# Properties

 Label 57.2.i.a Level 57 Weight 2 Character orbit 57.i Analytic conductor 0.455 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$57 = 3 \cdot 19$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 57.i (of order $$9$$ and degree $$6$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.455147291521$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.939693 + 0.342020i −0.766044 − 0.642788i −0.766044 + 0.642788i −0.173648 − 0.984808i 0.939693 − 0.342020i −0.173648 + 0.984808i
1.43969 + 0.524005i −0.173648 0.984808i 0.266044 + 0.223238i −1.93969 + 1.62760i 0.266044 1.50881i −0.266044 + 0.460802i −1.26604 2.19285i −0.939693 + 0.342020i −3.64543 + 1.32683i
16.1 −0.266044 0.223238i 0.939693 0.342020i −0.326352 1.85083i −0.233956 + 1.32683i −0.326352 0.118782i 0.326352 + 0.565258i −0.673648 + 1.16679i 0.766044 0.642788i 0.358441 0.300767i
25.1 −0.266044 + 0.223238i 0.939693 + 0.342020i −0.326352 + 1.85083i −0.233956 1.32683i −0.326352 + 0.118782i 0.326352 0.565258i −0.673648 1.16679i 0.766044 + 0.642788i 0.358441 + 0.300767i
28.1 0.326352 + 1.85083i −0.766044 + 0.642788i −1.43969 + 0.524005i −0.826352 0.300767i −1.43969 1.20805i 1.43969 2.49362i 0.439693 + 0.761570i 0.173648 0.984808i 0.286989 1.62760i
43.1 1.43969 0.524005i −0.173648 + 0.984808i 0.266044 0.223238i −1.93969 1.62760i 0.266044 + 1.50881i −0.266044 0.460802i −1.26604 + 2.19285i −0.939693 0.342020i −3.64543 1.32683i
55.1 0.326352 1.85083i −0.766044 0.642788i −1.43969 0.524005i −0.826352 + 0.300767i −1.43969 + 1.20805i 1.43969 + 2.49362i 0.439693 0.761570i 0.173648 + 0.984808i 0.286989 + 1.62760i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.1 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.e Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3 T_{2}^{5} + 6 T_{2}^{4} - 8 T_{2}^{3} + 3 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(57, [\chi])$$.