Properties

Label 19.3.d.a
Level 19
Weight 3
Character orbit 19.d
Analytic conductor 0.518
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.6967728.1
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta_{5} ) q^{2} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{3} \) \( + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -3 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{6} \) \( + ( -1 - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{7} \) \( + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta_{5} ) q^{2} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{3} \) \( + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{4} \) \( + ( -3 + \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{6} \) \( + ( -1 - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{7} \) \( + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} \) \( + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{9} \) \( + ( -12 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - \beta_{4} ) q^{10} \) \( + ( 6 + 5 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{11} \) \( + ( 4 + 2 \beta_{1} + \beta_{2} + 14 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{12} \) \( + ( 4 + 2 \beta_{3} - 4 \beta_{4} ) q^{13} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 4 \beta_{5} ) q^{14} \) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + 10 \beta_{4} ) q^{15} \) \( + ( 7 + 4 \beta_{1} + 3 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{16} \) \( + ( -16 - 15 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{17} \) \( + ( -10 - 6 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{18} \) \( + ( 14 - 5 \beta_{1} - 3 \beta_{2} + 13 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} ) q^{19} \) \( + ( 17 - 3 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{20} \) \( + ( -7 - \beta_{1} - 2 \beta_{2} + 14 \beta_{3} + 7 \beta_{5} ) q^{21} \) \( + ( -4 - \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{5} ) q^{22} \) \( + ( -4 + \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{23} \) \( + ( -26 - 5 \beta_{1} - 26 \beta_{3} ) q^{24} \) \( + ( 2 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{25} \) \( + ( -22 + 8 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{26} \) \( + ( 2 - 10 \beta_{1} - 5 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{27} \) \( + ( 7 \beta_{1} + 7 \beta_{2} - 13 \beta_{3} ) q^{28} \) \( + ( 9 \beta_{1} - 9 \beta_{2} + 13 \beta_{4} ) q^{29} \) \( + ( 45 - 17 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{30} \) \( + ( 17 - 2 \beta_{1} - \beta_{2} + 42 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{31} \) \( + ( 16 - 4 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 3 \beta_{4} ) q^{32} \) \( + ( 17 + 6 \beta_{1} + 12 \beta_{2} - 15 \beta_{3} + 2 \beta_{5} ) q^{33} \) \( + ( -10 - 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} - 15 \beta_{4} ) q^{34} \) \( + ( -26 - 14 \beta_{1} - 17 \beta_{3} - 18 \beta_{4} - 9 \beta_{5} ) q^{35} \) \( + ( -14 - 6 \beta_{1} - 15 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{36} \) \( + ( -3 + 2 \beta_{1} + \beta_{2} - 24 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} ) q^{37} \) \( + ( 18 + 18 \beta_{1} + 7 \beta_{2} + 33 \beta_{3} + 8 \beta_{4} + \beta_{5} ) q^{38} \) \( + ( -4 + 10 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} ) q^{39} \) \( + ( -23 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 19 \beta_{5} ) q^{40} \) \( + ( 3 - 10 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} - \beta_{5} ) q^{41} \) \( + ( 26 + 17 \beta_{1} + 17 \beta_{2} + 38 \beta_{3} + 13 \beta_{4} + 26 \beta_{5} ) q^{42} \) \( + ( -22 + 8 \beta_{1} - 18 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} ) q^{43} \) \( + ( -8 - 15 \beta_{1} - 15 \beta_{2} + 24 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} ) q^{44} \) \( + ( -4 + 2 \beta_{2} - 10 \beta_{4} + 10 \beta_{5} ) q^{45} \) \( + ( -15 - 10 \beta_{1} - 5 \beta_{2} - 22 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{46} \) \( + ( 12 - \beta_{1} - \beta_{2} - 13 \beta_{3} + 6 \beta_{4} + 12 \beta_{5} ) q^{47} \) \( + ( -46 + \beta_{1} - \beta_{2} - 23 \beta_{3} - 19 \beta_{4} ) q^{48} \) \( + ( 5 + 6 \beta_{2} + 7 \beta_{4} - 7 \beta_{5} ) q^{49} \) \( + ( 9 - 12 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} ) q^{50} \) \( + ( 34 + 14 \beta_{1} - 14 \beta_{2} + 17 \beta_{3} + 21 \beta_{4} ) q^{51} \) \( + ( 22 - 12 \beta_{1} - 24 \beta_{2} - 10 \beta_{3} + 12 \beta_{5} ) q^{52} \) \( + ( -8 - 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} ) q^{53} \) \( + ( 4 + 11 \beta_{1} + 5 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{54} \) \( + ( 1 + 17 \beta_{1} - 7 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} ) q^{55} \) \( + ( 31 - 6 \beta_{1} - 3 \beta_{2} + 42 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} ) q^{56} \) \( + ( -31 - 12 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} - 37 \beta_{4} - 26 \beta_{5} ) q^{57} \) \( + ( 29 + \beta_{2} + 9 \beta_{4} - 9 \beta_{5} ) q^{58} \) \( + ( 2 + 11 \beta_{1} + 22 \beta_{2} + 20 \beta_{3} + 22 \beta_{5} ) q^{59} \) \( + ( -35 + 17 \beta_{1} + 34 \beta_{2} + 11 \beta_{3} - 24 \beta_{5} ) q^{60} \) \( + ( -30 - 13 \beta_{1} - 13 \beta_{2} - 30 \beta_{3} - 15 \beta_{4} - 30 \beta_{5} ) q^{61} \) \( + ( 3 - 5 \beta_{1} - 17 \beta_{3} + 40 \beta_{4} + 20 \beta_{5} ) q^{62} \) \( + ( -24 - 8 \beta_{1} - 8 \beta_{2} - 38 \beta_{3} - 12 \beta_{4} - 24 \beta_{5} ) q^{63} \) \( + ( 27 - 2 \beta_{2} - 12 \beta_{4} + 12 \beta_{5} ) q^{64} \) \( + ( 18 + 28 \beta_{1} + 14 \beta_{2} + 40 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} \) \( + ( -18 - 14 \beta_{1} - 14 \beta_{2} - 8 \beta_{3} - 9 \beta_{4} - 18 \beta_{5} ) q^{66} \) \( + ( 48 - 13 \beta_{1} + 13 \beta_{2} + 24 \beta_{3} - 8 \beta_{4} ) q^{67} \) \( + ( -6 + 24 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} ) q^{68} \) \( + ( 25 + 6 \beta_{1} + 3 \beta_{2} + 12 \beta_{3} + 19 \beta_{4} + 19 \beta_{5} ) q^{69} \) \( + ( -62 - 4 \beta_{1} + 4 \beta_{2} - 31 \beta_{3} - 31 \beta_{4} ) q^{70} \) \( + ( -10 + 4 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} ) q^{71} \) \( + ( 46 + 8 \beta_{1} - 8 \beta_{2} + 23 \beta_{3} + 7 \beta_{4} ) q^{72} \) \( + ( 25 - 2 \beta_{1} + 14 \beta_{3} + 22 \beta_{4} + 11 \beta_{5} ) q^{73} \) \( + ( 31 + 15 \beta_{1} + 42 \beta_{3} - 22 \beta_{4} - 11 \beta_{5} ) q^{74} \) \( + ( -24 - 6 \beta_{1} - 3 \beta_{2} - 60 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{75} \) \( + ( -30 - 11 \beta_{1} - 18 \beta_{2} - 36 \beta_{3} + 31 \beta_{4} + 11 \beta_{5} ) q^{76} \) \( + ( -73 - 31 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} ) q^{77} \) \( + ( 48 - 22 \beta_{1} - 44 \beta_{2} - 40 \beta_{3} + 8 \beta_{5} ) q^{78} \) \( + ( 26 + 10 \beta_{1} + 20 \beta_{2} + 8 \beta_{3} + 34 \beta_{5} ) q^{79} \) \( + ( -30 - 29 \beta_{1} - 29 \beta_{2} - 50 \beta_{3} - 15 \beta_{4} - 30 \beta_{5} ) q^{80} \) \( + ( 81 + 4 \beta_{1} + 82 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{81} \) \( + ( -28 + 28 \beta_{1} + 28 \beta_{2} + 25 \beta_{3} - 14 \beta_{4} - 28 \beta_{5} ) q^{82} \) \( + ( 12 - 37 \beta_{2} + 13 \beta_{4} - 13 \beta_{5} ) q^{83} \) \( + ( 19 + 26 \beta_{1} + 13 \beta_{2} - 16 \beta_{3} + 27 \beta_{4} + 27 \beta_{5} ) q^{84} \) \( + ( 32 - 6 \beta_{1} - 6 \beta_{2} + 48 \beta_{3} + 16 \beta_{4} + 32 \beta_{5} ) q^{85} \) \( + ( -56 - 16 \beta_{1} + 16 \beta_{2} - 28 \beta_{3} - 10 \beta_{4} ) q^{86} \) \( + ( -99 - 13 \beta_{2} - 8 \beta_{4} + 8 \beta_{5} ) q^{87} \) \( + ( -20 + 22 \beta_{1} + 11 \beta_{2} - 50 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{88} \) \( + ( 6 - 24 \beta_{1} + 24 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} ) q^{89} \) \( + ( -32 + 18 \beta_{1} + 36 \beta_{2} + 48 \beta_{3} + 16 \beta_{5} ) q^{90} \) \( + ( -52 + 2 \beta_{1} - 2 \beta_{2} - 26 \beta_{3} - 4 \beta_{4} ) q^{91} \) \( + ( 7 + 3 \beta_{1} + 15 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} ) q^{92} \) \( + ( -61 - 67 \beta_{1} - 46 \beta_{3} - 30 \beta_{4} - 15 \beta_{5} ) q^{93} \) \( + ( 17 + 26 \beta_{1} + 13 \beta_{2} + 62 \beta_{3} - 14 \beta_{4} - 14 \beta_{5} ) q^{94} \) \( + ( 67 + 10 \beta_{1} - 13 \beta_{2} - 26 \beta_{3} - 4 \beta_{4} - 10 \beta_{5} ) q^{95} \) \( + ( 28 + 21 \beta_{2} - 22 \beta_{4} + 22 \beta_{5} ) q^{96} \) \( + ( -29 - 2 \beta_{1} - 4 \beta_{2} - 17 \beta_{3} - 46 \beta_{5} ) q^{97} \) \( + ( 35 - 20 \beta_{1} - 40 \beta_{2} - 41 \beta_{3} - 6 \beta_{5} ) q^{98} \) \( + ( 10 + 2 \beta_{1} + 2 \beta_{2} + 23 \beta_{3} + 5 \beta_{4} + 10 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 9q^{3} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut +\mathstrut q^{6} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut -\mathstrut 60q^{10} \) \(\mathstrut +\mathstrut 26q^{11} \) \(\mathstrut +\mathstrut 30q^{13} \) \(\mathstrut +\mathstrut 54q^{14} \) \(\mathstrut -\mathstrut 18q^{15} \) \(\mathstrut +\mathstrut q^{16} \) \(\mathstrut -\mathstrut 42q^{17} \) \(\mathstrut +\mathstrut 25q^{19} \) \(\mathstrut +\mathstrut 108q^{20} \) \(\mathstrut -\mathstrut 102q^{21} \) \(\mathstrut -\mathstrut 39q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 83q^{24} \) \(\mathstrut -\mathstrut 17q^{25} \) \(\mathstrut -\mathstrut 148q^{26} \) \(\mathstrut +\mathstrut 32q^{28} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 304q^{30} \) \(\mathstrut +\mathstrut 51q^{32} \) \(\mathstrut +\mathstrut 123q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 38q^{35} \) \(\mathstrut -\mathstrut 54q^{36} \) \(\mathstrut -\mathstrut 14q^{38} \) \(\mathstrut -\mathstrut 44q^{39} \) \(\mathstrut -\mathstrut 96q^{40} \) \(\mathstrut +\mathstrut 63q^{41} \) \(\mathstrut -\mathstrut 92q^{42} \) \(\mathstrut -\mathstrut 34q^{43} \) \(\mathstrut -\mathstrut 69q^{44} \) \(\mathstrut -\mathstrut 28q^{45} \) \(\mathstrut +\mathstrut 58q^{47} \) \(\mathstrut -\mathstrut 147q^{48} \) \(\mathstrut +\mathstrut 18q^{49} \) \(\mathstrut +\mathstrut 132q^{51} \) \(\mathstrut +\mathstrut 162q^{52} \) \(\mathstrut -\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 29q^{54} \) \(\mathstrut -\mathstrut 28q^{55} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 172q^{58} \) \(\mathstrut -\mathstrut 147q^{59} \) \(\mathstrut -\mathstrut 222q^{60} \) \(\mathstrut +\mathstrut 58q^{61} \) \(\mathstrut -\mathstrut 116q^{62} \) \(\mathstrut +\mathstrut 86q^{63} \) \(\mathstrut +\mathstrut 166q^{64} \) \(\mathstrut +\mathstrut 11q^{66} \) \(\mathstrut +\mathstrut 201q^{67} \) \(\mathstrut -\mathstrut 84q^{68} \) \(\mathstrut -\mathstrut 198q^{70} \) \(\mathstrut -\mathstrut 102q^{71} \) \(\mathstrut +\mathstrut 210q^{72} \) \(\mathstrut +\mathstrut 7q^{73} \) \(\mathstrut +\mathstrut 174q^{74} \) \(\mathstrut -\mathstrut 173q^{76} \) \(\mathstrut -\mathstrut 376q^{77} \) \(\mathstrut +\mathstrut 450q^{78} \) \(\mathstrut +\mathstrut 134q^{80} \) \(\mathstrut +\mathstrut 253q^{81} \) \(\mathstrut -\mathstrut 145q^{82} \) \(\mathstrut +\mathstrut 146q^{83} \) \(\mathstrut -\mathstrut 90q^{85} \) \(\mathstrut -\mathstrut 270q^{86} \) \(\mathstrut -\mathstrut 568q^{87} \) \(\mathstrut -\mathstrut 72q^{89} \) \(\mathstrut -\mathstrut 438q^{90} \) \(\mathstrut -\mathstrut 216q^{91} \) \(\mathstrut +\mathstrut 72q^{92} \) \(\mathstrut -\mathstrut 160q^{93} \) \(\mathstrut +\mathstrut 558q^{95} \) \(\mathstrut +\mathstrut 126q^{96} \) \(\mathstrut +\mathstrut 21q^{97} \) \(\mathstrut +\mathstrut 411q^{98} \) \(\mathstrut -\mathstrut 56q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut +\mathstrut \) \(8\) \(x^{4}\mathstrut +\mathstrut \) \(5\) \(x^{3}\mathstrut +\mathstrut \) \(50\) \(x^{2}\mathstrut -\mathstrut \) \(7\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 8 \nu^{4} + 64 \nu^{3} - 50 \nu^{2} + 7 \nu - 56 \)\()/393\)
\(\beta_{3}\)\(=\)\((\)\( 56 \nu^{5} - 55 \nu^{4} + 440 \nu^{3} + 344 \nu^{2} + 2750 \nu - 385 \)\()/393\)
\(\beta_{4}\)\(=\)\((\)\( 70 \nu^{5} - 36 \nu^{4} + 550 \nu^{3} + 561 \nu^{2} + 3634 \nu + 534 \)\()/393\)
\(\beta_{5}\)\(=\)\((\)\( 77 \nu^{5} - 92 \nu^{4} + 605 \nu^{3} + 211 \nu^{2} + 3683 \nu - 1430 \)\()/393\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(8\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(31\) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut -\mathstrut \) \(23\)
\(\nu^{5}\)\(=\)\(28\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\) \(\beta_{4}\mathstrut -\mathstrut \) \(48\) \(\beta_{3}\mathstrut -\mathstrut \) \(55\) \(\beta_{2}\mathstrut -\mathstrut \) \(55\) \(\beta_{1}\mathstrut +\mathstrut \) \(28\)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
1.56632 2.71294i
−1.13654 + 1.96854i
0.0702177 0.121621i
1.56632 + 2.71294i
−1.13654 1.96854i
0.0702177 + 0.121621i
−2.90671 1.67819i −3.29225 1.90078i 3.63264 + 6.29191i 3.47303 6.01546i 6.37974 + 11.0500i −1.22892 10.9595i 2.72593 + 4.72145i −20.1902 + 11.6568i
8.2 −0.583430 0.336844i 2.49304 + 1.43936i −1.77307 3.07105i −1.55311 + 2.69006i −0.969676 1.67953i −8.15294 5.08374i −0.356503 0.617481i 1.81226 1.04631i
8.3 1.99014 + 1.14901i −3.70079 2.13665i 0.640435 + 1.10927i −2.91992 + 5.05745i −4.91006 8.50447i 9.38186 6.24860i 4.63057 + 8.02039i −11.6221 + 6.71002i
12.1 −2.90671 + 1.67819i −3.29225 + 1.90078i 3.63264 6.29191i 3.47303 + 6.01546i 6.37974 11.0500i −1.22892 10.9595i 2.72593 4.72145i −20.1902 11.6568i
12.2 −0.583430 + 0.336844i 2.49304 1.43936i −1.77307 + 3.07105i −1.55311 2.69006i −0.969676 + 1.67953i −8.15294 5.08374i −0.356503 + 0.617481i 1.81226 + 1.04631i
12.3 1.99014 1.14901i −3.70079 + 2.13665i 0.640435 1.10927i −2.91992 5.05745i −4.91006 + 8.50447i 9.38186 6.24860i 4.63057 8.02039i −11.6221 6.71002i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.3
Significant digits:
Format:

Inner twists

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

Hecke kernels

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