# Properties

 Label 77.2.e.a Level $77$ Weight $2$ Character orbit 77.e Analytic conductor $0.615$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(23,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("77.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$77 = 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 77.e (of order $$3$$, 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: $$0.614848095564$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/77\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-\zeta_{18}^{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 −0.766044 − 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 + 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i
−0.939693 + 1.62760i −0.326352 0.565258i −0.766044 1.32683i −1.76604 + 3.05888i 1.22668 0.418748 + 2.61240i −0.879385 1.28699 2.22913i −3.31908 5.74881i
23.2 0.173648 0.300767i 0.266044 + 0.460802i 0.939693 + 1.62760i −0.0603074 + 0.104455i 0.184793 −2.47178 0.943555i 1.34730 1.35844 2.35289i 0.0209445 + 0.0362770i
23.3 0.766044 1.32683i −1.43969 2.49362i −0.173648 0.300767i −1.17365 + 2.03282i −4.41147 2.05303 1.66885i 2.53209 −2.64543 + 4.58202i 1.79813 + 3.11446i
67.1 −0.939693 1.62760i −0.326352 + 0.565258i −0.766044 + 1.32683i −1.76604 3.05888i 1.22668 0.418748 2.61240i −0.879385 1.28699 + 2.22913i −3.31908 + 5.74881i
67.2 0.173648 + 0.300767i 0.266044 0.460802i 0.939693 1.62760i −0.0603074 0.104455i 0.184793 −2.47178 + 0.943555i 1.34730 1.35844 + 2.35289i 0.0209445 0.0362770i
67.3 0.766044 + 1.32683i −1.43969 + 2.49362i −0.173648 + 0.300767i −1.17365 2.03282i −4.41147 2.05303 + 1.66885i 2.53209 −2.64543 4.58202i 1.79813 3.11446i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 67.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.e.a 6
3.b odd 2 1 693.2.i.h 6
4.b odd 2 1 1232.2.q.m 6
7.b odd 2 1 539.2.e.m 6
7.c even 3 1 inner 77.2.e.a 6
7.c even 3 1 539.2.a.j 3
7.d odd 6 1 539.2.a.g 3
7.d odd 6 1 539.2.e.m 6
11.b odd 2 1 847.2.e.c 6
11.c even 5 4 847.2.n.g 24
11.d odd 10 4 847.2.n.f 24
21.g even 6 1 4851.2.a.bk 3
21.h odd 6 1 693.2.i.h 6
21.h odd 6 1 4851.2.a.bj 3
28.f even 6 1 8624.2.a.co 3
28.g odd 6 1 1232.2.q.m 6
28.g odd 6 1 8624.2.a.ch 3
77.h odd 6 1 847.2.e.c 6
77.h odd 6 1 5929.2.a.x 3
77.i even 6 1 5929.2.a.u 3
77.m even 15 4 847.2.n.g 24
77.o odd 30 4 847.2.n.f 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.a 6 1.a even 1 1 trivial
77.2.e.a 6 7.c even 3 1 inner
539.2.a.g 3 7.d odd 6 1
539.2.a.j 3 7.c even 3 1
539.2.e.m 6 7.b odd 2 1
539.2.e.m 6 7.d odd 6 1
693.2.i.h 6 3.b odd 2 1
693.2.i.h 6 21.h odd 6 1
847.2.e.c 6 11.b odd 2 1
847.2.e.c 6 77.h odd 6 1
847.2.n.f 24 11.d odd 10 4
847.2.n.f 24 77.o odd 30 4
847.2.n.g 24 11.c even 5 4
847.2.n.g 24 77.m even 15 4
1232.2.q.m 6 4.b odd 2 1
1232.2.q.m 6 28.g odd 6 1
4851.2.a.bj 3 21.h odd 6 1
4851.2.a.bk 3 21.g even 6 1
5929.2.a.u 3 77.i even 6 1
5929.2.a.x 3 77.h odd 6 1
8624.2.a.ch 3 28.g odd 6 1
8624.2.a.co 3 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 3T_{2}^{4} - 2T_{2}^{3} + 9T_{2}^{2} - 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(77, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$3$ $$T^{6} + 3 T^{5} + 9 T^{4} + 2 T^{3} + \cdots + 1$$
$5$ $$T^{6} + 6 T^{5} + 27 T^{4} + 52 T^{3} + \cdots + 1$$
$7$ $$T^{6} + 17T^{3} + 343$$
$11$ $$(T^{2} + T + 1)^{3}$$
$13$ $$(T^{3} - 3 T^{2} - 6 T - 1)^{2}$$
$17$ $$T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 16129$$
$19$ $$T^{6} + 9 T^{5} + 63 T^{4} + 144 T^{3} + \cdots + 81$$
$23$ $$T^{6} + 57 T^{4} - 214 T^{3} + \cdots + 11449$$
$29$ $$(T^{3} + 3 T^{2} - 36 T + 51)^{2}$$
$31$ $$T^{6} + 9 T^{5} + 57 T^{4} + 178 T^{3} + \cdots + 361$$
$37$ $$T^{6} + 36 T^{4} + 144 T^{3} + \cdots + 5184$$
$41$ $$(T^{3} - 9 T^{2} + 6 T - 1)^{2}$$
$43$ $$(T^{3} - 9 T + 9)^{2}$$
$47$ $$T^{6} - 3 T^{5} + 87 T^{4} + \cdots + 104329$$
$53$ $$T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 210681$$
$59$ $$T^{6} + 93 T^{4} + 38 T^{3} + \cdots + 361$$
$61$ $$T^{6} + 12 T^{5} + 180 T^{4} + \cdots + 576$$
$67$ $$T^{6} + 12 T^{4} - 16 T^{3} + 144 T^{2} + \cdots + 64$$
$71$ $$(T^{3} + 9 T^{2} - 90 T - 801)^{2}$$
$73$ $$T^{6} + 6 T^{5} + 27 T^{4} + 52 T^{3} + \cdots + 1$$
$79$ $$T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369$$
$83$ $$(T^{3} + 15 T^{2} + 18 T - 267)^{2}$$
$89$ $$T^{6} - 15 T^{5} + 153 T^{4} + \cdots + 12321$$
$97$ $$(T^{3} - 45 T^{2} + 672 T - 3329)^{2}$$