Properties

Label 6039.2.a.g
Level 6039
Weight 2
Character orbit 6039.a
Self dual Yes
Analytic conductor 48.222
Analytic rank 1
Dimension 13
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: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{6} ) q^{5} \) \( -\beta_{9} q^{7} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{4} \) \( + ( -1 + \beta_{6} ) q^{5} \) \( -\beta_{9} q^{7} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} \) \( + ( \beta_{1} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{10} \) \(+ q^{11}\) \( + ( -\beta_{6} - \beta_{7} - \beta_{12} ) q^{13} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} ) q^{14} \) \( + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{16} \) \( + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{11} ) q^{17} \) \( + ( -\beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{19} \) \( + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{11} ) q^{20} \) \( -\beta_{1} q^{22} \) \( + ( -1 - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} ) q^{23} \) \( + ( 1 - \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{12} ) q^{25} \) \( + ( -2 + \beta_{1} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} + \beta_{12} ) q^{26} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{12} ) q^{28} \) \( + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{29} \) \( + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{31} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{32} \) \( + ( 1 - \beta_{1} - 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{34} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{35} \) \( + ( 2 - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{10} - \beta_{12} ) q^{37} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{38} \) \( + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{40} \) \( + ( -1 - \beta_{2} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{41} \) \( + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{9} ) q^{43} \) \( + ( 1 + \beta_{1} + \beta_{2} ) q^{44} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{46} \) \( + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{47} \) \( + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{49} \) \( + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - 4 \beta_{9} - \beta_{10} + \beta_{11} - 3 \beta_{12} ) q^{50} \) \( + ( 1 - 3 \beta_{2} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{52} \) \( + ( -3 + 2 \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} ) q^{53} \) \( + ( -1 + \beta_{6} ) q^{55} \) \( + ( -3 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{56} \) \( + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{12} ) q^{58} \) \( + ( -2 - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} ) q^{59} \) \(- q^{61}\) \( + ( 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{12} ) q^{62} \) \( + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{64} \) \( + ( -2 + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{11} ) q^{65} \) \( + ( 2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{67} \) \( + ( 2 + \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{68} \) \( + ( -1 - \beta_{2} - \beta_{5} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{70} \) \( + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{10} + 4 \beta_{12} ) q^{71} \) \( + ( -4 + \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{8} + \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{73} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{74} \) \( + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 4 \beta_{7} + \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{76} \) \( -\beta_{9} q^{77} \) \( + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} - 2 \beta_{12} ) q^{79} \) \( + ( 1 - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{80} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} - 3 \beta_{9} - \beta_{12} ) q^{82} \) \( + ( -3 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} ) q^{83} \) \( + ( -2 - 3 \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{85} \) \( + ( 2 + 2 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{86} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{88} \) \( + ( -2 - \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} ) q^{89} \) \( + ( 1 - \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{91} \) \( + ( -1 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{12} ) q^{92} \) \( + ( 4 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 4 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{12} ) q^{94} \) \( + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{95} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} + 3 \beta_{9} + \beta_{10} + 3 \beta_{11} + 2 \beta_{12} ) q^{97} \) \( + ( 2 + 2 \beta_{1} - 3 \beta_{2} + \beta_{4} - 2 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut -\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 8q^{10} \) \(\mathstrut +\mathstrut 13q^{11} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 19q^{14} \) \(\mathstrut +\mathstrut 18q^{16} \) \(\mathstrut -\mathstrut 7q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 15q^{20} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut -\mathstrut 23q^{23} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 8q^{26} \) \(\mathstrut +\mathstrut 9q^{28} \) \(\mathstrut -\mathstrut 16q^{29} \) \(\mathstrut +\mathstrut 9q^{31} \) \(\mathstrut -\mathstrut 29q^{32} \) \(\mathstrut +\mathstrut 2q^{34} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut +\mathstrut 14q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 16q^{40} \) \(\mathstrut -\mathstrut 19q^{41} \) \(\mathstrut +\mathstrut 7q^{43} \) \(\mathstrut +\mathstrut 12q^{44} \) \(\mathstrut +\mathstrut 4q^{46} \) \(\mathstrut -\mathstrut 26q^{47} \) \(\mathstrut +\mathstrut 8q^{49} \) \(\mathstrut +\mathstrut 15q^{50} \) \(\mathstrut -\mathstrut 17q^{52} \) \(\mathstrut -\mathstrut 18q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 44q^{56} \) \(\mathstrut -\mathstrut q^{58} \) \(\mathstrut -\mathstrut 31q^{59} \) \(\mathstrut -\mathstrut 13q^{61} \) \(\mathstrut +\mathstrut 5q^{62} \) \(\mathstrut -\mathstrut 17q^{64} \) \(\mathstrut -\mathstrut 31q^{65} \) \(\mathstrut +\mathstrut 14q^{67} \) \(\mathstrut +\mathstrut 32q^{68} \) \(\mathstrut -\mathstrut 20q^{70} \) \(\mathstrut -\mathstrut 37q^{71} \) \(\mathstrut -\mathstrut 16q^{73} \) \(\mathstrut +\mathstrut 6q^{74} \) \(\mathstrut -\mathstrut 7q^{76} \) \(\mathstrut +\mathstrut 5q^{77} \) \(\mathstrut -\mathstrut 17q^{79} \) \(\mathstrut +\mathstrut 2q^{80} \) \(\mathstrut -\mathstrut 2q^{82} \) \(\mathstrut -\mathstrut 30q^{83} \) \(\mathstrut -\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 22q^{86} \) \(\mathstrut -\mathstrut 15q^{88} \) \(\mathstrut -\mathstrut 35q^{89} \) \(\mathstrut -\mathstrut q^{91} \) \(\mathstrut -\mathstrut 24q^{92} \) \(\mathstrut -\mathstrut 11q^{94} \) \(\mathstrut -\mathstrut 13q^{95} \) \(\mathstrut -\mathstrut q^{97} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(11\) \(x^{11}\mathstrut +\mathstrut \) \(55\) \(x^{10}\mathstrut +\mathstrut \) \(32\) \(x^{9}\mathstrut -\mathstrut \) \(266\) \(x^{8}\mathstrut +\mathstrut \) \(13\) \(x^{7}\mathstrut +\mathstrut \) \(534\) \(x^{6}\mathstrut -\mathstrut \) \(141\) \(x^{5}\mathstrut -\mathstrut \) \(404\) \(x^{4}\mathstrut +\mathstrut \) \(98\) \(x^{3}\mathstrut +\mathstrut \) \(118\) \(x^{2}\mathstrut -\mathstrut \) \(16\) \(x\mathstrut -\mathstrut \) \(11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 2 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} - 5 \nu^{11} - 11 \nu^{10} + 73 \nu^{9} + 29 \nu^{8} - 374 \nu^{7} + 46 \nu^{6} + 770 \nu^{5} - 255 \nu^{4} - 495 \nu^{3} + 228 \nu^{2} + 14 \nu - 47 \)\()/17\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{12} + \nu^{11} + 20 \nu^{10} - 17 \nu^{9} - 150 \nu^{8} + 108 \nu^{7} + 506 \nu^{6} - 320 \nu^{5} - 695 \nu^{4} + 441 \nu^{3} + 201 \nu^{2} - 184 \nu + 8 \)\()/17\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{12} + 8 \nu^{11} + 18 \nu^{10} - 101 \nu^{9} - 8 \nu^{8} + 411 \nu^{7} - 292 \nu^{6} - 516 \nu^{5} + 749 \nu^{4} - 142 \nu^{3} - 386 \nu^{2} + 176 \nu + 66 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{12} - 7 \nu^{11} - 33 \nu^{10} + 104 \nu^{9} + 221 \nu^{8} - 540 \nu^{7} - 764 \nu^{6} + 1126 \nu^{5} + 1350 \nu^{4} - 788 \nu^{3} - 992 \nu^{2} + 164 \nu + 186 \)\()/17\)
\(\beta_{8}\)\(=\)\((\)\( -5 \nu^{12} + 14 \nu^{11} + 67 \nu^{10} - 194 \nu^{9} - 312 \nu^{8} + 960 \nu^{7} + 591 \nu^{6} - 2043 \nu^{5} - 411 \nu^{4} + 1740 \nu^{3} + 129 \nu^{2} - 427 \nu - 21 \)\()/17\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{12} + 11 \nu^{11} - 11 \nu^{10} - 140 \nu^{9} + 263 \nu^{8} + 586 \nu^{7} - 1384 \nu^{6} - 833 \nu^{5} + 2700 \nu^{4} + 66 \nu^{3} - 1696 \nu^{2} + 122 \nu + 267 \)\()/17\)
\(\beta_{10}\)\(=\)\((\)\( -6 \nu^{12} + 26 \nu^{11} + 58 \nu^{10} - 348 \nu^{9} - 91 \nu^{8} + 1604 \nu^{7} - 489 \nu^{6} - 2912 \nu^{5} + 1362 \nu^{4} + 1675 \nu^{3} - 463 \nu^{2} - 305 \nu - 12 \)\()/17\)
\(\beta_{11}\)\(=\)\((\)\( 6 \nu^{12} - 26 \nu^{11} - 58 \nu^{10} + 348 \nu^{9} + 91 \nu^{8} - 1604 \nu^{7} + 506 \nu^{6} + 2895 \nu^{5} - 1532 \nu^{4} - 1556 \nu^{3} + 905 \nu^{2} + 152 \nu - 141 \)\()/17\)
\(\beta_{12}\)\(=\)\((\)\( 7 \nu^{12} - 33 \nu^{11} - 56 \nu^{10} + 432 \nu^{9} - 51 \nu^{8} - 1924 \nu^{7} + 1321 \nu^{6} + 3278 \nu^{5} - 3112 \nu^{4} - 1551 \nu^{3} + 1747 \nu^{2} + 149 \nu - 233 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(9\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(36\) \(\beta_{2}\mathstrut +\mathstrut \) \(57\) \(\beta_{1}\mathstrut +\mathstrut \) \(84\)
\(\nu^{7}\)\(=\)\(-\)\(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(57\) \(\beta_{3}\mathstrut +\mathstrut \) \(68\) \(\beta_{2}\mathstrut +\mathstrut \) \(178\) \(\beta_{1}\mathstrut +\mathstrut \) \(83\)
\(\nu^{8}\)\(=\)\(-\)\(17\) \(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(79\) \(\beta_{10}\mathstrut +\mathstrut \) \(61\) \(\beta_{9}\mathstrut +\mathstrut \) \(76\) \(\beta_{8}\mathstrut -\mathstrut \) \(76\) \(\beta_{7}\mathstrut -\mathstrut \) \(89\) \(\beta_{5}\mathstrut +\mathstrut \) \(59\) \(\beta_{4}\mathstrut +\mathstrut \) \(94\) \(\beta_{3}\mathstrut +\mathstrut \) \(226\) \(\beta_{2}\mathstrut +\mathstrut \) \(392\) \(\beta_{1}\mathstrut +\mathstrut \) \(498\)
\(\nu^{9}\)\(=\)\(-\)\(123\) \(\beta_{12}\mathstrut +\mathstrut \) \(32\) \(\beta_{11}\mathstrut -\mathstrut \) \(124\) \(\beta_{10}\mathstrut -\mathstrut \) \(75\) \(\beta_{9}\mathstrut +\mathstrut \) \(31\) \(\beta_{8}\mathstrut -\mathstrut \) \(33\) \(\beta_{7}\mathstrut -\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(122\) \(\beta_{5}\mathstrut -\mathstrut \) \(75\) \(\beta_{4}\mathstrut +\mathstrut \) \(394\) \(\beta_{3}\mathstrut +\mathstrut \) \(486\) \(\beta_{2}\mathstrut +\mathstrut \) \(1130\) \(\beta_{1}\mathstrut +\mathstrut \) \(636\)
\(\nu^{10}\)\(=\)\(-\)\(193\) \(\beta_{12}\mathstrut +\mathstrut \) \(155\) \(\beta_{11}\mathstrut -\mathstrut \) \(581\) \(\beta_{10}\mathstrut +\mathstrut \) \(369\) \(\beta_{9}\mathstrut +\mathstrut \) \(526\) \(\beta_{8}\mathstrut -\mathstrut \) \(530\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(650\) \(\beta_{5}\mathstrut +\mathstrut \) \(340\) \(\beta_{4}\mathstrut +\mathstrut \) \(732\) \(\beta_{3}\mathstrut +\mathstrut \) \(1465\) \(\beta_{2}\mathstrut +\mathstrut \) \(2657\) \(\beta_{1}\mathstrut +\mathstrut \) \(3068\)
\(\nu^{11}\)\(=\)\(-\)\(1028\) \(\beta_{12}\mathstrut +\mathstrut \) \(348\) \(\beta_{11}\mathstrut -\mathstrut \) \(1044\) \(\beta_{10}\mathstrut -\mathstrut \) \(508\) \(\beta_{9}\mathstrut +\mathstrut \) \(328\) \(\beta_{8}\mathstrut -\mathstrut \) \(365\) \(\beta_{7}\mathstrut -\mathstrut \) \(106\) \(\beta_{6}\mathstrut -\mathstrut \) \(1011\) \(\beta_{5}\mathstrut -\mathstrut \) \(517\) \(\beta_{4}\mathstrut +\mathstrut \) \(2696\) \(\beta_{3}\mathstrut +\mathstrut \) \(3391\) \(\beta_{2}\mathstrut +\mathstrut \) \(7329\) \(\beta_{1}\mathstrut +\mathstrut \) \(4676\)
\(\nu^{12}\)\(=\)\(-\)\(1837\) \(\beta_{12}\mathstrut +\mathstrut \) \(1376\) \(\beta_{11}\mathstrut -\mathstrut \) \(4155\) \(\beta_{10}\mathstrut +\mathstrut \) \(2096\) \(\beta_{9}\mathstrut +\mathstrut \) \(3502\) \(\beta_{8}\mathstrut -\mathstrut \) \(3585\) \(\beta_{7}\mathstrut -\mathstrut \) \(50\) \(\beta_{6}\mathstrut -\mathstrut \) \(4559\) \(\beta_{5}\mathstrut +\mathstrut \) \(1807\) \(\beta_{4}\mathstrut +\mathstrut \) \(5446\) \(\beta_{3}\mathstrut +\mathstrut \) \(9681\) \(\beta_{2}\mathstrut +\mathstrut \) \(17907\) \(\beta_{1}\mathstrut +\mathstrut \) \(19419\)

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
2.62425
2.60074
2.38255
1.88022
1.03857
0.684638
0.468970
−0.329361
−0.534142
−0.762797
−1.74673
−1.96783
−2.33909
−2.62425 0 4.88668 −0.233370 0 1.03556 −7.57538 0 0.612421
1.2 −2.60074 0 4.76385 0.540623 0 4.31275 −7.18805 0 −1.40602
1.3 −2.38255 0 3.67655 −3.76064 0 −1.09133 −3.99445 0 8.95992
1.4 −1.88022 0 1.53524 −1.27827 0 0.0298223 0.873847 0 2.40344
1.5 −1.03857 0 −0.921370 −2.59019 0 2.45800 3.03405 0 2.69009
1.6 −0.684638 0 −1.53127 1.50791 0 2.85777 2.41764 0 −1.03237
1.7 −0.468970 0 −1.78007 3.25579 0 −3.43403 1.77274 0 −1.52687
1.8 0.329361 0 −1.89152 −2.08793 0 −4.32416 −1.28171 0 −0.687681
1.9 0.534142 0 −1.71469 −3.62612 0 4.31243 −1.98417 0 −1.93686
1.10 0.762797 0 −1.41814 1.72725 0 1.91723 −2.60735 0 1.31754
1.11 1.74673 0 1.05105 2.68410 0 −0.749083 −1.65755 0 4.68839
1.12 1.96783 0 1.87234 −3.39588 0 0.824946 −0.251204 0 −6.68250
1.13 2.33909 0 3.47134 0.256721 0 −3.14990 3.44160 0 0.600493
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 \(T_{2}^{13} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\).