Properties

Label 201.4.a.b
Level 201
Weight 4
Character orbit 201.a
Self dual yes
Analytic conductor 11.859
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 201 = 3 \cdot 67 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 201.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8593839112\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + ( -10 - \beta_{1} - \beta_{2} - 3 \beta_{5} ) q^{7} + ( -13 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + 3 q^{3} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{4} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{5} + ( -3 + 3 \beta_{1} ) q^{6} + ( -10 - \beta_{1} - \beta_{2} - 3 \beta_{5} ) q^{7} + ( -13 - \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{8} + 9 q^{9} + ( -17 + 4 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{10} + ( -12 + \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{11} + ( 6 - 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} ) q^{12} + ( -34 + 2 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{13} + ( 4 - 16 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - \beta_{5} ) q^{14} + ( -3 - 6 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{15} + ( 2 - 11 \beta_{1} + 2 \beta_{2} - 11 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -24 - 3 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} + 15 \beta_{4} + 7 \beta_{5} ) q^{17} + ( -9 + 9 \beta_{1} ) q^{18} + ( -44 + 15 \beta_{1} - 13 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} + 11 \beta_{5} ) q^{19} + ( 50 - 30 \beta_{1} + 11 \beta_{2} - 13 \beta_{3} + 6 \beta_{5} ) q^{20} + ( -30 - 3 \beta_{1} - 3 \beta_{2} - 9 \beta_{5} ) q^{21} + ( 14 - 29 \beta_{1} + 17 \beta_{2} - 11 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{22} + ( 7 - 17 \beta_{1} + 11 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 11 \beta_{5} ) q^{23} + ( -39 - 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} ) q^{24} + ( -11 - 6 \beta_{1} + 24 \beta_{2} + 21 \beta_{4} + 15 \beta_{5} ) q^{25} + ( 78 - 34 \beta_{1} - 4 \beta_{2} + \beta_{3} + 20 \beta_{4} + 13 \beta_{5} ) q^{26} + 27 q^{27} + ( -57 + 37 \beta_{1} - 22 \beta_{2} + 10 \beta_{3} - 13 \beta_{4} - 4 \beta_{5} ) q^{28} + ( -42 + 7 \beta_{1} + 25 \beta_{2} + 4 \beta_{3} - 15 \beta_{4} - 6 \beta_{5} ) q^{29} + ( -51 + 12 \beta_{1} - 27 \beta_{2} + 6 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{30} + ( -68 + 38 \beta_{1} - 48 \beta_{2} - 11 \beta_{3} + \beta_{4} - 14 \beta_{5} ) q^{31} + ( 13 + 37 \beta_{1} - 52 \beta_{2} - 2 \beta_{3} - 12 \beta_{4} - 16 \beta_{5} ) q^{32} + ( -36 + 3 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 15 \beta_{5} ) q^{33} + ( -38 - 3 \beta_{1} + 53 \beta_{2} + 11 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{34} + ( -65 + 31 \beta_{1} + 17 \beta_{2} + 22 \beta_{3} + 20 \beta_{4} - 8 \beta_{5} ) q^{35} + ( 18 - 18 \beta_{1} + 9 \beta_{2} + 9 \beta_{4} + 9 \beta_{5} ) q^{36} + ( -100 + 52 \beta_{1} - 30 \beta_{2} - 10 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{37} + ( 132 - 67 \beta_{1} + 11 \beta_{2} - 19 \beta_{3} - 17 \beta_{4} + 37 \beta_{5} ) q^{38} + ( -102 + 6 \beta_{1} + 24 \beta_{2} + 9 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{39} + ( -163 + 97 \beta_{1} - 14 \beta_{2} + 27 \beta_{3} + 5 \beta_{4} - 37 \beta_{5} ) q^{40} + ( -11 - \beta_{1} + 81 \beta_{2} + 20 \beta_{4} + 8 \beta_{5} ) q^{41} + ( 12 - 48 \beta_{1} + 9 \beta_{2} - 12 \beta_{3} - 3 \beta_{5} ) q^{42} + ( -54 - 37 \beta_{1} + 43 \beta_{2} - 37 \beta_{3} + 16 \beta_{4} - 20 \beta_{5} ) q^{43} + ( -128 + 77 \beta_{1} - 51 \beta_{2} + 31 \beta_{3} - 17 \beta_{4} - 19 \beta_{5} ) q^{44} + ( -9 - 18 \beta_{1} - 9 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{45} + ( -157 + 55 \beta_{1} - 18 \beta_{2} + 20 \beta_{3} - 3 \beta_{4} - 12 \beta_{5} ) q^{46} + ( -78 + 32 \beta_{1} - 58 \beta_{2} - 53 \beta_{3} - 60 \beta_{4} - 49 \beta_{5} ) q^{47} + ( 6 - 33 \beta_{1} + 6 \beta_{2} - 33 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} ) q^{48} + ( -45 + 55 \beta_{1} - 13 \beta_{2} - 17 \beta_{3} - 20 \beta_{4} + 58 \beta_{5} ) q^{49} + ( -43 + 70 \beta_{1} + 18 \beta_{2} + 60 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} ) q^{50} + ( -72 - 9 \beta_{1} + 9 \beta_{2} + 21 \beta_{3} + 45 \beta_{4} + 21 \beta_{5} ) q^{51} + ( -190 + 136 \beta_{1} - 44 \beta_{2} + 3 \beta_{3} - 38 \beta_{4} - 59 \beta_{5} ) q^{52} + ( 135 - 25 \beta_{1} - 19 \beta_{2} - 24 \beta_{3} - 12 \beta_{4} + 64 \beta_{5} ) q^{53} + ( -27 + 27 \beta_{1} ) q^{54} + ( -196 + 73 \beta_{1} + 25 \beta_{2} + 33 \beta_{3} + 11 \beta_{4} - 19 \beta_{5} ) q^{55} + ( 326 - 29 \beta_{1} + 30 \beta_{2} - 27 \beta_{3} + 7 \beta_{4} + 88 \beta_{5} ) q^{56} + ( -132 + 45 \beta_{1} - 39 \beta_{2} + 15 \beta_{3} - 21 \beta_{4} + 33 \beta_{5} ) q^{57} + ( 222 - 59 \beta_{1} - 45 \beta_{2} - 4 \beta_{3} + 67 \beta_{4} + 34 \beta_{5} ) q^{58} + ( -63 + 35 \beta_{1} - 103 \beta_{2} + 12 \beta_{3} + 22 \beta_{4} - 58 \beta_{5} ) q^{59} + ( 150 - 90 \beta_{1} + 33 \beta_{2} - 39 \beta_{3} + 18 \beta_{5} ) q^{60} + ( -240 - 44 \beta_{1} + 58 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} + 39 \beta_{5} ) q^{61} + ( 292 - 159 \beta_{1} + 70 \beta_{2} - 39 \beta_{3} - 55 \beta_{4} + 4 \beta_{5} ) q^{62} + ( -90 - 9 \beta_{1} - 9 \beta_{2} - 27 \beta_{5} ) q^{63} + ( 204 - 28 \beta_{1} + 47 \beta_{2} + 12 \beta_{3} - 21 \beta_{4} + 27 \beta_{5} ) q^{64} + ( -64 + 85 \beta_{1} - 105 \beta_{2} + 29 \beta_{3} - \beta_{4} - 77 \beta_{5} ) q^{65} + ( 42 - 87 \beta_{1} + 51 \beta_{2} - 33 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} ) q^{66} -67 q^{67} + ( 350 + 29 \beta_{1} - 41 \beta_{2} - 19 \beta_{3} - 9 \beta_{4} - 29 \beta_{5} ) q^{68} + ( 21 - 51 \beta_{1} + 33 \beta_{2} + 9 \beta_{3} + 12 \beta_{4} + 33 \beta_{5} ) q^{69} + ( 371 - 119 \beta_{1} + 148 \beta_{2} - 15 \beta_{3} + 95 \beta_{4} + 77 \beta_{5} ) q^{70} + ( -110 - 74 \beta_{1} - 16 \beta_{2} - 9 \beta_{3} - 98 \beta_{4} - 61 \beta_{5} ) q^{71} + ( -117 - 9 \beta_{2} + 27 \beta_{3} - 9 \beta_{4} - 27 \beta_{5} ) q^{72} + ( 12 - 109 \beta_{1} + 11 \beta_{2} + 15 \beta_{3} - 98 \beta_{4} + 58 \beta_{5} ) q^{73} + ( 486 - 171 \beta_{1} + 60 \beta_{2} - 14 \beta_{3} - 13 \beta_{4} + 21 \beta_{5} ) q^{74} + ( -33 - 18 \beta_{1} + 72 \beta_{2} + 63 \beta_{4} + 45 \beta_{5} ) q^{75} + ( -390 + 185 \beta_{1} - 119 \beta_{2} + 29 \beta_{3} - 45 \beta_{4} - 195 \beta_{5} ) q^{76} + ( 468 + 29 \beta_{1} - \beta_{2} - 35 \beta_{3} - 21 \beta_{4} + 83 \beta_{5} ) q^{77} + ( 234 - 102 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} + 60 \beta_{4} + 39 \beta_{5} ) q^{78} + ( -194 - 87 \beta_{1} + 75 \beta_{2} + 102 \beta_{3} + 85 \beta_{4} + 16 \beta_{5} ) q^{79} + ( 658 - 157 \beta_{1} + 156 \beta_{2} + 4 \beta_{3} + 133 \beta_{4} + 125 \beta_{5} ) q^{80} + 81 q^{81} + ( 189 + 107 \beta_{1} - 30 \beta_{2} + 109 \beta_{3} + 153 \beta_{4} - 21 \beta_{5} ) q^{82} + ( 293 + 101 \beta_{1} - 41 \beta_{2} + 12 \beta_{3} + 142 \beta_{4} - 24 \beta_{5} ) q^{83} + ( -171 + 111 \beta_{1} - 66 \beta_{2} + 30 \beta_{3} - 39 \beta_{4} - 12 \beta_{5} ) q^{84} + ( -126 - 83 \beta_{1} - 31 \beta_{2} - 77 \beta_{3} - 91 \beta_{4} - 85 \beta_{5} ) q^{85} + ( -142 + 76 \beta_{1} - 123 \beta_{2} + 113 \beta_{3} + 32 \beta_{4} - 164 \beta_{5} ) q^{86} + ( -126 + 21 \beta_{1} + 75 \beta_{2} + 12 \beta_{3} - 45 \beta_{4} - 18 \beta_{5} ) q^{87} + ( 628 - 141 \beta_{1} + 53 \beta_{2} - 61 \beta_{3} + 49 \beta_{4} + 163 \beta_{5} ) q^{88} + ( 82 + 19 \beta_{1} + 89 \beta_{2} - 15 \beta_{3} - 69 \beta_{4} - 191 \beta_{5} ) q^{89} + ( -153 + 36 \beta_{1} - 81 \beta_{2} + 18 \beta_{3} - 36 \beta_{4} - 36 \beta_{5} ) q^{90} + ( 228 - 77 \beta_{1} + 31 \beta_{2} - 27 \beta_{3} - 5 \beta_{4} + 109 \beta_{5} ) q^{91} + ( 572 - 161 \beta_{1} + 48 \beta_{2} - 97 \beta_{3} + 19 \beta_{4} + 30 \beta_{5} ) q^{92} + ( -204 + 114 \beta_{1} - 144 \beta_{2} - 33 \beta_{3} + 3 \beta_{4} - 42 \beta_{5} ) q^{93} + ( 410 - 220 \beta_{1} - 200 \beta_{2} - 61 \beta_{3} - 88 \beta_{4} - 67 \beta_{5} ) q^{94} + ( 96 + 353 \beta_{1} - 73 \beta_{2} + 51 \beta_{3} - 11 \beta_{4} - 85 \beta_{5} ) q^{95} + ( 39 + 111 \beta_{1} - 156 \beta_{2} - 6 \beta_{3} - 36 \beta_{4} - 48 \beta_{5} ) q^{96} + ( 284 - 51 \beta_{1} - 47 \beta_{2} - 71 \beta_{3} - 87 \beta_{4} + 117 \beta_{5} ) q^{97} + ( 425 + 17 \beta_{1} - 101 \beta_{2} + 59 \beta_{3} - 46 \beta_{4} + 24 \beta_{5} ) q^{98} + ( -108 + 9 \beta_{1} - 45 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} - 45 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 5q^{2} + 18q^{3} + 13q^{4} - 12q^{5} - 15q^{6} - 62q^{7} - 75q^{8} + 54q^{9} + O(q^{10}) \) \( 6q - 5q^{2} + 18q^{3} + 13q^{4} - 12q^{5} - 15q^{6} - 62q^{7} - 75q^{8} + 54q^{9} - 111q^{10} - 72q^{11} + 39q^{12} - 192q^{13} + 3q^{14} - 36q^{15} - 27q^{16} - 100q^{17} - 45q^{18} - 266q^{19} + 255q^{20} - 186q^{21} + 44q^{22} + 50q^{23} - 225q^{24} - 6q^{25} + 472q^{26} + 162q^{27} - 333q^{28} - 242q^{29} - 333q^{30} - 438q^{31} + 35q^{32} - 216q^{33} - 150q^{34} - 258q^{35} + 117q^{36} - 596q^{37} + 664q^{38} - 576q^{39} - 831q^{40} + 54q^{41} + 9q^{42} - 360q^{43} - 714q^{44} - 108q^{45} - 871q^{46} - 720q^{47} - 81q^{48} - 302q^{49} + 4q^{50} - 300q^{51} - 1118q^{52} + 694q^{53} - 135q^{54} - 990q^{55} + 1917q^{56} - 798q^{57} + 1354q^{58} - 378q^{59} + 765q^{60} - 1396q^{61} + 1475q^{62} - 558q^{63} + 1225q^{64} - 348q^{65} + 132q^{66} - 402q^{67} + 2032q^{68} + 150q^{69} + 2415q^{70} - 964q^{71} - 675q^{72} - 192q^{73} + 2751q^{74} - 18q^{75} - 2306q^{76} + 2724q^{77} + 1416q^{78} - 802q^{79} + 4221q^{80} + 486q^{81} + 1735q^{82} + 2126q^{83} - 999q^{84} - 1206q^{85} - 609q^{86} - 726q^{87} + 3656q^{88} + 432q^{89} - 999q^{90} + 1258q^{91} + 3163q^{92} - 1314q^{93} + 1742q^{94} + 936q^{95} + 105q^{96} + 1290q^{97} + 2492q^{98} - 648q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 28 x^{4} + 22 x^{3} + 202 x^{2} - 96 x - 384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 9 \nu^{4} + 20 \nu^{3} - 182 \nu^{2} - 26 \nu + 560 \)\()/64\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} - 5 \nu^{4} + 92 \nu^{3} + 94 \nu^{2} - 590 \nu - 240 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{5} - 3 \nu^{4} - 252 \nu^{3} + 82 \nu^{2} + 990 \nu - 336 \)\()/128\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{5} - 15 \nu^{4} + 212 \nu^{3} + 410 \nu^{2} - 938 \nu - 1936 \)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{2} + 9\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 13 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(20 \beta_{5} + 22 \beta_{4} + \beta_{3} + 28 \beta_{2} - 3 \beta_{1} + 119\)
\(\nu^{5}\)\(=\)\(-2 \beta_{5} + 56 \beta_{4} + 69 \beta_{3} + 46 \beta_{2} + 207 \beta_{1} - 27\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.17870
−2.57710
−1.43201
2.04345
2.81400
4.33036
−5.17870 3.00000 18.8190 17.1025 −15.5361 −19.8930 −56.0282 9.00000 −88.5686
1.2 −3.57710 3.00000 4.79568 −7.96859 −10.7313 0.508092 11.4622 9.00000 28.5045
1.3 −2.43201 3.00000 −2.08534 −2.10555 −7.29602 −2.84912 24.5276 9.00000 5.12071
1.4 1.04345 3.00000 −6.91121 −6.71512 3.13035 8.77104 −15.5591 9.00000 −7.00690
1.5 1.81400 3.00000 −4.70941 5.30362 5.44200 −31.2342 −23.0548 9.00000 9.62075
1.6 3.33036 3.00000 3.09132 −17.6168 9.99109 −17.3029 −16.3477 9.00000 −58.6704
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.a.b 6
3.b odd 2 1 603.4.a.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.a.b 6 1.a even 1 1 trivial
603.4.a.b 6 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(67\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 5 T_{2}^{5} - 18 T_{2}^{4} - 80 T_{2}^{3} + 105 T_{2}^{2} + 263 T_{2} - 284 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(201))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 30 T^{2} + 120 T^{3} + 489 T^{4} + 1543 T^{5} + 4724 T^{6} + 12344 T^{7} + 31296 T^{8} + 61440 T^{9} + 122880 T^{10} + 163840 T^{11} + 262144 T^{12} \)
$3$ \( ( 1 - 3 T )^{6} \)
$5$ \( 1 + 12 T + 450 T^{2} + 3702 T^{3} + 84996 T^{4} + 551406 T^{5} + 11272786 T^{6} + 68925750 T^{7} + 1328062500 T^{8} + 7230468750 T^{9} + 109863281250 T^{10} + 366210937500 T^{11} + 3814697265625 T^{12} \)
$7$ \( 1 + 62 T + 3102 T^{2} + 105906 T^{3} + 3095740 T^{4} + 72288890 T^{5} + 1474626964 T^{6} + 24795089270 T^{7} + 364210715260 T^{8} + 4273689102942 T^{9} + 42935672897502 T^{10} + 294348813616466 T^{11} + 1628413597910449 T^{12} \)
$11$ \( 1 + 72 T + 7730 T^{2} + 383004 T^{3} + 24191519 T^{4} + 914116548 T^{5} + 42016412684 T^{6} + 1216689125388 T^{7} + 42856751591159 T^{8} + 903103397443764 T^{9} + 24260051352053330 T^{10} + 300761868197926872 T^{11} + 5559917313492231481 T^{12} \)
$13$ \( 1 + 192 T + 24192 T^{2} + 2178520 T^{3} + 159997191 T^{4} + 9607642896 T^{5} + 491628309456 T^{6} + 21107991442512 T^{7} + 772275881493519 T^{8} + 23102113974067960 T^{9} + 563627275283060352 T^{10} + 9827691458705425344 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} \)
$17$ \( 1 + 100 T + 17406 T^{2} + 1324680 T^{3} + 167225871 T^{4} + 10618927604 T^{5} + 1002763591076 T^{6} + 52170791318452 T^{7} + 4036425999847599 T^{8} + 157090988238045960 T^{9} + 10141122661221219966 T^{10} + \)\(28\!\cdots\!00\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \)
$19$ \( 1 + 266 T + 42266 T^{2} + 4236138 T^{3} + 296092135 T^{4} + 13402949752 T^{5} + 719366719084 T^{6} + 91930832348968 T^{7} + 13929915348245935 T^{8} + 1366949618694137502 T^{9} + 93547968369250360826 T^{10} + \)\(40\!\cdots\!34\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \)
$23$ \( 1 - 50 T + 59580 T^{2} - 2453988 T^{3} + 1563496296 T^{4} - 52610439040 T^{5} + 24008793796256 T^{6} - 640111211799680 T^{7} + 231453564126567144 T^{8} - 4420007017398264444 T^{9} + \)\(13\!\cdots\!80\)\( T^{10} - \)\(13\!\cdots\!50\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \)
$29$ \( 1 + 242 T + 110184 T^{2} + 21008874 T^{3} + 5348102751 T^{4} + 825044232844 T^{5} + 158910112031072 T^{6} + 20122003794832316 T^{7} + 3181176239399056071 T^{8} + \)\(30\!\cdots\!06\)\( T^{9} + \)\(38\!\cdots\!44\)\( T^{10} + \)\(20\!\cdots\!58\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \)
$31$ \( 1 + 438 T + 167178 T^{2} + 47886674 T^{3} + 11737875204 T^{4} + 2476948325742 T^{5} + 452905537959060 T^{6} + 73790767572179922 T^{7} + 10417407450668625924 T^{8} + \)\(12\!\cdots\!54\)\( T^{9} + \)\(13\!\cdots\!58\)\( T^{10} + \)\(10\!\cdots\!38\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \)
$37$ \( 1 + 596 T + 371988 T^{2} + 141686280 T^{3} + 51370265512 T^{4} + 13921624565720 T^{5} + 3556511420813206 T^{6} + 705172049127415160 T^{7} + \)\(13\!\cdots\!08\)\( T^{8} + \)\(18\!\cdots\!60\)\( T^{9} + \)\(24\!\cdots\!28\)\( T^{10} + \)\(19\!\cdots\!28\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} \)
$41$ \( 1 - 54 T + 161642 T^{2} - 8481930 T^{3} + 10616368448 T^{4} - 1249043400036 T^{5} + 591220917557702 T^{6} - 86085320173881156 T^{7} + 50428856788863387968 T^{8} - \)\(27\!\cdots\!30\)\( T^{9} + \)\(36\!\cdots\!02\)\( T^{10} - \)\(83\!\cdots\!54\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} \)
$43$ \( 1 + 360 T + 165516 T^{2} + 47995920 T^{3} + 20642421372 T^{4} + 5792529905856 T^{5} + 2041194096923078 T^{6} + 460546675224892992 T^{7} + \)\(13\!\cdots\!28\)\( T^{8} + \)\(24\!\cdots\!60\)\( T^{9} + \)\(66\!\cdots\!16\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 + 720 T + 397004 T^{2} + 125695416 T^{3} + 26888555879 T^{4} + 2268069480336 T^{5} - 180545365290088 T^{6} + 235477777656924528 T^{7} + \)\(28\!\cdots\!91\)\( T^{8} + \)\(14\!\cdots\!72\)\( T^{9} + \)\(46\!\cdots\!64\)\( T^{10} + \)\(86\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$53$ \( 1 - 694 T + 664990 T^{2} - 406886710 T^{3} + 231435834356 T^{4} - 103855499414956 T^{5} + 45785711849399818 T^{6} - 15461695186400404412 T^{7} + \)\(51\!\cdots\!24\)\( T^{8} - \)\(13\!\cdots\!30\)\( T^{9} + \)\(32\!\cdots\!90\)\( T^{10} - \)\(50\!\cdots\!58\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \)
$59$ \( 1 + 378 T + 704502 T^{2} + 120948282 T^{3} + 212924962980 T^{4} + 17102539469628 T^{5} + 48061764657521098 T^{6} + 3512502453732729012 T^{7} + \)\(89\!\cdots\!80\)\( T^{8} + \)\(10\!\cdots\!98\)\( T^{9} + \)\(12\!\cdots\!62\)\( T^{10} + \)\(13\!\cdots\!22\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \)
$61$ \( 1 + 1396 T + 1977224 T^{2} + 1627616532 T^{3} + 1297419985615 T^{4} + 740967503983568 T^{5} + 408223248984371776 T^{6} + \)\(16\!\cdots\!08\)\( T^{7} + \)\(66\!\cdots\!15\)\( T^{8} + \)\(19\!\cdots\!12\)\( T^{9} + \)\(52\!\cdots\!04\)\( T^{10} + \)\(84\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \)
$67$ \( ( 1 + 67 T )^{6} \)
$71$ \( 1 + 964 T + 1676964 T^{2} + 1220182788 T^{3} + 1146291805335 T^{4} + 691552394370056 T^{5} + 483190046029632872 T^{6} + \)\(24\!\cdots\!16\)\( T^{7} + \)\(14\!\cdots\!35\)\( T^{8} + \)\(55\!\cdots\!28\)\( T^{9} + \)\(27\!\cdots\!24\)\( T^{10} + \)\(56\!\cdots\!64\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} \)
$73$ \( 1 + 192 T + 835608 T^{2} + 301915460 T^{3} + 533817926664 T^{4} + 150219534958800 T^{5} + 258543945814639926 T^{6} + 58437952831067499600 T^{7} + \)\(80\!\cdots\!96\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + \)\(19\!\cdots\!68\)\( T^{10} + \)\(17\!\cdots\!44\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} \)
$79$ \( 1 + 802 T + 1938044 T^{2} + 1472087886 T^{3} + 2006062790071 T^{4} + 1237567282690604 T^{5} + 1248751754152488280 T^{6} + \)\(61\!\cdots\!56\)\( T^{7} + \)\(48\!\cdots\!91\)\( T^{8} + \)\(17\!\cdots\!34\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} + \)\(23\!\cdots\!98\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \)
$83$ \( 1 - 2126 T + 3463794 T^{2} - 4147751538 T^{3} + 4417404883020 T^{4} - 3937769393691868 T^{5} + 3191365359439157882 T^{6} - \)\(22\!\cdots\!16\)\( T^{7} + \)\(14\!\cdots\!80\)\( T^{8} - \)\(77\!\cdots\!14\)\( T^{9} + \)\(37\!\cdots\!34\)\( T^{10} - \)\(12\!\cdots\!82\)\( T^{11} + \)\(34\!\cdots\!09\)\( T^{12} \)
$89$ \( 1 - 432 T + 1092362 T^{2} - 1355191524 T^{3} + 1512520351031 T^{4} - 1154979184953420 T^{5} + 1362263626157437724 T^{6} - \)\(81\!\cdots\!80\)\( T^{7} + \)\(75\!\cdots\!91\)\( T^{8} - \)\(47\!\cdots\!16\)\( T^{9} + \)\(26\!\cdots\!02\)\( T^{10} - \)\(75\!\cdots\!68\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \)
$97$ \( 1 - 1290 T + 3920922 T^{2} - 4779758798 T^{3} + 7817886084711 T^{4} - 7502808852969672 T^{5} + 9271207523010554172 T^{6} - \)\(68\!\cdots\!56\)\( T^{7} + \)\(65\!\cdots\!19\)\( T^{8} - \)\(36\!\cdots\!66\)\( T^{9} + \)\(27\!\cdots\!02\)\( T^{10} - \)\(81\!\cdots\!70\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} \)
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