# Properties

 Label 38.2.e.a Level $38$ Weight $2$ Character orbit 38.e Analytic conductor $0.303$ 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] = [38,2,Mod(5,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, 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("38.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 38.e (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: $$0.303431527681$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$\zeta_{18}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.939693 − 0.342020i −0.173648 − 0.984808i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 + 0.642788i −0.766044 − 0.642788i
−0.939693 + 0.342020i −0.326352 + 1.85083i 0.766044 0.642788i 1.53209 + 1.28558i −0.326352 1.85083i −2.53209 4.38571i −0.500000 + 0.866025i −0.500000 0.181985i −1.87939 0.684040i
9.1 0.173648 + 0.984808i 0.266044 0.223238i −0.939693 + 0.342020i −1.87939 0.684040i 0.266044 + 0.223238i 0.879385 1.52314i −0.500000 0.866025i −0.500000 + 2.83564i 0.347296 1.96962i
17.1 0.173648 0.984808i 0.266044 + 0.223238i −0.939693 0.342020i −1.87939 + 0.684040i 0.266044 0.223238i 0.879385 + 1.52314i −0.500000 + 0.866025i −0.500000 2.83564i 0.347296 + 1.96962i
23.1 −0.939693 0.342020i −0.326352 1.85083i 0.766044 + 0.642788i 1.53209 1.28558i −0.326352 + 1.85083i −2.53209 + 4.38571i −0.500000 0.866025i −0.500000 + 0.181985i −1.87939 + 0.684040i
25.1 0.766044 0.642788i −1.43969 0.524005i 0.173648 0.984808i 0.347296 + 1.96962i −1.43969 + 0.524005i −1.34730 + 2.33359i −0.500000 0.866025i −0.500000 0.419550i 1.53209 + 1.28558i
35.1 0.766044 + 0.642788i −1.43969 + 0.524005i 0.173648 + 0.984808i 0.347296 1.96962i −1.43969 0.524005i −1.34730 2.33359i −0.500000 + 0.866025i −0.500000 + 0.419550i 1.53209 1.28558i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.1 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 38.2.e.a 6
3.b odd 2 1 342.2.u.c 6
4.b odd 2 1 304.2.u.c 6
5.b even 2 1 950.2.l.d 6
5.c odd 4 2 950.2.u.b 12
19.b odd 2 1 722.2.e.k 6
19.c even 3 1 722.2.e.b 6
19.c even 3 1 722.2.e.m 6
19.d odd 6 1 722.2.e.a 6
19.d odd 6 1 722.2.e.l 6
19.e even 9 1 inner 38.2.e.a 6
19.e even 9 1 722.2.a.l 3
19.e even 9 2 722.2.c.k 6
19.e even 9 1 722.2.e.b 6
19.e even 9 1 722.2.e.m 6
19.f odd 18 1 722.2.a.k 3
19.f odd 18 2 722.2.c.l 6
19.f odd 18 1 722.2.e.a 6
19.f odd 18 1 722.2.e.k 6
19.f odd 18 1 722.2.e.l 6
57.j even 18 1 6498.2.a.bq 3
57.l odd 18 1 342.2.u.c 6
57.l odd 18 1 6498.2.a.bl 3
76.k even 18 1 5776.2.a.bo 3
76.l odd 18 1 304.2.u.c 6
76.l odd 18 1 5776.2.a.bn 3
95.p even 18 1 950.2.l.d 6
95.q odd 36 2 950.2.u.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.e.a 6 1.a even 1 1 trivial
38.2.e.a 6 19.e even 9 1 inner
304.2.u.c 6 4.b odd 2 1
304.2.u.c 6 76.l odd 18 1
342.2.u.c 6 3.b odd 2 1
342.2.u.c 6 57.l odd 18 1
722.2.a.k 3 19.f odd 18 1
722.2.a.l 3 19.e even 9 1
722.2.c.k 6 19.e even 9 2
722.2.c.l 6 19.f odd 18 2
722.2.e.a 6 19.d odd 6 1
722.2.e.a 6 19.f odd 18 1
722.2.e.b 6 19.c even 3 1
722.2.e.b 6 19.e even 9 1
722.2.e.k 6 19.b odd 2 1
722.2.e.k 6 19.f odd 18 1
722.2.e.l 6 19.d odd 6 1
722.2.e.l 6 19.f odd 18 1
722.2.e.m 6 19.c even 3 1
722.2.e.m 6 19.e even 9 1
950.2.l.d 6 5.b even 2 1
950.2.l.d 6 95.p even 18 1
950.2.u.b 12 5.c odd 4 2
950.2.u.b 12 95.q odd 36 2
5776.2.a.bn 3 76.l odd 18 1
5776.2.a.bo 3 76.k even 18 1
6498.2.a.bl 3 57.l odd 18 1
6498.2.a.bq 3 57.j even 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$5$ $$T^{6} + 8T^{3} + 64$$
$7$ $$T^{6} + 6 T^{5} + 36 T^{4} + 48 T^{3} + \cdots + 576$$
$11$ $$T^{6} + 6 T^{5} + 33 T^{4} + 56 T^{3} + \cdots + 361$$
$13$ $$T^{6} - 12 T^{5} + 48 T^{4} - 64 T^{3} + \cdots + 64$$
$17$ $$T^{6} + 12 T^{5} + 108 T^{4} + \cdots + 12321$$
$19$ $$T^{6} - 18 T^{5} + 162 T^{4} + \cdots + 6859$$
$23$ $$T^{6} + 12 T^{5} + 60 T^{4} + 152 T^{3} + \cdots + 64$$
$29$ $$T^{6} + 18 T^{5} + 144 T^{4} + \cdots + 23104$$
$31$ $$T^{6} - 6 T^{5} + 60 T^{4} + 160 T^{3} + \cdots + 64$$
$37$ $$(T^{3} + 6 T^{2} - 24 T - 136)^{2}$$
$41$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$
$43$ $$T^{6} + 6 T^{5} + 42 T^{4} + 271 T^{3} + \cdots + 289$$
$47$ $$T^{6} - 30 T^{5} + 372 T^{4} + \cdots + 87616$$
$53$ $$T^{6} - 24 T^{5} + 276 T^{4} + \cdots + 18496$$
$59$ $$T^{6} + 3 T^{5} + 54 T^{4} + 24 T^{3} + \cdots + 9$$
$61$ $$T^{6} - 6 T^{5} - 12 T^{4} + \cdots + 23104$$
$67$ $$T^{6} + 9 T^{5} + 162 T^{4} + \cdots + 6561$$
$71$ $$T^{6} + 18 T^{5} + 216 T^{4} + \cdots + 23104$$
$73$ $$T^{6} + 30 T^{5} + 279 T^{4} + \cdots + 3249$$
$79$ $$T^{6} - 6 T^{5} - 48 T^{4} + \cdots + 18496$$
$83$ $$T^{6} + 6 T^{5} + 63 T^{4} + \cdots + 2601$$
$89$ $$T^{6} + 36 T^{4} + 315 T^{3} + \cdots + 962361$$
$97$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$