# Properties

 Label 57.2.i.b Level $57$ Weight $2$ Character orbit 57.i Analytic conductor $0.455$ 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] = [57,2,Mod(4,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("57.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.i (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: $$0.455147291521$$ 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.

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

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$1$$ $$-\beta_{3} + \beta_{9}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 1.80139i 0.5 − 0.168222i 0.5 + 1.74095i 0.5 − 2.42499i 0.5 − 1.74095i 0.5 + 2.42499i 0.5 − 0.677980i 0.5 + 1.96356i 0.5 + 1.80139i 0.5 + 0.168222i 0.5 + 0.677980i 0.5 − 1.96356i
−2.23121 0.812094i 0.173648 + 0.984808i 2.78672 + 2.33833i 2.16379 1.81563i 0.412311 2.33833i 1.05756 1.83175i −1.94440 3.36780i −0.939693 + 0.342020i −6.30233 + 2.29386i
4.2 0.791518 + 0.288089i 0.173648 + 0.984808i −0.988583 0.829520i 1.30800 1.09754i −0.146267 + 0.829520i −1.96517 + 3.40377i −1.38582 2.40031i −0.939693 + 0.342020i 1.35149 0.491903i
16.1 −0.958464 0.804247i −0.939693 + 0.342020i −0.0754558 0.427931i 0.755623 4.28535i 1.17573 + 0.427931i 0.898157 + 1.55565i −1.52303 + 2.63796i 0.766044 0.642788i −4.17072 + 3.49965i
16.2 1.22451 + 1.02748i −0.939693 + 0.342020i 0.0964003 + 0.546713i −0.174371 + 0.988909i −1.50208 0.546713i −1.28482 2.22537i 1.15479 2.00015i 0.766044 0.642788i −1.22961 + 1.03176i
25.1 −0.958464 + 0.804247i −0.939693 0.342020i −0.0754558 + 0.427931i 0.755623 + 4.28535i 1.17573 0.427931i 0.898157 1.55565i −1.52303 2.63796i 0.766044 + 0.642788i −4.17072 3.49965i
25.2 1.22451 1.02748i −0.939693 0.342020i 0.0964003 0.546713i −0.174371 0.988909i −1.50208 + 0.546713i −1.28482 + 2.22537i 1.15479 + 2.00015i 0.766044 + 0.642788i −1.22961 1.03176i
28.1 −0.458021 2.59757i 0.766044 0.642788i −4.65819 + 1.69544i 1.91800 + 0.698096i −2.02055 1.69544i −1.30802 + 2.26556i 3.89993 + 6.75488i 0.173648 0.984808i 0.934866 5.30189i
28.2 0.131669 + 0.746734i 0.766044 0.642788i 1.33911 0.487396i −2.97104 1.08137i 0.580856 + 0.487396i −1.89771 + 3.28694i 1.29853 + 2.24912i 0.173648 0.984808i 0.416301 2.36096i
43.1 −2.23121 + 0.812094i 0.173648 0.984808i 2.78672 2.33833i 2.16379 + 1.81563i 0.412311 + 2.33833i 1.05756 + 1.83175i −1.94440 + 3.36780i −0.939693 0.342020i −6.30233 2.29386i
43.2 0.791518 0.288089i 0.173648 0.984808i −0.988583 + 0.829520i 1.30800 + 1.09754i −0.146267 0.829520i −1.96517 3.40377i −1.38582 + 2.40031i −0.939693 0.342020i 1.35149 + 0.491903i
55.1 −0.458021 + 2.59757i 0.766044 + 0.642788i −4.65819 1.69544i 1.91800 0.698096i −2.02055 + 1.69544i −1.30802 2.26556i 3.89993 6.75488i 0.173648 + 0.984808i 0.934866 + 5.30189i
55.2 0.131669 0.746734i 0.766044 + 0.642788i 1.33911 + 0.487396i −2.97104 + 1.08137i 0.580856 0.487396i −1.89771 3.28694i 1.29853 2.24912i 0.173648 + 0.984808i 0.416301 + 2.36096i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 57.2.i.b 12
3.b odd 2 1 171.2.u.e 12
4.b odd 2 1 912.2.bo.j 12
19.e even 9 1 inner 57.2.i.b 12
19.e even 9 1 1083.2.a.q 6
19.f odd 18 1 1083.2.a.p 6
57.j even 18 1 3249.2.a.bg 6
57.l odd 18 1 171.2.u.e 12
57.l odd 18 1 3249.2.a.bh 6
76.l odd 18 1 912.2.bo.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.i.b 12 1.a even 1 1 trivial
57.2.i.b 12 19.e even 9 1 inner
171.2.u.e 12 3.b odd 2 1
171.2.u.e 12 57.l odd 18 1
912.2.bo.j 12 4.b odd 2 1
912.2.bo.j 12 76.l odd 18 1
1083.2.a.p 6 19.f odd 18 1
1083.2.a.q 6 19.e even 9 1
3249.2.a.bg 6 57.j even 18 1
3249.2.a.bh 6 57.l odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 3 T_{2}^{11} + 6 T_{2}^{10} + 8 T_{2}^{9} - 9 T_{2}^{8} - 9 T_{2}^{7} + 99 T_{2}^{6} + 18 T_{2}^{5} - 36 T_{2}^{4} - 64 T_{2}^{3} + 96 T_{2}^{2} - 96 T_{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 3 T^{11} + 6 T^{10} + 8 T^{9} + \cdots + 64$$
$3$ $$(T^{6} + T^{3} + 1)^{2}$$
$5$ $$T^{12} - 6 T^{11} + 24 T^{10} + \cdots + 18496$$
$7$ $$T^{12} + 9 T^{11} + 66 T^{10} + \cdots + 145161$$
$11$ $$T^{12} + 9 T^{11} + 75 T^{10} + \cdots + 207936$$
$13$ $$T^{12} + 3 T^{11} - 6 T^{10} - 21 T^{9} + \cdots + 3249$$
$17$ $$T^{12} - 12 T^{11} + 18 T^{10} + \cdots + 41062464$$
$19$ $$T^{12} + 9 T^{11} + 72 T^{10} + \cdots + 47045881$$
$23$ $$T^{12} - 9 T^{11} - 18 T^{10} + \cdots + 2483776$$
$29$ $$T^{12} - 6 T^{10} - 125 T^{9} + \cdots + 87616$$
$31$ $$T^{12} + 36 T^{10} - 20 T^{9} + \cdots + 185761$$
$37$ $$(T^{6} + 6 T^{5} - 84 T^{4} - 862 T^{3} + \cdots + 2467)^{2}$$
$41$ $$T^{12} + 18 T^{11} + 183 T^{10} + \cdots + 87616$$
$43$ $$T^{12} - 15 T^{11} + \cdots + 587917009$$
$47$ $$T^{12} + 9 T^{11} + 63 T^{10} + \cdots + 46656$$
$53$ $$T^{12} + 30 T^{11} + 531 T^{10} + \cdots + 1871424$$
$59$ $$T^{12} + 30 T^{11} + 498 T^{10} + \cdots + 1617984$$
$61$ $$T^{12} - 21 T^{11} + 285 T^{10} + \cdots + 5329$$
$67$ $$T^{12} - 3 T^{11} + \cdots + 1254293056$$
$71$ $$T^{12} + 30 T^{11} + \cdots + 241118784$$
$73$ $$T^{12} + 24 T^{11} + \cdots + 1056705049$$
$79$ $$T^{12} - 24 T^{11} + \cdots + 4606201161$$
$83$ $$T^{12} - 3 T^{11} + \cdots + 75809912896$$
$89$ $$T^{12} - 3 T^{11} + \cdots + 21486869056$$
$97$ $$T^{12} + 27 T^{11} + 606 T^{10} + \cdots + 128881$$