Properties

Label 847.2.a.n
Level 847
Weight 2
Character orbit 847.a
Self dual yes
Analytic conductor 6.763
Analytic rank 0
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 3 \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 6 \nu^{2} + 4 \nu - 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 8\)

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
2.38595
1.82356
0.879640
−0.276564
−1.10939
−1.70320
−1.38595 −0.122479 −0.0791355 −0.133004 0.169750 1.00000 2.88158 −2.98500 0.184338
1.2 −0.823556 −0.960649 −1.32176 2.98565 0.791148 1.00000 2.73565 −2.07715 −2.45885
1.3 0.120360 2.76784 −1.98551 −2.80853 0.333137 1.00000 −0.479696 4.66094 −0.338034
1.4 1.27656 −2.57603 −0.370384 −4.09144 −3.28847 1.00000 −3.02595 3.63595 −5.22298
1.5 2.10939 1.69851 2.44952 0.492391 3.58282 1.00000 0.948212 −0.115054 1.03864
1.6 2.70320 −2.80719 5.30727 −0.445072 −7.58839 1.00000 8.94020 4.88032 −1.20312
\(n\): 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 847.2.a.n yes 6
3.b odd 2 1 7623.2.a.cp 6
7.b odd 2 1 5929.2.a.bm 6
11.b odd 2 1 847.2.a.m 6
11.c even 5 4 847.2.f.y 24
11.d odd 10 4 847.2.f.z 24
33.d even 2 1 7623.2.a.cs 6
77.b even 2 1 5929.2.a.bj 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.a.m 6 11.b odd 2 1
847.2.a.n yes 6 1.a even 1 1 trivial
847.2.f.y 24 11.c even 5 4
847.2.f.z 24 11.d odd 10 4
5929.2.a.bj 6 77.b even 2 1
5929.2.a.bm 6 7.b odd 2 1
7623.2.a.cp 6 3.b odd 2 1
7623.2.a.cs 6 33.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2}^{6} - 4 T_{2}^{5} + 12 T_{2}^{3} - 4 T_{2}^{2} - 8 T_{2} + 1 \)
\( T_{3}^{6} + 2 T_{3}^{5} - 11 T_{3}^{4} - 20 T_{3}^{3} + 25 T_{3}^{2} + 36 T_{3} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 12 T^{2} - 28 T^{3} + 56 T^{4} - 96 T^{5} + 145 T^{6} - 192 T^{7} + 224 T^{8} - 224 T^{9} + 192 T^{10} - 128 T^{11} + 64 T^{12} \)
$3$ \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 28 T^{4} + 36 T^{5} + 100 T^{6} + 108 T^{7} + 252 T^{8} + 270 T^{9} + 567 T^{10} + 486 T^{11} + 729 T^{12} \)
$5$ \( 1 + 4 T + 21 T^{2} + 64 T^{3} + 194 T^{4} + 468 T^{5} + 1141 T^{6} + 2340 T^{7} + 4850 T^{8} + 8000 T^{9} + 13125 T^{10} + 12500 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 - T )^{6} \)
$11$ \( \)
$13$ \( 1 - 4 T + 59 T^{2} - 188 T^{3} + 1511 T^{4} - 3984 T^{5} + 23754 T^{6} - 51792 T^{7} + 255359 T^{8} - 413036 T^{9} + 1685099 T^{10} - 1485172 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 22 T + 284 T^{2} - 2536 T^{3} + 17485 T^{4} - 96354 T^{5} + 437477 T^{6} - 1638018 T^{7} + 5053165 T^{8} - 12459368 T^{9} + 23719964 T^{10} - 31236854 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 6 T + 64 T^{2} - 428 T^{3} + 2540 T^{4} - 12238 T^{5} + 64622 T^{6} - 232522 T^{7} + 916940 T^{8} - 2935652 T^{9} + 8340544 T^{10} - 14856594 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 2 T + 53 T^{2} + 88 T^{3} + 1540 T^{4} + 3306 T^{5} + 47756 T^{6} + 76038 T^{7} + 814660 T^{8} + 1070696 T^{9} + 14831573 T^{10} - 12872686 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 12 T + 202 T^{2} - 1648 T^{3} + 15591 T^{4} - 93140 T^{5} + 613773 T^{6} - 2701060 T^{7} + 13112031 T^{8} - 40193072 T^{9} + 142870762 T^{10} - 246133788 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 2 T + 68 T^{2} + 28 T^{3} + 3104 T^{4} + 2730 T^{5} + 128070 T^{6} + 84630 T^{7} + 2982944 T^{8} + 834148 T^{9} + 62799428 T^{10} + 57258302 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 14 T + 187 T^{2} - 1538 T^{3} + 11951 T^{4} - 73832 T^{5} + 476570 T^{6} - 2731784 T^{7} + 16360919 T^{8} - 77904314 T^{9} + 350468107 T^{10} - 970815398 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 - 26 T + 493 T^{2} - 6330 T^{3} + 66918 T^{4} - 558026 T^{5} + 3963849 T^{6} - 22879066 T^{7} + 112489158 T^{8} - 436269930 T^{9} + 1393100173 T^{10} - 3012261226 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 4 T + 137 T^{2} + 608 T^{3} + 8804 T^{4} + 41340 T^{5} + 407244 T^{6} + 1777620 T^{7} + 16278596 T^{8} + 48340256 T^{9} + 468375737 T^{10} + 588033772 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 16 T + 255 T^{2} + 2338 T^{3} + 22256 T^{4} + 151482 T^{5} + 1182208 T^{6} + 7119654 T^{7} + 49163504 T^{8} + 242738174 T^{9} + 1244318655 T^{10} + 3669520112 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 4 T + 238 T^{2} - 1038 T^{3} + 27039 T^{4} - 104266 T^{5} + 1829061 T^{6} - 5526098 T^{7} + 75952551 T^{8} - 154534326 T^{9} + 1877934478 T^{10} - 1672781972 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + 4 T + 67 T^{2} + 234 T^{3} + 3960 T^{4} + 16822 T^{5} + 97728 T^{6} + 992498 T^{7} + 13784760 T^{8} + 48058686 T^{9} + 811863187 T^{10} + 2859697196 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 8 T + 43 T^{2} + 374 T^{3} + 8672 T^{4} + 41834 T^{5} + 220388 T^{6} + 2551874 T^{7} + 32268512 T^{8} + 84890894 T^{9} + 595371163 T^{10} + 6756770408 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 6 T + 241 T^{2} - 1340 T^{3} + 30488 T^{4} - 149518 T^{5} + 2474324 T^{6} - 10017706 T^{7} + 136860632 T^{8} - 403022420 T^{9} + 4856420161 T^{10} - 8100750642 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 - 22 T + 533 T^{2} - 7336 T^{3} + 101080 T^{4} - 997626 T^{5} + 9693428 T^{6} - 70831446 T^{7} + 509544280 T^{8} - 2625635096 T^{9} + 13544425973 T^{10} - 39693045722 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 - 14 T + 316 T^{2} - 3752 T^{3} + 49196 T^{4} - 458618 T^{5} + 4569698 T^{6} - 33479114 T^{7} + 262165484 T^{8} - 1459591784 T^{9} + 8973844156 T^{10} - 29023002302 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 28 T + 601 T^{2} + 8392 T^{3} + 100724 T^{4} + 981724 T^{5} + 9215300 T^{6} + 77556196 T^{7} + 628618484 T^{8} + 4137583288 T^{9} + 23408998681 T^{10} + 86157579172 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 22 T + 540 T^{2} - 6988 T^{3} + 99116 T^{4} - 932526 T^{5} + 10112698 T^{6} - 77399658 T^{7} + 682810124 T^{8} - 3995647556 T^{9} + 25627493340 T^{10} - 86658894146 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 280 T^{2} + 1400 T^{3} + 32241 T^{4} + 335650 T^{5} + 2775213 T^{6} + 29872850 T^{7} + 255380961 T^{8} + 986956600 T^{9} + 17567827480 T^{10} + 496981290961 T^{12} \)
$97$ \( 1 + 4 T + 429 T^{2} + 1520 T^{3} + 87938 T^{4} + 265044 T^{5} + 10749637 T^{6} + 25709268 T^{7} + 827408642 T^{8} + 1387262960 T^{9} + 37979061549 T^{10} + 34349361028 T^{11} + 832972004929 T^{12} \)
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