Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.t (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: 12.0.951588245534976.1
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.

\(f(q)\) \(=\) \( q -\beta_{10} q^{3} + ( -\beta_{4} + \beta_{7} ) q^{5} + ( \beta_{8} - \beta_{9} ) q^{7} + ( -\beta_{1} + \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})\) \( q -\beta_{10} q^{3} + ( -\beta_{4} + \beta_{7} ) q^{5} + ( \beta_{8} - \beta_{9} ) q^{7} + ( -\beta_{1} + \beta_{8} - \beta_{9} ) q^{9} + ( \beta_{10} - \beta_{11} ) q^{11} + ( -\beta_{6} - \beta_{11} ) q^{13} + ( 2 - \beta_{1} ) q^{15} + ( \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{17} + ( -2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{19} + ( -\beta_{2} + \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{21} -\beta_{8} q^{23} + ( -2 \beta_{3} + 2 \beta_{8} ) q^{25} + ( \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{27} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{11} ) q^{29} + ( 2 \beta_{5} - \beta_{9} ) q^{31} + ( -3 + 4 \beta_{1} - 3 \beta_{5} - 2 \beta_{8} + 4 \beta_{9} ) q^{33} + ( -2 \beta_{2} - \beta_{7} + \beta_{11} ) q^{35} + ( \beta_{4} - \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{3} - \beta_{8} + 3 \beta_{9} ) q^{39} + ( -3 \beta_{1} + \beta_{3} ) q^{41} + ( 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{7} - 2 \beta_{10} ) q^{43} + ( -2 \beta_{2} + \beta_{11} ) q^{45} + ( -6 + 3 \beta_{1} - 6 \beta_{5} + 3 \beta_{9} ) q^{47} + ( -1 + 3 \beta_{1} + \beta_{3} + 2 \beta_{5} - 2 \beta_{8} ) q^{49} + ( 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{51} + ( \beta_{2} + 3 \beta_{10} + \beta_{11} ) q^{53} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{55} + ( 1 - 4 \beta_{1} ) q^{57} + ( -4 \beta_{2} + 3 \beta_{10} ) q^{59} + ( -3 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{61} + ( -2 + \beta_{1} + 4 \beta_{5} - \beta_{8} - \beta_{9} ) q^{63} + ( 2 + \beta_{1} + 2 \beta_{5} - 3 \beta_{8} + \beta_{9} ) q^{65} + ( 2 \beta_{2} - \beta_{10} ) q^{67} + ( \beta_{2} + \beta_{4} + \beta_{7} + \beta_{10} ) q^{69} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -4 \beta_{3} + \beta_{5} + 4 \beta_{8} - 2 \beta_{9} ) q^{73} + ( -2 \beta_{4} - 2 \beta_{7} ) q^{75} + ( -3 \beta_{2} - 2 \beta_{4} - \beta_{6} + 6 \beta_{7} + 3 \beta_{10} - \beta_{11} ) q^{77} + ( 4 - 2 \beta_{1} + 4 \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{79} + ( 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{8} + \beta_{9} ) q^{81} + ( 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{10} ) q^{83} + ( 3 \beta_{2} + 3 \beta_{4} - \beta_{6} - 3 \beta_{7} - 3 \beta_{10} - \beta_{11} ) q^{85} + ( \beta_{3} - 2 \beta_{5} - \beta_{8} - 3 \beta_{9} ) q^{87} + ( -3 - 3 \beta_{5} + 4 \beta_{8} ) q^{89} + ( -2 \beta_{2} - 4 \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{10} - \beta_{11} ) q^{91} + ( \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{93} + ( -\beta_{3} - 6 \beta_{5} + \beta_{8} ) q^{95} + ( 2 + 3 \beta_{1} + \beta_{3} ) q^{97} + ( -2 \beta_{2} - 2 \beta_{4} - \beta_{6} + 6 \beta_{7} + 6 \beta_{10} - \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{7} + O(q^{10}) \) \( 12q + 4q^{7} + 20q^{15} - 2q^{17} - 2q^{23} - 4q^{25} - 10q^{31} - 14q^{33} - 4q^{39} - 8q^{41} - 30q^{47} - 12q^{49} - 4q^{55} - 4q^{57} - 44q^{63} + 8q^{65} - 32q^{71} - 10q^{73} + 22q^{79} + 22q^{81} + 20q^{87} - 10q^{89} + 34q^{95} + 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} + 2 x^{10} - 9 x^{9} + 8 x^{8} - 13 x^{7} + 35 x^{6} - 26 x^{5} + 32 x^{4} - 72 x^{3} + 32 x^{2} - 32 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{11} + 3 \nu^{10} + 9 \nu^{8} - 18 \nu^{7} - 7 \nu^{6} - 29 \nu^{5} + 44 \nu^{4} + 24 \nu^{3} + 32 \nu^{2} - 48 \nu - 64 \)\()/32\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{11} - 5 \nu^{10} - 6 \nu^{9} + 11 \nu^{8} + 24 \nu^{7} + 15 \nu^{6} - 25 \nu^{5} - 82 \nu^{4} - 32 \nu^{3} + 64 \nu^{2} + 160 \nu + 96 \)\()/32\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{10} + 2 \nu^{9} - \nu^{8} + 5 \nu^{7} - 9 \nu^{6} + 3 \nu^{5} - 12 \nu^{4} + 19 \nu^{3} + 16 \nu - 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{11} - 2 \nu^{10} - 5 \nu^{9} + 5 \nu^{8} + 3 \nu^{7} + 27 \nu^{6} - 24 \nu^{5} + 7 \nu^{4} - 80 \nu^{3} + 48 \nu^{2} - 8 \nu + 128 \)\()/16\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{11} - \nu^{10} - 9 \nu^{9} + 20 \nu^{8} - 3 \nu^{7} + 42 \nu^{6} - 63 \nu^{5} + 7 \nu^{4} - 100 \nu^{3} + 104 \nu^{2} + 32 \nu + 144 \)\()/16\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{11} + 7 \nu^{10} + 26 \nu^{9} - 25 \nu^{8} - 4 \nu^{7} - 141 \nu^{6} + 123 \nu^{5} - 10 \nu^{4} + 420 \nu^{3} - 248 \nu^{2} - 48 \nu - 672 \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -9 \nu^{11} - 7 \nu^{10} - 22 \nu^{9} + 45 \nu^{8} + 8 \nu^{7} + 105 \nu^{6} - 147 \nu^{5} - 18 \nu^{4} - 260 \nu^{3} + 248 \nu^{2} + 96 \nu + 416 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{11} - 7 \nu^{10} - 22 \nu^{9} + 45 \nu^{8} + 8 \nu^{7} + 105 \nu^{6} - 147 \nu^{5} - 18 \nu^{4} - 260 \nu^{3} + 248 \nu^{2} + 160 \nu + 416 \)\()/32\)
\(\beta_{9}\)\(=\)\((\)\( 13 \nu^{11} + 5 \nu^{10} + 28 \nu^{9} - 69 \nu^{8} - 2 \nu^{7} - 125 \nu^{6} + 221 \nu^{5} + 16 \nu^{4} + 296 \nu^{3} - 352 \nu^{2} - 160 \nu - 448 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( -11 \nu^{11} - \nu^{10} - 34 \nu^{9} + 67 \nu^{8} - 28 \nu^{7} + 167 \nu^{6} - 229 \nu^{5} + 82 \nu^{4} - 400 \nu^{3} + 392 \nu^{2} + 32 \nu + 544 \)\()/32\)
\(\beta_{11}\)\(=\)\((\)\( 7 \nu^{11} + 2 \nu^{10} + 15 \nu^{9} - 39 \nu^{8} + 3 \nu^{7} - 69 \nu^{6} + 112 \nu^{5} - 5 \nu^{4} + 164 \nu^{3} - 180 \nu^{2} - 64 \nu - 240 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} - \beta_{7}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} + \beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{1} + 4\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{8} - 3 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 1\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{11} + \beta_{10} + 4 \beta_{9} - \beta_{8} + 2 \beta_{5} + \beta_{3} + 2 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 7 \beta_{1} + 3\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-5 \beta_{9} - 9 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 10 \beta_{4} - 5 \beta_{1} + 2\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-9 \beta_{11} + \beta_{10} + 4 \beta_{9} - 11 \beta_{5} + \beta_{2}\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-2 \beta_{11} + 3 \beta_{10} + 3 \beta_{7} - 2 \beta_{6} - 16 \beta_{4} + 11 \beta_{3} - 16 \beta_{2} - 18 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-17 \beta_{9} - 11 \beta_{8} - 27 \beta_{7} + 15 \beta_{5} + 9 \beta_{4} - 17 \beta_{1} + 15\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-13 \beta_{11} + 15 \beta_{10} - 9 \beta_{9} + 4 \beta_{8} - 72 \beta_{5} - 4 \beta_{3} + 4 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
1.41417 0.0105323i
1.26950 + 0.623187i
−0.390636 1.35919i
−0.981777 1.01790i
−0.0950561 + 1.41102i
−0.716208 + 1.21944i
1.41417 + 0.0105323i
1.26950 0.623187i
−0.390636 + 1.35919i
−0.981777 + 1.01790i
−0.0950561 1.41102i
−0.716208 1.21944i
0 −2.13038 1.22998i 0 −1.28690 + 0.742990i 0 −0.129755 + 2.64257i 0 1.52569 + 2.64257i 0
81.2 0 −1.36456 0.787829i 0 0.476087 0.274869i 0 2.60755 0.447998i 0 −0.258652 0.447998i 0
81.3 0 −0.591141 0.341295i 0 −2.80486 + 1.61939i 0 −1.47779 2.19457i 0 −1.26704 2.19457i 0
81.4 0 0.591141 + 0.341295i 0 2.80486 1.61939i 0 −1.47779 2.19457i 0 −1.26704 2.19457i 0
81.5 0 1.36456 + 0.787829i 0 −0.476087 + 0.274869i 0 2.60755 0.447998i 0 −0.258652 0.447998i 0
81.6 0 2.13038 + 1.22998i 0 1.28690 0.742990i 0 −0.129755 + 2.64257i 0 1.52569 + 2.64257i 0
177.1 0 −2.13038 + 1.22998i 0 −1.28690 0.742990i 0 −0.129755 2.64257i 0 1.52569 2.64257i 0
177.2 0 −1.36456 + 0.787829i 0 0.476087 + 0.274869i 0 2.60755 + 0.447998i 0 −0.258652 + 0.447998i 0
177.3 0 −0.591141 + 0.341295i 0 −2.80486 1.61939i 0 −1.47779 + 2.19457i 0 −1.26704 + 2.19457i 0
177.4 0 0.591141 0.341295i 0 2.80486 + 1.61939i 0 −1.47779 + 2.19457i 0 −1.26704 + 2.19457i 0
177.5 0 1.36456 0.787829i 0 −0.476087 0.274869i 0 2.60755 + 0.447998i 0 −0.258652 + 0.447998i 0
177.6 0 2.13038 1.22998i 0 1.28690 + 0.742990i 0 −0.129755 2.64257i 0 1.52569 2.64257i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p 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.t.a 12
3.b odd 2 1 2016.2.cr.c 12
4.b odd 2 1 56.2.p.a 12
7.b odd 2 1 1568.2.t.g 12
7.c even 3 1 inner 224.2.t.a 12
7.c even 3 1 1568.2.b.f 6
7.d odd 6 1 1568.2.b.e 6
7.d odd 6 1 1568.2.t.g 12
8.b even 2 1 inner 224.2.t.a 12
8.d odd 2 1 56.2.p.a 12
12.b even 2 1 504.2.cj.c 12
21.h odd 6 1 2016.2.cr.c 12
24.f even 2 1 504.2.cj.c 12
24.h odd 2 1 2016.2.cr.c 12
28.d even 2 1 392.2.p.g 12
28.f even 6 1 392.2.b.f 6
28.f even 6 1 392.2.p.g 12
28.g odd 6 1 56.2.p.a 12
28.g odd 6 1 392.2.b.e 6
56.e even 2 1 392.2.p.g 12
56.h odd 2 1 1568.2.t.g 12
56.j odd 6 1 1568.2.b.e 6
56.j odd 6 1 1568.2.t.g 12
56.k odd 6 1 56.2.p.a 12
56.k odd 6 1 392.2.b.e 6
56.m even 6 1 392.2.b.f 6
56.m even 6 1 392.2.p.g 12
56.p even 6 1 inner 224.2.t.a 12
56.p even 6 1 1568.2.b.f 6
84.n even 6 1 504.2.cj.c 12
168.s odd 6 1 2016.2.cr.c 12
168.v even 6 1 504.2.cj.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.p.a 12 4.b odd 2 1
56.2.p.a 12 8.d odd 2 1
56.2.p.a 12 28.g odd 6 1
56.2.p.a 12 56.k odd 6 1
224.2.t.a 12 1.a even 1 1 trivial
224.2.t.a 12 7.c even 3 1 inner
224.2.t.a 12 8.b even 2 1 inner
224.2.t.a 12 56.p even 6 1 inner
392.2.b.e 6 28.g odd 6 1
392.2.b.e 6 56.k odd 6 1
392.2.b.f 6 28.f even 6 1
392.2.b.f 6 56.m even 6 1
392.2.p.g 12 28.d even 2 1
392.2.p.g 12 28.f even 6 1
392.2.p.g 12 56.e even 2 1
392.2.p.g 12 56.m even 6 1
504.2.cj.c 12 12.b even 2 1
504.2.cj.c 12 24.f even 2 1
504.2.cj.c 12 84.n even 6 1
504.2.cj.c 12 168.v even 6 1
1568.2.b.e 6 7.d odd 6 1
1568.2.b.e 6 56.j odd 6 1
1568.2.b.f 6 7.c even 3 1
1568.2.b.f 6 56.p even 6 1
1568.2.t.g 12 7.b odd 2 1
1568.2.t.g 12 7.d odd 6 1
1568.2.t.g 12 56.h odd 2 1
1568.2.t.g 12 56.j odd 6 1
2016.2.cr.c 12 3.b odd 2 1
2016.2.cr.c 12 21.h odd 6 1
2016.2.cr.c 12 24.h odd 2 1
2016.2.cr.c 12 168.s odd 6 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 9 T^{2} + 35 T^{4} + 92 T^{6} + 253 T^{8} + 683 T^{10} + 1774 T^{12} + 6147 T^{14} + 20493 T^{16} + 67068 T^{18} + 229635 T^{20} + 531441 T^{22} + 531441 T^{24} \)
$5$ \( 1 + 17 T^{2} + 147 T^{4} + 788 T^{6} + 2793 T^{8} + 5371 T^{10} + 7494 T^{12} + 134275 T^{14} + 1745625 T^{16} + 12312500 T^{18} + 57421875 T^{20} + 166015625 T^{22} + 244140625 T^{24} \)
$7$ \( ( 1 - 2 T + 5 T^{2} - 32 T^{3} + 35 T^{4} - 98 T^{5} + 343 T^{6} )^{2} \)
$11$ \( 1 + 29 T^{2} + 327 T^{4} + 2696 T^{6} + 24957 T^{8} - 44693 T^{10} - 3466650 T^{12} - 5407853 T^{14} + 365395437 T^{16} + 4776128456 T^{18} + 70095354087 T^{20} + 752185313429 T^{22} + 3138428376721 T^{24} \)
$13$ \( ( 1 - 46 T^{2} + 1191 T^{4} - 18804 T^{6} + 201279 T^{8} - 1313806 T^{10} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 + T - 33 T^{2} + 8 T^{3} + 565 T^{4} - 425 T^{5} - 9946 T^{6} - 7225 T^{7} + 163285 T^{8} + 39304 T^{9} - 2756193 T^{10} + 1419857 T^{11} + 24137569 T^{12} )^{2} \)
$19$ \( 1 + 53 T^{2} + 1215 T^{4} + 18608 T^{6} + 219741 T^{8} - 1587413 T^{10} - 104375754 T^{12} - 573056093 T^{14} + 28636866861 T^{16} + 875429753648 T^{18} + 20635029094815 T^{20} + 324946511663453 T^{22} + 2213314919066161 T^{24} \)
$23$ \( ( 1 + T - 61 T^{2} - 12 T^{3} + 2381 T^{4} - 29 T^{5} - 63238 T^{6} - 667 T^{7} + 1259549 T^{8} - 146004 T^{9} - 17070301 T^{10} + 6436343 T^{11} + 148035889 T^{12} )^{2} \)
$29$ \( ( 1 - 102 T^{2} + 4567 T^{4} - 141988 T^{6} + 3840847 T^{8} - 72142662 T^{10} + 594823321 T^{12} )^{2} \)
$31$ \( ( 1 + 5 T - 71 T^{2} - 126 T^{3} + 4725 T^{4} + 4133 T^{5} - 158330 T^{6} + 128123 T^{7} + 4540725 T^{8} - 3753666 T^{9} - 65569991 T^{10} + 143145755 T^{11} + 887503681 T^{12} )^{2} \)
$37$ \( 1 + 161 T^{2} + 13515 T^{4} + 840428 T^{6} + 43791513 T^{8} + 2009438227 T^{10} + 80476513014 T^{12} + 2750920932763 T^{14} + 82072345795593 T^{16} + 2156308314463052 T^{18} + 47471159819742315 T^{20} + 774182083959273689 T^{22} + 6582952005840035281 T^{24} \)
$41$ \( ( 1 + 2 T + 83 T^{2} + 80 T^{3} + 3403 T^{4} + 3362 T^{5} + 68921 T^{6} )^{4} \)
$43$ \( ( 1 - 94 T^{2} + 3543 T^{4} - 112068 T^{6} + 6551007 T^{8} - 321367294 T^{10} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 + 15 T + 57 T^{2} + 78 T^{3} + 2013 T^{4} + 1383 T^{5} - 125066 T^{6} + 65001 T^{7} + 4446717 T^{8} + 8098194 T^{9} + 278141817 T^{10} + 3440175105 T^{11} + 10779215329 T^{12} )^{2} \)
$53$ \( 1 + 161 T^{2} + 12171 T^{4} + 564332 T^{6} + 17660601 T^{8} + 173268691 T^{10} - 11132063562 T^{12} + 486711753019 T^{14} + 139350636639081 T^{16} + 12508058244650828 T^{18} + 757762691996674731 T^{20} + 28156882728847600889 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 177 T^{2} + 16739 T^{4} + 834316 T^{6} + 7372221 T^{8} - 3081613133 T^{10} - 272281952338 T^{12} - 10727095315973 T^{14} + 89331863228781 T^{16} + 35191894105224556 T^{18} + 2457794695058729219 T^{20} + 90467665334213527977 T^{22} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 73 T^{2} - 3941 T^{4} - 121236 T^{6} + 22314201 T^{8} - 326849717 T^{10} - 131320604138 T^{12} - 1216207796957 T^{14} + 308958879088041 T^{16} - 6246124106030196 T^{18} - 755518520522284421 T^{20} + 52074032551390429873 T^{22} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 361 T^{2} + 73723 T^{4} + 10542804 T^{6} + 1164287325 T^{8} + 103688059123 T^{10} + 7623484120510 T^{12} + 465455697403147 T^{14} + 23461694764841325 T^{16} + 953684993364861876 T^{18} + 29936527392508244443 T^{20} + \)\(65\!\cdots\!89\)\( T^{22} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( ( 1 + 8 T + 157 T^{2} + 704 T^{3} + 11147 T^{4} + 40328 T^{5} + 357911 T^{6} )^{4} \)
$73$ \( ( 1 + 5 T - 101 T^{2} + 52 T^{3} + 5233 T^{4} - 28921 T^{5} - 430618 T^{6} - 2111233 T^{7} + 27886657 T^{8} + 20228884 T^{9} - 2868222341 T^{10} + 10365357965 T^{11} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 - 11 T - 137 T^{2} + 656 T^{3} + 26965 T^{4} - 70973 T^{5} - 1975630 T^{6} - 5606867 T^{7} + 168288565 T^{8} + 323433584 T^{9} - 5336161097 T^{10} - 33847620389 T^{11} + 243087455521 T^{12} )^{2} \)
$83$ \( ( 1 - 446 T^{2} + 86503 T^{4} - 9357636 T^{6} + 595919167 T^{8} - 21166411166 T^{10} + 326940373369 T^{12} )^{2} \)
$89$ \( ( 1 + 5 T - 133 T^{2} - 1452 T^{3} + 5297 T^{4} + 76103 T^{5} + 246758 T^{6} + 6773167 T^{7} + 41957537 T^{8} - 1023614988 T^{9} - 8344718053 T^{10} + 27920297245 T^{11} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 - 10 T + 255 T^{2} - 1968 T^{3} + 24735 T^{4} - 94090 T^{5} + 912673 T^{6} )^{4} \)
show more
show less