# Properties

 Label 91.2.f.c Level $91$ Weight $2$ Character orbit 91.f Analytic conductor $0.727$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.f (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 7 x^{6} + 38 x^{4} - 16 x^{3} + 15 x^{2} + 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{6} + ( 1 + \beta_{4} ) q^{7} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( -2 - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{6} + ( 1 + \beta_{4} ) q^{7} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( -2 - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + ( -\beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{10} + \beta_{5} q^{11} + ( 3 - \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{12} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} -\beta_{3} q^{14} + ( -\beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{15} + ( -4 \beta_{1} + 4 \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( -1 - 2 \beta_{3} - \beta_{6} ) q^{18} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{19} + ( 1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} -\beta_{6} q^{21} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{22} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{23} + ( 4 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{24} + ( 2 - \beta_{2} + 3 \beta_{3} ) q^{25} + ( -3 \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{26} + ( -7 + \beta_{3} + 3 \beta_{6} ) q^{27} + ( \beta_{1} - \beta_{4} - \beta_{7} ) q^{28} + ( -\beta_{5} - \beta_{7} ) q^{29} + ( 1 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{30} + ( -1 + \beta_{2} ) q^{31} + ( 7 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{32} + ( 5 + \beta_{2} + 5 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( -\beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{34} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{35} + ( -9 \beta_{4} - 2 \beta_{5} ) q^{36} + ( \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{37} + ( -5 + 3 \beta_{2} + 4 \beta_{3} + \beta_{6} ) q^{38} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{39} + ( -4 + \beta_{2} - \beta_{6} ) q^{40} + ( -2 \beta_{1} - 6 \beta_{4} + \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{4} ) q^{42} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{43} + ( -3 + \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{44} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{45} + ( -6 - \beta_{1} + \beta_{2} - \beta_{3} - 6 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{46} + ( 1 + 3 \beta_{3} + \beta_{6} ) q^{47} + ( -1 + 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{48} + \beta_{4} q^{49} + ( 2 \beta_{1} + 11 \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{50} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{51} + ( 8 + 4 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{52} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} ) q^{53} + ( -5 \beta_{1} + \beta_{7} ) q^{54} + ( \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{55} + ( -1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{56} + ( -5 - 3 \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{57} + ( 3 + 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{59} + ( -4 - 2 \beta_{3} + 3 \beta_{6} ) q^{60} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{62} + ( -2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{63} + ( 1 - 2 \beta_{2} - 10 \beta_{3} ) q^{64} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{65} + ( 1 - \beta_{3} + \beta_{6} ) q^{66} + ( 6 \beta_{1} - \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{67} + ( 4 \beta_{1} - 7 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{68} + ( 4 + 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 3 - \beta_{2} + \beta_{3} ) q^{70} + ( 4 - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{72} + ( -2 - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{73} + ( -4 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - \beta_{7} ) q^{74} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{75} + ( -11 \beta_{1} + 5 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{76} + \beta_{6} q^{77} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{78} + ( -6 + \beta_{2} - \beta_{3} - 3 \beta_{6} ) q^{79} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{80} + ( \beta_{1} + 8 \beta_{4} + 7 \beta_{5} ) q^{81} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{82} + ( -1 - \beta_{2} - 4 \beta_{3} ) q^{83} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{84} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{85} + ( 3 - \beta_{2} ) q^{86} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{87} + ( -4 \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{88} + ( -\beta_{1} + 2 \beta_{5} - \beta_{7} ) q^{89} + ( 7 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{6} ) q^{90} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{91} + ( 4 + \beta_{2} + 7 \beta_{3} + 3 \beta_{6} ) q^{92} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{93} + ( -\beta_{1} + 8 \beta_{4} + 3 \beta_{7} ) q^{94} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{95} + ( -13 + \beta_{2} + 6 \beta_{3} ) q^{96} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 5 \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{97} + ( -\beta_{1} - \beta_{3} ) q^{98} + ( 7 - \beta_{3} - 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + q^{2} - q^{3} - 5q^{4} - 14q^{5} + 5q^{6} + 4q^{7} - 12q^{8} - 7q^{9} + O(q^{10})$$ $$8q + q^{2} - q^{3} - 5q^{4} - 14q^{5} + 5q^{6} + 4q^{7} - 12q^{8} - 7q^{9} + 11q^{10} + q^{11} + 24q^{12} + 4q^{13} + 2q^{14} - 3q^{15} - 19q^{16} + 4q^{17} - 6q^{18} - q^{19} + 2q^{20} - 2q^{21} - 5q^{22} + 2q^{23} + 3q^{24} + 10q^{25} + 12q^{26} - 52q^{27} + 5q^{28} - q^{29} + 4q^{30} - 8q^{31} + 33q^{32} + 19q^{33} + 6q^{34} - 7q^{35} + 34q^{36} + 10q^{37} - 46q^{38} + 20q^{39} - 34q^{40} + 22q^{41} - 5q^{42} - 3q^{43} - 24q^{44} + 11q^{45} - 24q^{46} + 4q^{47} - 11q^{48} - 4q^{49} - 43q^{50} + 14q^{51} + 65q^{52} + 4q^{53} - 5q^{54} + 3q^{55} - 6q^{56} - 34q^{57} + 11q^{58} + 8q^{59} - 22q^{60} - 8q^{61} + 5q^{62} + 7q^{63} + 28q^{64} + 7q^{65} + 12q^{66} + 6q^{67} + 33q^{68} + 18q^{69} + 22q^{70} + 14q^{71} - 5q^{72} - 16q^{73} - 20q^{74} + 7q^{75} - 32q^{76} + 2q^{77} - q^{78} - 52q^{79} - 7q^{80} - 24q^{81} + 14q^{82} + 12q^{84} - 5q^{85} + 24q^{86} - 13q^{87} - 3q^{88} + q^{89} + 52q^{90} - 4q^{91} + 24q^{92} + 7q^{93} - 33q^{94} - 21q^{95} - 116q^{96} - 3q^{97} + q^{98} + 46q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 7 x^{6} + 38 x^{4} - 16 x^{3} + 15 x^{2} + 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-12 \nu^{7} - 7 \nu^{6} - 76 \nu^{5} - 44 \nu^{4} - 602 \nu^{3} - 36 \nu^{2} - 8 \nu + 1249$$$$)/458$$ $$\beta_{3}$$ $$=$$ $$($$$$57 \nu^{7} - 24 \nu^{6} + 361 \nu^{5} + 209 \nu^{4} + 2287 \nu^{3} + 171 \nu^{2} + 38 \nu + 193$$$$)/916$$ $$\beta_{4}$$ $$=$$ $$($$$$193 \nu^{7} - 250 \nu^{6} + 1375 \nu^{5} - 361 \nu^{4} + 7125 \nu^{3} - 5375 \nu^{2} + 2724 \nu - 375$$$$)/916$$ $$\beta_{5}$$ $$=$$ $$($$$$174 \nu^{7} - 242 \nu^{6} + 1331 \nu^{5} - 507 \nu^{4} + 6897 \nu^{3} - 5203 \nu^{2} + 5383 \nu - 363$$$$)/458$$ $$\beta_{6}$$ $$=$$ $$($$$$-375 \nu^{7} + 182 \nu^{6} - 2375 \nu^{5} - 1375 \nu^{4} - 13889 \nu^{3} - 1125 \nu^{2} - 250 \nu - 2933$$$$)/916$$ $$\beta_{7}$$ $$=$$ $$($$$$-273 \nu^{7} + 356 \nu^{6} - 1958 \nu^{5} + 602 \nu^{4} - 10146 \nu^{3} + 7654 \nu^{2} - 3617 \nu + 534$$$$)/458$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + 7 \beta_{3} + \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{7} + \beta_{5} + 18 \beta_{4} - 10 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$10 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} + 15 \beta_{4} - 48 \beta_{3} - 10 \beta_{2} - 48 \beta_{1} + 15$$ $$\nu^{6}$$ $$=$$ $$-10 \beta_{6} - 86 \beta_{3} - 48 \beta_{2} + 117$$ $$\nu^{7}$$ $$=$$ $$-86 \beta_{7} - 48 \beta_{5} - 152 \beta_{4} + 337 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-1 - \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 −1.11000 + 1.92258i −0.115680 + 0.200364i 0.355143 − 0.615126i 1.37054 − 2.37385i −1.11000 − 1.92258i −0.115680 − 0.200364i 0.355143 + 0.615126i 1.37054 + 2.37385i
−1.11000 + 1.92258i −0.274776 + 0.475925i −1.46422 2.53610i −4.22001 −0.610004 1.05656i 0.500000 + 0.866025i 2.06113 1.34900 + 2.33653i 4.68423 8.11332i
22.2 −0.115680 + 0.200364i 1.66113 2.87716i 0.973236 + 1.68569i −2.23136 0.384320 + 0.665661i 0.500000 + 0.866025i −0.913059 −4.01868 6.96056i 0.258125 0.447085i
22.3 0.355143 0.615126i −1.20394 + 2.08529i 0.747746 + 1.29513i −1.28971 0.855143 + 1.48115i 0.500000 + 0.866025i 2.48280 −1.39895 2.42305i −0.458033 + 0.793337i
22.4 1.37054 2.37385i −0.682410 + 1.18197i −2.75677 4.77486i 0.741082 1.87054 + 3.23987i 0.500000 + 0.866025i −9.63087 0.568634 + 0.984903i 1.01568 1.75921i
29.1 −1.11000 1.92258i −0.274776 0.475925i −1.46422 + 2.53610i −4.22001 −0.610004 + 1.05656i 0.500000 0.866025i 2.06113 1.34900 2.33653i 4.68423 + 8.11332i
29.2 −0.115680 0.200364i 1.66113 + 2.87716i 0.973236 1.68569i −2.23136 0.384320 0.665661i 0.500000 0.866025i −0.913059 −4.01868 + 6.96056i 0.258125 + 0.447085i
29.3 0.355143 + 0.615126i −1.20394 2.08529i 0.747746 1.29513i −1.28971 0.855143 1.48115i 0.500000 0.866025i 2.48280 −1.39895 + 2.42305i −0.458033 0.793337i
29.4 1.37054 + 2.37385i −0.682410 1.18197i −2.75677 + 4.77486i 0.741082 1.87054 3.23987i 0.500000 0.866025i −9.63087 0.568634 0.984903i 1.01568 + 1.75921i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.f.c 8
3.b odd 2 1 819.2.o.h 8
4.b odd 2 1 1456.2.s.q 8
7.b odd 2 1 637.2.f.i 8
7.c even 3 1 637.2.g.k 8
7.c even 3 1 637.2.h.h 8
7.d odd 6 1 637.2.g.j 8
7.d odd 6 1 637.2.h.i 8
13.c even 3 1 inner 91.2.f.c 8
13.c even 3 1 1183.2.a.k 4
13.e even 6 1 1183.2.a.l 4
13.f odd 12 2 1183.2.c.g 8
39.i odd 6 1 819.2.o.h 8
52.j odd 6 1 1456.2.s.q 8
91.g even 3 1 637.2.h.h 8
91.h even 3 1 637.2.g.k 8
91.m odd 6 1 637.2.h.i 8
91.n odd 6 1 637.2.f.i 8
91.n odd 6 1 8281.2.a.bp 4
91.t odd 6 1 8281.2.a.bt 4
91.v odd 6 1 637.2.g.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 1.a even 1 1 trivial
91.2.f.c 8 13.c even 3 1 inner
637.2.f.i 8 7.b odd 2 1
637.2.f.i 8 91.n odd 6 1
637.2.g.j 8 7.d odd 6 1
637.2.g.j 8 91.v odd 6 1
637.2.g.k 8 7.c even 3 1
637.2.g.k 8 91.h even 3 1
637.2.h.h 8 7.c even 3 1
637.2.h.h 8 91.g even 3 1
637.2.h.i 8 7.d odd 6 1
637.2.h.i 8 91.m odd 6 1
819.2.o.h 8 3.b odd 2 1
819.2.o.h 8 39.i odd 6 1
1183.2.a.k 4 13.c even 3 1
1183.2.a.l 4 13.e even 6 1
1183.2.c.g 8 13.f odd 12 2
1456.2.s.q 8 4.b odd 2 1
1456.2.s.q 8 52.j odd 6 1
8281.2.a.bp 4 91.n odd 6 1
8281.2.a.bt 4 91.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - T_{2}^{7} + 7 T_{2}^{6} + 38 T_{2}^{4} - 16 T_{2}^{3} + 15 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 15 T^{2} - 16 T^{3} + 38 T^{4} + 7 T^{6} - T^{7} + T^{8}$$
$3$ $$36 + 96 T + 202 T^{2} + 156 T^{3} + 103 T^{4} + 23 T^{5} + 10 T^{6} + T^{7} + T^{8}$$
$5$ $$( -9 - T + 12 T^{2} + 7 T^{3} + T^{4} )^{2}$$
$7$ $$( 1 - T + T^{2} )^{4}$$
$11$ $$36 - 96 T + 202 T^{2} - 156 T^{3} + 103 T^{4} - 23 T^{5} + 10 T^{6} - T^{7} + T^{8}$$
$13$ $$28561 - 8788 T + 2704 T^{2} - 416 T^{3} + 17 T^{4} - 32 T^{5} + 16 T^{6} - 4 T^{7} + T^{8}$$
$17$ $$2809 - 3180 T + 2964 T^{2} - 1144 T^{3} + 437 T^{4} - 72 T^{5} + 28 T^{6} - 4 T^{7} + T^{8}$$
$19$ $$250000 + 27500 T^{2} - 1000 T^{3} + 2525 T^{4} - 55 T^{5} + 56 T^{6} + T^{7} + T^{8}$$
$23$ $$5184 + 4032 T + 5872 T^{2} - 1840 T^{3} + 1484 T^{4} - 36 T^{5} + 42 T^{6} - 2 T^{7} + T^{8}$$
$29$ $$25 - 105 T + 331 T^{2} - 452 T^{3} + 468 T^{4} - 64 T^{5} + 23 T^{6} + T^{7} + T^{8}$$
$31$ $$( 54 - 26 T - 13 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$37$ $$256 - 704 T + 2208 T^{2} + 428 T^{3} + 745 T^{4} - 258 T^{5} + 83 T^{6} - 10 T^{7} + T^{8}$$
$41$ $$318096 + 127464 T + 135112 T^{2} - 58490 T^{3} + 17793 T^{4} - 2826 T^{5} + 335 T^{6} - 22 T^{7} + T^{8}$$
$43$ $$4 + 16 T + 58 T^{2} + 36 T^{3} + 35 T^{4} + 7 T^{5} + 12 T^{6} + 3 T^{7} + T^{8}$$
$47$ $$( 100 - 12 T - 51 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$53$ $$( -1389 + 890 T - 140 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$59$ $$498436 - 388300 T + 248138 T^{2} - 53646 T^{3} + 11035 T^{4} - 484 T^{5} + 141 T^{6} - 8 T^{7} + T^{8}$$
$61$ $$10000 + 17600 T + 29476 T^{2} + 4240 T^{3} + 1733 T^{4} + 232 T^{5} + 79 T^{6} + 8 T^{7} + T^{8}$$
$67$ $$121220100 + 6980340 T + 2725066 T^{2} - 1654 T^{3} + 37315 T^{4} - 2 T^{5} + 247 T^{6} - 6 T^{7} + T^{8}$$
$71$ $$4129024 - 1446784 T + 474432 T^{2} - 68288 T^{3} + 12256 T^{4} - 1200 T^{5} + 212 T^{6} - 14 T^{7} + T^{8}$$
$73$ $$( 1772 - 88 T - 119 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$79$ $$( -7680 - 1024 T + 134 T^{2} + 26 T^{3} + T^{4} )^{2}$$
$83$ $$( -426 - 442 T - 97 T^{2} + T^{4} )^{2}$$
$89$ $$11664 - 19872 T + 26188 T^{2} - 13280 T^{3} + 5333 T^{4} - 297 T^{5} + 72 T^{6} - T^{7} + T^{8}$$
$97$ $$246238864 - 10482256 T + 5232284 T^{2} + 109588 T^{3} + 79337 T^{4} + 421 T^{5} + 314 T^{6} + 3 T^{7} + T^{8}$$