Properties

Label 3456.2.i.l
Level $3456$
Weight $2$
Character orbit 3456.i
Analytic conductor $27.596$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3456.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5962989386\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{6} q^{5} + ( 1 - \beta_{7} - \beta_{10} ) q^{7} +O(q^{10})\) \( q -\beta_{6} q^{5} + ( 1 - \beta_{7} - \beta_{10} ) q^{7} + ( -1 + \beta_{7} + \beta_{8} ) q^{11} + ( -\beta_{1} - \beta_{3} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{13} + ( -1 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} + ( -\beta_{1} + \beta_{2} - \beta_{5} ) q^{19} + ( -\beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{23} + ( -3 + \beta_{2} + \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{25} + ( -\beta_{9} + \beta_{10} ) q^{29} + ( -\beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{11} ) q^{31} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -\beta_{1} - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{41} + ( -1 + \beta_{2} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{43} + ( 3 - 2 \beta_{2} - 2 \beta_{6} - 3 \beta_{7} + \beta_{10} ) q^{47} + ( -\beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{49} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{53} + ( -1 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{55} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{59} + ( 2 - 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{61} + ( 2 - 4 \beta_{2} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{65} + ( 2 \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{11} ) q^{67} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{71} + ( 5 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{77} + ( 3 - \beta_{2} - \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{79} + ( -3 + 3 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{83} + ( \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{85} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{91} + ( -2 \beta_{1} - 2 \beta_{3} + 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{95} + ( -1 - 3 \beta_{2} - 3 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{5} + 6q^{7} + O(q^{10}) \) \( 12q + 2q^{5} + 6q^{7} - 4q^{11} + 10q^{13} - 4q^{17} + 4q^{19} - 8q^{23} - 14q^{25} + 2q^{29} + 8q^{31} - 8q^{35} + 2q^{41} - 2q^{43} + 14q^{47} - 18q^{49} - 24q^{53} - 16q^{55} - 6q^{59} + 14q^{61} + 8q^{65} + 4q^{67} + 28q^{71} + 60q^{73} - 2q^{77} + 16q^{79} - 24q^{83} + 16q^{85} + 48q^{89} - 52q^{91} + 20q^{95} - 14q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{11} + \nu^{10} - 3 \nu^{9} - 26 \nu^{8} + 25 \nu^{7} - 3 \nu^{6} + 141 \nu^{5} - 270 \nu^{4} + 117 \nu^{3} - 27 \nu^{2} + 1701 \nu - 972 \)\()/486\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{11} - 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} + 13 \nu^{7} - 24 \nu^{6} + 51 \nu^{5} - 108 \nu^{4} + 81 \nu^{3} - 54 \nu^{2} + 135 \nu - 162 \)\()/162\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{11} - \nu^{10} + 20 \nu^{8} - 34 \nu^{7} + 63 \nu^{6} - 240 \nu^{5} + 252 \nu^{4} - 630 \nu^{3} + 567 \nu^{2} - 1620 \nu + 1458 \)\()/486\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{11} + 5 \nu^{10} - 3 \nu^{9} + 56 \nu^{8} - 109 \nu^{7} + 123 \nu^{6} - 195 \nu^{5} + 720 \nu^{4} - 549 \nu^{3} + 621 \nu^{2} - 1377 \nu + 2430 \)\()/486\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{11} - 2 \nu^{10} - 2 \nu^{8} + 4 \nu^{7} - 15 \nu^{5} + 18 \nu^{4} + 9 \nu^{3} + 162 \nu^{2} - 216 \nu + 81 \)\()/81\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{11} - 2 \nu^{10} - 12 \nu^{9} + 16 \nu^{8} - 131 \nu^{7} + 42 \nu^{6} - 120 \nu^{5} + 810 \nu^{4} - 315 \nu^{3} + 540 \nu^{2} + 5346 \)\()/486\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{11} + 4 \nu^{10} - 33 \nu^{9} + 52 \nu^{8} - 179 \nu^{7} + 192 \nu^{6} - 327 \nu^{5} + 954 \nu^{4} - 693 \nu^{3} + 1404 \nu^{2} - 567 \nu + 4374 \)\()/486\)
\(\beta_{8}\)\(=\)\((\)\( -17 \nu^{11} + 13 \nu^{10} - 45 \nu^{9} + 64 \nu^{8} - 287 \nu^{7} + 243 \nu^{6} - 453 \nu^{5} + 1314 \nu^{4} - 1233 \nu^{3} + 2025 \nu^{2} - 81 \nu + 7290 \)\()/486\)
\(\beta_{9}\)\(=\)\((\)\( -31 \nu^{11} - \nu^{10} - 66 \nu^{9} + 95 \nu^{8} - 421 \nu^{7} + 357 \nu^{6} - 708 \nu^{5} + 2313 \nu^{4} - 1089 \nu^{3} + 3375 \nu^{2} - 1458 \nu + 12393 \)\()/486\)
\(\beta_{10}\)\(=\)\((\)\( 43 \nu^{11} - 17 \nu^{10} + 117 \nu^{9} - 173 \nu^{8} + 745 \nu^{7} - 729 \nu^{6} + 1311 \nu^{5} - 3735 \nu^{4} + 2925 \nu^{3} - 5589 \nu^{2} + 2025 \nu - 19197 \)\()/486\)
\(\beta_{11}\)\(=\)\((\)\( 64 \nu^{11} - 20 \nu^{10} + 165 \nu^{9} - 215 \nu^{8} + 1030 \nu^{7} - 960 \nu^{6} + 1617 \nu^{5} - 5229 \nu^{4} + 3816 \nu^{3} - 7506 \nu^{2} + 2349 \nu - 27945 \)\()/486\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} - 3 \beta_{10} - \beta_{8} - 4 \beta_{7} - \beta_{6} + \beta_{5} - 4 \beta_{3} - \beta_{2} - 2 \beta_{1} + 7\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{9} + \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} - 5 \beta_{2} + \beta_{1} - 7\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + \beta_{8} + 7 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta_{1} + 3\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{11} - 6 \beta_{10} - 20 \beta_{8} - 14 \beta_{7} - 8 \beta_{6} - 4 \beta_{5} + 6 \beta_{4} + \beta_{3} - 5 \beta_{2} + 11 \beta_{1} + 35\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-6 \beta_{11} + 30 \beta_{10} + 14 \beta_{9} + 9 \beta_{8} + 23 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} + 23 \beta_{3} - 34 \beta_{2} + 29 \beta_{1} + 46\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(30 \beta_{11} - 8 \beta_{10} - 2 \beta_{9} + 14 \beta_{8} + 113 \beta_{7} + 14 \beta_{6} - 5 \beta_{5} + 13 \beta_{4} - 6 \beta_{3} + 45 \beta_{2} - 5 \beta_{1} + 72\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(37 \beta_{11} - 69 \beta_{10} + 30 \beta_{9} - 13 \beta_{8} - 160 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 63 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} + 58 \beta_{1} + 1\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-48 \beta_{11} + 45 \beta_{10} - 5 \beta_{9} + 126 \beta_{8} - 257 \beta_{7} - 83 \beta_{6} + 63 \beta_{5} + 37 \beta_{4} - 95 \beta_{3} - 167 \beta_{2} - 44 \beta_{1} + 263\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(105 \beta_{11} - 277 \beta_{10} - 256 \beta_{9} - 56 \beta_{8} + 121 \beta_{7} + 139 \beta_{6} + 152 \beta_{5} - 25 \beta_{4} - 267 \beta_{3} + 81 \beta_{2} - 175 \beta_{1} + 351\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(2431\) \(2945\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
−0.433633 + 1.67689i
1.73202 0.0102491i
1.19051 1.25805i
−1.28252 + 1.16410i
0.952418 + 1.44669i
−1.15879 1.28733i
−0.433633 1.67689i
1.73202 + 0.0102491i
1.19051 + 1.25805i
−1.28252 1.16410i
0.952418 1.44669i
−1.15879 + 1.28733i
0 0 0 −2.22043 3.84590i 0 1.45488 2.51992i 0 0 0
1153.2 0 0 0 −0.551563 0.955334i 0 −1.62490 + 2.81442i 0 0 0
1153.3 0 0 0 −0.268104 0.464369i 0 2.35014 4.07056i 0 0 0
1153.4 0 0 0 1.05471 + 1.82681i 0 −1.43914 + 2.49267i 0 0 0
1153.5 0 0 0 1.24278 + 2.15256i 0 0.909142 1.57468i 0 0 0
1153.6 0 0 0 1.74260 + 3.01828i 0 1.34988 2.33807i 0 0 0
2305.1 0 0 0 −2.22043 + 3.84590i 0 1.45488 + 2.51992i 0 0 0
2305.2 0 0 0 −0.551563 + 0.955334i 0 −1.62490 2.81442i 0 0 0
2305.3 0 0 0 −0.268104 + 0.464369i 0 2.35014 + 4.07056i 0 0 0
2305.4 0 0 0 1.05471 1.82681i 0 −1.43914 2.49267i 0 0 0
2305.5 0 0 0 1.24278 2.15256i 0 0.909142 + 1.57468i 0 0 0
2305.6 0 0 0 1.74260 3.01828i 0 1.34988 + 2.33807i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2305.6
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 3456.2.i.l 12
3.b odd 2 1 1152.2.i.k yes 12
4.b odd 2 1 3456.2.i.k 12
8.b even 2 1 3456.2.i.j 12
8.d odd 2 1 3456.2.i.i 12
9.c even 3 1 inner 3456.2.i.l 12
9.d odd 6 1 1152.2.i.k yes 12
12.b even 2 1 1152.2.i.i 12
24.f even 2 1 1152.2.i.l yes 12
24.h odd 2 1 1152.2.i.j yes 12
36.f odd 6 1 3456.2.i.k 12
36.h even 6 1 1152.2.i.i 12
72.j odd 6 1 1152.2.i.j yes 12
72.l even 6 1 1152.2.i.l yes 12
72.n even 6 1 3456.2.i.j 12
72.p odd 6 1 3456.2.i.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.i 12 12.b even 2 1
1152.2.i.i 12 36.h even 6 1
1152.2.i.j yes 12 24.h odd 2 1
1152.2.i.j yes 12 72.j odd 6 1
1152.2.i.k yes 12 3.b odd 2 1
1152.2.i.k yes 12 9.d odd 6 1
1152.2.i.l yes 12 24.f even 2 1
1152.2.i.l yes 12 72.l even 6 1
3456.2.i.i 12 8.d odd 2 1
3456.2.i.i 12 72.p odd 6 1
3456.2.i.j 12 8.b even 2 1
3456.2.i.j 12 72.n even 6 1
3456.2.i.k 12 4.b odd 2 1
3456.2.i.k 12 36.f odd 6 1
3456.2.i.l 12 1.a even 1 1 trivial
3456.2.i.l 12 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3456, [\chi])\):

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( 2304 + 4224 T + 9424 T^{2} + 1720 T^{3} + 4665 T^{4} - 1674 T^{5} + 2928 T^{6} - 948 T^{7} + 465 T^{8} - 60 T^{9} + 24 T^{10} - 2 T^{11} + T^{12} \)
$7$ \( 394384 - 324048 T + 294516 T^{2} - 117452 T^{3} + 67353 T^{4} - 21192 T^{5} + 10164 T^{6} - 2400 T^{7} + 861 T^{8} - 152 T^{9} + 48 T^{10} - 6 T^{11} + T^{12} \)
$11$ \( 229441 + 288358 T + 330311 T^{2} + 161042 T^{3} + 95190 T^{4} + 30798 T^{5} + 16503 T^{6} + 3972 T^{7} + 1398 T^{8} + 128 T^{9} + 47 T^{10} + 4 T^{11} + T^{12} \)
$13$ \( 6533136 - 1216656 T + 2437516 T^{2} - 1295668 T^{3} + 927657 T^{4} - 322086 T^{5} + 104988 T^{6} - 20448 T^{7} + 4269 T^{8} - 588 T^{9} + 108 T^{10} - 10 T^{11} + T^{12} \)
$17$ \( ( 1812 + 3652 T + 1424 T^{2} - 176 T^{3} - 83 T^{4} + 2 T^{5} + T^{6} )^{2} \)
$19$ \( ( -3408 - 448 T + 1160 T^{2} + 80 T^{3} - 65 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$23$ \( 204304 + 2778896 T + 37480148 T^{2} + 4772236 T^{3} + 3584841 T^{4} + 440466 T^{5} + 244716 T^{6} + 26772 T^{7} + 7377 T^{8} + 484 T^{9} + 128 T^{10} + 8 T^{11} + T^{12} \)
$29$ \( 229704336 + 4183056 T + 32858604 T^{2} + 1767348 T^{3} + 3426993 T^{4} + 152658 T^{5} + 156912 T^{6} + 1824 T^{7} + 5049 T^{8} + 12 T^{9} + 88 T^{10} - 2 T^{11} + T^{12} \)
$31$ \( 1021953024 - 495631872 T + 228769632 T^{2} - 52045488 T^{3} + 13689369 T^{4} - 2224782 T^{5} + 493116 T^{6} - 61968 T^{7} + 10629 T^{8} - 876 T^{9} + 136 T^{10} - 8 T^{11} + T^{12} \)
$37$ \( ( -128 - 1728 T + 876 T^{2} + 68 T^{3} - 60 T^{4} + T^{6} )^{2} \)
$41$ \( 2259009 + 2065122 T + 2667933 T^{2} + 80478 T^{3} + 510354 T^{4} - 82566 T^{5} + 105981 T^{6} - 20682 T^{7} + 6570 T^{8} - 366 T^{9} + 85 T^{10} - 2 T^{11} + T^{12} \)
$43$ \( 16621929 - 27837756 T + 56418615 T^{2} + 14401800 T^{3} + 7050474 T^{4} + 752652 T^{5} + 276603 T^{6} + 21570 T^{7} + 7890 T^{8} + 294 T^{9} + 103 T^{10} + 2 T^{11} + T^{12} \)
$47$ \( 2178576 - 5957136 T + 16361620 T^{2} - 1703324 T^{3} + 2631105 T^{4} - 213660 T^{5} + 360204 T^{6} - 18288 T^{7} + 9465 T^{8} - 1008 T^{9} + 216 T^{10} - 14 T^{11} + T^{12} \)
$53$ \( ( 1728 + 1440 T - 852 T^{2} - 516 T^{3} - 24 T^{4} + 12 T^{5} + T^{6} )^{2} \)
$59$ \( 4100737369 - 617828976 T + 445991811 T^{2} - 19319756 T^{3} + 27186894 T^{4} - 898488 T^{5} + 878379 T^{6} + 19926 T^{7} + 16110 T^{8} + 322 T^{9} + 171 T^{10} + 6 T^{11} + T^{12} \)
$61$ \( 60715264 + 44071552 T + 32605904 T^{2} + 7501016 T^{3} + 2859633 T^{4} + 104130 T^{5} + 165648 T^{6} - 5484 T^{7} + 7077 T^{8} - 964 T^{9} + 200 T^{10} - 14 T^{11} + T^{12} \)
$67$ \( 3020711521 - 277553050 T + 682121567 T^{2} + 149369170 T^{3} + 125064942 T^{4} + 12391614 T^{5} + 3517287 T^{6} - 99444 T^{7} + 52302 T^{8} - 632 T^{9} + 263 T^{10} - 4 T^{11} + T^{12} \)
$71$ \( ( 1728 - 3744 T - 4716 T^{2} + 1608 T^{3} - 72 T^{4} - 14 T^{5} + T^{6} )^{2} \)
$73$ \( ( 39892 + 6612 T - 7884 T^{2} + 488 T^{3} + 225 T^{4} - 30 T^{5} + T^{6} )^{2} \)
$79$ \( 674337024 - 939626112 T + 891690448 T^{2} - 449230360 T^{3} + 164418801 T^{4} - 35734362 T^{5} + 5956416 T^{6} - 652080 T^{7} + 61569 T^{8} - 4020 T^{9} + 324 T^{10} - 16 T^{11} + T^{12} \)
$83$ \( 734843664 + 354247344 T + 245346732 T^{2} + 40494492 T^{3} + 24855345 T^{4} + 3431790 T^{5} + 1844226 T^{6} + 85896 T^{7} + 38355 T^{8} + 3828 T^{9} + 534 T^{10} + 24 T^{11} + T^{12} \)
$89$ \( ( -2864 + 7776 T - 3324 T^{2} + 68 T^{3} + 156 T^{4} - 24 T^{5} + T^{6} )^{2} \)
$97$ \( 78140934369 + 46797289170 T + 25477010197 T^{2} + 3355901918 T^{3} + 696053802 T^{4} + 52089210 T^{5} + 11046045 T^{6} + 674454 T^{7} + 90978 T^{8} + 3282 T^{9} + 429 T^{10} + 14 T^{11} + T^{12} \)
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