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 \(9\), degree \(6\), minimal)

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
Defining polynomial: \(x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6q^{2} - 6q^{3} - 6q^{4} - 3q^{5} - 6q^{7} + 6q^{8} + O(q^{10}) \) \( 12q - 6q^{2} - 6q^{3} - 6q^{4} - 3q^{5} - 6q^{7} + 6q^{8} - 3q^{10} + 3q^{11} + 12q^{12} - 6q^{13} + 15q^{14} + 9q^{15} + 9q^{17} + 9q^{18} - 3q^{19} - 3q^{20} - 12q^{21} + 3q^{22} - 12q^{23} - 18q^{24} + 3q^{25} - 30q^{26} - 9q^{27} - 12q^{28} - 6q^{29} - 9q^{30} + 3q^{31} + 9q^{34} + 12q^{35} + 18q^{36} - 3q^{37} + 42q^{38} + 33q^{39} + 21q^{40} + 15q^{41} + 18q^{42} + 3q^{43} + 3q^{44} - 9q^{45} - 3q^{46} - 15q^{47} - 15q^{48} + 12q^{49} - 33q^{50} - 18q^{51} + 9q^{52} - 18q^{53} - 54q^{54} - 12q^{55} - 33q^{56} - 3q^{57} + 21q^{58} - 12q^{59} + 12q^{61} - 12q^{62} + 9q^{63} + 12q^{64} + 3q^{65} - 9q^{66} - 15q^{67} + 9q^{68} + 9q^{69} - 15q^{70} + 27q^{71} + 18q^{72} + 6q^{73} + 33q^{74} + 39q^{75} - 48q^{76} + 15q^{77} + 18q^{78} - 42q^{79} + 42q^{80} + 36q^{81} - 12q^{82} + 39q^{83} + 6q^{84} - 27q^{85} + 51q^{86} + 9q^{87} - 30q^{88} + 9q^{89} + 18q^{90} + 6q^{91} - 39q^{92} - 39q^{93} - 15q^{94} - 33q^{95} + 3q^{97} - 45q^{98} - 27q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.2.e.a 12
3.b odd 2 1 81.2.e.a 12
4.b odd 2 1 432.2.u.c 12
5.b even 2 1 675.2.l.c 12
5.c odd 4 2 675.2.u.b 24
9.c even 3 1 243.2.e.c 12
9.c even 3 1 243.2.e.d 12
9.d odd 6 1 243.2.e.a 12
9.d odd 6 1 243.2.e.b 12
27.e even 9 1 inner 27.2.e.a 12
27.e even 9 1 243.2.e.c 12
27.e even 9 1 243.2.e.d 12
27.e even 9 1 729.2.a.a 6
27.e even 9 2 729.2.c.e 12
27.f odd 18 1 81.2.e.a 12
27.f odd 18 1 243.2.e.a 12
27.f odd 18 1 243.2.e.b 12
27.f odd 18 1 729.2.a.d 6
27.f odd 18 2 729.2.c.b 12
108.j odd 18 1 432.2.u.c 12
135.p even 18 1 675.2.l.c 12
135.r odd 36 2 675.2.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.e.a 12 1.a even 1 1 trivial
27.2.e.a 12 27.e even 9 1 inner
81.2.e.a 12 3.b odd 2 1
81.2.e.a 12 27.f odd 18 1
243.2.e.a 12 9.d odd 6 1
243.2.e.a 12 27.f odd 18 1
243.2.e.b 12 9.d odd 6 1
243.2.e.b 12 27.f odd 18 1
243.2.e.c 12 9.c even 3 1
243.2.e.c 12 27.e even 9 1
243.2.e.d 12 9.c even 3 1
243.2.e.d 12 27.e even 9 1
432.2.u.c 12 4.b odd 2 1
432.2.u.c 12 108.j odd 18 1
675.2.l.c 12 5.b even 2 1
675.2.l.c 12 135.p even 18 1
675.2.u.b 24 5.c odd 4 2
675.2.u.b 24 135.r odd 36 2
729.2.a.a 6 27.e even 9 1
729.2.a.d 6 27.f odd 18 1
729.2.c.b 12 27.f odd 18 2
729.2.c.e 12 27.e even 9 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(27, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 27 T + 27 T^{2} - 45 T^{3} - 18 T^{4} - 9 T^{5} + 6 T^{6} + 54 T^{7} + 72 T^{8} + 48 T^{9} + 21 T^{10} + 6 T^{11} + T^{12} \)
$3$ \( 729 + 1458 T + 1458 T^{2} + 1053 T^{3} + 567 T^{4} + 243 T^{5} + 117 T^{6} + 81 T^{7} + 63 T^{8} + 39 T^{9} + 18 T^{10} + 6 T^{11} + T^{12} \)
$5$ \( 9 - 135 T + 837 T^{2} + 1287 T^{3} + 1521 T^{4} + 1008 T^{5} + 303 T^{6} - 63 T^{7} - 63 T^{8} - 12 T^{9} + 3 T^{10} + 3 T^{11} + T^{12} \)
$7$ \( 289 - 816 T + 888 T^{2} - 758 T^{3} + 639 T^{4} + 225 T^{5} + 273 T^{6} + 225 T^{7} + 18 T^{8} - 11 T^{9} + 12 T^{10} + 6 T^{11} + T^{12} \)
$11$ \( 9 - 27 T + 108 T^{2} - 531 T^{3} + 495 T^{4} + 1341 T^{5} + 1491 T^{6} + 657 T^{7} + 261 T^{8} - 6 T^{9} - 15 T^{10} - 3 T^{11} + T^{12} \)
$13$ \( 1 + 84 T + 2994 T^{2} - 2675 T^{3} - 3546 T^{4} + 873 T^{5} + 3729 T^{6} + 2223 T^{7} + 747 T^{8} + 214 T^{9} + 48 T^{10} + 6 T^{11} + T^{12} \)
$17$ \( 729 - 2187 T + 8019 T^{2} + 1458 T^{3} + 7047 T^{4} - 1701 T^{5} + 2727 T^{6} - 567 T^{7} + 621 T^{8} - 189 T^{9} + 72 T^{10} - 9 T^{11} + T^{12} \)
$19$ \( 361 + 2280 T + 11208 T^{2} + 21604 T^{3} + 33354 T^{4} + 873 T^{5} + 6162 T^{6} + 252 T^{7} + 846 T^{8} - 14 T^{9} + 39 T^{10} + 3 T^{11} + T^{12} \)
$23$ \( 106929 + 264870 T + 322812 T^{2} + 288216 T^{3} + 170838 T^{4} + 32220 T^{5} + 22713 T^{6} - 774 T^{7} - 1440 T^{8} - 192 T^{9} + 48 T^{10} + 12 T^{11} + T^{12} \)
$29$ \( 45369 - 203202 T + 341037 T^{2} - 619236 T^{3} + 1278324 T^{4} + 188802 T^{5} - 4044 T^{6} - 10206 T^{7} + 423 T^{8} - 96 T^{9} + 21 T^{10} + 6 T^{11} + T^{12} \)
$31$ \( 26569 - 77262 T + 73617 T^{2} - 6158 T^{3} - 29034 T^{4} + 14481 T^{5} + 3810 T^{6} - 4365 T^{7} + 1881 T^{8} - 434 T^{9} + 84 T^{10} - 3 T^{11} + T^{12} \)
$37$ \( 24334489 + 19105509 T + 12573093 T^{2} + 4411480 T^{3} + 1506987 T^{4} + 331353 T^{5} + 90807 T^{6} + 15399 T^{7} + 3519 T^{8} + 337 T^{9} + 66 T^{10} + 3 T^{11} + T^{12} \)
$41$ \( 11229201 + 15350931 T + 16475103 T^{2} + 10695744 T^{3} + 2682513 T^{4} - 178299 T^{5} - 57801 T^{6} + 8208 T^{7} + 4797 T^{8} - 705 T^{9} + 93 T^{10} - 15 T^{11} + T^{12} \)
$43$ \( 3308761 - 4709391 T + 49059204 T^{2} - 4380266 T^{3} - 386325 T^{4} + 386325 T^{5} + 10965 T^{6} - 963 T^{7} + 2799 T^{8} + 16 T^{9} - 60 T^{10} - 3 T^{11} + T^{12} \)
$47$ \( 42732369 - 17296902 T + 4549149 T^{2} + 1182393 T^{3} + 1192761 T^{4} + 676449 T^{5} + 29409 T^{6} - 30357 T^{7} - 927 T^{8} - 114 T^{9} + 111 T^{10} + 15 T^{11} + T^{12} \)
$53$ \( ( -12393 - 2916 T + 4617 T^{2} - 513 T^{3} - 108 T^{4} + 9 T^{5} + T^{6} )^{2} \)
$59$ \( 176384961 - 170448354 T + 29313684 T^{2} + 10115676 T^{3} + 2890305 T^{4} - 2031741 T^{5} + 256911 T^{6} - 36189 T^{7} + 5796 T^{8} + 933 T^{9} + 192 T^{10} + 12 T^{11} + T^{12} \)
$61$ \( 273670849 - 12754653 T + 226908411 T^{2} + 81354967 T^{3} - 9276372 T^{4} - 1547091 T^{5} + 1238790 T^{6} - 134964 T^{7} + 19512 T^{8} + 34 T^{9} - 51 T^{10} - 12 T^{11} + T^{12} \)
$67$ \( 8288641 - 13275069 T + 5116533 T^{2} + 7604980 T^{3} - 2980269 T^{4} - 1780641 T^{5} + 660099 T^{6} + 86274 T^{7} + 23913 T^{8} + 2365 T^{9} + 255 T^{10} + 15 T^{11} + T^{12} \)
$71$ \( 729 + 6561 T + 78003 T^{2} - 144342 T^{3} + 604827 T^{4} + 231093 T^{5} + 400761 T^{6} - 147015 T^{7} + 38205 T^{8} - 5103 T^{9} + 504 T^{10} - 27 T^{11} + T^{12} \)
$73$ \( 185761 + 619347 T + 3782073 T^{2} - 5509508 T^{3} + 16156512 T^{4} + 498510 T^{5} + 746232 T^{6} + 2871 T^{7} + 27792 T^{8} + 544 T^{9} + 210 T^{10} - 6 T^{11} + T^{12} \)
$79$ \( 3508129 - 1848651 T - 6043107 T^{2} + 15614746 T^{3} + 26778888 T^{4} + 7412832 T^{5} + 2309097 T^{6} + 487872 T^{7} + 78291 T^{8} + 9520 T^{9} + 813 T^{10} + 42 T^{11} + T^{12} \)
$83$ \( 6951057129 - 10578532986 T + 7328037465 T^{2} - 3047714604 T^{3} + 853146414 T^{4} - 170563383 T^{5} + 25416024 T^{6} - 2898477 T^{7} + 256059 T^{8} - 17196 T^{9} + 912 T^{10} - 39 T^{11} + T^{12} \)
$89$ \( 1062042921 - 992530584 T + 765665784 T^{2} - 218178036 T^{3} + 61794900 T^{4} - 6160293 T^{5} + 1737990 T^{6} - 125712 T^{7} + 36666 T^{8} - 432 T^{9} + 261 T^{10} - 9 T^{11} + T^{12} \)
$97$ \( 66765241 + 21154719 T + 532756806 T^{2} - 381413396 T^{3} + 112050423 T^{4} - 14303619 T^{5} + 35103 T^{6} + 185823 T^{7} - 11349 T^{8} - 1010 T^{9} + 102 T^{10} - 3 T^{11} + T^{12} \)
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