# Properties

 Label 546.2.j.d Level $546$ Weight $2$ Character orbit 546.j Analytic conductor $4.360$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.j (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.447703281.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + 2 x^{5} + 3 x^{4} + 4 x^{3} - 8 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 2 \nu^{5} + 2 \nu^{4} + 3 \nu^{3} + 4 \nu^{2} - 8 \nu - 8$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + \nu^{5} - 2 \nu^{3} - \nu^{2} + 6 \nu + 4$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + 2 \nu^{4} + \nu^{3} - 6 \nu^{2} - 4 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 6 \nu^{5} + 2 \nu^{4} - 3 \nu^{3} - 12 \nu^{2} + 4 \nu + 32$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{5} + \nu^{4} - \nu^{3} - 5 \nu^{2} - \nu + 8$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{6} - 5 \nu^{5} - 4 \nu^{4} + 4 \nu^{3} + 17 \nu^{2} + 2 \nu - 24$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} - \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + \beta_{1} + 1$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 5$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 5 \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$-\beta_{7} - 4 \beta_{6} + 4 \beta_{5} - \beta_{3} - 2 \beta_{1} - 5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 1.26359 + 0.635098i −0.571299 + 1.29368i −1.38232 − 0.298668i 1.19003 − 0.764088i 1.26359 − 0.635098i −0.571299 − 1.29368i −1.38232 + 0.298668i 1.19003 + 0.764088i
1.00000 0.500000 0.866025i 1.00000 −1.97513 + 3.42102i 0.500000 0.866025i −1.15207 + 2.38175i 1.00000 −0.500000 0.866025i −1.97513 + 3.42102i
289.2 1.00000 0.500000 0.866025i 1.00000 −0.228205 + 0.395262i 0.500000 0.866025i −0.369922 2.61976i 1.00000 −0.500000 0.866025i −0.228205 + 0.395262i
289.3 1.00000 0.500000 0.866025i 1.00000 1.14553 1.98411i 0.500000 0.866025i 2.63641 0.222079i 1.00000 −0.500000 0.866025i 1.14553 1.98411i
289.4 1.00000 0.500000 0.866025i 1.00000 2.05781 3.56422i 0.500000 0.866025i −2.61442 0.405935i 1.00000 −0.500000 0.866025i 2.05781 3.56422i
529.1 1.00000 0.500000 + 0.866025i 1.00000 −1.97513 3.42102i 0.500000 + 0.866025i −1.15207 2.38175i 1.00000 −0.500000 + 0.866025i −1.97513 3.42102i
529.2 1.00000 0.500000 + 0.866025i 1.00000 −0.228205 0.395262i 0.500000 + 0.866025i −0.369922 + 2.61976i 1.00000 −0.500000 + 0.866025i −0.228205 0.395262i
529.3 1.00000 0.500000 + 0.866025i 1.00000 1.14553 + 1.98411i 0.500000 + 0.866025i 2.63641 + 0.222079i 1.00000 −0.500000 + 0.866025i 1.14553 + 1.98411i
529.4 1.00000 0.500000 + 0.866025i 1.00000 2.05781 + 3.56422i 0.500000 + 0.866025i −2.61442 + 0.405935i 1.00000 −0.500000 + 0.866025i 2.05781 + 3.56422i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.j.d 8
3.b odd 2 1 1638.2.m.g 8
7.c even 3 1 546.2.k.b yes 8
13.c even 3 1 546.2.k.b yes 8
21.h odd 6 1 1638.2.p.i 8
39.i odd 6 1 1638.2.p.i 8
91.h even 3 1 inner 546.2.j.d 8
273.s odd 6 1 1638.2.m.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.d 8 1.a even 1 1 trivial
546.2.j.d 8 91.h even 3 1 inner
546.2.k.b yes 8 7.c even 3 1
546.2.k.b yes 8 13.c even 3 1
1638.2.m.g 8 3.b odd 2 1
1638.2.m.g 8 273.s odd 6 1
1638.2.p.i 8 21.h odd 6 1
1638.2.p.i 8 39.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{8}$$
$3$ $$( 1 - T + T^{2} )^{4}$$
$5$ $$289 + 510 T + 1189 T^{2} - 442 T^{3} + 332 T^{4} - 26 T^{5} + 21 T^{6} - 2 T^{7} + T^{8}$$
$7$ $$2401 + 1029 T + 98 T^{2} - 147 T^{3} - 117 T^{4} - 21 T^{5} + 2 T^{6} + 3 T^{7} + T^{8}$$
$11$ $$4489 + 5963 T + 7251 T^{2} + 1694 T^{3} + 701 T^{4} + 118 T^{5} + 46 T^{6} + 6 T^{7} + T^{8}$$
$13$ $$28561 + 24167 T + 10478 T^{2} + 3471 T^{3} + 1031 T^{4} + 267 T^{5} + 62 T^{6} + 11 T^{7} + T^{8}$$
$17$ $$( -1 + 9 T - 14 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$19$ $$169 - 208 T + 321 T^{2} - 76 T^{3} + 134 T^{4} - 62 T^{5} + 31 T^{6} - 6 T^{7} + T^{8}$$
$23$ $$( -167 + 155 T - 10 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$29$ $$4489 + 7437 T + 8971 T^{2} + 5282 T^{3} + 2345 T^{4} + 322 T^{5} + 54 T^{6} - 2 T^{7} + T^{8}$$
$31$ $$210681 + 37179 T + 28593 T^{2} + 1620 T^{3} + 2331 T^{4} + 126 T^{5} + 84 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$( 211 - 823 T + 260 T^{2} - 28 T^{3} + T^{4} )^{2}$$
$41$ $$1912689 - 136917 T + 137037 T^{2} + 9108 T^{3} + 7081 T^{4} + 198 T^{5} + 92 T^{6} + T^{8}$$
$43$ $$4489 + 1005 T + 2771 T^{2} - 1374 T^{3} + 1287 T^{4} - 258 T^{5} + 74 T^{6} + 6 T^{7} + T^{8}$$
$47$ $$2601 + 5814 T + 15852 T^{2} - 6282 T^{3} + 3199 T^{4} - 172 T^{5} + 57 T^{6} - T^{7} + T^{8}$$
$53$ $$6561 - 1458 T + 3888 T^{2} + 1926 T^{3} + 1729 T^{4} + 344 T^{5} + 93 T^{6} - 7 T^{7} + T^{8}$$
$59$ $$( 311 - 660 T - 143 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$61$ $$17438976 - 1904256 T + 775872 T^{2} - 138432 T^{3} + 33616 T^{4} - 4176 T^{5} + 440 T^{6} - 24 T^{7} + T^{8}$$
$67$ $$27889 - 33233 T + 34758 T^{2} - 10781 T^{3} + 3659 T^{4} + 833 T^{5} + 196 T^{6} + 15 T^{7} + T^{8}$$
$71$ $$674041 - 502452 T + 296549 T^{2} - 67992 T^{3} + 13518 T^{4} - 654 T^{5} + 131 T^{6} - 6 T^{7} + T^{8}$$
$73$ $$1 + 5 T + 32 T^{2} - 33 T^{3} + 53 T^{4} - 3 T^{5} + 8 T^{6} - T^{7} + T^{8}$$
$79$ $$6985449 - 1538226 T + 542235 T^{2} - 18618 T^{3} + 10270 T^{4} + 240 T^{5} + 221 T^{6} + 12 T^{7} + T^{8}$$
$83$ $$( 1919 - 1329 T - 86 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$89$ $$( 27 + 180 T + 170 T^{2} + 25 T^{3} + T^{4} )^{2}$$
$97$ $$45873529 + 853398 T + 1289200 T^{2} - 37234 T^{3} + 28445 T^{4} - 440 T^{5} + 189 T^{6} + T^{7} + T^{8}$$
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