# Properties

 Label 66.2.e.a Level $66$ Weight $2$ Character orbit 66.e Analytic conductor $0.527$ 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] = [66,2,Mod(25,66)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(66, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 8]))

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

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

chi := DirichletCharacter("66.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 66.e (of order $$5$$, 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: $$0.527012653340$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/66\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.809017 − 0.587785i −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i
−0.809017 + 0.587785i −0.309017 0.951057i 0.309017 0.951057i 3.11803 + 2.26538i 0.809017 + 0.587785i 0.736068 2.26538i 0.309017 + 0.951057i −0.809017 + 0.587785i −3.85410
31.1 0.309017 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.881966 + 2.71441i −0.309017 0.951057i −3.73607 2.71441i −0.809017 + 0.587785i 0.309017 0.951057i 2.85410
37.1 −0.809017 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 3.11803 2.26538i 0.809017 0.587785i 0.736068 + 2.26538i 0.309017 0.951057i −0.809017 0.587785i −3.85410
49.1 0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.881966 2.71441i −0.309017 + 0.951057i −3.73607 + 2.71441i −0.809017 0.587785i 0.309017 + 0.951057i 2.85410
 $$n$$: 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.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.2.e.a 4
3.b odd 2 1 198.2.f.c 4
4.b odd 2 1 528.2.y.d 4
11.b odd 2 1 726.2.e.r 4
11.c even 5 1 inner 66.2.e.a 4
11.c even 5 1 726.2.a.l 2
11.c even 5 2 726.2.e.f 4
11.d odd 10 1 726.2.a.j 2
11.d odd 10 2 726.2.e.n 4
11.d odd 10 1 726.2.e.r 4
33.f even 10 1 2178.2.a.bb 2
33.h odd 10 1 198.2.f.c 4
33.h odd 10 1 2178.2.a.t 2
44.g even 10 1 5808.2.a.cg 2
44.h odd 10 1 528.2.y.d 4
44.h odd 10 1 5808.2.a.cb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.a 4 1.a even 1 1 trivial
66.2.e.a 4 11.c even 5 1 inner
198.2.f.c 4 3.b odd 2 1
198.2.f.c 4 33.h odd 10 1
528.2.y.d 4 4.b odd 2 1
528.2.y.d 4 44.h odd 10 1
726.2.a.j 2 11.d odd 10 1
726.2.a.l 2 11.c even 5 1
726.2.e.f 4 11.c even 5 2
726.2.e.n 4 11.d odd 10 2
726.2.e.r 4 11.b odd 2 1
726.2.e.r 4 11.d odd 10 1
2178.2.a.t 2 33.h odd 10 1
2178.2.a.bb 2 33.f even 10 1
5808.2.a.cb 2 44.h odd 10 1
5808.2.a.cg 2 44.g even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 8T_{5}^{3} + 34T_{5}^{2} - 77T_{5} + 121$$ acting on $$S_{2}^{\mathrm{new}}(66, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121$$
$7$ $$T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121$$
$11$ $$T^{4} + 4 T^{3} + 6 T^{2} + 44 T + 121$$
$13$ $$T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16$$
$17$ $$T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 256$$
$19$ $$T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16$$
$23$ $$(T^{2} + 2 T - 4)^{2}$$
$29$ $$T^{4} + T^{3} + 6 T^{2} - 4 T + 1$$
$31$ $$T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025$$
$37$ $$T^{4} - 4 T^{3} + 96 T^{2} + 256 T + 256$$
$41$ $$T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16$$
$43$ $$(T^{2} + 2 T - 4)^{2}$$
$47$ $$T^{4} + 8 T^{3} + 64 T^{2} + 192 T + 256$$
$53$ $$T^{4} + 7 T^{3} + 34 T^{2} + 88 T + 121$$
$59$ $$T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121$$
$61$ $$T^{4} - 14 T^{3} + 96 T^{2} + \cdots + 5776$$
$67$ $$(T - 4)^{4}$$
$71$ $$T^{4} - 2 T^{3} + 124 T^{2} + \cdots + 1936$$
$73$ $$T^{4} - 21 T^{3} + 196 T^{2} + \cdots + 2401$$
$79$ $$T^{4} + 16 T^{3} + 186 T^{2} + \cdots + 3481$$
$83$ $$T^{4} - 26 T^{3} + 256 T^{2} + \cdots + 121$$
$89$ $$(T^{2} + 10 T - 20)^{2}$$
$97$ $$T^{4} - 14 T^{3} + 196 T^{2} + \cdots + 2401$$