# Properties

 Label 378.2.u.a Level $378$ Weight $2$ Character orbit 378.u Analytic conductor $3.018$ 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] = [378,2,Mod(43,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("378.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.u (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: $$3.01834519640$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$\zeta_{18}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.766044 − 0.642788i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 + 0.984808i −0.173648 − 0.984808i
0.766044 + 0.642788i 1.11334 1.32683i 0.173648 + 0.984808i 2.37939 + 0.866025i 1.70574 0.300767i 0.173648 0.984808i −0.500000 + 0.866025i −0.520945 2.95442i 1.26604 + 2.19285i
85.1 −0.939693 0.342020i 0.592396 1.62760i 0.766044 + 0.642788i 0.152704 0.866025i −1.11334 + 1.32683i 0.766044 0.642788i −0.500000 0.866025i −2.29813 1.92836i −0.439693 + 0.761570i
169.1 −0.939693 + 0.342020i 0.592396 + 1.62760i 0.766044 0.642788i 0.152704 + 0.866025i −1.11334 1.32683i 0.766044 + 0.642788i −0.500000 + 0.866025i −2.29813 + 1.92836i −0.439693 0.761570i
211.1 0.766044 0.642788i 1.11334 + 1.32683i 0.173648 0.984808i 2.37939 0.866025i 1.70574 + 0.300767i 0.173648 + 0.984808i −0.500000 0.866025i −0.520945 + 2.95442i 1.26604 2.19285i
295.1 0.173648 0.984808i −1.70574 0.300767i −0.939693 0.342020i −1.03209 + 0.866025i −0.592396 + 1.62760i −0.939693 + 0.342020i −0.500000 + 0.866025i 2.81908 + 1.02606i 0.673648 + 1.16679i
337.1 0.173648 + 0.984808i −1.70574 + 0.300767i −0.939693 + 0.342020i −1.03209 0.866025i −0.592396 1.62760i −0.939693 0.342020i −0.500000 0.866025i 2.81908 1.02606i 0.673648 1.16679i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.u.a 6
27.e even 9 1 inner 378.2.u.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.u.a 6 1.a even 1 1 trivial
378.2.u.a 6 27.e even 9 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 3T_{5}^{5} + 3T_{5}^{3} + 9T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{3} + 1$$
$3$ $$T^{6} + 9T^{3} + 27$$
$5$ $$T^{6} - 3 T^{5} + 3 T^{3} + 9 T^{2} + \cdots + 9$$
$7$ $$T^{6} + T^{3} + 1$$
$11$ $$T^{6} - 6 T^{5} + 18 T^{4} - 3 T^{3} + \cdots + 9$$
$13$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$17$ $$T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81$$
$19$ $$T^{6} - 3 T^{5} + 33 T^{4} + \cdots + 2809$$
$23$ $$T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 3249$$
$29$ $$T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 3249$$
$31$ $$T^{6} + 3 T^{5} + 24 T^{4} + 17 T^{3} + \cdots + 361$$
$37$ $$T^{6} - 12 T^{5} + 132 T^{4} + \cdots + 64$$
$41$ $$T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 36864$$
$43$ $$T^{6} + 6 T^{5} - 6 T^{4} - 361 T^{3} + \cdots + 5329$$
$47$ $$T^{6} - 24 T^{5} + 306 T^{4} + \cdots + 47961$$
$53$ $$(T^{3} - 117 T + 153)^{2}$$
$59$ $$T^{6} + 24 T^{5} + 252 T^{4} + \cdots + 2601$$
$61$ $$T^{6} + 3 T^{5} + 114 T^{4} + \cdots + 16129$$
$67$ $$T^{6} - 3 T^{5} - 96 T^{4} + \cdots + 94249$$
$71$ $$T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 106929$$
$73$ $$T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 1369$$
$79$ $$T^{6} - 33 T^{5} + 492 T^{4} + \cdots + 73441$$
$83$ $$T^{6} + 144 T^{4} - 720 T^{3} + \cdots + 5184$$
$89$ $$T^{6} - 21 T^{5} + 405 T^{4} + \cdots + 12321$$
$97$ $$T^{6} + 15 T^{5} + 120 T^{4} + \cdots + 130321$$