# Properties

 Label 392.3.k.j Level 392 Weight 3 Character orbit 392.k Analytic conductor 10.681 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$392 = 2^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 392.k (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.6812263629$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.796594176.2 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{2} + ( 2 + 2 \beta_{1} + \beta_{3} - \beta_{6} ) q^{3} + ( 3 + 3 \beta_{1} + \beta_{5} - \beta_{7} ) q^{4} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{5} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{6} + ( -2 \beta_{2} - 6 \beta_{3} ) q^{8} + ( -3 \beta_{1} - 4 \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{3} - \beta_{4} ) q^{2} + ( 2 + 2 \beta_{1} + \beta_{3} - \beta_{6} ) q^{3} + ( 3 + 3 \beta_{1} + \beta_{5} - \beta_{7} ) q^{4} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{5} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{6} + ( -2 \beta_{2} - 6 \beta_{3} ) q^{8} + ( -3 \beta_{1} - 4 \beta_{6} ) q^{9} + ( -7 - 7 \beta_{1} + 8 \beta_{3} + 2 \beta_{4} - \beta_{5} - 8 \beta_{6} + \beta_{7} ) q^{10} + ( -4 - 4 \beta_{1} + 6 \beta_{3} - 6 \beta_{6} ) q^{11} + ( 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{12} + ( 2 \beta_{2} + \beta_{3} - 2 \beta_{7} ) q^{13} -2 \beta_{7} q^{15} + ( 2 \beta_{1} + 6 \beta_{5} ) q^{16} + ( 18 + 18 \beta_{1} + 4 \beta_{3} - 4 \beta_{6} ) q^{17} + ( -4 - 4 \beta_{1} - 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{7} ) q^{18} + ( -2 \beta_{1} - 19 \beta_{6} ) q^{19} + ( -14 + 6 \beta_{2} + 10 \beta_{3} - 6 \beta_{7} ) q^{20} + ( -6 + 4 \beta_{2} + 4 \beta_{3} - 6 \beta_{7} ) q^{22} + ( 16 \beta_{2} + 16 \beta_{3} + 16 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} ) q^{23} + ( -10 - 10 \beta_{1} - 8 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 8 \beta_{6} - 2 \beta_{7} ) q^{24} + ( 17 + 17 \beta_{1} - 28 \beta_{3} + 28 \beta_{6} ) q^{25} + ( -7 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 8 \beta_{6} ) q^{26} + ( 16 + 4 \beta_{3} ) q^{27} + ( 12 \beta_{2} + 6 \beta_{3} ) q^{29} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 8 \beta_{6} ) q^{30} + ( -4 \beta_{3} - 8 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} + 12 \beta_{7} ) q^{31} + ( -24 \beta_{3} - 4 \beta_{4} + 24 \beta_{6} ) q^{32} + ( 4 \beta_{1} - 8 \beta_{6} ) q^{33} + ( -4 - 18 \beta_{2} - 18 \beta_{3} - 4 \beta_{7} ) q^{34} + ( 9 + 8 \beta_{2} - 8 \beta_{3} - 3 \beta_{7} ) q^{36} + ( -20 \beta_{2} - 20 \beta_{3} - 20 \beta_{4} - 8 \beta_{5} + 10 \beta_{6} ) q^{37} + ( -19 - 19 \beta_{1} - 2 \beta_{4} + 19 \beta_{5} - 19 \beta_{7} ) q^{38} + ( 2 \beta_{5} - 2 \beta_{7} ) q^{39} + ( -14 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} + 20 \beta_{4} - 10 \beta_{5} - 24 \beta_{6} ) q^{40} + ( 10 + 12 \beta_{3} ) q^{41} + ( -20 + 2 \beta_{3} ) q^{43} + ( -12 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} + 12 \beta_{4} - 4 \beta_{5} - 24 \beta_{6} ) q^{44} + ( 11 \beta_{3} + 22 \beta_{4} - 14 \beta_{5} - 11 \beta_{6} + 14 \beta_{7} ) q^{45} + ( -56 - 56 \beta_{1} - 8 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{46} + ( 8 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} ) q^{47} + ( -4 + 12 \beta_{2} + 4 \beta_{3} + 12 \beta_{7} ) q^{48} + ( 28 - 17 \beta_{2} - 17 \beta_{3} + 28 \beta_{7} ) q^{50} + ( 44 \beta_{1} - 26 \beta_{6} ) q^{51} + ( -14 - 14 \beta_{1} + 4 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{52} + ( -12 \beta_{3} - 24 \beta_{4} - 20 \beta_{5} + 12 \beta_{6} + 20 \beta_{7} ) q^{53} + ( 4 \beta_{1} - 16 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} - 4 \beta_{5} ) q^{54} + ( -32 \beta_{2} - 16 \beta_{3} + 20 \beta_{7} ) q^{55} + ( -34 - 36 \beta_{3} ) q^{57} + ( -42 \beta_{1} - 6 \beta_{5} ) q^{58} + ( 46 + 46 \beta_{1} + 11 \beta_{3} - 11 \beta_{6} ) q^{59} + ( -14 - 14 \beta_{1} + 6 \beta_{5} - 6 \beta_{7} ) q^{60} + ( 6 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} + 10 \beta_{5} - 3 \beta_{6} ) q^{61} + ( 28 - 12 \beta_{2} + 36 \beta_{3} - 4 \beta_{7} ) q^{62} + ( 36 + 20 \beta_{7} ) q^{64} + ( 42 \beta_{1} + 28 \beta_{6} ) q^{65} + ( -8 - 8 \beta_{1} + 4 \beta_{4} + 8 \beta_{5} - 8 \beta_{7} ) q^{66} + ( 56 + 56 \beta_{1} - 16 \beta_{3} + 16 \beta_{6} ) q^{67} + ( 54 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} + 18 \beta_{5} - 16 \beta_{6} ) q^{68} + ( 36 \beta_{2} + 18 \beta_{3} + 20 \beta_{7} ) q^{69} + ( -32 \beta_{2} - 16 \beta_{3} - 16 \beta_{7} ) q^{71} + ( -40 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} - 12 \beta_{6} ) q^{72} + ( 58 + 58 \beta_{1} + 8 \beta_{3} - 8 \beta_{6} ) q^{73} + ( 70 + 70 \beta_{1} + 32 \beta_{3} + 8 \beta_{4} + 10 \beta_{5} - 32 \beta_{6} - 10 \beta_{7} ) q^{74} + ( -22 \beta_{1} + 39 \beta_{6} ) q^{75} + ( 6 + 38 \beta_{2} - 38 \beta_{3} - 2 \beta_{7} ) q^{76} + ( 2 \beta_{2} - 6 \beta_{3} ) q^{78} + ( 32 \beta_{2} + 32 \beta_{3} + 32 \beta_{4} - 8 \beta_{5} - 16 \beta_{6} ) q^{79} + ( -84 - 84 \beta_{1} + 40 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 40 \beta_{6} - 4 \beta_{7} ) q^{80} + ( 13 + 13 \beta_{1} + 60 \beta_{3} - 60 \beta_{6} ) q^{81} + ( 12 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} - 10 \beta_{4} - 12 \beta_{5} ) q^{82} + ( -22 + 13 \beta_{3} ) q^{83} + ( 20 \beta_{2} + 10 \beta_{3} - 28 \beta_{7} ) q^{85} + ( 2 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} + 20 \beta_{4} - 2 \beta_{5} ) q^{86} + ( -12 \beta_{3} - 24 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{87} + ( -60 - 60 \beta_{1} + 16 \beta_{3} - 8 \beta_{4} + 12 \beta_{5} - 16 \beta_{6} - 12 \beta_{7} ) q^{88} + ( -78 \beta_{1} - 24 \beta_{6} ) q^{89} + ( -77 - 14 \beta_{2} + 42 \beta_{3} + 11 \beta_{7} ) q^{90} + ( 14 + 48 \beta_{2} + 80 \beta_{3} + 6 \beta_{7} ) q^{92} + ( -40 \beta_{2} - 40 \beta_{3} - 40 \beta_{4} - 32 \beta_{5} + 20 \beta_{6} ) q^{93} + ( -28 - 28 \beta_{1} - 32 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} + 32 \beta_{6} + 4 \beta_{7} ) q^{94} + ( 40 \beta_{3} + 80 \beta_{4} - 42 \beta_{5} - 40 \beta_{6} + 42 \beta_{7} ) q^{95} + ( -44 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} - 4 \beta_{5} + 48 \beta_{6} ) q^{96} + ( 34 - 92 \beta_{3} ) q^{97} + ( -60 + 34 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{3} + 12q^{4} - 8q^{6} + 12q^{9} + O(q^{10})$$ $$8q + 8q^{3} + 12q^{4} - 8q^{6} + 12q^{9} - 28q^{10} - 16q^{11} - 24q^{12} - 8q^{16} + 72q^{17} - 16q^{18} + 8q^{19} - 112q^{20} - 48q^{22} - 40q^{24} + 68q^{25} + 28q^{26} + 128q^{27} - 16q^{33} - 32q^{34} + 72q^{36} - 76q^{38} + 56q^{40} + 80q^{41} - 160q^{43} + 48q^{44} - 224q^{46} - 32q^{48} + 224q^{50} - 176q^{51} - 56q^{52} - 16q^{54} - 272q^{57} + 168q^{58} + 184q^{59} - 56q^{60} + 224q^{62} + 288q^{64} - 168q^{65} - 32q^{66} + 224q^{67} - 216q^{68} + 160q^{72} + 232q^{73} + 280q^{74} + 88q^{75} + 48q^{76} - 336q^{80} + 52q^{81} - 48q^{82} - 176q^{83} - 8q^{86} - 240q^{88} + 312q^{89} - 616q^{90} + 112q^{92} - 112q^{94} + 176q^{96} + 272q^{97} - 480q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 18 x^{6} - 40 x^{5} + 83 x^{4} - 104 x^{3} + 22 x^{2} + 24 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} - 6 \nu^{5} + 31 \nu^{4} - 52 \nu^{3} + 125 \nu^{2} - 100 \nu - 34$$$$)/18$$ $$\beta_{2}$$ $$=$$ $$($$$$-23 \nu^{7} - 176 \nu^{6} + 410 \nu^{5} - 3150 \nu^{4} + 5009 \nu^{3} - 13942 \nu^{2} + 13100 \nu + 2464$$$$)/1026$$ $$\beta_{3}$$ $$=$$ $$($$$$46 \nu^{7} - 161 \nu^{6} + 719 \nu^{5} - 1395 \nu^{4} + 2807 \nu^{3} - 2896 \nu^{2} - 1576 \nu + 1228$$$$)/1026$$ $$\beta_{4}$$ $$=$$ $$($$$$101 \nu^{7} - 211 \nu^{6} + 1285 \nu^{5} - 1317 \nu^{4} + 4276 \nu^{3} - 614 \nu^{2} - 4184 \nu - 2860$$$$)/1026$$ $$\beta_{5}$$ $$=$$ $$($$$$176 \nu^{7} - 616 \nu^{6} + 2974 \nu^{5} - 5895 \nu^{4} + 13684 \nu^{3} - 14939 \nu^{2} + 5836 \nu - 610$$$$)/1026$$ $$\beta_{6}$$ $$=$$ $$($$$$124 \nu^{7} - 434 \nu^{6} + 2072 \nu^{5} - 4095 \nu^{4} + 9128 \nu^{3} - 9814 \nu^{2} + 1127 \nu + 946$$$$)/513$$ $$\beta_{7}$$ $$=$$ $$($$$$248 \nu^{7} - 868 \nu^{6} + 4144 \nu^{5} - 8190 \nu^{4} + 18256 \nu^{3} - 19628 \nu^{2} + 3280 \nu + 1379$$$$)/513$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{6} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{7} + 12 \beta_{6} + 6 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-15 \beta_{7} + 16 \beta_{6} + 12 \beta_{5} + 8 \beta_{4} + 24 \beta_{3} + 24 \beta_{2} + 44 \beta_{1} + 37$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$19 \beta_{7} - 12 \beta_{6} - 40 \beta_{5} + 30 \beta_{4} - 18 \beta_{3} + 70 \beta_{2} + 120 \beta_{1} + 111$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$95 \beta_{7} - 72 \beta_{6} - 150 \beta_{5} + 60 \beta_{4} - 228 \beta_{3} - 68 \beta_{2} - 210 \beta_{1} - 5$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$105 \beta_{7} - 380 \beta_{6} + 98 \beta_{5} + 98 \beta_{4} + 126 \beta_{3} - 490 \beta_{2} - 1162 \beta_{1} - 419$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$295$$ $$297$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.207107 − 2.54762i 1.20711 + 2.54762i −0.207107 + 0.0981308i 1.20711 − 0.0981308i −0.207107 + 2.54762i 1.20711 − 2.54762i −0.207107 − 0.0981308i 1.20711 + 0.0981308i
−1.97374 + 0.323042i 1.70711 2.95680i 3.79129 1.27520i −1.34221 + 0.774923i −2.41421 + 6.38741i 0 −7.07107 + 3.74166i −1.32843 2.30090i 2.39883 1.96308i
67.2 −1.26663 + 1.54779i 0.292893 0.507306i −0.791288 3.92095i 7.82295 4.51658i 0.414214 + 1.09591i 0 7.07107 + 3.74166i 4.32843 + 7.49706i −2.91809 + 17.8291i
67.3 1.26663 1.54779i 1.70711 2.95680i −0.791288 3.92095i 1.34221 0.774923i −2.41421 6.38741i 0 −7.07107 3.74166i −1.32843 2.30090i 0.500665 3.05899i
67.4 1.97374 0.323042i 0.292893 0.507306i 3.79129 1.27520i −7.82295 + 4.51658i 0.414214 1.09591i 0 7.07107 3.74166i 4.32843 + 7.49706i −13.9814 + 11.4417i
275.1 −1.97374 0.323042i 1.70711 + 2.95680i 3.79129 + 1.27520i −1.34221 0.774923i −2.41421 6.38741i 0 −7.07107 3.74166i −1.32843 + 2.30090i 2.39883 + 1.96308i
275.2 −1.26663 1.54779i 0.292893 + 0.507306i −0.791288 + 3.92095i 7.82295 + 4.51658i 0.414214 1.09591i 0 7.07107 3.74166i 4.32843 7.49706i −2.91809 17.8291i
275.3 1.26663 + 1.54779i 1.70711 + 2.95680i −0.791288 + 3.92095i 1.34221 + 0.774923i −2.41421 + 6.38741i 0 −7.07107 + 3.74166i −1.32843 + 2.30090i 0.500665 + 3.05899i
275.4 1.97374 + 0.323042i 0.292893 + 0.507306i 3.79129 + 1.27520i −7.82295 4.51658i 0.414214 + 1.09591i 0 7.07107 + 3.74166i 4.32843 7.49706i −13.9814 11.4417i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 275.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.j 8
7.b odd 2 1 392.3.k.i 8
7.c even 3 1 392.3.g.h 4
7.c even 3 1 inner 392.3.k.j 8
7.d odd 6 1 56.3.g.a 4
7.d odd 6 1 392.3.k.i 8
8.d odd 2 1 inner 392.3.k.j 8
21.g even 6 1 504.3.g.a 4
28.f even 6 1 224.3.g.a 4
28.g odd 6 1 1568.3.g.h 4
56.e even 2 1 392.3.k.i 8
56.j odd 6 1 224.3.g.a 4
56.k odd 6 1 392.3.g.h 4
56.k odd 6 1 inner 392.3.k.j 8
56.m even 6 1 56.3.g.a 4
56.m even 6 1 392.3.k.i 8
56.p even 6 1 1568.3.g.h 4
84.j odd 6 1 2016.3.g.a 4
112.v even 12 2 1792.3.d.g 8
112.x odd 12 2 1792.3.d.g 8
168.ba even 6 1 2016.3.g.a 4
168.be odd 6 1 504.3.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a 4 7.d odd 6 1
56.3.g.a 4 56.m even 6 1
224.3.g.a 4 28.f even 6 1
224.3.g.a 4 56.j odd 6 1
392.3.g.h 4 7.c even 3 1
392.3.g.h 4 56.k odd 6 1
392.3.k.i 8 7.b odd 2 1
392.3.k.i 8 7.d odd 6 1
392.3.k.i 8 56.e even 2 1
392.3.k.i 8 56.m even 6 1
392.3.k.j 8 1.a even 1 1 trivial
392.3.k.j 8 7.c even 3 1 inner
392.3.k.j 8 8.d odd 2 1 inner
392.3.k.j 8 56.k odd 6 1 inner
504.3.g.a 4 21.g even 6 1
504.3.g.a 4 168.be odd 6 1
1568.3.g.h 4 28.g odd 6 1
1568.3.g.h 4 56.p even 6 1
1792.3.d.g 8 112.v even 12 2
1792.3.d.g 8 112.x odd 12 2
2016.3.g.a 4 84.j odd 6 1
2016.3.g.a 4 168.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(392, [\chi])$$:

 $$T_{3}^{4} - 4 T_{3}^{3} + 14 T_{3}^{2} - 8 T_{3} + 4$$ $$T_{5}^{8} - 84 T_{5}^{6} + 6860 T_{5}^{4} - 16464 T_{5}^{2} + 38416$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T^{2} + 20 T^{4} - 96 T^{6} + 256 T^{8}$$
$3$ $$( 1 - 4 T - 4 T^{2} - 8 T^{3} + 175 T^{4} - 72 T^{5} - 324 T^{6} - 2916 T^{7} + 6561 T^{8} )^{2}$$
$5$ $$1 + 16 T^{2} + 510 T^{4} - 24064 T^{6} - 486109 T^{8} - 15040000 T^{10} + 199218750 T^{12} + 3906250000 T^{14} + 152587890625 T^{16}$$
$7$ 1
$11$ $$( 1 + 8 T - 122 T^{2} - 448 T^{3} + 12211 T^{4} - 54208 T^{5} - 1786202 T^{6} + 14172488 T^{7} + 214358881 T^{8} )^{2}$$
$13$ $$( 1 - 592 T^{2} + 143170 T^{4} - 16908112 T^{6} + 815730721 T^{8} )^{2}$$
$17$ $$( 1 - 36 T + 426 T^{2} - 10512 T^{3} + 298835 T^{4} - 3037968 T^{5} + 35579946 T^{6} - 868952484 T^{7} + 6975757441 T^{8} )^{2}$$
$19$ $$( 1 - 4 T + 12 T^{2} + 2872 T^{3} - 136081 T^{4} + 1036792 T^{5} + 1563852 T^{6} - 188183524 T^{7} + 16983563041 T^{8} )^{2}$$
$23$ $$1 + 268 T^{2} - 405462 T^{4} - 22082128 T^{6} + 129391640531 T^{8} - 6179484781648 T^{10} - 31752128714004822 T^{12} + 5873119347781446028 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$( 1 - 1178 T^{2} + 707281 T^{4} )^{4}$$
$31$ $$1 + 1380 T^{2} + 484426 T^{4} - 589353840 T^{6} - 595318269165 T^{8} - 544280647670640 T^{10} + 413162593703393866 T^{12} +$$$$10\!\cdots\!80$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 1780 T^{2} + 1136778 T^{4} - 3055726000 T^{6} - 5323083215437 T^{8} - 5726922495886000 T^{10} + 3992909368669406538 T^{12} +$$$$11\!\cdots\!80$$$$T^{14} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$( 1 - 20 T + 3174 T^{2} - 33620 T^{3} + 2825761 T^{4} )^{4}$$
$43$ $$( 1 + 40 T + 4090 T^{2} + 73960 T^{3} + 3418801 T^{4} )^{4}$$
$47$ $$1 + 7492 T^{2} + 32739594 T^{4} + 102124261136 T^{6} + 249405993329555 T^{8} + 498333816704377616 T^{10} +$$$$77\!\cdots\!34$$$$T^{12} +$$$$87\!\cdots\!72$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 + 1604 T^{2} + 8727850 T^{4} - 35185337584 T^{6} - 44676002634701 T^{8} - 277629237685137904 T^{10} +$$$$54\!\cdots\!50$$$$T^{12} +$$$$78\!\cdots\!64$$$$T^{14} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$( 1 - 92 T - 372 T^{2} - 172408 T^{3} + 36494351 T^{4} - 600152248 T^{5} - 4507658292 T^{6} - 3880609094972 T^{7} + 146830437604321 T^{8} )^{2}$$
$61$ $$1 + 13232 T^{2} + 103975486 T^{4} + 574515656192 T^{6} + 2440762377018979 T^{8} + 7954652427645097472 T^{10} +$$$$19\!\cdots\!66$$$$T^{12} +$$$$35\!\cdots\!72$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$( 1 - 112 T + 942 T^{2} - 293888 T^{3} + 58145267 T^{4} - 1319263232 T^{5} + 18982355982 T^{6} - 10131338802928 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 9412 T^{2} + 47279686 T^{4} - 239174741572 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 116 T - 438 T^{2} - 375376 T^{3} + 92937971 T^{4} - 2000378704 T^{5} - 12438429558 T^{6} - 17554770249524 T^{7} + 806460091894081 T^{8} )^{2}$$
$79$ $$1 + 16900 T^{2} + 142729866 T^{4} + 1098161526800 T^{6} + 7773091247541395 T^{8} + 42773480419943670800 T^{10} +$$$$21\!\cdots\!26$$$$T^{12} +$$$$99\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$( 1 + 44 T + 13924 T^{2} + 303116 T^{3} + 47458321 T^{4} )^{4}$$
$89$ $$( 1 - 156 T + 3562 T^{2} - 769392 T^{3} + 176051379 T^{4} - 6094354032 T^{5} + 223487862442 T^{6} - 77529081389916 T^{7} + 3936588805702081 T^{8} )^{2}$$
$97$ $$( 1 - 68 T + 3046 T^{2} - 639812 T^{3} + 88529281 T^{4} )^{4}$$