Properties

 Label 1120.2.h.a Level $1120$ Weight $2$ Character orbit 1120.h Analytic conductor $8.943$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1120 = 2^{5} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1120.h (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.94324502638$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - x^{15} - 2 x^{12} + 6 x^{11} - 12 x^{9} + 8 x^{8} - 24 x^{7} + 48 x^{5} - 32 x^{4} - 128 x + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - q^{5} -\beta_{8} q^{7} + ( -1 - \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} - q^{5} -\beta_{8} q^{7} + ( -1 - \beta_{12} ) q^{9} + \beta_{10} q^{11} -\beta_{5} q^{13} -\beta_{2} q^{15} + ( -\beta_{2} + \beta_{13} ) q^{17} + ( -\beta_{7} + \beta_{11} + \beta_{13} ) q^{19} + ( \beta_{7} - \beta_{15} ) q^{21} + ( -\beta_{2} - \beta_{14} ) q^{23} + q^{25} + ( -\beta_{2} + \beta_{4} + \beta_{7} + \beta_{9} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{27} + ( -\beta_{2} + \beta_{11} ) q^{29} + ( -2 - \beta_{1} + \beta_{3} + \beta_{6} + \beta_{10} - \beta_{15} ) q^{31} + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{14} ) q^{33} + \beta_{8} q^{35} + ( \beta_{9} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{37} + ( -\beta_{1} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{13} - \beta_{15} ) q^{39} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{41} + ( -\beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{12} ) q^{43} + ( 1 + \beta_{12} ) q^{45} + ( \beta_{1} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} - \beta_{12} ) q^{47} + ( -1 + \beta_{3} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{49} + ( 2 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{12} - \beta_{15} ) q^{51} + ( \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{53} -\beta_{10} q^{55} + ( -2 - 2 \beta_{1} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{57} + ( \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{9} + \beta_{12} - \beta_{15} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{13} ) q^{63} + \beta_{5} q^{65} + ( -2 + \beta_{3} - \beta_{5} + \beta_{8} + \beta_{10} - \beta_{12} - \beta_{15} ) q^{67} + ( 2 - \beta_{5} + 2 \beta_{10} + \beta_{12} ) q^{69} + ( \beta_{2} + \beta_{4} + \beta_{9} + \beta_{11} - \beta_{15} ) q^{71} + ( -\beta_{1} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{13} - \beta_{14} ) q^{73} + \beta_{2} q^{75} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{77} + ( \beta_{1} + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{11} - \beta_{13} + 2 \beta_{14} ) q^{79} + ( 1 + 2 \beta_{3} + 2 \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{15} ) q^{81} + ( -2 \beta_{2} + \beta_{4} - \beta_{7} - \beta_{9} - 2 \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{83} + ( \beta_{2} - \beta_{13} ) q^{85} + ( 4 - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{12} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} - \beta_{14} ) q^{89} + ( 2 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{8} + 2 \beta_{9} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{91} + ( -\beta_{1} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{11} - 2 \beta_{13} + \beta_{14} ) q^{93} + ( \beta_{7} - \beta_{11} - \beta_{13} ) q^{95} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{8} + 2 \beta_{11} - \beta_{13} ) q^{97} + ( -2 - \beta_{1} - \beta_{3} + \beta_{5} + 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{5} - 16q^{9} + O(q^{10})$$ $$16q - 16q^{5} - 16q^{9} + 4q^{11} + 4q^{21} + 16q^{25} - 16q^{31} + 4q^{43} + 16q^{45} - 8q^{49} + 40q^{51} - 4q^{55} - 16q^{57} + 8q^{61} + 28q^{63} - 20q^{67} + 40q^{69} + 4q^{77} + 24q^{81} + 72q^{87} + 32q^{91} - 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - x^{15} - 2 x^{12} + 6 x^{11} - 12 x^{9} + 8 x^{8} - 24 x^{7} + 48 x^{5} - 32 x^{4} - 128 x + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} + 2 \nu^{8} - 2 \nu^{7} - 2 \nu^{6} - 8 \nu^{2} - 32 \nu - 16$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{15} + \nu^{14} + 2 \nu^{13} + 4 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + 4 \nu^{9} - 4 \nu^{8} + 16 \nu^{7} - 8 \nu^{6} - 48 \nu^{5} - 16 \nu^{4} - 64 \nu^{3} - 64 \nu^{2} + 128 \nu$$$$)/128$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{14} + 3 \nu^{12} - 6 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} - 16 \nu^{5} + 8 \nu^{4} - 16 \nu^{3} - 96 \nu^{2} + 32$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - 6 \nu^{13} + 2 \nu^{11} - 10 \nu^{10} + 20 \nu^{9} - 20 \nu^{8} - 16 \nu^{7} + 24 \nu^{6} - 16 \nu^{5} + 112 \nu^{4} + 128 \nu^{3}$$$$)/128$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{14} - 2 \nu^{13} + 10 \nu^{11} - 2 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} - 48 \nu^{7} - 8 \nu^{6} + 48 \nu^{5} - 80 \nu^{4} - 128 \nu^{2} - 384 \nu + 256$$$$)/128$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + 32 \nu^{3} + 128 \nu + 128$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} + 2 \nu^{11} - 6 \nu^{10} + 12 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 48 \nu^{4} + 32 \nu^{3} - 128 \nu + 128$$$$)/64$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{14} + 6 \nu^{11} - 10 \nu^{10} - 8 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 32 \nu^{6} + 16 \nu^{5} - 80 \nu^{4} - 32 \nu^{2} + 320$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{15} + \nu^{14} + 2 \nu^{13} + 14 \nu^{11} + 10 \nu^{10} - 12 \nu^{9} + 4 \nu^{8} - 16 \nu^{7} + 40 \nu^{6} + 112 \nu^{5} + 16 \nu^{4} - 64 \nu^{3} - 128 \nu^{2} - 128 \nu$$$$)/128$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{15} + \nu^{14} - 2 \nu^{13} + 4 \nu^{12} + 14 \nu^{11} - 6 \nu^{10} - 20 \nu^{9} + 12 \nu^{8} - 16 \nu^{7} + 56 \nu^{6} + 112 \nu^{5} - 80 \nu^{4} - 64 \nu^{2} - 256 \nu + 512$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{14} - \nu^{13} - \nu^{12} + 2 \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 2 \nu^{8} + 12 \nu^{6} + 32 \nu^{5} + 8 \nu^{4} + 16 \nu^{3} - 16 \nu^{2} - 32 \nu + 64$$$$)/32$$ $$\beta_{12}$$ $$=$$ $$($$$$3 \nu^{15} + 3 \nu^{14} - 6 \nu^{13} - 8 \nu^{12} - 6 \nu^{11} - 2 \nu^{10} + 20 \nu^{9} + 12 \nu^{8} - 16 \nu^{7} - 40 \nu^{6} - 80 \nu^{5} + 48 \nu^{4} + 256 \nu^{3} + 128 \nu^{2} - 256$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$\nu^{15} + \nu^{12} + 4 \nu^{10} + 8 \nu^{9} - 2 \nu^{8} - 12 \nu^{6} - 16 \nu^{5} + 8 \nu^{4} + 16 \nu^{3} - 48 \nu^{2} - 64 \nu - 128$$$$)/32$$ $$\beta_{14}$$ $$=$$ $$($$$$3 \nu^{15} - \nu^{14} - 6 \nu^{13} + 4 \nu^{12} + 2 \nu^{11} + 22 \nu^{10} + 36 \nu^{9} - 28 \nu^{8} - 16 \nu^{7} + 8 \nu^{6} - 48 \nu^{5} + 144 \nu^{4} - 192 \nu^{2} - 256 \nu - 512$$$$)/128$$ $$\beta_{15}$$ $$=$$ $$($$$$-5 \nu^{15} - \nu^{14} - 2 \nu^{13} + 8 \nu^{12} + 18 \nu^{11} - 26 \nu^{10} - 20 \nu^{9} + 12 \nu^{8} - 48 \nu^{7} + 152 \nu^{6} + 144 \nu^{5} - 80 \nu^{4} + 64 \nu^{3} - 384 \nu^{2} - 128 \nu + 1024$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 4$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_{1} - 4$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} - 4$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-6 \beta_{15} + 2 \beta_{13} + 5 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 11 \beta_{2} - 5 \beta_{1} + 12$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$6 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} - 7 \beta_{9} - 10 \beta_{8} + 6 \beta_{6} - \beta_{5} - 7 \beta_{4} - 5 \beta_{3} + 11 \beta_{2} + 15 \beta_{1} + 4$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-2 \beta_{15} + 4 \beta_{14} + 10 \beta_{13} - 11 \beta_{12} + 7 \beta_{11} - 5 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 9 \beta_{3} + 7 \beta_{2} - 5 \beta_{1} + 36$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$-10 \beta_{15} + 4 \beta_{14} + 2 \beta_{13} + \beta_{12} - 5 \beta_{11} + 7 \beta_{10} + 9 \beta_{9} - 2 \beta_{8} + 14 \beta_{6} - \beta_{5} - 7 \beta_{4} + 11 \beta_{3} - 29 \beta_{2} - 9 \beta_{1} + 52$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-10 \beta_{15} - 4 \beta_{14} + 10 \beta_{13} + \beta_{12} + 3 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} + 30 \beta_{8} - 16 \beta_{7} - 2 \beta_{6} - 25 \beta_{5} + 17 \beta_{4} - 5 \beta_{3} + 19 \beta_{2} + 39 \beta_{1} + 20$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$14 \beta_{15} - 12 \beta_{14} + 10 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} + 47 \beta_{10} - 31 \beta_{9} - 42 \beta_{8} - 16 \beta_{7} + 22 \beta_{6} - \beta_{5} + 25 \beta_{4} - 5 \beta_{3} - 13 \beta_{2} + 15 \beta_{1} - 60$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$30 \beta_{15} - 44 \beta_{14} + 42 \beta_{13} - 23 \beta_{12} - 37 \beta_{11} - 49 \beta_{10} + 33 \beta_{9} + 6 \beta_{8} + 24 \beta_{7} - 34 \beta_{6} - 33 \beta_{5} + 25 \beta_{4} - 21 \beta_{3} - 29 \beta_{2} + 15 \beta_{1} + 68$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-18 \beta_{15} + 4 \beta_{14} - 22 \beta_{13} + 57 \beta_{12} - 21 \beta_{11} + 31 \beta_{10} + 81 \beta_{9} - 26 \beta_{8} + 32 \beta_{7} - 10 \beta_{6} + 31 \beta_{5} - 7 \beta_{4} + 11 \beta_{3} + 83 \beta_{2} - 81 \beta_{1} + 36$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$94 \beta_{15} + 52 \beta_{14} + 10 \beta_{13} + 33 \beta_{12} - 13 \beta_{11} + 7 \beta_{10} - 71 \beta_{9} + 54 \beta_{8} - 64 \beta_{7} - 90 \beta_{6} - 89 \beta_{5} - 79 \beta_{4} - 93 \beta_{3} + 75 \beta_{2} + 71 \beta_{1} - 60$$$$)/4$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$421$$ $$801$$ $$897$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
111.1
 −0.275585 − 1.38710i 1.07046 + 0.924187i 1.41214 − 0.0765298i 0.244064 + 1.39299i −1.24098 + 0.678208i −1.38133 − 0.303194i 1.14218 − 0.833926i −0.470943 − 1.33350i −0.470943 + 1.33350i 1.14218 + 0.833926i −1.38133 + 0.303194i −1.24098 − 0.678208i 0.244064 − 1.39299i 1.41214 + 0.0765298i 1.07046 − 0.924187i −0.275585 + 1.38710i
0 3.19977i 0 −1.00000 0 −2.59303 0.525543i 0 −7.23851 0
111.2 0 2.99734i 0 −1.00000 0 0.183359 + 2.63939i 0 −5.98405 0
111.3 0 2.21915i 0 −1.00000 0 1.20923 2.35325i 0 −1.92464 0
111.4 0 1.68420i 0 −1.00000 0 0.695780 2.55262i 0 0.163484 0
111.5 0 1.61069i 0 −1.00000 0 2.13463 + 1.56312i 0 0.405694 0
111.6 0 1.34113i 0 −1.00000 0 −1.28003 + 2.31550i 0 1.20136 0
111.7 0 0.586834i 0 −1.00000 0 −2.52442 + 0.792014i 0 2.65563 0
111.8 0 0.528177i 0 −1.00000 0 2.17448 1.50719i 0 2.72103 0
111.9 0 0.528177i 0 −1.00000 0 2.17448 + 1.50719i 0 2.72103 0
111.10 0 0.586834i 0 −1.00000 0 −2.52442 0.792014i 0 2.65563 0
111.11 0 1.34113i 0 −1.00000 0 −1.28003 2.31550i 0 1.20136 0
111.12 0 1.61069i 0 −1.00000 0 2.13463 1.56312i 0 0.405694 0
111.13 0 1.68420i 0 −1.00000 0 0.695780 + 2.55262i 0 0.163484 0
111.14 0 2.21915i 0 −1.00000 0 1.20923 + 2.35325i 0 −1.92464 0
111.15 0 2.99734i 0 −1.00000 0 0.183359 2.63939i 0 −5.98405 0
111.16 0 3.19977i 0 −1.00000 0 −2.59303 + 0.525543i 0 −7.23851 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 111.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.e even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.h.a 16
4.b odd 2 1 280.2.h.a 16
7.b odd 2 1 1120.2.h.b 16
8.b even 2 1 280.2.h.b yes 16
8.d odd 2 1 1120.2.h.b 16
28.d even 2 1 280.2.h.b yes 16
56.e even 2 1 inner 1120.2.h.a 16
56.h odd 2 1 280.2.h.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.h.a 16 4.b odd 2 1
280.2.h.a 16 56.h odd 2 1
280.2.h.b yes 16 8.b even 2 1
280.2.h.b yes 16 28.d even 2 1
1120.2.h.a 16 1.a even 1 1 trivial
1120.2.h.a 16 56.e even 2 1 inner
1120.2.h.b 16 7.b odd 2 1
1120.2.h.b 16 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{8} - 62 T_{13}^{6} - 12 T_{13}^{5} + 1213 T_{13}^{4} + 228 T_{13}^{3} - 7792 T_{13}^{2} + 1232 T_{13} + 10032$$ acting on $$S_{2}^{\mathrm{new}}(1120, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$576 + 4720 T^{2} + 13024 T^{4} + 14676 T^{6} + 8217 T^{8} + 2468 T^{10} + 398 T^{12} + 32 T^{14} + T^{16}$$
$5$ $$( 1 + T )^{16}$$
$7$ $$5764801 + 470596 T^{2} + 605052 T^{3} + 9604 T^{4} + 69972 T^{5} + 34300 T^{6} + 6328 T^{7} + 5814 T^{8} + 904 T^{9} + 700 T^{10} + 204 T^{11} + 4 T^{12} + 36 T^{13} + 4 T^{14} + T^{16}$$
$11$ $$( 2400 - 1960 T - 3932 T^{2} - 138 T^{3} + 761 T^{4} + 52 T^{5} - 50 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$13$ $$( 10032 + 1232 T - 7792 T^{2} + 228 T^{3} + 1213 T^{4} - 12 T^{5} - 62 T^{6} + T^{8} )^{2}$$
$17$ $$2359296 + 27167680 T^{2} + 34001456 T^{4} + 13596164 T^{6} + 2255465 T^{8} + 178476 T^{10} + 7086 T^{12} + 136 T^{14} + T^{16}$$
$19$ $$1327104 + 26099712 T^{2} + 81127424 T^{4} + 49375232 T^{6} + 6949696 T^{8} + 423168 T^{10} + 12736 T^{12} + 184 T^{14} + T^{16}$$
$23$ $$7573504 + 585268224 T^{2} + 289346560 T^{4} + 57190400 T^{6} + 5807360 T^{8} + 325872 T^{10} + 10096 T^{12} + 160 T^{14} + T^{16}$$
$29$ $$65536 + 2408448 T^{2} + 15058432 T^{4} + 9795008 T^{6} + 2200529 T^{8} + 217548 T^{10} + 9718 T^{12} + 172 T^{14} + T^{16}$$
$31$ $$( 98304 - 2816 T - 33920 T^{2} + 3040 T^{3} + 3496 T^{4} - 440 T^{5} - 92 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$37$ $$4063297536 + 30390333440 T^{2} + 7832304640 T^{4} + 825741568 T^{6} + 45832768 T^{8} + 1448768 T^{10} + 26192 T^{12} + 252 T^{14} + T^{16}$$
$41$ $$6423183360000 + 1819708211200 T^{2} + 201056874496 T^{4} + 11370832896 T^{6} + 364052032 T^{8} + 6823040 T^{10} + 73760 T^{12} + 424 T^{14} + T^{16}$$
$43$ $$( -69504 + 91360 T - 17664 T^{2} - 14912 T^{3} + 4232 T^{4} + 332 T^{5} - 128 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$47$ $$( -10368 + 5820 T + 36188 T^{2} - 28292 T^{3} + 5461 T^{4} + 424 T^{5} - 166 T^{6} + T^{8} )^{2}$$
$53$ $$262144 + 27312128 T^{2} + 158445568 T^{4} + 251149312 T^{6} + 69431040 T^{8} + 3445312 T^{10} + 58128 T^{12} + 404 T^{14} + T^{16}$$
$59$ $$1090584576 + 2931687424 T^{2} + 2362048512 T^{4} + 734793728 T^{6} + 82300416 T^{8} + 3287040 T^{10} + 55552 T^{12} + 400 T^{14} + T^{16}$$
$61$ $$( 10368 + 51712 T + 26432 T^{2} - 85216 T^{3} + 15176 T^{4} + 1256 T^{5} - 268 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$67$ $$( 165632 - 197184 T - 55616 T^{2} + 42240 T^{3} + 7712 T^{4} - 1204 T^{5} - 180 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$71$ $$2621440000 + 5845811200 T^{2} + 4596301824 T^{4} + 1494106112 T^{6} + 180376576 T^{8} + 6017280 T^{10} + 82752 T^{12} + 492 T^{14} + T^{16}$$
$73$ $$28179280429056 + 12310919774208 T^{2} + 1248988233728 T^{4} + 57276760064 T^{6} + 1406521344 T^{8} + 19483584 T^{10} + 151344 T^{12} + 612 T^{14} + T^{16}$$
$79$ $$39806206534656 + 12666930453504 T^{2} + 1373008608512 T^{4} + 61858049132 T^{6} + 1444670401 T^{8} + 19139020 T^{10} + 145478 T^{12} + 592 T^{14} + T^{16}$$
$83$ $$88794464256 + 165451399168 T^{2} + 51045990400 T^{4} + 6096578560 T^{6} + 337953344 T^{8} + 8985456 T^{10} + 109632 T^{12} + 564 T^{14} + T^{16}$$
$89$ $$347892350976 + 245081571328 T^{2} + 61230546944 T^{4} + 6628880384 T^{6} + 331338304 T^{8} + 8073856 T^{10} + 95808 T^{12} + 520 T^{14} + T^{16}$$
$97$ $$17227945230336 + 27670537460160 T^{2} + 4314469148336 T^{4} + 212781536324 T^{6} + 4708474873 T^{8} + 52726332 T^{10} + 307550 T^{12} + 888 T^{14} + T^{16}$$