Properties

Label 128.4.e.b
Level 128
Weight 4
Character orbit 128.e
Analytic conductor 7.552
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.e (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - x^{8} + 6 x^{7} + 14 x^{6} - 80 x^{5} + 56 x^{4} + 96 x^{3} - 64 x^{2} - 512 x + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} -\beta_{3} q^{5} + ( -3 \beta_{1} + \beta_{4} ) q^{7} + ( 5 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} -\beta_{3} q^{5} + ( -3 \beta_{1} + \beta_{4} ) q^{7} + ( 5 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{7} + \beta_{9} ) q^{11} + ( \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{13} + ( -13 - 2 \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{15} + ( -4 + 5 \beta_{2} - 2 \beta_{3} + 5 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} ) q^{17} + ( 1 + \beta_{1} + 2 \beta_{4} + \beta_{5} + 6 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{19} + ( -8 + 8 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{21} + ( 23 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + \beta_{4} + 4 \beta_{5} + 6 \beta_{6} + 2 \beta_{9} ) q^{23} + ( -\beta_{1} + 10 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} ) q^{25} + ( -19 + 19 \beta_{1} - 6 \beta_{3} + 2 \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{27} + ( 16 + 16 \beta_{1} - 2 \beta_{4} + 20 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{29} + ( 38 + 4 \beta_{2} + 6 \beta_{3} + 4 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{31} + ( 4 - 15 \beta_{2} - 4 \beta_{3} - 15 \beta_{5} + 4 \beta_{6} + \beta_{7} + \beta_{8} ) q^{33} + ( -46 - 46 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 8 \beta_{6} - 2 \beta_{8} ) q^{35} + ( 8 - 8 \beta_{1} - 30 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{37} + ( -71 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} ) q^{39} + ( -8 \beta_{1} - 24 \beta_{2} - 2 \beta_{3} + 24 \beta_{5} - 2 \beta_{6} - 4 \beta_{9} ) q^{41} + ( 84 - 84 \beta_{1} + 3 \beta_{2} + 8 \beta_{3} + 4 \beta_{7} - 4 \beta_{9} ) q^{43} + ( -8 - 8 \beta_{1} - \beta_{4} - 46 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - 5 \beta_{9} ) q^{45} + ( -98 - 6 \beta_{3} + 6 \beta_{6} + 10 \beta_{7} + 4 \beta_{8} ) q^{47} + ( 7 + 26 \beta_{2} + 12 \beta_{3} + 26 \beta_{5} - 12 \beta_{6} - 6 \beta_{7} + 10 \beta_{8} ) q^{49} + ( 153 + 153 \beta_{1} - 12 \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} + 12 \beta_{8} + 5 \beta_{9} ) q^{51} + ( 32 - 32 \beta_{1} + 42 \beta_{2} + 7 \beta_{3} + 13 \beta_{4} + 7 \beta_{7} + 13 \beta_{8} - 7 \beta_{9} ) q^{53} + ( 157 \beta_{1} + 28 \beta_{2} - 12 \beta_{3} - 3 \beta_{4} - 28 \beta_{5} - 12 \beta_{6} - 12 \beta_{9} ) q^{55} + ( -4 \beta_{1} + 25 \beta_{2} + 16 \beta_{3} - 7 \beta_{4} - 25 \beta_{5} + 16 \beta_{6} - 9 \beta_{9} ) q^{57} + ( -174 + 174 \beta_{1} - \beta_{2} - 14 \beta_{4} + 4 \beta_{7} - 14 \beta_{8} - 4 \beta_{9} ) q^{59} + ( -96 - 96 \beta_{1} + 9 \beta_{4} + 30 \beta_{5} - 3 \beta_{6} - 11 \beta_{7} - 9 \beta_{8} - 11 \beta_{9} ) q^{61} + ( 271 - 32 \beta_{2} - 6 \beta_{3} - 32 \beta_{5} + 6 \beta_{6} + 10 \beta_{7} - \beta_{8} ) q^{63} + ( -22 - 40 \beta_{2} + 26 \beta_{3} - 40 \beta_{5} - 26 \beta_{6} - 4 \beta_{7} ) q^{65} + ( -189 - 189 \beta_{1} - 12 \beta_{4} - 11 \beta_{5} - 2 \beta_{6} - \beta_{7} + 12 \beta_{8} - \beta_{9} ) q^{67} + ( -48 + 48 \beta_{1} - 50 \beta_{2} + 6 \beta_{3} - \beta_{4} + 5 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{69} + ( -337 \beta_{1} + 28 \beta_{2} + 6 \beta_{3} - 7 \beta_{4} - 28 \beta_{5} + 6 \beta_{6} + 2 \beta_{9} ) q^{71} + ( -50 \beta_{1} - 31 \beta_{2} + 12 \beta_{3} + 17 \beta_{4} + 31 \beta_{5} + 12 \beta_{6} - \beta_{9} ) q^{73} + ( 288 - 288 \beta_{1} + 9 \beta_{2} - 28 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{75} + ( 32 + 32 \beta_{1} - 15 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 15 \beta_{8} + 5 \beta_{9} ) q^{77} + ( -428 + 4 \beta_{2} + 32 \beta_{3} + 4 \beta_{5} - 32 \beta_{6} - 8 \beta_{7} + 4 \beta_{8} ) q^{79} + ( 35 + 9 \beta_{2} - 24 \beta_{3} + 9 \beta_{5} + 24 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} ) q^{81} + ( 260 + 260 \beta_{1} + 22 \beta_{4} - 11 \beta_{5} - 28 \beta_{6} - 2 \beta_{7} - 22 \beta_{8} - 2 \beta_{9} ) q^{83} + ( 8 - 8 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} - 7 \beta_{4} - 17 \beta_{7} - 7 \beta_{8} + 17 \beta_{9} ) q^{85} + ( 575 \beta_{1} - 76 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} + 76 \beta_{5} - 10 \beta_{6} + 18 \beta_{9} ) q^{87} + ( 18 \beta_{1} + 15 \beta_{2} - 52 \beta_{3} - \beta_{4} - 15 \beta_{5} - 52 \beta_{6} + \beta_{9} ) q^{89} + ( -320 + 320 \beta_{1} + 2 \beta_{2} + 36 \beta_{3} + 34 \beta_{4} + 2 \beta_{7} + 34 \beta_{8} - 2 \beta_{9} ) q^{91} + ( 216 + 216 \beta_{1} - 10 \beta_{4} + 20 \beta_{5} - 8 \beta_{6} + 14 \beta_{7} + 10 \beta_{8} + 14 \beta_{9} ) q^{93} + ( 641 + 104 \beta_{2} - 24 \beta_{3} + 104 \beta_{5} + 24 \beta_{6} - 8 \beta_{7} + 3 \beta_{8} ) q^{95} + ( -36 + 25 \beta_{2} - 58 \beta_{3} + 25 \beta_{5} + 58 \beta_{6} - 3 \beta_{7} - 15 \beta_{8} ) q^{97} + ( -518 - 518 \beta_{1} + 22 \beta_{4} + 21 \beta_{5} + 64 \beta_{6} + 4 \beta_{7} - 22 \beta_{8} + 4 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 2q^{3} + 2q^{5} + O(q^{10}) \) \( 10q + 2q^{3} + 2q^{5} - 18q^{11} + 2q^{13} - 124q^{15} - 4q^{17} + 26q^{19} - 52q^{21} - 184q^{27} + 202q^{29} + 368q^{31} - 4q^{33} - 476q^{35} + 10q^{37} + 838q^{43} - 194q^{45} - 944q^{47} + 94q^{49} + 1500q^{51} + 378q^{53} - 1706q^{59} - 910q^{61} + 2628q^{63} - 492q^{65} - 1942q^{67} - 580q^{69} + 2954q^{75} + 268q^{77} - 4416q^{79} + 482q^{81} + 2562q^{83} + 12q^{85} - 3332q^{91} + 2192q^{93} + 6900q^{95} - 4q^{97} - 4958q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - x^{8} + 6 x^{7} + 14 x^{6} - 80 x^{5} + 56 x^{4} + 96 x^{3} - 64 x^{2} - 512 x + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{9} + 14 \nu^{8} - 7 \nu^{7} - 82 \nu^{6} + 170 \nu^{5} + 120 \nu^{4} - 536 \nu^{3} - 384 \nu^{2} + 2752 \nu - 3072 \)\()/1280\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{9} + 2 \nu^{8} - 101 \nu^{7} + 114 \nu^{6} - 210 \nu^{5} + 120 \nu^{4} - 8 \nu^{3} + 3008 \nu^{2} - 3264 \nu + 7424 \)\()/1280\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{9} - 122 \nu^{8} + 341 \nu^{7} - 634 \nu^{6} + 130 \nu^{5} + 120 \nu^{4} + 3848 \nu^{3} - 9728 \nu^{2} + 16064 \nu - 2304 \)\()/1280\)
\(\beta_{4}\)\(=\)\((\)\( 7 \nu^{9} - 18 \nu^{8} + 49 \nu^{7} + 174 \nu^{6} - 870 \nu^{5} + 1240 \nu^{4} + 2152 \nu^{3} - 5632 \nu^{2} - 15424 \nu + 31744 \)\()/1280\)
\(\beta_{5}\)\(=\)\((\)\( -17 \nu^{9} + 38 \nu^{8} - 39 \nu^{7} + 86 \nu^{6} + 90 \nu^{5} + 520 \nu^{4} - 792 \nu^{3} + 2752 \nu^{2} - 1856 \nu - 4864 \)\()/1280\)
\(\beta_{6}\)\(=\)\((\)\( -37 \nu^{9} + 198 \nu^{8} - 419 \nu^{7} + 166 \nu^{6} + 50 \nu^{5} + 1560 \nu^{4} - 7992 \nu^{3} + 11392 \nu^{2} - 6976 \nu + 256 \)\()/1280\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{9} + 2 \nu^{8} + \nu^{7} - 6 \nu^{6} - 14 \nu^{5} + 80 \nu^{4} - 56 \nu^{3} - 96 \nu^{2} + 320 \nu + 416 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -3 \nu^{9} + 10 \nu^{8} + 3 \nu^{7} - 6 \nu^{6} - 90 \nu^{5} + 184 \nu^{4} - 56 \nu^{3} - 128 \nu^{2} - 896 \nu + 1728 \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( 39 \nu^{9} - 66 \nu^{8} - 47 \nu^{7} + 414 \nu^{6} - 294 \nu^{5} - 1288 \nu^{4} + 1704 \nu^{3} + 2944 \nu^{2} - 11072 \nu + 11264 \)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 3\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9} + \beta_{8} + \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 5 \beta_{1} + 9\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{8} - \beta_{7} - 7 \beta_{6} + 5 \beta_{5} - \beta_{4} - 3 \beta_{3} + 15 \beta_{2} + 33 \beta_{1} - 9\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{9} - \beta_{8} + 4 \beta_{7} + 3 \beta_{6} + 23 \beta_{5} + 6 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 43 \beta_{1} - 117\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-6 \beta_{8} - 7 \beta_{7} - \beta_{6} + 43 \beta_{5} + 13 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} + 67 \beta_{1} + 301\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(-3 \beta_{9} - 15 \beta_{8} + 28 \beta_{7} - 27 \beta_{6} + 17 \beta_{5} + 2 \beta_{4} - 49 \beta_{3} - 43 \beta_{2} - 99 \beta_{1} + 117\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(24 \beta_{9} + 22 \beta_{8} + 15 \beta_{7} + 33 \beta_{6} + 133 \beta_{5} - 45 \beta_{4} + 61 \beta_{3} - 233 \beta_{2} - 155 \beta_{1} + 859\)\()/16\)
\(\nu^{8}\)\(=\)\((\)\(11 \beta_{9} + 119 \beta_{8} - 4 \beta_{7} - 197 \beta_{6} - 17 \beta_{5} - 122 \beta_{4} - 191 \beta_{3} + 91 \beta_{2} + 1043 \beta_{1} + 987\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(208 \beta_{9} + 146 \beta_{8} + 121 \beta_{7} + 31 \beta_{6} + 155 \beta_{5} - 75 \beta_{4} + 211 \beta_{3} - 71 \beta_{2} + 3019 \beta_{1} - 4187\)\()/16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.28199 + 1.53509i
0.932438 1.76934i
−1.62580 + 1.16481i
−1.56339 1.24732i
1.97476 + 0.316760i
1.28199 1.53509i
0.932438 + 1.76934i
−1.62580 1.16481i
−1.56339 + 1.24732i
1.97476 0.316760i
0 −5.49618 5.49618i 0 4.66372 4.66372i 0 24.8965i 0 33.4160i 0
33.2 0 −1.98356 1.98356i 0 0.596848 0.596848i 0 29.0828i 0 19.1310i 0
33.3 0 −0.756776 0.756776i 0 −8.22587 + 8.22587i 0 2.67171i 0 25.8546i 0
33.4 0 3.27139 + 3.27139i 0 12.6449 12.6449i 0 13.8754i 0 5.59607i 0
33.5 0 5.96513 + 5.96513i 0 −8.67959 + 8.67959i 0 1.63924i 0 44.1656i 0
97.1 0 −5.49618 + 5.49618i 0 4.66372 + 4.66372i 0 24.8965i 0 33.4160i 0
97.2 0 −1.98356 + 1.98356i 0 0.596848 + 0.596848i 0 29.0828i 0 19.1310i 0
97.3 0 −0.756776 + 0.756776i 0 −8.22587 8.22587i 0 2.67171i 0 25.8546i 0
97.4 0 3.27139 3.27139i 0 12.6449 + 12.6449i 0 13.8754i 0 5.59607i 0
97.5 0 5.96513 5.96513i 0 −8.67959 8.67959i 0 1.63924i 0 44.1656i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.4.e.b 10
4.b odd 2 1 128.4.e.a 10
8.b even 2 1 16.4.e.a 10
8.d odd 2 1 64.4.e.a 10
16.e even 4 1 16.4.e.a 10
16.e even 4 1 inner 128.4.e.b 10
16.f odd 4 1 64.4.e.a 10
16.f odd 4 1 128.4.e.a 10
24.f even 2 1 576.4.k.a 10
24.h odd 2 1 144.4.k.a 10
32.g even 8 2 1024.4.a.n 10
32.g even 8 2 1024.4.b.j 10
32.h odd 8 2 1024.4.a.m 10
32.h odd 8 2 1024.4.b.k 10
48.i odd 4 1 144.4.k.a 10
48.k even 4 1 576.4.k.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a 10 8.b even 2 1
16.4.e.a 10 16.e even 4 1
64.4.e.a 10 8.d odd 2 1
64.4.e.a 10 16.f odd 4 1
128.4.e.a 10 4.b odd 2 1
128.4.e.a 10 16.f odd 4 1
128.4.e.b 10 1.a even 1 1 trivial
128.4.e.b 10 16.e even 4 1 inner
144.4.k.a 10 24.h odd 2 1
144.4.k.a 10 48.i odd 4 1
576.4.k.a 10 24.f even 2 1
576.4.k.a 10 48.k even 4 1
1024.4.a.m 10 32.h odd 8 2
1024.4.a.n 10 32.g even 8 2
1024.4.b.j 10 32.g even 8 2
1024.4.b.k 10 32.h odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{10} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(128, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 2 T + 2 T^{2} + 42 T^{3} - 571 T^{4} - 760 T^{5} + 3544 T^{6} - 66792 T^{7} + 57874 T^{8} + 3046228 T^{9} - 6701044 T^{10} + 82248156 T^{11} + 42190146 T^{12} - 1314666936 T^{13} + 1883426904 T^{14} - 10905169320 T^{15} - 221217099219 T^{16} + 439334834526 T^{17} + 564859072962 T^{18} - 15251194969974 T^{19} + 205891132094649 T^{20} \)
$5$ \( 1 - 2 T + 2 T^{2} + 966 T^{3} - 13723 T^{4} - 18040 T^{5} + 530104 T^{6} + 11981288 T^{7} - 28535006 T^{8} - 2301854348 T^{9} + 23672040908 T^{10} - 287731793500 T^{11} - 445859468750 T^{12} + 23400953125000 T^{13} + 129419921875000 T^{14} - 550537109375000 T^{15} - 52349090576171875 T^{16} + 460624694824218750 T^{17} + 119209289550781250 T^{18} - 14901161193847656250 T^{19} + \)\(93\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 - 1762 T^{2} + 1539965 T^{4} - 932087576 T^{6} + 440869947922 T^{8} - 168121217547916 T^{10} + 51867908503075378 T^{12} - 12901291835899914776 T^{14} + \)\(25\!\cdots\!85\)\( T^{16} - \)\(33\!\cdots\!62\)\( T^{18} + \)\(22\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 + 18 T + 162 T^{2} + 122934 T^{3} + 4077397 T^{4} - 79597000 T^{5} + 5463099864 T^{6} + 313798751208 T^{7} - 6887886337838 T^{8} - 101615185776500 T^{9} + 18755914132083020 T^{10} - 135249812268521500 T^{11} - 12202310808546625118 T^{12} + \)\(73\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!44\)\( T^{14} - \)\(33\!\cdots\!00\)\( T^{15} + \)\(22\!\cdots\!57\)\( T^{16} + \)\(90\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!42\)\( T^{18} + \)\(23\!\cdots\!78\)\( T^{19} + \)\(17\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 - 2 T + 2 T^{2} + 45206 T^{3} - 2401451 T^{4} - 29395960 T^{5} + 1085386040 T^{6} + 384228102440 T^{7} - 9400034983966 T^{8} - 1531455561616908 T^{9} + 23489689415409228 T^{10} - 3364607868872346876 T^{11} - 45372173460921944494 T^{12} + \)\(40\!\cdots\!20\)\( T^{13} + \)\(25\!\cdots\!40\)\( T^{14} - \)\(15\!\cdots\!20\)\( T^{15} - \)\(27\!\cdots\!79\)\( T^{16} + \)\(11\!\cdots\!78\)\( T^{17} + \)\(10\!\cdots\!22\)\( T^{18} - \)\(23\!\cdots\!34\)\( T^{19} + \)\(26\!\cdots\!49\)\( T^{20} \)
$17$ \( ( 1 + 2 T + 12653 T^{2} + 102520 T^{3} + 98460610 T^{4} + 354493580 T^{5} + 483736976930 T^{6} + 2474583573880 T^{7} + 1500492401316541 T^{8} + 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \)
$19$ \( 1 - 26 T + 338 T^{2} + 339906 T^{3} - 64153371 T^{4} - 2461461784 T^{5} + 143449890200 T^{6} - 36783398837960 T^{7} + 1011857007777554 T^{8} + 259766590630759364 T^{9} - 7366645907488092948 T^{10} + \)\(17\!\cdots\!76\)\( T^{11} + \)\(47\!\cdots\!74\)\( T^{12} - \)\(11\!\cdots\!40\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} - \)\(37\!\cdots\!16\)\( T^{15} - \)\(66\!\cdots\!11\)\( T^{16} + \)\(24\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!98\)\( T^{18} - \)\(87\!\cdots\!14\)\( T^{19} + \)\(23\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 - 76386 T^{2} + 2913757597 T^{4} - 73253961622040 T^{6} + 1342371312768300946 T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(19\!\cdots\!94\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{14} + \)\(94\!\cdots\!93\)\( T^{16} - \)\(36\!\cdots\!26\)\( T^{18} + \)\(71\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 - 202 T + 20402 T^{2} + 1177934 T^{3} + 398569397 T^{4} - 239164019416 T^{5} + 40873283338616 T^{6} - 2529271278095288 T^{7} + 194871598558001506 T^{8} - \)\(12\!\cdots\!32\)\( T^{9} + \)\(30\!\cdots\!36\)\( T^{10} - \)\(30\!\cdots\!48\)\( T^{11} + \)\(11\!\cdots\!26\)\( T^{12} - \)\(36\!\cdots\!72\)\( T^{13} + \)\(14\!\cdots\!56\)\( T^{14} - \)\(20\!\cdots\!84\)\( T^{15} + \)\(83\!\cdots\!17\)\( T^{16} + \)\(60\!\cdots\!86\)\( T^{17} + \)\(25\!\cdots\!62\)\( T^{18} - \)\(61\!\cdots\!18\)\( T^{19} + \)\(74\!\cdots\!01\)\( T^{20} \)
$31$ \( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \)
$37$ \( 1 - 10 T + 50 T^{2} - 1972962 T^{3} + 1630465317 T^{4} + 153991562664 T^{5} + 324850634232 T^{6} + 5252842710654600 T^{7} + 3474549392106364962 T^{8} - 27985624577691139772 T^{9} + \)\(10\!\cdots\!24\)\( T^{10} - \)\(14\!\cdots\!16\)\( T^{11} + \)\(89\!\cdots\!58\)\( T^{12} + \)\(68\!\cdots\!00\)\( T^{13} + \)\(21\!\cdots\!92\)\( T^{14} + \)\(51\!\cdots\!52\)\( T^{15} + \)\(27\!\cdots\!93\)\( T^{16} - \)\(16\!\cdots\!94\)\( T^{17} + \)\(21\!\cdots\!50\)\( T^{18} - \)\(21\!\cdots\!30\)\( T^{19} + \)\(11\!\cdots\!49\)\( T^{20} \)
$41$ \( 1 - 441018 T^{2} + 97166156061 T^{4} - 13934678680622904 T^{6} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(68\!\cdots\!74\)\( T^{12} - \)\(31\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!81\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(24\!\cdots\!01\)\( T^{20} \)
$43$ \( 1 - 838 T + 351122 T^{2} - 132133650 T^{3} + 56398378005 T^{4} - 20936462157416 T^{5} + 6471694737204248 T^{6} - 1952595983380873720 T^{7} + \)\(59\!\cdots\!10\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{9} + \)\(49\!\cdots\!52\)\( T^{10} - \)\(13\!\cdots\!76\)\( T^{11} + \)\(37\!\cdots\!90\)\( T^{12} - \)\(98\!\cdots\!60\)\( T^{13} + \)\(25\!\cdots\!48\)\( T^{14} - \)\(66\!\cdots\!12\)\( T^{15} + \)\(14\!\cdots\!45\)\( T^{16} - \)\(26\!\cdots\!50\)\( T^{17} + \)\(56\!\cdots\!22\)\( T^{18} - \)\(10\!\cdots\!66\)\( T^{19} + \)\(10\!\cdots\!49\)\( T^{20} \)
$47$ \( ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \)
$53$ \( 1 - 378 T + 71442 T^{2} + 52753550 T^{3} + 3016286341 T^{4} + 526686651752 T^{5} + 976891435665272 T^{6} + 1674349213754452168 T^{7} - 1581505854923305054 T^{8} + \)\(14\!\cdots\!20\)\( T^{9} + \)\(29\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} - \)\(35\!\cdots\!66\)\( T^{12} + \)\(55\!\cdots\!44\)\( T^{13} + \)\(47\!\cdots\!52\)\( T^{14} + \)\(38\!\cdots\!64\)\( T^{15} + \)\(32\!\cdots\!49\)\( T^{16} + \)\(85\!\cdots\!50\)\( T^{17} + \)\(17\!\cdots\!02\)\( T^{18} - \)\(13\!\cdots\!86\)\( T^{19} + \)\(53\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 + 1706 T + 1455218 T^{2} + 989315358 T^{3} + 555128806581 T^{4} + 232658117164632 T^{5} + 78453755015006616 T^{6} + 20092577710244830152 T^{7} + \)\(57\!\cdots\!98\)\( T^{8} - \)\(22\!\cdots\!08\)\( T^{9} - \)\(12\!\cdots\!80\)\( T^{10} - \)\(45\!\cdots\!32\)\( T^{11} + \)\(24\!\cdots\!18\)\( T^{12} + \)\(17\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!96\)\( T^{14} + \)\(85\!\cdots\!68\)\( T^{15} + \)\(41\!\cdots\!01\)\( T^{16} + \)\(15\!\cdots\!22\)\( T^{17} + \)\(46\!\cdots\!98\)\( T^{18} + \)\(11\!\cdots\!14\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 + 910 T + 414050 T^{2} - 45940410 T^{3} + 20471098485 T^{4} + 72823214590920 T^{5} + 58848327585507000 T^{6} + 6001380717052735080 T^{7} + \)\(59\!\cdots\!50\)\( T^{8} + \)\(32\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!00\)\( T^{10} + \)\(73\!\cdots\!00\)\( T^{11} + \)\(30\!\cdots\!50\)\( T^{12} + \)\(70\!\cdots\!80\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} + \)\(43\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!85\)\( T^{16} - \)\(14\!\cdots\!10\)\( T^{17} + \)\(29\!\cdots\!50\)\( T^{18} + \)\(14\!\cdots\!10\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 + 1942 T + 1885682 T^{2} + 1530965298 T^{3} + 1338066168261 T^{4} + 1068751594464168 T^{5} + 724275679990789656 T^{6} + \)\(46\!\cdots\!12\)\( T^{7} + \)\(29\!\cdots\!06\)\( T^{8} + \)\(17\!\cdots\!64\)\( T^{9} + \)\(97\!\cdots\!68\)\( T^{10} + \)\(52\!\cdots\!32\)\( T^{11} + \)\(26\!\cdots\!14\)\( T^{12} + \)\(12\!\cdots\!64\)\( T^{13} + \)\(59\!\cdots\!16\)\( T^{14} + \)\(26\!\cdots\!24\)\( T^{15} + \)\(99\!\cdots\!49\)\( T^{16} + \)\(34\!\cdots\!66\)\( T^{17} + \)\(12\!\cdots\!22\)\( T^{18} + \)\(39\!\cdots\!66\)\( T^{19} + \)\(60\!\cdots\!49\)\( T^{20} \)
$71$ \( 1 - 2500418 T^{2} + 3072516920573 T^{4} - 2420243241413642648 T^{6} + \)\(13\!\cdots\!66\)\( T^{8} - \)\(55\!\cdots\!32\)\( T^{10} + \)\(17\!\cdots\!86\)\( T^{12} - \)\(39\!\cdots\!68\)\( T^{14} + \)\(64\!\cdots\!53\)\( T^{16} - \)\(67\!\cdots\!58\)\( T^{18} + \)\(34\!\cdots\!01\)\( T^{20} \)
$73$ \( 1 - 3134282 T^{2} + 4627160201821 T^{4} - 4242124717558516472 T^{6} + \)\(26\!\cdots\!74\)\( T^{8} - \)\(12\!\cdots\!92\)\( T^{10} + \)\(40\!\cdots\!86\)\( T^{12} - \)\(97\!\cdots\!12\)\( T^{14} + \)\(16\!\cdots\!49\)\( T^{16} - \)\(16\!\cdots\!62\)\( T^{18} + \)\(79\!\cdots\!49\)\( T^{20} \)
$79$ \( ( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \)
$83$ \( 1 - 2562 T + 3281922 T^{2} - 2891460918 T^{3} + 1934799974629 T^{4} - 1191348439341176 T^{5} + 882645219418437336 T^{6} - \)\(79\!\cdots\!28\)\( T^{7} + \)\(89\!\cdots\!74\)\( T^{8} - \)\(97\!\cdots\!88\)\( T^{9} + \)\(82\!\cdots\!76\)\( T^{10} - \)\(55\!\cdots\!56\)\( T^{11} + \)\(29\!\cdots\!06\)\( T^{12} - \)\(14\!\cdots\!84\)\( T^{13} + \)\(94\!\cdots\!96\)\( T^{14} - \)\(72\!\cdots\!32\)\( T^{15} + \)\(67\!\cdots\!61\)\( T^{16} - \)\(57\!\cdots\!94\)\( T^{17} + \)\(37\!\cdots\!62\)\( T^{18} - \)\(16\!\cdots\!74\)\( T^{19} + \)\(37\!\cdots\!49\)\( T^{20} \)
$89$ \( 1 - 3643178 T^{2} + 6505439011133 T^{4} - 7720859292932177528 T^{6} + \)\(69\!\cdots\!98\)\( T^{8} - \)\(52\!\cdots\!28\)\( T^{10} + \)\(34\!\cdots\!78\)\( T^{12} - \)\(19\!\cdots\!88\)\( T^{14} + \)\(79\!\cdots\!73\)\( T^{16} - \)\(22\!\cdots\!98\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \)
$97$ \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \)
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