Properties

Label 4018.2.a.bp
Level 4018
Weight 2
Character orbit 4018.a
Self dual Yes
Analytic conductor 32.084
Analytic rank 1
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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.5163008.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

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
−1.14577
0.218114
0.545336
2.56754
1.60043
−1.78566
1.00000 −3.30757 1.00000 −3.87277 −3.30757 0 1.00000 7.94005 −3.87277
1.2 1.00000 −3.26089 1.00000 1.58475 −3.26089 0 1.00000 7.63338 1.58475
1.3 1.00000 −1.93139 1.00000 −1.16627 −1.93139 0 1.00000 0.730254 −1.16627
1.4 1.00000 −0.0387751 1.00000 −2.61052 −0.0387751 0 1.00000 −2.99850 −2.61052
1.5 1.00000 1.82475 1.00000 −2.37517 1.82475 0 1.00000 0.329701 −2.37517
1.6 1.00000 2.71387 1.00000 −3.56002 2.71387 0 1.00000 4.36512 −3.56002
\(n\): 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.

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

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