Properties

Label 462.2.k.g
Level $462$
Weight $2$
Character orbit 462.k
Analytic conductor $3.689$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 6 x^{19} + 19 x^{18} - 42 x^{17} + 62 x^{16} - 42 x^{15} - 25 x^{14} + 6 x^{13} + 445 x^{12} - 1764 x^{11} + 3864 x^{10} - 5292 x^{9} + 4005 x^{8} + 162 x^{7} - 2025 x^{6} - 10206 x^{5} + 45198 x^{4} - 91854 x^{3} + 124659 x^{2} - 118098 x + 59049\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{7} q^{2} -\beta_{12} q^{3} -\beta_{3} q^{4} + ( 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{5} + ( \beta_{5} + \beta_{9} ) q^{6} + ( -\beta_{1} + \beta_{8} + \beta_{15} ) q^{7} + \beta_{6} q^{8} + ( -\beta_{1} - \beta_{2} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} - \beta_{19} ) q^{9} +O(q^{10})\) \( q -\beta_{7} q^{2} -\beta_{12} q^{3} -\beta_{3} q^{4} + ( 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{5} + ( \beta_{5} + \beta_{9} ) q^{6} + ( -\beta_{1} + \beta_{8} + \beta_{15} ) q^{7} + \beta_{6} q^{8} + ( -\beta_{1} - \beta_{2} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} - \beta_{19} ) q^{9} + ( -2 - \beta_{2} - \beta_{3} + \beta_{12} ) q^{10} + ( \beta_{6} + \beta_{7} ) q^{11} -\beta_{1} q^{12} + ( -1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{15} + \beta_{17} ) q^{13} + ( \beta_{4} + \beta_{13} ) q^{14} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{15} + ( -1 - \beta_{3} ) q^{16} + ( \beta_{1} + \beta_{2} - \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{12} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{18} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} - \beta_{15} - 2 \beta_{17} ) q^{19} + ( -\beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{20} + ( 2 + \beta_{3} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{21} - q^{22} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - 3 \beta_{7} + 2 \beta_{12} ) q^{23} + \beta_{5} q^{24} + ( 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{15} + \beta_{17} ) q^{25} + ( \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{26} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{15} - \beta_{17} ) q^{27} + ( -\beta_{5} - \beta_{10} - \beta_{17} ) q^{28} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} + \beta_{17} ) q^{29} + ( -3 + \beta_{2} - 3 \beta_{3} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} + \beta_{19} ) q^{30} + ( 2 - \beta_{1} - 4 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 3 \beta_{13} - 2 \beta_{14} - \beta_{15} + 2 \beta_{16} + \beta_{17} + 4 \beta_{18} ) q^{31} + ( \beta_{6} + \beta_{7} ) q^{32} -\beta_{9} q^{33} + ( -2 \beta_{1} - \beta_{2} + \beta_{5} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{15} + \beta_{17} ) q^{34} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - 2 \beta_{19} ) q^{35} + ( -\beta_{2} + \beta_{8} - \beta_{11} - \beta_{14} ) q^{36} + ( 2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{13} + \beta_{14} + 2 \beta_{15} - 2 \beta_{18} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{38} + ( 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} - \beta_{19} ) q^{39} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{40} + ( \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{10} ) q^{41} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{42} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} + 2 \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{43} + \beta_{7} q^{44} + ( 2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{13} - \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} ) q^{45} + ( -3 \beta_{3} - \beta_{5} + \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{46} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} + 4 \beta_{11} - \beta_{12} - \beta_{15} - 2 \beta_{17} - 2 \beta_{19} ) q^{47} + ( -\beta_{1} + \beta_{12} ) q^{48} + ( -2 - 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{12} + \beta_{13} + \beta_{15} + 2 \beta_{16} + 3 \beta_{17} ) q^{49} + ( -2 \beta_{5} - 3 \beta_{6} + 3 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{14} ) q^{50} + ( -\beta_{1} + \beta_{2} - \beta_{7} + 2 \beta_{8} + 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{51} + ( -2 - \beta_{3} - \beta_{4} + \beta_{8} + \beta_{9} + \beta_{15} - \beta_{17} ) q^{52} + ( -2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 5 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{15} - \beta_{17} - 2 \beta_{19} ) q^{53} + ( -1 + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{13} - \beta_{14} ) q^{54} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{12} ) q^{55} + ( -\beta_{2} + \beta_{4} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{56} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{11} - 2 \beta_{13} + \beta_{14} - 2 \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{57} + ( 4 + 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{18} ) q^{58} + ( -2 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{10} - 2 \beta_{12} ) q^{59} + ( -\beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{6} + 3 \beta_{7} - \beta_{9} + \beta_{12} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{60} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{8} + 3 \beta_{9} ) q^{61} + ( -1 - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 4 \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{17} + 4 \beta_{19} ) q^{62} + ( -3 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{16} + \beta_{17} + \beta_{19} ) q^{63} - q^{64} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 4 \beta_{7} + \beta_{10} - \beta_{12} - \beta_{15} - 2 \beta_{19} ) q^{65} + \beta_{12} q^{66} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - 2 \beta_{12} - \beta_{13} - 2 \beta_{16} ) q^{67} + ( \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} ) q^{68} + ( 3 + 2 \beta_{1} + \beta_{2} + 6 \beta_{3} + 3 \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{14} + 2 \beta_{19} ) q^{69} + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{13} - \beta_{15} - 2 \beta_{16} - \beta_{17} ) q^{70} + ( \beta_{5} - 3 \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 3 \beta_{14} ) q^{71} + ( -\beta_{5} + \beta_{7} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{18} ) q^{72} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 3 \beta_{7} + 4 \beta_{8} - \beta_{9} + 3 \beta_{10} + \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 4 \beta_{18} ) q^{73} + ( \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - \beta_{12} + 2 \beta_{13} - \beta_{15} - \beta_{17} - 2 \beta_{19} ) q^{74} + ( -5 + 3 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} + 4 \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{17} - 2 \beta_{19} ) q^{75} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{8} - \beta_{12} - 2 \beta_{15} - \beta_{17} ) q^{76} + ( -\beta_{2} - \beta_{12} - \beta_{14} ) q^{77} + ( 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} - \beta_{18} ) q^{78} + ( -\beta_{1} - \beta_{2} + 3 \beta_{5} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + \beta_{12} + \beta_{15} ) q^{79} + ( -\beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} ) q^{80} + ( -3 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} - 4 \beta_{9} + \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} - \beta_{16} + 2 \beta_{17} + 2 \beta_{19} ) q^{81} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{82} + ( 5 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{12} - 6 \beta_{13} + 3 \beta_{14} ) q^{83} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{17} - \beta_{18} ) q^{84} + ( -1 + 2 \beta_{2} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{13} + \beta_{14} - 2 \beta_{16} - 2 \beta_{18} ) q^{85} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + 2 \beta_{19} ) q^{86} + ( -2 \beta_{2} - 3 \beta_{5} - 2 \beta_{6} - \beta_{9} + \beta_{11} - \beta_{13} - \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{87} + \beta_{3} q^{88} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 5 \beta_{9} - 4 \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + 2 \beta_{17} + 2 \beta_{19} ) q^{89} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + \beta_{14} + 2 \beta_{15} + \beta_{17} + 2 \beta_{19} ) q^{90} + ( -4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{12} - 2 \beta_{15} ) q^{91} + ( \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{6} + \beta_{12} ) q^{92} + ( -3 + 6 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - \beta_{5} + 2 \beta_{7} - \beta_{8} - 5 \beta_{9} + 5 \beta_{10} - 5 \beta_{13} + 5 \beta_{14} - 2 \beta_{15} - \beta_{18} + \beta_{19} ) q^{93} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} + 2 \beta_{12} - 3 \beta_{13} + 2 \beta_{14} - 2 \beta_{16} - 4 \beta_{18} ) q^{94} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 6 \beta_{9} + \beta_{10} - 4 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} + 2 \beta_{17} + 4 \beta_{19} ) q^{95} -\beta_{9} q^{96} + ( -3 - 6 \beta_{3} - 2 \beta_{5} - 2 \beta_{9} - 2 \beta_{10} ) q^{97} + ( -1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{11} - 2 \beta_{13} + 3 \beta_{14} - \beta_{17} ) q^{98} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{13} - \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 6q^{3} + 10q^{4} - 6q^{7} - 2q^{9} + O(q^{10}) \) \( 20q - 6q^{3} + 10q^{4} - 6q^{7} - 2q^{9} - 18q^{10} - 6q^{12} - 8q^{15} - 10q^{16} + 4q^{18} + 36q^{19} + 24q^{21} - 20q^{22} - 12q^{25} - 22q^{30} + 36q^{31} - 4q^{36} + 16q^{37} + 4q^{39} - 18q^{40} + 32q^{42} + 32q^{43} + 24q^{45} + 30q^{46} - 42q^{49} - 24q^{52} - 36q^{54} - 24q^{57} + 32q^{58} - 4q^{60} + 42q^{61} - 10q^{63} - 20q^{64} + 6q^{66} - 10q^{67} - 36q^{70} - 4q^{72} + 12q^{73} - 108q^{75} + 6q^{79} + 42q^{81} + 18q^{82} + 18q^{84} - 28q^{85} + 36q^{87} - 10q^{88} - 112q^{91} - 36q^{93} + 42q^{94} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 6 x^{19} + 19 x^{18} - 42 x^{17} + 62 x^{16} - 42 x^{15} - 25 x^{14} + 6 x^{13} + 445 x^{12} - 1764 x^{11} + 3864 x^{10} - 5292 x^{9} + 4005 x^{8} + 162 x^{7} - 2025 x^{6} - 10206 x^{5} + 45198 x^{4} - 91854 x^{3} + 124659 x^{2} - 118098 x + 59049\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{19} - 6 \nu^{18} + 19 \nu^{17} - 42 \nu^{16} + 62 \nu^{15} - 42 \nu^{14} - 25 \nu^{13} + 6 \nu^{12} + 445 \nu^{11} - 1764 \nu^{10} + 3864 \nu^{9} - 5292 \nu^{8} + 4005 \nu^{7} + 162 \nu^{6} - 2025 \nu^{5} - 10206 \nu^{4} + 45198 \nu^{3} - 91854 \nu^{2} + 124659 \nu - 118098 \)\()/19683\)
\(\beta_{3}\)\(=\)\((\)\(-1501 \nu^{19} - 11100 \nu^{18} + 79886 \nu^{17} - 186672 \nu^{16} + 236878 \nu^{15} + 4362 \nu^{14} - 634019 \nu^{13} + 876018 \nu^{12} + 1843352 \nu^{11} - 8896860 \nu^{10} + 16839312 \nu^{9} - 15200136 \nu^{8} - 6755409 \nu^{7} + 33358068 \nu^{6} - 16693290 \nu^{5} - 73085652 \nu^{4} + 229184478 \nu^{3} - 345007998 \nu^{2} + 245178009 \nu + 31768362\)\()/25351704\)
\(\beta_{4}\)\(=\)\((\)\(-538 \nu^{19} + 1727 \nu^{18} - 21322 \nu^{17} + 102482 \nu^{16} - 220028 \nu^{15} + 259474 \nu^{14} + 17812 \nu^{13} - 637247 \nu^{12} + 636608 \nu^{11} + 2792384 \nu^{10} - 10975692 \nu^{9} + 19686408 \nu^{8} - 17354826 \nu^{7} - 6842565 \nu^{6} + 34447518 \nu^{5} - 11202462 \nu^{4} - 97402176 \nu^{3} + 278601930 \nu^{2} - 412074540 \nu + 308714733\)\()/8450568\)
\(\beta_{5}\)\(=\)\((\)\(-3941 \nu^{19} + 57282 \nu^{18} - 277064 \nu^{17} + 662019 \nu^{16} - 914977 \nu^{15} + 320778 \nu^{14} + 1525880 \nu^{13} - 2317281 \nu^{12} - 5701892 \nu^{11} + 29395380 \nu^{10} - 61555452 \nu^{9} + 69171228 \nu^{8} - 11854773 \nu^{7} - 85305366 \nu^{6} + 78093396 \nu^{5} + 205478127 \nu^{4} - 788081805 \nu^{3} + 1366284510 \nu^{2} - 1341278352 \nu + 510911631\)\()/33802272\)
\(\beta_{6}\)\(=\)\((\)\(-1232 \nu^{19} + 5597 \nu^{18} - 13781 \nu^{17} + 29807 \nu^{16} - 38128 \nu^{15} + 20905 \nu^{14} + 9539 \nu^{13} + 48103 \nu^{12} - 377696 \nu^{11} + 1323236 \nu^{10} - 2673888 \nu^{9} + 3415020 \nu^{8} - 2422512 \nu^{7} - 480339 \nu^{6} + 1561923 \nu^{5} + 9271179 \nu^{4} - 34327152 \nu^{3} + 64977957 \nu^{2} - 83383749 \nu + 71849511\)\()/4828896\)
\(\beta_{7}\)\(=\)\((\)\(-25957 \nu^{19} + 143919 \nu^{18} - 321337 \nu^{17} + 259002 \nu^{16} + 376723 \nu^{15} - 1654737 \nu^{14} + 1611259 \nu^{13} + 4421898 \nu^{12} - 18502708 \nu^{11} + 28682472 \nu^{10} - 12111708 \nu^{9} - 47301912 \nu^{8} + 103555899 \nu^{7} - 39769353 \nu^{6} - 203353173 \nu^{5} + 499197330 \nu^{4} - 556770105 \nu^{3} + 20008863 \nu^{2} + 863079867 \nu - 958365270\)\()/ 101406816 \)
\(\beta_{8}\)\(=\)\((\)\(-27967 \nu^{19} + 12315 \nu^{18} + 252995 \nu^{17} - 657600 \nu^{16} + 920893 \nu^{15} - 97281 \nu^{14} - 2948021 \nu^{13} + 5266932 \nu^{12} + 3850868 \nu^{11} - 29969184 \nu^{10} + 61346796 \nu^{9} - 56788704 \nu^{8} - 38767311 \nu^{7} + 153518787 \nu^{6} - 92936241 \nu^{5} - 226227168 \nu^{4} + 817623801 \nu^{3} - 1264040073 \nu^{2} + 667299627 \nu + 638910180\)\()/ 101406816 \)
\(\beta_{9}\)\(=\)\((\)\(-1795 \nu^{19} + 9627 \nu^{18} - 21937 \nu^{17} + 38256 \nu^{16} - 30839 \nu^{15} - 21261 \nu^{14} + 55495 \nu^{13} + 170544 \nu^{12} - 850012 \nu^{11} + 2086560 \nu^{10} - 3104724 \nu^{9} + 2511648 \nu^{8} - 280755 \nu^{7} - 932877 \nu^{6} - 3302613 \nu^{5} + 21356784 \nu^{4} - 48186171 \nu^{3} + 70196139 \nu^{2} - 73647225 \nu + 72748368\)\()/4828896\)
\(\beta_{10}\)\(=\)\((\)\(-16230 \nu^{19} + 123337 \nu^{18} - 452289 \nu^{17} + 1002997 \nu^{16} - 1265262 \nu^{15} + 304937 \nu^{14} + 2060487 \nu^{13} - 1708639 \nu^{12} - 11644248 \nu^{11} + 47132428 \nu^{10} - 91395192 \nu^{9} + 98000868 \nu^{8} - 17699238 \nu^{7} - 106185159 \nu^{6} + 72635103 \nu^{5} + 368996553 \nu^{4} - 1232760870 \nu^{3} + 2047560525 \nu^{2} - 2043224433 \nu + 1053650673\)\()/33802272\)
\(\beta_{11}\)\(=\)\((\)\(-77773 \nu^{19} + 134496 \nu^{18} + 62150 \nu^{17} - 578403 \nu^{16} + 1140403 \nu^{15} - 583692 \nu^{14} - 3963446 \nu^{13} + 10593261 \nu^{12} - 7037044 \nu^{11} - 17331972 \nu^{10} + 57440292 \nu^{9} - 66235932 \nu^{8} - 46075725 \nu^{7} + 193247640 \nu^{6} - 224192286 \nu^{5} - 22349439 \nu^{4} + 557608455 \nu^{3} - 1196901360 \nu^{2} + 321659586 \nu + 977831757\)\()/ 101406816 \)
\(\beta_{12}\)\(=\)\((\)\(-2234 \nu^{19} + 12045 \nu^{18} - 27746 \nu^{17} + 36660 \nu^{16} - 6520 \nu^{15} - 74616 \nu^{14} + 98336 \nu^{13} + 279033 \nu^{12} - 1282736 \nu^{11} + 2515464 \nu^{10} - 2571492 \nu^{9} - 82656 \nu^{8} + 3733470 \nu^{7} - 2192535 \nu^{6} - 9822762 \nu^{5} + 33002964 \nu^{4} - 53653428 \nu^{3} + 48032352 \nu^{2} - 13349448 \nu + 9848061\)\()/2816856\)
\(\beta_{13}\)\(=\)\((\)\(-85589 \nu^{19} + 304506 \nu^{18} - 485684 \nu^{17} - 119673 \nu^{16} + 1927079 \nu^{15} - 4048698 \nu^{14} + 701660 \nu^{13} + 14980239 \nu^{12} - 37754276 \nu^{11} + 32169732 \nu^{10} + 42981204 \nu^{9} - 175929012 \nu^{8} + 211271067 \nu^{7} + 53283474 \nu^{6} - 576670104 \nu^{5} + 864944163 \nu^{4} - 349427925 \nu^{3} - 1450422774 \nu^{2} + 2881302516 \nu - 2300135697\)\()/ 101406816 \)
\(\beta_{14}\)\(=\)\((\)\(93578 \nu^{19} - 620469 \nu^{18} + 2151005 \nu^{17} - 4614381 \nu^{16} + 5525050 \nu^{15} - 826701 \nu^{14} - 9383507 \nu^{13} + 5032575 \nu^{12} + 59906504 \nu^{11} - 221733708 \nu^{10} + 412578600 \nu^{9} - 419791428 \nu^{8} + 52577226 \nu^{7} + 475914555 \nu^{6} - 225610515 \nu^{5} - 1854776961 \nu^{4} + 5685256674 \nu^{3} - 9138902193 \nu^{2} + 8846780229 \nu - 4479437457\)\()/ 101406816 \)
\(\beta_{15}\)\(=\)\((\)\(125911 \nu^{19} - 566055 \nu^{18} + 1259389 \nu^{17} - 1667448 \nu^{16} + 437723 \nu^{15} + 3023385 \nu^{14} - 3231763 \nu^{13} - 13469688 \nu^{12} + 56722540 \nu^{11} - 112828800 \nu^{10} + 121783620 \nu^{9} - 22194144 \nu^{8} - 115379577 \nu^{7} + 70354305 \nu^{6} + 413911377 \nu^{5} - 1428453144 \nu^{4} + 2440417167 \nu^{3} - 2449643391 \nu^{2} + 1660595661 \nu - 1121931000\)\()/ 101406816 \)
\(\beta_{16}\)\(=\)\((\)\(152435 \nu^{19} - 609981 \nu^{18} + 1552223 \nu^{17} - 2696802 \nu^{16} + 2474215 \nu^{15} + 796455 \nu^{14} - 2943185 \nu^{13} - 9418854 \nu^{12} + 55561436 \nu^{11} - 145135704 \nu^{10} + 230624772 \nu^{9} - 204881544 \nu^{8} + 47527875 \nu^{7} + 124498107 \nu^{6} + 153565875 \nu^{5} - 1416627306 \nu^{4} + 3546173115 \nu^{3} - 5145807609 \nu^{2} + 5730633279 \nu - 3856962582\)\()/ 101406816 \)
\(\beta_{17}\)\(=\)\((\)\(-160799 \nu^{19} + 572295 \nu^{18} - 946949 \nu^{17} + 498360 \nu^{16} + 1645421 \nu^{15} - 4756569 \nu^{14} + 356867 \nu^{13} + 22815672 \nu^{12} - 59172428 \nu^{11} + 73261440 \nu^{10} - 5090820 \nu^{9} - 148635648 \nu^{8} + 190314801 \nu^{7} + 96843519 \nu^{6} - 738808857 \nu^{5} + 1365724152 \nu^{4} - 1265543271 \nu^{3} - 370711809 \nu^{2} + 1424334051 \nu - 523252872\)\()/ 101406816 \)
\(\beta_{18}\)\(=\)\((\)\(17987 \nu^{19} - 75384 \nu^{18} + 186410 \nu^{17} - 310323 \nu^{16} + 249247 \nu^{15} + 185688 \nu^{14} - 415238 \nu^{13} - 1313943 \nu^{12} + 7216412 \nu^{11} - 17571540 \nu^{10} + 25338180 \nu^{9} - 19008444 \nu^{8} - 1256013 \nu^{7} + 14806224 \nu^{6} + 32159862 \nu^{5} - 189421119 \nu^{4} + 412395219 \nu^{3} - 548061228 \nu^{2} + 560678274 \nu - 375510087\)\()/11267424\)
\(\beta_{19}\)\(=\)\((\)\(-83249 \nu^{19} + 337060 \nu^{18} - 747026 \nu^{17} + 1038097 \nu^{16} - 458641 \nu^{15} - 1388020 \nu^{14} + 1361690 \nu^{13} + 8009669 \nu^{12} - 32524772 \nu^{11} + 66684268 \nu^{10} - 79654428 \nu^{9} + 34860756 \nu^{8} + 35400591 \nu^{7} - 30538260 \nu^{6} - 217437318 \nu^{5} + 799358949 \nu^{4} - 1500670557 \nu^{3} + 1690294392 \nu^{2} - 1524903246 \nu + 1122121269\)\()/33802272\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{19} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{17} - 2 \beta_{15} - 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{19} - \beta_{18} - 2 \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{5} + 6 \beta_{3} + \beta_{2} + \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(-4 \beta_{18} - \beta_{17} - 2 \beta_{16} + \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + 7 \beta_{12} + \beta_{10} - 4 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - \beta_{4} + 9 \beta_{2} + 3 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-2 \beta_{18} + \beta_{17} - 2 \beta_{16} + 5 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 6 \beta_{11} + 6 \beta_{10} - 9 \beta_{9} + 6 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - 4 \beta_{2} + 3 \beta_{1} - 12\)
\(\nu^{7}\)\(=\)\(4 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} + 4 \beta_{16} - \beta_{15} + 5 \beta_{14} + 9 \beta_{13} - \beta_{12} - 8 \beta_{11} + 4 \beta_{10} - 12 \beta_{9} + 2 \beta_{8} + 12 \beta_{7} + 20 \beta_{6} - 12 \beta_{5} + 16 \beta_{4} - 14 \beta_{3} - 5 \beta_{2} - 12 \beta_{1} + 14\)
\(\nu^{8}\)\(=\)\(-10 \beta_{19} - 8 \beta_{17} + 13 \beta_{16} - 8 \beta_{15} + 37 \beta_{13} - 20 \beta_{12} + 10 \beta_{11} + 22 \beta_{10} + 23 \beta_{9} - 6 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} - 11 \beta_{5} + 16 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + 16 \beta_{1}\)
\(\nu^{9}\)\(=\)\(24 \beta_{19} - 24 \beta_{18} + 13 \beta_{17} + 24 \beta_{16} + 26 \beta_{15} + 22 \beta_{14} - 20 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} + 21 \beta_{10} - 12 \beta_{9} - 9 \beta_{8} - 66 \beta_{7} - 21 \beta_{6} + 22 \beta_{5} - 17 \beta_{4} + 22 \beta_{3} + 40 \beta_{2} + 12 \beta_{1} + 11\)
\(\nu^{10}\)\(=\)\(7 \beta_{19} - 24 \beta_{18} - 45 \beta_{15} + 61 \beta_{14} - 61 \beta_{13} + 67 \beta_{12} + 38 \beta_{10} - 108 \beta_{9} - 34 \beta_{8} - 42 \beta_{7} - 79 \beta_{5} + 36 \beta_{4} - 129 \beta_{3} + 4 \beta_{2} + 50 \beta_{1} - 129\)
\(\nu^{11}\)\(=\)\(-56 \beta_{19} + 64 \beta_{18} + 108 \beta_{17} + 32 \beta_{16} - 108 \beta_{15} - 68 \beta_{14} + 66 \beta_{13} - 100 \beta_{12} - 56 \beta_{11} - 2 \beta_{10} - 42 \beta_{9} - 76 \beta_{8} - 22 \beta_{7} + 86 \beta_{6} - 74 \beta_{5} + 108 \beta_{4} + 86 \beta_{3} - 89 \beta_{2} - 34 \beta_{1} + 172\)
\(\nu^{12}\)\(=\)\(20 \beta_{18} + 80 \beta_{17} + 20 \beta_{16} + 38 \beta_{14} + 20 \beta_{13} + 72 \beta_{12} - 8 \beta_{11} - 96 \beta_{10} + 10 \beta_{9} - 310 \beta_{8} + 20 \beta_{7} + 90 \beta_{6} - 56 \beta_{5} + 248 \beta_{4} + 140 \beta_{2} - 8 \beta_{1} + 69\)
\(\nu^{13}\)\(=\)\(72 \beta_{19} + 56 \beta_{18} + 216 \beta_{17} + 112 \beta_{16} + 108 \beta_{15} - 82 \beta_{14} + 30 \beta_{13} + 108 \beta_{12} - 144 \beta_{11} + 26 \beta_{10} + 64 \beta_{9} - 86 \beta_{8} - 330 \beta_{7} - 772 \beta_{6} + 62 \beta_{5} + 240 \beta_{4} + 510 \beta_{3} - 236 \beta_{2} - 83 \beta_{1} - 510\)
\(\nu^{14}\)\(=\)\(159 \beta_{19} + 4 \beta_{17} + 64 \beta_{16} + 4 \beta_{15} - 189 \beta_{13} + 616 \beta_{12} - 159 \beta_{11} - 81 \beta_{10} - 989 \beta_{9} - 168 \beta_{8} - 90 \beta_{7} - 90 \beta_{6} - 398 \beta_{5} + 436 \beta_{4} + 42 \beta_{3} - 220 \beta_{2} - 979 \beta_{1}\)
\(\nu^{15}\)\(=\)\(-1224 \beta_{19} + 432 \beta_{18} - 651 \beta_{17} - 432 \beta_{16} - 1302 \beta_{15} - 950 \beta_{14} + 1468 \beta_{13} - 788 \beta_{12} + 612 \beta_{11} + 790 \beta_{10} + 419 \beta_{9} + 299 \beta_{8} + 50 \beta_{7} - 191 \beta_{6} - 1143 \beta_{5} - 174 \beta_{4} + 1046 \beta_{3} - 219 \beta_{2} - 461 \beta_{1} + 523\)
\(\nu^{16}\)\(=\)\(-514 \beta_{19} + 29 \beta_{18} + 1150 \beta_{15} - 1557 \beta_{14} + 1557 \beta_{13} - 490 \beta_{12} + 115 \beta_{10} + 1879 \beta_{9} + 1411 \beta_{8} - 2740 \beta_{7} + 508 \beta_{5} - 1836 \beta_{4} + 1596 \beta_{3} + 1979 \beta_{2} - 1887 \beta_{1} + 1596\)
\(\nu^{17}\)\(=\)\(820 \beta_{19} - 372 \beta_{18} - 1879 \beta_{17} - 186 \beta_{16} + 1879 \beta_{15} - 74 \beta_{14} - 149 \beta_{13} + 225 \beta_{12} + 820 \beta_{11} + 1829 \beta_{10} + 1730 \beta_{9} + 3745 \beta_{8} + 780 \beta_{7} - 1152 \beta_{6} - 2554 \beta_{5} - 1879 \beta_{4} - 4858 \beta_{3} + 1675 \beta_{2} - 37 \beta_{1} - 9716\)
\(\nu^{18}\)\(=\)\(3390 \beta_{18} + 771 \beta_{17} + 3390 \beta_{16} - 4307 \beta_{14} + 3390 \beta_{13} - 5768 \beta_{12} + 1654 \beta_{11} + 558 \beta_{10} + 5011 \beta_{9} + 4652 \beta_{8} + 3390 \beta_{7} + 4929 \beta_{6} + 950 \beta_{5} - 1689 \beta_{4} - 4196 \beta_{2} - 6539 \beta_{1} - 4578\)
\(\nu^{19}\)\(=\)\(-5768 \beta_{19} - 950 \beta_{18} - 3830 \beta_{17} - 1900 \beta_{16} - 1915 \beta_{15} + 757 \beta_{14} - 1143 \beta_{13} - 1915 \beta_{12} + 11536 \beta_{11} - 1158 \beta_{10} + 11368 \beta_{9} - 9012 \beta_{8} + 7650 \beta_{7} + 17200 \beta_{6} + 4046 \beta_{5} - 8792 \beta_{4} - 2384 \beta_{3} + 8633 \beta_{2} - 4862 \beta_{1} + 2384\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(-1\) \(1 + \beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−0.106512 + 1.72877i
1.68119 + 0.416664i
1.69321 0.364755i
−1.53534 + 0.801712i
−0.232547 1.71637i
0.530717 + 1.64874i
1.20144 + 1.24762i
1.44390 0.956629i
−1.60269 + 0.656793i
−0.0733649 1.73050i
−0.106512 1.72877i
1.68119 0.416664i
1.69321 + 0.364755i
−1.53534 0.801712i
−0.232547 + 1.71637i
0.530717 1.64874i
1.20144 1.24762i
1.44390 + 0.956629i
−1.60269 0.656793i
−0.0733649 + 1.73050i
−0.866025 0.500000i −1.44390 0.956629i 0.500000 + 0.866025i 0.0938814 0.162607i 0.772144 + 1.55042i −2.64498 0.0638828i 1.00000i 1.16972 + 2.76256i −0.162607 + 0.0938814i
89.2 −0.866025 0.500000i −1.20144 + 1.24762i 0.500000 + 0.866025i −0.798258 + 1.38262i 1.66428 0.479752i 0.157053 2.64109i 1.00000i −0.113106 2.99787i 1.38262 0.798258i
89.3 −0.866025 0.500000i −0.530717 + 1.64874i 0.500000 + 0.866025i −0.417958 + 0.723925i 1.28398 1.16249i −1.42670 + 2.22812i 1.00000i −2.43668 1.75003i 0.723925 0.417958i
89.4 −0.866025 0.500000i 0.0733649 1.73050i 0.500000 + 0.866025i 1.79481 3.10870i −0.928784 + 1.46197i 0.833981 2.51087i 1.00000i −2.98924 0.253915i −3.10870 + 1.79481i
89.5 −0.866025 0.500000i 1.60269 + 0.656793i 0.500000 + 0.866025i 1.92560 3.33524i −1.05958 1.37015i 1.58064 + 2.12169i 1.00000i 2.13725 + 2.10527i −3.33524 + 1.92560i
89.6 0.866025 + 0.500000i −1.69321 0.364755i 0.500000 + 0.866025i 0.417958 0.723925i −1.28398 1.16249i −1.42670 + 2.22812i 1.00000i 2.73391 + 1.23521i 0.723925 0.417958i
89.7 0.866025 + 0.500000i −1.68119 + 0.416664i 0.500000 + 0.866025i 0.798258 1.38262i −1.66428 0.479752i 0.157053 2.64109i 1.00000i 2.65278 1.40098i 1.38262 0.798258i
89.8 0.866025 + 0.500000i 0.106512 + 1.72877i 0.500000 + 0.866025i −0.0938814 + 0.162607i −0.772144 + 1.55042i −2.64498 0.0638828i 1.00000i −2.97731 + 0.368271i −0.162607 + 0.0938814i
89.9 0.866025 + 0.500000i 0.232547 1.71637i 0.500000 + 0.866025i −1.92560 + 3.33524i 1.05958 1.37015i 1.58064 + 2.12169i 1.00000i −2.89184 0.798273i −3.33524 + 1.92560i
89.10 0.866025 + 0.500000i 1.53534 + 0.801712i 0.500000 + 0.866025i −1.79481 + 3.10870i 0.928784 + 1.46197i 0.833981 2.51087i 1.00000i 1.71451 + 2.46180i −3.10870 + 1.79481i
353.1 −0.866025 + 0.500000i −1.44390 + 0.956629i 0.500000 0.866025i 0.0938814 + 0.162607i 0.772144 1.55042i −2.64498 + 0.0638828i 1.00000i 1.16972 2.76256i −0.162607 0.0938814i
353.2 −0.866025 + 0.500000i −1.20144 1.24762i 0.500000 0.866025i −0.798258 1.38262i 1.66428 + 0.479752i 0.157053 + 2.64109i 1.00000i −0.113106 + 2.99787i 1.38262 + 0.798258i
353.3 −0.866025 + 0.500000i −0.530717 1.64874i 0.500000 0.866025i −0.417958 0.723925i 1.28398 + 1.16249i −1.42670 2.22812i 1.00000i −2.43668 + 1.75003i 0.723925 + 0.417958i
353.4 −0.866025 + 0.500000i 0.0733649 + 1.73050i 0.500000 0.866025i 1.79481 + 3.10870i −0.928784 1.46197i 0.833981 + 2.51087i 1.00000i −2.98924 + 0.253915i −3.10870 1.79481i
353.5 −0.866025 + 0.500000i 1.60269 0.656793i 0.500000 0.866025i 1.92560 + 3.33524i −1.05958 + 1.37015i 1.58064 2.12169i 1.00000i 2.13725 2.10527i −3.33524 1.92560i
353.6 0.866025 0.500000i −1.69321 + 0.364755i 0.500000 0.866025i 0.417958 + 0.723925i −1.28398 + 1.16249i −1.42670 2.22812i 1.00000i 2.73391 1.23521i 0.723925 + 0.417958i
353.7 0.866025 0.500000i −1.68119 0.416664i 0.500000 0.866025i 0.798258 + 1.38262i −1.66428 + 0.479752i 0.157053 + 2.64109i 1.00000i 2.65278 + 1.40098i 1.38262 + 0.798258i
353.8 0.866025 0.500000i 0.106512 1.72877i 0.500000 0.866025i −0.0938814 0.162607i −0.772144 1.55042i −2.64498 + 0.0638828i 1.00000i −2.97731 0.368271i −0.162607 0.0938814i
353.9 0.866025 0.500000i 0.232547 + 1.71637i 0.500000 0.866025i −1.92560 3.33524i 1.05958 + 1.37015i 1.58064 2.12169i 1.00000i −2.89184 + 0.798273i −3.33524 1.92560i
353.10 0.866025 0.500000i 1.53534 0.801712i 0.500000 0.866025i −1.79481 3.10870i 0.928784 1.46197i 0.833981 + 2.51087i 1.00000i 1.71451 2.46180i −3.10870 1.79481i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.k.g 20
3.b odd 2 1 inner 462.2.k.g 20
7.d odd 6 1 inner 462.2.k.g 20
21.g even 6 1 inner 462.2.k.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.k.g 20 1.a even 1 1 trivial
462.2.k.g 20 3.b odd 2 1 inner
462.2.k.g 20 7.d odd 6 1 inner
462.2.k.g 20 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):

\(T_{5}^{20} + \cdots\)
\(T_{17}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{5} \)
$3$ \( 59049 + 118098 T + 124659 T^{2} + 91854 T^{3} + 45198 T^{4} + 10206 T^{5} - 2025 T^{6} - 162 T^{7} + 4005 T^{8} + 5292 T^{9} + 3864 T^{10} + 1764 T^{11} + 445 T^{12} - 6 T^{13} - 25 T^{14} + 42 T^{15} + 62 T^{16} + 42 T^{17} + 19 T^{18} + 6 T^{19} + T^{20} \)
$5$ \( 144 + 4368 T^{2} + 124336 T^{4} + 240704 T^{6} + 358652 T^{8} + 170564 T^{10} + 59212 T^{12} + 7444 T^{14} + 677 T^{16} + 31 T^{18} + T^{20} \)
$7$ \( ( 16807 + 7203 T + 5145 T^{2} + 2940 T^{3} + 1113 T^{4} + 521 T^{5} + 159 T^{6} + 60 T^{7} + 15 T^{8} + 3 T^{9} + T^{10} )^{2} \)
$11$ \( ( 1 - T^{2} + T^{4} )^{5} \)
$13$ \( ( 9408 + 72745 T^{2} + 18292 T^{4} + 1694 T^{6} + 68 T^{8} + T^{10} )^{2} \)
$17$ \( 241864704 + 248832000 T^{2} + 165425152 T^{4} + 64568320 T^{6} + 18281408 T^{8} + 3485632 T^{10} + 486784 T^{12} + 42632 T^{14} + 2561 T^{16} + 59 T^{18} + T^{20} \)
$19$ \( ( 363312 - 39672 T - 111308 T^{2} + 12312 T^{3} + 28652 T^{4} + 396 T^{5} - 1978 T^{6} + 36 T^{7} + 106 T^{8} - 18 T^{9} + T^{10} )^{2} \)
$23$ \( 2379503694096 - 1179006346224 T^{2} + 398589987888 T^{4} - 71409434112 T^{6} + 9167131260 T^{8} - 587283372 T^{10} + 26627292 T^{12} - 698436 T^{14} + 13221 T^{16} - 141 T^{18} + T^{20} \)
$29$ \( ( 52881984 + 8727049 T^{2} + 524212 T^{4} + 14718 T^{6} + 196 T^{8} + T^{10} )^{2} \)
$31$ \( ( 845107968 - 128498304 T - 29337920 T^{2} + 5451072 T^{3} + 922688 T^{4} - 197424 T^{5} - 1912 T^{6} + 2088 T^{7} - 8 T^{8} - 18 T^{9} + T^{10} )^{2} \)
$37$ \( ( 256 + 2272 T + 17348 T^{2} + 26464 T^{3} + 37636 T^{4} - 5808 T^{5} + 3382 T^{6} + 16 T^{7} + 110 T^{8} - 8 T^{9} + T^{10} )^{2} \)
$41$ \( ( -12 + 364 T^{2} - 680 T^{4} + 284 T^{6} - 31 T^{8} + T^{10} )^{2} \)
$43$ \( ( -3712 - 936 T + 848 T^{2} - 84 T^{3} - 8 T^{4} + T^{5} )^{4} \)
$47$ \( 17175891154447171584 + 1265531795367002112 T^{2} + 60163835630657536 T^{4} + 1664951401226240 T^{6} + 33188473955072 T^{8} + 443338974464 T^{10} + 4397741344 T^{12} + 30542320 T^{14} + 155801 T^{16} + 499 T^{18} + T^{20} \)
$53$ \( 1116762968166564096 - 133417207898759232 T^{2} + 11496017683721104 T^{4} - 402973492606240 T^{6} + 9618335709760 T^{8} - 154336472464 T^{10} + 1849836412 T^{12} - 15783280 T^{14} + 99520 T^{16} - 400 T^{18} + T^{20} \)
$59$ \( 435854640545424 + 90202258447500 T^{2} + 11933505675649 T^{4} + 934648986584 T^{6} + 52958296670 T^{8} + 2080894724 T^{10} + 61065715 T^{12} + 1245832 T^{14} + 18590 T^{16} + 172 T^{18} + T^{20} \)
$61$ \( ( 29862075 - 8452245 T - 4521881 T^{2} + 1505598 T^{3} + 804479 T^{4} - 135531 T^{5} - 5299 T^{6} + 1806 T^{7} + 61 T^{8} - 21 T^{9} + T^{10} )^{2} \)
$67$ \( ( 6215049 - 1503279 T + 2846637 T^{2} + 1318572 T^{3} + 892719 T^{4} + 151947 T^{5} + 26319 T^{6} + 1272 T^{7} + 169 T^{8} + 5 T^{9} + T^{10} )^{2} \)
$71$ \( ( 1679616 + 2636388 T^{2} + 493992 T^{4} + 24048 T^{6} + 300 T^{8} + T^{10} )^{2} \)
$73$ \( ( 119675568 - 7996056 T - 15814028 T^{2} + 1068504 T^{3} + 2003740 T^{4} - 553284 T^{5} + 45590 T^{6} + 1356 T^{7} - 214 T^{8} - 6 T^{9} + T^{10} )^{2} \)
$79$ \( ( 94848121 - 25292183 T + 10133581 T^{2} - 1121956 T^{3} + 361975 T^{4} - 30349 T^{5} + 9263 T^{6} - 384 T^{7} + 113 T^{8} - 3 T^{9} + T^{10} )^{2} \)
$83$ \( ( -3378820800 + 419274496 T^{2} - 12749696 T^{4} + 147848 T^{6} - 667 T^{8} + T^{10} )^{2} \)
$89$ \( 3472494098448384 + 7021495431671808 T^{2} + 13875049082941696 T^{4} + 642536762476544 T^{6} + 19984447367552 T^{8} + 339716221184 T^{10} + 4158120112 T^{12} + 30527104 T^{14} + 162392 T^{16} + 496 T^{18} + T^{20} \)
$97$ \( ( 240267 + 812629 T^{2} + 590782 T^{4} + 22106 T^{6} + 263 T^{8} + T^{10} )^{2} \)
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