# Properties

 Label 2738.2.a.q Level $2738$ Weight $2$ Character orbit 2738.a Self dual yes Analytic conductor $21.863$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2738,2,Mod(1,2738)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2738, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2738.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2738 = 2 \cdot 37^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2738.a (trivial)

## 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: yes Analytic conductor: $$21.8630400734$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.37902897.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64$$ x^6 - 15*x^4 - x^3 + 60*x^2 - 64 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{4} - \beta_{2} - 1) q^{7} - q^{8} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10})$$ q - q^2 - b1 * q^3 + q^4 - b3 * q^5 + b1 * q^6 + (-b4 - b2 - 1) * q^7 - q^8 + (b5 + b4 + b3 + 2) * q^9 $$q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{4} - \beta_{2} - 1) q^{7} - q^{8} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{9} + \beta_{3} q^{10} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{11} - \beta_1 q^{12} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{13} + (\beta_{4} + \beta_{2} + 1) q^{14} + (\beta_{4} + \beta_{2} + 1) q^{15} + q^{16} + ( - \beta_{4} - 1) q^{17} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2) q^{18} + (\beta_{4} + \beta_{2} - 3) q^{19} - \beta_{3} q^{20} + (\beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1 + 1) q^{21} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{22} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{23} + \beta_1 q^{24} + ( - \beta_{3} + \beta_{2} - 3) q^{25} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{26} + ( - \beta_{4} - 5 \beta_{2} - 1) q^{27} + ( - \beta_{4} - \beta_{2} - 1) q^{28} + (\beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{29} + ( - \beta_{4} - \beta_{2} - 1) q^{30} + (\beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{31} - q^{32} + ( - \beta_{5} - 5 \beta_{3} + 5 \beta_{2} + \beta_1) q^{33} + (\beta_{4} + 1) q^{34} + (\beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{35} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{36} + ( - \beta_{4} - \beta_{2} + 3) q^{38} + ( - \beta_{5} - \beta_{3} + 5 \beta_{2} + \beta_1 - 4) q^{39} + \beta_{3} q^{40} + ( - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{41} + ( - \beta_{5} - \beta_{4} - 5 \beta_{3} - \beta_1 - 1) q^{42} + (\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 - 3) q^{43} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{44} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 - 1) q^{45} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 4) q^{46} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{47} - \beta_1 q^{48} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + \beta_1 + 3) q^{49} + (\beta_{3} - \beta_{2} + 3) q^{50} + (4 \beta_{3} + \beta_1) q^{51} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{52} + (\beta_1 + 3) q^{53} + (\beta_{4} + 5 \beta_{2} + 1) q^{54} + (\beta_{4} + \beta_{2} - \beta_1 + 1) q^{55} + (\beta_{4} + \beta_{2} + 1) q^{56} + ( - \beta_{5} - \beta_{4} - 5 \beta_{3} + 3 \beta_1 - 1) q^{57} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{58} + ( - \beta_{4} - \beta_{2} - 5) q^{59} + (\beta_{4} + \beta_{2} + 1) q^{60} + ( - \beta_{5} + \beta_{4} + 5 \beta_{2} - \beta_1 + 5) q^{61} + ( - \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{62} + ( - \beta_{5} - 3 \beta_{4} - \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 7) q^{63} + q^{64} + (\beta_{5} - \beta_{2} + \beta_1) q^{65} + (\beta_{5} + 5 \beta_{3} - 5 \beta_{2} - \beta_1) q^{66} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - \beta_1 + 2) q^{67} + ( - \beta_{4} - 1) q^{68} + ( - \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 5) q^{69} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{70} + ( - 4 \beta_{2} + 2 \beta_1) q^{71} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2) q^{72} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 + 4) q^{73} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1) q^{75} + (\beta_{4} + \beta_{2} - 3) q^{76} + (4 \beta_{3} + \beta_1 - 4) q^{77} + (\beta_{5} + \beta_{3} - 5 \beta_{2} - \beta_1 + 4) q^{78} + (\beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{79} - \beta_{3} q^{80} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + \beta_1 - 1) q^{81} + (2 \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{82} + (\beta_{5} + \beta_{3} - \beta_{2} + 4) q^{83} + (\beta_{5} + \beta_{4} + 5 \beta_{3} + \beta_1 + 1) q^{84} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{85} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 3) q^{86} + (\beta_{5} - \beta_{4} + \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 1) q^{87} + (\beta_{5} + \beta_{3} - \beta_{2}) q^{88} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{89} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{90} + ( - 4 \beta_{3} + 4 \beta_{2} + \beta_1 + 4) q^{91} + (\beta_{5} + \beta_{3} - \beta_{2} + \beta_1 - 4) q^{92} + (\beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 11) q^{93} + (2 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{94} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{95} + \beta_1 q^{96} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 7) q^{97} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - \beta_1 - 3) q^{98} + ( - 3 \beta_{5} - \beta_{4} - 7 \beta_{3} + 6 \beta_{2} + \beta_1 - 5) q^{99}+O(q^{100})$$ q - q^2 - b1 * q^3 + q^4 - b3 * q^5 + b1 * q^6 + (-b4 - b2 - 1) * q^7 - q^8 + (b5 + b4 + b3 + 2) * q^9 + b3 * q^10 + (-b5 - b3 + b2) * q^11 - b1 * q^12 + (-b5 - b4 - b3 + b1) * q^13 + (b4 + b2 + 1) * q^14 + (b4 + b2 + 1) * q^15 + q^16 + (-b4 - 1) * q^17 + (-b5 - b4 - b3 - 2) * q^18 + (b4 + b2 - 3) * q^19 - b3 * q^20 + (b5 + b4 + 5*b3 + b1 + 1) * q^21 + (b5 + b3 - b2) * q^22 + (b5 + b3 - b2 + b1 - 4) * q^23 + b1 * q^24 + (-b3 + b2 - 3) * q^25 + (b5 + b4 + b3 - b1) * q^26 + (-b4 - 5*b2 - 1) * q^27 + (-b4 - b2 - 1) * q^28 + (b5 + b4 + 2*b3 - b2 + 1) * q^29 + (-b4 - b2 - 1) * q^30 + (b4 + 4*b3 - 3*b2 + 2*b1 - 3) * q^31 - q^32 + (-b5 - 5*b3 + 5*b2 + b1) * q^33 + (b4 + 1) * q^34 + (b5 + b3 - b2 + 2*b1) * q^35 + (b5 + b4 + b3 + 2) * q^36 + (-b4 - b2 + 3) * q^38 + (-b5 - b3 + 5*b2 + b1 - 4) * q^39 + b3 * q^40 + (-2*b5 + 2*b3 + b2 - b1 + 4) * q^41 + (-b5 - b4 - 5*b3 - b1 - 1) * q^42 + (b5 + b4 + b3 - 2*b1 - 3) * q^43 + (-b5 - b3 + b2) * q^44 + (-b5 - b4 - 2*b3 - b1 - 1) * q^45 + (-b5 - b3 + b2 - b1 + 4) * q^46 + (-2*b5 - b4 + 2*b3 + b2 + b1 - 1) * q^47 - b1 * q^48 + (b5 + 2*b4 - 3*b3 + 5*b2 + b1 + 3) * q^49 + (b3 - b2 + 3) * q^50 + (4*b3 + b1) * q^51 + (-b5 - b4 - b3 + b1) * q^52 + (b1 + 3) * q^53 + (b4 + 5*b2 + 1) * q^54 + (b4 + b2 - b1 + 1) * q^55 + (b4 + b2 + 1) * q^56 + (-b5 - b4 - 5*b3 + 3*b1 - 1) * q^57 + (-b5 - b4 - 2*b3 + b2 - 1) * q^58 + (-b4 - b2 - 5) * q^59 + (b4 + b2 + 1) * q^60 + (-b5 + b4 + 5*b2 - b1 + 5) * q^61 + (-b4 - 4*b3 + 3*b2 - 2*b1 + 3) * q^62 + (-b5 - 3*b4 - b3 - 6*b2 - 2*b1 - 7) * q^63 + q^64 + (b5 - b2 + b1) * q^65 + (b5 + 5*b3 - 5*b2 - b1) * q^66 + (b5 + 2*b4 - 3*b3 + 5*b2 - b1 + 2) * q^67 + (-b4 - 1) * q^68 + (-b4 + 4*b3 - 5*b2 + 3*b1 - 5) * q^69 + (-b5 - b3 + b2 - 2*b1) * q^70 + (-4*b2 + 2*b1) * q^71 + (-b5 - b4 - b3 - 2) * q^72 + (2*b5 + b4 + 2*b3 - 5*b2 + 2*b1 + 4) * q^73 + (-b5 - b3 + b2 + 3*b1) * q^75 + (b4 + b2 - 3) * q^76 + (4*b3 + b1 - 4) * q^77 + (b5 + b3 - 5*b2 - b1 + 4) * q^78 + (b4 - 4*b3 + b2 + b1 - 3) * q^79 - b3 * q^80 + (2*b5 + 2*b4 + 6*b3 + b1 - 1) * q^81 + (2*b5 - 2*b3 - b2 + b1 - 4) * q^82 + (b5 + b3 - b2 + 4) * q^83 + (b5 + b4 + 5*b3 + b1 + 1) * q^84 + (b5 + b3 - 2*b2 + 2*b1 - 1) * q^85 + (-b5 - b4 - b3 + 2*b1 + 3) * q^86 + (b5 - b4 + b3 - 6*b2 - 2*b1 - 1) * q^87 + (b5 + b3 - b2) * q^88 + (b5 - 2*b4 + b3 - 2*b2 - 2*b1 + 2) * q^89 + (b5 + b4 + 2*b3 + b1 + 1) * q^90 + (-4*b3 + 4*b2 + b1 + 4) * q^91 + (b5 + b3 - b2 + b1 - 4) * q^92 + (b5 - 3*b4 - 3*b3 - 4*b2 + 3*b1 - 11) * q^93 + (2*b5 + b4 - 2*b3 - b2 - b1 + 1) * q^94 + (-b5 + 3*b3 + b2 - 2*b1) * q^95 + b1 * q^96 + (b5 + b4 + 2*b3 + 3*b2 - 3*b1 - 7) * q^97 + (-b5 - 2*b4 + 3*b3 - 5*b2 - b1 - 3) * q^98 + (-3*b5 - b4 - 7*b3 + 6*b2 + b1 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{2} + 6 q^{4} - 3 q^{7} - 6 q^{8} + 12 q^{9}+O(q^{10})$$ 6 * q - 6 * q^2 + 6 * q^4 - 3 * q^7 - 6 * q^8 + 12 * q^9 $$6 q - 6 q^{2} + 6 q^{4} - 3 q^{7} - 6 q^{8} + 12 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{15} + 6 q^{16} - 3 q^{17} - 12 q^{18} - 21 q^{19} + 6 q^{21} + 3 q^{22} - 21 q^{23} - 18 q^{25} - 3 q^{27} - 3 q^{28} + 6 q^{29} - 3 q^{30} - 21 q^{31} - 6 q^{32} - 3 q^{33} + 3 q^{34} + 3 q^{35} + 12 q^{36} + 21 q^{38} - 27 q^{39} + 18 q^{41} - 6 q^{42} - 18 q^{43} - 3 q^{44} - 6 q^{45} + 21 q^{46} - 9 q^{47} + 15 q^{49} + 18 q^{50} + 18 q^{53} + 3 q^{54} + 3 q^{55} + 3 q^{56} - 6 q^{57} - 6 q^{58} - 27 q^{59} + 3 q^{60} + 24 q^{61} + 21 q^{62} - 36 q^{63} + 6 q^{64} + 3 q^{65} + 3 q^{66} + 9 q^{67} - 3 q^{68} - 27 q^{69} - 3 q^{70} - 12 q^{72} + 27 q^{73} - 3 q^{75} - 21 q^{76} - 24 q^{77} + 27 q^{78} - 21 q^{79} - 6 q^{81} - 18 q^{82} + 27 q^{83} + 6 q^{84} - 3 q^{85} + 18 q^{86} + 3 q^{88} + 21 q^{89} + 6 q^{90} + 24 q^{91} - 21 q^{92} - 54 q^{93} + 9 q^{94} - 3 q^{95} - 42 q^{97} - 15 q^{98} - 36 q^{99}+O(q^{100})$$ 6 * q - 6 * q^2 + 6 * q^4 - 3 * q^7 - 6 * q^8 + 12 * q^9 - 3 * q^11 + 3 * q^14 + 3 * q^15 + 6 * q^16 - 3 * q^17 - 12 * q^18 - 21 * q^19 + 6 * q^21 + 3 * q^22 - 21 * q^23 - 18 * q^25 - 3 * q^27 - 3 * q^28 + 6 * q^29 - 3 * q^30 - 21 * q^31 - 6 * q^32 - 3 * q^33 + 3 * q^34 + 3 * q^35 + 12 * q^36 + 21 * q^38 - 27 * q^39 + 18 * q^41 - 6 * q^42 - 18 * q^43 - 3 * q^44 - 6 * q^45 + 21 * q^46 - 9 * q^47 + 15 * q^49 + 18 * q^50 + 18 * q^53 + 3 * q^54 + 3 * q^55 + 3 * q^56 - 6 * q^57 - 6 * q^58 - 27 * q^59 + 3 * q^60 + 24 * q^61 + 21 * q^62 - 36 * q^63 + 6 * q^64 + 3 * q^65 + 3 * q^66 + 9 * q^67 - 3 * q^68 - 27 * q^69 - 3 * q^70 - 12 * q^72 + 27 * q^73 - 3 * q^75 - 21 * q^76 - 24 * q^77 + 27 * q^78 - 21 * q^79 - 6 * q^81 - 18 * q^82 + 27 * q^83 + 6 * q^84 - 3 * q^85 + 18 * q^86 + 3 * q^88 + 21 * q^89 + 6 * q^90 + 24 * q^91 - 21 * q^92 - 54 * q^93 + 9 * q^94 - 3 * q^95 - 42 * q^97 - 15 * q^98 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 15x^{4} - x^{3} + 60x^{2} - 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 15\nu^{3} + \nu^{2} - 44\nu ) / 16$$ (-v^5 + 15*v^3 + v^2 - 44*v) / 16 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 11\nu^{2} - \nu + 20 ) / 4$$ (v^4 - 11*v^2 - v + 20) / 4 $$\beta_{4}$$ $$=$$ $$( 5\nu^{5} - 59\nu^{3} - 5\nu^{2} + 124\nu - 16 ) / 16$$ (5*v^5 - 59*v^3 - 5*v^2 + 124*v - 16) / 16 $$\beta_{5}$$ $$=$$ $$( -5\nu^{5} - 4\nu^{4} + 59\nu^{3} + 65\nu^{2} - 120\nu - 144 ) / 16$$ (-5*v^5 - 4*v^4 + 59*v^3 + 65*v^2 - 120*v - 144) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + 5$$ b5 + b4 + b3 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} + 5\beta_{2} + 6\beta _1 + 1$$ b4 + 5*b2 + 6*b1 + 1 $$\nu^{4}$$ $$=$$ $$11\beta_{5} + 11\beta_{4} + 15\beta_{3} + \beta _1 + 35$$ 11*b5 + 11*b4 + 15*b3 + b1 + 35 $$\nu^{5}$$ $$=$$ $$\beta_{5} + 16\beta_{4} + \beta_{3} + 59\beta_{2} + 46\beta _1 + 20$$ b5 + 16*b4 + b3 + 59*b2 + 46*b1 + 20

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.14945 1.83388 1.37564 −1.27006 −2.18117 −2.90773
−1.00000 −3.14945 1.00000 −1.53209 3.14945 −4.82524 −1.00000 6.91903 1.53209
1.2 −1.00000 −1.83388 1.00000 1.87939 1.83388 3.44656 −1.00000 0.363102 −1.87939
1.3 −1.00000 −1.37564 1.00000 −0.347296 1.37564 −0.477756 −1.00000 −1.10761 0.347296
1.4 −1.00000 1.27006 1.00000 −1.53209 −1.27006 1.94585 −1.00000 −1.38694 1.53209
1.5 −1.00000 2.18117 1.00000 1.87939 −2.18117 −4.09926 −1.00000 1.75751 −1.87939
1.6 −1.00000 2.90773 1.00000 −0.347296 −2.90773 1.00984 −1.00000 5.45490 0.347296
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$37$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2738.2.a.q 6
37.b even 2 1 2738.2.a.t 6
37.h even 18 2 74.2.f.b 12
111.n odd 18 2 666.2.x.g 12
148.o odd 18 2 592.2.bc.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.b 12 37.h even 18 2
592.2.bc.d 12 148.o odd 18 2
666.2.x.g 12 111.n odd 18 2
2738.2.a.q 6 1.a even 1 1 trivial
2738.2.a.t 6 37.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2738))$$:

 $$T_{3}^{6} - 15T_{3}^{4} + T_{3}^{3} + 60T_{3}^{2} - 64$$ T3^6 - 15*T3^4 + T3^3 + 60*T3^2 - 64 $$T_{5}^{3} - 3T_{5} - 1$$ T5^3 - 3*T5 - 1 $$T_{7}^{6} + 3T_{7}^{5} - 24T_{7}^{4} - 37T_{7}^{3} + 168T_{7}^{2} - 48T_{7} - 64$$ T7^6 + 3*T7^5 - 24*T7^4 - 37*T7^3 + 168*T7^2 - 48*T7 - 64 $$T_{13}^{6} - 42T_{13}^{4} + 43T_{13}^{3} + 324T_{13}^{2} - 183T_{13} - 719$$ T13^6 - 42*T13^4 + 43*T13^3 + 324*T13^2 - 183*T13 - 719 $$T_{17}^{6} + 3T_{17}^{5} - 24T_{17}^{4} - 69T_{17}^{3} + 120T_{17}^{2} + 390T_{17} + 163$$ T17^6 + 3*T17^5 - 24*T17^4 - 69*T17^3 + 120*T17^2 + 390*T17 + 163

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{6}$$
$3$ $$T^{6} - 15 T^{4} + T^{3} + 60 T^{2} + \cdots - 64$$
$5$ $$(T^{3} - 3 T - 1)^{2}$$
$7$ $$T^{6} + 3 T^{5} - 24 T^{4} - 37 T^{3} + \cdots - 64$$
$11$ $$T^{6} + 3 T^{5} - 24 T^{4} - 37 T^{3} + \cdots - 64$$
$13$ $$T^{6} - 42 T^{4} + 43 T^{3} + \cdots - 719$$
$17$ $$T^{6} + 3 T^{5} - 24 T^{4} - 69 T^{3} + \cdots + 163$$
$19$ $$T^{6} + 21 T^{5} + 156 T^{4} + \cdots + 64$$
$23$ $$T^{6} + 21 T^{5} + 144 T^{4} + \cdots + 64$$
$29$ $$T^{6} - 6 T^{5} - 24 T^{4} + 84 T^{3} + \cdots + 37$$
$31$ $$T^{6} + 21 T^{5} + 24 T^{4} + \cdots + 130112$$
$37$ $$T^{6}$$
$41$ $$T^{6} - 18 T^{5} - 27 T^{4} + \cdots + 21608$$
$43$ $$T^{6} + 18 T^{5} + 51 T^{4} + \cdots + 8704$$
$47$ $$T^{6} + 9 T^{5} - 141 T^{4} + \cdots - 83008$$
$53$ $$T^{6} - 18 T^{5} + 120 T^{4} + \cdots + 17$$
$59$ $$T^{6} + 27 T^{5} + 276 T^{4} + \cdots + 1088$$
$61$ $$T^{6} - 24 T^{5} + 69 T^{4} + \cdots + 52928$$
$67$ $$T^{6} - 9 T^{5} - 144 T^{4} + \cdots - 512$$
$71$ $$T^{6} - 108 T^{4} + 72 T^{3} + \cdots - 512$$
$73$ $$T^{6} - 27 T^{5} + 108 T^{4} + \cdots + 216289$$
$79$ $$T^{6} + 21 T^{5} + 63 T^{4} + \cdots + 9792$$
$83$ $$T^{6} - 27 T^{5} + 276 T^{4} + \cdots + 1088$$
$89$ $$T^{6} - 21 T^{5} - 75 T^{4} + \cdots - 219419$$
$97$ $$T^{6} + 42 T^{5} + 549 T^{4} + \cdots + 2744$$