# Properties

 Label 252.2.bf.g Level 252 Weight 2 Character orbit 252.bf Analytic conductor 2.012 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.562828176.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} + 2 \nu^{3} + 8 \nu^{2} + 4 \nu - 8$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 4 \nu - 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} - 4 \nu^{4} - 2 \nu^{3} + 8 \nu^{2} - 4 \nu + 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{5} + 2 \nu^{3} - 4 \nu + 8$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 4 \nu^{6} - 3 \nu^{5} - 8 \nu^{4} + 10 \nu^{3} - 12 \nu + 40$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{7} + 2 \nu^{6} + \nu^{5} - 6 \nu^{4} + 10 \nu^{3} + 4 \nu^{2} - 12 \nu + 32$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} - 2 \beta_{3} + \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$4 \beta_{6} - 2 \beta_{5} - \beta_{4} + 6 \beta_{3} + \beta_{2} - 4$$ $$\nu^{7}$$ $$=$$ $$-2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} - \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} + 8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{3}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.0777157 + 1.41208i 1.40376 + 0.171630i 0.856419 − 1.12541i −1.33790 − 0.458297i 0.0777157 − 1.41208i 1.40376 − 0.171630i 0.856419 + 1.12541i −1.33790 + 0.458297i
−1.26175 0.638735i 0 1.18404 + 1.61185i −0.380152 0.219481i 0 −2.02350 1.70453i −0.464416 2.79004i 0 0.339468 + 0.519747i
19.2 −0.850516 + 1.12988i 0 −0.553244 1.92196i −0.834598 0.481855i 0 −1.20103 + 2.35744i 2.64212 + 1.00956i 0 1.25428 0.533167i
19.3 0.546424 + 1.30439i 0 −1.40284 + 1.42549i 3.33878 + 1.92764i 0 1.59285 2.11254i −2.62594 1.05092i 0 −0.690004 + 5.40836i
19.4 1.06584 0.929502i 0 0.272050 1.98141i −2.12403 1.22631i 0 2.63169 0.272415i −1.55176 2.36475i 0 −3.40374 + 0.667235i
199.1 −1.26175 + 0.638735i 0 1.18404 1.61185i −0.380152 + 0.219481i 0 −2.02350 + 1.70453i −0.464416 + 2.79004i 0 0.339468 0.519747i
199.2 −0.850516 1.12988i 0 −0.553244 + 1.92196i −0.834598 + 0.481855i 0 −1.20103 2.35744i 2.64212 1.00956i 0 1.25428 + 0.533167i
199.3 0.546424 1.30439i 0 −1.40284 1.42549i 3.33878 1.92764i 0 1.59285 + 2.11254i −2.62594 + 1.05092i 0 −0.690004 5.40836i
199.4 1.06584 + 0.929502i 0 0.272050 + 1.98141i −2.12403 + 1.22631i 0 2.63169 + 0.272415i −1.55176 + 2.36475i 0 −3.40374 0.667235i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bf.g 8
3.b odd 2 1 84.2.o.a 8
4.b odd 2 1 252.2.bf.f 8
7.c even 3 1 1764.2.b.i 8
7.d odd 6 1 252.2.bf.f 8
7.d odd 6 1 1764.2.b.j 8
12.b even 2 1 84.2.o.b yes 8
21.c even 2 1 588.2.o.d 8
21.g even 6 1 84.2.o.b yes 8
21.g even 6 1 588.2.b.a 8
21.h odd 6 1 588.2.b.b 8
21.h odd 6 1 588.2.o.b 8
24.f even 2 1 1344.2.bl.i 8
24.h odd 2 1 1344.2.bl.j 8
28.f even 6 1 inner 252.2.bf.g 8
28.f even 6 1 1764.2.b.i 8
28.g odd 6 1 1764.2.b.j 8
84.h odd 2 1 588.2.o.b 8
84.j odd 6 1 84.2.o.a 8
84.j odd 6 1 588.2.b.b 8
84.n even 6 1 588.2.b.a 8
84.n even 6 1 588.2.o.d 8
168.ba even 6 1 1344.2.bl.i 8
168.be odd 6 1 1344.2.bl.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 3.b odd 2 1
84.2.o.a 8 84.j odd 6 1
84.2.o.b yes 8 12.b even 2 1
84.2.o.b yes 8 21.g even 6 1
252.2.bf.f 8 4.b odd 2 1
252.2.bf.f 8 7.d odd 6 1
252.2.bf.g 8 1.a even 1 1 trivial
252.2.bf.g 8 28.f even 6 1 inner
588.2.b.a 8 21.g even 6 1
588.2.b.a 8 84.n even 6 1
588.2.b.b 8 21.h odd 6 1
588.2.b.b 8 84.j odd 6 1
588.2.o.b 8 21.h odd 6 1
588.2.o.b 8 84.h odd 2 1
588.2.o.d 8 21.c even 2 1
588.2.o.d 8 84.n even 6 1
1344.2.bl.i 8 24.f even 2 1
1344.2.bl.i 8 168.ba even 6 1
1344.2.bl.j 8 24.h odd 2 1
1344.2.bl.j 8 168.be odd 6 1
1764.2.b.i 8 7.c even 3 1
1764.2.b.i 8 28.f even 6 1
1764.2.b.j 8 7.d odd 6 1
1764.2.b.j 8 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}^{8} - 11 T_{5}^{6} + 125 T_{5}^{4} + 264 T_{5}^{3} + 236 T_{5}^{2} + 96 T_{5} + 16$$ $$T_{11}^{8} + \cdots$$ $$T_{19}^{8} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + 2 T^{3} + 6 T^{4} + 4 T^{5} + 4 T^{6} + 8 T^{7} + 16 T^{8}$$
$3$ 
$5$ $$1 + 9 T^{2} + 45 T^{4} - 96 T^{5} + 66 T^{6} - 864 T^{7} - 394 T^{8} - 4320 T^{9} + 1650 T^{10} - 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16}$$
$7$ $$1 - 2 T - 16 T^{3} + 65 T^{4} - 112 T^{5} - 686 T^{7} + 2401 T^{8}$$
$11$ $$( 1 - 2 T + 25 T^{2} - 58 T^{3} + 372 T^{4} - 638 T^{5} + 3025 T^{6} - 2662 T^{7} + 14641 T^{8} )( 1 + 8 T + 34 T^{2} + 112 T^{3} + 339 T^{4} + 1232 T^{5} + 4114 T^{6} + 10648 T^{7} + 14641 T^{8} )$$
$13$ $$1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16}$$
$17$ $$1 + 40 T^{2} + 786 T^{4} - 1104 T^{5} + 10208 T^{6} - 39072 T^{7} + 129731 T^{8} - 664224 T^{9} + 2950112 T^{10} - 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16}$$
$19$ $$1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 225492 T^{9} - 12339702 T^{10} + 13498512 T^{11} + 151823965 T^{12} - 371414850 T^{13} - 1552514073 T^{14} + 5363230434 T^{15} + 16983563041 T^{16}$$
$23$ $$1 + 52 T^{2} + 1242 T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 3228096 T^{9} + 12738320 T^{10} - 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 - 8 T + 71 T^{2} - 344 T^{3} + 1924 T^{4} - 9976 T^{5} + 59711 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2}$$
$31$ $$1 - 6 T - 4 T^{2} + 336 T^{3} - 2729 T^{4} + 10764 T^{5} + 2216 T^{6} - 444234 T^{7} + 3877768 T^{8} - 13771254 T^{9} + 2129576 T^{10} + 320670324 T^{11} - 2520288809 T^{12} + 9619394736 T^{13} - 3550014724 T^{14} - 165075684666 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16}$$
$43$ $$1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16}$$
$47$ $$1 + 4 T - 144 T^{2} - 456 T^{3} + 12722 T^{4} + 27948 T^{5} - 805600 T^{6} - 556940 T^{7} + 41968563 T^{8} - 26176180 T^{9} - 1779570400 T^{10} + 2901645204 T^{11} + 62079301682 T^{12} - 104581323192 T^{13} - 1552207007376 T^{14} + 2026492481852 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 4 T - 135 T^{2} + 900 T^{3} + 9413 T^{4} - 70368 T^{5} - 312982 T^{6} + 1938608 T^{7} + 10598262 T^{8} + 102746224 T^{9} - 879166438 T^{10} - 10476176736 T^{11} + 74273097653 T^{12} + 376375943700 T^{13} - 2992188752415 T^{14} - 4698844559348 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 14 T - 13 T^{2} + 1110 T^{3} - 3463 T^{4} - 11848 T^{5} - 87914 T^{6} - 1416852 T^{7} + 32978194 T^{8} - 83594268 T^{9} - 306028634 T^{10} - 2433330392 T^{11} - 41962421143 T^{12} + 793565971890 T^{13} - 548346937333 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 - 42 T + 1023 T^{2} - 18270 T^{3} + 261141 T^{4} - 3133152 T^{5} + 32837970 T^{6} - 307895844 T^{7} + 2631022010 T^{8} - 20629021548 T^{9} + 147409647330 T^{10} - 942336194976 T^{11} + 5262283889061 T^{12} - 24666785704890 T^{13} + 92538924958887 T^{14} - 254549887423566 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16}$$
$73$ $$1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 6 T + 132 T^{2} + 720 T^{3} + 9999 T^{4} + 52644 T^{5} - 314880 T^{6} + 1496442 T^{7} - 48671848 T^{8} + 118218918 T^{9} - 1965166080 T^{10} + 25955545116 T^{11} + 389461859919 T^{12} + 2215480607280 T^{13} + 32087544128772 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16}$$
$83$ $$( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 66566 T^{5} + 1577581 T^{6} + 1143574 T^{7} + 47458321 T^{8} )^{2}$$
$89$ $$1 + 264 T^{2} + 38610 T^{4} - 50400 T^{5} + 4052064 T^{6} - 8952768 T^{7} + 353995811 T^{8} - 796796352 T^{9} + 32096398944 T^{10} - 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16}$$
$97$ $$1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16}$$