# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.k (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.

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

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 $$\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 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
−0.500000 0.866025i −1.00000 −0.500000 + 0.866025i −1.97513 + 3.42102i 0.500000 + 0.866025i −1.48662 2.18860i 1.00000 1.00000 3.95025
373.2 −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i −0.228205 + 0.395262i 0.500000 + 0.866025i 2.45374 + 0.989520i 1.00000 1.00000 0.456409
373.3 −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 1.14553 1.98411i 0.500000 + 0.866025i −1.12588 + 2.39424i 1.00000 1.00000 −2.29105
373.4 −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 2.05781 3.56422i 0.500000 + 0.866025i 1.65876 2.06119i 1.00000 1.00000 −4.11561
445.1 −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i −1.97513 3.42102i 0.500000 0.866025i −1.48662 + 2.18860i 1.00000 1.00000 3.95025
445.2 −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i −0.228205 0.395262i 0.500000 0.866025i 2.45374 0.989520i 1.00000 1.00000 0.456409
445.3 −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 1.14553 + 1.98411i 0.500000 0.866025i −1.12588 2.39424i 1.00000 1.00000 −2.29105
445.4 −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 2.05781 + 3.56422i 0.500000 0.866025i 1.65876 + 2.06119i 1.00000 1.00000 −4.11561
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 445.4 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.g 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.k.b yes 8
3.b odd 2 1 1638.2.p.i 8
7.c even 3 1 546.2.j.d 8
13.c even 3 1 546.2.j.d 8
21.h odd 6 1 1638.2.m.g 8
39.i odd 6 1 1638.2.m.g 8
91.g even 3 1 inner 546.2.k.b yes 8
273.bm odd 6 1 1638.2.p.i 8

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

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{4}$$
$3$ $$( 1 + T )^{8}$$
$5$ $$289 + 510 T + 1189 T^{2} - 442 T^{3} + 332 T^{4} - 26 T^{5} + 21 T^{6} - 2 T^{7} + T^{8}$$
$7$ $$2401 - 1029 T + 392 T^{2} - 231 T^{3} + 123 T^{4} - 33 T^{5} + 8 T^{6} - 3 T^{7} + T^{8}$$
$11$ $$( -67 + 89 T - 10 T^{2} - 6 T^{3} + T^{4} )^{2}$$
$13$ $$28561 + 24167 T + 10478 T^{2} + 3471 T^{3} + 1031 T^{4} + 267 T^{5} + 62 T^{6} + 11 T^{7} + T^{8}$$
$17$ $$1 + 9 T + 67 T^{2} + 118 T^{3} + 161 T^{4} + 74 T^{5} + 30 T^{6} - 4 T^{7} + T^{8}$$
$19$ $$( -13 - 16 T + 5 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$23$ $$27889 + 25885 T + 22355 T^{2} + 4890 T^{3} + 1817 T^{4} + 210 T^{5} + 110 T^{6} + 10 T^{7} + T^{8}$$
$29$ $$4489 + 7437 T + 8971 T^{2} + 5282 T^{3} + 2345 T^{4} + 322 T^{5} + 54 T^{6} - 2 T^{7} + T^{8}$$
$31$ $$210681 + 37179 T + 28593 T^{2} + 1620 T^{3} + 2331 T^{4} + 126 T^{5} + 84 T^{6} - 6 T^{7} + T^{8}$$
$37$ $$44521 + 173653 T + 622469 T^{2} + 202164 T^{3} + 44345 T^{4} + 5634 T^{5} + 524 T^{6} + 28 T^{7} + T^{8}$$
$41$ $$1912689 - 136917 T + 137037 T^{2} + 9108 T^{3} + 7081 T^{4} + 198 T^{5} + 92 T^{6} + T^{8}$$
$43$ $$4489 + 1005 T + 2771 T^{2} - 1374 T^{3} + 1287 T^{4} - 258 T^{5} + 74 T^{6} + 6 T^{7} + T^{8}$$
$47$ $$2601 + 5814 T + 15852 T^{2} - 6282 T^{3} + 3199 T^{4} - 172 T^{5} + 57 T^{6} - T^{7} + T^{8}$$
$53$ $$6561 - 1458 T + 3888 T^{2} + 1926 T^{3} + 1729 T^{4} + 344 T^{5} + 93 T^{6} - 7 T^{7} + T^{8}$$
$59$ $$96721 + 205260 T + 480073 T^{2} - 93136 T^{3} + 21458 T^{4} - 1034 T^{5} + 147 T^{6} - 2 T^{7} + T^{8}$$
$61$ $$( -4176 - 456 T + 136 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$67$ $$( 167 + 199 T + 29 T^{2} - 15 T^{3} + T^{4} )^{2}$$
$71$ $$674041 - 502452 T + 296549 T^{2} - 67992 T^{3} + 13518 T^{4} - 654 T^{5} + 131 T^{6} - 6 T^{7} + T^{8}$$
$73$ $$1 + 5 T + 32 T^{2} - 33 T^{3} + 53 T^{4} - 3 T^{5} + 8 T^{6} - T^{7} + T^{8}$$
$79$ $$6985449 - 1538226 T + 542235 T^{2} - 18618 T^{3} + 10270 T^{4} + 240 T^{5} + 221 T^{6} + 12 T^{7} + T^{8}$$
$83$ $$( 1919 - 1329 T - 86 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$89$ $$729 - 4860 T + 27810 T^{2} - 29250 T^{3} + 24373 T^{4} - 3890 T^{5} + 455 T^{6} - 25 T^{7} + T^{8}$$
$97$ $$45873529 + 853398 T + 1289200 T^{2} - 37234 T^{3} + 28445 T^{4} - 440 T^{5} + 189 T^{6} + T^{7} + T^{8}$$