Properties

Label 630.2.i.f
Level 630
Weight 2
Character orbit 630.i
Analytic conductor 5.031
Analytic rank 0
Dimension 12
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: 12.0.91830304992969.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + x^{10} + 4 x^{9} - 7 x^{8} + x^{7} + 7 x^{6} + 2 x^{5} - 28 x^{4} + 32 x^{3} + 16 x^{2} - 64 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + 2 \nu^{10} - 17 \nu^{9} + 76 \nu^{8} - 73 \nu^{7} + 15 \nu^{6} + 73 \nu^{5} - 2 \nu^{4} - 100 \nu^{3} - 112 \nu^{2} + 784 \nu - 960 \)\()/288\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} + 10 \nu^{10} - 43 \nu^{9} + 56 \nu^{8} - 11 \nu^{7} - 3 \nu^{6} - \nu^{5} + 98 \nu^{4} + 16 \nu^{3} - 344 \nu^{2} + 992 \nu - 672 \)\()/288\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{10} - 4 \nu^{9} + 5 \nu^{8} - 2 \nu^{7} + \nu^{6} - 5 \nu^{5} + 13 \nu^{4} + 8 \nu^{3} - 36 \nu^{2} + 60 \nu - 56 \)\()/24\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{11} - 4 \nu^{10} - 9 \nu^{9} + 22 \nu^{8} - 21 \nu^{7} - 19 \nu^{6} + 31 \nu^{5} + 36 \nu^{4} - 128 \nu^{3} + 16 \nu^{2} + 208 \nu - 288 \)\()/96\)
\(\beta_{5}\)\(=\)\((\)\( 9 \nu^{11} + 10 \nu^{10} - 71 \nu^{9} + 104 \nu^{8} - 7 \nu^{7} - 131 \nu^{6} + 51 \nu^{5} + 182 \nu^{4} - 4 \nu^{3} - 488 \nu^{2} + 1264 \nu - 640 \)\()/288\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{11} - 11 \nu^{10} + 13 \nu^{9} - \nu^{8} - \nu^{7} - 2 \nu^{6} + 24 \nu^{5} + 11 \nu^{4} - 94 \nu^{3} + 172 \nu^{2} - 152 \nu + 128 \)\()/144\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{11} + 15 \nu^{10} - 32 \nu^{9} + 27 \nu^{8} + 2 \nu^{7} - 7 \nu^{6} + 5 \nu^{5} + 9 \nu^{4} + 62 \nu^{3} - 300 \nu^{2} + 352 \nu - 320 \)\()/144\)
\(\beta_{8}\)\(=\)\((\)\( -10 \nu^{11} + 11 \nu^{10} - 14 \nu^{9} - 5 \nu^{8} - 4 \nu^{7} - 3 \nu^{6} - 5 \nu^{5} - 41 \nu^{4} + 188 \nu^{3} - 160 \nu^{2} + 88 \nu \)\()/144\)
\(\beta_{9}\)\(=\)\((\)\( 7 \nu^{11} - 20 \nu^{10} + 17 \nu^{9} + 14 \nu^{8} - 35 \nu^{7} - 3 \nu^{6} + 41 \nu^{5} - 10 \nu^{4} - 266 \nu^{3} + 208 \nu^{2} - 16 \nu - 288 \)\()/144\)
\(\beta_{10}\)\(=\)\((\)\( 7 \nu^{11} - 16 \nu^{10} + 31 \nu^{9} - 38 \nu^{8} + 11 \nu^{7} + 13 \nu^{6} + 3 \nu^{5} - 36 \nu^{4} - 84 \nu^{3} + 376 \nu^{2} - 496 \nu + 448 \)\()/96\)
\(\beta_{11}\)\(=\)\((\)\( -25 \nu^{11} + 54 \nu^{10} - 25 \nu^{9} - 24 \nu^{8} + 55 \nu^{7} + 67 \nu^{6} - 131 \nu^{5} - 150 \nu^{4} + 436 \nu^{3} - 504 \nu^{2} + 32 \nu - 64 \)\()/288\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - \beta_{3} + 2 \beta_{2} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{1} - 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_{1} - 4\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - 1\)
\(\nu^{5}\)\(=\)\((\)\(-7 \beta_{11} - 3 \beta_{10} - 7 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 7 \beta_{3} + 2 \beta_{2} + 3 \beta_{1} - 5\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 7 \beta_{7} + 7 \beta_{6} - 7 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 2 \beta_{1} + 4\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(6 \beta_{11} + 3 \beta_{9} + 5 \beta_{7} + 23 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} - 6 \beta_{1} - 17\)\()/3\)
\(\nu^{8}\)\(=\)\(4 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} + 9 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 3 \beta_{1} + 2\)
\(\nu^{9}\)\(=\)\((\)\(-9 \beta_{11} - 19 \beta_{10} - 9 \beta_{9} - 24 \beta_{8} - 23 \beta_{7} - 23 \beta_{6} - 13 \beta_{4} - 18 \beta_{3} - 4 \beta_{2} + 13 \beta_{1} - 9\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(2 \beta_{11} - 4 \beta_{10} - 43 \beta_{9} - 38 \beta_{8} - 16 \beta_{7} - 46 \beta_{6} - 5 \beta_{5} + 32 \beta_{4} - 17 \beta_{3} - 27 \beta_{2} + 28 \beta_{1} + 14\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-28 \beta_{11} + 21 \beta_{10} - 13 \beta_{9} - 35 \beta_{8} + 26 \beta_{7} - 85 \beta_{6} + 7 \beta_{5} - 21 \beta_{4} - \beta_{3} - 25 \beta_{2} + 30 \beta_{1} - 5\)\()/3\)

Character Values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.989378 + 1.01051i
0.683706 1.23796i
1.04029 + 0.958022i
1.36982 0.351572i
−1.41396 0.0268737i
0.309529 + 1.37992i
−0.989378 1.01051i
0.683706 + 1.23796i
1.04029 0.958022i
1.36982 + 0.351572i
−1.41396 + 0.0268737i
0.309529 1.37992i
−1.00000 −1.56899 0.733675i 1.00000 −0.500000 0.866025i 1.56899 + 0.733675i −2.56238 0.658939i −1.00000 1.92344 + 2.30225i 0.500000 + 0.866025i
121.2 −1.00000 −0.554872 1.64077i 1.00000 −0.500000 0.866025i 0.554872 + 1.64077i 2.32383 + 1.26483i −1.00000 −2.38423 + 1.82083i 0.500000 + 0.866025i
121.3 −1.00000 −0.433986 + 1.67680i 1.00000 −0.500000 0.866025i 0.433986 1.67680i 1.23855 2.33795i −1.00000 −2.62331 1.45541i 0.500000 + 0.866025i
121.4 −1.00000 0.478015 + 1.66478i 1.00000 −0.500000 0.866025i −0.478015 1.66478i −2.56238 0.658939i −1.00000 −2.54300 + 1.59158i 0.500000 + 0.866025i
121.5 −1.00000 1.35166 1.08306i 1.00000 −0.500000 0.866025i −1.35166 + 1.08306i 2.32383 + 1.26483i −1.00000 0.653981 2.92785i 0.500000 + 0.866025i
121.6 −1.00000 1.72817 + 0.115916i 1.00000 −0.500000 0.866025i −1.72817 0.115916i 1.23855 2.33795i −1.00000 2.97313 + 0.400645i 0.500000 + 0.866025i
151.1 −1.00000 −1.56899 + 0.733675i 1.00000 −0.500000 + 0.866025i 1.56899 0.733675i −2.56238 + 0.658939i −1.00000 1.92344 2.30225i 0.500000 0.866025i
151.2 −1.00000 −0.554872 + 1.64077i 1.00000 −0.500000 + 0.866025i 0.554872 1.64077i 2.32383 1.26483i −1.00000 −2.38423 1.82083i 0.500000 0.866025i
151.3 −1.00000 −0.433986 1.67680i 1.00000 −0.500000 + 0.866025i 0.433986 + 1.67680i 1.23855 + 2.33795i −1.00000 −2.62331 + 1.45541i 0.500000 0.866025i
151.4 −1.00000 0.478015 1.66478i 1.00000 −0.500000 + 0.866025i −0.478015 + 1.66478i −2.56238 + 0.658939i −1.00000 −2.54300 1.59158i 0.500000 0.866025i
151.5 −1.00000 1.35166 + 1.08306i 1.00000 −0.500000 + 0.866025i −1.35166 1.08306i 2.32383 1.26483i −1.00000 0.653981 + 2.92785i 0.500000 0.866025i
151.6 −1.00000 1.72817 0.115916i 1.00000 −0.500000 + 0.866025i −1.72817 + 0.115916i 1.23855 + 2.33795i −1.00000 2.97313 0.400645i 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.6
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 \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\(T_{11}^{12} + \cdots\)
\(T_{13}^{12} + \cdots\)