# Properties

 Label 50.3.f.a Level $50$ Weight $3$ Character orbit 50.f Analytic conductor $1.362$ Analytic rank $0$ Dimension $16$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,3,Mod(3,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([7]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("50.3");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 50.f (of order $$20$$, degree $$8$$, 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: $$1.36240132180$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 30x^{14} + 351x^{12} + 2130x^{10} + 7341x^{8} + 14480x^{6} + 15196x^{4} + 6560x^{2} + 16$$ x^16 + 30*x^14 + 351*x^12 + 2130*x^10 + 7341*x^8 + 14480*x^6 + 15196*x^4 + 6560*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{9} + \beta_1) q^{2} + (\beta_{15} - \beta_{13} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + (\beta_{15} + \beta_{14} - 3 \beta_{9} + \cdots - 2) q^{9}+O(q^{10})$$ q + (b9 + b1) * q^2 + (b15 - b13 - b10 - b9 - b8 + b7 - b3 + b2 - 1) * q^3 + 2*b3 * q^4 + (-b13 + b12 + b10 - 2*b9 - b8 + b7 + b6 + b5 - 2*b4 + b3 - 2*b2 + 3*b1 - 1) * q^5 + (b14 + b13 + b9 - b7 + b6 + b3 + b2 + 1) * q^6 + (b15 - 2*b13 + 2*b12 - 3*b8 + 2*b6 - 2*b5 + 3*b4 - 4*b3 + b2 - 2*b1 - 2) * q^7 + (2*b7 - 2*b4) * q^8 + (b15 + b14 - 3*b9 + 2*b8 - 2*b7 + b6 + 2*b3 - b2 + 2*b1 - 2) * q^9 $$q + (\beta_{9} + \beta_1) q^{2} + (\beta_{15} - \beta_{13} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - 9 \beta_{15} + 9 \beta_{14} + \cdots - 6) q^{99}+O(q^{100})$$ q + (b9 + b1) * q^2 + (b15 - b13 - b10 - b9 - b8 + b7 - b3 + b2 - 1) * q^3 + 2*b3 * q^4 + (-b13 + b12 + b10 - 2*b9 - b8 + b7 + b6 + b5 - 2*b4 + b3 - 2*b2 + 3*b1 - 1) * q^5 + (b14 + b13 + b9 - b7 + b6 + b3 + b2 + 1) * q^6 + (b15 - 2*b13 + 2*b12 - 3*b8 + 2*b6 - 2*b5 + 3*b4 - 4*b3 + b2 - 2*b1 - 2) * q^7 + (2*b7 - 2*b4) * q^8 + (b15 + b14 - 3*b9 + 2*b8 - 2*b7 + b6 + 2*b3 - b2 + 2*b1 - 2) * q^9 + (-3*b15 + b13 + b11 + 3*b9 - b8 - b6 + b5 - b4 + b3 - 3*b2 - b1 + 1) * q^10 + (b15 + b14 + 2*b13 + b12 - b11 + b10 + 2*b9 + 4*b8 - 2*b7 + b6 - 2*b5 + 7*b4 - 9*b3 + 5*b2 - 5*b1 + 4) * q^11 - 2*b5 * q^12 + (b15 - 2*b14 - b13 - 5*b12 + 3*b11 - 6*b10 + b9 + 2*b8 - 8*b6 + b5 - 5*b4 - 2*b2) * q^13 + (-2*b15 - 2*b14 + 4*b13 - 3*b12 - 2*b11 + 2*b10 + 3*b9 + 5*b8 + 2*b7 - 3*b6 + 4*b5 + b4 + 7*b3 - 3*b2 + 2*b1 + 2) * q^14 + (-2*b15 - b12 - 2*b11 + b10 - 7*b9 - 2*b8 + b7 - b6 + 2*b5 - 2*b4 + 2*b3 + 2*b2 - 4*b1 - 2) * q^15 + (4*b9 + 4*b8 - 4*b7 + 4) * q^16 + (-3*b15 - 4*b14 + 3*b13 + 6*b10 + 2*b9 - 5*b7 - b6 - 3*b4 + 4*b3 - b2 - 2*b1) * q^17 + (-b12 - b10 - 4*b9 - 7*b8 + b7 - b5 - 2*b4 - 3*b3 + b2 - 5*b1 - 4) * q^18 + (-5*b15 - 4*b14 + 3*b13 - 6*b12 - 3*b11 + 2*b10 + 4*b9 + 3*b8 - b7 - b6 + b5 + 5*b4 + 2*b3 - 2*b2 + 7) * q^19 + (2*b14 - 2*b13 + 4*b12 + 2*b11 - 2*b9 + 2*b8 + 2*b7 - 2*b5 + 2*b4 - 4*b3 + 2*b2 - 4) * q^20 + (b14 + 3*b13 + 2*b12 - b11 + b10 + 9*b9 + 5*b8 - 8*b7 + b5 + 2*b4 - b3 + 5*b2 - 2*b1 + 1) * q^21 + (b15 - 4*b14 + b13 - 4*b12 - b11 + b10 - 2*b9 + 2*b8 - 7*b7 - 2*b6 + 4*b4 + 2*b3 + b2 + 6*b1 + 1) * q^22 + (6*b14 - 3*b13 + 5*b12 + 10*b11 - 3*b10 - 16*b9 - 11*b8 + 20*b7 + b6 - 4*b5 + 2*b4 - 2*b3 - 9*b2 + b1 - 16) * q^23 + (2*b15 - 2*b14 - 2*b12 - 2*b11 + 2*b9 + 2*b8 - 2*b6 + 2*b5 + 2*b3 + 2) * q^24 + (-b15 - 3*b14 + 2*b13 - 6*b12 - 7*b11 - b10 + 10*b9 + 14*b8 + 6*b7 - 7*b6 + 9*b5 - 8*b4 + 20*b3 - 8*b2 + 6*b1 + 11) * q^25 + (10*b15 + 10*b14 - 8*b13 + 6*b12 + 2*b11 - 4*b10 - 10*b9 - 9*b8 - b7 + 12*b6 - 4*b5 + 4*b4 - 10*b3 + 15*b2 - 5*b1 - 11) * q^26 + (-4*b15 + 2*b14 + 10*b12 + 4*b10 - 5*b9 - 7*b8 + 3*b7 + 4*b6 - 2*b5 - 5*b4 - 5*b3 - 4*b2 - 10) * q^27 + (-4*b13 + 4*b12 + 2*b11 - 6*b9 - 4*b8 + 4*b6 - 4*b5 - 8*b3 + 6*b2 - 4*b1 - 2) * q^28 + (5*b15 + 4*b14 + 3*b13 - 6*b12 - 3*b11 - 3*b10 + 12*b9 + b8 - b7 + 8*b6 - 2*b5 + 3*b4 - 8*b3 + 9*b2 - 3*b1 + 13) * q^29 + (-b14 - b13 + 3*b12 + 2*b11 - 3*b10 - 3*b9 - 2*b8 + b7 - 3*b6 - b5 - 3*b4 - 13*b3 - 6*b2 + b1 + 3) * q^30 + (5*b15 + 3*b14 - 7*b13 + 5*b12 + 3*b11 - 7*b10 + 7*b9 - 10*b8 - 12*b7 + 6*b6 - 10*b5 - 4*b4 - 7*b3 + 5*b2 + 8*b1 + 5) * q^31 + (-4*b4 + 4*b3 - 4*b2 + 4*b1 - 4) * q^32 + (3*b15 + 4*b11 + 7*b9 - 4*b8 - 7*b7 + 3*b6 + 4*b5 - 4*b4 + 11*b3 - 12*b2 + 9*b1 + 2) * q^33 + (-3*b15 - 6*b14 - 3*b13 - 3*b12 - 3*b11 + 4*b10 - 2*b9 + 5*b8 + 10*b7 - 2*b6 + 7*b5 + 3*b3 - 20*b2 + 3*b1 + 7) * q^34 + (-b15 + 5*b14 + 4*b13 + 4*b12 - 3*b11 + 5*b10 + 4*b9 - 2*b8 - 10*b7 + 5*b6 + b5 + 11*b4 + 15*b3 + 3*b2 - 5*b1 + 13) * q^35 + (2*b15 - 2*b11 + 4*b8 + 10*b4 - 8*b3 - 4*b1 + 4) * q^36 + (-4*b15 - 6*b13 + b12 + b11 - b10 + 17*b9 + 11*b8 - 2*b7 + b6 + 6*b5 - 14*b4 + 9*b3 + 4*b2 + 9*b1 + 15) * q^37 + (2*b15 - 4*b14 - 2*b13 + 4*b12 - 8*b11 + 2*b10 + b9 - 3*b8 + 2*b7 - 2*b6 + 2*b5 + 4*b4 - b3 - 2*b2 + 5*b1 - 2) * q^38 + (5*b15 + 4*b14 - 4*b13 + 6*b12 + 5*b11 - 4*b10 - 14*b9 + 2*b8 + 14*b7 + 6*b6 - 4*b5 + 22*b4 - 10*b3 + 31*b2 + 5*b1 - 30) * q^39 + (-2*b12 + 4*b11 - 2*b10 - 4*b9 - 2*b7 - 4*b6 + 2*b5 - 8*b4 + 4*b3 + 2*b2 + 2*b1) * q^40 + (3*b15 - 5*b14 + 2*b13 - 2*b12 + b11 + 2*b10 - 23*b9 - 17*b8 + 21*b7 - 6*b6 - 28*b4 + 15*b3 - 16*b2 + 17*b1 - 19) * q^41 + (-2*b15 - b14 + b13 + 4*b11 + 2*b10 - 10*b9 + 2*b8 + b7 + b6 - 4*b5 - 5*b4 + 8*b3 - 10*b2 + 4*b1 - 2) * q^42 + (-3*b15 - b14 - 5*b12 + b11 + 5*b10 + 10*b9 - 7*b8 + b7 - b6 - 5*b5 - 2*b4 + 13*b2 - 6*b1 - 1) * q^43 + (2*b15 - 2*b14 - 2*b13 - 2*b12 - 6*b11 + 2*b10 - 10*b9 - 20*b8 + 12*b7 + 4*b6 + 4*b5 + 2*b4 + 6*b3 - 6*b2 - 6) * q^44 + (5*b15 + 3*b13 - 7*b12 - 10*b11 - b10 + 5*b9 + 6*b8 + 9*b7 - 4*b6 + 3*b5 + 10*b4 - 8*b3 - 4*b2 - b1 + 2) * q^45 + (4*b15 + 9*b14 - 7*b13 - b12 + 6*b11 - 6*b10 - 6*b9 - 5*b8 - 2*b5 + 11*b4 - 23*b3 + 47*b2 - 23*b1 - 6) * q^46 + (-6*b15 + 2*b14 - b13 + 2*b12 + 5*b11 - 6*b10 - 28*b9 - 8*b8 - 2*b7 - 5*b6 + 22*b4 - 8*b3 + 7*b2 + 3*b1 - 14) * q^47 + 4*b6 * q^48 + (2*b15 - 2*b14 - 2*b13 - 2*b10 + 16*b9 - 6*b8 - 22*b7 - b6 - b5 - 23*b4 + 21*b3 - b2 + b1 + 7) * q^49 + (b15 + 3*b14 - 2*b13 + 11*b12 + 7*b11 - 4*b10 + 10*b9 + 6*b8 - 11*b7 + 7*b6 - 9*b5 - 17*b4 - 10*b3 + 8*b2 - b1 - 1) * q^50 + (-11*b15 - 11*b14 + 5*b13 - 8*b12 - 6*b11 + 3*b10 + 11*b9 + 11*b7 - 14*b6 + 5*b5 + 6*b4 + 19*b3 - 38*b2 - 16*b1 - 8) * q^51 + (-4*b15 + 2*b14 + 10*b13 - 10*b12 - 6*b10 - 2*b7 - 6*b6 - 2*b5 - 8*b4 + 10*b3 - 2*b2 - 2*b1 + 2) * q^52 + (-10*b15 - 10*b14 + 16*b13 - 8*b12 - 9*b11 + 10*b10 + 35*b9 + 69*b8 - 28*b7 - 16*b6 + 18*b5 + 10*b4 - 19*b3 + 8*b2 - 12*b1 + 52) * q^53 + (-10*b15 - 6*b14 + 8*b13 - 2*b12 + 6*b11 + 6*b10 + 10*b9 + 17*b8 - 6*b7 - 12*b6 + 4*b5 + 2*b4 + 3*b3 - 16*b2 - 2*b1 + 16) * q^54 + (b15 - 6*b14 + 9*b13 - 9*b12 - 3*b11 + 11*b10 - 8*b9 - 20*b8 - 5*b7 + 4*b6 + 2*b5 - 2*b4 + 15*b3 - 35*b2 + 21*b1 + 22) * q^55 + (-4*b15 - 2*b14 + 6*b13 - 4*b12 - 4*b11 + 4*b10 + 12*b9 + 8*b8 - 2*b7 - 6*b6 + 8*b5 - 6*b4 + 12*b3 - 16*b2 + 12*b1 + 12) * q^56 + (3*b15 - 6*b13 + 2*b12 - b10 + 24*b9 + 10*b8 + 22*b7 + 4*b6 - 2*b5 + 11*b4 - 51*b3 - 27*b1 + 19) * q^57 + (4*b15 - 2*b14 + 4*b13 - 4*b12 - 12*b11 + 19*b9 + 14*b8 - 22*b7 + 2*b6 - 8*b5 + 15*b4 + 11*b3 + 7*b2 + 4*b1 + 7) * q^58 + (-4*b15 + 4*b13 + 4*b12 + 5*b11 - 11*b10 - 20*b9 + 23*b8 - 27*b7 - 10*b6 - 15*b5 - 34*b3 + 9*b2 - 34*b1 - 23) * q^59 + (4*b14 + 2*b13 + 2*b12 + 4*b11 + 4*b10 + 8*b9 - 12*b7 + 4*b6 + 16*b4 - 4*b3 + 6*b2 + 2*b1 - 2) * q^60 + (-17*b15 - b14 + 3*b13 + 3*b12 + 17*b11 - b10 - 33*b9 + 23*b8 - 3*b7 - 5*b6 - 3*b5 - 12*b4 + 4*b3 - 10*b2 + 2*b1 + 19) * q^61 + (2*b15 + 12*b13 - 8*b12 - 8*b11 + 5*b10 + 22*b9 + 6*b8 + 8*b7 - 5*b6 + 7*b5 - 2*b4 + 42*b3 - 18*b2 + 14*b1 + 2) * q^62 + (9*b15 + 16*b14 - 9*b13 + 13*b12 + 3*b11 + 3*b10 - 21*b9 - 6*b8 - 8*b7 + 19*b6 - 8*b5 - 2*b4 + 4*b3 + 8*b2 - 2*b1 - 7) * q^63 - 8*b1 * q^64 + (2*b15 + 6*b14 - 15*b13 + 2*b12 + 8*b11 - 7*b10 - 37*b9 - 46*b8 + 9*b7 + 15*b6 - 14*b5 - 10*b4 - 30*b3 + 31*b2 - 39*b1 - 84) * q^65 + (-7*b15 + 8*b14 - b13 + b12 + b11 + 3*b10 - 3*b9 - 10*b8 + 19*b7 + 11*b6 - 4*b5 + 8*b4 + 3*b3 + 5*b2 - 13*b1 - 20) * q^66 + (20*b15 + 15*b14 - 7*b13 - 9*b11 - 14*b10 - 3*b9 - 11*b8 + 13*b7 - 5*b6 + 9*b5 + b4 - 2*b3 + 19*b2 - 6*b1 - 18) * q^67 + (-8*b15 + 6*b12 + 14*b9 + 2*b8 - 6*b7 + 6*b5 + 6*b4 - 16*b3 + 10*b2 - 4*b1 - 10) * q^68 + (11*b15 + 16*b14 - 3*b13 + 24*b12 + 13*b11 - 43*b9 - 65*b8 + 26*b7 + 3*b6 - 13*b5 - 7*b4 - 14*b3 + 14*b2 - 46) * q^69 + (-5*b15 - 7*b14 + 2*b13 - 3*b12 + 5*b11 - 2*b10 - 7*b9 - 6*b8 + 31*b7 - 9*b6 - 5*b5 - 13*b4 - 4*b3 - 6*b2 + 12*b1 - 13) * q^70 + (-b15 - 29*b14 - 3*b13 - 17*b12 - 14*b11 + 14*b10 + 4*b9 - 21*b8 + 10*b7 + 13*b5 + 21*b4 - 4*b3 + 6*b2 - 15*b1 + 14) * q^71 + (-2*b14 + 2*b13 - 2*b12 - 4*b9 - 4*b8 - 4*b7 + 6*b4 - 4*b3 + 10*b2 + 2*b1 - 8) * q^72 + (-9*b14 + 8*b13 - 9*b12 - 18*b11 + 8*b10 - 26*b9 + 6*b8 + 21*b7 + 3*b6 + 9*b5 + 11*b4 + 41*b3 - 27*b2 + 41*b1 - 18) * q^73 + (-8*b15 + 8*b14 + 2*b13 + 11*b12 + 6*b11 - 3*b10 + 24*b9 + 15*b8 - 9*b7 - 2*b6 - b5 - 41*b4 + 32*b3 - 30*b2 + 30*b1 + 5) * q^74 + (9*b15 - 8*b14 - 3*b13 - 6*b12 - 2*b11 - b10 + 10*b9 + 59*b8 - 24*b7 + 3*b6 - 6*b5 + 22*b4 - 15*b3 + 32*b2 + 11*b1 + 21) * q^75 + (-8*b15 - 8*b14 + 12*b13 - 2*b12 + 4*b11 + 6*b10 + 8*b9 + 6*b8 + 2*b7 - 4*b6 + 6*b5 + 14*b3 - 12*b2 + 4*b1 + 10) * q^76 + (14*b15 - b14 - 9*b13 - 7*b12 - 5*b10 + 4*b9 + 15*b8 - 27*b7 + b6 + b5 + 23*b4 - 8*b3 - 10*b1 + 15) * q^77 + (-4*b15 + 5*b14 + 7*b13 - 5*b12 + 4*b10 - 25*b9 - 20*b8 - 5*b7 + 2*b6 - 32*b4 + 34*b3 + 11*b2 + 2*b1 + 22) * q^78 + (-6*b15 + 4*b14 - 20*b13 + 26*b12 + 6*b11 + 6*b10 + 44*b9 + 27*b8 - 12*b7 + 8*b6 + 3*b4 - 34*b3 - 2*b2 - 3*b1 + 56) * q^79 + (4*b15 + 8*b14 - 8*b13 + 4*b12 + 4*b11 - 4*b10 - 8*b9 - 12*b8 + 4*b7 + 8*b6 - 4*b5 - 4*b4 - 8*b3 - 4) * q^80 + (-18*b15 - 9*b14 + 14*b13 - 5*b12 - 9*b11 + 14*b10 + 19*b9 + 23*b8 - 27*b7 - 18*b6 + 23*b5 - 37*b4 + 64*b3 - 56*b2 + 74*b1 + 15) * q^81 + (3*b15 - b14 - 6*b13 - 5*b12 - b11 + 2*b10 + 23*b9 + 5*b8 - 10*b7 + 9*b6 + 5*b5 + 27*b4 - 22*b3 + 11*b2 - 37*b1 + 36) * q^82 + (-9*b15 + b14 + 7*b13 + 2*b12 + 3*b11 - 20*b8 + 2*b7 - 8*b6 + b5 - 5*b4 + b3 - 30*b2 + 1) * q^83 + (2*b15 - 2*b14 - 4*b13 - 4*b12 - 6*b11 + 2*b10 + 6*b9 - 6*b8 + 4*b7 - 2*b6 + 6*b5 - 10*b3 + 4*b2 - 10*b1) * q^84 + (13*b15 - 10*b14 - 15*b13 - 3*b12 + 10*b11 - 17*b10 + 22*b9 - 10*b8 + 3*b7 - 7*b6 - 13*b5 + 40*b4 + 14*b3 + 18*b2 - 11*b1 - 6) * q^85 + (10*b15 - 9*b14 - 11*b13 - 7*b12 - 10*b11 - 9*b10 - 4*b9 + 18*b8 + 11*b7 - 11*b6 + 11*b5 - 4*b4 + 18*b3 - 17*b2 + 11*b1 + 16) * q^86 + (17*b15 + b13 - 6*b12 - 6*b11 - 8*b10 - 4*b9 - 22*b8 + 23*b7 + 8*b6 + 5*b5 - 56*b4 + 20*b3 - 7*b2 + b1 + 39) * q^87 + (-8*b15 - 4*b14 + 8*b13 - 4*b11 + 2*b10 + 14*b9 + 6*b8 + 10*b7 - 2*b6 + 2*b5 + 20*b4 - 12*b3 + 6*b2 - 12*b1 + 10) * q^88 + (17*b15 + 18*b14 - 29*b13 + 11*b12 + 17*b11 - 18*b10 - 24*b9 - 49*b8 - 15*b7 + 11*b6 - 29*b5 - 2*b4 - 40*b3 + 33*b2 + 21*b1 + 6) * q^89 + (8*b15 - 6*b14 - b12 - 7*b10 - 6*b9 + b8 - 23*b7 + 2*b6 - 7*b5 + 24*b4 - 17*b3 + 33*b2 - 17*b1 - 6) * q^90 + (-2*b15 - b14 + 3*b13 - 3*b12 - 3*b11 - 11*b10 + 13*b9 + 11*b8 + 66*b7 - 12*b6 + 14*b5 - 56*b4 - 4*b3 - 31*b2 + 66*b1 - 63) * q^91 + (12*b15 + 10*b14 - 10*b13 + 6*b11 - 20*b10 - 34*b9 - 8*b8 - 2*b6 - 6*b5 - 28*b4 + 8*b3 - 8*b2 + 34*b1 + 6) * q^92 + (10*b15 + 14*b14 + b12 - 14*b11 - 12*b10 + 34*b9 - 9*b8 - 16*b7 + 14*b6 + b5 - 6*b4 - 6*b3 + 33*b2 + 11*b1 + 10) * q^93 + (5*b15 + 11*b14 - 4*b13 + 12*b12 + 9*b11 - 2*b10 - 49*b9 - 65*b8 + 29*b7 + 2*b6 - 7*b5 - b4 - 24*b3 + 24*b2 - 30) * q^94 + (2*b15 + 3*b14 - 4*b13 + 11*b12 + 7*b11 + 7*b10 + 26*b9 - 4*b8 + 10*b7 + 26*b6 - 4*b5 + 55*b4 - 75*b3 + 75*b2 - 20*b1 + 6) * q^95 + (-4*b15 + 4*b13 - 4*b11 + 4*b10 + 8*b9 + 4*b8 - 4*b7 + 4*b3 - 4*b2 + 4) * q^96 + (10*b15 + 9*b14 + 2*b13 + 9*b12 - 3*b11 + 10*b10 - 62*b9 - 21*b8 + 32*b7 + 21*b6 + 67*b4 - 21*b3 + 56*b2 - 34*b1 - 44) * q^97 + (b14 + 3*b13 - b12 - 2*b11 + 3*b10 + 13*b9 + 18*b8 + 27*b7 + 3*b6 + 3*b5 - 16*b4 + 22*b3 - 45*b2 + 8*b1 + 2) * q^98 + (-9*b15 + 9*b14 - b13 + 3*b12 + 10*b11 + 6*b10 - 15*b9 - 8*b8 + 7*b7 - b6 + 7*b5 - 75*b4 + 71*b3 - 46*b2 + 46*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 4 q^{2} + 2 q^{3} + 4 q^{6} - 2 q^{7} + 8 q^{8} - 40 q^{9}+O(q^{10})$$ 16 * q - 4 * q^2 + 2 * q^3 + 4 * q^6 - 2 * q^7 + 8 * q^8 - 40 * q^9 $$16 q - 4 q^{2} + 2 q^{3} + 4 q^{6} - 2 q^{7} + 8 q^{8} - 40 q^{9} + 10 q^{10} + 32 q^{11} + 4 q^{12} - 8 q^{13} - 30 q^{14} + 16 q^{16} - 62 q^{17} - 16 q^{18} + 30 q^{19} - 20 q^{20} - 68 q^{21} - 48 q^{22} - 18 q^{23} + 70 q^{25} - 56 q^{26} - 40 q^{27} + 44 q^{28} + 100 q^{29} + 120 q^{30} + 132 q^{31} - 64 q^{32} - 36 q^{33} + 100 q^{34} + 150 q^{35} + 48 q^{36} + 138 q^{37} + 20 q^{38} - 320 q^{39} - 88 q^{41} - 8 q^{42} - 78 q^{43} + 40 q^{44} - 20 q^{45} - 26 q^{46} - 22 q^{47} - 8 q^{48} - 20 q^{50} - 168 q^{51} - 16 q^{52} + 182 q^{53} + 80 q^{54} + 280 q^{55} + 48 q^{56} + 280 q^{57} - 120 q^{58} - 350 q^{59} - 140 q^{60} + 372 q^{61} - 158 q^{62} + 22 q^{63} - 910 q^{65} - 202 q^{66} - 112 q^{67} - 196 q^{68} - 30 q^{69} - 20 q^{70} + 122 q^{71} - 132 q^{72} - 248 q^{73} - 80 q^{75} + 40 q^{76} + 16 q^{77} + 438 q^{78} + 760 q^{79} + 80 q^{80} - 144 q^{81} + 352 q^{82} + 132 q^{83} - 20 q^{84} - 30 q^{85} + 264 q^{86} + 770 q^{87} + 116 q^{88} + 550 q^{89} - 140 q^{90} - 798 q^{91} + 384 q^{92} + 54 q^{93} + 190 q^{94} + 40 q^{95} - 16 q^{96} - 292 q^{97} - 24 q^{98}+O(q^{100})$$ 16 * q - 4 * q^2 + 2 * q^3 + 4 * q^6 - 2 * q^7 + 8 * q^8 - 40 * q^9 + 10 * q^10 + 32 * q^11 + 4 * q^12 - 8 * q^13 - 30 * q^14 + 16 * q^16 - 62 * q^17 - 16 * q^18 + 30 * q^19 - 20 * q^20 - 68 * q^21 - 48 * q^22 - 18 * q^23 + 70 * q^25 - 56 * q^26 - 40 * q^27 + 44 * q^28 + 100 * q^29 + 120 * q^30 + 132 * q^31 - 64 * q^32 - 36 * q^33 + 100 * q^34 + 150 * q^35 + 48 * q^36 + 138 * q^37 + 20 * q^38 - 320 * q^39 - 88 * q^41 - 8 * q^42 - 78 * q^43 + 40 * q^44 - 20 * q^45 - 26 * q^46 - 22 * q^47 - 8 * q^48 - 20 * q^50 - 168 * q^51 - 16 * q^52 + 182 * q^53 + 80 * q^54 + 280 * q^55 + 48 * q^56 + 280 * q^57 - 120 * q^58 - 350 * q^59 - 140 * q^60 + 372 * q^61 - 158 * q^62 + 22 * q^63 - 910 * q^65 - 202 * q^66 - 112 * q^67 - 196 * q^68 - 30 * q^69 - 20 * q^70 + 122 * q^71 - 132 * q^72 - 248 * q^73 - 80 * q^75 + 40 * q^76 + 16 * q^77 + 438 * q^78 + 760 * q^79 + 80 * q^80 - 144 * q^81 + 352 * q^82 + 132 * q^83 - 20 * q^84 - 30 * q^85 + 264 * q^86 + 770 * q^87 + 116 * q^88 + 550 * q^89 - 140 * q^90 - 798 * q^91 + 384 * q^92 + 54 * q^93 + 190 * q^94 + 40 * q^95 - 16 * q^96 - 292 * q^97 - 24 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 30x^{14} + 351x^{12} + 2130x^{10} + 7341x^{8} + 14480x^{6} + 15196x^{4} + 6560x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( - 61 \nu^{15} + 364 \nu^{14} - 2016 \nu^{13} + 10504 \nu^{12} - 25551 \nu^{11} + 114532 \nu^{10} + \cdots + 6528 ) / 5728$$ (-61*v^15 + 364*v^14 - 2016*v^13 + 10504*v^12 - 25551*v^11 + 114532*v^10 - 158776*v^9 + 614968*v^8 - 514181*v^7 + 1726068*v^6 - 840674*v^5 + 2403480*v^4 - 589924*v^3 + 1294616*v^2 - 95008*v + 6528) / 5728 $$\beta_{2}$$ $$=$$ $$( 61 \nu^{15} + 364 \nu^{14} + 2016 \nu^{13} + 10504 \nu^{12} + 25551 \nu^{11} + 114532 \nu^{10} + \cdots + 6528 ) / 5728$$ (61*v^15 + 364*v^14 + 2016*v^13 + 10504*v^12 + 25551*v^11 + 114532*v^10 + 158776*v^9 + 614968*v^8 + 514181*v^7 + 1726068*v^6 + 840674*v^5 + 2403480*v^4 + 589924*v^3 + 1294616*v^2 + 95008*v + 6528) / 5728 $$\beta_{3}$$ $$=$$ $$( 297 \nu^{15} + 550 \nu^{14} + 8366 \nu^{13} + 14644 \nu^{12} + 88651 \nu^{11} + 143378 \nu^{10} + \cdots - 3040 ) / 5728$$ (297*v^15 + 550*v^14 + 8366*v^13 + 14644*v^12 + 88651*v^11 + 143378*v^10 + 463786*v^9 + 681348*v^8 + 1279041*v^7 + 1683462*v^6 + 1778096*v^5 + 2066448*v^4 + 994584*v^3 + 986600*v^2 + 38168*v - 3040) / 5728 $$\beta_{4}$$ $$=$$ $$( - 297 \nu^{15} + 550 \nu^{14} - 8366 \nu^{13} + 14644 \nu^{12} - 88651 \nu^{11} + 143378 \nu^{10} + \cdots - 3040 ) / 5728$$ (-297*v^15 + 550*v^14 - 8366*v^13 + 14644*v^12 - 88651*v^11 + 143378*v^10 - 463786*v^9 + 681348*v^8 - 1279041*v^7 + 1683462*v^6 - 1778096*v^5 + 2066448*v^4 - 994584*v^3 + 986600*v^2 - 38168*v - 3040) / 5728 $$\beta_{5}$$ $$=$$ $$( - 219 \nu^{15} - 1228 \nu^{14} - 5348 \nu^{13} - 33084 \nu^{12} - 45697 \nu^{11} - 329596 \nu^{10} + \cdots - 21488 ) / 5728$$ (-219*v^15 - 1228*v^14 - 5348*v^13 - 33084*v^12 - 45697*v^11 - 329596*v^10 - 180164*v^9 - 1601412*v^8 - 353451*v^7 - 4061780*v^6 - 349602*v^5 - 5137492*v^4 - 183644*v^3 - 2551696*v^2 - 49248*v - 21488) / 5728 $$\beta_{6}$$ $$=$$ $$( 219 \nu^{15} - 1228 \nu^{14} + 5348 \nu^{13} - 33084 \nu^{12} + 45697 \nu^{11} - 329596 \nu^{10} + \cdots - 21488 ) / 5728$$ (219*v^15 - 1228*v^14 + 5348*v^13 - 33084*v^12 + 45697*v^11 - 329596*v^10 + 180164*v^9 - 1601412*v^8 + 353451*v^7 - 4061780*v^6 + 349602*v^5 - 5137492*v^4 + 183644*v^3 - 2551696*v^2 + 49248*v - 21488) / 5728 $$\beta_{7}$$ $$=$$ $$( 347 \nu^{15} - 1584 \nu^{14} + 9502 \nu^{13} - 42948 \nu^{12} + 97129 \nu^{11} - 432232 \nu^{10} + \cdots - 10720 ) / 5728$$ (347*v^15 - 1584*v^14 + 9502*v^13 - 42948*v^12 + 97129*v^11 - 432232*v^10 + 490122*v^9 - 2131516*v^8 + 1314659*v^7 - 5519864*v^6 + 1815660*v^5 - 7173124*v^4 + 1073600*v^3 - 3666240*v^2 + 112616*v - 10720) / 5728 $$\beta_{8}$$ $$=$$ $$( 347 \nu^{15} + 1584 \nu^{14} + 9502 \nu^{13} + 42948 \nu^{12} + 97129 \nu^{11} + 432232 \nu^{10} + \cdots + 10720 ) / 5728$$ (347*v^15 + 1584*v^14 + 9502*v^13 + 42948*v^12 + 97129*v^11 + 432232*v^10 + 490122*v^9 + 2131516*v^8 + 1314659*v^7 + 5519864*v^6 + 1815660*v^5 + 7173124*v^4 + 1073600*v^3 + 3666240*v^2 + 112616*v + 10720) / 5728 $$\beta_{9}$$ $$=$$ $$( - 1259 \nu^{15} - 1584 \nu^{14} - 34132 \nu^{13} - 42948 \nu^{12} - 343473 \nu^{11} - 432232 \nu^{10} + \cdots - 13584 ) / 5728$$ (-1259*v^15 - 1584*v^14 - 34132*v^13 - 42948*v^12 - 343473*v^11 - 432232*v^10 - 1693980*v^9 - 2131516*v^8 - 4390483*v^7 - 5519864*v^6 - 5726794*v^5 - 7173124*v^4 - 2983660*v^3 - 3666240*v^2 - 75264*v - 13584) / 5728 $$\beta_{10}$$ $$=$$ $$( - 583 \nu^{15} - 1480 \nu^{14} - 15494 \nu^{13} - 39998 \nu^{12} - 151637 \nu^{11} - 400736 \nu^{10} + \cdots + 760 ) / 2864$$ (-583*v^15 - 1480*v^14 - 15494*v^13 - 39998*v^12 - 151637*v^11 - 400736*v^10 - 724606*v^9 - 1965886*v^8 - 1827487*v^7 - 5062604*v^6 - 2377540*v^5 - 6538530*v^4 - 1366564*v^3 - 3309340*v^2 - 181792*v + 760) / 2864 $$\beta_{11}$$ $$=$$ $$( - 165 \nu^{15} - 1998 \nu^{14} - 4429 \nu^{13} - 54946 \nu^{12} - 44159 \nu^{11} - 563834 \nu^{10} + \cdots - 11504 ) / 2864$$ (-165*v^15 - 1998*v^14 - 4429*v^13 - 54946*v^12 - 44159*v^11 - 563834*v^10 - 218259*v^9 - 2843346*v^8 - 585517*v^7 - 7534298*v^6 - 856823*v^5 - 10001882*v^4 - 631426*v^3 - 5199540*v^2 - 174508*v - 11504) / 2864 $$\beta_{12}$$ $$=$$ $$( 622 \nu^{15} - 1819 \nu^{14} + 17003 \nu^{13} - 49218 \nu^{12} + 173114 \nu^{11} - 493845 \nu^{10} + \cdots - 8640 ) / 2864$$ (622*v^15 - 1819*v^14 + 17003*v^13 - 49218*v^12 + 173114*v^11 - 493845*v^10 + 866417*v^9 - 2425918*v^8 + 2290282*v^7 - 6251763*v^6 + 3091787*v^5 - 8074052*v^4 + 1772034*v^3 - 4091888*v^2 + 176252*v - 8640) / 2864 $$\beta_{13}$$ $$=$$ $$( 33 \nu^{15} - 4546 \nu^{14} + 492 \nu^{13} - 124536 \nu^{12} - 333 \nu^{11} - 1271046 \nu^{10} + \cdots - 19968 ) / 5728$$ (33*v^15 - 4546*v^14 + 492*v^13 - 124536*v^12 - 333*v^11 - 1271046*v^10 - 27268*v^9 - 6368040*v^8 - 108007*v^7 - 16752058*v^6 - 64450*v^5 - 22070212*v^4 + 268268*v^3 - 11385680*v^2 + 310848*v - 19968) / 5728 $$\beta_{14}$$ $$=$$ $$( - 1820 \nu^{15} + 1365 \nu^{14} - 49298 \nu^{13} + 36884 \nu^{12} - 495332 \nu^{11} + 369351 \nu^{10} + \cdots + 7296 ) / 2864$$ (-1820*v^15 + 1365*v^14 - 49298*v^13 + 36884*v^12 - 495332*v^11 + 369351*v^10 - 2436526*v^9 + 1809584*v^8 - 6284724*v^7 + 4647313*v^6 - 8110546*v^5 + 5974346*v^4 - 4070900*v^3 + 3010036*v^2 + 44688*v + 7296) / 2864 $$\beta_{15}$$ $$=$$ $$( 1551 \nu^{15} - 5454 \nu^{14} + 42456 \nu^{13} - 149204 \nu^{12} + 432565 \nu^{11} - 1520034 \nu^{10} + \cdots - 25520 ) / 5728$$ (1551*v^15 - 5454*v^14 + 42456*v^13 - 149204*v^12 + 432565*v^11 - 1520034*v^10 + 2158784*v^9 - 7600708*v^8 + 5627871*v^7 - 19960958*v^6 + 7237574*v^5 - 26269624*v^4 + 3397972*v^3 - 13549384*v^2 - 305856*v - 25520) / 5728
 $$\nu$$ $$=$$ $$( - \beta_{15} + \beta_{14} - 3 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - \beta_{10} - 3 \beta_{9} + \cdots - 3 ) / 5$$ (-b15 + b14 - 3*b13 + 2*b12 + 4*b11 - b10 - 3*b9 - 3*b8 + 2*b6 - 3*b5 - 5*b3 + 5*b2 - 5*b1 - 3) / 5 $$\nu^{2}$$ $$=$$ $$( - 2 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots - 17 ) / 5$$ (-2*b15 - 2*b14 + 4*b13 - 4*b12 + 2*b11 - 2*b10 + 2*b9 + b8 + b7 - 5*b6 + b5 - 8*b4 + 2*b2 + 6*b1 - 17) / 5 $$\nu^{3}$$ $$=$$ $$( 2 \beta_{15} - 2 \beta_{14} + 21 \beta_{13} - 14 \beta_{12} - 23 \beta_{11} + 12 \beta_{10} + 6 \beta_{9} + \cdots + 16 ) / 5$$ (2*b15 - 2*b14 + 21*b13 - 14*b12 - 23*b11 + 12*b10 + 6*b9 + 31*b8 + 25*b7 - 14*b6 + 26*b5 - 10*b4 + 45*b3 - 55*b2 + 55*b1 + 16) / 5 $$\nu^{4}$$ $$=$$ $$8 \beta_{15} + 8 \beta_{14} - 11 \beta_{13} + 13 \beta_{12} - 3 \beta_{11} + 5 \beta_{10} - 8 \beta_{9} + \cdots + 21$$ 8*b15 + 8*b14 - 11*b13 + 13*b12 - 3*b11 + 5*b10 - 8*b9 + b8 - 9*b7 + 15*b6 - 6*b5 + 20*b4 - 4*b3 + 2*b2 - 14*b1 + 21 $$\nu^{5}$$ $$=$$ $$( 9 \beta_{15} - 9 \beta_{14} - 208 \beta_{13} + 137 \beta_{12} + 199 \beta_{11} - 146 \beta_{10} + \cdots - 133 ) / 5$$ (9*b15 - 9*b14 - 208*b13 + 137*b12 + 199*b11 - 146*b10 + 17*b9 - 338*b8 - 355*b7 + 122*b6 - 268*b5 + 195*b4 - 540*b3 + 655*b2 - 655*b1 - 133) / 5 $$\nu^{6}$$ $$=$$ $$( - 542 \beta_{15} - 542 \beta_{14} + 669 \beta_{13} - 819 \beta_{12} + 127 \beta_{11} - 277 \beta_{10} + \cdots - 912 ) / 5$$ (-542*b15 - 542*b14 + 669*b13 - 819*b12 + 127*b11 - 277*b10 + 542*b9 - 114*b8 + 656*b7 - 950*b6 + 411*b5 - 1188*b4 + 300*b3 - 298*b2 + 786*b1 - 912) / 5 $$\nu^{7}$$ $$=$$ $$( - 233 \beta_{15} + 233 \beta_{14} + 2321 \beta_{13} - 1509 \beta_{12} - 2088 \beta_{11} + 1742 \beta_{10} + \cdots + 1351 ) / 5$$ (-233*b15 + 233*b14 + 2321*b13 - 1509*b12 - 2088*b11 + 1742*b10 - 549*b9 + 3751*b8 + 4300*b7 - 1234*b6 + 2976*b5 - 2685*b4 + 6515*b3 - 7730*b2 + 7730*b1 + 1351) / 5 $$\nu^{8}$$ $$=$$ $$1320 \beta_{15} + 1320 \beta_{14} - 1568 \beta_{13} + 1948 \beta_{12} - 248 \beta_{11} + 628 \beta_{10} + \cdots + 1901$$ 1320*b15 + 1320*b14 - 1568*b13 + 1948*b12 - 248*b11 + 628*b10 - 1320*b9 + 323*b8 - 1643*b7 + 2275*b6 - 993*b5 + 2782*b4 - 734*b3 + 860*b2 - 1780*b1 + 1901 $$\nu^{9}$$ $$=$$ $$( 3264 \beta_{15} - 3264 \beta_{14} - 26693 \beta_{13} + 17232 \beta_{12} + 23429 \beta_{11} + \cdots - 14848 ) / 5$$ (3264*b15 - 3264*b14 - 26693*b13 + 17232*b12 + 23429*b11 - 20496*b10 + 8032*b9 - 42363*b8 - 50395*b7 + 13462*b6 - 33958*b5 + 33210*b4 - 77135*b3 + 90335*b2 - 90335*b1 - 14848) / 5 $$\nu^{10}$$ $$=$$ $$( - 77742 \beta_{15} - 77742 \beta_{14} + 90989 \beta_{13} - 113869 \beta_{12} + 13247 \beta_{11} + \cdots - 106197 ) / 5$$ (-77742*b15 - 77742*b14 + 90989*b13 - 113869*b12 + 13247*b11 - 36127*b10 + 77742*b9 - 20399*b8 + 98141*b7 - 133465*b6 + 58146*b5 - 161958*b4 + 42900*b3 - 53688*b2 + 101796*b1 - 106197) / 5 $$\nu^{11}$$ $$=$$ $$( - 40423 \beta_{15} + 40423 \beta_{14} + 309006 \beta_{13} - 198849 \beta_{12} - 268583 \beta_{11} + \cdots + 168331 ) / 5$$ (-40423*b15 + 40423*b14 + 309006*b13 - 198849*b12 - 268583*b11 + 239272*b10 - 101459*b9 + 484306*b8 + 585765*b7 - 151934*b6 + 391206*b5 - 394925*b4 + 902780*b3 - 1050465*b2 + 1050465*b1 + 168331) / 5 $$\nu^{12}$$ $$=$$ $$181168 \beta_{15} + 181168 \beta_{14} - 210773 \beta_{13} + 264731 \beta_{12} - 29605 \beta_{11} + \cdots + 242974$$ 181168*b15 + 181168*b14 - 210773*b13 + 264731*b12 - 29605*b11 + 83563*b10 - 181168*b9 + 49164*b8 - 230332*b7 + 310806*b6 - 135093*b5 + 376166*b4 - 99338*b3 + 127998*b2 - 234338*b1 + 242974 $$\nu^{13}$$ $$=$$ $$( 480759 \beta_{15} - 480759 \beta_{14} - 3581023 \beta_{13} + 2301347 \beta_{12} + 3100264 \beta_{11} + \cdots - 1932503 ) / 5$$ (480759*b15 - 480759*b14 - 3581023*b13 + 2301347*b12 + 3100264*b11 - 2782106*b10 + 1218447*b9 - 5574663*b8 - 6793110*b7 + 1740232*b6 - 4522338*b5 + 4624595*b4 - 10506965*b3 + 12188870*b2 - 12188870*b1 - 1932503) / 5 $$\nu^{14}$$ $$=$$ $$( - 10515952 \beta_{15} - 10515952 \beta_{14} + 12204034 \beta_{13} - 15354984 \beta_{12} + \cdots - 14009687 ) / 5$$ (-10515952*b15 - 10515952*b14 + 12204034*b13 - 15354984*b12 + 1688082*b11 - 4839032*b10 + 10515952*b9 - 2899639*b8 + 13415591*b7 - 18040665*b6 + 7830271*b5 - 21814538*b4 + 5744480*b3 - 7500178*b2 + 13531726*b1 - 14009687) / 5 $$\nu^{15}$$ $$=$$ $$( - 5630388 \beta_{15} + 5630388 \beta_{14} + 41499681 \beta_{13} - 26655034 \beta_{12} - 35869293 \beta_{11} + \cdots + 22301706 ) / 5$$ (-5630388*b15 + 5630388*b14 + 41499681*b13 - 26655034*b12 - 35869293*b11 + 32285422*b10 - 14337044*b9 + 64388491*b8 + 78725535*b7 - 20060304*b6 + 52345726*b5 - 53817410*b4 + 121972125*b3 - 141301605*b2 + 141301605*b1 + 22301706) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$\chi(n)$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 2.26402i − 1.64599i 1.78563i − 3.40366i − 2.26402i 1.64599i − 0.0495271i − 1.56851i − 1.78563i 3.40366i 1.84816i − 1.23012i 0.0495271i 1.56851i − 1.84816i 1.23012i
−0.642040 1.26007i −0.711697 4.49348i −1.17557 + 1.61803i −4.53886 + 2.09733i −5.20517 + 3.78178i 3.58690 3.58690i 2.79360 + 0.442463i −11.1253 + 3.61484i 5.55691 + 4.37272i
3.2 −0.642040 1.26007i 0.569657 + 3.59668i −1.17557 + 1.61803i 2.10649 + 4.53461i 4.16633 3.02702i 0.635779 0.635779i 2.79360 + 0.442463i −4.05206 + 1.31659i 4.36149 5.56574i
13.1 −1.39680 + 0.221232i −1.14941 + 2.25584i 1.90211 0.618034i −4.68874 + 1.73657i 1.10643 3.40526i −6.58346 + 6.58346i −2.52015 + 1.28408i 1.52238 + 2.09537i 6.16506 3.46295i
13.2 −1.39680 + 0.221232i 0.252608 0.495771i 1.90211 0.618034i 3.68015 + 3.38474i −0.243163 + 0.748380i 7.20385 7.20385i −2.52015 + 1.28408i 5.10809 + 7.03068i −5.88926 3.91365i
17.1 −0.642040 + 1.26007i −0.711697 + 4.49348i −1.17557 1.61803i −4.53886 2.09733i −5.20517 3.78178i 3.58690 + 3.58690i 2.79360 0.442463i −11.1253 3.61484i 5.55691 4.37272i
17.2 −0.642040 + 1.26007i 0.569657 3.59668i −1.17557 1.61803i 2.10649 4.53461i 4.16633 + 3.02702i 0.635779 + 0.635779i 2.79360 0.442463i −4.05206 1.31659i 4.36149 + 5.56574i
23.1 −0.221232 + 1.39680i −2.40328 + 1.22453i −1.90211 0.618034i −1.59895 + 4.73744i −1.17875 3.62781i −3.03568 + 3.03568i 1.28408 2.52015i −1.01379 + 1.39536i −6.26353 3.28149i
23.2 −0.221232 + 1.39680i 2.68205 1.36657i −1.90211 0.618034i 4.84361 1.24072i 1.31548 + 4.04862i −3.67488 + 3.67488i 1.28408 2.52015i 0.0358006 0.0492753i 0.661483 + 7.04006i
27.1 −1.39680 0.221232i −1.14941 2.25584i 1.90211 + 0.618034i −4.68874 1.73657i 1.10643 + 3.40526i −6.58346 6.58346i −2.52015 1.28408i 1.52238 2.09537i 6.16506 + 3.46295i
27.2 −1.39680 0.221232i 0.252608 + 0.495771i 1.90211 + 0.618034i 3.68015 3.38474i −0.243163 0.748380i 7.20385 + 7.20385i −2.52015 1.28408i 5.10809 7.03068i −5.88926 + 3.91365i
33.1 1.26007 + 0.642040i −0.742607 0.117617i 1.17557 + 1.61803i 4.02861 + 2.96147i −0.860225 0.624990i 2.71368 2.71368i 0.442463 + 2.79360i −8.02188 2.60647i 3.17497 + 6.31819i
33.2 1.26007 + 0.642040i 2.50268 + 0.396386i 1.17557 + 1.61803i −3.83232 3.21144i 2.89907 + 2.10629i −1.84619 + 1.84619i 0.442463 + 2.79360i −2.45322 0.797099i −2.76713 6.50715i
37.1 −0.221232 1.39680i −2.40328 1.22453i −1.90211 + 0.618034i −1.59895 4.73744i −1.17875 + 3.62781i −3.03568 3.03568i 1.28408 + 2.52015i −1.01379 1.39536i −6.26353 + 3.28149i
37.2 −0.221232 1.39680i 2.68205 + 1.36657i −1.90211 + 0.618034i 4.84361 + 1.24072i 1.31548 4.04862i −3.67488 3.67488i 1.28408 + 2.52015i 0.0358006 + 0.0492753i 0.661483 7.04006i
47.1 1.26007 0.642040i −0.742607 + 0.117617i 1.17557 1.61803i 4.02861 2.96147i −0.860225 + 0.624990i 2.71368 + 2.71368i 0.442463 2.79360i −8.02188 + 2.60647i 3.17497 6.31819i
47.2 1.26007 0.642040i 2.50268 0.396386i 1.17557 1.61803i −3.83232 + 3.21144i 2.89907 2.10629i −1.84619 1.84619i 0.442463 2.79360i −2.45322 + 0.797099i −2.76713 + 6.50715i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.3.f.a 16
4.b odd 2 1 400.3.bg.a 16
5.b even 2 1 250.3.f.b 16
5.c odd 4 1 250.3.f.a 16
5.c odd 4 1 250.3.f.c 16
25.d even 5 1 250.3.f.a 16
25.e even 10 1 250.3.f.c 16
25.f odd 20 1 inner 50.3.f.a 16
25.f odd 20 1 250.3.f.b 16
100.l even 20 1 400.3.bg.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.f.a 16 1.a even 1 1 trivial
50.3.f.a 16 25.f odd 20 1 inner
250.3.f.a 16 5.c odd 4 1
250.3.f.a 16 25.d even 5 1
250.3.f.b 16 5.b even 2 1
250.3.f.b 16 25.f odd 20 1
250.3.f.c 16 5.c odd 4 1
250.3.f.c 16 25.e even 10 1
400.3.bg.a 16 4.b odd 2 1
400.3.bg.a 16 100.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{16} - 2 T_{3}^{15} + 22 T_{3}^{14} - 76 T_{3}^{13} + 76 T_{3}^{12} - 322 T_{3}^{11} + \cdots + 130321$$ acting on $$S_{3}^{\mathrm{new}}(50, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 16)^{2}$$
$3$ $$T^{16} - 2 T^{15} + \cdots + 130321$$
$5$ $$T^{16} + \cdots + 152587890625$$
$7$ $$T^{16} + \cdots + 9353984656$$
$11$ $$T^{16} + \cdots + 21080623380496$$
$13$ $$T^{16} + \cdots + 79\!\cdots\!21$$
$17$ $$T^{16} + \cdots + 12\!\cdots\!16$$
$19$ $$T^{16} + \cdots + 15\!\cdots\!25$$
$23$ $$T^{16} + \cdots + 58\!\cdots\!81$$
$29$ $$T^{16} + \cdots + 52\!\cdots\!25$$
$31$ $$T^{16} + \cdots + 26\!\cdots\!21$$
$37$ $$T^{16} + \cdots + 11\!\cdots\!81$$
$41$ $$T^{16} + \cdots + 10\!\cdots\!36$$
$43$ $$T^{16} + \cdots + 47\!\cdots\!56$$
$47$ $$T^{16} + \cdots + 93\!\cdots\!21$$
$53$ $$T^{16} + \cdots + 17\!\cdots\!61$$
$59$ $$T^{16} + \cdots + 38\!\cdots\!25$$
$61$ $$T^{16} + \cdots + 17\!\cdots\!01$$
$67$ $$T^{16} + \cdots + 73\!\cdots\!36$$
$71$ $$T^{16} + \cdots + 15\!\cdots\!01$$
$73$ $$T^{16} + \cdots + 21\!\cdots\!61$$
$79$ $$T^{16} + \cdots + 27\!\cdots\!25$$
$83$ $$T^{16} + \cdots + 84\!\cdots\!01$$
$89$ $$T^{16} + \cdots + 10\!\cdots\!00$$
$97$ $$T^{16} + \cdots + 71\!\cdots\!81$$