Properties

Label 1323.2.o.d
Level $1323$
Weight $2$
Character orbit 1323.o
Analytic conductor $10.564$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
Defining polynomial: \(x^{10} - x^{9} + 7 x^{8} - 4 x^{7} + 34 x^{6} - 19 x^{5} + 64 x^{4} - x^{3} + 64 x^{2} - 21 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{4} + \beta_{7} + \beta_{8} ) q^{2} + ( 1 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{8} +O(q^{10})\) \( q + ( \beta_{4} + \beta_{7} + \beta_{8} ) q^{2} + ( 1 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{4} + ( -\beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{5} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{8} + ( -1 + \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{10} + ( -2 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{11} + ( -\beta_{1} + \beta_{3} + \beta_{7} ) q^{13} + ( 2 \beta_{3} + 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{16} + ( -3 - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{17} + ( 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{20} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{22} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{23} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{25} + ( 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} + ( 2 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{7} ) q^{31} + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{6} ) q^{32} + ( \beta_{1} + \beta_{2} - 3 \beta_{4} - 3 \beta_{7} - 3 \beta_{8} ) q^{34} + ( -2 - 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} ) q^{37} + ( -3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{38} + ( 1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{40} + ( 5 \beta_{1} - 4 \beta_{2} - 4 \beta_{5} + 4 \beta_{8} - 4 \beta_{9} ) q^{41} + ( 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{43} + ( 1 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{44} + ( 3 + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{46} + ( 2 \beta_{1} - 3 \beta_{2} - \beta_{5} + 3 \beta_{6} - \beta_{8} ) q^{47} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{50} + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} ) q^{52} + ( 1 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{53} + ( -1 + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{55} + ( 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 7 \beta_{5} + 3 \beta_{7} + 7 \beta_{8} - 3 \beta_{9} ) q^{58} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{59} + ( -2 - 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{6} + 4 \beta_{7} + 4 \beta_{8} - \beta_{9} ) q^{61} + ( -3 + \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{62} + ( -1 - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{64} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{65} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{67} + ( -6 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 6 \beta_{8} - 3 \beta_{9} ) q^{68} + ( 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{71} + ( 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{5} ) q^{73} + ( -4 - 4 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{74} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{76} + ( 3 \beta_{1} - \beta_{2} + 8 \beta_{3} + 11 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} + 6 \beta_{8} - 7 \beta_{9} ) q^{79} + ( -6 - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{80} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} ) q^{82} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 3 - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{85} + ( 5 + 3 \beta_{1} - 5 \beta_{3} - 3 \beta_{4} - \beta_{5} - 5 \beta_{6} - 5 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{86} + ( -3 \beta_{1} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} ) q^{88} + ( 6 - 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{89} + ( -6 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{5} - 3 \beta_{6} + 6 \beta_{8} ) q^{92} + ( 1 + 2 \beta_{1} - 8 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - \beta_{6} - 8 \beta_{7} - 6 \beta_{8} ) q^{94} + ( \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} - 4 \beta_{8} + \beta_{9} ) q^{95} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{4} - 6 \beta_{5} - 2 \beta_{7} + 5 \beta_{8} + \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 4q^{4} + O(q^{10}) \) \( 10q + 4q^{4} - 12q^{11} + 6q^{13} - 6q^{16} - 24q^{17} + 3q^{20} + 5q^{22} - 15q^{23} + 7q^{25} + 6q^{26} + 15q^{29} + 9q^{31} + 48q^{32} + 3q^{34} - 12q^{37} + 18q^{38} + 15q^{40} + 9q^{41} + 3q^{43} + 26q^{46} - 15q^{47} - 3q^{50} - 12q^{52} + 8q^{58} + 18q^{59} - 12q^{61} - 12q^{62} + 6q^{64} - 3q^{65} - 10q^{67} - 27q^{68} - 30q^{74} + 9q^{76} + 20q^{79} - 60q^{80} + 15q^{83} + 18q^{85} + 54q^{86} - 8q^{88} + 48q^{89} - 39q^{92} + 3q^{94} + 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 7 x^{8} - 4 x^{7} + 34 x^{6} - 19 x^{5} + 64 x^{4} - x^{3} + 64 x^{2} - 21 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709 \)\()/72795\)
\(\beta_{3}\)\(=\)\((\)\( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232 \)\()/72795\)
\(\beta_{4}\)\(=\)\((\)\( -4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583 \)\()/218385\)
\(\beta_{5}\)\(=\)\((\)\( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546 \)\()/218385\)
\(\beta_{6}\)\(=\)\((\)\( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156 \)\()/72795\)
\(\beta_{7}\)\(=\)\((\)\( -840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514 \)\()/14559\)
\(\beta_{8}\)\(=\)\((\)\( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856 \)\()/43677\)
\(\beta_{9}\)\(=\)\((\)\( -18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101 \)\()/218385\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{9} + 5 \beta_{8} + \beta_{7} - 12 \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_{2} + 5 \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(-5 \beta_{9} - \beta_{8} - \beta_{7} - 7 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(6 \beta_{9} - 8 \beta_{8} - 14 \beta_{7} + 16 \beta_{5} + 9 \beta_{4} - 7 \beta_{3} + 22 \beta_{2} + 51\)
\(\nu^{7}\)\(=\)\(\beta_{9} - 31 \beta_{8} - 8 \beta_{7} + 31 \beta_{5} - 30 \beta_{4} + 8 \beta_{3} + \beta_{2} + 43 \beta_{1}\)
\(\nu^{8}\)\(=\)\(75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + 76 \beta_{3} + 8 \beta_{2} - 112 \beta_{1}\)
\(\nu^{9}\)\(=\)\(95 \beta_{9} + 189 \beta_{8} + 94 \beta_{7} + 37 \beta_{5} + 84 \beta_{4} + 47 \beta_{3} - 194 \beta_{2} - 3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
440.1
1.07065 + 1.85442i
0.187540 + 0.324828i
−0.539982 0.935277i
−1.04536 1.81062i
0.827154 + 1.43267i
1.07065 1.85442i
0.187540 0.324828i
−0.539982 + 0.935277i
−1.04536 + 1.81062i
0.827154 1.43267i
−2.24607 1.29677i 0 2.36322 + 4.09323i 0.626493 + 1.08512i 0 0 7.07116i 0 3.24967i
440.2 −0.621951 0.359083i 0 −0.742118 1.28539i −0.723774 1.25361i 0 0 2.50226i 0 1.03958i
440.3 −0.254498 0.146935i 0 −0.956820 1.65726i 1.53014 + 2.65027i 0 0 1.15010i 0 0.899320i
440.4 1.30778 + 0.755047i 0 0.140193 + 0.242822i −0.387938 0.671929i 0 0 2.59678i 0 1.17165i
440.5 1.81474 + 1.04774i 0 1.19552 + 2.07070i −1.04492 1.80985i 0 0 0.819421i 0 4.37920i
881.1 −2.24607 + 1.29677i 0 2.36322 4.09323i 0.626493 1.08512i 0 0 7.07116i 0 3.24967i
881.2 −0.621951 + 0.359083i 0 −0.742118 + 1.28539i −0.723774 + 1.25361i 0 0 2.50226i 0 1.03958i
881.3 −0.254498 + 0.146935i 0 −0.956820 + 1.65726i 1.53014 2.65027i 0 0 1.15010i 0 0.899320i
881.4 1.30778 0.755047i 0 0.140193 0.242822i −0.387938 + 0.671929i 0 0 2.59678i 0 1.17165i
881.5 1.81474 1.04774i 0 1.19552 2.07070i −1.04492 + 1.80985i 0 0 0.819421i 0 4.37920i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.o.d 10
3.b odd 2 1 441.2.o.c 10
7.b odd 2 1 1323.2.o.c 10
7.c even 3 1 189.2.s.b 10
7.c even 3 1 1323.2.i.b 10
7.d odd 6 1 189.2.i.b 10
7.d odd 6 1 1323.2.s.b 10
9.c even 3 1 441.2.o.d 10
9.d odd 6 1 1323.2.o.c 10
21.c even 2 1 441.2.o.d 10
21.g even 6 1 63.2.i.b 10
21.g even 6 1 441.2.s.b 10
21.h odd 6 1 63.2.s.b yes 10
21.h odd 6 1 441.2.i.b 10
28.f even 6 1 3024.2.ca.b 10
28.g odd 6 1 3024.2.df.b 10
63.g even 3 1 63.2.i.b 10
63.h even 3 1 441.2.s.b 10
63.h even 3 1 567.2.p.d 10
63.i even 6 1 189.2.s.b 10
63.j odd 6 1 567.2.p.c 10
63.j odd 6 1 1323.2.s.b 10
63.k odd 6 1 441.2.i.b 10
63.k odd 6 1 567.2.p.c 10
63.l odd 6 1 441.2.o.c 10
63.n odd 6 1 189.2.i.b 10
63.o even 6 1 inner 1323.2.o.d 10
63.s even 6 1 567.2.p.d 10
63.s even 6 1 1323.2.i.b 10
63.t odd 6 1 63.2.s.b yes 10
84.j odd 6 1 1008.2.ca.b 10
84.n even 6 1 1008.2.df.b 10
252.o even 6 1 3024.2.ca.b 10
252.r odd 6 1 3024.2.df.b 10
252.bj even 6 1 1008.2.df.b 10
252.bl odd 6 1 1008.2.ca.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b 10 21.g even 6 1
63.2.i.b 10 63.g even 3 1
63.2.s.b yes 10 21.h odd 6 1
63.2.s.b yes 10 63.t odd 6 1
189.2.i.b 10 7.d odd 6 1
189.2.i.b 10 63.n odd 6 1
189.2.s.b 10 7.c even 3 1
189.2.s.b 10 63.i even 6 1
441.2.i.b 10 21.h odd 6 1
441.2.i.b 10 63.k odd 6 1
441.2.o.c 10 3.b odd 2 1
441.2.o.c 10 63.l odd 6 1
441.2.o.d 10 9.c even 3 1
441.2.o.d 10 21.c even 2 1
441.2.s.b 10 21.g even 6 1
441.2.s.b 10 63.h even 3 1
567.2.p.c 10 63.j odd 6 1
567.2.p.c 10 63.k odd 6 1
567.2.p.d 10 63.h even 3 1
567.2.p.d 10 63.s even 6 1
1008.2.ca.b 10 84.j odd 6 1
1008.2.ca.b 10 252.bl odd 6 1
1008.2.df.b 10 84.n even 6 1
1008.2.df.b 10 252.bj even 6 1
1323.2.i.b 10 7.c even 3 1
1323.2.i.b 10 63.s even 6 1
1323.2.o.c 10 7.b odd 2 1
1323.2.o.c 10 9.d odd 6 1
1323.2.o.d 10 1.a even 1 1 trivial
1323.2.o.d 10 63.o even 6 1 inner
1323.2.s.b 10 7.d odd 6 1
1323.2.s.b 10 63.j odd 6 1
3024.2.ca.b 10 28.f even 6 1
3024.2.ca.b 10 252.o even 6 1
3024.2.df.b 10 28.g odd 6 1
3024.2.df.b 10 252.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + 21 T + 52 T^{2} + 21 T^{3} - 46 T^{4} - 24 T^{5} + 42 T^{6} - 7 T^{8} + T^{10} \)
$3$ \( T^{10} \)
$5$ \( 81 + 108 T + 198 T^{2} + 90 T^{3} + 144 T^{4} + 63 T^{5} + 69 T^{6} + 12 T^{7} + 9 T^{8} + T^{10} \)
$7$ \( T^{10} \)
$11$ \( 2883 + 11904 T + 20011 T^{2} + 14976 T^{3} + 4679 T^{4} - 141 T^{5} - 336 T^{6} + 24 T^{7} + 50 T^{8} + 12 T^{9} + T^{10} \)
$13$ \( 3267 - 297 T - 2070 T^{2} + 189 T^{3} + 1098 T^{4} - 468 T^{5} - 48 T^{6} + 54 T^{7} + 3 T^{8} - 6 T^{9} + T^{10} \)
$17$ \( ( -513 - 504 T - 90 T^{2} + 33 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$19$ \( 2187 + 12717 T^{2} + 6264 T^{4} + 999 T^{6} + 57 T^{8} + T^{10} \)
$23$ \( 27 + 882 T + 10045 T^{2} + 14406 T^{3} + 6758 T^{4} - 726 T^{5} - 612 T^{6} + 75 T^{7} + 80 T^{8} + 15 T^{9} + T^{10} \)
$29$ \( 186003 - 1621737 T + 5085994 T^{2} - 3249987 T^{3} + 902708 T^{4} - 105537 T^{5} - 414 T^{6} + 1050 T^{7} + 5 T^{8} - 15 T^{9} + T^{10} \)
$31$ \( 16875 - 84375 T + 170325 T^{2} - 148500 T^{3} + 65322 T^{4} - 13293 T^{5} + 276 T^{6} + 297 T^{7} - 6 T^{8} - 9 T^{9} + T^{10} \)
$37$ \( ( 608 - 384 T - 392 T^{2} - 52 T^{3} + 6 T^{4} + T^{5} )^{2} \)
$41$ \( 40487769 - 9869013 T + 6127956 T^{2} - 238005 T^{3} + 424548 T^{4} - 31095 T^{5} + 11814 T^{6} - 360 T^{7} + 171 T^{8} - 9 T^{9} + T^{10} \)
$43$ \( 12243001 + 6707583 T + 4255723 T^{2} + 570524 T^{3} + 281512 T^{4} + 36083 T^{5} + 13714 T^{6} + 713 T^{7} + 136 T^{8} - 3 T^{9} + T^{10} \)
$47$ \( 321489 + 472878 T + 564579 T^{2} + 236880 T^{3} + 94392 T^{4} + 15444 T^{5} + 5820 T^{6} + 1047 T^{7} + 186 T^{8} + 15 T^{9} + T^{10} \)
$53$ \( 871563 + 814465 T^{2} + 126560 T^{4} + 6543 T^{6} + 137 T^{8} + T^{10} \)
$59$ \( 4100625 - 2490750 T + 1373175 T^{2} - 425070 T^{3} + 144531 T^{4} - 36459 T^{5} + 9528 T^{6} - 1650 T^{7} + 240 T^{8} - 18 T^{9} + T^{10} \)
$61$ \( 826875 - 897750 T + 197325 T^{2} + 138510 T^{3} - 32157 T^{4} - 23211 T^{5} + 10806 T^{6} - 1224 T^{7} - 54 T^{8} + 12 T^{9} + T^{10} \)
$67$ \( 361 - 1520 T + 5526 T^{2} - 5694 T^{3} + 6546 T^{4} + 819 T^{5} + 3189 T^{6} - 438 T^{7} + 153 T^{8} + 10 T^{9} + T^{10} \)
$71$ \( 46216875 + 17533075 T^{2} + 1560557 T^{4} + 40104 T^{6} + 359 T^{8} + T^{10} \)
$73$ \( 789507 + 992979 T^{2} + 202365 T^{4} + 12960 T^{6} + 219 T^{8} + T^{10} \)
$79$ \( 1067655625 - 584555750 T + 226863000 T^{2} - 48342930 T^{3} + 8053914 T^{4} - 799857 T^{5} + 76611 T^{6} - 4884 T^{7} + 441 T^{8} - 20 T^{9} + T^{10} \)
$83$ \( 340734681 + 31509513 T + 28498023 T^{2} - 4359474 T^{3} + 1551933 T^{4} - 144513 T^{5} + 25413 T^{6} - 1962 T^{7} + 279 T^{8} - 15 T^{9} + T^{10} \)
$89$ \( ( -5697 - 2448 T + 864 T^{2} + 87 T^{3} - 24 T^{4} + T^{5} )^{2} \)
$97$ \( 9687627 - 15202620 T + 8518455 T^{2} - 888300 T^{3} - 502227 T^{4} + 63981 T^{5} + 32406 T^{6} + 1116 T^{7} - 174 T^{8} - 6 T^{9} + T^{10} \)
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