# Properties

 Label 441.2.g.d Level $441$ Weight $2$ Character orbit 441.g Analytic conductor $3.521$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$\beta_{4}$$ $$-1 - \beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i 0.5 − 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i
−0.849814 + 1.47192i 1.64400 + 0.545231i −0.444368 0.769668i 3.58836 −2.19963 + 1.95649i 0 −1.88874 2.40545 + 1.79272i −3.04944 + 5.28179i
67.2 0.119562 0.207087i −1.71053 0.272169i 0.971410 + 1.68253i −1.18194 −0.260877 + 0.321688i 0 0.942820 2.85185 + 0.931107i −0.141315 + 0.244765i
67.3 1.23025 2.13086i −0.933463 + 1.45899i −2.02704 3.51094i 2.59358 1.96050 + 3.78400i 0 −5.05408 −1.25729 2.72382i 3.19076 5.52655i
79.1 −0.849814 1.47192i 1.64400 0.545231i −0.444368 + 0.769668i 3.58836 −2.19963 1.95649i 0 −1.88874 2.40545 1.79272i −3.04944 5.28179i
79.2 0.119562 + 0.207087i −1.71053 + 0.272169i 0.971410 1.68253i −1.18194 −0.260877 0.321688i 0 0.942820 2.85185 0.931107i −0.141315 0.244765i
79.3 1.23025 + 2.13086i −0.933463 1.45899i −2.02704 + 3.51094i 2.59358 1.96050 3.78400i 0 −5.05408 −1.25729 + 2.72382i 3.19076 + 5.52655i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.g.d 6
3.b odd 2 1 1323.2.g.b 6
7.b odd 2 1 441.2.g.e 6
7.c even 3 1 441.2.f.d 6
7.c even 3 1 441.2.h.b 6
7.d odd 6 1 63.2.f.b 6
7.d odd 6 1 441.2.h.c 6
9.c even 3 1 441.2.h.b 6
9.d odd 6 1 1323.2.h.e 6
21.c even 2 1 1323.2.g.c 6
21.g even 6 1 189.2.f.a 6
21.g even 6 1 1323.2.h.d 6
21.h odd 6 1 1323.2.f.c 6
21.h odd 6 1 1323.2.h.e 6
28.f even 6 1 1008.2.r.k 6
63.g even 3 1 inner 441.2.g.d 6
63.g even 3 1 3969.2.a.m 3
63.h even 3 1 441.2.f.d 6
63.i even 6 1 189.2.f.a 6
63.j odd 6 1 1323.2.f.c 6
63.k odd 6 1 441.2.g.e 6
63.k odd 6 1 567.2.a.d 3
63.l odd 6 1 441.2.h.c 6
63.n odd 6 1 1323.2.g.b 6
63.n odd 6 1 3969.2.a.p 3
63.o even 6 1 1323.2.h.d 6
63.s even 6 1 567.2.a.g 3
63.s even 6 1 1323.2.g.c 6
63.t odd 6 1 63.2.f.b 6
84.j odd 6 1 3024.2.r.g 6
252.n even 6 1 9072.2.a.bq 3
252.r odd 6 1 3024.2.r.g 6
252.bj even 6 1 1008.2.r.k 6
252.bn odd 6 1 9072.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 7.d odd 6 1
63.2.f.b 6 63.t odd 6 1
189.2.f.a 6 21.g even 6 1
189.2.f.a 6 63.i even 6 1
441.2.f.d 6 7.c even 3 1
441.2.f.d 6 63.h even 3 1
441.2.g.d 6 1.a even 1 1 trivial
441.2.g.d 6 63.g even 3 1 inner
441.2.g.e 6 7.b odd 2 1
441.2.g.e 6 63.k odd 6 1
441.2.h.b 6 7.c even 3 1
441.2.h.b 6 9.c even 3 1
441.2.h.c 6 7.d odd 6 1
441.2.h.c 6 63.l odd 6 1
567.2.a.d 3 63.k odd 6 1
567.2.a.g 3 63.s even 6 1
1008.2.r.k 6 28.f even 6 1
1008.2.r.k 6 252.bj even 6 1
1323.2.f.c 6 21.h odd 6 1
1323.2.f.c 6 63.j odd 6 1
1323.2.g.b 6 3.b odd 2 1
1323.2.g.b 6 63.n odd 6 1
1323.2.g.c 6 21.c even 2 1
1323.2.g.c 6 63.s even 6 1
1323.2.h.d 6 21.g even 6 1
1323.2.h.d 6 63.o even 6 1
1323.2.h.e 6 9.d odd 6 1
1323.2.h.e 6 21.h odd 6 1
3024.2.r.g 6 84.j odd 6 1
3024.2.r.g 6 252.r odd 6 1
3969.2.a.m 3 63.g even 3 1
3969.2.a.p 3 63.n odd 6 1
9072.2.a.bq 3 252.n even 6 1
9072.2.a.cd 3 252.bn odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{6} - T_{2}^{5} + 5 T_{2}^{4} + 2 T_{2}^{3} + 17 T_{2}^{2} - 4 T_{2} + 1$$ $$T_{5}^{3} - 5 T_{5}^{2} + 2 T_{5} + 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T + 17 T^{2} + 2 T^{3} + 5 T^{4} - T^{5} + T^{6}$$
$3$ $$27 + 18 T - 6 T^{2} - 9 T^{3} - 2 T^{4} + 2 T^{5} + T^{6}$$
$5$ $$( 11 + 2 T - 5 T^{2} + T^{3} )^{2}$$
$7$ $$T^{6}$$
$11$ $$( -47 - 19 T + 2 T^{2} + T^{3} )^{2}$$
$13$ $$( 1 - T + T^{2} )^{3}$$
$17$ $$729 + 1053 T + 1197 T^{2} + 414 T^{3} + 105 T^{4} + 12 T^{5} + T^{6}$$
$19$ $$49 + 42 T + 57 T^{2} - 4 T^{3} + 15 T^{4} + 3 T^{5} + T^{6}$$
$23$ $$( -9 - 33 T + T^{3} )^{2}$$
$29$ $$1 + 4 T + 17 T^{2} - 2 T^{3} + 5 T^{4} + T^{5} + T^{6}$$
$31$ $$729 - 648 T + 495 T^{2} - 126 T^{3} + 33 T^{4} + 3 T^{5} + T^{6}$$
$37$ $$6561 + 4374 T + 2673 T^{2} + 324 T^{3} + 63 T^{4} - 3 T^{5} + T^{6}$$
$41$ $$124609 + 54715 T + 16259 T^{2} + 2704 T^{3} + 329 T^{4} + 22 T^{5} + T^{6}$$
$43$ $$14641 + 7986 T + 3993 T^{2} + 440 T^{3} + 75 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$35721 + 10206 T + 4617 T^{2} - 108 T^{3} + 135 T^{4} + 9 T^{5} + T^{6}$$
$53$ $$81 - 675 T + 5463 T^{2} - 1332 T^{3} + 249 T^{4} - 18 T^{5} + T^{6}$$
$59$ $$3969 + 378 T + 603 T^{2} + 72 T^{3} + 87 T^{4} + 9 T^{5} + T^{6}$$
$61$ $$4489 + 1407 T + 843 T^{2} + 8 T^{3} + 57 T^{4} + 6 T^{5} + T^{6}$$
$67$ $$466489 + 141381 T + 42849 T^{2} + 1366 T^{3} + 207 T^{4} + T^{6}$$
$71$ $$( 81 - 6 T - 9 T^{2} + T^{3} )^{2}$$
$73$ $$59049 + 40824 T + 27495 T^{2} + 990 T^{3} + 177 T^{4} - 3 T^{5} + T^{6}$$
$79$ $$591361 + 36912 T + 13839 T^{2} + 818 T^{3} + 273 T^{4} + 15 T^{5} + T^{6}$$
$83$ $$729 + 1053 T + 1197 T^{2} + 414 T^{3} + 105 T^{4} + 12 T^{5} + T^{6}$$
$89$ $$143641 - 57229 T + 22043 T^{2} - 1060 T^{3} + 155 T^{4} + 2 T^{5} + T^{6}$$
$97$ $$363609 - 68742 T + 14805 T^{2} - 864 T^{3} + 123 T^{4} - 3 T^{5} + T^{6}$$