# Properties

 Label 441.2.h.c Level $441$ Weight $2$ Character orbit 441.h 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.h (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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

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

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 $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
214.1
 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i
−2.46050 0.796790 1.53790i 4.05408 1.29679 2.24611i −1.96050 + 3.78400i 0 −5.05408 −1.73025 2.45076i −3.19076 + 5.52655i
214.2 −0.239123 −1.09097 1.34528i −1.94282 −0.590972 + 1.02359i 0.260877 + 0.321688i 0 0.942820 −0.619562 + 2.93533i 0.141315 0.244765i
214.3 1.69963 1.29418 + 1.15113i 0.888736 1.79418 3.10761i 2.19963 + 1.95649i 0 −1.88874 0.349814 + 2.97954i 3.04944 5.28179i
373.1 −2.46050 0.796790 + 1.53790i 4.05408 1.29679 + 2.24611i −1.96050 3.78400i 0 −5.05408 −1.73025 + 2.45076i −3.19076 5.52655i
373.2 −0.239123 −1.09097 + 1.34528i −1.94282 −0.590972 1.02359i 0.260877 0.321688i 0 0.942820 −0.619562 2.93533i 0.141315 + 0.244765i
373.3 1.69963 1.29418 1.15113i 0.888736 1.79418 + 3.10761i 2.19963 1.95649i 0 −1.88874 0.349814 2.97954i 3.04944 + 5.28179i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 373.3 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.h 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.h.c 6
3.b odd 2 1 1323.2.h.d 6
7.b odd 2 1 441.2.h.b 6
7.c even 3 1 63.2.f.b 6
7.c even 3 1 441.2.g.e 6
7.d odd 6 1 441.2.f.d 6
7.d odd 6 1 441.2.g.d 6
9.c even 3 1 441.2.g.e 6
9.d odd 6 1 1323.2.g.c 6
21.c even 2 1 1323.2.h.e 6
21.g even 6 1 1323.2.f.c 6
21.g even 6 1 1323.2.g.b 6
21.h odd 6 1 189.2.f.a 6
21.h odd 6 1 1323.2.g.c 6
28.g odd 6 1 1008.2.r.k 6
63.g even 3 1 63.2.f.b 6
63.h even 3 1 inner 441.2.h.c 6
63.h even 3 1 567.2.a.d 3
63.i even 6 1 1323.2.h.e 6
63.i even 6 1 3969.2.a.p 3
63.j odd 6 1 567.2.a.g 3
63.j odd 6 1 1323.2.h.d 6
63.k odd 6 1 441.2.f.d 6
63.l odd 6 1 441.2.g.d 6
63.n odd 6 1 189.2.f.a 6
63.o even 6 1 1323.2.g.b 6
63.s even 6 1 1323.2.f.c 6
63.t odd 6 1 441.2.h.b 6
63.t odd 6 1 3969.2.a.m 3
84.n even 6 1 3024.2.r.g 6
252.o even 6 1 3024.2.r.g 6
252.u odd 6 1 9072.2.a.bq 3
252.bb even 6 1 9072.2.a.cd 3
252.bl odd 6 1 1008.2.r.k 6

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

## Hecke kernels

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

 $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 1$$ $$T_{5}^{6} - 5 T_{5}^{5} + 23 T_{5}^{4} - 32 T_{5}^{3} + 59 T_{5}^{2} + 22 T_{5} + 121$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 - 4 T + T^{2} + T^{3} )^{2}$$
$3$ $$27 - 18 T + 12 T^{2} - 3 T^{3} + 4 T^{4} - 2 T^{5} + T^{6}$$
$5$ $$121 + 22 T + 59 T^{2} - 32 T^{3} + 23 T^{4} - 5 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$2209 - 893 T + 455 T^{2} - 56 T^{3} + 23 T^{4} - 2 T^{5} + T^{6}$$
$13$ $$( 1 + T + T^{2} )^{3}$$
$17$ $$729 - 1053 T + 1197 T^{2} - 414 T^{3} + 105 T^{4} - 12 T^{5} + T^{6}$$
$19$ $$49 - 42 T + 57 T^{2} + 4 T^{3} + 15 T^{4} - 3 T^{5} + T^{6}$$
$23$ $$81 - 297 T + 1089 T^{2} - 18 T^{3} + 33 T^{4} + T^{6}$$
$29$ $$1 + 4 T + 17 T^{2} - 2 T^{3} + 5 T^{4} + T^{5} + T^{6}$$
$31$ $$( 27 - 24 T + 3 T^{2} + T^{3} )^{2}$$
$37$ $$6561 + 4374 T + 2673 T^{2} + 324 T^{3} + 63 T^{4} - 3 T^{5} + T^{6}$$
$41$ $$124609 - 54715 T + 16259 T^{2} - 2704 T^{3} + 329 T^{4} - 22 T^{5} + T^{6}$$
$43$ $$14641 + 7986 T + 3993 T^{2} + 440 T^{3} + 75 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$( -189 - 54 T + 9 T^{2} + T^{3} )^{2}$$
$53$ $$81 - 675 T + 5463 T^{2} - 1332 T^{3} + 249 T^{4} - 18 T^{5} + T^{6}$$
$59$ $$( -63 - 6 T + 9 T^{2} + T^{3} )^{2}$$
$61$ $$( -67 - 21 T + 6 T^{2} + T^{3} )^{2}$$
$67$ $$( 683 - 207 T + T^{3} )^{2}$$
$71$ $$( 81 - 6 T - 9 T^{2} + T^{3} )^{2}$$
$73$ $$59049 - 40824 T + 27495 T^{2} - 990 T^{3} + 177 T^{4} + 3 T^{5} + T^{6}$$
$79$ $$( 769 - 48 T - 15 T^{2} + T^{3} )^{2}$$
$83$ $$729 - 1053 T + 1197 T^{2} - 414 T^{3} + 105 T^{4} - 12 T^{5} + T^{6}$$
$89$ $$143641 + 57229 T + 22043 T^{2} + 1060 T^{3} + 155 T^{4} - 2 T^{5} + T^{6}$$
$97$ $$363609 + 68742 T + 14805 T^{2} + 864 T^{3} + 123 T^{4} + 3 T^{5} + T^{6}$$