# Properties

 Label 252.2.bf.f 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 Defining polynomial: $$x^{8} - 2 x^{7} + x^{6} + 2 x^{5} - 6 x^{4} + 4 x^{3} + 4 x^{2} - 16 x + 16$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{4} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{7} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})$$ $$q + \beta_{5} q^{2} + ( -\beta_{4} + \beta_{6} - \beta_{7} ) q^{4} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{5} + ( \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_{4} - \beta_{6} + \beta_{7} ) q^{10} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{13} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{14} + ( 2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{16} + ( 2 \beta_{1} + 2 \beta_{5} - 2 \beta_{6} ) q^{17} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} - \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_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{23} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{25} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{26} + ( 2 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{28} + ( 2 - 3 \beta_{1} + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + \beta_{7} ) q^{29} + ( -3 + 4 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{31} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{32} + ( 4 - 2 \beta_{2} - 2 \beta_{4} ) q^{34} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{35} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{38} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \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} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{44} + ( -4 \beta_{3} - 4 \beta_{4} ) q^{46} + ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{47} + ( 1 - 5 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \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_{4} - 2 \beta_{6} ) q^{52} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( 2 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{55} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{56} + ( -6 + \beta_{2} + 6 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{58} + ( -2 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{59} + ( 4 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} ) 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 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( -3 - 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{67} + ( 4 + 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{68} + ( 2 - 2 \beta_{1} - \beta_{2} + 6 \beta_{3} - 2 \beta_{5} + 3 \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} + ( -1 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{73} + ( 6 + \beta_{1} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{74} + ( -4 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{76} + ( -2 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{79} + ( 2 + 2 \beta_{1} + 4 \beta_{3} - \beta_{4} + 5 \beta_{6} - 3 \beta_{7} ) q^{80} + ( 4 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) 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} + ( 6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{86} + ( -2 + 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{88} + ( 2 \beta_{1} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 4 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 8 - 4 \beta_{1} - 8 \beta_{3} - 4 \beta_{5} ) q^{92} + ( -4 + 2 \beta_{1} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{94} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{95} + ( 2 - \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{97} + ( 2 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 5 \beta_{6} - 3 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - q^{2} - q^{4} - 2q^{7} - 4q^{8} + O(q^{10})$$ $$8q - q^{2} - q^{4} - 2q^{7} - 4q^{8} - 13q^{10} + 6q^{11} + 10q^{14} + 7q^{16} + 6q^{19} + 22q^{20} - 6q^{22} + 2q^{25} - 12q^{26} - 7q^{28} + 16q^{29} - 6q^{31} - 21q^{32} + 28q^{34} - 12q^{35} + 6q^{37} - 8q^{38} - 13q^{40} - 19q^{44} - 12q^{46} + 4q^{47} + 4q^{49} - 2q^{50} + 20q^{52} + 4q^{53} + 8q^{55} + q^{56} - 23q^{58} - 14q^{59} + 12q^{61} - 48q^{62} + 2q^{64} - 4q^{65} - 42q^{67} + 10q^{68} + 35q^{70} - 18q^{73} + 28q^{74} - 44q^{76} - 8q^{77} + 6q^{79} + 33q^{80} - 14q^{82} + 4q^{83} - 32q^{85} + 42q^{86} + 11q^{88} + 34q^{91} + 28q^{92} - 16q^{94} - 24q^{95} + 19q^{98} + O(q^{100})$$

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 $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

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

## 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
 0.856419 + 1.12541i 1.40376 − 0.171630i −1.33790 + 0.458297i 0.0777157 − 1.41208i 0.856419 − 1.12541i 1.40376 + 0.171630i −1.33790 − 0.458297i 0.0777157 + 1.41208i
−1.40284 + 0.178976i 0 1.93594 0.502151i 3.33878 + 1.92764i 0 −1.59285 + 2.11254i −2.62594 + 1.05092i 0 −5.02878 2.10662i
19.2 −0.553244 + 1.30151i 0 −1.38784 1.44010i −0.834598 0.481855i 0 1.20103 2.35744i 2.64212 1.00956i 0 1.08887 0.819652i
19.3 0.272050 1.38780i 0 −1.85198 0.755103i −2.12403 1.22631i 0 −2.63169 + 0.272415i −1.55176 + 2.36475i 0 −2.27971 + 2.61411i
19.4 1.18404 + 0.773342i 0 0.803884 + 1.83133i −0.380152 0.219481i 0 2.02350 + 1.70453i −0.464416 + 2.79004i 0 −0.280380 0.553861i
199.1 −1.40284 0.178976i 0 1.93594 + 0.502151i 3.33878 1.92764i 0 −1.59285 2.11254i −2.62594 1.05092i 0 −5.02878 + 2.10662i
199.2 −0.553244 1.30151i 0 −1.38784 + 1.44010i −0.834598 + 0.481855i 0 1.20103 + 2.35744i 2.64212 + 1.00956i 0 1.08887 + 0.819652i
199.3 0.272050 + 1.38780i 0 −1.85198 + 0.755103i −2.12403 + 1.22631i 0 −2.63169 0.272415i −1.55176 2.36475i 0 −2.27971 2.61411i
199.4 1.18404 0.773342i 0 0.803884 1.83133i −0.380152 + 0.219481i 0 2.02350 1.70453i −0.464416 2.79004i 0 −0.280380 + 0.553861i
 $$n$$: 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.f 8
3.b odd 2 1 84.2.o.b yes 8
4.b odd 2 1 252.2.bf.g 8
7.c even 3 1 1764.2.b.j 8
7.d odd 6 1 252.2.bf.g 8
7.d odd 6 1 1764.2.b.i 8
12.b even 2 1 84.2.o.a 8
21.c even 2 1 588.2.o.b 8
21.g even 6 1 84.2.o.a 8
21.g even 6 1 588.2.b.b 8
21.h odd 6 1 588.2.b.a 8
21.h odd 6 1 588.2.o.d 8
24.f even 2 1 1344.2.bl.j 8
24.h odd 2 1 1344.2.bl.i 8
28.f even 6 1 inner 252.2.bf.f 8
28.f even 6 1 1764.2.b.j 8
28.g odd 6 1 1764.2.b.i 8
84.h odd 2 1 588.2.o.d 8
84.j odd 6 1 84.2.o.b yes 8
84.j odd 6 1 588.2.b.a 8
84.n even 6 1 588.2.b.b 8
84.n even 6 1 588.2.o.b 8
168.ba even 6 1 1344.2.bl.j 8
168.be odd 6 1 1344.2.bl.i 8

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

$p$ $F_p(T)$
$2$ $$16 + 8 T + 4 T^{2} + 4 T^{3} + 2 T^{5} + T^{6} + T^{7} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$16 + 96 T + 236 T^{2} + 264 T^{3} + 125 T^{4} - 11 T^{6} + T^{8}$$
$7$ $$2401 + 686 T + 112 T^{3} + 65 T^{4} + 16 T^{5} + 2 T^{7} + T^{8}$$
$11$ $$400 + 240 T - 212 T^{2} - 156 T^{3} + 125 T^{4} + 78 T^{5} - T^{6} - 6 T^{7} + T^{8}$$
$13$ $$256 + 1936 T^{2} + 473 T^{4} + 38 T^{6} + T^{8}$$
$17$ $$1024 - 1536 T - 128 T^{2} + 1344 T^{3} + 752 T^{4} - 28 T^{6} + T^{8}$$
$19$ $$16 + 240 T + 3628 T^{2} - 372 T^{3} + 405 T^{4} - 78 T^{5} + 43 T^{6} - 6 T^{7} + T^{8}$$
$23$ $$16384 + 12288 T - 2048 T^{2} - 3840 T^{3} + 1472 T^{4} - 40 T^{6} + T^{8}$$
$29$ $$( -512 + 352 T - 45 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$4173849 - 551610 T + 244512 T^{2} - 1836 T^{3} + 6633 T^{4} + 36 T^{5} + 120 T^{6} + 6 T^{7} + T^{8}$$
$37$ $$355216 - 214560 T + 103972 T^{2} - 22632 T^{3} + 4605 T^{4} - 462 T^{5} + 79 T^{6} - 6 T^{7} + T^{8}$$
$41$ $$350464 + 311552 T^{2} + 14048 T^{4} + 208 T^{6} + T^{8}$$
$43$ $$1073296 + 140152 T^{2} + 6593 T^{4} + 134 T^{6} + T^{8}$$
$47$ $$4096 + 1024 T + 2048 T^{2} + 64 T^{3} + 784 T^{4} + 80 T^{5} + 44 T^{6} - 4 T^{7} + T^{8}$$
$53$ $$64 + 928 T + 12968 T^{2} + 7012 T^{3} + 3265 T^{4} + 476 T^{5} + 77 T^{6} - 4 T^{7} + T^{8}$$
$59$ $$1420864 - 548320 T + 243784 T^{2} - 20956 T^{3} + 5977 T^{4} + 542 T^{5} + 223 T^{6} + 14 T^{7} + T^{8}$$
$61$ $$1048576 - 1572864 T + 901120 T^{2} - 172032 T^{3} + 7424 T^{4} + 1344 T^{5} - 64 T^{6} - 12 T^{7} + T^{8}$$
$67$ $$4129024 - 146304 T - 337616 T^{2} + 12024 T^{3} + 30929 T^{4} + 7014 T^{5} + 755 T^{6} + 42 T^{7} + T^{8}$$
$71$ $$200704 + 173312 T^{2} + 19856 T^{4} + 280 T^{6} + T^{8}$$
$73$ $$952576 + 1124352 T + 494096 T^{2} + 61056 T^{3} - 3127 T^{4} - 954 T^{5} + 55 T^{6} + 18 T^{7} + T^{8}$$
$79$ $$241081 + 67758 T - 89888 T^{2} - 27048 T^{3} + 37649 T^{4} + 1176 T^{5} - 184 T^{6} - 6 T^{7} + T^{8}$$
$83$ $$( 196 - 304 T - 103 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$89$ $$4096 + 18432 T + 33536 T^{2} + 26496 T^{3} + 8528 T^{4} - 92 T^{6} + T^{8}$$
$97$ $$246016 + 86176 T^{2} + 8249 T^{4} + 182 T^{6} + T^{8}$$