Properties

Label 4018.2.a.w
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: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−1.00000 1.00000 1.00000 −2.23607 −1.00000 0 −1.00000 −2.00000 2.23607
1.2 −1.00000 1.00000 1.00000 2.23607 −1.00000 0 −1.00000 −2.00000 −2.23607
\(n\): 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
\(2\) \(1\)
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{5}^{2} - 5 \)
\( T_{11}^{2} - 20 \)