# Properties

 Label 363.3.h.o Level $363$ Weight $3$ Character orbit 363.h Analytic conductor $9.891$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 363.h (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.89103359628$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561$$ x^16 - 12*x^14 + 180*x^12 - 2562*x^10 + 25179*x^8 - 96398*x^6 + 239275*x^4 - 536393*x^2 + 1771561 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( - \beta_{12} + \beta_{6} - \beta_{2}) q^{3} + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{8} + 3 \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + (\beta_{10} + 3 \beta_{8} + \beta_{6} - 3 \beta_{3} - 3 \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b4 * q^2 + (-b12 + b6 - b2) * q^3 + (-b15 + b14 + b13 - b12 + b10 - 2*b8 + 3*b6 - b5 + b2 - b1 + 1) * q^4 + (b13 + b12 - b11 + b10 - b9 - b8 - b7 - 2*b5 + 2*b4 - 2*b1) * q^5 + (2*b14 - b13 - b10 + b8 + b5 + 3*b3 + 2*b2 + 3) * q^6 + (-b13 + b12 - b10 - b9 - 3*b6 - 2*b2 - 2) * q^7 + (-b15 - b14 + 2*b13 + 2*b12 + b11 + b9 + 2*b8 + 2*b7 - b5 + 2*b4 + 2*b1) * q^8 + (b10 + 3*b8 + b6 - 3*b3 - 3*b2 - 1) * q^9 $$q + \beta_{4} q^{2} + ( - \beta_{12} + \beta_{6} - \beta_{2}) q^{3} + ( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - 2 \beta_{8} + 3 \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + (8 \beta_{15} + 8 \beta_{14} + 31 \beta_{13} + 31 \beta_{12} + 38 \beta_{11} + \cdots + 22 \beta_1) q^{98}+O(q^{100})$$ q + b4 * q^2 + (-b12 + b6 - b2) * q^3 + (-b15 + b14 + b13 - b12 + b10 - 2*b8 + 3*b6 - b5 + b2 - b1 + 1) * q^4 + (b13 + b12 - b11 + b10 - b9 - b8 - b7 - 2*b5 + 2*b4 - 2*b1) * q^5 + (2*b14 - b13 - b10 + b8 + b5 + 3*b3 + 2*b2 + 3) * q^6 + (-b13 + b12 - b10 - b9 - 3*b6 - 2*b2 - 2) * q^7 + (-b15 - b14 + 2*b13 + 2*b12 + b11 + b9 + 2*b8 + 2*b7 - b5 + 2*b4 + 2*b1) * q^8 + (b10 + 3*b8 + b6 - 3*b3 - 3*b2 - 1) * q^9 + (b15 - b14 + 2*b13 - 2*b12 - b10 - 2*b8 + 4*b7 + 3*b6 + b5 - 3*b3 + b1 - 1) * q^10 + (3*b14 + 6*b13 + b12 + 3*b11 + 4*b10 - 4*b9 - 3*b8 - b7 + 7*b6 - 7*b5 + 4*b4 - 7*b3 - b1 + 4) * q^12 + (-3*b8 + 3*b7 + 3*b6 - 5*b3 - 5*b2 - 3) * q^13 + (-2*b15 - 2*b14 - b13 - b12 - 5*b11 + 2*b9 - 3*b8 - 3*b7 + 5*b5 - 3*b4 - 3*b1) * q^14 + (-3*b15 - b14 - b13 - b12 - 3*b11 - 2*b10 + 3*b9 - 2*b8 + 9*b6 + 4*b5 - 5*b2 - 5) * q^15 + (-b15 + b14 + 3*b13 - 3*b12 + 4*b10 + 3*b9 - 4*b8 + 2*b7 - b5 + 11*b3 + 5*b2 - b1 + 11) * q^16 + (-b15 - b14 - 4*b11 + b10 - b8 - b7 - b5 - b4) * q^17 + (-6*b15 + b14 - 2*b13 - 10*b12 - 3*b11 + 7*b10 - 7*b8 - 2*b6 + 2*b5 + 6*b2 - 9*b1 + 6) * q^18 + (b15 - b14 - 2*b13 + 2*b12 + b9 + b8 + b7 + 9*b6 + b5 + 5*b3 - 9*b2 + b1) * q^19 + (-5*b10 + 5*b9 + 3*b8 + 3*b7 + b5 - 2*b4 + b1) * q^20 + (3*b15 - 3*b14 - 3*b13 + 3*b12 - 3*b10 + 3*b8 + 3*b7 - 9*b6 + 6*b5 + 9*b3 + 3*b1 + 3) * q^21 + (-6*b15 - 6*b14 - b13 - b12 + 3*b11 + b10 + 5*b9 + 4*b8 + 4*b7 + 2*b5 - 5*b4 + 2*b1) * q^23 + (-2*b10 + 7*b9 + 4*b7 + 6*b6 + b5 + 8*b4 + b3 + b2 + b1 - 6) * q^24 + (b15 - b14 + b13 - b12 + b9 + b8 + b7 - 15*b6 + b5 + 13*b3 + 15*b2 + b1) * q^25 + (-2*b15 - 2*b14 + 7*b13 + 7*b12 + 4*b11 - 3*b10 + 5*b9 - 2*b5 + b1) * q^26 + (-12*b15 - 9*b14 - 9*b13 - 8*b12 - 9*b11 + 3*b10 + 8*b9 - 3*b8 - 4*b7 + 5*b5 + b4 - 7*b3 - 6*b2 - 13*b1 - 7) * q^27 + (5*b15 - 5*b14 - 5*b13 + 5*b12 - 10*b10 - 5*b9 + 10*b8 + 5*b5 - 23*b3 - 30*b2 + 5*b1 - 23) * q^28 + (-2*b15 - 2*b14 - 2*b13 - 2*b12 - 2*b11 + 2*b9 + 4*b5 - 2*b1) * q^29 + (12*b15 + 8*b14 + 17*b13 + 11*b12 + 9*b11 - 8*b9 + b8 + b7 + 13*b6 - 16*b5 + 8*b4 - 5*b3 - 13*b2 + b1) * q^30 + (5*b10 + 5*b9 + 5*b8 - 5*b7 + 9*b6 + 6*b3 + 6*b2 - 9) * q^31 + (8*b15 + 8*b14 + 9*b13 + 9*b12 + 5*b11 + 5*b10 - 13*b9 - 4*b8 - 4*b7 - 19*b5 + 13*b4) * q^32 + (3*b15 - 3*b14 - b13 + b12 - 3*b10 + b8 + 5*b7 - 26*b6 + 3*b5 + 26*b3 + 3*b1 + 5) * q^34 + (4*b10 - 4*b9 - 3*b8 - 3*b7 - 8*b5 - 3*b4 - 8*b1) * q^35 + (-3*b15 + 10*b14 + 7*b13 + 4*b12 + 6*b11 - 10*b9 - 4*b8 - 4*b7 + 23*b6 - 20*b5 + 10*b4 - 16*b3 - 23*b2 - 4*b1) * q^36 + (9*b13 - 9*b12 + 9*b10 + 9*b9 - 5*b6 - 16*b2 - 16) * q^37 + (13*b15 + 13*b14 + 18*b13 + 18*b12 + 20*b11 + 5*b10 - 18*b9 - 5*b8 - 5*b7 - 23*b5 + 9*b4 + 4*b1) * q^38 + (9*b15 + b14 + 10*b13 + 12*b12 + 9*b11 + b10 - 12*b9 - b8 - 3*b7 - 13*b5 + 12*b4 + 15*b3 + 10*b2 - 3*b1 + 15) * q^39 + (-3*b15 + 3*b14 + 3*b13 - 3*b12 + 3*b10 - 6*b8 - 20*b6 - 3*b5 + 27*b2 - 3*b1 + 27) * q^40 + (-2*b15 - 2*b14 - 3*b13 - 3*b12 - 21*b11 + 2*b9 - 19*b8 - 19*b7 - 8*b5 + 10*b4 - 19*b1) * q^41 + (-6*b10 - 6*b8 - 6*b7 + 21*b6 + 9*b5 - 12*b4 - 24*b3 - 24*b2 + 9*b1 - 21) * q^42 + (-5*b15 + 5*b14 - 2*b13 + 2*b12 + 5*b10 + 2*b8 - 12*b7 - 6*b6 - 5*b5 + 6*b3 - 5*b1 - 9) * q^43 + (6*b15 + 10*b14 + 7*b13 + 7*b12 + 6*b11 + 2*b10 - 8*b9 - b8 - 5*b7 + 14*b6 - 3*b5 + 8*b4 - 14*b3 + 4*b1 - 12) * q^45 + (11*b10 + 11*b9 - 44*b6 + 33*b3 + 33*b2 + 44) * q^46 + (-2*b15 - 2*b14 - 5*b13 - 5*b12 + 12*b11 + 2*b9 + 14*b8 + 14*b7 + 25*b5 - 23*b4 + 14*b1) * q^47 + (-3*b15 + 9*b13 - 6*b12 + 9*b11 + 15*b10 - 3*b8 + 33*b6 - 9*b5 + 12*b2 + 12) * q^48 + (-5*b15 + 5*b14 + 4*b13 - 4*b12 + 9*b10 + 4*b9 - 9*b8 - b7 - 5*b5 + 12*b3 - 8*b2 - 5*b1 + 12) * q^49 + (-3*b15 - 3*b14 - 7*b13 - 7*b12 - 2*b11 - 4*b10 + 7*b9 + 4*b8 + 4*b7 + 11*b5 - 21*b4 + 18*b1) * q^50 + (-9*b15 - 9*b14 + 6*b13 + b12 + 3*b11 + 2*b10 + 12*b9 + b6 + 6*b5 - 14*b2 + 9*b1 - 14) * q^51 + (7*b15 - 7*b14 + 5*b13 - 5*b12 + 7*b9 + 7*b8 + 7*b7 - 25*b6 + 7*b5 + 2*b3 + 25*b2 + 7*b1) * q^52 + (-9*b10 + 9*b9 + 3*b5 + 7*b4 + 3*b1) * q^53 + (-12*b15 - b14 + 2*b13 - 8*b12 - 6*b11 + 12*b10 - 8*b8 - 9*b7 + 3*b6 - 2*b5 - 3*b3 - 18*b1 + 65) * q^54 + (-10*b15 - 10*b14 - 23*b13 - 23*b12 - 37*b11 + b10 + 9*b9 - 14*b8 - 14*b7 + 26*b5 - 9*b4 - 38*b1) * q^56 + (16*b10 - 12*b9 - 6*b8 - 15*b7 + 13*b6 - 15*b5 + 12*b4 + 15*b3 + 15*b2 - 15*b1 - 13) * q^57 + (-2*b15 + 2*b14 + 2*b13 - 2*b12 - 2*b9 - 2*b8 - 2*b7 + 16*b6 - 2*b5 - 18*b3 - 16*b2 - 2*b1) * q^58 + (11*b15 + 11*b14 - 7*b13 - 7*b12 - 8*b11 - b10 - 10*b9 - 3*b5 - 28*b1) * q^59 + (6*b15 - 10*b14 + 2*b13 - 12*b12 - 6*b11 - 4*b10 + 12*b9 + 4*b8 + 18*b7 + 16*b5 + 3*b4 - 15*b3 + 23*b2 + 3*b1 - 15) * q^60 + (-2*b15 + 2*b14 + 11*b13 - 11*b12 + 13*b10 + 11*b9 - 13*b8 + 9*b7 - 2*b5 + 17*b3 + 15*b2 - 2*b1 + 17) * q^61 + (10*b15 + 10*b14 + 5*b13 + 5*b12 + 25*b11 + 20*b10 - 30*b9 - 35*b5 + b1) * q^62 + (9*b15 + 3*b14 - 6*b13 - 3*b9 - 3*b8 - 3*b7 + 3*b5 - 6*b4 + 33*b3 - 3*b1) * q^63 + (-14*b10 - 14*b9 - 10*b8 + 10*b7 - b6 + 15*b3 + 15*b2 + 1) * q^64 + (-12*b15 - 12*b14 - 15*b13 - 15*b12 - 14*b11 - 4*b10 + 16*b9 + b8 + b7 + 7*b5 - 16*b4 - 10*b1) * q^65 + (-12*b15 + 12*b14 + 15*b13 - 15*b12 + 12*b10 - 15*b8 - 9*b7 - 21*b6 - 12*b5 + 21*b3 - 12*b1 + 38) * q^67 + (5*b10 - 5*b9 - 12*b8 - 12*b7 - 2*b5 - 11*b4 - 2*b1) * q^68 + (12*b15 - 10*b14 + 2*b13 + 2*b12 - 6*b11 + 10*b9 + 4*b8 + 4*b7 - 11*b6 - 7*b5 + 17*b4 - 11*b3 + 11*b2 + 4*b1) * q^69 + (5*b15 - 5*b14 - 8*b13 + 8*b12 - 8*b10 - 3*b9 + 10*b8 - 55*b6 + 5*b5 + 11*b2 + 5*b1 + 11) * q^70 + (10*b15 + 10*b14 + b13 + b12 + 5*b11 - 9*b10 - b9 + 9*b8 + 9*b7 + 8*b5 - 6*b4 + 16*b1) * q^71 + (6*b15 + 17*b14 + 2*b13 + b12 + 3*b11 - 4*b10 - b9 + 4*b8 + 5*b7 + 3*b5 + 10*b4 + 32*b3 + 80*b2 - 4*b1 + 32) * q^72 + (-3*b15 + 3*b14 + b13 - b12 + b10 - 2*b9 - 6*b8 + 38*b6 - 3*b5 + 18*b2 - 3*b1 + 18) * q^73 + (-14*b15 - 14*b14 - 23*b13 - 23*b12 + 13*b11 + 14*b9 + 27*b8 + 27*b7 + 53*b5 - 39*b4 + 27*b1) * q^74 + (-3*b10 + 18*b9 + 3*b8 + 15*b7 - 9*b6 + 15*b5 - 18*b4 + 9*b3 + 9*b2 + 15*b1 + 9) * q^75 + (11*b15 - 11*b14 - 5*b13 + 5*b12 - 11*b10 + 5*b8 + 17*b7 - 6*b6 + 11*b5 + 6*b3 + 11*b1 - 53) * q^76 + (27*b15 + 5*b14 - b13 + 5*b12 + 3*b11 - 26*b10 - b9 + 4*b8 + 20*b7 + 43*b6 + 9*b5 + b4 - 43*b3 + 29*b1 - 51) * q^78 + (-8*b10 - 8*b9 + 9*b8 - 9*b7 - 49*b6 + b3 + b2 + 49) * q^79 + (-4*b15 - 4*b14 + 13*b13 + 13*b12 - 10*b11 + 4*b9 - 6*b8 - 6*b7 - 17*b5 + 21*b4 - 6*b1) * q^80 + (-17*b14 - 20*b13 - 9*b12 - 30*b11 - 21*b10 + 27*b9 + 17*b8 - 19*b6 + 47*b5 + 34*b2 + 9*b1 + 34) * q^81 + (24*b15 - 24*b14 - 10*b13 + 10*b12 - 34*b10 - 10*b9 + 34*b8 + 14*b7 + 24*b5 + 5*b3 - 37*b2 + 24*b1 + 5) * q^82 + (-8*b15 - 8*b14 - 5*b13 - 5*b12 + 5*b11 + 3*b10 + 5*b9 - 3*b8 - 3*b7 + 2*b5 - 3*b4 - 5*b1) * q^83 + (15*b15 - 15*b14 - 30*b13 + 18*b12 - 15*b11 - 33*b10 + 30*b8 - 36*b6 + 30*b5 - 57*b2 - 57) * q^84 + (-7*b15 + 7*b14 - 8*b13 + 8*b12 - 7*b9 - 7*b8 - 7*b7 - 28*b6 - 7*b5 - 7*b3 + 28*b2 - 7*b1) * q^85 + (33*b10 - 33*b9 - 11*b8 - 11*b7 - 22*b5 + 9*b4 - 22*b1) * q^86 + (2*b14 - 4*b13 - 4*b12 - 6*b11 - 2*b10 + 2*b9 - 2*b8 - 4*b7 + 4*b6 + 6*b5 - 2*b4 - 4*b3 - 4*b1 + 6) * q^87 + (5*b15 + 5*b14 + 6*b13 + 6*b12 + 3*b11 + 4*b10 - 9*b9 - 3*b8 - 3*b7 - 29*b5 + 9*b4 - b1) * q^89 + (b10 - 19*b9 - 9*b8 + 5*b7 + 95*b6 - 22*b5 + 10*b4 - 52*b3 - 52*b2 - 22*b1 - 95) * q^90 + (-3*b15 + 3*b14 + 4*b13 - 4*b12 - 3*b9 - 3*b8 - 3*b7 - 28*b6 - 3*b5 - 7*b3 + 28*b2 - 3*b1) * q^91 + (2*b15 + 2*b14 + 4*b13 + 4*b12 + 21*b11 + 17*b10 - 19*b9 - 23*b5 + 32*b1) * q^92 + (-30*b15 - 15*b13 - 11*b12 - 15*b11 + 15*b10 + 11*b9 - 15*b8 - 19*b7 - 4*b5 - 11*b4 - 19*b3 + 45*b2 - 19*b1 - 19) * q^93 + (-13*b15 + 13*b14 - 12*b13 + 12*b12 + b10 - 12*b9 - b8 - 25*b7 - 13*b5 - 82*b3 - 95*b2 - 13*b1 - 82) * q^94 + (4*b15 + 4*b14 + 18*b13 + 18*b12 - 5*b11 - 23*b10 + 19*b9 + b5 + 18*b1) * q^95 + (-12*b15 + 10*b14 + b13 + 14*b12 + 12*b11 - 10*b9 + 2*b8 + 2*b7 + 10*b6 - 14*b5 + 4*b4 + 41*b3 - 10*b2 + 2*b1) * q^96 + (8*b10 + 8*b9 - 2*b8 + 2*b7 - 46*b6 + 41*b3 + 41*b2 + 46) * q^97 + (8*b15 + 8*b14 + 31*b13 + 31*b12 + 38*b11 + 16*b10 - 24*b9 + 7*b8 + 7*b7 - 30*b5 + 24*b4 + 22*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 5 q^{3} + 18 q^{4} + 32 q^{6} - 34 q^{7} + 17 q^{9}+O(q^{10})$$ 16 * q + 5 * q^3 + 18 * q^4 + 32 * q^6 - 34 * q^7 + 17 * q^9 $$16 q + 5 q^{3} + 18 q^{4} + 32 q^{6} - 34 q^{7} + 17 q^{9} - 12 q^{10} + 106 q^{12} - 2 q^{13} - 28 q^{15} + 102 q^{16} + 42 q^{18} + 66 q^{19} - 12 q^{21} - 74 q^{24} - 176 q^{25} - 55 q^{27} - 146 q^{28} + 110 q^{30} - 126 q^{31} - 132 q^{34} + 226 q^{36} - 230 q^{37} + 136 q^{39} + 226 q^{40} - 72 q^{42} - 156 q^{43} - 72 q^{45} + 308 q^{46} + 255 q^{48} + 170 q^{49} - 169 q^{51} - 224 q^{52} + 1046 q^{54} - 259 q^{57} + 184 q^{58} - 316 q^{60} + 104 q^{61} - 108 q^{63} - 184 q^{64} + 368 q^{67} - 22 q^{69} - 52 q^{70} + 73 q^{72} + 354 q^{73} + 54 q^{75} - 900 q^{76} - 492 q^{78} + 566 q^{79} + 377 q^{81} + 200 q^{82} - 720 q^{84} - 162 q^{85} + 132 q^{87} - 774 q^{90} - 226 q^{91} - 370 q^{93} - 530 q^{94} - 67 q^{96} + 252 q^{97}+O(q^{100})$$ 16 * q + 5 * q^3 + 18 * q^4 + 32 * q^6 - 34 * q^7 + 17 * q^9 - 12 * q^10 + 106 * q^12 - 2 * q^13 - 28 * q^15 + 102 * q^16 + 42 * q^18 + 66 * q^19 - 12 * q^21 - 74 * q^24 - 176 * q^25 - 55 * q^27 - 146 * q^28 + 110 * q^30 - 126 * q^31 - 132 * q^34 + 226 * q^36 - 230 * q^37 + 136 * q^39 + 226 * q^40 - 72 * q^42 - 156 * q^43 - 72 * q^45 + 308 * q^46 + 255 * q^48 + 170 * q^49 - 169 * q^51 - 224 * q^52 + 1046 * q^54 - 259 * q^57 + 184 * q^58 - 316 * q^60 + 104 * q^61 - 108 * q^63 - 184 * q^64 + 368 * q^67 - 22 * q^69 - 52 * q^70 + 73 * q^72 + 354 * q^73 + 54 * q^75 - 900 * q^76 - 492 * q^78 + 566 * q^79 + 377 * q^81 + 200 * q^82 - 720 * q^84 - 162 * q^85 + 132 * q^87 - 774 * q^90 - 226 * q^91 - 370 * q^93 - 530 * q^94 - 67 * q^96 + 252 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 127407047787804 \nu^{14} - 322126035973969 \nu^{12} + \cdots + 37\!\cdots\!56 ) / 21\!\cdots\!19$$ (-127407047787804*v^14 - 322126035973969*v^12 - 40525263758459540*v^10 + 344558428769805180*v^8 - 4303642725224589076*v^6 + 44698619213114763312*v^4 - 744861315136512930500*v^2 + 374807322561273688856) / 2181808456934777493119 $$\beta_{3}$$ $$=$$ $$( 234959048766210 \nu^{14} + \cdots - 77\!\cdots\!99 ) / 21\!\cdots\!19$$ (234959048766210*v^14 + 11921441073701127*v^12 - 85177596039692545*v^10 + 1520211191539599580*v^8 - 23265517903780782960*v^6 + 247688922335353991145*v^4 - 208024183527244179835*v^2 - 77838063410538248499) / 2181808456934777493119 $$\beta_{4}$$ $$=$$ $$( 127407047787804 \nu^{15} + 322126035973969 \nu^{13} + \cdots - 37\!\cdots\!56 \nu ) / 21\!\cdots\!19$$ (127407047787804*v^15 + 322126035973969*v^13 + 40525263758459540*v^11 - 344558428769805180*v^9 + 4303642725224589076*v^7 - 44698619213114763312*v^5 + 744861315136512930500*v^3 - 374807322561273688856*v) / 2181808456934777493119 $$\beta_{5}$$ $$=$$ $$( 234959048766210 \nu^{15} + \cdots - 77\!\cdots\!99 \nu ) / 21\!\cdots\!19$$ (234959048766210*v^15 + 11921441073701127*v^13 - 85177596039692545*v^11 + 1520211191539599580*v^9 - 23265517903780782960*v^7 + 247688922335353991145*v^5 - 208024183527244179835*v^3 - 77838063410538248499*v) / 2181808456934777493119 $$\beta_{6}$$ $$=$$ $$( 44\!\cdots\!44 \nu^{14} + \cdots + 20\!\cdots\!70 ) / 21\!\cdots\!19$$ (4438276679147244*v^14 - 38143852521781896*v^12 + 631568334130072500*v^10 - 8766582639358512750*v^8 + 76663977785573994959*v^6 - 74027125217335407442*v^4 - 82997600383267449850*v^2 + 207581537941818757970) / 2181808456934777493119 $$\beta_{7}$$ $$=$$ $$( - 29\!\cdots\!01 \nu^{15} + \cdots + 38\!\cdots\!99 ) / 47\!\cdots\!18$$ (-2949643553271501*v^15 - 107478072222941431*v^14 + 31035928814102023*v^13 + 742877598465063798*v^12 - 1694622520719673097*v^11 - 11853231551733086559*v^10 + 18173418136790321913*v^9 + 184172957827569504013*v^8 - 247225326229442937739*v^7 - 1221194376663844125048*v^6 + 2596346362793508526818*v^5 - 3066965541789538832905*v^4 - 21420285449145050092157*v^3 + 17059214158261006460234*v^2 + 21898844749963893603650*v + 38060055284176600406699) / 47999786052565104848618 $$\beta_{8}$$ $$=$$ $$( - 29\!\cdots\!01 \nu^{15} + \cdots - 38\!\cdots\!99 ) / 47\!\cdots\!18$$ (-2949643553271501*v^15 + 107478072222941431*v^14 + 31035928814102023*v^13 - 742877598465063798*v^12 - 1694622520719673097*v^11 + 11853231551733086559*v^10 + 18173418136790321913*v^9 - 184172957827569504013*v^8 - 247225326229442937739*v^7 + 1221194376663844125048*v^6 + 2596346362793508526818*v^5 + 3066965541789538832905*v^4 - 21420285449145050092157*v^3 - 17059214158261006460234*v^2 + 21898844749963893603650*v - 38060055284176600406699) / 47999786052565104848618 $$\beta_{9}$$ $$=$$ $$( 55\!\cdots\!28 \nu^{15} + \cdots - 91\!\cdots\!65 ) / 47\!\cdots\!18$$ (5530938613687128*v^15 + 166012584300402708*v^14 - 33630130225371761*v^13 - 2117280218216450276*v^12 + 2005487528427821201*v^11 + 31045509051468975822*v^10 - 18276913998033521354*v^9 - 453165291912414021743*v^8 + 234870744257407535442*v^7 + 4437259853751563534641*v^6 - 2448733533544500813279*v^5 - 18835563517351609927800*v^4 + 17382546759243455724562*v^3 + 48992445765202921573548*v^2 - 20574790310578950257680*v - 91659310975868063351165) / 47999786052565104848618 $$\beta_{10}$$ $$=$$ $$( - 55\!\cdots\!28 \nu^{15} + \cdots - 91\!\cdots\!65 ) / 47\!\cdots\!18$$ (-5530938613687128*v^15 + 166012584300402708*v^14 + 33630130225371761*v^13 - 2117280218216450276*v^12 - 2005487528427821201*v^11 + 31045509051468975822*v^10 + 18276913998033521354*v^9 - 453165291912414021743*v^8 - 234870744257407535442*v^7 + 4437259853751563534641*v^6 + 2448733533544500813279*v^5 - 18835563517351609927800*v^4 - 17382546759243455724562*v^3 + 48992445765202921573548*v^2 + 20574790310578950257680*v - 91659310975868063351165) / 47999786052565104848618 $$\beta_{11}$$ $$=$$ $$( 30\!\cdots\!44 \nu^{15} + \cdots - 55\!\cdots\!46 \nu ) / 23\!\cdots\!09$$ (30054413983928844*v^15 - 201987708092060081*v^13 + 4502519471732524742*v^11 - 60964388315221160139*v^9 + 517159654487638123533*v^7 - 1439208800755732819107*v^5 + 15876462984469736097197*v^3 - 55056646013161149255646*v) / 23999893026282552424309 $$\beta_{12}$$ $$=$$ $$( - 72\!\cdots\!54 \nu^{15} + \cdots + 25\!\cdots\!82 ) / 47\!\cdots\!18$$ (-72618660380064154*v^15 + 49907241408562835*v^14 + 904789569200787956*v^13 - 1271613859106383353*v^12 - 13142255289684674994*v^11 + 14689100732554831748*v^10 + 193257387972027534305*v^9 - 221948869246166066510*v^8 - 1861216681368690697007*v^7 + 2581424220326366958345*v^6 + 7651271435780790370144*v^5 - 15244472422698155686654*v^4 - 19622760602151650003666*v^3 + 22215491198933656595906*v^2 + 38836150383671216498131*v + 2527463022467253365682) / 47999786052565104848618 $$\beta_{13}$$ $$=$$ $$( - 72\!\cdots\!54 \nu^{15} + \cdots - 25\!\cdots\!82 ) / 47\!\cdots\!18$$ (-72618660380064154*v^15 - 49907241408562835*v^14 + 904789569200787956*v^13 + 1271613859106383353*v^12 - 13142255289684674994*v^11 - 14689100732554831748*v^10 + 193257387972027534305*v^9 + 221948869246166066510*v^8 - 1861216681368690697007*v^7 - 2581424220326366958345*v^6 + 7651271435780790370144*v^5 + 15244472422698155686654*v^4 - 19622760602151650003666*v^3 - 22215491198933656595906*v^2 + 38836150383671216498131*v - 2527463022467253365682) / 47999786052565104848618 $$\beta_{14}$$ $$=$$ $$( 99\!\cdots\!41 \nu^{15} + \cdots - 84\!\cdots\!42 ) / 47\!\cdots\!18$$ (99135684000027241*v^15 - 20810066889315561*v^14 - 862706736638173675*v^13 - 476043899992241295*v^12 + 14023945388217859866*v^11 + 1701287476880316347*v^10 - 195384258337122023097*v^9 - 55882176278914975803*v^8 + 1663164806667153890606*v^7 + 920555929379994081361*v^6 - 1429591891788951514170*v^5 - 9875787159646622730582*v^4 - 5780946714034994365846*v^3 + 20270682318663355761963*v^2 + 18115509591203040974177*v - 84217831418990427698342) / 47999786052565104848618 $$\beta_{15}$$ $$=$$ $$( 94\!\cdots\!95 \nu^{15} + \cdots - 74\!\cdots\!23 ) / 47\!\cdots\!18$$ (94334933420026495*v^15 + 186822651189718269*v^14 - 1153420167662430754*v^13 - 1641236318224208981*v^12 + 17281610014102620849*v^11 + 29344221574588659475*v^10 - 246898826826540336329*v^9 - 397283115633499045940*v^8 + 2434586108751809455762*v^7 + 3516703924371569453280*v^6 - 9622707375209255559717*v^5 - 8959776357704987197218*v^4 + 24253609462611022050276*v^3 + 28721763446539565811585*v^2 - 51394738255679059357083*v - 7441479556877635652823) / 47999786052565104848618
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{3} - 8\beta_{2} - 1$$ -b13 + b12 - b10 - b9 + b8 - b7 - b3 - 8*b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} - \beta_{5} + 10\beta_{4} - \beta_1$$ b10 - b9 + 2*b8 + 2*b7 - b5 + 10*b4 - b1 $$\nu^{4}$$ $$=$$ $$\beta_{15} - \beta_{14} - 15 \beta_{13} + 15 \beta_{12} + \beta_{9} + \beta_{8} + \beta_{7} - 23 \beta_{6} + \beta_{5} + 85 \beta_{3} + 23 \beta_{2} + \beta_1$$ b15 - b14 - 15*b13 + 15*b12 + b9 + b8 + b7 - 23*b6 + b5 + 85*b3 + 23*b2 + b1 $$\nu^{5}$$ $$=$$ $$- 8 \beta_{15} - 8 \beta_{14} - 41 \beta_{13} - 41 \beta_{12} - 85 \beta_{11} + 11 \beta_{10} - 3 \beta_{9} - 44 \beta_{8} - 44 \beta_{7} + 99 \beta_{5} + 3 \beta_{4} - 96 \beta_1$$ -8*b15 - 8*b14 - 41*b13 - 41*b12 - 85*b11 + 11*b10 - 3*b9 - 44*b8 - 44*b7 + 99*b5 + 3*b4 - 96*b1 $$\nu^{6}$$ $$=$$ $$- 190 \beta_{15} + 190 \beta_{14} + 220 \beta_{13} - 220 \beta_{12} + 220 \beta_{10} + 30 \beta_{9} - 380 \beta_{8} + 633 \beta_{6} - 190 \beta_{5} + 378 \beta_{2} - 190 \beta _1 + 378$$ -190*b15 + 190*b14 + 220*b13 - 220*b12 + 220*b10 + 30*b9 - 380*b8 + 633*b6 - 190*b5 + 378*b2 - 190*b1 + 378 $$\nu^{7}$$ $$=$$ $$603 \beta_{15} + 603 \beta_{14} + 1143 \beta_{13} + 1143 \beta_{12} + 1073 \beta_{11} + 540 \beta_{10} - 1143 \beta_{9} - 540 \beta_{8} - 540 \beta_{7} - 1683 \beta_{5} - 118 \beta_{4} + 721 \beta_1$$ 603*b15 + 603*b14 + 1143*b13 + 1143*b12 + 1073*b11 + 540*b10 - 1143*b9 - 540*b8 - 540*b7 - 1683*b5 - 118*b4 + 721*b1 $$\nu^{8}$$ $$=$$ $$- 2341 \beta_{10} - 2341 \beta_{9} + 588 \beta_{8} - 588 \beta_{7} + 6969 \beta_{6} - 12506 \beta_{3} - 12506 \beta_{2} - 6969$$ -2341*b10 - 2341*b9 + 588*b8 - 588*b7 + 6969*b6 - 12506*b3 - 12506*b2 - 6969 $$\nu^{9}$$ $$=$$ $$6381 \beta_{15} + 6381 \beta_{14} + 12816 \beta_{13} + 12816 \beta_{12} + 12827 \beta_{11} - 6381 \beta_{9} + 6446 \beta_{8} + 6446 \beta_{7} - 22393 \beta_{5} + 16012 \beta_{4} + 6446 \beta_1$$ 6381*b15 + 6381*b14 + 12816*b13 + 12816*b12 + 12827*b11 - 6381*b9 + 6446*b8 + 6446*b7 - 22393*b5 + 16012*b4 + 6446*b1 $$\nu^{10}$$ $$=$$ $$28893 \beta_{15} - 28893 \beta_{14} - 38546 \beta_{13} + 38546 \beta_{12} - 28893 \beta_{10} + 38546 \beta_{8} + 19240 \beta_{7} - 77049 \beta_{6} + 28893 \beta_{5} + 77049 \beta_{3} + 28893 \beta _1 - 80060$$ 28893*b15 - 28893*b14 - 38546*b13 + 38546*b12 - 28893*b10 + 38546*b8 + 19240*b7 - 77049*b6 + 28893*b5 + 77049*b3 + 28893*b1 - 80060 $$\nu^{11}$$ $$=$$ $$- 48156 \beta_{15} - 48156 \beta_{14} - 134901 \beta_{13} - 134901 \beta_{12} - 211927 \beta_{11} - 77026 \beta_{10} + 125182 \beta_{9} + 260083 \beta_{5} - 205308 \beta_1$$ -48156*b15 - 48156*b14 - 134901*b13 - 134901*b12 - 211927*b11 - 77026*b10 + 125182*b9 + 260083*b5 - 205308*b1 $$\nu^{12}$$ $$=$$ $$- 144620 \beta_{15} + 144620 \beta_{14} + 359360 \beta_{13} - 359360 \beta_{12} + 503980 \beta_{10} + 359360 \beta_{9} - 503980 \beta_{8} + 214740 \beta_{7} - 144620 \beta_{5} + \cdots + 1043432$$ -144620*b15 + 144620*b14 + 359360*b13 - 359360*b12 + 503980*b10 + 359360*b9 - 503980*b8 + 214740*b7 - 144620*b5 + 1043432*b3 + 1989005*b2 - 144620*b1 + 1043432 $$\nu^{13}$$ $$=$$ $$219120 \beta_{10} - 219120 \beta_{9} - 1152580 \beta_{8} - 1152580 \beta_{7} + 464952 \beta_{5} - 2418485 \beta_{4} + 464952 \beta_1$$ 219120*b10 - 219120*b9 - 1152580*b8 - 1152580*b7 + 464952*b5 - 2418485*b4 + 464952*b1 $$\nu^{14}$$ $$=$$ $$- 2055772 \beta_{15} + 2055772 \beta_{14} + 4504525 \beta_{13} - 4504525 \beta_{12} - 2055772 \beta_{9} - 2055772 \beta_{8} - 2055772 \beta_{7} + 13903349 \beta_{6} + \cdots - 2055772 \beta_1$$ -2055772*b15 + 2055772*b14 + 4504525*b13 - 4504525*b12 - 2055772*b9 - 2055772*b8 - 2055772*b7 + 13903349*b6 - 2055772*b5 - 25299632*b3 - 13903349*b2 - 2055772*b1 $$\nu^{15}$$ $$=$$ $$9398824 \beta_{15} + 9398824 \beta_{14} + 24575190 \beta_{13} + 24575190 \beta_{12} + 36032993 \beta_{11} + 3718563 \beta_{10} - 13117387 \beta_{9} + 11457803 \beta_{8} + \cdots + 32314430 \beta_1$$ 9398824*b15 + 9398824*b14 + 24575190*b13 + 24575190*b12 + 36032993*b11 + 3718563*b10 - 13117387*b9 + 11457803*b8 + 11457803*b7 - 47033088*b5 + 13117387*b4 + 32314430*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$$.

 $$n$$ $$122$$ $$244$$ $$\chi(n)$$ $$-1$$ $$-\beta_{6}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
245.1
 −2.91048 − 0.945671i −1.90610 − 0.619331i 1.90610 + 0.619331i 2.91048 + 0.945671i 2.10855 − 2.90217i 0.974642 − 1.34148i −0.974642 + 1.34148i −2.10855 + 2.90217i 2.10855 + 2.90217i 0.974642 + 1.34148i −0.974642 − 1.34148i −2.10855 − 2.90217i −2.91048 + 0.945671i −1.90610 + 0.619331i 1.90610 − 0.619331i 2.91048 − 0.945671i
−1.79877 2.47580i −2.81156 + 1.04648i −1.65793 + 5.10257i 3.90250 5.37133i 7.64823 + 5.07849i −0.946512 + 2.91306i 3.97326 1.29099i 6.80977 5.88447i −20.3180
245.2 −1.17804 1.62143i 1.88824 + 2.33121i −0.00519352 + 0.0159840i −3.22364 + 4.43696i 1.55548 5.80790i 1.72766 5.31721i −7.59238 + 2.46692i −1.86912 + 8.80377i 10.9918
245.3 1.17804 + 1.62143i −0.157363 + 2.99587i −0.00519352 + 0.0159840i 3.22364 4.43696i −5.04297 + 3.27409i 1.72766 5.31721i 7.59238 2.46692i −8.95047 0.942880i 10.9918
245.4 1.79877 + 2.47580i 2.88971 0.805978i −1.65793 + 5.10257i −3.90250 + 5.37133i 7.19336 + 5.70456i −0.946512 + 2.91306i −3.97326 + 1.29099i 7.70080 4.65808i −20.3180
251.1 −3.41170 1.10853i 1.50565 2.59480i 7.17480 + 5.21280i −1.98428 + 0.644731i −8.01326 + 7.18363i −7.17034 5.20956i −10.2655 14.1293i −4.46601 7.81376i 7.48447
251.2 −1.57700 0.512399i −2.99453 0.181006i −1.01168 0.735029i 0.664316 0.215849i 4.62964 + 1.81985i −2.11081 1.53360i 5.11736 + 7.04345i 8.93447 + 1.08406i −1.15823
251.3 1.57700 + 0.512399i −0.753215 + 2.90391i −1.01168 0.735029i −0.664316 + 0.215849i −2.67578 + 4.19352i −2.11081 1.53360i −5.11736 7.04345i −7.86534 4.37453i −1.15823
251.4 3.41170 + 1.10853i 2.93308 0.630124i 7.17480 + 5.21280i 1.98428 0.644731i 10.7053 + 1.10160i −7.17034 5.20956i 10.2655 + 14.1293i 8.20589 3.69641i 7.48447
269.1 −3.41170 + 1.10853i 1.50565 + 2.59480i 7.17480 5.21280i −1.98428 0.644731i −8.01326 7.18363i −7.17034 + 5.20956i −10.2655 + 14.1293i −4.46601 + 7.81376i 7.48447
269.2 −1.57700 + 0.512399i −2.99453 + 0.181006i −1.01168 + 0.735029i 0.664316 + 0.215849i 4.62964 1.81985i −2.11081 + 1.53360i 5.11736 7.04345i 8.93447 1.08406i −1.15823
269.3 1.57700 0.512399i −0.753215 2.90391i −1.01168 + 0.735029i −0.664316 0.215849i −2.67578 4.19352i −2.11081 + 1.53360i −5.11736 + 7.04345i −7.86534 + 4.37453i −1.15823
269.4 3.41170 1.10853i 2.93308 + 0.630124i 7.17480 5.21280i 1.98428 + 0.644731i 10.7053 1.10160i −7.17034 + 5.20956i 10.2655 14.1293i 8.20589 + 3.69641i 7.48447
323.1 −1.79877 + 2.47580i −2.81156 1.04648i −1.65793 5.10257i 3.90250 + 5.37133i 7.64823 5.07849i −0.946512 2.91306i 3.97326 + 1.29099i 6.80977 + 5.88447i −20.3180
323.2 −1.17804 + 1.62143i 1.88824 2.33121i −0.00519352 0.0159840i −3.22364 4.43696i 1.55548 + 5.80790i 1.72766 + 5.31721i −7.59238 2.46692i −1.86912 8.80377i 10.9918
323.3 1.17804 1.62143i −0.157363 2.99587i −0.00519352 0.0159840i 3.22364 + 4.43696i −5.04297 3.27409i 1.72766 + 5.31721i 7.59238 + 2.46692i −8.95047 + 0.942880i 10.9918
323.4 1.79877 2.47580i 2.88971 + 0.805978i −1.65793 5.10257i −3.90250 5.37133i 7.19336 5.70456i −0.946512 2.91306i −3.97326 1.29099i 7.70080 + 4.65808i −20.3180
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 323.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 1 inner
33.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.o 16
3.b odd 2 1 inner 363.3.h.o 16
11.b odd 2 1 363.3.h.n 16
11.c even 5 2 33.3.h.b 16
11.c even 5 1 363.3.b.m 8
11.c even 5 1 inner 363.3.h.o 16
11.d odd 10 1 363.3.b.l 8
11.d odd 10 2 363.3.h.j 16
11.d odd 10 1 363.3.h.n 16
33.d even 2 1 363.3.h.n 16
33.f even 10 1 363.3.b.l 8
33.f even 10 2 363.3.h.j 16
33.f even 10 1 363.3.h.n 16
33.h odd 10 2 33.3.h.b 16
33.h odd 10 1 363.3.b.m 8
33.h odd 10 1 inner 363.3.h.o 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.h.b 16 11.c even 5 2
33.3.h.b 16 33.h odd 10 2
363.3.b.l 8 11.d odd 10 1
363.3.b.l 8 33.f even 10 1
363.3.b.m 8 11.c even 5 1
363.3.b.m 8 33.h odd 10 1
363.3.h.j 16 11.d odd 10 2
363.3.h.j 16 33.f even 10 2
363.3.h.n 16 11.b odd 2 1
363.3.h.n 16 11.d odd 10 1
363.3.h.n 16 33.d even 2 1
363.3.h.n 16 33.f even 10 1
363.3.h.o 16 1.a even 1 1 trivial
363.3.h.o 16 3.b odd 2 1 inner
363.3.h.o 16 11.c even 5 1 inner
363.3.h.o 16 33.h odd 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(363, [\chi])$$:

 $$T_{2}^{16} - 17 T_{2}^{14} + 175 T_{2}^{12} - 1372 T_{2}^{10} + 18829 T_{2}^{8} - 48408 T_{2}^{6} + 245930 T_{2}^{4} - 875798 T_{2}^{2} + 1771561$$ T2^16 - 17*T2^14 + 175*T2^12 - 1372*T2^10 + 18829*T2^8 - 48408*T2^6 + 245930*T2^4 - 875798*T2^2 + 1771561 $$T_{5}^{16} + 38 T_{5}^{14} + 3020 T_{5}^{12} + 35613 T_{5}^{10} + 1364209 T_{5}^{8} - 12321408 T_{5}^{6} + 42508800 T_{5}^{4} - 28971008 T_{5}^{2} + 7929856$$ T5^16 + 38*T5^14 + 3020*T5^12 + 35613*T5^10 + 1364209*T5^8 - 12321408*T5^6 + 42508800*T5^4 - 28971008*T5^2 + 7929856 $$T_{7}^{8} + 17T_{7}^{7} + 151T_{7}^{6} + 861T_{7}^{5} + 5629T_{7}^{4} + 23148T_{7}^{3} + 72504T_{7}^{2} + 140184T_{7} + 156816$$ T7^8 + 17*T7^7 + 151*T7^6 + 861*T7^5 + 5629*T7^4 + 23148*T7^3 + 72504*T7^2 + 140184*T7 + 156816

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 17 T^{14} + 175 T^{12} + \cdots + 1771561$$
$3$ $$T^{16} - 5 T^{15} + 4 T^{14} + \cdots + 43046721$$
$5$ $$T^{16} + 38 T^{14} + 3020 T^{12} + \cdots + 7929856$$
$7$ $$(T^{8} + 17 T^{7} + 151 T^{6} + \cdots + 156816)^{2}$$
$11$ $$T^{16}$$
$13$ $$(T^{8} + T^{7} + 129 T^{6} + \cdots + 19749136)^{2}$$
$17$ $$T^{16} - 472 T^{14} + \cdots + 69002444446441$$
$19$ $$(T^{8} - 33 T^{7} + 1060 T^{6} + \cdots + 4224870001)^{2}$$
$23$ $$(T^{8} + 1848 T^{6} + \cdots + 25255373616)^{2}$$
$29$ $$T^{16} + 68 T^{14} + \cdots + 116101021696$$
$31$ $$(T^{8} + 63 T^{7} + 1558 T^{6} + \cdots + 9448617616)^{2}$$
$37$ $$(T^{8} + 115 T^{7} + \cdots + 487254257296)^{2}$$
$41$ $$T^{16} - 1714 T^{14} + \cdots + 15\!\cdots\!81$$
$43$ $$(T^{4} + 39 T^{3} - 1528 T^{2} + \cdots - 684409)^{4}$$
$47$ $$T^{16} - 1406 T^{14} + \cdots + 21\!\cdots\!96$$
$53$ $$T^{16} - 4259 T^{14} + \cdots + 30\!\cdots\!36$$
$59$ $$T^{16} - 7753 T^{14} + \cdots + 26\!\cdots\!01$$
$61$ $$(T^{8} - 52 T^{7} + \cdots + 1393523586576)^{2}$$
$67$ $$(T^{4} - 92 T^{3} - 3927 T^{2} + \cdots - 11977619)^{4}$$
$71$ $$T^{16} - 2046 T^{14} + \cdots + 27\!\cdots\!96$$
$73$ $$(T^{8} - 177 T^{7} + \cdots + 5458559358736)^{2}$$
$79$ $$(T^{8} - 283 T^{7} + \cdots + 743504776736656)^{2}$$
$83$ $$T^{16} - 5211 T^{14} + \cdots + 21\!\cdots\!81$$
$89$ $$(T^{8} + 8529 T^{6} + \cdots + 3248250664131)^{2}$$
$97$ $$(T^{8} - 126 T^{7} + \cdots + 9061117408561)^{2}$$