Properties

Label 784.2.m.h
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26027151847\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{10} + \beta_{2}) q^{3} + \beta_1 q^{4} + ( - \beta_{9} + \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{11} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{11} + \beta_{6} - \beta_{4} - 1) q^{8} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{10} + \beta_{2}) q^{3} + \beta_1 q^{4} + ( - \beta_{9} + \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{5} + (\beta_{11} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{6} + ( - \beta_{11} + \beta_{6} - \beta_{4} - 1) q^{8} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{8} + \beta_{6} + \beta_{4} + \beta_1 + 1) q^{10} + (\beta_{10} - \beta_{9} + \beta_{5} - 2 \beta_1 - 2) q^{11} + (2 \beta_{10} + \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{12} + ( - \beta_{9} + 2 \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{13} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 2 \beta_{2} + \beta_1 - 1) q^{15} + ( - \beta_{11} - 2 \beta_{10} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{16} + (\beta_{10} - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2}) q^{17} + (2 \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{18} + (2 \beta_{10} + \beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_{2}) q^{19} + ( - \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 3 \beta_{3} + \beta_{2} + 2) q^{20} + (\beta_{11} - \beta_{8} - \beta_{6} + \beta_{5} + 2) q^{22} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_1 - 1) q^{23} + ( - \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_1) q^{24} + ( - \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{25} + ( - 4 \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - 1) q^{26}+ \cdots + ( - \beta_{10} - \beta_{9} + 4 \beta_{8} + 2 \beta_{7} + \beta_{6} - 2 \beta_{4} - 4 \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 4 q^{3} - 6 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 4 q^{3} - 6 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{10} - 8 q^{12} - 24 q^{15} + 10 q^{16} + 8 q^{17} + 20 q^{20} + 14 q^{22} + 8 q^{24} + 20 q^{26} - 4 q^{27} - 4 q^{29} - 28 q^{30} + 8 q^{31} + 12 q^{32} - 8 q^{34} - 16 q^{36} - 20 q^{37} - 16 q^{38} + 8 q^{40} + 16 q^{43} + 14 q^{44} - 40 q^{45} - 28 q^{46} - 16 q^{47} - 16 q^{48} + 44 q^{50} - 16 q^{51} + 16 q^{52} + 4 q^{53} - 64 q^{54} + 14 q^{58} + 16 q^{59} + 60 q^{60} + 20 q^{61} - 8 q^{62} - 18 q^{64} + 32 q^{65} - 12 q^{66} + 24 q^{67} + 28 q^{68} + 4 q^{69} + 6 q^{72} - 38 q^{74} + 40 q^{75} - 48 q^{76} - 76 q^{78} + 24 q^{79} - 24 q^{80} - 44 q^{81} + 16 q^{82} + 20 q^{83} - 8 q^{85} + 38 q^{86} - 14 q^{88} + 40 q^{90} + 32 q^{92} - 48 q^{93} + 24 q^{94} + 16 q^{96} - 48 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 3\nu^{8} - 2\nu^{7} + 2\nu^{6} + 4\nu^{5} + 2\nu^{4} + 8\nu^{3} + 8\nu^{2} - 16\nu - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 10 \nu^{10} + 15 \nu^{9} + 12 \nu^{8} - 10 \nu^{7} - 32 \nu^{6} - 34 \nu^{5} - 52 \nu^{4} - 32 \nu^{3} + 144 \nu^{2} + 272 \nu + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 3\nu^{9} - 2\nu^{8} + 2\nu^{7} + 4\nu^{6} + 2\nu^{5} + 8\nu^{4} + 8\nu^{3} - 16\nu^{2} - 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + 64 \nu^{3} + 112 \nu^{2} + 48 \nu + 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} - 12 \nu^{4} - 96 \nu^{3} - 144 \nu^{2} - 80 \nu + 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} - 12 \nu^{4} - 96 \nu^{3} - 144 \nu^{2} + 48 \nu + 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + \nu^{10} - 3 \nu^{9} - 5 \nu^{8} - 4 \nu^{7} + 2 \nu^{6} + 6 \nu^{5} + 10 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 48 \nu - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13 \nu^{11} + 14 \nu^{10} - 19 \nu^{9} - 44 \nu^{8} - 46 \nu^{7} + 26 \nu^{5} + 196 \nu^{4} + 256 \nu^{3} + 112 \nu^{2} - 336 \nu - 480 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} - 14 \nu^{10} + 9 \nu^{9} + 32 \nu^{8} + 34 \nu^{7} + 8 \nu^{6} + 2 \nu^{5} - 52 \nu^{4} - 192 \nu^{3} - 144 \nu^{2} + 176 \nu + 352 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} + 26 \nu^{10} - 17 \nu^{9} - 68 \nu^{8} - 58 \nu^{7} - 32 \nu^{6} + 30 \nu^{5} + 172 \nu^{4} + 384 \nu^{3} + 272 \nu^{2} - 368 \nu - 672 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{11} - 8 \nu^{10} + 9 \nu^{9} + 30 \nu^{8} + 22 \nu^{7} + 4 \nu^{6} - 6 \nu^{5} - 72 \nu^{4} - 144 \nu^{3} - 96 \nu^{2} + 208 \nu + 256 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + 2\beta_{7} - \beta_{6} - \beta_{5} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} - \beta_{6} - \beta_{5} + 2\beta_{4} - 4\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{9} + 2\beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{11} + 4\beta_{10} + 2\beta_{9} + 2\beta_{7} + \beta_{6} + \beta_{5} + 4\beta_{4} + 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 6 \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} + 8 \beta_{10} + 10 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} + 12 \beta_{3} - 4 \beta_{2} + 2 \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 8 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + 20 \beta_{4} - 4 \beta_{3} - 12 \beta_{2} + 14 \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} - \beta_{6} + 3 \beta_{5} - 12 \beta_{4} + 28 \beta_{3} + 20 \beta_{2} + 10 \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 5 \beta_{5} + 52 \beta_{4} + 20 \beta_{3} + 12 \beta_{2} + 6 \beta _1 - 58 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−0.605558 1.27801i
1.35309 0.411286i
−1.40471 + 0.163666i
1.37925 + 0.312504i
−1.12465 + 0.857418i
0.402577 + 1.35570i
−0.605558 + 1.27801i
1.35309 + 0.411286i
−1.40471 0.163666i
1.37925 0.312504i
−1.12465 0.857418i
0.402577 1.35570i
−1.27801 0.605558i −1.39123 + 1.39123i 1.26660 + 1.54781i −2.16478 2.16478i 2.62048 0.935533i 0 −0.681431 2.74511i 0.871066i 1.45570 + 4.07750i
197.2 −0.411286 + 1.35309i −2.21570 + 2.21570i −1.66169 1.11301i 0.393125 + 0.393125i −2.08675 3.90932i 0 2.18943 1.79064i 6.81864i −0.693620 + 0.370246i
197.3 0.163666 1.40471i 2.05500 2.05500i −1.94643 0.459808i −2.72766 2.72766i −2.55034 3.22301i 0 −0.964462 + 2.65891i 5.44602i −4.27801 + 3.38515i
197.4 0.312504 + 1.37925i 0.599978 0.599978i −1.80468 + 0.862045i −0.974969 0.974969i 1.01502 + 0.640026i 0 −1.75295 2.21972i 2.28005i 1.04005 1.64941i
197.5 0.857418 1.12465i −0.416854 + 0.416854i −0.529667 1.92859i 1.13169 + 1.13169i 0.111396 + 0.826233i 0 −2.62313 1.05792i 2.65247i 2.24309 0.302422i
197.6 1.35570 + 0.402577i −0.631188 + 0.631188i 1.67586 + 1.09155i 2.34259 + 2.34259i −1.10981 + 0.601602i 0 1.83254 + 2.15448i 2.20320i 2.23279 + 4.11894i
589.1 −1.27801 + 0.605558i −1.39123 1.39123i 1.26660 1.54781i −2.16478 + 2.16478i 2.62048 + 0.935533i 0 −0.681431 + 2.74511i 0.871066i 1.45570 4.07750i
589.2 −0.411286 1.35309i −2.21570 2.21570i −1.66169 + 1.11301i 0.393125 0.393125i −2.08675 + 3.90932i 0 2.18943 + 1.79064i 6.81864i −0.693620 0.370246i
589.3 0.163666 + 1.40471i 2.05500 + 2.05500i −1.94643 + 0.459808i −2.72766 + 2.72766i −2.55034 + 3.22301i 0 −0.964462 2.65891i 5.44602i −4.27801 3.38515i
589.4 0.312504 1.37925i 0.599978 + 0.599978i −1.80468 0.862045i −0.974969 + 0.974969i 1.01502 0.640026i 0 −1.75295 + 2.21972i 2.28005i 1.04005 + 1.64941i
589.5 0.857418 + 1.12465i −0.416854 0.416854i −0.529667 + 1.92859i 1.13169 1.13169i 0.111396 0.826233i 0 −2.62313 + 1.05792i 2.65247i 2.24309 + 0.302422i
589.6 1.35570 0.402577i −0.631188 0.631188i 1.67586 1.09155i 2.34259 2.34259i −1.10981 0.601602i 0 1.83254 2.15448i 2.20320i 2.23279 4.11894i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.6
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 784.2.m.h 12
7.b odd 2 1 112.2.m.d 12
7.c even 3 2 784.2.x.m 24
7.d odd 6 2 784.2.x.l 24
16.e even 4 1 inner 784.2.m.h 12
28.d even 2 1 448.2.m.d 12
56.e even 2 1 896.2.m.h 12
56.h odd 2 1 896.2.m.g 12
112.j even 4 1 448.2.m.d 12
112.j even 4 1 896.2.m.h 12
112.l odd 4 1 112.2.m.d 12
112.l odd 4 1 896.2.m.g 12
112.w even 12 2 784.2.x.m 24
112.x odd 12 2 784.2.x.l 24
224.v odd 8 2 7168.2.a.bj 12
224.x even 8 2 7168.2.a.bi 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d 12 7.b odd 2 1
112.2.m.d 12 112.l odd 4 1
448.2.m.d 12 28.d even 2 1
448.2.m.d 12 112.j even 4 1
784.2.m.h 12 1.a even 1 1 trivial
784.2.m.h 12 16.e even 4 1 inner
784.2.x.l 24 7.d odd 6 2
784.2.x.l 24 112.x odd 12 2
784.2.x.m 24 7.c even 3 2
784.2.x.m 24 112.w even 12 2
896.2.m.g 12 56.h odd 2 1
896.2.m.g 12 112.l odd 4 1
896.2.m.h 12 56.e even 2 1
896.2.m.h 12 112.j even 4 1
7168.2.a.bi 12 224.x even 8 2
7168.2.a.bj 12 224.v odd 8 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{12} + 4 T_{3}^{11} + 8 T_{3}^{10} + 4 T_{3}^{9} + 76 T_{3}^{8} + 288 T_{3}^{7} + 552 T_{3}^{6} + 376 T_{3}^{5} + 164 T_{3}^{4} + 144 T_{3}^{3} + 288 T_{3}^{2} + 192 T_{3} + 64 \) Copy content Toggle raw display
\( T_{5}^{12} + 4 T_{5}^{11} + 8 T_{5}^{10} - 12 T_{5}^{9} + 140 T_{5}^{8} + 504 T_{5}^{7} + 968 T_{5}^{6} - 1288 T_{5}^{5} + 1828 T_{5}^{4} + 3072 T_{5}^{3} + 4608 T_{5}^{2} - 4608 T_{5} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + 5 T^{10} - 6 T^{9} + \cdots + 64 \) 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
$7$ \( T^{12} \) 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
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