# Properties

 Label 209.2.j.a Level $209$ Weight $2$ Character orbit 209.j Analytic conductor $1.669$ 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] = [209,2,Mod(23,209)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(209, base_ring=CyclotomicField(18))

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

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

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

chi := DirichletCharacter("209.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$209 = 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 209.j (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.66887340224$$ Analytic rank: $$0$$ Dimension: $$6$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/209\mathbb{Z}\right)^\times$$.

 $$n$$ $$78$$ $$134$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$ $$1$$

## 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
 −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 − 0.342020i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.766044 + 0.642788i
1.26604 + 0.460802i −0.347296 1.96962i −0.141559 0.118782i −0.358441 + 0.300767i 0.467911 2.65366i 1.17365 2.03282i −1.47178 2.54920i −0.939693 + 0.342020i −0.592396 + 0.215615i
100.1 1.26604 0.460802i −0.347296 + 1.96962i −0.141559 + 0.118782i −0.358441 0.300767i 0.467911 + 2.65366i 1.17365 + 2.03282i −1.47178 + 2.54920i −0.939693 0.342020i −0.592396 0.215615i
111.1 0.673648 + 0.565258i 1.87939 0.684040i −0.213011 1.20805i −0.286989 + 1.62760i 1.65270 + 0.601535i 0.0603074 + 0.104455i 1.41875 2.45734i 0.766044 0.642788i −1.11334 + 0.934204i
177.1 0.673648 0.565258i 1.87939 + 0.684040i −0.213011 + 1.20805i −0.286989 1.62760i 1.65270 0.601535i 0.0603074 0.104455i 1.41875 + 2.45734i 0.766044 + 0.642788i −1.11334 0.934204i
188.1 −0.439693 + 2.49362i −1.53209 1.28558i −4.14543 1.50881i 3.64543 1.32683i 3.87939 3.25519i 1.76604 + 3.05888i 3.05303 5.28801i 0.173648 + 0.984808i 1.70574 + 9.67372i
199.1 −0.439693 2.49362i −1.53209 + 1.28558i −4.14543 + 1.50881i 3.64543 + 1.32683i 3.87939 + 3.25519i 1.76604 3.05888i 3.05303 + 5.28801i 0.173648 0.984808i 1.70574 9.67372i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 209.2.j.a 6
19.e even 9 1 inner 209.2.j.a 6
19.e even 9 1 3971.2.a.e 3
19.f odd 18 1 3971.2.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.j.a 6 1.a even 1 1 trivial
209.2.j.a 6 19.e even 9 1 inner
3971.2.a.e 3 19.e even 9 1
3971.2.a.f 3 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 24T_{2}^{3} + 36T_{2}^{2} - 27T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(209, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + 9 T^{4} - 24 T^{3} + \cdots + 9$$
$3$ $$T^{6} - 8T^{3} + 64$$
$5$ $$T^{6} - 6 T^{5} + 9 T^{4} - 3 T^{3} + \cdots + 9$$
$7$ $$T^{6} - 6 T^{5} + 27 T^{4} - 52 T^{3} + \cdots + 1$$
$11$ $$(T^{2} + T + 1)^{3}$$
$13$ $$T^{6} - 12 T^{5} + 75 T^{4} + \cdots + 5041$$
$17$ $$T^{6} + 3 T^{5} - 9 T^{4} + \cdots + 12321$$
$19$ $$T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 9 T^{5} + 36 T^{4} - 153 T^{3} + \cdots + 81$$
$29$ $$T^{6} + 6 T^{5} - 240 T^{3} + \cdots + 576$$
$31$ $$T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 16129$$
$37$ $$(T^{3} + 9 T^{2} + 24 T + 19)^{2}$$
$41$ $$T^{6} + 9 T^{5} + 36 T^{4} + 72 T^{3} + \cdots + 81$$
$43$ $$T^{6} + 12 T^{5} + 159 T^{4} + \cdots + 5329$$
$47$ $$T^{6} - 27 T^{5} + 324 T^{4} + \cdots + 210681$$
$53$ $$T^{6} - 18 T^{5} + 270 T^{4} + \cdots + 263169$$
$59$ $$T^{6} + 6 T^{5} + 126 T^{4} + \cdots + 45369$$
$61$ $$T^{6} + 21 T^{5} + 276 T^{4} + \cdots + 11449$$
$67$ $$T^{6} - 12 T^{5} + 246 T^{4} + \cdots + 128881$$
$71$ $$T^{6} + 42 T^{5} + 693 T^{4} + \cdots + 3249$$
$73$ $$T^{6} + 15 T^{5} + 228 T^{4} + \cdots + 1129969$$
$79$ $$T^{6} + 9 T^{5} + 18 T^{4} - 388 T^{3} + \cdots + 289$$
$83$ $$T^{6} - 6 T^{5} + 135 T^{4} + \cdots + 47961$$
$89$ $$T^{6} - 21 T^{5} + 207 T^{4} + \cdots + 12321$$
$97$ $$T^{6} - 27 T^{5} + 378 T^{4} + \cdots + 157609$$