Properties

Label 4018.2.a.bg
Level 4018
Weight 2
Character orbit 4018.a
Self dual Yes
Analytic conductor 32.084
Analytic rank 0
Dimension 3
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: \(3\)
Coefficient field: 3.3.568.1
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \( + \beta_{2} q^{3} \) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( + \beta_{2} q^{6} \) \(+ q^{8}\) \( + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \( + \beta_{2} q^{3} \) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( + \beta_{2} q^{6} \) \(+ q^{8}\) \( + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} \) \( -\beta_{1} q^{10} \) \( + ( 2 + \beta_{1} + \beta_{2} ) q^{11} \) \( + \beta_{2} q^{12} \) \( + ( 2 + \beta_{2} ) q^{15} \) \(+ q^{16}\) \( + ( 2 - \beta_{2} ) q^{17} \) \( + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{18} \) \( + ( -2 - 2 \beta_{2} ) q^{19} \) \( -\beta_{1} q^{20} \) \( + ( 2 + \beta_{1} + \beta_{2} ) q^{22} \) \( + ( 2 - 2 \beta_{1} ) q^{23} \) \( + \beta_{2} q^{24} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{25} \) \( + ( -2 + 2 \beta_{1} + 3 \beta_{2} ) q^{27} \) \( + ( 2 - \beta_{1} ) q^{29} \) \( + ( 2 + \beta_{2} ) q^{30} \) \( + ( -2 + \beta_{2} ) q^{31} \) \(+ q^{32}\) \( + ( 4 - 2 \beta_{1} ) q^{33} \) \( + ( 2 - \beta_{2} ) q^{34} \) \( + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{36} \) \( + 2 q^{37} \) \( + ( -2 - 2 \beta_{2} ) q^{38} \) \( -\beta_{1} q^{40} \) \(+ q^{41}\) \( + ( -2 + 3 \beta_{2} ) q^{43} \) \( + ( 2 + \beta_{1} + \beta_{2} ) q^{44} \) \( + ( 6 + \beta_{1} + \beta_{2} ) q^{45} \) \( + ( 2 - 2 \beta_{1} ) q^{46} \) \( -2 q^{47} \) \( + \beta_{2} q^{48} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{50} \) \( + ( -6 + 2 \beta_{1} + 3 \beta_{2} ) q^{51} \) \( + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{53} \) \( + ( -2 + 2 \beta_{1} + 3 \beta_{2} ) q^{54} \) \( + ( -2 - 4 \beta_{1} ) q^{55} \) \( + ( -12 + 4 \beta_{1} ) q^{57} \) \( + ( 2 - \beta_{1} ) q^{58} \) \( + ( 3 \beta_{1} + \beta_{2} ) q^{59} \) \( + ( 2 + \beta_{2} ) q^{60} \) \( + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{61} \) \( + ( -2 + \beta_{2} ) q^{62} \) \(+ q^{64}\) \( + ( 4 - 2 \beta_{1} ) q^{66} \) \( + ( 6 + \beta_{1} - 3 \beta_{2} ) q^{67} \) \( + ( 2 - \beta_{2} ) q^{68} \) \( + ( 4 + 4 \beta_{2} ) q^{69} \) \( + ( 2 + 4 \beta_{1} + 3 \beta_{2} ) q^{71} \) \( + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{72} \) \( + ( 10 + 2 \beta_{2} ) q^{73} \) \( + 2 q^{74} \) \( + ( 2 - 2 \beta_{1} - 4 \beta_{2} ) q^{75} \) \( + ( -2 - 2 \beta_{2} ) q^{76} \) \( + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{79} \) \( -\beta_{1} q^{80} \) \( + ( 5 - 4 \beta_{2} ) q^{81} \) \(+ q^{82}\) \( + ( 4 + \beta_{1} - \beta_{2} ) q^{83} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{85} \) \( + ( -2 + 3 \beta_{2} ) q^{86} \) \( + ( 2 + 3 \beta_{2} ) q^{87} \) \( + ( 2 + \beta_{1} + \beta_{2} ) q^{88} \) \( + ( -8 + 2 \beta_{1} + \beta_{2} ) q^{89} \) \( + ( 6 + \beta_{1} + \beta_{2} ) q^{90} \) \( + ( 2 - 2 \beta_{1} ) q^{92} \) \( + ( 6 - 2 \beta_{1} - 3 \beta_{2} ) q^{93} \) \( -2 q^{94} \) \( + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} \) \( + \beta_{2} q^{96} \) \( + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{97} \) \( + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut -\mathstrut q^{6} \) \(\mathstrut +\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut -\mathstrut q^{10} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut q^{12} \) \(\mathstrut +\mathstrut 5q^{15} \) \(\mathstrut +\mathstrut 3q^{16} \) \(\mathstrut +\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut +\mathstrut 6q^{22} \) \(\mathstrut +\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut q^{24} \) \(\mathstrut -\mathstrut 2q^{25} \) \(\mathstrut -\mathstrut 7q^{27} \) \(\mathstrut +\mathstrut 5q^{29} \) \(\mathstrut +\mathstrut 5q^{30} \) \(\mathstrut -\mathstrut 7q^{31} \) \(\mathstrut +\mathstrut 3q^{32} \) \(\mathstrut +\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 7q^{34} \) \(\mathstrut +\mathstrut 8q^{36} \) \(\mathstrut +\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut q^{40} \) \(\mathstrut +\mathstrut 3q^{41} \) \(\mathstrut -\mathstrut 9q^{43} \) \(\mathstrut +\mathstrut 6q^{44} \) \(\mathstrut +\mathstrut 18q^{45} \) \(\mathstrut +\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 6q^{47} \) \(\mathstrut -\mathstrut q^{48} \) \(\mathstrut -\mathstrut 2q^{50} \) \(\mathstrut -\mathstrut 19q^{51} \) \(\mathstrut -\mathstrut 7q^{53} \) \(\mathstrut -\mathstrut 7q^{54} \) \(\mathstrut -\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 32q^{57} \) \(\mathstrut +\mathstrut 5q^{58} \) \(\mathstrut +\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 5q^{60} \) \(\mathstrut +\mathstrut 13q^{61} \) \(\mathstrut -\mathstrut 7q^{62} \) \(\mathstrut +\mathstrut 3q^{64} \) \(\mathstrut +\mathstrut 10q^{66} \) \(\mathstrut +\mathstrut 22q^{67} \) \(\mathstrut +\mathstrut 7q^{68} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 7q^{71} \) \(\mathstrut +\mathstrut 8q^{72} \) \(\mathstrut +\mathstrut 28q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut +\mathstrut 8q^{75} \) \(\mathstrut -\mathstrut 4q^{76} \) \(\mathstrut +\mathstrut 19q^{79} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut +\mathstrut 19q^{81} \) \(\mathstrut +\mathstrut 3q^{82} \) \(\mathstrut +\mathstrut 14q^{83} \) \(\mathstrut -\mathstrut 7q^{85} \) \(\mathstrut -\mathstrut 9q^{86} \) \(\mathstrut +\mathstrut 3q^{87} \) \(\mathstrut +\mathstrut 6q^{88} \) \(\mathstrut -\mathstrut 23q^{89} \) \(\mathstrut +\mathstrut 18q^{90} \) \(\mathstrut +\mathstrut 4q^{92} \) \(\mathstrut +\mathstrut 19q^{93} \) \(\mathstrut -\mathstrut 6q^{94} \) \(\mathstrut -\mathstrut 8q^{95} \) \(\mathstrut -\mathstrut q^{96} \) \(\mathstrut +\mathstrut 25q^{97} \) \(\mathstrut -\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(6\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)

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
−0.363328
3.12489
−1.76156
1.00000 −3.14134 1.00000 0.363328 −3.14134 0 1.00000 6.86799 0.363328
1.2 1.00000 −0.484862 1.00000 −3.12489 −0.484862 0 1.00000 −2.76491 −3.12489
1.3 1.00000 2.62620 1.00000 1.76156 2.62620 0 1.00000 3.89692 1.76156
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\(T_{3}^{3} \) \(\mathstrut +\mathstrut T_{3}^{2} \) \(\mathstrut -\mathstrut 8 T_{3} \) \(\mathstrut -\mathstrut 4 \)
\(T_{5}^{3} \) \(\mathstrut +\mathstrut T_{5}^{2} \) \(\mathstrut -\mathstrut 6 T_{5} \) \(\mathstrut +\mathstrut 2 \)
\(T_{11}^{3} \) \(\mathstrut -\mathstrut 6 T_{11}^{2} \) \(\mathstrut +\mathstrut 2 T_{11} \) \(\mathstrut +\mathstrut 20 \)