# Properties

 Label 114.2.l.a Level $114$ Weight $2$ Character orbit 114.l Analytic conductor $0.910$ Analytic rank $0$ Dimension $18$ 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(29,114)]

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([9, 17]))

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

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

chi := DirichletCharacter("114.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.l (of order $$18$$, 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: $$18$$ Relative dimension: $$3$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + 162 x^{7} + 270 x^{6} + 2916 x^{5} - 4374 x^{4} - 729 x^{3} + 19683$$ x^18 - x^15 - 18*x^14 + 36*x^13 + 10*x^12 + 18*x^11 + 90*x^10 - 567*x^9 + 270*x^8 + 162*x^7 + 270*x^6 + 2916*x^5 - 4374*x^4 - 729*x^3 + 19683 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-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 basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} - \beta_{6}) q^{2} + ( - \beta_{17} + \beta_1) q^{3} + (\beta_{8} + \beta_{4}) q^{4} + ( - \beta_{15} + \beta_{14} + \beta_{2}) q^{5} + \beta_{5} q^{6} + (\beta_{16} - \beta_{15} + \beta_{6} + \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{11} + 1) q^{8} + ( - \beta_{16} - 2 \beta_{9} + 2 \beta_{8} + \beta_{4} + \beta_{3}) q^{9}+O(q^{10})$$ q + (-b7 - b6) * q^2 + (-b17 + b1) * q^3 + (b8 + b4) * q^4 + (-b15 + b14 + b2) * q^5 + b5 * q^6 + (b16 - b15 + b6 + b4 - b3) * q^7 + (-b11 + 1) * q^8 + (-b16 - 2*b9 + 2*b8 + b4 + b3) * q^9 $$q + ( - \beta_{7} - \beta_{6}) q^{2} + ( - \beta_{17} + \beta_1) q^{3} + (\beta_{8} + \beta_{4}) q^{4} + ( - \beta_{15} + \beta_{14} + \beta_{2}) q^{5} + \beta_{5} q^{6} + (\beta_{16} - \beta_{15} + \beta_{6} + \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{11} + 1) q^{8} + ( - \beta_{16} - 2 \beta_{9} + 2 \beta_{8} + \beta_{4} + \beta_{3}) q^{9} + ( - \beta_{16} - \beta_1) q^{10} + (\beta_{17} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{4}) q^{11} + \beta_{15} q^{12} + (\beta_{13} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{17} - \beta_{10} - \beta_{8} - \beta_1 + 1) q^{14} + ( - 2 \beta_{14} + \beta_{13} + 3 \beta_{11} - \beta_{7} + \beta_{6} - \beta_{2} - 3) q^{15} - \beta_{6} q^{16} + (\beta_{17} + \beta_{16} - \beta_{12} - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{17} + 2 \beta_{12} - 2 \beta_{11} + \beta_{10} + 1) q^{18} + (\beta_{16} - 2 \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{19}+ \cdots + (\beta_{17} + 2 \beta_{16} - \beta_{14} + \beta_{13} + 3 \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots - 3) q^{99}+O(q^{100})$$ q + (-b7 - b6) * q^2 + (-b17 + b1) * q^3 + (b8 + b4) * q^4 + (-b15 + b14 + b2) * q^5 + b5 * q^6 + (b16 - b15 + b6 + b4 - b3) * q^7 + (-b11 + 1) * q^8 + (-b16 - 2*b9 + 2*b8 + b4 + b3) * q^9 + (-b16 - b1) * q^10 + (b17 + b14 - b13 - b12 - b8 + b7 - b6 - 2*b4) * q^11 + b15 * q^12 + (b13 + b9 - b8 + b7 + b6 + b5 - 2*b4 + b2 + b1 - 1) * q^13 + (b17 - b10 - b8 - b1 + 1) * q^14 + (-2*b14 + b13 + 3*b11 - b7 + b6 - b2 - 3) * q^15 - b6 * q^16 + (b17 + b16 - b12 - b10 + b9 - b7 + b6 - b5 - b3 - b2 - b1) * q^17 + (-b17 + 2*b12 - 2*b11 + b10 + 1) * q^18 + (b16 - 2*b13 - b11 - b10 + b9 - 2*b8 - b5 - b4 - b3 - b2 + 1) * q^19 + (-b13 + b10 - b5) * q^20 + (b15 - b14 + b7 - b6 + b3 + b2) * q^21 + (-b16 + b15 - b14 + b13 + b11 - b9 + b8 - b4 + b3 - 2) * q^22 + (-b17 + b16 + b15 - b14 + b12 + b10 + 2*b9 - b8 - b5 + b4 - b2 - b1) * q^23 + b1 * q^24 + (-b17 - b16 + b15 + b12 - b11 + 2*b10 - b9 + b8 - b7 - 2*b6 + 2*b3 + b1 - 1) * q^25 + (b13 - b12 + b11 + b9 - b8 + b7 + b6 + b5 - b4 - b3 - 2) * q^26 + (b15 - b14 + 4*b7 + 2*b6 - b5 - b3 - 2*b2) * q^27 + (b11 - b7 - b6 - b5 - b2) * q^28 + (-b16 + b15 + b14 + b13 - 2*b12 - b10 + b5 + 2*b3 + b2 + b1) * q^29 + (2*b16 - b15 + b9 - b8 + 3*b6 + b4 - b3) * q^30 + (b17 + b16 + b15 + 2*b14 + b12 + b10 - b8 + b7 - b5 + b3 + b2 - 2*b1) * q^31 + b8 * q^32 + (-b16 - b15 + 2*b13 + b12 - b11 - b10 + b9 - b8 + b5 + b4 - b3 - b1 - 1) * q^33 + (b17 - b15 - b14 - b12 - b10 - b9 - b8 - b5 + b4 - b2) * q^34 + (2*b17 + 2*b12 + b11 + b10 - b9 + b8 - b5 + 2*b4 - b1 + 1) * q^35 + (2*b14 - b13 + b7 - b6 + b2) * q^36 + (b15 - b13 + 2*b12 + b10 - b9 + b8 - b6 + b5 - b2 + b1) * q^37 + (b17 + b15 - b12 + 2*b11 - b10 - b9 - b6 + 2*b3 - b2 - 1) * q^38 + (-2*b16 - b15 + b12 + 2*b11 + 2*b10 - b9 + b8 - 3*b7 - b5 - b4 - 1) * q^39 + (b3 + b2) * q^40 + (-2*b15 - 2*b14 - 2*b11 - b10 - b9 + 2*b8 + b5 + b4 - 2*b3 + b1 + 1) * q^41 + (-b17 + b16 + 2*b9 + b8 - b4 + b1) * q^42 + (-2*b16 + b15 + b14 + b13 - 2*b12 + b11 - 2*b10 - b9 + b8 - 3*b7 + 2*b5 + b4 + b2 - 2*b1 - 1) * q^43 + (-b17 + b16 - b15 + b12 - b11 + b10 + b9 + b7 + 2*b6 - b3 + b1 - 1) * q^44 + (-2*b13 - b12 - 2*b11 + b10 - b5 - 2*b1 + 4) * q^45 + (b16 - b15 + b14 - 2*b13 - 2*b12 + b11 - b10 - b5 + b2 + b1 + 1) * q^46 + (-2*b17 - 2*b16 + b14 - b13 - b11 - 2*b9 - 4*b8 - 2*b7 - b6 - 2*b4 - 1) * q^47 + (b13 + b5) * q^48 + (-b14 + b13 - b12 - 2*b11 - b10 - b7 + b5 - b4 - b2 + b1 + 2) * q^49 + (-2*b17 + b14 + b12 - b11 + b10 + 2*b8 + 2*b7 + b6 + b5 + b4 + 2*b2 + b1) * q^50 + (b17 + b16 + b14 - 2*b13 - 2*b12 - b11 - b10 + 2*b9 + b8 - b7 - 2*b6 + 2*b4 - b3 + 2*b2 - 1) * q^51 + (b17 - b14 - b12 + b11 - b8 + b7 + 2*b6 - b4 - b3 - 1) * q^52 + (-b17 - b15 - b14 + b13 + 2*b12 - 2*b11 + b10 - b9 - 2*b7 - 4*b6 + b5 + b3 + 4) * q^53 + (b17 + b16 - b15 - b9 - 2*b8 - 4*b4 + b1) * q^54 + (-2*b17 + 2*b16 - b15 + b14 + b13 - b12 - b11 - 2*b10 + 3*b9 + b8 + 3*b7 + 3*b6 + 3*b5 + 2*b4 + b1 + 2) * q^55 + (-b15 - b9 + b8 - b7 + b4) * q^56 + (b17 - b15 + b14 - 3*b11 - 3*b10 + 3*b8 - b7 - 2*b6 + b5 + b3 + 2*b2 - b1 + 3) * q^57 + (-2*b17 - b16 - 2*b14 + b13 + b10 + b5 - b3 - b2 + b1) * q^58 + (b15 - b14 + b12 + 2*b11 + b8 + 3*b7 + 3*b6 - b4 - 2*b3 + 2*b2 - b1 - 1) * q^59 + (b17 - b12 + b11 - 2*b10 - 3*b8 - b1 + 1) * q^60 + (b17 - 2*b15 - b14 + b13 + 2*b11 + 2*b10 + b9 + 2*b7 - b6 + b5 - 2*b3 + 1) * q^61 + (-b17 - 2*b16 - b15 + b14 - b13 + b11 - b10 - b9 - 2*b5 - b4 + b2 + b1) * q^62 + (b14 + b13 + 3*b11 - b7 - 2*b6 - b2 + 3) * q^63 - b11 * q^64 + (-b17 + b16 + 2*b15 - 2*b14 + b13 - b12 + 3*b11 - b10 - b9 + 2*b8 - 4*b7 - 2*b6 + b5 - 2*b4 + b3 - 3*b2 + 3*b1 - 3) * q^65 + (b17 - b15 + b14 - b13 - b12 + b11 + b10 + 2*b7 + b6 - b5 - 2*b3 - b2 - b1 + 1) * q^66 + (b17 + b16 - 3*b15 + 3*b14 - 3*b13 - 3*b12 - b11 + b9 + 2*b8 - b7 - b6 - 2*b5 + b4 - 3*b3 + 2*b2) * q^67 + (b16 - b15 - b14 + b13 + b12 + b11 - b2 - b1 + 1) * q^68 + (b17 + 3*b16 - b15 + 2*b14 - b13 - 2*b12 - b11 - b10 - 3*b8 - 5*b7 - 4*b6 + b3 + b2 + 2) * q^69 + (-b15 + 2*b14 + b13 + b12 - b11 - 2*b7 - b6 - b5 + b2 + 2) * q^70 + (-b17 - 3*b16 + b15 - b14 + b12 + 2*b10 + 3*b8 + 3*b5 + 2*b3 + b2 + 2*b1 - 3) * q^71 + (-2*b16 + b15 - b9 + b8 - b4 + b3) * q^72 + (-3*b17 - b16 + b15 - b13 + 3*b12 + 2*b11 + 2*b10 + b9 - 2*b8 + 3*b6 - 2*b5 + b4 - b2 + b1 - 3) * q^73 + (2*b15 + 2*b14 + b13 + b12 - b11 - b9 + b8 + b5 + b3 + b2 + b1) * q^74 + (2*b17 - b16 - 2*b9 - 4*b8 - 2*b4 - 3*b2 - b1) * q^75 + (-2*b17 - b14 + b13 + b12 - b9 + b8 - b7 + b6 - b2 + b1) * q^76 + (-b17 - b15 + b14 - b13 + 2*b12 - 4*b11 + b10 + b9 + 2*b8 + 4*b7 + 2*b6 + b5 + 4*b4 + b2 + 2) * q^77 + (-b15 + b14 + b12 - b11 + 2*b10 - b7 + b6 + 3*b4 + 2*b2 - b1 - 1) * q^78 + (b17 + 2*b15 - 2*b14 + b12 + b11 - b10 - 2*b9 + b8 - 2*b5 + b4 + 2*b3 - 2*b1) * q^79 + (-b17 + b9) * q^80 + (b17 + 3*b14 + b13 - 2*b12 - 4*b11 - b10 + 3*b7 - b5 + 3*b2 + 2) * q^81 + (2*b17 + 2*b16 + b15 + b13 + b12 - 2*b11 + 2*b9 + b7 - b6 + b5 - b2 - 2*b1 + 1) * q^82 + (-3*b17 + b16 + b15 - 2*b14 + 2*b13 - b12 + 2*b11 + b10 - 2*b9 + 3*b8 - 3*b7 - 3*b6 + b5 + 3*b4 + 3*b3 - 3*b2 + 2*b1 - 4) * q^83 + (-2*b12 - b11 - b10 + b5 - 1) * q^84 + (3*b17 - 2*b16 - b15 + b14 + b13 - b10 - 3*b9 + b5 + 3*b4 + b2 - b1) * q^85 + (-b16 + b15 - 2*b14 - 2*b13 + b12 - b11 + 2*b10 + b6 - 2*b5 + 3*b4 - b3 - 2*b2 + b1 + 1) * q^86 + (-b17 - 2*b16 + b15 - b13 + b12 + 2*b11 + 2*b10 - b9 + 4*b8 - 3*b7 - 2*b5 + 5*b4 + b3 - 3*b2 + b1 - 4) * q^87 + (b17 + b14 - b12 - b10 - 2*b8 + 2*b7 + b6 + b5 - b4 + b2 - b1) * q^88 + (b15 - 2*b13 + b12 + b11 - b10 + 2*b9 - 3*b8 + 4*b7 + 5*b6 - 2*b5 - b3 + 2*b2 - 2*b1 + 4) * q^89 + (b15 - b14 - 2*b13 - 2*b7 - 4*b6 - 2*b5 + 2*b3 + b2) * q^90 + (2*b15 - b14 - 2*b13 + b12 - 4*b11 + 2*b10 + 2*b8 + 4*b7 + b6 - b5 - 2*b4 + b3 + b2 + 1) * q^91 + (-b16 + b15 - 2*b14 + b13 - b10 - 2*b7 - b6 + b5 + 2*b3 - b2 - b1) * q^92 + (b17 + 3*b16 - b15 + b14 - 2*b12 + 5*b11 - b10 + 3*b9 - 6*b8 - b7 - 5*b6 - b5 - 3*b4 - b3 - b2 - 7) * q^93 + (-b16 + b15 - 2*b13 + 2*b12 + 4*b11 + 2*b10 - b9 + b8 + 2*b7 + b6 + 2*b4 + b3 - 2) * q^94 + (b17 + b16 - b15 - b14 + 3*b13 + b12 + 4*b11 - 2*b10 + 3*b9 - 4*b8 + 2*b7 + 4*b6 + b4 - b3 - 2) * q^95 - b3 * q^96 + (-b17 + 3*b16 - 2*b15 - 2*b14 + 3*b13 + 2*b12 - 2*b11 + b10 + b8 - 2*b7 + 2*b6 - b4 - b3 - b2 + 2*b1 + 1) * q^97 + (b16 - b14 + b13 - 2*b6 + b5 + b4 - b3 - b2 - 1) * q^98 + (b17 + 2*b16 - b14 + b13 + 3*b10 + b9 + 2*b8 - 2*b7 + 5*b6 + b5 - 2*b4 + b3 + b2 + b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 3 q^{6} + 9 q^{8}+O(q^{10})$$ 18 * q + 3 * q^6 + 9 * q^8 $$18 q + 3 q^{6} + 9 q^{8} - 12 q^{13} + 12 q^{14} - 24 q^{15} - 6 q^{17} - 6 q^{19} - 24 q^{22} - 18 q^{25} - 18 q^{26} - 3 q^{27} + 6 q^{28} + 6 q^{29} - 27 q^{33} - 6 q^{34} + 24 q^{35} - 3 q^{36} - 3 q^{38} + 6 q^{39} - 3 q^{41} - 6 q^{43} - 24 q^{44} + 54 q^{45} + 18 q^{46} - 30 q^{47} + 6 q^{48} + 21 q^{49} - 3 q^{50} - 33 q^{51} - 6 q^{52} + 60 q^{53} + 30 q^{55} + 12 q^{57} + 12 q^{58} - 3 q^{59} + 18 q^{60} + 54 q^{61} - 6 q^{62} + 84 q^{63} - 9 q^{64} - 24 q^{65} + 30 q^{66} - 15 q^{67} + 27 q^{68} + 24 q^{69} + 24 q^{70} - 36 q^{71} - 42 q^{73} - 6 q^{74} - 18 q^{78} - 6 q^{79} + 3 q^{82} - 36 q^{83} - 24 q^{84} + 6 q^{86} - 54 q^{87} + 60 q^{89} - 12 q^{90} - 18 q^{91} - 84 q^{93} - 6 q^{95} + 9 q^{97} - 12 q^{98} - 30 q^{99}+O(q^{100})$$ 18 * q + 3 * q^6 + 9 * q^8 - 12 * q^13 + 12 * q^14 - 24 * q^15 - 6 * q^17 - 6 * q^19 - 24 * q^22 - 18 * q^25 - 18 * q^26 - 3 * q^27 + 6 * q^28 + 6 * q^29 - 27 * q^33 - 6 * q^34 + 24 * q^35 - 3 * q^36 - 3 * q^38 + 6 * q^39 - 3 * q^41 - 6 * q^43 - 24 * q^44 + 54 * q^45 + 18 * q^46 - 30 * q^47 + 6 * q^48 + 21 * q^49 - 3 * q^50 - 33 * q^51 - 6 * q^52 + 60 * q^53 + 30 * q^55 + 12 * q^57 + 12 * q^58 - 3 * q^59 + 18 * q^60 + 54 * q^61 - 6 * q^62 + 84 * q^63 - 9 * q^64 - 24 * q^65 + 30 * q^66 - 15 * q^67 + 27 * q^68 + 24 * q^69 + 24 * q^70 - 36 * q^71 - 42 * q^73 - 6 * q^74 - 18 * q^78 - 6 * q^79 + 3 * q^82 - 36 * q^83 - 24 * q^84 + 6 * q^86 - 54 * q^87 + 60 * q^89 - 12 * q^90 - 18 * q^91 - 84 * q^93 - 6 * q^95 + 9 * q^97 - 12 * q^98 - 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + 162 x^{7} + 270 x^{6} + 2916 x^{5} - 4374 x^{4} - 729 x^{3} + 19683$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - \nu^{17} + \nu^{14} + 18 \nu^{13} - 36 \nu^{12} - 10 \nu^{11} - 18 \nu^{10} - 90 \nu^{9} + 567 \nu^{8} - 270 \nu^{7} - 162 \nu^{6} - 270 \nu^{5} - 2916 \nu^{4} + 4374 \nu^{3} + 729 \nu^{2} ) / 6561$$ (-v^17 + v^14 + 18*v^13 - 36*v^12 - 10*v^11 - 18*v^10 - 90*v^9 + 567*v^8 - 270*v^7 - 162*v^6 - 270*v^5 - 2916*v^4 + 4374*v^3 + 729*v^2) / 6561 $$\beta_{3}$$ $$=$$ $$( 2 \nu^{17} + 21 \nu^{16} - 81 \nu^{15} + 97 \nu^{14} - 57 \nu^{13} - 198 \nu^{12} + 1568 \nu^{11} - 3075 \nu^{10} + 2313 \nu^{9} - 117 \nu^{8} - 9261 \nu^{7} + 29268 \nu^{6} - 31374 \nu^{5} + \cdots - 65610 ) / 4374$$ (2*v^17 + 21*v^16 - 81*v^15 + 97*v^14 - 57*v^13 - 198*v^12 + 1568*v^11 - 3075*v^10 + 2313*v^9 - 117*v^8 - 9261*v^7 + 29268*v^6 - 31374*v^5 + 11421*v^4 + 11421*v^3 - 82377*v^2 + 150903*v - 65610) / 4374 $$\beta_{4}$$ $$=$$ $$( - 5 \nu^{17} + 99 \nu^{16} - 153 \nu^{15} - 22 \nu^{14} + 207 \nu^{13} - 1485 \nu^{12} + 4675 \nu^{11} - 2637 \nu^{10} - 4167 \nu^{9} + 10098 \nu^{8} - 38475 \nu^{7} + 51273 \nu^{6} + \cdots + 334611 ) / 13122$$ (-5*v^17 + 99*v^16 - 153*v^15 - 22*v^14 + 207*v^13 - 1485*v^12 + 4675*v^11 - 2637*v^10 - 4167*v^9 + 10098*v^8 - 38475*v^7 + 51273*v^6 + 24003*v^5 - 60507*v^4 + 91125*v^3 - 237654*v^2 + 63423*v + 334611) / 13122 $$\beta_{5}$$ $$=$$ $$( \nu^{17} + 2 \nu^{16} - 27 \nu^{15} + 50 \nu^{14} - 47 \nu^{13} + 9 \nu^{12} + 409 \nu^{11} - 1366 \nu^{10} + 1629 \nu^{9} - 822 \nu^{8} - 1809 \nu^{7} + 11025 \nu^{6} - 19359 \nu^{5} + 12420 \nu^{4} + \cdots - 72171 ) / 1458$$ (v^17 + 2*v^16 - 27*v^15 + 50*v^14 - 47*v^13 + 9*v^12 + 409*v^11 - 1366*v^10 + 1629*v^9 - 822*v^8 - 1809*v^7 + 11025*v^6 - 19359*v^5 + 12420*v^4 - 81*v^3 - 24786*v^2 + 75087*v - 72171) / 1458 $$\beta_{6}$$ $$=$$ $$( - 16 \nu^{17} + 81 \nu^{16} + 36 \nu^{15} - 335 \nu^{14} + 477 \nu^{13} - 1422 \nu^{12} + 1406 \nu^{11} + 6165 \nu^{10} - 12420 \nu^{9} + 12609 \nu^{8} - 20493 \nu^{7} - 25272 \nu^{6} + \cdots + 603612 ) / 13122$$ (-16*v^17 + 81*v^16 + 36*v^15 - 335*v^14 + 477*v^13 - 1422*v^12 + 1406*v^11 + 6165*v^10 - 12420*v^9 + 12609*v^8 - 20493*v^7 - 25272*v^6 + 127062*v^5 - 103761*v^4 + 67554*v^3 - 41553*v^2 - 373977*v + 603612) / 13122 $$\beta_{7}$$ $$=$$ $$( - 11 \nu^{17} - 3 \nu^{16} - 6 \nu^{15} + 92 \nu^{14} + 48 \nu^{13} - 255 \nu^{12} - 137 \nu^{11} - 1425 \nu^{10} + 3108 \nu^{9} + 1350 \nu^{8} - 504 \nu^{7} + 3645 \nu^{6} - 36045 \nu^{5} + \cdots - 225261 ) / 4374$$ (-11*v^17 - 3*v^16 - 6*v^15 + 92*v^14 + 48*v^13 - 255*v^12 - 137*v^11 - 1425*v^10 + 3108*v^9 + 1350*v^8 - 504*v^7 + 3645*v^6 - 36045*v^5 + 26001*v^4 + 10854*v^3 + 8262*v^2 + 74358*v - 225261) / 4374 $$\beta_{8}$$ $$=$$ $$( - 5 \nu^{17} - 105 \nu^{16} + 90 \nu^{15} + 275 \nu^{14} - 318 \nu^{13} + 1296 \nu^{12} - 4181 \nu^{11} - 2247 \nu^{10} + 12492 \nu^{9} - 11367 \nu^{8} + 36126 \nu^{7} - 25110 \nu^{6} + \cdots - 787320 ) / 13122$$ (-5*v^17 - 105*v^16 + 90*v^15 + 275*v^14 - 318*v^13 + 1296*v^12 - 4181*v^11 - 2247*v^10 + 12492*v^9 - 11367*v^8 + 36126*v^7 - 25110*v^6 - 114507*v^5 + 125469*v^4 - 81648*v^3 + 210681*v^2 + 183708*v - 787320) / 13122 $$\beta_{9}$$ $$=$$ $$( - 11 \nu^{17} + 147 \nu^{16} - 216 \nu^{15} - 43 \nu^{14} + 375 \nu^{13} - 2259 \nu^{12} + 6451 \nu^{11} - 3183 \nu^{10} - 6660 \nu^{9} + 15525 \nu^{8} - 52947 \nu^{7} + 64233 \nu^{6} + \cdots + 452709 ) / 13122$$ (-11*v^17 + 147*v^16 - 216*v^15 - 43*v^14 + 375*v^13 - 2259*v^12 + 6451*v^11 - 3183*v^10 - 6660*v^9 + 15525*v^8 - 52947*v^7 + 64233*v^6 + 40041*v^5 - 88857*v^4 + 125388*v^3 - 292329*v^2 + 50301*v + 452709) / 13122 $$\beta_{10}$$ $$=$$ $$( - 17 \nu^{17} - 5 \nu^{16} + 99 \nu^{15} - 136 \nu^{14} + 284 \nu^{13} - 405 \nu^{12} - 1655 \nu^{11} + 4369 \nu^{10} - 4167 \nu^{9} + 5472 \nu^{8} + 5508 \nu^{7} - 41229 \nu^{6} + \cdots + 63423 ) / 4374$$ (-17*v^17 - 5*v^16 + 99*v^15 - 136*v^14 + 284*v^13 - 405*v^12 - 1655*v^11 + 4369*v^10 - 4167*v^9 + 5472*v^8 + 5508*v^7 - 41229*v^6 + 46683*v^5 - 25569*v^4 + 13851*v^3 + 103518*v^2 - 237654*v + 63423) / 4374 $$\beta_{11}$$ $$=$$ $$( - 4 \nu^{17} + 17 \nu^{16} - 10 \nu^{15} - 23 \nu^{14} + 82 \nu^{13} - 332 \nu^{12} + 536 \nu^{11} + 197 \nu^{10} - 1234 \nu^{9} + 2619 \nu^{8} - 5760 \nu^{7} + 2430 \nu^{6} + 9666 \nu^{5} + \cdots + 56862 ) / 1458$$ (-4*v^17 + 17*v^16 - 10*v^15 - 23*v^14 + 82*v^13 - 332*v^12 + 536*v^11 + 197*v^10 - 1234*v^9 + 2619*v^8 - 5760*v^7 + 2430*v^6 + 9666*v^5 - 12825*v^4 + 18198*v^3 - 24057*v^2 - 20412*v + 56862) / 1458 $$\beta_{12}$$ $$=$$ $$( - 23 \nu^{17} + 10 \nu^{16} + 6 \nu^{15} + 86 \nu^{14} + 161 \nu^{13} - 717 \nu^{12} - 41 \nu^{11} - 908 \nu^{10} + 2814 \nu^{9} + 4716 \nu^{8} - 4941 \nu^{7} - 1377 \nu^{6} - 32373 \nu^{5} + \cdots - 247131 ) / 4374$$ (-23*v^17 + 10*v^16 + 6*v^15 + 86*v^14 + 161*v^13 - 717*v^12 - 41*v^11 - 908*v^10 + 2814*v^9 + 4716*v^8 - 4941*v^7 - 1377*v^6 - 32373*v^5 + 23436*v^4 + 35640*v^3 + 7290*v^2 + 26973*v - 247131) / 4374 $$\beta_{13}$$ $$=$$ $$( - 9 \nu^{17} - 4 \nu^{16} + 39 \nu^{15} - 21 \nu^{14} + 94 \nu^{13} - 174 \nu^{12} - 717 \nu^{11} + 1220 \nu^{10} - 393 \nu^{9} + 1797 \nu^{8} + 2520 \nu^{7} - 14598 \nu^{6} + 6345 \nu^{5} + \cdots - 34992 ) / 1458$$ (-9*v^17 - 4*v^16 + 39*v^15 - 21*v^14 + 94*v^13 - 174*v^12 - 717*v^11 + 1220*v^10 - 393*v^9 + 1797*v^8 + 2520*v^7 - 14598*v^6 + 6345*v^5 + 270*v^4 + 5913*v^3 + 41553*v^2 - 67068*v - 34992) / 1458 $$\beta_{14}$$ $$=$$ $$( - 76 \nu^{17} - 108 \nu^{16} + 459 \nu^{15} - 194 \nu^{14} + 747 \nu^{13} - 522 \nu^{12} - 9724 \nu^{11} + 13104 \nu^{10} - 1521 \nu^{9} + 9774 \nu^{8} + 50193 \nu^{7} - 167832 \nu^{6} + \cdots - 551124 ) / 13122$$ (-76*v^17 - 108*v^16 + 459*v^15 - 194*v^14 + 747*v^13 - 522*v^12 - 9724*v^11 + 13104*v^10 - 1521*v^9 + 9774*v^8 + 50193*v^7 - 167832*v^6 + 45090*v^5 + 39366*v^4 - 13851*v^3 + 546750*v^2 - 649539*v - 551124) / 13122 $$\beta_{15}$$ $$=$$ $$( 11 \nu^{17} - 17 \nu^{16} - 3 \nu^{15} + 13 \nu^{14} - 145 \nu^{13} + 525 \nu^{12} - 283 \nu^{11} - 413 \nu^{10} + 807 \nu^{9} - 4125 \nu^{8} + 5787 \nu^{7} + 2817 \nu^{6} - 5103 \nu^{5} + \cdots + 10935 ) / 1458$$ (11*v^17 - 17*v^16 - 3*v^15 + 13*v^14 - 145*v^13 + 525*v^12 - 283*v^11 - 413*v^10 + 807*v^9 - 4125*v^8 + 5787*v^7 + 2817*v^6 - 5103*v^5 + 7695*v^4 - 26811*v^3 + 7047*v^2 + 37179*v + 10935) / 1458 $$\beta_{16}$$ $$=$$ $$( 103 \nu^{17} - 99 \nu^{16} - 27 \nu^{15} - 157 \nu^{14} - 1026 \nu^{13} + 4140 \nu^{12} - 1265 \nu^{11} + 621 \nu^{10} - 3555 \nu^{9} - 30429 \nu^{8} + 39960 \nu^{7} + 12150 \nu^{6} + \cdots + 669222 ) / 13122$$ (103*v^17 - 99*v^16 - 27*v^15 - 157*v^14 - 1026*v^13 + 4140*v^12 - 1265*v^11 + 621*v^10 - 3555*v^9 - 30429*v^8 + 39960*v^7 + 12150*v^6 + 60615*v^5 - 24057*v^4 - 216513*v^3 + 22599*v^2 + 74358*v + 669222) / 13122 $$\beta_{17}$$ $$=$$ $$( - 17 \nu^{17} + 10 \nu^{16} + 27 \nu^{15} - 10 \nu^{14} + 188 \nu^{13} - 576 \nu^{12} - 269 \nu^{11} + 874 \nu^{10} - 351 \nu^{9} + 4680 \nu^{8} - 3078 \nu^{7} - 10746 \nu^{6} + 1161 \nu^{5} + \cdots - 78732 ) / 1458$$ (-17*v^17 + 10*v^16 + 27*v^15 - 10*v^14 + 188*v^13 - 576*v^12 - 269*v^11 + 874*v^10 - 351*v^9 + 4680*v^8 - 3078*v^7 - 10746*v^6 + 1161*v^5 - 702*v^4 + 26973*v^3 + 20412*v^2 - 55404*v - 78732) / 1458
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{16} + \beta_{15} + \beta_{9} - \beta_{8} - 2\beta_{4}$$ -b16 + b15 + b9 - b8 - 2*b4 $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{14} - 4\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2}$$ -b15 + b14 - 4*b7 - 2*b6 + b5 + b3 + 2*b2 $$\nu^{4}$$ $$=$$ $$- \beta_{17} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 3 \beta_{2} + \beta _1 + 2$$ -b17 + b13 + b12 + 2*b11 + 2*b10 - 3*b7 - 3*b6 + 2*b5 - 3*b2 + b1 + 2 $$\nu^{5}$$ $$=$$ $$- 3 \beta_{17} - 2 \beta_{16} + \beta_{15} + 3 \beta_{13} + 3 \beta_{10} - \beta_{9} - 2 \beta_{8} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 6 \beta _1 - 12$$ -3*b17 - 2*b16 + b15 + 3*b13 + 3*b10 - b9 - 2*b8 + 3*b5 + 2*b4 + b3 + 6*b1 - 12 $$\nu^{6}$$ $$=$$ $$- 9 \beta_{16} + 10 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 2 \beta_{7} + 2 \beta_{6} + 5 \beta_{3} + \beta_{2} - 9 \beta _1 - 6$$ -9*b16 + 10*b15 + 2*b14 - b13 - 3*b12 + 3*b11 + 3*b10 - 2*b7 + 2*b6 + 5*b3 + b2 - 9*b1 - 6 $$\nu^{7}$$ $$=$$ $$- 4 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} + 15 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 9 \beta_{9} + 18 \beta_{8} - 12 \beta_{7} + 7 \beta_{5} + 36 \beta_{4} + 15 \beta_{2} - 3 \beta _1 - 2$$ -4*b17 + 9*b16 - 9*b15 + 15*b14 - 7*b13 + 2*b12 + 4*b11 + b10 - 9*b9 + 18*b8 - 12*b7 + 7*b5 + 36*b4 + 15*b2 - 3*b1 - 2 $$\nu^{8}$$ $$=$$ $$- 18 \beta_{17} - \beta_{16} + 15 \beta_{15} - 9 \beta_{14} + 21 \beta_{12} + 24 \beta_{11} + 21 \beta_{10} - 11 \beta_{9} + 5 \beta_{8} + 36 \beta_{7} + 18 \beta_{6} + 3 \beta_{5} + 7 \beta_{4} - 14 \beta_{3} - 18 \beta_{2} + 9 \beta_1$$ -18*b17 - b16 + 15*b15 - 9*b14 + 21*b12 + 24*b11 + 21*b10 - 11*b9 + 5*b8 + 36*b7 + 18*b6 + 3*b5 + 7*b4 - 14*b3 - 18*b2 + 9*b1 $$\nu^{9}$$ $$=$$ $$- 27 \beta_{17} - 27 \beta_{16} + 11 \beta_{15} + \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - 30 \beta_{11} + 6 \beta_{10} - 27 \beta_{9} - 36 \beta_{8} + 74 \beta_{7} + 76 \beta_{6} - 7 \beta_{5} + 13 \beta_{3} + 44 \beta_{2} + 27 \beta _1 - 30$$ -27*b17 - 27*b16 + 11*b15 + b14 - 4*b13 + 3*b12 - 30*b11 + 6*b10 - 27*b9 - 36*b8 + 74*b7 + 76*b6 - 7*b5 + 13*b3 + 44*b2 + 27*b1 - 30 $$\nu^{10}$$ $$=$$ $$33 \beta_{17} - 54 \beta_{16} + 63 \beta_{15} + 24 \beta_{14} - 89 \beta_{13} + \beta_{12} + 2 \beta_{11} - 10 \beta_{10} - 27 \beta_{9} + 18 \beta_{8} + 9 \beta_{7} + 66 \beta_{6} - 76 \beta_{5} - 18 \beta_{4} + 63 \beta_{3} + 9 \beta_{2} - 67 \beta _1 + 140$$ 33*b17 - 54*b16 + 63*b15 + 24*b14 - 89*b13 + b12 + 2*b11 - 10*b10 - 27*b9 + 18*b8 + 9*b7 + 66*b6 - 76*b5 - 18*b4 + 63*b3 + 9*b2 - 67*b1 + 140 $$\nu^{11}$$ $$=$$ $$- 15 \beta_{17} + 100 \beta_{16} - 103 \beta_{15} + 126 \beta_{14} - 129 \beta_{13} + 102 \beta_{12} - 30 \beta_{11} - 63 \beta_{10} - \beta_{9} - 2 \beta_{8} - 18 \beta_{7} + 18 \beta_{6} - 9 \beta_{5} + 68 \beta_{4} - 51 \beta_{3} + 63 \beta_{2} + \cdots + 120$$ -15*b17 + 100*b16 - 103*b15 + 126*b14 - 129*b13 + 102*b12 - 30*b11 - 63*b10 - b9 - 2*b8 - 18*b7 + 18*b6 - 9*b5 + 68*b4 - 51*b3 + 63*b2 + 66*b1 + 120 $$\nu^{12}$$ $$=$$ $$- 90 \beta_{17} - 81 \beta_{16} + 136 \beta_{15} - 154 \beta_{14} + 33 \beta_{13} + 249 \beta_{12} + 129 \beta_{11} + 111 \beta_{10} + 36 \beta_{9} - 342 \beta_{8} + 148 \beta_{7} + 2 \beta_{6} - 34 \beta_{5} - 612 \beta_{4} + 17 \beta_{3} + \cdots - 51$$ -90*b17 - 81*b16 + 136*b15 - 154*b14 + 33*b13 + 249*b12 + 129*b11 + 111*b10 + 36*b9 - 342*b8 + 148*b7 + 2*b6 - 34*b5 - 612*b4 + 17*b3 - 155*b2 + 81*b1 - 51 $$\nu^{13}$$ $$=$$ $$- 62 \beta_{17} - 171 \beta_{16} - 189 \beta_{15} + 153 \beta_{14} - 19 \beta_{13} - 100 \beta_{12} - 362 \beta_{11} - 101 \beta_{10} - 99 \beta_{9} - 648 \beta_{8} - 96 \beta_{7} - 6 \beta_{6} - 29 \beta_{5} - 486 \beta_{4} + 432 \beta_{3} + \cdots - 2$$ -62*b17 - 171*b16 - 189*b15 + 153*b14 - 19*b13 - 100*b12 - 362*b11 - 101*b10 - 99*b9 - 648*b8 - 96*b7 - 6*b6 - 29*b5 - 486*b4 + 432*b3 + 255*b2 + 44*b1 - 2 $$\nu^{14}$$ $$=$$ $$372 \beta_{17} - 106 \beta_{16} + 206 \beta_{15} + 243 \beta_{14} - 426 \beta_{13} + 9 \beta_{12} + 450 \beta_{11} - 39 \beta_{10} - 80 \beta_{9} + 380 \beta_{8} - 1026 \beta_{7} - 756 \beta_{6} - 525 \beta_{5} - 29 \beta_{4} + 710 \beta_{3} + \cdots + 354$$ 372*b17 - 106*b16 + 206*b15 + 243*b14 - 426*b13 + 9*b12 + 450*b11 - 39*b10 - 80*b9 + 380*b8 - 1026*b7 - 756*b6 - 525*b5 - 29*b4 + 710*b3 - 810*b2 - 393*b1 + 354 $$\nu^{15}$$ $$=$$ $$- 351 \beta_{17} + 765 \beta_{16} - 802 \beta_{15} + 916 \beta_{14} + 46 \beta_{13} + 327 \beta_{12} + 402 \beta_{11} - 624 \beta_{10} + 351 \beta_{9} - 378 \beta_{8} - 376 \beta_{7} - 974 \beta_{6} + 522 \beta_{5} + 270 \beta_{4} + \cdots - 2208$$ -351*b17 + 765*b16 - 802*b15 + 916*b14 + 46*b13 + 327*b12 + 402*b11 - 624*b10 + 351*b9 - 378*b8 - 376*b7 - 974*b6 + 522*b5 + 270*b4 - 752*b3 - 298*b2 + 279*b1 - 2208 $$\nu^{16}$$ $$=$$ $$- 878 \beta_{17} - 630 \beta_{16} + 927 \beta_{15} - 1554 \beta_{14} + 1726 \beta_{13} + 430 \beta_{12} + 2318 \beta_{11} + 998 \beta_{10} - 423 \beta_{9} - 558 \beta_{8} + 390 \beta_{7} - 351 \beta_{6} + 326 \beta_{5} + \cdots - 4210$$ -878*b17 - 630*b16 + 927*b15 - 1554*b14 + 1726*b13 + 430*b12 + 2318*b11 + 998*b10 - 423*b9 - 558*b8 + 390*b7 - 351*b6 + 326*b5 - 3060*b4 + 729*b3 - 852*b2 - 1284*b1 - 4210 $$\nu^{17}$$ $$=$$ $$- 918 \beta_{17} + 406 \beta_{16} - 3786 \beta_{15} + 1656 \beta_{14} - 378 \beta_{13} - 3126 \beta_{12} - 1536 \beta_{11} - 1074 \beta_{10} - 3040 \beta_{9} + 1750 \beta_{8} + 504 \beta_{7} + 738 \beta_{6} - 192 \beta_{5} + \cdots + 54$$ -918*b17 + 406*b16 - 3786*b15 + 1656*b14 - 378*b13 - 3126*b12 - 1536*b11 - 1074*b10 - 3040*b9 + 1750*b8 + 504*b7 + 738*b6 - 192*b5 + 5150*b4 + 1436*b3 + 1125*b2 - 1566*b1 + 54

## Character values

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

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$-1$$ $$-\beta_{4} - \beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 −1.72388 + 0.168030i 1.47158 − 0.913487i 0.0786547 + 1.73026i −0.442647 + 1.67453i −1.73189 − 0.0237018i 1.40849 − 1.00804i −0.363139 − 1.69356i 1.69944 − 0.334495i −0.396613 + 1.68603i −1.72388 − 0.168030i 1.47158 + 0.913487i 0.0786547 − 1.73026i −0.363139 + 1.69356i 1.69944 + 0.334495i −0.396613 − 1.68603i −0.442647 − 1.67453i −1.73189 + 0.0237018i 1.40849 + 1.00804i
−0.173648 0.984808i −0.716422 + 1.57694i −0.939693 + 0.342020i −1.14133 + 3.13578i 1.67739 + 0.431705i −1.07356 + 1.85947i 0.500000 + 0.866025i −1.97348 2.25951i 3.28633 + 0.579469i
29.2 −0.173648 0.984808i −0.0553136 1.73117i −0.939693 + 0.342020i 0.882820 2.42553i −1.69526 + 0.355087i −1.58376 + 2.74316i 0.500000 + 0.866025i −2.99388 + 0.191514i −2.54198 0.448219i
29.3 −0.173648 0.984808i 1.53778 + 0.797015i −0.939693 + 0.342020i 0.258510 0.710252i 0.517874 1.65282i 0.777943 1.34744i 0.500000 + 0.866025i 1.72953 + 2.45127i −0.744351 0.131249i
41.1 −0.766044 0.642788i −1.67151 + 0.453924i 0.173648 + 0.984808i 1.96615 + 0.346685i 1.57223 + 0.726702i 0.910931 + 1.57778i 0.500000 0.866025i 2.58791 1.51748i −1.28331 1.52939i
41.2 −0.766044 0.642788i −0.845418 1.51171i 0.173648 + 0.984808i −2.22841 0.392929i −0.324081 + 1.70146i −1.16829 2.02354i 0.500000 0.866025i −1.57054 + 2.55605i 1.45449 + 1.73339i
41.3 −0.766044 0.642788i 1.57724 + 0.715766i 0.173648 + 0.984808i 0.262261 + 0.0462437i −0.748148 1.56214i 0.604656 + 1.04730i 0.500000 0.866025i 1.97536 + 2.25787i −0.171179 0.204003i
53.1 0.939693 + 0.342020i −1.64823 0.532290i 0.766044 + 0.642788i 2.20556 + 2.62849i −1.36678 1.06392i 1.68651 2.92113i 0.500000 + 0.866025i 2.43333 + 1.75467i 1.17355 + 3.22432i
53.2 0.939693 + 0.342020i 0.560041 1.63901i 0.766044 + 0.642788i −0.343148 0.408948i 1.08684 1.34862i −0.716507 + 1.24103i 0.500000 + 0.866025i −2.37271 1.83583i −0.182585 0.501649i
53.3 0.939693 + 0.342020i 1.26184 + 1.18649i 0.766044 + 0.642788i −1.86241 2.21954i 0.779936 + 1.54651i 0.562083 0.973556i 0.500000 + 0.866025i 0.184473 + 2.99432i −0.990970 2.72267i
59.1 −0.173648 + 0.984808i −0.716422 1.57694i −0.939693 0.342020i −1.14133 3.13578i 1.67739 0.431705i −1.07356 1.85947i 0.500000 0.866025i −1.97348 + 2.25951i 3.28633 0.579469i
59.2 −0.173648 + 0.984808i −0.0553136 + 1.73117i −0.939693 0.342020i 0.882820 + 2.42553i −1.69526 0.355087i −1.58376 2.74316i 0.500000 0.866025i −2.99388 0.191514i −2.54198 + 0.448219i
59.3 −0.173648 + 0.984808i 1.53778 0.797015i −0.939693 0.342020i 0.258510 + 0.710252i 0.517874 + 1.65282i 0.777943 + 1.34744i 0.500000 0.866025i 1.72953 2.45127i −0.744351 + 0.131249i
71.1 0.939693 0.342020i −1.64823 + 0.532290i 0.766044 0.642788i 2.20556 2.62849i −1.36678 + 1.06392i 1.68651 + 2.92113i 0.500000 0.866025i 2.43333 1.75467i 1.17355 3.22432i
71.2 0.939693 0.342020i 0.560041 + 1.63901i 0.766044 0.642788i −0.343148 + 0.408948i 1.08684 + 1.34862i −0.716507 1.24103i 0.500000 0.866025i −2.37271 + 1.83583i −0.182585 + 0.501649i
71.3 0.939693 0.342020i 1.26184 1.18649i 0.766044 0.642788i −1.86241 + 2.21954i 0.779936 1.54651i 0.562083 + 0.973556i 0.500000 0.866025i 0.184473 2.99432i −0.990970 + 2.72267i
89.1 −0.766044 + 0.642788i −1.67151 0.453924i 0.173648 0.984808i 1.96615 0.346685i 1.57223 0.726702i 0.910931 1.57778i 0.500000 + 0.866025i 2.58791 + 1.51748i −1.28331 + 1.52939i
89.2 −0.766044 + 0.642788i −0.845418 + 1.51171i 0.173648 0.984808i −2.22841 + 0.392929i −0.324081 1.70146i −1.16829 + 2.02354i 0.500000 + 0.866025i −1.57054 2.55605i 1.45449 1.73339i
89.3 −0.766044 + 0.642788i 1.57724 0.715766i 0.173648 0.984808i 0.262261 0.0462437i −0.748148 + 1.56214i 0.604656 1.04730i 0.500000 + 0.866025i 1.97536 2.25787i −0.171179 + 0.204003i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.l.a 18
3.b odd 2 1 114.2.l.b yes 18
4.b odd 2 1 912.2.cc.d 18
12.b even 2 1 912.2.cc.c 18
19.f odd 18 1 114.2.l.b yes 18
57.j even 18 1 inner 114.2.l.a 18
76.k even 18 1 912.2.cc.c 18
228.u odd 18 1 912.2.cc.d 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.l.a 18 1.a even 1 1 trivial
114.2.l.a 18 57.j even 18 1 inner
114.2.l.b yes 18 3.b odd 2 1
114.2.l.b yes 18 19.f odd 18 1
912.2.cc.c 18 12.b even 2 1
912.2.cc.c 18 76.k even 18 1
912.2.cc.d 18 4.b odd 2 1
912.2.cc.d 18 228.u odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{18} + 9 T_{5}^{16} + 108 T_{5}^{14} + 162 T_{5}^{13} + 46 T_{5}^{12} - 324 T_{5}^{11} - 1458 T_{5}^{10} + 1458 T_{5}^{9} - 30258 T_{5}^{8} - 5202 T_{5}^{7} + 141193 T_{5}^{6} - 60552 T_{5}^{5} + 70344 T_{5}^{4} + \cdots + 1728$$ acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - T^{3} + 1)^{3}$$
$3$ $$T^{18} + T^{15} + 18 T^{13} + 10 T^{12} + \cdots + 19683$$
$5$ $$T^{18} + 9 T^{16} + 108 T^{14} + \cdots + 1728$$
$7$ $$T^{18} + 21 T^{16} - 4 T^{15} + \cdots + 87616$$
$11$ $$T^{18} - 51 T^{16} + 1884 T^{14} + \cdots + 35769627$$
$13$ $$T^{18} + 12 T^{17} + 27 T^{16} + \cdots + 2365632$$
$17$ $$T^{18} + 6 T^{17} + 21 T^{16} + \cdots + 3878307$$
$19$ $$T^{18} + 6 T^{17} + \cdots + 322687697779$$
$23$ $$T^{18} - 105 T^{16} + \cdots + 123187392$$
$29$ $$T^{18} - 6 T^{17} + \cdots + 52719833664$$
$31$ $$T^{18} - 165 T^{16} + \cdots + 6231379854528$$
$37$ $$T^{18} + 342 T^{16} + \cdots + 7868768303808$$
$41$ $$T^{18} + 3 T^{17} + \cdots + 1846709769969$$
$43$ $$T^{18} + 6 T^{17} + \cdots + 390621250009$$
$47$ $$T^{18} + 30 T^{17} + \cdots + 3499077312$$
$53$ $$T^{18} - 60 T^{17} + \cdots + 3426463296$$
$59$ $$T^{18} + 3 T^{17} + \cdots + 38983402581561$$
$61$ $$T^{18} - 54 T^{17} + \cdots + 65033160256$$
$67$ $$T^{18} + 15 T^{17} + \cdots + 56\!\cdots\!23$$
$71$ $$T^{18} + 36 T^{17} + \cdots + 404099233344$$
$73$ $$T^{18} + 42 T^{17} + \cdots + 192753487369$$
$79$ $$T^{18} + 6 T^{17} + \cdots + 21259626441408$$
$83$ $$T^{18} + \cdots + 176145902499843$$
$89$ $$T^{18} + \cdots + 381874169643201$$
$97$ $$T^{18} - 9 T^{17} + \cdots + 17447631785307$$