↑

Properties

Label 448.2.m.d
Level $448$
Weight $2$
Character orbit 448.m
Analytic conductor $3.577$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.20138089353117696.1
Defining polynomial: \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{5} - \beta_{3} q^{7} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{9} + \beta_{8} q^{11} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_1) q^{13} + (\beta_{11} + \beta_{6} + \beta_{5} + 3) q^{15} + (\beta_{10} - 1) q^{17} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{19} - \beta_{5} q^{21} + ( - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{3} - \beta_1) q^{23} + ( - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{3} - \beta_1) q^{25} + (\beta_{10} - \beta_{8} + \beta_{7} - 2 \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{27} + (\beta_{11} + \beta_1) q^{29} + ( - \beta_{6} - \beta_{5}) q^{31} + (\beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 1) q^{33} + (\beta_{6} - \beta_{4} - \beta_{3} + 1) q^{35} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{37} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_1) q^{39} + (\beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2}) q^{41} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 2 \beta_{3} + \beta_1 - 2) q^{43} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} + 3 \beta_{6} - 5 \beta_{3} + \beta_1 + 5) q^{45} + ( - 2 \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 2) q^{47} - q^{49} + ( - \beta_{11} - \beta_{3} - \beta_1 + 1) q^{51} + ( - \beta_{10} + 2 \beta_{8} - \beta_{7} - 2 \beta_{2}) q^{53} + (\beta_{9} - \beta_{8} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{55} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - \beta_1) q^{57} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{59} + ( - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - 2) q^{61} + ( - \beta_{11} + \beta_{4} - \beta_{2} - 2) q^{63} + (\beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{65} + (\beta_{11} + \beta_{9} - 2 \beta_{4} + \beta_{3} + \beta_1 - 1) q^{67} + (\beta_{11} - \beta_{8} + 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{69} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_1) q^{71} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{73} + (2 \beta_{11} - \beta_{8} + 3 \beta_{5} + 6 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 6) q^{75} + \beta_{9} q^{77} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - 1) q^{79} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - 3) q^{81} + (2 \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_1 + 3) q^{83} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{7} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{85} + (\beta_{9} - \beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - \beta_{4} - 6 \beta_{3} - \beta_{2}) q^{87} + (\beta_{9} - \beta_{8} + \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{89} + (\beta_{11} + \beta_{8} - \beta_{5} - \beta_1) q^{91} + ( - \beta_{11} + 2 \beta_{4} + 5 \beta_{3} - \beta_1 - 5) q^{93} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \cdots - 2) q^{95}+ \cdots + ( - 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 3 \beta_{3} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 4 q^{5} + 24 q^{15} - 8 q^{17} + 4 q^{21} - 4 q^{27} - 4 q^{29} + 8 q^{31} + 4 q^{35} - 20 q^{37} - 16 q^{43} + 40 q^{45} - 16 q^{47} - 12 q^{49} + 16 q^{51} + 4 q^{53} + 16 q^{59} - 20 q^{61} - 12 q^{63} + 32 q^{65} - 24 q^{67} - 4 q^{69} + 40 q^{75} - 24 q^{79} - 44 q^{81} + 20 q^{83} - 8 q^{85} - 48 q^{93} + 48 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 14 \nu^{10} + \nu^{9} - 20 \nu^{8} - 54 \nu^{7} + 16 \nu^{6} + 34 \nu^{5} + 36 \nu^{4} + 192 \nu^{3} + 240 \nu^{2} - 144 \nu - 608 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3 \nu^{11} + 10 \nu^{10} + 3 \nu^{9} - 28 \nu^{8} - 34 \nu^{7} - 16 \nu^{6} - 26 \nu^{5} + 44 \nu^{4} + 192 \nu^{3} + 208 \nu^{2} - 176 \nu - 416 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + 64 \nu^{3} + 112 \nu^{2} + 48 \nu + 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{11} - 10 \nu^{10} + 21 \nu^{9} + 44 \nu^{8} + 18 \nu^{7} - 16 \nu^{6} + 10 \nu^{5} - 108 \nu^{4} - 256 \nu^{3} - 80 \nu^{2} + 304 \nu + 416 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{11} - 14 \nu^{10} + 3 \nu^{9} + 28 \nu^{8} + 38 \nu^{7} + 16 \nu^{6} + 6 \nu^{5} - 36 \nu^{4} - 176 \nu^{3} - 176 \nu^{2} + 80 \nu + 352 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} - \nu^{10} + 2\nu^{9} + 5\nu^{8} + 3\nu^{7} - 4\nu^{5} - 14\nu^{4} - 26\nu^{3} - 8\nu^{2} + 40\nu + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21 \nu^{11} + 54 \nu^{10} + 21 \nu^{9} - 52 \nu^{8} - 78 \nu^{7} - 112 \nu^{6} - 118 \nu^{5} + 84 \nu^{4} + 448 \nu^{3} + 752 \nu^{2} + 176 \nu - 480 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23 \nu^{11} - 18 \nu^{10} + 41 \nu^{9} + 60 \nu^{8} + 58 \nu^{7} + 48 \nu^{6} - 14 \nu^{5} - 220 \nu^{4} - 448 \nu^{3} - 144 \nu^{2} + 624 \nu + 544 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23 \nu^{11} - 34 \nu^{10} + 41 \nu^{9} + 108 \nu^{8} + 90 \nu^{7} + 16 \nu^{6} - 78 \nu^{5} - 252 \nu^{4} - 576 \nu^{3} - 528 \nu^{2} + 880 \nu + 1312 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{11} - 22 \nu^{10} + 31 \nu^{9} + 76 \nu^{8} + 54 \nu^{7} - 34 \nu^{5} - 180 \nu^{4} - 416 \nu^{3} - 176 \nu^{2} + 720 \nu + 800 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19 \nu^{11} + 18 \nu^{10} - 29 \nu^{9} - 68 \nu^{8} - 66 \nu^{7} - 16 \nu^{6} + 38 \nu^{5} + 220 \nu^{4} + 352 \nu^{3} + 240 \nu^{2} - 560 \nu - 736 ) / 64 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - 2\beta_{9} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + \beta_{10} - 2\beta_{8} - 3\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - 8 \beta_{3} + \beta_{2} + 2 \beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{11} + \beta_{10} - 2\beta_{9} - 7\beta_{6} + \beta_{5} + 5\beta_{4} + 6\beta_{3} - 5\beta_{2} + 2\beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{11} - \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 6 \beta _1 + 11 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6 \beta_{11} + \beta_{10} - 6 \beta_{9} + 4 \beta_{7} - 7 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta _1 - 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 12 \beta_{8} + 4 \beta_{7} + 7 \beta_{6} + 15 \beta_{5} - \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 18 \beta _1 - 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2 \beta_{11} - 7 \beta_{10} + 14 \beta_{9} + 12 \beta_{8} + 4 \beta_{7} - 23 \beta_{6} + 17 \beta_{5} + 17 \beta_{4} + 30 \beta_{3} + 11 \beta_{2} + 6 \beta _1 - 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2 \beta_{11} - 21 \beta_{10} + 10 \beta_{9} + 4 \beta_{8} + 16 \beta_{7} + 15 \beta_{6} - \beta_{5} + 7 \beta_{4} - 50 \beta_{3} - 35 \beta_{2} + 10 \beta _1 + 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10 \beta_{11} - 7 \beta_{10} + 6 \beta_{9} + 12 \beta_{8} - 4 \beta_{7} - 7 \beta_{6} + 17 \beta_{5} + \beta_{4} + 110 \beta_{3} - 29 \beta_{2} - 2 \beta _1 - 111 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.35309 − 0.411286i
−0.605558 − 1.27801i
0.402577 + 1.35570i
−1.12465 + 0.857418i
1.37925 + 0.312504i
−1.40471 + 0.163666i
1.35309 + 0.411286i
−0.605558 + 1.27801i
0.402577 − 1.35570i
−1.12465 − 0.857418i
1.37925 − 0.312504i
−1.40471 − 0.163666i
0 −2.21570 + 2.21570i 0 −0.393125 − 0.393125i 0 − 1.00000i 0 − 6.81864i 0
113.2 0 −1.39123 + 1.39123i 0 2.16478 + 2.16478i 0 − 1.00000i 0 − 0.871066i 0
113.3 0 −0.631188 + 0.631188i 0 −2.34259 − 2.34259i 0 − 1.00000i 0 2.20320i 0
113.4 0 −0.416854 + 0.416854i 0 −1.13169 − 1.13169i 0 − 1.00000i 0 2.65247i 0
113.5 0 0.599978 − 0.599978i 0 0.974969 + 0.974969i 0 − 1.00000i 0 2.28005i 0
113.6 0 2.05500 − 2.05500i 0 2.72766 + 2.72766i 0 − 1.00000i 0 − 5.44602i 0
337.1 0 −2.21570 − 2.21570i 0 −0.393125 + 0.393125i 0 1.00000i 0 6.81864i 0
337.2 0 −1.39123 − 1.39123i 0 2.16478 − 2.16478i 0 1.00000i 0 0.871066i 0
337.3 0 −0.631188 − 0.631188i 0 −2.34259 + 2.34259i 0 1.00000i 0 − 2.20320i 0
337.4 0 −0.416854 − 0.416854i 0 −1.13169 + 1.13169i 0 1.00000i 0 − 2.65247i 0
337.5 0 0.599978 + 0.599978i 0 0.974969 − 0.974969i 0 1.00000i 0 − 2.28005i 0
337.6 0 2.05500 + 2.05500i 0 2.72766 − 2.72766i 0 1.00000i 0 5.44602i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.6
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 448.2.m.d 12
4.b odd 2 1 112.2.m.d ✓ 12
8.b even 2 1 896.2.m.h 12
8.d odd 2 1 896.2.m.g 12
16.e even 4 1 inner 448.2.m.d 12
16.e even 4 1 896.2.m.h 12
16.f odd 4 1 112.2.m.d ✓ 12
16.f odd 4 1 896.2.m.g 12
28.d even 2 1 784.2.m.h 12
28.f even 6 2 784.2.x.m 24
28.g odd 6 2 784.2.x.l 24
32.g even 8 2 7168.2.a.bi 12
32.h odd 8 2 7168.2.a.bj 12
112.j even 4 1 784.2.m.h 12
112.u odd 12 2 784.2.x.l 24
112.v even 12 2 784.2.x.m 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d ✓ 12 4.b odd 2 1
112.2.m.d ✓ 12 16.f odd 4 1
448.2.m.d 12 1.a even 1 1 trivial
448.2.m.d 12 16.e even 4 1 inner
784.2.m.h 12 28.d even 2 1
784.2.m.h 12 112.j even 4 1
784.2.x.l 24 28.g odd 6 2
784.2.x.l 24 112.u odd 12 2
784.2.x.m 24 28.f even 6 2
784.2.x.m 24 112.v even 12 2
896.2.m.g 12 8.d odd 2 1
896.2.m.g 12 16.f odd 4 1
896.2.m.h 12 8.b even 2 1
896.2.m.h 12 16.e even 4 1
7168.2.a.bi 12 32.g even 8 2
7168.2.a.bj 12 32.h odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 4 T_{3}^{11} + 8 T_{3}^{10} + 4 T_{3}^{9} + 76 T_{3}^{8} + 288 T_{3}^{7} + 552 T_{3}^{6} + 376 T_{3}^{5} + 164 T_{3}^{4} + 144 T_{3}^{3} + 288 T_{3}^{2} + 192 T_{3} + 64 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + 8 T^{10} + 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + 8 T^{10} + 12 T^{9} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} - 32 T^{9} + 740 T^{8} + \cdots + 541696 \) Copy content Toggle raw display
$13$ \( T^{12} + 20 T^{9} + 1724 T^{8} + \cdots + 3211264 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} - 44 T^{4} - 200 T^{3} + \cdots - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 76 T^{9} + 2444 T^{8} + \cdots + 2849344 \) Copy content Toggle raw display
$23$ \( T^{12} + 136 T^{10} + \cdots + 10137856 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + 8 T^{10} + \cdots + 8620096 \) Copy content Toggle raw display
$31$ \( (T^{6} - 4 T^{5} - 16 T^{4} + 72 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 20 T^{11} + 200 T^{10} + \cdots + 5053504 \) Copy content Toggle raw display
$41$ \( T^{12} + 120 T^{10} + 4704 T^{8} + \cdots + 25600 \) Copy content Toggle raw display
$43$ \( T^{12} + 16 T^{11} + 128 T^{10} + \cdots + 23040000 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{5} - 104 T^{4} - 1032 T^{3} + \cdots + 19776)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 4 T^{11} + 8 T^{10} + \cdots + 78400 \) Copy content Toggle raw display
$59$ \( T^{12} - 16 T^{11} + \cdots + 119596096 \) Copy content Toggle raw display
$61$ \( T^{12} + 20 T^{11} + 200 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{12} + 24 T^{11} + 288 T^{10} + \cdots + 3686400 \) Copy content Toggle raw display
$71$ \( T^{12} + 432 T^{10} + 60960 T^{8} + \cdots + 6553600 \) Copy content Toggle raw display
$73$ \( T^{12} + 616 T^{10} + \cdots + 3050573824 \) Copy content Toggle raw display
$79$ \( (T^{6} + 12 T^{5} - 44 T^{4} - 576 T^{3} + \cdots - 2240)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 20 T^{11} + \cdots + 320093429824 \) Copy content Toggle raw display
$89$ \( T^{12} + 552 T^{10} + \cdots + 35476475904 \) Copy content Toggle raw display
$97$ \( (T^{6} - 24 T^{5} - 60 T^{4} + 2568 T^{3} + \cdots - 39008)^{2} \) Copy content Toggle raw display
show more
show less