# Properties

 Label 11.3.d.a Level $11$ Weight $3$ Character orbit 11.d Analytic conductor $0.300$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [11,3,Mod(2,11)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("11.2");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 11.d (of order $$10$$, degree $$4$$, 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: $$0.299728290796$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ 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: $\mathrm{SU}(2)[C_{10}]$

## $q$-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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{2} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{3} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{4} + 4 \zeta_{10}^{2} q^{5} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} - 7 \zeta_{10} + 6) q^{6} + ( - 8 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{7} + (3 \zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} + 3) q^{8} + (4 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{9}+O(q^{10})$$ q + (-z^3 + 2*z^2 - 2*z) * q^2 + (2*z^3 - 3*z^2 + 3*z - 2) * q^3 + (-4*z^2 + 3*z - 4) * q^4 + 4*z^2 * q^5 + (z^3 + 3*z^2 - 7*z + 6) * q^6 + (-8*z^3 + 2*z^2 - 4*z + 6) * q^7 + (3*z^3 + z^2 + z + 3) * q^8 + (4*z^3 + 5*z - 5) * q^9 $$q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{2} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{3} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{4} + 4 \zeta_{10}^{2} q^{5} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} - 7 \zeta_{10} + 6) q^{6} + ( - 8 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{7} + (3 \zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} + 3) q^{8} + (4 \zeta_{10}^{3} + 5 \zeta_{10} - 5) q^{9} + ( - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 4) q^{10} + (7 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{11} + ( - 11 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 2) q^{12} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 6 \zeta_{10} - 2) q^{13} + (6 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 6) q^{14} + (4 \zeta_{10}^{2} - 12 \zeta_{10} + 4) q^{15} + (8 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 8 \zeta_{10}) q^{16} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}) q^{17} + (2 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + \zeta_{10} + 5) q^{18} + (5 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10} + 5) q^{19} + ( - 4 \zeta_{10}^{3} - 16 \zeta_{10} + 16) q^{20} + (8 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{21} + ( - 14 \zeta_{10}^{3} + 29 \zeta_{10}^{2} - 16 \zeta_{10} + 6) q^{22} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4) q^{23} + (8 \zeta_{10}^{3} - 16 \zeta_{10}^{2} + 9 \zeta_{10} - 7) q^{24} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 9 \zeta_{10} + 9) q^{25} + ( - 10 \zeta_{10}^{2} + 20 \zeta_{10} - 10) q^{26} + ( - 2 \zeta_{10}^{3} - 19 \zeta_{10}^{2} - 2 \zeta_{10}) q^{27} + (22 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 2 \zeta_{10} - 24) q^{28} + (20 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 10 \zeta_{10} - 18) q^{29} + ( - 16 \zeta_{10}^{3} + 12 \zeta_{10}^{2} + 12 \zeta_{10} - 16) q^{30} + ( - 4 \zeta_{10}^{3} + 18 \zeta_{10} - 18) q^{31} + ( - 24 \zeta_{10}^{3} + 22 \zeta_{10}^{2} - 46 \zeta_{10} + 23) q^{32} + (4 \zeta_{10}^{3} - 24 \zeta_{10}^{2} + 25 \zeta_{10} + 3) q^{33} + (39 \zeta_{10}^{3} - 39 \zeta_{10}^{2} + 13) q^{34} + ( - 8 \zeta_{10}^{3} + 16 \zeta_{10}^{2} + 8 \zeta_{10} + 24) q^{35} + ( - 24 \zeta_{10}^{3} + 23 \zeta_{10}^{2} - 23 \zeta_{10} + 24) q^{36} + (36 \zeta_{10}^{2} - 18 \zeta_{10} + 36) q^{37} + ( - 17 \zeta_{10}^{3} + 26 \zeta_{10}^{2} - 17 \zeta_{10}) q^{38} + (10 \zeta_{10}^{2} - 20 \zeta_{10} + 20) q^{39} + (8 \zeta_{10}^{3} + 8 \zeta_{10}^{2} + 4 \zeta_{10} - 16) q^{40} + ( - 16 \zeta_{10}^{3} - 27 \zeta_{10}^{2} - 27 \zeta_{10} - 16) q^{41} - 10 \zeta_{10}^{3} q^{42} + (19 \zeta_{10}^{3} - 17 \zeta_{10}^{2} + 36 \zeta_{10} - 18) q^{43} + (14 \zeta_{10}^{3} - 18 \zeta_{10}^{2} + 60 \zeta_{10} - 17) q^{44} + (20 \zeta_{10}^{3} - 20 \zeta_{10}^{2} - 16) q^{45} + (8 \zeta_{10}^{3} - 16 \zeta_{10}^{2} + 14 \zeta_{10} - 2) q^{46} + (26 \zeta_{10}^{3} - 24 \zeta_{10}^{2} + 24 \zeta_{10} - 26) q^{47} + ( - 19 \zeta_{10}^{2} + 17 \zeta_{10} - 19) q^{48} + ( - 44 \zeta_{10}^{3} + 21 \zeta_{10}^{2} - 44 \zeta_{10}) q^{49} + ( - 9 \zeta_{10}^{2} + 18 \zeta_{10} - 18) q^{50} + ( - 26 \zeta_{10}^{3} + 52 \zeta_{10}^{2} - 13 \zeta_{10} - 26) q^{51} + (22 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 4 \zeta_{10} + 22) q^{52} + (30 \zeta_{10}^{3} - 30 \zeta_{10} + 30) q^{53} + (2 \zeta_{10}^{3} + 36 \zeta_{10}^{2} - 34 \zeta_{10} + 17) q^{54} + ( - 48 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 36 \zeta_{10} + 8) q^{55} + ( - 26 \zeta_{10}^{3} + 26 \zeta_{10}^{2} + 38) q^{56} + (17 \zeta_{10}^{3} - 34 \zeta_{10}^{2} + 26 \zeta_{10} - 8) q^{57} + ( - 12 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 12) q^{58} + (15 \zeta_{10}^{2} - 22 \zeta_{10} + 15) q^{59} + (44 \zeta_{10}^{3} - 52 \zeta_{10}^{2} + 44 \zeta_{10}) q^{60} + ( - 18 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 14 \zeta_{10} + 4) q^{61} + (44 \zeta_{10}^{3} - 62 \zeta_{10}^{2} + 22 \zeta_{10} + 18) q^{62} + (18 \zeta_{10}^{3} + 26 \zeta_{10}^{2} + 26 \zeta_{10} + 18) q^{63} + ( - 41 \zeta_{10}^{3} + 36 \zeta_{10} - 36) q^{64} + ( - 8 \zeta_{10}^{3} - 24 \zeta_{10}^{2} + 16 \zeta_{10} - 8) q^{65} + (14 \zeta_{10}^{3} + 37 \zeta_{10}^{2} - 83 \zeta_{10} + 49) q^{66} + ( - 17 \zeta_{10}^{3} + 17 \zeta_{10}^{2} - 49) q^{67} + ( - 39 \zeta_{10}^{3} + 78 \zeta_{10}^{2} - 91 \zeta_{10} - 13) q^{68} + ( - 10 \zeta_{10}^{3} + 20 \zeta_{10}^{2} - 20 \zeta_{10} + 10) q^{69} + ( - 8 \zeta_{10}^{2} - 16 \zeta_{10} - 8) q^{70} + (62 \zeta_{10}^{3} - 24 \zeta_{10}^{2} + 62 \zeta_{10}) q^{71} + (21 \zeta_{10}^{3} - 19 \zeta_{10}^{2} + 17 \zeta_{10} - 38) q^{72} + ( - 34 \zeta_{10}^{3} - 43 \zeta_{10}^{2} - 17 \zeta_{10} + 77) q^{73} + ( - 54 \zeta_{10}^{3} + 18 \zeta_{10}^{2} + 18 \zeta_{10} - 54) q^{74} + (18 \zeta_{10}^{3} - 9 \zeta_{10} + 9) q^{75} + (5 \zeta_{10}^{3} - 41 \zeta_{10}^{2} + 46 \zeta_{10} - 23) q^{76} + (74 \zeta_{10}^{3} - 26 \zeta_{10}^{2} + 28 \zeta_{10} - 82) q^{77} + ( - 40 \zeta_{10}^{3} + 40 \zeta_{10}^{2} - 30) q^{78} + (26 \zeta_{10}^{3} - 52 \zeta_{10}^{2} + 40 \zeta_{10} - 12) q^{79} + (20 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} - 20) q^{80} + ( - 60 \zeta_{10}^{2} + 64 \zeta_{10} - 60) q^{81} + (21 \zeta_{10}^{3} + 17 \zeta_{10}^{2} + 21 \zeta_{10}) q^{82} + ( - 43 \zeta_{10}^{3} + 45 \zeta_{10}^{2} - 47 \zeta_{10} + 90) q^{83} + ( - 28 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 14 \zeta_{10} + 32) q^{84} + (52 \zeta_{10}^{3} + 52) q^{85} + (16 \zeta_{10}^{3} - 53 \zeta_{10} + 53) q^{86} + ( - 26 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 28 \zeta_{10} + 14) q^{87} + ( - 11 \zeta_{10}^{3} - 22 \zeta_{10}^{2} - 44 \zeta_{10} + 22) q^{88} + ( - 83 \zeta_{10}^{3} + 83 \zeta_{10}^{2} + 72) q^{89} + ( - 24 \zeta_{10}^{3} + 48 \zeta_{10}^{2} - 28 \zeta_{10} + 20) q^{90} + (40 \zeta_{10}^{3} - 20 \zeta_{10}^{2} + 20 \zeta_{10} - 40) q^{91} + (22 \zeta_{10}^{2} - 26 \zeta_{10} + 22) q^{92} + ( - 58 \zeta_{10}^{3} + 84 \zeta_{10}^{2} - 58 \zeta_{10}) q^{93} + ( - 2 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 46 \zeta_{10} + 48) q^{94} + ( - 16 \zeta_{10}^{3} + 28 \zeta_{10}^{2} - 8 \zeta_{10} - 12) q^{95} + (68 \zeta_{10}^{3} - 21 \zeta_{10}^{2} - 21 \zeta_{10} + 68) q^{96} + (36 \zeta_{10}^{3} + 67 \zeta_{10} - 67) q^{97} + (44 \zeta_{10}^{3} - 86 \zeta_{10}^{2} + 130 \zeta_{10} - 65) q^{98} + ( - 69 \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 5 \zeta_{10} + 28) q^{99} +O(q^{100})$$ q + (-z^3 + 2*z^2 - 2*z) * q^2 + (2*z^3 - 3*z^2 + 3*z - 2) * q^3 + (-4*z^2 + 3*z - 4) * q^4 + 4*z^2 * q^5 + (z^3 + 3*z^2 - 7*z + 6) * q^6 + (-8*z^3 + 2*z^2 - 4*z + 6) * q^7 + (3*z^3 + z^2 + z + 3) * q^8 + (4*z^3 + 5*z - 5) * q^9 + (-8*z^2 + 8*z - 4) * q^10 + (7*z^3 - 9*z^2 - 3*z - 3) * q^11 + (-11*z^3 + 11*z^2 - 2) * q^12 + (-2*z^3 + 4*z^2 - 6*z - 2) * q^13 + (6*z^3 - 4*z^2 + 4*z - 6) * q^14 + (4*z^2 - 12*z + 4) * q^15 + (8*z^3 - 3*z^2 + 8*z) * q^16 + (-13*z^3 + 13*z) * q^17 + (2*z^3 - 7*z^2 + z + 5) * q^18 + (5*z^3 - 2*z^2 - 2*z + 5) * q^19 + (-4*z^3 - 16*z + 16) * q^20 + (8*z^3 + 4*z^2 + 4*z - 2) * q^21 + (-14*z^3 + 29*z^2 - 16*z + 6) * q^22 + (-2*z^3 + 2*z^2 - 4) * q^23 + (8*z^3 - 16*z^2 + 9*z - 7) * q^24 + (-9*z^3 + 9*z^2 - 9*z + 9) * q^25 + (-10*z^2 + 20*z - 10) * q^26 + (-2*z^3 - 19*z^2 - 2*z) * q^27 + (22*z^3 - 12*z^2 + 2*z - 24) * q^28 + (20*z^3 - 2*z^2 + 10*z - 18) * q^29 + (-16*z^3 + 12*z^2 + 12*z - 16) * q^30 + (-4*z^3 + 18*z - 18) * q^31 + (-24*z^3 + 22*z^2 - 46*z + 23) * q^32 + (4*z^3 - 24*z^2 + 25*z + 3) * q^33 + (39*z^3 - 39*z^2 + 13) * q^34 + (-8*z^3 + 16*z^2 + 8*z + 24) * q^35 + (-24*z^3 + 23*z^2 - 23*z + 24) * q^36 + (36*z^2 - 18*z + 36) * q^37 + (-17*z^3 + 26*z^2 - 17*z) * q^38 + (10*z^2 - 20*z + 20) * q^39 + (8*z^3 + 8*z^2 + 4*z - 16) * q^40 + (-16*z^3 - 27*z^2 - 27*z - 16) * q^41 - 10*z^3 * q^42 + (19*z^3 - 17*z^2 + 36*z - 18) * q^43 + (14*z^3 - 18*z^2 + 60*z - 17) * q^44 + (20*z^3 - 20*z^2 - 16) * q^45 + (8*z^3 - 16*z^2 + 14*z - 2) * q^46 + (26*z^3 - 24*z^2 + 24*z - 26) * q^47 + (-19*z^2 + 17*z - 19) * q^48 + (-44*z^3 + 21*z^2 - 44*z) * q^49 + (-9*z^2 + 18*z - 18) * q^50 + (-26*z^3 + 52*z^2 - 13*z - 26) * q^51 + (22*z^3 - 4*z^2 - 4*z + 22) * q^52 + (30*z^3 - 30*z + 30) * q^53 + (2*z^3 + 36*z^2 - 34*z + 17) * q^54 + (-48*z^3 + 24*z^2 - 36*z + 8) * q^55 + (-26*z^3 + 26*z^2 + 38) * q^56 + (17*z^3 - 34*z^2 + 26*z - 8) * q^57 + (-12*z^3 - 2*z^2 + 2*z + 12) * q^58 + (15*z^2 - 22*z + 15) * q^59 + (44*z^3 - 52*z^2 + 44*z) * q^60 + (-18*z^3 + 2*z^2 + 14*z + 4) * q^61 + (44*z^3 - 62*z^2 + 22*z + 18) * q^62 + (18*z^3 + 26*z^2 + 26*z + 18) * q^63 + (-41*z^3 + 36*z - 36) * q^64 + (-8*z^3 - 24*z^2 + 16*z - 8) * q^65 + (14*z^3 + 37*z^2 - 83*z + 49) * q^66 + (-17*z^3 + 17*z^2 - 49) * q^67 + (-39*z^3 + 78*z^2 - 91*z - 13) * q^68 + (-10*z^3 + 20*z^2 - 20*z + 10) * q^69 + (-8*z^2 - 16*z - 8) * q^70 + (62*z^3 - 24*z^2 + 62*z) * q^71 + (21*z^3 - 19*z^2 + 17*z - 38) * q^72 + (-34*z^3 - 43*z^2 - 17*z + 77) * q^73 + (-54*z^3 + 18*z^2 + 18*z - 54) * q^74 + (18*z^3 - 9*z + 9) * q^75 + (5*z^3 - 41*z^2 + 46*z - 23) * q^76 + (74*z^3 - 26*z^2 + 28*z - 82) * q^77 + (-40*z^3 + 40*z^2 - 30) * q^78 + (26*z^3 - 52*z^2 + 40*z - 12) * q^79 + (20*z^3 + 12*z^2 - 12*z - 20) * q^80 + (-60*z^2 + 64*z - 60) * q^81 + (21*z^3 + 17*z^2 + 21*z) * q^82 + (-43*z^3 + 45*z^2 - 47*z + 90) * q^83 + (-28*z^3 - 4*z^2 - 14*z + 32) * q^84 + (52*z^3 + 52) * q^85 + (16*z^3 - 53*z + 53) * q^86 + (-26*z^3 + 2*z^2 - 28*z + 14) * q^87 + (-11*z^3 - 22*z^2 - 44*z + 22) * q^88 + (-83*z^3 + 83*z^2 + 72) * q^89 + (-24*z^3 + 48*z^2 - 28*z + 20) * q^90 + (40*z^3 - 20*z^2 + 20*z - 40) * q^91 + (22*z^2 - 26*z + 22) * q^92 + (-58*z^3 + 84*z^2 - 58*z) * q^93 + (-2*z^3 + 24*z^2 - 46*z + 48) * q^94 + (-16*z^3 + 28*z^2 - 8*z - 12) * q^95 + (68*z^3 - 21*z^2 - 21*z + 68) * q^96 + (36*z^3 + 67*z - 67) * q^97 + (44*z^3 - 86*z^2 + 130*z - 65) * q^98 + (-69*z^3 + 7*z^2 - 5*z + 28) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} - 9 q^{4} - 4 q^{5} + 15 q^{6} + 10 q^{7} + 15 q^{8} - 11 q^{9}+O(q^{10})$$ 4 * q - 5 * q^2 - 9 * q^4 - 4 * q^5 + 15 * q^6 + 10 * q^7 + 15 * q^8 - 11 * q^9 $$4 q - 5 q^{2} - 9 q^{4} - 4 q^{5} + 15 q^{6} + 10 q^{7} + 15 q^{8} - 11 q^{9} + q^{11} - 30 q^{12} - 20 q^{13} - 10 q^{14} + 19 q^{16} + 30 q^{18} + 25 q^{19} + 44 q^{20} - 35 q^{22} - 20 q^{23} + 5 q^{24} + 9 q^{25} - 10 q^{26} + 15 q^{27} - 60 q^{28} - 40 q^{29} - 80 q^{30} - 58 q^{31} + 65 q^{33} + 130 q^{34} + 80 q^{35} + 26 q^{36} + 90 q^{37} - 60 q^{38} + 50 q^{39} - 60 q^{40} - 80 q^{41} - 10 q^{42} + 24 q^{44} - 24 q^{45} + 30 q^{46} - 30 q^{47} - 40 q^{48} - 109 q^{49} - 45 q^{50} - 195 q^{51} + 110 q^{52} + 120 q^{53} - 76 q^{55} + 100 q^{56} + 45 q^{57} + 40 q^{58} + 23 q^{59} + 140 q^{60} + 10 q^{61} + 200 q^{62} + 90 q^{63} - 149 q^{64} + 90 q^{66} - 230 q^{67} - 260 q^{68} - 10 q^{69} - 40 q^{70} + 148 q^{71} - 95 q^{72} + 300 q^{73} - 270 q^{74} + 45 q^{75} - 200 q^{77} - 200 q^{78} + 70 q^{79} - 84 q^{80} - 116 q^{81} + 25 q^{82} + 225 q^{83} + 90 q^{84} + 260 q^{85} + 175 q^{86} + 55 q^{88} + 122 q^{89} - 20 q^{90} - 80 q^{91} + 40 q^{92} - 200 q^{93} + 120 q^{94} - 100 q^{95} + 340 q^{96} - 165 q^{97} + 31 q^{99}+O(q^{100})$$ 4 * q - 5 * q^2 - 9 * q^4 - 4 * q^5 + 15 * q^6 + 10 * q^7 + 15 * q^8 - 11 * q^9 + q^11 - 30 * q^12 - 20 * q^13 - 10 * q^14 + 19 * q^16 + 30 * q^18 + 25 * q^19 + 44 * q^20 - 35 * q^22 - 20 * q^23 + 5 * q^24 + 9 * q^25 - 10 * q^26 + 15 * q^27 - 60 * q^28 - 40 * q^29 - 80 * q^30 - 58 * q^31 + 65 * q^33 + 130 * q^34 + 80 * q^35 + 26 * q^36 + 90 * q^37 - 60 * q^38 + 50 * q^39 - 60 * q^40 - 80 * q^41 - 10 * q^42 + 24 * q^44 - 24 * q^45 + 30 * q^46 - 30 * q^47 - 40 * q^48 - 109 * q^49 - 45 * q^50 - 195 * q^51 + 110 * q^52 + 120 * q^53 - 76 * q^55 + 100 * q^56 + 45 * q^57 + 40 * q^58 + 23 * q^59 + 140 * q^60 + 10 * q^61 + 200 * q^62 + 90 * q^63 - 149 * q^64 + 90 * q^66 - 230 * q^67 - 260 * q^68 - 10 * q^69 - 40 * q^70 + 148 * q^71 - 95 * q^72 + 300 * q^73 - 270 * q^74 + 45 * q^75 - 200 * q^77 - 200 * q^78 + 70 * q^79 - 84 * q^80 - 116 * q^81 + 25 * q^82 + 225 * q^83 + 90 * q^84 + 260 * q^85 + 175 * q^86 + 55 * q^88 + 122 * q^89 - 20 * q^90 - 80 * q^91 + 40 * q^92 - 200 * q^93 + 120 * q^94 - 100 * q^95 + 340 * q^96 - 165 * q^97 + 31 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/11\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i
−0.690983 0.224514i −1.11803 + 0.812299i −2.80902 2.04087i 1.23607 + 3.80423i 0.954915 0.310271i 5.85410 8.05748i 3.19098 + 4.39201i −2.19098 + 6.74315i 2.90617i
6.1 −0.690983 + 0.224514i −1.11803 0.812299i −2.80902 + 2.04087i 1.23607 3.80423i 0.954915 + 0.310271i 5.85410 + 8.05748i 3.19098 4.39201i −2.19098 6.74315i 2.90617i
7.1 −1.80902 + 2.48990i 1.11803 3.44095i −1.69098 5.20431i −3.23607 + 2.35114i 6.54508 + 9.00854i −0.854102 + 0.277515i 4.30902 + 1.40008i −3.30902 2.40414i 12.3107i
8.1 −1.80902 2.48990i 1.11803 + 3.44095i −1.69098 + 5.20431i −3.23607 2.35114i 6.54508 9.00854i −0.854102 0.277515i 4.30902 1.40008i −3.30902 + 2.40414i 12.3107i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.3.d.a 4
3.b odd 2 1 99.3.k.a 4
4.b odd 2 1 176.3.n.a 4
5.b even 2 1 275.3.x.e 4
5.c odd 4 2 275.3.q.d 8
11.b odd 2 1 121.3.d.d 4
11.c even 5 1 121.3.b.b 4
11.c even 5 1 121.3.d.a 4
11.c even 5 1 121.3.d.c 4
11.c even 5 1 121.3.d.d 4
11.d odd 10 1 inner 11.3.d.a 4
11.d odd 10 1 121.3.b.b 4
11.d odd 10 1 121.3.d.a 4
11.d odd 10 1 121.3.d.c 4
33.f even 10 1 99.3.k.a 4
33.f even 10 1 1089.3.c.e 4
33.h odd 10 1 1089.3.c.e 4
44.g even 10 1 176.3.n.a 4
55.h odd 10 1 275.3.x.e 4
55.l even 20 2 275.3.q.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a 4 1.a even 1 1 trivial
11.3.d.a 4 11.d odd 10 1 inner
99.3.k.a 4 3.b odd 2 1
99.3.k.a 4 33.f even 10 1
121.3.b.b 4 11.c even 5 1
121.3.b.b 4 11.d odd 10 1
121.3.d.a 4 11.c even 5 1
121.3.d.a 4 11.d odd 10 1
121.3.d.c 4 11.c even 5 1
121.3.d.c 4 11.d odd 10 1
121.3.d.d 4 11.b odd 2 1
121.3.d.d 4 11.c even 5 1
176.3.n.a 4 4.b odd 2 1
176.3.n.a 4 44.g even 10 1
275.3.q.d 8 5.c odd 4 2
275.3.q.d 8 55.l even 20 2
275.3.x.e 4 5.b even 2 1
275.3.x.e 4 55.h odd 10 1
1089.3.c.e 4 33.f even 10 1
1089.3.c.e 4 33.h odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(11, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 5 T^{3} + 15 T^{2} + 15 T + 5$$
$3$ $$T^{4} + 10 T^{2} + 25 T + 25$$
$5$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$7$ $$T^{4} - 10 T^{3} + 80 T^{2} + 160 T + 80$$
$11$ $$T^{4} - T^{3} - 209 T^{2} + \cdots + 14641$$
$13$ $$T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 2000$$
$17$ $$T^{4} - 10985 T + 142805$$
$19$ $$T^{4} - 25 T^{3} + 200 T^{2} + \cdots + 605$$
$23$ $$(T^{2} + 10 T + 20)^{2}$$
$29$ $$T^{4} + 40 T^{3} + 1040 T^{2} + \cdots + 9680$$
$31$ $$T^{4} + 58 T^{3} + 1384 T^{2} + \cdots + 55696$$
$37$ $$T^{4} - 90 T^{3} + 4860 T^{2} + \cdots + 2624400$$
$41$ $$T^{4} + 80 T^{3} + 4720 T^{2} + \cdots + 8405$$
$43$ $$T^{4} + 1625 T^{2} + 581405$$
$47$ $$T^{4} + 30 T^{3} + 640 T^{2} + \cdots + 384400$$
$53$ $$T^{4} - 120 T^{3} + 5400 T^{2} + \cdots + 810000$$
$59$ $$T^{4} - 23 T^{3} + 1054 T^{2} + \cdots + 5041$$
$61$ $$T^{4} - 10 T^{3} - 120 T^{2} + \cdots + 403280$$
$67$ $$(T^{2} + 115 T + 2945)^{2}$$
$71$ $$T^{4} - 148 T^{3} + \cdots + 22619536$$
$73$ $$T^{4} - 300 T^{3} + \cdots + 93787805$$
$79$ $$T^{4} - 70 T^{3} + 3780 T^{2} + \cdots + 67280$$
$83$ $$T^{4} - 225 T^{3} + \cdots + 22281605$$
$89$ $$(T^{2} - 61 T - 7681)^{2}$$
$97$ $$T^{4} + 165 T^{3} + \cdots + 31416025$$