Properties

Label 4017.2.a.f
Level 4017
Weight 2
Character orbit 4017.a
Self dual yes
Analytic conductor 32.076
Analytic rank 1
Dimension 19
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: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{19} - 4 x^{18} - 16 x^{17} + 77 x^{16} + 88 x^{15} - 594 x^{14} - 154 x^{13} + 2388 x^{12} - 278 x^{11} - 5460 x^{10} + 1491 x^{9} + 7285 x^{8} - 2223 x^{7} - 5579 x^{6} + 1430 x^{5} + 2261 x^{4} - 352 x^{3} - 378 x^{2} + 11 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-94088 \nu^{18} - 626428 \nu^{17} + 3886444 \nu^{16} + 9779883 \nu^{15} - 46409280 \nu^{14} - 53503587 \nu^{13} + 230717285 \nu^{12} + 115056639 \nu^{11} - 445048447 \nu^{10} - 44364492 \nu^{9} - 136140833 \nu^{8} - 125435006 \nu^{7} + 1628375777 \nu^{6} + 70345533 \nu^{5} - 1910156217 \nu^{4} + 67933382 \nu^{3} + 670807958 \nu^{2} - 65076335 \nu - 30277713\)\()/8327001\)
\(\beta_{4}\)\(=\)\((\)\(193859 \nu^{18} - 2142369 \nu^{17} - 4314 \nu^{16} + 40884782 \nu^{15} - 39656466 \nu^{14} - 310348080 \nu^{13} + 377207518 \nu^{12} + 1213875850 \nu^{11} - 1520636534 \nu^{10} - 2649906620 \nu^{9} + 3112759541 \nu^{8} + 3263812942 \nu^{7} - 3299395776 \nu^{6} - 2166951074 \nu^{5} + 1726428678 \nu^{4} + 685986403 \nu^{3} - 385111717 \nu^{2} - 92926590 \nu + 8487263\)\()/8327001\)
\(\beta_{5}\)\(=\)\((\)\(-301002 \nu^{18} + 2086417 \nu^{17} + 830096 \nu^{16} - 36097151 \nu^{15} + 49187691 \nu^{14} + 235624707 \nu^{13} - 523591869 \nu^{12} - 720016087 \nu^{11} + 2280606051 \nu^{10} + 1009320407 \nu^{9} - 5111196996 \nu^{8} - 464378927 \nu^{7} + 6113602981 \nu^{6} - 87045064 \nu^{5} - 3694969428 \nu^{4} - 38526093 \nu^{3} + 900642533 \nu^{2} + 74082323 \nu - 18473720\)\()/8327001\)
\(\beta_{6}\)\(=\)\((\)\(-505639 \nu^{18} + 943097 \nu^{17} + 11786836 \nu^{16} - 20247569 \nu^{15} - 112671033 \nu^{14} + 173915817 \nu^{13} + 571733266 \nu^{12} - 761598784 \nu^{11} - 1668257465 \nu^{10} + 1786828862 \nu^{9} + 2842754969 \nu^{8} - 2137689600 \nu^{7} - 2757965584 \nu^{6} + 1068552050 \nu^{5} + 1414007301 \nu^{4} - 61613093 \nu^{3} - 298043445 \nu^{2} - 50846708 \nu - 7102352\)\()/8327001\)
\(\beta_{7}\)\(=\)\((\)\(522452 \nu^{18} - 173763 \nu^{17} - 14520768 \nu^{16} + 6574721 \nu^{15} + 162869580 \nu^{14} - 84744672 \nu^{13} - 963497294 \nu^{12} + 519460555 \nu^{11} + 3292783051 \nu^{10} - 1680536573 \nu^{9} - 6682423192 \nu^{8} + 2902932841 \nu^{7} + 7923527727 \nu^{6} - 2506162403 \nu^{5} - 5081746752 \nu^{4} + 883967110 \nu^{3} + 1412281769 \nu^{2} - 49334292 \nu - 73754683\)\()/8327001\)
\(\beta_{8}\)\(=\)\((\)\(683216 \nu^{18} + 1334850 \nu^{17} - 24023172 \nu^{16} - 16582591 \nu^{15} + 300102768 \nu^{14} + 37272603 \nu^{13} - 1828942592 \nu^{12} + 288782608 \nu^{11} + 6037093933 \nu^{10} - 1718391077 \nu^{9} - 11061827209 \nu^{8} + 3367642315 \nu^{7} + 10924134189 \nu^{6} - 2586060788 \nu^{5} - 5271494415 \nu^{4} + 508159444 \nu^{3} + 919589882 \nu^{2} + 90613854 \nu + 8797055\)\()/8327001\)
\(\beta_{9}\)\(=\)\((\)\(878636 \nu^{18} - 2477135 \nu^{17} - 17742577 \nu^{16} + 51743466 \nu^{15} + 142470306 \nu^{14} - 438788274 \nu^{13} - 579561356 \nu^{12} + 1949589303 \nu^{11} + 1244174596 \nu^{10} - 4869475809 \nu^{9} - 1281877039 \nu^{8} + 6779610320 \nu^{7} + 352120375 \nu^{6} - 4845433314 \nu^{5} + 273029682 \nu^{4} + 1416269452 \nu^{3} - 137691452 \nu^{2} - 46280392 \nu + 9751737\)\()/8327001\)
\(\beta_{10}\)\(=\)\((\)\(-898500 \nu^{18} + 445742 \nu^{17} + 25408444 \nu^{16} - 16785145 \nu^{15} - 284915496 \nu^{14} + 214341540 \nu^{13} + 1654523163 \nu^{12} - 1301382665 \nu^{11} - 5429734446 \nu^{10} + 4181996197 \nu^{9} + 10277505300 \nu^{8} - 7250315182 \nu^{7} - 10915609351 \nu^{6} + 6490474048 \nu^{5} + 5904177015 \nu^{4} - 2646783813 \nu^{3} - 1196579108 \nu^{2} + 361699234 \nu - 11793478\)\()/8327001\)
\(\beta_{11}\)\(=\)\((\)\(977686 \nu^{18} - 2109846 \nu^{17} - 19576350 \nu^{16} + 40045288 \nu^{15} + 159886860 \nu^{14} - 304111437 \nu^{13} - 701472610 \nu^{12} + 1199138972 \nu^{11} + 1835184089 \nu^{10} - 2664298648 \nu^{9} - 2982909800 \nu^{8} + 3357982253 \nu^{7} + 2953197273 \nu^{6} - 2203868323 \nu^{5} - 1578545622 \nu^{4} + 499205258 \nu^{3} + 333663343 \nu^{2} + 79078185 \nu - 3170015\)\()/8327001\)
\(\beta_{12}\)\(=\)\((\)\(-1006596 \nu^{18} + 4839121 \nu^{17} + 13408784 \nu^{16} - 90912362 \nu^{15} - 38552868 \nu^{14} + 676127358 \nu^{13} - 210763494 \nu^{12} - 2570195272 \nu^{11} + 1636153110 \nu^{10} + 5380497584 \nu^{9} - 4231425600 \nu^{8} - 6214385801 \nu^{7} + 5188235767 \nu^{6} + 3726959093 \nu^{5} - 2999440299 \nu^{4} - 1002901935 \nu^{3} + 671237324 \nu^{2} + 94367681 \nu - 20840648\)\()/8327001\)
\(\beta_{13}\)\(=\)\((\)\(340719 \nu^{18} - 2041117 \nu^{17} - 2297555 \nu^{16} + 35421487 \nu^{15} - 28351011 \nu^{14} - 233045118 \nu^{13} + 371781427 \nu^{12} + 725834826 \nu^{11} - 1656310546 \nu^{10} - 1066464572 \nu^{9} + 3628535304 \nu^{8} + 562746675 \nu^{7} - 4127234417 \nu^{6} + 120589701 \nu^{5} + 2302393775 \nu^{4} - 140104997 \nu^{3} - 480758306 \nu^{2} + 7664027 \nu - 1711033\)\()/2775667\)
\(\beta_{14}\)\(=\)\((\)\(-1173795 \nu^{18} + 4091147 \nu^{17} + 20237422 \nu^{16} - 78194119 \nu^{15} - 131370000 \nu^{14} + 595242228 \nu^{13} + 398312019 \nu^{12} - 2332250702 \nu^{11} - 555378540 \nu^{10} + 5073257425 \nu^{9} + 286658199 \nu^{8} - 6164640280 \nu^{7} - 112710592 \nu^{6} + 4020429766 \nu^{5} + 306252117 \nu^{4} - 1293543924 \nu^{3} - 227428052 \nu^{2} + 163481722 \nu + 33234527\)\()/8327001\)
\(\beta_{15}\)\(=\)\((\)\(-1656344 \nu^{18} + 2850571 \nu^{17} + 37818140 \nu^{16} - 60186352 \nu^{15} - 356551428 \nu^{14} + 518525517 \nu^{13} + 1803133061 \nu^{12} - 2357316008 \nu^{11} - 5303138830 \nu^{10} + 6102042916 \nu^{9} + 9166163902 \nu^{8} - 9045153015 \nu^{7} - 8918685770 \nu^{6} + 7316823022 \nu^{5} + 4354535292 \nu^{4} - 2833548478 \nu^{3} - 808126617 \nu^{2} + 355144607 \nu + 21257525\)\()/8327001\)
\(\beta_{16}\)\(=\)\((\)\(2105177 \nu^{18} - 11038000 \nu^{17} - 22037027 \nu^{16} + 196981321 \nu^{15} - 25058403 \nu^{14} - 1357359756 \nu^{13} + 1150153099 \nu^{12} + 4592467277 \nu^{11} - 5703282992 \nu^{10} - 7991010202 \nu^{9} + 12419678693 \nu^{8} + 6739037490 \nu^{7} - 13149766900 \nu^{6} - 2090427664 \nu^{5} + 6362522892 \nu^{4} - 218917406 \nu^{3} - 1069751136 \nu^{2} + 159048874 \nu + 6865267\)\()/8327001\)
\(\beta_{17}\)\(=\)\((\)\(-855796 \nu^{18} + 3359640 \nu^{17} + 13123579 \nu^{16} - 62837734 \nu^{15} - 64697412 \nu^{14} + 464512414 \nu^{13} + 54323182 \nu^{12} - 1750271618 \nu^{11} + 521784388 \nu^{10} + 3618039992 \nu^{9} - 1830878338 \nu^{8} - 4115548654 \nu^{7} + 2467406610 \nu^{6} + 2447706825 \nu^{5} - 1492732591 \nu^{4} - 668587229 \nu^{3} + 345567051 \nu^{2} + 64549695 \nu - 10034449\)\()/2775667\)
\(\beta_{18}\)\(=\)\((\)\(-2617292 \nu^{18} + 11645805 \nu^{17} + 34882692 \nu^{16} - 210313979 \nu^{15} - 106884618 \nu^{14} + 1474350357 \nu^{13} - 434695399 \nu^{12} - 5118043786 \nu^{11} + 3503256218 \nu^{10} + 9280859786 \nu^{9} - 8597176955 \nu^{8} - 8469958429 \nu^{7} + 9654354819 \nu^{6} + 3352119782 \nu^{5} - 4978722603 \nu^{4} - 328728832 \nu^{3} + 963132781 \nu^{2} - 24618681 \nu - 25074710\)\()/8327001\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{17} + \beta_{14} + \beta_{12} - \beta_{11} + \beta_{9} - \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{17} + \beta_{14} - \beta_{13} - \beta_{8} - \beta_{6} - \beta_{4} + 7 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{18} - 9 \beta_{17} - \beta_{16} + 8 \beta_{14} - 2 \beta_{13} + 9 \beta_{12} - 7 \beta_{11} + \beta_{10} + 10 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 10 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} + 19 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(-14 \beta_{17} - 2 \beta_{16} - \beta_{15} + 12 \beta_{14} - 11 \beta_{13} + 2 \beta_{12} + 3 \beta_{10} + \beta_{9} - 12 \beta_{8} + \beta_{7} - 12 \beta_{6} + \beta_{5} - 11 \beta_{4} + 2 \beta_{3} + 46 \beta_{2} + \beta_{1} + 77\)
\(\nu^{7}\)\(=\)\(-12 \beta_{18} - 72 \beta_{17} - 14 \beta_{16} + 58 \beta_{14} - 26 \beta_{13} + 67 \beta_{12} - 44 \beta_{11} + 14 \beta_{10} + 78 \beta_{9} - 25 \beta_{8} - 10 \beta_{7} - 13 \beta_{6} - \beta_{5} - 81 \beta_{4} + 69 \beta_{3} + 93 \beta_{2} + 101 \beta_{1} + 84\)
\(\nu^{8}\)\(=\)\(-4 \beta_{18} - 138 \beta_{17} - 29 \beta_{16} - 11 \beta_{15} + 109 \beta_{14} - 98 \beta_{13} + 33 \beta_{12} - 2 \beta_{11} + 40 \beta_{10} + 21 \beta_{9} - 108 \beta_{8} + 11 \beta_{7} - 104 \beta_{6} + 11 \beta_{5} - 100 \beta_{4} + 32 \beta_{3} + 315 \beta_{2} + 17 \beta_{1} + 470\)
\(\nu^{9}\)\(=\)\(-106 \beta_{18} - 557 \beta_{17} - 135 \beta_{16} - 2 \beta_{15} + 424 \beta_{14} - 247 \beta_{13} + 479 \beta_{12} - 280 \beta_{11} + 138 \beta_{10} + 569 \beta_{9} - 233 \beta_{8} - 74 \beta_{7} - 127 \beta_{6} - 13 \beta_{5} - 615 \beta_{4} + 505 \beta_{3} + 732 \beta_{2} + 581 \beta_{1} + 673\)
\(\nu^{10}\)\(=\)\(-69 \beta_{18} - 1198 \beta_{17} - 294 \beta_{16} - 90 \beta_{15} + 905 \beta_{14} - 809 \beta_{13} + 372 \beta_{12} - 47 \beta_{11} + 385 \beta_{10} + 275 \beta_{9} - 881 \beta_{8} + 88 \beta_{7} - 811 \beta_{6} + 88 \beta_{5} - 859 \beta_{4} + 358 \beta_{3} + 2244 \beta_{2} + 197 \beta_{1} + 3082\)
\(\nu^{11}\)\(=\)\(-842 \beta_{18} - 4248 \beta_{17} - 1138 \beta_{16} - 39 \beta_{15} + 3141 \beta_{14} - 2099 \beta_{13} + 3406 \beta_{12} - 1834 \beta_{11} + 1194 \beta_{10} + 4067 \beta_{9} - 1964 \beta_{8} - 495 \beta_{7} - 1120 \beta_{6} - 117 \beta_{5} - 4562 \beta_{4} + 3638 \beta_{3} + 5625 \beta_{2} + 3549 \beta_{1} + 5298\)
\(\nu^{12}\)\(=\)\(-795 \beta_{18} - 9808 \beta_{17} - 2598 \beta_{16} - 675 \beta_{15} + 7229 \beta_{14} - 6434 \beta_{13} + 3585 \beta_{12} - 655 \beta_{11} + 3293 \beta_{10} + 2928 \beta_{9} - 6887 \beta_{8} + 618 \beta_{7} - 6073 \beta_{6} + 627 \beta_{5} - 7156 \beta_{4} + 3462 \beta_{3} + 16405 \beta_{2} + 1961 \beta_{1} + 21211\)
\(\nu^{13}\)\(=\)\(-6394 \beta_{18} - 32191 \beta_{17} - 9047 \beta_{16} - 487 \beta_{15} + 23457 \beta_{14} - 16956 \beta_{13} + 24330 \beta_{12} - 12333 \beta_{11} + 9727 \beta_{10} + 28948 \beta_{9} - 15834 \beta_{8} - 3182 \beta_{7} - 9401 \beta_{6} - 900 \beta_{5} - 33585 \beta_{4} + 26118 \beta_{3} + 42845 \beta_{2} + 22727 \beta_{1} + 41335\)
\(\nu^{14}\)\(=\)\(-7734 \beta_{18} - 77881 \beta_{17} - 21490 \beta_{16} - 4932 \beta_{15} + 56619 \beta_{14} - 50140 \beta_{13} + 31827 \beta_{12} - 7254 \beta_{11} + 26683 \beta_{10} + 27859 \beta_{9} - 52759 \beta_{8} + 4032 \beta_{7} - 44793 \beta_{6} + 4228 \beta_{5} - 58332 \beta_{4} + 30980 \beta_{3} + 121774 \beta_{2} + 18008 \beta_{1} + 150690\)
\(\nu^{15}\)\(=\)\(-47601 \beta_{18} - 243283 \beta_{17} - 69979 \beta_{16} - 5003 \beta_{15} + 175946 \beta_{14} - 133442 \beta_{13} + 175055 \beta_{12} - 84696 \beta_{11} + 76823 \beta_{10} + 206548 \beta_{9} - 124793 \beta_{8} - 20156 \beta_{7} - 76699 \beta_{6} - 6336 \beta_{5} - 247030 \beta_{4} + 187922 \beta_{3} + 325246 \beta_{2} + 151046 \beta_{1} + 320588\)
\(\nu^{16}\)\(=\)\(-68849 \beta_{18} - 607884 \beta_{17} - 171603 \beta_{16} - 35856 \beta_{15} + 438532 \beta_{14} - 386211 \beta_{13} + 269172 \beta_{12} - 70882 \beta_{11} + 210265 \beta_{10} + 247615 \beta_{9} - 400233 \beta_{8} + 24997 \beta_{7} - 329050 \beta_{6} + 27674 \beta_{5} - 467720 \beta_{4} + 264410 \beta_{3} + 911826 \beta_{2} + 157234 \beta_{1} + 1092865\)
\(\nu^{17}\)\(=\)\(-351573 \beta_{18} - 1836887 \beta_{17} - 534255 \beta_{16} - 46235 \beta_{15} + 1323034 \beta_{14} - 1035008 \beta_{13} + 1268980 \beta_{12} - 591374 \beta_{11} + 596476 \beta_{10} + 1481566 \beta_{9} - 971076 \beta_{8} - 127320 \beta_{7} - 614363 \beta_{6} - 42098 \beta_{5} - 1820672 \beta_{4} + 1358569 \beta_{3} + 2465860 \beta_{2} + 1033344 \beta_{1} + 2475244\)
\(\nu^{18}\)\(=\)\(-581913 \beta_{18} - 4696666 \beta_{17} - 1343260 \beta_{16} - 261250 \beta_{15} + 3373548 \beta_{14} - 2954518 \beta_{13} + 2207094 \beta_{12} - 641881 \beta_{11} + 1631484 \beta_{10} + 2105746 \beta_{9} - 3022494 \beta_{8} + 148509 \beta_{7} - 2419455 \beta_{6} + 178143 \beta_{5} - 3703210 \beta_{4} + 2187142 \beta_{3} + 6861491 \beta_{2} + 1326141 \beta_{1} + 8033073\)

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
2.75283
2.17652
2.10013
2.02307
1.82121
1.44458
1.21153
0.950625
0.725598
0.116858
−0.0952190
−0.569180
−0.810100
−0.929067
−1.09685
−1.49588
−1.78092
−2.08649
−2.45923
−2.75283 1.00000 5.57808 1.28742 −2.75283 −1.78893 −9.84984 1.00000 −3.54405
1.2 −2.17652 1.00000 2.73724 −0.898923 −2.17652 0.611535 −1.60461 1.00000 1.95652
1.3 −2.10013 1.00000 2.41053 −2.56145 −2.10013 −4.38780 −0.862160 1.00000 5.37937
1.4 −2.02307 1.00000 2.09279 −0.298497 −2.02307 0.750849 −0.187724 1.00000 0.603878
1.5 −1.82121 1.00000 1.31679 3.98129 −1.82121 −0.917766 1.24426 1.00000 −7.25075
1.6 −1.44458 1.00000 0.0867978 −1.93878 −1.44458 1.10404 2.76376 1.00000 2.80071
1.7 −1.21153 1.00000 −0.532186 2.05730 −1.21153 −4.52276 3.06783 1.00000 −2.49249
1.8 −0.950625 1.00000 −1.09631 −0.0164301 −0.950625 3.21492 2.94343 1.00000 0.0156189
1.9 −0.725598 1.00000 −1.47351 −3.68550 −0.725598 −0.281235 2.52037 1.00000 2.67419
1.10 −0.116858 1.00000 −1.98634 −1.39349 −0.116858 −4.81215 0.465837 1.00000 0.162841
1.11 0.0952190 1.00000 −1.99093 2.07279 0.0952190 −1.92020 −0.380013 1.00000 0.197369
1.12 0.569180 1.00000 −1.67603 1.84267 0.569180 1.78912 −2.09233 1.00000 1.04881
1.13 0.810100 1.00000 −1.34374 −3.50747 0.810100 −0.645139 −2.70876 1.00000 −2.84140
1.14 0.929067 1.00000 −1.13683 3.14855 0.929067 −4.69509 −2.91433 1.00000 2.92521
1.15 1.09685 1.00000 −0.796918 0.842996 1.09685 0.628687 −3.06780 1.00000 0.924641
1.16 1.49588 1.00000 0.237660 −0.502630 1.49588 −1.16688 −2.63625 1.00000 −0.751875
1.17 1.78092 1.00000 1.17168 −1.00354 1.78092 −0.242679 −1.47518 1.00000 −1.78722
1.18 2.08649 1.00000 2.35342 0.146235 2.08649 −2.80287 0.737414 1.00000 0.305116
1.19 2.45923 1.00000 4.04782 −2.57255 2.45923 −2.91567 5.03608 1.00000 −6.32649
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4017.2.a.f 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.f 19 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

Hecke kernels

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

\(T_{2}^{19} + \cdots\)
\(T_{23}^{19} + \cdots\)