# Properties

 Label 171.2.u.c Level $171$ Weight $2$ Character orbit 171.u Analytic conductor $1.365$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 171.u (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1
 −0.173648 − 0.984808i −0.173648 + 0.984808i −0.766044 − 0.642788i −0.766044 + 0.642788i 0.939693 − 0.342020i 0.939693 + 0.342020i
0.233956 + 1.32683i 0 0.173648 0.0632028i 0.826352 + 0.300767i 0 −0.173648 + 0.300767i 1.47178 + 2.54920i 0 −0.205737 + 1.16679i
55.1 0.233956 1.32683i 0 0.173648 + 0.0632028i 0.826352 0.300767i 0 −0.173648 0.300767i 1.47178 2.54920i 0 −0.205737 1.16679i
73.1 1.93969 + 1.62760i 0 0.766044 + 4.34445i 0.233956 1.32683i 0 −0.766044 1.32683i −3.05303 + 5.28801i 0 2.61334 2.19285i
82.1 1.93969 1.62760i 0 0.766044 4.34445i 0.233956 + 1.32683i 0 −0.766044 + 1.32683i −3.05303 5.28801i 0 2.61334 + 2.19285i
100.1 0.826352 0.300767i 0 −0.939693 + 0.788496i 1.93969 + 1.62760i 0 0.939693 + 1.62760i −1.41875 + 2.45734i 0 2.09240 + 0.761570i
118.1 0.826352 + 0.300767i 0 −0.939693 0.788496i 1.93969 1.62760i 0 0.939693 1.62760i −1.41875 2.45734i 0 2.09240 0.761570i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 118.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.u.c 6
3.b odd 2 1 19.2.e.a 6
12.b even 2 1 304.2.u.b 6
15.d odd 2 1 475.2.l.a 6
15.e even 4 2 475.2.u.a 12
19.e even 9 1 inner 171.2.u.c 6
19.e even 9 1 3249.2.a.z 3
19.f odd 18 1 3249.2.a.s 3
21.c even 2 1 931.2.w.a 6
21.g even 6 1 931.2.v.a 6
21.g even 6 1 931.2.x.b 6
21.h odd 6 1 931.2.v.b 6
21.h odd 6 1 931.2.x.a 6
57.d even 2 1 361.2.e.h 6
57.f even 6 1 361.2.e.a 6
57.f even 6 1 361.2.e.b 6
57.h odd 6 1 361.2.e.f 6
57.h odd 6 1 361.2.e.g 6
57.j even 18 1 361.2.a.h 3
57.j even 18 2 361.2.c.h 6
57.j even 18 1 361.2.e.a 6
57.j even 18 1 361.2.e.b 6
57.j even 18 1 361.2.e.h 6
57.l odd 18 1 19.2.e.a 6
57.l odd 18 1 361.2.a.g 3
57.l odd 18 2 361.2.c.i 6
57.l odd 18 1 361.2.e.f 6
57.l odd 18 1 361.2.e.g 6
228.u odd 18 1 5776.2.a.bi 3
228.v even 18 1 304.2.u.b 6
228.v even 18 1 5776.2.a.br 3
285.bd odd 18 1 475.2.l.a 6
285.bd odd 18 1 9025.2.a.bd 3
285.bf even 18 1 9025.2.a.x 3
285.bi even 36 2 475.2.u.a 12
399.ca odd 18 1 931.2.v.b 6
399.cb even 18 1 931.2.v.a 6
399.ch odd 18 1 931.2.x.a 6
399.ci even 18 1 931.2.x.b 6
399.cj even 18 1 931.2.w.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 3.b odd 2 1
19.2.e.a 6 57.l odd 18 1
171.2.u.c 6 1.a even 1 1 trivial
171.2.u.c 6 19.e even 9 1 inner
304.2.u.b 6 12.b even 2 1
304.2.u.b 6 228.v even 18 1
361.2.a.g 3 57.l odd 18 1
361.2.a.h 3 57.j even 18 1
361.2.c.h 6 57.j even 18 2
361.2.c.i 6 57.l odd 18 2
361.2.e.a 6 57.f even 6 1
361.2.e.a 6 57.j even 18 1
361.2.e.b 6 57.f even 6 1
361.2.e.b 6 57.j even 18 1
361.2.e.f 6 57.h odd 6 1
361.2.e.f 6 57.l odd 18 1
361.2.e.g 6 57.h odd 6 1
361.2.e.g 6 57.l odd 18 1
361.2.e.h 6 57.d even 2 1
361.2.e.h 6 57.j even 18 1
475.2.l.a 6 15.d odd 2 1
475.2.l.a 6 285.bd odd 18 1
475.2.u.a 12 15.e even 4 2
475.2.u.a 12 285.bi even 36 2
931.2.v.a 6 21.g even 6 1
931.2.v.a 6 399.cb even 18 1
931.2.v.b 6 21.h odd 6 1
931.2.v.b 6 399.ca odd 18 1
931.2.w.a 6 21.c even 2 1
931.2.w.a 6 399.cj even 18 1
931.2.x.a 6 21.h odd 6 1
931.2.x.a 6 399.ch odd 18 1
931.2.x.b 6 21.g even 6 1
931.2.x.b 6 399.ci even 18 1
3249.2.a.s 3 19.f odd 18 1
3249.2.a.z 3 19.e even 9 1
5776.2.a.bi 3 228.u odd 18 1
5776.2.a.br 3 228.v even 18 1
9025.2.a.x 3 285.bf even 18 1
9025.2.a.bd 3 285.bd odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 6T_{2}^{5} + 18T_{2}^{4} - 30T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(171, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9$$
$7$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$11$ $$T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81$$
$13$ $$T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1369$$
$17$ $$T^{6} + 3 T^{5} - 30 T^{3} + 36 T^{2} + \cdots + 9$$
$19$ $$T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859$$
$23$ $$T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576$$
$29$ $$T^{6} - 3 T^{5} + 36 T^{4} + \cdots + 12321$$
$31$ $$T^{6} - 9 T^{5} + 75 T^{4} + \cdots + 2809$$
$37$ $$(T^{3} - 21 T - 17)^{2}$$
$41$ $$T^{6} + 21 T^{5} + 162 T^{4} + \cdots + 12321$$
$43$ $$T^{6} + 3 T^{5} + 60 T^{4} + \cdots + 26569$$
$47$ $$T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9$$
$53$ $$T^{6} - 3 T^{5} + 84 T^{3} + \cdots + 2601$$
$59$ $$T^{6} + 12 T^{5} + 18 T^{4} + \cdots + 71289$$
$61$ $$T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761$$
$67$ $$T^{6} + 30 T^{5} + 348 T^{4} + \cdots + 179776$$
$71$ $$T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 788544$$
$73$ $$T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096$$
$79$ $$T^{6} + 39 T^{5} + 708 T^{4} + \cdots + 654481$$
$83$ $$T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681$$
$89$ $$T^{6} - 12 T^{5} + 54 T^{4} + \cdots + 3249$$
$97$ $$T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 16129$$