Properties

Label 224.2.q.a
Level $224$
Weight $2$
Character orbit 224.q
Analytic conductor $1.789$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.144054149089536.2
Defining polynomial: \( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{8} q^{3} + ( - \beta_{7} - \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{7} + \beta_{5}) q^{7} + ( - \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{3} + ( - \beta_{7} - \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{7} + \beta_{5}) q^{7} + ( - \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_1 - 1) q^{9} + ( - \beta_{6} + \beta_{3} + 1) q^{11} + ( - \beta_{11} - \beta_{7} + 2 \beta_{5}) q^{13} + ( - \beta_{11} + \beta_{10} - \beta_{7} - \beta_{4}) q^{15} + ( - \beta_{8} + \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{9} - \beta_{6} - \beta_{3} + \beta_1) q^{19} + (\beta_{11} - 2 \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4}) q^{21} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{23} + 2 \beta_{3} q^{25} + (2 \beta_{6} - 2 \beta_{3} - \beta_1 - 1) q^{27} + ( - \beta_{11} + 2 \beta_{10} + \beta_{7}) q^{29} + (2 \beta_{11} + \beta_{10} - \beta_{5} + \beta_{4} + \beta_{2}) q^{31} + (2 \beta_{9} + \beta_{6}) q^{33} + (3 \beta_{9} - \beta_{8} + \beta_{6} + \beta_{3} - 2 \beta_1) q^{35} + (3 \beta_{11} + \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{37} + ( - 2 \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{39} + (3 \beta_{9} - 3 \beta_{8} + 2 \beta_{6} - 2 \beta_{3} - \beta_1 + 2) q^{41} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{45} + ( - \beta_{11} + 2 \beta_{10} + 3 \beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{47} + ( - 3 \beta_{9} + \beta_{8} - 2 \beta_{3} - 3 \beta_1 - 3) q^{49} + (2 \beta_{9} - 4 \beta_{8} + \beta_{6} + \beta_{3} + \beta_1 + 2) q^{51} + ( - 4 \beta_{10} - \beta_{5} - \beta_{4} + 4 \beta_{2}) q^{53} + (\beta_{10} + \beta_{4} - 2 \beta_{2}) q^{55} + ( - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_1 - 3) q^{57} + ( - \beta_{8} - 2 \beta_{6} + 2 \beta_{3} + 4 \beta_1 - 2) q^{59} + (\beta_{11} + 4 \beta_{10} + \beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{61} + (\beta_{11} - 2 \beta_{10} + 2 \beta_{7} - \beta_{5} - \beta_{4}) q^{63} + (\beta_{9} - 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_1 + 1) q^{65} + ( - 2 \beta_{9} + \beta_{8} + 4 \beta_{6} + 2 \beta_{3} - 6) q^{67} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} - 4 \beta_{5} - \beta_{4} - 4 \beta_{2}) q^{69} + (2 \beta_{11} + 2 \beta_{7} + 2 \beta_{4}) q^{71} + (\beta_{6} + 2 \beta_{3} + 4 \beta_1 + 1) q^{73} + (2 \beta_{9} + 2 \beta_{6} - 2) q^{75} + ( - 3 \beta_{11} + 2 \beta_{5}) q^{77} + ( - 2 \beta_{11} - 3 \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{79} + (2 \beta_{9} - \beta_{8} + 3 \beta_{3} + 2) q^{81} + ( - 2 \beta_{9} + 2 \beta_{8} - 8 \beta_{6} - 4 \beta_{3} - 2 \beta_1 + 2) q^{83} + (\beta_{11} - 4 \beta_{10} - 3 \beta_{7} - \beta_{4}) q^{85} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{87} + ( - \beta_{6} + 2) q^{89} + ( - 3 \beta_{9} + \beta_{8} - 2 \beta_{3} - 3 \beta_1 + 4) q^{91} + ( - 5 \beta_{11} - 3 \beta_{7} + 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{93} + ( - 5 \beta_{10} - 3 \beta_{7} + 2 \beta_{5} - 4 \beta_{4} + 5 \beta_{2}) q^{95} + ( - 7 \beta_{9} + 7 \beta_{8} - 2 \beta_{6} - 2 \beta_{3} - \beta_1 - 6) q^{97} + (\beta_{9} + \beta_{8} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 6 q^{11} - 6 q^{17} + 6 q^{19} - 6 q^{33} - 18 q^{35} - 12 q^{49} - 6 q^{51} - 36 q^{57} - 42 q^{59} - 12 q^{65} - 30 q^{67} + 18 q^{73} - 24 q^{75} + 6 q^{81} + 18 q^{89} + 72 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6789 \nu^{11} + 407977 \nu^{10} - 701704 \nu^{9} - 1014429 \nu^{8} - 865016 \nu^{7} + 19540967 \nu^{6} - 53823631 \nu^{5} + 103674982 \nu^{4} + \cdots - 9570816 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 76837 \nu^{11} - 171179 \nu^{10} + 509986 \nu^{9} + 1428477 \nu^{8} - 2520360 \nu^{7} - 4710591 \nu^{6} + 8507441 \nu^{5} - 20646084 \nu^{4} + \cdots - 7508164 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 54041 \nu^{11} - 302383 \nu^{10} + 439891 \nu^{9} + 184300 \nu^{8} + 2261772 \nu^{7} - 17461376 \nu^{6} + 49851815 \nu^{5} - 91525793 \nu^{4} + \cdots - 178398 ) / 2750174 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 127195 \nu^{11} + 82977 \nu^{10} + 563109 \nu^{9} + 592838 \nu^{8} - 5748844 \nu^{7} + 9850368 \nu^{6} - 14441625 \nu^{5} + 3585319 \nu^{4} + \cdots - 2572044 ) / 2750174 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 295642 \nu^{11} - 410251 \nu^{10} - 714509 \nu^{9} - 877952 \nu^{8} + 12614519 \nu^{7} - 34529048 \nu^{6} + 71188533 \nu^{5} - 75428613 \nu^{4} + \cdots + 15787424 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2236 \nu^{11} - 4785 \nu^{10} - 6307 \nu^{9} + 1316 \nu^{8} + 113135 \nu^{7} - 323728 \nu^{6} + 580379 \nu^{5} - 648191 \nu^{4} + 212358 \nu^{3} + 208368 \nu^{2} + \cdots + 27316 ) / 41356 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 588552 \nu^{11} + 1471277 \nu^{10} + 959613 \nu^{9} - 594282 \nu^{8} - 29122539 \nu^{7} + 96716674 \nu^{6} - 193796667 \nu^{5} + 255549677 \nu^{4} + \cdots + 3770204 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 658600 \nu^{11} - 892121 \nu^{10} - 2171303 \nu^{9} - 1848624 \nu^{8} + 30777883 \nu^{7} - 72465116 \nu^{6} + 131465595 \nu^{5} - 131228611 \nu^{4} + \cdots - 5832856 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 721311 \nu^{11} + 2509218 \nu^{10} - 349445 \nu^{9} - 2200105 \nu^{8} - 35847041 \nu^{7} + 152603387 \nu^{6} - 342514582 \nu^{5} + 520339833 \nu^{4} + \cdots - 2329388 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 944653 \nu^{11} - 1864021 \nu^{10} - 2371448 \nu^{9} - 304551 \nu^{8} + 45642708 \nu^{7} - 132130403 \nu^{6} + 248926171 \nu^{5} - 285355566 \nu^{4} + \cdots + 938824 ) / 5500348 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4510 \nu^{11} + 15163 \nu^{10} - 1205 \nu^{9} - 9766 \nu^{8} - 224935 \nu^{7} + 920210 \nu^{6} - 2085585 \nu^{5} + 3227207 \nu^{4} - 3042430 \nu^{3} + \cdots + 76880 ) / 26068 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 2\beta_{10} - 2\beta_{9} + 3\beta_{8} - 2\beta_{7} - 4\beta_{6} + \beta_{4} - 3\beta_{3} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 2\beta_{10} + 7\beta_{7} + 14\beta_{6} + 7\beta_{4} + 7\beta_{3} - \beta_{2} + 5\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} - 20 \beta_{10} - 8 \beta_{9} + 16 \beta_{8} - 12 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} - 21 \beta_{4} - 14 \beta_{3} + 14 \beta_{2} - 22 \beta _1 - 25 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} - 3 \beta_{8} + 8 \beta_{7} - 5 \beta_{6} - 13 \beta_{5} + 62 \beta_{4} - 13 \beta_{3} - 51 \beta_{2} + 55 \beta _1 + 56 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11 \beta_{11} + 24 \beta_{10} + 54 \beta_{9} - 9 \beta_{8} + 136 \beta_{7} + 217 \beta_{6} + 9 \beta_{5} - 68 \beta_{4} + 119 \beta_{3} + 78 \beta_{2} - 62 \beta _1 - 278 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 114 \beta_{11} - 291 \beta_{10} - 326 \beta_{9} + 194 \beta_{8} - 542 \beta_{7} - 901 \beta_{6} - 55 \beta_{5} - 141 \beta_{4} - 580 \beta_{3} - 148 \beta_{2} - 165 \beta _1 + 247 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 113 \beta_{11} + 1276 \beta_{10} + 404 \beta_{9} - 940 \beta_{8} + 1655 \beta_{7} + 2527 \beta_{6} - 195 \beta_{5} + 1312 \beta_{4} + 1588 \beta_{3} - 652 \beta_{2} + 1500 \beta _1 + 96 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25 \beta_{11} - 2766 \beta_{10} + 72 \beta_{9} + 1773 \beta_{8} - 2716 \beta_{7} - 3969 \beta_{6} + 525 \beta_{5} - 5278 \beta_{4} - 2352 \beta_{3} + 2895 \beta_{2} - 5001 \beta _1 - 4406 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1301 \beta_{11} + 4480 \beta_{10} - 4337 \beta_{9} - 3460 \beta_{8} - 1298 \beta_{7} - 4210 \beta_{6} - 2532 \beta_{5} + 12294 \beta_{4} - 2171 \beta_{3} - 10904 \beta_{2} + 12175 \beta _1 + 18488 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7243 \beta_{11} + 10057 \beta_{10} + 22468 \beta_{9} - 7768 \beta_{8} + 29731 \beta_{7} + 51343 \beta_{6} + 4507 \beta_{5} - 15000 \beta_{4} + 33291 \beta_{3} + 21451 \beta_{2} - 10568 \beta _1 - 47282 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.09935 + 0.468876i
−0.0263223 + 0.217464i
−2.37165 1.78079i
0.186445 + 1.54034i
2.00233 0.854000i
0.609850 0.457915i
1.09935 0.468876i
−0.0263223 0.217464i
−2.37165 + 1.78079i
0.186445 1.54034i
2.00233 + 0.854000i
0.609850 + 0.457915i
0 −1.18878 + 0.686340i 0 −0.345107 + 0.597743i 0 −2.63639 + 0.222310i 0 −0.557875 + 0.966267i 0
47.2 0 −1.18878 + 0.686340i 0 0.345107 0.597743i 0 2.63639 0.222310i 0 −0.557875 + 0.966267i 0
47.3 0 0.416472 0.240450i 0 −1.59713 + 2.76632i 0 0.694153 + 2.55307i 0 −1.38437 + 2.39779i 0
47.4 0 0.416472 0.240450i 0 1.59713 2.76632i 0 −0.694153 2.55307i 0 −1.38437 + 2.39779i 0
47.5 0 2.27230 1.31191i 0 −1.03926 + 1.80005i 0 1.25203 2.33076i 0 1.94224 3.36406i 0
47.6 0 2.27230 1.31191i 0 1.03926 1.80005i 0 −1.25203 + 2.33076i 0 1.94224 3.36406i 0
143.1 0 −1.18878 0.686340i 0 −0.345107 0.597743i 0 −2.63639 0.222310i 0 −0.557875 0.966267i 0
143.2 0 −1.18878 0.686340i 0 0.345107 + 0.597743i 0 2.63639 + 0.222310i 0 −0.557875 0.966267i 0
143.3 0 0.416472 + 0.240450i 0 −1.59713 2.76632i 0 0.694153 2.55307i 0 −1.38437 2.39779i 0
143.4 0 0.416472 + 0.240450i 0 1.59713 + 2.76632i 0 −0.694153 + 2.55307i 0 −1.38437 2.39779i 0
143.5 0 2.27230 + 1.31191i 0 −1.03926 1.80005i 0 1.25203 + 2.33076i 0 1.94224 + 3.36406i 0
143.6 0 2.27230 + 1.31191i 0 1.03926 + 1.80005i 0 −1.25203 2.33076i 0 1.94224 + 3.36406i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.q.a 12
3.b odd 2 1 2016.2.bs.a 12
4.b odd 2 1 56.2.m.a 12
7.b odd 2 1 1568.2.q.g 12
7.c even 3 1 1568.2.e.e 12
7.c even 3 1 1568.2.q.g 12
7.d odd 6 1 inner 224.2.q.a 12
7.d odd 6 1 1568.2.e.e 12
8.b even 2 1 56.2.m.a 12
8.d odd 2 1 inner 224.2.q.a 12
12.b even 2 1 504.2.bk.a 12
21.g even 6 1 2016.2.bs.a 12
24.f even 2 1 2016.2.bs.a 12
24.h odd 2 1 504.2.bk.a 12
28.d even 2 1 392.2.m.g 12
28.f even 6 1 56.2.m.a 12
28.f even 6 1 392.2.e.e 12
28.g odd 6 1 392.2.e.e 12
28.g odd 6 1 392.2.m.g 12
56.e even 2 1 1568.2.q.g 12
56.h odd 2 1 392.2.m.g 12
56.j odd 6 1 56.2.m.a 12
56.j odd 6 1 392.2.e.e 12
56.k odd 6 1 1568.2.e.e 12
56.k odd 6 1 1568.2.q.g 12
56.m even 6 1 inner 224.2.q.a 12
56.m even 6 1 1568.2.e.e 12
56.p even 6 1 392.2.e.e 12
56.p even 6 1 392.2.m.g 12
84.j odd 6 1 504.2.bk.a 12
168.ba even 6 1 504.2.bk.a 12
168.be odd 6 1 2016.2.bs.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.m.a 12 4.b odd 2 1
56.2.m.a 12 8.b even 2 1
56.2.m.a 12 28.f even 6 1
56.2.m.a 12 56.j odd 6 1
224.2.q.a 12 1.a even 1 1 trivial
224.2.q.a 12 7.d odd 6 1 inner
224.2.q.a 12 8.d odd 2 1 inner
224.2.q.a 12 56.m even 6 1 inner
392.2.e.e 12 28.f even 6 1
392.2.e.e 12 28.g odd 6 1
392.2.e.e 12 56.j odd 6 1
392.2.e.e 12 56.p even 6 1
392.2.m.g 12 28.d even 2 1
392.2.m.g 12 28.g odd 6 1
392.2.m.g 12 56.h odd 2 1
392.2.m.g 12 56.p even 6 1
504.2.bk.a 12 12.b even 2 1
504.2.bk.a 12 24.h odd 2 1
504.2.bk.a 12 84.j odd 6 1
504.2.bk.a 12 168.ba even 6 1
1568.2.e.e 12 7.c even 3 1
1568.2.e.e 12 7.d odd 6 1
1568.2.e.e 12 56.k odd 6 1
1568.2.e.e 12 56.m even 6 1
1568.2.q.g 12 7.b odd 2 1
1568.2.q.g 12 7.c even 3 1
1568.2.q.g 12 56.e even 2 1
1568.2.q.g 12 56.k odd 6 1
2016.2.bs.a 12 3.b odd 2 1
2016.2.bs.a 12 21.g even 6 1
2016.2.bs.a 12 24.f even 2 1
2016.2.bs.a 12 168.be odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 3 T^{5} + 9 T^{3} + 6 T^{2} - 9 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + 174 T^{8} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{10} - 33 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + 12 T^{4} - 5 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 36 T^{4} + 240 T^{2} - 336)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 3 T^{5} - 12 T^{4} - 45 T^{3} + \cdots + 1083)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 3 T^{5} - 18 T^{4} + 63 T^{3} + \cdots + 147)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 69 T^{10} + 3450 T^{8} + \cdots + 45252529 \) Copy content Toggle raw display
$29$ \( (T^{6} + 48 T^{4} + 576 T^{2} + 112)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 99 T^{10} + 8238 T^{8} + \cdots + 441 \) Copy content Toggle raw display
$37$ \( T^{12} - 117 T^{10} + \cdots + 3074369809 \) Copy content Toggle raw display
$41$ \( (T^{6} + 144 T^{4} + 5184 T^{2} + \cdots + 52272)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} + 123 T^{10} + 12990 T^{8} + \cdots + 6456681 \) Copy content Toggle raw display
$53$ \( T^{12} - 213 T^{10} + \cdots + 108651322129 \) Copy content Toggle raw display
$59$ \( (T^{6} + 21 T^{5} + 132 T^{4} + \cdots + 133563)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 207 T^{10} + \cdots + 42362284041 \) Copy content Toggle raw display
$67$ \( (T^{6} + 15 T^{5} + 198 T^{4} + \cdots + 52441)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 28)^{6} \) Copy content Toggle raw display
$73$ \( (T^{6} - 9 T^{5} - 36 T^{4} + 567 T^{3} + \cdots + 1323)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} - 213 T^{10} + \cdots + 31317319089 \) Copy content Toggle raw display
$83$ \( (T^{6} + 228 T^{4} + 816 T^{2} + 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 3 T + 3)^{6} \) Copy content Toggle raw display
$97$ \( (T^{6} + 360 T^{4} + 19248 T^{2} + \cdots + 134832)^{2} \) Copy content Toggle raw display
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