Properties

Label 224.3.r.d
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.

\(f(q)\) \(=\) \( q + ( \beta_{7} + \beta_{9} - \beta_{10} ) q^{3} + ( -3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{5} + ( -2 \beta_{8} - \beta_{9} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( \beta_{7} + \beta_{9} - \beta_{10} ) q^{3} + ( -3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{5} + ( -2 \beta_{8} - \beta_{9} ) q^{7} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{9} + ( 4 \beta_{7} - \beta_{8} + 4 \beta_{9} - 3 \beta_{10} ) q^{11} + ( -\beta_{2} - \beta_{3} ) q^{13} + ( 3 \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{15} + ( -1 - \beta_{1} - 5 \beta_{5} ) q^{17} + ( 3 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} - 9 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} ) q^{19} + ( 10 + 12 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{21} + ( -4 \beta_{6} + 6 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{23} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 8 \beta_{5} ) q^{25} + ( 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} - 4 \beta_{11} ) q^{27} + 7 \beta_{4} q^{29} + ( \beta_{6} - 4 \beta_{7} - \beta_{8} - 4 \beta_{9} + 5 \beta_{10} ) q^{31} + ( -3 - 28 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{33} + ( -7 \beta_{6} + 7 \beta_{7} - 8 \beta_{8} + 10 \beta_{9} - 7 \beta_{10} ) q^{35} + ( 2 + 17 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{37} + ( 4 \beta_{6} + 9 \beta_{7} + 3 \beta_{8} + 9 \beta_{9} - 12 \beta_{10} ) q^{39} + ( -2 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} ) q^{41} + ( -16 \beta_{7} - 16 \beta_{8} + 4 \beta_{9} + 4 \beta_{10} - 4 \beta_{11} ) q^{43} + ( -2 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 9 \beta_{5} ) q^{45} + ( 7 \beta_{6} - 7 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} + 7 \beta_{11} ) q^{47} + ( 16 + 15 \beta_{1} + 8 \beta_{2} - 5 \beta_{3} - 13 \beta_{4} + 8 \beta_{5} ) q^{49} + ( 5 \beta_{6} - 5 \beta_{7} + 6 \beta_{8} - \beta_{9} + 6 \beta_{10} + 5 \beta_{11} ) q^{51} + ( 2 - 8 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} + 10 \beta_{4} + 2 \beta_{5} ) q^{53} + ( 13 \beta_{7} + 13 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + 8 \beta_{11} ) q^{55} + ( -57 + 6 \beta_{2} + 6 \beta_{3} ) q^{57} + ( -12 \beta_{6} - 19 \beta_{7} + 12 \beta_{8} - 19 \beta_{9} + 7 \beta_{10} ) q^{59} + ( -2 - 51 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} + 10 \beta_{5} ) q^{61} + ( -21 \beta_{7} - 7 \beta_{8} - 7 \beta_{10} + 7 \beta_{11} ) q^{63} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 11 \beta_{4} + 9 \beta_{5} ) q^{65} + ( 7 \beta_{6} + 3 \beta_{7} - 6 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} ) q^{67} + ( -19 + 7 \beta_{2} + 7 \beta_{3} + 10 \beta_{4} ) q^{69} + ( -20 \beta_{7} - 20 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 19 \beta_{11} ) q^{71} + ( 58 + 56 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{73} + ( 11 \beta_{6} - 10 \beta_{7} - 2 \beta_{8} + 12 \beta_{9} - 2 \beta_{10} + 11 \beta_{11} ) q^{75} + ( 53 + 37 \beta_{1} - 5 \beta_{2} + 11 \beta_{3} + 16 \beta_{4} - 5 \beta_{5} ) q^{77} + ( -20 \beta_{6} - 10 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} - 20 \beta_{11} ) q^{79} + ( -29 - 33 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 5 \beta_{5} ) q^{81} + ( -4 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} - 6 \beta_{10} - 15 \beta_{11} ) q^{83} + ( -83 - 5 \beta_{2} - 5 \beta_{3} - 21 \beta_{4} ) q^{85} + ( -7 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} ) q^{87} + ( 8 - 33 \beta_{1} - 8 \beta_{2} + 16 \beta_{3} + 8 \beta_{4} ) q^{89} + ( -7 \beta_{6} - 28 \beta_{7} - 7 \beta_{9} + 21 \beta_{10} ) q^{91} + ( 4 + 33 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 4 \beta_{5} ) q^{93} + ( 3 \beta_{6} - 27 \beta_{7} + 12 \beta_{8} - 27 \beta_{9} + 15 \beta_{10} ) q^{95} + ( -\beta_{2} - \beta_{3} ) q^{97} + ( 40 \beta_{7} + 40 \beta_{8} - 17 \beta_{9} - 17 \beta_{10} - 11 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 18q^{5} + O(q^{10}) \) \( 12q + 18q^{5} - 6q^{17} + 18q^{21} + 186q^{33} - 114q^{37} + 180q^{49} - 18q^{53} - 684q^{57} + 318q^{61} - 228q^{69} + 342q^{73} + 318q^{77} - 186q^{81} - 996q^{85} + 150q^{89} - 222q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 15 x^{10} + 90 x^{8} - 247 x^{6} + 270 x^{4} + 21 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -2 \nu^{10} + 83 \nu^{8} - 906 \nu^{6} + 3874 \nu^{4} - 5950 \nu^{2} - 1505 \)\()/2947\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{10} - 338 \nu^{8} + 3304 \nu^{6} - 10019 \nu^{4} - 898 \nu^{2} + 16177 \)\()/2947\)
\(\beta_{3}\)\(=\)\((\)\( 52 \nu^{10} - 895 \nu^{8} + 5032 \nu^{6} - 8946 \nu^{4} - 1912 \nu^{2} - 2128 \)\()/2947\)
\(\beta_{4}\)\(=\)\((\)\( 58 \nu^{10} - 723 \nu^{8} + 3540 \nu^{6} - 6675 \nu^{4} - 902 \nu^{2} + 14175 \)\()/2947\)
\(\beta_{5}\)\(=\)\((\)\( -89 \nu^{10} + 1378 \nu^{8} - 8321 \nu^{6} + 23780 \nu^{4} - 30699 \nu^{2} + 8176 \)\()/2947\)
\(\beta_{6}\)\(=\)\((\)\( 80 \nu^{11} - 794 \nu^{9} - 808 \nu^{7} + 34490 \nu^{5} - 122376 \nu^{3} + 113246 \nu \)\()/20629\)
\(\beta_{7}\)\(=\)\((\)\( 488 \nu^{11} - 7201 \nu^{9} + 43402 \nu^{7} - 131463 \nu^{5} + 222480 \nu^{3} - 142611 \nu \)\()/41258\)
\(\beta_{8}\)\(=\)\((\)\( -97 \nu^{11} + 1710 \nu^{9} - 11945 \nu^{7} + 39276 \nu^{5} - 54499 \nu^{3} + 8050 \nu \)\()/5894\)
\(\beta_{9}\)\(=\)\((\)\( -151 \nu^{11} + 2267 \nu^{9} - 13673 \nu^{7} + 38203 \nu^{5} - 41697 \nu^{3} - 14903 \nu \)\()/5894\)
\(\beta_{10}\)\(=\)\((\)\( -1137 \nu^{11} + 16663 \nu^{9} - 94903 \nu^{7} + 232931 \nu^{5} - 169503 \nu^{3} - 135051 \nu \)\()/41258\)
\(\beta_{11}\)\(=\)\((\)\( -1002 \nu^{11} + 14218 \nu^{9} - 80058 \nu^{7} + 197934 \nu^{5} - 177090 \nu^{3} - 90930 \nu \)\()/20629\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{10} - 2 \beta_{9} + \beta_{6}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_{1} + 10\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{11} + 4 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{5} - \beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 11 \beta_{1} + 13\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-21 \beta_{11} + 34 \beta_{10} + 16 \beta_{9} - 18 \beta_{8} - 2 \beta_{7} + 14 \beta_{6}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(34 \beta_{5} + 7 \beta_{4} + 42 \beta_{3} - 33 \beta_{2} - 185 \beta_{1} - 11\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-76 \beta_{11} + 69 \beta_{10} + 85 \beta_{9} - 54 \beta_{8} - 44 \beta_{7} + 5 \beta_{6}\)\()/2\)
\(\nu^{8}\)\(=\)\(52 \beta_{5} + 31 \beta_{4} + 42 \beta_{3} - 43 \beta_{2} - 280 \beta_{1} - 170\)
\(\nu^{9}\)\(=\)\((\)\(-893 \beta_{11} + 480 \beta_{10} + 1050 \beta_{9} - 426 \beta_{8} - 762 \beta_{7} - 321 \beta_{6}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(978 \beta_{5} + 1076 \beta_{4} + 467 \beta_{3} - 836 \beta_{2} - 5233 \beta_{1} - 5635\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(-2202 \beta_{11} + 453 \beta_{10} + 2506 \beta_{9} - 309 \beta_{8} - 2265 \beta_{7} - 1693 \beta_{6}\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
1.75780 0.500000i
0.385124 + 0.500000i
2.23871 + 0.500000i
−2.23871 0.500000i
−0.385124 0.500000i
−1.75780 + 0.500000i
1.75780 + 0.500000i
0.385124 0.500000i
2.23871 0.500000i
−2.23871 + 0.500000i
−0.385124 + 0.500000i
−1.75780 0.500000i
0 −4.23009 2.44224i 0 1.16012 + 2.00939i 0 4.65898 + 5.22436i 0 7.42909 + 12.8676i 0
95.2 0 −1.53177 0.884367i 0 4.10168 + 7.10432i 0 −5.46804 + 4.37041i 0 −2.93579 5.08494i 0
95.3 0 −0.100242 0.0578747i 0 −0.761802 1.31948i 0 6.66292 2.14605i 0 −4.49330 7.78263i 0
95.4 0 0.100242 + 0.0578747i 0 −0.761802 1.31948i 0 −6.66292 + 2.14605i 0 −4.49330 7.78263i 0
95.5 0 1.53177 + 0.884367i 0 4.10168 + 7.10432i 0 5.46804 4.37041i 0 −2.93579 5.08494i 0
95.6 0 4.23009 + 2.44224i 0 1.16012 + 2.00939i 0 −4.65898 5.22436i 0 7.42909 + 12.8676i 0
191.1 0 −4.23009 + 2.44224i 0 1.16012 2.00939i 0 4.65898 5.22436i 0 7.42909 12.8676i 0
191.2 0 −1.53177 + 0.884367i 0 4.10168 7.10432i 0 −5.46804 4.37041i 0 −2.93579 + 5.08494i 0
191.3 0 −0.100242 + 0.0578747i 0 −0.761802 + 1.31948i 0 6.66292 + 2.14605i 0 −4.49330 + 7.78263i 0
191.4 0 0.100242 0.0578747i 0 −0.761802 + 1.31948i 0 −6.66292 2.14605i 0 −4.49330 + 7.78263i 0
191.5 0 1.53177 0.884367i 0 4.10168 7.10432i 0 5.46804 + 4.37041i 0 −2.93579 + 5.08494i 0
191.6 0 4.23009 2.44224i 0 1.16012 2.00939i 0 −4.65898 + 5.22436i 0 7.42909 12.8676i 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.d 12
4.b odd 2 1 inner 224.3.r.d 12
7.c even 3 1 inner 224.3.r.d 12
7.c even 3 1 1568.3.d.i 6
7.d odd 6 1 1568.3.d.l 6
8.b even 2 1 448.3.r.f 12
8.d odd 2 1 448.3.r.f 12
28.f even 6 1 1568.3.d.l 6
28.g odd 6 1 inner 224.3.r.d 12
28.g odd 6 1 1568.3.d.i 6
56.k odd 6 1 448.3.r.f 12
56.p even 6 1 448.3.r.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.d 12 1.a even 1 1 trivial
224.3.r.d 12 4.b odd 2 1 inner
224.3.r.d 12 7.c even 3 1 inner
224.3.r.d 12 28.g odd 6 1 inner
448.3.r.f 12 8.b even 2 1
448.3.r.f 12 8.d odd 2 1
448.3.r.f 12 56.k odd 6 1
448.3.r.f 12 56.p even 6 1
1568.3.d.i 6 7.c even 3 1
1568.3.d.i 6 28.g odd 6 1
1568.3.d.l 6 7.d odd 6 1
1568.3.d.l 6 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 27 T_{3}^{10} + 654 T_{3}^{8} - 2023 T_{3}^{6} + 5598 T_{3}^{4} - 75 T_{3}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 27 T^{2} + 411 T^{4} + 2972 T^{6} - 423 T^{8} - 321159 T^{10} - 4031882 T^{12} - 26013879 T^{14} - 2775303 T^{16} + 1579442652 T^{18} + 17692202331 T^{20} + 94143178827 T^{22} + 282429536481 T^{24} \)
$5$ \( ( 1 - 9 T + 3 T^{2} + 140 T^{3} + 345 T^{4} - 3363 T^{5} + 5766 T^{6} - 84075 T^{7} + 215625 T^{8} + 2187500 T^{9} + 1171875 T^{10} - 87890625 T^{11} + 244140625 T^{12} )^{2} \)
$7$ \( 1 - 90 T^{2} + 7791 T^{4} - 412972 T^{6} + 18706191 T^{8} - 518832090 T^{10} + 13841287201 T^{12} \)
$11$ \( 1 + 363 T^{2} + 55851 T^{4} + 5892268 T^{6} + 719654457 T^{8} + 93270469209 T^{10} + 11000493019158 T^{12} + 1365572939688969 T^{14} + 154264324109182617 T^{16} + 18492461094445093228 T^{18} + \)\(25\!\cdots\!11\)\( T^{20} + \)\(24\!\cdots\!63\)\( T^{22} + \)\(98\!\cdots\!41\)\( T^{24} \)
$13$ \( ( 1 + 339 T^{2} - 784 T^{3} + 57291 T^{4} + 4826809 T^{6} )^{4} \)
$17$ \( ( 1 + 3 T - 261 T^{2} - 5460 T^{3} - 14535 T^{4} + 628833 T^{5} + 36283574 T^{6} + 181732737 T^{7} - 1213977735 T^{8} - 131791126740 T^{9} - 1820672692101 T^{10} + 6047981701347 T^{11} + 582622237229761 T^{12} )^{2} \)
$19$ \( 1 + 411 T^{2} - 128661 T^{4} - 89218804 T^{6} + 6305474169 T^{8} + 6575128475913 T^{10} + 1202874182390166 T^{12} + 856877318109458073 T^{14} + \)\(10\!\cdots\!29\)\( T^{16} - \)\(19\!\cdots\!44\)\( T^{18} - \)\(37\!\cdots\!41\)\( T^{20} + \)\(15\!\cdots\!11\)\( T^{22} + \)\(48\!\cdots\!21\)\( T^{24} \)
$23$ \( 1 + 579 T^{2} + 62859 T^{4} - 35156516 T^{6} - 57866315751 T^{8} - 16658355952143 T^{10} + 6113539249932246 T^{12} - 4661690988003649263 T^{14} - \)\(45\!\cdots\!31\)\( T^{16} - \)\(77\!\cdots\!36\)\( T^{18} + \)\(38\!\cdots\!99\)\( T^{20} + \)\(99\!\cdots\!79\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 + 1347 T^{2} - 5488 T^{3} + 1132827 T^{4} + 594823321 T^{6} )^{4} \)
$31$ \( 1 + 5235 T^{2} + 15533691 T^{4} + 32305589068 T^{6} + 51858557764857 T^{8} + 66810392242083729 T^{10} + 70681395526523023542 T^{12} + \)\(61\!\cdots\!09\)\( T^{14} + \)\(44\!\cdots\!37\)\( T^{16} + \)\(25\!\cdots\!48\)\( T^{18} + \)\(11\!\cdots\!71\)\( T^{20} + \)\(35\!\cdots\!35\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( ( 1 + 57 T - 1269 T^{2} - 55332 T^{3} + 5310729 T^{4} + 108467211 T^{5} - 4432135786 T^{6} + 148491611859 T^{7} + 9953161173369 T^{8} - 141966773662788 T^{9} - 4457336427025749 T^{10} + 274089309227817393 T^{11} + 6582952005840035281 T^{12} )^{2} \)
$41$ \( ( 1 + 3195 T^{2} + 22736 T^{3} + 5370795 T^{4} + 4750104241 T^{6} )^{4} \)
$43$ \( ( 1 - 5718 T^{2} + 18746463 T^{4} - 40352107060 T^{6} + 64090426450863 T^{8} - 66833129187322518 T^{10} + 39959630797262576401 T^{12} )^{2} \)
$47$ \( 1 + 8307 T^{2} + 34424571 T^{4} + 100893075532 T^{6} + 253028343980217 T^{8} + 616841824418715921 T^{10} + \)\(14\!\cdots\!38\)\( T^{12} + \)\(30\!\cdots\!01\)\( T^{14} + \)\(60\!\cdots\!37\)\( T^{16} + \)\(11\!\cdots\!12\)\( T^{18} + \)\(19\!\cdots\!91\)\( T^{20} + \)\(22\!\cdots\!07\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( ( 1 + 9 T - 2997 T^{2} - 116484 T^{3} + 179145 T^{4} + 137286747 T^{5} + 22274291990 T^{6} + 385638472323 T^{7} + 1413540218745 T^{8} - 2581793441750436 T^{9} - 186592292162848917 T^{10} + 1573987233289617441 T^{11} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \)
$59$ \( 1 + 1707 T^{2} - 9128709 T^{4} - 112178055524 T^{6} - 114189887648679 T^{8} + 551687617569890985 T^{10} + \)\(57\!\cdots\!54\)\( T^{12} + \)\(66\!\cdots\!85\)\( T^{14} - \)\(16\!\cdots\!59\)\( T^{16} - \)\(19\!\cdots\!44\)\( T^{18} - \)\(19\!\cdots\!69\)\( T^{20} + \)\(44\!\cdots\!07\)\( T^{22} + \)\(31\!\cdots\!61\)\( T^{24} \)
$61$ \( ( 1 - 159 T + 7899 T^{2} - 401828 T^{3} + 49683657 T^{4} - 2161110333 T^{5} + 23418423222 T^{6} - 8041491549093 T^{7} + 687912015120537 T^{8} - 20702328988731908 T^{9} + 1514296065365522619 T^{10} - \)\(11\!\cdots\!59\)\( T^{11} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \)
$67$ \( 1 + 23547 T^{2} + 312061755 T^{4} + 2891423739964 T^{6} + 20714504811186393 T^{8} + \)\(12\!\cdots\!29\)\( T^{10} + \)\(59\!\cdots\!86\)\( T^{12} + \)\(24\!\cdots\!09\)\( T^{14} + \)\(84\!\cdots\!13\)\( T^{16} + \)\(23\!\cdots\!04\)\( T^{18} + \)\(51\!\cdots\!55\)\( T^{20} + \)\(78\!\cdots\!47\)\( T^{22} + \)\(66\!\cdots\!21\)\( T^{24} \)
$71$ \( ( 1 - 10086 T^{2} + 56255151 T^{4} - 272956768468 T^{6} + 1429537951818831 T^{8} - 6513070116144745446 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 171 T + 4275 T^{2} - 355908 T^{3} + 155672073 T^{4} - 8204564097 T^{5} - 20860196506 T^{6} - 43722122072913 T^{7} + 4420813046023593 T^{8} - 53861061810065412 T^{9} + 3447616892847196275 T^{10} - \)\(73\!\cdots\!79\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \)
$79$ \( 1 + 16371 T^{2} + 121567851 T^{4} + 486780511900 T^{6} + 101914373627865 T^{8} - 22023984316104222399 T^{10} - \)\(21\!\cdots\!94\)\( T^{12} - \)\(85\!\cdots\!19\)\( T^{14} + \)\(15\!\cdots\!65\)\( T^{16} + \)\(28\!\cdots\!00\)\( T^{18} + \)\(27\!\cdots\!71\)\( T^{20} + \)\(14\!\cdots\!71\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( ( 1 - 29238 T^{2} + 403545087 T^{4} - 3424551164404 T^{6} + 19151572276818927 T^{8} - 65852520283281280758 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \)
$89$ \( ( 1 - 75 T - 7725 T^{2} + 660828 T^{3} + 10149225 T^{4} + 61763775 T^{5} - 80037126682 T^{6} + 489230861775 T^{7} + 636785120913225 T^{8} + 328419152543175708 T^{9} - 30410148524048575725 T^{10} - \)\(23\!\cdots\!75\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \)
$97$ \( ( 1 + 28059 T^{2} - 784 T^{3} + 264007131 T^{4} + 832972004929 T^{6} )^{4} \)
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