Properties

Label 1900.2.i.e
Level $1900$
Weight $2$
Character orbit 1900.i
Analytic conductor $15.172$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{6} ) q^{3} + \beta_{4} q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{6} ) q^{3} + \beta_{4} q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} + \beta_{10} ) q^{9} -\beta_{5} q^{11} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{6} + \beta_{8} + \beta_{11} ) q^{13} + ( \beta_{7} + \beta_{9} - \beta_{10} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + ( \beta_{7} + \beta_{9} + \beta_{10} ) q^{21} + ( \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{23} + ( 2 + 2 \beta_{3} - \beta_{4} ) q^{27} + ( -1 + \beta_{1} - \beta_{2} - \beta_{6} - \beta_{8} - \beta_{11} ) q^{29} + ( 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( -2 + \beta_{1} - 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{33} + ( -2 - 2 \beta_{5} ) q^{37} + ( 2 - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} ) q^{39} + ( -2 + \beta_{1} - 2 \beta_{6} + 2 \beta_{9} - 3 \beta_{10} ) q^{41} + ( -5 + 3 \beta_{1} - 5 \beta_{6} + \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + \beta_{9} + 2 \beta_{10} ) q^{47} + ( -2 \beta_{2} - \beta_{3} - \beta_{8} ) q^{49} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{53} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{57} + ( -1 + \beta_{1} - \beta_{6} + 2 \beta_{7} - \beta_{9} - 2 \beta_{11} ) q^{59} + ( 3 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{61} + ( -3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{6} + \beta_{8} - 3 \beta_{10} + \beta_{11} ) q^{63} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{67} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{69} + ( 1 + \beta_{1} + \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{71} + ( 2 + 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{73} + ( 1 + 3 \beta_{2} + \beta_{8} ) q^{77} + ( 2 - 4 \beta_{1} + 2 \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{79} + ( -1 + 3 \beta_{1} - \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{81} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{8} ) q^{83} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{8} ) q^{87} + ( \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{91} + ( -4 + 5 \beta_{1} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{93} + ( 3 - \beta_{1} + 3 \beta_{6} + 2 \beta_{7} - \beta_{10} + 2 \beta_{11} ) q^{97} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{6} + \beta_{7} + 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 3q^{3} - 3q^{9} + O(q^{10}) \) \( 12q - 3q^{3} - 3q^{9} + 2q^{11} - 7q^{13} + q^{17} + q^{21} + 2q^{23} + 24q^{27} + q^{29} + 2q^{31} - 10q^{33} - 20q^{37} + 36q^{39} - 7q^{41} - 19q^{43} + 14q^{47} + 8q^{49} + 11q^{51} - 6q^{53} + 28q^{57} - 5q^{61} + 11q^{63} - 14q^{67} - 14q^{69} + 8q^{71} + 9q^{73} - 2q^{77} + q^{79} + 2q^{81} + 26q^{83} - 30q^{87} - 8q^{89} + 3q^{91} - 9q^{93} + 11q^{97} - 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-638495 \nu^{11} - 5059670 \nu^{10} + 2185864 \nu^{9} - 61632615 \nu^{8} - 86473310 \nu^{7} - 313073128 \nu^{6} - 445497855 \nu^{5} - 1233567830 \nu^{4} - 3200881200 \nu^{3} - 892972135 \nu^{2} - 207837960 \nu - 2681192682\)\()/ 2135068398 \)
\(\beta_{3}\)\(=\)\((\)\(-2112220 \nu^{11} + 5607653 \nu^{10} - 24678133 \nu^{9} + 8533470 \nu^{8} - 89367331 \nu^{7} + 19282261 \nu^{6} - 259923180 \nu^{5} - 606181345 \nu^{4} + 772282665 \nu^{3} - 452450816 \nu^{2} - 105738663 \nu - 1951111485\)\()/ 711689466 \)
\(\beta_{4}\)\(=\)\((\)\(3389210 \nu^{11} + 4511687 \nu^{10} + 20306405 \nu^{9} + 114731760 \nu^{8} + 262313951 \nu^{7} + 606863995 \nu^{6} + 1150918890 \nu^{5} + 3073317005 \nu^{4} + 4917790269 \nu^{3} + 2238395086 \nu^{2} + 521414583 \nu + 1619981121\)\()/ 711689466 \)
\(\beta_{5}\)\(=\)\((\)\(-35281678 \nu^{11} + 70497029 \nu^{10} - 385595812 \nu^{9} - 77727516 \nu^{8} - 1696717843 \nu^{7} - 1136318216 \nu^{6} - 5600322462 \nu^{5} - 13728352537 \nu^{4} - 422891238 \nu^{3} - 10151561138 \nu^{2} - 2369519571 \nu - 5196018312\)\()/ 2135068398 \)
\(\beta_{6}\)\(=\)\((\)\(-60680476 \nu^{11} + 182679923 \nu^{10} - 905147470 \nu^{9} + 968701752 \nu^{8} - 4732124989 \nu^{7} + 4030704250 \nu^{6} - 17769708720 \nu^{5} - 343348333 \nu^{4} - 12904983078 \nu^{3} + 2351354536 \nu^{2} - 7905696885 \nu - 1794617748\)\()/ 2135068398 \)
\(\beta_{7}\)\(=\)\((\)\(62867497 \nu^{11} - 203695424 \nu^{10} + 980933620 \nu^{9} - 1201195899 \nu^{8} + 5086161106 \nu^{7} - 5139259390 \nu^{6} + 19505876175 \nu^{5} - 3449941442 \nu^{4} + 14173622286 \nu^{3} - 2952954247 \nu^{2} + 17820058356 \nu - 397480590\)\()/ 2135068398 \)
\(\beta_{8}\)\(=\)\((\)\(102677425 \nu^{11} - 177003953 \nu^{10} + 1095480130 \nu^{9} + 493877715 \nu^{8} + 5185680031 \nu^{7} + 4686279104 \nu^{6} + 17827730625 \nu^{5} + 44330893465 \nu^{4} + 9517460142 \nu^{3} + 32695561541 \nu^{2} + 7628969433 \nu + 17785843362\)\()/ 2135068398 \)
\(\beta_{9}\)\(=\)\((\)\(-131028322 \nu^{11} + 444681245 \nu^{10} - 2136716515 \nu^{9} + 2925836544 \nu^{8} - 11360311729 \nu^{7} + 12770142025 \nu^{6} - 42996086940 \nu^{5} + 14406379883 \nu^{4} - 31258077591 \nu^{3} + 7259257192 \nu^{2} - 16532556735 \nu + 924096825\)\()/ 2135068398 \)
\(\beta_{10}\)\(=\)\((\)\(175066273 \nu^{11} - 536276480 \nu^{10} + 2643593875 \nu^{9} - 2942137461 \nu^{8} + 13841799664 \nu^{7} - 12347339095 \nu^{6} + 52083858765 \nu^{5} - 2022066866 \nu^{4} + 37830916029 \nu^{3} - 7169319793 \nu^{2} + 21056968308 \nu - 1015605495\)\()/ 2135068398 \)
\(\beta_{11}\)\(=\)\((\)\(137992489 \nu^{11} - 469201302 \nu^{10} + 2243299983 \nu^{9} - 3049379221 \nu^{8} + 11871320512 \nu^{7} - 13293030539 \nu^{6} + 45104060701 \nu^{5} - 14256401492 \nu^{4} + 32789448329 \nu^{3} - 7561467697 \nu^{2} + 19342466968 \nu - 965971179\)\()/ 711689466 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + 3 \beta_{6} - \beta_{3} - \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{4} - \beta_{3} - 6 \beta_{2} - 8\)
\(\nu^{4}\)\(=\)\(-7 \beta_{10} - \beta_{9} + \beta_{7} - 17 \beta_{6} - 10 \beta_{1} - 17\)
\(\nu^{5}\)\(=\)\(-\beta_{11} - 10 \beta_{10} - 3 \beta_{9} - \beta_{8} + 8 \beta_{7} - 21 \beta_{6} - 3 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} + 42 \beta_{2} - 42 \beta_{1} + 42\)
\(\nu^{6}\)\(=\)\(-3 \beta_{8} - 15 \beta_{5} + 13 \beta_{4} + 47 \beta_{3} + 87 \beta_{2} + 197\)
\(\nu^{7}\)\(=\)\(15 \beta_{11} + 85 \beta_{10} + 46 \beta_{9} - 62 \beta_{7} + 184 \beta_{6} + 309 \beta_{1} + 184\)
\(\nu^{8}\)\(=\)\(46 \beta_{11} + 325 \beta_{10} + 169 \beta_{9} + 46 \beta_{8} - 131 \beta_{7} + 750 \beta_{6} + 169 \beta_{5} - 131 \beta_{4} - 325 \beta_{3} - 724 \beta_{2} + 724 \beta_{1} - 724\)
\(\nu^{9}\)\(=\)\(169 \beta_{8} + 515 \beta_{5} - 494 \beta_{4} - 686 \beta_{3} - 2339 \beta_{2} - 3848\)
\(\nu^{10}\)\(=\)\(-515 \beta_{11} - 2318 \beta_{10} - 1693 \beta_{9} + 1201 \beta_{7} - 5301 \beta_{6} - 5904 \beta_{1} - 5301\)
\(\nu^{11}\)\(=\)\(-1693 \beta_{11} - 5412 \beta_{10} - 5102 \beta_{9} - 1693 \beta_{8} + 4011 \beta_{7} - 12008 \beta_{6} - 5102 \beta_{5} + 4011 \beta_{4} + 5412 \beta_{3} + 18049 \beta_{2} - 18049 \beta_{1} + 18049\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
−1.08347 + 1.87662i
−0.432807 + 0.749643i
−0.126563 + 0.219213i
0.430479 0.745611i
1.28913 2.23284i
1.42323 2.46510i
−1.08347 1.87662i
−0.432807 0.749643i
−0.126563 0.219213i
0.430479 + 0.745611i
1.28913 + 2.23284i
1.42323 + 2.46510i
0 −1.58347 + 2.74265i 0 0 0 −3.03607 0 −3.51475 6.08773i 0
201.2 0 −0.932807 + 1.61567i 0 0 0 3.93019 0 −0.240257 0.416137i 0
201.3 0 −0.626563 + 1.08524i 0 0 0 2.18534 0 0.714838 + 1.23814i 0
201.4 0 −0.0695210 + 0.120414i 0 0 0 −3.40785 0 1.49033 + 2.58133i 0
201.5 0 0.789132 1.36682i 0 0 0 −1.39989 0 0.254541 + 0.440878i 0
201.6 0 0.923228 1.59908i 0 0 0 1.72830 0 −0.204702 0.354554i 0
501.1 0 −1.58347 2.74265i 0 0 0 −3.03607 0 −3.51475 + 6.08773i 0
501.2 0 −0.932807 1.61567i 0 0 0 3.93019 0 −0.240257 + 0.416137i 0
501.3 0 −0.626563 1.08524i 0 0 0 2.18534 0 0.714838 1.23814i 0
501.4 0 −0.0695210 0.120414i 0 0 0 −3.40785 0 1.49033 2.58133i 0
501.5 0 0.789132 + 1.36682i 0 0 0 −1.39989 0 0.254541 0.440878i 0
501.6 0 0.923228 + 1.59908i 0 0 0 1.72830 0 −0.204702 + 0.354554i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.i.e 12
5.b even 2 1 1900.2.i.f yes 12
5.c odd 4 2 1900.2.s.e 24
19.c even 3 1 inner 1900.2.i.e 12
95.i even 6 1 1900.2.i.f yes 12
95.m odd 12 2 1900.2.s.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.i.e 12 1.a even 1 1 trivial
1900.2.i.e 12 19.c even 3 1 inner
1900.2.i.f yes 12 5.b even 2 1
1900.2.i.f yes 12 95.i even 6 1
1900.2.s.e 24 5.c odd 4 2
1900.2.s.e 24 95.m odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 9 + 69 T + 505 T^{2} + 286 T^{3} + 473 T^{4} + 149 T^{5} + 274 T^{6} + 77 T^{7} + 79 T^{8} + 16 T^{9} + 15 T^{10} + 3 T^{11} + T^{12} \)
$5$ \( T^{12} \)
$7$ \( ( -215 - 10 T + 141 T^{2} - 2 T^{3} - 23 T^{4} + T^{6} )^{2} \)
$11$ \( ( 27 - 159 T + 66 T^{2} + 43 T^{3} - 22 T^{4} - T^{5} + T^{6} )^{2} \)
$13$ \( 199809 + 169413 T + 238405 T^{2} + 95770 T^{3} + 128547 T^{4} + 60053 T^{5} + 32810 T^{6} + 7287 T^{7} + 1991 T^{8} + 254 T^{9} + 69 T^{10} + 7 T^{11} + T^{12} \)
$17$ \( 762129 + 112617 T + 435681 T^{2} - 63666 T^{3} + 188367 T^{4} - 11991 T^{5} + 21166 T^{6} + 879 T^{7} + 1823 T^{8} + 50 T^{9} + 49 T^{10} - T^{11} + T^{12} \)
$19$ \( 47045881 + 4300593 T^{2} + 864234 T^{3} + 238260 T^{4} + 93024 T^{5} + 10519 T^{6} + 4896 T^{7} + 660 T^{8} + 126 T^{9} + 33 T^{10} + T^{12} \)
$23$ \( 455625 + 486000 T + 702675 T^{2} + 103140 T^{3} + 186444 T^{4} - 40284 T^{5} + 65877 T^{6} - 15390 T^{7} + 5212 T^{8} - 302 T^{9} + 75 T^{10} - 2 T^{11} + T^{12} \)
$29$ \( 729 - 8667 T + 97533 T^{2} - 69102 T^{3} + 64203 T^{4} - 12039 T^{5} + 12382 T^{6} - 2593 T^{7} + 1463 T^{8} - 94 T^{9} + 41 T^{10} - T^{11} + T^{12} \)
$31$ \( ( 2213 + 4937 T + 2880 T^{2} + 53 T^{3} - 120 T^{4} - T^{5} + T^{6} )^{2} \)
$37$ \( ( -2880 + 480 T + 1088 T^{2} - 280 T^{3} - 48 T^{4} + 10 T^{5} + T^{6} )^{2} \)
$41$ \( 303282225 + 610482825 T + 1070655165 T^{2} + 355394250 T^{3} + 123056091 T^{4} + 3944901 T^{5} + 2659294 T^{6} + 111591 T^{7} + 35207 T^{8} + 778 T^{9} + 241 T^{10} + 7 T^{11} + T^{12} \)
$43$ \( 1154300625 - 923950125 T + 687042675 T^{2} - 149064720 T^{3} + 44542741 T^{4} + 2877225 T^{5} + 1865050 T^{6} + 117443 T^{7} + 31871 T^{8} + 2770 T^{9} + 381 T^{10} + 19 T^{11} + T^{12} \)
$47$ \( 2546514369 - 1011884076 T + 483984153 T^{2} - 78272352 T^{3} + 26518356 T^{4} - 3972384 T^{5} + 965751 T^{6} - 106122 T^{7} + 18364 T^{8} - 1678 T^{9} + 233 T^{10} - 14 T^{11} + T^{12} \)
$53$ \( 19210689 - 24167862 T + 50824593 T^{2} + 22604094 T^{3} + 22994142 T^{4} - 23094 T^{5} + 776245 T^{6} + 2054 T^{7} + 19654 T^{8} - 190 T^{9} + 185 T^{10} + 6 T^{11} + T^{12} \)
$59$ \( 624650049 - 1078048062 T + 1609287327 T^{2} - 464717394 T^{3} + 133565568 T^{4} - 13329870 T^{5} + 2718901 T^{6} - 184328 T^{7} + 41476 T^{8} - 1244 T^{9} + 227 T^{10} + T^{12} \)
$61$ \( 3932289 - 4071099 T + 36454423 T^{2} + 30732352 T^{3} + 265196165 T^{4} - 9827501 T^{5} + 4495158 T^{6} + 6223 T^{7} + 49577 T^{8} + 84 T^{9} + 275 T^{10} + 5 T^{11} + T^{12} \)
$67$ \( 5089536 + 11334144 T + 18147712 T^{2} + 12709248 T^{3} + 6439296 T^{4} + 2078720 T^{5} + 521104 T^{6} + 85744 T^{7} + 12736 T^{8} + 1312 T^{9} + 200 T^{10} + 14 T^{11} + T^{12} \)
$71$ \( 531441 + 2690010 T + 14716161 T^{2} - 5157054 T^{3} + 3246948 T^{4} - 284490 T^{5} + 194919 T^{6} - 6900 T^{7} + 10156 T^{8} + 212 T^{9} + 161 T^{10} - 8 T^{11} + T^{12} \)
$73$ \( 352125225 - 193373325 T + 216643815 T^{2} - 34633440 T^{3} + 64524861 T^{4} - 19194219 T^{5} + 5225878 T^{6} - 618975 T^{7} + 67941 T^{8} - 3296 T^{9} + 279 T^{10} - 9 T^{11} + T^{12} \)
$79$ \( 526105969 + 105900129 T + 150360251 T^{2} - 19598756 T^{3} + 28164979 T^{4} - 857169 T^{5} + 981510 T^{6} - 18385 T^{7} + 26913 T^{8} - 98 T^{9} + 181 T^{10} - T^{11} + T^{12} \)
$83$ \( ( 27345 - 17355 T - 5378 T^{2} + 2185 T^{3} - 122 T^{4} - 13 T^{5} + T^{6} )^{2} \)
$89$ \( 3908529 + 12217860 T + 47600943 T^{2} - 30754980 T^{3} + 20232538 T^{4} - 3599116 T^{5} + 917767 T^{6} - 28264 T^{7} + 23242 T^{8} - 592 T^{9} + 223 T^{10} + 8 T^{11} + T^{12} \)
$97$ \( 22325625 - 38059875 T + 78179175 T^{2} + 4702320 T^{3} + 24015501 T^{4} - 8075649 T^{5} + 3067522 T^{6} - 385529 T^{7} + 51281 T^{8} - 1976 T^{9} + 287 T^{10} - 11 T^{11} + T^{12} \)
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