Properties

Label 19.3.f.a
Level 19
Weight 3
Character orbit 19.f
Analytic conductor 0.518
Analytic rank 0
Dimension 12
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 19.f (of order \(18\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{10} q^{2} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{3} \) \( + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{4} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{5} \) \( + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{6} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{7} \) \( + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} + \beta_{10} ) q^{8} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{10} q^{2} \) \( + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{3} \) \( + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{4} \) \( + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{5} \) \( + ( -4 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{6} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{7} \) \( + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} + \beta_{10} ) q^{8} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{9} \) \( + ( 5 - 5 \beta_{1} - \beta_{2} - 5 \beta_{4} + 2 \beta_{5} + 5 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{10} \) \( + ( 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 5 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{11} \) \( + ( 4 + 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{12} \) \( + ( 5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{13} \) \( + ( -\beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 7 \beta_{11} ) q^{14} \) \( + ( 5 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} + 7 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{15} \) \( + ( 4 + 6 \beta_{1} - \beta_{2} + 5 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 7 \beta_{11} ) q^{16} \) \( + ( -7 + 3 \beta_{1} - \beta_{3} + 3 \beta_{4} + 6 \beta_{5} - 7 \beta_{6} + 4 \beta_{7} + \beta_{8} + 4 \beta_{9} - 9 \beta_{11} ) q^{17} \) \( + ( -7 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 9 \beta_{6} + 5 \beta_{9} - 5 \beta_{10} + 10 \beta_{11} ) q^{18} \) \( + ( -8 - 3 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} + 9 \beta_{4} - 8 \beta_{6} - \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 7 \beta_{11} ) q^{19} \) \( + ( -6 - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{20} \) \( + ( 7 + 2 \beta_{1} + 4 \beta_{3} - 15 \beta_{4} + 7 \beta_{5} + 12 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} - 5 \beta_{9} - 7 \beta_{10} + 9 \beta_{11} ) q^{21} \) \( + ( -11 + 10 \beta_{1} + 4 \beta_{2} - 7 \beta_{4} - 10 \beta_{5} + 15 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 6 \beta_{11} ) q^{22} \) \( + ( -4 - 16 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 10 \beta_{11} ) q^{23} \) \( + ( -7 + 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 9 \beta_{11} ) q^{24} \) \( + ( -15 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 11 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - 4 \beta_{11} ) q^{25} \) \( + ( 5 + 19 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 18 \beta_{4} - \beta_{5} - 16 \beta_{6} + 3 \beta_{9} + 3 \beta_{10} + 8 \beta_{11} ) q^{26} \) \( + ( 3 - 8 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} + 11 \beta_{4} + 12 \beta_{5} - 4 \beta_{6} - 7 \beta_{7} - 7 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{27} \) \( + ( -2 - 4 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 17 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} - 5 \beta_{11} ) q^{28} \) \( + ( 13 - 5 \beta_{1} + \beta_{2} - 12 \beta_{4} + 3 \beta_{5} + 13 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{29} \) \( + ( 4 + 10 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 5 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} + 6 \beta_{10} - 3 \beta_{11} ) q^{30} \) \( + ( 3 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} - \beta_{6} + 10 \beta_{8} + 5 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} ) q^{31} \) \( + ( 20 + 2 \beta_{1} + 3 \beta_{2} - 17 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} - 3 \beta_{10} + 17 \beta_{11} ) q^{32} \) \( + ( 2 - 11 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 12 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} ) q^{33} \) \( + ( 16 - 5 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} ) q^{34} \) \( + ( 4 + 2 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} + 27 \beta_{4} - 13 \beta_{5} - 9 \beta_{6} - 2 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} + 10 \beta_{10} - 3 \beta_{11} ) q^{35} \) \( + ( 11 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 12 \beta_{5} + 13 \beta_{6} - 5 \beta_{7} - 3 \beta_{8} - 5 \beta_{9} + 3 \beta_{10} - 17 \beta_{11} ) q^{36} \) \( + ( 23 + 18 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} + 9 \beta_{4} - \beta_{5} + 8 \beta_{6} - 9 \beta_{9} + 9 \beta_{10} + 28 \beta_{11} ) q^{37} \) \( + ( 3 - 30 \beta_{1} - 13 \beta_{2} - \beta_{3} - 15 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - 7 \beta_{9} - 9 \beta_{10} - 10 \beta_{11} ) q^{38} \) \( + ( -11 + 12 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 14 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{39} \) \( + ( -10 - 8 \beta_{1} - 4 \beta_{3} - 24 \beta_{4} + 14 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + \beta_{10} + 6 \beta_{11} ) q^{40} \) \( + ( -23 + 23 \beta_{1} - 9 \beta_{2} - \beta_{3} - 10 \beta_{4} - 29 \beta_{5} + 12 \beta_{6} - 7 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} - 32 \beta_{11} ) q^{41} \) \( + ( -15 - 7 \beta_{1} + 5 \beta_{2} + 8 \beta_{3} + 23 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} + 8 \beta_{9} + 10 \beta_{10} + 6 \beta_{11} ) q^{42} \) \( + ( 3 - 7 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} + \beta_{5} - 10 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 5 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{43} \) \( + ( -4 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 11 \beta_{4} - 16 \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 12 \beta_{11} ) q^{44} \) \( + ( -3 + 9 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 11 \beta_{5} - 11 \beta_{6} + 12 \beta_{7} + 6 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{45} \) \( + ( -20 + 14 \beta_{1} + 16 \beta_{2} - 14 \beta_{3} + 16 \beta_{4} + 14 \beta_{5} - 28 \beta_{6} + 8 \beta_{7} + 16 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{46} \) \( + ( -14 - 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 24 \beta_{4} - 9 \beta_{5} + 8 \beta_{6} + 3 \beta_{7} + 17 \beta_{8} + 3 \beta_{9} - 17 \beta_{10} + 10 \beta_{11} ) q^{47} \) \( + ( -4 - 31 \beta_{1} - 3 \beta_{2} + 8 \beta_{4} + 14 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 8 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{48} \) \( + ( -11 - 4 \beta_{1} - 11 \beta_{2} + 13 \beta_{3} + 7 \beta_{4} + 17 \beta_{5} + 13 \beta_{6} + 6 \beta_{7} + 12 \beta_{8} + 11 \beta_{9} - 24 \beta_{10} + 5 \beta_{11} ) q^{49} \) \( + ( 9 - 8 \beta_{1} - 6 \beta_{2} + 16 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 4 \beta_{8} - 10 \beta_{9} - 16 \beta_{10} + \beta_{11} ) q^{50} \) \( + ( -3 + 12 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 10 \beta_{5} + 7 \beta_{6} + 3 \beta_{7} + 10 \beta_{8} - 3 \beta_{10} + 4 \beta_{11} ) q^{51} \) \( + ( -4 - 15 \beta_{1} - \beta_{2} - 10 \beta_{3} - \beta_{4} + 14 \beta_{5} + \beta_{6} + 10 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} - \beta_{10} - 14 \beta_{11} ) q^{52} \) \( + ( 13 + 15 \beta_{1} + 24 \beta_{2} - 3 \beta_{3} + 16 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} - 15 \beta_{7} - 24 \beta_{8} + 3 \beta_{9} + 9 \beta_{10} + 15 \beta_{11} ) q^{53} \) \( + ( 8 + 16 \beta_{1} - \beta_{2} - 14 \beta_{3} - 26 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 18 \beta_{7} + 4 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + 17 \beta_{11} ) q^{54} \) \( + ( 15 - 16 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} - 5 \beta_{4} + 15 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} - 6 \beta_{8} - 11 \beta_{9} - 6 \beta_{10} - 11 \beta_{11} ) q^{55} \) \( + ( -9 - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 15 \beta_{5} - 10 \beta_{6} - 6 \beta_{7} - 6 \beta_{8} - 5 \beta_{9} + 5 \beta_{10} - 28 \beta_{11} ) q^{56} \) \( + ( -12 + \beta_{1} - 15 \beta_{2} + 7 \beta_{3} + 27 \beta_{4} - 4 \beta_{5} - 21 \beta_{6} - 9 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 13 \beta_{11} ) q^{57} \) \( + ( -11 + 4 \beta_{1} + 7 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 11 \beta_{9} + 11 \beta_{10} ) q^{58} \) \( + ( 2 + 20 \beta_{1} - 13 \beta_{3} + 13 \beta_{4} - 14 \beta_{5} - 22 \beta_{6} - 18 \beta_{7} - 13 \beta_{8} + 18 \beta_{9} + 22 \beta_{10} + 6 \beta_{11} ) q^{59} \) \( + ( 2 - 22 \beta_{1} + 3 \beta_{2} - \beta_{3} + 6 \beta_{4} + 16 \beta_{5} + 13 \beta_{7} + 14 \beta_{8} + 14 \beta_{9} + 25 \beta_{11} ) q^{60} \) \( + ( 20 + 6 \beta_{1} + 9 \beta_{2} - 11 \beta_{3} - 31 \beta_{4} + 3 \beta_{5} + 15 \beta_{6} - \beta_{7} + 9 \beta_{8} - 11 \beta_{9} + 10 \beta_{10} + 4 \beta_{11} ) q^{61} \) \( + ( 13 + 10 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 33 \beta_{4} - 11 \beta_{5} + 40 \beta_{6} + 7 \beta_{7} - 4 \beta_{8} - 11 \beta_{9} + 3 \beta_{10} - 11 \beta_{11} ) q^{62} \) \( + ( 23 + 8 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} - 30 \beta_{4} + 43 \beta_{6} + 9 \beta_{7} - 5 \beta_{8} - 20 \beta_{9} + 9 \beta_{10} - 20 \beta_{11} ) q^{63} \) \( + ( 24 - \beta_{1} - 6 \beta_{2} + 11 \beta_{3} + 10 \beta_{4} + 11 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} - 17 \beta_{9} + 11 \beta_{10} + 7 \beta_{11} ) q^{64} \) \( + ( 18 - 34 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 10 \beta_{4} - 6 \beta_{5} + 40 \beta_{6} - 11 \beta_{7} - 8 \beta_{9} - 4 \beta_{10} + 5 \beta_{11} ) q^{65} \) \( + ( 23 + 6 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} + 28 \beta_{4} + 16 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} - 8 \beta_{8} - 9 \beta_{9} + 8 \beta_{10} - 5 \beta_{11} ) q^{66} \) \( + ( 6 + 37 \beta_{1} + 13 \beta_{2} - 5 \beta_{4} - 12 \beta_{5} + 6 \beta_{6} + 13 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + \beta_{11} ) q^{67} \) \( + ( 4 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 8 \beta_{4} - 10 \beta_{5} - 14 \beta_{6} - 5 \beta_{7} - 10 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} + 8 \beta_{11} ) q^{68} \) \( + ( -4 + 4 \beta_{1} + 4 \beta_{2} - 20 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} - 16 \beta_{6} - 14 \beta_{8} + 16 \beta_{9} + 20 \beta_{10} - 12 \beta_{11} ) q^{69} \) \( + ( 10 + 5 \beta_{1} - \beta_{2} - 20 \beta_{3} - 11 \beta_{4} - 12 \beta_{5} - 19 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{10} - 9 \beta_{11} ) q^{70} \) \( + ( -8 + 2 \beta_{1} - 20 \beta_{2} + 16 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 16 \beta_{7} - 6 \beta_{8} - 10 \beta_{9} - 20 \beta_{10} + 4 \beta_{11} ) q^{71} \) \( + ( -29 + 7 \beta_{1} - 2 \beta_{2} - 16 \beta_{3} - 13 \beta_{4} + 9 \beta_{5} + 2 \beta_{8} + 16 \beta_{9} - 2 \beta_{10} - 16 \beta_{11} ) q^{72} \) \( + ( -19 - 36 \beta_{1} - 15 \beta_{2} + 13 \beta_{3} - 10 \beta_{4} + 33 \beta_{5} + 14 \beta_{6} + \beta_{7} + 14 \beta_{8} - 14 \beta_{9} - 30 \beta_{10} - 21 \beta_{11} ) q^{73} \) \( + ( -11 + 7 \beta_{1} + 4 \beta_{2} - 17 \beta_{3} - 9 \beta_{4} - 36 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} + 17 \beta_{8} + 8 \beta_{9} + 2 \beta_{10} + 33 \beta_{11} ) q^{74} \) \( + ( 2 + \beta_{2} + \beta_{3} + 9 \beta_{4} + 8 \beta_{5} - \beta_{6} + 23 \beta_{7} + 23 \beta_{8} + 9 \beta_{9} - 9 \beta_{10} + 22 \beta_{11} ) q^{75} \) \( + ( 11 + 18 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + 25 \beta_{6} - 5 \beta_{7} - \beta_{8} + 12 \beta_{9} + 9 \beta_{10} + 18 \beta_{11} ) q^{76} \) \( + ( 16 - 16 \beta_{1} + 9 \beta_{2} - 9 \beta_{3} + 28 \beta_{4} + 24 \beta_{5} - 36 \beta_{6} + 12 \beta_{9} + 12 \beta_{10} ) q^{77} \) \( + ( -26 - 21 \beta_{1} + 11 \beta_{3} + 14 \beta_{4} + 5 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} + 11 \beta_{8} + 5 \beta_{9} - 5 \beta_{10} - 16 \beta_{11} ) q^{78} \) \( + ( 2 - 19 \beta_{1} - 4 \beta_{2} + 13 \beta_{3} + 44 \beta_{4} + 9 \beta_{5} - 55 \beta_{6} + 20 \beta_{7} + 7 \beta_{8} + 7 \beta_{9} + 15 \beta_{11} ) q^{79} \) \( + ( -19 + 27 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + 24 \beta_{4} - 29 \beta_{5} - 34 \beta_{6} + 9 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} - 11 \beta_{10} - 29 \beta_{11} ) q^{80} \) \( + ( -3 - 27 \beta_{1} + 11 \beta_{2} + 4 \beta_{3} + 19 \beta_{4} + 42 \beta_{5} - 15 \beta_{6} + 4 \beta_{7} + 11 \beta_{8} + 7 \beta_{9} - 11 \beta_{10} + 42 \beta_{11} ) q^{81} \) \( + ( -24 - 3 \beta_{1} - 22 \beta_{2} - 9 \beta_{3} + 14 \beta_{4} - 29 \beta_{6} - 22 \beta_{7} + 18 \beta_{9} - 22 \beta_{10} + 5 \beta_{11} ) q^{82} \) \( + ( -32 - 28 \beta_{1} - 4 \beta_{3} - 25 \beta_{4} + 3 \beta_{5} + 24 \beta_{6} + 12 \beta_{7} + 6 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} - 28 \beta_{11} ) q^{83} \) \( + ( 5 + 23 \beta_{1} - 23 \beta_{2} - 6 \beta_{3} - 48 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} - \beta_{7} - 23 \beta_{9} - 29 \beta_{10} - 9 \beta_{11} ) q^{84} \) \( + ( -31 + 8 \beta_{1} + 8 \beta_{2} - 36 \beta_{3} - 32 \beta_{4} - 5 \beta_{6} - 8 \beta_{7} - 27 \beta_{8} + 18 \beta_{9} + 27 \beta_{10} + \beta_{11} ) q^{85} \) \( + ( -10 + 26 \beta_{1} + 8 \beta_{2} + 14 \beta_{4} - 9 \beta_{5} - 10 \beta_{6} + 8 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{11} ) q^{86} \) \( + ( 5 + 17 \beta_{1} + 5 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} - 23 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} - 24 \beta_{8} - 5 \beta_{9} + 15 \beta_{10} - 14 \beta_{11} ) q^{87} \) \( + ( -13 + 24 \beta_{1} + 13 \beta_{2} + 6 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} + 30 \beta_{6} + 12 \beta_{8} - 19 \beta_{9} - 6 \beta_{10} + 32 \beta_{11} ) q^{88} \) \( + ( -57 - 33 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} + 36 \beta_{4} + 52 \beta_{5} + 30 \beta_{6} + 2 \beta_{7} - 5 \beta_{8} - 2 \beta_{10} - 27 \beta_{11} ) q^{89} \) \( + ( 9 + 8 \beta_{1} + \beta_{2} + 25 \beta_{3} + 25 \beta_{4} + \beta_{5} + 34 \beta_{6} - 25 \beta_{7} - 8 \beta_{8} - 17 \beta_{9} + \beta_{10} - \beta_{11} ) q^{90} \) \( + ( -37 - 21 \beta_{1} - 15 \beta_{2} + 12 \beta_{3} - 49 \beta_{4} - 6 \beta_{5} + 32 \beta_{6} + 11 \beta_{7} + 15 \beta_{8} - 12 \beta_{9} - 4 \beta_{10} - 20 \beta_{11} ) q^{91} \) \( + ( 26 + 20 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} - 46 \beta_{4} - 14 \beta_{5} + 12 \beta_{6} - 8 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} + 8 \beta_{10} + 16 \beta_{11} ) q^{92} \) \( + ( 2 - 23 \beta_{1} - 12 \beta_{2} - 11 \beta_{3} - 2 \beta_{4} - 17 \beta_{5} - 19 \beta_{6} + 9 \beta_{7} + 11 \beta_{8} + 9 \beta_{9} - 6 \beta_{10} + 28 \beta_{11} ) q^{93} \) \( + ( -14 - 26 \beta_{1} + 19 \beta_{2} + 19 \beta_{3} - 35 \beta_{4} + 37 \beta_{5} + 46 \beta_{6} - 29 \beta_{7} - 29 \beta_{8} - 9 \beta_{9} + 9 \beta_{10} - 46 \beta_{11} ) q^{94} \) \( + ( 7 + 23 \beta_{1} + 29 \beta_{2} - 10 \beta_{3} - 14 \beta_{4} + 37 \beta_{5} + 31 \beta_{7} + 8 \beta_{8} + 19 \beta_{9} + 11 \beta_{10} + 53 \beta_{11} ) q^{95} \) \( + ( 39 + 14 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} - 21 \beta_{4} - 24 \beta_{5} + 31 \beta_{6} + 20 \beta_{7} - 20 \beta_{8} - 7 \beta_{9} - 7 \beta_{10} ) q^{96} \) \( + ( 33 + 9 \beta_{1} + 3 \beta_{3} + 81 \beta_{4} - 47 \beta_{5} - 27 \beta_{6} + 24 \beta_{7} + 3 \beta_{8} - 24 \beta_{9} + \beta_{10} - 38 \beta_{11} ) q^{97} \) \( + ( 70 - 17 \beta_{1} + 3 \beta_{2} - 14 \beta_{3} - 7 \beta_{4} + 71 \beta_{5} - 13 \beta_{6} - 13 \beta_{7} + \beta_{8} + \beta_{9} + 20 \beta_{11} ) q^{98} \) \( + ( 40 + 5 \beta_{1} - 19 \beta_{2} + 3 \beta_{3} - 37 \beta_{4} - 24 \beta_{5} + 4 \beta_{6} - 7 \beta_{7} - 19 \beta_{8} + 3 \beta_{9} - 12 \beta_{10} + 7 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 36q^{6} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 36q^{6} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut -\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut 51q^{10} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 63q^{12} \) \(\mathstrut +\mathstrut 21q^{13} \) \(\mathstrut +\mathstrut 9q^{14} \) \(\mathstrut +\mathstrut 63q^{15} \) \(\mathstrut -\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 3q^{17} \) \(\mathstrut -\mathstrut 24q^{19} \) \(\mathstrut -\mathstrut 90q^{20} \) \(\mathstrut +\mathstrut 30q^{21} \) \(\mathstrut -\mathstrut 78q^{22} \) \(\mathstrut -\mathstrut 102q^{23} \) \(\mathstrut -\mathstrut 12q^{24} \) \(\mathstrut -\mathstrut 156q^{25} \) \(\mathstrut +\mathstrut 21q^{26} \) \(\mathstrut -\mathstrut 27q^{27} \) \(\mathstrut +\mathstrut 12q^{28} \) \(\mathstrut +\mathstrut 147q^{29} \) \(\mathstrut +\mathstrut 24q^{30} \) \(\mathstrut +\mathstrut 99q^{31} \) \(\mathstrut +\mathstrut 165q^{32} \) \(\mathstrut +\mathstrut 84q^{33} \) \(\mathstrut +\mathstrut 132q^{34} \) \(\mathstrut +\mathstrut 96q^{35} \) \(\mathstrut +\mathstrut 63q^{36} \) \(\mathstrut +\mathstrut 72q^{38} \) \(\mathstrut -\mathstrut 108q^{39} \) \(\mathstrut -\mathstrut 138q^{40} \) \(\mathstrut -\mathstrut 144q^{41} \) \(\mathstrut -\mathstrut 237q^{42} \) \(\mathstrut -\mathstrut 27q^{43} \) \(\mathstrut -\mathstrut 123q^{44} \) \(\mathstrut -\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 54q^{46} \) \(\mathstrut -\mathstrut 99q^{47} \) \(\mathstrut -\mathstrut 51q^{48} \) \(\mathstrut -\mathstrut 24q^{49} \) \(\mathstrut +\mathstrut 72q^{50} \) \(\mathstrut -\mathstrut 42q^{51} \) \(\mathstrut +\mathstrut 93q^{52} \) \(\mathstrut +\mathstrut 111q^{53} \) \(\mathstrut +\mathstrut 21q^{54} \) \(\mathstrut +\mathstrut 162q^{55} \) \(\mathstrut -\mathstrut 168q^{57} \) \(\mathstrut -\mathstrut 132q^{58} \) \(\mathstrut +\mathstrut 3q^{59} \) \(\mathstrut -\mathstrut 30q^{60} \) \(\mathstrut +\mathstrut 150q^{61} \) \(\mathstrut +\mathstrut 108q^{62} \) \(\mathstrut +\mathstrut 234q^{63} \) \(\mathstrut +\mathstrut 27q^{64} \) \(\mathstrut +\mathstrut 126q^{65} \) \(\mathstrut +\mathstrut 168q^{66} \) \(\mathstrut +\mathstrut 135q^{67} \) \(\mathstrut -\mathstrut 30q^{68} \) \(\mathstrut +\mathstrut 72q^{69} \) \(\mathstrut +\mathstrut 225q^{70} \) \(\mathstrut -\mathstrut 168q^{71} \) \(\mathstrut -\mathstrut 102q^{72} \) \(\mathstrut -\mathstrut 90q^{73} \) \(\mathstrut -\mathstrut 231q^{74} \) \(\mathstrut +\mathstrut 42q^{76} \) \(\mathstrut +\mathstrut 246q^{77} \) \(\mathstrut -\mathstrut 189q^{78} \) \(\mathstrut -\mathstrut 75q^{79} \) \(\mathstrut +\mathstrut 21q^{80} \) \(\mathstrut -\mathstrut 159q^{81} \) \(\mathstrut -\mathstrut 117q^{82} \) \(\mathstrut -\mathstrut 156q^{83} \) \(\mathstrut +\mathstrut 99q^{84} \) \(\mathstrut -\mathstrut 300q^{85} \) \(\mathstrut -\mathstrut 144q^{86} \) \(\mathstrut +\mathstrut 69q^{87} \) \(\mathstrut -\mathstrut 405q^{88} \) \(\mathstrut -\mathstrut 558q^{89} \) \(\mathstrut -\mathstrut 66q^{90} \) \(\mathstrut -\mathstrut 453q^{91} \) \(\mathstrut +\mathstrut 48q^{92} \) \(\mathstrut -\mathstrut 57q^{93} \) \(\mathstrut -\mathstrut 69q^{95} \) \(\mathstrut +\mathstrut 558q^{96} \) \(\mathstrut +\mathstrut 465q^{97} \) \(\mathstrut +\mathstrut 777q^{98} \) \(\mathstrut +\mathstrut 462q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut +\mathstrut \) \(24\) \(x^{10}\mathstrut +\mathstrut \) \(216\) \(x^{8}\mathstrut +\mathstrut \) \(905\) \(x^{6}\mathstrut +\mathstrut \) \(1770\) \(x^{4}\mathstrut +\mathstrut \) \(1395\) \(x^{2}\mathstrut +\mathstrut \) \(361\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} - 774 \nu^{4} - 2566 \nu^{3} - 2742 \nu^{2} - 1177 \nu - 1520 \)\()/304\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} - 5 \nu^{10} - \nu^{9} - 89 \nu^{8} + 239 \nu^{7} - 437 \nu^{6} + 2078 \nu^{5} - 22 \nu^{4} + 5308 \nu^{3} + 2960 \nu^{2} + 2697 \nu + 1881 \)\()/304\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{11} + 5 \nu^{10} - \nu^{9} + 89 \nu^{8} + 239 \nu^{7} + 437 \nu^{6} + 2078 \nu^{5} + 22 \nu^{4} + 5308 \nu^{3} - 2960 \nu^{2} + 2697 \nu - 1881 \)\()/304\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + 774 \nu^{4} - 2566 \nu^{3} + 2742 \nu^{2} - 1177 \nu + 1520 \)\()/304\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + 5334 \nu^{4} - 24 \nu^{3} + 3464 \nu^{2} + 417 \nu + 171 \)\()/608\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} - 5334 \nu^{4} - 24 \nu^{3} - 3464 \nu^{2} + 417 \nu - 171 \)\()/608\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{10} + 21 \nu^{8} + 153 \nu^{6} + 454 \nu^{4} + 504 \nu^{2} + 8 \nu + 171 \)\()/16\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{10} - 21 \nu^{8} - 153 \nu^{6} - 454 \nu^{4} - 504 \nu^{2} + 8 \nu - 171 \)\()/16\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{11} + \nu^{10} - 56 \nu^{9} + 14 \nu^{8} - 329 \nu^{7} + 19 \nu^{6} - 549 \nu^{5} - 387 \nu^{4} + 411 \nu^{3} - 1295 \nu^{2} + 650 \nu - 380 \)\()/152\)
\(\beta_{10}\)\(=\)\((\)\( 17 \nu^{11} + 17 \nu^{10} + 333 \nu^{9} + 333 \nu^{8} + 2185 \nu^{7} + 2185 \nu^{6} + 5334 \nu^{5} + 5334 \nu^{4} + 3464 \nu^{3} + 3768 \nu^{2} + 171 \nu + 1083 \)\()/608\)
\(\beta_{11}\)\(=\)\((\)\( 9 \nu^{11} + 197 \nu^{9} + 1545 \nu^{7} + 5238 \nu^{5} + 7304 \nu^{3} + 2979 \nu - 152 \)\()/304\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\)
\(\nu^{2}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(5\) \(\beta_{8}\mathstrut -\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(7\) \(\beta_{10}\mathstrut -\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\(2\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(32\) \(\beta_{8}\mathstrut +\mathstrut \) \(32\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut -\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\) \(\beta_{2}\mathstrut -\mathstrut \) \(26\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(50\) \(\beta_{10}\mathstrut +\mathstrut \) \(50\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut -\mathstrut \) \(30\) \(\beta_{6}\mathstrut -\mathstrut \) \(20\) \(\beta_{5}\mathstrut +\mathstrut \) \(102\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\) \(\beta_{3}\mathstrut -\mathstrut \) \(23\) \(\beta_{2}\mathstrut -\mathstrut \) \(52\) \(\beta_{1}\mathstrut -\mathstrut \) \(153\)
\(\nu^{7}\)\(=\)\(-\)\(38\) \(\beta_{11}\mathstrut +\mathstrut \) \(44\) \(\beta_{10}\mathstrut -\mathstrut \) \(44\) \(\beta_{9}\mathstrut -\mathstrut \) \(226\) \(\beta_{8}\mathstrut -\mathstrut \) \(226\) \(\beta_{7}\mathstrut +\mathstrut \) \(158\) \(\beta_{6}\mathstrut +\mathstrut \) \(114\) \(\beta_{5}\mathstrut +\mathstrut \) \(200\) \(\beta_{4}\mathstrut +\mathstrut \) \(130\) \(\beta_{3}\mathstrut +\mathstrut \) \(130\) \(\beta_{2}\mathstrut +\mathstrut \) \(244\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)
\(\nu^{8}\)\(=\)\(-\)\(384\) \(\beta_{10}\mathstrut -\mathstrut \) \(384\) \(\beta_{9}\mathstrut -\mathstrut \) \(149\) \(\beta_{8}\mathstrut +\mathstrut \) \(149\) \(\beta_{7}\mathstrut +\mathstrut \) \(334\) \(\beta_{6}\mathstrut +\mathstrut \) \(50\) \(\beta_{5}\mathstrut -\mathstrut \) \(888\) \(\beta_{4}\mathstrut -\mathstrut \) \(200\) \(\beta_{3}\mathstrut +\mathstrut \) \(200\) \(\beta_{2}\mathstrut +\mathstrut \) \(504\) \(\beta_{1}\mathstrut +\mathstrut \) \(1104\)
\(\nu^{9}\)\(=\)\(494\) \(\beta_{11}\mathstrut -\mathstrut \) \(483\) \(\beta_{10}\mathstrut +\mathstrut \) \(483\) \(\beta_{9}\mathstrut +\mathstrut \) \(1688\) \(\beta_{8}\mathstrut +\mathstrut \) \(1688\) \(\beta_{7}\mathstrut -\mathstrut \) \(1400\) \(\beta_{6}\mathstrut -\mathstrut \) \(917\) \(\beta_{5}\mathstrut -\mathstrut \) \(1589\) \(\beta_{4}\mathstrut -\mathstrut \) \(1186\) \(\beta_{3}\mathstrut -\mathstrut \) \(1186\) \(\beta_{2}\mathstrut -\mathstrut \) \(2072\) \(\beta_{1}\mathstrut -\mathstrut \) \(236\)
\(\nu^{10}\)\(=\)\(3088\) \(\beta_{10}\mathstrut +\mathstrut \) \(3088\) \(\beta_{9}\mathstrut +\mathstrut \) \(1433\) \(\beta_{8}\mathstrut -\mathstrut \) \(1433\) \(\beta_{7}\mathstrut -\mathstrut \) \(3332\) \(\beta_{6}\mathstrut +\mathstrut \) \(244\) \(\beta_{5}\mathstrut +\mathstrut \) \(7532\) \(\beta_{4}\mathstrut +\mathstrut \) \(1589\) \(\beta_{3}\mathstrut -\mathstrut \) \(1589\) \(\beta_{2}\mathstrut -\mathstrut \) \(4444\) \(\beta_{1}\mathstrut -\mathstrut \) \(8372\)
\(\nu^{11}\)\(=\)\(-\)\(5420\) \(\beta_{11}\mathstrut +\mathstrut \) \(4765\) \(\beta_{10}\mathstrut -\mathstrut \) \(4765\) \(\beta_{9}\mathstrut -\mathstrut \) \(13049\) \(\beta_{8}\mathstrut -\mathstrut \) \(13049\) \(\beta_{7}\mathstrut +\mathstrut \) \(12374\) \(\beta_{6}\mathstrut +\mathstrut \) \(7609\) \(\beta_{5}\mathstrut +\mathstrut \) \(12211\) \(\beta_{4}\mathstrut +\mathstrut \) \(10398\) \(\beta_{3}\mathstrut +\mathstrut \) \(10398\) \(\beta_{2}\mathstrut +\mathstrut \) \(16976\) \(\beta_{1}\mathstrut +\mathstrut \) \(2055\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1} - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
2.88811i
0.918492i
2.01431i
0.728740i
2.88811i
0.918492i
2.01431i
0.728740i
2.57727i
1.89323i
2.57727i
1.89323i
−2.84423 0.501515i 3.28392 3.91363i 4.07936 + 1.48477i −3.00117 + 1.09234i −11.3030 + 9.48432i 3.87208 + 6.70664i −0.853313 0.492661i −2.96949 16.8408i 9.08386 1.60173i
2.2 0.904538 + 0.159494i −0.464845 + 0.553981i −2.96602 1.07954i 0.295437 0.107530i −0.508827 + 0.426957i −0.328846 0.569578i −5.69245 3.28654i 1.47202 + 8.34824i 0.284385 0.0501447i
3.1 −1.29478 + 1.54305i 0.621128 + 1.70654i −0.00997859 0.0565914i 1.24445 7.05761i −3.43750 1.25115i 0.422527 + 0.731838i −6.87755 3.97075i 4.36793 3.66513i 9.27900 + 11.0583i
3.2 0.468425 0.558247i −1.14207 3.13782i 0.602375 + 3.41624i −1.13111 + 6.41483i −2.28665 0.832274i −5.47943 9.49065i 4.71370 + 2.72146i −1.64718 + 1.38215i 3.05122 + 3.63630i
10.1 −2.84423 + 0.501515i 3.28392 + 3.91363i 4.07936 1.48477i −3.00117 1.09234i −11.3030 9.48432i 3.87208 6.70664i −0.853313 + 0.492661i −2.96949 + 16.8408i 9.08386 + 1.60173i
10.2 0.904538 0.159494i −0.464845 0.553981i −2.96602 + 1.07954i 0.295437 + 0.107530i −0.508827 0.426957i −0.328846 + 0.569578i −5.69245 + 3.28654i 1.47202 8.34824i 0.284385 + 0.0501447i
13.1 −1.29478 1.54305i 0.621128 1.70654i −0.00997859 + 0.0565914i 1.24445 + 7.05761i −3.43750 + 1.25115i 0.422527 0.731838i −6.87755 + 3.97075i 4.36793 + 3.66513i 9.27900 11.0583i
13.2 0.468425 + 0.558247i −1.14207 + 3.13782i 0.602375 3.41624i −1.13111 6.41483i −2.28665 + 0.832274i −5.47943 + 9.49065i 4.71370 2.72146i −1.64718 1.38215i 3.05122 3.63630i
14.1 −0.881480 2.42185i −0.384565 0.0678091i −2.02415 + 1.69847i 1.73199 + 1.45331i 0.174763 + 0.991129i 5.72163 + 9.91015i −3.03027 1.74952i −8.31394 3.02603i 1.99298 5.47566i
14.2 0.647524 + 1.77906i −1.91357 0.337414i 0.318417 0.267183i −2.13959 1.79533i −0.638803 3.62283i −1.20796 2.09224i 7.23987 + 4.17994i −4.90934 1.78685i 1.80856 4.96897i
15.1 −0.881480 + 2.42185i −0.384565 + 0.0678091i −2.02415 1.69847i 1.73199 1.45331i 0.174763 0.991129i 5.72163 9.91015i −3.03027 + 1.74952i −8.31394 + 3.02603i 1.99298 + 5.47566i
15.2 0.647524 1.77906i −1.91357 + 0.337414i 0.318417 + 0.267183i −2.13959 + 1.79533i −0.638803 + 3.62283i −1.20796 + 2.09224i 7.23987 4.17994i −4.90934 + 1.78685i 1.80856 + 4.96897i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.2
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(19, [\chi])\).