Properties

Label 4018.2.a.bn
Level 4018
Weight 2
Character orbit 4018.a
Self dual yes
Analytic conductor 32.084
Analytic rank 0
Dimension 6
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: \(6\)
Coefficient field: 6.6.52046292.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 574)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{3} q^{3} + q^{4} + \beta_{4} q^{5} -\beta_{3} q^{6} - q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{3} q^{3} + q^{4} + \beta_{4} q^{5} -\beta_{3} q^{6} - q^{8} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{9} -\beta_{4} q^{10} + ( 1 + \beta_{2} + \beta_{5} ) q^{11} + \beta_{3} q^{12} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{13} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{15} + q^{16} + ( \beta_{3} - \beta_{5} ) q^{17} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{18} + ( \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{19} + \beta_{4} q^{20} + ( -1 - \beta_{2} - \beta_{5} ) q^{22} + ( 3 - \beta_{2} - \beta_{4} ) q^{23} -\beta_{3} q^{24} + ( -\beta_{2} + \beta_{3} + \beta_{5} ) q^{25} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{26} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{27} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{30} + ( \beta_{3} + \beta_{4} + \beta_{5} ) q^{31} - q^{32} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{33} + ( -\beta_{3} + \beta_{5} ) q^{34} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{36} + ( 1 + 3 \beta_{2} - \beta_{3} ) q^{37} + ( -\beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{38} + ( -2 - \beta_{3} ) q^{39} -\beta_{4} q^{40} - q^{41} + ( 3 - 2 \beta_{1} + 2 \beta_{5} ) q^{43} + ( 1 + \beta_{2} + \beta_{5} ) q^{44} + ( -4 + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{45} + ( -3 + \beta_{2} + \beta_{4} ) q^{46} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + \beta_{3} q^{48} + ( \beta_{2} - \beta_{3} - \beta_{5} ) q^{50} + ( 2 - \beta_{3} - \beta_{4} ) q^{51} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{52} + ( 3 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{53} + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{54} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{55} + ( 2 \beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{57} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{58} + ( -4 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{59} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{60} + ( 4 + 2 \beta_{3} + \beta_{5} ) q^{61} + ( -\beta_{3} - \beta_{4} - \beta_{5} ) q^{62} + q^{64} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( -3 - 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{66} + ( -1 + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( \beta_{3} - \beta_{5} ) q^{68} + ( -1 - 4 \beta_{1} + \beta_{2} + 6 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{69} + ( 2 - 5 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{71} + ( -1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{72} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{73} + ( -1 - 3 \beta_{2} + \beta_{3} ) q^{74} + ( 5 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{75} + ( \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{76} + ( 2 + \beta_{3} ) q^{78} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{79} + \beta_{4} q^{80} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{81} + q^{82} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{83} + ( -1 - \beta_{2} - \beta_{3} ) q^{85} + ( -3 + 2 \beta_{1} - 2 \beta_{5} ) q^{86} + ( -2 + 2 \beta_{1} + \beta_{4} ) q^{87} + ( -1 - \beta_{2} - \beta_{5} ) q^{88} + ( 4 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{89} + ( 4 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{90} + ( 3 - \beta_{2} - \beta_{4} ) q^{92} + ( 6 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{93} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{94} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{95} -\beta_{3} q^{96} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{97} + ( 2 - 4 \beta_{1} + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} - q^{3} + 6q^{4} + q^{6} - 6q^{8} + 5q^{9} + O(q^{10}) \) \( 6q - 6q^{2} - q^{3} + 6q^{4} + q^{6} - 6q^{8} + 5q^{9} + q^{11} - q^{12} - 4q^{13} + 2q^{15} + 6q^{16} + q^{17} - 5q^{18} + 3q^{19} - q^{22} + 21q^{23} + q^{24} + 4q^{26} - 13q^{27} + 5q^{29} - 2q^{30} - 3q^{31} - 6q^{32} + 19q^{33} - q^{34} + 5q^{36} - 2q^{37} - 3q^{38} - 11q^{39} - 6q^{41} + 12q^{43} + q^{44} - 28q^{45} - 21q^{46} + 18q^{47} - q^{48} + 13q^{51} - 4q^{52} + 14q^{53} + 13q^{54} + 7q^{55} + 5q^{57} - 5q^{58} - 16q^{59} + 2q^{60} + 20q^{61} + 3q^{62} + 6q^{64} + 18q^{65} - 19q^{66} - 13q^{67} + q^{68} - 15q^{69} + 11q^{71} - 5q^{72} - q^{73} + 2q^{74} + 23q^{75} + 3q^{76} + 11q^{78} + 19q^{79} - 6q^{81} + 6q^{82} - 15q^{83} - 2q^{85} - 12q^{86} - 10q^{87} - q^{88} + 14q^{89} + 28q^{90} + 21q^{92} + 35q^{93} - 18q^{94} + 24q^{95} + q^{96} - q^{97} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 10 x^{4} + 9 x^{3} + 24 x^{2} - 18 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + \nu^{3} - 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + \nu^{4} - 7 \nu^{3} - 5 \nu^{2} + 8 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - \beta_{3} + 6 \beta_{2} + 21\)
\(\nu^{5}\)\(=\)\(\beta_{5} - \beta_{4} + 8 \beta_{3} - \beta_{2} + 27 \beta_{1} + 1\)

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
0.643548
−2.43859
0.0605469
2.28320
2.38442
−1.93313
−1.00000 −2.95121 1.00000 −2.26461 2.95121 0 −1.00000 5.70965 2.26461
1.2 −1.00000 −2.30861 1.00000 0.374437 2.30861 0 −1.00000 2.32969 −0.374437
1.3 −1.00000 −0.302513 1.00000 2.67551 0.302513 0 −1.00000 −2.90849 −2.67551
1.4 −1.00000 0.486321 1.00000 −0.616311 −0.486321 0 −1.00000 −2.76349 0.616311
1.5 −1.00000 1.63447 1.00000 2.84627 −1.63447 0 −1.00000 −0.328504 −2.84627
1.6 −1.00000 2.44154 1.00000 −3.01529 −2.44154 0 −1.00000 2.96114 3.01529
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4018.2.a.bn 6
7.b odd 2 1 4018.2.a.bo 6
7.d odd 6 2 574.2.e.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.g 12 7.d odd 6 2
4018.2.a.bn 6 1.a even 1 1 trivial
4018.2.a.bo 6 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(41\) \(1\)

Hecke kernels

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

\( T_{3}^{6} + T_{3}^{5} - 11 T_{3}^{4} - 5 T_{3}^{3} + 30 T_{3}^{2} - 4 T_{3} - 4 \)
\( T_{5}^{6} - 15 T_{5}^{4} - T_{5}^{3} + 56 T_{5}^{2} + 12 T_{5} - 12 \)
\( T_{11}^{6} - T_{11}^{5} - 38 T_{11}^{4} + 7 T_{11}^{3} + 320 T_{11}^{2} - 48 T_{11} - 684 \)