# Properties

 Label 1680.3.s.c Level 1680 Weight 3 Character orbit 1680.s Analytic conductor 45.777 Analytic rank 0 Dimension 12 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1680.s (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$45.7766844125$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: no (minimal twist has level 105) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} + \beta_{8} q^{5} + ( 1 + \beta_{3} + \beta_{8} - \beta_{10} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} + \beta_{8} q^{5} + ( 1 + \beta_{3} + \beta_{8} - \beta_{10} ) q^{7} -3 q^{9} + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{11} + ( -1 + \beta_{7} + 4 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{13} -\beta_{6} q^{15} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{17} + ( -1 - \beta_{2} - 6 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{19} + ( 2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{21} + ( 3 + 3 \beta_{1} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} ) q^{23} -5 q^{25} + 3 \beta_{3} q^{27} + ( 6 + 5 \beta_{1} + 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} ) q^{29} + ( 1 + 2 \beta_{3} - \beta_{7} - 10 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{31} + ( -\beta_{2} - \beta_{3} + \beta_{5} + 3 \beta_{9} - 3 \beta_{10} ) q^{33} + ( -6 + \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{35} + ( 6 - 5 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{37} + ( 3 + \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{11} ) q^{39} + ( 1 + 21 \beta_{3} - 3 \beta_{5} - \beta_{7} - 3 \beta_{8} - 5 \beta_{11} ) q^{41} + ( -8 - 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 5 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{43} -3 \beta_{8} q^{45} + ( 1 + 7 \beta_{2} - 11 \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{47} + ( 2 + \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} + 3 \beta_{11} ) q^{49} + ( -3 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{11} ) q^{51} + ( 21 - 9 \beta_{1} + 6 \beta_{6} - 6 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{53} + ( 2 + 3 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{55} + ( -24 + 3 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{57} + ( -7 - 2 \beta_{2} - 11 \beta_{3} - 7 \beta_{5} + 7 \beta_{7} - 7 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + 5 \beta_{11} ) q^{59} + ( 4 + 9 \beta_{2} - 22 \beta_{3} - 4 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{61} + ( -3 - 3 \beta_{3} - 3 \beta_{8} + 3 \beta_{10} ) q^{63} + ( -22 + 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{65} + ( -18 + 9 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 12 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{67} + ( 4 + \beta_{2} - 5 \beta_{3} + 5 \beta_{5} - 4 \beta_{7} - 2 \beta_{11} ) q^{69} + ( 1 + 4 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - \beta_{11} ) q^{71} + ( -3 - 2 \beta_{2} - 16 \beta_{3} - 10 \beta_{5} + 3 \beta_{7} + 14 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} ) q^{73} + 5 \beta_{3} q^{75} + ( 11 - 2 \beta_{1} + 3 \beta_{2} - 23 \beta_{3} - 2 \beta_{4} - \beta_{5} - 14 \beta_{6} - 7 \beta_{7} + 12 \beta_{8} - 7 \beta_{9} - 3 \beta_{10} - 10 \beta_{11} ) q^{77} + ( -10 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 12 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 4 \beta_{10} - 2 \beta_{11} ) q^{79} + 9 q^{81} + ( 6 - 12 \beta_{3} + 12 \beta_{5} - 6 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} ) q^{83} + ( 8 + 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{85} + ( 5 - \beta_{2} - 7 \beta_{3} + 7 \beta_{5} - 5 \beta_{7} - 4 \beta_{11} ) q^{87} + ( -5 - 4 \beta_{2} + 5 \beta_{3} + \beta_{5} + 5 \beta_{7} - 31 \beta_{8} + 4 \beta_{9} - 4 \beta_{10} + 7 \beta_{11} ) q^{89} + ( -10 + 11 \beta_{1} + 5 \beta_{2} - 20 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 14 \beta_{6} + 3 \beta_{7} + 19 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{91} + ( 3 + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{93} + ( 2 + 3 \beta_{1} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 8 \beta_{6} - 5 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{95} + ( 1 + 6 \beta_{2} - 14 \beta_{3} - \beta_{7} - 4 \beta_{8} + 10 \beta_{9} - 10 \beta_{10} + 4 \beta_{11} ) q^{97} + ( -3 - 3 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 3 \beta_{8} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{7} - 36q^{9} + O(q^{10})$$ $$12q + 8q^{7} - 36q^{9} + 16q^{11} + 36q^{21} + 64q^{23} - 60q^{25} + 104q^{29} - 60q^{35} + 32q^{37} + 24q^{39} - 152q^{43} + 60q^{49} - 24q^{51} + 176q^{53} - 240q^{57} - 24q^{63} - 240q^{65} - 168q^{67} - 32q^{71} + 8q^{77} - 120q^{79} + 108q^{81} + 120q^{85} - 24q^{91} + 48q^{93} - 48q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} - 5472 x^{3} + 4950 x^{2} - 1638 x + 441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1163071964 \nu^{11} + 179172244 \nu^{10} + 20630949354 \nu^{9} + 16671982336 \nu^{8} + 315532034086 \nu^{7} + 213549867516 \nu^{6} + 1552126862414 \nu^{5} + 1781364901824 \nu^{4} + 8663666606754 \nu^{3} + 2969181679896 \nu^{2} - 1028895680526 \nu + 2151653525805$$$$)/ 5604734688861$$ $$\beta_{2}$$ $$=$$ $$($$$$19870505890 \nu^{11} - 84005908649 \nu^{10} + 518349481172 \nu^{9} - 1428356575517 \nu^{8} + 7194121908415 \nu^{7} - 19979390806744 \nu^{6} + 52938005623695 \nu^{5} - 89943465015027 \nu^{4} + 167210840887752 \nu^{3} - 323219534885292 \nu^{2} + 411966888798087 \nu - 144857546720712$$$$)/ 16814204066583$$ $$\beta_{3}$$ $$=$$ $$($$$$-7830116016 \nu^{11} + 14497160068 \nu^{10} - 164611608580 \nu^{9} + 182952067062 \nu^{8} - 2326556207056 \nu^{7} + 2597271123866 \nu^{6} - 13556067558780 \nu^{5} + 6857417738770 \nu^{4} - 45731806099632 \nu^{3} + 34182728232798 \nu^{2} - 41728255959096 \nu + 8249891025873$$$$)/ 5604734688861$$ $$\beta_{4}$$ $$=$$ $$($$$$28642872442 \nu^{11} - 134337649452 \nu^{10} + 684289167827 \nu^{9} - 2290451892108 \nu^{8} + 9154276278382 \nu^{7} - 32964375662714 \nu^{6} + 58795985026140 \nu^{5} - 154323934459524 \nu^{4} + 153524430515955 \nu^{3} - 606435948666924 \nu^{2} + 210065305235184 \nu - 231105440938869$$$$)/ 16814204066583$$ $$\beta_{5}$$ $$=$$ $$($$$$39871638046 \nu^{11} - 128034905585 \nu^{10} + 950291458850 \nu^{9} - 2027228723711 \nu^{8} + 13227194427325 \nu^{7} - 28411853780890 \nu^{6} + 88949827712793 \nu^{5} - 115859812936809 \nu^{4} + 278415259366386 \nu^{3} - 434675264623374 \nu^{2} + 472981527450621 \nu - 159247976309163$$$$)/ 16814204066583$$ $$\beta_{6}$$ $$=$$ $$($$$$420436 \nu^{11} - 161001 \nu^{10} + 7746906 \nu^{9} + 2597287 \nu^{8} + 112253475 \nu^{7} + 32874720 \nu^{6} + 544174569 \nu^{5} + 552012621 \nu^{4} + 2082413232 \nu^{3} + 966776562 \nu^{2} - 335831265 \nu + 794649726$$$$)/ 133884909$$ $$\beta_{7}$$ $$=$$ $$($$$$-11187033068 \nu^{11} + 5210631312 \nu^{10} - 205981302354 \nu^{9} - 55032700892 \nu^{8} - 2979440990802 \nu^{7} - 696798378966 \nu^{6} - 14410123013778 \nu^{5} - 14310718180788 \nu^{4} - 55318057600752 \nu^{3} - 25286831481312 \nu^{2} + 8787673541682 \nu - 18282636368199$$$$)/ 2402029152369$$ $$\beta_{8}$$ $$=$$ $$($$$$5888 \nu^{11} - 11357 \nu^{10} + 121362 \nu^{9} - 142983 \nu^{8} + 1697439 \nu^{7} - 2055140 \nu^{6} + 9463719 \nu^{5} - 5419563 \nu^{4} + 30382920 \nu^{3} - 29629158 \nu^{2} + 18996579 \nu - 4306806$$$$)/1130283$$ $$\beta_{9}$$ $$=$$ $$($$$$10465136770 \nu^{11} - 21796565040 \nu^{10} + 217694819151 \nu^{9} - 281471772670 \nu^{8} + 3036253477328 \nu^{7} - 4036087599626 \nu^{6} + 17185537551444 \nu^{5} - 11089514438346 \nu^{4} + 55017188023641 \nu^{3} - 57941854444554 \nu^{2} + 45520828453800 \nu - 2470563074763$$$$)/ 1528564006053$$ $$\beta_{10}$$ $$=$$ $$($$$$-4771832146 \nu^{11} + 7993408932 \nu^{10} - 97232846541 \nu^{9} + 92976169234 \nu^{8} - 1370545304570 \nu^{7} + 1347840842486 \nu^{6} - 7600600799214 \nu^{5} + 2916220839078 \nu^{4} - 25482799710645 \nu^{3} + 19858677257310 \nu^{2} - 16532688498102 \nu + 6309118974753$$$$)/ 579800140227$$ $$\beta_{11}$$ $$=$$ $$($$$$-151294336430 \nu^{11} + 350115328056 \nu^{10} - 3210228154956 \nu^{9} + 4809676764430 \nu^{8} - 44563758346308 \nu^{7} + 68851916591280 \nu^{6} - 256948296495024 \nu^{5} + 222892955737464 \nu^{4} - 792860368623084 \nu^{3} + 1053296354400858 \nu^{2} - 646628005813608 \nu + 212231462685480$$$$)/ 16814204066583$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} - \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 12 \beta_{3} - \beta_{2} + \beta_{1} - 15$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{11} + \beta_{8} + 2 \beta_{7} - \beta_{5} - 2 \beta_{4} + \beta_{3} + 11 \beta_{1} - 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$23 \beta_{11} + 15 \beta_{10} + 9 \beta_{9} + 59 \beta_{8} + 10 \beta_{7} + 32 \beta_{6} + 27 \beta_{5} + 2 \beta_{4} + 125 \beta_{3} + 10 \beta_{2} + 14 \beta_{1} - 148$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$96 \beta_{11} - 48 \beta_{10} + 54 \beta_{9} + 16 \beta_{8} + 6 \beta_{7} - 8 \beta_{6} + 145 \beta_{5} + 34 \beta_{4} + 103 \beta_{3} - 137 \beta_{2} - 131 \beta_{1} + 111$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-13 \beta_{11} - 144 \beta_{10} - 144 \beta_{9} + 13 \beta_{8} - 251 \beta_{7} - 452 \beta_{6} - 13 \beta_{5} - 26 \beta_{4} + 13 \beta_{3} - 194 \beta_{1} + 1895$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-511 \beta_{11} + 381 \beta_{10} - 336 \beta_{9} - 541 \beta_{8} - 653 \beta_{7} - 118 \beta_{6} - 687 \beta_{5} + 239 \beta_{4} - 964 \beta_{3} + 874 \beta_{2} - 829 \beta_{1} + 1475$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-5220 \beta_{11} + 1416 \beta_{10} + 2142 \beta_{9} - 13222 \beta_{8} + 3558 \beta_{7} + 6146 \beta_{6} - 4423 \beta_{5} + 242 \beta_{4} - 19045 \beta_{3} - 793 \beta_{2} + 2765 \beta_{1} - 24261$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-3194 \beta_{11} - 429 \beta_{10} - 429 \beta_{9} + 3194 \beta_{8} + 9158 \beta_{7} + 4808 \beta_{6} - 3194 \beta_{5} - 6388 \beta_{4} + 3194 \beta_{3} + 21683 \beta_{1} - 33449$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$77615 \beta_{11} + 24843 \beta_{10} + 20103 \beta_{9} + 179477 \beta_{8} + 54352 \beta_{7} + 82562 \beta_{6} + 63342 \beta_{5} + 1580 \beta_{4} + 245402 \beta_{3} + 4867 \beta_{2} + 40079 \beta_{1} - 323017$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$312327 \beta_{11} - 124221 \beta_{10} + 128205 \beta_{9} + 218377 \beta_{8} + 3984 \beta_{7} - 84824 \beta_{6} + 328747 \beta_{5} + 84142 \beta_{4} + 326107 \beta_{3} - 292652 \beta_{2} - 288668 \beta_{1} + 362202$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1441.1
 0.198184 − 0.343264i −1.01714 + 1.76174i 1.31896 − 2.28450i 0.378061 − 0.654821i 1.86875 − 3.23677i −1.74681 + 3.02556i 0.378061 + 0.654821i 1.86875 + 3.23677i −1.74681 − 3.02556i 0.198184 + 0.343264i −1.01714 − 1.76174i 1.31896 + 2.28450i
0 1.73205i 0 2.23607i 0 −4.15782 5.63139i 0 −3.00000 0
1441.2 0 1.73205i 0 2.23607i 0 3.33344 + 6.15534i 0 −3.00000 0
1441.3 0 1.73205i 0 2.23607i 0 6.69736 2.03600i 0 −3.00000 0
1441.4 0 1.73205i 0 2.23607i 0 −6.13981 + 3.36195i 0 −3.00000 0
1441.5 0 1.73205i 0 2.23607i 0 −2.44621 + 6.55866i 0 −3.00000 0
1441.6 0 1.73205i 0 2.23607i 0 6.71303 + 1.98374i 0 −3.00000 0
1441.7 0 1.73205i 0 2.23607i 0 −6.13981 3.36195i 0 −3.00000 0
1441.8 0 1.73205i 0 2.23607i 0 −2.44621 6.55866i 0 −3.00000 0
1441.9 0 1.73205i 0 2.23607i 0 6.71303 1.98374i 0 −3.00000 0
1441.10 0 1.73205i 0 2.23607i 0 −4.15782 + 5.63139i 0 −3.00000 0
1441.11 0 1.73205i 0 2.23607i 0 3.33344 6.15534i 0 −3.00000 0
1441.12 0 1.73205i 0 2.23607i 0 6.69736 + 2.03600i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1441.12 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.3.s.c 12
4.b odd 2 1 105.3.h.a 12
7.b odd 2 1 inner 1680.3.s.c 12
12.b even 2 1 315.3.h.d 12
20.d odd 2 1 525.3.h.d 12
20.e even 4 2 525.3.e.c 24
28.d even 2 1 105.3.h.a 12
84.h odd 2 1 315.3.h.d 12
140.c even 2 1 525.3.h.d 12
140.j odd 4 2 525.3.e.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.h.a 12 4.b odd 2 1
105.3.h.a 12 28.d even 2 1
315.3.h.d 12 12.b even 2 1
315.3.h.d 12 84.h odd 2 1
525.3.e.c 24 20.e even 4 2
525.3.e.c 24 140.j odd 4 2
525.3.h.d 12 20.d odd 2 1
525.3.h.d 12 140.c even 2 1
1680.3.s.c 12 1.a even 1 1 trivial
1680.3.s.c 12 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{6} - 8 T_{11}^{5} - 528 T_{11}^{4} + 3424 T_{11}^{3} + 66832 T_{11}^{2} - 336576 T_{11} + 388416$$ acting on $$S_{3}^{\mathrm{new}}(1680, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 3 T^{2} )^{6}$$
$5$ $$( 1 + 5 T^{2} )^{6}$$
$7$ $$1 - 8 T + 2 T^{2} - 312 T^{3} + 4255 T^{4} - 13888 T^{5} + 43708 T^{6} - 680512 T^{7} + 10216255 T^{8} - 36706488 T^{9} + 11529602 T^{10} - 2259801992 T^{11} + 13841287201 T^{12}$$
$11$ $$( 1 - 8 T + 198 T^{2} - 1416 T^{3} + 30895 T^{4} - 264944 T^{5} + 5610292 T^{6} - 32058224 T^{7} + 452333695 T^{8} - 2508530376 T^{9} + 42443058438 T^{10} - 207499396808 T^{11} + 3138428376721 T^{12} )^{2}$$
$13$ $$1 - 852 T^{2} + 436674 T^{4} - 159409348 T^{6} + 45042421839 T^{8} - 10253858128680 T^{10} + 1907667001388316 T^{12} - 292860442013229480 T^{14} + 36742487242313615919 T^{16} -$$$$37\!\cdots\!88$$$$T^{18} +$$$$29\!\cdots\!34$$$$T^{20} -$$$$16\!\cdots\!52$$$$T^{22} +$$$$54\!\cdots\!61$$$$T^{24}$$
$17$ $$1 - 2172 T^{2} + 2401794 T^{4} - 1757595148 T^{6} + 941667483759 T^{8} - 387997181982840 T^{10} + 125896935467491356 T^{12} - 32405912636388779640 T^{14} +$$$$65\!\cdots\!19$$$$T^{16} -$$$$10\!\cdots\!28$$$$T^{18} +$$$$11\!\cdots\!14$$$$T^{20} -$$$$88\!\cdots\!72$$$$T^{22} +$$$$33\!\cdots\!21$$$$T^{24}$$
$19$ $$1 - 1404 T^{2} + 1342050 T^{4} - 896154316 T^{6} + 500154418383 T^{8} - 227690692547448 T^{10} + 90028611036772572 T^{12} - 29672878743475970808 T^{14} +$$$$84\!\cdots\!03$$$$T^{16} -$$$$19\!\cdots\!76$$$$T^{18} +$$$$38\!\cdots\!50$$$$T^{20} -$$$$52\!\cdots\!04$$$$T^{22} +$$$$48\!\cdots\!21$$$$T^{24}$$
$23$ $$( 1 - 32 T + 2418 T^{2} - 60144 T^{3} + 2694751 T^{4} - 53735792 T^{5} + 1782680284 T^{6} - 28426233968 T^{7} + 754101814591 T^{8} - 8903470508016 T^{9} + 189355962409458 T^{10} - 1325648358836768 T^{11} + 21914624432020321 T^{12} )^{2}$$
$29$ $$( 1 - 52 T + 3990 T^{2} - 132324 T^{3} + 6311983 T^{4} - 164304136 T^{5} + 6334525012 T^{6} - 138179778376 T^{7} + 4464345648223 T^{8} - 78709401128004 T^{9} + 1995983187714390 T^{10} - 21876776131610452 T^{11} + 353814783205469041 T^{12} )^{2}$$
$31$ $$1 - 6228 T^{2} + 19622274 T^{4} - 42341564932 T^{6} + 69647410996719 T^{8} - 91080823295899560 T^{10} + 96735996796679061276 T^{12} -$$$$84\!\cdots\!60$$$$T^{14} +$$$$59\!\cdots\!79$$$$T^{16} -$$$$33\!\cdots\!52$$$$T^{18} +$$$$14\!\cdots\!94$$$$T^{20} -$$$$41\!\cdots\!28$$$$T^{22} +$$$$62\!\cdots\!21$$$$T^{24}$$
$37$ $$( 1 - 16 T + 4622 T^{2} + 32784 T^{3} + 7136719 T^{4} + 278279680 T^{5} + 7429424740 T^{6} + 380964881920 T^{7} + 13375360417759 T^{8} + 84114774592656 T^{9} + 16234680036022862 T^{10} - 76937349958685584 T^{11} + 6582952005840035281 T^{12} )^{2}$$
$41$ $$1 - 7524 T^{2} + 32464386 T^{4} - 93240551380 T^{6} + 206850565610415 T^{8} - 379921653079011144 T^{10} +$$$$65\!\cdots\!56$$$$T^{12} -$$$$10\!\cdots\!84$$$$T^{14} +$$$$16\!\cdots\!15$$$$T^{16} -$$$$21\!\cdots\!80$$$$T^{18} +$$$$20\!\cdots\!26$$$$T^{20} -$$$$13\!\cdots\!24$$$$T^{22} +$$$$50\!\cdots\!61$$$$T^{24}$$
$43$ $$( 1 + 76 T + 9590 T^{2} + 631788 T^{3} + 42146383 T^{4} + 2188827368 T^{5} + 102971644948 T^{6} + 4047141803432 T^{7} + 144090096346783 T^{8} + 3993761318001612 T^{9} + 112089840662193590 T^{10} + 1642472655809602924 T^{11} + 39959630797262576401 T^{12} )^{2}$$
$47$ $$1 - 6132 T^{2} + 27456354 T^{4} - 92403901348 T^{6} + 277727516070639 T^{8} - 754801362269230440 T^{10} +$$$$17\!\cdots\!96$$$$T^{12} -$$$$36\!\cdots\!40$$$$T^{14} +$$$$66\!\cdots\!79$$$$T^{16} -$$$$10\!\cdots\!68$$$$T^{18} +$$$$15\!\cdots\!34$$$$T^{20} -$$$$16\!\cdots\!32$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$( 1 - 88 T + 10434 T^{2} - 574440 T^{3} + 50272015 T^{4} - 2575931008 T^{5} + 181692199804 T^{6} - 7235790201472 T^{7} + 396670379189215 T^{8} - 12732095606942760 T^{9} + 649617609752140674 T^{10} - 15390097392165148312 T^{11} +$$$$49\!\cdots\!41$$$$T^{12} )^{2}$$
$59$ $$1 - 15948 T^{2} + 137050242 T^{4} - 864152710108 T^{6} + 4419517662043791 T^{8} - 18954945757384385304 T^{10} +$$$$70\!\cdots\!16$$$$T^{12} -$$$$22\!\cdots\!44$$$$T^{14} +$$$$64\!\cdots\!11$$$$T^{16} -$$$$15\!\cdots\!48$$$$T^{18} +$$$$29\!\cdots\!22$$$$T^{20} -$$$$41\!\cdots\!48$$$$T^{22} +$$$$31\!\cdots\!61$$$$T^{24}$$
$61$ $$1 - 13716 T^{2} + 109224834 T^{4} - 648593370628 T^{6} + 3203041286245839 T^{8} - 13994290873734287016 T^{10} +$$$$55\!\cdots\!56$$$$T^{12} -$$$$19\!\cdots\!56$$$$T^{14} +$$$$61\!\cdots\!59$$$$T^{16} -$$$$17\!\cdots\!88$$$$T^{18} +$$$$40\!\cdots\!74$$$$T^{20} -$$$$69\!\cdots\!16$$$$T^{22} +$$$$70\!\cdots\!41$$$$T^{24}$$
$67$ $$( 1 + 84 T + 15366 T^{2} + 763540 T^{3} + 102124911 T^{4} + 4033313976 T^{5} + 514098700788 T^{6} + 18105546438264 T^{7} + 2057931438675231 T^{8} + 69068593121318260 T^{9} + 6239635933335345606 T^{10} +$$$$15\!\cdots\!16$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12} )^{2}$$
$71$ $$( 1 + 16 T + 16698 T^{2} + 42336 T^{3} + 131112895 T^{4} - 1041328304 T^{5} + 716701578892 T^{6} - 5249335980464 T^{7} + 3331799062726495 T^{8} + 5423253620079456 T^{9} + 10782792464741717178 T^{10} + 52083896816158099216 T^{11} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$
$73$ $$1 - 19284 T^{2} + 216036930 T^{4} - 1973102936452 T^{6} + 14880325491672495 T^{8} - 96746229105564519336 T^{10} +$$$$55\!\cdots\!08$$$$T^{12} -$$$$27\!\cdots\!76$$$$T^{14} +$$$$12\!\cdots\!95$$$$T^{16} -$$$$45\!\cdots\!92$$$$T^{18} +$$$$14\!\cdots\!30$$$$T^{20} -$$$$35\!\cdots\!84$$$$T^{22} +$$$$52\!\cdots\!41$$$$T^{24}$$
$79$ $$( 1 + 60 T + 22242 T^{2} + 662476 T^{3} + 201003183 T^{4} + 1192106616 T^{5} + 1261845815388 T^{6} + 7439937390456 T^{7} + 7829090259107823 T^{8} + 161039605183729996 T^{9} + 33743534149941729762 T^{10} +$$$$56\!\cdots\!60$$$$T^{11} +$$$$59\!\cdots\!41$$$$T^{12} )^{2}$$
$83$ $$1 - 45420 T^{2} + 1066844514 T^{4} - 17098817034364 T^{6} + 206563644037003983 T^{8} -$$$$19\!\cdots\!08$$$$T^{10} +$$$$15\!\cdots\!04$$$$T^{12} -$$$$93\!\cdots\!68$$$$T^{14} +$$$$46\!\cdots\!03$$$$T^{16} -$$$$18\!\cdots\!04$$$$T^{18} +$$$$54\!\cdots\!34$$$$T^{20} -$$$$10\!\cdots\!20$$$$T^{22} +$$$$11\!\cdots\!21$$$$T^{24}$$
$89$ $$1 - 46020 T^{2} + 1126013634 T^{4} - 19221077499316 T^{6} + 252845241317038383 T^{8} -$$$$26\!\cdots\!72$$$$T^{10} +$$$$23\!\cdots\!84$$$$T^{12} -$$$$16\!\cdots\!52$$$$T^{14} +$$$$99\!\cdots\!23$$$$T^{16} -$$$$47\!\cdots\!36$$$$T^{18} +$$$$17\!\cdots\!74$$$$T^{20} -$$$$44\!\cdots\!20$$$$T^{22} +$$$$61\!\cdots\!41$$$$T^{24}$$
$97$ $$1 - 51924 T^{2} + 1396974978 T^{4} - 26352399780484 T^{6} + 389015604150559215 T^{8} -$$$$47\!\cdots\!84$$$$T^{10} +$$$$48\!\cdots\!52$$$$T^{12} -$$$$41\!\cdots\!04$$$$T^{14} +$$$$30\!\cdots\!15$$$$T^{16} -$$$$18\!\cdots\!44$$$$T^{18} +$$$$85\!\cdots\!38$$$$T^{20} -$$$$28\!\cdots\!24$$$$T^{22} +$$$$48\!\cdots\!81$$$$T^{24}$$