Properties

Label 4029.2.a.e
Level 4029
Weight 2
Character orbit 4029.a
Self dual yes
Analytic conductor 32.172
Analytic rank 1
Dimension 18
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 6 x^{17} - 10 x^{16} + 120 x^{15} - 56 x^{14} - 921 x^{13} + 1181 x^{12} + 3316 x^{11} - 6280 x^{10} - 5249 x^{9} + 15005 x^{8} + 1809 x^{7} - 16711 x^{6} + 2434 x^{5} + 8758 x^{4} - 1858 x^{3} - 1942 x^{2} + 318 x + 138\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 14 \nu^{17} - 113 \nu^{16} + 69 \nu^{15} + 1603 \nu^{14} - 3539 \nu^{13} - 6970 \nu^{12} + 25822 \nu^{11} + 4682 \nu^{10} - 73431 \nu^{9} + 30548 \nu^{8} + 84496 \nu^{7} - 51518 \nu^{6} - 39620 \nu^{5} + 26524 \nu^{4} + 3936 \nu^{3} - 5801 \nu^{2} + 550 \nu + 579 \)\()/65\)
\(\beta_{4}\)\(=\)\((\)\(-43 \nu^{17} + 309 \nu^{16} + 49 \nu^{15} - 4878 \nu^{14} + 7019 \nu^{13} + 27126 \nu^{12} - 60752 \nu^{11} - 60627 \nu^{10} + 200020 \nu^{9} + 40802 \nu^{8} - 292315 \nu^{7} + 2453 \nu^{6} + 202368 \nu^{5} - 263 \nu^{4} - 58891 \nu^{3} - 3355 \nu^{2} + 5717 \nu + 752\)\()/65\)
\(\beta_{5}\)\(=\)\((\)\(2 \nu^{17} - 44 \nu^{16} + 268 \nu^{15} - 57 \nu^{14} - 4166 \nu^{13} + 8732 \nu^{12} + 19508 \nu^{11} - 69325 \nu^{10} - 19774 \nu^{9} + 212910 \nu^{8} - 65151 \nu^{7} - 282150 \nu^{6} + 127434 \nu^{5} + 175573 \nu^{4} - 67372 \nu^{3} - 46221 \nu^{2} + 10356 \nu + 3901\)\()/65\)
\(\beta_{6}\)\(=\)\((\)\(29 \nu^{17} - 14 \nu^{16} - 1249 \nu^{15} + 2391 \nu^{14} + 14200 \nu^{13} - 37290 \nu^{12} - 61335 \nu^{11} + 222852 \nu^{10} + 75782 \nu^{9} - 603297 \nu^{8} + 117768 \nu^{7} + 733177 \nu^{6} - 264772 \nu^{5} - 417145 \nu^{4} + 142289 \nu^{3} + 100910 \nu^{2} - 22153 \nu - 8052\)\()/65\)
\(\beta_{7}\)\(=\)\((\)\(-135 \nu^{17} + 747 \nu^{16} + 1306 \nu^{15} - 13046 \nu^{14} + 2862 \nu^{13} + 86395 \nu^{12} - 73626 \nu^{11} - 273530 \nu^{10} + 308629 \nu^{9} + 444545 \nu^{8} - 532954 \nu^{7} - 409050 \nu^{6} + 427849 \nu^{5} + 219953 \nu^{4} - 148926 \nu^{3} - 58277 \nu^{2} + 18011 \nu + 5672\)\()/65\)
\(\beta_{8}\)\(=\)\((\)\(112 \nu^{17} - 553 \nu^{16} - 1554 \nu^{15} + 10770 \nu^{14} + 4760 \nu^{13} - 82553 \nu^{12} + 24680 \nu^{11} + 317060 \nu^{10} - 193847 \nu^{9} - 647065 \nu^{8} + 463002 \nu^{7} + 701215 \nu^{6} - 461754 \nu^{5} - 396078 \nu^{4} + 185785 \nu^{3} + 99153 \nu^{2} - 24551 \nu - 8173\)\()/65\)
\(\beta_{9}\)\(=\)\((\)\(-122 \nu^{17} + 604 \nu^{16} + 1683 \nu^{15} - 11759 \nu^{14} - 4938 \nu^{13} + 89970 \nu^{12} - 29231 \nu^{11} - 343886 \nu^{10} + 221633 \nu^{9} + 694431 \nu^{8} - 526493 \nu^{7} - 738496 \nu^{6} + 522775 \nu^{5} + 410208 \nu^{4} - 210078 \nu^{3} - 103387 \nu^{2} + 27605 \nu + 9013\)\()/65\)
\(\beta_{10}\)\(=\)\((\)\(-155 \nu^{17} + 862 \nu^{16} + 1603 \nu^{15} - 15648 \nu^{14} + 2259 \nu^{13} + 109679 \nu^{12} - 85432 \nu^{11} - 377687 \nu^{10} + 393867 \nu^{9} + 687867 \nu^{8} - 755317 \nu^{7} - 698762 \nu^{6} + 664122 \nu^{5} + 395880 \nu^{4} - 244741 \nu^{3} - 102963 \nu^{2} + 30528 \nu + 9380\)\()/65\)
\(\beta_{11}\)\(=\)\((\)\(144 \nu^{17} - 620 \nu^{16} - 2609 \nu^{15} + 13641 \nu^{14} + 15402 \nu^{13} - 118549 \nu^{12} - 15921 \nu^{11} + 513814 \nu^{10} - 165393 \nu^{9} - 1156469 \nu^{8} + 611373 \nu^{7} + 1307034 \nu^{6} - 715802 \nu^{5} - 734691 \nu^{4} + 312192 \nu^{3} + 178552 \nu^{2} - 43208 \nu - 14371\)\()/65\)
\(\beta_{12}\)\(=\)\((\)\(-216 \nu^{17} + 982 \nu^{16} + 3465 \nu^{15} - 20078 \nu^{14} - 16733 \nu^{13} + 162347 \nu^{12} - 3776 \nu^{11} - 658414 \nu^{10} + 265503 \nu^{9} + 1406109 \nu^{8} - 788223 \nu^{7} - 1549049 \nu^{6} + 853444 \nu^{5} + 864650 \nu^{4} - 357294 \nu^{3} - 210979 \nu^{2} + 48341 \nu + 16948\)\()/65\)
\(\beta_{13}\)\(=\)\((\)\(-196 \nu^{17} + 880 \nu^{16} + 3311 \nu^{15} - 18659 \nu^{14} - 17313 \nu^{13} + 156626 \nu^{12} + 2869 \nu^{11} - 658621 \nu^{10} + 258772 \nu^{9} + 1450321 \nu^{8} - 829032 \nu^{7} - 1627926 \nu^{6} + 933448 \nu^{5} + 921384 \nu^{4} - 403218 \nu^{3} - 228693 \nu^{2} + 56247 \nu + 18934\)\()/65\)
\(\beta_{14}\)\(=\)\((\)\(22 \nu^{17} - 100 \nu^{16} - 359 \nu^{15} + 2075 \nu^{14} + 1769 \nu^{13} - 17036 \nu^{12} + 363 \nu^{11} + 70104 \nu^{10} - 28892 \nu^{9} - 151444 \nu^{8} + 87357 \nu^{7} + 167684 \nu^{6} - 95454 \nu^{5} - 93920 \nu^{4} + 40201 \nu^{3} + 23037 \nu^{2} - 5501 \nu - 1881\)\()/5\)
\(\beta_{15}\)\(=\)\((\)\(-277 \nu^{17} + 1206 \nu^{16} + 4872 \nu^{15} - 26003 \nu^{14} - 27548 \nu^{13} + 221606 \nu^{12} + 21694 \nu^{11} - 942989 \nu^{10} + 322714 \nu^{9} + 2087964 \nu^{8} - 1129034 \nu^{7} - 2330124 \nu^{6} + 1296721 \nu^{5} + 1301219 \nu^{4} - 562199 \nu^{3} - 319398 \nu^{2} + 77984 \nu + 26492\)\()/65\)
\(\beta_{16}\)\(=\)\((\)\(-432 \nu^{17} + 2146 \nu^{16} + 5890 \nu^{15} - 41521 \nu^{14} - 16670 \nu^{13} + 315542 \nu^{12} - 105000 \nu^{11} - 1198437 \nu^{10} + 771181 \nu^{9} + 2411792 \nu^{8} - 1803881 \nu^{7} - 2573717 \nu^{6} + 1778193 \nu^{5} + 1435357 \nu^{4} - 712924 \nu^{3} - 358232 \nu^{2} + 95122 \nu + 30464\)\()/65\)
\(\beta_{17}\)\(=\)\((\)\(428 \nu^{17} - 1759 \nu^{16} - 8233 \nu^{15} + 39932 \nu^{14} + 53407 \nu^{13} - 356679 \nu^{12} - 90601 \nu^{11} + 1580811 \nu^{10} - 389681 \nu^{9} - 3612191 \nu^{8} + 1731721 \nu^{7} + 4100966 \nu^{6} - 2109774 \nu^{5} - 2302226 \nu^{4} + 935821 \nu^{3} + 560667 \nu^{2} - 131226 \nu - 45468\)\()/65\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{14} + \beta_{7} + \beta_{2} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{17} - \beta_{15} + \beta_{14} + \beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} + 8 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-9 \beta_{17} + \beta_{16} - 8 \beta_{15} + 9 \beta_{14} + 2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{7} + \beta_{6} - \beta_{4} + 11 \beta_{2} + 18 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-11 \beta_{17} - \beta_{16} - 11 \beta_{15} + 13 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + 11 \beta_{10} - 2 \beta_{9} + 10 \beta_{8} + 12 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} + \beta_{3} + 58 \beta_{2} + 5 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(-68 \beta_{17} + 9 \beta_{16} - 55 \beta_{15} + 72 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 24 \beta_{11} + 14 \beta_{10} - 14 \beta_{9} + 12 \beta_{8} + 46 \beta_{7} + 10 \beta_{6} - 3 \beta_{5} - 9 \beta_{4} + \beta_{3} + 95 \beta_{2} + 91 \beta_{1} + 31\)
\(\nu^{8}\)\(=\)\(-97 \beta_{17} - 14 \beta_{16} - 91 \beta_{15} + 128 \beta_{14} + 33 \beta_{13} + 31 \beta_{12} + 8 \beta_{11} + 95 \beta_{10} - 31 \beta_{9} + 83 \beta_{8} + 106 \beta_{7} - 23 \beta_{6} - 29 \beta_{5} - 51 \beta_{4} + 11 \beta_{3} + 413 \beta_{2} + 68 \beta_{1} + 423\)
\(\nu^{9}\)\(=\)\(-496 \beta_{17} + 59 \beta_{16} - 369 \beta_{15} + 564 \beta_{14} + 74 \beta_{13} + 68 \beta_{12} + 220 \beta_{11} + 142 \beta_{10} - 145 \beta_{9} + 121 \beta_{8} + 313 \beta_{7} + 77 \beta_{6} - 51 \beta_{5} - 59 \beta_{4} + 9 \beta_{3} + 762 \beta_{2} + 503 \beta_{1} + 320\)
\(\nu^{10}\)\(=\)\(-808 \beta_{17} - 135 \beta_{16} - 691 \beta_{15} + 1146 \beta_{14} + 380 \beta_{13} + 339 \beta_{12} + 158 \beta_{11} + 755 \beta_{10} - 341 \beta_{9} + 667 \beta_{8} + 845 \beta_{7} - 179 \beta_{6} - 300 \beta_{5} - 298 \beta_{4} + 76 \beta_{3} + 2941 \beta_{2} + 635 \beta_{1} + 2563\)
\(\nu^{11}\)\(=\)\(-3610 \beta_{17} + 329 \beta_{16} - 2486 \beta_{15} + 4402 \beta_{14} + 917 \beta_{13} + 797 \beta_{12} + 1853 \beta_{11} + 1278 \beta_{10} - 1331 \beta_{9} + 1156 \beta_{8} + 2212 \beta_{7} + 556 \beta_{6} - 592 \beta_{5} - 330 \beta_{4} + 28 \beta_{3} + 5927 \beta_{2} + 2965 \beta_{1} + 2811\)
\(\nu^{12}\)\(=\)\(-6584 \beta_{17} - 1131 \beta_{16} - 5089 \beta_{15} + 9796 \beta_{14} + 3786 \beta_{13} + 3242 \beta_{12} + 2049 \beta_{11} + 5792 \beta_{10} - 3264 \beta_{9} + 5352 \beta_{8} + 6450 \beta_{7} - 1147 \beta_{6} - 2732 \beta_{5} - 1624 \beta_{4} + 364 \beta_{3} + 21058 \beta_{2} + 5097 \beta_{1} + 16233\)
\(\nu^{13}\)\(=\)\(-26459 \beta_{17} + 1550 \beta_{16} - 16953 \beta_{15} + 34359 \beta_{14} + 9597 \beta_{13} + 8023 \beta_{12} + 15112 \beta_{11} + 10842 \beta_{10} - 11482 \beta_{9} + 10629 \beta_{8} + 16051 \beta_{7} + 3997 \beta_{6} - 5882 \beta_{5} - 1533 \beta_{4} - 334 \beta_{3} + 45407 \beta_{2} + 18285 \beta_{1} + 22809\)
\(\nu^{14}\)\(=\)\(-52969 \beta_{17} - 8906 \beta_{16} - 37087 \beta_{15} + 81465 \beta_{14} + 34992 \beta_{13} + 29017 \beta_{12} + 22094 \beta_{11} + 43760 \beta_{10} - 29041 \beta_{9} + 43080 \beta_{8} + 48256 \beta_{7} - 6221 \beta_{6} - 23424 \beta_{5} - 8004 \beta_{4} + 593 \beta_{3} + 151811 \beta_{2} + 37924 \beta_{1} + 106330\)
\(\nu^{15}\)\(=\)\(-195665 \beta_{17} + 5142 \beta_{16} - 117226 \beta_{15} + 268286 \beta_{14} + 91728 \beta_{13} + 74535 \beta_{12} + 121651 \beta_{11} + 88902 \beta_{10} - 95548 \beta_{9} + 94623 \beta_{8} + 118320 \beta_{7} + 29284 \beta_{6} - 53895 \beta_{5} - 4358 \beta_{4} - 7528 \beta_{3} + 345008 \beta_{2} + 116336 \beta_{1} + 177289\)
\(\nu^{16}\)\(=\)\(-422173 \beta_{17} - 68279 \beta_{16} - 269753 \beta_{15} + 665226 \beta_{14} + 309301 \beta_{13} + 250081 \beta_{12} + 215125 \beta_{11} + 328704 \beta_{10} - 247336 \beta_{9} + 347441 \beta_{8} + 357734 \beta_{7} - 26075 \beta_{6} - 194520 \beta_{5} - 31734 \beta_{4} - 13195 \beta_{3} + 1101869 \beta_{2} + 270608 \beta_{1} + 714456\)
\(\nu^{17}\)\(=\)\(-1459248 \beta_{17} - 4804 \beta_{16} - 821669 \beta_{15} + 2094864 \beta_{14} + 829624 \beta_{13} + 659726 \beta_{12} + 973948 \beta_{11} + 713477 \beta_{10} - 777349 \beta_{9} + 820353 \beta_{8} + 879716 \beta_{7} + 220389 \beta_{6} - 470656 \beta_{5} + 19846 \beta_{4} - 96717 \beta_{3} + 2609286 \beta_{2} + 756043 \beta_{1} + 1344775\)

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
2.77821
2.62992
2.53461
1.99378
1.64567
1.59625
1.47280
1.44853
0.615823
0.498176
−0.239152
−0.540222
−0.823313
−0.926931
−2.08696
−2.08785
−2.10151
−2.40783
−2.77821 1.00000 5.71847 1.71746 −2.77821 −2.08361 −10.3307 1.00000 −4.77148
1.2 −2.62992 1.00000 4.91649 0.484241 −2.62992 0.944316 −7.67014 1.00000 −1.27352
1.3 −2.53461 1.00000 4.42426 −2.65587 −2.53461 1.73204 −6.14455 1.00000 6.73160
1.4 −1.99378 1.00000 1.97517 1.24008 −1.99378 −4.06333 0.0495043 1.00000 −2.47246
1.5 −1.64567 1.00000 0.708234 −0.419467 −1.64567 1.49188 2.12582 1.00000 0.690305
1.6 −1.59625 1.00000 0.548008 −4.15667 −1.59625 −4.26341 2.31774 1.00000 6.63507
1.7 −1.47280 1.00000 0.169132 4.03939 −1.47280 0.912411 2.69650 1.00000 −5.94920
1.8 −1.44853 1.00000 0.0982249 1.34868 −1.44853 −1.91511 2.75477 1.00000 −1.95360
1.9 −0.615823 1.00000 −1.62076 −3.91500 −0.615823 4.24139 2.22975 1.00000 2.41095
1.10 −0.498176 1.00000 −1.75182 2.85615 −0.498176 −0.756391 1.86907 1.00000 −1.42286
1.11 0.239152 1.00000 −1.94281 −0.863011 0.239152 −3.18045 −0.942930 1.00000 −0.206391
1.12 0.540222 1.00000 −1.70816 3.09244 0.540222 −3.94821 −2.00323 1.00000 1.67061
1.13 0.823313 1.00000 −1.32216 0.963284 0.823313 4.26360 −2.73517 1.00000 0.793084
1.14 0.926931 1.00000 −1.14080 −2.07435 0.926931 1.24239 −2.91130 1.00000 −1.92278
1.15 2.08696 1.00000 2.35542 −2.36854 2.08696 −0.791085 0.741746 1.00000 −4.94306
1.16 2.08785 1.00000 2.35911 −3.16874 2.08785 0.308283 0.749773 1.00000 −6.61585
1.17 2.10151 1.00000 2.41637 −0.968130 2.10151 −2.79477 0.874998 1.00000 −2.03454
1.18 2.40783 1.00000 3.79762 −0.151954 2.40783 −4.33995 4.32836 1.00000 −0.365880
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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 4029.2.a.e 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.e 18 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(-1\)
\(79\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4029))\):

\(T_{2}^{18} + \cdots\)
\(T_{5}^{18} + \cdots\)