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: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{3} + 10q^{4} - 6q^{5} - 13q^{7} - 9q^{8} + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{3} + 10q^{4} - 6q^{5} - 13q^{7} - 9q^{8} + 16q^{9} - 8q^{10} - 5q^{11} + 10q^{12} - 16q^{13} - 8q^{14} - 6q^{15} - 14q^{16} - q^{17} + 6q^{19} - 4q^{20} - 13q^{21} - 11q^{22} - 21q^{23} - 9q^{24} - 10q^{25} + 16q^{27} - 10q^{28} - 17q^{29} - 8q^{30} - 33q^{31} - 18q^{32} - 5q^{33} - 5q^{34} - 4q^{35} + 10q^{36} - 23q^{37} - 28q^{38} - 16q^{39} - 12q^{40} + 7q^{41} - 8q^{42} - 33q^{43} + 11q^{44} - 6q^{45} - 15q^{46} - 13q^{47} - 14q^{48} - 17q^{49} + 35q^{50} - q^{51} - 10q^{52} - 20q^{53} - 54q^{55} + 12q^{56} + 6q^{57} - 33q^{58} + 6q^{59} - 4q^{60} - 49q^{61} - 13q^{62} - 13q^{63} - 35q^{64} + 6q^{65} - 11q^{66} - 4q^{67} - 14q^{68} - 21q^{69} - 33q^{70} - 29q^{71} - 9q^{72} - 21q^{73} + 22q^{74} - 10q^{75} + 10q^{76} - 21q^{77} - 70q^{79} - 8q^{80} + 16q^{81} - 10q^{82} + 5q^{83} - 10q^{84} + 14q^{85} + 29q^{86} - 17q^{87} - 45q^{88} - 8q^{89} - 8q^{90} + 13q^{91} - 29q^{92} - 33q^{93} + 12q^{94} - 45q^{95} - 18q^{96} - 30q^{97} + 15q^{98} - 5q^{99} + O(q^{100}) \)

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

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

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.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4017.2.a.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.e 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

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