# Properties

 Label 460.2.i.b Level $460$ Weight $2$ Character orbit 460.i Analytic conductor $3.673$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 18 x^{14} + 146 x^{12} - 798 x^{10} + 3934 x^{8} - 19950 x^{6} + 91250 x^{4} - 281250 x^{2} + 390625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} -\beta_{3} q^{5} + ( -\beta_{3} + \beta_{15} ) q^{7} + ( -1 + \beta_{2} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} -\beta_{3} q^{5} + ( -\beta_{3} + \beta_{15} ) q^{7} + ( -1 + \beta_{2} - \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{12} ) q^{9} + ( \beta_{1} - \beta_{4} - \beta_{7} + \beta_{14} + \beta_{15} ) q^{11} + ( 1 - \beta_{6} + \beta_{9} + \beta_{10} ) q^{13} + ( -\beta_{4} + 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{15} + ( \beta_{4} - \beta_{11} ) q^{17} + ( \beta_{3} + \beta_{11} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} - \beta_{13} + 2 \beta_{14} ) q^{21} + ( 1 - \beta_{3} - \beta_{6} + \beta_{9} + \beta_{10} + \beta_{14} ) q^{23} + ( 3 + \beta_{8} - \beta_{10} - \beta_{12} ) q^{25} + ( 4 + \beta_{5} + 3 \beta_{8} - 4 \beta_{9} ) q^{27} + ( 1 - \beta_{2} + 3 \beta_{9} - \beta_{12} ) q^{29} + ( -1 + \beta_{6} - \beta_{8} ) q^{31} + ( -2 \beta_{4} - \beta_{7} - 2 \beta_{11} ) q^{33} + ( 1 - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{12} ) q^{35} + ( -\beta_{1} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{37} + ( \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{39} + ( -3 + \beta_{5} - \beta_{10} ) q^{41} + ( -\beta_{4} + \beta_{7} - \beta_{11} ) q^{43} + ( 4 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{7} + \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{45} + ( -1 + 2 \beta_{2} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{47} + ( -\beta_{5} - \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{49} + ( -\beta_{1} - 2 \beta_{4} + \beta_{7} + 2 \beta_{14} - \beta_{15} ) q^{51} + ( -\beta_{1} - 3 \beta_{4} - 2 \beta_{7} - 3 \beta_{11} - \beta_{13} ) q^{53} + ( \beta_{2} - 2 \beta_{5} - 2 \beta_{9} + 2 \beta_{12} ) q^{55} + ( \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{11} - \beta_{13} + \beta_{15} ) q^{57} + ( 1 - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{12} ) q^{59} + ( -2 \beta_{1} - \beta_{3} + 3 \beta_{4} + \beta_{7} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{61} + ( 2 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{11} + 2 \beta_{13} - 5 \beta_{14} ) q^{63} + ( \beta_{1} + \beta_{3} + 3 \beta_{4} + \beta_{7} - \beta_{11} - \beta_{14} - 2 \beta_{15} ) q^{65} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{11} + 2 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{67} + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - 3 \beta_{9} + \beta_{10} + 4 \beta_{11} - \beta_{13} + \beta_{14} ) q^{69} + ( -\beta_{2} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{71} + ( -3 - \beta_{6} - 3 \beta_{9} + 2 \beta_{10} ) q^{73} + ( -3 - \beta_{2} - \beta_{5} + 3 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} ) q^{75} + ( -3 - 2 \beta_{2} - \beta_{5} + 5 \beta_{9} + \beta_{10} ) q^{77} + ( -\beta_{1} - 2 \beta_{3} + 2 \beta_{7} - \beta_{11} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{79} + ( -12 + \beta_{2} + 2 \beta_{5} + 5 \beta_{6} - 5 \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{81} + ( -4 \beta_{1} - \beta_{3} + 2 \beta_{7} - 4 \beta_{13} + \beta_{14} ) q^{83} + ( 2 - 3 \beta_{6} + \beta_{10} + \beta_{12} ) q^{85} + ( 3 - 2 \beta_{2} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{87} + ( -\beta_{3} - 2 \beta_{7} - 3 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{89} + ( 3 \beta_{1} + 3 \beta_{3} + 5 \beta_{4} + 4 \beta_{13} - \beta_{14} ) q^{91} + ( 5 - \beta_{5} - 3 \beta_{6} + 5 \beta_{9} + \beta_{10} + 2 \beta_{12} ) q^{93} + ( -5 + \beta_{6} - 2 \beta_{8} ) q^{95} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{11} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{97} + ( -3 \beta_{1} + 2 \beta_{3} + \beta_{7} + 3 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 4q^{3} + O(q^{10})$$ $$16q + 4q^{3} + 12q^{13} + 12q^{23} + 36q^{25} + 52q^{27} - 8q^{31} + 16q^{35} - 48q^{41} + 4q^{47} + 24q^{55} + 8q^{71} - 52q^{73} - 56q^{75} - 64q^{77} - 152q^{81} + 28q^{85} + 28q^{87} + 84q^{93} - 68q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 18 x^{14} + 146 x^{12} - 798 x^{10} + 3934 x^{8} - 19950 x^{6} + 91250 x^{4} - 281250 x^{2} + 390625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-28 \nu^{14} + 9729 \nu^{12} - 78013 \nu^{10} + 385944 \nu^{8} - 1646452 \nu^{6} + 8912125 \nu^{4} - 43639625 \nu^{2} + 123225000$$$$)/11356250$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} - 18 \nu^{13} + 146 \nu^{11} - 798 \nu^{9} + 3934 \nu^{7} - 19950 \nu^{5} + 91250 \nu^{3} - 281250 \nu$$$$)/78125$$ $$\beta_{4}$$ $$=$$ $$($$$$6206 \nu^{15} - 145108 \nu^{13} + 1091651 \nu^{11} - 5206913 \nu^{9} + 25292604 \nu^{7} - 120536550 \nu^{5} + 599261875 \nu^{3} - 1562984375 \nu$$$$)/ 283906250$$ $$\beta_{5}$$ $$=$$ $$($$$$-9539 \nu^{14} + 106427 \nu^{12} - 622119 \nu^{10} + 2938847 \nu^{8} - 15928851 \nu^{6} + 75874325 \nu^{4} - 260211875 \nu^{2} + 434828125$$$$)/56781250$$ $$\beta_{6}$$ $$=$$ $$($$$$-9679 \nu^{14} + 155072 \nu^{12} - 1012184 \nu^{10} + 4868567 \nu^{8} - 24161111 \nu^{6} + 120434950 \nu^{4} - 535191250 \nu^{2} + 1164515625$$$$)/56781250$$ $$\beta_{7}$$ $$=$$ $$($$$$1768 \nu^{15} - 26839 \nu^{13} + 159273 \nu^{11} - 818804 \nu^{9} + 3607532 \nu^{7} - 21256985 \nu^{5} + 87834625 \nu^{3} - 119118750 \nu$$$$)/56781250$$ $$\beta_{8}$$ $$=$$ $$($$$$13152 \nu^{14} - 165036 \nu^{12} + 932717 \nu^{10} - 4530221 \nu^{8} + 23029618 \nu^{6} - 120333350 \nu^{4} + 471120625 \nu^{2} - 766046875$$$$)/56781250$$ $$\beta_{9}$$ $$=$$ $$($$$$-14766 \nu^{14} + 182013 \nu^{12} - 1107886 \nu^{10} + 5441493 \nu^{8} - 26865744 \nu^{6} + 137700225 \nu^{4} - 545257500 \nu^{2} + 983109375$$$$)/56781250$$ $$\beta_{10}$$ $$=$$ $$($$$$-37979 \nu^{14} + 527497 \nu^{12} - 3336559 \nu^{10} + 15924867 \nu^{8} - 79211011 \nu^{6} + 411359675 \nu^{4} - 1676681875 \nu^{2} + 3273890625$$$$)/56781250$$ $$\beta_{11}$$ $$=$$ $$($$$$-39617 \nu^{15} + 550556 \nu^{13} - 3408182 \nu^{11} + 16828941 \nu^{9} - 84516503 \nu^{7} + 433599950 \nu^{5} - 1809035000 \nu^{3} + 3434765625 \nu$$$$)/ 283906250$$ $$\beta_{12}$$ $$=$$ $$($$$$10953 \nu^{14} - 151589 \nu^{12} + 959968 \nu^{10} - 4671004 \nu^{8} + 23307357 \nu^{6} - 120838265 \nu^{4} + 495881000 \nu^{2} - 978331250$$$$)/11356250$$ $$\beta_{13}$$ $$=$$ $$($$$$62919 \nu^{15} - 763392 \nu^{13} + 4635849 \nu^{11} - 22512212 \nu^{9} + 111486021 \nu^{7} - 583590450 \nu^{5} + 2298853125 \nu^{3} - 4064531250 \nu$$$$)/ 283906250$$ $$\beta_{14}$$ $$=$$ $$($$$$14766 \nu^{15} - 182013 \nu^{13} + 1107886 \nu^{11} - 5441493 \nu^{9} + 26865744 \nu^{7} - 137700225 \nu^{5} + 545257500 \nu^{3} - 983109375 \nu$$$$)/56781250$$ $$\beta_{15}$$ $$=$$ $$($$$$99216 \nu^{15} - 1413363 \nu^{13} + 9018211 \nu^{11} - 43634468 \nu^{9} + 219432044 \nu^{7} - 1129291475 \nu^{5} + 4663698125 \nu^{3} - 9530000000 \nu$$$$)/ 283906250$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} + \beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{15} - 3 \beta_{14} + 4 \beta_{13} - 4 \beta_{11} - \beta_{7} + 3 \beta_{4} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{12} + \beta_{10} - 5 \beta_{9} - \beta_{8} - 9 \beta_{6} + 2 \beta_{2} + 6$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{15} + 3 \beta_{14} + \beta_{13} - 13 \beta_{11} - 9 \beta_{7} + 16 \beta_{4} - 7 \beta_{3} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{12} + 18 \beta_{10} + 28 \beta_{9} - \beta_{8} - 24 \beta_{6} - 29 \beta_{5} + 8 \beta_{2} + 39$$ $$\nu^{7}$$ $$=$$ $$7 \beta_{15} + 22 \beta_{14} - 49 \beta_{13} - 45 \beta_{11} - 89 \beta_{7} - 35 \beta_{4} - 4 \beta_{3} + 65 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-6 \beta_{12} - 109 \beta_{10} + 294 \beta_{9} + 6 \beta_{8} - 38 \beta_{6} - 11 \beta_{5} + 154 \beta_{2} - 49$$ $$\nu^{9}$$ $$=$$ $$63 \beta_{15} - 426 \beta_{14} + 389 \beta_{13} - 129 \beta_{11} - 381 \beta_{7} + 3 \beta_{4} - 285 \beta_{3} + 60 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-360 \beta_{12} - 867 \beta_{10} + 164 \beta_{9} - 744 \beta_{8} - 3 \beta_{6} + 102 \beta_{5} + 441 \beta_{2} + 594$$ $$\nu^{11}$$ $$=$$ $$471 \beta_{15} - 809 \beta_{14} + 1593 \beta_{13} + 2235 \beta_{11} - 612 \beta_{7} + 1380 \beta_{4} - 507 \beta_{3} + 2058 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$1042 \beta_{12} - 512 \beta_{10} - 1083 \beta_{9} - 5507 \beta_{8} + 1173 \beta_{6} + 875 \beta_{5} + 2670 \beta_{2} + 4014$$ $$\nu^{13}$$ $$=$$ $$-437 \beta_{15} - 3337 \beta_{14} + 8644 \beta_{13} + 6996 \beta_{11} - 4706 \beta_{7} + 2753 \beta_{4} + 13080 \beta_{3} + 19038 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-1021 \beta_{12} - 4853 \beta_{10} - 18728 \beta_{9} - 17355 \beta_{8} - 2656 \beta_{6} + 14916 \beta_{5} + 23744 \beta_{2} - 35425$$ $$\nu^{15}$$ $$=$$ $$-30996 \beta_{15} - 34848 \beta_{14} + 81152 \beta_{13} - 20644 \beta_{11} - 37568 \beta_{7} + 33608 \beta_{4} + 36243 \beta_{3} + 4436 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$$.

 $$n$$ $$231$$ $$277$$ $$281$$ $$\chi(n)$$ $$1$$ $$-\beta_{9}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1
 −2.19448 + 0.429243i 2.19448 − 0.429243i −2.23266 − 0.123447i 2.23266 + 0.123447i −1.88618 − 1.20098i 1.88618 + 1.20098i −1.06856 + 1.96422i 1.06856 − 1.96422i −2.19448 − 0.429243i 2.19448 + 0.429243i −2.23266 + 0.123447i 2.23266 − 0.123447i −1.88618 + 1.20098i 1.88618 − 1.20098i −1.06856 − 1.96422i 1.06856 + 1.96422i
0 −2.36693 + 2.36693i 0 −2.19448 0.429243i 0 −2.93008 + 2.93008i 0 8.20472i 0
137.2 0 −2.36693 + 2.36693i 0 2.19448 + 0.429243i 0 2.93008 2.93008i 0 8.20472i 0
137.3 0 0.237125 0.237125i 0 −2.23266 + 0.123447i 0 1.69421 1.69421i 0 2.88754i 0
137.4 0 0.237125 0.237125i 0 2.23266 0.123447i 0 −1.69421 + 1.69421i 0 2.88754i 0
137.5 0 1.09396 1.09396i 0 −1.88618 + 1.20098i 0 −2.67082 + 2.67082i 0 0.606488i 0
137.6 0 1.09396 1.09396i 0 1.88618 1.20098i 0 2.67082 2.67082i 0 0.606488i 0
137.7 0 2.03584 2.03584i 0 −1.06856 1.96422i 0 0.641104 0.641104i 0 5.28931i 0
137.8 0 2.03584 2.03584i 0 1.06856 + 1.96422i 0 −0.641104 + 0.641104i 0 5.28931i 0
413.1 0 −2.36693 2.36693i 0 −2.19448 + 0.429243i 0 −2.93008 2.93008i 0 8.20472i 0
413.2 0 −2.36693 2.36693i 0 2.19448 0.429243i 0 2.93008 + 2.93008i 0 8.20472i 0
413.3 0 0.237125 + 0.237125i 0 −2.23266 0.123447i 0 1.69421 + 1.69421i 0 2.88754i 0
413.4 0 0.237125 + 0.237125i 0 2.23266 + 0.123447i 0 −1.69421 1.69421i 0 2.88754i 0
413.5 0 1.09396 + 1.09396i 0 −1.88618 1.20098i 0 −2.67082 2.67082i 0 0.606488i 0
413.6 0 1.09396 + 1.09396i 0 1.88618 + 1.20098i 0 2.67082 + 2.67082i 0 0.606488i 0
413.7 0 2.03584 + 2.03584i 0 −1.06856 + 1.96422i 0 0.641104 + 0.641104i 0 5.28931i 0
413.8 0 2.03584 + 2.03584i 0 1.06856 1.96422i 0 −0.641104 0.641104i 0 5.28931i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.i.b 16
5.b even 2 1 2300.2.i.d 16
5.c odd 4 1 inner 460.2.i.b 16
5.c odd 4 1 2300.2.i.d 16
23.b odd 2 1 inner 460.2.i.b 16
115.c odd 2 1 2300.2.i.d 16
115.e even 4 1 inner 460.2.i.b 16
115.e even 4 1 2300.2.i.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.i.b 16 1.a even 1 1 trivial
460.2.i.b 16 5.c odd 4 1 inner
460.2.i.b 16 23.b odd 2 1 inner
460.2.i.b 16 115.e even 4 1 inner
2300.2.i.d 16 5.b even 2 1
2300.2.i.d 16 5.c odd 4 1
2300.2.i.d 16 115.c odd 2 1
2300.2.i.d 16 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(460, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 25 - 130 T + 338 T^{2} - 270 T^{3} + 110 T^{4} - 6 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$5$ $$390625 - 281250 T^{2} + 91250 T^{4} - 19950 T^{6} + 3934 T^{8} - 798 T^{10} + 146 T^{12} - 18 T^{14} + T^{16}$$
$7$ $$1336336 + 2029264 T^{4} + 76792 T^{8} + 532 T^{12} + T^{16}$$
$11$ $$( 2500 + 2596 T^{2} + 672 T^{4} + 50 T^{6} + T^{8} )^{2}$$
$13$ $$( 13225 - 15870 T + 9522 T^{2} - 2898 T^{3} + 486 T^{4} - 42 T^{5} + 18 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$17$ $$16 + 10288 T^{4} + 71020 T^{8} + 880 T^{12} + T^{16}$$
$19$ $$( 100 - 244 T^{2} + 178 T^{4} - 36 T^{6} + T^{8} )^{2}$$
$23$ $$78310985281 - 40857905364 T + 10658584008 T^{2} - 3939041916 T^{3} + 1120483364 T^{4} - 218275980 T^{5} + 60445656 T^{6} - 13578372 T^{7} + 2322310 T^{8} - 590364 T^{9} + 114264 T^{10} - 17940 T^{11} + 4004 T^{12} - 612 T^{13} + 72 T^{14} - 12 T^{15} + T^{16}$$
$29$ $$( 28561 + 16680 T^{2} + 2110 T^{4} + 88 T^{6} + T^{8} )^{2}$$
$31$ $$( 11 + 10 T - 20 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$37$ $$456976 + 21869584 T^{4} + 458296 T^{8} + 1348 T^{12} + T^{16}$$
$41$ $$( -107 - 188 T + 10 T^{2} + 12 T^{3} + T^{4} )^{4}$$
$43$ $$127880620816 + 3092373184 T^{4} + 11231212 T^{8} + 6892 T^{12} + T^{16}$$
$47$ $$( 24025 + 49910 T + 51842 T^{2} + 24162 T^{3} + 6086 T^{4} + 474 T^{5} + 2 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$53$ $$533794816 + 2911069184 T^{4} + 584919936 T^{8} + 54928 T^{12} + T^{16}$$
$59$ $$( 110224 + 167728 T^{2} + 9244 T^{4} + 168 T^{6} + T^{8} )^{2}$$
$61$ $$( 100 + 3044 T^{2} + 5698 T^{4} + 224 T^{6} + T^{8} )^{2}$$
$67$ $$3675002150265616 + 2504414676496 T^{4} + 498843384 T^{8} + 38308 T^{12} + T^{16}$$
$71$ $$( 169 - 162 T - 74 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$73$ $$( 25 - 390 T + 3042 T^{2} - 3874 T^{3} + 2294 T^{4} + 1326 T^{5} + 338 T^{6} + 26 T^{7} + T^{8} )^{2}$$
$79$ $$( 115562500 - 4549996 T^{2} + 66346 T^{4} - 424 T^{6} + T^{8} )^{2}$$
$83$ $$17799252215056 + 1094073550768 T^{4} + 527018796 T^{8} + 62144 T^{12} + T^{16}$$
$89$ $$( 1144900 - 257316 T^{2} + 16786 T^{4} - 260 T^{6} + T^{8} )^{2}$$
$97$ $$583506543376 + 8112583184 T^{4} + 22546936 T^{8} + 15908 T^{12} + T^{16}$$