Properties

Label 6018.2.a.bc
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 0
Dimension 14
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(2\) \(x^{13}\mathstrut -\mathstrut \) \(49\) \(x^{12}\mathstrut +\mathstrut \) \(79\) \(x^{11}\mathstrut +\mathstrut \) \(956\) \(x^{10}\mathstrut -\mathstrut \) \(1179\) \(x^{9}\mathstrut -\mathstrut \) \(9396\) \(x^{8}\mathstrut +\mathstrut \) \(8315\) \(x^{7}\mathstrut +\mathstrut \) \(48570\) \(x^{6}\mathstrut -\mathstrut \) \(28124\) \(x^{5}\mathstrut -\mathstrut \) \(125592\) \(x^{4}\mathstrut +\mathstrut \) \(40576\) \(x^{3}\mathstrut +\mathstrut \) \(138096\) \(x^{2}\mathstrut -\mathstrut \) \(22032\) \(x\mathstrut -\mathstrut \) \(43744\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(16618726754\) \(\nu^{13}\mathstrut -\mathstrut \) \(2880670567959\) \(\nu^{12}\mathstrut -\mathstrut \) \(19196372187770\) \(\nu^{11}\mathstrut +\mathstrut \) \(218934418638321\) \(\nu^{10}\mathstrut +\mathstrut \) \(664075790819685\) \(\nu^{9}\mathstrut -\mathstrut \) \(5146738100625936\) \(\nu^{8}\mathstrut -\mathstrut \) \(8545314250127135\) \(\nu^{7}\mathstrut +\mathstrut \) \(50968867163271240\) \(\nu^{6}\mathstrut +\mathstrut \) \(48833287977907775\) \(\nu^{5}\mathstrut -\mathstrut \) \(217396661630863236\) \(\nu^{4}\mathstrut -\mathstrut \) \(114630360428554364\) \(\nu^{3}\mathstrut +\mathstrut \) \(335998399721244300\) \(\nu^{2}\mathstrut +\mathstrut \) \(82447262577467504\) \(\nu\mathstrut -\mathstrut \) \(121204003017027824\)\()/\)\(7894197520302920\)
\(\beta_{3}\)\(=\)\((\)\(186379188393\) \(\nu^{13}\mathstrut -\mathstrut \) \(14081190935493\) \(\nu^{12}\mathstrut +\mathstrut \) \(28753457488965\) \(\nu^{11}\mathstrut +\mathstrut \) \(579978221320162\) \(\nu^{10}\mathstrut -\mathstrut \) \(1128332602443005\) \(\nu^{9}\mathstrut -\mathstrut \) \(9178698567078167\) \(\nu^{8}\mathstrut +\mathstrut \) \(14347689427204105\) \(\nu^{7}\mathstrut +\mathstrut \) \(69366955830930035\) \(\nu^{6}\mathstrut -\mathstrut \) \(77170871949262535\) \(\nu^{5}\mathstrut -\mathstrut \) \(252075259966176062\) \(\nu^{4}\mathstrut +\mathstrut \) \(175862488069791692\) \(\nu^{3}\mathstrut +\mathstrut \) \(386591439015569500\) \(\nu^{2}\mathstrut -\mathstrut \) \(160331873586642872\) \(\nu\mathstrut -\mathstrut \) \(147416272648448328\)\()/\)\(7894197520302920\)
\(\beta_{4}\)\(=\)\((\)\(202997915147\) \(\nu^{13}\mathstrut -\mathstrut \) \(16961861503452\) \(\nu^{12}\mathstrut +\mathstrut \) \(9557085301195\) \(\nu^{11}\mathstrut +\mathstrut \) \(798912639958483\) \(\nu^{10}\mathstrut -\mathstrut \) \(464256811623320\) \(\nu^{9}\mathstrut -\mathstrut \) \(14325436667704103\) \(\nu^{8}\mathstrut +\mathstrut \) \(5802375177076970\) \(\nu^{7}\mathstrut +\mathstrut \) \(120335822994201275\) \(\nu^{6}\mathstrut -\mathstrut \) \(28337583971354760\) \(\nu^{5}\mathstrut -\mathstrut \) \(469471921597039298\) \(\nu^{4}\mathstrut +\mathstrut \) \(61232127641237328\) \(\nu^{3}\mathstrut +\mathstrut \) \(730484036257116720\) \(\nu^{2}\mathstrut -\mathstrut \) \(77884611009175368\) \(\nu\mathstrut -\mathstrut \) \(323879658307596592\)\()/\)\(7894197520302920\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(236276976171\) \(\nu^{13}\mathstrut +\mathstrut \) \(2329023024781\) \(\nu^{12}\mathstrut +\mathstrut \) \(3942662986183\) \(\nu^{11}\mathstrut -\mathstrut \) \(87670549177272\) \(\nu^{10}\mathstrut +\mathstrut \) \(50692387805607\) \(\nu^{9}\mathstrut +\mathstrut \) \(1220653172685771\) \(\nu^{8}\mathstrut -\mathstrut \) \(1404708310869837\) \(\nu^{7}\mathstrut -\mathstrut \) \(7517982884554071\) \(\nu^{6}\mathstrut +\mathstrut \) \(9313167069656711\) \(\nu^{5}\mathstrut +\mathstrut \) \(17549153084715332\) \(\nu^{4}\mathstrut -\mathstrut \) \(20706135037061272\) \(\nu^{3}\mathstrut -\mathstrut \) \(1669417898377500\) \(\nu^{2}\mathstrut +\mathstrut \) \(7626075980681936\) \(\nu\mathstrut -\mathstrut \) \(13207959668528272\)\()/\)\(3157679008121168\)
\(\beta_{6}\)\(=\)\((\)\(443751573941\) \(\nu^{13}\mathstrut -\mathstrut \) \(3008483462819\) \(\nu^{12}\mathstrut -\mathstrut \) \(15414700154749\) \(\nu^{11}\mathstrut +\mathstrut \) \(114215106567488\) \(\nu^{10}\mathstrut +\mathstrut \) \(218559476116059\) \(\nu^{9}\mathstrut -\mathstrut \) \(1599349359996641\) \(\nu^{8}\mathstrut -\mathstrut \) \(1794366343522957\) \(\nu^{7}\mathstrut +\mathstrut \) \(9814685899795205\) \(\nu^{6}\mathstrut +\mathstrut \) \(9382257151775639\) \(\nu^{5}\mathstrut -\mathstrut \) \(22696271950686584\) \(\nu^{4}\mathstrut -\mathstrut \) \(25513830497582096\) \(\nu^{3}\mathstrut +\mathstrut \) \(2824971170499412\) \(\nu^{2}\mathstrut +\mathstrut \) \(19999582304517216\) \(\nu\mathstrut +\mathstrut \) \(13827833046029584\)\()/\)\(3157679008121168\)
\(\beta_{7}\)\(=\)\((\)\(2226419771871\) \(\nu^{13}\mathstrut -\mathstrut \) \(14073288977621\) \(\nu^{12}\mathstrut -\mathstrut \) \(88902889968565\) \(\nu^{11}\mathstrut +\mathstrut \) \(621826637080384\) \(\nu^{10}\mathstrut +\mathstrut \) \(1279836923435835\) \(\nu^{9}\mathstrut -\mathstrut \) \(10443761348377309\) \(\nu^{8}\mathstrut -\mathstrut \) \(7309843756060875\) \(\nu^{7}\mathstrut +\mathstrut \) \(82123225867424865\) \(\nu^{6}\mathstrut +\mathstrut \) \(4465199891261865\) \(\nu^{5}\mathstrut -\mathstrut \) \(295403685028325434\) \(\nu^{4}\mathstrut +\mathstrut \) \(86934350501483764\) \(\nu^{3}\mathstrut +\mathstrut \) \(380724393788637740\) \(\nu^{2}\mathstrut -\mathstrut \) \(151910773512655624\) \(\nu\mathstrut -\mathstrut \) \(79394039693874016\)\()/\)\(15788395040605840\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(5727717858944\) \(\nu^{13}\mathstrut +\mathstrut \) \(7373884997349\) \(\nu^{12}\mathstrut +\mathstrut \) \(260235001126690\) \(\nu^{11}\mathstrut -\mathstrut \) \(207291278819761\) \(\nu^{10}\mathstrut -\mathstrut \) \(4646880167421965\) \(\nu^{9}\mathstrut +\mathstrut \) \(1817846619873516\) \(\nu^{8}\mathstrut +\mathstrut \) \(40551053846921025\) \(\nu^{7}\mathstrut -\mathstrut \) \(4934053698709940\) \(\nu^{6}\mathstrut -\mathstrut \) \(174746025060120125\) \(\nu^{5}\mathstrut -\mathstrut \) \(3134468239806414\) \(\nu^{4}\mathstrut +\mathstrut \) \(333908596796940704\) \(\nu^{3}\mathstrut +\mathstrut \) \(48280623164166500\) \(\nu^{2}\mathstrut -\mathstrut \) \(213409310594817704\) \(\nu\mathstrut -\mathstrut \) \(90032763346833376\)\()/\)\(15788395040605840\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(3246512395886\) \(\nu^{13}\mathstrut +\mathstrut \) \(10334588586211\) \(\nu^{12}\mathstrut +\mathstrut \) \(126833659135550\) \(\nu^{11}\mathstrut -\mathstrut \) \(358168627839709\) \(\nu^{10}\mathstrut -\mathstrut \) \(1863261897938825\) \(\nu^{9}\mathstrut +\mathstrut \) \(4435000064547344\) \(\nu^{8}\mathstrut +\mathstrut \) \(12733747965022935\) \(\nu^{7}\mathstrut -\mathstrut \) \(23909630494110700\) \(\nu^{6}\mathstrut -\mathstrut \) \(41721469365383055\) \(\nu^{5}\mathstrut +\mathstrut \) \(56781193842740004\) \(\nu^{4}\mathstrut +\mathstrut \) \(72685524970824776\) \(\nu^{3}\mathstrut -\mathstrut \) \(60806142225794540\) \(\nu^{2}\mathstrut -\mathstrut \) \(81782923919894696\) \(\nu\mathstrut +\mathstrut \) \(17065631601733976\)\()/\)\(7894197520302920\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10198052289283\) \(\nu^{13}\mathstrut +\mathstrut \) \(32343651313568\) \(\nu^{12}\mathstrut +\mathstrut \) \(416444703490635\) \(\nu^{11}\mathstrut -\mathstrut \) \(1121038637019707\) \(\nu^{10}\mathstrut -\mathstrut \) \(6666675407303450\) \(\nu^{9}\mathstrut +\mathstrut \) \(13997174731938057\) \(\nu^{8}\mathstrut +\mathstrut \) \(52688098277426130\) \(\nu^{7}\mathstrut -\mathstrut \) \(76968570525750485\) \(\nu^{6}\mathstrut -\mathstrut \) \(210583446468153260\) \(\nu^{5}\mathstrut +\mathstrut \) \(182706845683949412\) \(\nu^{4}\mathstrut +\mathstrut \) \(389401116803341508\) \(\nu^{3}\mathstrut -\mathstrut \) \(142957183349222400\) \(\nu^{2}\mathstrut -\mathstrut \) \(242528671429568608\) \(\nu\mathstrut -\mathstrut \) \(25869449120688432\)\()/\)\(15788395040605840\)
\(\beta_{11}\)\(=\)\((\)\(12229460566967\) \(\nu^{13}\mathstrut -\mathstrut \) \(60219666399172\) \(\nu^{12}\mathstrut -\mathstrut \) \(443909983325205\) \(\nu^{11}\mathstrut +\mathstrut \) \(2249746437989783\) \(\nu^{10}\mathstrut +\mathstrut \) \(6115066006045500\) \(\nu^{9}\mathstrut -\mathstrut \) \(31255545769888503\) \(\nu^{8}\mathstrut -\mathstrut \) \(41404355245907430\) \(\nu^{7}\mathstrut +\mathstrut \) \(199267414681539515\) \(\nu^{6}\mathstrut +\mathstrut \) \(152264418850087960\) \(\nu^{5}\mathstrut -\mathstrut \) \(579220686043196398\) \(\nu^{4}\mathstrut -\mathstrut \) \(295590582482895472\) \(\nu^{3}\mathstrut +\mathstrut \) \(676467191148702720\) \(\nu^{2}\mathstrut +\mathstrut \) \(158405026529980472\) \(\nu\mathstrut -\mathstrut \) \(260037911487082352\)\()/\)\(15788395040605840\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(17924326795869\) \(\nu^{13}\mathstrut +\mathstrut \) \(23455557245524\) \(\nu^{12}\mathstrut +\mathstrut \) \(879153061774715\) \(\nu^{11}\mathstrut -\mathstrut \) \(953369861904061\) \(\nu^{10}\mathstrut -\mathstrut \) \(16369355249611440\) \(\nu^{9}\mathstrut +\mathstrut \) \(14477216382795401\) \(\nu^{8}\mathstrut +\mathstrut \) \(143065720146530890\) \(\nu^{7}\mathstrut -\mathstrut \) \(103215181842607525\) \(\nu^{6}\mathstrut -\mathstrut \) \(586122971969355440\) \(\nu^{5}\mathstrut +\mathstrut \) \(354134323447064406\) \(\nu^{4}\mathstrut +\mathstrut \) \(966238739011843464\) \(\nu^{3}\mathstrut -\mathstrut \) \(500374230332802640\) \(\nu^{2}\mathstrut -\mathstrut \) \(384292349947742024\) \(\nu\mathstrut +\mathstrut \) \(166761480409701264\)\()/\)\(15788395040605840\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(21287880786177\) \(\nu^{13}\mathstrut +\mathstrut \) \(51088675909607\) \(\nu^{12}\mathstrut +\mathstrut \) \(889326633697785\) \(\nu^{11}\mathstrut -\mathstrut \) \(1730523496563728\) \(\nu^{10}\mathstrut -\mathstrut \) \(14193063964416095\) \(\nu^{9}\mathstrut +\mathstrut \) \(21117514546830813\) \(\nu^{8}\mathstrut +\mathstrut \) \(106085745505489865\) \(\nu^{7}\mathstrut -\mathstrut \) \(116847168938051585\) \(\nu^{6}\mathstrut -\mathstrut \) \(361806925655129755\) \(\nu^{5}\mathstrut +\mathstrut \) \(320335811846240008\) \(\nu^{4}\mathstrut +\mathstrut \) \(460439267349737952\) \(\nu^{3}\mathstrut -\mathstrut \) \(452152041773406660\) \(\nu^{2}\mathstrut -\mathstrut \) \(163056024996551552\) \(\nu\mathstrut +\mathstrut \) \(182105343465589472\)\()/\)\(15788395040605840\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(71\)
\(\nu^{5}\)\(=\)\(19\) \(\beta_{13}\mathstrut -\mathstrut \) \(18\) \(\beta_{12}\mathstrut -\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(19\) \(\beta_{10}\mathstrut -\mathstrut \) \(8\) \(\beta_{9}\mathstrut -\mathstrut \) \(20\) \(\beta_{8}\mathstrut -\mathstrut \) \(19\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(45\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(62\) \(\beta_{2}\mathstrut +\mathstrut \) \(140\) \(\beta_{1}\mathstrut +\mathstrut \) \(58\)
\(\nu^{6}\)\(=\)\(71\) \(\beta_{13}\mathstrut -\mathstrut \) \(25\) \(\beta_{12}\mathstrut -\mathstrut \) \(24\) \(\beta_{11}\mathstrut -\mathstrut \) \(46\) \(\beta_{10}\mathstrut -\mathstrut \) \(52\) \(\beta_{9}\mathstrut -\mathstrut \) \(91\) \(\beta_{8}\mathstrut +\mathstrut \) \(44\) \(\beta_{7}\mathstrut -\mathstrut \) \(45\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(243\) \(\beta_{4}\mathstrut -\mathstrut \) \(191\) \(\beta_{3}\mathstrut -\mathstrut \) \(292\) \(\beta_{2}\mathstrut +\mathstrut \) \(193\) \(\beta_{1}\mathstrut +\mathstrut \) \(860\)
\(\nu^{7}\)\(=\)\(310\) \(\beta_{13}\mathstrut -\mathstrut \) \(281\) \(\beta_{12}\mathstrut -\mathstrut \) \(405\) \(\beta_{11}\mathstrut -\mathstrut \) \(292\) \(\beta_{10}\mathstrut -\mathstrut \) \(208\) \(\beta_{9}\mathstrut -\mathstrut \) \(346\) \(\beta_{8}\mathstrut -\mathstrut \) \(272\) \(\beta_{7}\mathstrut +\mathstrut \) \(144\) \(\beta_{6}\mathstrut -\mathstrut \) \(222\) \(\beta_{5}\mathstrut +\mathstrut \) \(819\) \(\beta_{4}\mathstrut -\mathstrut \) \(180\) \(\beta_{3}\mathstrut -\mathstrut \) \(1093\) \(\beta_{2}\mathstrut +\mathstrut \) \(1955\) \(\beta_{1}\mathstrut +\mathstrut \) \(1252\)
\(\nu^{8}\)\(=\)\(1320\) \(\beta_{13}\mathstrut -\mathstrut \) \(501\) \(\beta_{12}\mathstrut -\mathstrut \) \(473\) \(\beta_{11}\mathstrut -\mathstrut \) \(784\) \(\beta_{10}\mathstrut -\mathstrut \) \(1058\) \(\beta_{9}\mathstrut -\mathstrut \) \(1652\) \(\beta_{8}\mathstrut +\mathstrut \) \(768\) \(\beta_{7}\mathstrut -\mathstrut \) \(692\) \(\beta_{6}\mathstrut -\mathstrut \) \(38\) \(\beta_{5}\mathstrut +\mathstrut \) \(3752\) \(\beta_{4}\mathstrut -\mathstrut \) \(2706\) \(\beta_{3}\mathstrut -\mathstrut \) \(4713\) \(\beta_{2}\mathstrut +\mathstrut \) \(3995\) \(\beta_{1}\mathstrut +\mathstrut \) \(11634\)
\(\nu^{9}\)\(=\)\(4995\) \(\beta_{13}\mathstrut -\mathstrut \) \(4282\) \(\beta_{12}\mathstrut -\mathstrut \) \(6505\) \(\beta_{11}\mathstrut -\mathstrut \) \(4184\) \(\beta_{10}\mathstrut -\mathstrut \) \(4174\) \(\beta_{9}\mathstrut -\mathstrut \) \(5947\) \(\beta_{8}\mathstrut -\mathstrut \) \(3500\) \(\beta_{7}\mathstrut +\mathstrut \) \(2936\) \(\beta_{6}\mathstrut -\mathstrut \) \(2530\) \(\beta_{5}\mathstrut +\mathstrut \) \(13971\) \(\beta_{4}\mathstrut -\mathstrut \) \(3942\) \(\beta_{3}\mathstrut -\mathstrut \) \(18446\) \(\beta_{2}\mathstrut +\mathstrut \) \(28994\) \(\beta_{1}\mathstrut +\mathstrut \) \(24063\)
\(\nu^{10}\)\(=\)\(22756\) \(\beta_{13}\mathstrut -\mathstrut \) \(9271\) \(\beta_{12}\mathstrut -\mathstrut \) \(8770\) \(\beta_{11}\mathstrut -\mathstrut \) \(11800\) \(\beta_{10}\mathstrut -\mathstrut \) \(19844\) \(\beta_{9}\mathstrut -\mathstrut \) \(28501\) \(\beta_{8}\mathstrut +\mathstrut \) \(12532\) \(\beta_{7}\mathstrut -\mathstrut \) \(8865\) \(\beta_{6}\mathstrut +\mathstrut \) \(1536\) \(\beta_{5}\mathstrut +\mathstrut \) \(59188\) \(\beta_{4}\mathstrut -\mathstrut \) \(39780\) \(\beta_{3}\mathstrut -\mathstrut \) \(76621\) \(\beta_{2}\mathstrut +\mathstrut \) \(74525\) \(\beta_{1}\mathstrut +\mathstrut \) \(169394\)
\(\nu^{11}\)\(=\)\(81353\) \(\beta_{13}\mathstrut -\mathstrut \) \(65487\) \(\beta_{12}\mathstrut -\mathstrut \) \(100665\) \(\beta_{11}\mathstrut -\mathstrut \) \(57174\) \(\beta_{10}\mathstrut -\mathstrut \) \(77614\) \(\beta_{9}\mathstrut -\mathstrut \) \(103428\) \(\beta_{8}\mathstrut -\mathstrut \) \(41640\) \(\beta_{7}\mathstrut +\mathstrut \) \(52937\) \(\beta_{6}\mathstrut -\mathstrut \) \(24732\) \(\beta_{5}\mathstrut +\mathstrut \) \(232684\) \(\beta_{4}\mathstrut -\mathstrut \) \(76579\) \(\beta_{3}\mathstrut -\mathstrut \) \(307057\) \(\beta_{2}\mathstrut +\mathstrut \) \(446895\) \(\beta_{1}\mathstrut +\mathstrut \) \(437040\)
\(\nu^{12}\)\(=\)\(380736\) \(\beta_{13}\mathstrut -\mathstrut \) \(164714\) \(\beta_{12}\mathstrut -\mathstrut \) \(156937\) \(\beta_{11}\mathstrut -\mathstrut \) \(162943\) \(\beta_{10}\mathstrut -\mathstrut \) \(358722\) \(\beta_{9}\mathstrut -\mathstrut \) \(487858\) \(\beta_{8}\mathstrut +\mathstrut \) \(200505\) \(\beta_{7}\mathstrut -\mathstrut \) \(95762\) \(\beta_{6}\mathstrut +\mathstrut \) \(56233\) \(\beta_{5}\mathstrut +\mathstrut \) \(949565\) \(\beta_{4}\mathstrut -\mathstrut \) \(602482\) \(\beta_{3}\mathstrut -\mathstrut \) \(1255126\) \(\beta_{2}\mathstrut +\mathstrut \) \(1324332\) \(\beta_{1}\mathstrut +\mathstrut \) \(2589789\)
\(\nu^{13}\)\(=\)\(1339890\) \(\beta_{13}\mathstrut -\mathstrut \) \(1013394\) \(\beta_{12}\mathstrut -\mathstrut \) \(1533614\) \(\beta_{11}\mathstrut -\mathstrut \) \(739948\) \(\beta_{10}\mathstrut -\mathstrut \) \(1402056\) \(\beta_{9}\mathstrut -\mathstrut \) \(1815204\) \(\beta_{8}\mathstrut -\mathstrut \) \(447344\) \(\beta_{7}\mathstrut +\mathstrut \) \(907917\) \(\beta_{6}\mathstrut -\mathstrut \) \(167693\) \(\beta_{5}\mathstrut +\mathstrut \) \(3840807\) \(\beta_{4}\mathstrut -\mathstrut \) \(1403512\) \(\beta_{3}\mathstrut -\mathstrut \) \(5093009\) \(\beta_{2}\mathstrut +\mathstrut \) \(7061050\) \(\beta_{1}\mathstrut +\mathstrut \) \(7704906\)

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
−3.63246
−3.15319
−3.00862
−2.67453
−1.69357
−1.44259
−0.625391
0.800423
1.26062
2.33625
2.45834
3.24808
3.97677
4.14985
1.00000 −1.00000 1.00000 −3.63246 −1.00000 4.65920 1.00000 1.00000 −3.63246
1.2 1.00000 −1.00000 1.00000 −3.15319 −1.00000 −5.11963 1.00000 1.00000 −3.15319
1.3 1.00000 −1.00000 1.00000 −3.00862 −1.00000 −0.900066 1.00000 1.00000 −3.00862
1.4 1.00000 −1.00000 1.00000 −2.67453 −1.00000 −1.12425 1.00000 1.00000 −2.67453
1.5 1.00000 −1.00000 1.00000 −1.69357 −1.00000 3.25998 1.00000 1.00000 −1.69357
1.6 1.00000 −1.00000 1.00000 −1.44259 −1.00000 2.70884 1.00000 1.00000 −1.44259
1.7 1.00000 −1.00000 1.00000 −0.625391 −1.00000 −1.55533 1.00000 1.00000 −0.625391
1.8 1.00000 −1.00000 1.00000 0.800423 −1.00000 −4.52948 1.00000 1.00000 0.800423
1.9 1.00000 −1.00000 1.00000 1.26062 −1.00000 3.87711 1.00000 1.00000 1.26062
1.10 1.00000 −1.00000 1.00000 2.33625 −1.00000 −0.683869 1.00000 1.00000 2.33625
1.11 1.00000 −1.00000 1.00000 2.45834 −1.00000 −3.89101 1.00000 1.00000 2.45834
1.12 1.00000 −1.00000 1.00000 3.24808 −1.00000 4.67729 1.00000 1.00000 3.24808
1.13 1.00000 −1.00000 1.00000 3.97677 −1.00000 1.13692 1.00000 1.00000 3.97677
1.14 1.00000 −1.00000 1.00000 4.14985 −1.00000 −1.51570 1.00000 1.00000 4.14985
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{14} - \cdots\)
\(T_{7}^{14} - \cdots\)