# Properties

 Label 384.3.h.g Level $384$ Weight $3$ Character orbit 384.h Analytic conductor $10.463$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 28 x^{14} + 274 x^{12} + 1236 x^{10} + 2703 x^{8} + 2676 x^{6} + 946 x^{4} + 64 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{46}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{3} + \beta_{2} q^{5} + \beta_{1} q^{7} + ( -1 - \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{8} q^{3} + \beta_{2} q^{5} + \beta_{1} q^{7} + ( -1 - \beta_{5} ) q^{9} -\beta_{3} q^{11} + \beta_{15} q^{13} -\beta_{13} q^{15} + ( \beta_{4} - \beta_{5} - \beta_{11} ) q^{17} -\beta_{9} q^{19} + ( -\beta_{7} - \beta_{15} ) q^{21} + ( \beta_{12} - \beta_{13} ) q^{23} + ( 3 + \beta_{4} + \beta_{5} ) q^{25} + ( -\beta_{3} - \beta_{9} + \beta_{10} ) q^{27} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{29} + ( -\beta_{13} - \beta_{14} ) q^{31} + ( -4 - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{11} ) q^{33} + ( -3 \beta_{3} + 3 \beta_{8} + \beta_{10} ) q^{35} + ( \beta_{6} + \beta_{7} - \beta_{15} ) q^{37} + ( 5 \beta_{1} + \beta_{12} ) q^{39} + ( -4 \beta_{4} + 4 \beta_{5} - \beta_{11} ) q^{41} + ( 6 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{43} + ( -6 \beta_{2} + 3 \beta_{6} ) q^{45} + ( -\beta_{1} - \beta_{12} - 4 \beta_{13} + \beta_{14} ) q^{47} + ( -5 + \beta_{4} + \beta_{5} ) q^{49} + ( -3 \beta_{3} + \beta_{8} - \beta_{10} ) q^{51} + ( \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( 11 \beta_{1} + \beta_{13} + \beta_{14} ) q^{55} + ( 2 + 6 \beta_{4} - \beta_{5} + 3 \beta_{11} ) q^{57} + ( 9 \beta_{8} + 3 \beta_{10} ) q^{59} + ( -3 \beta_{6} - 3 \beta_{7} - 3 \beta_{15} ) q^{61} + ( -4 \beta_{1} - \beta_{12} - \beta_{14} ) q^{63} + ( 2 \beta_{4} - 2 \beta_{5} - 7 \beta_{11} ) q^{65} + ( -3 \beta_{8} + \beta_{10} ) q^{67} + ( -9 \beta_{2} - 5 \beta_{7} + 4 \beta_{15} ) q^{69} + ( \beta_{1} + 2 \beta_{12} + 3 \beta_{13} - \beta_{14} ) q^{71} + ( -18 - 6 \beta_{4} - 6 \beta_{5} ) q^{73} + ( \beta_{3} + \beta_{8} + \beta_{9} - 4 \beta_{10} ) q^{75} + ( 18 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{77} + ( -12 \beta_{1} - \beta_{13} - \beta_{14} ) q^{79} + ( 25 + 3 \beta_{4} + \beta_{5} + 6 \beta_{11} ) q^{81} + ( 5 \beta_{3} - 6 \beta_{8} - 2 \beta_{10} ) q^{83} + ( 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{15} ) q^{85} + ( -5 \beta_{1} + \beta_{12} + \beta_{13} + \beta_{14} ) q^{87} + ( 7 \beta_{4} - 7 \beta_{5} - 8 \beta_{11} ) q^{89} + ( -21 \beta_{8} + \beta_{9} + 7 \beta_{10} ) q^{91} + ( 9 \beta_{2} + 9 \beta_{6} + \beta_{7} + \beta_{15} ) q^{93} + ( -3 \beta_{12} + 3 \beta_{13} ) q^{95} + ( 14 - 5 \beta_{4} - 5 \beta_{5} ) q^{97} + ( 8 \beta_{3} - 6 \beta_{8} - \beta_{9} + 4 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{9} + O(q^{10})$$ $$16q - 16q^{9} + 48q^{25} - 64q^{33} - 80q^{49} + 32q^{57} - 288q^{73} + 400q^{81} + 224q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 28 x^{14} + 274 x^{12} + 1236 x^{10} + 2703 x^{8} + 2676 x^{6} + 946 x^{4} + 64 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-30 \nu^{14} - 691 \nu^{12} - 3961 \nu^{10} + 5346 \nu^{8} + 107233 \nu^{6} + 290346 \nu^{4} + 234257 \nu^{2} + 31667$$$$)/3462$$ $$\beta_{2}$$ $$=$$ $$($$$$-546 \nu^{14} - 15115 \nu^{12} - 145023 \nu^{10} - 634108 \nu^{8} - 1316949 \nu^{6} - 1189412 \nu^{4} - 361293 \nu^{2} - 30203$$$$)/3462$$ $$\beta_{3}$$ $$=$$ $$($$$$2984 \nu^{14} + 83695 \nu^{12} + 820929 \nu^{10} + 3712317 \nu^{8} + 8150361 \nu^{6} + 8172537 \nu^{4} + 3022631 \nu^{2} + 157630$$$$)/5193$$ $$\beta_{4}$$ $$=$$ $$($$$$-1015 \nu^{15} + 2208 \nu^{14} - 28091 \nu^{13} + 61936 \nu^{12} - 269128 \nu^{11} + 607264 \nu^{10} - 1170461 \nu^{9} + 2738952 \nu^{8} - 2389964 \nu^{7} + 5950112 \nu^{6} - 2023591 \nu^{5} + 5733840 \nu^{4} - 386663 \nu^{3} + 1819072 \nu^{2} + 76556 \nu + 63628$$$$)/3462$$ $$\beta_{5}$$ $$=$$ $$($$$$1015 \nu^{15} + 2208 \nu^{14} + 28091 \nu^{13} + 61936 \nu^{12} + 269128 \nu^{11} + 607264 \nu^{10} + 1170461 \nu^{9} + 2738952 \nu^{8} + 2389964 \nu^{7} + 5950112 \nu^{6} + 2023591 \nu^{5} + 5733840 \nu^{4} + 386663 \nu^{3} + 1819072 \nu^{2} - 76556 \nu + 63628$$$$)/3462$$ $$\beta_{6}$$ $$=$$ $$($$$$-5936 \nu^{15} + 9902 \nu^{14} - 166576 \nu^{13} + 275467 \nu^{12} - 1637556 \nu^{11} + 2666883 \nu^{10} - 7457340 \nu^{9} + 11841816 \nu^{8} - 16653828 \nu^{7} + 25268433 \nu^{6} - 17351100 \nu^{5} + 23926968 \nu^{4} - 7071932 \nu^{3} + 7513421 \nu^{2} - 702316 \nu + 209431$$$$)/10386$$ $$\beta_{7}$$ $$=$$ $$($$$$-5936 \nu^{15} - 9902 \nu^{14} - 166576 \nu^{13} - 275467 \nu^{12} - 1637556 \nu^{11} - 2666883 \nu^{10} - 7457340 \nu^{9} - 11841816 \nu^{8} - 16653828 \nu^{7} - 25268433 \nu^{6} - 17351100 \nu^{5} - 23926968 \nu^{4} - 7071932 \nu^{3} - 7513421 \nu^{2} - 702316 \nu - 209431$$$$)/10386$$ $$\beta_{8}$$ $$=$$ $$($$$$21559 \nu^{15} + 3128 \nu^{14} + 602609 \nu^{13} + 87358 \nu^{12} + 5877930 \nu^{11} + 850674 \nu^{10} + 26362059 \nu^{9} + 3804018 \nu^{8} + 57016878 \nu^{7} + 8191986 \nu^{6} + 55099233 \nu^{5} + 7872522 \nu^{4} + 18148993 \nu^{3} + 2546630 \nu^{2} + 789812 \nu + 67252$$$$)/20772$$ $$\beta_{9}$$ $$=$$ $$($$$$-18625 \nu^{15} - 520835 \nu^{13} - 5081682 \nu^{11} - 22763733 \nu^{9} - 48910902 \nu^{7} - 45951759 \nu^{5} - 12957235 \nu^{3} + 618256 \nu$$$$)/10386$$ $$\beta_{10}$$ $$=$$ $$($$$$-21559 \nu^{15} + 3128 \nu^{14} - 602609 \nu^{13} + 87358 \nu^{12} - 5877930 \nu^{11} + 850674 \nu^{10} - 26362059 \nu^{9} + 3804018 \nu^{8} - 57016878 \nu^{7} + 8191986 \nu^{6} - 55099233 \nu^{5} + 7872522 \nu^{4} - 18148993 \nu^{3} + 2546630 \nu^{2} - 789812 \nu + 67252$$$$)/6924$$ $$\beta_{11}$$ $$=$$ $$($$$$-2892 \nu^{15} - 80922 \nu^{13} - 790818 \nu^{11} - 3557804 \nu^{9} - 7735302 \nu^{7} - 7541844 \nu^{5} - 2513894 \nu^{3} - 96798 \nu$$$$)/577$$ $$\beta_{12}$$ $$=$$ $$($$$$17434 \nu^{15} + 2016 \nu^{14} + 486536 \nu^{13} + 56475 \nu^{12} + 4732924 \nu^{11} + 553179 \nu^{10} + 21136724 \nu^{9} + 2498976 \nu^{8} + 45372032 \nu^{7} + 5465901 \nu^{6} + 43087192 \nu^{5} + 5358372 \nu^{4} + 13227938 \nu^{3} + 1774611 \nu^{2} + 73060 \nu + 51999$$$$)/3462$$ $$\beta_{13}$$ $$=$$ $$($$$$7002 \nu^{15} + 672 \nu^{14} + 195784 \nu^{13} + 18825 \nu^{12} + 1911052 \nu^{11} + 184393 \nu^{10} + 8582708 \nu^{9} + 832992 \nu^{8} + 18609520 \nu^{7} + 1821967 \nu^{6} + 18059224 \nu^{5} + 1786124 \nu^{4} + 5985234 \nu^{3} + 591537 \nu^{2} + 273444 \nu + 17333$$$$)/1154$$ $$\beta_{14}$$ $$=$$ $$($$$$-10503 \nu^{15} + 5025 \nu^{14} - 293676 \nu^{13} + 140842 \nu^{12} - 2866578 \nu^{11} + 1380967 \nu^{10} - 12874062 \nu^{9} + 6250113 \nu^{8} - 27914280 \nu^{7} + 13718369 \nu^{6} - 27088836 \nu^{5} + 13541103 \nu^{4} - 8977851 \nu^{3} + 4553656 \nu^{2} - 410166 \nu + 145831$$$$)/1731$$ $$\beta_{15}$$ $$=$$ $$($$$$-58 \nu^{15} - 1622 \nu^{13} - 15837 \nu^{11} - 71163 \nu^{9} - 154461 \nu^{7} - 150243 \nu^{5} - 50005 \nu^{3} - 1925 \nu$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{11} + \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + 3 \beta_{6}$$$$)/48$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{14} - 2 \beta_{13} - 8 \beta_{10} - 24 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} + 6 \beta_{4} - 6 \beta_{2} + 14 \beta_{1} - 168$$$$)/48$$ $$\nu^{3}$$ $$=$$ $$($$$$12 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} + 6 \beta_{12} + 16 \beta_{11} - 8 \beta_{10} - 16 \beta_{9} + 24 \beta_{8} - 17 \beta_{7} - 17 \beta_{6} - 12 \beta_{5} + 12 \beta_{4} + 2 \beta_{1}$$$$)/32$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{14} + 14 \beta_{13} + 58 \beta_{10} + 174 \beta_{8} + 27 \beta_{7} - 27 \beta_{6} - 39 \beta_{5} - 39 \beta_{4} - 6 \beta_{3} - 18 \beta_{2} - 74 \beta_{1} + 708$$$$)/24$$ $$\nu^{5}$$ $$=$$ $$($$$$-540 \beta_{15} + 90 \beta_{14} - 120 \beta_{13} - 330 \beta_{12} - 534 \beta_{11} + 326 \beta_{10} + 522 \beta_{9} - 978 \beta_{8} + 561 \beta_{7} + 561 \beta_{6} + 660 \beta_{5} - 660 \beta_{4} - 90 \beta_{1}$$$$)/96$$ $$\nu^{6}$$ $$=$$ $$($$$$-62 \beta_{14} - 62 \beta_{13} - 273 \beta_{10} - 819 \beta_{8} - 130 \beta_{7} + 130 \beta_{6} + 169 \beta_{5} + 169 \beta_{4} + 47 \beta_{3} + 160 \beta_{2} + 286 \beta_{1} - 2644$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$7308 \beta_{15} - 1225 \beta_{14} + 1316 \beta_{13} + 4809 \beta_{12} + 6810 \beta_{11} - 4470 \beta_{10} - 6450 \beta_{9} + 13410 \beta_{8} - 6915 \beta_{7} - 6915 \beta_{6} - 9744 \beta_{5} + 9744 \beta_{4} + 1225 \beta_{1}$$$$)/96$$ $$\nu^{8}$$ $$=$$ $$($$$$2468 \beta_{14} + 2468 \beta_{13} + 11236 \beta_{10} + 33708 \beta_{8} + 5346 \beta_{7} - 5346 \beta_{6} - 6705 \beta_{5} - 6705 \beta_{4} - 2220 \beta_{3} - 7452 \beta_{2} - 10748 \beta_{1} + 99492$$$$)/24$$ $$\nu^{9}$$ $$=$$ $$($$$$-32556 \beta_{15} + 5484 \beta_{14} - 5376 \beta_{13} - 22044 \beta_{12} - 30008 \beta_{11} + 20148 \beta_{10} + 27940 \beta_{9} - 60444 \beta_{8} + 29843 \beta_{7} + 29843 \beta_{6} + 44988 \beta_{5} - 44988 \beta_{4} - 5484 \beta_{1}$$$$)/32$$ $$\nu^{10}$$ $$=$$ $$($$$$-65722 \beta_{14} - 65722 \beta_{13} - 303766 \beta_{10} - 911298 \beta_{8} - 144237 \beta_{7} + 144237 \beta_{6} + 178584 \beta_{5} + 178584 \beta_{4} + 62706 \beta_{3} + 208686 \beta_{2} + 280078 \beta_{1} - 2601168$$$$)/48$$ $$\nu^{11}$$ $$=$$ $$($$$$1304424 \beta_{15} - 220363 \beta_{14} + 208736 \beta_{13} + 893079 \beta_{12} + 1200666 \beta_{11} - 811542 \beta_{10} - 1109514 \beta_{9} + 2434626 \beta_{8} - 1182660 \beta_{7} - 1182660 \beta_{6} - 1828332 \beta_{5} + 1828332 \beta_{4} + 220363 \beta_{1}$$$$)/96$$ $$\nu^{12}$$ $$=$$ $$($$$$73098 \beta_{14} + 73098 \beta_{13} + 340002 \beta_{10} + 1020006 \beta_{8} + 161271 \beta_{7} - 161271 \beta_{6} - 198704 \beta_{5} - 198704 \beta_{4} - 71226 \beta_{3} - 236214 \beta_{2} - 309106 \beta_{1} + 2875588$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$-8716032 \beta_{15} + 1474590 \beta_{14} - 1379820 \beta_{13} - 5993130 \beta_{12} - 8024325 \beta_{11} + 5433755 \beta_{10} + 7391985 \beta_{9} - 16301265 \beta_{8} + 7872387 \beta_{7} + 7872387 \beta_{6} + 12284376 \beta_{5} - 12284376 \beta_{4} - 1474590 \beta_{1}$$$$)/48$$ $$\nu^{14}$$ $$=$$ $$($$$$-11721514 \beta_{14} - 11721514 \beta_{13} - 54659524 \beta_{10} - 163978572 \beta_{8} - 25914759 \beta_{7} + 25914759 \beta_{6} + 31869510 \beta_{5} + 31869510 \beta_{4} + 11508228 \beta_{3} + 38116338 \beta_{2} + 49431814 \beta_{1} - 460174344$$$$)/48$$ $$\nu^{15}$$ $$=$$ $$($$$$77686908 \beta_{15} - 13152080 \beta_{14} + 12254364 \beta_{13} + 53506036 \beta_{12} + 71541968 \beta_{11} - 48467984 \beta_{10} - 65821112 \beta_{9} + 145403952 \beta_{8} - 70074937 \beta_{7} - 70074937 \beta_{6} - 109723892 \beta_{5} + 109723892 \beta_{4} + 13152080 \beta_{1}$$$$)/32$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.242964i 2.24296i − 0.242964i − 2.24296i − 0.736993i 1.26301i 0.736993i − 1.26301i − 0.151206i − 2.15121i 0.151206i 2.15121i 3.65718i 1.65718i − 3.65718i − 1.65718i
0 −2.75782 1.18087i 0 −3.68481 0 5.43855 0 6.21110 + 6.51323i 0
65.2 0 −2.75782 1.18087i 0 3.68481 0 −5.43855 0 6.21110 + 6.51323i 0
65.3 0 −2.75782 + 1.18087i 0 −3.68481 0 5.43855 0 6.21110 6.51323i 0
65.4 0 −2.75782 + 1.18087i 0 3.68481 0 −5.43855 0 6.21110 6.51323i 0
65.5 0 −0.628052 2.93352i 0 −6.51323 0 −7.64344 0 −8.21110 + 3.68481i 0
65.6 0 −0.628052 2.93352i 0 6.51323 0 7.64344 0 −8.21110 + 3.68481i 0
65.7 0 −0.628052 + 2.93352i 0 −6.51323 0 −7.64344 0 −8.21110 3.68481i 0
65.8 0 −0.628052 + 2.93352i 0 6.51323 0 7.64344 0 −8.21110 3.68481i 0
65.9 0 0.628052 2.93352i 0 −6.51323 0 7.64344 0 −8.21110 3.68481i 0
65.10 0 0.628052 2.93352i 0 6.51323 0 −7.64344 0 −8.21110 3.68481i 0
65.11 0 0.628052 + 2.93352i 0 −6.51323 0 7.64344 0 −8.21110 + 3.68481i 0
65.12 0 0.628052 + 2.93352i 0 6.51323 0 −7.64344 0 −8.21110 + 3.68481i 0
65.13 0 2.75782 1.18087i 0 −3.68481 0 −5.43855 0 6.21110 6.51323i 0
65.14 0 2.75782 1.18087i 0 3.68481 0 5.43855 0 6.21110 6.51323i 0
65.15 0 2.75782 + 1.18087i 0 −3.68481 0 −5.43855 0 6.21110 + 6.51323i 0
65.16 0 2.75782 + 1.18087i 0 3.68481 0 5.43855 0 6.21110 + 6.51323i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.16 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.h.g 16
3.b odd 2 1 inner 384.3.h.g 16
4.b odd 2 1 inner 384.3.h.g 16
8.b even 2 1 inner 384.3.h.g 16
8.d odd 2 1 inner 384.3.h.g 16
12.b even 2 1 inner 384.3.h.g 16
16.e even 4 1 768.3.e.o 8
16.e even 4 1 768.3.e.p 8
16.f odd 4 1 768.3.e.o 8
16.f odd 4 1 768.3.e.p 8
24.f even 2 1 inner 384.3.h.g 16
24.h odd 2 1 inner 384.3.h.g 16
48.i odd 4 1 768.3.e.o 8
48.i odd 4 1 768.3.e.p 8
48.k even 4 1 768.3.e.o 8
48.k even 4 1 768.3.e.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.h.g 16 1.a even 1 1 trivial
384.3.h.g 16 3.b odd 2 1 inner
384.3.h.g 16 4.b odd 2 1 inner
384.3.h.g 16 8.b even 2 1 inner
384.3.h.g 16 8.d odd 2 1 inner
384.3.h.g 16 12.b even 2 1 inner
384.3.h.g 16 24.f even 2 1 inner
384.3.h.g 16 24.h odd 2 1 inner
768.3.e.o 8 16.e even 4 1
768.3.e.o 8 16.f odd 4 1
768.3.e.o 8 48.i odd 4 1
768.3.e.o 8 48.k even 4 1
768.3.e.p 8 16.e even 4 1
768.3.e.p 8 16.f odd 4 1
768.3.e.p 8 48.i odd 4 1
768.3.e.p 8 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{4} - 56 T_{5}^{2} + 576$$ $$T_{11}^{4} - 320 T_{11}^{2} + 432$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 6561 + 324 T^{2} - 42 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$5$ $$( 576 - 56 T^{2} + T^{4} )^{4}$$
$7$ $$( 1728 - 88 T^{2} + T^{4} )^{4}$$
$11$ $$( 432 - 320 T^{2} + T^{4} )^{4}$$
$13$ $$( 208 + T^{2} )^{8}$$
$17$ $$( 1024 + 352 T^{2} + T^{4} )^{4}$$
$19$ $$( 139968 + 1000 T^{2} + T^{4} )^{4}$$
$23$ $$( 995328 + 2048 T^{2} + T^{4} )^{4}$$
$29$ $$( 10816 - 1144 T^{2} + T^{4} )^{4}$$
$31$ $$( 3195072 - 4120 T^{2} + T^{4} )^{4}$$
$37$ $$( 2304 + 928 T^{2} + T^{4} )^{4}$$
$41$ $$( 1327104 + 4352 T^{2} + T^{4} )^{4}$$
$43$ $$( 914112 + 2344 T^{2} + T^{4} )^{4}$$
$47$ $$( 28311552 + 11264 T^{2} + T^{4} )^{4}$$
$53$ $$( 2096704 - 3832 T^{2} + T^{4} )^{4}$$
$59$ $$( 314928 - 2592 T^{2} + T^{4} )^{4}$$
$61$ $$( 186624 + 8352 T^{2} + T^{4} )^{4}$$
$67$ $$( 15552 + 360 T^{2} + T^{4} )^{4}$$
$71$ $$( 31961088 + 12800 T^{2} + T^{4} )^{4}$$
$73$ $$( -7164 + 36 T + T^{2} )^{8}$$
$79$ $$( 1453248 - 15448 T^{2} + T^{4} )^{4}$$
$83$ $$( 10112688 - 8192 T^{2} + T^{4} )^{4}$$
$89$ $$( 9216 + 20192 T^{2} + T^{4} )^{4}$$
$97$ $$( -5004 - 28 T + T^{2} )^{8}$$