# Properties

 Label 420.2.l.c Level $420$ Weight $2$ Character orbit 420.l Analytic conductor $3.354$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.386672896.3 Defining polynomial: $$x^{8} - x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} - 4 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{6} - 2 x^{5} + 2 x^{4} - 4 x^{3} - 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} + 2 \nu^{4} + 2 \nu^{3} - 4 \nu$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} - 2 \nu^{4} + 2 \nu^{3} - 4 \nu^{2} - 4 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - 2 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} + 4 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{6} - \nu^{4} + 2 \nu^{3} + 2 \nu^{2} - 4 \nu - 8$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - \nu^{5} - 2 \nu^{3} - 4 \nu^{2} - 12 \nu$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} + 3 \nu^{5} + 2 \nu^{3} - 4 \nu^{2} - 12 \nu - 24$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{4} + 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{4} - 2 \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 4$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_{1} + 4$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{7} + \beta_{6} - 3 \beta_{5} + 5 \beta_{4} + 4 \beta_{3} + 6 \beta_{2} + 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 −1.19503 + 0.756243i −1.19503 − 0.756243i −0.835949 + 1.14070i −0.835949 − 1.14070i 0.621372 + 1.27039i 0.621372 − 1.27039i 1.40961 + 0.114062i 1.40961 − 0.114062i
−1.19503 0.756243i 0.356193 1.69503i 0.856193 + 1.80747i −1.00000 + 2.00000i −1.70752 + 1.75624i −1.00000 0.343707 2.80747i −2.74625 1.20752i 2.70752 1.63382i
239.2 −1.19503 + 0.756243i 0.356193 + 1.69503i 0.856193 1.80747i −1.00000 2.00000i −1.70752 1.75624i −1.00000 0.343707 + 2.80747i −2.74625 + 1.20752i 2.70752 + 1.63382i
239.3 −0.835949 1.14070i −1.10238 + 1.33595i −0.602380 + 1.90713i −1.00000 2.00000i 2.44545 + 0.140697i −1.00000 2.67901 0.907128i −0.569517 2.94545i −1.44545 + 2.81259i
239.4 −0.835949 + 1.14070i −1.10238 1.33595i −0.602380 1.90713i −1.00000 + 2.00000i 2.44545 0.140697i −1.00000 2.67901 + 0.907128i −0.569517 + 2.94545i −1.44545 2.81259i
239.5 0.621372 1.27039i −1.72779 + 0.121372i −1.22779 1.57877i −1.00000 + 2.00000i −0.919412 + 2.27039i −1.00000 −2.76858 + 0.578773i 2.97054 0.419412i 1.91941 + 2.51314i
239.6 0.621372 + 1.27039i −1.72779 0.121372i −1.22779 + 1.57877i −1.00000 2.00000i −0.919412 2.27039i −1.00000 −2.76858 0.578773i 2.97054 + 0.419412i 1.91941 2.51314i
239.7 1.40961 0.114062i 1.47398 + 0.909606i 1.97398 0.321565i −1.00000 + 2.00000i 2.18148 + 1.11406i −1.00000 2.74586 0.678435i 1.34523 + 2.68148i −1.18148 + 2.93327i
239.8 1.40961 + 0.114062i 1.47398 0.909606i 1.97398 + 0.321565i −1.00000 2.00000i 2.18148 1.11406i −1.00000 2.74586 + 0.678435i 1.34523 2.68148i −1.18148 2.93327i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 239.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.l.c 8
3.b odd 2 1 420.2.l.d yes 8
4.b odd 2 1 420.2.l.e yes 8
5.b even 2 1 420.2.l.f yes 8
12.b even 2 1 420.2.l.f yes 8
15.d odd 2 1 420.2.l.e yes 8
20.d odd 2 1 420.2.l.d yes 8
60.h even 2 1 inner 420.2.l.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.l.c 8 1.a even 1 1 trivial
420.2.l.c 8 60.h even 2 1 inner
420.2.l.d yes 8 3.b odd 2 1
420.2.l.d yes 8 20.d odd 2 1
420.2.l.e yes 8 4.b odd 2 1
420.2.l.e yes 8 15.d odd 2 1
420.2.l.f yes 8 5.b even 2 1
420.2.l.f yes 8 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$:

 $$T_{11}^{4} - 2 T_{11}^{3} - 11 T_{11}^{2} + 16 T_{11} + 16$$ $$T_{17}^{4} - 2 T_{17}^{3} - 35 T_{17}^{2} + 52 T_{17} + 100$$ $$T_{43}^{4} - 4 T_{43}^{3} - 56 T_{43}^{2} + 192 T_{43} - 128$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T^{2} - 4 T^{3} + 2 T^{4} - 2 T^{5} - T^{6} + T^{8}$$
$3$ $$81 + 54 T + 9 T^{2} + 6 T^{3} + 4 T^{4} + 2 T^{5} + T^{6} + 2 T^{7} + T^{8}$$
$5$ $$( 5 + 2 T + T^{2} )^{4}$$
$7$ $$( 1 + T )^{8}$$
$11$ $$( 16 + 16 T - 11 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$13$ $$6400 + 5664 T^{2} + 1321 T^{4} + 70 T^{6} + T^{8}$$
$17$ $$( 100 + 52 T - 35 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$19$ $$1024 + 2048 T^{2} + 1104 T^{4} + 88 T^{6} + T^{8}$$
$23$ $$6400 + 18944 T^{2} + 2656 T^{4} + 96 T^{6} + T^{8}$$
$29$ $$6400 + 5664 T^{2} + 1321 T^{4} + 70 T^{6} + T^{8}$$
$31$ $$12544 + 209920 T^{2} + 11488 T^{4} + 192 T^{6} + T^{8}$$
$37$ $$4096 + 25088 T^{2} + 3984 T^{4} + 136 T^{6} + T^{8}$$
$41$ $$65536 + 38912 T^{2} + 3472 T^{4} + 104 T^{6} + T^{8}$$
$43$ $$( -128 + 192 T - 56 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$47$ $$16 + 280 T^{2} + 145 T^{4} + 22 T^{6} + T^{8}$$
$53$ $$( 5392 + 256 T - 200 T^{2} + T^{4} )^{2}$$
$59$ $$( -2240 + 1152 T - 140 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$61$ $$( -1312 - 1232 T - 148 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$67$ $$( -6848 + 2112 T - 124 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$71$ $$( 8 + T )^{8}$$
$73$ $$( 64 + T^{2} )^{4}$$
$79$ $$753424 + 895192 T^{2} + 66289 T^{4} + 502 T^{6} + T^{8}$$
$83$ $$29073664 + 2034432 T^{2} + 48224 T^{4} + 432 T^{6} + T^{8}$$
$89$ $$6553600 + 712704 T^{2} + 26176 T^{4} + 352 T^{6} + T^{8}$$
$97$ $$16777216 + 2859008 T^{2} + 128601 T^{4} + 694 T^{6} + T^{8}$$