# Properties

 Label 115.2.g.a Level $115$ Weight $2$ Character orbit 115.g Analytic conductor $0.918$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(6,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 18]))

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

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

chi := DirichletCharacter("115.6");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.g (of order $$11$$, degree $$10$$, 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: $$0.918279623245$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-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 $$\zeta_{22}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$1$$ $$\zeta_{22}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 0.959493 − 0.281733i −0.841254 − 0.540641i −0.415415 − 0.909632i −0.415415 + 0.909632i −0.841254 + 0.540641i 0.654861 + 0.755750i 0.142315 − 0.989821i 0.142315 + 0.989821i 0.959493 + 0.281733i 0.654861 − 0.755750i
0.925839 2.02730i 0.959493 + 0.281733i −1.94306 2.24241i −0.841254 + 0.540641i 1.45949 1.68434i −0.0125459 + 0.0872586i −2.06815 + 0.607265i −1.68251 1.08128i 0.317178 + 2.20602i
16.1 1.57028 1.81219i −0.841254 + 0.540641i −0.533654 3.71165i −0.415415 0.909632i −0.341254 + 2.37347i 1.31718 0.386758i −3.52977 2.26844i −0.830830 + 1.81926i −2.30075 0.675560i
26.1 −0.0125459 0.0872586i −0.415415 + 0.909632i 1.91153 0.561276i 0.654861 0.755750i 0.0845850 + 0.0248364i −1.30075 + 0.835939i −0.146201 0.320135i 1.30972 + 1.51150i −0.0741615 0.0476607i
31.1 −0.0125459 + 0.0872586i −0.415415 0.909632i 1.91153 + 0.561276i 0.654861 + 0.755750i 0.0845850 0.0248364i −1.30075 0.835939i −0.146201 + 0.320135i 1.30972 1.51150i −0.0741615 + 0.0476607i
36.1 1.57028 + 1.81219i −0.841254 0.540641i −0.533654 + 3.71165i −0.415415 + 0.909632i −0.341254 2.37347i 1.31718 + 0.386758i −3.52977 + 2.26844i −0.830830 1.81926i −2.30075 + 0.675560i
41.1 1.31718 + 0.386758i 0.654861 0.755750i −0.0971309 0.0624222i 0.142315 0.989821i 1.15486 0.742184i 0.925839 + 2.02730i −1.90176 2.19475i 0.284630 + 1.97964i 0.570276 1.24873i
71.1 −1.30075 0.835939i 0.142315 + 0.989821i 0.162317 + 0.355426i 0.959493 + 0.281733i 0.642315 1.40647i 1.57028 1.81219i −0.354114 + 2.46292i 1.91899 0.563465i −1.01255 1.16854i
81.1 −1.30075 + 0.835939i 0.142315 0.989821i 0.162317 0.355426i 0.959493 0.281733i 0.642315 + 1.40647i 1.57028 + 1.81219i −0.354114 2.46292i 1.91899 + 0.563465i −1.01255 + 1.16854i
96.1 0.925839 + 2.02730i 0.959493 0.281733i −1.94306 + 2.24241i −0.841254 0.540641i 1.45949 + 1.68434i −0.0125459 0.0872586i −2.06815 0.607265i −1.68251 + 1.08128i 0.317178 2.20602i
101.1 1.31718 0.386758i 0.654861 + 0.755750i −0.0971309 + 0.0624222i 0.142315 + 0.989821i 1.15486 + 0.742184i 0.925839 2.02730i −1.90176 + 2.19475i 0.284630 1.97964i 0.570276 + 1.24873i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.g.a 10
5.b even 2 1 575.2.k.a 10
5.c odd 4 2 575.2.p.a 20
23.c even 11 1 inner 115.2.g.a 10
23.c even 11 1 2645.2.a.n 5
23.d odd 22 1 2645.2.a.o 5
115.j even 22 1 575.2.k.a 10
115.k odd 44 2 575.2.p.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.a 10 1.a even 1 1 trivial
115.2.g.a 10 23.c even 11 1 inner
575.2.k.a 10 5.b even 2 1
575.2.k.a 10 115.j even 22 1
575.2.p.a 20 5.c odd 4 2
575.2.p.a 20 115.k odd 44 2
2645.2.a.n 5 23.c even 11 1
2645.2.a.o 5 23.d odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} - 5T_{2}^{9} + 14T_{2}^{8} - 15T_{2}^{7} - 2T_{2}^{6} + 21T_{2}^{5} + 38T_{2}^{4} - 157T_{2}^{3} + 125T_{2}^{2} + 2T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(115, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} - 5 T^{9} + 14 T^{8} - 15 T^{7} + \cdots + 1$$
$3$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$5$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$7$ $$T^{10} - 5 T^{9} + 14 T^{8} - 15 T^{7} + \cdots + 1$$
$11$ $$T^{10} - 8 T^{9} + 31 T^{8} + \cdots + 7921$$
$13$ $$T^{10} + 11 T^{8} + 11 T^{7} + \cdots + 121$$
$17$ $$T^{10} + 23 T^{9} + 265 T^{8} + \cdots + 466489$$
$19$ $$T^{10} + 13 T^{9} + 48 T^{8} + \cdots + 11881$$
$23$ $$T^{10} + 21 T^{9} + 243 T^{8} + \cdots + 6436343$$
$29$ $$T^{10} - 2 T^{9} + 70 T^{8} + \cdots + 127893481$$
$31$ $$T^{10} + 20 T^{9} + 158 T^{8} + \cdots + 7921$$
$37$ $$T^{10} - 13 T^{9} + 169 T^{8} + \cdots + 11485321$$
$41$ $$T^{10} + 5 T^{9} + 91 T^{8} + \cdots + 193293409$$
$43$ $$T^{10} - 15 T^{9} + 247 T^{8} + \cdots + 4280761$$
$47$ $$(T^{5} + 26 T^{4} + 189 T^{3} - 82 T^{2} + \cdots - 9811)^{2}$$
$53$ $$T^{10} - 6 T^{9} + 179 T^{8} + \cdots + 1408969$$
$59$ $$T^{10} - 5 T^{9} + 201 T^{8} + \cdots + 437688241$$
$61$ $$T^{10} + 3 T^{9} + 75 T^{8} + \cdots + 436921$$
$67$ $$T^{10} - 20 T^{9} + 290 T^{8} + \cdots + 25593481$$
$71$ $$T^{10} + 22 T^{9} + 275 T^{8} + \cdots + 1437601$$
$73$ $$T^{10} - 43 T^{9} + 870 T^{8} + \cdots + 78623689$$
$79$ $$T^{10} - 51 T^{9} + \cdots + 246772681$$
$83$ $$T^{10} + 17 T^{9} + 36 T^{8} + \cdots + 10029889$$
$89$ $$T^{10} - 57 T^{9} + 1544 T^{8} + \cdots + 43256929$$
$97$ $$T^{10} - 52 T^{9} + 1318 T^{8} + \cdots + 62047129$$