Properties

Label 6018.2.a.t
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 8
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( -1 - \beta_{4} ) q^{5} \) \(- q^{6}\) \( + ( -\beta_{1} + \beta_{4} ) q^{7} \) \(- q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + ( -1 - \beta_{4} ) q^{5} \) \(- q^{6}\) \( + ( -\beta_{1} + \beta_{4} ) q^{7} \) \(- q^{8}\) \(+ q^{9}\) \( + ( 1 + \beta_{4} ) q^{10} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} ) q^{11} \) \(+ q^{12}\) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{13} \) \( + ( \beta_{1} - \beta_{4} ) q^{14} \) \( + ( -1 - \beta_{4} ) q^{15} \) \(+ q^{16}\) \(+ q^{17}\) \(- q^{18}\) \( + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} \) \( + ( -1 - \beta_{4} ) q^{20} \) \( + ( -\beta_{1} + \beta_{4} ) q^{21} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} ) q^{22} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{23} \) \(- q^{24}\) \( + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{25} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{26} \) \(+ q^{27}\) \( + ( -\beta_{1} + \beta_{4} ) q^{28} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{29} \) \( + ( 1 + \beta_{4} ) q^{30} \) \( + ( \beta_{3} + \beta_{4} - \beta_{6} ) q^{31} \) \(- q^{32}\) \( + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} ) q^{33} \) \(- q^{34}\) \( + ( -4 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{35} \) \(+ q^{36}\) \( + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{37} \) \( + ( -\beta_{4} - \beta_{6} - \beta_{7} ) q^{38} \) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{39} \) \( + ( 1 + \beta_{4} ) q^{40} \) \( + ( -2 + 2 \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} \) \( + ( \beta_{1} - \beta_{4} ) q^{42} \) \( + ( 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{43} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} ) q^{44} \) \( + ( -1 - \beta_{4} ) q^{45} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{46} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{47} \) \(+ q^{48}\) \( + ( 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{49} \) \( + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{50} \) \(+ q^{51}\) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{52} \) \( + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{53} \) \(- q^{54}\) \( + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{55} \) \( + ( \beta_{1} - \beta_{4} ) q^{56} \) \( + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{57} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{58} \) \(- q^{59}\) \( + ( -1 - \beta_{4} ) q^{60} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{7} ) q^{61} \) \( + ( -\beta_{3} - \beta_{4} + \beta_{6} ) q^{62} \) \( + ( -\beta_{1} + \beta_{4} ) q^{63} \) \(+ q^{64}\) \( + ( 1 - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{65} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} ) q^{66} \) \( + ( 3 - \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{67} \) \(+ q^{68}\) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{69} \) \( + ( 4 - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{70} \) \( + ( -3 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{71} \) \(- q^{72}\) \( + ( 2 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} \) \( + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{7} ) q^{74} \) \( + ( 1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{75} \) \( + ( \beta_{4} + \beta_{6} + \beta_{7} ) q^{76} \) \( + ( -1 - 4 \beta_{1} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{77} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{78} \) \( + ( -7 - \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{79} \) \( + ( -1 - \beta_{4} ) q^{80} \) \(+ q^{81}\) \( + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{82} \) \( + ( -2 - \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{83} \) \( + ( -\beta_{1} + \beta_{4} ) q^{84} \) \( + ( -1 - \beta_{4} ) q^{85} \) \( + ( -2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{86} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{87} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} ) q^{88} \) \( + ( 1 - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{89} \) \( + ( 1 + \beta_{4} ) q^{90} \) \( + ( 2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{91} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} ) q^{92} \) \( + ( \beta_{3} + \beta_{4} - \beta_{6} ) q^{93} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} ) q^{94} \) \( + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{95} \) \(- q^{96}\) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{97} \) \( + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{98} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 8q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 8q^{3} \) \(\mathstrut +\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 8q^{8} \) \(\mathstrut +\mathstrut 8q^{9} \) \(\mathstrut +\mathstrut 6q^{10} \) \(\mathstrut +\mathstrut q^{11} \) \(\mathstrut +\mathstrut 8q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 8q^{16} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 8q^{18} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 6q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut -\mathstrut q^{22} \) \(\mathstrut -\mathstrut 11q^{23} \) \(\mathstrut -\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 6q^{25} \) \(\mathstrut +\mathstrut 2q^{26} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut -\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 6q^{30} \) \(\mathstrut -\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 8q^{32} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut -\mathstrut 8q^{34} \) \(\mathstrut -\mathstrut 28q^{35} \) \(\mathstrut +\mathstrut 8q^{36} \) \(\mathstrut -\mathstrut 22q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 6q^{40} \) \(\mathstrut -\mathstrut 19q^{41} \) \(\mathstrut +\mathstrut 4q^{42} \) \(\mathstrut -\mathstrut 5q^{43} \) \(\mathstrut +\mathstrut q^{44} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 11q^{46} \) \(\mathstrut -\mathstrut 26q^{47} \) \(\mathstrut +\mathstrut 8q^{48} \) \(\mathstrut -\mathstrut 6q^{50} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut -\mathstrut 2q^{52} \) \(\mathstrut -\mathstrut 21q^{53} \) \(\mathstrut -\mathstrut 8q^{54} \) \(\mathstrut -\mathstrut 13q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut +\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 8q^{59} \) \(\mathstrut -\mathstrut 6q^{60} \) \(\mathstrut +\mathstrut 9q^{61} \) \(\mathstrut +\mathstrut 9q^{62} \) \(\mathstrut -\mathstrut 4q^{63} \) \(\mathstrut +\mathstrut 8q^{64} \) \(\mathstrut +\mathstrut 14q^{65} \) \(\mathstrut -\mathstrut q^{66} \) \(\mathstrut +\mathstrut 26q^{67} \) \(\mathstrut +\mathstrut 8q^{68} \) \(\mathstrut -\mathstrut 11q^{69} \) \(\mathstrut +\mathstrut 28q^{70} \) \(\mathstrut -\mathstrut 14q^{71} \) \(\mathstrut -\mathstrut 8q^{72} \) \(\mathstrut +\mathstrut 17q^{73} \) \(\mathstrut +\mathstrut 22q^{74} \) \(\mathstrut +\mathstrut 6q^{75} \) \(\mathstrut +\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 18q^{77} \) \(\mathstrut +\mathstrut 2q^{78} \) \(\mathstrut -\mathstrut 39q^{79} \) \(\mathstrut -\mathstrut 6q^{80} \) \(\mathstrut +\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut 19q^{82} \) \(\mathstrut -\mathstrut 11q^{83} \) \(\mathstrut -\mathstrut 4q^{84} \) \(\mathstrut -\mathstrut 6q^{85} \) \(\mathstrut +\mathstrut 5q^{86} \) \(\mathstrut -\mathstrut 12q^{87} \) \(\mathstrut -\mathstrut q^{88} \) \(\mathstrut +\mathstrut 6q^{90} \) \(\mathstrut +\mathstrut 11q^{91} \) \(\mathstrut -\mathstrut 11q^{92} \) \(\mathstrut -\mathstrut 9q^{93} \) \(\mathstrut +\mathstrut 26q^{94} \) \(\mathstrut -\mathstrut 15q^{95} \) \(\mathstrut -\mathstrut 8q^{96} \) \(\mathstrut +\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(15\) \(x^{6}\mathstrut +\mathstrut \) \(14\) \(x^{5}\mathstrut +\mathstrut \) \(84\) \(x^{4}\mathstrut +\mathstrut \) \(9\) \(x^{3}\mathstrut -\mathstrut \) \(158\) \(x^{2}\mathstrut -\mathstrut \) \(142\) \(x\mathstrut -\mathstrut \) \(35\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{7} - 4 \nu^{6} - 8 \nu^{5} + 33 \nu^{4} + 27 \nu^{3} - 66 \nu^{2} - 53 \nu - 6 \)
\(\beta_{4}\)\(=\)\( 3 \nu^{7} - 11 \nu^{6} - 27 \nu^{5} + 89 \nu^{4} + 105 \nu^{3} - 166 \nu^{2} - 200 \nu - 50 \)
\(\beta_{5}\)\(=\)\( 4 \nu^{7} - 14 \nu^{6} - 39 \nu^{5} + 115 \nu^{4} + 163 \nu^{3} - 214 \nu^{2} - 311 \nu - 87 \)
\(\beta_{6}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} + 78 \nu^{5} - 229 \nu^{4} - 328 \nu^{3} + 420 \nu^{2} + 630 \nu + 188 \)
\(\beta_{7}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} + 78 \nu^{5} - 229 \nu^{4} - 329 \nu^{3} + 422 \nu^{2} + 635 \nu + 185 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)
\(\nu^{5}\)\(=\)\(-\)\(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(60\) \(\beta_{1}\mathstrut +\mathstrut \) \(64\)
\(\nu^{6}\)\(=\)\(-\)\(35\) \(\beta_{7}\mathstrut +\mathstrut \) \(54\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(148\) \(\beta_{2}\mathstrut +\mathstrut \) \(161\) \(\beta_{1}\mathstrut +\mathstrut \) \(256\)
\(\nu^{7}\)\(=\)\(-\)\(151\) \(\beta_{7}\mathstrut +\mathstrut \) \(218\) \(\beta_{6}\mathstrut +\mathstrut \) \(114\) \(\beta_{5}\mathstrut +\mathstrut \) \(44\) \(\beta_{4}\mathstrut -\mathstrut \) \(51\) \(\beta_{3}\mathstrut +\mathstrut \) \(496\) \(\beta_{2}\mathstrut +\mathstrut \) \(592\) \(\beta_{1}\mathstrut +\mathstrut \) \(747\)

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
2.12828
−1.50609
−0.502246
3.51784
−1.48295
2.70183
−2.13380
−0.722864
−1.00000 1.00000 1.00000 −3.78816 −1.00000 0.659879 −1.00000 1.00000 3.78816
1.2 −1.00000 1.00000 1.00000 −3.00979 −1.00000 3.51588 −1.00000 1.00000 3.00979
1.3 −1.00000 1.00000 1.00000 −2.59811 −1.00000 2.10035 −1.00000 1.00000 2.59811
1.4 −1.00000 1.00000 1.00000 −1.98195 −1.00000 −2.53589 −1.00000 1.00000 1.98195
1.5 −1.00000 1.00000 1.00000 0.137387 −1.00000 0.345564 −1.00000 1.00000 −0.137387
1.6 −1.00000 1.00000 1.00000 0.876977 −1.00000 −4.57880 −1.00000 1.00000 −0.876977
1.7 −1.00000 1.00000 1.00000 1.28626 −1.00000 −0.152464 −1.00000 1.00000 −1.28626
1.8 −1.00000 1.00000 1.00000 3.07738 −1.00000 −3.35452 −1.00000 1.00000 −3.07738
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(-1\)
\(59\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)