Properties

Label 1152.2.i.g
Level $1152$
Weight $2$
Character orbit 1152.i
Analytic conductor $9.199$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8528759163648.1
Defining polynomial: \(x^{10} - 2 x^{9} + x^{8} + 9 x^{6} - 36 x^{5} + 27 x^{4} + 27 x^{2} - 162 x + 243\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{5} + \beta_{7} ) q^{3} + \beta_{8} q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{7} + ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( \beta_{5} + \beta_{7} ) q^{3} + \beta_{8} q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{7} + ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{9} + ( -\beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} ) q^{11} + ( -\beta_{2} + \beta_{4} - \beta_{5} ) q^{13} + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{15} + ( -1 + \beta_{1} + \beta_{3} - \beta_{7} + \beta_{9} ) q^{17} + ( 2 - \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{9} ) q^{19} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} ) q^{21} + ( \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{23} + ( -\beta_{1} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{25} + ( -2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{27} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{29} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{31} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{9} ) q^{33} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} ) q^{35} + ( 3 - 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{9} ) q^{37} + ( -3 \beta_{2} - \beta_{5} - \beta_{8} - \beta_{9} ) q^{39} + ( 4 \beta_{1} - 2 \beta_{2} - \beta_{4} - 3 \beta_{5} - 4 \beta_{7} - \beta_{8} ) q^{41} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{43} + ( 3 \beta_{2} - 3 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{45} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{47} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{49} + ( 2 \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{8} ) q^{51} + ( -1 + 3 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{53} + ( 1 - \beta_{1} + \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{57} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} ) q^{59} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{61} + ( -3 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{8} - \beta_{9} ) q^{63} + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 3 \beta_{8} + \beta_{9} ) q^{65} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{67} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{69} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{7} - \beta_{9} ) q^{71} + ( -3 - \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{7} + \beta_{9} ) q^{73} + ( -4 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{75} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} - 2 \beta_{7} - 3 \beta_{8} ) q^{77} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{79} + ( 3 + 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{81} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{83} + ( -2 \beta_{1} - 3 \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{7} - \beta_{8} ) q^{85} + ( 1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{87} + ( 3 + 5 \beta_{1} + \beta_{3} - 3 \beta_{4} + 5 \beta_{5} + 3 \beta_{7} ) q^{89} + ( 4 + \beta_{1} - 4 \beta_{3} - 4 \beta_{4} + \beta_{5} + 4 \beta_{7} ) q^{91} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} ) q^{95} + ( 4 - \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 3 \beta_{7} - \beta_{8} ) q^{97} + ( 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + q^{3} - 4q^{7} - q^{9} + O(q^{10}) \) \( 10q + q^{3} - 4q^{7} - q^{9} - q^{11} - 6q^{13} + 12q^{15} - 6q^{17} + 18q^{19} - 16q^{21} + 4q^{23} + q^{25} - 2q^{27} + 4q^{29} - 8q^{31} - 13q^{33} - 24q^{35} + 20q^{37} - 18q^{39} - 5q^{41} - 13q^{43} + 12q^{45} - 6q^{47} + 3q^{49} + 3q^{51} + 12q^{55} + 27q^{57} - 13q^{59} - 10q^{61} - 20q^{63} - 17q^{67} + 10q^{69} + 8q^{71} - 34q^{73} - 29q^{75} - 8q^{77} - 6q^{79} - q^{81} + 12q^{83} - 18q^{85} + 10q^{87} + 44q^{89} + 36q^{91} - 26q^{93} - 6q^{95} + 27q^{97} + 34q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + x^{8} + 9 x^{6} - 36 x^{5} + 27 x^{4} + 27 x^{2} - 162 x + 243\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} + \nu^{8} + 4 \nu^{7} + 12 \nu^{6} + 45 \nu^{5} - 63 \nu^{4} + 27 \nu - 567 \)\()/486\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{9} - \nu^{8} - 4 \nu^{7} + 15 \nu^{6} - 18 \nu^{5} + 9 \nu^{4} + 81 \nu^{3} + 162 \nu^{2} - 270 \nu + 81 \)\()/243\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{9} - \nu^{8} - 4 \nu^{7} - 12 \nu^{6} + 9 \nu^{5} + 9 \nu^{4} + 297 \nu + 81 \)\()/162\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{9} + 2 \nu^{8} - \nu^{7} - 9 \nu^{5} + 36 \nu^{4} - 27 \nu^{3} - 27 \nu + 162 \)\()/81\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{9} + 7 \nu^{8} + 28 \nu^{7} - 24 \nu^{6} + 45 \nu^{5} - 63 \nu^{4} + 162 \nu^{3} - 648 \nu^{2} - 297 \nu - 567 \)\()/486\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{9} + 11 \nu^{8} - 10 \nu^{7} - 12 \nu^{6} - 99 \nu^{5} + 117 \nu^{4} - 189 \nu + 1053 \)\()/486\)
\(\beta_{8}\)\(=\)\((\)\( -4 \nu^{9} - 4 \nu^{8} + 11 \nu^{7} + 6 \nu^{6} - 45 \nu^{5} + 36 \nu^{4} + 81 \nu^{3} - 81 \nu^{2} - 108 \nu + 324 \)\()/243\)
\(\beta_{9}\)\(=\)\((\)\( -17 \nu^{9} + 19 \nu^{8} + 22 \nu^{7} - 60 \nu^{6} - 171 \nu^{5} + 369 \nu^{4} + 162 \nu^{3} - 324 \nu^{2} - 459 \nu + 2349 \)\()/486\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + \beta_{1} - 3\)
\(\nu^{5}\)\(=\)\(-3 \beta_{7} + \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 5 \beta_{1} + 6\)
\(\nu^{6}\)\(=\)\(-6 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} + 4 \beta_{6} + 9 \beta_{5} + \beta_{4} + 5 \beta_{3} + 6 \beta_{2} + 7 \beta_{1} + 6\)
\(\nu^{7}\)\(=\)\(15 \beta_{9} - 6 \beta_{8} - 18 \beta_{7} - 11 \beta_{6} - 12 \beta_{5} - 20 \beta_{4} - 10 \beta_{3} + 6 \beta_{2} + 19 \beta_{1} + 6\)
\(\nu^{8}\)\(=\)\(9 \beta_{9} - 18 \beta_{8} + 15 \beta_{7} + \beta_{6} - 6 \beta_{5} - 14 \beta_{4} + 2 \beta_{3} + 33 \beta_{2} + 31 \beta_{1} + 6\)
\(\nu^{9}\)\(=\)\(3 \beta_{9} - 30 \beta_{8} - 33 \beta_{7} + 13 \beta_{6} - 35 \beta_{4} + 14 \beta_{3} - 102 \beta_{2} + 70 \beta_{1} + 6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
−1.41743 + 0.995434i
0.756905 + 1.55791i
−1.13593 1.30754i
1.72806 + 0.117480i
1.06839 1.36328i
−1.41743 0.995434i
0.756905 1.55791i
−1.13593 + 1.30754i
1.72806 0.117480i
1.06839 + 1.36328i
0 −1.57079 + 0.729814i 0 −0.115851 0.200661i 0 −0.230793 + 0.399745i 0 1.93474 2.29277i 0
385.2 0 −0.970741 1.43446i 0 1.07447 + 1.86104i 0 0.153174 0.265305i 0 −1.11533 + 2.78497i 0
385.3 0 0.564403 + 1.63751i 0 −1.59327 2.75962i 0 −0.607060 + 1.05146i 0 −2.36290 + 1.84844i 0
385.4 0 0.762291 1.55529i 0 −0.705463 1.22190i 0 1.17123 2.02864i 0 −1.83783 2.37116i 0
385.5 0 1.71483 0.243611i 0 1.34011 + 2.32114i 0 −2.48656 + 4.30684i 0 2.88131 0.835506i 0
769.1 0 −1.57079 0.729814i 0 −0.115851 + 0.200661i 0 −0.230793 0.399745i 0 1.93474 + 2.29277i 0
769.2 0 −0.970741 + 1.43446i 0 1.07447 1.86104i 0 0.153174 + 0.265305i 0 −1.11533 2.78497i 0
769.3 0 0.564403 1.63751i 0 −1.59327 + 2.75962i 0 −0.607060 1.05146i 0 −2.36290 1.84844i 0
769.4 0 0.762291 + 1.55529i 0 −0.705463 + 1.22190i 0 1.17123 + 2.02864i 0 −1.83783 + 2.37116i 0
769.5 0 1.71483 + 0.243611i 0 1.34011 2.32114i 0 −2.48656 4.30684i 0 2.88131 + 0.835506i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.i.g yes 10
3.b odd 2 1 3456.2.i.f 10
4.b odd 2 1 1152.2.i.f yes 10
8.b even 2 1 1152.2.i.e 10
8.d odd 2 1 1152.2.i.h yes 10
9.c even 3 1 inner 1152.2.i.g yes 10
9.d odd 6 1 3456.2.i.f 10
12.b even 2 1 3456.2.i.g 10
24.f even 2 1 3456.2.i.h 10
24.h odd 2 1 3456.2.i.e 10
36.f odd 6 1 1152.2.i.f yes 10
36.h even 6 1 3456.2.i.g 10
72.j odd 6 1 3456.2.i.e 10
72.l even 6 1 3456.2.i.h 10
72.n even 6 1 1152.2.i.e 10
72.p odd 6 1 1152.2.i.h yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.e 10 8.b even 2 1
1152.2.i.e 10 72.n even 6 1
1152.2.i.f yes 10 4.b odd 2 1
1152.2.i.f yes 10 36.f odd 6 1
1152.2.i.g yes 10 1.a even 1 1 trivial
1152.2.i.g yes 10 9.c even 3 1 inner
1152.2.i.h yes 10 8.d odd 2 1
1152.2.i.h yes 10 72.p odd 6 1
3456.2.i.e 10 24.h odd 2 1
3456.2.i.e 10 72.j odd 6 1
3456.2.i.f 10 3.b odd 2 1
3456.2.i.f 10 9.d odd 6 1
3456.2.i.g 10 12.b even 2 1
3456.2.i.g 10 36.h even 6 1
3456.2.i.h 10 24.f even 2 1
3456.2.i.h 10 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( 243 - 81 T + 27 T^{2} - 18 T^{5} + T^{8} - T^{9} + T^{10} \)
$5$ \( 36 + 162 T + 717 T^{2} + 198 T^{3} + 328 T^{4} - 18 T^{5} + 117 T^{6} - 4 T^{7} + 12 T^{8} + T^{10} \)
$7$ \( 4 - 2 T + 33 T^{2} + 48 T^{3} + 240 T^{4} + 138 T^{5} + 129 T^{6} + 24 T^{8} + 4 T^{9} + T^{10} \)
$11$ \( 961 - 341 T + 1671 T^{2} + 1790 T^{3} + 2249 T^{4} + 1053 T^{5} + 461 T^{6} + 80 T^{7} + 21 T^{8} + T^{9} + T^{10} \)
$13$ \( 2304 - 1008 T + 2265 T^{2} + 798 T^{3} + 1156 T^{4} + 300 T^{5} + 249 T^{6} + 76 T^{7} + 36 T^{8} + 6 T^{9} + T^{10} \)
$17$ \( ( 108 - 80 T^{2} - 24 T^{3} + 3 T^{4} + T^{5} )^{2} \)
$19$ \( ( -144 - 48 T + 104 T^{2} - 9 T^{4} + T^{5} )^{2} \)
$23$ \( 1024 + 1376 T + 2553 T^{2} + 1102 T^{3} + 1988 T^{4} + 1080 T^{5} + 893 T^{6} + 172 T^{7} + 48 T^{8} - 4 T^{9} + T^{10} \)
$29$ \( 34596 - 241614 T + 1706373 T^{2} + 102738 T^{3} + 113580 T^{4} + 2046 T^{5} + 5509 T^{6} + 116 T^{7} + 96 T^{8} - 4 T^{9} + T^{10} \)
$31$ \( 22810176 + 6146712 T + 3690945 T^{2} + 101274 T^{3} + 230784 T^{4} + 13152 T^{5} + 6745 T^{6} + 308 T^{7} + 132 T^{8} + 8 T^{9} + T^{10} \)
$37$ \( ( -344 - 1228 T + 796 T^{2} - 68 T^{3} - 10 T^{4} + T^{5} )^{2} \)
$41$ \( 319587129 + 86614065 T + 33449391 T^{2} + 2516574 T^{3} + 929349 T^{4} + 50895 T^{5} + 19261 T^{6} + 386 T^{7} + 171 T^{8} + 5 T^{9} + T^{10} \)
$43$ \( 720801 - 2361069 T + 8620317 T^{2} + 2988264 T^{3} + 939849 T^{4} + 125355 T^{5} + 18853 T^{6} + 1438 T^{7} + 219 T^{8} + 13 T^{9} + T^{10} \)
$47$ \( 12194064 - 7217964 T + 3839481 T^{2} - 842964 T^{3} + 209956 T^{4} - 17880 T^{5} + 5733 T^{6} - 256 T^{7} + 120 T^{8} + 6 T^{9} + T^{10} \)
$53$ \( ( -9648 + 4716 T + 132 T^{2} - 144 T^{3} + T^{5} )^{2} \)
$59$ \( 2802325969 + 104127079 T + 108260853 T^{2} + 10308192 T^{3} + 3464181 T^{4} + 266043 T^{5} + 41625 T^{6} + 2202 T^{7} + 303 T^{8} + 13 T^{9} + T^{10} \)
$61$ \( 2249415184 + 301404940 T + 111622881 T^{2} + 7149446 T^{3} + 2900204 T^{4} + 184680 T^{5} + 39641 T^{6} + 1244 T^{7} + 276 T^{8} + 10 T^{9} + T^{10} \)
$67$ \( 63001 - 177457 T + 586695 T^{2} + 224542 T^{3} + 143729 T^{4} + 10449 T^{5} + 8189 T^{6} + 1372 T^{7} + 249 T^{8} + 17 T^{9} + T^{10} \)
$71$ \( ( -14472 + 1620 T + 1080 T^{2} - 164 T^{3} - 4 T^{4} + T^{5} )^{2} \)
$73$ \( ( -2972 - 1696 T - 140 T^{2} + 76 T^{3} + 17 T^{4} + T^{5} )^{2} \)
$79$ \( 6986619396 + 813709710 T + 290528637 T^{2} + 25346166 T^{3} + 7787128 T^{4} + 641262 T^{5} + 87261 T^{6} + 2956 T^{7} + 324 T^{8} + 6 T^{9} + T^{10} \)
$83$ \( 8088336 + 3045924 T + 2802249 T^{2} - 725706 T^{3} + 285318 T^{4} - 39024 T^{5} + 8379 T^{6} - 948 T^{7} + 162 T^{8} - 12 T^{9} + T^{10} \)
$89$ \( ( -27464 - 7396 T + 3388 T^{2} - 80 T^{3} - 22 T^{4} + T^{5} )^{2} \)
$97$ \( 10710387081 + 1172656521 T + 487091367 T^{2} - 33063786 T^{3} + 9558829 T^{4} - 611385 T^{5} + 105813 T^{6} - 7742 T^{7} + 699 T^{8} - 27 T^{9} + T^{10} \)
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