Properties

 Label 546.2.bk.b Level $546$ Weight $2$ Character orbit 546.bk Analytic conductor $4.360$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.bk (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 15 x^{10} + 90 x^{8} - 247 x^{6} + 270 x^{4} + 21 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 15 x^{10} + 90 x^{8} - 247 x^{6} + 270 x^{4} + 21 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{10} - 83 \nu^{8} + 906 \nu^{6} - 3874 \nu^{4} + 5950 \nu^{2} + 1505$$$$)/2947$$ $$\beta_{3}$$ $$=$$ $$($$$$-25 \nu^{10} + 406 \nu^{8} - 2063 \nu^{6} + 2536 \nu^{4} + 3931 \nu^{2} + 3290$$$$)/5894$$ $$\beta_{4}$$ $$=$$ $$($$$$-29 \nu^{10} + 151 \nu^{8} + 335 \nu^{6} - 3609 \nu^{4} + 2977 \nu^{2} + 3227$$$$)/5894$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{11} - 83 \nu^{9} + 906 \nu^{7} - 3874 \nu^{5} + 5950 \nu^{3} - 1442 \nu$$$$)/2947$$ $$\beta_{6}$$ $$=$$ $$($$$$-40 \nu^{11} + 397 \nu^{9} + 404 \nu^{7} - 17245 \nu^{5} + 61188 \nu^{3} - 56623 \nu$$$$)/41258$$ $$\beta_{7}$$ $$=$$ $$($$$$-58 \nu^{10} + 723 \nu^{8} - 3540 \nu^{6} + 6675 \nu^{4} + 902 \nu^{2} - 14175$$$$)/5894$$ $$\beta_{8}$$ $$=$$ $$($$$$-55 \nu^{11} + 1651 \nu^{9} - 15653 \nu^{7} + 69487 \nu^{5} - 156047 \nu^{3} + 151641 \nu$$$$)/41258$$ $$\beta_{9}$$ $$=$$ $$($$$$-89 \nu^{10} + 1378 \nu^{8} - 8321 \nu^{6} + 23780 \nu^{4} - 30699 \nu^{2} + 8176$$$$)/5894$$ $$\beta_{10}$$ $$=$$ $$($$$$-501 \nu^{11} + 7109 \nu^{9} - 40029 \nu^{7} + 98967 \nu^{5} - 88545 \nu^{3} - 45465 \nu$$$$)/41258$$ $$\beta_{11}$$ $$=$$ $$($$$$-332 \nu^{11} + 5358 \nu^{9} - 33779 \nu^{7} + 94100 \nu^{5} - 82129 \nu^{3} - 75957 \nu$$$$)/20629$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{10} - \beta_{8} - 2 \beta_{6} + \beta_{5} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{9} + 4 \beta_{7} - 6 \beta_{4} - 8 \beta_{3} + 9 \beta_{2} + 10$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{11} - 8 \beta_{10} - 4 \beta_{8} - 18 \beta_{6} + 13 \beta_{5} + 17 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$17 \beta_{9} + 13 \beta_{7} - 33 \beta_{4} - 42 \beta_{3} + 65 \beta_{2} + 16$$ $$\nu^{7}$$ $$=$$ $$50 \beta_{11} - 53 \beta_{10} - 5 \beta_{8} - 90 \beta_{6} + 99 \beta_{5} + 69 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$104 \beta_{9} + 24 \beta_{7} - 172 \beta_{4} - 168 \beta_{3} + 365 \beta_{2} - 85$$ $$\nu^{9}$$ $$=$$ $$276 \beta_{11} - 320 \beta_{10} + 84 \beta_{8} - 288 \beta_{6} + 573 \beta_{5} + 240 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$489 \beta_{9} - 120 \beta_{7} - 836 \beta_{4} - 467 \beta_{3} + 1634 \beta_{2} - 1083$$ $$\nu^{11}$$ $$=$$ $$1325 \beta_{11} - 1792 \beta_{10} + 978 \beta_{8} - 98 \beta_{6} + 2612 \beta_{5} + 453 \beta_{1}$$

Character values

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

 $$n$$ $$157$$ $$365$$ $$379$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −0.385124 + 0.500000i 1.75780 + 0.500000i −2.23871 + 0.500000i 2.23871 − 0.500000i −1.75780 − 0.500000i 0.385124 − 0.500000i −0.385124 − 0.500000i 1.75780 − 0.500000i −2.23871 − 0.500000i 2.23871 + 0.500000i −1.75780 + 0.500000i 0.385124 + 0.500000i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −2.25312 + 1.30084i 1.00000i 1.49160 2.18521i 1.00000i −0.500000 0.866025i 1.30084 2.25312i
25.2 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.294342 0.169938i 1.00000i 0.420136 + 2.61218i 1.00000i −0.500000 0.866025i −0.169938 + 0.294342i
25.3 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 1.95878 1.13090i 1.00000i 2.41839 + 1.07303i 1.00000i −0.500000 0.866025i −1.13090 + 1.95878i
25.4 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −1.95878 + 1.13090i 1.00000i −2.41839 1.07303i 1.00000i −0.500000 0.866025i −1.13090 + 1.95878i
25.5 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.294342 + 0.169938i 1.00000i −0.420136 2.61218i 1.00000i −0.500000 0.866025i −0.169938 + 0.294342i
25.6 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 2.25312 1.30084i 1.00000i −1.49160 + 2.18521i 1.00000i −0.500000 0.866025i 1.30084 2.25312i
415.1 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −2.25312 1.30084i 1.00000i 1.49160 + 2.18521i 1.00000i −0.500000 + 0.866025i 1.30084 + 2.25312i
415.2 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.294342 + 0.169938i 1.00000i 0.420136 2.61218i 1.00000i −0.500000 + 0.866025i −0.169938 0.294342i
415.3 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 1.95878 + 1.13090i 1.00000i 2.41839 1.07303i 1.00000i −0.500000 + 0.866025i −1.13090 1.95878i
415.4 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −1.95878 1.13090i 1.00000i −2.41839 + 1.07303i 1.00000i −0.500000 + 0.866025i −1.13090 1.95878i
415.5 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.294342 0.169938i 1.00000i −0.420136 + 2.61218i 1.00000i −0.500000 + 0.866025i −0.169938 0.294342i
415.6 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 2.25312 + 1.30084i 1.00000i −1.49160 2.18521i 1.00000i −0.500000 + 0.866025i 1.30084 + 2.25312i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 415.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bk.b 12
3.b odd 2 1 1638.2.dm.c 12
7.c even 3 1 inner 546.2.bk.b 12
7.c even 3 1 3822.2.c.k 6
7.d odd 6 1 3822.2.c.j 6
13.b even 2 1 inner 546.2.bk.b 12
21.h odd 6 1 1638.2.dm.c 12
39.d odd 2 1 1638.2.dm.c 12
91.r even 6 1 inner 546.2.bk.b 12
91.r even 6 1 3822.2.c.k 6
91.s odd 6 1 3822.2.c.j 6
273.w odd 6 1 1638.2.dm.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bk.b 12 1.a even 1 1 trivial
546.2.bk.b 12 7.c even 3 1 inner
546.2.bk.b 12 13.b even 2 1 inner
546.2.bk.b 12 91.r even 6 1 inner
1638.2.dm.c 12 3.b odd 2 1
1638.2.dm.c 12 21.h odd 6 1
1638.2.dm.c 12 39.d odd 2 1
1638.2.dm.c 12 273.w odd 6 1
3822.2.c.j 6 7.d odd 6 1
3822.2.c.j 6 91.s odd 6 1
3822.2.c.k 6 7.c even 3 1
3822.2.c.k 6 91.r even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{12} - 12 T_{5}^{10} + 108 T_{5}^{8} - 424 T_{5}^{6} + 1248 T_{5}^{4} - 144 T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{3}$$
$3$ $$( 1 + T + T^{2} )^{6}$$
$5$ $$16 - 144 T^{2} + 1248 T^{4} - 424 T^{6} + 108 T^{8} - 12 T^{10} + T^{12}$$
$7$ $$117649 + 21609 T^{2} + 2058 T^{4} + 245 T^{6} + 42 T^{8} + 9 T^{10} + T^{12}$$
$11$ $$81 - 243 T^{2} + 594 T^{4} - 387 T^{6} + 198 T^{8} - 15 T^{10} + T^{12}$$
$13$ $$( 2197 - 1014 T + 351 T^{2} - 132 T^{3} + 27 T^{4} - 6 T^{5} + T^{6} )^{2}$$
$17$ $$( 49 + 105 T + 288 T^{2} - 149 T^{3} + 66 T^{4} - 9 T^{5} + T^{6} )^{2}$$
$19$ $$15752961 - 9537507 T^{2} + 5381478 T^{4} - 229959 T^{6} + 7398 T^{8} - 99 T^{10} + T^{12}$$
$23$ $$( 9604 + 4116 T + 1764 T^{2} + 196 T^{3} + 42 T^{4} + T^{6} )^{2}$$
$29$ $$( -77 - 45 T - 3 T^{2} + T^{3} )^{4}$$
$31$ $$( 256 - 16 T^{2} + T^{4} )^{3}$$
$37$ $$16 - 576 T^{2} + 20640 T^{4} - 3448 T^{6} + 432 T^{8} - 24 T^{10} + T^{12}$$
$41$ $$( 676 + 984 T^{2} + 108 T^{4} + T^{6} )^{2}$$
$43$ $$( 104 - 36 T - 6 T^{2} + T^{3} )^{4}$$
$47$ $$8428892481 - 960964803 T^{2} + 91655334 T^{4} - 1857447 T^{6} + 27558 T^{8} - 195 T^{10} + T^{12}$$
$53$ $$( 9 + 3 T + T^{2} )^{6}$$
$59$ $$1982119441 - 235738695 T^{2} + 20423934 T^{4} - 816403 T^{6} + 23946 T^{8} - 171 T^{10} + T^{12}$$
$61$ $$( 4489 + 3015 T + 1824 T^{2} + 269 T^{3} + 54 T^{4} - 3 T^{5} + T^{6} )^{2}$$
$67$ $$163047361 - 101781699 T^{2} + 61046886 T^{4} - 1528807 T^{6} + 30054 T^{8} - 195 T^{10} + T^{12}$$
$71$ $$( 441 + 603 T^{2} + 51 T^{4} + T^{6} )^{2}$$
$73$ $$1474777075216 - 41955229392 T^{2} + 800097408 T^{4} - 8764744 T^{6} + 70428 T^{8} - 324 T^{10} + T^{12}$$
$79$ $$( 278784 - 25344 T + 8640 T^{2} - 480 T^{3} + 192 T^{4} - 12 T^{5} + T^{6} )^{2}$$
$83$ $$( 712336 + 30960 T^{2} + 372 T^{4} + T^{6} )^{2}$$
$89$ $$29376588816 - 2147249088 T^{2} + 119929248 T^{4} - 2363256 T^{6} + 34128 T^{8} - 216 T^{10} + T^{12}$$
$97$ $$( 324 + 16740 T^{2} + 324 T^{4} + T^{6} )^{2}$$