# Properties

 Label 375.2.i.c Level $375$ Weight $2$ Character orbit 375.i Analytic conductor $2.994$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(49,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 7]))

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

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

chi := DirichletCharacter("375.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.i (of order $$10$$, 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: $$2.99439007580$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1$$ x^16 + 20*x^14 + 156*x^12 + 610*x^10 + 1286*x^8 + 1440*x^6 + 761*x^4 + 130*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( 9 \nu^{15} - 11 \nu^{14} + 158 \nu^{13} - 194 \nu^{12} + 1016 \nu^{11} - 1257 \nu^{10} + 2975 \nu^{9} - 3731 \nu^{8} + 4097 \nu^{7} - 5277 \nu^{6} + 2332 \nu^{5} - 3223 \nu^{4} + 270 \nu^{3} - 565 \nu^{2} + \cdots - 2 ) / 4$$ (9*v^15 - 11*v^14 + 158*v^13 - 194*v^12 + 1016*v^11 - 1257*v^10 + 2975*v^9 - 3731*v^8 + 4097*v^7 - 5277*v^6 + 2332*v^5 - 3223*v^4 + 270*v^3 - 565*v^2 - 34*v - 2) / 4 $$\beta_{2}$$ $$=$$ $$( 15 \nu^{15} - 6 \nu^{14} + 263 \nu^{13} - 106 \nu^{12} + 1688 \nu^{11} - 689 \nu^{10} + 4929 \nu^{9} - 2058 \nu^{8} + 6764 \nu^{7} - 2948 \nu^{6} + 3837 \nu^{5} - 1847 \nu^{4} + 444 \nu^{3} - 339 \nu^{2} + \cdots + 2 ) / 4$$ (15*v^15 - 6*v^14 + 263*v^13 - 106*v^12 + 1688*v^11 - 689*v^10 + 4929*v^9 - 2058*v^8 + 6764*v^7 - 2948*v^6 + 3837*v^5 - 1847*v^4 + 444*v^3 - 339*v^2 - 64*v + 2) / 4 $$\beta_{3}$$ $$=$$ $$( - 9 \nu^{15} - 11 \nu^{14} - 158 \nu^{13} - 194 \nu^{12} - 1016 \nu^{11} - 1257 \nu^{10} - 2975 \nu^{9} - 3731 \nu^{8} - 4097 \nu^{7} - 5277 \nu^{6} - 2332 \nu^{5} - 3223 \nu^{4} - 270 \nu^{3} + \cdots - 2 ) / 4$$ (-9*v^15 - 11*v^14 - 158*v^13 - 194*v^12 - 1016*v^11 - 1257*v^10 - 2975*v^9 - 3731*v^8 - 4097*v^7 - 5277*v^6 - 2332*v^5 - 3223*v^4 - 270*v^3 - 565*v^2 + 34*v - 2) / 4 $$\beta_{4}$$ $$=$$ $$( - 15 \nu^{15} - 6 \nu^{14} - 263 \nu^{13} - 106 \nu^{12} - 1688 \nu^{11} - 689 \nu^{10} - 4929 \nu^{9} - 2058 \nu^{8} - 6764 \nu^{7} - 2948 \nu^{6} - 3837 \nu^{5} - 1847 \nu^{4} - 444 \nu^{3} + \cdots + 2 ) / 4$$ (-15*v^15 - 6*v^14 - 263*v^13 - 106*v^12 - 1688*v^11 - 689*v^10 - 4929*v^9 - 2058*v^8 - 6764*v^7 - 2948*v^6 - 3837*v^5 - 1847*v^4 - 444*v^3 - 339*v^2 + 64*v + 2) / 4 $$\beta_{5}$$ $$=$$ $$( -24\nu^{14} - 423\nu^{12} - 2739\nu^{10} - 8128\nu^{8} - 11513\nu^{6} - 7066\nu^{4} - 1247\nu^{2} - 2\nu - 2 ) / 4$$ (-24*v^14 - 423*v^12 - 2739*v^10 - 8128*v^8 - 11513*v^6 - 7066*v^4 - 1247*v^2 - 2*v - 2) / 4 $$\beta_{6}$$ $$=$$ $$( -24\nu^{14} - 423\nu^{12} - 2739\nu^{10} - 8128\nu^{8} - 11513\nu^{6} - 7066\nu^{4} - 1247\nu^{2} + 2\nu - 2 ) / 4$$ (-24*v^14 - 423*v^12 - 2739*v^10 - 8128*v^8 - 11513*v^6 - 7066*v^4 - 1247*v^2 + 2*v - 2) / 4 $$\beta_{7}$$ $$=$$ $$( 7 \nu^{15} - 32 \nu^{14} + 124 \nu^{13} - 564 \nu^{12} + 810 \nu^{11} - 3652 \nu^{10} + 2444 \nu^{9} - 10838 \nu^{8} + 3583 \nu^{7} - 15360 \nu^{6} + 2400 \nu^{5} - 9462 \nu^{4} + 595 \nu^{3} + \cdots - 12 ) / 4$$ (7*v^15 - 32*v^14 + 124*v^13 - 564*v^12 + 810*v^11 - 3652*v^10 + 2444*v^9 - 10838*v^8 + 3583*v^7 - 15360*v^6 + 2400*v^5 - 9462*v^4 + 595*v^3 - 1716*v^2 + 47*v - 12) / 4 $$\beta_{8}$$ $$=$$ $$( - 7 \nu^{15} - 32 \nu^{14} - 124 \nu^{13} - 564 \nu^{12} - 810 \nu^{11} - 3652 \nu^{10} - 2444 \nu^{9} - 10838 \nu^{8} - 3583 \nu^{7} - 15360 \nu^{6} - 2400 \nu^{5} - 9462 \nu^{4} - 595 \nu^{3} + \cdots - 12 ) / 4$$ (-7*v^15 - 32*v^14 - 124*v^13 - 564*v^12 - 810*v^11 - 3652*v^10 - 2444*v^9 - 10838*v^8 - 3583*v^7 - 15360*v^6 - 2400*v^5 - 9462*v^4 - 595*v^3 - 1716*v^2 - 47*v - 12) / 4 $$\beta_{9}$$ $$=$$ $$( 12 \nu^{15} - 45 \nu^{14} + 208 \nu^{13} - 794 \nu^{12} + 1308 \nu^{11} - 5150 \nu^{10} + 3668 \nu^{9} - 15324 \nu^{8} + 4594 \nu^{7} - 21803 \nu^{6} + 1920 \nu^{5} - 13514 \nu^{4} - 330 \nu^{3} + \cdots - 17 ) / 4$$ (12*v^15 - 45*v^14 + 208*v^13 - 794*v^12 + 1308*v^11 - 5150*v^10 + 3668*v^9 - 15324*v^8 + 4594*v^7 - 21803*v^6 + 1920*v^5 - 13514*v^4 - 330*v^3 - 2485*v^2 - 156*v - 17) / 4 $$\beta_{10}$$ $$=$$ $$( - 10 \nu^{15} + 45 \nu^{14} - 177 \nu^{13} + 794 \nu^{12} - 1154 \nu^{11} + 5150 \nu^{10} - 3465 \nu^{9} + 15324 \nu^{8} - 5013 \nu^{7} + 21803 \nu^{6} - 3225 \nu^{5} + 13514 \nu^{4} + \cdots + 23 ) / 4$$ (-10*v^15 + 45*v^14 - 177*v^13 + 794*v^12 - 1154*v^11 + 5150*v^10 - 3465*v^9 + 15324*v^8 - 5013*v^7 + 21803*v^6 - 3225*v^5 + 13514*v^4 - 675*v^3 + 2487*v^2 - 19*v + 23) / 4 $$\beta_{11}$$ $$=$$ $$( 6 \nu^{15} - 45 \nu^{14} + 108 \nu^{13} - 794 \nu^{12} + 724 \nu^{11} - 5150 \nu^{10} + 2282 \nu^{9} - 15324 \nu^{8} + 3600 \nu^{7} - 21803 \nu^{6} + 2744 \nu^{5} - 13514 \nu^{4} + 870 \nu^{3} + \cdots - 17 ) / 4$$ (6*v^15 - 45*v^14 + 108*v^13 - 794*v^12 + 724*v^11 - 5150*v^10 + 2282*v^9 - 15324*v^8 + 3600*v^7 - 21803*v^6 + 2744*v^5 - 13514*v^4 + 870*v^3 - 2485*v^2 + 88*v - 17) / 4 $$\beta_{12}$$ $$=$$ $$( - 22 \nu^{15} - 29 \nu^{14} - 387 \nu^{13} - 512 \nu^{12} - 2498 \nu^{11} - 3324 \nu^{10} - 7373 \nu^{9} - 9905 \nu^{8} - 10347 \nu^{7} - 14123 \nu^{6} - 6237 \nu^{5} - 8782 \nu^{4} - 1041 \nu^{3} + \cdots - 8 ) / 4$$ (-22*v^15 - 29*v^14 - 387*v^13 - 512*v^12 - 2498*v^11 - 3324*v^10 - 7373*v^9 - 9905*v^8 - 10347*v^7 - 14123*v^6 - 6237*v^5 - 8782*v^4 - 1041*v^3 - 1620*v^2 + 7*v - 8) / 4 $$\beta_{13}$$ $$=$$ $$( 10 \nu^{15} + 45 \nu^{14} + 177 \nu^{13} + 794 \nu^{12} + 1154 \nu^{11} + 5150 \nu^{10} + 3465 \nu^{9} + 15324 \nu^{8} + 5013 \nu^{7} + 21803 \nu^{6} + 3225 \nu^{5} + 13514 \nu^{4} + 675 \nu^{3} + \cdots + 23 ) / 4$$ (10*v^15 + 45*v^14 + 177*v^13 + 794*v^12 + 1154*v^11 + 5150*v^10 + 3465*v^9 + 15324*v^8 + 5013*v^7 + 21803*v^6 + 3225*v^5 + 13514*v^4 + 675*v^3 + 2487*v^2 + 19*v + 23) / 4 $$\beta_{14}$$ $$=$$ $$( 29 \nu^{15} + 32 \nu^{14} + 512 \nu^{13} + 564 \nu^{12} + 3325 \nu^{11} + 3652 \nu^{10} + 9921 \nu^{9} + 10838 \nu^{8} + 14211 \nu^{7} + 15360 \nu^{6} + 8977 \nu^{5} + 9462 \nu^{4} + 1785 \nu^{3} + \cdots + 10 ) / 4$$ (29*v^15 + 32*v^14 + 512*v^13 + 564*v^12 + 3325*v^11 + 3652*v^10 + 9921*v^9 + 10838*v^8 + 14211*v^7 + 15360*v^6 + 8977*v^5 + 9462*v^4 + 1785*v^3 + 1716*v^2 + 40*v + 10) / 4 $$\beta_{15}$$ $$=$$ $$( - 45 \nu^{15} - 17 \nu^{14} - 794 \nu^{13} - 299 \nu^{12} - 5150 \nu^{11} - 1930 \nu^{10} - 15324 \nu^{9} - 5701 \nu^{8} - 21803 \nu^{7} - 8032 \nu^{6} - 13514 \nu^{5} - 4923 \nu^{4} - 2487 \nu^{3} + \cdots - 8 ) / 4$$ (-45*v^15 - 17*v^14 - 794*v^13 - 299*v^12 - 5150*v^11 - 1930*v^10 - 15324*v^9 - 5701*v^8 - 21803*v^7 - 8032*v^6 - 13514*v^5 - 4923*v^4 - 2487*v^3 - 904*v^2 - 25*v - 8) / 4
 $$\nu$$ $$=$$ $$\beta_{6} - \beta_{5}$$ b6 - b5 $$\nu^{2}$$ $$=$$ $$\beta_{13} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{3} - \beta _1 - 3$$ b13 + b11 + b10 + b9 + b3 - b1 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} + 2\beta_{8} - 5\beta_{6} + 4\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 1$$ -b15 - b13 - b12 + b10 + 2*b8 - 5*b6 + 4*b5 + b4 + b3 - b1 + 1 $$\nu^{4}$$ $$=$$ $$- 2 \beta_{15} - 9 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 5 \beta_{10} - 9 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} - 2 \beta_{5} - 7 \beta_{3} + 2 \beta_{2} + 7 \beta _1 + 20$$ -2*b15 - 9*b13 + 2*b12 - 9*b11 - 5*b10 - 9*b9 + 3*b8 + 3*b7 - 2*b5 - 7*b3 + 2*b2 + 7*b1 + 20 $$\nu^{5}$$ $$=$$ $$8 \beta_{15} + 7 \beta_{13} + 8 \beta_{12} - 7 \beta_{10} - 17 \beta_{8} + \beta_{7} + 28 \beta_{6} - 20 \beta_{5} - 7 \beta_{4} - 10 \beta_{3} - \beta_{2} + 10 \beta _1 - 8$$ 8*b15 + 7*b13 + 8*b12 - 7*b10 - 17*b8 + b7 + 28*b6 - 20*b5 - 7*b4 - 10*b3 - b2 + 10*b1 - 8 $$\nu^{6}$$ $$=$$ $$19 \beta_{15} + 63 \beta_{13} - 19 \beta_{12} + 62 \beta_{11} + 25 \beta_{10} + 62 \beta_{9} - 28 \beta_{8} - 28 \beta_{7} + 19 \beta_{5} + 3 \beta_{4} + 44 \beta_{3} - 16 \beta_{2} - 42 \beta _1 - 130$$ 19*b15 + 63*b13 - 19*b12 + 62*b11 + 25*b10 + 62*b9 - 28*b8 - 28*b7 + 19*b5 + 3*b4 + 44*b3 - 16*b2 - 42*b1 - 130 $$\nu^{7}$$ $$=$$ $$- 57 \beta_{15} - 8 \beta_{14} - 43 \beta_{13} - 57 \beta_{12} + \beta_{11} + 43 \beta_{10} - \beta_{9} + 116 \beta_{8} - 10 \beta_{7} - 167 \beta_{6} + 110 \beta_{5} + 43 \beta_{4} + 78 \beta_{3} + 14 \beta_{2} - 78 \beta _1 + 53$$ -57*b15 - 8*b14 - 43*b13 - 57*b12 + b11 + 43*b10 - b9 + 116*b8 - 10*b7 - 167*b6 + 110*b5 + 43*b4 + 78*b3 + 14*b2 - 78*b1 + 53 $$\nu^{8}$$ $$=$$ $$- 144 \beta_{15} - 413 \beta_{13} + 144 \beta_{12} - 399 \beta_{11} - 125 \beta_{10} - 399 \beta_{9} + 206 \beta_{8} + 206 \beta_{7} + 4 \beta_{6} - 140 \beta_{5} - 42 \beta_{4} - 269 \beta_{3} + 102 \beta_{2} + 241 \beta _1 + 826$$ -144*b15 - 413*b13 + 144*b12 - 399*b11 - 125*b10 - 399*b9 + 206*b8 + 206*b7 + 4*b6 - 140*b5 - 42*b4 - 269*b3 + 102*b2 + 241*b1 + 826 $$\nu^{9}$$ $$=$$ $$395 \beta_{15} + 120 \beta_{14} + 259 \beta_{13} + 395 \beta_{12} - 18 \beta_{11} - 259 \beta_{10} + 18 \beta_{9} - 752 \beta_{8} + 82 \beta_{7} + 1024 \beta_{6} - 629 \beta_{5} - 255 \beta_{4} - 563 \beta_{3} - 140 \beta_{2} + \cdots - 335$$ 395*b15 + 120*b14 + 259*b13 + 395*b12 - 18*b11 - 259*b10 + 18*b9 - 752*b8 + 82*b7 + 1024*b6 - 629*b5 - 255*b4 - 563*b3 - 140*b2 + 563*b1 - 335 $$\nu^{10}$$ $$=$$ $$1018 \beta_{15} + 2654 \beta_{13} - 1018 \beta_{12} + 2518 \beta_{11} + 618 \beta_{10} + 2518 \beta_{9} - 1415 \beta_{8} - 1415 \beta_{7} - 60 \beta_{6} + 958 \beta_{5} + 406 \beta_{4} + 1640 \beta_{3} - 612 \beta_{2} + \cdots - 5209$$ 1018*b15 + 2654*b13 - 1018*b12 + 2518*b11 + 618*b10 + 2518*b9 - 1415*b8 - 1415*b7 - 60*b6 + 958*b5 + 406*b4 + 1640*b3 - 612*b2 - 1360*b1 - 5209 $$\nu^{11}$$ $$=$$ $$- 2699 \beta_{15} - 1212 \beta_{14} - 1564 \beta_{13} - 2699 \beta_{12} + 200 \beta_{11} + 1564 \beta_{10} - 200 \beta_{9} + 4811 \beta_{8} - 625 \beta_{7} - 6359 \beta_{6} + 3660 \beta_{5} + 1496 \beta_{4} + \cdots + 2093$$ -2699*b15 - 1212*b14 - 1564*b13 - 2699*b12 + 200*b11 + 1564*b10 - 200*b9 + 4811*b8 - 625*b7 - 6359*b6 + 3660*b5 + 1496*b4 + 3931*b3 + 1203*b2 - 3931*b1 + 2093 $$\nu^{12}$$ $$=$$ $$- 6987 \beta_{15} - 16954 \beta_{13} + 6987 \beta_{12} - 15819 \beta_{11} - 2980 \beta_{10} - 15819 \beta_{9} + 9477 \beta_{8} + 9477 \beta_{7} + 606 \beta_{6} - 6381 \beta_{5} - 3379 \beta_{4} + \cdots + 32816$$ -6987*b15 - 16954*b13 + 6987*b12 - 15819*b11 - 2980*b10 - 15819*b9 + 9477*b8 + 9477*b7 + 606*b6 - 6381*b5 - 3379*b4 - 10035*b3 + 3608*b2 + 7629*b1 + 32816 $$\nu^{13}$$ $$=$$ $$18251 \beta_{15} + 10376 \beta_{14} + 9506 \beta_{13} + 18251 \beta_{12} - 1809 \beta_{11} - 9506 \beta_{10} + 1809 \beta_{9} - 30691 \beta_{8} + 4565 \beta_{7} + 39759 \beta_{6} - 21508 \beta_{5} + \cdots - 13063$$ 18251*b15 + 10376*b14 + 9506*b13 + 18251*b12 - 1809*b11 - 9506*b10 + 1809*b9 - 30691*b8 + 4565*b7 + 39759*b6 - 21508*b5 - 8764*b4 - 26944*b3 - 9487*b2 + 26944*b1 - 13063 $$\nu^{14}$$ $$=$$ $$47209 \beta_{15} + 108172 \beta_{13} - 47209 \beta_{12} + 99427 \beta_{11} + 13754 \beta_{10} + 99427 \beta_{9} - 62762 \beta_{8} - 62762 \beta_{7} - 5188 \beta_{6} + 42021 \beta_{5} + 26005 \beta_{4} + \cdots - 207014$$ 47209*b15 + 108172*b13 - 47209*b12 + 99427*b11 + 13754*b10 + 99427*b9 - 62762*b8 - 62762*b7 - 5188*b6 + 42021*b5 + 26005*b4 + 61705*b3 - 21204*b2 - 42731*b1 - 207014 $$\nu^{15}$$ $$=$$ $$- 122384 \beta_{15} - 81360 \beta_{14} - 58148 \beta_{13} - 122384 \beta_{12} + 14675 \beta_{11} + 58148 \beta_{10} - 14675 \beta_{9} + 195806 \beta_{8} - 32398 \beta_{7} - 249699 \beta_{6} + \cdots + 81704$$ -122384*b15 - 81360*b14 - 58148*b13 - 122384*b12 + 14675*b11 + 58148*b10 - 14675*b9 + 195806*b8 - 32398*b7 - 249699*b6 + 127315*b5 + 51476*b4 + 182407*b3 + 70908*b2 - 182407*b1 + 81704

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$-\beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 2.53767i 0.0898194i − 1.08982i − 1.53767i − 2.53767i − 0.0898194i 1.08982i 1.53767i 1.53655i 1.35083i − 0.536547i − 2.35083i − 1.53655i − 1.35083i 0.536547i 2.35083i
−1.49161 + 2.05302i 0.951057 0.309017i −1.37197 4.22249i 0 −0.784184 + 2.41347i 1.04054i 5.88835 + 1.91324i 0.809017 0.587785i 0
49.2 −0.0527945 + 0.0726655i −0.951057 + 0.309017i 0.615541 + 1.89444i 0 0.0277557 0.0854234i 4.36070i −0.341004 0.110799i 0.809017 0.587785i 0
49.3 0.640580 0.881682i −0.951057 + 0.309017i 0.251013 + 0.772537i 0 −0.336773 + 1.03648i 3.08724i 2.91489 + 0.947104i 0.809017 0.587785i 0
49.4 0.903822 1.24400i 0.951057 0.309017i −0.112618 0.346603i 0 0.475167 1.46241i 1.68601i 2.39187 + 0.777165i 0.809017 0.587785i 0
199.1 −1.49161 2.05302i 0.951057 + 0.309017i −1.37197 + 4.22249i 0 −0.784184 2.41347i 1.04054i 5.88835 1.91324i 0.809017 + 0.587785i 0
199.2 −0.0527945 0.0726655i −0.951057 0.309017i 0.615541 1.89444i 0 0.0277557 + 0.0854234i 4.36070i −0.341004 + 0.110799i 0.809017 + 0.587785i 0
199.3 0.640580 + 0.881682i −0.951057 0.309017i 0.251013 0.772537i 0 −0.336773 1.03648i 3.08724i 2.91489 0.947104i 0.809017 + 0.587785i 0
199.4 0.903822 + 1.24400i 0.951057 + 0.309017i −0.112618 + 0.346603i 0 0.475167 + 1.46241i 1.68601i 2.39187 0.777165i 0.809017 + 0.587785i 0
274.1 −1.46134 0.474819i −0.587785 + 0.809017i 0.292036 + 0.212177i 0 1.24309 0.903160i 1.49550i 1.48030 + 2.03746i −0.309017 0.951057i 0
274.2 −1.28472 0.417429i 0.587785 0.809017i −0.141788 0.103015i 0 −1.09284 + 0.793998i 1.59580i 1.72715 + 2.37722i −0.309017 0.951057i 0
274.3 0.510286 + 0.165802i −0.587785 + 0.809017i −1.38513 1.00636i 0 −0.434076 + 0.315374i 2.57318i −1.17071 1.61134i −0.309017 0.951057i 0
274.4 2.23577 + 0.726446i 0.587785 0.809017i 2.85292 + 2.07277i 0 1.90186 1.38178i 3.48189i 2.10915 + 2.90300i −0.309017 0.951057i 0
349.1 −1.46134 + 0.474819i −0.587785 0.809017i 0.292036 0.212177i 0 1.24309 + 0.903160i 1.49550i 1.48030 2.03746i −0.309017 + 0.951057i 0
349.2 −1.28472 + 0.417429i 0.587785 + 0.809017i −0.141788 + 0.103015i 0 −1.09284 0.793998i 1.59580i 1.72715 2.37722i −0.309017 + 0.951057i 0
349.3 0.510286 0.165802i −0.587785 0.809017i −1.38513 + 1.00636i 0 −0.434076 0.315374i 2.57318i −1.17071 + 1.61134i −0.309017 + 0.951057i 0
349.4 2.23577 0.726446i 0.587785 + 0.809017i 2.85292 2.07277i 0 1.90186 + 1.38178i 3.48189i 2.10915 2.90300i −0.309017 + 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.i.c 16
5.b even 2 1 75.2.i.a 16
5.c odd 4 1 375.2.g.d 16
5.c odd 4 1 375.2.g.e 16
15.d odd 2 1 225.2.m.b 16
25.d even 5 1 75.2.i.a 16
25.d even 5 1 1875.2.b.h 16
25.e even 10 1 inner 375.2.i.c 16
25.e even 10 1 1875.2.b.h 16
25.f odd 20 1 375.2.g.d 16
25.f odd 20 1 375.2.g.e 16
25.f odd 20 1 1875.2.a.m 8
25.f odd 20 1 1875.2.a.p 8
75.j odd 10 1 225.2.m.b 16
75.l even 20 1 5625.2.a.t 8
75.l even 20 1 5625.2.a.bd 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.i.a 16 5.b even 2 1
75.2.i.a 16 25.d even 5 1
225.2.m.b 16 15.d odd 2 1
225.2.m.b 16 75.j odd 10 1
375.2.g.d 16 5.c odd 4 1
375.2.g.d 16 25.f odd 20 1
375.2.g.e 16 5.c odd 4 1
375.2.g.e 16 25.f odd 20 1
375.2.i.c 16 1.a even 1 1 trivial
375.2.i.c 16 25.e even 10 1 inner
1875.2.a.m 8 25.f odd 20 1
1875.2.a.p 8 25.f odd 20 1
1875.2.b.h 16 25.d even 5 1
1875.2.b.h 16 25.e even 10 1
5625.2.a.t 8 75.l even 20 1
5625.2.a.bd 8 75.l even 20 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 5 T_{2}^{14} - 10 T_{2}^{13} + 26 T_{2}^{12} + 50 T_{2}^{11} - 95 T_{2}^{10} - 20 T_{2}^{9} + 381 T_{2}^{8} - 30 T_{2}^{7} - 365 T_{2}^{6} + 410 T_{2}^{5} + 226 T_{2}^{4} - 360 T_{2}^{3} + 85 T_{2}^{2} + 10 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 5 T^{14} - 10 T^{13} + 26 T^{12} + \cdots + 1$$
$3$ $$(T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{16}$$
$7$ $$T^{16} + 56 T^{14} + 1236 T^{12} + \cdots + 255025$$
$11$ $$T^{16} + 6 T^{15} + 48 T^{14} + \cdots + 27888961$$
$13$ $$T^{16} - 30 T^{14} - 210 T^{13} + \cdots + 78961$$
$17$ $$T^{16} + 10 T^{15} - 13 T^{14} + \cdots + 53860921$$
$19$ $$T^{16} + 2 T^{15} + 23 T^{14} + \cdots + 6375625$$
$23$ $$T^{16} - 20 T^{15} + 89 T^{14} + \cdots + 4389025$$
$29$ $$T^{16} - 16 T^{15} + 217 T^{14} + \cdots + 156025$$
$31$ $$T^{16} - 6 T^{15} + 97 T^{14} + \cdots + 15625$$
$37$ $$T^{16} - 10 T^{15} + \cdots + 8653650625$$
$41$ $$T^{16} + 14 T^{15} + 82 T^{14} + \cdots + 22137025$$
$43$ $$T^{16} + 388 T^{14} + \cdots + 527207521$$
$47$ $$T^{16} + 40 T^{15} + \cdots + 36687479166361$$
$53$ $$T^{16} + 10 T^{15} + \cdots + 40398990025$$
$59$ $$T^{16} - 12 T^{15} + 188 T^{14} + \cdots + 12924025$$
$61$ $$T^{16} + \cdots + 275701483356121$$
$67$ $$T^{16} - 40 T^{15} + 762 T^{14} + \cdots + 13980121$$
$71$ $$T^{16} + 8 T^{15} + \cdots + 25529328841$$
$73$ $$T^{16} - 20 T^{15} + \cdots + 757413387025$$
$79$ $$T^{16} + 20 T^{15} + \cdots + 3940125750625$$
$83$ $$T^{16} + 10 T^{15} + \cdots + 2356228681$$
$89$ $$T^{16} - 18 T^{15} + 68 T^{14} + 1034 T^{13} + \cdots + 25$$
$97$ $$T^{16} + 40 T^{15} + \cdots + 216648961$$