# Properties

 Label 69.2.e.a Level $69$ Weight $2$ Character orbit 69.e Analytic conductor $0.551$ 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] = [69,2,Mod(4,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(22))

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

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

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

chi := DirichletCharacter("69.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$69 = 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 69.e (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: $$0.550967773947$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$47$$ $$\chi(n)$$ $$-\zeta_{22}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −0.415415 − 0.909632i 0.654861 + 0.755750i 0.654861 − 0.755750i −0.841254 − 0.540641i 0.142315 − 0.989821i 0.142315 + 0.989821i −0.415415 + 0.909632i 0.959493 − 0.281733i −0.841254 + 0.540641i 0.959493 + 0.281733i
−0.662317 1.45027i 0.959493 0.281733i −0.354905 + 0.409583i −0.438384 0.281733i −1.04408 1.20493i 0.188515 + 1.31115i −2.23047 0.654925i 0.841254 0.540641i −0.118239 + 0.822373i
13.1 1.44306 + 1.66538i −0.841254 0.540641i −0.406440 + 2.82685i 0.246902 0.540641i −0.313607 2.18119i −2.76921 0.813115i −1.58671 + 1.01971i 0.415415 + 0.909632i 1.25667 0.368991i
16.1 1.44306 1.66538i −0.841254 + 0.540641i −0.406440 2.82685i 0.246902 + 0.540641i −0.313607 + 2.18119i −2.76921 + 0.813115i −1.58671 1.01971i 0.415415 0.909632i 1.25667 + 0.368991i
25.1 −0.402869 0.258908i 0.142315 + 0.989821i −0.735560 1.61065i 3.37102 + 0.989821i 0.198939 0.435615i 0.527646 0.608936i −0.256983 + 1.78736i −0.959493 + 0.281733i −1.10181 1.27155i
31.1 0.0336545 0.234072i −0.415415 0.909632i 1.86533 + 0.547710i −0.788201 0.909632i −0.226900 + 0.0666238i 0.0566239 + 0.0363899i 0.387454 0.848406i −0.654861 + 0.755750i −0.239446 + 0.153882i
49.1 0.0336545 + 0.234072i −0.415415 + 0.909632i 1.86533 0.547710i −0.788201 + 0.909632i −0.226900 0.0666238i 0.0566239 0.0363899i 0.387454 + 0.848406i −0.654861 0.755750i −0.239446 0.153882i
52.1 −0.662317 + 1.45027i 0.959493 + 0.281733i −0.354905 0.409583i −0.438384 + 0.281733i −1.04408 + 1.20493i 0.188515 1.31115i −2.23047 + 0.654925i 0.841254 + 0.540641i −0.118239 0.822373i
55.1 −2.41153 + 0.708089i 0.654861 + 0.755750i 3.63158 2.33387i 0.108660 + 0.755750i −2.11435 1.35881i −2.00357 + 4.38721i −3.81329 + 4.40077i −0.142315 + 0.989821i −0.797176 1.74557i
58.1 −0.402869 + 0.258908i 0.142315 0.989821i −0.735560 + 1.61065i 3.37102 0.989821i 0.198939 + 0.435615i 0.527646 + 0.608936i −0.256983 1.78736i −0.959493 0.281733i −1.10181 + 1.27155i
64.1 −2.41153 0.708089i 0.654861 0.755750i 3.63158 + 2.33387i 0.108660 0.755750i −2.11435 + 1.35881i −2.00357 4.38721i −3.81329 4.40077i −0.142315 0.989821i −0.797176 + 1.74557i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.1 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 69.2.e.a 10
3.b odd 2 1 207.2.i.b 10
23.c even 11 1 inner 69.2.e.a 10
23.c even 11 1 1587.2.a.o 5
23.d odd 22 1 1587.2.a.p 5
69.g even 22 1 4761.2.a.bq 5
69.h odd 22 1 207.2.i.b 10
69.h odd 22 1 4761.2.a.br 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.2.e.a 10 1.a even 1 1 trivial
69.2.e.a 10 23.c even 11 1 inner
207.2.i.b 10 3.b odd 2 1
207.2.i.b 10 69.h odd 22 1
1587.2.a.o 5 23.c even 11 1
1587.2.a.p 5 23.d odd 22 1
4761.2.a.bq 5 69.g even 22 1
4761.2.a.br 5 69.h odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 4T_{2}^{9} + 5T_{2}^{8} + 9T_{2}^{7} + 36T_{2}^{6} + 78T_{2}^{5} + 125T_{2}^{4} + 71T_{2}^{3} + 20T_{2}^{2} + 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(69, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 4 T^{9} + 5 T^{8} + 9 T^{7} + \cdots + 1$$
$3$ $$T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1$$
$5$ $$T^{10} - 5 T^{9} + 3 T^{8} + 7 T^{7} + \cdots + 1$$
$7$ $$T^{10} + 8 T^{9} + 42 T^{8} + 105 T^{7} + \cdots + 1$$
$11$ $$T^{10} - 7 T^{9} + 16 T^{8} + 31 T^{7} + \cdots + 529$$
$13$ $$T^{10} + 30 T^{9} + 438 T^{8} + \cdots + 2042041$$
$17$ $$T^{10} + 2 T^{9} - 18 T^{8} + \cdots + 17161$$
$19$ $$T^{10} - 10 T^{9} + 56 T^{8} - 186 T^{7} + \cdots + 1$$
$23$ $$T^{10} + T^{9} + T^{8} - 21 T^{7} + \cdots + 6436343$$
$29$ $$T^{10} + 14 T^{9} + 108 T^{8} + \cdots + 734449$$
$31$ $$T^{10} + 28 T^{9} + 300 T^{8} + \cdots + 4489$$
$37$ $$T^{10} - 19 T^{9} + 152 T^{8} + \cdots + 14645929$$
$41$ $$T^{10} - 19 T^{9} + 196 T^{8} + \cdots + 38809$$
$43$ $$T^{10} + 24 T^{9} + 301 T^{8} + \cdots + 1739761$$
$47$ $$(T^{5} - 13 T^{4} - 27 T^{3} + 486 T^{2} + \cdots - 2507)^{2}$$
$53$ $$T^{10} + T^{9} - 54 T^{8} - 615 T^{7} + \cdots + 94249$$
$59$ $$T^{10} - 2 T^{9} + 37 T^{8} + \cdots + 1515361$$
$61$ $$T^{10} - 30 T^{9} + \cdots + 519520849$$
$67$ $$T^{10} - 4 T^{9} + 16 T^{8} + \cdots + 24930049$$
$71$ $$T^{10} + 14 T^{9} + \cdots + 4663114369$$
$73$ $$T^{10} + 26 T^{9} + 302 T^{8} + \cdots + 12243001$$
$79$ $$T^{10} - 20 T^{9} + 323 T^{8} + \cdots + 326041$$
$83$ $$T^{10} - 18 T^{9} + 192 T^{8} + \cdots + 529$$
$89$ $$T^{10} + 5 T^{9} + \cdots + 3373402561$$
$97$ $$T^{10} - 15 T^{9} + 379 T^{8} + \cdots + 83740801$$