Properties

Label 4018.2.a.be
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: 3.3.404.1
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.

\(f(q)\) \(=\) \( q - q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{5} + ( \beta_{1} - \beta_{2} ) q^{6} - q^{8} + ( 3 - 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -\beta_{1} + \beta_{2} ) q^{3} + q^{4} + ( 1 - \beta_{1} ) q^{5} + ( \beta_{1} - \beta_{2} ) q^{6} - q^{8} + ( 3 - 2 \beta_{1} ) q^{9} + ( -1 + \beta_{1} ) q^{10} + ( 1 - \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{12} + ( -\beta_{1} - \beta_{2} ) q^{13} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{15} + q^{16} + ( 4 + \beta_{1} - \beta_{2} ) q^{17} + ( -3 + 2 \beta_{1} ) q^{18} + ( 2 - \beta_{1} - \beta_{2} ) q^{19} + ( 1 - \beta_{1} ) q^{20} + ( -1 + \beta_{2} ) q^{22} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{23} + ( \beta_{1} - \beta_{2} ) q^{24} + ( -1 - \beta_{1} + \beta_{2} ) q^{25} + ( \beta_{1} + \beta_{2} ) q^{26} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( 1 - 4 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{30} + ( 4 - 2 \beta_{2} ) q^{31} - q^{32} + ( -4 + 2 \beta_{2} ) q^{33} + ( -4 - \beta_{1} + \beta_{2} ) q^{34} + ( 3 - 2 \beta_{1} ) q^{36} + ( 6 + 2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( -2 + \beta_{1} + \beta_{2} ) q^{38} + ( -2 + 2 \beta_{2} ) q^{39} + ( -1 + \beta_{1} ) q^{40} - q^{41} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 1 - \beta_{2} ) q^{44} + ( 9 - 3 \beta_{1} + 2 \beta_{2} ) q^{45} + ( 2 - \beta_{1} + 3 \beta_{2} ) q^{46} + ( 8 - \beta_{1} - \beta_{2} ) q^{47} + ( -\beta_{1} + \beta_{2} ) q^{48} + ( 1 + \beta_{1} - \beta_{2} ) q^{50} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{51} + ( -\beta_{1} - \beta_{2} ) q^{52} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{53} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{54} + ( 2 + \beta_{1} - \beta_{2} ) q^{55} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{57} + ( -1 + 4 \beta_{1} - \beta_{2} ) q^{58} + ( 3 + 3 \beta_{1} ) q^{59} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{60} + ( -7 + \beta_{1} - 4 \beta_{2} ) q^{61} + ( -4 + 2 \beta_{2} ) q^{62} + q^{64} + ( 4 + 2 \beta_{1} ) q^{65} + ( 4 - 2 \beta_{2} ) q^{66} + ( 3 - 2 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 4 + \beta_{1} - \beta_{2} ) q^{68} + ( -14 + 6 \beta_{1} ) q^{69} + ( -2 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -3 + 2 \beta_{1} ) q^{72} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -6 - 2 \beta_{1} + 2 \beta_{2} ) q^{74} + ( 6 - \beta_{1} - \beta_{2} ) q^{75} + ( 2 - \beta_{1} - \beta_{2} ) q^{76} + ( 2 - 2 \beta_{2} ) q^{78} + ( 4 - 4 \beta_{1} ) q^{79} + ( 1 - \beta_{1} ) q^{80} + ( 3 - 2 \beta_{1} + 4 \beta_{2} ) q^{81} + q^{82} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 4 + 2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 12 - 6 \beta_{1} + 4 \beta_{2} ) q^{87} + ( -1 + \beta_{2} ) q^{88} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -9 + 3 \beta_{1} - 2 \beta_{2} ) q^{90} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{92} + ( -8 - 2 \beta_{1} + 6 \beta_{2} ) q^{93} + ( -8 + \beta_{1} + \beta_{2} ) q^{94} + 6 q^{95} + ( \beta_{1} - \beta_{2} ) q^{96} + ( -6 + 4 \beta_{1} - 4 \beta_{2} ) q^{97} + ( 5 + 2 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + 7q^{9} - 2q^{10} + 2q^{11} - 2q^{13} + 6q^{15} + 3q^{16} + 12q^{17} - 7q^{18} + 4q^{19} + 2q^{20} - 2q^{22} - 8q^{23} - 3q^{25} + 2q^{26} + 12q^{27} - 6q^{30} + 10q^{31} - 3q^{32} - 10q^{33} - 12q^{34} + 7q^{36} + 18q^{37} - 4q^{38} - 4q^{39} - 2q^{40} - 3q^{41} - 12q^{43} + 2q^{44} + 26q^{45} + 8q^{46} + 22q^{47} + 3q^{50} - 16q^{51} - 2q^{52} - 10q^{53} - 12q^{54} + 6q^{55} - 4q^{57} + 12q^{59} + 6q^{60} - 24q^{61} - 10q^{62} + 3q^{64} + 14q^{65} + 10q^{66} + 4q^{67} + 12q^{68} - 36q^{69} - 4q^{71} - 7q^{72} + 12q^{73} - 18q^{74} + 16q^{75} + 4q^{76} + 4q^{78} + 8q^{79} + 2q^{80} + 11q^{81} + 3q^{82} - 24q^{83} + 2q^{85} + 12q^{86} + 34q^{87} - 2q^{88} + 6q^{89} - 26q^{90} - 8q^{92} - 20q^{93} - 22q^{94} + 18q^{95} - 18q^{97} + 14q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)

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
−0.210756
2.86620
−1.65544
−1.00000 −2.53407 1.00000 1.21076 2.53407 0 −1.00000 3.42151 −1.21076
1.2 −1.00000 −0.517304 1.00000 −1.86620 0.517304 0 −1.00000 −2.73240 1.86620
1.3 −1.00000 3.05137 1.00000 2.65544 −3.05137 0 −1.00000 6.31088 −2.65544
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

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