Properties

Label 39.2.k.b
Level $39$
Weight $2$
Character orbit 39.k
Analytic conductor $0.311$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.56070144.2
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{6} - 2 \beta_{5} - 2) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 2) q^{6} + ( - \beta_{6} - \beta_{5}) q^{7} + ( - 2 \beta_{7} - 3 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{8} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{6} - 2 \beta_{5} - 2) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{5} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 2) q^{6} + ( - \beta_{6} - \beta_{5}) q^{7} + ( - 2 \beta_{7} - 3 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{8} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{9} + (\beta_{6} + 4 \beta_{4} + 1) q^{10} + ( - 2 \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_1 + 1) q^{11} + (2 \beta_{6} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{12} + ( - 2 \beta_{6} - 3 \beta_{4} + 2) q^{13} + ( - \beta_{5} + \beta_{4} + 2 \beta_1 - 1) q^{14} + (2 \beta_{6} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{15} + (\beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 1) q^{16} + (3 \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{17} + (2 \beta_{7} - 5 \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 1) q^{18} + (2 \beta_{4} - 2) q^{19} + (3 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{20} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{21} + (\beta_{6} + 3 \beta_{5} - 6 \beta_{4}) q^{22} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{24} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} + 1) q^{25} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 5 \beta_{2}) q^{26} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_1 + 2) q^{27} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} + 3) q^{28} + ( - \beta_{7} + 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 2) q^{29} + ( - 4 \beta_{7} - 4 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{30} + (4 \beta_{6} + \beta_{5} + 3 \beta_{4} - 1) q^{31} + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{32} + (\beta_{7} - 4 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{33} + ( - \beta_{6} - 4 \beta_{5} + 5 \beta_{4} - 4) q^{34} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2}) q^{35} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{36} + ( - 3 \beta_{6} + 5 \beta_{5} - 3 \beta_{4} - 2) q^{37} + ( - 2 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{38} + (3 \beta_{7} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 3) q^{39} + ( - 4 \beta_{5} - 4 \beta_{4} - 3) q^{40} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{41} + ( - \beta_{7} + 5 \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 4) q^{42} + ( - 3 \beta_{6} + 3 \beta_{5} + 6) q^{43} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 1) q^{44} + (2 \beta_{6} + 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{45} + (4 \beta_{3} + 4 \beta_1 - 4) q^{47} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} - \beta_{2} - 2) q^{48} - 5 \beta_{4} q^{49} + (2 \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_1 - 1) q^{50} + (3 \beta_{6} + 9 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{51} + (4 \beta_{6} - \beta_{5} + 7 \beta_{4} + 4) q^{52} + (2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{53} + (2 \beta_{7} + \beta_{6} + 8 \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{54} + ( - \beta_{6} - 6 \beta_{5} + 3 \beta_{4} + 1) q^{55} + (\beta_{6} + 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 4 \beta_1 + 2) q^{56} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 4) q^{57} + (9 \beta_{6} - 9 \beta_{5} + \beta_{4} - 1) q^{58} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{2}) q^{59} + (\beta_{7} - 6 \beta_{6} + 10 \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_1 + 9) q^{60} + 7 \beta_{6} q^{61} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \cdots + 1) q^{62}+ \cdots + (4 \beta_{6} + 3 \beta_{5} - 7 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 12 q^{4} - 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} - 12 q^{4} - 2 q^{6} - 4 q^{7} + 4 q^{9} + 12 q^{10} + 8 q^{13} - 14 q^{15} - 4 q^{16} + 4 q^{18} - 16 q^{19} + 4 q^{21} + 4 q^{22} + 18 q^{24} + 4 q^{27} + 20 q^{28} + 18 q^{30} + 8 q^{31} + 16 q^{33} - 36 q^{34} - 36 q^{36} - 28 q^{37} - 14 q^{39} - 24 q^{40} - 16 q^{42} + 36 q^{43} - 20 q^{45} - 14 q^{48} + 48 q^{52} + 46 q^{54} + 4 q^{55} + 16 q^{57} + 28 q^{58} + 44 q^{60} + 28 q^{61} - 8 q^{63} - 40 q^{66} - 40 q^{67} - 32 q^{70} + 12 q^{72} - 28 q^{73} + 12 q^{75} - 8 q^{76} - 80 q^{78} + 16 q^{79} + 4 q^{81} - 24 q^{82} + 4 q^{84} + 12 q^{85} - 34 q^{87} - 12 q^{88} + 8 q^{91} + 4 q^{93} + 64 q^{94} + 16 q^{96} + 20 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(\beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.500000 1.19293i
0.500000 + 2.19293i
0.500000 1.56488i
0.500000 + 0.564882i
0.500000 + 1.19293i
0.500000 2.19293i
0.500000 + 1.56488i
0.500000 0.564882i
−0.619657 + 2.31259i 1.64914 0.529480i −3.23205 1.86603i −1.69293 1.69293i 0.202571 + 4.14187i −1.36603 + 0.366025i 2.93225 2.93225i 2.43930 1.74637i 4.96410 2.86603i
2.2 0.619657 2.31259i −1.28311 + 1.16345i −3.23205 1.86603i 1.69293 + 1.69293i 1.89551 + 3.68825i −1.36603 + 0.366025i −2.93225 + 2.93225i 0.292748 2.98568i 4.96410 2.86603i
11.1 −1.45466 0.389774i 0.239203 1.71545i 0.232051 + 0.133975i 1.06488 1.06488i −1.01660 + 2.40216i 0.366025 + 1.36603i 1.84443 + 1.84443i −2.88556 0.820682i −1.96410 + 1.13397i
11.2 1.45466 + 0.389774i −1.60523 0.650571i 0.232051 + 0.133975i −1.06488 + 1.06488i −2.08148 1.57203i 0.366025 + 1.36603i −1.84443 1.84443i 2.15351 + 2.08863i −1.96410 + 1.13397i
20.1 −0.619657 2.31259i 1.64914 + 0.529480i −3.23205 + 1.86603i −1.69293 + 1.69293i 0.202571 4.14187i −1.36603 0.366025i 2.93225 + 2.93225i 2.43930 + 1.74637i 4.96410 + 2.86603i
20.2 0.619657 + 2.31259i −1.28311 1.16345i −3.23205 + 1.86603i 1.69293 1.69293i 1.89551 3.68825i −1.36603 0.366025i −2.93225 2.93225i 0.292748 + 2.98568i 4.96410 + 2.86603i
32.1 −1.45466 + 0.389774i 0.239203 + 1.71545i 0.232051 0.133975i 1.06488 + 1.06488i −1.01660 2.40216i 0.366025 1.36603i 1.84443 1.84443i −2.88556 + 0.820682i −1.96410 1.13397i
32.2 1.45466 0.389774i −1.60523 + 0.650571i 0.232051 0.133975i −1.06488 1.06488i −2.08148 + 1.57203i 0.366025 1.36603i −1.84443 + 1.84443i 2.15351 2.08863i −1.96410 1.13397i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.k.b 8
3.b odd 2 1 inner 39.2.k.b 8
4.b odd 2 1 624.2.cn.c 8
5.b even 2 1 975.2.bo.d 8
5.c odd 4 1 975.2.bp.e 8
5.c odd 4 1 975.2.bp.f 8
12.b even 2 1 624.2.cn.c 8
13.b even 2 1 507.2.k.d 8
13.c even 3 1 507.2.f.f 8
13.c even 3 1 507.2.k.e 8
13.d odd 4 1 507.2.k.e 8
13.d odd 4 1 507.2.k.f 8
13.e even 6 1 507.2.f.e 8
13.e even 6 1 507.2.k.f 8
13.f odd 12 1 inner 39.2.k.b 8
13.f odd 12 1 507.2.f.e 8
13.f odd 12 1 507.2.f.f 8
13.f odd 12 1 507.2.k.d 8
15.d odd 2 1 975.2.bo.d 8
15.e even 4 1 975.2.bp.e 8
15.e even 4 1 975.2.bp.f 8
39.d odd 2 1 507.2.k.d 8
39.f even 4 1 507.2.k.e 8
39.f even 4 1 507.2.k.f 8
39.h odd 6 1 507.2.f.e 8
39.h odd 6 1 507.2.k.f 8
39.i odd 6 1 507.2.f.f 8
39.i odd 6 1 507.2.k.e 8
39.k even 12 1 inner 39.2.k.b 8
39.k even 12 1 507.2.f.e 8
39.k even 12 1 507.2.f.f 8
39.k even 12 1 507.2.k.d 8
52.l even 12 1 624.2.cn.c 8
65.o even 12 1 975.2.bp.f 8
65.s odd 12 1 975.2.bo.d 8
65.t even 12 1 975.2.bp.e 8
156.v odd 12 1 624.2.cn.c 8
195.bc odd 12 1 975.2.bp.e 8
195.bh even 12 1 975.2.bo.d 8
195.bn odd 12 1 975.2.bp.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.b 8 1.a even 1 1 trivial
39.2.k.b 8 3.b odd 2 1 inner
39.2.k.b 8 13.f odd 12 1 inner
39.2.k.b 8 39.k even 12 1 inner
507.2.f.e 8 13.e even 6 1
507.2.f.e 8 13.f odd 12 1
507.2.f.e 8 39.h odd 6 1
507.2.f.e 8 39.k even 12 1
507.2.f.f 8 13.c even 3 1
507.2.f.f 8 13.f odd 12 1
507.2.f.f 8 39.i odd 6 1
507.2.f.f 8 39.k even 12 1
507.2.k.d 8 13.b even 2 1
507.2.k.d 8 13.f odd 12 1
507.2.k.d 8 39.d odd 2 1
507.2.k.d 8 39.k even 12 1
507.2.k.e 8 13.c even 3 1
507.2.k.e 8 13.d odd 4 1
507.2.k.e 8 39.f even 4 1
507.2.k.e 8 39.i odd 6 1
507.2.k.f 8 13.d odd 4 1
507.2.k.f 8 13.e even 6 1
507.2.k.f 8 39.f even 4 1
507.2.k.f 8 39.h odd 6 1
624.2.cn.c 8 4.b odd 2 1
624.2.cn.c 8 12.b even 2 1
624.2.cn.c 8 52.l even 12 1
624.2.cn.c 8 156.v odd 12 1
975.2.bo.d 8 5.b even 2 1
975.2.bo.d 8 15.d odd 2 1
975.2.bo.d 8 65.s odd 12 1
975.2.bo.d 8 195.bh even 12 1
975.2.bp.e 8 5.c odd 4 1
975.2.bp.e 8 15.e even 4 1
975.2.bp.e 8 65.t even 12 1
975.2.bp.e 8 195.bc odd 12 1
975.2.bp.f 8 5.c odd 4 1
975.2.bp.f 8 15.e even 4 1
975.2.bp.f 8 65.o even 12 1
975.2.bp.f 8 195.bn odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 6T_{2}^{6} - T_{2}^{4} - 78T_{2}^{2} + 169 \) acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{6} - T^{4} - 78 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} - 4 T^{5} - 5 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 38T^{4} + 169 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 24 T^{6} + 140 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 30 T^{6} + 783 T^{4} + \cdots + 13689 \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} + 20 T^{2} + 16 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 82 T^{6} + 5151 T^{4} + \cdots + 2474329 \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + 8 T^{2} + 88 T + 484)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 14 T^{3} + 113 T^{2} + 592 T + 1369)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 54 T^{6} + 959 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$43$ \( (T^{4} - 18 T^{3} + 126 T^{2} - 324 T + 324)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 9728 T^{4} + \cdots + 11075584 \) Copy content Toggle raw display
$53$ \( (T^{4} + 22 T^{2} + 13)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{6} - 16 T^{4} + \cdots + 43264 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 20 T^{3} + 164 T^{2} + 832 T + 2704)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 24 T^{6} - 16 T^{4} + \cdots + 43264 \) Copy content Toggle raw display
$73$ \( (T^{4} + 14 T^{3} + 98 T^{2} + 154 T + 121)^{2} \) Copy content Toggle raw display
$79$ \( (T - 2)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 296T^{4} + 2704 \) Copy content Toggle raw display
$89$ \( T^{8} - 24 T^{6} - 8596 T^{4} + \cdots + 77228944 \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + 194 T^{2} - 572 T + 484)^{2} \) Copy content Toggle raw display
show more
show less