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Properties

Label 57.2.i.b
Level 57
Weight 2
Character orbit 57.i
Analytic conductor 0.455
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.i (of order \(9\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(6\) \(x^{11}\mathstrut +\mathstrut \) \(33\) \(x^{10}\mathstrut -\mathstrut \) \(110\) \(x^{9}\mathstrut +\mathstrut \) \(318\) \(x^{8}\mathstrut -\mathstrut \) \(678\) \(x^{7}\mathstrut +\mathstrut \) \(1225\) \(x^{6}\mathstrut -\mathstrut \) \(1698\) \(x^{5}\mathstrut +\mathstrut \) \(1905\) \(x^{4}\mathstrut -\mathstrut \) \(1584\) \(x^{3}\mathstrut +\mathstrut \) \(936\) \(x^{2}\mathstrut -\mathstrut \) \(342\) \(x\mathstrut +\mathstrut \) \(57\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 \)\()/218\)
\(\beta_{2}\)\(=\)\((\)\( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 \)\()/218\)
\(\beta_{3}\)\(=\)\((\)\( -27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 \)\()/218\)
\(\beta_{4}\)\(=\)\((\)\( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 \)\()/218\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 \)\()/218\)
\(\beta_{6}\)\(=\)\((\)\( -27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 \)\()/218\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 \)\()/218\)
\(\beta_{9}\)\(=\)\((\)\( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 \)\()/218\)
\(\beta_{10}\)\(=\)\((\)\( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 \)\()/218\)
\(\beta_{11}\)\(=\)\((\)\( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 \)\()/218\)
Display \(\nu^j\) in terms of \(\beta_i\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(7\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(5\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut +\mathstrut \) \(8\) \(\beta_{9}\mathstrut -\mathstrut \) \(8\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(31\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(23\) \(\beta_{11}\mathstrut +\mathstrut \) \(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(34\) \(\beta_{9}\mathstrut +\mathstrut \) \(25\) \(\beta_{8}\mathstrut -\mathstrut \) \(26\) \(\beta_{7}\mathstrut +\mathstrut \) \(65\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut -\mathstrut \) \(16\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(77\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(110\) \(\beta_{11}\mathstrut -\mathstrut \) \(49\) \(\beta_{10}\mathstrut -\mathstrut \) \(88\) \(\beta_{9}\mathstrut +\mathstrut \) \(67\) \(\beta_{8}\mathstrut +\mathstrut \) \(85\) \(\beta_{7}\mathstrut +\mathstrut \) \(101\) \(\beta_{6}\mathstrut -\mathstrut \) \(94\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(161\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(104\) \(\beta_{1}\mathstrut -\mathstrut \) \(121\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(25\) \(\beta_{11}\mathstrut -\mathstrut \) \(187\) \(\beta_{10}\mathstrut +\mathstrut \) \(158\) \(\beta_{9}\mathstrut -\mathstrut \) \(101\) \(\beta_{8}\mathstrut +\mathstrut \) \(229\) \(\beta_{7}\mathstrut -\mathstrut \) \(271\) \(\beta_{6}\mathstrut -\mathstrut \) \(46\) \(\beta_{5}\mathstrut -\mathstrut \) \(56\) \(\beta_{4}\mathstrut +\mathstrut \) \(212\) \(\beta_{3}\mathstrut +\mathstrut \) \(149\) \(\beta_{2}\mathstrut -\mathstrut \) \(82\) \(\beta_{1}\mathstrut -\mathstrut \) \(535\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(652\) \(\beta_{11}\mathstrut +\mathstrut \) \(41\) \(\beta_{10}\mathstrut +\mathstrut \) \(761\) \(\beta_{9}\mathstrut -\mathstrut \) \(509\) \(\beta_{8}\mathstrut -\mathstrut \) \(263\) \(\beta_{7}\mathstrut -\mathstrut \) \(826\) \(\beta_{6}\mathstrut +\mathstrut \) \(515\) \(\beta_{5}\mathstrut +\mathstrut \) \(151\) \(\beta_{4}\mathstrut -\mathstrut \) \(724\) \(\beta_{3}\mathstrut -\mathstrut \) \(19\) \(\beta_{2}\mathstrut -\mathstrut \) \(829\) \(\beta_{1}\mathstrut +\mathstrut \) \(257\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(478\) \(\beta_{11}\mathstrut +\mathstrut \) \(1304\) \(\beta_{10}\mathstrut -\mathstrut \) \(268\) \(\beta_{9}\mathstrut +\mathstrut \) \(160\) \(\beta_{8}\mathstrut -\mathstrut \) \(1571\) \(\beta_{7}\mathstrut +\mathstrut \) \(821\) \(\beta_{6}\mathstrut +\mathstrut \) \(812\) \(\beta_{5}\mathstrut +\mathstrut \) \(640\) \(\beta_{4}\mathstrut -\mathstrut \) \(1951\) \(\beta_{3}\mathstrut -\mathstrut \) \(1108\) \(\beta_{2}\mathstrut -\mathstrut \) \(406\) \(\beta_{1}\mathstrut +\mathstrut \) \(3359\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(3353\) \(\beta_{11}\mathstrut +\mathstrut \) \(1451\) \(\beta_{10}\mathstrut -\mathstrut \) \(5224\) \(\beta_{9}\mathstrut +\mathstrut \) \(3352\) \(\beta_{8}\mathstrut +\mathstrut \) \(25\) \(\beta_{7}\mathstrut +\mathstrut \) \(5630\) \(\beta_{6}\mathstrut -\mathstrut \) \(2308\) \(\beta_{5}\mathstrut -\mathstrut \) \(521\) \(\beta_{4}\mathstrut +\mathstrut \) \(2375\) \(\beta_{3}\mathstrut -\mathstrut \) \(913\) \(\beta_{2}\mathstrut +\mathstrut \) \(4949\) \(\beta_{1}\mathstrut +\mathstrut \) \(1451\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(6041\) \(\beta_{11}\mathstrut -\mathstrut \) \(6547\) \(\beta_{10}\mathstrut -\mathstrut \) \(3685\) \(\beta_{9}\mathstrut +\mathstrut \) \(2323\) \(\beta_{8}\mathstrut +\mathstrut \) \(9241\) \(\beta_{7}\mathstrut +\mathstrut \) \(326\) \(\beta_{6}\mathstrut -\mathstrut \) \(7273\) \(\beta_{5}\mathstrut -\mathstrut \) \(5168\) \(\beta_{4}\mathstrut +\mathstrut \) \(14021\) \(\beta_{3}\mathstrut +\mathstrut \) \(6587\) \(\beta_{2}\mathstrut +\mathstrut \) \(7973\) \(\beta_{1}\mathstrut -\mathstrut \) \(18736\)\()/3\)
Display \(\beta_i\) in terms of \(\nu^j\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(-\beta_{3} + \beta_{9}\)

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 − 1.80139i
0.500000 − 0.168222i
0.500000 + 1.74095i
0.500000 − 2.42499i
0.500000 − 1.74095i
0.500000 + 2.42499i
0.500000 − 0.677980i
0.500000 + 1.96356i
0.500000 + 1.80139i
0.500000 + 0.168222i
0.500000 + 0.677980i
0.500000 − 1.96356i
−2.23121 − 0.812094i 0.173648 + 0.984808i 2.78672 + 2.33833i 2.16379 − 1.81563i 0.412311 − 2.33833i 1.05756 − 1.83175i −1.94440 − 3.36780i −0.939693 + 0.342020i −6.30233 + 2.29386i
4.2 0.791518 + 0.288089i 0.173648 + 0.984808i −0.988583 − 0.829520i 1.30800 − 1.09754i −0.146267 + 0.829520i −1.96517 + 3.40377i −1.38582 − 2.40031i −0.939693 + 0.342020i 1.35149 − 0.491903i
16.1 −0.958464 − 0.804247i −0.939693 + 0.342020i −0.0754558 − 0.427931i 0.755623 − 4.28535i 1.17573 + 0.427931i 0.898157 + 1.55565i −1.52303 + 2.63796i 0.766044 − 0.642788i −4.17072 + 3.49965i
16.2 1.22451 + 1.02748i −0.939693 + 0.342020i 0.0964003 + 0.546713i −0.174371 + 0.988909i −1.50208 − 0.546713i −1.28482 − 2.22537i 1.15479 − 2.00015i 0.766044 − 0.642788i −1.22961 + 1.03176i
25.1 −0.958464 + 0.804247i −0.939693 − 0.342020i −0.0754558 + 0.427931i 0.755623 + 4.28535i 1.17573 − 0.427931i 0.898157 − 1.55565i −1.52303 − 2.63796i 0.766044 + 0.642788i −4.17072 − 3.49965i
25.2 1.22451 − 1.02748i −0.939693 − 0.342020i 0.0964003 − 0.546713i −0.174371 − 0.988909i −1.50208 + 0.546713i −1.28482 + 2.22537i 1.15479 + 2.00015i 0.766044 + 0.642788i −1.22961 − 1.03176i
28.1 −0.458021 − 2.59757i 0.766044 − 0.642788i −4.65819 + 1.69544i 1.91800 + 0.698096i −2.02055 − 1.69544i −1.30802 + 2.26556i 3.89993 + 6.75488i 0.173648 − 0.984808i 0.934866 − 5.30189i
28.2 0.131669 + 0.746734i 0.766044 − 0.642788i 1.33911 − 0.487396i −2.97104 − 1.08137i 0.580856 + 0.487396i −1.89771 + 3.28694i 1.29853 + 2.24912i 0.173648 − 0.984808i 0.416301 − 2.36096i
43.1 −2.23121 + 0.812094i 0.173648 − 0.984808i 2.78672 − 2.33833i 2.16379 + 1.81563i 0.412311 + 2.33833i 1.05756 + 1.83175i −1.94440 + 3.36780i −0.939693 − 0.342020i −6.30233 − 2.29386i
43.2 0.791518 − 0.288089i 0.173648 − 0.984808i −0.988583 + 0.829520i 1.30800 + 1.09754i −0.146267 − 0.829520i −1.96517 − 3.40377i −1.38582 + 2.40031i −0.939693 − 0.342020i 1.35149 + 0.491903i
55.1 −0.458021 + 2.59757i 0.766044 + 0.642788i −4.65819 − 1.69544i 1.91800 − 0.698096i −2.02055 + 1.69544i −1.30802 − 2.26556i 3.89993 − 6.75488i 0.173648 + 0.984808i 0.934866 + 5.30189i
55.2 0.131669 − 0.746734i 0.766044 + 0.642788i 1.33911 + 0.487396i −2.97104 + 1.08137i 0.580856 − 0.487396i −1.89771 − 3.28694i 1.29853 − 2.24912i 0.173648 + 0.984808i 0.416301 + 2.36096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.2
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).