Properties

Label 8037.2.a.n
Level 8037
Weight 2
Character orbit 8037.a
Self dual Yes
Analytic conductor 64.176
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8037.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(4\) \(x^{15}\mathstrut -\mathstrut \) \(13\) \(x^{14}\mathstrut +\mathstrut \) \(65\) \(x^{13}\mathstrut +\mathstrut \) \(47\) \(x^{12}\mathstrut -\mathstrut \) \(390\) \(x^{11}\mathstrut +\mathstrut \) \(4\) \(x^{10}\mathstrut +\mathstrut \) \(1115\) \(x^{9}\mathstrut -\mathstrut \) \(320\) \(x^{8}\mathstrut -\mathstrut \) \(1639\) \(x^{7}\mathstrut +\mathstrut \) \(618\) \(x^{6}\mathstrut +\mathstrut \) \(1250\) \(x^{5}\mathstrut -\mathstrut \) \(487\) \(x^{4}\mathstrut -\mathstrut \) \(456\) \(x^{3}\mathstrut +\mathstrut \) \(179\) \(x^{2}\mathstrut +\mathstrut \) \(62\) \(x\mathstrut -\mathstrut \) \(25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} - 5 \nu^{14} - 7 \nu^{13} + 71 \nu^{12} - 42 \nu^{11} - 332 \nu^{10} + 459 \nu^{9} + 560 \nu^{8} - 1290 \nu^{7} - 64 \nu^{6} + 1404 \nu^{5} - 602 \nu^{4} - 584 \nu^{3} + 456 \nu^{2} + 81 \nu - 82 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 5 \nu^{14} + 6 \nu^{13} - 69 \nu^{12} + 59 \nu^{11} + 301 \nu^{10} - 569 \nu^{9} - 389 \nu^{8} + 1643 \nu^{7} - 355 \nu^{6} - 2000 \nu^{5} + 1064 \nu^{4} + 1063 \nu^{3} - 673 \nu^{2} - 198 \nu + 125 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{15} + 7 \nu^{14} + 29 \nu^{13} - 116 \nu^{12} - 140 \nu^{11} + 715 \nu^{10} + 245 \nu^{9} - 2110 \nu^{8} - 3 \nu^{7} + 3181 \nu^{6} - 399 \nu^{5} - 2407 \nu^{4} + 350 \nu^{3} + 816 \nu^{2} - 88 \nu - 89 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{15} - 21 \nu^{14} - 60 \nu^{13} + 331 \nu^{12} + 166 \nu^{11} - 1891 \nu^{10} + 321 \nu^{9} + 5006 \nu^{8} - 1989 \nu^{7} - 6552 \nu^{6} + 2735 \nu^{5} + 4250 \nu^{4} - 1371 \nu^{3} - 1217 \nu^{2} + 222 \nu + 112 \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 7 \nu^{15} - 24 \nu^{14} - 105 \nu^{13} + 397 \nu^{12} + 558 \nu^{11} - 2444 \nu^{10} - 1357 \nu^{9} + 7228 \nu^{8} + 1746 \nu^{7} - 11035 \nu^{6} - 1520 \nu^{5} + 8648 \nu^{4} + 931 \nu^{3} - 3127 \nu^{2} - 214 \nu + 400 \)\()/3\)
\(\beta_{8}\)\(=\)\((\)\( -7 \nu^{15} + 24 \nu^{14} + 105 \nu^{13} - 397 \nu^{12} - 558 \nu^{11} + 2444 \nu^{10} + 1357 \nu^{9} - 7228 \nu^{8} - 1746 \nu^{7} + 11035 \nu^{6} + 1520 \nu^{5} - 8651 \nu^{4} - 931 \nu^{3} + 3148 \nu^{2} + 214 \nu - 418 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 8 \nu^{15} - 27 \nu^{14} - 123 \nu^{13} + 451 \nu^{12} + 685 \nu^{11} - 2815 \nu^{10} - 1821 \nu^{9} + 8474 \nu^{8} + 2709 \nu^{7} - 13188 \nu^{6} - 2653 \nu^{5} + 10475 \nu^{4} + 1596 \nu^{3} - 3791 \nu^{2} - 336 \nu + 493 \)\()/3\)
\(\beta_{10}\)\(=\)\((\)\( 10 \nu^{15} - 39 \nu^{14} - 131 \nu^{13} + 624 \nu^{12} + 501 \nu^{11} - 3649 \nu^{10} - 283 \nu^{9} + 9997 \nu^{8} - 1700 \nu^{7} - 13687 \nu^{6} + 2835 \nu^{5} + 9274 \nu^{4} - 1358 \nu^{3} - 2736 \nu^{2} + 204 \nu + 268 \)\()/3\)
\(\beta_{11}\)\(=\)\((\)\( 11 \nu^{15} - 41 \nu^{14} - 153 \nu^{13} + 666 \nu^{12} + 689 \nu^{11} - 3987 \nu^{10} - 1074 \nu^{9} + 11313 \nu^{8} + 25 \nu^{7} - 16288 \nu^{6} + 933 \nu^{5} + 11809 \nu^{4} - 383 \nu^{3} - 3843 \nu^{2} + 37 \nu + 433 \)\()/3\)
\(\beta_{12}\)\(=\)\((\)\( 12 \nu^{15} - 46 \nu^{14} - 159 \nu^{13} + 735 \nu^{12} + 630 \nu^{11} - 4288 \nu^{10} - 505 \nu^{9} + 11702 \nu^{8} - 1618 \nu^{7} - 15933 \nu^{6} + 2933 \nu^{5} + 10745 \nu^{4} - 1443 \nu^{3} - 3170 \nu^{2} + 220 \nu + 308 \)\()/3\)
\(\beta_{13}\)\(=\)\((\)\( 15 \nu^{15} - 50 \nu^{14} - 230 \nu^{13} + 832 \nu^{12} + 1273 \nu^{11} - 5164 \nu^{10} - 3336 \nu^{9} + 15424 \nu^{8} + 4820 \nu^{7} - 23761 \nu^{6} - 4548 \nu^{5} + 18658 \nu^{4} + 2683 \nu^{3} - 6687 \nu^{2} - 568 \nu + 864 \)\()/3\)
\(\beta_{14}\)\(=\)\((\)\( -14 \nu^{15} + 45 \nu^{14} + 225 \nu^{13} - 767 \nu^{12} - 1344 \nu^{11} + 4924 \nu^{10} + 3941 \nu^{9} - 15365 \nu^{8} - 6438 \nu^{7} + 24914 \nu^{6} + 6334 \nu^{5} - 20617 \nu^{4} - 3545 \nu^{3} + 7802 \nu^{2} + 731 \nu - 1055 \)\()/3\)
\(\beta_{15}\)\(=\)\((\)\( -26 \nu^{15} + 93 \nu^{14} + 375 \nu^{13} - 1521 \nu^{12} - 1841 \nu^{11} + 9205 \nu^{10} + 3766 \nu^{9} - 26570 \nu^{8} - 3339 \nu^{7} + 39289 \nu^{6} + 2038 \nu^{5} - 29676 \nu^{4} - 1615 \nu^{3} + 10328 \nu^{2} + 466 \nu - 1299 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(8\) \(\beta_{12}\mathstrut -\mathstrut \) \(8\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(9\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(45\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(86\)
\(\nu^{7}\)\(=\)\(\beta_{15}\mathstrut +\mathstrut \) \(10\) \(\beta_{14}\mathstrut -\mathstrut \) \(10\) \(\beta_{13}\mathstrut +\mathstrut \) \(57\) \(\beta_{12}\mathstrut -\mathstrut \) \(55\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(19\) \(\beta_{9}\mathstrut -\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(20\) \(\beta_{5}\mathstrut +\mathstrut \) \(68\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(167\) \(\beta_{1}\mathstrut +\mathstrut \) \(19\)
\(\nu^{8}\)\(=\)\(12\) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(15\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(28\) \(\beta_{10}\mathstrut -\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(63\) \(\beta_{8}\mathstrut -\mathstrut \) \(78\) \(\beta_{7}\mathstrut +\mathstrut \) \(27\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(287\) \(\beta_{2}\mathstrut -\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(520\)
\(\nu^{9}\)\(=\)\(14\) \(\beta_{15}\mathstrut +\mathstrut \) \(78\) \(\beta_{14}\mathstrut -\mathstrut \) \(70\) \(\beta_{13}\mathstrut +\mathstrut \) \(392\) \(\beta_{12}\mathstrut -\mathstrut \) \(362\) \(\beta_{11}\mathstrut -\mathstrut \) \(18\) \(\beta_{10}\mathstrut +\mathstrut \) \(134\) \(\beta_{9}\mathstrut -\mathstrut \) \(87\) \(\beta_{8}\mathstrut +\mathstrut \) \(102\) \(\beta_{7}\mathstrut -\mathstrut \) \(71\) \(\beta_{6}\mathstrut -\mathstrut \) \(153\) \(\beta_{5}\mathstrut +\mathstrut \) \(486\) \(\beta_{4}\mathstrut +\mathstrut \) \(104\) \(\beta_{3}\mathstrut -\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(1032\) \(\beta_{1}\mathstrut +\mathstrut \) \(132\)
\(\nu^{10}\)\(=\)\(104\) \(\beta_{15}\mathstrut -\mathstrut \) \(102\) \(\beta_{14}\mathstrut +\mathstrut \) \(157\) \(\beta_{13}\mathstrut +\mathstrut \) \(103\) \(\beta_{12}\mathstrut +\mathstrut \) \(153\) \(\beta_{11}\mathstrut -\mathstrut \) \(271\) \(\beta_{10}\mathstrut -\mathstrut \) \(173\) \(\beta_{9}\mathstrut -\mathstrut \) \(408\) \(\beta_{8}\mathstrut -\mathstrut \) \(562\) \(\beta_{7}\mathstrut +\mathstrut \) \(257\) \(\beta_{6}\mathstrut +\mathstrut \) \(105\) \(\beta_{5}\mathstrut +\mathstrut \) \(36\) \(\beta_{4}\mathstrut +\mathstrut \) \(135\) \(\beta_{3}\mathstrut +\mathstrut \) \(1836\) \(\beta_{2}\mathstrut -\mathstrut \) \(156\) \(\beta_{1}\mathstrut +\mathstrut \) \(3231\)
\(\nu^{11}\)\(=\)\(135\) \(\beta_{15}\mathstrut +\mathstrut \) \(562\) \(\beta_{14}\mathstrut -\mathstrut \) \(419\) \(\beta_{13}\mathstrut +\mathstrut \) \(2653\) \(\beta_{12}\mathstrut -\mathstrut \) \(2344\) \(\beta_{11}\mathstrut -\mathstrut \) \(210\) \(\beta_{10}\mathstrut +\mathstrut \) \(845\) \(\beta_{9}\mathstrut -\mathstrut \) \(605\) \(\beta_{8}\mathstrut +\mathstrut \) \(763\) \(\beta_{7}\mathstrut -\mathstrut \) \(437\) \(\beta_{6}\mathstrut -\mathstrut \) \(1073\) \(\beta_{5}\mathstrut +\mathstrut \) \(3390\) \(\beta_{4}\mathstrut +\mathstrut \) \(804\) \(\beta_{3}\mathstrut -\mathstrut \) \(326\) \(\beta_{2}\mathstrut +\mathstrut \) \(6514\) \(\beta_{1}\mathstrut +\mathstrut \) \(805\)
\(\nu^{12}\)\(=\)\(804\) \(\beta_{15}\mathstrut -\mathstrut \) \(763\) \(\beta_{14}\mathstrut +\mathstrut \) \(1408\) \(\beta_{13}\mathstrut +\mathstrut \) \(782\) \(\beta_{12}\mathstrut +\mathstrut \) \(1331\) \(\beta_{11}\mathstrut -\mathstrut \) \(2266\) \(\beta_{10}\mathstrut -\mathstrut \) \(1577\) \(\beta_{9}\mathstrut -\mathstrut \) \(2568\) \(\beta_{8}\mathstrut -\mathstrut \) \(3911\) \(\beta_{7}\mathstrut +\mathstrut \) \(2137\) \(\beta_{6}\mathstrut +\mathstrut \) \(819\) \(\beta_{5}\mathstrut +\mathstrut \) \(418\) \(\beta_{4}\mathstrut +\mathstrut \) \(1125\) \(\beta_{3}\mathstrut +\mathstrut \) \(11802\) \(\beta_{2}\mathstrut -\mathstrut \) \(1389\) \(\beta_{1}\mathstrut +\mathstrut \) \(20410\)
\(\nu^{13}\)\(=\)\(1125\) \(\beta_{15}\mathstrut +\mathstrut \) \(3911\) \(\beta_{14}\mathstrut -\mathstrut \) \(2262\) \(\beta_{13}\mathstrut +\mathstrut \) \(17817\) \(\beta_{12}\mathstrut -\mathstrut \) \(15089\) \(\beta_{11}\mathstrut -\mathstrut \) \(2024\) \(\beta_{10}\mathstrut +\mathstrut \) \(5032\) \(\beta_{9}\mathstrut -\mathstrut \) \(3954\) \(\beta_{8}\mathstrut +\mathstrut \) \(5381\) \(\beta_{7}\mathstrut -\mathstrut \) \(2467\) \(\beta_{6}\mathstrut -\mathstrut \) \(7263\) \(\beta_{5}\mathstrut +\mathstrut \) \(23348\) \(\beta_{4}\mathstrut +\mathstrut \) \(5905\) \(\beta_{3}\mathstrut -\mathstrut \) \(2911\) \(\beta_{2}\mathstrut +\mathstrut \) \(41671\) \(\beta_{1}\mathstrut +\mathstrut \) \(4547\)
\(\nu^{14}\)\(=\)\(5905\) \(\beta_{15}\mathstrut -\mathstrut \) \(5381\) \(\beta_{14}\mathstrut +\mathstrut \) \(11585\) \(\beta_{13}\mathstrut +\mathstrut \) \(5620\) \(\beta_{12}\mathstrut +\mathstrut \) \(10646\) \(\beta_{11}\mathstrut -\mathstrut \) \(17598\) \(\beta_{10}\mathstrut -\mathstrut \) \(13085\) \(\beta_{9}\mathstrut -\mathstrut \) \(16013\) \(\beta_{8}\mathstrut -\mathstrut \) \(26735\) \(\beta_{7}\mathstrut +\mathstrut \) \(16592\) \(\beta_{6}\mathstrut +\mathstrut \) \(6043\) \(\beta_{5}\mathstrut +\mathstrut \) \(4007\) \(\beta_{4}\mathstrut +\mathstrut \) \(8722\) \(\beta_{3}\mathstrut +\mathstrut \) \(76222\) \(\beta_{2}\mathstrut -\mathstrut \) \(11351\) \(\beta_{1}\mathstrut +\mathstrut \) \(130369\)
\(\nu^{15}\)\(=\)\(8722\) \(\beta_{15}\mathstrut +\mathstrut \) \(26735\) \(\beta_{14}\mathstrut -\mathstrut \) \(11053\) \(\beta_{13}\mathstrut +\mathstrut \) \(119217\) \(\beta_{12}\mathstrut -\mathstrut \) \(97026\) \(\beta_{11}\mathstrut -\mathstrut \) \(17540\) \(\beta_{10}\mathstrut +\mathstrut \) \(28912\) \(\beta_{9}\mathstrut -\mathstrut \) \(25032\) \(\beta_{8}\mathstrut +\mathstrut \) \(36831\) \(\beta_{7}\mathstrut -\mathstrut \) \(12965\) \(\beta_{6}\mathstrut -\mathstrut \) \(48402\) \(\beta_{5}\mathstrut +\mathstrut \) \(159599\) \(\beta_{4}\mathstrut +\mathstrut \) \(42225\) \(\beta_{3}\mathstrut -\mathstrut \) \(23782\) \(\beta_{2}\mathstrut +\mathstrut \) \(268962\) \(\beta_{1}\mathstrut +\mathstrut \) \(24228\)

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
−2.57826
−1.97762
−1.60180
−1.04691
−0.880178
−0.788915
−0.601130
0.469922
0.489985
0.640778
1.22346
1.38504
1.88684
2.38924
2.39915
2.59041
−2.57826 0 4.64740 3.07310 0 −1.66345 −6.82569 0 −7.92325
1.2 −1.97762 0 1.91096 2.84370 0 −3.61101 0.176083 0 −5.62374
1.3 −1.60180 0 0.565755 −0.438857 0 0.237690 2.29737 0 0.702960
1.4 −1.04691 0 −0.903974 −3.78031 0 −0.354270 3.04021 0 3.95766
1.5 −0.880178 0 −1.22529 −0.815813 0 −4.34804 2.83883 0 0.718060
1.6 −0.788915 0 −1.37761 2.33649 0 4.65174 2.66465 0 −1.84329
1.7 −0.601130 0 −1.63864 0.0865384 0 1.51992 2.18730 0 −0.0520209
1.8 0.469922 0 −1.77917 −0.709772 0 −4.48221 −1.77592 0 −0.333538
1.9 0.489985 0 −1.75991 2.01274 0 0.633596 −1.84230 0 0.986215
1.10 0.640778 0 −1.58940 −3.12262 0 −2.58301 −2.30001 0 −2.00091
1.11 1.22346 0 −0.503155 −1.49488 0 1.92629 −3.06250 0 −1.82892
1.12 1.38504 0 −0.0816767 2.56770 0 0.393019 −2.88320 0 3.55636
1.13 1.88684 0 1.56016 2.70381 0 1.30906 −0.829905 0 5.10166
1.14 2.38924 0 3.70845 −0.579664 0 −3.82951 4.08190 0 −1.38495
1.15 2.39915 0 3.75590 −2.64565 0 2.83947 4.21266 0 −6.34730
1.16 2.59041 0 4.71020 −1.03652 0 −1.63930 7.02052 0 −2.68500
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(1\)
\(47\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\(T_{2}^{16} - \cdots\)
\(T_{5}^{16} - \cdots\)