Properties

Label 27.2.e.a
Level 27
Weight 2
Character orbit 27.e
Analytic conductor 0.216
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 27.e (of order \(9\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.215596085457\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(6\) \(x^{11}\mathstrut +\mathstrut \) \(27\) \(x^{10}\mathstrut -\mathstrut \) \(80\) \(x^{9}\mathstrut +\mathstrut \) \(186\) \(x^{8}\mathstrut -\mathstrut \) \(330\) \(x^{7}\mathstrut +\mathstrut \) \(463\) \(x^{6}\mathstrut -\mathstrut \) \(504\) \(x^{5}\mathstrut +\mathstrut \) \(420\) \(x^{4}\mathstrut -\mathstrut \) \(258\) \(x^{3}\mathstrut +\mathstrut \) \(108\) \(x^{2}\mathstrut -\mathstrut \) \(27\) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{10} - 5 \nu^{9} + 22 \nu^{8} - 58 \nu^{7} + 127 \nu^{6} - 199 \nu^{5} + 249 \nu^{4} - 224 \nu^{3} + 145 \nu^{2} - 58 \nu + 9 \)
\(\beta_{2}\)\(=\)\( 3 \nu^{11} - 16 \nu^{10} + 71 \nu^{9} - 197 \nu^{8} + 445 \nu^{7} - 747 \nu^{6} + 1006 \nu^{5} - 1030 \nu^{4} + 803 \nu^{3} - 445 \nu^{2} + 155 \nu - 25 \)
\(\beta_{3}\)\(=\)\( -6 \nu^{11} + 32 \nu^{10} - 140 \nu^{9} + 384 \nu^{8} - 849 \nu^{7} + 1390 \nu^{6} - 1805 \nu^{5} + 1762 \nu^{4} - 1285 \nu^{3} + 649 \nu^{2} - 195 \nu + 25 \)
\(\beta_{4}\)\(=\)\( -9 \nu^{11} + 49 \nu^{10} - 216 \nu^{9} + 601 \nu^{8} - 1344 \nu^{7} + 2232 \nu^{6} - 2942 \nu^{5} + 2918 \nu^{4} - 2170 \nu^{3} + 1118 \nu^{2} - 348 \nu + 49 \)
\(\beta_{5}\)\(=\)\( 9 \nu^{11} - 50 \nu^{10} + 221 \nu^{9} - 623 \nu^{8} + 1402 \nu^{7} - 2360 \nu^{6} + 3144 \nu^{5} - 3178 \nu^{4} + 2411 \nu^{3} - 1286 \nu^{2} + 421 \nu - 62 \)
\(\beta_{6}\)\(=\)\( 11 \nu^{11} - 60 \nu^{10} + 265 \nu^{9} - 739 \nu^{8} + 1657 \nu^{7} - 2761 \nu^{6} + 3653 \nu^{5} - 3643 \nu^{4} + 2724 \nu^{3} - 1417 \nu^{2} + 442 \nu - 61 \)
\(\beta_{7}\)\(=\)\( -16 \nu^{11} + 87 \nu^{10} - 383 \nu^{9} + 1064 \nu^{8} - 2375 \nu^{7} + 3936 \nu^{6} - 5176 \nu^{5} + 5122 \nu^{4} - 3802 \nu^{3} + 1958 \nu^{2} - 610 \nu + 85 \)
\(\beta_{8}\)\(=\)\( -16 \nu^{11} + 89 \nu^{10} - 393 \nu^{9} + 1108 \nu^{8} - 2491 \nu^{7} + 4191 \nu^{6} - 5577 \nu^{5} + 5631 \nu^{4} - 4267 \nu^{3} + 2272 \nu^{2} - 742 \nu + 110 \)
\(\beta_{9}\)\(=\)\( 36 \nu^{11} - 198 \nu^{10} + 873 \nu^{9} - 2443 \nu^{8} + 5472 \nu^{7} - 9134 \nu^{6} + 12076 \nu^{5} - 12058 \nu^{4} + 9024 \nu^{3} - 4708 \nu^{2} + 1486 \nu - 209 \)
\(\beta_{10}\)\(=\)\( -36 \nu^{11} + 198 \nu^{10} - 873 \nu^{9} + 2444 \nu^{8} - 5476 \nu^{7} + 9150 \nu^{6} - 12110 \nu^{5} + 12120 \nu^{4} - 9096 \nu^{3} + 4772 \nu^{2} - 1519 \nu + 217 \)
\(\beta_{11}\)\(=\)\( -42 \nu^{11} + 231 \nu^{10} - 1019 \nu^{9} + 2853 \nu^{8} - 6396 \nu^{7} + 10689 \nu^{6} - 14157 \nu^{5} + 14172 \nu^{4} - 10648 \nu^{3} + 5589 \nu^{2} - 1785 \nu + 257 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(5\) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(8\) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(18\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(11\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut -\mathstrut \) \(20\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(19\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(31\) \(\beta_{9}\mathstrut -\mathstrut \) \(32\) \(\beta_{8}\mathstrut +\mathstrut \) \(43\) \(\beta_{7}\mathstrut -\mathstrut \) \(20\) \(\beta_{6}\mathstrut -\mathstrut \) \(41\) \(\beta_{5}\mathstrut -\mathstrut \) \(44\) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(25\) \(\beta_{2}\mathstrut +\mathstrut \) \(40\) \(\beta_{1}\mathstrut +\mathstrut \) \(87\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(85\) \(\beta_{11}\mathstrut -\mathstrut \) \(97\) \(\beta_{10}\mathstrut +\mathstrut \) \(55\) \(\beta_{9}\mathstrut +\mathstrut \) \(64\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(101\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{5}\mathstrut -\mathstrut \) \(62\) \(\beta_{4}\mathstrut +\mathstrut \) \(91\) \(\beta_{3}\mathstrut +\mathstrut \) \(70\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(60\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(20\) \(\beta_{11}\mathstrut -\mathstrut \) \(118\) \(\beta_{10}\mathstrut -\mathstrut \) \(134\) \(\beta_{9}\mathstrut +\mathstrut \) \(244\) \(\beta_{8}\mathstrut -\mathstrut \) \(218\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(232\) \(\beta_{5}\mathstrut +\mathstrut \) \(157\) \(\beta_{4}\mathstrut +\mathstrut \) \(52\) \(\beta_{3}\mathstrut -\mathstrut \) \(74\) \(\beta_{2}\mathstrut -\mathstrut \) \(179\) \(\beta_{1}\mathstrut -\mathstrut \) \(357\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(503\) \(\beta_{11}\mathstrut +\mathstrut \) \(386\) \(\beta_{10}\mathstrut -\mathstrut \) \(440\) \(\beta_{9}\mathstrut -\mathstrut \) \(47\) \(\beta_{8}\mathstrut -\mathstrut \) \(233\) \(\beta_{7}\mathstrut +\mathstrut \) \(514\) \(\beta_{6}\mathstrut +\mathstrut \) \(163\) \(\beta_{5}\mathstrut +\mathstrut \) \(466\) \(\beta_{4}\mathstrut -\mathstrut \) \(431\) \(\beta_{3}\mathstrut -\mathstrut \) \(461\) \(\beta_{2}\mathstrut -\mathstrut \) \(329\) \(\beta_{1}\mathstrut -\mathstrut \) \(639\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(425\) \(\beta_{11}\mathstrut +\mathstrut \) \(1076\) \(\beta_{10}\mathstrut +\mathstrut \) \(319\) \(\beta_{9}\mathstrut -\mathstrut \) \(1313\) \(\beta_{8}\mathstrut +\mathstrut \) \(955\) \(\beta_{7}\mathstrut +\mathstrut \) \(502\) \(\beta_{6}\mathstrut -\mathstrut \) \(1013\) \(\beta_{5}\mathstrut -\mathstrut \) \(332\) \(\beta_{4}\mathstrut -\mathstrut \) \(743\) \(\beta_{3}\mathstrut -\mathstrut \) \(59\) \(\beta_{2}\mathstrut +\mathstrut \) \(631\) \(\beta_{1}\mathstrut +\mathstrut \) \(1164\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(2299\) \(\beta_{11}\mathstrut -\mathstrut \) \(862\) \(\beta_{10}\mathstrut +\mathstrut \) \(2725\) \(\beta_{9}\mathstrut -\mathstrut \) \(1193\) \(\beta_{8}\mathstrut +\mathstrut \) \(2104\) \(\beta_{7}\mathstrut -\mathstrut \) \(2135\) \(\beta_{6}\mathstrut -\mathstrut \) \(1907\) \(\beta_{5}\mathstrut -\mathstrut \) \(2705\) \(\beta_{4}\mathstrut +\mathstrut \) \(1495\) \(\beta_{3}\mathstrut +\mathstrut \) \(2425\) \(\beta_{2}\mathstrut +\mathstrut \) \(2344\) \(\beta_{1}\mathstrut +\mathstrut \) \(4356\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(4708\) \(\beta_{11}\mathstrut -\mathstrut \) \(6628\) \(\beta_{10}\mathstrut +\mathstrut \) \(985\) \(\beta_{9}\mathstrut +\mathstrut \) \(5506\) \(\beta_{8}\mathstrut -\mathstrut \) \(3107\) \(\beta_{7}\mathstrut -\mathstrut \) \(4679\) \(\beta_{6}\mathstrut +\mathstrut \) \(3238\) \(\beta_{5}\mathstrut -\mathstrut \) \(992\) \(\beta_{4}\mathstrut +\mathstrut \) \(5476\) \(\beta_{3}\mathstrut +\mathstrut \) \(2770\) \(\beta_{2}\mathstrut -\mathstrut \) \(1043\) \(\beta_{1}\mathstrut -\mathstrut \) \(1698\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{10}\)

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
0.500000 1.27297i
0.500000 0.0126039i
0.500000 + 1.27297i
0.500000 + 0.0126039i
0.500000 1.68614i
0.500000 + 1.00210i
0.500000 + 0.258654i
0.500000 2.22827i
0.500000 0.258654i
0.500000 + 2.22827i
0.500000 + 1.68614i
0.500000 1.00210i
−1.57954 0.574906i 1.45446 0.940501i 0.632343 + 0.530599i −0.196143 + 1.11238i −2.83808 + 0.649381i −2.99441 + 2.51261i 0.987144 + 1.70978i 1.23092 2.73584i 0.949332 1.64429i
4.2 0.753189 + 0.274138i −1.68842 0.386327i −1.03995 0.872619i −0.477505 + 2.70806i −1.16579 0.753837i 1.82076 1.52780i −1.34559 2.33062i 2.70150 + 1.30456i −1.10204 + 1.90878i
7.1 −1.57954 + 0.574906i 1.45446 + 0.940501i 0.632343 0.530599i −0.196143 1.11238i −2.83808 0.649381i −2.99441 2.51261i 0.987144 1.70978i 1.23092 + 2.73584i 0.949332 + 1.64429i
7.2 0.753189 0.274138i −1.68842 + 0.386327i −1.03995 + 0.872619i −0.477505 2.70806i −1.16579 + 0.753837i 1.82076 + 1.52780i −1.34559 + 2.33062i 2.70150 1.30456i −1.10204 1.90878i
13.1 −0.417037 2.36514i −0.210069 + 1.71926i −3.54056 + 1.28866i 0.0713060 + 0.0598329i 4.15390 0.220155i 0.544891 + 0.198324i 2.12277 + 3.67675i −2.91174 0.722330i 0.111776 0.193601i
13.2 0.183082 + 1.03831i −1.72962 0.0916693i 0.834822 0.303850i −1.33735 1.12217i −0.221481 1.81266i −2.31094 0.841112i 1.52266 + 2.63732i 2.98319 + 0.317107i 0.920313 1.59403i
16.1 −1.62143 1.36054i −0.986166 1.42389i 0.430663 + 2.44241i 2.52129 + 0.917674i −0.338267 + 3.65046i 0.168844 0.957561i 0.508086 0.880031i −1.05495 + 2.80839i −2.83955 4.91825i
16.2 −0.318266 0.267057i 0.159815 + 1.72466i −0.317323 1.79963i −2.08159 0.757639i 0.409719 0.591580i −0.229151 + 1.29958i −0.795075 + 1.37711i −2.94892 + 0.551252i 0.460168 + 0.797034i
22.1 −1.62143 + 1.36054i −0.986166 + 1.42389i 0.430663 2.44241i 2.52129 0.917674i −0.338267 3.65046i 0.168844 + 0.957561i 0.508086 + 0.880031i −1.05495 2.80839i −2.83955 + 4.91825i
22.2 −0.318266 + 0.267057i 0.159815 1.72466i −0.317323 + 1.79963i −2.08159 + 0.757639i 0.409719 + 0.591580i −0.229151 1.29958i −0.795075 1.37711i −2.94892 0.551252i 0.460168 0.797034i
25.1 −0.417037 + 2.36514i −0.210069 1.71926i −3.54056 1.28866i 0.0713060 0.0598329i 4.15390 + 0.220155i 0.544891 0.198324i 2.12277 3.67675i −2.91174 + 0.722330i 0.111776 + 0.193601i
25.2 0.183082 1.03831i −1.72962 + 0.0916693i 0.834822 + 0.303850i −1.33735 + 1.12217i −0.221481 + 1.81266i −2.31094 + 0.841112i 1.52266 2.63732i 2.98319 0.317107i 0.920313 + 1.59403i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
27.e Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(27, [\chi])\).