Properties

Label 4018.2.a.bq
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: 6.6.5163008.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{3} + q^{4} + ( 2 + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{6} + q^{8} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{3} + q^{4} + ( 2 + \beta_{4} ) q^{5} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{6} + q^{8} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{9} + ( 2 + \beta_{4} ) q^{10} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{12} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{13} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{15} + q^{16} + ( 3 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{17} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{18} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{19} + ( 2 + \beta_{4} ) q^{20} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{22} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{24} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{25} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{26} + ( 6 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{27} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{29} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{30} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{31} + q^{32} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 4 \beta_{5} ) q^{33} + ( 3 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{34} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{36} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{38} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{39} + ( 2 + \beta_{4} ) q^{40} - q^{41} + ( 1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{43} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{44} + ( 7 - 5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{45} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{46} + ( -3 + \beta_{1} - 4 \beta_{3} - \beta_{4} ) q^{47} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{48} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{50} + ( -1 + 2 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{51} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{52} + ( -3 - 3 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 6 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{54} + ( -5 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{55} + ( -5 - \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{57} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{58} + ( 2 - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + 6 \beta_{5} ) q^{59} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{60} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{61} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{62} + q^{64} + ( -2 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -4 - 2 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + 4 \beta_{5} ) q^{66} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 3 - \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{68} + ( 10 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{69} + ( 3 - 3 \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{71} + ( 4 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{72} + ( 3 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} ) q^{74} + ( 1 + 3 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{75} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{76} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{78} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 2 + \beta_{4} ) q^{80} + ( 10 + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 5 \beta_{5} ) q^{81} - q^{82} + ( 4 + 4 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 9 - 4 \beta_{1} - \beta_{2} + 5 \beta_{4} - 3 \beta_{5} ) q^{85} + ( 1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{86} + ( 4 - 2 \beta_{1} + 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{87} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{88} + ( 2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{89} + ( 7 - 5 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{90} + ( 2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{92} + ( -5 + 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{93} + ( -3 + \beta_{1} - 4 \beta_{3} - \beta_{4} ) q^{94} + ( 2 + 5 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} ) q^{95} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{96} + ( 8 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{97} + ( -16 + 3 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 4q^{3} + 6q^{4} + 12q^{5} + 4q^{6} + 6q^{8} + 18q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 4q^{3} + 6q^{4} + 12q^{5} + 4q^{6} + 6q^{8} + 18q^{9} + 12q^{10} + 4q^{12} + 8q^{13} - 4q^{15} + 6q^{16} + 16q^{17} + 18q^{18} + 12q^{19} + 12q^{20} + 12q^{23} + 4q^{24} + 14q^{25} + 8q^{26} + 28q^{27} - 4q^{30} - 4q^{31} + 6q^{32} - 20q^{33} + 16q^{34} + 18q^{36} - 24q^{37} + 12q^{38} + 4q^{39} + 12q^{40} - 6q^{41} + 4q^{43} + 28q^{45} + 12q^{46} - 16q^{47} + 4q^{48} + 14q^{50} - 16q^{51} + 8q^{52} - 16q^{53} + 28q^{54} - 12q^{55} - 28q^{57} + 16q^{59} - 4q^{60} + 32q^{61} - 4q^{62} + 6q^{64} - 4q^{65} - 20q^{66} - 20q^{67} + 16q^{68} + 56q^{69} + 12q^{71} + 18q^{72} + 16q^{73} - 24q^{74} - 4q^{75} + 12q^{76} + 4q^{78} + 12q^{80} + 42q^{81} - 6q^{82} + 32q^{83} + 48q^{85} + 4q^{86} + 20q^{87} + 8q^{89} + 28q^{90} + 12q^{92} - 12q^{93} - 16q^{94} + 28q^{95} + 4q^{96} + 8q^{97} - 76q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 5 x^{2} - 6 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 7 \nu^{2} + 6 \nu - 3 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} + 5 \nu - 5 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{5} - 3 \nu^{4} - 11 \nu^{3} + 10 \nu^{2} + 13 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{5} + 5 \beta_{4} - 7 \beta_{3} + \beta_{2} + 8 \beta_{1} + 8\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + 8 \beta_{4} - 11 \beta_{3} + 7 \beta_{2} + 28 \beta_{1} + 10\)

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
−1.78566
1.60043
2.56754
0.545336
0.218114
−1.14577
1.00000 −2.71387 1.00000 3.56002 −2.71387 0 1.00000 4.36512 3.56002
1.2 1.00000 −1.82475 1.00000 2.37517 −1.82475 0 1.00000 0.329701 2.37517
1.3 1.00000 0.0387751 1.00000 2.61052 0.0387751 0 1.00000 −2.99850 2.61052
1.4 1.00000 1.93139 1.00000 1.16627 1.93139 0 1.00000 0.730254 1.16627
1.5 1.00000 3.26089 1.00000 −1.58475 3.26089 0 1.00000 7.63338 −1.58475
1.6 1.00000 3.30757 1.00000 3.87277 3.30757 0 1.00000 7.94005 3.87277
\(n\): 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.bq yes 6
7.b odd 2 1 4018.2.a.bp 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4018.2.a.bp 6 7.b odd 2 1
4018.2.a.bq yes 6 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

\( T_{3}^{6} - 4 T_{3}^{5} - 10 T_{3}^{4} + 44 T_{3}^{3} + 20 T_{3}^{2} - 104 T_{3} + 4 \)
\( T_{5}^{6} - 12 T_{5}^{5} + 50 T_{5}^{4} - 68 T_{5}^{3} - 68 T_{5}^{2} + 248 T_{5} - 158 \)
\( T_{11}^{6} - 50 T_{11}^{4} + 8 T_{11}^{3} + 498 T_{11}^{2} - 300 T_{11} - 398 \)