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
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-6789 \nu^{11} + 407977 \nu^{10} - 701704 \nu^{9} - 1014429 \nu^{8} - 865016 \nu^{7} + 19540967 \nu^{6} - 53823631 \nu^{5} + 103674982 \nu^{4} - 118229724 \nu^{3} + 58499210 \nu^{2} - 2665028 \nu - 9570816\)\()/5500348\)
\(\beta_{2}\)\(=\)\((\)\(-76837 \nu^{11} - 171179 \nu^{10} + 509986 \nu^{9} + 1428477 \nu^{8} - 2520360 \nu^{7} - 4710591 \nu^{6} + 8507441 \nu^{5} - 20646084 \nu^{4} + 15569686 \nu^{3} + 23756054 \nu^{2} - 19439928 \nu - 7508164\)\()/5500348\)
\(\beta_{3}\)\(=\)\((\)\(54041 \nu^{11} - 302383 \nu^{10} + 439891 \nu^{9} + 184300 \nu^{8} + 2261772 \nu^{7} - 17461376 \nu^{6} + 49851815 \nu^{5} - 91525793 \nu^{4} + 115305627 \nu^{3} - 80402588 \nu^{2} + 26100310 \nu - 178398\)\()/2750174\)
\(\beta_{4}\)\(=\)\((\)\(-127195 \nu^{11} + 82977 \nu^{10} + 563109 \nu^{9} + 592838 \nu^{8} - 5748844 \nu^{7} + 9850368 \nu^{6} - 14441625 \nu^{5} + 3585319 \nu^{4} + 16374455 \nu^{3} - 12968760 \nu^{2} + 70472 \nu - 2572044\)\()/2750174\)
\(\beta_{5}\)\(=\)\((\)\(295642 \nu^{11} - 410251 \nu^{10} - 714509 \nu^{9} - 877952 \nu^{8} + 12614519 \nu^{7} - 34529048 \nu^{6} + 71188533 \nu^{5} - 75428613 \nu^{4} + 41121960 \nu^{3} + 4078592 \nu^{2} - 31162314 \nu + 15787424\)\()/5500348\)
\(\beta_{6}\)\(=\)\((\)\( 2236 \nu^{11} - 4785 \nu^{10} - 6307 \nu^{9} + 1316 \nu^{8} + 113135 \nu^{7} - 323728 \nu^{6} + 580379 \nu^{5} - 648191 \nu^{4} + 212358 \nu^{3} + 208368 \nu^{2} - 142154 \nu + 27316 \)\()/41356\)
\(\beta_{7}\)\(=\)\((\)\(-588552 \nu^{11} + 1471277 \nu^{10} + 959613 \nu^{9} - 594282 \nu^{8} - 29122539 \nu^{7} + 96716674 \nu^{6} - 193796667 \nu^{5} + 255549677 \nu^{4} - 178977012 \nu^{3} + 33607448 \nu^{2} + 7943010 \nu + 3770204\)\()/5500348\)
\(\beta_{8}\)\(=\)\((\)\(658600 \nu^{11} - 892121 \nu^{10} - 2171303 \nu^{9} - 1848624 \nu^{8} + 30777883 \nu^{7} - 72465116 \nu^{6} + 131465595 \nu^{5} - 131228611 \nu^{4} + 45177602 \nu^{3} + 1135708 \nu^{2} + 19832586 \nu - 5832856\)\()/5500348\)
\(\beta_{9}\)\(=\)\((\)\(-721311 \nu^{11} + 2509218 \nu^{10} - 349445 \nu^{9} - 2200105 \nu^{8} - 35847041 \nu^{7} + 152603387 \nu^{6} - 342514582 \nu^{5} + 520339833 \nu^{4} - 478822416 \nu^{3} + 211570982 \nu^{2} - 29719866 \nu - 2329388\)\()/5500348\)
\(\beta_{10}\)\(=\)\((\)\(944653 \nu^{11} - 1864021 \nu^{10} - 2371448 \nu^{9} - 304551 \nu^{8} + 45642708 \nu^{7} - 132130403 \nu^{6} + 248926171 \nu^{5} - 285355566 \nu^{4} + 136789536 \nu^{3} + 36475726 \nu^{2} - 33967380 \nu + 938824\)\()/5500348\)
\(\beta_{11}\)\(=\)\((\)\( -4510 \nu^{11} + 15163 \nu^{10} - 1205 \nu^{9} - 9766 \nu^{8} - 224935 \nu^{7} + 920210 \nu^{6} - 2085585 \nu^{5} + 3227207 \nu^{4} - 3042430 \nu^{3} + 1622152 \nu^{2} - 452518 \nu + 76880 \)\()/26068\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} + \beta_{7} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + \beta_{4} - 3 \beta_{3} - 2 \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + 2 \beta_{10} + 7 \beta_{7} + 14 \beta_{6} + 7 \beta_{4} + 7 \beta_{3} - \beta_{2} + 5 \beta_{1} - 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{11} - 20 \beta_{10} - 8 \beta_{9} + 16 \beta_{8} - 12 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} - 21 \beta_{4} - 14 \beta_{3} + 14 \beta_{2} - 22 \beta_{1} - 25\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(14 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} - 3 \beta_{8} + 8 \beta_{7} - 5 \beta_{6} - 13 \beta_{5} + 62 \beta_{4} - 13 \beta_{3} - 51 \beta_{2} + 55 \beta_{1} + 56\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-11 \beta_{11} + 24 \beta_{10} + 54 \beta_{9} - 9 \beta_{8} + 136 \beta_{7} + 217 \beta_{6} + 9 \beta_{5} - 68 \beta_{4} + 119 \beta_{3} + 78 \beta_{2} - 62 \beta_{1} - 278\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(114 \beta_{11} - 291 \beta_{10} - 326 \beta_{9} + 194 \beta_{8} - 542 \beta_{7} - 901 \beta_{6} - 55 \beta_{5} - 141 \beta_{4} - 580 \beta_{3} - 148 \beta_{2} - 165 \beta_{1} + 247\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-113 \beta_{11} + 1276 \beta_{10} + 404 \beta_{9} - 940 \beta_{8} + 1655 \beta_{7} + 2527 \beta_{6} - 195 \beta_{5} + 1312 \beta_{4} + 1588 \beta_{3} - 652 \beta_{2} + 1500 \beta_{1} + 96\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-25 \beta_{11} - 2766 \beta_{10} + 72 \beta_{9} + 1773 \beta_{8} - 2716 \beta_{7} - 3969 \beta_{6} + 525 \beta_{5} - 5278 \beta_{4} - 2352 \beta_{3} + 2895 \beta_{2} - 5001 \beta_{1} - 4406\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(1301 \beta_{11} + 4480 \beta_{10} - 4337 \beta_{9} - 3460 \beta_{8} - 1298 \beta_{7} - 4210 \beta_{6} - 2532 \beta_{5} + 12294 \beta_{4} - 2171 \beta_{3} - 10904 \beta_{2} + 12175 \beta_{1} + 18488\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-7243 \beta_{11} + 10057 \beta_{10} + 22468 \beta_{9} - 7768 \beta_{8} + 29731 \beta_{7} + 51343 \beta_{6} + 4507 \beta_{5} - 15000 \beta_{4} + 33291 \beta_{3} + 21451 \beta_{2} - 10568 \beta_{1} - 47282\)\()/2\)

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

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + 9 T^{2} - 18 T^{3} + 33 T^{4} - 63 T^{5} + 102 T^{6} - 189 T^{7} + 297 T^{8} - 486 T^{9} + 729 T^{10} - 729 T^{11} + 729 T^{12} )^{2} \)
$5$ \( 1 - 15 T^{2} + 99 T^{4} - 412 T^{6} + 1641 T^{8} - 7989 T^{10} + 39846 T^{12} - 199725 T^{14} + 1025625 T^{16} - 6437500 T^{18} + 38671875 T^{20} - 146484375 T^{22} + 244140625 T^{24} \)
$7$ \( 1 + 6 T^{2} - 33 T^{4} - 700 T^{6} - 1617 T^{8} + 14406 T^{10} + 117649 T^{12} \)
$11$ \( ( 1 - 3 T - 21 T^{2} + 28 T^{3} + 393 T^{4} - 153 T^{5} - 4846 T^{6} - 1683 T^{7} + 47553 T^{8} + 37268 T^{9} - 307461 T^{10} - 483153 T^{11} + 1771561 T^{12} )^{2} \)
$13$ \( ( 1 + 42 T^{2} + 903 T^{4} + 13340 T^{6} + 152607 T^{8} + 1199562 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 + 3 T + 39 T^{2} + 108 T^{3} + 729 T^{4} + 753 T^{5} + 12014 T^{6} + 12801 T^{7} + 210681 T^{8} + 530604 T^{9} + 3257319 T^{10} + 4259571 T^{11} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 - 3 T + 39 T^{2} - 108 T^{3} + 705 T^{4} - 2265 T^{5} + 12706 T^{6} - 43035 T^{7} + 254505 T^{8} - 740772 T^{9} + 5082519 T^{10} - 7428297 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( 1 + 69 T^{2} + 1863 T^{4} + 44136 T^{6} + 1489365 T^{8} + 33541107 T^{10} + 591428630 T^{12} + 17743245603 T^{14} + 416785390965 T^{16} + 6533711996904 T^{18} + 145893365578503 T^{20} + 2858429273741781 T^{22} + 21914624432020321 T^{24} \)
$29$ \( ( 1 - 126 T^{2} + 7623 T^{4} - 278868 T^{6} + 6410943 T^{8} - 89117406 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( 1 - 87 T^{2} + 3867 T^{4} - 78332 T^{6} - 455595 T^{8} + 87513459 T^{10} - 3532899090 T^{12} + 84100434099 T^{14} - 420751549995 T^{16} - 69519938340092 T^{18} + 3298129641784347 T^{20} - 71307660967329687 T^{22} + 787662783788549761 T^{24} \)
$37$ \( 1 + 105 T^{2} + 3339 T^{4} + 152764 T^{6} + 14203497 T^{8} + 501483003 T^{10} + 10611705558 T^{12} + 686530231107 T^{14} + 26619640141017 T^{16} + 391950629144476 T^{18} + 11728168896642219 T^{20} + 504901359103874145 T^{22} + 6582952005840035281 T^{24} \)
$41$ \( ( 1 - 102 T^{2} + 6783 T^{4} - 298852 T^{6} + 11402223 T^{8} - 288227622 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 + 43 T^{2} )^{12} \)
$47$ \( 1 - 159 T^{2} + 13131 T^{4} - 642364 T^{6} + 18269013 T^{8} - 74583429 T^{10} - 12209351154 T^{12} - 164754794661 T^{14} + 89146955624853 T^{16} - 6924179875597756 T^{18} + 312666005155583691 T^{20} - 8363262025496977791 T^{22} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 105 T^{2} - 693 T^{4} - 10308 T^{6} + 39259017 T^{8} + 1185419067 T^{10} - 43539019882 T^{12} + 3329842159203 T^{14} + 309772527717177 T^{16} - 228470234517732 T^{18} - 43145965455073173 T^{20} + 18363184388378870145 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 + 21 T + 309 T^{2} + 3402 T^{3} + 29601 T^{4} + 225789 T^{5} + 1760606 T^{6} + 13321551 T^{7} + 103041081 T^{8} + 698699358 T^{9} + 3744264549 T^{10} + 15013410279 T^{11} + 42180533641 T^{12} )^{2} \)
$61$ \( 1 - 159 T^{2} + 7419 T^{4} - 352004 T^{6} + 54786249 T^{8} - 3286587357 T^{10} + 113418284406 T^{12} - 12229391555397 T^{14} + 758561692640409 T^{16} - 18135377856569444 T^{18} + 1422276555126827739 T^{20} - \)\(11\!\cdots\!59\)\( T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( ( 1 + 15 T - 3 T^{2} - 142 T^{3} + 9993 T^{4} + 15123 T^{5} - 719466 T^{6} + 1013241 T^{7} + 44858577 T^{8} - 42708346 T^{9} - 60453363 T^{10} + 20251876605 T^{11} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 16 T + 71 T^{2} )^{6}( 1 + 16 T + 71 T^{2} )^{6} \)
$73$ \( ( 1 - 9 T + 183 T^{2} - 1404 T^{3} + 16701 T^{4} - 147195 T^{5} + 1376278 T^{6} - 10745235 T^{7} + 88999629 T^{8} - 546179868 T^{9} + 5196878103 T^{10} - 18657644337 T^{11} + 151334226289 T^{12} )^{2} \)
$79$ \( 1 + 261 T^{2} + 30519 T^{4} + 2829272 T^{6} + 267713397 T^{8} + 18988265187 T^{10} + 1205673992502 T^{12} + 118505763032067 T^{14} + 10427458497935157 T^{16} + 687760531456810712 T^{18} + 46300643769538335159 T^{20} + \)\(24\!\cdots\!61\)\( T^{22} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( ( 1 - 270 T^{2} + 28455 T^{4} - 2146852 T^{6} + 196026495 T^{8} - 12813746670 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 - 3 T + 92 T^{2} - 267 T^{3} + 7921 T^{4} )^{6} \)
$97$ \( ( 1 - 222 T^{2} + 20703 T^{4} - 1529300 T^{6} + 194794527 T^{8} - 19653500382 T^{10} + 832972004929 T^{12} )^{2} \)
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