Properties

Label 9075.2.a.dy
Level $9075$
Weight $2$
Character orbit 9075.a
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} - 148 x^{3} + 71 x^{2} + 10 x - 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} - 148 x^{3} + 71 x^{2} + 10 x - 5\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{11} + 46 \nu^{10} - 212 \nu^{9} - 784 \nu^{8} + 1980 \nu^{7} + 5252 \nu^{6} - 6883 \nu^{5} - 16686 \nu^{4} + 7961 \nu^{3} + 22435 \nu^{2} + 130 \nu - 5310 \)\()/1033\)
\(\beta_{4}\)\(=\)\((\)\( 29 \nu^{11} - 122 \nu^{10} - 336 \nu^{9} + 1720 \nu^{8} + 1306 \nu^{7} - 8360 \nu^{6} - 2450 \nu^{5} + 15420 \nu^{4} + 3184 \nu^{3} - 7088 \nu^{2} - 749 \nu + 160 \)\()/1033\)
\(\beta_{5}\)\(=\)\((\)\( 32 \nu^{11} - 99 \nu^{10} - 442 \nu^{9} + 1328 \nu^{8} + 2296 \nu^{7} - 5734 \nu^{6} - 6408 \nu^{5} + 8110 \nu^{4} + 10780 \nu^{3} - 1552 \nu^{2} - 4816 \nu + 604 \)\()/1033\)
\(\beta_{6}\)\(=\)\((\)\( -102 \nu^{11} + 251 \nu^{10} + 1538 \nu^{9} - 3200 \nu^{8} - 8868 \nu^{7} + 12983 \nu^{6} + 25074 \nu^{5} - 16941 \nu^{4} - 35136 \nu^{3} + 1848 \nu^{2} + 14318 \nu + 1432 \)\()/1033\)
\(\beta_{7}\)\(=\)\((\)\( -241 \nu^{11} + 907 \nu^{10} + 3006 \nu^{9} - 12584 \nu^{8} - 12385 \nu^{7} + 59002 \nu^{6} + 21144 \nu^{5} - 105562 \nu^{4} - 22435 \nu^{3} + 60756 \nu^{2} + 7863 \nu - 8810 \)\()/1033\)
\(\beta_{8}\)\(=\)\((\)\( -563 \nu^{11} + 2226 \nu^{10} + 5775 \nu^{9} - 29046 \nu^{8} - 12246 \nu^{7} + 123544 \nu^{6} - 23357 \nu^{5} - 186265 \nu^{4} + 57872 \nu^{3} + 80505 \nu^{2} - 22184 \nu - 2750 \)\()/1033\)
\(\beta_{9}\)\(=\)\((\)\( -662 \nu^{11} + 2500 \nu^{10} + 7207 \nu^{9} - 32638 \nu^{8} - 20124 \nu^{7} + 139153 \nu^{6} - 2241 \nu^{5} - 211549 \nu^{4} + 31365 \nu^{3} + 95120 \nu^{2} - 17098 \nu - 8105 \)\()/1033\)
\(\beta_{10}\)\(=\)\((\)\( 665 \nu^{11} - 2477 \nu^{10} - 7313 \nu^{9} + 32246 \nu^{8} + 21114 \nu^{7} - 136527 \nu^{6} - 1717 \nu^{5} + 204239 \nu^{4} - 22736 \nu^{3} - 89584 \nu^{2} + 6833 \nu + 7516 \)\()/1033\)
\(\beta_{11}\)\(=\)\((\)\( 849 \nu^{11} - 2788 \nu^{10} - 10371 \nu^{9} + 36783 \nu^{8} + 39481 \nu^{7} - 159684 \nu^{6} - 48893 \nu^{5} + 252421 \nu^{4} + 23754 \nu^{3} - 124333 \nu^{2} - 3298 \nu + 13055 \)\()/1033\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{8} + \beta_{6} + 7 \beta_{2} + \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(9 \beta_{10} + 7 \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - 9 \beta_{5} + 9 \beta_{4} + \beta_{3} + 3 \beta_{2} + 36 \beta_{1} + 10\)
\(\nu^{6}\)\(=\)\(13 \beta_{10} + 2 \beta_{9} + 11 \beta_{8} + 12 \beta_{6} - \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 47 \beta_{2} + 15 \beta_{1} + 85\)
\(\nu^{7}\)\(=\)\(69 \beta_{10} + 45 \beta_{9} + 24 \beta_{8} + \beta_{7} + 24 \beta_{6} - 62 \beta_{5} + 70 \beta_{4} + 10 \beta_{3} + 38 \beta_{2} + 224 \beta_{1} + 81\)
\(\nu^{8}\)\(=\)\(\beta_{11} + 121 \beta_{10} + 29 \beta_{9} + 93 \beta_{8} + \beta_{7} + 108 \beta_{6} - 12 \beta_{5} + 60 \beta_{4} + 24 \beta_{3} + 316 \beta_{2} + 160 \beta_{1} + 510\)
\(\nu^{9}\)\(=\)\(2 \beta_{11} + 506 \beta_{10} + 293 \beta_{9} + 214 \beta_{8} + 18 \beta_{7} + 221 \beta_{6} - 393 \beta_{5} + 526 \beta_{4} + 77 \beta_{3} + 357 \beta_{2} + 1444 \beta_{1} + 624\)
\(\nu^{10}\)\(=\)\(20 \beta_{11} + 1000 \beta_{10} + 298 \beta_{9} + 720 \beta_{8} + 28 \beta_{7} + 876 \beta_{6} - 95 \beta_{5} + 640 \beta_{4} + 206 \beta_{3} + 2138 \beta_{2} + 1484 \beta_{1} + 3180\)
\(\nu^{11}\)\(=\)\(48 \beta_{11} + 3652 \beta_{10} + 1960 \beta_{9} + 1720 \beta_{8} + 222 \beta_{7} + 1856 \beta_{6} - 2388 \beta_{5} + 3915 \beta_{4} + 546 \beta_{3} + 3002 \beta_{2} + 9597 \beta_{1} + 4702\)

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
2.72592
2.59201
2.10651
1.90743
0.803366
0.488299
0.298064
−0.288813
−0.784131
−1.27093
−2.21395
−2.36377
−2.72592 1.00000 5.43063 0 −2.72592 −0.486331 −9.35163 1.00000 0
1.2 −2.59201 1.00000 4.71851 0 −2.59201 −3.41835 −7.04639 1.00000 0
1.3 −2.10651 1.00000 2.43738 0 −2.10651 3.31633 −0.921348 1.00000 0
1.4 −1.90743 1.00000 1.63829 0 −1.90743 −4.45734 0.689943 1.00000 0
1.5 −0.803366 1.00000 −1.35460 0 −0.803366 0.508505 2.69497 1.00000 0
1.6 −0.488299 1.00000 −1.76156 0 −0.488299 −5.10895 1.83677 1.00000 0
1.7 −0.298064 1.00000 −1.91116 0 −0.298064 2.32107 1.16578 1.00000 0
1.8 0.288813 1.00000 −1.91659 0 0.288813 3.66740 −1.13116 1.00000 0
1.9 0.784131 1.00000 −1.38514 0 0.784131 −1.51801 −2.65439 1.00000 0
1.10 1.27093 1.00000 −0.384732 0 1.27093 0.483291 −3.03083 1.00000 0
1.11 2.21395 1.00000 2.90157 0 2.21395 −1.59339 1.99602 1.00000 0
1.12 2.36377 1.00000 3.58741 0 2.36377 −1.71423 3.75227 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 9075.2.a.dy 12
5.b even 2 1 9075.2.a.dz 12
5.c odd 4 2 1815.2.c.j 24
11.b odd 2 1 9075.2.a.ea 12
11.c even 5 2 825.2.n.p 24
55.d odd 2 1 9075.2.a.dx 12
55.e even 4 2 1815.2.c.k 24
55.j even 10 2 825.2.n.o 24
55.k odd 20 4 165.2.s.a 48
165.v even 20 4 495.2.ba.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.s.a 48 55.k odd 20 4
495.2.ba.c 48 165.v even 20 4
825.2.n.o 24 55.j even 10 2
825.2.n.p 24 11.c even 5 2
1815.2.c.j 24 5.c odd 4 2
1815.2.c.k 24 55.e even 4 2
9075.2.a.dx 12 55.d odd 2 1
9075.2.a.dy 12 1.a even 1 1 trivial
9075.2.a.dz 12 5.b even 2 1
9075.2.a.ea 12 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9075))\):

\(T_{2}^{12} + \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{13}^{12} + \cdots\)
\(T_{17}^{12} + \cdots\)
\(T_{19}^{12} + \cdots\)
\(T_{23}^{12} - \cdots\)
\(T_{37}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 - 10 T + 71 T^{2} + 148 T^{3} - 145 T^{4} - 330 T^{5} + 61 T^{6} + 220 T^{7} + 18 T^{8} - 52 T^{9} - 10 T^{10} + 4 T^{11} + T^{12} \)
$3$ \( ( -1 + T )^{12} \)
$5$ \( T^{12} \)
$7$ \( -1089 + 438 T + 7573 T^{2} - 484 T^{3} - 13299 T^{4} - 6282 T^{5} + 3233 T^{6} + 2248 T^{7} - 135 T^{8} - 240 T^{9} - 16 T^{10} + 8 T^{11} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( -21101 - 64010 T + 11122 T^{2} + 195208 T^{3} + 225555 T^{4} + 95756 T^{5} + 2239 T^{6} - 11538 T^{7} - 3561 T^{8} - 186 T^{9} + 93 T^{10} + 18 T^{11} + T^{12} \)
$17$ \( -66481 - 348248 T - 554099 T^{2} - 228418 T^{3} + 143429 T^{4} + 146596 T^{5} + 30964 T^{6} - 7728 T^{7} - 4133 T^{8} - 380 T^{9} + 79 T^{10} + 18 T^{11} + T^{12} \)
$19$ \( 734525 + 1279690 T + 51211 T^{2} - 947592 T^{3} - 402597 T^{4} + 137110 T^{5} + 106375 T^{6} + 10358 T^{7} - 4470 T^{8} - 942 T^{9} + 16 T^{10} + 16 T^{11} + T^{12} \)
$23$ \( 41299351 - 17065710 T - 22549211 T^{2} + 3184940 T^{3} + 3688110 T^{4} - 210060 T^{5} - 268150 T^{6} + 5890 T^{7} + 9560 T^{8} - 60 T^{9} - 161 T^{10} + T^{12} \)
$29$ \( 1084880 + 2300720 T - 2240856 T^{2} - 2292676 T^{3} + 1200861 T^{4} + 317270 T^{5} - 156430 T^{6} - 14046 T^{7} + 7752 T^{8} + 190 T^{9} - 155 T^{10} + T^{12} \)
$31$ \( 11486981 + 42667160 T - 5221310 T^{2} - 25666330 T^{3} + 7741451 T^{4} + 1567750 T^{5} - 621447 T^{6} - 32170 T^{7} + 18219 T^{8} + 210 T^{9} - 226 T^{10} + T^{12} \)
$37$ \( 1618831 - 11311680 T + 22684779 T^{2} - 14616160 T^{3} + 1725088 T^{4} + 1388010 T^{5} - 372590 T^{6} - 34920 T^{7} + 15718 T^{8} + 190 T^{9} - 231 T^{10} + T^{12} \)
$41$ \( -87119 + 1902 T + 326391 T^{2} + 233930 T^{3} - 120693 T^{4} - 154488 T^{5} - 25957 T^{6} + 13228 T^{7} + 3801 T^{8} - 300 T^{9} - 132 T^{10} + T^{12} \)
$43$ \( -972401 - 4684218 T + 971563 T^{2} + 13469606 T^{3} + 9982651 T^{4} + 1756834 T^{5} - 634005 T^{6} - 319862 T^{7} - 50373 T^{8} - 2200 T^{9} + 256 T^{10} + 32 T^{11} + T^{12} \)
$47$ \( 1113195775 + 2052240880 T + 1342441434 T^{2} + 322916684 T^{3} - 17186472 T^{4} - 19073208 T^{5} - 1607235 T^{6} + 319952 T^{7} + 42995 T^{8} - 1998 T^{9} - 363 T^{10} + 4 T^{11} + T^{12} \)
$53$ \( -47459081 - 22102754 T + 48854588 T^{2} + 14683866 T^{3} - 11457544 T^{4} - 3859880 T^{5} + 406379 T^{6} + 223878 T^{7} + 8187 T^{8} - 3080 T^{9} - 213 T^{10} + 12 T^{11} + T^{12} \)
$59$ \( -965103975 - 2217444600 T + 978070310 T^{2} + 184654180 T^{3} - 128097099 T^{4} + 9090400 T^{5} + 3341501 T^{6} - 506110 T^{7} - 16319 T^{8} + 6400 T^{9} - 194 T^{10} - 20 T^{11} + T^{12} \)
$61$ \( -48108995 + 8410050 T + 47654511 T^{2} + 2029144 T^{3} - 12598237 T^{4} - 1672106 T^{5} + 980351 T^{6} + 187010 T^{7} - 12070 T^{8} - 4014 T^{9} - 96 T^{10} + 20 T^{11} + T^{12} \)
$67$ \( 474528995 - 1005562710 T - 308196999 T^{2} + 192173858 T^{3} + 72465139 T^{4} - 1389440 T^{5} - 2637777 T^{6} - 104996 T^{7} + 40094 T^{8} + 1968 T^{9} - 302 T^{10} - 10 T^{11} + T^{12} \)
$71$ \( 517162301 - 562609146 T - 22303572 T^{2} + 141624384 T^{3} - 25649029 T^{4} - 8050366 T^{5} + 2650521 T^{6} - 65090 T^{7} - 47993 T^{8} + 4374 T^{9} + 156 T^{10} - 32 T^{11} + T^{12} \)
$73$ \( -306704 - 2866704 T - 2006664 T^{2} + 21166692 T^{3} - 15290929 T^{4} - 1794872 T^{5} + 1722666 T^{6} + 248348 T^{7} - 28014 T^{8} - 5658 T^{9} - 49 T^{10} + 26 T^{11} + T^{12} \)
$79$ \( -1150699275 + 1095518520 T + 786126316 T^{2} - 208717684 T^{3} - 172670996 T^{4} - 14993432 T^{5} + 5185466 T^{6} + 738528 T^{7} - 44148 T^{8} - 8972 T^{9} - 36 T^{10} + 32 T^{11} + T^{12} \)
$83$ \( 119614891 + 857253988 T + 746588195 T^{2} + 81250886 T^{3} - 73628818 T^{4} - 14771224 T^{5} + 1933640 T^{6} + 501390 T^{7} - 13848 T^{8} - 6194 T^{9} - 89 T^{10} + 26 T^{11} + T^{12} \)
$89$ \( -619209755 - 122202140 T + 583076316 T^{2} + 125558680 T^{3} - 119629276 T^{4} - 43349214 T^{5} - 1562726 T^{6} + 817780 T^{7} + 64224 T^{8} - 5004 T^{9} - 476 T^{10} + 10 T^{11} + T^{12} \)
$97$ \( 8411891531 + 8955782172 T - 1753647562 T^{2} - 4175795144 T^{3} - 1419399388 T^{4} - 135614302 T^{5} + 12014015 T^{6} + 2542148 T^{7} + 27027 T^{8} - 13204 T^{9} - 457 T^{10} + 22 T^{11} + T^{12} \)
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