Properties

Label 4018.2.a.bo
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: \( 1 \)
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.

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

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

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

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
−1.93313
2.38442
2.28320
0.0605469
−2.43859
0.643548
−1.00000 −2.44154 1.00000 3.01529 2.44154 0 −1.00000 2.96114 −3.01529
1.2 −1.00000 −1.63447 1.00000 −2.84627 1.63447 0 −1.00000 −0.328504 2.84627
1.3 −1.00000 −0.486321 1.00000 0.616311 0.486321 0 −1.00000 −2.76349 −0.616311
1.4 −1.00000 0.302513 1.00000 −2.67551 −0.302513 0 −1.00000 −2.90849 2.67551
1.5 −1.00000 2.30861 1.00000 −0.374437 −2.30861 0 −1.00000 2.32969 0.374437
1.6 −1.00000 2.95121 1.00000 2.26461 −2.95121 0 −1.00000 5.70965 −2.26461
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4018))\):

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