Properties

Label 63.2.i.b
Level 63
Weight 2
Character orbit 63.i
Analytic conductor 0.503
Analytic rank 0
Dimension 10
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 63.i (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 7 x^{8} - 4 x^{7} + 34 x^{6} - 19 x^{5} + 64 x^{4} - x^{3} + 64 x^{2} - 21 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709 \)\()/72795\)
\(\beta_{3}\)\(=\)\((\)\( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232 \)\()/72795\)
\(\beta_{4}\)\(=\)\((\)\( -4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583 \)\()/218385\)
\(\beta_{5}\)\(=\)\((\)\( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546 \)\()/218385\)
\(\beta_{6}\)\(=\)\((\)\( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156 \)\()/72795\)
\(\beta_{7}\)\(=\)\((\)\( -840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514 \)\()/14559\)
\(\beta_{8}\)\(=\)\((\)\( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856 \)\()/43677\)
\(\beta_{9}\)\(=\)\((\)\( -18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101 \)\()/218385\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{9} + 5 \beta_{8} + \beta_{7} - 12 \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_{2} + 5 \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(-5 \beta_{9} - \beta_{8} - \beta_{7} - 7 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(6 \beta_{9} - 8 \beta_{8} - 14 \beta_{7} + 16 \beta_{5} + 9 \beta_{4} - 7 \beta_{3} + 22 \beta_{2} + 51\)
\(\nu^{7}\)\(=\)\(\beta_{9} - 31 \beta_{8} - 8 \beta_{7} + 31 \beta_{5} - 30 \beta_{4} + 8 \beta_{3} + \beta_{2} + 43 \beta_{1}\)
\(\nu^{8}\)\(=\)\(75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + 76 \beta_{3} + 8 \beta_{2} - 112 \beta_{1}\)
\(\nu^{9}\)\(=\)\(95 \beta_{9} + 189 \beta_{8} + 94 \beta_{7} + 37 \beta_{5} + 84 \beta_{4} + 47 \beta_{3} - 194 \beta_{2} - 3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1 + \beta_{6}\) \(1 + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.827154 1.43267i
−1.04536 + 1.81062i
−0.539982 + 0.935277i
0.187540 0.324828i
1.07065 1.85442i
1.07065 + 1.85442i
0.187540 + 0.324828i
−0.539982 0.935277i
−1.04536 1.81062i
0.827154 + 1.43267i
2.09548i −1.72861 0.109097i −2.39104 −1.04492 1.80985i −0.228612 + 3.62227i 2.60068 + 0.486271i 0.819421i 2.97620 + 0.377174i −3.79250 + 2.18960i
5.2 1.51009i 0.811070 + 1.53041i −0.280386 −0.387938 0.671929i 2.31107 1.22479i −2.46849 + 0.952131i 2.59678i −1.68433 + 2.48254i −1.01468 + 0.585823i
5.3 0.293869i −1.65249 + 0.518912i 1.91364 1.53014 + 2.65027i −0.152492 0.485617i −1.41763 2.23391i 1.15010i 2.46146 1.71499i −0.778834 + 0.449660i
5.4 0.718167i −0.271473 1.71064i 1.48424 −0.723774 1.25361i 1.22853 0.194963i 0.182786 + 2.63943i 2.50226i −2.85261 + 0.928786i 0.900304 0.519791i
5.5 2.59354i 1.34151 1.09561i −4.72645 0.626493 + 1.08512i 2.84151 + 3.47925i −1.89735 1.84393i 7.07116i 0.599280 2.93953i −2.81429 + 1.62483i
38.1 2.59354i 1.34151 + 1.09561i −4.72645 0.626493 1.08512i 2.84151 3.47925i −1.89735 + 1.84393i 7.07116i 0.599280 + 2.93953i −2.81429 1.62483i
38.2 0.718167i −0.271473 + 1.71064i 1.48424 −0.723774 + 1.25361i 1.22853 + 0.194963i 0.182786 2.63943i 2.50226i −2.85261 0.928786i 0.900304 + 0.519791i
38.3 0.293869i −1.65249 0.518912i 1.91364 1.53014 2.65027i −0.152492 + 0.485617i −1.41763 + 2.23391i 1.15010i 2.46146 + 1.71499i −0.778834 0.449660i
38.4 1.51009i 0.811070 1.53041i −0.280386 −0.387938 + 0.671929i 2.31107 + 1.22479i −2.46849 0.952131i 2.59678i −1.68433 2.48254i −1.01468 0.585823i
38.5 2.09548i −1.72861 + 0.109097i −2.39104 −1.04492 + 1.80985i −0.228612 3.62227i 2.60068 0.486271i 0.819421i 2.97620 0.377174i −3.79250 2.18960i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.5
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
63.i Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{10} + 14 T_{2}^{8} + 63 T_{2}^{6} + 101 T_{2}^{4} + 43 T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\).