# Properties

 Label 169.3.f.a Level $169$ Weight $3$ Character orbit 169.f Analytic conductor $4.605$ 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] = [169,3,Mod(19,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([5]))

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

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

chi := DirichletCharacter("169.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 169.f (of order $$12$$, degree $$4$$, not 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: $$4.60491646769$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{4} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 4) q^{5} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{7} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{8} + (4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 2 \zeta_{12} + 5) q^{9}+O(q^{10})$$ q + (-z - 1) * q^2 + (-z^3 - z^2 - z) * q^3 + (z^2 - 2*z + 1) * q^4 + (-4*z^3 - z^2 + z + 4) * q^5 + (2*z^3 + 3*z^2 + z - 1) * q^6 + (-6*z^3 + 2*z^2 + 4*z + 4) * q^7 + (-z^3 + 5*z^2 + 5*z - 1) * q^8 + (4*z^3 - 5*z^2 - 2*z + 5) * q^9 $$q + ( - \zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{4} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 4) q^{5} + (2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + ( - 6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{7} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{8} + (4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 2 \zeta_{12} + 5) q^{9} + (5 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 5 \zeta_{12} - 8) q^{10} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 10 \zeta_{12} - 4) q^{11} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{12} + (4 \zeta_{12}^{3} - 8 \zeta_{12} - 10) q^{14} + (\zeta_{12}^{3} - \zeta_{12}^{2} - 9 \zeta_{12} - 8) q^{15} + ( - 8 \zeta_{12}^{3} - \zeta_{12}^{2} - 8 \zeta_{12}) q^{16} + (2 \zeta_{12}^{2} - 15 \zeta_{12} + 2) q^{17} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{18} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{19} + ( - 5 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{20} + ( - 6 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} - 6) q^{21} + (12 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12} - 2) q^{22} + ( - 15 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 15 \zeta_{12} - 6) q^{23} + ( - 13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} + 5 \zeta_{12} + 8) q^{24} + ( - \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 7) q^{25} + (12 \zeta_{12}^{3} - 24 \zeta_{12} - 8) q^{27} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 2 \zeta_{12} - 10) q^{28} + (6 \zeta_{12}^{3} + \zeta_{12}^{2} + 6 \zeta_{12}) q^{29} + (9 \zeta_{12}^{2} + 17 \zeta_{12} + 9) q^{30} + ( - \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} + 1) q^{31} + ( - 11 \zeta_{12}^{3} + \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{32} + (12 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 8 \zeta_{12} - 8) q^{33} + ( - 2 \zeta_{12}^{3} + 13 \zeta_{12}^{2} + 13 \zeta_{12} - 2) q^{34} + ( - 44 \zeta_{12}^{3} - 16 \zeta_{12}^{2} + 22 \zeta_{12} + 16) q^{35} + (16 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - 16 \zeta_{12} + 18) q^{36} + (29 \zeta_{12}^{3} + 29 \zeta_{12}^{2} - 24 \zeta_{12} - 5) q^{37} + ( - 5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{38} + ( - 19 \zeta_{12}^{3} + 38 \zeta_{12} + 18) q^{40} + (23 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 4 \zeta_{12} + 27) q^{41} + (14 \zeta_{12}^{3} + 22 \zeta_{12}^{2} + 14 \zeta_{12}) q^{42} + ( - 30 \zeta_{12}^{2} - 30) q^{43} + (10 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 24 \zeta_{12} - 10) q^{44} + (9 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 19 \zeta_{12} + 19) q^{45} + (12 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 9 \zeta_{12} - 9) q^{46} + (27 \zeta_{12}^{3} - 20 \zeta_{12}^{2} - 20 \zeta_{12} + 27) q^{47} + (18 \zeta_{12}^{3} + 25 \zeta_{12}^{2} - 9 \zeta_{12} - 25) q^{48} + ( - 7 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 7 \zeta_{12} + 24) q^{49} + (15 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 7 \zeta_{12} - 8) q^{50} + (9 \zeta_{12}^{3} + 26 \zeta_{12}^{2} - 13) q^{51} + ( - 27 \zeta_{12}^{3} + 54 \zeta_{12} + 32) q^{53} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 32 \zeta_{12} + 20) q^{54} + (6 \zeta_{12}^{3} - 50 \zeta_{12}^{2} + 6 \zeta_{12}) q^{55} + (14 \zeta_{12}^{2} + 48 \zeta_{12} + 14) q^{56} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{57} + ( - 7 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{58} + (22 \zeta_{12}^{3} - 60 \zeta_{12}^{2} + 38 \zeta_{12} + 38) q^{59} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{60} + (42 \zeta_{12}^{3} + 62 \zeta_{12}^{2} - 21 \zeta_{12} - 62) q^{61} + ( - 7 \zeta_{12}^{3} + \zeta_{12}^{2} + 7 \zeta_{12} - 2) q^{62} + ( - 26 \zeta_{12} + 26) q^{63} + (14 \zeta_{12}^{3} + 62 \zeta_{12}^{2} - 31) q^{64} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12} + 20) q^{66} + (11 \zeta_{12}^{3} - 11 \zeta_{12}^{2} + 37 \zeta_{12} + 48) q^{67} + ( - 19 \zeta_{12}^{3} + 36 \zeta_{12}^{2} - 19 \zeta_{12}) q^{68} + ( - 12 \zeta_{12}^{2} - 6 \zeta_{12} - 12) q^{69} + (60 \zeta_{12}^{3} + 38 \zeta_{12}^{2} - 38 \zeta_{12} - 60) q^{70} + ( - 32 \zeta_{12}^{3} - 11 \zeta_{12}^{2} + 21 \zeta_{12} - 21) q^{71} + ( - 19 \zeta_{12}^{3} + 17 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{72} + (5 \zeta_{12}^{3} - 39 \zeta_{12}^{2} - 39 \zeta_{12} + 5) q^{73} + ( - 58 \zeta_{12}^{3} - 34 \zeta_{12}^{2} + 29 \zeta_{12} + 34) q^{74} + (22 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 22 \zeta_{12} - 16) q^{75} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 13 \zeta_{12} + 10) q^{76} + (20 \zeta_{12}^{3} - 88 \zeta_{12}^{2} + 44) q^{77} + (42 \zeta_{12}^{3} - 84 \zeta_{12} - 10) q^{79} + ( - 13 \zeta_{12}^{3} + 13 \zeta_{12}^{2} - 44 \zeta_{12} - 57) q^{80} + (38 \zeta_{12}^{3} - \zeta_{12}^{2} + 38 \zeta_{12}) q^{81} + ( - 4 \zeta_{12}^{2} - 31 \zeta_{12} - 4) q^{82} + ( - 60 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26 \zeta_{12} + 60) q^{83} + ( - 4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 10 \zeta_{12} - 10) q^{84} + (\zeta_{12}^{3} + 49 \zeta_{12}^{2} - 50 \zeta_{12} - 50) q^{85} + (30 \zeta_{12}^{3} + 30 \zeta_{12}^{2} + 30 \zeta_{12} + 30) q^{86} + ( - 14 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 7 \zeta_{12} + 19) q^{87} + (6 \zeta_{12}^{3} - 34 \zeta_{12}^{2} - 6 \zeta_{12} + 68) q^{88} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 45 \zeta_{12} - 50) q^{89} + (\zeta_{12}^{3} + 20 \zeta_{12}^{2} - 10) q^{90} + ( - 21 \zeta_{12}^{3} + 42 \zeta_{12} - 39) q^{92} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{93} + ( - 7 \zeta_{12}^{3} + 13 \zeta_{12}^{2} - 7 \zeta_{12}) q^{94} + (13 \zeta_{12}^{2} - \zeta_{12} + 13) q^{95} + (9 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 9) q^{96} + (54 \zeta_{12}^{3} + 35 \zeta_{12}^{2} - 19 \zeta_{12} + 19) q^{97} + (19 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 31 \zeta_{12} - 31) q^{98} + ( - 78 \zeta_{12}^{3} + 52 \zeta_{12}^{2} + 52 \zeta_{12} - 78) q^{99}+O(q^{100})$$ q + (-z - 1) * q^2 + (-z^3 - z^2 - z) * q^3 + (z^2 - 2*z + 1) * q^4 + (-4*z^3 - z^2 + z + 4) * q^5 + (2*z^3 + 3*z^2 + z - 1) * q^6 + (-6*z^3 + 2*z^2 + 4*z + 4) * q^7 + (-z^3 + 5*z^2 + 5*z - 1) * q^8 + (4*z^3 - 5*z^2 - 2*z + 5) * q^9 + (5*z^3 + 4*z^2 - 5*z - 8) * q^10 + (-6*z^3 - 6*z^2 + 10*z - 4) * q^11 + (-z^3 + 2*z^2 - 1) * q^12 + (4*z^3 - 8*z - 10) * q^14 + (z^3 - z^2 - 9*z - 8) * q^15 + (-8*z^3 - z^2 - 8*z) * q^16 + (2*z^2 - 15*z + 2) * q^17 + (z^3 + 3*z^2 - 3*z - 1) * q^18 + (4*z^3 + z^2 - 3*z + 3) * q^19 + (-5*z^3 + 8*z^2 - 3*z - 3) * q^20 + (-6*z^3 - 8*z^2 - 8*z - 6) * q^21 + (12*z^3 + 2*z^2 - 6*z - 2) * q^22 + (-15*z^3 + 3*z^2 + 15*z - 6) * q^23 + (-13*z^3 - 13*z^2 + 5*z + 8) * q^24 + (-z^3 - 14*z^2 + 7) * q^25 + (12*z^3 - 24*z - 8) * q^27 + (-12*z^3 + 12*z^2 + 2*z - 10) * q^28 + (6*z^3 + z^2 + 6*z) * q^29 + (9*z^2 + 17*z + 9) * q^30 + (-z^3 + 8*z^2 - 8*z + 1) * q^31 + (-11*z^3 + z^2 + 12*z - 12) * q^32 + (12*z^3 - 4*z^2 - 8*z - 8) * q^33 + (-2*z^3 + 13*z^2 + 13*z - 2) * q^34 + (-44*z^3 - 16*z^2 + 22*z + 16) * q^35 + (16*z^3 - 9*z^2 - 16*z + 18) * q^36 + (29*z^3 + 29*z^2 - 24*z - 5) * q^37 + (-5*z^3 - 2*z^2 + 1) * q^38 + (-19*z^3 + 38*z + 18) * q^40 + (23*z^3 - 23*z^2 + 4*z + 27) * q^41 + (14*z^3 + 22*z^2 + 14*z) * q^42 + (-30*z^2 - 30) * q^43 + (10*z^3 - 24*z^2 + 24*z - 10) * q^44 + (9*z^3 - 10*z^2 - 19*z + 19) * q^45 + (12*z^3 - 3*z^2 - 9*z - 9) * q^46 + (27*z^3 - 20*z^2 - 20*z + 27) * q^47 + (18*z^3 + 25*z^2 - 9*z - 25) * q^48 + (-7*z^3 - 12*z^2 + 7*z + 24) * q^49 + (15*z^3 + 15*z^2 - 7*z - 8) * q^50 + (9*z^3 + 26*z^2 - 13) * q^51 + (-27*z^3 + 54*z + 32) * q^53 + (-12*z^3 + 12*z^2 + 32*z + 20) * q^54 + (6*z^3 - 50*z^2 + 6*z) * q^55 + (14*z^2 + 48*z + 14) * q^56 + (-6*z^3 - 2*z^2 + 2*z + 6) * q^57 + (-7*z^3 - 13*z^2 - 6*z + 6) * q^58 + (22*z^3 - 60*z^2 + 38*z + 38) * q^59 + (-5*z^3 + 6*z^2 + 6*z - 5) * q^60 + (42*z^3 + 62*z^2 - 21*z - 62) * q^61 + (-7*z^3 + z^2 + 7*z - 2) * q^62 + (-26*z + 26) * q^63 + (14*z^3 + 62*z^2 - 31) * q^64 + (-8*z^3 + 16*z + 20) * q^66 + (11*z^3 - 11*z^2 + 37*z + 48) * q^67 + (-19*z^3 + 36*z^2 - 19*z) * q^68 + (-12*z^2 - 6*z - 12) * q^69 + (60*z^3 + 38*z^2 - 38*z - 60) * q^70 + (-32*z^3 - 11*z^2 + 21*z - 21) * q^71 + (-19*z^3 + 17*z^2 + 2*z + 2) * q^72 + (5*z^3 - 39*z^2 - 39*z + 5) * q^73 + (-58*z^3 - 34*z^2 + 29*z + 34) * q^74 + (22*z^3 + 8*z^2 - 22*z - 16) * q^75 + (3*z^3 + 3*z^2 - 13*z + 10) * q^76 + (20*z^3 - 88*z^2 + 44) * q^77 + (42*z^3 - 84*z - 10) * q^79 + (-13*z^3 + 13*z^2 - 44*z - 57) * q^80 + (38*z^3 - z^2 + 38*z) * q^81 + (-4*z^2 - 31*z - 4) * q^82 + (-60*z^3 - 26*z^2 + 26*z + 60) * q^83 + (-4*z^3 + 6*z^2 + 10*z - 10) * q^84 + (z^3 + 49*z^2 - 50*z - 50) * q^85 + (30*z^3 + 30*z^2 + 30*z + 30) * q^86 + (-14*z^3 - 19*z^2 + 7*z + 19) * q^87 + (6*z^3 - 34*z^2 - 6*z + 68) * q^88 + (5*z^3 + 5*z^2 + 45*z - 50) * q^89 + (z^3 + 20*z^2 - 10) * q^90 + (-21*z^3 + 42*z - 39) * q^92 + (-8*z^3 + 8*z^2 + 6*z - 2) * q^93 + (-7*z^3 + 13*z^2 - 7*z) * q^94 + (13*z^2 - z + 13) * q^95 + (9*z^3 - 2*z^2 + 2*z - 9) * q^96 + (54*z^3 + 35*z^2 - 19*z + 19) * q^97 + (19*z^3 + 12*z^2 - 31*z - 31) * q^98 + (-78*z^3 + 52*z^2 + 52*z - 78) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 2 q^{3} + 6 q^{4} + 14 q^{5} + 2 q^{6} + 20 q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 2 * q^3 + 6 * q^4 + 14 * q^5 + 2 * q^6 + 20 * q^7 + 6 * q^8 + 10 * q^9 $$4 q - 4 q^{2} - 2 q^{3} + 6 q^{4} + 14 q^{5} + 2 q^{6} + 20 q^{7} + 6 q^{8} + 10 q^{9} - 24 q^{10} - 28 q^{11} - 40 q^{14} - 34 q^{15} - 2 q^{16} + 12 q^{17} + 2 q^{18} + 14 q^{19} + 4 q^{20} - 40 q^{21} - 4 q^{22} - 18 q^{23} + 6 q^{24} - 32 q^{27} - 16 q^{28} + 2 q^{29} + 54 q^{30} + 20 q^{31} - 46 q^{32} - 40 q^{33} + 18 q^{34} + 32 q^{35} + 54 q^{36} + 38 q^{37} + 72 q^{40} + 62 q^{41} + 44 q^{42} - 180 q^{43} - 88 q^{44} + 56 q^{45} - 42 q^{46} + 68 q^{47} - 50 q^{48} + 72 q^{49} - 2 q^{50} + 128 q^{53} + 104 q^{54} - 100 q^{55} + 84 q^{56} + 20 q^{57} - 2 q^{58} + 32 q^{59} - 8 q^{60} - 124 q^{61} - 6 q^{62} + 104 q^{63} + 80 q^{66} + 170 q^{67} + 72 q^{68} - 72 q^{69} - 164 q^{70} - 106 q^{71} + 42 q^{72} - 58 q^{73} + 68 q^{74} - 48 q^{75} + 46 q^{76} - 40 q^{79} - 202 q^{80} - 2 q^{81} - 24 q^{82} + 188 q^{83} - 28 q^{84} - 102 q^{85} + 180 q^{86} + 38 q^{87} + 204 q^{88} - 190 q^{89} - 156 q^{92} + 8 q^{93} + 26 q^{94} + 78 q^{95} - 40 q^{96} + 146 q^{97} - 100 q^{98} - 208 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 - 2 * q^3 + 6 * q^4 + 14 * q^5 + 2 * q^6 + 20 * q^7 + 6 * q^8 + 10 * q^9 - 24 * q^10 - 28 * q^11 - 40 * q^14 - 34 * q^15 - 2 * q^16 + 12 * q^17 + 2 * q^18 + 14 * q^19 + 4 * q^20 - 40 * q^21 - 4 * q^22 - 18 * q^23 + 6 * q^24 - 32 * q^27 - 16 * q^28 + 2 * q^29 + 54 * q^30 + 20 * q^31 - 46 * q^32 - 40 * q^33 + 18 * q^34 + 32 * q^35 + 54 * q^36 + 38 * q^37 + 72 * q^40 + 62 * q^41 + 44 * q^42 - 180 * q^43 - 88 * q^44 + 56 * q^45 - 42 * q^46 + 68 * q^47 - 50 * q^48 + 72 * q^49 - 2 * q^50 + 128 * q^53 + 104 * q^54 - 100 * q^55 + 84 * q^56 + 20 * q^57 - 2 * q^58 + 32 * q^59 - 8 * q^60 - 124 * q^61 - 6 * q^62 + 104 * q^63 + 80 * q^66 + 170 * q^67 + 72 * q^68 - 72 * q^69 - 164 * q^70 - 106 * q^71 + 42 * q^72 - 58 * q^73 + 68 * q^74 - 48 * q^75 + 46 * q^76 - 40 * q^79 - 202 * q^80 - 2 * q^81 - 24 * q^82 + 188 * q^83 - 28 * q^84 - 102 * q^85 + 180 * q^86 + 38 * q^87 + 204 * q^88 - 190 * q^89 - 156 * q^92 + 8 * q^93 + 26 * q^94 + 78 * q^95 - 40 * q^96 + 146 * q^97 - 100 * q^98 - 208 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
−0.133975 0.500000i 0.366025 0.633975i 3.23205 1.86603i 2.63397 2.63397i −0.366025 0.0980762i 1.53590 5.73205i −2.83013 2.83013i 4.23205 + 7.33013i −1.66987 0.964102i
80.1 −1.86603 0.500000i −1.36603 2.36603i −0.232051 0.133975i 4.36603 4.36603i 1.36603 + 5.09808i 8.46410 2.26795i 5.83013 + 5.83013i 0.767949 1.33013i −10.3301 + 5.96410i
89.1 −0.133975 + 0.500000i 0.366025 + 0.633975i 3.23205 + 1.86603i 2.63397 + 2.63397i −0.366025 + 0.0980762i 1.53590 + 5.73205i −2.83013 + 2.83013i 4.23205 7.33013i −1.66987 + 0.964102i
150.1 −1.86603 + 0.500000i −1.36603 + 2.36603i −0.232051 + 0.133975i 4.36603 + 4.36603i 1.36603 5.09808i 8.46410 + 2.26795i 5.83013 5.83013i 0.767949 + 1.33013i −10.3301 5.96410i
 $$n$$: 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
13.f odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.f.a 4
13.b even 2 1 169.3.f.c 4
13.c even 3 1 169.3.d.c 4
13.c even 3 1 169.3.f.b 4
13.d odd 4 1 13.3.f.a 4
13.d odd 4 1 169.3.f.b 4
13.e even 6 1 13.3.f.a 4
13.e even 6 1 169.3.d.a 4
13.f odd 12 1 169.3.d.a 4
13.f odd 12 1 169.3.d.c 4
13.f odd 12 1 inner 169.3.f.a 4
13.f odd 12 1 169.3.f.c 4
39.f even 4 1 117.3.bd.b 4
39.h odd 6 1 117.3.bd.b 4
52.f even 4 1 208.3.bd.d 4
52.i odd 6 1 208.3.bd.d 4
65.f even 4 1 325.3.w.b 4
65.g odd 4 1 325.3.t.a 4
65.k even 4 1 325.3.w.a 4
65.l even 6 1 325.3.t.a 4
65.r odd 12 1 325.3.w.a 4
65.r odd 12 1 325.3.w.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.f.a 4 13.d odd 4 1
13.3.f.a 4 13.e even 6 1
117.3.bd.b 4 39.f even 4 1
117.3.bd.b 4 39.h odd 6 1
169.3.d.a 4 13.e even 6 1
169.3.d.a 4 13.f odd 12 1
169.3.d.c 4 13.c even 3 1
169.3.d.c 4 13.f odd 12 1
169.3.f.a 4 1.a even 1 1 trivial
169.3.f.a 4 13.f odd 12 1 inner
169.3.f.b 4 13.c even 3 1
169.3.f.b 4 13.d odd 4 1
169.3.f.c 4 13.b even 2 1
169.3.f.c 4 13.f odd 12 1
208.3.bd.d 4 52.f even 4 1
208.3.bd.d 4 52.i odd 6 1
325.3.t.a 4 65.g odd 4 1
325.3.t.a 4 65.l even 6 1
325.3.w.a 4 65.k even 4 1
325.3.w.a 4 65.r odd 12 1
325.3.w.b 4 65.f even 4 1
325.3.w.b 4 65.r odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 4T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$5$ $$T^{4} - 14 T^{3} + 98 T^{2} + \cdots + 529$$
$7$ $$T^{4} - 20 T^{3} + 164 T^{2} + \cdots + 2704$$
$11$ $$T^{4} + 28 T^{3} + 200 T^{2} + \cdots + 10816$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 12 T^{3} - 165 T^{2} + \cdots + 45369$$
$19$ $$T^{4} - 14 T^{3} + 74 T^{2} + \cdots + 484$$
$23$ $$T^{4} + 18 T^{3} - 90 T^{2} + \cdots + 39204$$
$29$ $$T^{4} - 2 T^{3} + 111 T^{2} + \cdots + 11449$$
$31$ $$T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 2116$$
$37$ $$T^{4} - 38 T^{3} + 1517 T^{2} + \cdots + 1868689$$
$41$ $$T^{4} - 62 T^{3} + 3461 T^{2} + \cdots + 833569$$
$43$ $$(T^{2} + 90 T + 2700)^{2}$$
$47$ $$T^{4} - 68 T^{3} + 2312 T^{2} + \cdots + 484$$
$53$ $$(T^{2} - 64 T - 1163)^{2}$$
$59$ $$T^{4} - 32 T^{3} + 6980 T^{2} + \cdots + 16613776$$
$61$ $$T^{4} + 124 T^{3} + 12855 T^{2} + \cdots + 6355441$$
$67$ $$T^{4} - 170 T^{3} + 10706 T^{2} + \cdots + 9721924$$
$71$ $$T^{4} + 106 T^{3} + 4658 T^{2} + \cdots + 2208196$$
$73$ $$T^{4} + 58 T^{3} + 1682 T^{2} + \cdots + 3463321$$
$79$ $$(T^{2} + 20 T - 5192)^{2}$$
$83$ $$T^{4} - 188 T^{3} + \cdots + 11587216$$
$89$ $$T^{4} + 190 T^{3} + 12050 T^{2} + \cdots + 8702500$$
$97$ $$T^{4} - 146 T^{3} + \cdots + 18028516$$