# Properties

 Label 288.2.v.b Level 288 Weight 2 Character orbit 288.v Analytic conductor 2.300 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 288.v (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: 8.0.18939904.2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{4} + ( \beta_{6} + \beta_{7} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7} + ( \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( \beta_{2} - \beta_{3} + \beta_{6} ) q^{4} + ( \beta_{6} + \beta_{7} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{7} + ( \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{8} + ( \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{10} + ( -1 + \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{13} + ( -2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{14} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{16} + ( -2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{20} + ( -\beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{22} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{23} + ( -1 + 3 \beta_{5} + \beta_{7} ) q^{25} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{26} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} - \beta_{7} ) q^{28} + ( 2 - 4 \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( 4 - 2 \beta_{5} - 2 \beta_{6} ) q^{31} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{32} + ( -2 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{34} + ( -1 - 2 \beta_{2} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{35} + ( -2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 3 \beta_{5} ) q^{38} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{40} + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{41} + ( -1 - \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{43} + ( -2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{7} ) q^{44} + ( 2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{46} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{7} ) q^{47} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{49} + ( 3 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{50} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} ) q^{52} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{53} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{55} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{56} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{58} + ( 3 - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{59} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{61} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{62} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{64} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( -6 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} ) q^{67} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 6 \beta_{7} ) q^{68} + ( -\beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{7} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{71} + ( -5 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} - 3 \beta_{7} ) q^{73} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} + 7 \beta_{7} ) q^{74} + ( -2 + \beta_{1} - \beta_{3} + 5 \beta_{5} + \beta_{6} ) q^{76} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{77} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{79} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{7} ) q^{80} + ( -4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{82} + ( -4 - 3 \beta_{1} + 3 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{83} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{85} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{86} + ( 4 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{88} + ( -1 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{89} + ( 5 + 6 \beta_{3} - 5 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 2 + \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{92} + ( -4 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{94} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{95} + ( 4 + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) q^{97} + ( 4 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} + 4q^{4} - 8q^{7} + 4q^{8} + O(q^{10})$$ $$8q + 4q^{2} + 4q^{4} - 8q^{7} + 4q^{8} - 4q^{11} - 8q^{13} - 12q^{14} + 4q^{19} - 4q^{20} + 4q^{22} + 8q^{23} - 8q^{25} + 20q^{26} - 16q^{28} + 32q^{31} + 24q^{32} - 16q^{35} - 8q^{37} - 8q^{38} + 16q^{40} - 8q^{41} - 12q^{43} - 20q^{44} + 12q^{46} - 16q^{50} + 12q^{52} - 8q^{53} - 16q^{55} - 8q^{56} - 12q^{58} + 20q^{59} + 24q^{61} + 24q^{62} - 8q^{64} - 36q^{67} - 16q^{68} - 8q^{70} + 24q^{71} - 32q^{73} - 8q^{74} - 20q^{76} - 16q^{77} - 8q^{80} - 20q^{82} - 20q^{83} + 8q^{85} - 4q^{86} + 8q^{88} + 16q^{89} + 40q^{91} + 16q^{92} - 24q^{94} + 8q^{95} + 32q^{97} + 24q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{7} - 7 \nu^{6} + 24 \nu^{5} - 42 \nu^{4} + 59 \nu^{3} - 48 \nu^{2} + 24 \nu - 5$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{7} + 7 \nu^{6} - 24 \nu^{5} + 43 \nu^{4} - 61 \nu^{3} + 54 \nu^{2} - 29 \nu + 8$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{7} + 10 \nu^{6} - 35 \nu^{5} + 60 \nu^{4} - 87 \nu^{3} + 73 \nu^{2} - 42 \nu + 11$$ $$\beta_{4}$$ $$=$$ $$-3 \nu^{7} + 11 \nu^{6} - 38 \nu^{5} + 70 \nu^{4} - 102 \nu^{3} + 91 \nu^{2} - 53 \nu + 13$$ $$\beta_{5}$$ $$=$$ $$5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19$$ $$\beta_{6}$$ $$=$$ $$-5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23$$ $$\beta_{7}$$ $$=$$ $$-8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} - \beta_{6} + 7 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} + 11 \beta_{4} - \beta_{3} - 5 \beta_{2} + 9 \beta_{1} + 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-13 \beta_{7} - 11 \beta_{6} - 29 \beta_{5} + 17 \beta_{4} - 23 \beta_{3} + 13 \beta_{2} - 3 \beta_{1} + 18$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-67 \beta_{7} + 59 \beta_{6} - 41 \beta_{5} - 29 \beta_{4} - 15 \beta_{3} + 37 \beta_{2} - 47 \beta_{1} - 48$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{7} + 113 \beta_{6} + 97 \beta_{5} - 105 \beta_{4} + 91 \beta_{3} - 31 \beta_{2} - 39 \beta_{1} - 122$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-\beta_{5}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.691860i 0.5 + 2.10607i 0.5 + 0.691860i 0.5 − 2.10607i 0.5 − 1.44392i 0.5 + 0.0297061i 0.5 + 1.44392i 0.5 − 0.0297061i
0.443806 1.34277i 0 −1.60607 1.19186i −0.707107 0.292893i 0 −2.27133 2.27133i −2.31318 + 1.62764i 0 −0.707107 + 0.819496i
37.2 1.26330 + 0.635665i 0 1.19186 + 1.60607i −0.707107 0.292893i 0 1.68554 + 1.68554i 0.484753 + 2.78658i 0 −0.707107 0.819496i
109.1 0.443806 + 1.34277i 0 −1.60607 + 1.19186i −0.707107 + 0.292893i 0 −2.27133 + 2.27133i −2.31318 1.62764i 0 −0.707107 0.819496i
109.2 1.26330 0.635665i 0 1.19186 1.60607i −0.707107 + 0.292893i 0 1.68554 1.68554i 0.484753 2.78658i 0 −0.707107 + 0.819496i
181.1 −1.11137 + 0.874559i 0 0.470294 1.94392i 0.707107 1.70711i 0 −0.665096 0.665096i 1.17740 + 2.57172i 0 0.707107 + 2.51564i
181.2 1.40426 0.167452i 0 1.94392 0.470294i 0.707107 1.70711i 0 −2.74912 2.74912i 2.65103 0.985930i 0 0.707107 2.51564i
253.1 −1.11137 0.874559i 0 0.470294 + 1.94392i 0.707107 + 1.70711i 0 −0.665096 + 0.665096i 1.17740 2.57172i 0 0.707107 2.51564i
253.2 1.40426 + 0.167452i 0 1.94392 + 0.470294i 0.707107 + 1.70711i 0 −2.74912 + 2.74912i 2.65103 + 0.985930i 0 0.707107 + 2.51564i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.v.b 8
3.b odd 2 1 32.2.g.b 8
4.b odd 2 1 1152.2.v.b 8
12.b even 2 1 128.2.g.b 8
15.d odd 2 1 800.2.y.b 8
15.e even 4 1 800.2.ba.c 8
15.e even 4 1 800.2.ba.d 8
24.f even 2 1 256.2.g.c 8
24.h odd 2 1 256.2.g.d 8
32.g even 8 1 inner 288.2.v.b 8
32.h odd 8 1 1152.2.v.b 8
48.i odd 4 1 512.2.g.e 8
48.i odd 4 1 512.2.g.h 8
48.k even 4 1 512.2.g.f 8
48.k even 4 1 512.2.g.g 8
96.o even 8 1 128.2.g.b 8
96.o even 8 1 256.2.g.c 8
96.o even 8 1 512.2.g.f 8
96.o even 8 1 512.2.g.g 8
96.p odd 8 1 32.2.g.b 8
96.p odd 8 1 256.2.g.d 8
96.p odd 8 1 512.2.g.e 8
96.p odd 8 1 512.2.g.h 8
192.q odd 16 2 4096.2.a.k 8
192.s even 16 2 4096.2.a.q 8
480.br even 8 1 800.2.ba.c 8
480.bu odd 8 1 800.2.y.b 8
480.cb even 8 1 800.2.ba.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 3.b odd 2 1
32.2.g.b 8 96.p odd 8 1
128.2.g.b 8 12.b even 2 1
128.2.g.b 8 96.o even 8 1
256.2.g.c 8 24.f even 2 1
256.2.g.c 8 96.o even 8 1
256.2.g.d 8 24.h odd 2 1
256.2.g.d 8 96.p odd 8 1
288.2.v.b 8 1.a even 1 1 trivial
288.2.v.b 8 32.g even 8 1 inner
512.2.g.e 8 48.i odd 4 1
512.2.g.e 8 96.p odd 8 1
512.2.g.f 8 48.k even 4 1
512.2.g.f 8 96.o even 8 1
512.2.g.g 8 48.k even 4 1
512.2.g.g 8 96.o even 8 1
512.2.g.h 8 48.i odd 4 1
512.2.g.h 8 96.p odd 8 1
800.2.y.b 8 15.d odd 2 1
800.2.y.b 8 480.bu odd 8 1
800.2.ba.c 8 15.e even 4 1
800.2.ba.c 8 480.br even 8 1
800.2.ba.d 8 15.e even 4 1
800.2.ba.d 8 480.cb even 8 1
1152.2.v.b 8 4.b odd 2 1
1152.2.v.b 8 32.h odd 8 1
4096.2.a.k 8 192.q odd 16 2
4096.2.a.q 8 192.s even 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 2 T_{5}^{2} + 4 T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 8 T^{5} + 24 T^{6} - 32 T^{7} + 16 T^{8}$$
$3$ 1
$5$ $$( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 80 T^{5} + 50 T^{6} + 625 T^{8} )^{2}$$
$7$ $$1 + 8 T + 32 T^{2} + 104 T^{3} + 252 T^{4} + 392 T^{5} + 480 T^{6} + 168 T^{7} - 1274 T^{8} + 1176 T^{9} + 23520 T^{10} + 134456 T^{11} + 605052 T^{12} + 1747928 T^{13} + 3764768 T^{14} + 6588344 T^{15} + 5764801 T^{16}$$
$11$ $$1 + 4 T + 8 T^{2} - 68 T^{3} - 304 T^{4} - 804 T^{5} + 1304 T^{6} + 7396 T^{7} + 43982 T^{8} + 81356 T^{9} + 157784 T^{10} - 1070124 T^{11} - 4450864 T^{12} - 10951468 T^{13} + 14172488 T^{14} + 77948684 T^{15} + 214358881 T^{16}$$
$13$ $$1 + 8 T + 36 T^{2} + 104 T^{3} + 200 T^{4} - 72 T^{5} - 4084 T^{6} - 34216 T^{7} - 153618 T^{8} - 444808 T^{9} - 690196 T^{10} - 158184 T^{11} + 5712200 T^{12} + 38614472 T^{13} + 173765124 T^{14} + 501988136 T^{15} + 815730721 T^{16}$$
$17$ $$1 - 72 T^{2} + 2620 T^{4} - 64376 T^{6} + 1215110 T^{8} - 18604664 T^{10} + 218825020 T^{12} - 1737904968 T^{14} + 6975757441 T^{16}$$
$19$ $$1 - 4 T - 8 T^{2} - 28 T^{3} + 336 T^{4} - 1308 T^{5} + 14056 T^{6} - 28228 T^{7} - 115058 T^{8} - 536332 T^{9} + 5074216 T^{10} - 8971572 T^{11} + 43787856 T^{12} - 69330772 T^{13} - 376367048 T^{14} - 3575486956 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 8 T + 32 T^{2} - 200 T^{3} + 2108 T^{4} - 11560 T^{5} + 45024 T^{6} - 277224 T^{7} + 1704454 T^{8} - 6376152 T^{9} + 23817696 T^{10} - 140650520 T^{11} + 589904828 T^{12} - 1287268600 T^{13} + 4737148448 T^{14} - 27238603576 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 12 T^{2} + 64 T^{3} + 72 T^{4} - 4416 T^{5} + 14140 T^{6} + 57344 T^{7} - 693650 T^{8} + 1662976 T^{9} + 11891740 T^{10} - 107701824 T^{11} + 50924232 T^{12} + 1312713536 T^{13} - 7137879852 T^{14} + 500246412961 T^{16}$$
$31$ $$( 1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4} )^{4}$$
$37$ $$1 + 8 T - 44 T^{2} - 760 T^{3} - 1272 T^{4} + 22872 T^{5} + 112156 T^{6} - 219496 T^{7} - 4000082 T^{8} - 8121352 T^{9} + 153541564 T^{10} + 1158535416 T^{11} - 2383932792 T^{12} - 52701407320 T^{13} - 112891961996 T^{14} + 759455017064 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 + 8 T + 32 T^{2} + 88 T^{3} + 1788 T^{4} + 23048 T^{5} + 131040 T^{6} + 748824 T^{7} + 3829318 T^{8} + 30701784 T^{9} + 220278240 T^{10} + 1588491208 T^{11} + 5052460668 T^{12} + 10195345688 T^{13} + 152003335712 T^{14} + 1558034191048 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 12 T + 56 T^{2} - 604 T^{3} - 8560 T^{4} - 59644 T^{5} - 32984 T^{6} + 2234540 T^{7} + 25213774 T^{8} + 96085220 T^{9} - 60987416 T^{10} - 4742115508 T^{11} - 29264936560 T^{12} - 88793099572 T^{13} + 353996330744 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 312 T^{2} + 44348 T^{4} - 3794696 T^{6} + 215798406 T^{8} - 8382483464 T^{10} + 216404092988 T^{12} - 3363115182648 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 + 8 T + 100 T^{2} + 424 T^{3} + 3016 T^{4} + 19768 T^{5} + 118540 T^{6} + 2296472 T^{7} + 11779502 T^{8} + 121713016 T^{9} + 332978860 T^{10} + 2943000536 T^{11} + 23797690696 T^{12} + 177314889032 T^{13} + 2216436112900 T^{14} + 9397689118696 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 20 T + 136 T^{2} + 1124 T^{3} - 27632 T^{4} + 217412 T^{5} - 194408 T^{6} - 12195572 T^{7} + 142178894 T^{8} - 719538748 T^{9} - 676734248 T^{10} + 44651859148 T^{11} - 334826919152 T^{12} + 803574912076 T^{13} + 5736552575176 T^{14} - 49773029696380 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 24 T + 132 T^{2} + 648 T^{3} + 72 T^{4} - 138600 T^{5} + 1067820 T^{6} - 2497416 T^{7} + 88430 T^{8} - 152342376 T^{9} + 3973358220 T^{10} - 31459566600 T^{11} + 996900552 T^{12} + 547298403048 T^{13} + 6800689415652 T^{14} - 75425828064504 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 36 T + 504 T^{2} + 2652 T^{3} - 10800 T^{4} - 217380 T^{5} - 1006488 T^{6} - 637596 T^{7} + 11635982 T^{8} - 42718932 T^{9} - 4518124632 T^{10} - 65379860940 T^{11} - 217632106800 T^{12} + 3580531783764 T^{13} + 45591024613176 T^{14} + 218185617791628 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 24 T + 288 T^{2} - 2904 T^{3} + 21308 T^{4} - 62712 T^{5} - 415008 T^{6} + 11818440 T^{7} - 144298362 T^{8} + 839109240 T^{9} - 2092055328 T^{10} - 22445314632 T^{11} + 541472098748 T^{12} - 5239482035304 T^{13} + 36892881769248 T^{14} - 218282883801384 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 32 T + 512 T^{2} + 7008 T^{3} + 95708 T^{4} + 1109024 T^{5} + 11042304 T^{6} + 107800160 T^{7} + 989395590 T^{8} + 7869411680 T^{9} + 58844438016 T^{10} + 431429189408 T^{11} + 2717938849628 T^{12} + 14528085723744 T^{13} + 77483123859968 T^{14} + 353516752611104 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 120 T^{2} + 10940 T^{4} + 170552 T^{6} - 23710074 T^{8} + 1064415032 T^{10} + 426113886140 T^{12} - 29170494662520 T^{14} + 1517108809906561 T^{16}$$
$83$ $$1 + 20 T + 184 T^{2} + 2684 T^{3} + 37008 T^{4} + 337916 T^{5} + 3624872 T^{6} + 38211284 T^{7} + 335969678 T^{8} + 3171536572 T^{9} + 24971743208 T^{10} + 193215975892 T^{11} + 1756337543568 T^{12} + 10572385085812 T^{13} + 60157028699896 T^{14} + 542721019792540 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 - 16 T + 128 T^{2} - 2096 T^{3} + 18652 T^{4} + 22640 T^{5} - 553088 T^{6} + 12396240 T^{7} - 224072442 T^{8} + 1103265360 T^{9} - 4381010048 T^{10} + 15960498160 T^{11} + 1170268279132 T^{12} - 11704188605104 T^{13} + 63613605243008 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16}$$
$97$ $$( 1 - 16 T + 428 T^{2} - 4368 T^{3} + 63222 T^{4} - 423696 T^{5} + 4027052 T^{6} - 14602768 T^{7} + 88529281 T^{8} )^{2}$$