# Properties

 Label 54.2.e.b Level $54$ Weight $2$ Character orbit 54.e Analytic conductor $0.431$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,2,Mod(7,54)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 54.e (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.431192170915$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$ x^12 - 6*x^11 + 33*x^10 - 110*x^9 + 318*x^8 - 678*x^7 + 1225*x^6 - 1698*x^5 + 1905*x^4 - 1584*x^3 + 936*x^2 - 342*x + 57 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_{4}) q^{2} - \beta_{11} q^{3} + \beta_{3} q^{4} + (\beta_{10} + \beta_{8} - \beta_{7} - \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{5} - \beta_{2} + 1) q^{6} + (\beta_{11} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{7} - \beta_1 q^{8} + ( - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{9}+O(q^{10})$$ q + (b6 - b4) * q^2 - b11 * q^3 + b3 * q^4 + (b10 + b8 - b7 - b5) * q^5 + (-b9 + b5 - b2 + 1) * q^6 + (b11 - b8 + b7 - 2*b6 + b4 + b2 - 1) * q^7 - b1 * q^8 + (-b10 + b9 - b8 - b3 + b2 + 2*b1 - 3) * q^9 $$q + (\beta_{6} - \beta_{4}) q^{2} - \beta_{11} q^{3} + \beta_{3} q^{4} + (\beta_{10} + \beta_{8} - \beta_{7} - \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{5} - \beta_{2} + 1) q^{6} + (\beta_{11} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{7} - \beta_1 q^{8} + ( - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{9} + (\beta_{11} - \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} - 1) q^{10} + (\beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{11} + (\beta_{10} + \beta_{8} + \beta_1) q^{12} + ( - \beta_{11} + 2 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{3} + 2) q^{13} + (2 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{3} + \beta_{2} - 1) q^{14} + (\beta_{11} + \beta_{10} + 2 \beta_{9} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{15} + \beta_{4} q^{16} + ( - \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{17}+ \cdots + ( - 3 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 3 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + \cdots + 1) q^{99}+O(q^{100})$$ q + (b6 - b4) * q^2 - b11 * q^3 + b3 * q^4 + (b10 + b8 - b7 - b5) * q^5 + (-b9 + b5 - b2 + 1) * q^6 + (b11 - b8 + b7 - 2*b6 + b4 + b2 - 1) * q^7 - b1 * q^8 + (-b10 + b9 - b8 - b3 + b2 + 2*b1 - 3) * q^9 + (b11 - b10 + b7 - b5 + b4 - 1) * q^10 + (b11 - b10 - 2*b9 + b7 + b5 + b4 + b3 - b2 - b1) * q^11 + (b10 + b8 + b1) * q^12 + (-b11 + 2*b9 + b8 - b7 + b5 - b4 - 2*b3 + 2) * q^13 + (2*b9 - b8 - b7 - 2*b3 + b2 - 1) * q^14 + (b11 + b10 + 2*b9 + 3*b4 - 2*b3 - b2 - 2*b1) * q^15 + b4 * q^16 + (-b11 - b10 - b9 - b8 - b7 + b6 - 3*b4 + 2*b3 - b2 - 1) * q^17 + (-b11 - b8 - b7 - 2*b6 - b4) * q^18 + (b11 - b9 + b8 + b6 - b4 - b3 - b2 - 2*b1 + 1) * q^19 + (-b11 - b10 - b4 + b2) * q^20 + (b8 + b7 + 2*b6 - b5 - 2*b4 + 4*b3 + 2) * q^21 + (-b11 + b10 + b8 - b6 + b4 + b2 + b1 - 1) * q^22 + (b10 + b7 - 2*b6 - b5 + 4*b4 + b2 + 2*b1 + 1) * q^23 + (b11 + b7 - b6 + b4) * q^24 + (-b11 + b10 - 4*b9 - b8 + b7 + b6 - 2*b4 + 2*b3 + 4*b1 - 3) * q^25 + (b10 + b7 - b2 + 2) * q^26 + (-b10 - 2*b9 - b7 + 4*b6 - 4*b4 + 2*b3 + b2 - b1) * q^27 + (-b8 - b7 - b5) * q^28 + (b11 + b10 + b8 + b7 + 2*b5 + b4 + 3) * q^29 + (b11 - 2*b9 + b8 - b5 + 3*b4 + b2 + 2) * q^30 + (-b11 - 2*b7 + 4*b6 + b5 - 3*b4 + 2*b3 - 2*b2 - 4*b1 + 3) * q^31 - b9 * q^32 + (b11 + b10 - 2*b9 - 3*b6 - b5 + b2 + b1 + 2) * q^33 + (-b11 + b9 + b8 - b7 - b4 + b3 - b2 - b1) * q^34 + (-2*b11 - b10 + 4*b9 - b8 - 4*b6 + b5 - 2*b3) * q^35 + (2*b9 - b7 + b6 - b4 - 2*b3 - b2) * q^36 + (-2*b11 + 4*b9 - b8 - 2*b7 + 2*b6 + b5 - 2*b3 - b2 + 2*b1 - 2) * q^37 + (b9 + b8 + b7 - b5 + 2*b4 + b3 + b2 + 2*b1 - 1) * q^38 + (-b11 - 2*b10 - b8 - b5 - 3*b4 - 2*b3 - 2*b1 - 4) * q^39 + (-b11 - b8 + b5 - b4 - b2) * q^40 + (b11 + 2*b8 + b7 + 2*b6 - b5 + b1 - 1) * q^41 + (-b10 + b7 + 2*b6 + b5 - 2*b4 + 2*b3 - 4*b1 + 1) * q^42 + (b9 - 3*b5 + 2*b4 - 2*b3 + 3*b2 - b1 - 1) * q^43 + (b11 - b9 - b8 + b7 - 2*b6 + b5 + 2*b4 - b3 - b2) * q^44 + (2*b11 - b10 + 4*b9 + b8 + b7 - 4*b6 + b5 + 4*b4 - 2*b3 + b2 - 4*b1 + 4) * q^45 + (b11 - b10 - 2*b9 - b8 + 2*b6 + b5 - 3*b4 - 2*b3) * q^46 + (2*b11 + b10 + 2*b9 - b8 + b7 + 2*b4 - 4*b3 + 4*b1 - 3) * q^47 + (b9 - b3 + b2) * q^48 + (2*b11 + b10 - 3*b9 + b8 - 2*b6 + 6*b4 - 2*b2 + 2*b1 - 3) * q^49 + (b11 - b7 - 2*b6 + 2*b5 - b4 + b3 - b2 + 2*b1 - 2) * q^50 + (2*b10 - 4*b9 - b7 - 2*b6 + b5 + 2*b4 + 3*b3 - b2 + 2*b1 - 2) * q^51 + (b11 + b8 + 2*b6 + b5 - b4 + 2) * q^52 + (-2*b10 - 2*b8 - b7 - 2*b5 + 2*b2 - 4) * q^53 + (-b11 - b8 - b5 + 4*b3 - 4) * q^54 + (-b10 - 2*b9 + b8 - 4*b6 + b5 + 2*b4 + 4*b3 + b2 + 2) * q^55 + (-b10 - b7 + 2*b6 - b5 - 2*b4 - 1) * q^56 + (2*b11 - 2*b9 - b8 + b7 - 4*b6 + 4*b4 - b3 - 2*b2 - 3*b1 + 3) * q^57 + (b11 + 2*b10 + b7 + b4 + b2) * q^58 + (-b11 - b10 - b8 + 2*b7 - b6 + 2*b5 - 2*b4 + b2 - 2*b1 + 2) * q^59 + (-b10 - 2*b9 - b8 + b7 + 2*b6 - b5 - 2*b4 + b2 + 2*b1 - 4) * q^60 + (-b11 - 2*b10 + 2*b9 - b8 + b7 - 2*b6 - b4 + 2*b3 - 2*b1 - 3) * q^61 + (b10 - 2*b9 + 2*b8 + 2*b6 - b5 + 2*b4 + 4*b3 - b2 - 2*b1 + 1) * q^62 + (-b11 + 3*b10 - 2*b9 + 4*b8 + b5 + 3*b4 - 2*b3 - 2*b2 + 6*b1 + 1) * q^63 + (b1 - 1) * q^64 + (-2*b11 + 2*b10 - 2*b9 - b8 - 3*b7 - b5 - 2*b4 - 2*b3 + b2 - 4*b1) * q^65 + (b11 - b10 + 4*b9 - b8 + 3*b6 - b5 - 3*b4 - 3*b3 + b2 + 2*b1 - 4) * q^66 + (3*b11 + 2*b9 + 3*b7 - 3*b6 - 3*b5 - 2*b3 + 3*b1) * q^67 + (b8 + b7 - b6 + 2*b4 - b2 - 2*b1 + 2) * q^68 + (-3*b11 - b10 + 6*b9 - b7 + 4*b6 - b5 - b4 - 2*b3 + 3*b2 + 2*b1 - 1) * q^69 + (-b11 + b10 + 2*b9 - b7 + 2*b5 - b4 - 4*b3 - 2*b2 - 2*b1 + 6) * q^70 + (-b11 + 2*b10 + 2*b9 + 2*b8 - b7 + 4*b6 - 3*b4 + 2*b3 + 2*b2 + 2) * q^71 + (b8 - b5 + b3 + 2) * q^72 + (-3*b11 + b10 + 2*b9 - b8 - 2*b6 - b5 - 2*b4 - b3 + 2*b2 - 2*b1 - 2) * q^73 + (b10 - 4*b9 + b8 - b7 - 2*b6 + 2*b3 - 2*b2 - 2*b1 + 5) * q^74 + (-2*b11 + b9 - 3*b8 - 4*b7 + 4*b6 + 3*b5 - 4*b4 + 2*b3 - 2*b2 + 6*b1) * q^75 + (-b10 - 2*b9 - b8 + b7 - b6 + b5 - b4 - 2*b1 + 1) * q^76 + (-2*b11 - 2*b10 - 3*b7 + 8*b6 + b5 - 6*b4 - 3*b2 - 8*b1 + 5) * q^77 + (-2*b11 - b10 + 2*b9 - b7 - 2*b6 + b5 + 2*b4 - b2 + 2*b1 + 1) * q^78 + (2*b11 - b10 - 2*b9 + 2*b8 - b7 + 2*b6 - 2*b5 + 4*b3 + 2*b1 + 2) * q^79 + (b10 + b8 - b7 + b5 - b2 + 2) * q^80 + (b11 + 3*b10 - 6*b9 + 2*b7 + b6 + 3*b5 - b4 - 3*b2 + 3) * q^81 + (-b10 - b9 + 2*b7 - 2*b6 + b4 + 2*b3 + b2) * q^82 + (-b11 - 2*b10 + 2*b9 - b8 - 2*b7 - 3*b6 - 3*b5 + 2*b4 + 2*b3 - 2*b1) * q^83 + (-b11 + b10 + b5 + 3*b4 + 2*b3 - 2*b1 + 1) * q^84 + (2*b11 - 3*b10 + 4*b7 + 2*b6 - 2*b5 + 4*b4 + 4*b2 - 2*b1 + 2) * q^85 + (-3*b10 - 2*b9 - 3*b8 + 2*b6 - b4 + b1 - 2) * q^86 + (-2*b11 + 2*b9 + b8 - 6*b4 - 2*b3 - b2 + 3) * q^87 + (b10 + b9 + b8 - b7 - 2*b3 + b2 + 2*b1) * q^88 + (4*b11 + b10 - 4*b9 + 4*b6 - b5 - 4*b4 - 4*b3 + b2 + 3*b1 - 1) * q^89 + (-b11 + b10 + 2*b9 - b8 + b7 + 2*b6 - b5 + b4 - 4*b3 + 2*b2 - 2*b1 + 2) * q^90 + (3*b11 + 3*b10 + 2*b9 + 3*b8 + 3*b7 + 4*b6 + b4 + 2*b3 + 3*b2 + 4*b1 - 1) * q^91 + (-b11 + b10 + 2*b9 - b7 - b5 - b4 + 2*b3 + b2 + 4*b1 - 3) * q^92 + (b11 + 3*b10 - 2*b9 + b8 + 2*b7 - 8*b6 + 4*b5 + 5*b4 - 2*b3 - 2*b2 + 1) * q^93 + (b11 - b7 - 2*b6 - b5 - b4 + 2*b2 + 2*b1 + 1) * q^94 + (-6*b9 - 2*b8 + b7 + 2*b6 - 4*b4 + 6*b3 - b2 + 4*b1 - 4) * q^95 - b8 * q^96 + (2*b11 - 2*b10 + 2*b7 + 5*b5 + 4*b4 - 3*b3 - 5*b2 - 3*b1 + 8) * q^97 + (b11 - 2*b9 + 2*b8 + b7 - 4*b6 - 2*b5 + 3*b4 - 2*b3 + 2*b2 + 3*b1 - 3) * q^98 + (-3*b11 + b10 + b9 - b8 + 3*b6 - 2*b5 + 4*b3 - 2*b2 - 2*b1 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{5} + 3 q^{6} - 3 q^{7} - 6 q^{8} - 12 q^{9}+O(q^{10})$$ 12 * q - 3 * q^5 + 3 * q^6 - 3 * q^7 - 6 * q^8 - 12 * q^9 $$12 q - 3 q^{5} + 3 q^{6} - 3 q^{7} - 6 q^{8} - 12 q^{9} - 3 q^{10} - 12 q^{11} - 3 q^{12} + 12 q^{13} - 3 q^{14} - 18 q^{15} - 6 q^{17} + 6 q^{18} - 9 q^{19} + 6 q^{20} + 24 q^{21} - 12 q^{22} + 30 q^{23} - 9 q^{25} + 18 q^{26} + 12 q^{28} + 15 q^{29} + 27 q^{30} + 36 q^{33} - 15 q^{34} + 3 q^{35} - 3 q^{36} - 15 q^{37} + 3 q^{38} - 42 q^{39} - 3 q^{40} - 12 q^{41} - 15 q^{42} + 9 q^{43} - 3 q^{44} + 18 q^{45} + 3 q^{46} - 9 q^{47} + 3 q^{48} - 39 q^{49} - 27 q^{50} - 27 q^{51} + 12 q^{52} - 12 q^{53} - 36 q^{54} + 18 q^{55} - 3 q^{56} + 18 q^{57} - 3 q^{58} + 12 q^{59} - 18 q^{60} - 36 q^{61} - 12 q^{62} + 3 q^{63} - 6 q^{64} - 15 q^{65} - 18 q^{66} + 36 q^{67} + 3 q^{68} + 18 q^{69} + 39 q^{70} + 12 q^{71} + 24 q^{72} - 21 q^{73} + 33 q^{74} + 30 q^{75} + 3 q^{76} + 3 q^{77} + 18 q^{78} + 39 q^{79} + 6 q^{80} + 6 q^{82} + 18 q^{83} - 9 q^{84} + 45 q^{85} + 9 q^{86} + 27 q^{87} + 6 q^{88} + 12 q^{89} + 27 q^{90} - 6 q^{91} - 6 q^{92} - 33 q^{93} + 36 q^{94} - 15 q^{95} + 6 q^{96} + 39 q^{97} - 12 q^{98} + 9 q^{99}+O(q^{100})$$ 12 * q - 3 * q^5 + 3 * q^6 - 3 * q^7 - 6 * q^8 - 12 * q^9 - 3 * q^10 - 12 * q^11 - 3 * q^12 + 12 * q^13 - 3 * q^14 - 18 * q^15 - 6 * q^17 + 6 * q^18 - 9 * q^19 + 6 * q^20 + 24 * q^21 - 12 * q^22 + 30 * q^23 - 9 * q^25 + 18 * q^26 + 12 * q^28 + 15 * q^29 + 27 * q^30 + 36 * q^33 - 15 * q^34 + 3 * q^35 - 3 * q^36 - 15 * q^37 + 3 * q^38 - 42 * q^39 - 3 * q^40 - 12 * q^41 - 15 * q^42 + 9 * q^43 - 3 * q^44 + 18 * q^45 + 3 * q^46 - 9 * q^47 + 3 * q^48 - 39 * q^49 - 27 * q^50 - 27 * q^51 + 12 * q^52 - 12 * q^53 - 36 * q^54 + 18 * q^55 - 3 * q^56 + 18 * q^57 - 3 * q^58 + 12 * q^59 - 18 * q^60 - 36 * q^61 - 12 * q^62 + 3 * q^63 - 6 * q^64 - 15 * q^65 - 18 * q^66 + 36 * q^67 + 3 * q^68 + 18 * q^69 + 39 * q^70 + 12 * q^71 + 24 * q^72 - 21 * q^73 + 33 * q^74 + 30 * q^75 + 3 * q^76 + 3 * q^77 + 18 * q^78 + 39 * q^79 + 6 * q^80 + 6 * q^82 + 18 * q^83 - 9 * q^84 + 45 * q^85 + 9 * q^86 + 27 * q^87 + 6 * q^88 + 12 * q^89 + 27 * q^90 - 6 * q^91 - 6 * q^92 - 33 * q^93 + 36 * q^94 - 15 * q^95 + 6 * q^96 + 39 * q^97 - 12 * q^98 + 9 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 6 x^{11} + 33 x^{10} - 110 x^{9} + 318 x^{8} - 678 x^{7} + 1225 x^{6} - 1698 x^{5} + 1905 x^{4} - 1584 x^{3} + 936 x^{2} - 342 x + 57$$ :

 $$\beta_{1}$$ $$=$$ $$( 6 \nu^{11} - 33 \nu^{10} + 127 \nu^{9} - 324 \nu^{8} + 438 \nu^{7} - 252 \nu^{6} - 1278 \nu^{5} + 3234 \nu^{4} - 5701 \nu^{3} + 5358 \nu^{2} - 3477 \nu + 1060 ) / 218$$ (6*v^11 - 33*v^10 + 127*v^9 - 324*v^8 + 438*v^7 - 252*v^6 - 1278*v^5 + 3234*v^4 - 5701*v^3 + 5358*v^2 - 3477*v + 1060) / 218 $$\beta_{2}$$ $$=$$ $$( 36 \nu^{11} - 89 \nu^{10} + 544 \nu^{9} - 745 \nu^{8} + 2301 \nu^{7} - 1512 \nu^{6} + 3777 \nu^{5} - 2069 \nu^{4} + 5579 \nu^{3} - 6002 \nu^{2} + 5080 \nu - 1706 ) / 218$$ (36*v^11 - 89*v^10 + 544*v^9 - 745*v^8 + 2301*v^7 - 1512*v^6 + 3777*v^5 - 2069*v^4 + 5579*v^3 - 6002*v^2 + 5080*v - 1706) / 218 $$\beta_{3}$$ $$=$$ $$( - 27 \nu^{11} + 94 \nu^{10} - 408 \nu^{9} + 586 \nu^{8} - 445 \nu^{7} - 2572 \nu^{6} + 9021 \nu^{5} - 18150 \nu^{4} + 24401 \nu^{3} - 20623 \nu^{2} + 10469 \nu - 2263 ) / 218$$ (-27*v^11 + 94*v^10 - 408*v^9 + 586*v^8 - 445*v^7 - 2572*v^6 + 9021*v^5 - 18150*v^4 + 24401*v^3 - 20623*v^2 + 10469*v - 2263) / 218 $$\beta_{4}$$ $$=$$ $$( 26 \nu^{11} - 34 \nu^{10} + 187 \nu^{9} + 449 \nu^{8} - 1590 \nu^{7} + 6865 \nu^{6} - 12623 \nu^{5} + 20118 \nu^{4} - 19981 \nu^{3} + 12972 \nu^{2} - 4712 \nu + 524 ) / 218$$ (26*v^11 - 34*v^10 + 187*v^9 + 449*v^8 - 1590*v^7 + 6865*v^6 - 12623*v^5 + 20118*v^4 - 19981*v^3 + 12972*v^2 - 4712*v + 524) / 218 $$\beta_{5}$$ $$=$$ $$( - 2 \nu^{11} + 120 \nu^{10} - 551 \nu^{9} + 2615 \nu^{8} - 6686 \nu^{7} + 15780 \nu^{6} - 23990 \nu^{5} + 31404 \nu^{4} - 25822 \nu^{3} + 15545 \nu^{2} - 4618 \nu + 555 ) / 218$$ (-2*v^11 + 120*v^10 - 551*v^9 + 2615*v^8 - 6686*v^7 + 15780*v^6 - 23990*v^5 + 31404*v^4 - 25822*v^3 + 15545*v^2 - 4618*v + 555) / 218 $$\beta_{6}$$ $$=$$ $$( - 27 \nu^{11} + 203 \nu^{10} - 953 \nu^{9} + 3311 \nu^{8} - 8075 \nu^{7} + 16285 \nu^{6} - 23134 \nu^{5} + 26758 \nu^{4} - 19635 \nu^{3} + 9570 \nu^{2} - 1957 \nu - 83 ) / 218$$ (-27*v^11 + 203*v^10 - 953*v^9 + 3311*v^8 - 8075*v^7 + 16285*v^6 - 23134*v^5 + 26758*v^4 - 19635*v^3 + 9570*v^2 - 1957*v - 83) / 218 $$\beta_{7}$$ $$=$$ $$( \nu^{10} - 5 \nu^{9} + 25 \nu^{8} - 70 \nu^{7} + 173 \nu^{6} - 295 \nu^{5} + 412 \nu^{4} - 404 \nu^{3} + 279 \nu^{2} - 116 \nu + 26 ) / 2$$ (v^10 - 5*v^9 + 25*v^8 - 70*v^7 + 173*v^6 - 295*v^5 + 412*v^4 - 404*v^3 + 279*v^2 - 116*v + 26) / 2 $$\beta_{8}$$ $$=$$ $$( 2 \nu^{11} + 98 \nu^{10} - 539 \nu^{9} + 2726 \nu^{8} - 8138 \nu^{7} + 20190 \nu^{6} - 36614 \nu^{5} + 52308 \nu^{4} - 55710 \nu^{3} + 41571 \nu^{2} - 19689 \nu + 4350 ) / 218$$ (2*v^11 + 98*v^10 - 539*v^9 + 2726*v^8 - 8138*v^7 + 20190*v^6 - 36614*v^5 + 52308*v^4 - 55710*v^3 + 41571*v^2 - 19689*v + 4350) / 218 $$\beta_{9}$$ $$=$$ $$( 26 \nu^{11} - 252 \nu^{10} + 1277 \nu^{9} - 4892 \nu^{8} + 13234 \nu^{7} - 28887 \nu^{6} + 47327 \nu^{5} - 60760 \nu^{4} + 56973 \nu^{3} - 36296 \nu^{2} + 13927 \nu - 2201 ) / 218$$ (26*v^11 - 252*v^10 + 1277*v^9 - 4892*v^8 + 13234*v^7 - 28887*v^6 + 47327*v^5 - 60760*v^4 + 56973*v^3 - 36296*v^2 + 13927*v - 2201) / 218 $$\beta_{10}$$ $$=$$ $$( 36 \nu^{11} - 307 \nu^{10} + 1634 \nu^{9} - 6086 \nu^{8} + 17125 \nu^{7} - 37373 \nu^{6} + 64054 \nu^{5} - 84255 \nu^{4} + 84604 \nu^{3} - 58431 \nu^{2} + 25899 \nu - 5194 ) / 218$$ (36*v^11 - 307*v^10 + 1634*v^9 - 6086*v^8 + 17125*v^7 - 37373*v^6 + 64054*v^5 - 84255*v^4 + 84604*v^3 - 58431*v^2 + 25899*v - 5194) / 218 $$\beta_{11}$$ $$=$$ $$( 91 \nu^{11} - 664 \nu^{10} + 3325 \nu^{9} - 11563 \nu^{8} + 30405 \nu^{7} - 62791 \nu^{6} + 99754 \nu^{5} - 123498 \nu^{4} + 112914 \nu^{3} - 71337 \nu^{2} + 28743 \nu - 5469 ) / 218$$ (91*v^11 - 664*v^10 + 3325*v^9 - 11563*v^8 + 30405*v^7 - 62791*v^6 + 99754*v^5 - 123498*v^4 + 112914*v^3 - 71337*v^2 + 28743*v - 5469) / 218
 $$\nu$$ $$=$$ $$( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 3$$ (2*b11 - b10 - b9 + b8 + b7 + 2*b6 - b5 + b4 + 2*b3 - b2 - b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + 4 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} - \beta _1 - 7 ) / 3$$ (2*b11 - b10 - b9 + b8 + 4*b7 - b6 - b5 + b4 + 5*b3 - b2 - b1 - 7) / 3 $$\nu^{3}$$ $$=$$ $$( - 7 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + \beta_{7} - 13 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 4 \beta _1 - 10 ) / 3$$ (-7*b11 + 2*b10 + 5*b9 - 5*b8 + b7 - 13*b6 + 5*b5 - 2*b4 - 4*b3 + 2*b2 - 4*b1 - 10) / 3 $$\nu^{4}$$ $$=$$ $$( - 16 \beta_{11} + 11 \beta_{10} + 8 \beta_{9} - 8 \beta_{8} - 20 \beta_{7} - 7 \beta_{6} + 14 \beta_{5} - 2 \beta_{4} - 31 \beta_{3} - \beta_{2} - 7 \beta _1 + 32 ) / 3$$ (-16*b11 + 11*b10 + 8*b9 - 8*b8 - 20*b7 - 7*b6 + 14*b5 - 2*b4 - 31*b3 - b2 - 7*b1 + 32) / 3 $$\nu^{5}$$ $$=$$ $$( 23 \beta_{11} + 14 \beta_{10} - 34 \beta_{9} + 25 \beta_{8} - 26 \beta_{7} + 65 \beta_{6} - 10 \beta_{5} + 4 \beta_{4} - 10 \beta_{3} - 16 \beta_{2} + 29 \beta _1 + 77 ) / 3$$ (23*b11 + 14*b10 - 34*b9 + 25*b8 - 26*b7 + 65*b6 - 10*b5 + 4*b4 - 10*b3 - 16*b2 + 29*b1 + 77) / 3 $$\nu^{6}$$ $$=$$ $$( 110 \beta_{11} - 49 \beta_{10} - 88 \beta_{9} + 67 \beta_{8} + 85 \beta_{7} + 101 \beta_{6} - 94 \beta_{5} - 17 \beta_{4} + 161 \beta_{3} + 17 \beta_{2} + 104 \beta _1 - 121 ) / 3$$ (110*b11 - 49*b10 - 88*b9 + 67*b8 + 85*b7 + 101*b6 - 94*b5 - 17*b4 + 161*b3 + 17*b2 + 104*b1 - 121) / 3 $$\nu^{7}$$ $$=$$ $$( - 25 \beta_{11} - 187 \beta_{10} + 158 \beta_{9} - 101 \beta_{8} + 229 \beta_{7} - 271 \beta_{6} - 46 \beta_{5} - 56 \beta_{4} + 212 \beta_{3} + 149 \beta_{2} - 82 \beta _1 - 535 ) / 3$$ (-25*b11 - 187*b10 + 158*b9 - 101*b8 + 229*b7 - 271*b6 - 46*b5 - 56*b4 + 212*b3 + 149*b2 - 82*b1 - 535) / 3 $$\nu^{8}$$ $$=$$ $$( - 652 \beta_{11} + 41 \beta_{10} + 761 \beta_{9} - 509 \beta_{8} - 263 \beta_{7} - 826 \beta_{6} + 515 \beta_{5} + 151 \beta_{4} - 724 \beta_{3} - 19 \beta_{2} - 829 \beta _1 + 257 ) / 3$$ (-652*b11 + 41*b10 + 761*b9 - 509*b8 - 263*b7 - 826*b6 + 515*b5 + 151*b4 - 724*b3 - 19*b2 - 829*b1 + 257) / 3 $$\nu^{9}$$ $$=$$ $$( - 478 \beta_{11} + 1304 \beta_{10} - 268 \beta_{9} + 160 \beta_{8} - 1571 \beta_{7} + 821 \beta_{6} + 812 \beta_{5} + 640 \beta_{4} - 1951 \beta_{3} - 1108 \beta_{2} - 406 \beta _1 + 3359 ) / 3$$ (-478*b11 + 1304*b10 - 268*b9 + 160*b8 - 1571*b7 + 821*b6 + 812*b5 + 640*b4 - 1951*b3 - 1108*b2 - 406*b1 + 3359) / 3 $$\nu^{10}$$ $$=$$ $$( 3353 \beta_{11} + 1451 \beta_{10} - 5224 \beta_{9} + 3352 \beta_{8} + 25 \beta_{7} + 5630 \beta_{6} - 2308 \beta_{5} - 521 \beta_{4} + 2375 \beta_{3} - 913 \beta_{2} + 4949 \beta _1 + 1451 ) / 3$$ (3353*b11 + 1451*b10 - 5224*b9 + 3352*b8 + 25*b7 + 5630*b6 - 2308*b5 - 521*b4 + 2375*b3 - 913*b2 + 4949*b1 + 1451) / 3 $$\nu^{11}$$ $$=$$ $$( 6041 \beta_{11} - 6547 \beta_{10} - 3685 \beta_{9} + 2323 \beta_{8} + 9241 \beta_{7} + 326 \beta_{6} - 7273 \beta_{5} - 5168 \beta_{4} + 14021 \beta_{3} + 6587 \beta_{2} + 7973 \beta _1 - 18736 ) / 3$$ (6041*b11 - 6547*b10 - 3685*b9 + 2323*b8 + 9241*b7 + 326*b6 - 7273*b5 - 5168*b4 + 14021*b3 + 6587*b2 + 7973*b1 - 18736) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$\chi(n)$$ $$\beta_{3}$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 1.96356i 0.5 + 0.677980i 0.5 + 2.42499i 0.5 − 1.74095i 0.5 − 2.42499i 0.5 + 1.74095i 0.5 + 1.96356i 0.5 − 0.677980i 0.5 + 1.80139i 0.5 + 0.168222i 0.5 − 1.80139i 0.5 − 0.168222i
−0.939693 + 0.342020i −1.14517 + 1.29945i 0.766044 0.642788i 0.617090 + 3.49969i 0.631669 1.61276i −0.244752 0.205371i −0.500000 + 0.866025i −0.377165 2.97620i −1.77684 3.07758i
7.2 −0.939693 + 0.342020i 0.552775 1.64147i 0.766044 0.642788i −0.177398 1.00607i 0.0419788 + 1.73154i 2.04289 + 1.71418i −0.500000 + 0.866025i −2.38888 1.81473i 0.510796 + 0.884725i
13.1 0.173648 + 0.984808i 0.140451 1.72635i −0.939693 + 0.342020i 2.42692 + 2.03643i 1.72451 0.161460i −3.46344 1.26059i −0.500000 0.866025i −2.96055 0.484935i −1.58406 + 2.74367i
13.2 0.173648 + 0.984808i 1.56529 + 0.741539i −0.939693 + 0.342020i −3.10057 2.60168i −0.458464 + 1.67027i 0.144365 + 0.0525446i −0.500000 0.866025i 1.90024 + 2.32144i 2.02375 3.50524i
25.1 0.173648 0.984808i 0.140451 + 1.72635i −0.939693 0.342020i 2.42692 2.03643i 1.72451 + 0.161460i −3.46344 + 1.26059i −0.500000 + 0.866025i −2.96055 + 0.484935i −1.58406 2.74367i
25.2 0.173648 0.984808i 1.56529 0.741539i −0.939693 0.342020i −3.10057 + 2.60168i −0.458464 1.67027i 0.144365 0.0525446i −0.500000 + 0.866025i 1.90024 2.32144i 2.02375 + 3.50524i
31.1 −0.939693 0.342020i −1.14517 1.29945i 0.766044 + 0.642788i 0.617090 3.49969i 0.631669 + 1.61276i −0.244752 + 0.205371i −0.500000 0.866025i −0.377165 + 2.97620i −1.77684 + 3.07758i
31.2 −0.939693 0.342020i 0.552775 + 1.64147i 0.766044 + 0.642788i −0.177398 + 1.00607i 0.0419788 1.73154i 2.04289 1.71418i −0.500000 0.866025i −2.38888 + 1.81473i 0.510796 0.884725i
43.1 0.766044 + 0.642788i −1.36085 + 1.07149i 0.173648 + 0.984808i 0.696050 + 0.253341i −1.73121 0.0539310i 0.717657 4.07003i −0.500000 + 0.866025i 0.703829 2.91627i 0.370360 + 0.641483i
43.2 0.766044 + 0.642788i 0.247510 1.71428i 0.173648 + 0.984808i −1.96209 0.714144i 1.29152 1.15411i −0.696712 + 3.95125i −0.500000 + 0.866025i −2.87748 0.848600i −1.04401 1.80828i
49.1 0.766044 0.642788i −1.36085 1.07149i 0.173648 0.984808i 0.696050 0.253341i −1.73121 + 0.0539310i 0.717657 + 4.07003i −0.500000 0.866025i 0.703829 + 2.91627i 0.370360 0.641483i
49.2 0.766044 0.642788i 0.247510 + 1.71428i 0.173648 0.984808i −1.96209 + 0.714144i 1.29152 + 1.15411i −0.696712 3.95125i −0.500000 0.866025i −2.87748 + 0.848600i −1.04401 + 1.80828i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.2.e.b 12
3.b odd 2 1 162.2.e.b 12
4.b odd 2 1 432.2.u.b 12
9.c even 3 1 486.2.e.f 12
9.c even 3 1 486.2.e.h 12
9.d odd 6 1 486.2.e.e 12
9.d odd 6 1 486.2.e.g 12
27.e even 9 1 inner 54.2.e.b 12
27.e even 9 1 486.2.e.f 12
27.e even 9 1 486.2.e.h 12
27.e even 9 1 1458.2.a.g 6
27.e even 9 2 1458.2.c.f 12
27.f odd 18 1 162.2.e.b 12
27.f odd 18 1 486.2.e.e 12
27.f odd 18 1 486.2.e.g 12
27.f odd 18 1 1458.2.a.f 6
27.f odd 18 2 1458.2.c.g 12
108.j odd 18 1 432.2.u.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.e.b 12 1.a even 1 1 trivial
54.2.e.b 12 27.e even 9 1 inner
162.2.e.b 12 3.b odd 2 1
162.2.e.b 12 27.f odd 18 1
432.2.u.b 12 4.b odd 2 1
432.2.u.b 12 108.j odd 18 1
486.2.e.e 12 9.d odd 6 1
486.2.e.e 12 27.f odd 18 1
486.2.e.f 12 9.c even 3 1
486.2.e.f 12 27.e even 9 1
486.2.e.g 12 9.d odd 6 1
486.2.e.g 12 27.f odd 18 1
486.2.e.h 12 9.c even 3 1
486.2.e.h 12 27.e even 9 1
1458.2.a.f 6 27.f odd 18 1
1458.2.a.g 6 27.e even 9 1
1458.2.c.f 12 27.e even 9 2
1458.2.c.g 12 27.f odd 18 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} + 3 T_{5}^{11} + 9 T_{5}^{10} + 24 T_{5}^{9} + 162 T_{5}^{8} - 27 T_{5}^{7} + 1053 T_{5}^{6} + 5184 T_{5}^{5} + 3564 T_{5}^{4} - 3672 T_{5}^{3} + 2592 T_{5}^{2} - 7776 T_{5} + 5184$$ acting on $$S_{2}^{\mathrm{new}}(54, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + T^{3} + 1)^{2}$$
$3$ $$T^{12} + 6 T^{10} + 18 T^{8} - 18 T^{7} + \cdots + 729$$
$5$ $$T^{12} + 3 T^{11} + 9 T^{10} + 24 T^{9} + \cdots + 5184$$
$7$ $$T^{12} + 3 T^{11} + 24 T^{10} + 88 T^{9} + \cdots + 64$$
$11$ $$T^{12} + 12 T^{11} + 90 T^{10} + 537 T^{9} + \cdots + 81$$
$13$ $$T^{12} - 12 T^{11} + 48 T^{10} + \cdots + 23104$$
$17$ $$T^{12} + 6 T^{11} + 54 T^{10} + \cdots + 110889$$
$19$ $$T^{12} + 9 T^{11} + 78 T^{10} + \cdots + 94249$$
$23$ $$T^{12} - 30 T^{11} + 414 T^{10} + \cdots + 5184$$
$29$ $$T^{12} - 15 T^{11} + 81 T^{10} + \cdots + 5184$$
$31$ $$T^{12} + 81 T^{10} + 421 T^{9} + \cdots + 4032064$$
$37$ $$T^{12} + 15 T^{11} + \cdots + 142659136$$
$41$ $$T^{12} + 12 T^{11} + 117 T^{10} + \cdots + 2653641$$
$43$ $$T^{12} - 9 T^{11} + 36 T^{10} + \cdots + 49674304$$
$47$ $$T^{12} + 9 T^{11} + 99 T^{10} + \cdots + 419904$$
$53$ $$(T^{6} + 6 T^{5} - 63 T^{4} + 3 T^{3} + \cdots - 72)^{2}$$
$59$ $$T^{12} - 12 T^{11} + 9 T^{10} + \cdots + 82464561$$
$61$ $$T^{12} + 36 T^{11} + 531 T^{10} + \cdots + 1000000$$
$67$ $$T^{12} - 36 T^{11} + \cdots + 249393368449$$
$71$ $$T^{12} - 12 T^{11} + \cdots + 488586816$$
$73$ $$T^{12} + 21 T^{11} + 390 T^{10} + \cdots + 72361$$
$79$ $$T^{12} - 39 T^{11} + \cdots + 591851584$$
$83$ $$T^{12} - 18 T^{11} + \cdots + 13756474944$$
$89$ $$T^{12} - 12 T^{11} + \cdots + 126899100441$$
$97$ $$T^{12} - 39 T^{11} + \cdots + 373532435929$$