Properties

Label 6042.2.a.bd
Level 6042
Weight 2
Character orbit 6042.a
Self dual Yes
Analytic conductor 48.246
Analytic rank 1
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6042.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(2\) \(x^{10}\mathstrut -\mathstrut \) \(35\) \(x^{9}\mathstrut +\mathstrut \) \(77\) \(x^{8}\mathstrut +\mathstrut \) \(394\) \(x^{7}\mathstrut -\mathstrut \) \(994\) \(x^{6}\mathstrut -\mathstrut \) \(1477\) \(x^{5}\mathstrut +\mathstrut \) \(4683\) \(x^{4}\mathstrut +\mathstrut \) \(563\) \(x^{3}\mathstrut -\mathstrut \) \(6194\) \(x^{2}\mathstrut +\mathstrut \) \(1258\) \(x\mathstrut +\mathstrut \) \(1534\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(61365\) \(\nu^{10}\mathstrut -\mathstrut \) \(19979\) \(\nu^{9}\mathstrut +\mathstrut \) \(2101724\) \(\nu^{8}\mathstrut +\mathstrut \) \(162971\) \(\nu^{7}\mathstrut -\mathstrut \) \(23813959\) \(\nu^{6}\mathstrut +\mathstrut \) \(5613147\) \(\nu^{5}\mathstrut +\mathstrut \) \(103864180\) \(\nu^{4}\mathstrut -\mathstrut \) \(45917643\) \(\nu^{3}\mathstrut -\mathstrut \) \(141974290\) \(\nu^{2}\mathstrut +\mathstrut \) \(50557192\) \(\nu\mathstrut +\mathstrut \) \(40606678\)\()/34480\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(84083\) \(\nu^{10}\mathstrut -\mathstrut \) \(26993\) \(\nu^{9}\mathstrut +\mathstrut \) \(2879648\) \(\nu^{8}\mathstrut +\mathstrut \) \(209481\) \(\nu^{7}\mathstrut -\mathstrut \) \(32622537\) \(\nu^{6}\mathstrut +\mathstrut \) \(7851333\) \(\nu^{5}\mathstrut +\mathstrut \) \(142207024\) \(\nu^{4}\mathstrut -\mathstrut \) \(63537677\) \(\nu^{3}\mathstrut -\mathstrut \) \(194086526\) \(\nu^{2}\mathstrut +\mathstrut \) \(69706224\) \(\nu\mathstrut +\mathstrut \) \(55455562\)\()/34480\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(77556\) \(\nu^{10}\mathstrut -\mathstrut \) \(25031\) \(\nu^{9}\mathstrut +\mathstrut \) \(2656416\) \(\nu^{8}\mathstrut +\mathstrut \) \(198432\) \(\nu^{7}\mathstrut -\mathstrut \) \(30099019\) \(\nu^{6}\mathstrut +\mathstrut \) \(7177316\) \(\nu^{5}\mathstrut +\mathstrut \) \(131247773\) \(\nu^{4}\mathstrut -\mathstrut \) \(58352059\) \(\nu^{3}\mathstrut -\mathstrut \) \(179277372\) \(\nu^{2}\mathstrut +\mathstrut \) \(64132198\) \(\nu\mathstrut +\mathstrut \) \(51286814\)\()/17240\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(50505\) \(\nu^{10}\mathstrut -\mathstrut \) \(16163\) \(\nu^{9}\mathstrut +\mathstrut \) \(1730043\) \(\nu^{8}\mathstrut +\mathstrut \) \(124862\) \(\nu^{7}\mathstrut -\mathstrut \) \(19605108\) \(\nu^{6}\mathstrut +\mathstrut \) \(4717169\) \(\nu^{5}\mathstrut +\mathstrut \) \(85498490\) \(\nu^{4}\mathstrut -\mathstrut \) \(38143726\) \(\nu^{3}\mathstrut -\mathstrut \) \(116783530\) \(\nu^{2}\mathstrut +\mathstrut \) \(41822964\) \(\nu\mathstrut +\mathstrut \) \(33382236\)\()/8620\)
\(\beta_{6}\)\(=\)\((\)\(210451\) \(\nu^{10}\mathstrut +\mathstrut \) \(67875\) \(\nu^{9}\mathstrut -\mathstrut \) \(7208340\) \(\nu^{8}\mathstrut -\mathstrut \) \(537653\) \(\nu^{7}\mathstrut +\mathstrut \) \(81674783\) \(\nu^{6}\mathstrut -\mathstrut \) \(19471453\) \(\nu^{5}\mathstrut -\mathstrut \) \(356110518\) \(\nu^{4}\mathstrut +\mathstrut \) \(158228347\) \(\nu^{3}\mathstrut +\mathstrut \) \(486159382\) \(\nu^{2}\mathstrut -\mathstrut \) \(173556980\) \(\nu\mathstrut -\mathstrut \) \(138742022\)\()/34480\)
\(\beta_{7}\)\(=\)\((\)\(192480\) \(\nu^{10}\mathstrut +\mathstrut \) \(61607\) \(\nu^{9}\mathstrut -\mathstrut \) \(6593832\) \(\nu^{8}\mathstrut -\mathstrut \) \(477308\) \(\nu^{7}\mathstrut +\mathstrut \) \(74728307\) \(\nu^{6}\mathstrut -\mathstrut \) \(17946496\) \(\nu^{5}\mathstrut -\mathstrut \) \(325910105\) \(\nu^{4}\mathstrut +\mathstrut \) \(145209819\) \(\nu^{3}\mathstrut +\mathstrut \) \(445157380\) \(\nu^{2}\mathstrut -\mathstrut \) \(159257926\) \(\nu\mathstrut -\mathstrut \) \(127192774\)\()/17240\)
\(\beta_{8}\)\(=\)\((\)\(107137\) \(\nu^{10}\mathstrut +\mathstrut \) \(34407\) \(\nu^{9}\mathstrut -\mathstrut \) \(3669944\) \(\nu^{8}\mathstrut -\mathstrut \) \(269099\) \(\nu^{7}\mathstrut +\mathstrut \) \(41587847\) \(\nu^{6}\mathstrut -\mathstrut \) \(9957687\) \(\nu^{5}\mathstrut -\mathstrut \) \(181362420\) \(\nu^{4}\mathstrut +\mathstrut \) \(80709251\) \(\nu^{3}\mathstrut +\mathstrut \) \(247718074\) \(\nu^{2}\mathstrut -\mathstrut \) \(88503368\) \(\nu\mathstrut -\mathstrut \) \(70791910\)\()/6896\)
\(\beta_{9}\)\(=\)\((\)\(647795\) \(\nu^{10}\mathstrut +\mathstrut \) \(207847\) \(\nu^{9}\mathstrut -\mathstrut \) \(22190632\) \(\nu^{8}\mathstrut -\mathstrut \) \(1620833\) \(\nu^{7}\mathstrut +\mathstrut \) \(251475487\) \(\nu^{6}\mathstrut -\mathstrut \) \(60275181\) \(\nu^{5}\mathstrut -\mathstrut \) \(1096742850\) \(\nu^{4}\mathstrut +\mathstrut \) \(488288139\) \(\nu^{3}\mathstrut +\mathstrut \) \(1498189590\) \(\nu^{2}\mathstrut -\mathstrut \) \(535480556\) \(\nu\mathstrut -\mathstrut \) \(428293574\)\()/34480\)
\(\beta_{10}\)\(=\)\((\)\(365939\) \(\nu^{10}\mathstrut +\mathstrut \) \(117420\) \(\nu^{9}\mathstrut -\mathstrut \) \(12535140\) \(\nu^{8}\mathstrut -\mathstrut \) \(915157\) \(\nu^{7}\mathstrut +\mathstrut \) \(142049012\) \(\nu^{6}\mathstrut -\mathstrut \) \(34063257\) \(\nu^{5}\mathstrut -\mathstrut \) \(619474937\) \(\nu^{4}\mathstrut +\mathstrut \) \(275912908\) \(\nu^{3}\mathstrut +\mathstrut \) \(846120818\) \(\nu^{2}\mathstrut -\mathstrut \) \(302610630\) \(\nu\mathstrut -\mathstrut \) \(241753888\)\()/17240\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(14\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(80\)
\(\nu^{5}\)\(=\)\(19\) \(\beta_{10}\mathstrut -\mathstrut \) \(17\) \(\beta_{9}\mathstrut +\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(23\) \(\beta_{6}\mathstrut +\mathstrut \) \(48\) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut -\mathstrut \) \(26\) \(\beta_{3}\mathstrut -\mathstrut \) \(19\) \(\beta_{2}\mathstrut +\mathstrut \) \(143\) \(\beta_{1}\mathstrut -\mathstrut \) \(73\)
\(\nu^{6}\)\(=\)\(-\)\(201\) \(\beta_{10}\mathstrut +\mathstrut \) \(245\) \(\beta_{9}\mathstrut -\mathstrut \) \(64\) \(\beta_{8}\mathstrut +\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(38\) \(\beta_{6}\mathstrut +\mathstrut \) \(39\) \(\beta_{5}\mathstrut +\mathstrut \) \(72\) \(\beta_{4}\mathstrut -\mathstrut \) \(268\) \(\beta_{3}\mathstrut -\mathstrut \) \(57\) \(\beta_{2}\mathstrut -\mathstrut \) \(253\) \(\beta_{1}\mathstrut +\mathstrut \) \(1076\)
\(\nu^{7}\)\(=\)\(304\) \(\beta_{10}\mathstrut -\mathstrut \) \(218\) \(\beta_{9}\mathstrut +\mathstrut \) \(86\) \(\beta_{8}\mathstrut +\mathstrut \) \(257\) \(\beta_{7}\mathstrut -\mathstrut \) \(406\) \(\beta_{6}\mathstrut +\mathstrut \) \(896\) \(\beta_{5}\mathstrut +\mathstrut \) \(137\) \(\beta_{4}\mathstrut -\mathstrut \) \(516\) \(\beta_{3}\mathstrut -\mathstrut \) \(294\) \(\beta_{2}\mathstrut +\mathstrut \) \(2027\) \(\beta_{1}\mathstrut -\mathstrut \) \(997\)
\(\nu^{8}\)\(=\)\(-\)\(3020\) \(\beta_{10}\mathstrut +\mathstrut \) \(3714\) \(\beta_{9}\mathstrut -\mathstrut \) \(1050\) \(\beta_{8}\mathstrut +\mathstrut \) \(372\) \(\beta_{7}\mathstrut +\mathstrut \) \(593\) \(\beta_{6}\mathstrut +\mathstrut \) \(607\) \(\beta_{5}\mathstrut +\mathstrut \) \(1353\) \(\beta_{4}\mathstrut -\mathstrut \) \(4251\) \(\beta_{3}\mathstrut -\mathstrut \) \(1204\) \(\beta_{2}\mathstrut -\mathstrut \) \(3619\) \(\beta_{1}\mathstrut +\mathstrut \) \(15505\)
\(\nu^{9}\)\(=\)\(4602\) \(\beta_{10}\mathstrut -\mathstrut \) \(2352\) \(\beta_{9}\mathstrut +\mathstrut \) \(1003\) \(\beta_{8}\mathstrut +\mathstrut \) \(3834\) \(\beta_{7}\mathstrut -\mathstrut \) \(6611\) \(\beta_{6}\mathstrut +\mathstrut \) \(15577\) \(\beta_{5}\mathstrut +\mathstrut \) \(2417\) \(\beta_{4}\mathstrut -\mathstrut \) \(9378\) \(\beta_{3}\mathstrut -\mathstrut \) \(4348\) \(\beta_{2}\mathstrut +\mathstrut \) \(29997\) \(\beta_{1}\mathstrut -\mathstrut \) \(11958\)
\(\nu^{10}\)\(=\)\(-\)\(46515\) \(\beta_{10}\mathstrut +\mathstrut \) \(56273\) \(\beta_{9}\mathstrut -\mathstrut \) \(15844\) \(\beta_{8}\mathstrut +\mathstrut \) \(6913\) \(\beta_{7}\mathstrut +\mathstrut \) \(8667\) \(\beta_{6}\mathstrut +\mathstrut \) \(9242\) \(\beta_{5}\mathstrut +\mathstrut \) \(23602\) \(\beta_{4}\mathstrut -\mathstrut \) \(67999\) \(\beta_{3}\mathstrut -\mathstrut \) \(22857\) \(\beta_{2}\mathstrut -\mathstrut \) \(50937\) \(\beta_{1}\mathstrut +\mathstrut \) \(230908\)

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
−3.92287
−3.61806
−2.56652
−1.23681
−0.425735
0.864610
1.67126
2.32043
2.32145
2.52811
4.06414
−1.00000 −1.00000 1.00000 −3.92287 1.00000 −2.55808 −1.00000 1.00000 3.92287
1.2 −1.00000 −1.00000 1.00000 −3.61806 1.00000 4.03393 −1.00000 1.00000 3.61806
1.3 −1.00000 −1.00000 1.00000 −2.56652 1.00000 −0.973492 −1.00000 1.00000 2.56652
1.4 −1.00000 −1.00000 1.00000 −1.23681 1.00000 2.50061 −1.00000 1.00000 1.23681
1.5 −1.00000 −1.00000 1.00000 −0.425735 1.00000 −2.24729 −1.00000 1.00000 0.425735
1.6 −1.00000 −1.00000 1.00000 0.864610 1.00000 2.90002 −1.00000 1.00000 −0.864610
1.7 −1.00000 −1.00000 1.00000 1.67126 1.00000 −4.35376 −1.00000 1.00000 −1.67126
1.8 −1.00000 −1.00000 1.00000 2.32043 1.00000 −0.928685 −1.00000 1.00000 −2.32043
1.9 −1.00000 −1.00000 1.00000 2.32145 1.00000 −4.55788 −1.00000 1.00000 −2.32145
1.10 −1.00000 −1.00000 1.00000 2.52811 1.00000 0.198222 −1.00000 1.00000 −2.52811
1.11 −1.00000 −1.00000 1.00000 4.06414 1.00000 3.98638 −1.00000 1.00000 −4.06414
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(19\) \(-1\)
\(53\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

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