \( + ( -1 - \beta ) q^{22} \) \( -2 q^{23} \) \( -2 q^{24} \) \( + ( -1 + 2 \beta ) q^{25} \) \( + ( 2 + 2 \beta ) q^{26} \) \( + 4 q^{27} \) \( + ( 1 - 5 \beta ) q^{29} \) \( + ( 2 + 2 \beta ) q^{30} \) \( + ( 2 - 2 \beta ) q^{31} \) \(+ q^{32}\) \( + ( 2 + 2 \beta ) q^{33} \) \(+ q^{36}\) \( + ( -2 + 4 \beta ) q^{37} \) \( + 2 \beta q^{38} \) \( + ( -4 - 4 \beta ) q^{39} \) \( + ( -1 - \beta ) q^{40} \) \(- q^{41}\) \( + ( 2 + 6 \beta ) q^{43} \) \( + ( -1 - \beta ) q^{44} \) \( + ( -1 - \beta ) q^{45} \) \( -2 q^{46} \) \( + ( -4 + 2 \beta ) q^{47} \) \( -2 q^{48} \) \( + ( -1 + 2 \beta ) q^{50} \) \( + ( 2 + 2 \beta ) q^{52} \) \( + ( -1 + 5 \beta ) q^{53} \) \( + 4 q^{54} \) \( + ( 4 + 2 \beta ) q^{55} \) \( -4 \beta q^{57} \) \( + ( 1 - 5 \beta ) q^{58} \) \( + ( -1 - 7 \beta ) q^{59} \) \( + ( 2 + 2 \beta ) q^{60} \) \( + ( -5 - \beta ) q^{61} \) \( + ( 2 - 2 \beta ) q^{62} \) \(+ q^{64}\) \( + ( -8 - 4 \beta ) q^{65} \) \( + ( 2 + 2 \beta ) q^{66} \) \( + ( -5 + 3 \beta ) q^{67} \) \( + 4 q^{69} \) \( + ( 6 - 2 \beta ) q^{71} \) \(+ q^{72}\) \( + ( 2 + 4 \beta ) q^{73} \) \( + ( -2 + 4 \beta ) q^{74} \) \( + ( 2 - 4 \beta ) q^{75} \) \( + 2 \beta q^{76} \) \( + ( -4 - 4 \beta ) q^{78} \) \( + ( -1 - \beta ) q^{80} \) \( -11 q^{81} \) \(- q^{82}\) \( + ( -3 - \beta ) q^{83} \) \( + ( 2 + 6 \beta ) q^{86} \) \( + ( -2 + 10 \beta ) q^{87} \) \( + ( -1 - \beta ) q^{88} \) \( + ( -6 - 4 \beta ) q^{89} \) \( + ( -1 - \beta ) q^{90} \) \( -2 q^{92} \) \( + ( -4 + 4 \beta ) q^{93} \) \( + ( -4 + 2 \beta ) q^{94} \) \( + ( -6 - 2 \beta ) q^{95} \) \( -2 q^{96} \) \( -2 q^{97} \) \( + ( -1 - \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut +\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut -\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 4q^{12} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 2q^{18} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut -\mathstrut 2q^{22} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 4q^{24} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut +\mathstrut 4q^{26} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut +\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut +\mathstrut 2q^{36} \) \(\mathstrut -\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 2q^{40} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 4q^{43} \) \(\mathstrut -\mathstrut 2q^{44} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 8q^{47} \) \(\mathstrut -\mathstrut 4q^{48} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut -\mathstrut 2q^{53} \) \(\mathstrut +\mathstrut 8q^{54} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 2q^{58} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut -\mathstrut 10q^{61} \) \(\mathstrut +\mathstrut 4q^{62} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 16q^{65} \) \(\mathstrut +\mathstrut 4q^{66} \) \(\mathstrut -\mathstrut 10q^{67} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 2q^{72} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 4q^{74} \) \(\mathstrut +\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 8q^{78} \) \(\mathstrut -\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 22q^{81} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut -\mathstrut 6q^{83} \) \(\mathstrut +\mathstrut 4q^{86} \) \(\mathstrut -\mathstrut 4q^{87} \) \(\mathstrut -\mathstrut 2q^{88} \) \(\mathstrut -\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 2q^{90} \) \(\mathstrut -\mathstrut 4q^{92} \) \(\mathstrut -\mathstrut 8q^{93} \) \(\mathstrut -\mathstrut 8q^{94} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut -\mathstrut 4q^{96} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
1.73205
−1.73205
1.00000 −2.00000 1.00000 −2.73205 −2.00000 0 1.00000 1.00000 −2.73205
1.2 1.00000 −2.00000 1.00000 0.732051 −2.00000 0 1.00000 1.00000 0.732051
\(n\): 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
\(2\) \(-1\)
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

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