Properties

Label 6018.2.a.z
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 11
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(4\) \(x^{10}\mathstrut -\mathstrut \) \(27\) \(x^{9}\mathstrut +\mathstrut \) \(117\) \(x^{8}\mathstrut +\mathstrut \) \(200\) \(x^{7}\mathstrut -\mathstrut \) \(1023\) \(x^{6}\mathstrut -\mathstrut \) \(484\) \(x^{5}\mathstrut +\mathstrut \) \(3403\) \(x^{4}\mathstrut +\mathstrut \) \(562\) \(x^{3}\mathstrut -\mathstrut \) \(4372\) \(x^{2}\mathstrut -\mathstrut \) \(692\) \(x\mathstrut +\mathstrut \) \(1200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(68377\) \(\nu^{10}\mathstrut +\mathstrut \) \(100902\) \(\nu^{9}\mathstrut +\mathstrut \) \(1972995\) \(\nu^{8}\mathstrut -\mathstrut \) \(2438247\) \(\nu^{7}\mathstrut -\mathstrut \) \(16451050\) \(\nu^{6}\mathstrut +\mathstrut \) \(12213703\) \(\nu^{5}\mathstrut +\mathstrut \) \(41682174\) \(\nu^{4}\mathstrut -\mathstrut \) \(1530071\) \(\nu^{3}\mathstrut -\mathstrut \) \(16641720\) \(\nu^{2}\mathstrut -\mathstrut \) \(30704680\) \(\nu\mathstrut -\mathstrut \) \(6530016\)\()/9881348\)
\(\beta_{3}\)\(=\)\((\)\(478220\) \(\nu^{10}\mathstrut -\mathstrut \) \(806963\) \(\nu^{9}\mathstrut -\mathstrut \) \(14837488\) \(\nu^{8}\mathstrut +\mathstrut \) \(21958815\) \(\nu^{7}\mathstrut +\mathstrut \) \(147883587\) \(\nu^{6}\mathstrut -\mathstrut \) \(155628260\) \(\nu^{5}\mathstrut -\mathstrut \) \(599066657\) \(\nu^{4}\mathstrut +\mathstrut \) \(301247970\) \(\nu^{3}\mathstrut +\mathstrut \) \(966550127\) \(\nu^{2}\mathstrut +\mathstrut \) \(17946790\) \(\nu\mathstrut -\mathstrut \) \(277474058\)\()/4940674\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(690399\) \(\nu^{10}\mathstrut +\mathstrut \) \(1061578\) \(\nu^{9}\mathstrut +\mathstrut \) \(21837033\) \(\nu^{8}\mathstrut -\mathstrut \) \(28769331\) \(\nu^{7}\mathstrut -\mathstrut \) \(224589340\) \(\nu^{6}\mathstrut +\mathstrut \) \(200728579\) \(\nu^{5}\mathstrut +\mathstrut \) \(944194472\) \(\nu^{4}\mathstrut -\mathstrut \) \(362823799\) \(\nu^{3}\mathstrut -\mathstrut \) \(1555875710\) \(\nu^{2}\mathstrut -\mathstrut \) \(115397832\) \(\nu\mathstrut +\mathstrut \) \(437533206\)\()/4940674\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(3156869\) \(\nu^{10}\mathstrut +\mathstrut \) \(5380560\) \(\nu^{9}\mathstrut +\mathstrut \) \(98175115\) \(\nu^{8}\mathstrut -\mathstrut \) \(145767653\) \(\nu^{7}\mathstrut -\mathstrut \) \(981973340\) \(\nu^{6}\mathstrut +\mathstrut \) \(1024667347\) \(\nu^{5}\mathstrut +\mathstrut \) \(3999281264\) \(\nu^{4}\mathstrut -\mathstrut \) \(1934315723\) \(\nu^{3}\mathstrut -\mathstrut \) \(6482502258\) \(\nu^{2}\mathstrut -\mathstrut \) \(230680552\) \(\nu\mathstrut +\mathstrut \) \(1824506128\)\()/9881348\)
\(\beta_{6}\)\(=\)\((\)\(1183567\) \(\nu^{10}\mathstrut -\mathstrut \) \(2051153\) \(\nu^{9}\mathstrut -\mathstrut \) \(36518360\) \(\nu^{8}\mathstrut +\mathstrut \) \(55439219\) \(\nu^{7}\mathstrut +\mathstrut \) \(359948269\) \(\nu^{6}\mathstrut -\mathstrut \) \(387860873\) \(\nu^{5}\mathstrut -\mathstrut \) \(1432388663\) \(\nu^{4}\mathstrut +\mathstrut \) \(728576639\) \(\nu^{3}\mathstrut +\mathstrut \) \(2258150935\) \(\nu^{2}\mathstrut +\mathstrut \) \(65727972\) \(\nu\mathstrut -\mathstrut \) \(599619469\)\()/2470337\)
\(\beta_{7}\)\(=\)\((\)\(4973375\) \(\nu^{10}\mathstrut -\mathstrut \) \(8328764\) \(\nu^{9}\mathstrut -\mathstrut \) \(154431069\) \(\nu^{8}\mathstrut +\mathstrut \) \(224826723\) \(\nu^{7}\mathstrut +\mathstrut \) \(1539701372\) \(\nu^{6}\mathstrut -\mathstrut \) \(1563255225\) \(\nu^{5}\mathstrut -\mathstrut \) \(6227037308\) \(\nu^{4}\mathstrut +\mathstrut \) \(2853612557\) \(\nu^{3}\mathstrut +\mathstrut \) \(9978785870\) \(\nu^{2}\mathstrut +\mathstrut \) \(558006780\) \(\nu\mathstrut -\mathstrut \) \(2734225624\)\()/9881348\)
\(\beta_{8}\)\(=\)\((\)\(1707617\) \(\nu^{10}\mathstrut -\mathstrut \) \(2986333\) \(\nu^{9}\mathstrut -\mathstrut \) \(52746178\) \(\nu^{8}\mathstrut +\mathstrut \) \(80817374\) \(\nu^{7}\mathstrut +\mathstrut \) \(521178581\) \(\nu^{6}\mathstrut -\mathstrut \) \(567120079\) \(\nu^{5}\mathstrut -\mathstrut \) \(2085306979\) \(\nu^{4}\mathstrut +\mathstrut \) \(1067360641\) \(\nu^{3}\mathstrut +\mathstrut \) \(3317249635\) \(\nu^{2}\mathstrut +\mathstrut \) \(108639160\) \(\nu\mathstrut -\mathstrut \) \(894863368\)\()/2470337\)
\(\beta_{9}\)\(=\)\((\)\(10702739\) \(\nu^{10}\mathstrut -\mathstrut \) \(18943456\) \(\nu^{9}\mathstrut -\mathstrut \) \(330397033\) \(\nu^{8}\mathstrut +\mathstrut \) \(513011107\) \(\nu^{7}\mathstrut +\mathstrut \) \(3262430232\) \(\nu^{6}\mathstrut -\mathstrut \) \(3608727329\) \(\nu^{5}\mathstrut -\mathstrut \) \(13063793500\) \(\nu^{4}\mathstrut +\mathstrut \) \(6833246697\) \(\nu^{3}\mathstrut +\mathstrut \) \(20866262830\) \(\nu^{2}\mathstrut +\mathstrut \) \(615967628\) \(\nu\mathstrut -\mathstrut \) \(5650580696\)\()/9881348\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(20411677\) \(\nu^{10}\mathstrut +\mathstrut \) \(35419276\) \(\nu^{9}\mathstrut +\mathstrut \) \(632398343\) \(\nu^{8}\mathstrut -\mathstrut \) \(959470865\) \(\nu^{7}\mathstrut -\mathstrut \) \(6282554076\) \(\nu^{6}\mathstrut +\mathstrut \) \(6746297163\) \(\nu^{5}\mathstrut +\mathstrut \) \(25332890104\) \(\nu^{4}\mathstrut -\mathstrut \) \(12726418695\) \(\nu^{3}\mathstrut -\mathstrut \) \(40564710178\) \(\nu^{2}\mathstrut -\mathstrut \) \(1489912936\) \(\nu\mathstrut +\mathstrut \) \(10921660816\)\()/9881348\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(20\) \(\beta_{9}\mathstrut +\mathstrut \) \(36\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(20\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(66\)
\(\nu^{5}\)\(=\)\(19\) \(\beta_{10}\mathstrut +\mathstrut \) \(44\) \(\beta_{9}\mathstrut -\mathstrut \) \(26\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(37\) \(\beta_{6}\mathstrut +\mathstrut \) \(29\) \(\beta_{5}\mathstrut -\mathstrut \) \(83\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(124\) \(\beta_{1}\mathstrut -\mathstrut \) \(40\)
\(\nu^{6}\)\(=\)\(-\)\(25\) \(\beta_{10}\mathstrut -\mathstrut \) \(337\) \(\beta_{9}\mathstrut +\mathstrut \) \(572\) \(\beta_{8}\mathstrut +\mathstrut \) \(32\) \(\beta_{7}\mathstrut -\mathstrut \) \(251\) \(\beta_{6}\mathstrut -\mathstrut \) \(253\) \(\beta_{5}\mathstrut +\mathstrut \) \(355\) \(\beta_{4}\mathstrut -\mathstrut \) \(81\) \(\beta_{3}\mathstrut +\mathstrut \) \(181\) \(\beta_{2}\mathstrut +\mathstrut \) \(196\) \(\beta_{1}\mathstrut +\mathstrut \) \(901\)
\(\nu^{7}\)\(=\)\(312\) \(\beta_{10}\mathstrut +\mathstrut \) \(824\) \(\beta_{9}\mathstrut -\mathstrut \) \(587\) \(\beta_{8}\mathstrut -\mathstrut \) \(300\) \(\beta_{7}\mathstrut +\mathstrut \) \(643\) \(\beta_{6}\mathstrut +\mathstrut \) \(596\) \(\beta_{5}\mathstrut -\mathstrut \) \(1472\) \(\beta_{4}\mathstrut -\mathstrut \) \(144\) \(\beta_{3}\mathstrut +\mathstrut \) \(96\) \(\beta_{2}\mathstrut +\mathstrut \) \(1680\) \(\beta_{1}\mathstrut -\mathstrut \) \(937\)
\(\nu^{8}\)\(=\)\(-\)\(512\) \(\beta_{10}\mathstrut -\mathstrut \) \(5508\) \(\beta_{9}\mathstrut +\mathstrut \) \(8945\) \(\beta_{8}\mathstrut +\mathstrut \) \(708\) \(\beta_{7}\mathstrut -\mathstrut \) \(4004\) \(\beta_{6}\mathstrut -\mathstrut \) \(4024\) \(\beta_{5}\mathstrut +\mathstrut \) \(6100\) \(\beta_{4}\mathstrut -\mathstrut \) \(1282\) \(\beta_{3}\mathstrut +\mathstrut \) \(3188\) \(\beta_{2}\mathstrut +\mathstrut \) \(2422\) \(\beta_{1}\mathstrut +\mathstrut \) \(13390\)
\(\nu^{9}\)\(=\)\(4996\) \(\beta_{10}\mathstrut +\mathstrut \) \(14746\) \(\beta_{9}\mathstrut -\mathstrut \) \(12143\) \(\beta_{8}\mathstrut -\mathstrut \) \(4849\) \(\beta_{7}\mathstrut +\mathstrut \) \(11135\) \(\beta_{6}\mathstrut +\mathstrut \) \(11092\) \(\beta_{5}\mathstrut -\mathstrut \) \(25076\) \(\beta_{4}\mathstrut -\mathstrut \) \(1149\) \(\beta_{3}\mathstrut +\mathstrut \) \(199\) \(\beta_{2}\mathstrut +\mathstrut \) \(23706\) \(\beta_{1}\mathstrut -\mathstrut \) \(19052\)
\(\nu^{10}\)\(=\)\(-\)\(9750\) \(\beta_{10}\mathstrut -\mathstrut \) \(89608\) \(\beta_{9}\mathstrut +\mathstrut \) \(140335\) \(\beta_{8}\mathstrut +\mathstrut \) \(13689\) \(\beta_{7}\mathstrut -\mathstrut \) \(64588\) \(\beta_{6}\mathstrut -\mathstrut \) \(64478\) \(\beta_{5}\mathstrut +\mathstrut \) \(103301\) \(\beta_{4}\mathstrut -\mathstrut \) \(18981\) \(\beta_{3}\mathstrut +\mathstrut \) \(52165\) \(\beta_{2}\mathstrut +\mathstrut \) \(28182\) \(\beta_{1}\mathstrut +\mathstrut \) \(206442\)

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
−4.08681
−2.64826
−1.55818
−1.33314
−0.760310
0.504544
2.03178
2.25177
2.25950
3.59322
3.74589
1.00000 −1.00000 1.00000 −4.08681 −1.00000 −0.153752 1.00000 1.00000 −4.08681
1.2 1.00000 −1.00000 1.00000 −2.64826 −1.00000 3.55508 1.00000 1.00000 −2.64826
1.3 1.00000 −1.00000 1.00000 −1.55818 −1.00000 −1.66703 1.00000 1.00000 −1.55818
1.4 1.00000 −1.00000 1.00000 −1.33314 −1.00000 −3.20872 1.00000 1.00000 −1.33314
1.5 1.00000 −1.00000 1.00000 −0.760310 −1.00000 −1.62737 1.00000 1.00000 −0.760310
1.6 1.00000 −1.00000 1.00000 0.504544 −1.00000 2.39214 1.00000 1.00000 0.504544
1.7 1.00000 −1.00000 1.00000 2.03178 −1.00000 −2.78135 1.00000 1.00000 2.03178
1.8 1.00000 −1.00000 1.00000 2.25177 −1.00000 3.43625 1.00000 1.00000 2.25177
1.9 1.00000 −1.00000 1.00000 2.25950 −1.00000 3.73494 1.00000 1.00000 2.25950
1.10 1.00000 −1.00000 1.00000 3.59322 −1.00000 3.34551 1.00000 1.00000 3.59322
1.11 1.00000 −1.00000 1.00000 3.74589 −1.00000 −4.02570 1.00000 1.00000 3.74589
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)