# Properties

 Label 224.3.o.d Level 224 Weight 3 Character orbit 224.o Analytic conductor 6.104 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 224.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.10355792167$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{3} + ( -\beta_{1} - \beta_{4} - \beta_{8} ) q^{5} + ( 2 \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{7} + ( \beta_{3} + 4 \beta_{5} + 5 \beta_{6} - 4 \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{3} + ( -\beta_{1} - \beta_{4} - \beta_{8} ) q^{5} + ( 2 \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{7} + ( \beta_{3} + 4 \beta_{5} + 5 \beta_{6} - 4 \beta_{10} ) q^{9} + ( -6 - 6 \beta_{6} - 3 \beta_{10} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 5 \beta_{8} - 2 \beta_{9} - 5 \beta_{11} ) q^{13} + ( -3 \beta_{4} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} ) q^{15} + ( 6 - \beta_{3} + 6 \beta_{6} + \beta_{7} + 2 \beta_{10} ) q^{17} + ( 4 \beta_{3} - \beta_{5} + 12 \beta_{6} + \beta_{10} ) q^{19} + ( 2 \beta_{1} - \beta_{2} - 8 \beta_{4} - 7 \beta_{8} - 3 \beta_{9} + 7 \beta_{11} ) q^{21} + ( -7 \beta_{4} + 5 \beta_{8} ) q^{23} + ( -14 - 2 \beta_{3} - 14 \beta_{6} + 2 \beta_{7} + 2 \beta_{10} ) q^{25} + ( -8 + 13 \beta_{5} + 2 \beta_{7} ) q^{27} + ( 8 \beta_{1} + 8 \beta_{2} + 2 \beta_{4} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{29} + ( -9 \beta_{9} + 6 \beta_{11} ) q^{31} + ( 6 \beta_{3} + 6 \beta_{5} + 9 \beta_{6} - 6 \beta_{10} ) q^{33} + ( 23 - 2 \beta_{3} + 3 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - \beta_{10} ) q^{35} + ( -9 \beta_{1} + \beta_{4} - 4 \beta_{8} ) q^{37} + ( -14 \beta_{2} + 8 \beta_{9} - 17 \beta_{11} ) q^{39} + ( -25 - 18 \beta_{5} + \beta_{7} ) q^{41} + ( 10 - 10 \beta_{5} - 10 \beta_{7} ) q^{43} + ( 4 \beta_{2} + 2 \beta_{9} + 9 \beta_{11} ) q^{45} + ( -16 \beta_{1} + 5 \beta_{4} - 2 \beta_{8} ) q^{47} + ( 20 - 6 \beta_{3} + 2 \beta_{5} - 24 \beta_{6} + 9 \beta_{7} - 10 \beta_{10} ) q^{49} + ( -4 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{10} ) q^{51} + ( 3 \beta_{2} + 7 \beta_{9} - 20 \beta_{11} ) q^{53} + ( 12 \beta_{1} + 12 \beta_{2} + 9 \beta_{4} + 3 \beta_{8} + 9 \beta_{9} - 3 \beta_{11} ) q^{55} + ( 25 + 12 \beta_{5} - 20 \beta_{7} ) q^{57} + ( 17 - 5 \beta_{3} + 17 \beta_{6} + 5 \beta_{7} - 9 \beta_{10} ) q^{59} + ( 15 \beta_{1} + 25 \beta_{4} + 10 \beta_{8} ) q^{61} + ( 6 \beta_{2} + 4 \beta_{4} - 17 \beta_{8} - 4 \beta_{9} + 36 \beta_{11} ) q^{63} + ( -11 \beta_{3} - 18 \beta_{5} + 15 \beta_{6} + 18 \beta_{10} ) q^{65} + ( -73 + 3 \beta_{3} - 73 \beta_{6} - 3 \beta_{7} + \beta_{10} ) q^{67} + ( -31 \beta_{1} - 31 \beta_{2} + 3 \beta_{4} - 13 \beta_{8} + 3 \beta_{9} + 13 \beta_{11} ) q^{69} + ( 6 \beta_{1} + 6 \beta_{2} + 8 \beta_{4} + 22 \beta_{8} + 8 \beta_{9} - 22 \beta_{11} ) q^{71} + ( 27 - 12 \beta_{3} + 27 \beta_{6} + 12 \beta_{7} + 18 \beta_{10} ) q^{73} + ( 18 \beta_{3} - 24 \beta_{5} - 8 \beta_{6} + 24 \beta_{10} ) q^{75} + ( 24 \beta_{1} + 3 \beta_{2} + 12 \beta_{4} + 3 \beta_{8} + 3 \beta_{9} + 6 \beta_{11} ) q^{77} + ( 16 \beta_{1} + 25 \beta_{4} - 5 \beta_{8} ) q^{79} + ( -12 + 21 \beta_{3} - 12 \beta_{6} - 21 \beta_{7} + 26 \beta_{10} ) q^{81} + ( 36 - 12 \beta_{5} - 14 \beta_{7} ) q^{83} + ( -11 \beta_{1} - 11 \beta_{2} - 11 \beta_{4} - 4 \beta_{8} - 11 \beta_{9} + 4 \beta_{11} ) q^{85} + ( -8 \beta_{2} + 8 \beta_{9} + 23 \beta_{11} ) q^{87} + ( 36 \beta_{3} + 4 \beta_{5} - 49 \beta_{6} - 4 \beta_{10} ) q^{89} + ( -1 - 20 \beta_{3} - 26 \beta_{5} - 10 \beta_{6} + 23 \beta_{7} + 46 \beta_{10} ) q^{91} + ( 15 \beta_{1} + 21 \beta_{4} + 54 \beta_{8} ) q^{93} + ( -10 \beta_{2} - \beta_{9} + 11 \beta_{11} ) q^{95} + ( -15 + 8 \beta_{5} - 15 \beta_{7} ) q^{97} + ( -45 + 12 \beta_{5} - 21 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 6q^{3} - 40q^{9} + O(q^{10})$$ $$12q + 6q^{3} - 40q^{9} - 30q^{11} + 30q^{17} - 78q^{19} - 92q^{25} - 156q^{27} - 78q^{33} + 222q^{35} - 232q^{41} + 200q^{43} + 372q^{49} - 10q^{51} + 332q^{57} + 110q^{59} - 32q^{65} - 434q^{67} + 102q^{73} + 60q^{75} - 82q^{81} + 536q^{83} + 214q^{89} + 8q^{91} - 152q^{97} - 504q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} - 116 x^{3} + 60 x^{2} - 20 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2848753 \nu^{11} + 128409 \nu^{10} - 36949178 \nu^{9} - 33927641 \nu^{8} + 280693408 \nu^{7} + 324275527 \nu^{6} - 906643427 \nu^{5} + 266302456 \nu^{4} - 492489836 \nu^{3} + 675410750 \nu^{2} - 298885728 \nu + 114476732$$$$)/20513668$$ $$\beta_{2}$$ $$=$$ $$($$$$908144 \nu^{11} - 6494043 \nu^{10} + 3252536 \nu^{9} + 28087343 \nu^{8} + 91567337 \nu^{7} - 469374709 \nu^{6} + 376525973 \nu^{5} - 267919561 \nu^{4} + 416913921 \nu^{3} - 270088448 \nu^{2} + 111915639 \nu - 26684466$$$$)/5128417$$ $$\beta_{3}$$ $$=$$ $$($$$$775023 \nu^{11} - 2790290 \nu^{10} - 2275565 \nu^{9} + 5234055 \nu^{8} + 68919723 \nu^{7} - 163955841 \nu^{6} + 148312850 \nu^{5} - 164484671 \nu^{4} + 171431608 \nu^{3} - 108634476 \nu^{2} + 67339246 \nu - 23024344$$$$)/2930524$$ $$\beta_{4}$$ $$=$$ $$($$$$-894552 \nu^{11} + 880921 \nu^{10} + 8410783 \nu^{9} + 6680758 \nu^{8} - 80688819 \nu^{7} - 13990590 \nu^{6} + 103354841 \nu^{5} - 1783409 \nu^{4} + 46480180 \nu^{3} - 87465566 \nu^{2} + 40145506 \nu - 16276324$$$$)/2930524$$ $$\beta_{5}$$ $$=$$ $$($$$$8673537 \nu^{11} - 20159299 \nu^{10} - 48705482 \nu^{9} - 5607853 \nu^{8} + 742433540 \nu^{7} - 912321093 \nu^{6} + 701452097 \nu^{5} - 889048300 \nu^{4} + 667643040 \nu^{3} - 426711138 \nu^{2} + 154926652 \nu - 29235872$$$$)/20513668$$ $$\beta_{6}$$ $$=$$ $$($$$$8864783 \nu^{11} - 20455402 \nu^{10} - 48834261 \nu^{9} - 8785933 \nu^{8} + 751559539 \nu^{7} - 927115697 \nu^{6} + 797312818 \nu^{5} - 981317799 \nu^{4} + 836256216 \nu^{3} - 526622432 \nu^{2} + 236999870 \nu - 70375800$$$$)/20513668$$ $$\beta_{7}$$ $$=$$ $$($$$$-4885386 \nu^{11} + 11354967 \nu^{10} + 27428562 \nu^{9} + 3157071 \nu^{8} - 418178829 \nu^{7} + 513874517 \nu^{6} - 395095635 \nu^{5} + 500760792 \nu^{4} - 376052347 \nu^{3} + 240347478 \nu^{2} - 87263352 \nu - 20957728$$$$)/10256834$$ $$\beta_{8}$$ $$=$$ $$($$$$5752921 \nu^{11} - 9678616 \nu^{10} - 40188533 \nu^{9} - 25630209 \nu^{8} + 486470615 \nu^{7} - 292655047 \nu^{6} + 125254932 \nu^{5} - 345038603 \nu^{4} + 172274176 \nu^{3} + 19638430 \nu^{2} - 24222874 \nu + 18987776$$$$)/10256834$$ $$\beta_{9}$$ $$=$$ $$($$$$-1785337 \nu^{11} + 6633082 \nu^{10} + 5117731 \nu^{9} - 14794777 \nu^{8} - 159689841 \nu^{7} + 400659219 \nu^{6} - 329059530 \nu^{5} + 293285149 \nu^{4} - 357701700 \nu^{3} + 229568032 \nu^{2} - 91947870 \nu + 22371428$$$$)/2930524$$ $$\beta_{10}$$ $$=$$ $$($$$$-15221487 \nu^{11} + 28755150 \nu^{10} + 95322325 \nu^{9} + 55362465 \nu^{8} - 1263522623 \nu^{7} + 1071722261 \nu^{6} - 959373046 \nu^{5} + 1267262003 \nu^{4} - 975668172 \nu^{3} + 603936480 \nu^{2} - 194264578 \nu + 55637788$$$$)/20513668$$ $$\beta_{11}$$ $$=$$ $$($$$$-14292759 \nu^{11} + 36030049 \nu^{10} + 73017973 \nu^{9} - 5911956 \nu^{8} - 1221850366 \nu^{7} + 1754115116 \nu^{6} - 1489733269 \nu^{5} + 1698131925 \nu^{4} - 1577732703 \nu^{3} + 998782176 \nu^{2} - 379578618 \nu + 95337586$$$$)/10256834$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{6} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{10} - 7 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{11} + 9 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 3 \beta_{5} + 9 \beta_{4} + 8 \beta_{2} + 8 \beta_{1} + 11$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{11} - 16 \beta_{10} + 19 \beta_{9} - 14 \beta_{7} - 51 \beta_{6} + 14 \beta_{3} + 17 \beta_{2} - 51$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-45 \beta_{10} + 20 \beta_{8} - 146 \beta_{6} + 45 \beta_{5} + 61 \beta_{4} + 40 \beta_{3} + 54 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-75 \beta_{11} + 231 \beta_{9} + 75 \beta_{8} - 85 \beta_{7} - 96 \beta_{5} + 231 \beta_{4} + 205 \beta_{2} + 205 \beta_{1} - 310$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$101 \beta_{11} - 497 \beta_{10} - 311 \beta_{9} - 441 \beta_{7} - 1609 \beta_{6} + 441 \beta_{3} - 276 \beta_{2} - 1609$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$356 \beta_{10} + 768 \beta_{8} + 1153 \beta_{6} - 356 \beta_{5} + 2367 \beta_{4} - 316 \beta_{3} + 2101 \beta_{1}$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$130 \beta_{11} - 401 \beta_{9} - 130 \beta_{8} - 4242 \beta_{7} - 4779 \beta_{5} - 401 \beta_{4} - 356 \beta_{2} - 356 \beta_{1} - 15478$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$6965 \beta_{11} - 1610 \beta_{10} - 21471 \beta_{9} - 1429 \beta_{7} - 5214 \beta_{6} + 1429 \beta_{3} - 19059 \beta_{2} - 5214$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$40931 \beta_{10} + 5807 \beta_{8} + 132571 \beta_{6} - 40931 \beta_{5} + 17901 \beta_{4} - 36333 \beta_{3} + 15890 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.378279 + 0.358951i 0.121721 + 0.507075i −2.29733 + 1.90372i 2.79733 − 1.03769i 0.907369 + 0.0534805i −0.407369 + 0.812545i 0.378279 − 0.358951i 0.121721 − 0.507075i −2.29733 − 1.90372i 2.79733 + 1.03769i 0.907369 − 0.0534805i −0.407369 − 0.812545i
0 −1.99052 3.44767i 0 −1.63031 0.941260i 0 −5.14749 + 4.74377i 0 −3.42430 + 5.93106i 0
79.2 0 −1.99052 3.44767i 0 1.63031 + 0.941260i 0 5.14749 4.74377i 0 −3.42430 + 5.93106i 0
79.3 0 0.824388 + 1.42788i 0 −3.95004 2.28056i 0 −6.75545 + 1.83408i 0 3.14077 5.43997i 0
79.4 0 0.824388 + 1.42788i 0 3.95004 + 2.28056i 0 6.75545 1.83408i 0 3.14077 5.43997i 0
79.5 0 2.66613 + 4.61787i 0 −1.86796 1.07847i 0 −6.91861 1.06433i 0 −9.71647 + 16.8294i 0
79.6 0 2.66613 + 4.61787i 0 1.86796 + 1.07847i 0 6.91861 + 1.06433i 0 −9.71647 + 16.8294i 0
207.1 0 −1.99052 + 3.44767i 0 −1.63031 + 0.941260i 0 −5.14749 4.74377i 0 −3.42430 5.93106i 0
207.2 0 −1.99052 + 3.44767i 0 1.63031 0.941260i 0 5.14749 + 4.74377i 0 −3.42430 5.93106i 0
207.3 0 0.824388 1.42788i 0 −3.95004 + 2.28056i 0 −6.75545 1.83408i 0 3.14077 + 5.43997i 0
207.4 0 0.824388 1.42788i 0 3.95004 2.28056i 0 6.75545 + 1.83408i 0 3.14077 + 5.43997i 0
207.5 0 2.66613 4.61787i 0 −1.86796 + 1.07847i 0 −6.91861 + 1.06433i 0 −9.71647 16.8294i 0
207.6 0 2.66613 4.61787i 0 1.86796 1.07847i 0 6.91861 1.06433i 0 −9.71647 16.8294i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 207.6 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
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 224.3.o.d 12
4.b odd 2 1 56.3.k.d 12
7.c even 3 1 inner 224.3.o.d 12
7.c even 3 1 1568.3.g.j 6
7.d odd 6 1 1568.3.g.l 6
8.b even 2 1 56.3.k.d 12
8.d odd 2 1 inner 224.3.o.d 12
28.d even 2 1 392.3.k.l 12
28.f even 6 1 392.3.g.i 6
28.f even 6 1 392.3.k.l 12
28.g odd 6 1 56.3.k.d 12
28.g odd 6 1 392.3.g.j 6
56.h odd 2 1 392.3.k.l 12
56.j odd 6 1 392.3.g.i 6
56.j odd 6 1 392.3.k.l 12
56.k odd 6 1 inner 224.3.o.d 12
56.k odd 6 1 1568.3.g.j 6
56.m even 6 1 1568.3.g.l 6
56.p even 6 1 56.3.k.d 12
56.p even 6 1 392.3.g.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d 12 4.b odd 2 1
56.3.k.d 12 8.b even 2 1
56.3.k.d 12 28.g odd 6 1
56.3.k.d 12 56.p even 6 1
224.3.o.d 12 1.a even 1 1 trivial
224.3.o.d 12 7.c even 3 1 inner
224.3.o.d 12 8.d odd 2 1 inner
224.3.o.d 12 56.k odd 6 1 inner
392.3.g.i 6 28.f even 6 1
392.3.g.i 6 56.j odd 6 1
392.3.g.j 6 28.g odd 6 1
392.3.g.j 6 56.p even 6 1
392.3.k.l 12 28.d even 2 1
392.3.k.l 12 28.f even 6 1
392.3.k.l 12 56.h odd 2 1
392.3.k.l 12 56.j odd 6 1
1568.3.g.j 6 7.c even 3 1
1568.3.g.j 6 56.k odd 6 1
1568.3.g.l 6 7.d odd 6 1
1568.3.g.l 6 56.m even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(224, [\chi])$$:

 $$T_{3}^{6} - 3 T_{3}^{5} + 28 T_{3}^{4} - 13 T_{3}^{3} + 466 T_{3}^{2} - 665 T_{3} + 1225$$ $$T_{5}^{12} - 29 T_{5}^{10} + 654 T_{5}^{8} - 4737 T_{5}^{6} + 25022 T_{5}^{4} - 64141 T_{5}^{2} + 117649$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 3 T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 333 T^{7} - 5265 T^{8} + 10206 T^{9} + 6561 T^{10} - 177147 T^{11} + 531441 T^{12} )^{2}$$
$5$ $$1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 285640786875 T^{14} + 5654256640625 T^{16} + 92916992187500 T^{18} + 1217498779296875 T^{20} + 11539459228515625 T^{22} + 59604644775390625 T^{24}$$
$7$ $$1 - 186 T^{2} + 16527 T^{4} - 956284 T^{6} + 39681327 T^{8} - 1072252986 T^{10} + 13841287201 T^{12}$$
$11$ $$( 1 + 15 T - 129 T^{2} - 924 T^{3} + 35727 T^{4} + 38013 T^{5} - 5198650 T^{6} + 4599573 T^{7} + 523079007 T^{8} - 1636922364 T^{9} - 27652295649 T^{10} + 389061369015 T^{11} + 3138428376721 T^{12} )^{2}$$
$13$ $$( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12} )^{2}$$
$17$ $$( 1 - 15 T - 665 T^{2} + 3920 T^{3} + 405365 T^{4} - 1221665 T^{5} - 124870762 T^{6} - 353061185 T^{7} + 33856490165 T^{8} + 94619270480 T^{9} - 4638878698265 T^{10} - 30239908506735 T^{11} + 582622237229761 T^{12} )^{2}$$
$19$ $$( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 2101362589 T^{7} + 34293580187 T^{8} - 186489872284 T^{9} + 2564518019191 T^{10} + 239111584054239 T^{11} + 2213314919066161 T^{12} )^{2}$$
$23$ $$1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 7107938910998636023 T^{14} -$$$$95\!\cdots\!15$$$$T^{16} -$$$$69\!\cdots\!00$$$$T^{18} -$$$$95\!\cdots\!41$$$$T^{20} +$$$$11\!\cdots\!93$$$$T^{22} +$$$$48\!\cdots\!41$$$$T^{24}$$
$29$ $$( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12} )^{2}$$
$31$ $$1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} +$$$$12\!\cdots\!55$$$$T^{14} +$$$$10\!\cdots\!13$$$$T^{16} +$$$$65\!\cdots\!04$$$$T^{18} +$$$$42\!\cdots\!99$$$$T^{20} +$$$$23\!\cdots\!61$$$$T^{22} +$$$$62\!\cdots\!21$$$$T^{24}$$
$37$ $$1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} +$$$$11\!\cdots\!14$$$$T^{12} +$$$$16\!\cdots\!67$$$$T^{14} +$$$$22\!\cdots\!81$$$$T^{16} +$$$$25\!\cdots\!12$$$$T^{18} +$$$$22\!\cdots\!07$$$$T^{20} +$$$$13\!\cdots\!85$$$$T^{22} +$$$$43\!\cdots\!61$$$$T^{24}$$
$41$ $$( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 5276659 T^{4} + 163894138 T^{5} + 4750104241 T^{6} )^{4}$$
$43$ $$( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 7482903 T^{4} - 170940050 T^{5} + 6321363049 T^{6} )^{4}$$
$47$ $$1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} -$$$$24\!\cdots\!46$$$$T^{12} -$$$$48\!\cdots\!93$$$$T^{14} -$$$$68\!\cdots\!19$$$$T^{16} +$$$$24\!\cdots\!32$$$$T^{18} +$$$$37\!\cdots\!27$$$$T^{20} +$$$$10\!\cdots\!05$$$$T^{22} +$$$$13\!\cdots\!81$$$$T^{24}$$
$53$ $$1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} +$$$$12\!\cdots\!10$$$$T^{12} +$$$$59\!\cdots\!91$$$$T^{14} +$$$$30\!\cdots\!97$$$$T^{16} +$$$$10\!\cdots\!44$$$$T^{18} +$$$$23\!\cdots\!51$$$$T^{20} +$$$$32\!\cdots\!61$$$$T^{22} +$$$$24\!\cdots\!81$$$$T^{24}$$
$59$ $$( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 2082660463687 T^{7} + 619437155669327 T^{8} + 7640666224798022 T^{9} - 1081699833831032807 T^{10} - 28111421431535277055 T^{11} +$$$$17\!\cdots\!81$$$$T^{12} )^{2}$$
$61$ $$1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} -$$$$19\!\cdots\!14$$$$T^{12} -$$$$54\!\cdots\!09$$$$T^{14} -$$$$14\!\cdots\!35$$$$T^{16} +$$$$41\!\cdots\!00$$$$T^{18} +$$$$20\!\cdots\!71$$$$T^{20} +$$$$46\!\cdots\!01$$$$T^{22} +$$$$70\!\cdots\!41$$$$T^{24}$$
$67$ $$( 1 + 217 T + 18053 T^{2} + 1666970 T^{3} + 209974835 T^{4} + 15078221277 T^{5} + 815515698066 T^{6} + 67686135312453 T^{7} + 4231228307040035 T^{8} + 150791409324257930 T^{9} + 7330739782930039973 T^{10} +$$$$39\!\cdots\!33$$$$T^{11} +$$$$81\!\cdots\!61$$$$T^{12} )^{2}$$
$71$ $$( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} +$$$$16\!\cdots\!41$$$$T^{12} )^{2}$$
$73$ $$( 1 - 51 T - 9165 T^{2} + 579660 T^{3} + 45654729 T^{4} - 1993134921 T^{5} - 160997144218 T^{6} - 10621415994009 T^{7} + 1296513996931689 T^{8} + 87722397610681740 T^{9} - 7391206742209252365 T^{10} -$$$$21\!\cdots\!99$$$$T^{11} +$$$$22\!\cdots\!21$$$$T^{12} )^{2}$$
$79$ $$1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} -$$$$37\!\cdots\!74$$$$T^{12} -$$$$17\!\cdots\!25$$$$T^{14} -$$$$14\!\cdots\!39$$$$T^{16} +$$$$37\!\cdots\!92$$$$T^{18} +$$$$34\!\cdots\!43$$$$T^{20} +$$$$14\!\cdots\!93$$$$T^{22} +$$$$34\!\cdots\!81$$$$T^{24}$$
$83$ $$( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 155574287 T^{4} - 6359415014 T^{5} + 326940373369 T^{6} )^{4}$$
$89$ $$( 1 - 107 T + 1395 T^{2} + 497084 T^{3} - 62702935 T^{4} + 2422278015 T^{5} + 93422604838 T^{6} + 19186864156815 T^{7} - 3934122659177335 T^{8} + 247041448036057724 T^{9} + 5491541383954402995 T^{10} -$$$$33\!\cdots\!07$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} )^{2}$$
$97$ $$( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 237022119 T^{4} + 3364112678 T^{5} + 832972004929 T^{6} )^{4}$$