# Properties

 Label 15.4.e.a Level 15 Weight 4 Character orbit 15.e Analytic conductor 0.885 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 15.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.885028650086$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.28356903014400.8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( -1 - \beta_{2} + \beta_{5} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{5} + ( -1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{6} + ( -1 + \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} + ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( -1 - \beta_{2} + \beta_{5} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{5} + ( -1 + 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{6} + ( -1 + \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} + ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -3 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{9} + ( -15 - 10 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{10} + ( -7 \beta_{1} - 7 \beta_{3} + 2 \beta_{6} ) q^{11} + ( 17 - 17 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{12} + ( 8 + 8 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} + ( 7 \beta_{1} - 7 \beta_{3} - 8 \beta_{7} ) q^{14} + ( 10 + 9 \beta_{1} + 25 \beta_{2} - 12 \beta_{3} + 5 \beta_{5} - 6 \beta_{6} + 3 \beta_{7} ) q^{15} + ( 39 + 7 \beta_{4} - 7 \beta_{5} ) q^{16} + ( 20 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{17} + ( -30 - 21 \beta_{1} - 30 \beta_{2} - 6 \beta_{6} + 6 \beta_{7} ) q^{18} + ( -12 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{19} + ( 2 \beta_{1} + 19 \beta_{3} + 2 \beta_{6} - 6 \beta_{7} ) q^{20} + ( -62 + 6 \beta_{1} + 6 \beta_{3} - \beta_{4} + \beta_{5} - 11 \beta_{6} ) q^{21} + ( -65 + 65 \beta_{2} - 10 \beta_{4} + 5 \beta_{6} - 5 \beta_{7} ) q^{22} + ( 14 \beta_{1} + 9 \beta_{6} - 9 \beta_{7} ) q^{23} + ( -3 \beta_{1} + 39 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 21 \beta_{7} ) q^{24} + ( -20 - 85 \beta_{2} + 20 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{25} + ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{6} ) q^{26} + ( 87 - 87 \beta_{2} - 18 \beta_{3} - 3 \beta_{4} - 6 \beta_{6} - 9 \beta_{7} ) q^{27} + ( 63 + 63 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{28} + ( -3 \beta_{1} + 3 \beta_{3} + 22 \beta_{7} ) q^{29} + ( 85 - 15 \beta_{1} + 105 \beta_{2} - 10 \beta_{4} - 20 \beta_{5} + 25 \beta_{6} - 15 \beta_{7} ) q^{30} + ( 62 - 30 \beta_{4} + 30 \beta_{5} ) q^{31} + ( -43 \beta_{3} - 30 \beta_{6} - 30 \beta_{7} ) q^{32} + ( -25 + 36 \beta_{1} - 25 \beta_{2} - 20 \beta_{5} + 27 \beta_{6} - 27 \beta_{7} ) q^{33} + ( -166 \beta_{2} + 34 \beta_{4} + 34 \beta_{5} - 34 \beta_{6} + 34 \beta_{7} ) q^{34} + ( -33 \beta_{1} - 31 \beta_{3} + 17 \beta_{6} + 19 \beta_{7} ) q^{35} + ( -105 + 6 \beta_{1} + 6 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} + 24 \beta_{6} ) q^{36} + ( -146 + 146 \beta_{2} - 6 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{37} + ( -12 \beta_{1} - 12 \beta_{6} + 12 \beta_{7} ) q^{38} + ( -3 \beta_{1} + 16 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} - 6 \beta_{7} ) q^{39} + ( -70 - 55 \beta_{2} - 15 \beta_{4} - 35 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{40} + ( 32 \beta_{1} + 32 \beta_{3} - 52 \beta_{6} ) q^{41} + ( 65 - 65 \beta_{2} + 66 \beta_{3} - 10 \beta_{4} + 7 \beta_{6} - 3 \beta_{7} ) q^{42} + ( 78 + 78 \beta_{2} - 38 \beta_{5} + 19 \beta_{6} - 19 \beta_{7} ) q^{43} + ( -29 \beta_{1} + 29 \beta_{3} + 36 \beta_{7} ) q^{44} + ( 30 + 12 \beta_{1} + 90 \beta_{2} + 69 \beta_{3} + 45 \beta_{4} + 15 \beta_{5} - 18 \beta_{6} + 24 \beta_{7} ) q^{45} + ( 108 + 32 \beta_{4} - 32 \beta_{5} ) q^{46} + ( 2 \beta_{3} + 21 \beta_{6} + 21 \beta_{7} ) q^{47} + ( -151 + 21 \beta_{1} - 151 \beta_{2} + 46 \beta_{5} - 21 \beta_{6} + 21 \beta_{7} ) q^{48} + ( 85 \beta_{2} - 16 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} ) q^{49} + ( 105 \beta_{1} - 40 \beta_{3} - 20 \beta_{6} - 40 \beta_{7} ) q^{50} + ( -106 - 81 \beta_{1} - 81 \beta_{3} + \beta_{4} - \beta_{5} - 19 \beta_{6} ) q^{51} + ( 24 - 24 \beta_{2} - 16 \beta_{4} + 8 \beta_{6} - 8 \beta_{7} ) q^{52} + ( -130 \beta_{1} + \beta_{6} - \beta_{7} ) q^{53} + ( 81 \beta_{1} + 147 \beta_{2} - 81 \beta_{3} - 33 \beta_{4} - 33 \beta_{5} + 33 \beta_{6} - 27 \beta_{7} ) q^{54} + ( 35 - 145 \beta_{2} - 10 \beta_{4} + 80 \beta_{5} - 35 \beta_{6} + 35 \beta_{7} ) q^{55} + ( -3 \beta_{1} - 3 \beta_{3} + 68 \beta_{6} ) q^{56} + ( 84 - 84 \beta_{2} - 18 \beta_{3} - 6 \beta_{4} + 24 \beta_{6} + 18 \beta_{7} ) q^{57} + ( -5 - 5 \beta_{2} + 50 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{58} + ( 39 \beta_{1} - 39 \beta_{3} - 136 \beta_{7} ) q^{59} + ( 55 - 69 \beta_{1} - 70 \beta_{2} - 33 \beta_{3} - 35 \beta_{4} + 15 \beta_{5} - 34 \beta_{6} - 18 \beta_{7} ) q^{60} + 2 q^{61} + ( 58 \beta_{3} + 60 \beta_{6} + 60 \beta_{7} ) q^{62} + ( 183 - 6 \beta_{1} + 183 \beta_{2} - 18 \beta_{5} - 48 \beta_{6} + 48 \beta_{7} ) q^{63} + ( 15 \beta_{2} - 47 \beta_{4} - 47 \beta_{5} + 47 \beta_{6} - 47 \beta_{7} ) q^{64} + ( 8 \beta_{1} + 6 \beta_{3} - 17 \beta_{6} + 31 \beta_{7} ) q^{65} + ( 290 - 15 \beta_{1} - 15 \beta_{3} + 70 \beta_{4} - 70 \beta_{5} - 40 \beta_{6} ) q^{66} + ( 84 - 84 \beta_{2} + 134 \beta_{4} - 67 \beta_{6} + 67 \beta_{7} ) q^{67} + ( 142 \beta_{1} + 12 \beta_{6} - 12 \beta_{7} ) q^{68} + ( -69 \beta_{1} - 148 \beta_{2} + 69 \beta_{3} - 13 \beta_{4} - 13 \beta_{5} + 13 \beta_{6} - 12 \beta_{7} ) q^{69} + ( -295 + 315 \beta_{2} - 30 \beta_{4} + 40 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{70} + ( 116 \beta_{1} + 116 \beta_{3} + 74 \beta_{6} ) q^{71} + ( -210 + 210 \beta_{2} - 27 \beta_{3} + 60 \beta_{4} - 60 \beta_{6} ) q^{72} + ( -317 - 317 \beta_{2} + 12 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{73} + ( -158 \beta_{1} + 158 \beta_{3} + 12 \beta_{7} ) q^{74} + ( -385 + 45 \beta_{1} - 55 \beta_{2} + 15 \beta_{3} - 90 \beta_{4} - 5 \beta_{5} + 30 \beta_{6} + 30 \beta_{7} ) q^{75} + ( -180 + 12 \beta_{4} - 12 \beta_{5} ) q^{76} + ( -74 \beta_{3} - 78 \beta_{6} - 78 \beta_{7} ) q^{77} + ( -40 + 12 \beta_{1} - 40 \beta_{2} - 20 \beta_{5} + 24 \beta_{6} - 24 \beta_{7} ) q^{78} + ( 298 \beta_{2} - 56 \beta_{4} - 56 \beta_{5} + 56 \beta_{6} - 56 \beta_{7} ) q^{79} + ( -197 \beta_{1} + 46 \beta_{3} - 22 \beta_{6} + 46 \beta_{7} ) q^{80} + ( -9 + 72 \beta_{1} + 72 \beta_{3} - 90 \beta_{4} + 90 \beta_{5} + 108 \beta_{6} ) q^{81} + ( 340 - 340 \beta_{2} - 40 \beta_{4} + 20 \beta_{6} - 20 \beta_{7} ) q^{82} + ( 86 \beta_{1} - 7 \beta_{6} + 7 \beta_{7} ) q^{83} + ( -3 \beta_{1} - 94 \beta_{2} + 3 \beta_{3} + 62 \beta_{4} + 62 \beta_{5} - 62 \beta_{6} - 6 \beta_{7} ) q^{84} + ( 650 + 60 \beta_{2} + 30 \beta_{4} - 100 \beta_{5} + 35 \beta_{6} - 35 \beta_{7} ) q^{85} + ( -154 \beta_{1} - 154 \beta_{3} - 76 \beta_{6} ) q^{86} + ( -195 + 195 \beta_{2} - 84 \beta_{3} + 60 \beta_{4} - 3 \beta_{6} + 57 \beta_{7} ) q^{87} + ( 295 + 295 \beta_{2} + 50 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{88} + ( 276 \beta_{1} - 276 \beta_{3} + 126 \beta_{7} ) q^{89} + ( 90 + 30 \beta_{1} - 615 \beta_{2} - 90 \beta_{3} + 105 \beta_{4} + 45 \beta_{5} - 45 \beta_{6} - 15 \beta_{7} ) q^{90} + ( -144 + 38 \beta_{4} - 38 \beta_{5} ) q^{91} + ( -124 \beta_{3} + 8 \beta_{6} + 8 \beta_{7} ) q^{92} + ( 418 - 90 \beta_{1} + 418 \beta_{2} + 32 \beta_{5} + 90 \beta_{6} - 90 \beta_{7} ) q^{93} + ( 24 \beta_{2} + 44 \beta_{4} + 44 \beta_{5} - 44 \beta_{6} + 44 \beta_{7} ) q^{94} + ( -6 \beta_{1} + 78 \beta_{3} - 6 \beta_{6} - 72 \beta_{7} ) q^{95} + ( 497 + 219 \beta_{1} + 219 \beta_{3} - 47 \beta_{4} + 47 \beta_{5} - 4 \beta_{6} ) q^{96} + ( 179 - 179 \beta_{2} - 236 \beta_{4} + 118 \beta_{6} - 118 \beta_{7} ) q^{97} + ( -149 \beta_{1} - 32 \beta_{6} + 32 \beta_{7} ) q^{98} + ( -129 \beta_{1} - 540 \beta_{2} + 129 \beta_{3} - 30 \beta_{4} - 30 \beta_{5} + 30 \beta_{6} + 66 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{3} - 12q^{6} - 16q^{7} + O(q^{10})$$ $$8q - 6q^{3} - 12q^{6} - 16q^{7} - 100q^{10} + 132q^{12} + 68q^{13} + 90q^{15} + 284q^{16} - 240q^{18} - 492q^{21} - 500q^{22} - 220q^{25} + 702q^{27} + 508q^{28} + 660q^{30} + 616q^{31} - 240q^{33} - 804q^{36} - 1156q^{37} - 600q^{40} + 540q^{42} + 548q^{43} + 180q^{45} + 736q^{46} - 1116q^{48} - 852q^{51} + 224q^{52} + 460q^{55} + 684q^{57} + 60q^{58} + 540q^{60} + 16q^{61} + 1428q^{63} + 2040q^{66} + 404q^{67} - 2220q^{70} - 1800q^{72} - 2512q^{73} - 2910q^{75} - 1488q^{76} - 360q^{78} + 288q^{81} + 2800q^{82} + 4940q^{85} - 1680q^{87} + 2460q^{88} + 600q^{90} - 1304q^{91} + 3408q^{93} + 4164q^{96} + 1904q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 209 x^{4} + 1600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 249 \nu^{2}$$$$)/680$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 249 \nu^{3}$$$$)/680$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{6} - 20 \nu^{5} - 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu - 4520$$$$)/1360$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{6} - 20 \nu^{5} + 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu + 4520$$$$)/1360$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{7} - 40 \nu^{5} - 2557 \nu^{3} - 7240 \nu$$$$)/2720$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{7} + 40 \nu^{5} - 2557 \nu^{3} + 7240 \nu$$$$)/2720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 9 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{6} + 13 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$17 \beta_{5} - 17 \beta_{4} - 113$$ $$\nu^{5}$$ $$=$$ $$34 \beta_{7} - 34 \beta_{6} - 181 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-249 \beta_{7} + 249 \beta_{6} - 249 \beta_{5} - 249 \beta_{4} + 1561 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$-498 \beta_{7} - 498 \beta_{6} - 2557 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −2.66260 − 2.66260i −1.18766 − 1.18766i 1.18766 + 1.18766i 2.66260 + 2.66260i −2.66260 + 2.66260i −1.18766 + 1.18766i 1.18766 − 1.18766i 2.66260 − 2.66260i
−2.66260 + 2.66260i −4.37420 + 2.80471i 6.17891i 9.55729 + 5.80157i 4.17891 19.1146i 9.35782 + 9.35782i −4.84884 4.84884i 11.2672 24.5367i −40.8945 + 10.0000i
2.2 −1.18766 + 1.18766i 5.11173 + 0.932827i 5.17891i −2.48157 10.9015i −7.17891 + 4.96314i −13.3578 13.3578i −15.6521 15.6521i 25.2597 + 9.53673i 15.8945 + 10.0000i
2.3 1.18766 1.18766i −0.932827 5.11173i 5.17891i 2.48157 + 10.9015i −7.17891 4.96314i −13.3578 13.3578i 15.6521 + 15.6521i −25.2597 + 9.53673i 15.8945 + 10.0000i
2.4 2.66260 2.66260i −2.80471 + 4.37420i 6.17891i −9.55729 5.80157i 4.17891 + 19.1146i 9.35782 + 9.35782i 4.84884 + 4.84884i −11.2672 24.5367i −40.8945 + 10.0000i
8.1 −2.66260 2.66260i −4.37420 2.80471i 6.17891i 9.55729 5.80157i 4.17891 + 19.1146i 9.35782 9.35782i −4.84884 + 4.84884i 11.2672 + 24.5367i −40.8945 10.0000i
8.2 −1.18766 1.18766i 5.11173 0.932827i 5.17891i −2.48157 + 10.9015i −7.17891 4.96314i −13.3578 + 13.3578i −15.6521 + 15.6521i 25.2597 9.53673i 15.8945 10.0000i
8.3 1.18766 + 1.18766i −0.932827 + 5.11173i 5.17891i 2.48157 10.9015i −7.17891 + 4.96314i −13.3578 + 13.3578i 15.6521 15.6521i −25.2597 9.53673i 15.8945 10.0000i
8.4 2.66260 + 2.66260i −2.80471 4.37420i 6.17891i −9.55729 + 5.80157i 4.17891 19.1146i 9.35782 9.35782i 4.84884 4.84884i −11.2672 + 24.5367i −40.8945 10.0000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.4.e.a 8
3.b odd 2 1 inner 15.4.e.a 8
4.b odd 2 1 240.4.v.c 8
5.b even 2 1 75.4.e.c 8
5.c odd 4 1 inner 15.4.e.a 8
5.c odd 4 1 75.4.e.c 8
12.b even 2 1 240.4.v.c 8
15.d odd 2 1 75.4.e.c 8
15.e even 4 1 inner 15.4.e.a 8
15.e even 4 1 75.4.e.c 8
20.e even 4 1 240.4.v.c 8
60.l odd 4 1 240.4.v.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.e.a 8 1.a even 1 1 trivial
15.4.e.a 8 3.b odd 2 1 inner
15.4.e.a 8 5.c odd 4 1 inner
15.4.e.a 8 15.e even 4 1 inner
75.4.e.c 8 5.b even 2 1
75.4.e.c 8 5.c odd 4 1
75.4.e.c 8 15.d odd 2 1
75.4.e.c 8 15.e even 4 1
240.4.v.c 8 4.b odd 2 1
240.4.v.c 8 12.b even 2 1
240.4.v.c 8 20.e even 4 1
240.4.v.c 8 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 79 T^{4} + 2496 T^{8} - 323584 T^{12} + 16777216 T^{16}$$
$3$ $$1 + 6 T + 18 T^{2} - 198 T^{3} - 1422 T^{4} - 5346 T^{5} + 13122 T^{6} + 118098 T^{7} + 531441 T^{8}$$
$5$ $$1 + 110 T^{2} + 5250 T^{4} + 1718750 T^{6} + 244140625 T^{8}$$
$7$ $$( 1 + 8 T + 32 T^{2} + 744 T^{3} - 45202 T^{4} + 255192 T^{5} + 3764768 T^{6} + 322828856 T^{7} + 13841287201 T^{8} )^{2}$$
$11$ $$( 1 - 3334 T^{2} + 6292986 T^{4} - 5906384374 T^{6} + 3138428376721 T^{8} )^{2}$$
$13$ $$( 1 - 34 T + 578 T^{2} - 77418 T^{3} + 10363058 T^{4} - 170087346 T^{5} + 2789895602 T^{6} - 360552978682 T^{7} + 23298085122481 T^{8} )^{2}$$
$17$ $$1 - 10588864 T^{4} - 447530075229954 T^{8} -$$$$61\!\cdots\!04$$$$T^{12} +$$$$33\!\cdots\!21$$$$T^{16}$$
$19$ $$( 1 - 24664 T^{2} + 245125086 T^{4} - 1160339608984 T^{6} + 2213314919066161 T^{8} )^{2}$$
$23$ $$1 + 232150736 T^{4} + 54876630002867166 T^{8} +$$$$50\!\cdots\!56$$$$T^{12} +$$$$48\!\cdots\!41$$$$T^{16}$$
$29$ $$( 1 + 79166 T^{2} + 2712313506 T^{4} + 47089783030286 T^{6} + 353814783205469041 T^{8} )^{2}$$
$31$ $$( 1 - 154 T + 36486 T^{2} - 4587814 T^{3} + 887503681 T^{4} )^{4}$$
$37$ $$( 1 + 578 T + 167042 T^{2} + 53079474 T^{3} + 15170807378 T^{4} + 2688634596522 T^{5} + 428584070812178 T^{6} + 75117885601554506 T^{7} + 6582952005840035281 T^{8} )^{2}$$
$41$ $$( 1 - 115444 T^{2} + 7614039366 T^{4} - 548371033998004 T^{6} + 22563490300366186081 T^{8} )^{2}$$
$43$ $$( 1 - 274 T + 37538 T^{2} - 17976318 T^{3} + 8415346898 T^{4} - 1429243115226 T^{5} + 237291326133362 T^{6} - 137710375670694982 T^{7} + 39959630797262576401 T^{8} )^{2}$$
$47$ $$1 + 29984206736 T^{4} +$$$$42\!\cdots\!06$$$$T^{8} +$$$$34\!\cdots\!76$$$$T^{12} +$$$$13\!\cdots\!81$$$$T^{16}$$
$53$ $$1 - 24371904064 T^{4} +$$$$31\!\cdots\!06$$$$T^{8} -$$$$11\!\cdots\!24$$$$T^{12} +$$$$24\!\cdots\!81$$$$T^{16}$$
$59$ $$( 1 + 112106 T^{2} + 56049165066 T^{4} + 4728690904357946 T^{6} +$$$$17\!\cdots\!81$$$$T^{8} )^{2}$$
$61$ $$( 1 - 2 T + 226981 T^{2} )^{8}$$
$67$ $$( 1 - 202 T + 20402 T^{2} - 3297246 T^{3} - 80373232942 T^{4} - 991689598698 T^{5} + 1845531913011938 T^{6} - 5495719948051579294 T^{7} +$$$$81\!\cdots\!61$$$$T^{8} )^{2}$$
$71$ $$( 1 - 863584 T^{2} + 377860054206 T^{4} - 110625355589632864 T^{6} +$$$$16\!\cdots\!41$$$$T^{8} )^{2}$$
$73$ $$( 1 + 1256 T + 788768 T^{2} + 733362072 T^{3} + 643873740638 T^{4} + 285290313163224 T^{5} + 119367595001521952 T^{6} + 73942712905584498728 T^{7} +$$$$22\!\cdots\!21$$$$T^{8} )^{2}$$
$79$ $$( 1 - 1624084 T^{2} + 1116095863206 T^{4} - 394794447112367764 T^{6} +$$$$59\!\cdots\!41$$$$T^{8} )^{2}$$
$83$ $$1 + 1005705244496 T^{4} +$$$$46\!\cdots\!26$$$$T^{8} +$$$$10\!\cdots\!56$$$$T^{12} +$$$$11\!\cdots\!21$$$$T^{16}$$
$89$ $$( 1 - 806584 T^{2} + 1118488022286 T^{4} - 400857157588487224 T^{6} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$( 1 - 952 T + 453152 T^{2} - 121725576 T^{3} - 583228842562 T^{4} - 111095646624648 T^{5} + 377462929977586208 T^{6} -$$$$72\!\cdots\!84$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8} )^{2}$$