Properties

Label 630.2.i.i
Level 630
Weight 2
Character orbit 630.i
Analytic conductor 5.031
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.i (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + 2 x^{14} - 4 x^{13} + 5 x^{12} + 2 x^{11} - 35 x^{10} + 81 x^{9} - 66 x^{8} + 243 x^{7} - 315 x^{6} + 54 x^{5} + 405 x^{4} - 972 x^{3} + 1458 x^{2} - 2187 x + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{15} + 47 \nu^{14} - 161 \nu^{13} + 13 \nu^{12} - 287 \nu^{11} + 1486 \nu^{10} - 1109 \nu^{9} - 961 \nu^{8} - 408 \nu^{7} + 741 \nu^{6} + 3951 \nu^{5} - 36378 \nu^{4} + 38637 \nu^{3} + 7209 \nu^{2} + 96957 \nu - 188811 \)\()/43740\)
\(\beta_{2}\)\(=\)\((\)\(-193 \nu^{15} + 520 \nu^{14} - 2882 \nu^{13} + 5890 \nu^{12} - 14819 \nu^{11} + 17755 \nu^{10} - 22498 \nu^{9} + 10029 \nu^{8} + 46911 \nu^{7} - 149940 \nu^{6} + 260487 \nu^{5} - 621135 \nu^{4} + 664524 \nu^{3} - 767880 \nu^{2} + 412614 \nu - 448335\)\()/787320\)
\(\beta_{3}\)\(=\)\((\)\(119 \nu^{15} + 1900 \nu^{14} + 586 \nu^{13} + 1870 \nu^{12} - 22943 \nu^{11} + 10315 \nu^{10} - 1846 \nu^{9} + 40233 \nu^{8} - 5433 \nu^{7} - 121320 \nu^{6} + 522819 \nu^{5} - 514215 \nu^{4} - 164592 \nu^{3} - 1598940 \nu^{2} + 2188458 \nu + 3881925\)\()/787320\)
\(\beta_{4}\)\(=\)\((\)\( 53 \nu^{15} - 173 \nu^{14} + 37 \nu^{13} - 317 \nu^{12} + 1474 \nu^{11} - 899 \nu^{10} - 1447 \nu^{9} - 12 \nu^{8} - 717 \nu^{7} + 5841 \nu^{6} - 36702 \nu^{5} + 36207 \nu^{4} + 13041 \nu^{3} + 88209 \nu^{2} - 175689 \nu - 39366 \)\()/131220\)
\(\beta_{5}\)\(=\)\((\)\( -92 \nu^{15} + 341 \nu^{14} - 208 \nu^{13} + 209 \nu^{12} - 736 \nu^{11} - 2242 \nu^{10} + 4303 \nu^{9} - 8238 \nu^{8} + 16206 \nu^{7} - 19377 \nu^{6} + 23688 \nu^{5} + 32346 \nu^{4} - 67959 \nu^{3} - 50058 \nu^{2} - 226719 \nu + 428652 \)\()/196830\)
\(\beta_{6}\)\(=\)\((\)\(349 \nu^{15} - 532 \nu^{14} + 1106 \nu^{13} - 718 \nu^{12} + 767 \nu^{11} - 12511 \nu^{10} + 14734 \nu^{9} + 12471 \nu^{8} + 48093 \nu^{7} - 38736 \nu^{6} - 39411 \nu^{5} + 445203 \nu^{4} - 459432 \nu^{3} + 260496 \nu^{2} - 1296162 \nu + 3833811\)\()/787320\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} - 5 \nu^{11} - 2 \nu^{10} + 35 \nu^{9} - 81 \nu^{8} + 66 \nu^{7} - 243 \nu^{6} + 315 \nu^{5} - 54 \nu^{4} - 405 \nu^{3} + 972 \nu^{2} - 1458 \nu + 2187 \)\()/2187\)
\(\beta_{8}\)\(=\)\((\)\(-247 \nu^{15} + 1105 \nu^{14} - 3143 \nu^{13} + 3145 \nu^{12} - 7466 \nu^{11} + 20005 \nu^{10} - 19357 \nu^{9} - 24 \nu^{8} + 2739 \nu^{7} - 27765 \nu^{6} + 160938 \nu^{5} - 558765 \nu^{4} + 827091 \nu^{3} - 633015 \nu^{2} + 1508301 \nu - 2777490\)\()/393660\)
\(\beta_{9}\)\(=\)\((\)\( 229 \nu^{15} - 385 \nu^{14} - 349 \nu^{13} - 1345 \nu^{12} + 32 \nu^{11} + 4205 \nu^{10} - 3071 \nu^{9} + 14658 \nu^{8} - 17913 \nu^{7} + 6705 \nu^{6} - 66816 \nu^{5} - 80325 \nu^{4} + 132273 \nu^{3} + 239355 \nu^{2} + 748683 \nu - 174960 \)\()/393660\)
\(\beta_{10}\)\(=\)\((\)\(-727 \nu^{15} + 982 \nu^{14} - 4148 \nu^{13} + 9448 \nu^{12} - 3971 \nu^{11} - 899 \nu^{10} + 3608 \nu^{9} - 57327 \nu^{8} + 121413 \nu^{7} - 249354 \nu^{6} + 257463 \nu^{5} - 23193 \nu^{4} + 580446 \nu^{3} - 92826 \nu^{2} - 1892484 \nu + 234009\)\()/787320\)
\(\beta_{11}\)\(=\)\((\)\(-619 \nu^{15} - 1520 \nu^{14} + 514 \nu^{13} + 6550 \nu^{12} + 4903 \nu^{11} - 935 \nu^{10} - 33094 \nu^{9} + 11127 \nu^{8} - 26547 \nu^{7} - 103500 \nu^{6} - 1899 \nu^{5} + 397035 \nu^{4} + 1028052 \nu^{3} - 680400 \nu^{2} - 1853118 \nu - 2219805\)\()/787320\)
\(\beta_{12}\)\(=\)\((\)\(-503 \nu^{15} + 905 \nu^{14} + 1193 \nu^{13} - 55 \nu^{12} + 4346 \nu^{11} - 24115 \nu^{10} + 41647 \nu^{9} - 55416 \nu^{8} + 63771 \nu^{7} - 44865 \nu^{6} - 19458 \nu^{5} + 556335 \nu^{4} - 1062801 \nu^{3} + 786105 \nu^{2} - 2186271 \nu + 3171150\)\()/393660\)
\(\beta_{13}\)\(=\)\((\)\(499 \nu^{15} - 367 \nu^{14} + 1091 \nu^{13} - 3523 \nu^{12} + 5522 \nu^{11} - 7801 \nu^{10} + 15379 \nu^{9} + 636 \nu^{8} + 15513 \nu^{7} + 59139 \nu^{6} - 182646 \nu^{5} + 180333 \nu^{4} - 487377 \nu^{3} + 840051 \nu^{2} - 629127 \nu + 1570266\)\()/393660\)
\(\beta_{14}\)\(=\)\((\)\(1291 \nu^{15} - 88 \nu^{14} - 286 \nu^{13} - 2902 \nu^{12} - 11947 \nu^{11} + 33671 \nu^{10} - 37874 \nu^{9} + 106629 \nu^{8} - 84333 \nu^{7} + 119916 \nu^{6} + 163791 \nu^{5} - 967923 \nu^{4} + 577692 \nu^{3} - 1334556 \nu^{2} + 4547502 \nu - 640791\)\()/787320\)
\(\beta_{15}\)\(=\)\((\)\(376 \nu^{15} - 883 \nu^{14} + 269 \nu^{13} - 1717 \nu^{12} + 3413 \nu^{11} + 3176 \nu^{10} - 11159 \nu^{9} + 22839 \nu^{8} - 32178 \nu^{7} + 53361 \nu^{6} - 140499 \nu^{5} - 13338 \nu^{4} + 190107 \nu^{3} + 17739 \nu^{2} + 271917 \nu - 968841\)\()/196830\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} + \beta_{14} - \beta_{13} + \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + \beta_{3} - \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} - 5 \beta_{14} + 5 \beta_{13} - 6 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 2 \beta_{1} + 7\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{15} - \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - \beta_{10} + \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + 10 \beta_{6} - 4 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + 5 \beta_{1} - 7\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-7 \beta_{15} - 2 \beta_{14} - \beta_{13} - 9 \beta_{12} - 6 \beta_{11} - 2 \beta_{10} - 10 \beta_{9} + 7 \beta_{8} + 18 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} + 17 \beta_{4} + 2 \beta_{3} - 9 \beta_{2} - 8 \beta_{1} - 23\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-2 \beta_{15} - 22 \beta_{14} + \beta_{13} - 12 \beta_{12} - 18 \beta_{11} - 4 \beta_{10} - 23 \beta_{9} + 17 \beta_{8} - 6 \beta_{7} + 22 \beta_{6} - 34 \beta_{5} + \beta_{4} + 4 \beta_{3} - 6 \beta_{2} - 10 \beta_{1} + 32\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-40 \beta_{15} + 37 \beta_{14} + 2 \beta_{13} - 21 \beta_{12} + 15 \beta_{11} - 5 \beta_{10} + 17 \beta_{9} - 41 \beta_{8} - 27 \beta_{7} - 4 \beta_{6} + 67 \beta_{5} - 25 \beta_{4} - 52 \beta_{3} + 21 \beta_{2} + 13 \beta_{1} - 26\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-23 \beta_{15} + 47 \beta_{14} + 25 \beta_{13} + 27 \beta_{12} + 36 \beta_{11} - 52 \beta_{10} + 16 \beta_{9} + 68 \beta_{8} + 81 \beta_{7} + 25 \beta_{6} - \beta_{5} + 64 \beta_{4} - 20 \beta_{3} + 18 \beta_{2} - 70 \beta_{1} - 163\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(80 \beta_{15} - 59 \beta_{14} + 95 \beta_{13} - 60 \beta_{12} + 37 \beta_{10} - 55 \beta_{9} + 28 \beta_{8} + 117 \beta_{7} - 10 \beta_{6} + 37 \beta_{5} - 256 \beta_{4} + 11 \beta_{3} - 57 \beta_{2} - 47 \beta_{1} + 97\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(82 \beta_{15} + 197 \beta_{14} - 32 \beta_{13} + 147 \beta_{12} + 171 \beta_{11} - 91 \beta_{10} - 26 \beta_{9} - 34 \beta_{8} + 195 \beta_{7} + 127 \beta_{6} + 176 \beta_{5} - 224 \beta_{4} - 134 \beta_{3} + 42 \beta_{2} + 212 \beta_{1} + 101\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-106 \beta_{15} + 520 \beta_{14} + 8 \beta_{13} + 117 \beta_{12} + 228 \beta_{11} + 52 \beta_{10} + 80 \beta_{9} - 128 \beta_{8} + 279 \beta_{7} - 259 \beta_{6} + 394 \beta_{5} + 116 \beta_{4} - 259 \beta_{3} - 54 \beta_{2} - 35 \beta_{1} - 59\)\()/3\)
\(\nu^{12}\)\(=\)\((\)\(232 \beta_{15} - 4 \beta_{14} - 8 \beta_{13} + 483 \beta_{12} + 18 \beta_{11} + 482 \beta_{10} + 76 \beta_{9} - 37 \beta_{8} - 969 \beta_{7} - 248 \beta_{6} - 844 \beta_{5} - 449 \beta_{4} + 184 \beta_{3} + 66 \beta_{2} + 179 \beta_{1} + 743\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(-598 \beta_{15} + 478 \beta_{14} + 929 \beta_{13} - 435 \beta_{12} + 1032 \beta_{11} + 157 \beta_{10} + 431 \beta_{9} - 1283 \beta_{8} - 1008 \beta_{7} - 1354 \beta_{6} + 1444 \beta_{5} - 1573 \beta_{4} - 394 \beta_{3} + 273 \beta_{2} + 742 \beta_{1} + 532\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(-32 \beta_{15} + 713 \beta_{14} + 934 \beta_{13} + 1575 \beta_{12} + 1368 \beta_{11} - 403 \beta_{10} + 655 \beta_{9} + 1337 \beta_{8} - 585 \beta_{7} - 704 \beta_{6} - \beta_{5} + 1747 \beta_{4} + 970 \beta_{3} - 234 \beta_{2} + 119 \beta_{1} - 2152\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(2627 \beta_{15} - 2318 \beta_{14} + 617 \beta_{13} + 174 \beta_{12} - 1548 \beta_{11} + 3871 \beta_{10} - 1504 \beta_{9} - 350 \beta_{8} - 3375 \beta_{7} - 172 \beta_{6} - 3527 \beta_{5} - 805 \beta_{4} + 3854 \beta_{3} - 2397 \beta_{2} + 2005 \beta_{1} - 1532\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.579249 1.63232i
0.702194 1.58333i
−1.73198 0.0153002i
1.52536 0.820531i
−1.03843 + 1.38624i
−0.803168 + 1.53458i
1.67659 + 0.434811i
0.748691 + 1.56188i
−0.579249 + 1.63232i
0.702194 + 1.58333i
−1.73198 + 0.0153002i
1.52536 + 0.820531i
−1.03843 1.38624i
−0.803168 1.53458i
1.67659 0.434811i
0.748691 1.56188i
−1.00000 −1.70326 + 0.314515i 1.00000 0.500000 + 0.866025i 1.70326 0.314515i 2.48140 + 0.917950i −1.00000 2.80216 1.07140i −0.500000 0.866025i
121.2 −1.00000 −1.02010 + 1.39978i 1.00000 0.500000 + 0.866025i 1.02010 1.39978i −2.52336 + 0.795395i −1.00000 −0.918776 2.85585i −0.500000 0.866025i
121.3 −1.00000 −0.879242 1.49229i 1.00000 0.500000 + 0.866025i 0.879242 + 1.49229i 2.58337 0.571125i −1.00000 −1.45387 + 2.62417i −0.500000 0.866025i
121.4 −1.00000 0.0520808 + 1.73127i 1.00000 0.500000 + 0.866025i −0.0520808 1.73127i 0.226513 + 2.63604i −1.00000 −2.99458 + 0.180332i −0.500000 0.866025i
121.5 −1.00000 0.681302 1.59243i 1.00000 0.500000 + 0.866025i −0.681302 + 1.59243i −1.52280 + 2.16358i −1.00000 −2.07165 2.16985i −0.500000 0.866025i
121.6 −1.00000 0.927397 1.46285i 1.00000 0.500000 + 0.866025i −0.927397 + 1.46285i 0.832221 2.51146i −1.00000 −1.27987 2.71329i −0.500000 0.866025i
121.7 −1.00000 1.21485 + 1.23456i 1.00000 0.500000 + 0.866025i −1.21485 1.23456i −1.09594 2.40809i −1.00000 −0.0482768 + 2.99961i −0.500000 0.866025i
121.8 −1.00000 1.72697 0.132553i 1.00000 0.500000 + 0.866025i −1.72697 + 0.132553i 1.01860 + 2.44181i −1.00000 2.96486 0.457832i −0.500000 0.866025i
151.1 −1.00000 −1.70326 0.314515i 1.00000 0.500000 0.866025i 1.70326 + 0.314515i 2.48140 0.917950i −1.00000 2.80216 + 1.07140i −0.500000 + 0.866025i
151.2 −1.00000 −1.02010 1.39978i 1.00000 0.500000 0.866025i 1.02010 + 1.39978i −2.52336 0.795395i −1.00000 −0.918776 + 2.85585i −0.500000 + 0.866025i
151.3 −1.00000 −0.879242 + 1.49229i 1.00000 0.500000 0.866025i 0.879242 1.49229i 2.58337 + 0.571125i −1.00000 −1.45387 2.62417i −0.500000 + 0.866025i
151.4 −1.00000 0.0520808 1.73127i 1.00000 0.500000 0.866025i −0.0520808 + 1.73127i 0.226513 2.63604i −1.00000 −2.99458 0.180332i −0.500000 + 0.866025i
151.5 −1.00000 0.681302 + 1.59243i 1.00000 0.500000 0.866025i −0.681302 1.59243i −1.52280 2.16358i −1.00000 −2.07165 + 2.16985i −0.500000 + 0.866025i
151.6 −1.00000 0.927397 + 1.46285i 1.00000 0.500000 0.866025i −0.927397 1.46285i 0.832221 + 2.51146i −1.00000 −1.27987 + 2.71329i −0.500000 + 0.866025i
151.7 −1.00000 1.21485 1.23456i 1.00000 0.500000 0.866025i −1.21485 + 1.23456i −1.09594 + 2.40809i −1.00000 −0.0482768 2.99961i −0.500000 + 0.866025i
151.8 −1.00000 1.72697 + 0.132553i 1.00000 0.500000 0.866025i −1.72697 0.132553i 1.01860 2.44181i −1.00000 2.96486 + 0.457832i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.8
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\(T_{11}^{16} - \cdots\)
\(T_{13}^{16} - \cdots\)