# Properties

 Label 360.2.w.d Level $360$ Weight $2$ Character orbit 360.w Analytic conductor $2.875$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.w (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{15} q^{2} -\beta_{2} q^{4} + ( \beta_{10} - \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + \beta_{11} ) q^{7} + ( -\beta_{4} + \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{15} q^{2} -\beta_{2} q^{4} + ( \beta_{10} - \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + \beta_{11} ) q^{7} + ( -\beta_{4} + \beta_{5} ) q^{8} + ( \beta_{2} + 2 \beta_{8} + \beta_{9} ) q^{10} + ( -\beta_{1} + \beta_{14} + \beta_{15} ) q^{11} + ( -\beta_{3} - \beta_{7} ) q^{13} + ( -2 \beta_{4} + \beta_{5} - 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{14} + ( -1 - \beta_{2} - \beta_{3} - \beta_{9} + \beta_{11} ) q^{16} + ( \beta_{4} + \beta_{10} - 2 \beta_{13} - \beta_{14} ) q^{17} + 4 \beta_{8} q^{19} + ( \beta_{1} - \beta_{4} - \beta_{5} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{20} + ( 2 - \beta_{2} - \beta_{6} + 2 \beta_{8} ) q^{22} + ( 2 \beta_{4} + 2 \beta_{10} - 2 \beta_{14} ) q^{23} + ( -2 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{25} + ( -\beta_{1} - \beta_{5} - 4 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{26} + ( 4 - \beta_{2} - 2 \beta_{3} + \beta_{6} - 2 \beta_{8} ) q^{28} + ( -\beta_{1} + 2 \beta_{4} + 2 \beta_{12} - \beta_{14} - \beta_{15} ) q^{29} + ( 1 + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{31} + ( \beta_{4} - 2 \beta_{10} + \beta_{13} - \beta_{14} ) q^{32} + ( -2 + 2 \beta_{3} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{34} + ( \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{35} + ( -1 - \beta_{2} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{37} + 4 \beta_{1} q^{38} + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} ) q^{40} + ( -2 \beta_{5} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{41} + ( -4 + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{44} + ( 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{46} + ( 2 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{47} + ( 1 - 3 \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} - 4 \beta_{8} - \beta_{11} ) q^{49} + ( -2 \beta_{1} + \beta_{5} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{50} + ( -2 - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{52} + ( 2 \beta_{1} + \beta_{4} + \beta_{10} - \beta_{12} ) q^{53} + ( 2 + \beta_{2} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{55} + ( -4 \beta_{1} - \beta_{5} - 4 \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{14} + 4 \beta_{15} ) q^{56} + ( -2 + \beta_{2} - \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{58} + ( -\beta_{1} - \beta_{12} - \beta_{15} ) q^{59} + ( -3 + \beta_{2} + \beta_{3} + \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{61} + ( -2 \beta_{4} + 4 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{15} ) q^{62} + ( 1 - \beta_{3} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} - \beta_{9} - \beta_{11} ) q^{64} + ( -4 \beta_{1} + \beta_{4} + \beta_{10} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{65} + ( 2 + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{5} + 2 \beta_{10} - 4 \beta_{12} + 2 \beta_{14} ) q^{68} + ( -6 + 3 \beta_{2} + 4 \beta_{3} - \beta_{6} - 2 \beta_{7} - 4 \beta_{8} ) q^{70} + ( 5 - 5 \beta_{8} ) q^{73} + ( \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{74} + 4 \beta_{6} q^{76} + ( -2 \beta_{4} + 2 \beta_{10} - 3 \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{77} + ( -1 - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{79} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{80} + ( 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{9} - 4 \beta_{11} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{10} - \beta_{12} - \beta_{14} ) q^{83} + ( 2 + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{85} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{13} + 6 \beta_{14} - 2 \beta_{15} ) q^{86} + ( -3 \beta_{2} - 2 \beta_{3} + 3 \beta_{6} + 2 \beta_{8} ) q^{88} + ( 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{12} + 2 \beta_{13} + \beta_{14} + 4 \beta_{15} ) q^{89} + ( -6 + 2 \beta_{2} + 2 \beta_{3} - 6 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{11} ) q^{91} + ( -4 \beta_{4} + 4 \beta_{10} - 4 \beta_{12} - 4 \beta_{15} ) q^{92} + ( 4 \beta_{2} + 8 \beta_{8} ) q^{94} + ( -4 \beta_{1} - 4 \beta_{4} ) q^{95} + ( -3 + 2 \beta_{2} + 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{11} ) q^{97} + ( -5 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{12} - 2 \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{10} - 20q^{16} + 36q^{22} - 24q^{25} + 44q^{28} + 32q^{40} - 16q^{43} - 32q^{46} - 8q^{52} - 44q^{58} + 16q^{67} - 44q^{70} + 80q^{73} - 16q^{76} - 8q^{82} - 28q^{88} - 32q^{91} - 64q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5 x^{12} + 28 x^{8} + 80 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{14} - 4 \nu^{12} + 29 \nu^{10} - 36 \nu^{8} + 84 \nu^{6} - 256 \nu^{4} + 176 \nu^{2} - 384$$$$)/320$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{13} - 11 \nu^{11} - 36 \nu^{9} + 44 \nu^{7} + 64 \nu^{5} - 144 \nu^{3} - 384 \nu$$$$)/640$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{15} + 12 \nu^{13} - 11 \nu^{11} + 108 \nu^{9} + 44 \nu^{7} + 448 \nu^{5} - 144 \nu^{3} + 1152 \nu$$$$)/640$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{12} - 9 \nu^{8} - 24 \nu^{4} - 96$$$$)/40$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{14} - 8 \nu^{12} + 11 \nu^{10} + 8 \nu^{8} - 44 \nu^{6} - 112 \nu^{4} + 144 \nu^{2} - 128$$$$)/320$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{14} - 7 \nu^{10} - 12 \nu^{6} - 48 \nu^{2}$$$$)/320$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{14} - 2 \nu^{12} - \nu^{10} + 2 \nu^{8} + 14 \nu^{6} - 28 \nu^{4} - 24 \nu^{2} - 32$$$$)/80$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{15} + 2 \nu^{13} + 9 \nu^{11} - 2 \nu^{9} + 24 \nu^{7} + 28 \nu^{5} + 136 \nu^{3} - 48 \nu$$$$)/160$$ $$\beta_{11}$$ $$=$$ $$($$$$5 \nu^{14} - 12 \nu^{12} + 25 \nu^{10} - 28 \nu^{8} + 140 \nu^{6} - 48 \nu^{4} + 400 \nu^{2} - 192$$$$)/320$$ $$\beta_{12}$$ $$=$$ $$($$$$5 \nu^{15} + 12 \nu^{13} + 25 \nu^{11} + 28 \nu^{9} + 140 \nu^{7} + 48 \nu^{5} + 400 \nu^{3} + 192 \nu$$$$)/640$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{15} + 2 \nu^{13} - 9 \nu^{11} - 2 \nu^{9} - 24 \nu^{7} + 28 \nu^{5} + 24 \nu^{3} - 48 \nu$$$$)/160$$ $$\beta_{14}$$ $$=$$ $$($$$$5 \nu^{15} - 12 \nu^{13} + 25 \nu^{11} - 28 \nu^{9} + 140 \nu^{7} - 48 \nu^{5} + 400 \nu^{3} - 192 \nu$$$$)/640$$ $$\beta_{15}$$ $$=$$ $$($$$$3 \nu^{15} + 7 \nu^{11} + 12 \nu^{7} + 48 \nu^{3}$$$$)/320$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{14} + \beta_{13} - \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} - \beta_{9} - \beta_{3} - \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{12} + 2 \beta_{10} + \beta_{5} + \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} + \beta_{9} + 4 \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{3} - 1$$ $$\nu^{7}$$ $$=$$ $$-4 \beta_{15} + \beta_{14} - \beta_{13} + 4 \beta_{12} - 2 \beta_{10} + 3 \beta_{4}$$ $$\nu^{8}$$ $$=$$ $$-\beta_{11} + 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + \beta_{3} + \beta_{2} - 7$$ $$\nu^{9}$$ $$=$$ $$4 \beta_{14} + 2 \beta_{12} - 6 \beta_{10} + 3 \beta_{5} - 9 \beta_{4} - 12 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$3 \beta_{11} - 9 \beta_{9} - 8 \beta_{8} + 6 \beta_{7} - 3 \beta_{6} + 3 \beta_{3} - 12 \beta_{2} - 3$$ $$\nu^{11}$$ $$=$$ $$-4 \beta_{15} - 9 \beta_{14} - 15 \beta_{13} - 12 \beta_{12} + 18 \beta_{10} - 3 \beta_{4}$$ $$\nu^{12}$$ $$=$$ $$-15 \beta_{11} - 3 \beta_{9} - 18 \beta_{8} - 18 \beta_{7} - 4 \beta_{6} + 15 \beta_{3} + 15 \beta_{2} - 9$$ $$\nu^{13}$$ $$=$$ $$-36 \beta_{14} + 30 \beta_{12} + 6 \beta_{10} - 11 \beta_{5} + 17 \beta_{4} + 12 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-11 \beta_{11} + 17 \beta_{9} - 104 \beta_{8} - 6 \beta_{7} + 11 \beta_{6} - 11 \beta_{3} + 12 \beta_{2} + 11$$ $$\nu^{15}$$ $$=$$ $$132 \beta_{15} + \beta_{14} + 23 \beta_{13} + 12 \beta_{12} - 34 \beta_{10} + 11 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$-1$$ $$\beta_{8}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 −0.512386 + 1.31813i 0.788026 + 1.17431i 1.17431 + 0.788026i −1.31813 + 0.512386i 1.31813 − 0.512386i −1.17431 − 0.788026i −0.788026 − 1.17431i 0.512386 − 1.31813i −0.512386 − 1.31813i 0.788026 − 1.17431i 1.17431 − 0.788026i −1.31813 − 0.512386i 1.31813 + 0.512386i −1.17431 + 0.788026i −0.788026 + 1.17431i 0.512386 + 1.31813i
−1.31813 0.512386i 0 1.47492 + 1.35078i 1.83051 + 1.28422i 0 2.94984 + 2.94984i −1.25201 2.53623i 0 −1.75483 2.63069i
163.2 −1.17431 + 0.788026i 0 0.758030 1.85078i 0.386289 2.20245i 0 1.51606 + 1.51606i 0.568298 + 2.77075i 0 1.28196 + 2.89077i
163.3 −0.788026 + 1.17431i 0 −0.758030 1.85078i −0.386289 + 2.20245i 0 −1.51606 1.51606i 2.77075 + 0.568298i 0 −2.28196 2.18921i
163.4 −0.512386 1.31813i 0 −1.47492 + 1.35078i 1.83051 + 1.28422i 0 −2.94984 2.94984i 2.53623 + 1.25201i 0 0.754834 3.07087i
163.5 0.512386 + 1.31813i 0 −1.47492 + 1.35078i −1.83051 1.28422i 0 −2.94984 2.94984i −2.53623 1.25201i 0 0.754834 3.07087i
163.6 0.788026 1.17431i 0 −0.758030 1.85078i 0.386289 2.20245i 0 −1.51606 1.51606i −2.77075 0.568298i 0 −2.28196 2.18921i
163.7 1.17431 0.788026i 0 0.758030 1.85078i −0.386289 + 2.20245i 0 1.51606 + 1.51606i −0.568298 2.77075i 0 1.28196 + 2.89077i
163.8 1.31813 + 0.512386i 0 1.47492 + 1.35078i −1.83051 1.28422i 0 2.94984 + 2.94984i 1.25201 + 2.53623i 0 −1.75483 2.63069i
307.1 −1.31813 + 0.512386i 0 1.47492 1.35078i 1.83051 1.28422i 0 2.94984 2.94984i −1.25201 + 2.53623i 0 −1.75483 + 2.63069i
307.2 −1.17431 0.788026i 0 0.758030 + 1.85078i 0.386289 + 2.20245i 0 1.51606 1.51606i 0.568298 2.77075i 0 1.28196 2.89077i
307.3 −0.788026 1.17431i 0 −0.758030 + 1.85078i −0.386289 2.20245i 0 −1.51606 + 1.51606i 2.77075 0.568298i 0 −2.28196 + 2.18921i
307.4 −0.512386 + 1.31813i 0 −1.47492 1.35078i 1.83051 1.28422i 0 −2.94984 + 2.94984i 2.53623 1.25201i 0 0.754834 + 3.07087i
307.5 0.512386 1.31813i 0 −1.47492 1.35078i −1.83051 + 1.28422i 0 −2.94984 + 2.94984i −2.53623 + 1.25201i 0 0.754834 + 3.07087i
307.6 0.788026 + 1.17431i 0 −0.758030 + 1.85078i 0.386289 + 2.20245i 0 −1.51606 + 1.51606i −2.77075 + 0.568298i 0 −2.28196 + 2.18921i
307.7 1.17431 + 0.788026i 0 0.758030 + 1.85078i −0.386289 2.20245i 0 1.51606 1.51606i −0.568298 + 2.77075i 0 1.28196 2.89077i
307.8 1.31813 0.512386i 0 1.47492 1.35078i −1.83051 + 1.28422i 0 2.94984 2.94984i 1.25201 2.53623i 0 −1.75483 + 2.63069i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.d odd 2 1 inner
15.e even 4 1 inner
24.f even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.w.d 16
3.b odd 2 1 inner 360.2.w.d 16
4.b odd 2 1 1440.2.bi.d 16
5.c odd 4 1 inner 360.2.w.d 16
8.b even 2 1 1440.2.bi.d 16
8.d odd 2 1 inner 360.2.w.d 16
12.b even 2 1 1440.2.bi.d 16
15.e even 4 1 inner 360.2.w.d 16
20.e even 4 1 1440.2.bi.d 16
24.f even 2 1 inner 360.2.w.d 16
24.h odd 2 1 1440.2.bi.d 16
40.i odd 4 1 1440.2.bi.d 16
40.k even 4 1 inner 360.2.w.d 16
60.l odd 4 1 1440.2.bi.d 16
120.q odd 4 1 inner 360.2.w.d 16
120.w even 4 1 1440.2.bi.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.w.d 16 1.a even 1 1 trivial
360.2.w.d 16 3.b odd 2 1 inner
360.2.w.d 16 5.c odd 4 1 inner
360.2.w.d 16 8.d odd 2 1 inner
360.2.w.d 16 15.e even 4 1 inner
360.2.w.d 16 24.f even 2 1 inner
360.2.w.d 16 40.k even 4 1 inner
360.2.w.d 16 120.q odd 4 1 inner
1440.2.bi.d 16 4.b odd 2 1
1440.2.bi.d 16 8.b even 2 1
1440.2.bi.d 16 12.b even 2 1
1440.2.bi.d 16 20.e even 4 1
1440.2.bi.d 16 24.h odd 2 1
1440.2.bi.d 16 40.i odd 4 1
1440.2.bi.d 16 60.l odd 4 1
1440.2.bi.d 16 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 324 T_{7}^{4} + 6400$$ acting on $$S_{2}^{\mathrm{new}}(360, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + 80 T^{4} + 28 T^{8} + 5 T^{12} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} )^{2}$$
$7$ $$( 6400 + 324 T^{4} + T^{8} )^{2}$$
$11$ $$( 40 - 18 T^{2} + T^{4} )^{4}$$
$13$ $$( 102400 + 804 T^{4} + T^{8} )^{2}$$
$17$ $$( 25600 + 1796 T^{4} + T^{8} )^{2}$$
$19$ $$( 16 + T^{2} )^{8}$$
$23$ $$( 16384 + 2880 T^{4} + T^{8} )^{2}$$
$29$ $$( 800 - 58 T^{2} + T^{4} )^{4}$$
$31$ $$( 1280 + 76 T^{2} + T^{4} )^{4}$$
$37$ $$( 102400 + 804 T^{4} + T^{8} )^{2}$$
$41$ $$( 640 - 92 T^{2} + T^{4} )^{4}$$
$43$ $$( 6400 - 320 T + 8 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$47$ $$( 262144 + 11520 T^{4} + T^{8} )^{2}$$
$53$ $$( 16384 + 420 T^{4} + T^{8} )^{2}$$
$59$ $$( 40 + 18 T^{2} + T^{4} )^{4}$$
$61$ $$( 8000 + 220 T^{2} + T^{4} )^{4}$$
$67$ $$( 6400 + 320 T + 8 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$71$ $$T^{16}$$
$73$ $$( 50 - 10 T + T^{2} )^{8}$$
$79$ $$( 1280 - 76 T^{2} + T^{4} )^{4}$$
$83$ $$( 104857600 + 21136 T^{4} + T^{8} )^{2}$$
$89$ $$( 2560 + 348 T^{2} + T^{4} )^{4}$$
$97$ $$( 2500 - 800 T + 128 T^{2} + 16 T^{3} + T^{4} )^{4}$$