# Properties

 Label 114.2.i.a Level $114$ Weight $2$ Character orbit 114.i Analytic conductor $0.910$ 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] = [114,2,Mod(25,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

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

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

chi := DirichletCharacter("114.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.i (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: $$0.910294583043$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i 0.939693 + 0.342020i −0.766044 − 0.642788i −0.173648 − 0.984808i
−0.766044 + 0.642788i −0.939693 0.342020i 0.173648 0.984808i −0.386659 2.19285i 0.939693 0.342020i 1.32635 2.29731i 0.500000 + 0.866025i 0.766044 + 0.642788i 1.70574 + 1.43128i
43.1 0.939693 0.342020i 0.173648 0.984808i 0.766044 0.642788i −0.907604 0.761570i −0.173648 0.984808i 0.733956 + 1.27125i 0.500000 0.866025i −0.939693 0.342020i −1.11334 0.405223i
55.1 −0.173648 + 0.984808i 0.766044 + 0.642788i −0.939693 0.342020i −3.20574 + 1.16679i −0.766044 + 0.642788i 2.43969 + 4.22567i 0.500000 0.866025i 0.173648 + 0.984808i −0.592396 3.35965i
61.1 0.939693 + 0.342020i 0.173648 + 0.984808i 0.766044 + 0.642788i −0.907604 + 0.761570i −0.173648 + 0.984808i 0.733956 1.27125i 0.500000 + 0.866025i −0.939693 + 0.342020i −1.11334 + 0.405223i
73.1 −0.766044 0.642788i −0.939693 + 0.342020i 0.173648 + 0.984808i −0.386659 + 2.19285i 0.939693 + 0.342020i 1.32635 + 2.29731i 0.500000 0.866025i 0.766044 0.642788i 1.70574 1.43128i
85.1 −0.173648 0.984808i 0.766044 0.642788i −0.939693 + 0.342020i −3.20574 1.16679i −0.766044 0.642788i 2.43969 4.22567i 0.500000 + 0.866025i 0.173648 0.984808i −0.592396 + 3.35965i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 25.1 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 114.2.i.a 6
3.b odd 2 1 342.2.u.e 6
4.b odd 2 1 912.2.bo.a 6
19.e even 9 1 inner 114.2.i.a 6
19.e even 9 1 2166.2.a.q 3
19.f odd 18 1 2166.2.a.s 3
57.j even 18 1 6498.2.a.bm 3
57.l odd 18 1 342.2.u.e 6
57.l odd 18 1 6498.2.a.br 3
76.l odd 18 1 912.2.bo.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.a 6 1.a even 1 1 trivial
114.2.i.a 6 19.e even 9 1 inner
342.2.u.e 6 3.b odd 2 1
342.2.u.e 6 57.l odd 18 1
912.2.bo.a 6 4.b odd 2 1
912.2.bo.a 6 76.l odd 18 1
2166.2.a.q 3 19.e even 9 1
2166.2.a.s 3 19.f odd 18 1
6498.2.a.bm 3 57.j even 18 1
6498.2.a.br 3 57.l odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 9T_{5}^{5} + 36T_{5}^{4} + 90T_{5}^{3} + 162T_{5}^{2} + 162T_{5} + 81$$ acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6} + T^{3} + 1$$
$5$ $$T^{6} + 9 T^{5} + 36 T^{4} + 90 T^{3} + \cdots + 81$$
$7$ $$T^{6} - 9 T^{5} + 57 T^{4} - 178 T^{3} + \cdots + 361$$
$11$ $$T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81$$
$13$ $$T^{6} - 15 T^{5} + 96 T^{4} + \cdots + 1369$$
$17$ $$T^{6} + 9 T^{5} + 36 T^{4} + 72 T^{3} + \cdots + 81$$
$19$ $$T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 6859$$
$23$ $$T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81$$
$29$ $$T^{6} + 9 T^{5} - 18 T^{4} + \cdots + 29241$$
$31$ $$T^{6} + 9 T^{5} + 147 T^{4} + \cdots + 292681$$
$37$ $$(T^{3} - 9 T^{2} + 15 T + 1)^{2}$$
$41$ $$T^{6} - 27 T^{5} + 405 T^{4} + \cdots + 210681$$
$43$ $$T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 32041$$
$47$ $$T^{6} - 27 T^{5} + 333 T^{4} + \cdots + 23409$$
$53$ $$T^{6} + 90 T^{4} + 315 T^{3} + \cdots + 81$$
$59$ $$T^{6} - 9 T^{5} + 162 T^{4} + \cdots + 6561$$
$61$ $$T^{6} + 3 T^{5} - 12 T^{4} + \cdots + 94249$$
$67$ $$T^{6} + 3 T^{5} - 66 T^{4} + \cdots + 2809$$
$71$ $$T^{6} - 9 T^{5} - 27 T^{4} + \cdots + 263169$$
$73$ $$T^{6} + 12 T^{5} + 78 T^{4} + \cdots + 54289$$
$79$ $$T^{6} + 21 T^{5} + 231 T^{4} + \cdots + 2809$$
$83$ $$T^{6} + 9 T^{5} + 225 T^{4} + \cdots + 408321$$
$89$ $$T^{6} + 90 T^{4} + 900 T^{3} + \cdots + 431649$$
$97$ $$T^{6} + 54 T^{5} + 1323 T^{4} + \cdots + 2679769$$