Properties

Label 16.3.f.a
Level 16
Weight 3
Character orbit 16.f
Analytic conductor 0.436
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 16.f (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.435968422976\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -1 - \beta_{2} + \beta_{5} ) q^{3} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{4} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{6} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( -2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{8} \) \( + ( 2 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -1 - \beta_{2} + \beta_{5} ) q^{3} \) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{4} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( -2 - 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + \beta_{4} ) q^{6} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{7} \) \( + ( -2 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{8} \) \( + ( 2 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{9} \) \( + ( 6 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{10} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{11} \) \( + ( 8 + 6 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{12} \) \( + ( -2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{13} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} - 2 \beta_{5} ) q^{14} \) \( + ( 1 + \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{15} \) \( + ( -6 - 2 \beta_{1} + 10 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{16} \) \( + ( -2 + 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{17} \) \( + ( -12 + 4 \beta_{1} - 8 \beta_{3} - \beta_{4} ) q^{18} \) \( + ( 5 - 2 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} \) \( + ( -14 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} ) q^{20} \) \( + ( -6 - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} ) q^{21} \) \( + ( -8 + 4 \beta_{1} - \beta_{2} + 10 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{22} \) \( + ( 11 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{23} \) \( + ( 10 - 2 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{24} \) \( + ( -4 - 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{25} \) \( + ( 18 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 6 \beta_{5} ) q^{26} \) \( + ( 10 + 2 \beta_{1} - 2 \beta_{2} + 10 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{27} \) \( + ( 14 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 12 \beta_{4} + 2 \beta_{5} ) q^{28} \) \( + ( 7 \beta_{1} + \beta_{2} - 8 \beta_{3} - 9 \beta_{4} + \beta_{5} ) q^{29} \) \( + ( 6 + 2 \beta_{1} - 10 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} ) q^{30} \) \( + ( 4 \beta_{2} - 20 \beta_{3} + 4 \beta_{5} ) q^{31} \) \( + ( -4 + 4 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} + 12 \beta_{4} ) q^{32} \) \( + ( 2 - 8 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} ) q^{33} \) \( + ( -16 - 8 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{34} \) \( + ( -14 + 4 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{35} \) \( + ( -13 - 3 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{36} \) \( + ( 14 + 5 \beta_{1} + 17 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 7 \beta_{5} ) q^{37} \) \( + ( 10 + 2 \beta_{1} + 7 \beta_{2} + 8 \beta_{3} - 7 \beta_{4} + 4 \beta_{5} ) q^{38} \) \( + ( -33 + \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 13 \beta_{4} + 7 \beta_{5} ) q^{39} \) \( + ( 6 + 2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} - 10 \beta_{5} ) q^{40} \) \( + ( -12 \beta_{2} - 4 \beta_{3} - 12 \beta_{5} ) q^{41} \) \( + ( -8 - 8 \beta_{1} - 6 \beta_{2} + 16 \beta_{3} + 6 \beta_{4} ) q^{42} \) \( + ( -24 + \beta_{1} - 8 \beta_{2} - 17 \beta_{3} + 15 \beta_{4} - 8 \beta_{5} ) q^{43} \) \( + ( -6 - 2 \beta_{1} - 12 \beta_{2} - 16 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} ) q^{44} \) \( + ( 14 - \beta_{1} + 5 \beta_{2} + 10 \beta_{3} - 9 \beta_{4} + 5 \beta_{5} ) q^{45} \) \( + ( 10 + 6 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{46} \) \( + ( -8 - 8 \beta_{1} - 4 \beta_{2} + 36 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} ) q^{47} \) \( + ( 8 + 8 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{48} \) \( + ( -5 - 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 16 \beta_{4} + 4 \beta_{5} ) q^{49} \) \( + ( 8 - 8 \beta_{1} + 24 \beta_{3} + 7 \beta_{4} - 8 \beta_{5} ) q^{50} \) \( + ( 28 + 6 \beta_{1} + 12 \beta_{2} - 22 \beta_{3} + 6 \beta_{4} ) q^{51} \) \( + ( 24 + 8 \beta_{1} + 20 \beta_{2} - 14 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{52} \) \( + ( 10 + \beta_{1} - 3 \beta_{2} - 14 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{53} \) \( + ( 4 - 4 \beta_{1} + 8 \beta_{2} - 12 \beta_{3} + 8 \beta_{4} + 4 \beta_{5} ) q^{54} \) \( + ( 39 + 9 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 5 \beta_{4} + 7 \beta_{5} ) q^{55} \) \( + ( -28 + 12 \beta_{1} + 12 \beta_{2} + 8 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} ) q^{56} \) \( + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{57} \) \( + ( -34 + 2 \beta_{1} - 7 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} + 14 \beta_{5} ) q^{58} \) \( + ( 36 - 3 \beta_{1} + 4 \beta_{2} + 35 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} ) q^{59} \) \( + ( -22 - 10 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{60} \) \( + ( 2 + \beta_{1} - 5 \beta_{2} + 6 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} ) q^{61} \) \( + ( -16 + 8 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} ) q^{62} \) \( + ( 1 + \beta_{1} - 9 \beta_{2} - 51 \beta_{3} - \beta_{4} - 11 \beta_{5} ) q^{63} \) \( + ( 4 - 20 \beta_{1} - 8 \beta_{2} - 20 \beta_{3} + 16 \beta_{4} - 4 \beta_{5} ) q^{64} \) \( + ( -2 + 12 \beta_{1} + 12 \beta_{4} ) q^{65} \) \( + ( 40 + 16 \beta_{1} + 10 \beta_{2} - 4 \beta_{3} - 12 \beta_{5} ) q^{66} \) \( + ( -39 - 14 \beta_{1} - 23 \beta_{2} + 30 \beta_{3} - 14 \beta_{4} - 5 \beta_{5} ) q^{67} \) \( + ( 18 - 2 \beta_{1} - 8 \beta_{2} + 42 \beta_{3} - 6 \beta_{5} ) q^{68} \) \( + ( -26 - 4 \beta_{1} - 16 \beta_{2} + 14 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} ) q^{69} \) \( + ( -12 + 4 \beta_{1} - 18 \beta_{2} - 32 \beta_{3} + 18 \beta_{4} - 8 \beta_{5} ) q^{70} \) \( + ( -37 - 19 \beta_{1} + 17 \beta_{2} + 17 \beta_{3} + 15 \beta_{4} - 17 \beta_{5} ) q^{71} \) \( + ( 2 - 10 \beta_{1} - 10 \beta_{2} + 48 \beta_{3} - 6 \beta_{4} + 16 \beta_{5} ) q^{72} \) \( + ( -2 - 2 \beta_{1} + 20 \beta_{2} + 2 \beta_{4} + 24 \beta_{5} ) q^{73} \) \( + ( -6 + 14 \beta_{1} + 9 \beta_{2} - 58 \beta_{3} - 9 \beta_{4} - 10 \beta_{5} ) q^{74} \) \( + ( -32 + 7 \beta_{1} + 8 \beta_{2} - 47 \beta_{3} - 23 \beta_{4} + 8 \beta_{5} ) q^{75} \) \( + ( -32 + 8 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} ) q^{76} \) \( + ( -38 - 4 \beta_{2} - 34 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{77} \) \( + ( -30 - 2 \beta_{1} - 34 \beta_{2} + 38 \beta_{3} + 14 \beta_{5} ) q^{78} \) \( + ( 24 + 24 \beta_{1} + 32 \beta_{2} + 48 \beta_{3} - 24 \beta_{4} - 16 \beta_{5} ) q^{79} \) \( + ( 36 - 4 \beta_{1} + 4 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 24 \beta_{5} ) q^{80} \) \( + ( 9 + 16 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 28 \beta_{4} - 6 \beta_{5} ) q^{81} \) \( + ( 48 - 24 \beta_{3} - 16 \beta_{4} + 24 \beta_{5} ) q^{82} \) \( + ( 47 - 9 \beta_{2} - 56 \beta_{3} + 9 \beta_{5} ) q^{83} \) \( + ( 36 - 4 \beta_{1} + 24 \beta_{3} + 16 \beta_{4} - 8 \beta_{5} ) q^{84} \) \( + ( -26 - 4 \beta_{1} + 8 \beta_{2} + 38 \beta_{3} - 4 \beta_{4} - 16 \beta_{5} ) q^{85} \) \( + ( 44 - 16 \beta_{1} - 25 \beta_{2} - 34 \beta_{3} - 25 \beta_{4} + 2 \beta_{5} ) q^{86} \) \( + ( 75 - 3 \beta_{1} + 17 \beta_{2} + 17 \beta_{3} + 31 \beta_{4} - 17 \beta_{5} ) q^{87} \) \( + ( 2 - 26 \beta_{1} - 4 \beta_{2} + 34 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} ) q^{88} \) \( + ( 18 + 18 \beta_{1} + 28 \beta_{2} - 16 \beta_{3} - 18 \beta_{4} - 8 \beta_{5} ) q^{89} \) \( + ( -26 + 10 \beta_{1} + 15 \beta_{2} + 22 \beta_{3} + 15 \beta_{4} - 2 \beta_{5} ) q^{90} \) \( + ( 32 - 14 \beta_{1} + 8 \beta_{2} + 38 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} ) q^{91} \) \( + ( -26 + 10 \beta_{1} + 4 \beta_{2} - 38 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} ) q^{92} \) \( + ( -12 - 28 \beta_{1} + 4 \beta_{2} + 12 \beta_{3} + 20 \beta_{4} + 4 \beta_{5} ) q^{93} \) \( + ( -16 \beta_{1} + 56 \beta_{3} + 48 \beta_{4} - 24 \beta_{5} ) q^{94} \) \( + ( -7 - 7 \beta_{1} - 21 \beta_{2} - 39 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} ) q^{95} \) \( + ( -12 + 28 \beta_{1} - 60 \beta_{3} - 24 \beta_{4} + 20 \beta_{5} ) q^{96} \) \( + ( -2 + 4 \beta_{1} - 26 \beta_{2} - 26 \beta_{3} - 48 \beta_{4} + 26 \beta_{5} ) q^{97} \) \( + ( 16 \beta_{1} + 3 \beta_{2} + 56 \beta_{3} + 8 \beta_{5} ) q^{98} \) \( + ( -45 + 4 \beta_{1} - 5 \beta_{2} + 36 \beta_{3} + 4 \beta_{4} + 13 \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut +\mathstrut 36q^{10} \) \(\mathstrut -\mathstrut 18q^{11} \) \(\mathstrut +\mathstrut 52q^{12} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut +\mathstrut 12q^{14} \) \(\mathstrut -\mathstrut 40q^{16} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 74q^{18} \) \(\mathstrut +\mathstrut 30q^{19} \) \(\mathstrut -\mathstrut 84q^{20} \) \(\mathstrut -\mathstrut 20q^{21} \) \(\mathstrut -\mathstrut 52q^{22} \) \(\mathstrut +\mathstrut 60q^{23} \) \(\mathstrut +\mathstrut 48q^{24} \) \(\mathstrut +\mathstrut 96q^{26} \) \(\mathstrut +\mathstrut 64q^{27} \) \(\mathstrut +\mathstrut 56q^{28} \) \(\mathstrut -\mathstrut 18q^{29} \) \(\mathstrut +\mathstrut 52q^{30} \) \(\mathstrut +\mathstrut 8q^{32} \) \(\mathstrut -\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 76q^{34} \) \(\mathstrut -\mathstrut 100q^{35} \) \(\mathstrut -\mathstrut 52q^{36} \) \(\mathstrut +\mathstrut 46q^{37} \) \(\mathstrut +\mathstrut 40q^{38} \) \(\mathstrut -\mathstrut 196q^{39} \) \(\mathstrut +\mathstrut 40q^{40} \) \(\mathstrut -\mathstrut 24q^{42} \) \(\mathstrut -\mathstrut 114q^{43} \) \(\mathstrut +\mathstrut 20q^{44} \) \(\mathstrut +\mathstrut 66q^{45} \) \(\mathstrut +\mathstrut 28q^{46} \) \(\mathstrut -\mathstrut 24q^{48} \) \(\mathstrut -\mathstrut 46q^{49} \) \(\mathstrut +\mathstrut 46q^{50} \) \(\mathstrut +\mathstrut 156q^{51} \) \(\mathstrut +\mathstrut 100q^{52} \) \(\mathstrut +\mathstrut 78q^{53} \) \(\mathstrut +\mathstrut 32q^{54} \) \(\mathstrut +\mathstrut 252q^{55} \) \(\mathstrut -\mathstrut 168q^{56} \) \(\mathstrut -\mathstrut 176q^{58} \) \(\mathstrut +\mathstrut 206q^{59} \) \(\mathstrut -\mathstrut 160q^{60} \) \(\mathstrut +\mathstrut 30q^{61} \) \(\mathstrut -\mathstrut 144q^{62} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut +\mathstrut 12q^{65} \) \(\mathstrut +\mathstrut 196q^{66} \) \(\mathstrut -\mathstrut 226q^{67} \) \(\mathstrut +\mathstrut 112q^{68} \) \(\mathstrut -\mathstrut 116q^{69} \) \(\mathstrut -\mathstrut 16q^{70} \) \(\mathstrut -\mathstrut 260q^{71} \) \(\mathstrut +\mathstrut 52q^{72} \) \(\mathstrut -\mathstrut 92q^{74} \) \(\mathstrut -\mathstrut 238q^{75} \) \(\mathstrut -\mathstrut 188q^{76} \) \(\mathstrut -\mathstrut 212q^{77} \) \(\mathstrut -\mathstrut 84q^{78} \) \(\mathstrut +\mathstrut 232q^{80} \) \(\mathstrut +\mathstrut 86q^{81} \) \(\mathstrut +\mathstrut 304q^{82} \) \(\mathstrut +\mathstrut 318q^{83} \) \(\mathstrut +\mathstrut 232q^{84} \) \(\mathstrut -\mathstrut 212q^{85} \) \(\mathstrut +\mathstrut 268q^{86} \) \(\mathstrut +\mathstrut 444q^{87} \) \(\mathstrut -\mathstrut 8q^{88} \) \(\mathstrut -\mathstrut 160q^{90} \) \(\mathstrut +\mathstrut 188q^{91} \) \(\mathstrut -\mathstrut 168q^{92} \) \(\mathstrut -\mathstrut 32q^{93} \) \(\mathstrut +\mathstrut 48q^{94} \) \(\mathstrut -\mathstrut 80q^{96} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 10q^{98} \) \(\mathstrut -\mathstrut 226q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{3} + 6 \nu - 8 \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} - 3 \nu^{3} + 2 \nu + 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{3} - 4 \nu^{2} + 2 \nu - 8 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -3 \nu^{5} + 4 \nu^{4} - 9 \nu^{3} + 12 \nu^{2} - 10 \nu + 20 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 8 \nu^{2} - 4 \nu + 14 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)\()/2\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/16\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(15\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.40680 + 0.144584i
−0.671462 + 1.24464i
0.264658 1.38923i
1.40680 0.144584i
−0.671462 1.24464i
0.264658 + 1.38923i
−1.55139 1.26222i 2.10278 2.10278i 0.813607 + 3.91638i −4.62721 + 4.62721i −5.91638 + 0.608056i 3.04888 3.68111 7.10278i 0.156674i 13.0192 1.33804i
3.2 −0.573183 + 1.91611i 0.146365 0.146365i −3.34292 2.19656i 3.68585 3.68585i 0.196558 + 0.364346i −9.66442 6.12494 5.14637i 8.95715i 4.94981 + 9.17513i
3.3 1.12457 1.65389i −3.24914 + 3.24914i −1.47068 3.71982i −0.0586332 + 0.0586332i 1.71982 + 9.02760i 4.61555 −7.80605 1.75086i 12.1138i 0.0310355 + 0.162910i
11.1 −1.55139 + 1.26222i 2.10278 + 2.10278i 0.813607 3.91638i −4.62721 4.62721i −5.91638 0.608056i 3.04888 3.68111 + 7.10278i 0.156674i 13.0192 + 1.33804i
11.2 −0.573183 1.91611i 0.146365 + 0.146365i −3.34292 + 2.19656i 3.68585 + 3.68585i 0.196558 0.364346i −9.66442 6.12494 + 5.14637i 8.95715i 4.94981 9.17513i
11.3 1.12457 + 1.65389i −3.24914 3.24914i −1.47068 + 3.71982i −0.0586332 0.0586332i 1.71982 9.02760i 4.61555 −7.80605 + 1.75086i 12.1138i 0.0310355 0.162910i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{3}^{\mathrm{new}}(16, [\chi])\).