# Properties

 Label 74.2.f.a Level $74$ Weight $2$ Character orbit 74.f Analytic conductor $0.591$ 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] = [74,2,Mod(7,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("74.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.f (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: $$0.590892974957$$ 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: $$\Z[a_1, a_2]$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −0.173648 + 0.984808i −0.766044 + 0.642788i −0.766044 − 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i 0.939693 − 0.342020i
0.939693 0.342020i −0.326352 0.118782i 0.766044 0.642788i −0.0209445 0.118782i −0.347296 0.233956 + 1.32683i 0.500000 0.866025i −2.20574 1.85083i −0.0603074 0.104455i
9.1 −0.173648 0.984808i 0.266044 1.50881i −0.939693 + 0.342020i −1.79813 1.50881i −1.53209 1.93969 + 1.62760i 0.500000 + 0.866025i 0.613341 + 0.223238i −1.17365 + 2.03282i
33.1 −0.173648 + 0.984808i 0.266044 + 1.50881i −0.939693 0.342020i −1.79813 + 1.50881i −1.53209 1.93969 1.62760i 0.500000 0.866025i 0.613341 0.223238i −1.17365 2.03282i
49.1 −0.766044 + 0.642788i −1.43969 1.20805i 0.173648 0.984808i 3.31908 1.20805i 1.87939 0.826352 0.300767i 0.500000 + 0.866025i 0.0923963 + 0.524005i −1.76604 + 3.05888i
53.1 0.939693 + 0.342020i −0.326352 + 0.118782i 0.766044 + 0.642788i −0.0209445 + 0.118782i −0.347296 0.233956 1.32683i 0.500000 + 0.866025i −2.20574 + 1.85083i −0.0603074 + 0.104455i
71.1 −0.766044 0.642788i −1.43969 + 1.20805i 0.173648 + 0.984808i 3.31908 + 1.20805i 1.87939 0.826352 + 0.300767i 0.500000 0.866025i 0.0923963 0.524005i −1.76604 3.05888i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.f.a 6
3.b odd 2 1 666.2.x.c 6
4.b odd 2 1 592.2.bc.b 6
37.f even 9 1 inner 74.2.f.a 6
37.f even 9 1 2738.2.a.m 3
37.h even 18 1 2738.2.a.p 3
111.p odd 18 1 666.2.x.c 6
148.p odd 18 1 592.2.bc.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.a 6 1.a even 1 1 trivial
74.2.f.a 6 37.f even 9 1 inner
592.2.bc.b 6 4.b odd 2 1
592.2.bc.b 6 148.p odd 18 1
666.2.x.c 6 3.b odd 2 1
666.2.x.c 6 111.p odd 18 1
2738.2.a.m 3 37.f even 9 1
2738.2.a.p 3 37.h even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 3T_{3}^{5} + 6T_{3}^{4} + 8T_{3}^{3} + 12T_{3}^{2} + 6T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$5$ $$T^{6} - 3 T^{5} - 6 T^{4} + 8 T^{3} + \cdots + 1$$
$7$ $$T^{6} - 6 T^{5} + 18 T^{4} - 30 T^{3} + \cdots + 9$$
$11$ $$T^{6} + 9 T^{5} + 57 T^{4} + 182 T^{3} + \cdots + 289$$
$13$ $$T^{6} + 36 T^{4} - 90 T^{3} + 81 T + 81$$
$17$ $$T^{6} - 27T^{3} + 729$$
$19$ $$T^{6} - 9 T^{5} + 63 T^{4} + \cdots + 2809$$
$23$ $$T^{6} + 15 T^{5} + 171 T^{4} + 804 T^{3} + \cdots + 9$$
$29$ $$T^{6} + 63 T^{4} - 342 T^{3} + \cdots + 29241$$
$31$ $$(T^{3} + 9 T^{2} + 24 T + 17)^{2}$$
$37$ $$T^{6} - 9 T^{5} + 54 T^{4} + \cdots + 50653$$
$41$ $$T^{6} - 6 T^{5} + 36 T^{4} + \cdots + 207936$$
$43$ $$(T^{3} - 6 T^{2} - 81 T + 467)^{2}$$
$47$ $$T^{6} + 3 T^{5} + 54 T^{4} - 169 T^{3} + \cdots + 289$$
$53$ $$T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 81$$
$59$ $$T^{6} + 6 T^{5} + 120 T^{4} + \cdots + 687241$$
$61$ $$T^{6} + 12 T^{5} + 48 T^{4} + \cdots + 4096$$
$67$ $$T^{6} + 3 T^{5} - 36 T^{4} + \cdots + 103041$$
$71$ $$T^{6} + 6 T^{5} - 144 T^{4} + \cdots + 207936$$
$73$ $$(T^{3} + 18 T^{2} + 87 T + 53)^{2}$$
$79$ $$T^{6} - 30 T^{5} + 360 T^{4} + \cdots + 45369$$
$83$ $$T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 32041$$
$89$ $$T^{6} + 33 T^{5} + 516 T^{4} + \cdots + 687241$$
$97$ $$T^{6} - 42 T^{5} + 1188 T^{4} + \cdots + 6594624$$