# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{3}^{\mathrm{new}}(16, [\chi])$$.