Properties

Label 6018.2.a.w
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 9
CM No
Inner twists 1

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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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(x^{8}\mathstrut -\mathstrut \) \(22\) \(x^{7}\mathstrut +\mathstrut \) \(20\) \(x^{6}\mathstrut +\mathstrut \) \(129\) \(x^{5}\mathstrut -\mathstrut \) \(106\) \(x^{4}\mathstrut -\mathstrut \) \(126\) \(x^{3}\mathstrut +\mathstrut \) \(48\) \(x^{2}\mathstrut +\mathstrut \) \(24\) \(x\mathstrut -\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{8} - 4 \nu^{7} + 74 \nu^{6} + 91 \nu^{5} - 543 \nu^{4} - 533 \nu^{3} + 1171 \nu^{2} + 504 \nu - 300 \)\()/26\)
\(\beta_{3}\)\(=\)\((\)\( -45 \nu^{8} + 5 \nu^{7} + 1032 \nu^{6} - 26 \nu^{5} - 6637 \nu^{4} - 234 \nu^{3} + 9908 \nu^{2} + 1840 \nu - 1900 \)\()/52\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{8} - 28 \nu^{7} + 518 \nu^{6} + 637 \nu^{5} - 3827 \nu^{4} - 3653 \nu^{3} + 8405 \nu^{2} + 2774 \nu - 1866 \)\()/26\)
\(\beta_{5}\)\(=\)\((\)\( 41 \nu^{8} - 84 \nu^{7} - 838 \nu^{6} + 1729 \nu^{5} + 3989 \nu^{4} - 9139 \nu^{3} + 1633 \nu^{2} + 3434 \nu - 1152 \)\()/26\)
\(\beta_{6}\)\(=\)\((\)\( -137 \nu^{8} + 125 \nu^{7} + 2998 \nu^{6} - 2444 \nu^{5} - 17309 \nu^{4} + 12350 \nu^{3} + 15182 \nu^{2} - 1944 \nu - 1740 \)\()/52\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{8} - 9 \nu^{7} - 242 \nu^{6} + 174 \nu^{5} + 1415 \nu^{4} - 868 \nu^{3} - 1344 \nu^{2} + 68 \nu + 160 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( -134 \nu^{8} + 129 \nu^{7} + 2924 \nu^{6} - 2535 \nu^{5} - 16766 \nu^{4} + 12857 \nu^{3} + 14011 \nu^{2} - 2162 \nu - 1414 \)\()/26\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(52\)
\(\nu^{5}\)\(=\)\(-\)\(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(29\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\) \(\beta_{2}\mathstrut +\mathstrut \) \(125\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)
\(\nu^{6}\)\(=\)\(-\)\(61\) \(\beta_{8}\mathstrut +\mathstrut \) \(58\) \(\beta_{7}\mathstrut +\mathstrut \) \(186\) \(\beta_{6}\mathstrut +\mathstrut \) \(9\) \(\beta_{5}\mathstrut -\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(146\) \(\beta_{2}\mathstrut +\mathstrut \) \(150\) \(\beta_{1}\mathstrut +\mathstrut \) \(595\)
\(\nu^{7}\)\(=\)\(-\)\(248\) \(\beta_{8}\mathstrut +\mathstrut \) \(66\) \(\beta_{7}\mathstrut +\mathstrut \) \(398\) \(\beta_{6}\mathstrut -\mathstrut \) \(139\) \(\beta_{5}\mathstrut +\mathstrut \) \(68\) \(\beta_{4}\mathstrut +\mathstrut \) \(184\) \(\beta_{3}\mathstrut -\mathstrut \) \(193\) \(\beta_{2}\mathstrut +\mathstrut \) \(1469\) \(\beta_{1}\mathstrut +\mathstrut \) \(408\)
\(\nu^{8}\)\(=\)\(-\)\(969\) \(\beta_{8}\mathstrut +\mathstrut \) \(376\) \(\beta_{7}\mathstrut +\mathstrut \) \(2438\) \(\beta_{6}\mathstrut +\mathstrut \) \(195\) \(\beta_{5}\mathstrut -\mathstrut \) \(627\) \(\beta_{4}\mathstrut +\mathstrut \) \(244\) \(\beta_{3}\mathstrut +\mathstrut \) \(1791\) \(\beta_{2}\mathstrut +\mathstrut \) \(1965\) \(\beta_{1}\mathstrut +\mathstrut \) \(7062\)

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
−3.44871
−2.94492
−0.637772
−0.621481
0.354931
0.374254
1.44979
2.83873
3.63517
−1.00000 1.00000 1.00000 −3.44871 −1.00000 −0.188516 −1.00000 1.00000 3.44871
1.2 −1.00000 1.00000 1.00000 −2.94492 −1.00000 −3.69752 −1.00000 1.00000 2.94492
1.3 −1.00000 1.00000 1.00000 −0.637772 −1.00000 −2.55112 −1.00000 1.00000 0.637772
1.4 −1.00000 1.00000 1.00000 −0.621481 −1.00000 2.55275 −1.00000 1.00000 0.621481
1.5 −1.00000 1.00000 1.00000 0.354931 −1.00000 1.61788 −1.00000 1.00000 −0.354931
1.6 −1.00000 1.00000 1.00000 0.374254 −1.00000 −3.42020 −1.00000 1.00000 −0.374254
1.7 −1.00000 1.00000 1.00000 1.44979 −1.00000 4.37806 −1.00000 1.00000 −1.44979
1.8 −1.00000 1.00000 1.00000 2.83873 −1.00000 −0.0272276 −1.00000 1.00000 −2.83873
1.9 −1.00000 1.00000 1.00000 3.63517 −1.00000 1.33590 −1.00000 1.00000 −3.63517
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} \) \(\mathstrut -\mathstrut 31 T_{7}^{7} \) \(\mathstrut +\mathstrut 284 T_{7}^{5} \) \(\mathstrut -\mathstrut 93 T_{7}^{4} \) \(\mathstrut -\mathstrut 836 T_{7}^{3} \) \(\mathstrut +\mathstrut 605 T_{7}^{2} \) \(\mathstrut +\mathstrut 164 T_{7} \) \(\mathstrut +\mathstrut 4 \)