# Properties

 Label 57.2.f.a Level $57$ Weight $2$ Character orbit 57.f Analytic conductor $0.455$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(8,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("57.8");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.f (of order $$6$$, degree $$2$$, 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$20$$ $$40$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i −0.965926 + 0.258819i
−0.965926 + 1.67303i 1.57313 0.724745i −0.866025 1.50000i 1.22474 + 0.707107i −0.307007 + 3.33195i −3.73205 −0.517638 1.94949 2.28024i −2.36603 + 1.36603i
8.2 −0.258819 + 0.448288i −1.57313 + 0.724745i 0.866025 + 1.50000i 1.22474 + 0.707107i 0.0822623 0.892794i −0.267949 −1.93185 1.94949 2.28024i −0.633975 + 0.366025i
8.3 0.258819 0.448288i −0.158919 1.72474i 0.866025 + 1.50000i −1.22474 0.707107i −0.814313 0.375156i −0.267949 1.93185 −2.94949 + 0.548188i −0.633975 + 0.366025i
8.4 0.965926 1.67303i 0.158919 + 1.72474i −0.866025 1.50000i −1.22474 0.707107i 3.03906 + 1.40010i −3.73205 0.517638 −2.94949 + 0.548188i −2.36603 + 1.36603i
50.1 −0.965926 1.67303i 1.57313 + 0.724745i −0.866025 + 1.50000i 1.22474 0.707107i −0.307007 3.33195i −3.73205 −0.517638 1.94949 + 2.28024i −2.36603 1.36603i
50.2 −0.258819 0.448288i −1.57313 0.724745i 0.866025 1.50000i 1.22474 0.707107i 0.0822623 + 0.892794i −0.267949 −1.93185 1.94949 + 2.28024i −0.633975 0.366025i
50.3 0.258819 + 0.448288i −0.158919 + 1.72474i 0.866025 1.50000i −1.22474 + 0.707107i −0.814313 + 0.375156i −0.267949 1.93185 −2.94949 0.548188i −0.633975 0.366025i
50.4 0.965926 + 1.67303i 0.158919 1.72474i −0.866025 + 1.50000i −1.22474 + 0.707107i 3.03906 1.40010i −3.73205 0.517638 −2.94949 0.548188i −2.36603 1.36603i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.f.a 8
3.b odd 2 1 inner 57.2.f.a 8
4.b odd 2 1 912.2.bn.m 8
12.b even 2 1 912.2.bn.m 8
19.c even 3 1 1083.2.d.b 8
19.d odd 6 1 inner 57.2.f.a 8
19.d odd 6 1 1083.2.d.b 8
57.f even 6 1 inner 57.2.f.a 8
57.f even 6 1 1083.2.d.b 8
57.h odd 6 1 1083.2.d.b 8
76.f even 6 1 912.2.bn.m 8
228.n odd 6 1 912.2.bn.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.f.a 8 1.a even 1 1 trivial
57.2.f.a 8 3.b odd 2 1 inner
57.2.f.a 8 19.d odd 6 1 inner
57.2.f.a 8 57.f even 6 1 inner
912.2.bn.m 8 4.b odd 2 1
912.2.bn.m 8 12.b even 2 1
912.2.bn.m 8 76.f even 6 1
912.2.bn.m 8 228.n odd 6 1
1083.2.d.b 8 19.c even 3 1
1083.2.d.b 8 19.d odd 6 1
1083.2.d.b 8 57.f even 6 1
1083.2.d.b 8 57.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(57, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 4 T^{6} + 15 T^{4} + 4 T^{2} + \cdots + 1$$
$3$ $$T^{8} + 2 T^{6} - 5 T^{4} + 18 T^{2} + \cdots + 81$$
$5$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$7$ $$(T^{2} + 4 T + 1)^{4}$$
$11$ $$(T^{4} + 28 T^{2} + 4)^{2}$$
$13$ $$(T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1)^{2}$$
$17$ $$(T^{4} - 24 T^{2} + 576)^{2}$$
$19$ $$(T^{4} + 26 T^{2} + 361)^{2}$$
$23$ $$T^{8} - 28 T^{6} + 780 T^{4} + \cdots + 16$$
$29$ $$T^{8} + 64 T^{6} + 3840 T^{4} + \cdots + 65536$$
$31$ $$(T^{4} + 26 T^{2} + 121)^{2}$$
$37$ $$(T^{4} + 78 T^{2} + 1089)^{2}$$
$41$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$43$ $$(T^{4} + 8 T^{3} + 51 T^{2} + 104 T + 169)^{2}$$
$47$ $$T^{8} - 112 T^{6} + 12480 T^{4} + \cdots + 4096$$
$53$ $$T^{8} + 156 T^{6} + \cdots + 18974736$$
$59$ $$T^{8} + 196 T^{6} + \cdots + 78074896$$
$61$ $$(T^{4} - 14 T^{3} + 159 T^{2} - 518 T + 1369)^{2}$$
$67$ $$(T^{4} - T^{2} + 1)^{2}$$
$71$ $$T^{8} + 192 T^{6} + 34560 T^{4} + \cdots + 5308416$$
$73$ $$(T^{2} - 3 T + 9)^{4}$$
$79$ $$(T^{4} - 12 T^{3} + 11 T^{2} + 444 T + 1369)^{2}$$
$83$ $$(T^{4} + 64 T^{2} + 256)^{2}$$
$89$ $$(T^{4} + 54 T^{2} + 2916)^{2}$$
$97$ $$(T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16)^{2}$$
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