Properties

Label 2268.2.i.n
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 9 x^{14} + 31 x^{12} - 282 x^{10} + 1695 x^{8} - 3318 x^{6} + 4606 x^{4} - 4116 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{7} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{13} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q -\beta_{13} q^{5} + \beta_{3} q^{7} -\beta_{12} q^{11} + ( 2 - 2 \beta_{4} - \beta_{8} + \beta_{9} ) q^{13} + ( -\beta_{6} - \beta_{11} - \beta_{13} ) q^{17} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{19} + \beta_{11} q^{23} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{25} + ( 2 \beta_{5} + \beta_{6} + 2 \beta_{11} + 2 \beta_{13} ) q^{29} + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} ) q^{31} + ( \beta_{5} + \beta_{6} + 2 \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{35} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{37} + ( \beta_{6} + \beta_{11} - \beta_{12} + 3 \beta_{14} - \beta_{15} ) q^{41} + ( 1 - \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{43} + ( -\beta_{5} - \beta_{6} + 3 \beta_{10} - \beta_{12} - \beta_{14} + 2 \beta_{15} ) q^{47} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{49} + ( -\beta_{6} + 3 \beta_{13} ) q^{53} + ( -2 - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} ) q^{55} + ( -2 \beta_{5} - \beta_{6} + \beta_{10} - 2 \beta_{12} - \beta_{14} ) q^{59} + ( -3 + \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{7} + 2 \beta_{9} ) q^{61} + ( -3 \beta_{6} + 4 \beta_{10} - 3 \beta_{14} + \beta_{15} ) q^{65} + ( -1 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{9} ) q^{67} + ( \beta_{5} - 2 \beta_{6} + \beta_{12} - 2 \beta_{14} + 3 \beta_{15} ) q^{71} + ( -2 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{7} - \beta_{8} ) q^{73} + ( -\beta_{5} + 5 \beta_{6} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{77} + ( 5 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{9} ) q^{79} + ( 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{11} - 2 \beta_{13} ) q^{83} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{85} + ( -2 \beta_{6} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{89} + ( -2 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{8} - 3 \beta_{9} ) q^{91} + ( -2 \beta_{5} - 4 \beta_{6} + 5 \beta_{10} - 2 \beta_{12} - 4 \beta_{14} - \beta_{15} ) q^{95} + ( 3 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7} + O(q^{10}) \) \( 16 q - 6 q^{7} + 10 q^{13} + 8 q^{19} + 16 q^{31} - 4 q^{37} - 10 q^{43} + 10 q^{49} - 32 q^{55} - 56 q^{61} - 36 q^{67} + 40 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 9 x^{14} + 31 x^{12} - 282 x^{10} + 1695 x^{8} - 3318 x^{6} + 4606 x^{4} - 4116 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 92249 \nu^{14} - 22952 \nu^{12} - 2800495 \nu^{10} - 21569239 \nu^{8} - 19347710 \nu^{6} + 586034029 \nu^{4} + 1081706164 \nu^{2} + 170475459 \)\()/ 982062165 \)
\(\beta_{2}\)\(=\)\((\)\( 244351 \nu^{14} - 3103286 \nu^{12} + 15156266 \nu^{10} - 92194120 \nu^{8} + 665006344 \nu^{6} - 2126767580 \nu^{4} + 3126524037 \nu^{2} - 4446117606 \)\()/ 982062165 \)
\(\beta_{3}\)\(=\)\((\)\( 50804 \nu^{14} - 431490 \nu^{12} + 910001 \nu^{10} - 11273818 \nu^{8} + 75102634 \nu^{6} - 34785107 \nu^{4} - 145125260 \nu^{2} - 188652058 \)\()/ 140294595 \)
\(\beta_{4}\)\(=\)\((\)\( 95779 \nu^{14} - 763661 \nu^{12} + 2336349 \nu^{10} - 25351497 \nu^{8} + 138865326 \nu^{6} - 211337511 \nu^{4} + 367714228 \nu^{2} - 129671150 \)\()/ 196412433 \)
\(\beta_{5}\)\(=\)\((\)\( -784170 \nu^{15} + 214456 \nu^{13} + 30604253 \nu^{11} + 93415788 \nu^{9} + 434366602 \nu^{7} - 7116987318 \nu^{5} + 3388669823 \nu^{3} - 6997483661 \nu \)\()/ 6874435155 \)
\(\beta_{6}\)\(=\)\((\)\( 202439 \nu^{15} - 3779851 \nu^{13} + 20969988 \nu^{11} - 90097992 \nu^{9} + 810563151 \nu^{7} - 3169832022 \nu^{5} + 2394515438 \nu^{3} - 2189789518 \nu \)\()/ 1374887031 \)
\(\beta_{7}\)\(=\)\((\)\( -1700901 \nu^{14} + 13906205 \nu^{12} - 39713699 \nu^{10} + 439930867 \nu^{8} - 2478376396 \nu^{6} + 3214460858 \nu^{4} - 4135709270 \nu^{2} + 401714397 \)\()/ 982062165 \)
\(\beta_{8}\)\(=\)\((\)\( -53324 \nu^{14} + 366096 \nu^{12} - 722548 \nu^{10} + 12291821 \nu^{8} - 61209692 \nu^{6} + 7818769 \nu^{4} - 19057962 \nu^{2} + 6024452 \)\()/22838655\)
\(\beta_{9}\)\(=\)\((\)\( 160457 \nu^{14} - 1409701 \nu^{12} + 4498055 \nu^{10} - 43172212 \nu^{8} + 259237135 \nu^{6} - 437984288 \nu^{4} + 445839877 \nu^{2} - 305714528 \)\()/42698355\)
\(\beta_{10}\)\(=\)\((\)\( -635345 \nu^{15} + 5091886 \nu^{13} - 13365107 \nu^{11} + 152655738 \nu^{9} - 878690623 \nu^{7} + 875750037 \nu^{5} + 504845978 \nu^{3} - 2003509256 \nu \)\()/ 982062165 \)
\(\beta_{11}\)\(=\)\((\)\( 751964 \nu^{15} - 2837787 \nu^{13} - 5745335 \nu^{11} - 140264304 \nu^{9} + 307950965 \nu^{7} + 2566998474 \nu^{5} - 571100796 \nu^{3} + 3200801814 \nu \)\()/ 982062165 \)
\(\beta_{12}\)\(=\)\((\)\( 7006527 \nu^{15} - 54909609 \nu^{13} + 140871131 \nu^{11} - 1753147101 \nu^{9} + 9658554559 \nu^{7} - 8917693029 \nu^{5} + 11250666413 \nu^{3} - 6552594825 \nu \)\()/ 6874435155 \)
\(\beta_{13}\)\(=\)\((\)\(7841464 \nu^{15} - 71750653 \nu^{13} + 246634027 \nu^{11} - 2192423802 \nu^{9} + 13373548133 \nu^{7} - 26134593933 \nu^{5} + 29297334416 \nu^{3} - 8080133320 \nu\)\()/ 6874435155 \)
\(\beta_{14}\)\(=\)\((\)\( 1635500 \nu^{15} - 12087451 \nu^{13} + 27574530 \nu^{11} - 390459270 \nu^{9} + 2074772181 \nu^{7} - 1221325980 \nu^{5} + 1133172530 \nu^{3} - 504509782 \nu \)\()/ 1374887031 \)
\(\beta_{15}\)\(=\)\((\)\(-12267456 \nu^{15} + 107332095 \nu^{13} - 343863724 \nu^{11} + 3308249367 \nu^{9} - 19848020141 \nu^{7} + 33437101488 \nu^{5} - 35352763285 \nu^{3} + 38302053342 \nu\)\()/ 6874435155 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} + \beta_{13} + \beta_{12} - \beta_{6} + 2 \beta_{5}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + \beta_{8} - 3 \beta_{7} + 7 \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{15} + 9 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 6 \beta_{10} + 9 \beta_{6} + \beta_{5}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{9} - 6 \beta_{8} - 13 \beta_{7} - 31 \beta_{4} - 8 \beta_{3} + 20 \beta_{2} - 10 \beta_{1} + 25\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(10 \beta_{15} - 16 \beta_{14} - 21 \beta_{13} + 44 \beta_{12} + 17 \beta_{11} + 2 \beta_{10} + 46 \beta_{6} + 22 \beta_{5}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(11 \beta_{9} - 22 \beta_{8} + 71 \beta_{7} + 10 \beta_{4} + 31 \beta_{3} + 49 \beta_{2} + 21 \beta_{1} + 322\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(70 \beta_{15} - 12 \beta_{14} + 23 \beta_{13} + 178 \beta_{12} + 14 \beta_{11} + 33 \beta_{10} - 117 \beta_{6} + 356 \beta_{5}\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-234 \beta_{9} + 117 \beta_{8} - 169 \beta_{7} + 1409 \beta_{4} + 538 \beta_{3} + 26 \beta_{2} - 304 \beta_{1} + 187\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(804 \beta_{15} + 1814 \beta_{14} - 25 \beta_{13} - 467 \beta_{12} + 221 \beta_{11} + 902 \beta_{10} + 1691 \beta_{6} + 467 \beta_{5}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-1148 \beta_{9} - 1148 \beta_{8} - 2191 \beta_{7} - 3861 \beta_{4} - 2165 \beta_{3} + 3234 \beta_{2} - 3182 \beta_{1} + 2713\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(1322 \beta_{15} - 2061 \beta_{14} - 5174 \beta_{13} + 5600 \beta_{12} + 4591 \beta_{11} + 156 \beta_{10} + 12972 \beta_{6} + 2800 \beta_{5}\)\()/3\)
\(\nu^{12}\)\(=\)\(1284 \beta_{9} - 2568 \beta_{8} + 5956 \beta_{7} - 1481 \beta_{4} - 1678 \beta_{3} + 3388 \beta_{2} - 197 \beta_{1} + 19573\)
\(\nu^{13}\)\(=\)\((\)\(3218 \beta_{15} - 21675 \beta_{14} - 5929 \beta_{13} + 34676 \beta_{12} + 13164 \beta_{11} - 4017 \beta_{10} - 12365 \beta_{6} + 69352 \beta_{5}\)\()/3\)
\(\nu^{14}\)\(=\)\((\)\(-24730 \beta_{9} + 12365 \beta_{8} + 31071 \beta_{7} + 328079 \beta_{4} + 80961 \beta_{3} - 21718 \beta_{2} - 56231 \beta_{1} + 43866\)\()/3\)
\(\nu^{15}\)\(=\)\((\)\(108952 \beta_{15} + 307191 \beta_{14} - 7798 \beta_{13} - 141578 \beta_{12} + 66890 \beta_{11} + 138498 \beta_{10} + 269847 \beta_{6} + 141578 \beta_{5}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.817131 0.735533i
−2.40332 + 0.123797i
1.04556 0.339889i
1.30887 2.01944i
−1.30887 + 2.01944i
−1.04556 + 0.339889i
2.40332 0.123797i
−0.817131 + 0.735533i
0.817131 + 0.735533i
−2.40332 0.123797i
1.04556 + 0.339889i
1.30887 + 2.01944i
−1.30887 2.01944i
−1.04556 0.339889i
2.40332 + 0.123797i
−0.817131 0.735533i
0 0 0 −1.83843 + 3.18426i 0 −1.55575 + 2.14001i 0 0 0
865.2 0 0 0 −1.15101 + 1.99360i 0 2.41508 + 1.08045i 0 0 0
865.3 0 0 0 −0.515559 + 0.892975i 0 −2.63118 + 0.277320i 0 0 0
865.4 0 0 0 −0.171869 + 0.297685i 0 0.271847 2.63175i 0 0 0
865.5 0 0 0 0.171869 0.297685i 0 0.271847 2.63175i 0 0 0
865.6 0 0 0 0.515559 0.892975i 0 −2.63118 + 0.277320i 0 0 0
865.7 0 0 0 1.15101 1.99360i 0 2.41508 + 1.08045i 0 0 0
865.8 0 0 0 1.83843 3.18426i 0 −1.55575 + 2.14001i 0 0 0
2053.1 0 0 0 −1.83843 3.18426i 0 −1.55575 2.14001i 0 0 0
2053.2 0 0 0 −1.15101 1.99360i 0 2.41508 1.08045i 0 0 0
2053.3 0 0 0 −0.515559 0.892975i 0 −2.63118 0.277320i 0 0 0
2053.4 0 0 0 −0.171869 0.297685i 0 0.271847 + 2.63175i 0 0 0
2053.5 0 0 0 0.171869 + 0.297685i 0 0.271847 + 2.63175i 0 0 0
2053.6 0 0 0 0.515559 + 0.892975i 0 −2.63118 0.277320i 0 0 0
2053.7 0 0 0 1.15101 + 1.99360i 0 2.41508 1.08045i 0 0 0
2053.8 0 0 0 1.83843 + 3.18426i 0 −1.55575 2.14001i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2053.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.h even 3 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.n 16
3.b odd 2 1 inner 2268.2.i.n 16
7.c even 3 1 2268.2.l.n 16
9.c even 3 1 2268.2.k.g 16
9.c even 3 1 2268.2.l.n 16
9.d odd 6 1 2268.2.k.g 16
9.d odd 6 1 2268.2.l.n 16
21.h odd 6 1 2268.2.l.n 16
63.g even 3 1 2268.2.k.g 16
63.h even 3 1 inner 2268.2.i.n 16
63.j odd 6 1 inner 2268.2.i.n 16
63.n odd 6 1 2268.2.k.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.n 16 1.a even 1 1 trivial
2268.2.i.n 16 3.b odd 2 1 inner
2268.2.i.n 16 63.h even 3 1 inner
2268.2.i.n 16 63.j odd 6 1 inner
2268.2.k.g 16 9.c even 3 1
2268.2.k.g 16 9.d odd 6 1
2268.2.k.g 16 63.g even 3 1
2268.2.k.g 16 63.n odd 6 1
2268.2.l.n 16 7.c even 3 1
2268.2.l.n 16 9.c even 3 1
2268.2.l.n 16 9.d odd 6 1
2268.2.l.n 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\(T_{5}^{16} + \cdots\)
\(T_{13}^{8} - \cdots\)
\(T_{19}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 81 + 783 T^{2} + 6723 T^{4} + 7818 T^{6} + 7087 T^{8} + 1706 T^{10} + 306 T^{12} + 20 T^{14} + T^{16} \)
$7$ \( ( 2401 + 1029 T + 98 T^{2} - 21 T^{3} - 27 T^{4} - 3 T^{5} + 2 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$11$ \( 194481 + 629748 T^{2} + 1770174 T^{4} + 829626 T^{6} + 304543 T^{8} + 25814 T^{10} + 1599 T^{12} + 47 T^{14} + T^{16} \)
$13$ \( ( 21609 - 18522 T + 12936 T^{2} - 3990 T^{3} + 1177 T^{4} - 152 T^{5} + 45 T^{6} - 5 T^{7} + T^{8} )^{2} \)
$17$ \( 194481 + 3685878 T^{2} + 69302709 T^{4} + 10434606 T^{6} + 1056388 T^{8} + 61094 T^{10} + 2589 T^{12} + 62 T^{14} + T^{16} \)
$19$ \( ( 49 - 49 T + 217 T^{2} + 224 T^{3} + 541 T^{4} + 110 T^{5} + 40 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$23$ \( 50625 + 124875 T^{2} + 230175 T^{4} + 172230 T^{6} + 95071 T^{8} + 14114 T^{10} + 1590 T^{12} + 44 T^{14} + T^{16} \)
$29$ \( 10016218555281 + 1387681503588 T^{2} + 143040909474 T^{4} + 5476284816 T^{6} + 145682443 T^{8} + 2419664 T^{10} + 29394 T^{12} + 212 T^{14} + T^{16} \)
$31$ \( ( 63 + 42 T - 17 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$37$ \( ( 1 - 187 T + 35029 T^{2} + 11216 T^{3} + 3973 T^{4} + 254 T^{5} + 64 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$41$ \( 9845600625 + 4125775500 T^{2} + 1359184050 T^{4} + 132899130 T^{6} + 9168471 T^{8} + 330426 T^{10} + 8595 T^{12} + 111 T^{14} + T^{16} \)
$43$ \( ( 3200521 - 833674 T + 421102 T^{2} + 35234 T^{3} + 13537 T^{4} + 362 T^{5} + 139 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$47$ \( ( 62583921 - 3521961 T^{2} + 62541 T^{4} - 429 T^{6} + T^{8} )^{2} \)
$53$ \( 1315703055681 + 261666434043 T^{2} + 38722957119 T^{4} + 2208044286 T^{6} + 89845443 T^{8} + 1772874 T^{10} + 25254 T^{12} + 192 T^{14} + T^{16} \)
$59$ \( ( 3272481 - 385695 T^{2} + 13942 T^{4} - 200 T^{6} + T^{8} )^{2} \)
$61$ \( ( -2649 - 1596 T - 95 T^{2} + 14 T^{3} + T^{4} )^{4} \)
$67$ \( ( -917 - 672 T - 62 T^{2} + 9 T^{3} + T^{4} )^{4} \)
$71$ \( ( 505521 - 419085 T^{2} + 22657 T^{4} - 305 T^{6} + T^{8} )^{2} \)
$73$ \( ( 7458361 + 286755 T + 360593 T^{2} - 13440 T^{3} + 13653 T^{4} - 210 T^{5} + 128 T^{6} + T^{8} )^{2} \)
$79$ \( ( -1497 + 669 T - 50 T^{2} - 10 T^{3} + T^{4} )^{4} \)
$83$ \( 603060481873556721 + 15981999895181628 T^{2} + 276217524125802 T^{4} + 2769099363618 T^{6} + 20172185623 T^{8} + 97523354 T^{10} + 344643 T^{12} + 731 T^{14} + T^{16} \)
$89$ \( 4631487063921 + 1485257767083 T^{2} + 421566649983 T^{4} + 16261945398 T^{6} + 437692167 T^{8} + 6249906 T^{10} + 64566 T^{12} + 300 T^{14} + T^{16} \)
$97$ \( ( 8614225 + 3698100 T + 1611080 T^{2} + 113190 T^{3} + 23589 T^{4} - 2352 T^{5} + 449 T^{6} - 21 T^{7} + T^{8} )^{2} \)
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