Properties

Label 4018.2.a.ba
Level 4018
Weight 2
Character orbit 4018.a
Self dual Yes
Analytic conductor 32.084
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( + \beta q^{3} \) \(+ q^{4}\) \( -2 \beta q^{5} \) \( + \beta q^{6} \) \(+ q^{8}\) \(- q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( + \beta q^{3} \) \(+ q^{4}\) \( -2 \beta q^{5} \) \( + \beta q^{6} \) \(+ q^{8}\) \(- q^{9}\) \( -2 \beta q^{10} \) \( -3 \beta q^{11} \) \( + \beta q^{12} \) \( -4 q^{15} \) \(+ q^{16}\) \( + ( -2 + 4 \beta ) q^{17} \) \(- q^{18}\) \( + ( 4 - \beta ) q^{19} \) \( -2 \beta q^{20} \) \( -3 \beta q^{22} \) \( + ( 4 + 2 \beta ) q^{23} \) \( + \beta q^{24} \) \( + 3 q^{25} \) \( -4 \beta q^{27} \) \( + ( 4 + 4 \beta ) q^{29} \) \( -4 q^{30} \) \( + ( 4 + 2 \beta ) q^{31} \) \(+ q^{32}\) \( -6 q^{33} \) \( + ( -2 + 4 \beta ) q^{34} \) \(- q^{36}\) \( -6 \beta q^{37} \) \( + ( 4 - \beta ) q^{38} \) \( -2 \beta q^{40} \) \(+ q^{41}\) \( + ( 4 + 4 \beta ) q^{43} \) \( -3 \beta q^{44} \) \( + 2 \beta q^{45} \) \( + ( 4 + 2 \beta ) q^{46} \) \( + ( 2 - 5 \beta ) q^{47} \) \( + \beta q^{48} \) \( + 3 q^{50} \) \( + ( 8 - 2 \beta ) q^{51} \) \( + 12 q^{53} \) \( -4 \beta q^{54} \) \( + 12 q^{55} \) \( + ( -2 + 4 \beta ) q^{57} \) \( + ( 4 + 4 \beta ) q^{58} \) \( + ( 4 + 2 \beta ) q^{59} \) \( -4 q^{60} \) \( -6 q^{61} \) \( + ( 4 + 2 \beta ) q^{62} \) \(+ q^{64}\) \( -6 q^{66} \) \( + ( -4 + 3 \beta ) q^{67} \) \( + ( -2 + 4 \beta ) q^{68} \) \( + ( 4 + 4 \beta ) q^{69} \) \( + ( -2 - \beta ) q^{71} \) \(- q^{72}\) \( + ( 8 - 4 \beta ) q^{73} \) \( -6 \beta q^{74} \) \( + 3 \beta q^{75} \) \( + ( 4 - \beta ) q^{76} \) \( + ( -6 + 3 \beta ) q^{79} \) \( -2 \beta q^{80} \) \( -5 q^{81} \) \(+ q^{82}\) \( + ( -12 + 4 \beta ) q^{83} \) \( + ( -16 + 4 \beta ) q^{85} \) \( + ( 4 + 4 \beta ) q^{86} \) \( + ( 8 + 4 \beta ) q^{87} \) \( -3 \beta q^{88} \) \( + ( 6 - 4 \beta ) q^{89} \) \( + 2 \beta q^{90} \) \( + ( 4 + 2 \beta ) q^{92} \) \( + ( 4 + 4 \beta ) q^{93} \) \( + ( 2 - 5 \beta ) q^{94} \) \( + ( 4 - 8 \beta ) q^{95} \) \( + \beta q^{96} \) \( + ( 2 + 4 \beta ) q^{97} \) \( + 3 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 2q^{9} \) \(\mathstrut -\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 8q^{30} \) \(\mathstrut +\mathstrut 8q^{31} \) \(\mathstrut +\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 12q^{33} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut -\mathstrut 2q^{36} \) \(\mathstrut +\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 8q^{46} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 6q^{50} \) \(\mathstrut +\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 24q^{53} \) \(\mathstrut +\mathstrut 24q^{55} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 8q^{58} \) \(\mathstrut +\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 12q^{61} \) \(\mathstrut +\mathstrut 8q^{62} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 12q^{66} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 4q^{68} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 4q^{71} \) \(\mathstrut -\mathstrut 2q^{72} \) \(\mathstrut +\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 8q^{76} \) \(\mathstrut -\mathstrut 12q^{79} \) \(\mathstrut -\mathstrut 10q^{81} \) \(\mathstrut +\mathstrut 2q^{82} \) \(\mathstrut -\mathstrut 24q^{83} \) \(\mathstrut -\mathstrut 32q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 16q^{87} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut +\mathstrut 8q^{92} \) \(\mathstrut +\mathstrut 8q^{93} \) \(\mathstrut +\mathstrut 4q^{94} \) \(\mathstrut +\mathstrut 8q^{95} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.41421
1.41421
1.00000 −1.41421 1.00000 2.82843 −1.41421 0 1.00000 −1.00000 2.82843
1.2 1.00000 1.41421 1.00000 −2.82843 1.41421 0 1.00000 −1.00000 −2.82843
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\(T_{3}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{5}^{2} \) \(\mathstrut -\mathstrut 8 \)
\(T_{11}^{2} \) \(\mathstrut -\mathstrut 18 \)