Properties

Label 6039.2.a.j
Level 6039
Weight 2
Character orbit 6039.a
Self dual Yes
Analytic conductor 48.222
Analytic rank 0
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6039.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(x^{13}\mathstrut -\mathstrut \) \(21\) \(x^{12}\mathstrut +\mathstrut \) \(20\) \(x^{11}\mathstrut +\mathstrut \) \(167\) \(x^{10}\mathstrut -\mathstrut \) \(148\) \(x^{9}\mathstrut -\mathstrut \) \(627\) \(x^{8}\mathstrut +\mathstrut \) \(497\) \(x^{7}\mathstrut +\mathstrut \) \(1123\) \(x^{6}\mathstrut -\mathstrut \) \(745\) \(x^{5}\mathstrut -\mathstrut \) \(802\) \(x^{4}\mathstrut +\mathstrut \) \(386\) \(x^{3}\mathstrut +\mathstrut \) \(74\) \(x^{2}\mathstrut -\mathstrut \) \(15\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{13} - 71 \nu^{12} - 586 \nu^{11} + 2838 \nu^{10} + 10454 \nu^{9} - 35539 \nu^{8} - 62411 \nu^{7} + 186606 \nu^{6} + 140254 \nu^{5} - 411132 \nu^{4} - 83942 \nu^{3} + 307840 \nu^{2} - 27224 \nu - 13953 \)\()/4505\)
\(\beta_{5}\)\(=\)\((\)\( -82 \nu^{13} - 1317 \nu^{12} + 1503 \nu^{11} + 25486 \nu^{10} - 7732 \nu^{9} - 179663 \nu^{8} - 4527 \nu^{7} + 556827 \nu^{6} + 125703 \nu^{5} - 713699 \nu^{4} - 233864 \nu^{3} + 253645 \nu^{2} + 24637 \nu - 22401 \)\()/4505\)
\(\beta_{6}\)\(=\)\((\)\( -311 \nu^{13} + 444 \nu^{12} + 6964 \nu^{11} - 9372 \nu^{10} - 59981 \nu^{9} + 74721 \nu^{8} + 245494 \nu^{7} - 273749 \nu^{6} - 466936 \nu^{5} + 435843 \nu^{4} + 306853 \nu^{3} - 209250 \nu^{2} + 43281 \nu - 1068 \)\()/4505\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(349\) \(\nu^{13}\mathstrut -\mathstrut \) \(2254\) \(\nu^{12}\mathstrut +\mathstrut \) \(7221\) \(\nu^{11}\mathstrut +\mathstrut \) \(44412\) \(\nu^{10}\mathstrut -\mathstrut \) \(54664\) \(\nu^{9}\mathstrut -\mathstrut \) \(320701\) \(\nu^{8}\mathstrut +\mathstrut \) \(180436\) \(\nu^{7}\mathstrut +\mathstrut \) \(1032859\) \(\nu^{6}\mathstrut -\mathstrut \) \(232439\) \(\nu^{5}\mathstrut -\mathstrut \) \(1428903\) \(\nu^{4}\mathstrut +\mathstrut \) \(76842\) \(\nu^{3}\mathstrut +\mathstrut \) \(618105\) \(\nu^{2}\mathstrut -\mathstrut \) \(67706\) \(\nu\mathstrut -\mathstrut \) \(17712\)\()/4505\)
\(\beta_{8}\)\(=\)\((\)\( -251 \nu^{13} + 199 \nu^{12} + 5183 \nu^{11} - 3957 \nu^{10} - 40303 \nu^{9} + 29344 \nu^{8} + 146488 \nu^{7} - 100390 \nu^{6} - 248794 \nu^{5} + 158677 \nu^{4} + 158118 \nu^{3} - 92020 \nu^{2} - 2743 \nu + 885 \)\()/901\)
\(\beta_{9}\)\(=\)\((\)\(1423\) \(\nu^{13}\mathstrut -\mathstrut \) \(2582\) \(\nu^{12}\mathstrut -\mathstrut \) \(31082\) \(\nu^{11}\mathstrut +\mathstrut \) \(52066\) \(\nu^{10}\mathstrut +\mathstrut \) \(260758\) \(\nu^{9}\mathstrut -\mathstrut \) \(388563\) \(\nu^{8}\mathstrut -\mathstrut \) \(1050382\) \(\nu^{7}\mathstrut +\mathstrut \) \(1308832\) \(\nu^{6}\mathstrut +\mathstrut \) \(2048843\) \(\nu^{5}\mathstrut -\mathstrut \) \(1915614\) \(\nu^{4}\mathstrut -\mathstrut \) \(1595379\) \(\nu^{3}\mathstrut +\mathstrut \) \(883850\) \(\nu^{2}\mathstrut +\mathstrut \) \(113882\) \(\nu\mathstrut -\mathstrut \) \(20941\)\()/4505\)
\(\beta_{10}\)\(=\)\((\)\(1433\) \(\nu^{13}\mathstrut -\mathstrut \) \(1872\) \(\nu^{12}\mathstrut -\mathstrut \) \(29727\) \(\nu^{11}\mathstrut +\mathstrut \) \(37201\) \(\nu^{10}\mathstrut +\mathstrut \) \(232803\) \(\nu^{9}\mathstrut -\mathstrut \) \(271938\) \(\nu^{8}\mathstrut -\mathstrut \) \(858752\) \(\nu^{7}\mathstrut +\mathstrut \) \(893382\) \(\nu^{6}\mathstrut +\mathstrut \) \(1511263\) \(\nu^{5}\mathstrut -\mathstrut \) \(1286659\) \(\nu^{4}\mathstrut -\mathstrut \) \(1048784\) \(\nu^{3}\mathstrut +\mathstrut \) \(607560\) \(\nu^{2}\mathstrut +\mathstrut \) \(52752\) \(\nu\mathstrut -\mathstrut \) \(12056\)\()/4505\)
\(\beta_{11}\)\(=\)\((\)\( -303 \nu^{13} + 111 \nu^{12} + 6246 \nu^{11} - 2343 \nu^{10} - 48107 \nu^{9} + 18455 \nu^{8} + 170845 \nu^{7} - 67311 \nu^{6} - 277112 \nu^{5} + 115493 \nu^{4} + 162984 \nu^{3} - 77090 \nu^{2} - 5623 \nu + 3337 \)\()/901\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1576\) \(\nu^{13}\mathstrut +\mathstrut \) \(729\) \(\nu^{12}\mathstrut +\mathstrut \) \(31524\) \(\nu^{11}\mathstrut -\mathstrut \) \(14292\) \(\nu^{10}\mathstrut -\mathstrut \) \(233541\) \(\nu^{9}\mathstrut +\mathstrut \) \(104816\) \(\nu^{8}\mathstrut +\mathstrut \) \(786304\) \(\nu^{7}\mathstrut -\mathstrut \) \(360249\) \(\nu^{6}\mathstrut -\mathstrut \) \(1177831\) \(\nu^{5}\mathstrut +\mathstrut \) \(604778\) \(\nu^{4}\mathstrut +\mathstrut \) \(591393\) \(\nu^{3}\mathstrut -\mathstrut \) \(424595\) \(\nu^{2}\mathstrut +\mathstrut \) \(14111\) \(\nu\mathstrut +\mathstrut \) \(21502\)\()/4505\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(3393\) \(\nu^{13}\mathstrut +\mathstrut \) \(2367\) \(\nu^{12}\mathstrut +\mathstrut \) \(70487\) \(\nu^{11}\mathstrut -\mathstrut \) \(47406\) \(\nu^{10}\mathstrut -\mathstrut \) \(551558\) \(\nu^{9}\mathstrut +\mathstrut \) \(352898\) \(\nu^{8}\mathstrut +\mathstrut \) \(2019747\) \(\nu^{7}\mathstrut -\mathstrut \) \(1203902\) \(\nu^{6}\mathstrut -\mathstrut \) \(3477208\) \(\nu^{5}\mathstrut +\mathstrut \) \(1876454\) \(\nu^{4}\mathstrut +\mathstrut \) \(2324484\) \(\nu^{3}\mathstrut -\mathstrut \) \(1064330\) \(\nu^{2}\mathstrut -\mathstrut \) \(176207\) \(\nu\mathstrut +\mathstrut \) \(36556\)\()/4505\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(9\) \(\beta_{11}\mathstrut +\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(84\)
\(\nu^{7}\)\(=\)\(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(67\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(168\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{8}\)\(=\)\(16\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(64\) \(\beta_{11}\mathstrut +\mathstrut \) \(66\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(26\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(28\) \(\beta_{6}\mathstrut -\mathstrut \) \(79\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(326\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(504\)
\(\nu^{9}\)\(=\)\(35\) \(\beta_{13}\mathstrut -\mathstrut \) \(109\) \(\beta_{12}\mathstrut +\mathstrut \) \(79\) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(125\) \(\beta_{9}\mathstrut +\mathstrut \) \(57\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(114\) \(\beta_{6}\mathstrut -\mathstrut \) \(106\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(472\) \(\beta_{3}\mathstrut +\mathstrut \) \(33\) \(\beta_{2}\mathstrut +\mathstrut \) \(1046\) \(\beta_{1}\mathstrut -\mathstrut \) \(12\)
\(\nu^{10}\)\(=\)\(176\) \(\beta_{13}\mathstrut -\mathstrut \) \(37\) \(\beta_{12}\mathstrut +\mathstrut \) \(426\) \(\beta_{11}\mathstrut +\mathstrut \) \(464\) \(\beta_{10}\mathstrut +\mathstrut \) \(174\) \(\beta_{9}\mathstrut -\mathstrut \) \(243\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(282\) \(\beta_{6}\mathstrut -\mathstrut \) \(579\) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(141\) \(\beta_{3}\mathstrut +\mathstrut \) \(2204\) \(\beta_{2}\mathstrut -\mathstrut \) \(118\) \(\beta_{1}\mathstrut +\mathstrut \) \(3147\)
\(\nu^{11}\)\(=\)\(405\) \(\beta_{13}\mathstrut -\mathstrut \) \(898\) \(\beta_{12}\mathstrut +\mathstrut \) \(586\) \(\beta_{11}\mathstrut -\mathstrut \) \(56\) \(\beta_{10}\mathstrut +\mathstrut \) \(1066\) \(\beta_{9}\mathstrut +\mathstrut \) \(302\) \(\beta_{8}\mathstrut -\mathstrut \) \(75\) \(\beta_{7}\mathstrut +\mathstrut \) \(984\) \(\beta_{6}\mathstrut -\mathstrut \) \(839\) \(\beta_{5}\mathstrut -\mathstrut \) \(98\) \(\beta_{4}\mathstrut +\mathstrut \) \(3258\) \(\beta_{3}\mathstrut +\mathstrut \) \(370\) \(\beta_{2}\mathstrut +\mathstrut \) \(6644\) \(\beta_{1}\mathstrut -\mathstrut \) \(89\)
\(\nu^{12}\)\(=\)\(1652\) \(\beta_{13}\mathstrut -\mathstrut \) \(443\) \(\beta_{12}\mathstrut +\mathstrut \) \(2777\) \(\beta_{11}\mathstrut +\mathstrut \) \(3245\) \(\beta_{10}\mathstrut +\mathstrut \) \(1609\) \(\beta_{9}\mathstrut -\mathstrut \) \(2010\) \(\beta_{8}\mathstrut -\mathstrut \) \(256\) \(\beta_{7}\mathstrut +\mathstrut \) \(2491\) \(\beta_{6}\mathstrut -\mathstrut \) \(4113\) \(\beta_{5}\mathstrut -\mathstrut \) \(72\) \(\beta_{4}\mathstrut +\mathstrut \) \(1251\) \(\beta_{3}\mathstrut +\mathstrut \) \(14874\) \(\beta_{2}\mathstrut -\mathstrut \) \(923\) \(\beta_{1}\mathstrut +\mathstrut \) \(20093\)
\(\nu^{13}\)\(=\)\(3916\) \(\beta_{13}\mathstrut -\mathstrut \) \(7055\) \(\beta_{12}\mathstrut +\mathstrut \) \(4261\) \(\beta_{11}\mathstrut -\mathstrut \) \(128\) \(\beta_{10}\mathstrut +\mathstrut \) \(8540\) \(\beta_{9}\mathstrut +\mathstrut \) \(1339\) \(\beta_{8}\mathstrut -\mathstrut \) \(911\) \(\beta_{7}\mathstrut +\mathstrut \) \(8029\) \(\beta_{6}\mathstrut -\mathstrut \) \(6318\) \(\beta_{5}\mathstrut -\mathstrut \) \(681\) \(\beta_{4}\mathstrut +\mathstrut \) \(22304\) \(\beta_{3}\mathstrut +\mathstrut \) \(3528\) \(\beta_{2}\mathstrut +\mathstrut \) \(42680\) \(\beta_{1}\mathstrut -\mathstrut \) \(465\)

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
−2.57087
−2.54331
−1.76637
−1.69494
−1.18140
−0.231279
−0.0561655
0.179763
0.546298
1.61392
1.67203
1.93923
2.45909
2.63401
−2.57087 0 4.60938 −4.00721 0 2.04273 −6.70837 0 10.3020
1.2 −2.54331 0 4.46845 −0.329781 0 0.595463 −6.27804 0 0.838736
1.3 −1.76637 0 1.12006 2.50284 0 −4.08919 1.55430 0 −4.42094
1.4 −1.69494 0 0.872835 −1.13650 0 4.14659 1.91048 0 1.92630
1.5 −1.18140 0 −0.604292 2.90081 0 −0.528840 3.07671 0 −3.42702
1.6 −0.231279 0 −1.94651 −3.70694 0 −0.911108 0.912746 0 0.857338
1.7 −0.0561655 0 −1.99685 2.87016 0 3.53988 0.224485 0 −0.161204
1.8 0.179763 0 −1.96769 −2.20477 0 3.17177 −0.713244 0 −0.396336
1.9 0.546298 0 −1.70156 0.842631 0 −4.19208 −2.02216 0 0.460328
1.10 1.61392 0 0.604750 3.65776 0 0.985739 −2.25183 0 5.90335
1.11 1.67203 0 0.795679 1.45953 0 −0.113904 −2.01366 0 2.44037
1.12 1.93923 0 1.76061 −3.07794 0 −3.15900 −0.464237 0 −5.96883
1.13 2.45909 0 4.04715 1.85442 0 2.32343 5.03413 0 4.56020
1.14 2.63401 0 4.93799 −2.62502 0 5.18852 7.73867 0 −6.91432
\(n\): 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
\(3\) \(-1\)
\(11\) \(-1\)
\(61\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{14} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).