Properties

Label 6045.2.a.t
Level 6045
Weight 2
Character orbit 6045.a
Self dual yes
Analytic conductor 48.270
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 3 x^{8} - 7 x^{7} + 22 x^{6} + 14 x^{5} - 52 x^{4} - 5 x^{3} + 41 x^{2} - 4 x - 5\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{7} + ( -3 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{8} + q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + q^{3} + ( 1 - \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -1 + \beta_{1} ) q^{6} + ( -1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{7} + ( -3 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{8} + q^{9} + ( -1 + \beta_{1} ) q^{10} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} ) q^{12} + q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{14} + q^{15} + ( 5 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{16} + ( -1 + 2 \beta_{4} - \beta_{5} - \beta_{8} ) q^{17} + ( -1 + \beta_{1} ) q^{18} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( 1 - \beta_{1} + \beta_{2} ) q^{20} + ( -1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{21} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{22} + ( -3 + 2 \beta_{7} - \beta_{8} ) q^{23} + ( -3 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{24} + q^{25} + ( -1 + \beta_{1} ) q^{26} + q^{27} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{28} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{29} + ( -1 + \beta_{1} ) q^{30} + q^{31} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{32} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{34} + ( -1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{35} + ( 1 - \beta_{1} + \beta_{2} ) q^{36} + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{37} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{38} + q^{39} + ( -3 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{40} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} - \beta_{8} ) q^{41} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{42} + ( -1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{43} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 7 \beta_{4} + 3 \beta_{5} - \beta_{6} + 4 \beta_{7} + 3 \beta_{8} ) q^{44} + q^{45} + ( 5 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} ) q^{46} + ( -5 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{47} + ( 5 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{48} + ( 1 + 2 \beta_{1} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( -1 + 2 \beta_{4} - \beta_{5} - \beta_{8} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} ) q^{52} + ( -5 + \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{8} ) q^{53} + ( -1 + \beta_{1} ) q^{54} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( 5 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{8} ) q^{56} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{57} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{58} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{59} + ( 1 - \beta_{1} + \beta_{2} ) q^{60} + ( -2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{61} + ( -1 + \beta_{1} ) q^{62} + ( -1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{63} + ( 8 - 5 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{64} + q^{65} + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{66} + ( -4 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} ) q^{67} + ( 4 - 4 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} - 6 \beta_{5} - \beta_{6} - 3 \beta_{7} - 5 \beta_{8} ) q^{68} + ( -3 + 2 \beta_{7} - \beta_{8} ) q^{69} + ( -\beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{70} + ( -5 - 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{71} + ( -3 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{72} + ( -2 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} - \beta_{7} + \beta_{8} ) q^{73} + ( -3 + 4 \beta_{1} - \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{74} + q^{75} + ( -1 - 3 \beta_{1} - \beta_{2} + 3 \beta_{4} + \beta_{5} - 2 \beta_{8} ) q^{76} + ( 2 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 5 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} - 2 \beta_{8} ) q^{77} + ( -1 + \beta_{1} ) q^{78} + ( -\beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - \beta_{8} ) q^{79} + ( 5 - 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{80} + q^{81} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{82} + ( 1 - 4 \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{83} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{84} + ( -1 + 2 \beta_{4} - \beta_{5} - \beta_{8} ) q^{85} + ( -4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{86} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{87} + ( 7 - 5 \beta_{1} - \beta_{2} - 3 \beta_{3} + 13 \beta_{4} - 6 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} - 5 \beta_{8} ) q^{88} + ( -2 \beta_{1} + 4 \beta_{2} + \beta_{4} - 4 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} - \beta_{8} ) q^{89} + ( -1 + \beta_{1} ) q^{90} + ( -1 - \beta_{1} + \beta_{5} + \beta_{8} ) q^{91} + ( -8 + 7 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{92} + q^{93} + ( 8 - 6 \beta_{1} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{94} + ( 1 - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{95} + ( -5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{96} + ( -3 + 4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} + \beta_{8} ) q^{97} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{98} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 6q^{2} + 9q^{3} + 8q^{4} + 9q^{5} - 6q^{6} - 12q^{7} - 21q^{8} + 9q^{9} + O(q^{10}) \) \( 9q - 6q^{2} + 9q^{3} + 8q^{4} + 9q^{5} - 6q^{6} - 12q^{7} - 21q^{8} + 9q^{9} - 6q^{10} - 10q^{11} + 8q^{12} + 9q^{13} - 9q^{14} + 9q^{15} + 30q^{16} - 17q^{17} - 6q^{18} + 6q^{19} + 8q^{20} - 12q^{21} + 15q^{22} - 23q^{23} - 21q^{24} + 9q^{25} - 6q^{26} + 9q^{27} + 2q^{28} - 4q^{29} - 6q^{30} + 9q^{31} - 38q^{32} - 10q^{33} + 15q^{34} - 12q^{35} + 8q^{36} + 11q^{37} - 16q^{38} + 9q^{39} - 21q^{40} - 8q^{41} - 9q^{42} - 15q^{43} - q^{44} + 9q^{45} + 26q^{46} - 33q^{47} + 30q^{48} + 15q^{49} - 6q^{50} - 17q^{51} + 8q^{52} - 38q^{53} - 6q^{54} - 10q^{55} + 37q^{56} + 6q^{57} - 26q^{58} - 25q^{59} + 8q^{60} - 10q^{61} - 6q^{62} - 12q^{63} + 47q^{64} + 9q^{65} + 15q^{66} - 19q^{67} + 7q^{68} - 23q^{69} - 9q^{70} - 43q^{71} - 21q^{72} + q^{73} + 4q^{74} + 9q^{75} - 26q^{76} + 2q^{77} - 6q^{78} - 9q^{79} + 30q^{80} + 9q^{81} + 15q^{82} - 12q^{83} + 2q^{84} - 17q^{85} - 30q^{86} - 4q^{87} + q^{88} - 5q^{89} - 6q^{90} - 12q^{91} - 57q^{92} + 9q^{93} + 40q^{94} + 6q^{95} - 38q^{96} - 4q^{97} + 34q^{98} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 3 x^{8} - 7 x^{7} + 22 x^{6} + 14 x^{5} - 52 x^{4} - 5 x^{3} + 41 x^{2} - 4 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 6 \nu^{3} + 7 \nu^{2} + 8 \nu - 4 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 7 \nu^{3} + 9 \nu^{2} - 4 \nu - 3 \)
\(\beta_{5}\)\(=\)\( \nu^{8} - \nu^{7} - 10 \nu^{6} + 4 \nu^{5} + 29 \nu^{4} - 2 \nu^{3} - 26 \nu^{2} - 4 \nu + 3 \)
\(\beta_{6}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 8 \nu^{6} + 11 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} - 17 \nu^{2} + 4 \nu + 3 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - \nu^{7} - 10 \nu^{6} + 4 \nu^{5} + 30 \nu^{4} - 4 \nu^{3} - 31 \nu^{2} + \nu + 7 \)
\(\beta_{8}\)\(=\)\( -2 \nu^{8} + 3 \nu^{7} + 19 \nu^{6} - 17 \nu^{5} - 56 \nu^{4} + 28 \nu^{3} + 54 \nu^{2} - 11 \nu - 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 3 \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 11 \beta_{2} + 10 \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(10 \beta_{8} + 12 \beta_{7} + 10 \beta_{6} - 2 \beta_{5} - 10 \beta_{4} + \beta_{3} + 33 \beta_{2} + 35 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(25 \beta_{8} + 35 \beta_{7} + 25 \beta_{6} - 10 \beta_{5} - 24 \beta_{4} + 2 \beta_{3} + 102 \beta_{2} + 90 \beta_{1} + 25\)
\(\nu^{7}\)\(=\)\(88 \beta_{8} + 113 \beta_{7} + 87 \beta_{6} - 24 \beta_{5} - 86 \beta_{4} + 11 \beta_{3} + 303 \beta_{2} + 282 \beta_{1} + 28\)
\(\nu^{8}\)\(=\)\(242 \beta_{8} + 330 \beta_{7} + 241 \beta_{6} - 86 \beta_{5} - 230 \beta_{4} + 27 \beta_{3} + 904 \beta_{2} + 792 \beta_{1} + 145\)

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
−1.82576
−1.63414
−1.12407
−0.316933
0.493841
1.29080
1.50555
1.66264
2.94807
−2.82576 1.00000 5.98491 1.00000 −2.82576 −2.33173 −11.2604 1.00000 −2.82576
1.2 −2.63414 1.00000 4.93869 1.00000 −2.63414 −0.293212 −7.74092 1.00000 −2.63414
1.3 −2.12407 1.00000 2.51167 1.00000 −2.12407 2.73139 −1.08683 1.00000 −2.12407
1.4 −1.31693 1.00000 −0.265687 1.00000 −1.31693 1.83326 2.98376 1.00000 −1.31693
1.5 −0.506159 1.00000 −1.74380 1.00000 −0.506159 −4.78785 1.89496 1.00000 −0.506159
1.6 0.290796 1.00000 −1.91544 1.00000 0.290796 1.24204 −1.13859 1.00000 0.290796
1.7 0.505549 1.00000 −1.74442 1.00000 0.505549 −2.69242 −1.89299 1.00000 0.505549
1.8 0.662643 1.00000 −1.56090 1.00000 0.662643 −4.22758 −2.35961 1.00000 0.662643
1.9 1.94807 1.00000 1.79498 1.00000 1.94807 −3.47390 −0.399392 1.00000 1.94807
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)
\(31\) \(-1\)

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 6045.2.a.t 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.t 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 23 T^{2} + 69 T^{3} + 171 T^{4} + 367 T^{5} + 700 T^{6} + 1209 T^{7} + 1917 T^{8} + 2814 T^{9} + 3834 T^{10} + 4836 T^{11} + 5600 T^{12} + 5872 T^{13} + 5472 T^{14} + 4416 T^{15} + 2944 T^{16} + 1536 T^{17} + 512 T^{18} \)
$3$ \( ( 1 - T )^{9} \)
$5$ \( ( 1 - T )^{9} \)
$7$ \( 1 + 12 T + 96 T^{2} + 569 T^{3} + 2836 T^{4} + 12102 T^{5} + 45856 T^{6} + 154910 T^{7} + 474445 T^{8} + 1314649 T^{9} + 3321115 T^{10} + 7590590 T^{11} + 15728608 T^{12} + 29056902 T^{13} + 47664652 T^{14} + 66942281 T^{15} + 79060128 T^{16} + 69177612 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 10 T + 82 T^{2} + 422 T^{3} + 2013 T^{4} + 7684 T^{5} + 31505 T^{6} + 116662 T^{7} + 458426 T^{8} + 1530726 T^{9} + 5042686 T^{10} + 14116102 T^{11} + 41933155 T^{12} + 112501444 T^{13} + 324195663 T^{14} + 747598742 T^{15} + 1597948022 T^{16} + 2143588810 T^{17} + 2357947691 T^{18} \)
$13$ \( ( 1 - T )^{9} \)
$17$ \( 1 + 17 T + 198 T^{2} + 1579 T^{3} + 10164 T^{4} + 50783 T^{5} + 211606 T^{6} + 700783 T^{7} + 2157490 T^{8} + 7148926 T^{9} + 36677330 T^{10} + 202526287 T^{11} + 1039620278 T^{12} + 4241446943 T^{13} + 14431426548 T^{14} + 38113221451 T^{15} + 81247057254 T^{16} + 118587876497 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 - 6 T + 103 T^{2} - 564 T^{3} + 5257 T^{4} - 25844 T^{5} + 180543 T^{6} - 781626 T^{7} + 4579137 T^{8} - 17217784 T^{9} + 87003603 T^{10} - 282166986 T^{11} + 1238344437 T^{12} - 3368015924 T^{13} + 13016852443 T^{14} - 26533876884 T^{15} + 92068789117 T^{16} - 101901378246 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 23 T + 350 T^{2} + 3884 T^{3} + 35446 T^{4} + 275528 T^{5} + 1892253 T^{6} + 11617327 T^{7} + 64576901 T^{8} + 324785234 T^{9} + 1485268723 T^{10} + 6145565983 T^{11} + 23023042251 T^{12} + 77104031048 T^{13} + 228142613978 T^{14} + 574971392876 T^{15} + 1191688906450 T^{16} + 1801152661463 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 4 T + 45 T^{2} + 18 T^{3} + 1410 T^{4} + 3563 T^{5} + 51490 T^{6} + 137119 T^{7} + 1579520 T^{8} + 6166205 T^{9} + 45806080 T^{10} + 115317079 T^{11} + 1255789610 T^{12} + 2520042203 T^{13} + 28920720090 T^{14} + 10706819778 T^{15} + 776244433905 T^{16} + 2000985651844 T^{17} + 14507145975869 T^{18} \)
$31$ \( ( 1 - T )^{9} \)
$37$ \( 1 - 11 T + 210 T^{2} - 1959 T^{3} + 22319 T^{4} - 177600 T^{5} + 1546392 T^{6} - 10628146 T^{7} + 76648501 T^{8} - 458056038 T^{9} + 2835994537 T^{10} - 14549931874 T^{11} + 78329393976 T^{12} - 332850993600 T^{13} + 1547687776283 T^{14} - 5026258035231 T^{15} + 19935694197930 T^{16} - 38637273993131 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 8 T + 224 T^{2} + 1906 T^{3} + 27401 T^{4} + 217103 T^{5} + 2187165 T^{6} + 15627101 T^{7} + 123581568 T^{8} + 765435461 T^{9} + 5066844288 T^{10} + 26269156781 T^{11} + 150741598965 T^{12} + 613481190383 T^{13} + 3174575763601 T^{14} + 9053698683346 T^{15} + 43624957349344 T^{16} + 63879401832968 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 15 T + 268 T^{2} + 2695 T^{3} + 29711 T^{4} + 219751 T^{5} + 1843185 T^{6} + 11089462 T^{7} + 83173948 T^{8} + 466195437 T^{9} + 3576479764 T^{10} + 20504415238 T^{11} + 146546109795 T^{12} + 751284938551 T^{13} + 4367767849973 T^{14} + 17036073417055 T^{15} + 72847387776676 T^{16} + 175323004164015 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 33 T + 731 T^{2} + 11689 T^{3} + 155668 T^{4} + 1750831 T^{5} + 17423194 T^{6} + 153511633 T^{7} + 1224155805 T^{8} + 8791568684 T^{9} + 57535322835 T^{10} + 339107197297 T^{11} + 1808928270662 T^{12} + 8543496764911 T^{13} + 35701678549676 T^{14} + 125998247980681 T^{15} + 370341501058453 T^{16} + 785772459838113 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 38 T + 882 T^{2} + 15467 T^{3} + 222161 T^{4} + 2721796 T^{5} + 29162504 T^{6} + 277344973 T^{7} + 2362366855 T^{8} + 18113761198 T^{9} + 125205443315 T^{10} + 779062029157 T^{11} + 4341626108008 T^{12} + 21476279623876 T^{13} + 92906728920373 T^{14} + 342816173582243 T^{15} + 1036095225336234 T^{16} + 2365868235631718 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 25 T + 598 T^{2} + 8836 T^{3} + 125382 T^{4} + 1372240 T^{5} + 14739200 T^{6} + 132023743 T^{7} + 1178625939 T^{8} + 9043301631 T^{9} + 69538930401 T^{10} + 459574649383 T^{11} + 3027122156800 T^{12} + 16627927458640 T^{13} + 89638638457218 T^{14} + 372707195251876 T^{15} + 1488213587921762 T^{16} + 3670760940108025 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 10 T + 213 T^{2} + 1636 T^{3} + 24454 T^{4} + 163520 T^{5} + 2052270 T^{6} + 12263416 T^{7} + 147500303 T^{8} + 819376538 T^{9} + 8997518483 T^{10} + 45632170936 T^{11} + 465826296870 T^{12} + 2264071920320 T^{13} + 20653757944654 T^{14} + 84287332454596 T^{15} + 669404224072473 T^{16} + 1917073129972810 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 19 T + 364 T^{2} + 3968 T^{3} + 47546 T^{4} + 407038 T^{5} + 4235261 T^{6} + 32861357 T^{7} + 318758463 T^{8} + 2313027443 T^{9} + 21356817021 T^{10} + 147514631573 T^{11} + 1273809804143 T^{12} + 8202271989598 T^{13} + 64193048337422 T^{14} + 358938860446592 T^{15} + 2206099024337572 T^{16} + 7715285873576179 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + 43 T + 1250 T^{2} + 26537 T^{3} + 464310 T^{4} + 6823092 T^{5} + 87406863 T^{6} + 981563961 T^{7} + 9819184763 T^{8} + 87402239446 T^{9} + 697162118173 T^{10} + 4948063927401 T^{11} + 31283877743193 T^{12} + 173386237337652 T^{13} + 837721729962810 T^{14} + 3399397234411577 T^{15} + 11368900197988750 T^{16} + 27767401843567723 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - T + 201 T^{2} + 624 T^{3} + 27268 T^{4} + 76833 T^{5} + 3362478 T^{6} + 7658689 T^{7} + 282006259 T^{8} + 821117972 T^{9} + 20586456907 T^{10} + 40813153681 T^{11} + 1308061104126 T^{12} + 2181922050753 T^{13} + 56528516197924 T^{14} + 94432557204336 T^{15} + 2220527102338497 T^{16} - 806460091894081 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 + 9 T + 459 T^{2} + 4009 T^{3} + 98119 T^{4} + 855507 T^{5} + 13286490 T^{6} + 115060659 T^{7} + 1319858202 T^{8} + 10753106806 T^{9} + 104268797958 T^{10} + 718093572819 T^{11} + 6550757743110 T^{12} + 33322066946067 T^{13} + 301917696813481 T^{14} + 974537609183689 T^{15} + 8814594224646981 T^{16} + 13653979289159049 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 12 T + 403 T^{2} + 4312 T^{3} + 88615 T^{4} + 830672 T^{5} + 12887502 T^{6} + 107980426 T^{7} + 1390571815 T^{8} + 10298966981 T^{9} + 115417460645 T^{10} + 743877154714 T^{11} + 7368906106074 T^{12} + 39422298421712 T^{13} + 349058086579445 T^{14} + 1409766889967128 T^{15} + 10935828548819681 T^{16} + 27027506785668492 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 5 T + 333 T^{2} + 1789 T^{3} + 62420 T^{4} + 266030 T^{5} + 8358386 T^{6} + 27361400 T^{7} + 861715531 T^{8} + 2548208682 T^{9} + 76692682259 T^{10} + 216729649400 T^{11} + 5892403020034 T^{12} + 16691318373230 T^{13} + 348556990806580 T^{14} + 889099529529229 T^{15} + 14729034520211157 T^{16} + 19682944028510405 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 4 T + 501 T^{2} + 1151 T^{3} + 129781 T^{4} + 218247 T^{5} + 23001178 T^{6} + 31050897 T^{7} + 2957902923 T^{8} + 3288593497 T^{9} + 286916583531 T^{10} + 292157889873 T^{11} + 20992554128794 T^{12} + 19321249990407 T^{13} + 1114473605893717 T^{14} + 958750777673279 T^{15} + 40479940523534613 T^{16} + 31349734377507844 T^{17} + 760231058654565217 T^{18} \)
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