# Properties

 Label 76.2.f.a Level $76$ Weight $2$ Character orbit 76.f Analytic conductor $0.607$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,2,Mod(27,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, 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("76.27");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 76.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: $$0.606863055362$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256$$ x^16 - 3*x^15 + 6*x^14 - 9*x^13 + 12*x^12 - 9*x^11 + 3*x^10 + 6*x^9 - 10*x^8 + 12*x^7 + 12*x^6 - 72*x^5 + 192*x^4 - 288*x^3 + 384*x^2 - 384*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 3 x^{15} + 6 x^{14} - 9 x^{13} + 12 x^{12} - 9 x^{11} + 3 x^{10} + 6 x^{9} - 10 x^{8} + \cdots + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 26 \nu^{15} - 397 \nu^{14} + 335 \nu^{13} - 666 \nu^{12} + 419 \nu^{11} - 428 \nu^{10} + \cdots - 5056 ) / 40768$$ (-26*v^15 - 397*v^14 + 335*v^13 - 666*v^12 + 419*v^11 - 428*v^10 + 625*v^9 - 183*v^8 - 1340*v^7 - 730*v^6 + 1304*v^5 - 4308*v^4 + 17312*v^3 - 29216*v^2 + 15616*v - 5056) / 40768 $$\beta_{3}$$ $$=$$ $$( 79 \nu^{15} - 341 \nu^{14} - 1114 \nu^{13} + 629 \nu^{12} - 1716 \nu^{11} + 965 \nu^{10} + \cdots + 32128 ) / 81536$$ (79*v^15 - 341*v^14 - 1114*v^13 + 629*v^12 - 1716*v^11 + 965*v^10 - 1475*v^9 + 2974*v^8 - 1522*v^7 - 4412*v^6 - 1972*v^5 - 472*v^4 - 2064*v^3 + 46496*v^2 - 86528*v + 32128) / 81536 $$\beta_{4}$$ $$=$$ $$( 130 \nu^{15} - 269 \nu^{14} + 1363 \nu^{13} - 2 \nu^{12} - 37 \nu^{11} + 1552 \nu^{10} + \cdots + 62912 ) / 81536$$ (130*v^15 - 269*v^14 + 1363*v^13 - 2*v^12 - 37*v^11 + 1552*v^10 + 109*v^9 + 5325*v^8 + 7288*v^7 - 270*v^6 + 31896*v^5 - 13740*v^4 + 48288*v^3 + 8880*v^2 + 34816*v + 62912) / 81536 $$\beta_{5}$$ $$=$$ $$( 339 \nu^{15} + 2061 \nu^{14} + 1416 \nu^{13} - 2119 \nu^{12} - 1594 \nu^{11} + 3285 \nu^{10} + \cdots + 45056 ) / 163072$$ (339*v^15 + 2061*v^14 + 1416*v^13 - 2119*v^12 - 1594*v^11 + 3285*v^10 + 10699*v^9 + 6764*v^8 + 8742*v^7 + 1712*v^6 - 2468*v^5 + 37120*v^4 - 24656*v^3 + 17216*v^2 + 174400*v + 45056) / 163072 $$\beta_{6}$$ $$=$$ $$( - 425 \nu^{15} - 845 \nu^{14} - 666 \nu^{13} - 883 \nu^{12} - 1268 \nu^{11} - 5251 \nu^{10} + \cdots + 51840 ) / 163072$$ (-425*v^15 - 845*v^14 - 666*v^13 - 883*v^12 - 1268*v^11 - 5251*v^10 - 8251*v^9 + 2638*v^8 - 7234*v^7 - 1948*v^6 + 13596*v^5 - 6968*v^4 - 29168*v^3 + 4832*v^2 - 4544*v + 51840) / 163072 $$\beta_{7}$$ $$=$$ $$( - 230 \nu^{15} - 97 \nu^{14} - 949 \nu^{13} + 290 \nu^{12} + 367 \nu^{11} - 1404 \nu^{10} + \cdots - 52928 ) / 81536$$ (-230*v^15 - 97*v^14 - 949*v^13 + 290*v^12 + 367*v^11 - 1404*v^10 - 1791*v^9 + 1781*v^8 - 9924*v^7 + 7986*v^6 - 1280*v^5 - 6508*v^4 - 26512*v^3 - 20656*v^2 + 640*v - 52928) / 81536 $$\beta_{8}$$ $$=$$ $$( - 537 \nu^{15} + 2473 \nu^{14} - 5608 \nu^{13} + 1525 \nu^{12} - 1506 \nu^{11} + 97 \nu^{10} + \cdots - 147968 ) / 163072$$ (-537*v^15 + 2473*v^14 - 5608*v^13 + 1525*v^12 - 1506*v^11 + 97*v^10 + 7023*v^9 - 7988*v^8 + 4302*v^7 + 5808*v^6 - 39828*v^5 + 22208*v^4 - 122128*v^3 + 90176*v^2 - 960*v - 147968) / 163072 $$\beta_{9}$$ $$=$$ $$( - 361 \nu^{15} - 550 \nu^{14} + 835 \nu^{13} - 1671 \nu^{12} + 2921 \nu^{11} - 3225 \nu^{10} + \cdots - 95168 ) / 81536$$ (-361*v^15 - 550*v^14 + 835*v^13 - 1671*v^12 + 2921*v^11 - 3225*v^10 + 934*v^9 - 1559*v^8 - 11082*v^7 + 10938*v^6 + 3300*v^5 - 14652*v^4 + 10176*v^3 - 44048*v^2 + 36864*v - 95168) / 81536 $$\beta_{10}$$ $$=$$ $$( 885 \nu^{15} - 431 \nu^{14} + 114 \nu^{13} + 4223 \nu^{12} - 4660 \nu^{11} + 5903 \nu^{10} + \cdots + 273536 ) / 163072$$ (885*v^15 - 431*v^14 + 114*v^13 + 4223*v^12 - 4660*v^11 + 5903*v^10 + 759*v^9 + 4874*v^8 + 16498*v^7 + 18316*v^6 + 10916*v^5 + 7832*v^4 - 21296*v^3 + 141536*v^2 - 153536*v + 273536) / 163072 $$\beta_{11}$$ $$=$$ $$( - 621 \nu^{15} + 2879 \nu^{14} - 5566 \nu^{13} + 9309 \nu^{12} - 9892 \nu^{11} + 5725 \nu^{10} + \cdots + 352896 ) / 81536$$ (-621*v^15 + 2879*v^14 - 5566*v^13 + 9309*v^12 - 9892*v^11 + 5725*v^10 - 607*v^9 - 5790*v^8 + 14522*v^7 - 11356*v^6 - 7180*v^5 + 96520*v^4 - 160096*v^3 + 299616*v^2 - 308736*v + 352896) / 81536 $$\beta_{12}$$ $$=$$ $$( \nu^{15} - 3 \nu^{14} + 6 \nu^{13} - 9 \nu^{12} + 12 \nu^{11} - 9 \nu^{10} + 3 \nu^{9} + 6 \nu^{8} + \cdots - 384 ) / 128$$ (v^15 - 3*v^14 + 6*v^13 - 9*v^12 + 12*v^11 - 9*v^10 + 3*v^9 + 6*v^8 - 10*v^7 + 12*v^6 + 12*v^5 - 72*v^4 + 192*v^3 - 288*v^2 + 384*v - 384) / 128 $$\beta_{13}$$ $$=$$ $$( 1441 \nu^{15} + 1063 \nu^{14} + 2720 \nu^{13} - 1949 \nu^{12} + 522 \nu^{11} + 2391 \nu^{10} + \cdots + 125952 ) / 163072$$ (1441*v^15 + 1063*v^14 + 2720*v^13 - 1949*v^12 + 522*v^11 + 2391*v^10 + 3345*v^9 + 3420*v^8 + 4898*v^7 + 18096*v^6 + 62516*v^5 - 41344*v^4 + 54288*v^3 - 26496*v^2 + 175424*v + 125952) / 163072 $$\beta_{14}$$ $$=$$ $$( 475 \nu^{15} - 491 \nu^{14} + 900 \nu^{13} - 731 \nu^{12} + 662 \nu^{11} - 703 \nu^{10} + \cdots - 6656 ) / 40768$$ (475*v^15 - 491*v^14 + 900*v^13 - 731*v^12 + 662*v^11 - 703*v^10 + 27*v^9 + 1600*v^8 + 418*v^7 - 1616*v^6 + 6180*v^5 - 22304*v^4 + 36704*v^3 - 25600*v^2 + 15040*v - 6656) / 40768 $$\beta_{15}$$ $$=$$ $$( 1346 \nu^{15} - 2357 \nu^{14} + 3471 \nu^{13} - 6350 \nu^{12} + 7867 \nu^{11} - 3172 \nu^{10} + \cdots - 249664 ) / 81536$$ (1346*v^15 - 2357*v^14 + 3471*v^13 - 6350*v^12 + 7867*v^11 - 3172*v^10 + 821*v^9 + 2561*v^8 - 6044*v^7 - 9158*v^6 + 25216*v^5 - 79964*v^4 + 127856*v^3 - 164848*v^2 + 232000*v - 249664) / 81536
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - \beta_{7} + \beta_{3} - \beta_{2} + \beta_1$$ b9 - b7 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{3} + \beta_1$$ -b15 + b13 + b12 - b10 - b7 - b6 - b5 + 2*b3 + b1 $$\nu^{4}$$ $$=$$ $$-2\beta_{14} + \beta_{13} + 2\beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 1$$ -2*b14 + b13 + 2*b12 + b11 - b9 - b8 - b4 + b3 + b2 - b1 - 1 $$\nu^{5}$$ $$=$$ $$\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2\beta_{2} - 2\beta _1 - 2$$ b15 - b14 + b13 + b11 + b7 + b6 + b4 - b3 + 2*b2 - 2*b1 - 2 $$\nu^{6}$$ $$=$$ $$- 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{8} + \cdots - 1$$ -2*b15 - 2*b14 + 3*b13 + 2*b12 - b11 + 2*b10 + b8 - b7 - 2*b5 - b4 + 2*b3 - 1 $$\nu^{7}$$ $$=$$ $$- \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} + 3 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} + \cdots - 5 \beta_1$$ -b14 - 2*b13 + b12 - b11 + 3*b10 - 4*b9 + 4*b8 - 2*b7 + 4*b6 - b5 + 3*b4 - 5*b3 - 5*b1 $$\nu^{8}$$ $$=$$ $$2 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} - 3 \beta_{9} + 9 \beta_{7} + 6 \beta_{5} + 6 \beta_{4} + \cdots + 6$$ 2*b14 - 6*b13 + 2*b12 - 3*b9 + 9*b7 + 6*b5 + 6*b4 + b3 - 5*b2 - 5*b1 + 6 $$\nu^{9}$$ $$=$$ $$3 \beta_{15} + 8 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} + 4 \beta_{11} + 3 \beta_{10} + 8 \beta_{9} + \cdots - 3 \beta_1$$ 3*b15 + 8*b14 - 7*b13 - 3*b12 + 4*b11 + 3*b10 + 8*b9 + 4*b8 + 5*b7 - 3*b6 + 9*b5 + 4*b4 - 6*b3 + 12*b2 - 3*b1 $$\nu^{10}$$ $$=$$ $$4 \beta_{15} - 16 \beta_{14} + 5 \beta_{13} - 8 \beta_{12} - 7 \beta_{11} + 2 \beta_{10} - 5 \beta_{9} + \cdots + 1$$ 4*b15 - 16*b14 + 5*b13 - 8*b12 - 7*b11 + 2*b10 - 5*b9 - 5*b8 - 14*b6 - 4*b5 - b4 + 9*b3 - b2 + 5*b1 + 1 $$\nu^{11}$$ $$=$$ $$11 \beta_{15} - 13 \beta_{14} - 9 \beta_{13} + 8 \beta_{12} + 5 \beta_{11} + 9 \beta_{7} - 13 \beta_{6} + \cdots + 44$$ 11*b15 - 13*b14 - 9*b13 + 8*b12 + 5*b11 + 9*b7 - 13*b6 - 4*b5 + 9*b4 - 5*b3 - 24*b2 - 18*b1 + 44 $$\nu^{12}$$ $$=$$ $$24 \beta_{14} - 19 \beta_{13} + 11 \beta_{11} - 7 \beta_{8} + 31 \beta_{7} - 20 \beta_{6} - 2 \beta_{5} + \cdots - 5$$ 24*b14 - 19*b13 + 11*b11 - 7*b8 + 31*b7 - 20*b6 - 2*b5 + 11*b4 - 30*b3 + 18*b1 - 5 $$\nu^{13}$$ $$=$$ $$21 \beta_{14} - 18 \beta_{13} - 21 \beta_{12} - 5 \beta_{11} + 5 \beta_{10} + 24 \beta_{9} - 24 \beta_{8} + \cdots - 26$$ 21*b14 - 18*b13 - 21*b12 - 5*b11 + 5*b10 + 24*b9 - 24*b8 - 10*b7 + 2*b6 + 19*b5 - 5*b4 - 29*b3 - 22*b2 + 9*b1 - 26 $$\nu^{14}$$ $$=$$ $$- 26 \beta_{15} + 18 \beta_{14} + 28 \beta_{13} + 6 \beta_{12} - 8 \beta_{11} - 52 \beta_{10} + \cdots + 24$$ -26*b15 + 18*b14 + 28*b13 + 6*b12 - 8*b11 - 52*b10 - 7*b9 - 7*b7 - 22*b6 - 6*b5 + 2*b4 + 13*b3 - 55*b2 - 17*b1 + 24 $$\nu^{15}$$ $$=$$ $$21 \beta_{15} + 2 \beta_{14} + 43 \beta_{13} + 39 \beta_{12} + 32 \beta_{11} + 21 \beta_{10} - 40 \beta_{9} + \cdots - 24$$ 21*b15 + 2*b14 + 43*b13 + 39*b12 + 32*b11 + 21*b10 - 40*b9 - 20*b8 + 25*b7 + b6 + 5*b5 - 72*b4 + 10*b3 - 29*b1 - 24

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1
 1.05003 − 0.947334i −0.112075 − 1.40977i 1.34543 − 0.435684i −0.835469 − 1.14105i 1.16486 + 0.801943i −1.35532 − 0.403874i 0.570443 + 1.29406i −0.327894 + 1.37568i 1.05003 + 0.947334i −0.112075 + 1.40977i 1.34543 + 0.435684i −0.835469 + 1.14105i 1.16486 − 0.801943i −1.35532 + 0.403874i 0.570443 − 1.29406i −0.327894 − 1.37568i
−1.34543 0.435684i 0.982349 1.70148i 1.62036 + 1.17236i −0.349646 + 0.605604i −2.06299 + 1.86122i 3.80025i −1.66930 2.28330i −0.430019 0.744815i 0.734275 0.662462i
27.2 −1.16486 + 0.801943i 0.305055 0.528371i 0.713775 1.86829i 1.59295 2.75907i 0.0683782 + 0.860112i 2.36291i 0.666820 + 2.74870i 1.31388 + 2.27571i 0.357061 + 4.49138i
27.3 −1.05003 0.947334i −0.982349 + 1.70148i 0.205118 + 1.98945i −0.349646 + 0.605604i 2.64336 0.855988i 3.80025i 1.66930 2.28330i −0.430019 0.744815i 0.940847 0.304670i
27.4 −0.570443 + 1.29406i −0.637123 + 1.10353i −1.34919 1.47638i −1.60333 + 2.77705i −1.06459 1.45398i 1.25044i 2.68016 0.903746i 0.688149 + 1.19191i −2.67906 3.65895i
27.5 0.112075 1.40977i −0.305055 + 0.528371i −1.97488 0.316000i 1.59295 2.75907i 0.710690 + 0.489273i 2.36291i −0.666820 + 2.74870i 1.31388 + 2.27571i −3.71111 2.55491i
27.6 0.327894 + 1.37568i 1.42689 2.47144i −1.78497 + 0.902152i −0.139977 + 0.242447i 3.86777 + 1.15256i 1.55280i −1.82635 2.15973i −2.57201 4.45486i −0.379427 0.113066i
27.7 0.835469 1.14105i 0.637123 1.10353i −0.603985 1.90662i −1.60333 + 2.77705i −0.726885 1.64895i 1.25044i −2.68016 0.903746i 0.688149 + 1.19191i 1.82921 + 4.14961i
27.8 1.35532 0.403874i −1.42689 + 2.47144i 1.67377 1.09475i −0.139977 + 0.242447i −0.935735 + 3.92587i 1.55280i 1.82635 2.15973i −2.57201 4.45486i −0.0917953 + 0.385126i
31.1 −1.34543 + 0.435684i 0.982349 + 1.70148i 1.62036 1.17236i −0.349646 0.605604i −2.06299 1.86122i 3.80025i −1.66930 + 2.28330i −0.430019 + 0.744815i 0.734275 + 0.662462i
31.2 −1.16486 0.801943i 0.305055 + 0.528371i 0.713775 + 1.86829i 1.59295 + 2.75907i 0.0683782 0.860112i 2.36291i 0.666820 2.74870i 1.31388 2.27571i 0.357061 4.49138i
31.3 −1.05003 + 0.947334i −0.982349 1.70148i 0.205118 1.98945i −0.349646 0.605604i 2.64336 + 0.855988i 3.80025i 1.66930 + 2.28330i −0.430019 + 0.744815i 0.940847 + 0.304670i
31.4 −0.570443 1.29406i −0.637123 1.10353i −1.34919 + 1.47638i −1.60333 2.77705i −1.06459 + 1.45398i 1.25044i 2.68016 + 0.903746i 0.688149 1.19191i −2.67906 + 3.65895i
31.5 0.112075 + 1.40977i −0.305055 0.528371i −1.97488 + 0.316000i 1.59295 + 2.75907i 0.710690 0.489273i 2.36291i −0.666820 2.74870i 1.31388 2.27571i −3.71111 + 2.55491i
31.6 0.327894 1.37568i 1.42689 + 2.47144i −1.78497 0.902152i −0.139977 0.242447i 3.86777 1.15256i 1.55280i −1.82635 + 2.15973i −2.57201 + 4.45486i −0.379427 + 0.113066i
31.7 0.835469 + 1.14105i 0.637123 + 1.10353i −0.603985 + 1.90662i −1.60333 2.77705i −0.726885 + 1.64895i 1.25044i −2.68016 + 0.903746i 0.688149 1.19191i 1.82921 4.14961i
31.8 1.35532 + 0.403874i −1.42689 2.47144i 1.67377 + 1.09475i −0.139977 0.242447i −0.935735 3.92587i 1.55280i 1.82635 + 2.15973i −2.57201 + 4.45486i −0.0917953 0.385126i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.2.f.a 16
3.b odd 2 1 684.2.r.a 16
4.b odd 2 1 inner 76.2.f.a 16
8.b even 2 1 1216.2.n.f 16
8.d odd 2 1 1216.2.n.f 16
12.b even 2 1 684.2.r.a 16
19.d odd 6 1 inner 76.2.f.a 16
57.f even 6 1 684.2.r.a 16
76.f even 6 1 inner 76.2.f.a 16
152.l odd 6 1 1216.2.n.f 16
152.o even 6 1 1216.2.n.f 16
228.n odd 6 1 684.2.r.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.2.f.a 16 1.a even 1 1 trivial
76.2.f.a 16 4.b odd 2 1 inner
76.2.f.a 16 19.d odd 6 1 inner
76.2.f.a 16 76.f even 6 1 inner
684.2.r.a 16 3.b odd 2 1
684.2.r.a 16 12.b even 2 1
684.2.r.a 16 57.f even 6 1
684.2.r.a 16 228.n odd 6 1
1216.2.n.f 16 8.b even 2 1
1216.2.n.f 16 8.d odd 2 1
1216.2.n.f 16 152.l odd 6 1
1216.2.n.f 16 152.o even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + 3 T^{15} + \cdots + 256$$
$3$ $$T^{16} + 14 T^{14} + \cdots + 361$$
$5$ $$(T^{8} + T^{7} + 11 T^{6} + \cdots + 4)^{2}$$
$7$ $$(T^{8} + 24 T^{6} + \cdots + 304)^{2}$$
$11$ $$(T^{8} + 45 T^{6} + \cdots + 3724)^{2}$$
$13$ $$(T^{8} + 9 T^{7} + \cdots + 3136)^{2}$$
$17$ $$(T^{8} - T^{7} + 19 T^{6} + \cdots + 784)^{2}$$
$19$ $$T^{16} + \cdots + 16983563041$$
$23$ $$T^{16} + \cdots + 757115775376$$
$29$ $$(T^{8} + 3 T^{7} + \cdots + 98596)^{2}$$
$31$ $$(T^{8} - 124 T^{6} + \cdots + 7600)^{2}$$
$37$ $$(T^{8} + 124 T^{6} + \cdots + 784)^{2}$$
$41$ $$(T^{8} + 24 T^{7} + \cdots + 841)^{2}$$
$43$ $$T^{16} - 105 T^{14} + \cdots + 1478656$$
$47$ $$T^{16} - 69 T^{14} + \cdots + 37896336$$
$53$ $$(T^{8} - 3 T^{7} + \cdots + 802816)^{2}$$
$59$ $$T^{16} + \cdots + 1300683225625$$
$61$ $$(T^{8} + 13 T^{7} + \cdots + 209764)^{2}$$
$67$ $$T^{16} + 106 T^{14} + \cdots + 361$$
$71$ $$T^{16} + \cdots + 3550253056$$
$73$ $$(T^{8} - 8 T^{7} + \cdots + 1190281)^{2}$$
$79$ $$T^{16} + \cdots + 136386521399296$$
$83$ $$(T^{8} + 181 T^{6} + \cdots + 255664)^{2}$$
$89$ $$(T^{8} - 9 T^{7} + \cdots + 250000)^{2}$$
$97$ $$(T^{8} + 6 T^{7} + \cdots + 24649)^{2}$$