# Properties

 Label 72.2.n.b Level $72$ Weight $2$ Character orbit 72.n Analytic conductor $0.575$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,2,Mod(13,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("72.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.n (of order $$6$$, degree $$2$$, 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: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256$$ x^16 - x^15 + x^14 + 2*x^12 - 4*x^11 - 8*x^9 + 4*x^8 - 16*x^7 - 32*x^5 + 32*x^4 + 64*x^2 - 128*x + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( - \beta_{10} - \beta_{3}) q^{3} + ( - \beta_{15} - \beta_{13} + \beta_{8}) q^{4} + (\beta_{13} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{8} + \beta_1 - 1) q^{6} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - \beta_{2} - 1) q^{8} + (\beta_{11} - \beta_{9} + 1) q^{9}+O(q^{10})$$ q + b6 * q^2 + (-b10 - b3) * q^3 + (-b15 - b13 + b8) * q^4 + (b13 + b10 + b7 + b5 + b3) * q^5 + (-b11 - b8 + b1 - 1) * q^6 + (-b6 - b4 - b1) * q^7 + (b9 - b2 - 1) * q^8 + (b11 - b9 + 1) * q^9 $$q + \beta_{6} q^{2} + ( - \beta_{10} - \beta_{3}) q^{3} + ( - \beta_{15} - \beta_{13} + \beta_{8}) q^{4} + (\beta_{13} + \beta_{10} + \beta_{7} + \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{8} + \beta_1 - 1) q^{6} + ( - \beta_{6} - \beta_{4} - \beta_1) q^{7} + (\beta_{9} - \beta_{2} - 1) q^{8} + (\beta_{11} - \beta_{9} + 1) q^{9} + (\beta_{15} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{7} + \beta_{3} - \beta_1) q^{10} + ( - \beta_{13} - \beta_{12} + \beta_{2}) q^{11} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{12} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5}) q^{13} + (\beta_{15} + \beta_{13} - \beta_{8} - \beta_{5} + 2 \beta_{4} + 2) q^{14} + ( - \beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + \cdots - 1) q^{15}+ \cdots + ( - 4 \beta_{15} - 4 \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots + 2 \beta_1) q^{99}+O(q^{100})$$ q + b6 * q^2 + (-b10 - b3) * q^3 + (-b15 - b13 + b8) * q^4 + (b13 + b10 + b7 + b5 + b3) * q^5 + (-b11 - b8 + b1 - 1) * q^6 + (-b6 - b4 - b1) * q^7 + (b9 - b2 - 1) * q^8 + (b11 - b9 + 1) * q^9 + (b15 + b13 + b12 + b11 + b7 + b3 - b1) * q^10 + (-b13 - b12 + b2) * q^11 + (b15 - b14 - b13 - b8 - b7 - b6 - b5 - b4 + b2 + b1 - 1) * q^12 + (-b11 + b10 - b9 - b7 - b5) * q^13 + (b15 + b13 - b8 - b5 + 2*b4 + 2) * q^14 + (-b15 - b13 - b12 + b10 + 2*b8 - b7 + b5 + b4 - b2 - 1) * q^15 + (3*b15 - b14 + b13 - b11 - 2*b10 + b9 - 2*b8 + b6 + 2*b4 + b2 - b1) * q^16 + (-b15 + 2*b14 - b10 - b3 - 2) * q^17 + (-2*b15 + b14 + 2*b10 - b9 + 2*b8 - b7 + b5 - b4 + b1) * q^18 + (b12 + b11 + b9 - 2*b7 - 2*b6 - 2*b5 - b2 + 2*b1) * q^19 + (-2*b15 + 2*b14 + b11 - b9 + 3*b8 - b6 - 2*b3 - b2 - b1) * q^20 + (b15 + b13 + b11 - b10 + b9 + b6 - b3 - b1) * q^21 + (-2*b15 + b14 - b12 + b11 + b8 + b5 - b4 + b3 - b2 - 1) * q^22 + (2*b15 - 2*b14 + b13 + 2*b12 - b11 + b9 - 2*b8 + b7 - b5 + b4 + b3 + 2*b2 + 1) * q^23 + (-2*b15 + b14 - b12 - b7 - 2*b6 - b5 - 2*b4 - b3 - 1) * q^24 + (3*b15 - 2*b14 + 2*b13 + b12 - 2*b11 - b10 + 2*b9 - 4*b8 + 2*b6 + 2*b4 + b3 + b2 + 2*b1) * q^25 + (-2*b14 - 2*b13 + 2*b10 - b9 - b6 - b5 + b2 + 3) * q^26 + (b12 - b10 + 2*b7 + 2*b6 + 2*b5 + b3 - b2 - 2*b1) * q^27 + (-b14 - b9 + 2*b6 + 2*b5 + b2 + 1) * q^28 + (-b13 + b12 - b6 + 2*b3 - b2 + b1) * q^29 + (3*b15 - b14 + b13 - b11 + b9 - 2*b8 + b7 + b6 + 2*b5 + b4 + 2*b3 - 2*b1 + 1) * q^30 + (-2*b15 + 2*b14 - b13 - 2*b12 + b11 - b9 + 2*b8 + b7 - b5 - 3*b4 - b3 - 2*b2 - 3) * q^31 + (-2*b15 + b14 - 2*b13 - b12 - 2*b10 - b9 - b7 + b5 - b3 - b2) * q^32 + (b15 - 2*b14 + b10 + 2*b7 - 2*b5 - b4 + b3) * q^33 + (-b15 + b14 - b13 - b12 + b11 - b9 + 2*b8 - 2*b6 - b4 - b3 + 2*b1) * q^34 + (-b15 - b13 - b12 - b11 - b9 - b3 + b2) * q^35 + (b12 + b11 + 2*b10 + b8 + 3*b7 - b6 - b5 - 2*b4 + b3 - b1 + 2) * q^36 + (-b15 - 2*b13 - b12 - b11 + b10 - b9 + 2*b7 + 2*b6 + 2*b5 - b3 + b2 - 2*b1) * q^37 + (3*b15 - b14 + 3*b13 - b11 - 2*b10 + b9 - 2*b8 + b6 - 2*b4 - 2*b3 + b2 - b1) * q^38 + (-b15 + 2*b14 + b12 - b11 - b10 + b9 - b7 + 2*b6 + b5 + 3*b4 - b3 + b2 + 2*b1 + 1) * q^39 + (b15 + b13 - b11 + 2*b10 - b9 + 3*b7 + b5 + b4 + 1) * q^40 + (-4*b15 + 2*b14 - 3*b13 - 2*b12 + b11 + 2*b10 - b9 + 6*b8 - 2*b7 + 2*b5 - 3*b4 - b3 - 2*b2 - 3) * q^41 + (-b15 + b14 + b13 - b11 - 2*b10 + 2*b7 + b6 + b5 + 2*b4 - b1 + 1) * q^42 + (-b15 + b10) * q^43 + (3*b15 - b14 + 3*b13 + b12 + b11 + b9 - b7 + b6 + b5 + 3*b3 - b2 + b1 + 1) * q^44 + (-2*b15 - 2*b13 + b11 - b10 + b9 - 3*b7 - 2*b6 - 3*b5 + 2*b1) * q^45 + (-3*b15 + b14 - b13 + b12 + b11 - 2*b10 + b9 + b7 - 2*b6 - 2*b5 - 3*b3 - b2 - b1 - 1) * q^46 + (4*b15 - 4*b14 + 2*b13 - 4*b8 + b6 - b4 + 2*b3 + b1) * q^47 + (4*b15 - 3*b14 + 4*b13 + b12 + b9 - 4*b8 + b7 - b5 + 2*b4 + 3*b3 + b2 + 2*b1 + 4) * q^48 + (-b15 - b13 + b10 + 2*b8 - 2*b7 + 2*b5 + 2*b4 + 2) * q^49 + (-b14 - 2*b13 + b12 - 4*b10 + b9 - 2*b8 - b7 - b5 - 2*b4 - b3 + b2 - 2) * q^50 + (2*b15 + 2*b13 - b11 + 2*b10 - b9 + 2*b6 + 2*b3 - 2*b1) * q^51 + (-2*b15 - b11 + 2*b10 + b9 - b8 + b6 + 2*b4 + b2 - b1) * q^52 + (-b15 + 2*b13 + b12 + b11 - 3*b10 + b9 - b3 - b2) * q^53 + (-b15 + 2*b14 - 3*b13 + 3*b8 - 3*b7 - b5 + 3*b4 - 2*b3 - b2 + b1 - 1) * q^54 + (-b15 + 2*b14 + b12 + b11 - b10 - b9 - b7 + b6 + b5 - b3 + b2 + b1 + 1) * q^55 + (-4*b15 + 2*b14 - 2*b13 + b12 + 2*b10 + 2*b8 - b6 - b4 + b3 - b2 + b1) * q^56 + (-b15 + 2*b14 + b12 - b10 + 2*b7 - 4*b6 - 2*b5 - b3 + b2 - 4*b1 + 2) * q^57 + (b15 + b14 - b13 - b12 + b11 + 2*b8 - 4*b7 - b5 - 3*b4 - 3*b3 - b2 - 3) * q^58 + (2*b15 + b13 + b10 - 2*b7 - 2*b5 - b3) * q^59 + (2*b14 - 2*b13 - b12 + b11 - 2*b10 - b9 + b8 - 2*b7 + 2*b6 + 3*b4 - b3 - 2*b1 + 2) * q^60 + (2*b15 + b13 - b12 - 2*b10 - b6 - 2*b3 + b2 + b1) * q^61 + (3*b15 - 3*b14 + b13 - b12 - b11 + 2*b10 - b9 - b7 + 3*b3 + b2 + b1 - 3) * q^62 + (b15 + b13 - b12 - b10 - 2*b8 + 2*b7 - b6 - 2*b5 + b4 - b2 - b1) * q^63 + (b15 - b13 - b12 - b11 + 2*b10 - 5*b7 + b6 + b5 + b3 + 5*b1) * q^64 + (2*b15 - 2*b14 + b13 - 2*b8 - 2*b6 + b4 + b3 - 2*b1) * q^65 + (b15 - 3*b14 + b13 + b12 - b11 + b9 - 2*b8 - b5 + b4 + b3 - 2*b1 - 4) * q^66 + (2*b13 + b11 + b10 + b9 + 4*b7 + 4*b5 + 2*b3) * q^67 + (b14 + 2*b13 - b12 + 2*b10 - b9 + b7 + b5 - 6*b4 + b3 - b2 - 6) * q^68 + (3*b15 - b12 + 2*b10 + b7 - 2*b6 + b5 + b3 + b2 + 2*b1) * q^69 + (-b15 - b14 - b13 + 2*b10 - b8 - b6 + 2*b3 + 2*b1) * q^70 + (-b15 + 2*b14 + b12 + b11 - b10 - b9 - b3 + b2 + 6) * q^71 + (-b15 + b14 + b13 + b11 + 2*b8 + 2*b7 + b6 - 2*b5 + 2*b4 - 2*b3 - 3*b1 - 3) * q^72 + (b15 - 2*b14 + b10 + 2*b7 - 2*b6 - 2*b5 + b3 - 2*b1 - 2) * q^73 + (-4*b15 + 2*b14 - 4*b13 - b12 + 2*b10 + 2*b8 - b6 + 5*b4 + b3 + b2 + b1) * q^74 + (-b15 + 2*b13 - b11 + 3*b10 - b9 - 4*b6 + b3 + 4*b1) * q^75 + (-b14 + b12 - 2*b11 - 2*b10 - b9 - 4*b8 + 3*b7 - 3*b5 + 2*b4 + b3 + b2 + 2) * q^76 + (2*b15 - b13 - b10 - b7 - b5 - 3*b3) * q^77 + (-4*b15 + 3*b14 - 4*b13 - b12 + 2*b11 - 2*b10 + 4*b8 + 2*b7 + b5 - 5*b4 - 3*b3 - b2 + b1 - 2) * q^78 + (-b15 - b13 - b12 + 2*b11 + b10 - 2*b9 + 2*b8 - 3*b6 + b4 - b2 - 3*b1) * q^79 + (-2*b13 + 2*b10 - 2*b9 + 2*b7 + 2*b2 - 2*b1 - 4) * q^80 + (3*b15 - 2*b14 + 2*b13 - b11 - b10 + b9 - 4*b8 - 4*b7 + 2*b6 + 4*b5 - 4*b4 + b3 + 2*b1 - 3) * q^81 + (3*b15 + b13 - b12 - b11 + 2*b10 + b9 + 3*b7 + 3*b3 - b2 - 3*b1 + 1) * q^82 + (b12 + 2*b6 + b3 - b2 - 2*b1) * q^83 + (b15 - b14 - b13 + b12 + b11 - b7 + b6 + b5 + b3 - 2*b2 + 3*b1 - 2) * q^84 + (-2*b15 - 3*b13 + b11 - 4*b10 + b9 - 2*b7 - 2*b5 - b3) * q^85 + (b15 - b14 + b13 + b12 + b9 - b8 + b7 + 2*b4 + b3 + b2 + 2) * q^86 + (2*b15 - 2*b14 + b13 + 2*b12 - b11 + b9 - 2*b8 - 2*b7 - b6 + 2*b5 - 3*b4 + b3 + 2*b2 - b1 + 2) * q^87 + (b15 + b14 + b13 + b12 + b11 - 2*b10 - b9 + 2*b8 + 2*b6 - 5*b4 - b3 - 2*b2 - 4*b1) * q^88 + (-b15 + 2*b14 - b12 - b11 - b10 + b9 - b3 - b2 + 2) * q^89 + (2*b15 - 2*b14 + 4*b13 - 2*b10 + 3*b9 - 2*b8 - 2*b7 + b6 + b5 - 2*b4 - b2 + 2*b1 + 1) * q^90 + (2*b15 + 3*b13 - b10 + 2*b3) * q^91 + (4*b15 - 2*b14 + 4*b13 + b12 - 3*b11 - 2*b10 + 3*b9 - 5*b8 + 7*b4 - b3 + 2*b2 + 4*b1) * q^92 + (-5*b15 - 2*b13 - b12 - 2*b11 + 2*b10 - 2*b9 + b7 + 2*b6 + b5 - b3 + b2 - 2*b1) * q^93 + (b15 - 2*b14 + b13 + 2*b12 - 2*b11 - 3*b8 - 4*b7 - b5 + 2*b3 + 2*b2) * q^94 + (-6*b15 + 4*b14 - 4*b13 - 4*b12 + 2*b11 + 2*b10 - 2*b9 + 8*b8 + 2*b4 - 2*b3 - 4*b2 + 2) * q^95 + (b15 - 2*b14 + 3*b13 + 3*b12 + b11 - 2*b10 - 2*b8 - b7 + 3*b6 + b5 - 4*b4 + b3 - b1 - 2) * q^96 + (-5*b15 + 4*b14 - 3*b13 - b12 + 2*b11 + b10 - 2*b9 + 6*b8 + 2*b6 - b4 - 2*b3 - b2 + 2*b1) * q^97 + (2*b14 + b9 + 2*b7 + 2*b6 + 2*b5 - b2 - 2*b1 + 3) * q^98 + (-4*b15 - 4*b13 - 2*b12 - b11 + 2*b10 - b9 + 2*b7 - 2*b6 + 2*b5 - b3 + 2*b2 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + q^{2} - q^{4} - 7 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 16 * q + q^2 - q^4 - 7 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 $$16 q + q^{2} - q^{4} - 7 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 16 q^{10} - 16 q^{12} + 16 q^{14} - 10 q^{15} - 9 q^{16} - 28 q^{17} + 4 q^{18} - 8 q^{20} + q^{22} - 10 q^{23} + 7 q^{24} + 2 q^{25} + 28 q^{26} + 4 q^{28} + 22 q^{30} - 10 q^{31} + 11 q^{32} + q^{34} + 27 q^{36} + 23 q^{38} + 2 q^{39} + 6 q^{40} - 8 q^{41} + 8 q^{42} + 18 q^{44} - 20 q^{46} + 6 q^{47} + 39 q^{48} + 18 q^{49} - 23 q^{50} - 8 q^{52} - 29 q^{54} - 4 q^{55} + 10 q^{56} + 10 q^{57} - 14 q^{58} + 6 q^{60} - 52 q^{62} + 2 q^{63} + 26 q^{64} - 14 q^{65} - 72 q^{66} - 39 q^{68} + 72 q^{71} - 77 q^{72} - 44 q^{73} - 38 q^{74} + 5 q^{76} + 10 q^{78} - 30 q^{79} - 96 q^{80} + 10 q^{81} + 38 q^{82} - 28 q^{84} + 7 q^{86} + 42 q^{87} + 31 q^{88} + 64 q^{89} + 64 q^{90} - 30 q^{92} - 12 q^{94} + 44 q^{95} - 26 q^{96} + 66 q^{98}+O(q^{100})$$ 16 * q + q^2 - q^4 - 7 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 16 * q^10 - 16 * q^12 + 16 * q^14 - 10 * q^15 - 9 * q^16 - 28 * q^17 + 4 * q^18 - 8 * q^20 + q^22 - 10 * q^23 + 7 * q^24 + 2 * q^25 + 28 * q^26 + 4 * q^28 + 22 * q^30 - 10 * q^31 + 11 * q^32 + q^34 + 27 * q^36 + 23 * q^38 + 2 * q^39 + 6 * q^40 - 8 * q^41 + 8 * q^42 + 18 * q^44 - 20 * q^46 + 6 * q^47 + 39 * q^48 + 18 * q^49 - 23 * q^50 - 8 * q^52 - 29 * q^54 - 4 * q^55 + 10 * q^56 + 10 * q^57 - 14 * q^58 + 6 * q^60 - 52 * q^62 + 2 * q^63 + 26 * q^64 - 14 * q^65 - 72 * q^66 - 39 * q^68 + 72 * q^71 - 77 * q^72 - 44 * q^73 - 38 * q^74 + 5 * q^76 + 10 * q^78 - 30 * q^79 - 96 * q^80 + 10 * q^81 + 38 * q^82 - 28 * q^84 + 7 * q^86 + 42 * q^87 + 31 * q^88 + 64 * q^89 + 64 * q^90 - 30 * q^92 - 12 * q^94 + 44 * q^95 - 26 * q^96 + 66 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{15} - \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 8 \nu^{10} - 16 \nu^{9} - 48 \nu^{8} - 12 \nu^{7} + 16 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} + 256 \nu^{2} + 384 \nu - 128 ) / 384$$ (v^15 - v^14 - 3*v^13 - 8*v^12 + 2*v^11 + 8*v^10 - 16*v^9 - 48*v^8 - 12*v^7 + 16*v^6 + 16*v^5 - 80*v^4 + 256*v^2 + 384*v - 128) / 384 $$\beta_{3}$$ $$=$$ $$( - \nu^{15} - \nu^{14} - 3 \nu^{13} - 6 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + 24 \nu^{4} + 80 \nu^{3} + 32 \nu^{2} + 32 \nu + 64 ) / 192$$ (-v^15 - v^14 - 3*v^13 - 6*v^10 - 4*v^9 - 4*v^8 + 4*v^7 + 8*v^6 + 24*v^4 + 80*v^3 + 32*v^2 + 32*v + 64) / 192 $$\beta_{4}$$ $$=$$ $$( \nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} - 12 \nu^{7} - 24 \nu^{6} - 32 \nu^{5} - 72 \nu^{4} - 96 \nu^{3} - 128 \nu^{2} - 96 \nu - 256 ) / 192$$ (v^15 + v^14 + 3*v^13 + 4*v^12 + 8*v^11 + 6*v^10 + 8*v^9 + 4*v^8 - 12*v^7 - 24*v^6 - 32*v^5 - 72*v^4 - 96*v^3 - 128*v^2 - 96*v - 256) / 192 $$\beta_{5}$$ $$=$$ $$( - \nu^{15} + \nu^{14} - \nu^{13} - 2 \nu^{11} + 4 \nu^{10} + 8 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + 32 \nu^{4} - 32 \nu^{3} - 64 \nu + 128 ) / 128$$ (-v^15 + v^14 - v^13 - 2*v^11 + 4*v^10 + 8*v^8 - 4*v^7 + 16*v^6 + 32*v^4 - 32*v^3 - 64*v + 128) / 128 $$\beta_{6}$$ $$=$$ $$( - \nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} - 24 \nu^{8} - 20 \nu^{7} + 64 \nu^{6} + 96 \nu^{5} + 160 \nu^{4} + 256 \nu^{3} + 384 \nu^{2} + 448 \nu + 512 ) / 384$$ (-v^15 - 3*v^14 - 5*v^13 - 12*v^12 - 18*v^11 - 28*v^10 - 24*v^9 - 24*v^8 - 20*v^7 + 64*v^6 + 96*v^5 + 160*v^4 + 256*v^3 + 384*v^2 + 448*v + 512) / 384 $$\beta_{7}$$ $$=$$ $$( - \nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + 4 \nu^{7} + 16 \nu^{6} + 20 \nu^{5} + 64 \nu^{4} + 64 \nu^{3} + 80 \nu^{2} + 64 \nu + 128 ) / 96$$ (-v^15 - v^14 - 2*v^13 - 3*v^12 - 5*v^11 - 4*v^10 - 6*v^9 + 8*v^8 + 4*v^7 + 16*v^6 + 20*v^5 + 64*v^4 + 64*v^3 + 80*v^2 + 64*v + 128) / 96 $$\beta_{8}$$ $$=$$ $$( \nu^{15} + \nu^{14} + 7 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + 20 \nu^{10} + 24 \nu^{9} + 16 \nu^{8} + 20 \nu^{7} - 40 \nu^{6} - 160 \nu^{5} - 272 \nu^{4} - 160 \nu^{3} - 128 \nu^{2} - 384 \nu - 640 ) / 384$$ (v^15 + v^14 + 7*v^13 + 14*v^12 + 22*v^11 + 20*v^10 + 24*v^9 + 16*v^8 + 20*v^7 - 40*v^6 - 160*v^5 - 272*v^4 - 160*v^3 - 128*v^2 - 384*v - 640) / 384 $$\beta_{9}$$ $$=$$ $$( \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 2 \nu^{12} - 10 \nu^{11} - 4 \nu^{10} - 16 \nu^{9} + 36 \nu^{7} + 88 \nu^{6} + 64 \nu^{5} + 112 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} - 128 ) / 384$$ (v^15 - 7*v^14 - 9*v^13 - 2*v^12 - 10*v^11 - 4*v^10 - 16*v^9 + 36*v^7 + 88*v^6 + 64*v^5 + 112*v^4 + 192*v^3 + 448*v^2 - 128) / 384 $$\beta_{10}$$ $$=$$ $$( - \nu^{15} - \nu^{14} - 7 \nu^{13} - 14 \nu^{12} - 22 \nu^{11} - 20 \nu^{10} - 24 \nu^{9} - 16 \nu^{8} - 20 \nu^{7} + 40 \nu^{6} + 160 \nu^{5} + 272 \nu^{4} + 160 \nu^{3} + 512 \nu^{2} + 384 \nu + 640 ) / 384$$ (-v^15 - v^14 - 7*v^13 - 14*v^12 - 22*v^11 - 20*v^10 - 24*v^9 - 16*v^8 - 20*v^7 + 40*v^6 + 160*v^5 + 272*v^4 + 160*v^3 + 512*v^2 + 384*v + 640) / 384 $$\beta_{11}$$ $$=$$ $$( \nu^{15} - \nu^{14} + 4 \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} + 8 \nu^{7} + 20 \nu^{5} - 52 \nu^{4} - 56 \nu^{3} - 48 \nu^{2} + 16 \nu - 288 ) / 96$$ (v^15 - v^14 + 4*v^11 + v^10 + 4*v^9 - 2*v^8 + 8*v^7 + 20*v^5 - 52*v^4 - 56*v^3 - 48*v^2 + 16*v - 288) / 96 $$\beta_{12}$$ $$=$$ $$( - \nu^{15} + \nu^{14} - 4 \nu^{11} - \nu^{10} - 4 \nu^{9} + 2 \nu^{8} - 8 \nu^{7} - 20 \nu^{5} + 52 \nu^{4} - 40 \nu^{3} + 48 \nu^{2} - 16 \nu + 192 ) / 96$$ (-v^15 + v^14 - 4*v^11 - v^10 - 4*v^9 + 2*v^8 - 8*v^7 - 20*v^5 + 52*v^4 - 40*v^3 + 48*v^2 - 16*v + 192) / 96 $$\beta_{13}$$ $$=$$ $$( 3 \nu^{15} - \nu^{14} + 9 \nu^{13} + 6 \nu^{12} + 26 \nu^{11} + 12 \nu^{10} + 16 \nu^{9} - 16 \nu^{8} + 44 \nu^{7} - 40 \nu^{6} - 32 \nu^{5} - 240 \nu^{4} - 64 \nu^{3} - 256 \nu^{2} - 128 \nu - 896 ) / 384$$ (3*v^15 - v^14 + 9*v^13 + 6*v^12 + 26*v^11 + 12*v^10 + 16*v^9 - 16*v^8 + 44*v^7 - 40*v^6 - 32*v^5 - 240*v^4 - 64*v^3 - 256*v^2 - 128*v - 896) / 384 $$\beta_{14}$$ $$=$$ $$( - 2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} - 8 \nu^{8} - 32 \nu^{7} + 12 \nu^{6} + 64 \nu^{5} + 160 \nu^{4} + 96 \nu^{3} + 256 \nu^{2} + 256 \nu + 704 ) / 192$$ (-2*v^15 + v^14 - 5*v^13 - 5*v^12 - 16*v^11 - 10*v^10 - 28*v^9 - 8*v^8 - 32*v^7 + 12*v^6 + 64*v^5 + 160*v^4 + 96*v^3 + 256*v^2 + 256*v + 704) / 192 $$\beta_{15}$$ $$=$$ $$( - 9 \nu^{15} + \nu^{14} - 9 \nu^{13} - 8 \nu^{12} - 42 \nu^{11} - 12 \nu^{10} - 24 \nu^{9} + 40 \nu^{8} - 52 \nu^{7} + 48 \nu^{6} + 96 \nu^{5} + 384 \nu^{4} + 192 \nu^{3} + 448 \nu^{2} + 64 \nu + 1664 ) / 384$$ (-9*v^15 + v^14 - 9*v^13 - 8*v^12 - 42*v^11 - 12*v^10 - 24*v^9 + 40*v^8 - 52*v^7 + 48*v^6 + 96*v^5 + 384*v^4 + 192*v^3 + 448*v^2 + 64*v + 1664) / 384
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + \beta_{8}$$ b10 + b8 $$\nu^{3}$$ $$=$$ $$-\beta_{12} - \beta_{11} - 1$$ -b12 - b11 - 1 $$\nu^{4}$$ $$=$$ $$- \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{6} + 2 \beta_{4} + \beta_{3} + \beta_1$$ -b14 + b13 + b12 - b11 + b10 + b9 - 2*b8 - b6 + 2*b4 + b3 + b1 $$\nu^{5}$$ $$=$$ $$\beta_{15} + \beta_{14} + 2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{2}$$ b15 + b14 + 2*b13 - b12 + b11 + b10 - b7 + b5 - b2 $$\nu^{6}$$ $$=$$ $$-\beta_{15} + \beta_{13} - 2\beta_{10} + \beta_{9} - \beta_{7} + 5\beta_{6} + 5\beta_{5} - \beta_{3} - \beta_{2} + \beta_1$$ -b15 + b13 - 2*b10 + b9 - b7 + 5*b6 + 5*b5 - b3 - b2 + b1 $$\nu^{7}$$ $$=$$ $$- 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - 10 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1$$ -2*b15 + b14 + b13 + b12 + b11 + b10 - b9 + 2*b8 - 3*b6 - 10*b4 - b3 - 2*b2 + b1 $$\nu^{8}$$ $$=$$ $$- 3 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + 7 \beta_{7} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 3 \beta_{2} + 2$$ -3*b15 + 3*b14 - 4*b13 - 3*b12 + b11 + b10 - 2*b9 + 6*b8 + 7*b7 + b5 + 2*b4 - 4*b3 - 3*b2 + 2 $$\nu^{9}$$ $$=$$ $$5 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} - \beta_{9} - 7 \beta_{7} - \beta_{6} - \beta_{5} + 5 \beta_{3} + \beta_{2} + 7 \beta _1 + 2$$ 5*b15 - 12*b14 - 5*b13 - 2*b12 - 2*b11 + 10*b10 - b9 - 7*b7 - b6 - b5 + 5*b3 + b2 + 7*b1 + 2 $$\nu^{10}$$ $$=$$ $$6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} - 3 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} - 11 \beta_{6} + 6 \beta_{4} - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_1$$ 6*b15 - 3*b14 + 5*b13 - 3*b12 - 7*b11 - 3*b10 + 7*b9 - 10*b8 - 11*b6 + 6*b4 - 5*b3 + 10*b2 + 5*b1 $$\nu^{11}$$ $$=$$ $$- 7 \beta_{15} - \beta_{14} + 12 \beta_{13} + \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} + 13 \beta_{5} + 26 \beta_{4} + 20 \beta_{3} + \beta_{2} + 26$$ -7*b15 - b14 + 12*b13 + b12 - 3*b11 + 17*b10 - 2*b9 + 2*b8 + 11*b7 + 13*b5 + 26*b4 + 20*b3 + b2 + 26 $$\nu^{12}$$ $$=$$ $$9 \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 19 \beta_{9} - 27 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 9 \beta_{3} - 19 \beta_{2} + 27 \beta _1 - 10$$ 9*b15 + 12*b14 + 15*b13 + 2*b12 + 2*b11 - 6*b10 + 19*b9 - 27*b7 + 3*b6 + 3*b5 + 9*b3 - 19*b2 + 27*b1 - 10 $$\nu^{13}$$ $$=$$ $$14 \beta_{15} - 3 \beta_{14} + 45 \beta_{13} + 13 \beta_{12} - 7 \beta_{11} - 11 \beta_{10} + 7 \beta_{9} - 10 \beta_{8} + 29 \beta_{6} + 14 \beta_{4} - 29 \beta_{3} - 6 \beta_{2} + 5 \beta_1$$ 14*b15 - 3*b14 + 45*b13 + 13*b12 - 7*b11 - 11*b10 + 7*b9 - 10*b8 + 29*b6 + 14*b4 - 29*b3 - 6*b2 + 5*b1 $$\nu^{14}$$ $$=$$ $$- 47 \beta_{15} + 31 \beta_{14} - 28 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} + 41 \beta_{10} - 42 \beta_{9} + 58 \beta_{8} - 13 \beta_{7} + 53 \beta_{5} - 78 \beta_{4} - 12 \beta_{3} - 31 \beta_{2} - 78$$ -47*b15 + 31*b14 - 28*b13 - 31*b12 - 11*b11 + 41*b10 - 42*b9 + 58*b8 - 13*b7 + 53*b5 - 78*b4 - 12*b3 - 31*b2 - 78 $$\nu^{15}$$ $$=$$ $$- 55 \beta_{15} + 12 \beta_{14} - 65 \beta_{13} - 22 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + 19 \beta_{9} + 5 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} - 55 \beta_{3} - 19 \beta_{2} - 5 \beta _1 + 46$$ -55*b15 + 12*b14 - 65*b13 - 22*b12 - 22*b11 + 10*b10 + 19*b9 + 5*b7 - 13*b6 - 13*b5 - 55*b3 - 19*b2 - 5*b1 + 46

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 1.05026 − 0.947078i −0.179748 − 1.40274i 1.41411 − 0.0174668i −1.12494 − 0.857038i 0.820200 + 1.15207i 0.587625 + 1.28635i −1.34532 + 0.436011i −0.722180 + 1.21592i 1.05026 + 0.947078i −0.179748 + 1.40274i 1.41411 + 0.0174668i −1.12494 + 0.857038i 0.820200 − 1.15207i 0.587625 − 1.28635i −1.34532 − 0.436011i −0.722180 − 1.21592i
−1.34532 + 0.436011i 1.52768 + 0.816201i 1.61979 1.17315i −0.602794 0.348023i −2.41110 0.431967i 0.795065 + 1.37709i −1.66763 + 2.28452i 1.66763 + 2.49379i 0.962695 + 0.205379i
13.2 −1.12494 0.857038i −0.986088 + 1.42395i 0.530970 + 1.92823i 1.19115 + 0.687709i 2.32967 0.756738i 1.80469 + 3.12581i 1.05526 2.62420i −1.05526 2.80828i −0.750573 1.79449i
13.3 −0.722180 + 1.21592i 0.294546 1.70682i −0.956913 1.75622i 3.17262 + 1.83171i 1.86264 + 1.59078i −0.191926 0.332426i 2.82649 + 0.104780i −2.82649 1.00547i −4.51841 + 2.53482i
13.4 −0.179748 1.40274i 0.986088 1.42395i −1.93538 + 0.504281i −1.19115 0.687709i −2.17468 1.12728i 1.80469 + 3.12581i 1.05526 + 2.62420i −1.05526 2.80828i −0.750573 + 1.79449i
13.5 0.587625 + 1.28635i 1.69028 0.378078i −1.30939 + 1.51178i −1.97542 1.14051i 1.47959 + 1.95213i −0.907824 1.57240i −2.71411 0.795980i 2.71411 1.27812i 0.306290 3.21128i
13.6 0.820200 + 1.15207i −1.69028 + 0.378078i −0.654545 + 1.88986i 1.97542 + 1.14051i −1.82194 1.63723i −0.907824 1.57240i −2.71411 + 0.795980i 2.71411 1.27812i 0.306290 + 3.21128i
13.7 1.05026 0.947078i −1.52768 0.816201i 0.206086 1.98935i 0.602794 + 0.348023i −2.37747 + 0.589613i 0.795065 + 1.37709i −1.66763 2.28452i 1.66763 + 2.49379i 0.962695 0.205379i
13.8 1.41411 0.0174668i −0.294546 + 1.70682i 1.99939 0.0493999i −3.17262 1.83171i −0.386706 + 2.41877i −0.191926 0.332426i 2.82649 0.104780i −2.82649 1.00547i −4.51841 2.53482i
61.1 −1.34532 0.436011i 1.52768 0.816201i 1.61979 + 1.17315i −0.602794 + 0.348023i −2.41110 + 0.431967i 0.795065 1.37709i −1.66763 2.28452i 1.66763 2.49379i 0.962695 0.205379i
61.2 −1.12494 + 0.857038i −0.986088 1.42395i 0.530970 1.92823i 1.19115 0.687709i 2.32967 + 0.756738i 1.80469 3.12581i 1.05526 + 2.62420i −1.05526 + 2.80828i −0.750573 + 1.79449i
61.3 −0.722180 1.21592i 0.294546 + 1.70682i −0.956913 + 1.75622i 3.17262 1.83171i 1.86264 1.59078i −0.191926 + 0.332426i 2.82649 0.104780i −2.82649 + 1.00547i −4.51841 2.53482i
61.4 −0.179748 + 1.40274i 0.986088 + 1.42395i −1.93538 0.504281i −1.19115 + 0.687709i −2.17468 + 1.12728i 1.80469 3.12581i 1.05526 2.62420i −1.05526 + 2.80828i −0.750573 1.79449i
61.5 0.587625 1.28635i 1.69028 + 0.378078i −1.30939 1.51178i −1.97542 + 1.14051i 1.47959 1.95213i −0.907824 + 1.57240i −2.71411 + 0.795980i 2.71411 + 1.27812i 0.306290 + 3.21128i
61.6 0.820200 1.15207i −1.69028 0.378078i −0.654545 1.88986i 1.97542 1.14051i −1.82194 + 1.63723i −0.907824 + 1.57240i −2.71411 0.795980i 2.71411 + 1.27812i 0.306290 3.21128i
61.7 1.05026 + 0.947078i −1.52768 + 0.816201i 0.206086 + 1.98935i 0.602794 0.348023i −2.37747 0.589613i 0.795065 1.37709i −1.66763 + 2.28452i 1.66763 2.49379i 0.962695 + 0.205379i
61.8 1.41411 + 0.0174668i −0.294546 1.70682i 1.99939 + 0.0493999i −3.17262 + 1.83171i −0.386706 2.41877i −0.191926 + 0.332426i 2.82649 + 0.104780i −2.82649 + 1.00547i −4.51841 + 2.53482i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.n.b 16
3.b odd 2 1 216.2.n.b 16
4.b odd 2 1 288.2.r.b 16
8.b even 2 1 inner 72.2.n.b 16
8.d odd 2 1 288.2.r.b 16
9.c even 3 1 inner 72.2.n.b 16
9.c even 3 1 648.2.d.j 8
9.d odd 6 1 216.2.n.b 16
9.d odd 6 1 648.2.d.k 8
12.b even 2 1 864.2.r.b 16
24.f even 2 1 864.2.r.b 16
24.h odd 2 1 216.2.n.b 16
36.f odd 6 1 288.2.r.b 16
36.f odd 6 1 2592.2.d.j 8
36.h even 6 1 864.2.r.b 16
36.h even 6 1 2592.2.d.k 8
72.j odd 6 1 216.2.n.b 16
72.j odd 6 1 648.2.d.k 8
72.l even 6 1 864.2.r.b 16
72.l even 6 1 2592.2.d.k 8
72.n even 6 1 inner 72.2.n.b 16
72.n even 6 1 648.2.d.j 8
72.p odd 6 1 288.2.r.b 16
72.p odd 6 1 2592.2.d.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b 16 1.a even 1 1 trivial
72.2.n.b 16 8.b even 2 1 inner
72.2.n.b 16 9.c even 3 1 inner
72.2.n.b 16 72.n even 6 1 inner
216.2.n.b 16 3.b odd 2 1
216.2.n.b 16 9.d odd 6 1
216.2.n.b 16 24.h odd 2 1
216.2.n.b 16 72.j odd 6 1
288.2.r.b 16 4.b odd 2 1
288.2.r.b 16 8.d odd 2 1
288.2.r.b 16 36.f odd 6 1
288.2.r.b 16 72.p odd 6 1
648.2.d.j 8 9.c even 3 1
648.2.d.j 8 72.n even 6 1
648.2.d.k 8 9.d odd 6 1
648.2.d.k 8 72.j odd 6 1
864.2.r.b 16 12.b even 2 1
864.2.r.b 16 24.f even 2 1
864.2.r.b 16 36.h even 6 1
864.2.r.b 16 72.l even 6 1
2592.2.d.j 8 36.f odd 6 1
2592.2.d.j 8 72.p odd 6 1
2592.2.d.k 8 36.h even 6 1
2592.2.d.k 8 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{16} - 21T_{5}^{14} + 326T_{5}^{12} - 2049T_{5}^{10} + 9318T_{5}^{8} - 18357T_{5}^{6} + 26129T_{5}^{4} - 11712T_{5}^{2} + 4096$$ acting on $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - T^{15} + T^{14} + 2 T^{12} - 4 T^{11} + \cdots + 256$$
$3$ $$T^{16} - T^{14} - 2 T^{12} + 9 T^{10} + \cdots + 6561$$
$5$ $$T^{16} - 21 T^{14} + 326 T^{12} + \cdots + 4096$$
$7$ $$(T^{8} - 3 T^{7} + 14 T^{6} - 3 T^{5} + \cdots + 16)^{2}$$
$11$ $$T^{16} - 40 T^{14} + 1134 T^{12} + \cdots + 10556001$$
$13$ $$T^{16} - 53 T^{14} + 2094 T^{12} + \cdots + 20736$$
$17$ $$(T^{4} + 7 T^{3} - 2 T^{2} - 48 T + 36)^{4}$$
$19$ $$(T^{8} + 83 T^{6} + 1660 T^{4} + \cdots + 5184)^{2}$$
$23$ $$(T^{8} + 5 T^{7} + 60 T^{6} + 275 T^{5} + \cdots + 90000)^{2}$$
$29$ $$T^{16} - 109 T^{14} + \cdots + 4032758016$$
$31$ $$(T^{8} + 5 T^{7} + 76 T^{6} + 155 T^{5} + \cdots + 92416)^{2}$$
$37$ $$(T^{8} + 152 T^{6} + 8080 T^{4} + \cdots + 1327104)^{2}$$
$41$ $$(T^{8} + 4 T^{7} + 90 T^{6} + 616 T^{5} + \cdots + 335241)^{2}$$
$43$ $$T^{16} - 20 T^{14} + 294 T^{12} + \cdots + 81$$
$47$ $$(T^{8} - 3 T^{7} + 90 T^{6} + \cdots + 2178576)^{2}$$
$53$ $$(T^{8} + 160 T^{6} + 7744 T^{4} + \cdots + 451584)^{2}$$
$59$ $$T^{16} - 108 T^{14} + \cdots + 131079601$$
$61$ $$T^{16} - 165 T^{14} + \cdots + 176319369216$$
$67$ $$T^{16} - 240 T^{14} + \cdots + 9881774573841$$
$71$ $$(T^{4} - 18 T^{3} + 48 T^{2} + 252 T - 864)^{4}$$
$73$ $$(T^{4} + 11 T^{3} - 14 T^{2} - 84 T + 36)^{4}$$
$79$ $$(T^{8} + 15 T^{7} + 236 T^{6} + 1065 T^{5} + \cdots + 3136)^{2}$$
$83$ $$T^{16} - 105 T^{14} + \cdots + 31713911056$$
$89$ $$(T^{4} - 16 T^{3} + 52 T^{2} + 156 T - 504)^{4}$$
$97$ $$(T^{8} + 134 T^{6} + 1560 T^{5} + \cdots + 1324801)^{2}$$