Properties

Label 4017.2.a.e
Level 4017
Weight 2
Character orbit 4017.a
Self dual Yes
Analytic conductor 32.076
Analytic rank 1
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4017.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0759064919\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(21\) \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut +\mathstrut \) \(177\) \(x^{12}\mathstrut +\mathstrut \) \(45\) \(x^{11}\mathstrut -\mathstrut \) \(763\) \(x^{10}\mathstrut -\mathstrut \) \(251\) \(x^{9}\mathstrut +\mathstrut \) \(1771\) \(x^{8}\mathstrut +\mathstrut \) \(639\) \(x^{7}\mathstrut -\mathstrut \) \(2118\) \(x^{6}\mathstrut -\mathstrut \) \(710\) \(x^{5}\mathstrut +\mathstrut \) \(1113\) \(x^{4}\mathstrut +\mathstrut \) \(243\) \(x^{3}\mathstrut -\mathstrut \) \(183\) \(x^{2}\mathstrut -\mathstrut \) \(10\) \(x\mathstrut +\mathstrut \) \(7\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{15} + 8 \nu^{14} + 62 \nu^{13} - 106 \nu^{12} - 382 \nu^{11} + 488 \nu^{10} + 1224 \nu^{9} - 853 \nu^{8} - 2219 \nu^{7} + 127 \nu^{6} + 2215 \nu^{5} + 936 \nu^{4} - 963 \nu^{3} - 513 \nu^{2} + 75 \nu + 34 \)\()/3\)
\(\beta_{5}\)\(=\)\( 3 \nu^{15} - 3 \nu^{14} - 59 \nu^{13} + 50 \nu^{12} + 458 \nu^{11} - 318 \nu^{10} - 1773 \nu^{9} + 955 \nu^{8} + 3544 \nu^{7} - 1328 \nu^{6} - 3365 \nu^{5} + 659 \nu^{4} + 1142 \nu^{3} - 10 \nu^{2} - 71 \nu - 2 \)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{15} + 19 \nu^{14} + 193 \nu^{13} - 293 \nu^{12} - 1349 \nu^{11} + 1723 \nu^{10} + 4773 \nu^{9} - 4832 \nu^{8} - 8881 \nu^{7} + 6497 \nu^{6} + 7997 \nu^{5} - 3558 \nu^{4} - 2598 \nu^{3} + 468 \nu^{2} + 159 \nu - 19 \)\()/3\)
\(\beta_{7}\)\(=\)\( -3 \nu^{15} + 2 \nu^{14} + 63 \nu^{13} - 40 \nu^{12} - 512 \nu^{11} + 299 \nu^{10} + 2035 \nu^{9} - 1039 \nu^{8} - 4091 \nu^{7} + 1667 \nu^{6} + 3800 \nu^{5} - 995 \nu^{4} - 1188 \nu^{3} + 66 \nu^{2} + 58 \nu + 3 \)
\(\beta_{8}\)\(=\)\((\)\( -16 \nu^{15} + 32 \nu^{14} + 272 \nu^{13} - 499 \nu^{12} - 1831 \nu^{11} + 2990 \nu^{10} + 6195 \nu^{9} - 8665 \nu^{8} - 10901 \nu^{7} + 12397 \nu^{6} + 9055 \nu^{5} - 7776 \nu^{4} - 2484 \nu^{3} + 1488 \nu^{2} + 111 \nu - 59 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( 16 \nu^{15} - 29 \nu^{14} - 281 \nu^{13} + 457 \nu^{12} + 1960 \nu^{11} - 2765 \nu^{10} - 6882 \nu^{9} + 8056 \nu^{8} + 12578 \nu^{7} - 11446 \nu^{6} - 10882 \nu^{5} + 6885 \nu^{4} + 3162 \nu^{3} - 1113 \nu^{2} - 132 \nu + 32 \)\()/3\)
\(\beta_{10}\)\(=\)\((\)\( 14 \nu^{15} - 25 \nu^{14} - 253 \nu^{13} + 416 \nu^{12} + 1802 \nu^{11} - 2689 \nu^{10} - 6369 \nu^{9} + 8492 \nu^{8} + 11431 \nu^{7} - 13367 \nu^{6} - 9239 \nu^{5} + 9309 \nu^{4} + 2103 \nu^{3} - 1977 \nu^{2} + 82 \)\()/3\)
\(\beta_{11}\)\(=\)\((\)\( -23 \nu^{15} + 46 \nu^{14} + 388 \nu^{13} - 704 \nu^{12} - 2600 \nu^{11} + 4108 \nu^{10} + 8817 \nu^{9} - 11453 \nu^{8} - 15745 \nu^{7} + 15428 \nu^{6} + 13583 \nu^{5} - 8709 \nu^{4} - 4104 \nu^{3} + 1341 \nu^{2} + 207 \nu - 43 \)\()/3\)
\(\beta_{12}\)\(=\)\((\)\( -20 \nu^{15} + 34 \nu^{14} + 367 \nu^{13} - 572 \nu^{12} - 2654 \nu^{11} + 3739 \nu^{10} + 9525 \nu^{9} - 11933 \nu^{8} - 17398 \nu^{7} + 18947 \nu^{6} + 14477 \nu^{5} - 13257 \nu^{4} - 3672 \nu^{3} + 2805 \nu^{2} + 120 \nu - 115 \)\()/3\)
\(\beta_{13}\)\(=\)\( -12 \nu^{15} + 22 \nu^{14} + 212 \nu^{13} - 353 \nu^{12} - 1484 \nu^{11} + 2187 \nu^{10} + 5207 \nu^{9} - 6579 \nu^{8} - 9445 \nu^{7} + 9793 \nu^{6} + 8018 \nu^{5} - 6387 \nu^{4} - 2233 \nu^{3} + 1269 \nu^{2} + 101 \nu - 54 \)
\(\beta_{14}\)\(=\)\((\)\( -41 \nu^{15} + 73 \nu^{14} + 724 \nu^{13} - 1154 \nu^{12} - 5078 \nu^{11} + 7009 \nu^{10} + 17925 \nu^{9} - 20516 \nu^{8} - 32929 \nu^{7} + 29312 \nu^{6} + 28664 \nu^{5} - 17766 \nu^{4} - 8454 \nu^{3} + 2943 \nu^{2} + 378 \nu - 100 \)\()/3\)
\(\beta_{15}\)\(=\)\((\)\( -49 \nu^{15} + 77 \nu^{14} + 902 \nu^{13} - 1264 \nu^{12} - 6571 \nu^{11} + 8015 \nu^{10} + 23913 \nu^{9} - 24610 \nu^{8} - 44789 \nu^{7} + 37054 \nu^{6} + 39136 \nu^{5} - 23808 \nu^{4} - 11310 \nu^{3} + 4224 \nu^{2} + 516 \nu - 155 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(10\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{13}\mathstrut -\mathstrut \) \(10\) \(\beta_{12}\mathstrut -\mathstrut \) \(9\) \(\beta_{11}\mathstrut +\mathstrut \) \(9\) \(\beta_{10}\mathstrut -\mathstrut \) \(8\) \(\beta_{9}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(74\)
\(\nu^{7}\)\(=\)\(\beta_{15}\mathstrut -\mathstrut \) \(8\) \(\beta_{14}\mathstrut +\mathstrut \) \(11\) \(\beta_{13}\mathstrut -\mathstrut \) \(13\) \(\beta_{12}\mathstrut -\mathstrut \) \(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(20\) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(22\) \(\beta_{5}\mathstrut -\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(64\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(89\) \(\beta_{1}\mathstrut +\mathstrut \) \(78\)
\(\nu^{8}\)\(=\)\(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(74\) \(\beta_{13}\mathstrut -\mathstrut \) \(76\) \(\beta_{12}\mathstrut -\mathstrut \) \(64\) \(\beta_{11}\mathstrut +\mathstrut \) \(65\) \(\beta_{10}\mathstrut -\mathstrut \) \(51\) \(\beta_{9}\mathstrut +\mathstrut \) \(65\) \(\beta_{8}\mathstrut +\mathstrut \) \(76\) \(\beta_{7}\mathstrut -\mathstrut \) \(62\) \(\beta_{6}\mathstrut +\mathstrut \) \(78\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(141\) \(\beta_{3}\mathstrut +\mathstrut \) \(141\) \(\beta_{2}\mathstrut +\mathstrut \) \(65\) \(\beta_{1}\mathstrut +\mathstrut \) \(418\)
\(\nu^{9}\)\(=\)\(15\) \(\beta_{15}\mathstrut -\mathstrut \) \(47\) \(\beta_{14}\mathstrut +\mathstrut \) \(89\) \(\beta_{13}\mathstrut -\mathstrut \) \(116\) \(\beta_{12}\mathstrut -\mathstrut \) \(42\) \(\beta_{11}\mathstrut +\mathstrut \) \(92\) \(\beta_{10}\mathstrut -\mathstrut \) \(74\) \(\beta_{9}\mathstrut +\mathstrut \) \(90\) \(\beta_{8}\mathstrut +\mathstrut \) \(150\) \(\beta_{7}\mathstrut +\mathstrut \) \(45\) \(\beta_{6}\mathstrut +\mathstrut \) \(179\) \(\beta_{5}\mathstrut -\mathstrut \) \(74\) \(\beta_{4}\mathstrut +\mathstrut \) \(423\) \(\beta_{3}\mathstrut +\mathstrut \) \(30\) \(\beta_{2}\mathstrut +\mathstrut \) \(471\) \(\beta_{1}\mathstrut +\mathstrut \) \(555\)
\(\nu^{10}\)\(=\)\(21\) \(\beta_{15}\mathstrut +\mathstrut \) \(18\) \(\beta_{14}\mathstrut +\mathstrut \) \(491\) \(\beta_{13}\mathstrut -\mathstrut \) \(528\) \(\beta_{12}\mathstrut -\mathstrut \) \(425\) \(\beta_{11}\mathstrut +\mathstrut \) \(438\) \(\beta_{10}\mathstrut -\mathstrut \) \(302\) \(\beta_{9}\mathstrut +\mathstrut \) \(437\) \(\beta_{8}\mathstrut +\mathstrut \) \(517\) \(\beta_{7}\mathstrut -\mathstrut \) \(394\) \(\beta_{6}\mathstrut +\mathstrut \) \(555\) \(\beta_{5}\mathstrut -\mathstrut \) \(31\) \(\beta_{4}\mathstrut +\mathstrut \) \(962\) \(\beta_{3}\mathstrut +\mathstrut \) \(789\) \(\beta_{2}\mathstrut +\mathstrut \) \(441\) \(\beta_{1}\mathstrut +\mathstrut \) \(2455\)
\(\nu^{11}\)\(=\)\(155\) \(\beta_{15}\mathstrut -\mathstrut \) \(237\) \(\beta_{14}\mathstrut +\mathstrut \) \(642\) \(\beta_{13}\mathstrut -\mathstrut \) \(900\) \(\beta_{12}\mathstrut -\mathstrut \) \(404\) \(\beta_{11}\mathstrut +\mathstrut \) \(694\) \(\beta_{10}\mathstrut -\mathstrut \) \(486\) \(\beta_{9}\mathstrut +\mathstrut \) \(664\) \(\beta_{8}\mathstrut +\mathstrut \) \(1016\) \(\beta_{7}\mathstrut +\mathstrut \) \(198\) \(\beta_{6}\mathstrut +\mathstrut \) \(1304\) \(\beta_{5}\mathstrut -\mathstrut \) \(492\) \(\beta_{4}\mathstrut +\mathstrut \) \(2716\) \(\beta_{3}\mathstrut +\mathstrut \) \(307\) \(\beta_{2}\mathstrut +\mathstrut \) \(2610\) \(\beta_{1}\mathstrut +\mathstrut \) \(3786\)
\(\nu^{12}\)\(=\)\(264\) \(\beta_{15}\mathstrut +\mathstrut \) \(206\) \(\beta_{14}\mathstrut +\mathstrut \) \(3101\) \(\beta_{13}\mathstrut -\mathstrut \) \(3532\) \(\beta_{12}\mathstrut -\mathstrut \) \(2754\) \(\beta_{11}\mathstrut +\mathstrut \) \(2870\) \(\beta_{10}\mathstrut -\mathstrut \) \(1734\) \(\beta_{9}\mathstrut +\mathstrut \) \(2847\) \(\beta_{8}\mathstrut +\mathstrut \) \(3328\) \(\beta_{7}\mathstrut -\mathstrut \) \(2437\) \(\beta_{6}\mathstrut +\mathstrut \) \(3781\) \(\beta_{5}\mathstrut -\mathstrut \) \(317\) \(\beta_{4}\mathstrut +\mathstrut \) \(6326\) \(\beta_{3}\mathstrut +\mathstrut \) \(4522\) \(\beta_{2}\mathstrut +\mathstrut \) \(2923\) \(\beta_{1}\mathstrut +\mathstrut \) \(14762\)
\(\nu^{13}\)\(=\)\(1365\) \(\beta_{15}\mathstrut -\mathstrut \) \(1031\) \(\beta_{14}\mathstrut +\mathstrut \) \(4386\) \(\beta_{13}\mathstrut -\mathstrut \) \(6535\) \(\beta_{12}\mathstrut -\mathstrut \) \(3335\) \(\beta_{11}\mathstrut +\mathstrut \) \(4976\) \(\beta_{10}\mathstrut -\mathstrut \) \(3003\) \(\beta_{9}\mathstrut +\mathstrut \) \(4677\) \(\beta_{8}\mathstrut +\mathstrut \) \(6580\) \(\beta_{7}\mathstrut +\mathstrut \) \(564\) \(\beta_{6}\mathstrut +\mathstrut \) \(9025\) \(\beta_{5}\mathstrut -\mathstrut \) \(3130\) \(\beta_{4}\mathstrut +\mathstrut \) \(17236\) \(\beta_{3}\mathstrut +\mathstrut \) \(2662\) \(\beta_{2}\mathstrut +\mathstrut \) \(14911\) \(\beta_{1}\mathstrut +\mathstrut \) \(25272\)
\(\nu^{14}\)\(=\)\(2630\) \(\beta_{15}\mathstrut +\mathstrut \) \(1935\) \(\beta_{14}\mathstrut +\mathstrut \) \(19131\) \(\beta_{13}\mathstrut -\mathstrut \) \(23214\) \(\beta_{12}\mathstrut -\mathstrut \) \(17691\) \(\beta_{11}\mathstrut +\mathstrut \) \(18583\) \(\beta_{10}\mathstrut -\mathstrut \) \(9815\) \(\beta_{9}\mathstrut +\mathstrut \) \(18272\) \(\beta_{8}\mathstrut +\mathstrut \) \(20812\) \(\beta_{7}\mathstrut -\mathstrut \) \(14978\) \(\beta_{6}\mathstrut +\mathstrut \) \(25184\) \(\beta_{5}\mathstrut -\mathstrut \) \(2718\) \(\beta_{4}\mathstrut +\mathstrut \) \(40874\) \(\beta_{3}\mathstrut +\mathstrut \) \(26411\) \(\beta_{2}\mathstrut +\mathstrut \) \(19162\) \(\beta_{1}\mathstrut +\mathstrut \) \(90106\)
\(\nu^{15}\)\(=\)\(11003\) \(\beta_{15}\mathstrut -\mathstrut \) \(3451\) \(\beta_{14}\mathstrut +\mathstrut \) \(29116\) \(\beta_{13}\mathstrut -\mathstrut \) \(45771\) \(\beta_{12}\mathstrut -\mathstrut \) \(25421\) \(\beta_{11}\mathstrut +\mathstrut \) \(34660\) \(\beta_{10}\mathstrut -\mathstrut \) \(17919\) \(\beta_{9}\mathstrut +\mathstrut \) \(32144\) \(\beta_{8}\mathstrut +\mathstrut \) \(41725\) \(\beta_{7}\mathstrut -\mathstrut \) \(1023\) \(\beta_{6}\mathstrut +\mathstrut \) \(60832\) \(\beta_{5}\mathstrut -\mathstrut \) \(19571\) \(\beta_{4}\mathstrut +\mathstrut \) \(108933\) \(\beta_{3}\mathstrut +\mathstrut \) \(21059\) \(\beta_{2}\mathstrut +\mathstrut \) \(86967\) \(\beta_{1}\mathstrut +\mathstrut \) \(166649\)

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
2.54124
2.37644
1.91188
1.71723
1.56390
0.820556
0.291985
0.256478
−0.228474
−0.549353
−1.25986
−1.67713
−1.68093
−1.89635
−1.89812
−2.28950
−2.54124 1.00000 4.45789 −0.349136 −2.54124 −1.96840 −6.24610 1.00000 0.887239
1.2 −2.37644 1.00000 3.64749 1.54914 −2.37644 1.30434 −3.91517 1.00000 −3.68145
1.3 −1.91188 1.00000 1.65528 −0.339131 −1.91188 −3.26096 0.659059 1.00000 0.648377
1.4 −1.71723 1.00000 0.948894 −2.53727 −1.71723 −0.257645 1.80499 1.00000 4.35709
1.5 −1.56390 1.00000 0.445771 0.640434 −1.56390 3.30522 2.43065 1.00000 −1.00157
1.6 −0.820556 1.00000 −1.32669 3.24973 −0.820556 −1.69765 2.72973 1.00000 −2.66659
1.7 −0.291985 1.00000 −1.91474 −3.42004 −0.291985 −3.90865 1.14305 1.00000 0.998600
1.8 −0.256478 1.00000 −1.93422 −0.330746 −0.256478 1.25226 1.00904 1.00000 0.0848290
1.9 0.228474 1.00000 −1.94780 −1.93998 0.228474 1.64440 −0.901970 1.00000 −0.443234
1.10 0.549353 1.00000 −1.69821 1.27882 0.549353 −2.63834 −2.03162 1.00000 0.702521
1.11 1.25986 1.00000 −0.412758 −1.76649 1.25986 −1.24457 −3.03973 1.00000 −2.22553
1.12 1.67713 1.00000 0.812756 −1.82601 1.67713 2.67229 −1.99116 1.00000 −3.06245
1.13 1.68093 1.00000 0.825530 3.10294 1.68093 −2.88861 −1.97420 1.00000 5.21583
1.14 1.89635 1.00000 1.59615 1.10443 1.89635 −4.39855 −0.765840 1.00000 2.09438
1.15 1.89812 1.00000 1.60286 −0.521152 1.89812 0.438711 −0.753828 1.00000 −0.989209
1.16 2.28950 1.00000 3.24180 −3.89554 2.28950 −1.35384 2.84309 1.00000 −8.91883
\(n\): 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
\(3\) \(-1\)
\(13\) \(1\)
\(103\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\(T_{2}^{16} - \cdots\)
\(T_{23}^{16} + \cdots\)