Properties

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

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.r (of order \(6\), degree \(2\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( \beta_{5} + \beta_{11} ) q^{5} + ( \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{7} + ( 5 \beta_{4} + \beta_{5} + \beta_{11} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( \beta_{5} + \beta_{11} ) q^{5} + ( \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{7} + ( 5 \beta_{4} + \beta_{5} + \beta_{11} ) q^{9} + ( \beta_{1} + 2 \beta_{3} + \beta_{6} ) q^{11} + ( -11 + \beta_{5} ) q^{13} + ( 2 \beta_{3} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{15} + ( -2 + \beta_{2} + 2 \beta_{4} + \beta_{10} - 3 \beta_{11} ) q^{17} + ( -3 \beta_{1} + 3 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{19} + ( -10 + \beta_{2} + 9 \beta_{4} + 3 \beta_{5} + 4 \beta_{11} ) q^{21} + ( -\beta_{1} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{23} + ( -6 - \beta_{2} + 6 \beta_{4} - \beta_{10} - 2 \beta_{11} ) q^{25} + ( 2 \beta_{3} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{27} + ( 11 + 3 \beta_{2} - \beta_{5} ) q^{29} + ( -3 \beta_{1} - 2 \beta_{3} + 5 \beta_{6} ) q^{31} + ( -21 \beta_{4} - 4 \beta_{5} + 3 \beta_{10} - 4 \beta_{11} ) q^{33} + ( 3 \beta_{1} - 7 \beta_{3} + \beta_{6} - 7 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} ) q^{35} + ( -13 \beta_{4} + 6 \beta_{5} + \beta_{10} + 6 \beta_{11} ) q^{37} + ( 8 \beta_{1} + 2 \beta_{3} - 3 \beta_{6} ) q^{39} + ( 31 - \beta_{2} + 3 \beta_{5} ) q^{41} + ( 10 \beta_{3} - 4 \beta_{6} - 8 \beta_{7} + 10 \beta_{8} - 4 \beta_{9} ) q^{43} + ( -31 - \beta_{2} + 31 \beta_{4} - \beta_{10} + 3 \beta_{11} ) q^{45} + ( -11 \beta_{1} + 11 \beta_{7} - 8 \beta_{8} + 3 \beta_{9} ) q^{47} + ( -12 + 22 \beta_{4} - \beta_{5} - 3 \beta_{10} - 2 \beta_{11} ) q^{49} + ( -3 \beta_{1} + 3 \beta_{7} + 7 \beta_{8} + 13 \beta_{9} ) q^{51} + ( 1 - 4 \beta_{2} - \beta_{4} - 4 \beta_{10} + 4 \beta_{11} ) q^{53} + ( 3 \beta_{3} + 10 \beta_{6} + 7 \beta_{7} + 3 \beta_{8} + 10 \beta_{9} ) q^{55} + ( 41 - 4 \beta_{2} - 6 \beta_{5} ) q^{57} + ( -\beta_{1} - 2 \beta_{3} - 4 \beta_{6} ) q^{59} + ( -73 \beta_{4} + 2 \beta_{5} - \beta_{10} + 2 \beta_{11} ) q^{61} + ( 8 \beta_{1} - 7 \beta_{3} - 4 \beta_{6} - 12 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{63} + ( 31 \beta_{4} - 13 \beta_{5} - \beta_{10} - 13 \beta_{11} ) q^{65} + ( 19 \beta_{1} - 9 \beta_{3} + 6 \beta_{6} ) q^{67} + ( 12 - 3 \beta_{2} - 5 \beta_{5} ) q^{69} + ( -5 \beta_{3} + 14 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + 14 \beta_{9} ) q^{71} + ( -1 + 3 \beta_{2} + \beta_{4} + 3 \beta_{10} + 2 \beta_{11} ) q^{73} + ( -4 \beta_{1} + 4 \beta_{7} - 17 \beta_{8} + 2 \beta_{9} ) q^{75} + ( -19 + 2 \beta_{2} - 20 \beta_{4} - 11 \beta_{5} + 4 \beta_{10} - 11 \beta_{11} ) q^{77} + ( -\beta_{1} + \beta_{7} + 5 \beta_{8} - 10 \beta_{9} ) q^{79} + ( 70 - \beta_{2} - 70 \beta_{4} - \beta_{10} - \beta_{11} ) q^{81} + ( -15 \beta_{3} + 2 \beta_{6} + 12 \beta_{7} - 15 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 85 + 6 \beta_{2} ) q^{85} + ( 4 \beta_{1} - 41 \beta_{3} - 9 \beta_{6} ) q^{87} + ( -87 \beta_{4} + 2 \beta_{10} ) q^{89} + ( -4 \beta_{1} - 17 \beta_{3} + 13 \beta_{6} + 8 \beta_{7} - 10 \beta_{8} + \beta_{9} ) q^{91} + ( 55 \beta_{4} - 12 \beta_{5} + 3 \beta_{10} - 12 \beta_{11} ) q^{93} + ( 3 \beta_{1} + 26 \beta_{3} - 8 \beta_{6} ) q^{95} + ( -59 + 2 \beta_{2} - 15 \beta_{5} ) q^{97} + ( 13 \beta_{3} + 15 \beta_{6} + 12 \beta_{7} + 13 \beta_{8} + 15 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{5} + 32q^{9} + O(q^{10}) \) \( 12q + 2q^{5} + 32q^{9} - 128q^{13} - 6q^{17} - 62q^{21} - 32q^{25} + 128q^{29} - 134q^{33} - 66q^{37} + 384q^{41} - 192q^{45} - 12q^{49} - 2q^{53} + 468q^{57} - 434q^{61} + 160q^{65} + 124q^{69} - 10q^{73} - 370q^{77} + 422q^{81} + 1020q^{85} - 522q^{89} + 306q^{93} - 768q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 28 x^{9} - 100 x^{8} + 140 x^{7} + 392 x^{6} + 1400 x^{5} + 8040 x^{4} + 11256 x^{3} + 9800 x^{2} + 13720 x + 9604\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(185087 \nu^{11} - 998230 \nu^{10} + 1397522 \nu^{9} - 6636168 \nu^{8} + 8312290 \nu^{7} + 88185794 \nu^{6} - 136812774 \nu^{5} + 177107560 \nu^{4} + 592256728 \nu^{3} - 5486280500 \nu^{2} + 554967140 \nu + 776953996\)\()/ 2812715640 \)
\(\beta_{2}\)\(=\)\((\)\(9925 \nu^{11} - 13895 \nu^{10} + 52651 \nu^{9} - 374850 \nu^{8} - 467710 \nu^{7} + 1347136 \nu^{6} + 2329950 \nu^{5} + 17604650 \nu^{4} + 28459298 \nu^{3} + 23118200 \nu^{2} + 32365480 \nu + 532857500\)\()/78928245\)
\(\beta_{3}\)\(=\)\((\)\(1063307 \nu^{11} - 4971100 \nu^{10} + 6959540 \nu^{9} - 36262548 \nu^{8} + 39668650 \nu^{7} + 441573890 \nu^{6} - 757254414 \nu^{5} + 906142300 \nu^{4} + 3088145620 \nu^{3} - 22969617500 \nu^{2} + 2932821500 \nu + 4105950100\)\()/ 7734968010 \)
\(\beta_{4}\)\(=\)\((\)\(-47849 \nu^{11} + 43750 \nu^{10} - 61250 \nu^{9} + 1425522 \nu^{8} + 3439850 \nu^{7} - 9190790 \nu^{6} - 9143106 \nu^{5} - 61020400 \nu^{4} - 331811440 \nu^{3} - 260890672 \nu^{2} - 397130300 \nu - 328244140\)\()/ 227738280 \)
\(\beta_{5}\)\(=\)\((\)\(33515 \nu^{11} - 46921 \nu^{10} + 122126 \nu^{9} - 1464615 \nu^{8} - 1301039 \nu^{7} + 5328386 \nu^{6} + 10651200 \nu^{5} + 75432304 \nu^{4} + 118480348 \nu^{3} + 97549690 \nu^{2} + 136569566 \nu - 1032109070\)\()/ 157856490 \)
\(\beta_{6}\)\(=\)\((\)\(-2703596 \nu^{11} + 11908585 \nu^{10} - 16672019 \nu^{9} + 90419994 \nu^{8} - 93122680 \nu^{7} - 1060486748 \nu^{6} + 1897920192 \nu^{5} - 2197406470 \nu^{4} - 7551097666 \nu^{3} + 34749182480 \nu^{2} - 7212555980 \nu - 10097578372\)\()/ 7734968010 \)
\(\beta_{7}\)\(=\)\((\)\(63057475 \nu^{11} - 88280465 \nu^{10} + 95938717 \nu^{9} - 1880565750 \nu^{8} - 3672955450 \nu^{7} + 14550916744 \nu^{6} + 7789022850 \nu^{5} + 71568506870 \nu^{4} + 413550016298 \nu^{3} + 97780587800 \nu^{2} + 136892822920 \nu + 413603679776\)\()/ 108289552140 \)
\(\beta_{8}\)\(=\)\((\)\(42267851 \nu^{11} - 34797700 \nu^{10} + 16182740 \nu^{9} - 1206155664 \nu^{8} - 3204789650 \nu^{7} + 8649690350 \nu^{6} + 11125655898 \nu^{5} + 48265430500 \nu^{4} + 296825119540 \nu^{3} + 235663193500 \nu^{2} + 84296468900 \nu + 289957457860\)\()/ 54144776070 \)
\(\beta_{9}\)\(=\)\((\)\(-5716498 \nu^{11} + 2833565 \nu^{10} + 3870755 \nu^{9} + 154642887 \nu^{8} + 505900669 \nu^{7} - 1254655150 \nu^{6} - 1991859114 \nu^{5} - 6269750816 \nu^{4} - 40737809870 \nu^{3} - 32703518330 \nu^{2} - 10236754738 \nu - 39215176280\)\()/ 4922252370 \)
\(\beta_{10}\)\(=\)\((\)\(994220 \nu^{11} - 927136 \nu^{10} + 972650 \nu^{9} - 29199870 \nu^{8} - 68302394 \nu^{7} + 195169100 \nu^{6} + 214099620 \nu^{5} + 1160490016 \nu^{4} + 6923873500 \nu^{3} + 5462022580 \nu^{2} + 8284498796 \nu + 6820349200\)\()/ 773496801 \)
\(\beta_{11}\)\(=\)\((\)\(308950247 \nu^{11} - 255847270 \nu^{10} + 358186178 \nu^{9} - 8664707646 \nu^{8} - 24485870270 \nu^{7} + 59864945378 \nu^{6} + 47467063518 \nu^{5} + 418021281160 \nu^{4} + 2291747902264 \nu^{3} + 1791591829216 \nu^{2} + 2750637740900 \nu + 3850892837260\)\()/ 216579104280 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{10} + \beta_{8} - 4 \beta_{7} + 4 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{6} - 7 \beta_{3} + 4 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-12 \beta_{8} + 20 \beta_{7} - 12 \beta_{3} - 5 \beta_{2} + 28\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(20 \beta_{11} + 17 \beta_{10} + 20 \beta_{5} + 260 \beta_{4}\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(14 \beta_{11} + 57 \beta_{10} - 14 \beta_{6} + 462 \beta_{4} - 169 \beta_{3} + 57 \beta_{2} + 228 \beta_{1} - 462\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-100 \beta_{9} - 469 \beta_{8} + 480 \beta_{7} - 100 \beta_{6} - 469 \beta_{3}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(280 \beta_{11} + 689 \beta_{10} - 280 \beta_{9} - 2229 \beta_{8} + 2756 \beta_{7} + 280 \beta_{5} + 6440 \beta_{4} - 2756 \beta_{1}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(1098 \beta_{11} + 1599 \beta_{10} + 18194 \beta_{4} + 1599 \beta_{2} - 18194\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-2100 \beta_{9} - 14428 \beta_{8} + 17140 \beta_{7} - 2100 \beta_{6} - 2100 \beta_{5} - 14428 \beta_{3} + 4285 \beta_{2} - 42672\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-12940 \beta_{9} - 74333 \beta_{8} + 83252 \beta_{7} - 83252 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(28686 \beta_{11} + 54043 \beta_{10} + 28686 \beta_{6} + 554078 \beta_{4} + 185391 \beta_{3} + 54043 \beta_{2} - 216172 \beta_{1} - 554078\)\()/2\)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
0.630651 2.35362i
3.45178 + 0.924901i
0.294251 1.09816i
−1.09816 0.294251i
−0.924901 + 3.45178i
−2.35362 0.630651i
0.630651 + 2.35362i
3.45178 0.924901i
0.294251 + 1.09816i
−1.09816 + 0.294251i
−0.924901 3.45178i
−2.35362 + 0.630651i
0 −3.85030 2.22297i 0 2.88318 + 4.99382i 0 −5.85985 3.82912i 0 5.38318 + 9.32395i 0
95.2 0 −3.51065 2.02688i 0 1.21646 + 2.10698i 0 2.97300 6.33729i 0 3.71646 + 6.43710i 0
95.3 0 −2.25843 1.30391i 0 −3.59965 6.23477i 0 5.36875 + 4.49183i 0 −1.09965 1.90465i 0
95.4 0 2.25843 + 1.30391i 0 −3.59965 6.23477i 0 −5.36875 4.49183i 0 −1.09965 1.90465i 0
95.5 0 3.51065 + 2.02688i 0 1.21646 + 2.10698i 0 −2.97300 + 6.33729i 0 3.71646 + 6.43710i 0
95.6 0 3.85030 + 2.22297i 0 2.88318 + 4.99382i 0 5.85985 + 3.82912i 0 5.38318 + 9.32395i 0
191.1 0 −3.85030 + 2.22297i 0 2.88318 4.99382i 0 −5.85985 + 3.82912i 0 5.38318 9.32395i 0
191.2 0 −3.51065 + 2.02688i 0 1.21646 2.10698i 0 2.97300 + 6.33729i 0 3.71646 6.43710i 0
191.3 0 −2.25843 + 1.30391i 0 −3.59965 + 6.23477i 0 5.36875 4.49183i 0 −1.09965 + 1.90465i 0
191.4 0 2.25843 1.30391i 0 −3.59965 + 6.23477i 0 −5.36875 + 4.49183i 0 −1.09965 + 1.90465i 0
191.5 0 3.51065 2.02688i 0 1.21646 2.10698i 0 −2.97300 6.33729i 0 3.71646 6.43710i 0
191.6 0 3.85030 2.22297i 0 2.88318 4.99382i 0 5.85985 3.82912i 0 5.38318 9.32395i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g 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.r.c 12
4.b odd 2 1 inner 224.3.r.c 12
7.c even 3 1 inner 224.3.r.c 12
7.c even 3 1 1568.3.d.j 6
7.d odd 6 1 1568.3.d.k 6
8.b even 2 1 448.3.r.g 12
8.d odd 2 1 448.3.r.g 12
28.f even 6 1 1568.3.d.k 6
28.g odd 6 1 inner 224.3.r.c 12
28.g odd 6 1 1568.3.d.j 6
56.k odd 6 1 448.3.r.g 12
56.p even 6 1 448.3.r.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.c 12 1.a even 1 1 trivial
224.3.r.c 12 4.b odd 2 1 inner
224.3.r.c 12 7.c even 3 1 inner
224.3.r.c 12 28.g odd 6 1 inner
448.3.r.g 12 8.b even 2 1
448.3.r.g 12 8.d odd 2 1
448.3.r.g 12 56.k odd 6 1
448.3.r.g 12 56.p even 6 1
1568.3.d.j 6 7.c even 3 1
1568.3.d.j 6 28.g odd 6 1
1568.3.d.k 6 7.d odd 6 1
1568.3.d.k 6 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 43 T_{3}^{10} + 1278 T_{3}^{8} - 20135 T_{3}^{6} + 231054 T_{3}^{4} - 1261339 T_{3}^{2} + 4879681 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 11 T^{2} - 117 T^{4} - 884 T^{6} + 18105 T^{8} + 64793 T^{10} - 1253162 T^{12} + 5248233 T^{14} + 118786905 T^{16} - 469793844 T^{18} - 5036466357 T^{20} + 38354628411 T^{22} + 282429536481 T^{24} \)
$5$ \( ( 1 - T - 29 T^{2} - 132 T^{3} + 201 T^{4} + 2405 T^{5} + 10726 T^{6} + 60125 T^{7} + 125625 T^{8} - 2062500 T^{9} - 11328125 T^{10} - 9765625 T^{11} + 244140625 T^{12} )^{2} \)
$7$ \( 1 + 6 T^{2} + 4335 T^{4} + 71444 T^{6} + 10408335 T^{8} + 34588806 T^{10} + 13841287201 T^{12} \)
$11$ \( 1 + 283 T^{2} + 46555 T^{4} + 3528284 T^{6} - 210042215 T^{8} - 117264051335 T^{10} - 18469284955402 T^{12} - 1716862975595735 T^{14} - 45024414170161415 T^{16} + 11073266626730676764 T^{18} + \)\(21\!\cdots\!55\)\( T^{20} + \)\(19\!\cdots\!83\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \)
$13$ \( ( 1 + 32 T + 803 T^{2} + 11632 T^{3} + 135707 T^{4} + 913952 T^{5} + 4826809 T^{6} )^{4} \)
$17$ \( ( 1 + 3 T - 181 T^{2} + 11996 T^{3} - 1671 T^{4} - 1164367 T^{5} + 69728918 T^{6} - 336502063 T^{7} - 139563591 T^{8} + 289554277724 T^{9} - 1262612096821 T^{10} + 6047981701347 T^{11} + 582622237229761 T^{12} )^{2} \)
$19$ \( 1 + 843 T^{2} + 169755 T^{4} + 10759612 T^{6} + 34418758617 T^{8} + 15878060416809 T^{10} + 3783625584664182 T^{12} + 2069244711578965689 T^{14} + \)\(58\!\cdots\!97\)\( T^{16} + \)\(23\!\cdots\!32\)\( T^{18} + \)\(48\!\cdots\!55\)\( T^{20} + \)\(31\!\cdots\!43\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 + 2659 T^{2} + 3876571 T^{4} + 4127666252 T^{6} + 3503571279769 T^{8} + 2424695206910689 T^{10} + 1397846635041452342 T^{12} + \)\(67\!\cdots\!49\)\( T^{14} + \)\(27\!\cdots\!89\)\( T^{16} + \)\(90\!\cdots\!92\)\( T^{18} + \)\(23\!\cdots\!31\)\( T^{20} + \)\(45\!\cdots\!59\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 - 32 T - 109 T^{2} + 37136 T^{3} - 91669 T^{4} - 22632992 T^{5} + 594823321 T^{6} )^{4} \)
$31$ \( 1 + 2963 T^{2} + 4291147 T^{4} + 3752683836 T^{6} + 1856892008697 T^{8} - 613860419386879 T^{10} - 1734915478854125546 T^{12} - \)\(56\!\cdots\!59\)\( T^{14} + \)\(15\!\cdots\!77\)\( T^{16} + \)\(29\!\cdots\!96\)\( T^{18} + \)\(31\!\cdots\!07\)\( T^{20} + \)\(19\!\cdots\!63\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( ( 1 + 33 T - 1333 T^{2} - 122132 T^{3} - 783591 T^{4} + 87702163 T^{5} + 5266681334 T^{6} + 120064261147 T^{7} - 1468575692151 T^{8} - 313357297783988 T^{9} - 4682135112076693 T^{10} + 158683284289789017 T^{11} + 6582952005840035281 T^{12} )^{2} \)
$41$ \( ( 1 - 96 T + 7339 T^{2} - 328752 T^{3} + 12336859 T^{4} - 271273056 T^{5} + 4750104241 T^{6} )^{4} \)
$43$ \( ( 1 - 5206 T^{2} + 17138783 T^{4} - 37018566196 T^{6} + 58594088459183 T^{8} - 60848770645190806 T^{10} + 39959630797262576401 T^{12} )^{2} \)
$47$ \( 1 + 6995 T^{2} + 22181707 T^{4} + 50014251324 T^{6} + 105476166484281 T^{8} + 174006129657696449 T^{10} + \)\(27\!\cdots\!62\)\( T^{12} + \)\(84\!\cdots\!69\)\( T^{14} + \)\(25\!\cdots\!41\)\( T^{16} + \)\(58\!\cdots\!84\)\( T^{18} + \)\(12\!\cdots\!47\)\( T^{20} + \)\(19\!\cdots\!95\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( ( 1 + T - 2837 T^{2} - 53748 T^{3} + 54009 T^{4} + 72242131 T^{5} + 22496954230 T^{6} + 202928145979 T^{7} + 426156988329 T^{8} - 1191290081961492 T^{9} - 176630741697031157 T^{10} + 174887470365513049 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \)
$59$ \( 1 + 18811 T^{2} + 200120491 T^{4} + 1494527327372 T^{6} + 8628303999799801 T^{8} + 40087667805525876841 T^{10} + \)\(15\!\cdots\!62\)\( T^{12} + \)\(48\!\cdots\!01\)\( T^{14} + \)\(12\!\cdots\!21\)\( T^{16} + \)\(26\!\cdots\!32\)\( T^{18} + \)\(43\!\cdots\!31\)\( T^{20} + \)\(49\!\cdots\!11\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \)
$61$ \( ( 1 + 217 T + 20699 T^{2} + 1814604 T^{3} + 173780089 T^{4} + 12006150107 T^{5} + 685616953942 T^{6} + 44674884548147 T^{7} + 2406131481259849 T^{8} + 93489077396968044 T^{9} + 3968149671730719419 T^{10} + \)\(15\!\cdots\!17\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \)
$67$ \( 1 + 9899 T^{2} + 24620203 T^{4} - 79278218964 T^{6} - 82489071404871 T^{8} + 4300819315073397209 T^{10} + \)\(30\!\cdots\!86\)\( T^{12} + \)\(86\!\cdots\!89\)\( T^{14} - \)\(33\!\cdots\!11\)\( T^{16} - \)\(64\!\cdots\!04\)\( T^{18} + \)\(40\!\cdots\!43\)\( T^{20} + \)\(32\!\cdots\!99\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( ( 1 - 12262 T^{2} + 100517039 T^{4} - 635265930196 T^{6} + 2554306930132559 T^{8} - 7918229800135521382 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( ( 1 + 5 T - 12813 T^{2} - 167716 T^{3} + 95820777 T^{4} + 946458799 T^{5} - 562891511514 T^{6} + 5043678939871 T^{7} + 2721141518053257 T^{8} - 25381171096285924 T^{9} - 10333173157438859853 T^{10} + 21488129148517788245 T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \)
$79$ \( 1 + 28083 T^{2} + 421723515 T^{4} + 4556484807052 T^{6} + 39635396888713497 T^{8} + \)\(29\!\cdots\!89\)\( T^{10} + \)\(19\!\cdots\!22\)\( T^{12} + \)\(11\!\cdots\!09\)\( T^{14} + \)\(60\!\cdots\!17\)\( T^{16} + \)\(26\!\cdots\!32\)\( T^{18} + \)\(97\!\cdots\!15\)\( T^{20} + \)\(25\!\cdots\!83\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( ( 1 - 29238 T^{2} + 396631039 T^{4} - 3342660983284 T^{6} + 18823443167425519 T^{8} - 65852520283281280758 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \)
$89$ \( ( 1 + 261 T + 22931 T^{2} + 2459452 T^{3} + 465950601 T^{4} + 40453273231 T^{5} + 2330338704422 T^{6} + 320430377262751 T^{7} + 29234784902036841 T^{8} + 1222301630016613372 T^{9} + 90269917903554419411 T^{10} + \)\(81\!\cdots\!61\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \)
$97$ \( ( 1 + 192 T + 28555 T^{2} + 2621616 T^{3} + 268673995 T^{4} + 16997621952 T^{5} + 832972004929 T^{6} )^{4} \)
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