Properties

Label 63.2.g.b
Level $63$
Weight $2$
Character orbit 63.g
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.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
Defining polynomial: \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{9} + 9 \nu^{8} - 3 \nu^{7} + 95 \nu^{6} + 18 \nu^{5} + 402 \nu^{4} - 87 \nu^{3} + 936 \nu^{2} + 342 \nu + 72 \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{9} + \nu^{8} - 12 \nu^{7} - 8 \nu^{6} - 68 \nu^{5} - 30 \nu^{4} - 123 \nu^{3} - 204 \nu^{2} - 270 \nu - 63 \)\()/63\)
\(\beta_{4}\)\(=\)\((\)\( 17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} - 27 \nu^{2} + 1395 \nu + 639 \)\()/567\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 \)\()/567\)
\(\beta_{6}\)\(=\)\((\)\( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu + 504 \)\()/567\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{9} - 12 \nu^{8} + 69 \nu^{7} - 43 \nu^{6} + 330 \nu^{5} - 219 \nu^{4} + 732 \nu^{3} - 45 \nu^{2} + 477 \nu - 306 \)\()/189\)
\(\beta_{8}\)\(=\)\((\)\( -71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} - 1458 \nu^{2} - 1476 \nu - 234 \)\()/567\)
\(\beta_{9}\)\(=\)\((\)\( -82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 \)\()/567\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + 3 \beta_{6} + \beta_{4} - \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{9} + 4 \beta_{5} - \beta_{3} - 4 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{8} - 5 \beta_{7} - 13 \beta_{6} - \beta_{4} - \beta_{3} + 4 \beta_{2} - \beta_{1}\)
\(\nu^{5}\)\(=\)\(-7 \beta_{9} + \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 19 \beta_{5} - \beta_{4} + \beta_{2} + 7\)
\(\nu^{6}\)\(=\)\(-10 \beta_{9} + 9 \beta_{8} + 15 \beta_{7} - 10 \beta_{5} - 15 \beta_{4} + 10 \beta_{3} + 9 \beta_{2} + 10 \beta_{1} + 61\)
\(\nu^{7}\)\(=\)\(11 \beta_{8} + 11 \beta_{7} + 46 \beta_{6} + 19 \beta_{4} + 43 \beta_{3} + 8 \beta_{2} + 94 \beta_{1}\)
\(\nu^{8}\)\(=\)\(73 \beta_{9} + 56 \beta_{8} + 62 \beta_{7} + 298 \beta_{6} + 76 \beta_{5} + 118 \beta_{4} - 118 \beta_{2} - 298\)
\(\nu^{9}\)\(=\)\(253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} - 478 \beta_{1} - 295\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−1.02682 + 1.77851i
−0.335166 + 0.580525i
0.247934 0.429435i
0.920620 1.59456i
1.19343 2.06709i
−1.02682 1.77851i
−0.335166 0.580525i
0.247934 + 0.429435i
0.920620 + 1.59456i
1.19343 + 2.06709i
−1.02682 + 1.77851i 1.09995 + 1.33795i −1.10873 1.92038i −0.146246 −3.50901 + 0.582422i 0.0802402 2.64453i 0.446582 −0.580240 + 2.94335i 0.150168 0.260099i
4.2 −0.335166 + 0.580525i −0.377302 1.69046i 0.775327 + 1.34291i 1.42494 1.10781 + 0.347551i 2.21529 1.44655i −2.38012 −2.71529 + 1.27563i −0.477591 + 0.827212i
4.3 0.247934 0.429435i 1.59836 0.667278i 0.877057 + 1.51911i −3.69258 0.109735 0.851830i −2.60948 0.436591i 1.86155 2.10948 2.13309i −0.915516 + 1.58572i
4.4 0.920620 1.59456i −1.58800 0.691567i −0.695084 1.20392i 1.33475 −2.56469 + 1.89549i −2.54347 + 0.728536i 1.12285 2.04347 + 2.19641i 1.22880 2.12835i
4.5 1.19343 2.06709i 0.266999 + 1.71135i −1.84857 3.20182i −2.92087 3.85615 + 1.49047i 2.35742 + 1.20106i −4.05086 −2.85742 + 0.913855i −3.48586 + 6.03769i
16.1 −1.02682 1.77851i 1.09995 1.33795i −1.10873 + 1.92038i −0.146246 −3.50901 0.582422i 0.0802402 + 2.64453i 0.446582 −0.580240 2.94335i 0.150168 + 0.260099i
16.2 −0.335166 0.580525i −0.377302 + 1.69046i 0.775327 1.34291i 1.42494 1.10781 0.347551i 2.21529 + 1.44655i −2.38012 −2.71529 1.27563i −0.477591 0.827212i
16.3 0.247934 + 0.429435i 1.59836 + 0.667278i 0.877057 1.51911i −3.69258 0.109735 + 0.851830i −2.60948 + 0.436591i 1.86155 2.10948 + 2.13309i −0.915516 1.58572i
16.4 0.920620 + 1.59456i −1.58800 + 0.691567i −0.695084 + 1.20392i 1.33475 −2.56469 1.89549i −2.54347 0.728536i 1.12285 2.04347 2.19641i 1.22880 + 2.12835i
16.5 1.19343 + 2.06709i 0.266999 1.71135i −1.84857 + 3.20182i −2.92087 3.85615 1.49047i 2.35742 1.20106i −4.05086 −2.85742 0.913855i −3.48586 6.03769i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.g.b 10
3.b odd 2 1 189.2.g.b 10
4.b odd 2 1 1008.2.t.i 10
7.b odd 2 1 441.2.g.f 10
7.c even 3 1 63.2.h.b yes 10
7.c even 3 1 441.2.f.e 10
7.d odd 6 1 441.2.f.f 10
7.d odd 6 1 441.2.h.f 10
9.c even 3 1 63.2.h.b yes 10
9.c even 3 1 567.2.e.f 10
9.d odd 6 1 189.2.h.b 10
9.d odd 6 1 567.2.e.e 10
12.b even 2 1 3024.2.t.i 10
21.c even 2 1 1323.2.g.f 10
21.g even 6 1 1323.2.f.f 10
21.g even 6 1 1323.2.h.f 10
21.h odd 6 1 189.2.h.b 10
21.h odd 6 1 1323.2.f.e 10
28.g odd 6 1 1008.2.q.i 10
36.f odd 6 1 1008.2.q.i 10
36.h even 6 1 3024.2.q.i 10
63.g even 3 1 inner 63.2.g.b 10
63.g even 3 1 3969.2.a.z 5
63.h even 3 1 441.2.f.e 10
63.h even 3 1 567.2.e.f 10
63.i even 6 1 1323.2.f.f 10
63.j odd 6 1 567.2.e.e 10
63.j odd 6 1 1323.2.f.e 10
63.k odd 6 1 441.2.g.f 10
63.k odd 6 1 3969.2.a.ba 5
63.l odd 6 1 441.2.h.f 10
63.n odd 6 1 189.2.g.b 10
63.n odd 6 1 3969.2.a.bc 5
63.o even 6 1 1323.2.h.f 10
63.s even 6 1 1323.2.g.f 10
63.s even 6 1 3969.2.a.bb 5
63.t odd 6 1 441.2.f.f 10
84.n even 6 1 3024.2.q.i 10
252.o even 6 1 3024.2.t.i 10
252.bl odd 6 1 1008.2.t.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 1.a even 1 1 trivial
63.2.g.b 10 63.g even 3 1 inner
63.2.h.b yes 10 7.c even 3 1
63.2.h.b yes 10 9.c even 3 1
189.2.g.b 10 3.b odd 2 1
189.2.g.b 10 63.n odd 6 1
189.2.h.b 10 9.d odd 6 1
189.2.h.b 10 21.h odd 6 1
441.2.f.e 10 7.c even 3 1
441.2.f.e 10 63.h even 3 1
441.2.f.f 10 7.d odd 6 1
441.2.f.f 10 63.t odd 6 1
441.2.g.f 10 7.b odd 2 1
441.2.g.f 10 63.k odd 6 1
441.2.h.f 10 7.d odd 6 1
441.2.h.f 10 63.l odd 6 1
567.2.e.e 10 9.d odd 6 1
567.2.e.e 10 63.j odd 6 1
567.2.e.f 10 9.c even 3 1
567.2.e.f 10 63.h even 3 1
1008.2.q.i 10 28.g odd 6 1
1008.2.q.i 10 36.f odd 6 1
1008.2.t.i 10 4.b odd 2 1
1008.2.t.i 10 252.bl odd 6 1
1323.2.f.e 10 21.h odd 6 1
1323.2.f.e 10 63.j odd 6 1
1323.2.f.f 10 21.g even 6 1
1323.2.f.f 10 63.i even 6 1
1323.2.g.f 10 21.c even 2 1
1323.2.g.f 10 63.s even 6 1
1323.2.h.f 10 21.g even 6 1
1323.2.h.f 10 63.o even 6 1
3024.2.q.i 10 36.h even 6 1
3024.2.q.i 10 84.n even 6 1
3024.2.t.i 10 12.b even 2 1
3024.2.t.i 10 252.o even 6 1
3969.2.a.z 5 63.g even 3 1
3969.2.a.ba 5 63.k odd 6 1
3969.2.a.bb 5 63.s even 6 1
3969.2.a.bc 5 63.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\).