# Properties

 Label 66.2.e.b 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{10} q^{2} - \zeta_{10}^{3} q^{3} + \zeta_{10}^{2} q^{4} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{6} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{7} + \zeta_{10}^{3} q^{8} - \zeta_{10} q^{9} +O(q^{10})$$ q + z * q^2 - z^3 * q^3 + z^2 * q^4 + (2*z^3 - 3*z^2 + 3*z - 2) * q^5 + (-z^3 + z^2 - z + 1) * q^6 + (-z^3 - 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} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{5} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{6} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{7} + \zeta_{10}^{3} q^{8} - \zeta_{10} q^{9} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 2) q^{10} + (2 \zeta_{10}^{3} - 2 \zeta_{10} - 1) q^{11} + q^{12} + (2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{13} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{14} + ( - \zeta_{10}^{3} + 3 \zeta_{10}^{2} - \zeta_{10}) q^{15} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{16} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 4) q^{17} - \zeta_{10}^{2} q^{18} + ( - 4 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{19} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{20} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{21} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10} - 2) q^{22} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 4) q^{23} + \zeta_{10} q^{24} + (5 \zeta_{10} - 5) 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} + ( - \zeta_{10}^{3} + 1) q^{28} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 3 \zeta_{10}) q^{29} + (2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{30} + (\zeta_{10}^{2} + 4 \zeta_{10} + 1) q^{31} - q^{32} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{33} - 4 q^{34} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{35} - \zeta_{10}^{3} q^{36} + (8 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 8 \zeta_{10}) q^{37} + ( - 4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 4) q^{38} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{39} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + \zeta_{10}) q^{40} + ( - 4 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{41} + ( - \zeta_{10}^{2} - 1) q^{42} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 8) q^{43} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{44} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{45} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{46} - 4 \zeta_{10}^{3} q^{47} + \zeta_{10}^{2} q^{48} + ( - 5 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 5) q^{49} + (5 \zeta_{10}^{2} - 5 \zeta_{10}) q^{50} + 4 \zeta_{10}^{2} q^{51} + (2 \zeta_{10} - 2) q^{52} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{53} - q^{54} + (2 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - \zeta_{10} + 6) q^{55} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{56} + ( - 6 \zeta_{10}^{2} + 2 \zeta_{10} - 6) q^{57} + (2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{58} + ( - 7 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 7 \zeta_{10}) q^{59} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 2) q^{60} + (6 \zeta_{10}^{3} - 6) q^{61} + (\zeta_{10}^{3} + 4 \zeta_{10}^{2} + \zeta_{10}) q^{62} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{63} - \zeta_{10} q^{64} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{65} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10} - 3) q^{66} + (8 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 8) q^{67} - 4 \zeta_{10} q^{68} + ( - 4 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{69} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{70} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 6) q^{71} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{72} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 9 \zeta_{10}) q^{73} + (4 \zeta_{10}^{3} + 8 \zeta_{10} - 8) q^{74} + (5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{75} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 4) q^{76} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3 \zeta_{10}) q^{77} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{78} + ( - 3 \zeta_{10}^{2} + 5 \zeta_{10} - 3) q^{79} + ( - 2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{80} + \zeta_{10}^{2} q^{81} + ( - 4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 4) q^{82} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{83} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{84} + ( - 8 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{85} + ( - 2 \zeta_{10}^{2} - 6 \zeta_{10} - 2) q^{86} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 2) q^{87} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{88} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 10) q^{89} + ( - \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{90} - 2 \zeta_{10}^{3} q^{91} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 6 \zeta_{10}) q^{92} + ( - 5 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 5) q^{93} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 4) q^{94} + (14 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 14 \zeta_{10}) q^{95} + \zeta_{10}^{3} q^{96} + ( - \zeta_{10}^{2} + 9 \zeta_{10} - 1) q^{97} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 5) q^{98} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - \zeta_{10} + 2) q^{99} +O(q^{100})$$ q + z * q^2 - z^3 * q^3 + z^2 * q^4 + (2*z^3 - 3*z^2 + 3*z - 2) * q^5 + (-z^3 + z^2 - z + 1) * q^6 + (-z^3 - z) * q^7 + z^3 * q^8 - z * q^9 + (-z^3 + z^2 - 2) * q^10 + (2*z^3 - 2*z - 1) * q^11 + q^12 + (2*z^2 - 2*z + 2) * q^13 + (-z^3 - z + 1) * q^14 + (-z^3 + 3*z^2 - z) * q^15 + (z^3 - z^2 + z - 1) * q^16 + (4*z^3 - 4*z^2 + 4*z - 4) * q^17 - z^2 * q^18 + (-4*z^3 - 6*z + 6) * q^19 + (z^2 - 3*z + 1) * q^20 + (z^3 - z^2 - 1) * q^21 + (2*z^3 - 4*z^2 + z - 2) * q^22 + (-6*z^3 + 6*z^2 + 4) * q^23 + z * q^24 + (5*z - 5) * q^25 + (2*z^3 - 2*z^2 + 2*z) * q^26 + (z^3 - z^2 + z - 1) * q^27 + (-z^3 + 1) * q^28 + (-3*z^3 + 5*z^2 - 3*z) * q^29 + (2*z^3 - z + 1) * q^30 + (z^2 + 4*z + 1) * q^31 - q^32 + (3*z^3 - 2*z^2 + 4*z - 2) * q^33 - 4 * q^34 + (2*z^2 - z + 2) * q^35 - z^3 * q^36 + (8*z^3 - 4*z^2 + 8*z) * q^37 + (-4*z^3 - 2*z^2 + 2*z + 4) * q^38 + (-2*z^2 + 2*z) * q^39 + (z^3 - 3*z^2 + z) * q^40 + (-4*z^3 - 6*z + 6) * q^41 + (-z^2 - 1) * q^42 + (2*z^3 - 2*z^2 - 8) * q^43 + (-2*z^3 - z^2 - 2) * q^44 + (z^3 - z^2 + 2) * q^45 + (6*z^2 - 2*z + 6) * q^46 - 4*z^3 * q^47 + z^2 * q^48 + (-5*z^3 + 6*z^2 - 6*z + 5) * q^49 + (5*z^2 - 5*z) * q^50 + 4*z^2 * q^51 + (2*z - 2) * q^52 + (-z^2 - 2*z - 1) * q^53 - q^54 + (2*z^3 - 5*z^2 - z + 6) * q^55 + (-z^3 + z^2 + 1) * q^56 + (-6*z^2 + 2*z - 6) * q^57 + (2*z^3 - 3*z + 3) * q^58 + (-7*z^3 + 6*z^2 - 7*z) * q^59 + (2*z^3 - 3*z^2 + 3*z - 2) * q^60 + (6*z^3 - 6) * q^61 + (z^3 + 4*z^2 + z) * q^62 + (z^3 + z - 1) * q^63 - z * q^64 + (6*z^3 - 6*z^2 + 2) * q^65 + (z^3 + z^2 + z - 3) * q^66 + (8*z^3 - 8*z^2 - 8) * q^67 - 4*z * q^68 + (-4*z^3 - 6*z + 6) * q^69 + (2*z^3 - z^2 + 2*z) * q^70 + (6*z^3 - 6*z^2 + 6*z - 6) * q^71 + (-z^3 + z^2 - z + 1) * q^72 + (-9*z^3 + 9*z^2 - 9*z) * q^73 + (4*z^3 + 8*z - 8) * q^74 + (5*z^2 - 5*z + 5) * q^75 + (-6*z^3 + 6*z^2 + 4) * q^76 + (z^3 + 2*z^2 + 3*z) * q^77 + (-2*z^3 + 2*z^2) * q^78 + (-3*z^2 + 5*z - 3) * q^79 + (-2*z^3 + z - 1) * q^80 + z^2 * q^81 + (-4*z^3 - 2*z^2 + 2*z + 4) * q^82 + (-z^2 + z) * q^83 + (-z^3 - z) * q^84 + (-8*z^3 + 4*z - 4) * q^85 + (-2*z^2 - 6*z - 2) * q^86 + (3*z^3 - 3*z^2 + 2) * q^87 + (-3*z^3 + 2*z^2 - 4*z + 2) * q^88 + (-2*z^3 + 2*z^2 + 10) * q^89 + (-z^2 + 3*z - 1) * q^90 - 2*z^3 * q^91 + (6*z^3 - 2*z^2 + 6*z) * q^92 + (-5*z^3 + 4*z^2 - 4*z + 5) * q^93 + (-4*z^3 + 4*z^2 - 4*z + 4) * q^94 + (14*z^3 - 12*z^2 + 14*z) * q^95 + z^3 * q^96 + (-z^2 + 9*z - 1) * q^97 + (z^3 - z^2 + 5) * q^98 + (-2*z^3 + 4*z^2 - z + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 - q^3 - q^4 + q^6 - 2 * q^7 + q^8 - q^9 $$4 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} + q^{8} - q^{9} - 10 q^{10} - 4 q^{11} + 4 q^{12} + 4 q^{13} + 2 q^{14} - 5 q^{15} - q^{16} - 4 q^{17} + q^{18} + 14 q^{19} - 2 q^{21} - q^{22} + 4 q^{23} + q^{24} - 15 q^{25} + 6 q^{26} - q^{27} + 3 q^{28} - 11 q^{29} + 5 q^{30} + 7 q^{31} - 4 q^{32} + q^{33} - 16 q^{34} + 5 q^{35} - q^{36} + 20 q^{37} + 16 q^{38} + 4 q^{39} + 5 q^{40} + 14 q^{41} - 3 q^{42} - 28 q^{43} - 9 q^{44} + 10 q^{45} + 16 q^{46} - 4 q^{47} - q^{48} + 3 q^{49} - 10 q^{50} - 4 q^{51} - 6 q^{52} - 5 q^{53} - 4 q^{54} + 30 q^{55} + 2 q^{56} - 16 q^{57} + 11 q^{58} - 20 q^{59} - 18 q^{61} - 2 q^{62} - 2 q^{63} - q^{64} + 20 q^{65} - 11 q^{66} - 16 q^{67} - 4 q^{68} + 14 q^{69} + 5 q^{70} - 6 q^{71} + q^{72} - 27 q^{73} - 20 q^{74} + 10 q^{75} + 4 q^{76} + 2 q^{77} - 4 q^{78} - 4 q^{79} - 5 q^{80} - q^{81} + 16 q^{82} + 2 q^{83} - 2 q^{84} - 20 q^{85} - 12 q^{86} + 14 q^{87} - q^{88} + 36 q^{89} - 2 q^{91} + 14 q^{92} + 7 q^{93} + 4 q^{94} + 40 q^{95} + q^{96} + 6 q^{97} + 22 q^{98} + q^{99}+O(q^{100})$$ 4 * q + q^2 - q^3 - q^4 + q^6 - 2 * q^7 + q^8 - q^9 - 10 * q^10 - 4 * q^11 + 4 * q^12 + 4 * q^13 + 2 * q^14 - 5 * q^15 - q^16 - 4 * q^17 + q^18 + 14 * q^19 - 2 * q^21 - q^22 + 4 * q^23 + q^24 - 15 * q^25 + 6 * q^26 - q^27 + 3 * q^28 - 11 * q^29 + 5 * q^30 + 7 * q^31 - 4 * q^32 + q^33 - 16 * q^34 + 5 * q^35 - q^36 + 20 * q^37 + 16 * q^38 + 4 * q^39 + 5 * q^40 + 14 * q^41 - 3 * q^42 - 28 * q^43 - 9 * q^44 + 10 * q^45 + 16 * q^46 - 4 * q^47 - q^48 + 3 * q^49 - 10 * q^50 - 4 * q^51 - 6 * q^52 - 5 * q^53 - 4 * q^54 + 30 * q^55 + 2 * q^56 - 16 * q^57 + 11 * q^58 - 20 * q^59 - 18 * q^61 - 2 * q^62 - 2 * q^63 - q^64 + 20 * q^65 - 11 * q^66 - 16 * q^67 - 4 * q^68 + 14 * q^69 + 5 * q^70 - 6 * q^71 + q^72 - 27 * q^73 - 20 * q^74 + 10 * q^75 + 4 * q^76 + 2 * q^77 - 4 * q^78 - 4 * q^79 - 5 * q^80 - q^81 + 16 * q^82 + 2 * q^83 - 2 * q^84 - 20 * q^85 - 12 * q^86 + 14 * q^87 - q^88 + 36 * q^89 - 2 * q^91 + 14 * q^92 + 7 * q^93 + 4 * q^94 + 40 * q^95 + q^96 + 6 * q^97 + 22 * q^98 + 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 −1.11803 0.812299i 0.809017 + 0.587785i −0.500000 + 1.53884i −0.309017 0.951057i −0.809017 + 0.587785i −1.38197
31.1 −0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 1.11803 + 3.44095i −0.309017 0.951057i −0.500000 0.363271i 0.809017 0.587785i 0.309017 0.951057i −3.61803
37.1 0.809017 + 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −1.11803 + 0.812299i 0.809017 0.587785i −0.500000 1.53884i −0.309017 + 0.951057i −0.809017 0.587785i −1.38197
49.1 −0.309017 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 1.11803 3.44095i −0.309017 + 0.951057i −0.500000 + 0.363271i 0.809017 + 0.587785i 0.309017 + 0.951057i −3.61803
 $$n$$: 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.b 4
3.b odd 2 1 198.2.f.a 4
4.b odd 2 1 528.2.y.g 4
11.b odd 2 1 726.2.e.c 4
11.c even 5 1 inner 66.2.e.b 4
11.c even 5 1 726.2.a.k 2
11.c even 5 2 726.2.e.j 4
11.d odd 10 1 726.2.a.m 2
11.d odd 10 2 726.2.e.a 4
11.d odd 10 1 726.2.e.c 4
33.f even 10 1 2178.2.a.o 2
33.h odd 10 1 198.2.f.a 4
33.h odd 10 1 2178.2.a.v 2
44.g even 10 1 5808.2.a.by 2
44.h odd 10 1 528.2.y.g 4
44.h odd 10 1 5808.2.a.bz 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.e.b 4 1.a even 1 1 trivial
66.2.e.b 4 11.c even 5 1 inner
198.2.f.a 4 3.b odd 2 1
198.2.f.a 4 33.h odd 10 1
528.2.y.g 4 4.b odd 2 1
528.2.y.g 4 44.h odd 10 1
726.2.a.k 2 11.c even 5 1
726.2.a.m 2 11.d odd 10 1
726.2.e.a 4 11.d odd 10 2
726.2.e.c 4 11.b odd 2 1
726.2.e.c 4 11.d odd 10 1
726.2.e.j 4 11.c even 5 2
2178.2.a.o 2 33.f even 10 1
2178.2.a.v 2 33.h odd 10 1
5808.2.a.by 2 44.g even 10 1
5808.2.a.bz 2 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 10T_{5}^{2} + 25T_{5} + 25$$ acting on $$S_{2}^{\mathrm{new}}(66, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4} + T^{3} + T^{2} + T + 1$$
$5$ $$T^{4} + 10 T^{2} + 25 T + 25$$
$7$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$
$11$ $$T^{4} + 4 T^{3} + 6 T^{2} + 44 T + 121$$
$13$ $$T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16$$
$17$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$19$ $$T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936$$
$23$ $$(T^{2} - 2 T - 44)^{2}$$
$29$ $$T^{4} + 11 T^{3} + 46 T^{2} - 4 T + 1$$
$31$ $$T^{4} - 7 T^{3} + 24 T^{2} - 38 T + 361$$
$37$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$41$ $$T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936$$
$43$ $$(T^{2} + 14 T + 44)^{2}$$
$47$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$53$ $$T^{4} + 5 T^{3} + 10 T^{2} + 25$$
$59$ $$T^{4} + 20 T^{3} + 190 T^{2} + \cdots + 3025$$
$61$ $$T^{4} + 18 T^{3} + 144 T^{2} + \cdots + 1296$$
$67$ $$(T^{2} + 8 T - 64)^{2}$$
$71$ $$T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 1296$$
$73$ $$T^{4} + 27 T^{3} + 324 T^{2} + \cdots + 6561$$
$79$ $$T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1$$
$83$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$89$ $$(T^{2} - 18 T + 76)^{2}$$
$97$ $$T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041$$