# Properties

 Label 504.2.bl.a Level 504 Weight 2 Character orbit 504.bl Analytic conductor 4.024 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 504.bl (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.02446026187$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} - \beta_{10} ) q^{5} + ( -\beta_{3} - \beta_{9} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{4} - \beta_{10} ) q^{5} + ( -\beta_{3} - \beta_{9} ) q^{7} + ( -\beta_{2} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{11} + ( 1 - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{13} + ( -\beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{17} + ( 3 + 2 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{19} + ( \beta_{2} - \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{23} + ( 3 + \beta_{1} - 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{25} + ( -\beta_{2} + 2 \beta_{4} - \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{29} + ( 2 + \beta_{1} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{31} + ( \beta_{2} - \beta_{4} + 2 \beta_{10} - 2 \beta_{11} ) q^{35} + ( -2 - 3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{37} + ( -\beta_{2} - \beta_{10} + \beta_{11} - 4 \beta_{12} - \beta_{13} - \beta_{15} ) q^{41} + ( 1 + \beta_{6} - \beta_{9} ) q^{43} + ( -2 \beta_{2} + 2 \beta_{4} - 2 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{47} + ( -\beta_{1} - \beta_{3} + 3 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{49} + ( \beta_{2} + 2 \beta_{4} - 4 \beta_{10} - \beta_{11} + 4 \beta_{12} - \beta_{13} - \beta_{14} ) q^{53} + ( 2 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{55} + ( -3 \beta_{4} + 2 \beta_{11} - \beta_{12} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{61} + ( \beta_{2} + 3 \beta_{4} + 3 \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{65} + ( -4 + \beta_{3} + 3 \beta_{5} + \beta_{8} ) q^{67} + ( -4 \beta_{4} + 2 \beta_{10} + \beta_{11} ) q^{71} + ( -1 + \beta_{1} + 2 \beta_{3} - 4 \beta_{5} + \beta_{7} - 3 \beta_{8} ) q^{73} + ( 2 \beta_{2} - 3 \beta_{4} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{77} + ( -4 - 2 \beta_{1} - 3 \beta_{3} + \beta_{6} + \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{79} + ( -\beta_{2} + \beta_{10} - 2 \beta_{11} + 4 \beta_{12} + \beta_{14} - \beta_{15} ) q^{83} + ( -3 - \beta_{1} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{85} + ( -2 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{89} + ( -10 - 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} + 5 \beta_{7} + \beta_{8} + \beta_{9} ) q^{91} + ( 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{10} - 2 \beta_{11} - 5 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{95} + ( -3 + 3 \beta_{1} + 3 \beta_{3} + 6 \beta_{5} + 4 \beta_{6} + 4 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{7} + O(q^{10})$$ $$16q - 8q^{7} + 12q^{19} + 12q^{25} + 24q^{31} + 4q^{37} + 8q^{43} + 32q^{49} - 28q^{67} - 60q^{73} - 32q^{79} - 32q^{85} - 84q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10 x^{14} + 61 x^{12} + 266 x^{10} + 852 x^{8} + 1438 x^{6} + 1933 x^{4} + 3038 x^{2} + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$34158 \nu^{14} + 852503 \nu^{12} + 4914956 \nu^{10} + 18664744 \nu^{8} + 59406470 \nu^{6} + 70780144 \nu^{4} - 260640102 \nu^{2} + 139803027$$$$)/ 392984900$$ $$\beta_{2}$$ $$=$$ $$($$$$66287 \nu^{15} - 2125573 \nu^{13} - 20719966 \nu^{11} - 144889584 \nu^{9} - 625207850 \nu^{7} - 2285038534 \nu^{5} - 3180000603 \nu^{3} - 6339564777 \nu$$$$)/ 2750894300$$ $$\beta_{3}$$ $$=$$ $$($$$$-382234 \nu^{14} - 3667549 \nu^{12} - 21832848 \nu^{10} - 94498292 \nu^{8} - 297554910 \nu^{6} - 530245552 \nu^{4} - 674195374 \nu^{2} - 1050178241$$$$)/ 392984900$$ $$\beta_{4}$$ $$=$$ $$($$$$-88213 \nu^{15} - 1057753 \nu^{13} - 8202336 \nu^{11} - 41107094 \nu^{9} - 157143590 \nu^{7} - 407420164 \nu^{5} - 856246593 \nu^{3} - 905297057 \nu$$$$)/ 785969800$$ $$\beta_{5}$$ $$=$$ $$($$$$623003 \nu^{14} + 5366258 \nu^{12} + 30938216 \nu^{10} + 125675214 \nu^{8} + 373946420 \nu^{6} + 445540384 \nu^{4} + 843429583 \nu^{2} + 1273004222$$$$)/ 392984900$$ $$\beta_{6}$$ $$=$$ $$($$$$83051 \nu^{14} + 542488 \nu^{12} + 2797264 \nu^{10} + 9444519 \nu^{8} + 21581337 \nu^{6} - 16157524 \nu^{4} + 47574110 \nu^{2} + 3256785$$$$)/39298490$$ $$\beta_{7}$$ $$=$$ $$($$$$17628 \nu^{14} + 144183 \nu^{12} + 817216 \nu^{10} + 3201064 \nu^{8} + 9190570 \nu^{6} + 7363984 \nu^{4} + 12646508 \nu^{2} + 14486947$$$$)/8020100$$ $$\beta_{8}$$ $$=$$ $$($$$$3705 \nu^{14} + 31546 \nu^{12} + 171760 \nu^{10} + 672790 \nu^{8} + 1902192 \nu^{6} + 1547740 \nu^{4} + 2658005 \nu^{2} + 7562162$$$$)/1604020$$ $$\beta_{9}$$ $$=$$ $$($$$$1744691 \nu^{14} + 15899931 \nu^{12} + 90773542 \nu^{10} + 373226028 \nu^{8} + 1092704010 \nu^{6} + 1301735958 \nu^{4} + 1516055181 \nu^{2} + 3124455159$$$$)/ 392984900$$ $$\beta_{10}$$ $$=$$ $$($$$$-154017 \nu^{15} - 1553547 \nu^{13} - 9597064 \nu^{11} - 41588666 \nu^{9} - 133651610 \nu^{7} - 226254436 \nu^{5} - 303303017 \nu^{3} - 238292243 \nu$$$$)/ 239208200$$ $$\beta_{11}$$ $$=$$ $$($$$$-13169 \nu^{15} - 130759 \nu^{13} - 753868 \nu^{11} - 3141922 \nu^{9} - 9111910 \nu^{7} - 10856432 \nu^{5} - 5104409 \nu^{3} - 21443331 \nu$$$$)/17355800$$ $$\beta_{12}$$ $$=$$ $$($$$$7738601 \nu^{15} + 58163261 \nu^{13} + 313306372 \nu^{11} + 1167320938 \nu^{9} + 3102633990 \nu^{7} + 1535163128 \nu^{5} + 6928535561 \nu^{3} + 9719272549 \nu$$$$)/ 5501788600$$ $$\beta_{13}$$ $$=$$ $$($$$$-9919057 \nu^{15} - 78396587 \nu^{13} - 451981724 \nu^{11} - 1839736626 \nu^{9} - 5463047630 \nu^{7} - 6508976176 \nu^{5} - 15803364417 \nu^{3} - 7354565183 \nu$$$$)/ 5501788600$$ $$\beta_{14}$$ $$=$$ $$($$$$-438892 \nu^{15} - 3881427 \nu^{13} - 22712419 \nu^{11} - 93951256 \nu^{9} - 283749965 \nu^{7} - 373105681 \nu^{5} - 562359552 \nu^{3} - 752948308 \nu$$$$)/ 125040650$$ $$\beta_{15}$$ $$=$$ $$($$$$965736 \nu^{15} + 8386545 \nu^{13} + 48089618 \nu^{11} + 197972096 \nu^{9} + 592040102 \nu^{7} + 736276512 \nu^{5} + 1335892868 \nu^{3} + 1747120823 \nu$$$$)/ 275089430$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{14} + \beta_{13} + 3 \beta_{10} - 2 \beta_{4}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} + 2 \beta_{5} + \beta_{3} - 2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} + 5 \beta_{14} - \beta_{13} + 2 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} - 8 \beta_{4} + 2 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$-\beta_{8} - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 5 \beta_{3} + 2 \beta_{1} + 2$$ $$\nu^{5}$$ $$=$$ $$($$$$-16 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + 28 \beta_{12} - 41 \beta_{10} + 34 \beta_{4} - 28 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$-14 \beta_{9} + 19 \beta_{8} + 24 \beta_{7} - 14 \beta_{6} - 7 \beta_{5} - 7 \beta_{3} - 7 \beta_{1} - 14$$ $$\nu^{7}$$ $$=$$ $$($$$$98 \beta_{15} - 83 \beta_{14} + 91 \beta_{13} - 158 \beta_{12} + 256 \beta_{11} + 69 \beta_{10} + 112 \beta_{4} + 90 \beta_{2}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$62 \beta_{9} - 31 \beta_{8} - 37 \beta_{7} - 31 \beta_{5} + 68 \beta_{3} - 81 \beta_{1} + 31$$ $$\nu^{9}$$ $$=$$ $$($$$$-112 \beta_{15} + 235 \beta_{14} - 535 \beta_{13} - 112 \beta_{12} - 1092 \beta_{11} + 759 \beta_{10} - 658 \beta_{4} - 112 \beta_{2}$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$-404 \beta_{8} - 99 \beta_{7} + 198 \beta_{6} + 454 \beta_{5} + 32 \beta_{3} + 503 \beta_{1} + 503$$ $$\nu^{11}$$ $$=$$ $$($$$$-1526 \beta_{15} - 759 \beta_{14} + 759 \beta_{13} + 1942 \beta_{12} - 1671 \beta_{10} - 2132 \beta_{4} + 614 \beta_{2}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$-332 \beta_{9} + 1742 \beta_{8} - 788 \beta_{7} - 332 \beta_{6} - 166 \beta_{5} - 166 \beta_{3} - 166 \beta_{1} - 3969$$ $$\nu^{13}$$ $$=$$ $$($$$$7424 \beta_{15} + 7163 \beta_{14} + 3621 \beta_{13} + 1616 \beta_{12} + 5808 \beta_{11} - 14769 \beta_{10} + 15030 \beta_{4} - 3360 \beta_{2}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$-1244 \beta_{9} + 622 \beta_{8} + 6831 \beta_{7} - 10592 \beta_{5} - 7453 \beta_{3} - 4438 \beta_{1} + 10592$$ $$\nu^{15}$$ $$=$$ $$($$$$-15030 \beta_{15} - 29095 \beta_{14} - 13493 \beta_{13} - 15030 \beta_{12} + 22012 \beta_{11} + 43553 \beta_{10} + 30632 \beta_{4} - 15030 \beta_{2}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$127$$ $$253$$ $$281$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 1.45333 + 1.51725i 1.01089 + 0.750919i 0.587308 + 2.01725i 0.144868 + 1.25092i −0.144868 − 1.25092i −0.587308 − 2.01725i −1.01089 − 0.750919i −1.45333 − 1.51725i 1.45333 − 1.51725i 1.01089 − 0.750919i 0.587308 − 2.01725i 0.144868 − 1.25092i −0.144868 + 1.25092i −0.587308 + 2.01725i −1.01089 + 0.750919i −1.45333 + 1.51725i
0 0 0 −1.45333 2.51725i 0 −2.64571 0.0146827i 0 0 0
17.2 0 0 0 −1.01089 1.75092i 0 0.561961 + 2.58538i 0 0 0
17.3 0 0 0 −0.587308 1.01725i 0 2.35282 1.21006i 0 0 0
17.4 0 0 0 −0.144868 0.250919i 0 −2.26907 1.36064i 0 0 0
17.5 0 0 0 0.144868 + 0.250919i 0 −2.26907 1.36064i 0 0 0
17.6 0 0 0 0.587308 + 1.01725i 0 2.35282 1.21006i 0 0 0
17.7 0 0 0 1.01089 + 1.75092i 0 0.561961 + 2.58538i 0 0 0
17.8 0 0 0 1.45333 + 2.51725i 0 −2.64571 0.0146827i 0 0 0
89.1 0 0 0 −1.45333 + 2.51725i 0 −2.64571 + 0.0146827i 0 0 0
89.2 0 0 0 −1.01089 + 1.75092i 0 0.561961 2.58538i 0 0 0
89.3 0 0 0 −0.587308 + 1.01725i 0 2.35282 + 1.21006i 0 0 0
89.4 0 0 0 −0.144868 + 0.250919i 0 −2.26907 + 1.36064i 0 0 0
89.5 0 0 0 0.144868 0.250919i 0 −2.26907 + 1.36064i 0 0 0
89.6 0 0 0 0.587308 1.01725i 0 2.35282 + 1.21006i 0 0 0
89.7 0 0 0 1.01089 1.75092i 0 0.561961 2.58538i 0 0 0
89.8 0 0 0 1.45333 2.51725i 0 −2.64571 + 0.0146827i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.8 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
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.bl.a 16
3.b odd 2 1 inner 504.2.bl.a 16
4.b odd 2 1 1008.2.bt.d 16
7.b odd 2 1 3528.2.bl.a 16
7.c even 3 1 3528.2.k.b 16
7.c even 3 1 3528.2.bl.a 16
7.d odd 6 1 inner 504.2.bl.a 16
7.d odd 6 1 3528.2.k.b 16
12.b even 2 1 1008.2.bt.d 16
21.c even 2 1 3528.2.bl.a 16
21.g even 6 1 inner 504.2.bl.a 16
21.g even 6 1 3528.2.k.b 16
21.h odd 6 1 3528.2.k.b 16
21.h odd 6 1 3528.2.bl.a 16
28.f even 6 1 1008.2.bt.d 16
28.f even 6 1 7056.2.k.h 16
28.g odd 6 1 7056.2.k.h 16
84.j odd 6 1 1008.2.bt.d 16
84.j odd 6 1 7056.2.k.h 16
84.n even 6 1 7056.2.k.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bl.a 16 1.a even 1 1 trivial
504.2.bl.a 16 3.b odd 2 1 inner
504.2.bl.a 16 7.d odd 6 1 inner
504.2.bl.a 16 21.g even 6 1 inner
1008.2.bt.d 16 4.b odd 2 1
1008.2.bt.d 16 12.b even 2 1
1008.2.bt.d 16 28.f even 6 1
1008.2.bt.d 16 84.j odd 6 1
3528.2.k.b 16 7.c even 3 1
3528.2.k.b 16 7.d odd 6 1
3528.2.k.b 16 21.g even 6 1
3528.2.k.b 16 21.h odd 6 1
3528.2.bl.a 16 7.b odd 2 1
3528.2.bl.a 16 7.c even 3 1
3528.2.bl.a 16 21.c even 2 1
3528.2.bl.a 16 21.h odd 6 1
7056.2.k.h 16 28.f even 6 1
7056.2.k.h 16 28.g odd 6 1
7056.2.k.h 16 84.j odd 6 1
7056.2.k.h 16 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(504, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$1 - 26 T^{2} + 343 T^{4} - 3142 T^{6} + 22997 T^{8} - 145796 T^{10} + 840842 T^{12} - 4532272 T^{14} + 23186506 T^{16} - 113306800 T^{18} + 525526250 T^{20} - 2278062500 T^{22} + 8983203125 T^{24} - 30683593750 T^{26} + 83740234375 T^{28} - 158691406250 T^{30} + 152587890625 T^{32}$$
$7$ $$( 1 + 4 T - 4 T^{3} + 29 T^{4} - 28 T^{5} + 1372 T^{7} + 2401 T^{8} )^{2}$$
$11$ $$1 + 34 T^{2} + 631 T^{4} + 7742 T^{6} + 56309 T^{8} + 36244 T^{10} - 4803190 T^{12} - 80571856 T^{14} - 990277910 T^{16} - 9749194576 T^{18} - 70323504790 T^{20} + 64208456884 T^{22} + 12070334230229 T^{24} + 200807541260942 T^{26} + 1980348305710951 T^{28} + 12911494341830194 T^{30} + 45949729863572161 T^{32}$$
$13$ $$( 1 - 22 T^{2} + 597 T^{4} - 10874 T^{6} + 145448 T^{8} - 1837706 T^{10} + 17050917 T^{12} - 106189798 T^{14} + 815730721 T^{16} )^{2}$$
$17$ $$1 - 84 T^{2} + 3336 T^{4} - 99240 T^{6} + 2736514 T^{8} - 66944172 T^{10} + 1418515200 T^{12} - 27661974396 T^{14} + 497038747923 T^{16} - 7994310600444 T^{18} + 118475808019200 T^{20} - 1615869570797868 T^{22} + 19089257897900674 T^{24} - 200067234680558760 T^{26} + 1943627783398482696 T^{28} - 14143737430989678036 T^{30} + 48661191875666868481 T^{32}$$
$19$ $$( 1 - 6 T + 41 T^{2} - 174 T^{3} + 567 T^{4} + 1140 T^{5} - 10940 T^{6} + 109212 T^{7} - 514114 T^{8} + 2075028 T^{9} - 3949340 T^{10} + 7819260 T^{11} + 73892007 T^{12} - 430841226 T^{13} + 1928881121 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} )^{2}$$
$23$ $$1 + 84 T^{2} + 2840 T^{4} + 59304 T^{6} + 1536034 T^{8} + 51810444 T^{10} + 1287398720 T^{12} + 22322505660 T^{14} + 403780671955 T^{16} + 11808605494140 T^{18} + 360266945203520 T^{20} + 7669805137024716 T^{22} + 120288335965115554 T^{24} + 2456757821014240296 T^{26} + 62237533386937711640 T^{28} +$$$$97\!\cdots\!56$$$$T^{30} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$( 1 - 82 T^{2} + 3369 T^{4} - 137570 T^{6} + 4900772 T^{8} - 115696370 T^{10} + 2382829689 T^{12} - 48775512322 T^{14} + 500246412961 T^{16} )^{2}$$
$31$ $$( 1 - 8 T + 72 T^{2} - 172 T^{3} + 1373 T^{4} - 5332 T^{5} + 69192 T^{6} - 238328 T^{7} + 923521 T^{8} )^{2}( 1 - 4 T + 48 T^{2} - 404 T^{3} + 1502 T^{4} - 12524 T^{5} + 46128 T^{6} - 119164 T^{7} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 2 T - 39 T^{2} + 686 T^{3} - 133 T^{4} - 26040 T^{5} + 186136 T^{6} + 542152 T^{7} - 8048970 T^{8} + 20059624 T^{9} + 254820184 T^{10} - 1319004120 T^{11} - 249263413 T^{12} + 47569954502 T^{13} - 100063329951 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 + 204 T^{2} + 20760 T^{4} + 1374660 T^{6} + 65555246 T^{8} + 2310803460 T^{10} + 58662798360 T^{12} + 969021265164 T^{14} + 7984925229121 T^{16} )^{2}$$
$43$ $$( 1 - 2 T + 139 T^{2} - 134 T^{3} + 8158 T^{4} - 5762 T^{5} + 257011 T^{6} - 159014 T^{7} + 3418801 T^{8} )^{4}$$
$47$ $$1 - 200 T^{2} + 20236 T^{4} - 1289200 T^{6} + 54604010 T^{8} - 1402967000 T^{10} + 1590108464 T^{12} + 2082887856200 T^{14} - 134280636932141 T^{16} + 4601099274345800 T^{18} + 7759222059719984 T^{20} - 15122883392481143000 T^{22} +$$$$13\!\cdots\!10$$$$T^{24} -$$$$67\!\cdots\!00$$$$T^{26} +$$$$23\!\cdots\!76$$$$T^{28} -$$$$51\!\cdots\!00$$$$T^{30} +$$$$56\!\cdots\!21$$$$T^{32}$$
$53$ $$1 + 82 T^{2} + 1819 T^{4} - 293986 T^{6} - 25475671 T^{8} - 757240268 T^{10} + 15438451790 T^{12} + 2705625303848 T^{14} + 146204406396370 T^{16} + 7600101478509032 T^{18} + 121816810518410990 T^{20} - 16783746761372742572 T^{22} -$$$$15\!\cdots\!31$$$$T^{24} -$$$$51\!\cdots\!14$$$$T^{26} +$$$$89\!\cdots\!79$$$$T^{28} +$$$$11\!\cdots\!58$$$$T^{30} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$1 - 322 T^{2} + 52107 T^{4} - 6104798 T^{6} + 600323513 T^{8} - 51577574244 T^{10} + 3916097487742 T^{12} - 266299877019064 T^{14} + 16439030582633874 T^{16} - 926989871903361784 T^{18} + 47452766970162888862 T^{20} -$$$$21\!\cdots\!04$$$$T^{22} +$$$$88\!\cdots\!73$$$$T^{24} -$$$$31\!\cdots\!98$$$$T^{26} +$$$$92\!\cdots\!67$$$$T^{28} -$$$$19\!\cdots\!42$$$$T^{30} +$$$$21\!\cdots\!41$$$$T^{32}$$
$61$ $$( 1 + 70 T^{2} + 4306 T^{4} + 9720 T^{5} - 90560 T^{6} + 641520 T^{7} - 8009105 T^{8} + 39132720 T^{9} - 336973760 T^{10} + 2206255320 T^{11} + 59620191346 T^{12} + 3606426205270 T^{14} + 191707312997281 T^{16} )^{2}$$
$67$ $$( 1 + 14 T - 125 T^{2} - 1238 T^{3} + 28877 T^{4} + 158888 T^{5} - 2519254 T^{6} - 1335236 T^{7} + 243583474 T^{8} - 89460812 T^{9} - 11308931206 T^{10} + 47787631544 T^{11} + 581903921117 T^{12} - 1671454882466 T^{13} - 11307297771125 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 392 T^{2} + 75892 T^{4} - 9334040 T^{6} + 790096294 T^{8} - 47052895640 T^{10} + 1928543294452 T^{12} - 50215311297032 T^{14} + 645753531245761 T^{16} )^{2}$$
$73$ $$( 1 + 30 T + 591 T^{2} + 8730 T^{3} + 105765 T^{4} + 1168044 T^{5} + 12012906 T^{6} + 115920888 T^{7} + 1041481850 T^{8} + 8462224824 T^{9} + 64016776074 T^{10} + 454388972748 T^{11} + 3003539959365 T^{12} + 18097915006890 T^{13} + 89438527736799 T^{14} + 331421955572910 T^{15} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 + 16 T + 64 T^{2} + 56 T^{3} - 1765 T^{4} - 55364 T^{5} - 19696 T^{6} + 3181796 T^{7} + 14045608 T^{8} + 251361884 T^{9} - 122922736 T^{10} - 27296611196 T^{11} - 68746892965 T^{12} + 172315158344 T^{13} + 15557597153344 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$( 1 + 438 T^{2} + 94557 T^{4} + 13226610 T^{6} + 1298783528 T^{8} + 91118116290 T^{10} + 4487516458797 T^{12} + 143199883535622 T^{14} + 2252292232139041 T^{16} )^{2}$$
$89$ $$1 - 384 T^{2} + 64644 T^{4} - 8318208 T^{6} + 1128930634 T^{8} - 127988142720 T^{10} + 11460228290064 T^{12} - 1112004852903552 T^{14} + 109920677865950547 T^{16} - 8808190439849035392 T^{18} +$$$$71\!\cdots\!24$$$$T^{20} -$$$$63\!\cdots\!20$$$$T^{22} +$$$$44\!\cdots\!54$$$$T^{24} -$$$$25\!\cdots\!08$$$$T^{26} +$$$$15\!\cdots\!24$$$$T^{28} -$$$$75\!\cdots\!44$$$$T^{30} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$( 1 - 234 T^{2} + 40305 T^{4} - 5045730 T^{6} + 535964996 T^{8} - 47475273570 T^{10} + 3568172670705 T^{12} - 194915449153386 T^{14} + 7837433594376961 T^{16} )^{2}$$