# Properties

 Label 114.2.i.d 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{18} q^{2} - \zeta_{18}^{4} q^{3} + \zeta_{18}^{2} q^{4} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{5} - \zeta_{18}^{5} q^{6} + (\zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3} - \zeta_{18}^{2} + 1) q^{7} + \zeta_{18}^{3} q^{8} + (\zeta_{18}^{5} - \zeta_{18}^{2}) q^{9} +O(q^{10})$$ 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 - z^3 - z^2 + 1) * 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} - \zeta_{18}^{3} - \zeta_{18}^{2} + 1) 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} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{11} + ( - \zeta_{18}^{3} + 1) q^{12} + (2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{13} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} - 1) q^{14} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{3} + \zeta_{18} - 2) q^{15} + \zeta_{18}^{4} q^{16} + ( - 3 \zeta_{18}^{5} + 3 \zeta_{18}^{3} + \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{17} - q^{18} + (2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - \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} - \zeta_{18}) q^{21} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{22} + ( - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 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}^{3} - \zeta_{18} - 2) q^{26} + \zeta_{18}^{3} q^{27} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + \zeta_{18}^{2} - \zeta_{18} - 1) q^{28} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{29} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - 2 \zeta_{18}) q^{30} + ( - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3 \zeta_{18} + 3) q^{31} + \zeta_{18}^{5} q^{32} + (\zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 1) q^{33} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} + 3) q^{34} + (2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{2} + 4) q^{35} - \zeta_{18} q^{36} + ( - 2 \zeta_{18}^{5} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{2} + 6 \zeta_{18} - 3) q^{37} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{38} + ( - \zeta_{18}^{5} + \zeta_{18}^{2} + \zeta_{18} + 2) q^{39} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{40} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{41} + (\zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{42} + (\zeta_{18}^{5} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 5 \zeta_{18} - 1) q^{43} + ( - \zeta_{18}^{4} - \zeta_{18}^{3} + 1) q^{44} + (\zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{45} + ( - 2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18}) q^{46} + (\zeta_{18}^{5} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - \zeta_{18}^{2}) q^{47} + ( - \zeta_{18}^{5} + \zeta_{18}^{2}) q^{48} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} + 4 \zeta_{18}^{3} - \zeta_{18}^{2} + 3 \zeta_{18}) q^{49} + (3 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 1) q^{50} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{51} + (\zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2 \zeta_{18} - 1) q^{52} + ( - 2 \zeta_{18}^{4} + 8 \zeta_{18}^{3} + \zeta_{18}^{2} + 8 \zeta_{18} - 2) q^{53} + \zeta_{18}^{4} q^{54} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - 2) q^{55} + (\zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 1) q^{56} + ( - \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{57} + (3 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{2} + 2 \zeta_{18} + 2) q^{58} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 4 \zeta_{18} + 6) q^{59} + (\zeta_{18}^{5} - \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{60} + (\zeta_{18}^{4} - \zeta_{18}^{3} - 11 \zeta_{18}^{2} - \zeta_{18} + 1) q^{61} + ( - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 3 \zeta_{18}) q^{62} + (\zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}) q^{63} + (\zeta_{18}^{3} - 1) q^{64} + (3 \zeta_{18}^{5} + 3 \zeta_{18}) q^{65} + (\zeta_{18}^{4} - \zeta_{18} - 1) q^{66} + ( - 4 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - \zeta_{18}^{3} + 4 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{67} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 3 \zeta_{18}) q^{68} + (5 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{69} + (3 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 4 \zeta_{18} - 2) q^{70} + ( - 7 \zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 5 \zeta_{18} + 7) q^{71} - \zeta_{18}^{2} q^{72} + (5 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 7) q^{73} + ( - 4 \zeta_{18}^{5} + 4 \zeta_{18}^{3} + 6 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{74} + ( - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{2} + 3 \zeta_{18} - 1) q^{75} + ( - 2 \zeta_{18}^{5} + 2 \zeta_{18}^{4} + \zeta_{18}^{3} - 2 \zeta_{18}^{2} - 2 \zeta_{18} - 2) q^{76} + q^{77} + (\zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{78} + ( - 10 \zeta_{18}^{5} - 5 \zeta_{18}^{4} - 5 \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} - \zeta_{18} + 4) q^{82} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 9 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} - 9) q^{83} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}) q^{84} + ( - 9 \zeta_{18}^{5} - \zeta_{18}^{4} + 8 \zeta_{18}^{3} + 9 \zeta_{18}^{2} - 7 \zeta_{18} - 7) q^{85} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} - 1) q^{86} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 5 \zeta_{18}) q^{87} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}) q^{88} + ( - 5 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} + 3 \zeta_{18} - 7) q^{89} + (\zeta_{18}^{5} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{90} + ( - 2 \zeta_{18}^{4} + \zeta_{18}^{2} - 2) q^{91} + ( - 3 \zeta_{18}^{5} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{92} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 3) q^{93} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 1) q^{94} + (10 \zeta_{18}^{5} + \zeta_{18}^{4} - 6 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18} + 3) q^{95} + q^{96} + ( - \zeta_{18}^{2} - 4 \zeta_{18} - 1) q^{97} + ( - \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2} - 3) q^{98} + (\zeta_{18}^{3} + \zeta_{18}) 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 - z^3 - z^2 + 1) * q^7 + z^3 * q^8 + (z^5 - z^2) * q^9 + (z^4 - 2*z^3 + z + 1) * q^10 + (-z^4 - z^2) * q^11 + (-z^3 + 1) * q^12 + (2*z^5 + z^4 + z^3 - 1) * q^13 + (z^5 - z^4 + z - 1) * q^14 + (-2*z^4 + z^3 + z - 2) * q^15 + z^4 * q^16 + (-3*z^5 + 3*z^3 + z^2 - 2*z - 2) * q^17 - q^18 + (2*z^5 + 2*z^4 - 2*z^3 - z - 2) * q^19 + (z^5 - 2*z^4 + z^2 + z) * q^20 + (-z^5 + z^3 + z^2 - z) * q^21 + (-z^5 - z^3) * q^22 + (-2*z^4 - 3*z^3 + 2*z^2 - 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^3 - z - 2) * q^26 + z^3 * q^27 + (-z^5 + z^3 + z^2 - z - 1) * q^28 + (-2*z^5 + 3*z^4 - 5*z^3 + 2*z^2 + 2*z + 2) * q^29 + (-2*z^5 + z^4 + z^2 - 2*z) * q^30 + (-3*z^3 + 3*z^2 - 3*z + 3) * q^31 + z^5 * q^32 + (z^5 + z^3 - z^2 - 1) * q^33 + (3*z^4 - 2*z^3 - 2*z^2 - 2*z + 3) * q^34 + (2*z^5 + 3*z^4 - 2*z^3 - 4*z^2 + 4) * q^35 - z * q^36 + (-2*z^5 - 4*z^4 + 6*z^2 + 6*z - 3) * q^37 + (2*z^5 - 2*z^4 + 2*z^3 - z^2 - 2*z - 2) * q^38 + (-z^5 + z^2 + z + 2) * q^39 + (-2*z^5 + 2*z^3 + z^2 - 1) * q^40 + (-4*z^5 - 3*z^4 - 3*z^3 + z^2 - 1) * q^41 + (z^4 - z^2 + 1) * q^42 + (z^5 - 5*z^4 + 3*z^3 - 3*z^2 + 5*z - 1) * q^43 + (-z^4 - z^3 + 1) * q^44 + (z^5 + z^4 - 2*z^2 + z) * q^45 + (-2*z^5 - 3*z^4 + 2*z^3 - 3*z^2 - 2*z) * q^46 + (z^5 + 3*z^4 - 3*z^3 - z^2) * q^47 + (-z^5 + z^2) * q^48 + (3*z^5 - z^4 + 4*z^3 - z^2 + 3*z) * q^49 + (3*z^5 + 3*z^4 - z^3 - 3*z^2 + 1) * q^50 + (2*z^5 - z^4 - z^3 + 3*z - 2) * q^51 + (z^5 + 2*z^4 + z^3 - z^2 - 2*z - 1) * q^52 + (-2*z^4 + 8*z^3 + z^2 + 8*z - 2) * q^53 + z^4 * q^54 + (-z^5 + z^3 - z^2 - 2) * q^55 + (z^4 - z^2 - z + 1) * q^56 + (-z^5 + 4*z^4 + 2*z^2 - 2*z + 2) * q^57 + (3*z^5 - 5*z^4 + 2*z^2 + 2*z + 2) * q^58 + (3*z^5 - 3*z^3 + 3*z^2 - 4*z + 6) * q^59 + (z^5 - z^3 - 2*z^2 + 2) * q^60 + (z^4 - z^3 - 11*z^2 - z + 1) * q^61 + (-3*z^4 + 3*z^3 - 3*z^2 + 3*z) * q^62 + (z^5 - z^4 - z^3 + z) * q^63 + (z^3 - 1) * q^64 + (3*z^5 + 3*z) * q^65 + (z^4 - z - 1) * q^66 + (-4*z^5 - 3*z^4 - z^3 + 4*z^2 + 4*z + 4) * q^67 + (3*z^5 - 2*z^4 - 2*z^3 - 2*z^2 + 3*z) * q^68 + (5*z^5 + 5*z^4 - 2*z^3 - 2*z^2 - 3*z + 2) * q^69 + (3*z^5 - 2*z^4 - 2*z^3 + 4*z - 2) * q^70 + (-7*z^5 + 5*z^4 - z^3 + z^2 - 5*z + 7) * q^71 - z^2 * q^72 + (5*z^5 + z^4 - 2*z^3 - 7*z^2 + 7) * q^73 + (-4*z^5 + 4*z^3 + 6*z^2 - 3*z + 2) * q^74 + (-3*z^4 + 3*z^2 + 3*z - 1) * q^75 + (-2*z^5 + 2*z^4 + z^3 - 2*z^2 - 2*z - 2) * q^76 + q^77 + (z^2 + 2*z + 1) * q^78 + (-10*z^5 - 5*z^4 - 5*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 - z + 4) * q^82 + (z^5 + z^4 + 9*z^3 + 2*z^2 - 3*z - 9) * q^83 + (z^5 - z^3 + z) * q^84 + (-9*z^5 - z^4 + 8*z^3 + 9*z^2 - 7*z - 7) * q^85 + (-5*z^5 + 3*z^4 - 2*z^3 + 5*z^2 - z - 1) * q^86 + (-5*z^5 + 3*z^4 - 2*z^3 + 3*z^2 - 5*z) * q^87 + (-z^5 - z^4 + z) * q^88 + (-5*z^5 + 4*z^4 + 4*z^3 + 3*z - 7) * q^89 + (z^5 - z^3 + z^2 - 1) * q^90 + (-2*z^4 + z^2 - 2) * q^91 + (-3*z^5 + 2*z^4 - 5*z^3 - 2*z^2 + 2) * q^92 + (3*z^5 - 3*z^3 - 3*z + 3) * q^93 + (3*z^5 - 3*z^4 - 1) * q^94 + (10*z^5 + z^4 - 6*z^3 - 2*z^2 + z + 3) * q^95 + q^96 + (-z^2 - 4*z - 1) * q^97 + (-z^5 + 4*z^4 + 2*z^3 + 3*z^2 - 3) * q^98 + (z^3 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 9 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10})$$ 6 * q + 9 * q^5 + 3 * q^7 + 3 * q^8 $$6 q + 9 q^{5} + 3 q^{7} + 3 q^{8} + 3 q^{12} - 3 q^{13} - 6 q^{14} - 9 q^{15} - 3 q^{17} - 6 q^{18} - 18 q^{19} + 3 q^{21} - 3 q^{22} - 21 q^{23} + 9 q^{25} - 6 q^{26} + 3 q^{27} - 3 q^{28} - 3 q^{29} + 9 q^{31} - 3 q^{33} + 12 q^{34} + 18 q^{35} - 18 q^{37} - 6 q^{38} + 12 q^{39} - 15 q^{41} + 6 q^{42} + 3 q^{43} + 3 q^{44} + 6 q^{46} - 9 q^{47} + 12 q^{49} + 3 q^{50} - 15 q^{51} - 3 q^{52} + 12 q^{53} - 9 q^{55} + 6 q^{56} + 12 q^{57} + 12 q^{58} + 27 q^{59} + 9 q^{60} + 3 q^{61} + 9 q^{62} - 3 q^{63} - 3 q^{64} - 6 q^{66} + 21 q^{67} - 6 q^{68} + 6 q^{69} - 18 q^{70} + 39 q^{71} + 36 q^{73} + 24 q^{74} - 6 q^{75} - 9 q^{76} + 6 q^{77} + 6 q^{78} - 45 q^{79} + 9 q^{80} + 15 q^{82} - 27 q^{83} - 3 q^{84} - 18 q^{85} - 12 q^{86} - 6 q^{87} - 30 q^{89} - 9 q^{90} - 12 q^{91} - 3 q^{92} + 9 q^{93} - 6 q^{94} + 6 q^{96} - 6 q^{97} - 12 q^{98} + 3 q^{99}+O(q^{100})$$ 6 * q + 9 * q^5 + 3 * q^7 + 3 * q^8 + 3 * q^12 - 3 * q^13 - 6 * q^14 - 9 * q^15 - 3 * q^17 - 6 * q^18 - 18 * q^19 + 3 * q^21 - 3 * q^22 - 21 * q^23 + 9 * q^25 - 6 * q^26 + 3 * q^27 - 3 * q^28 - 3 * q^29 + 9 * q^31 - 3 * q^33 + 12 * q^34 + 18 * q^35 - 18 * q^37 - 6 * q^38 + 12 * q^39 - 15 * q^41 + 6 * q^42 + 3 * q^43 + 3 * q^44 + 6 * q^46 - 9 * q^47 + 12 * q^49 + 3 * q^50 - 15 * q^51 - 3 * q^52 + 12 * q^53 - 9 * q^55 + 6 * q^56 + 12 * q^57 + 12 * q^58 + 27 * q^59 + 9 * q^60 + 3 * q^61 + 9 * q^62 - 3 * q^63 - 3 * q^64 - 6 * q^66 + 21 * q^67 - 6 * q^68 + 6 * q^69 - 18 * q^70 + 39 * q^71 + 36 * q^73 + 24 * q^74 - 6 * q^75 - 9 * q^76 + 6 * q^77 + 6 * q^78 - 45 * q^79 + 9 * q^80 + 15 * q^82 - 27 * q^83 - 3 * q^84 - 18 * q^85 - 12 * q^86 - 6 * q^87 - 30 * q^89 - 9 * q^90 - 12 * q^91 - 3 * q^92 + 9 * q^93 - 6 * q^94 + 6 * q^96 - 6 * q^97 - 12 * q^98 + 3 * 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 0.326352 0.565258i 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 −0.266044 0.460802i 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 1.43969 + 2.49362i 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 −0.266044 + 0.460802i 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 0.326352 + 0.565258i 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 1.43969 2.49362i 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.d 6
3.b odd 2 1 342.2.u.a 6
4.b odd 2 1 912.2.bo.f 6
19.e even 9 1 inner 114.2.i.d 6
19.e even 9 1 2166.2.a.o 3
19.f odd 18 1 2166.2.a.u 3
57.j even 18 1 6498.2.a.bn 3
57.l odd 18 1 342.2.u.a 6
57.l odd 18 1 6498.2.a.bs 3
76.l odd 18 1 912.2.bo.f 6

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

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6} - T^{3} + 1$$
$5$ $$T^{6} - 9 T^{5} + 36 T^{4} - 90 T^{3} + \cdots + 81$$
$7$ $$T^{6} - 3 T^{5} + 9 T^{4} - 2 T^{3} + \cdots + 1$$
$11$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$13$ $$T^{6} + 3 T^{5} + 18 T^{4} + 24 T^{3} + \cdots + 9$$
$17$ $$T^{6} + 3 T^{5} + 48 T^{4} + 244 T^{3} + \cdots + 289$$
$19$ $$T^{6} + 18 T^{5} + 162 T^{4} + \cdots + 6859$$
$23$ $$T^{6} + 21 T^{5} + 210 T^{4} + \cdots + 72361$$
$29$ $$T^{6} + 3 T^{5} + 18 T^{4} + \cdots + 45369$$
$31$ $$T^{6} - 9 T^{5} + 81 T^{4} - 54 T^{3} + \cdots + 729$$
$37$ $$(T^{3} + 9 T^{2} - 57 T - 361)^{2}$$
$41$ $$T^{6} + 15 T^{5} + 177 T^{4} + \cdots + 11881$$
$43$ $$T^{6} - 3 T^{5} + 99 T^{4} + \cdots + 3249$$
$47$ $$T^{6} + 9 T^{5} + 63 T^{4} + \cdots + 2809$$
$53$ $$T^{6} - 12 T^{5} + 174 T^{4} + \cdots + 94249$$
$59$ $$T^{6} - 27 T^{5} + 324 T^{4} + \cdots + 289$$
$61$ $$T^{6} - 3 T^{5} - 60 T^{4} + \cdots + 1682209$$
$67$ $$T^{6} - 21 T^{5} + 126 T^{4} - 24 T^{3} + \cdots + 9$$
$71$ $$T^{6} - 39 T^{5} + 561 T^{4} + \cdots + 201601$$
$73$ $$T^{6} - 36 T^{5} + 558 T^{4} + \cdots + 1369$$
$79$ $$T^{6} + 45 T^{5} + 1125 T^{4} + \cdots + 4515625$$
$83$ $$T^{6} + 27 T^{5} + 507 T^{4} + \cdots + 253009$$
$89$ $$T^{6} + 30 T^{5} + 246 T^{4} + \cdots + 11449$$
$97$ $$T^{6} + 6 T^{5} + 3 T^{4} + 35 T^{3} + \cdots + 2809$$