# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{3}^{\mathrm{new}}(19, [\chi])$$.