# Properties

 Label 6048.2.c.e Level 6048 Weight 2 Character orbit 6048.c Analytic conductor 48.294 Analytic rank 0 Dimension 20 CM no Inner twists 4

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$6048 = 2^{5} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6048.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.2935231425$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{18}\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 1512) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{12} q^{5} + q^{7} +O(q^{10})$$ $$q + \beta_{12} q^{5} + q^{7} + ( -\beta_{4} - \beta_{10} + \beta_{12} ) q^{11} + \beta_{5} q^{13} -\beta_{1} q^{17} + \beta_{2} q^{19} + \beta_{11} q^{23} + ( -1 + \beta_{8} ) q^{25} + ( -\beta_{4} + \beta_{12} - \beta_{16} ) q^{29} + ( -2 - \beta_{7} ) q^{31} + \beta_{12} q^{35} + ( -\beta_{5} - \beta_{9} - \beta_{13} ) q^{37} -\beta_{17} q^{41} + ( \beta_{3} - \beta_{5} - \beta_{9} ) q^{43} + ( -\beta_{11} - \beta_{15} - \beta_{19} ) q^{47} + q^{49} + ( -\beta_{4} + \beta_{12} - \beta_{14} ) q^{53} + ( -1 - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{18} ) q^{55} + ( -\beta_{4} + \beta_{16} ) q^{59} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{13} ) q^{61} + ( \beta_{1} + \beta_{19} ) q^{65} + \beta_{9} q^{67} + ( \beta_{1} - \beta_{15} - \beta_{17} ) q^{71} + ( -2 \beta_{7} - \beta_{18} ) q^{73} + ( -\beta_{4} - \beta_{10} + \beta_{12} ) q^{77} + ( -3 - \beta_{6} ) q^{79} + ( -2 \beta_{10} + \beta_{12} + \beta_{14} - 2 \beta_{16} ) q^{83} + ( \beta_{2} - \beta_{3} + 3 \beta_{5} ) q^{85} + ( \beta_{1} + \beta_{15} - \beta_{17} ) q^{89} + \beta_{5} q^{91} + ( \beta_{11} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{95} + ( 3 - \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 20q^{7} + O(q^{10})$$ $$20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + x^{18} + 4 x^{16} + 8 x^{12} + 4 x^{10} + 32 x^{8} + 256 x^{4} + 256 x^{2} + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{17} + \nu^{15} - 2 \nu^{13} - 8 \nu^{11} + 8 \nu^{9} - 20 \nu^{7} + 40 \nu^{5} - 192 \nu$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{18} - 17 \nu^{16} + 4 \nu^{14} + 8 \nu^{12} + 8 \nu^{10} + 124 \nu^{8} + 160 \nu^{6} + 96 \nu^{4} - 64 \nu^{2} - 3072$$$$)/896$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{18} + 4 \nu^{16} - 3 \nu^{14} + 8 \nu^{12} + 64 \nu^{10} + 68 \nu^{8} + 244 \nu^{6} + 208 \nu^{4} + 160 \nu^{2} + 960$$$$)/448$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{19} - 3 \nu^{17} + 4 \nu^{15} - 6 \nu^{13} + 8 \nu^{11} - 100 \nu^{9} - 120 \nu^{7} + 40 \nu^{5} - 288 \nu^{3} - 832 \nu$$$$)/896$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{18} + \nu^{16} + 4 \nu^{14} + 8 \nu^{10} + 4 \nu^{8} + 32 \nu^{6} + 512 \nu^{2} + 256$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{18} + 3 \nu^{16} - 16 \nu^{12} - 40 \nu^{10} + 28 \nu^{8} + 112 \nu^{6} + 128 \nu^{4} - 384 \nu^{2} + 256$$$$)/256$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{18} - \nu^{16} - 4 \nu^{14} - 8 \nu^{10} - 4 \nu^{8} - 32 \nu^{6} - 256$$$$)/256$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{18} - 3 \nu^{16} - 16 \nu^{12} + 8 \nu^{10} - 28 \nu^{8} + 16 \nu^{6} + 128 \nu^{4} + 256 \nu^{2} - 512$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{18} + 3 \nu^{16} + 16 \nu^{12} - 8 \nu^{10} + 28 \nu^{8} - 16 \nu^{6} + 384 \nu^{4} - 256 \nu^{2} + 768$$$$)/256$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{19} + 3 \nu^{17} + 16 \nu^{13} - 8 \nu^{11} + 28 \nu^{9} - 16 \nu^{7} + 128 \nu^{5} + 256 \nu^{3} + 768 \nu$$$$)/512$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{19} - 3 \nu^{17} - 16 \nu^{13} + 8 \nu^{11} - 28 \nu^{9} + 16 \nu^{7} - 128 \nu^{5} + 768 \nu^{3} - 768 \nu$$$$)/512$$ $$\beta_{12}$$ $$=$$ $$($$$$9 \nu^{19} - 15 \nu^{17} + 20 \nu^{15} + 40 \nu^{13} + 40 \nu^{11} - 220 \nu^{9} + 128 \nu^{7} + 32 \nu^{5} + 2816 \nu^{3} - 1024 \nu$$$$)/3584$$ $$\beta_{13}$$ $$=$$ $$($$$$-15 \nu^{18} - 31 \nu^{16} + 4 \nu^{14} - 48 \nu^{12} + 8 \nu^{10} - 828 \nu^{8} + 608 \nu^{6} - 1472 \nu^{4} - 2304 \nu^{2} - 10240$$$$)/1792$$ $$\beta_{14}$$ $$=$$ $$($$$$-3 \nu^{19} - 2 \nu^{17} + 5 \nu^{15} - 4 \nu^{13} + 80 \nu^{11} + 92 \nu^{9} + 228 \nu^{7} + 736 \nu^{5} + 32 \nu^{3} + 640 \nu$$$$)/896$$ $$\beta_{15}$$ $$=$$ $$($$$$-\nu^{19} - \nu^{17} - 4 \nu^{15} - 8 \nu^{11} - 4 \nu^{9} - 32 \nu^{7} - 256 \nu^{3} + 256 \nu$$$$)/256$$ $$\beta_{16}$$ $$=$$ $$($$$$\nu^{19} + \nu^{17} + 4 \nu^{15} + 8 \nu^{11} + 4 \nu^{9} + 32 \nu^{7} + 256 \nu^{3} + 768 \nu$$$$)/256$$ $$\beta_{17}$$ $$=$$ $$($$$$-3 \nu^{19} - 11 \nu^{17} + 4 \nu^{15} - 8 \nu^{13} + 8 \nu^{11} - 12 \nu^{9} - 64 \nu^{7} - 416 \nu^{5} - 256 \nu^{3} - 2560 \nu$$$$)/512$$ $$\beta_{18}$$ $$=$$ $$($$$$\nu^{18} - \nu^{14} - 4 \nu^{8} + 28 \nu^{6} + 16 \nu^{4} + 224 \nu^{2} + 128$$$$)/64$$ $$\beta_{19}$$ $$=$$ $$($$$$-\nu^{19} - \nu^{17} + 2 \nu^{15} - 2 \nu^{13} - 8 \nu^{11} - 20 \nu^{9} + 48 \nu^{7} - 40 \nu^{5} - 160 \nu^{3} - 576 \nu$$$$)/128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{16} + \beta_{15}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{9} + \beta_{8} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{17} - \beta_{16} + \beta_{14} + \beta_{12} + \beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{18} + \beta_{13} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$2 \beta_{19} - \beta_{17} + \beta_{15} + \beta_{14} + 3 \beta_{12} - 2 \beta_{10} - 5 \beta_{4} - \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-\beta_{18} - 3 \beta_{13} - 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + 3 \beta_{2} - 3$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-2 \beta_{19} + 3 \beta_{17} - 3 \beta_{15} + \beta_{14} - 5 \beta_{12} - 2 \beta_{11} - 9 \beta_{4} + 3 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-3 \beta_{18} - \beta_{13} - 2 \beta_{9} + 5 \beta_{8} - 5 \beta_{7} - 5 \beta_{6} - 4 \beta_{5} + 7 \beta_{3} - 3 \beta_{2} - 5$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-6 \beta_{19} + 5 \beta_{17} - 2 \beta_{16} - 7 \beta_{15} + 7 \beta_{14} - 3 \beta_{12} - 2 \beta_{11} - 4 \beta_{10} + 9 \beta_{4} - 11 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$7 \beta_{18} + 5 \beta_{13} + 10 \beta_{9} - 17 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} - 3 \beta_{3} + 7 \beta_{2} - 15$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-2 \beta_{19} + 7 \beta_{17} - 14 \beta_{16} - 5 \beta_{15} - 3 \beta_{14} + 47 \beta_{12} - 26 \beta_{11} + 16 \beta_{10} - 13 \beta_{4} - 9 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-19 \beta_{18} + 7 \beta_{13} + 10 \beta_{9} + \beta_{8} - 49 \beta_{7} + 3 \beta_{6} + 48 \beta_{5} - 17 \beta_{3} + 13 \beta_{2} + 31$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$10 \beta_{19} + 17 \beta_{17} + 58 \beta_{16} - 39 \beta_{15} - 5 \beta_{14} - 31 \beta_{12} - 30 \beta_{11} + 16 \beta_{10} + 45 \beta_{4} + 25 \beta_{1}$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$-5 \beta_{18} - 7 \beta_{13} - 2 \beta_{9} - 17 \beta_{8} - 7 \beta_{7} + 13 \beta_{6} + 16 \beta_{5} + 17 \beta_{3} - 69 \beta_{2} - 367$$$$)/2$$ $$\nu^{17}$$ $$=$$ $$($$$$6 \beta_{19} - 33 \beta_{17} - 34 \beta_{16} - 113 \beta_{15} - 27 \beta_{14} - 161 \beta_{12} + 22 \beta_{11} + 56 \beta_{10} + 67 \beta_{4} - 41 \beta_{1}$$$$)/2$$ $$\nu^{18}$$ $$=$$ $$($$$$77 \beta_{18} - 33 \beta_{13} - 14 \beta_{9} + 9 \beta_{8} - 241 \beta_{7} - 21 \beta_{6} - 176 \beta_{5} - 41 \beta_{3} - 3 \beta_{2} - 249$$$$)/2$$ $$\nu^{19}$$ $$=$$ $$($$$$-54 \beta_{19} - 55 \beta_{17} - 54 \beta_{16} - 79 \beta_{15} - 45 \beta_{14} + 233 \beta_{12} - 134 \beta_{11} - 280 \beta_{10} - 123 \beta_{4} + 49 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$$.

 $$n$$ $$2593$$ $$3781$$ $$3809$$ $$4159$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3025.1
 −0.885915 + 1.10234i 0.885915 + 1.10234i 1.19566 − 0.755240i −1.19566 − 0.755240i 0.725842 − 1.21374i −0.725842 − 1.21374i 0.328272 − 1.37559i −0.328272 − 1.37559i 1.37874 − 0.314750i −1.37874 − 0.314750i −1.37874 + 0.314750i 1.37874 + 0.314750i −0.328272 + 1.37559i 0.328272 + 1.37559i −0.725842 + 1.21374i 0.725842 + 1.21374i −1.19566 + 0.755240i 1.19566 + 0.755240i 0.885915 − 1.10234i −0.885915 − 1.10234i
0 0 0 3.50133i 0 1.00000 0 0 0
3025.2 0 0 0 3.50133i 0 1.00000 0 0 0
3025.3 0 0 0 3.16969i 0 1.00000 0 0 0
3025.4 0 0 0 3.16969i 0 1.00000 0 0 0
3025.5 0 0 0 3.06888i 0 1.00000 0 0 0
3025.6 0 0 0 3.06888i 0 1.00000 0 0 0
3025.7 0 0 0 0.512447i 0 1.00000 0 0 0
3025.8 0 0 0 0.512447i 0 1.00000 0 0 0
3025.9 0 0 0 0.114591i 0 1.00000 0 0 0
3025.10 0 0 0 0.114591i 0 1.00000 0 0 0
3025.11 0 0 0 0.114591i 0 1.00000 0 0 0
3025.12 0 0 0 0.114591i 0 1.00000 0 0 0
3025.13 0 0 0 0.512447i 0 1.00000 0 0 0
3025.14 0 0 0 0.512447i 0 1.00000 0 0 0
3025.15 0 0 0 3.06888i 0 1.00000 0 0 0
3025.16 0 0 0 3.06888i 0 1.00000 0 0 0
3025.17 0 0 0 3.16969i 0 1.00000 0 0 0
3025.18 0 0 0 3.16969i 0 1.00000 0 0 0
3025.19 0 0 0 3.50133i 0 1.00000 0 0 0
3025.20 0 0 0 3.50133i 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3025.20 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
8.b 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 6048.2.c.e 20
3.b odd 2 1 inner 6048.2.c.e 20
4.b odd 2 1 1512.2.c.e 20
8.b even 2 1 inner 6048.2.c.e 20
8.d odd 2 1 1512.2.c.e 20
12.b even 2 1 1512.2.c.e 20
24.f even 2 1 1512.2.c.e 20
24.h odd 2 1 inner 6048.2.c.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.e 20 4.b odd 2 1
1512.2.c.e 20 8.d odd 2 1
1512.2.c.e 20 12.b even 2 1
1512.2.c.e 20 24.f even 2 1
6048.2.c.e 20 1.a even 1 1 trivial
6048.2.c.e 20 3.b odd 2 1 inner
6048.2.c.e 20 8.b even 2 1 inner
6048.2.c.e 20 24.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(6048, [\chi])$$:

 $$T_{5}^{10} + 32 T_{5}^{8} + 342 T_{5}^{6} + 1252 T_{5}^{4} + 321 T_{5}^{2} + 4$$ $$T_{17}^{10} - 131 T_{17}^{8} + 5266 T_{17}^{6} - 69022 T_{17}^{4} + 319757 T_{17}^{2} - 395839$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ 
$5$ $$( 1 - 18 T^{2} + 187 T^{4} - 1608 T^{6} + 10781 T^{8} - 57906 T^{10} + 269525 T^{12} - 1005000 T^{14} + 2921875 T^{16} - 7031250 T^{18} + 9765625 T^{20} )^{2}$$
$7$ $$( 1 - T )^{20}$$
$11$ $$( 1 - 34 T^{2} + 589 T^{4} - 8472 T^{6} + 122234 T^{8} - 1528172 T^{10} + 14790314 T^{12} - 124038552 T^{14} + 1043449429 T^{16} - 7288201954 T^{18} + 25937424601 T^{20} )^{2}$$
$13$ $$( 1 - 83 T^{2} + 3533 T^{4} - 98556 T^{6} + 1975738 T^{8} - 29558418 T^{10} + 333899722 T^{12} - 2814857916 T^{14} + 17053116197 T^{16} - 67705649843 T^{18} + 137858491849 T^{20} )^{2}$$
$17$ $$( 1 + 39 T^{2} + 455 T^{4} - 2382 T^{6} - 47987 T^{8} + 145305 T^{10} - 13868243 T^{12} - 198947022 T^{14} + 10982593895 T^{16} + 272054540199 T^{18} + 2015993900449 T^{20} )^{2}$$
$19$ $$( 1 - 70 T^{2} + 3325 T^{4} - 107448 T^{6} + 2823098 T^{8} - 58351460 T^{10} + 1019138378 T^{12} - 14002730808 T^{14} + 156427554325 T^{16} - 1188849412870 T^{18} + 6131066257801 T^{20} )^{2}$$
$23$ $$( 1 + 139 T^{2} + 10081 T^{4} + 485484 T^{6} + 16965806 T^{8} + 447075730 T^{10} + 8974911374 T^{12} + 135858328044 T^{14} + 1492349797009 T^{16} + 10885226954059 T^{18} + 41426511213649 T^{20} )^{2}$$
$29$ $$( 1 - 189 T^{2} + 17145 T^{4} - 999612 T^{6} + 42327774 T^{8} - 1383568894 T^{10} + 35597657934 T^{12} - 707006574972 T^{14} + 10198245838545 T^{16} - 94546572049629 T^{18} + 420707233300201 T^{20} )^{2}$$
$31$ $$( 1 + 9 T + 171 T^{2} + 1092 T^{3} + 11058 T^{4} + 50402 T^{5} + 342798 T^{6} + 1049412 T^{7} + 5094261 T^{8} + 8311689 T^{9} + 28629151 T^{10} )^{4}$$
$37$ $$( 1 - 22 T^{2} + 1771 T^{4} + 13920 T^{6} + 526445 T^{8} + 112600450 T^{10} + 720703205 T^{12} + 26088321120 T^{14} + 4543901470339 T^{16} - 77274547986262 T^{18} + 4808584372417849 T^{20} )^{2}$$
$41$ $$( 1 + 190 T^{2} + 16651 T^{4} + 899952 T^{6} + 36306749 T^{8} + 1400716342 T^{10} + 61031645069 T^{12} + 2543049263472 T^{14} + 79093985716891 T^{16} + 1517135793532990 T^{18} + 13422659310152401 T^{20} )^{2}$$
$43$ $$( 1 - 131 T^{2} + 5615 T^{4} - 93162 T^{6} + 5954173 T^{8} - 455929005 T^{10} + 11009265877 T^{12} - 318502338762 T^{14} + 35494453520135 T^{16} - 1531154236365731 T^{18} + 21611482313284249 T^{20} )^{2}$$
$47$ $$( 1 + 110 T^{2} + 11187 T^{4} + 742128 T^{6} + 47768349 T^{8} + 2326511142 T^{10} + 105520282941 T^{12} + 3621347901168 T^{14} + 120587081885523 T^{16} + 2619241532793710 T^{18} + 52599132235830049 T^{20} )^{2}$$
$53$ $$( 1 - 193 T^{2} + 20557 T^{4} - 1818204 T^{6} + 128583722 T^{8} - 7326778022 T^{10} + 361191675098 T^{12} - 14346504116124 T^{14} + 455632771728853 T^{16} - 12016120249392673 T^{18} + 174887470365513049 T^{20} )^{2}$$
$59$ $$( 1 - 449 T^{2} + 96439 T^{4} - 13053078 T^{6} + 1232082461 T^{8} - 84671750375 T^{10} + 4288879046741 T^{12} - 158168858287158 T^{14} + 4067848483804399 T^{16} - 65926866484340129 T^{18} + 511116753300641401 T^{20} )^{2}$$
$61$ $$( 1 - 186 T^{2} + 23517 T^{4} - 2152152 T^{6} + 175128378 T^{8} - 11500636220 T^{10} + 651652694538 T^{12} - 29798354399832 T^{14} + 1211604643847637 T^{16} - 35657560217494266 T^{18} + 713342911662882601 T^{20} )^{2}$$
$67$ $$( 1 - 495 T^{2} + 120093 T^{4} - 18487716 T^{6} + 1986970986 T^{8} - 155229602234 T^{10} + 8919512756154 T^{12} - 372548202129636 T^{14} + 10863418489821717 T^{16} - 201003500390537295 T^{18} + 1822837804551761449 T^{20} )^{2}$$
$71$ $$( 1 + 343 T^{2} + 52613 T^{4} + 4802116 T^{6} + 309151594 T^{8} + 19321038474 T^{10} + 1558433185354 T^{12} + 122029839916996 T^{14} + 6739740237935573 T^{16} + 221493461217296023 T^{18} + 3255243551009881201 T^{20} )^{2}$$
$73$ $$( 1 + 149 T^{2} + 432 T^{3} + 14410 T^{4} + 51408 T^{5} + 1051930 T^{6} + 2302128 T^{7} + 57963533 T^{8} + 2073071593 T^{10} )^{4}$$
$79$ $$( 1 + 16 T + 335 T^{2} + 3202 T^{3} + 40117 T^{4} + 296322 T^{5} + 3169243 T^{6} + 19983682 T^{7} + 165168065 T^{8} + 623201296 T^{9} + 3077056399 T^{10} )^{4}$$
$83$ $$( 1 - 138 T^{2} + 13627 T^{4} - 1173792 T^{6} + 82503245 T^{8} - 3517947618 T^{10} + 568364854805 T^{12} - 55706197523232 T^{14} + 4455216467899363 T^{16} - 310816328035187658 T^{18} + 15516041187205853449 T^{20} )^{2}$$
$89$ $$( 1 + 547 T^{2} + 153205 T^{4} + 28413396 T^{6} + 3828024122 T^{8} + 389779473874 T^{10} + 30321779070362 T^{12} + 1782720139460436 T^{14} + 76140018681680005 T^{16} + 2153314076719038307 T^{18} + 31181719929966183601 T^{20} )^{2}$$
$97$ $$( 1 - 14 T + 341 T^{2} - 3480 T^{3} + 53038 T^{4} - 434964 T^{5} + 5144686 T^{6} - 32743320 T^{7} + 311221493 T^{8} - 1239409934 T^{9} + 8587340257 T^{10} )^{4}$$
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