Properties

Label 576.2.r.f
Level $576$
Weight $2$
Character orbit 576.r
Analytic conductor $4.599$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4}) q^{3} + (\beta_{6} + 2) q^{5} + (\beta_{11} - \beta_{4} - \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{4}) q^{3} + (\beta_{6} + 2) q^{5} + (\beta_{11} - \beta_{4} - \beta_1) q^{7} + (\beta_{10} - \beta_{6} - \beta_{3} - 1) q^{9} + (\beta_{4} - \beta_1) q^{11} + ( - \beta_{10} + \beta_{7}) q^{13} + (\beta_{5} - 2 \beta_{4}) q^{15} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \beta_{3}) q^{17} - \beta_{8} q^{19} + (\beta_{10} + 4 \beta_{6} - 2 \beta_{3} + 2) q^{21} + (\beta_{11} - \beta_{8} + \beta_{5} - 2 \beta_{4} + \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{6} - 2) q^{25} + ( - \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1) q^{27} + (2 \beta_{10} + 2 \beta_{9} - \beta_{7} + \beta_{6} - \beta_{3} - 1) q^{29} + (\beta_{11} - \beta_{8} - 3 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1) q^{31} + (\beta_{10} + 4 \beta_{6} + \beta_{3} + 5) q^{33} + (2 \beta_{11} - \beta_{5} - \beta_{4} - \beta_1) q^{35} + ( - \beta_{10} + \beta_{9} - 2 \beta_{6} + \beta_{3} - 1) q^{37} + ( - 2 \beta_{11} + \beta_{5} - 2 \beta_{4} + 3 \beta_{2} + \beta_1) q^{39} + (\beta_{10} + \beta_{9} - 3 \beta_{6}) q^{41} + ( - \beta_{11} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1) q^{43} + (\beta_{10} - 2 \beta_{6} - 2 \beta_{3} - 1) q^{45} + (\beta_{11} + 2 \beta_{8} + 4 \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{47} + (\beta_{10} - 2 \beta_{9} + \beta_{7} + 5 \beta_{6} - 2 \beta_{3}) q^{49} + (\beta_{11} + 3 \beta_{2} + \beta_1) q^{51} + (\beta_{10} + \beta_{9} + \beta_{7} - 4 \beta_{6} + \beta_{3} - 2) q^{53} + (\beta_{11} + \beta_{5} + \beta_{4} - 2 \beta_1) q^{55} + ( - \beta_{7} + 1) q^{57} + (\beta_{11} - 3 \beta_{8} + \beta_{5} + 2 \beta_{4} - 3 \beta_{2}) q^{59} + ( - 2 \beta_{10} - 2 \beta_{9} + \beta_{7} + \beta_{6} + \beta_{3} - 1) q^{61} + ( - 2 \beta_{11} - 3 \beta_{8} - 3 \beta_{4} - 3 \beta_{2} + \beta_1) q^{63} + ( - 2 \beta_{10} + \beta_{9} + \beta_{7} + \beta_{3}) q^{65} + ( - \beta_{11} + 2 \beta_{8} - \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - \beta_1) q^{67} + ( - \beta_{9} - \beta_{7} - 2 \beta_{6} - 3 \beta_{3} - 7) q^{69} + (\beta_{11} - 2 \beta_{8} - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{2} + \beta_1) q^{71} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{3} - 1) q^{73} + 2 \beta_{4} q^{75} + (2 \beta_{10} - \beta_{9} - \beta_{7} + 6 \beta_{6} + 2 \beta_{3} + 12) q^{77} + (2 \beta_{8} - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{79} + (2 \beta_{9} - \beta_{7} - 6 \beta_{6} + 2 \beta_{3} - 4) q^{81} + (\beta_{11} - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + \beta_1) q^{83} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} + 2 \beta_{3}) q^{85} + ( - 3 \beta_{11} + 3 \beta_{8} - 2 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{87} + (\beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{3} - 3) q^{89} + ( - 3 \beta_{11} - 3 \beta_{5} + 3 \beta_{4}) q^{91} + ( - 2 \beta_{10} - \beta_{9} - \beta_{7} + 8 \beta_{6} + \beta_{3} + 7) q^{93} + ( - \beta_{8} + \beta_{2}) q^{95} + ( - 3 \beta_{7} - \beta_{6} - 3 \beta_{3} - 1) q^{97} + (\beta_{11} + 3 \beta_{8} - 3 \beta_{4} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{5} - 4 q^{9} + 6 q^{13} + 6 q^{21} - 12 q^{25} - 18 q^{29} + 30 q^{33} + 18 q^{41} + 6 q^{45} - 24 q^{49} + 8 q^{57} - 18 q^{61} + 6 q^{65} - 66 q^{69} + 90 q^{77} - 20 q^{81} - 48 q^{89} + 30 q^{93} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 778 \nu^{11} + 5496 \nu^{10} - 7293 \nu^{9} + 5400 \nu^{8} + 10120 \nu^{7} - 68622 \nu^{6} - 13980 \nu^{5} + 12828 \nu^{4} + 200318 \nu^{3} + 699840 \nu^{2} + \cdots + 73584 ) / 51972 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 962 \nu^{11} - 2392 \nu^{10} + 2887 \nu^{9} - 2220 \nu^{8} - 13990 \nu^{7} + 14442 \nu^{6} + 11380 \nu^{5} + 57708 \nu^{4} + 98118 \nu^{3} - 132080 \nu^{2} + 49284 \nu + 15984 ) / 51972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1360 \nu^{11} - 864 \nu^{10} - 400 \nu^{9} + 2902 \nu^{8} - 25539 \nu^{7} - 16320 \nu^{6} + 27692 \nu^{5} + 92050 \nu^{4} + 330320 \nu^{3} + 266532 \nu^{2} + 143634 \nu + 219540 ) / 51972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5984 \nu^{11} - 7522 \nu^{10} + 8393 \nu^{9} - 6600 \nu^{8} - 90816 \nu^{7} - 28498 \nu^{6} + 55616 \nu^{5} + 473748 \nu^{4} + 1167562 \nu^{3} + 792408 \nu^{2} + \cdots + 272448 ) / 103944 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6823 \nu^{11} - 186 \nu^{10} + 2391 \nu^{9} - 2376 \nu^{8} + 113464 \nu^{7} + 159834 \nu^{6} - 23754 \nu^{5} - 683700 \nu^{4} - 2089522 \nu^{3} - 2446620 \nu^{2} + \cdots - 548280 ) / 103944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + 10408 \nu^{4} + 27758 \nu^{3} + 22776 \nu^{2} + 12468 \nu + 3468 ) / 1704 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4689 \nu^{11} - 3876 \nu^{10} + 3854 \nu^{9} - 5534 \nu^{8} - 68264 \nu^{7} - 56268 \nu^{6} + 37850 \nu^{5} + 438544 \nu^{4} + 1102444 \nu^{3} + 911136 \nu^{2} + \cdots + 48876 ) / 51972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1688 \nu^{11} + 1054 \nu^{10} - 840 \nu^{9} + 684 \nu^{8} + 26378 \nu^{7} + 24587 \nu^{6} - 12648 \nu^{5} - 151968 \nu^{4} - 420176 \nu^{3} - 400856 \nu^{2} + \cdots - 105120 ) / 12993 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4502 \nu^{11} + 3246 \nu^{10} - 927 \nu^{9} - 1080 \nu^{8} + 73378 \nu^{7} + 54024 \nu^{6} - 56276 \nu^{5} - 403744 \nu^{4} - 1035370 \nu^{3} - 847128 \nu^{2} + \cdots - 228796 ) / 34648 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25054 \nu^{11} - 18618 \nu^{10} + 8389 \nu^{9} + 272 \nu^{8} - 404246 \nu^{7} - 300648 \nu^{6} + 310036 \nu^{5} + 2217896 \nu^{4} + 5869134 \nu^{3} + 4806792 \nu^{2} + \cdots + 1030836 ) / 103944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13366 \nu^{11} + 8843 \nu^{10} - 7504 \nu^{9} + 5514 \nu^{8} + 211707 \nu^{7} + 177716 \nu^{6} - 88792 \nu^{5} - 1193088 \nu^{4} - 3315152 \nu^{3} - 3104220 \nu^{2} + \cdots - 830016 ) / 51972 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{7} + 4\beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} + 4\beta_{5} - 2\beta_{4} - 9\beta_{2} - 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{11} + 4 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} - 12 \beta_{6} + 8 \beta_{5} - 16 \beta_{4} + 4 \beta_{3} - 6 \beta_{2} - 4 \beta _1 - 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{10} + 5\beta_{9} + 2\beta_{7} - 30\beta_{6} - 4\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 22 \beta_{11} + 38 \beta_{10} + 35 \beta_{9} + 42 \beta_{8} - 16 \beta_{7} - 39 \beta_{6} + 32 \beta_{5} + 38 \beta_{4} - 22 \beta_{3} + 21 \beta_{2} + 38 \beta _1 + 39 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -56\beta_{11} + 81\beta_{8} + 64\beta_{5} + 16\beta_{4} + 40\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 94 \beta_{11} - 35 \beta_{10} - 47 \beta_{9} + 60 \beta_{8} + 82 \beta_{7} - 108 \beta_{6} + 164 \beta_{5} - 70 \beta_{4} + 94 \beta_{3} - 60 \beta_{2} + 35 \beta _1 - 216 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 70\beta_{10} - 35\beta_{9} + 214\beta_{7} - 669\beta_{6} + 214\beta_{3} - 669 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 164 \beta_{11} + 308 \beta_{10} + 164 \beta_{9} + 321 \beta_{8} + 236 \beta_{7} - 1140 \beta_{6} - 472 \beta_{5} + 800 \beta_{4} + 164 \beta_{3} + 642 \beta_{2} + 308 \beta _1 - 570 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -328\beta_{11} + 1908\beta_{8} - 328\beta_{5} + 2228\beta_{4} + 1908\beta_{2} + 1442\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1586 \beta_{11} - 2386 \beta_{10} - 1993 \beta_{9} + 3342 \beta_{8} + 800 \beta_{7} + 2949 \beta_{6} + 1600 \beta_{5} + 2386 \beta_{4} + 1586 \beta_{3} + 1671 \beta_{2} + 2386 \beta _1 - 2949 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.583700 2.17840i
−1.50511 0.403293i
−0.673288 0.180407i
−0.180407 + 0.673288i
−0.403293 + 1.50511i
2.17840 + 0.583700i
0.583700 + 2.17840i
−1.50511 + 0.403293i
−0.673288 + 0.180407i
−0.180407 0.673288i
−0.403293 1.50511i
2.17840 0.583700i
0 −1.59470 0.675970i 0 1.50000 0.866025i 0 −1.80664 + 3.12920i 0 2.08613 + 2.15594i 0
97.2 0 −1.10182 1.33641i 0 1.50000 0.866025i 0 −0.495361 + 0.857990i 0 −0.571993 + 2.94497i 0
97.3 0 −0.492881 + 1.66044i 0 1.50000 0.866025i 0 −2.17731 + 3.77121i 0 −2.51414 1.63680i 0
97.4 0 0.492881 1.66044i 0 1.50000 0.866025i 0 2.17731 3.77121i 0 −2.51414 1.63680i 0
97.5 0 1.10182 + 1.33641i 0 1.50000 0.866025i 0 0.495361 0.857990i 0 −0.571993 + 2.94497i 0
97.6 0 1.59470 + 0.675970i 0 1.50000 0.866025i 0 1.80664 3.12920i 0 2.08613 + 2.15594i 0
481.1 0 −1.59470 + 0.675970i 0 1.50000 + 0.866025i 0 −1.80664 3.12920i 0 2.08613 2.15594i 0
481.2 0 −1.10182 + 1.33641i 0 1.50000 + 0.866025i 0 −0.495361 0.857990i 0 −0.571993 2.94497i 0
481.3 0 −0.492881 1.66044i 0 1.50000 + 0.866025i 0 −2.17731 3.77121i 0 −2.51414 + 1.63680i 0
481.4 0 0.492881 + 1.66044i 0 1.50000 + 0.866025i 0 2.17731 + 3.77121i 0 −2.51414 + 1.63680i 0
481.5 0 1.10182 1.33641i 0 1.50000 + 0.866025i 0 0.495361 + 0.857990i 0 −0.571993 2.94497i 0
481.6 0 1.59470 0.675970i 0 1.50000 + 0.866025i 0 1.80664 + 3.12920i 0 2.08613 2.15594i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
72.n even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.r.f yes 12
3.b odd 2 1 1728.2.r.e 12
4.b odd 2 1 inner 576.2.r.f yes 12
8.b even 2 1 576.2.r.e 12
8.d odd 2 1 576.2.r.e 12
9.c even 3 1 576.2.r.e 12
9.c even 3 1 5184.2.d.q 12
9.d odd 6 1 1728.2.r.f 12
9.d odd 6 1 5184.2.d.r 12
12.b even 2 1 1728.2.r.e 12
24.f even 2 1 1728.2.r.f 12
24.h odd 2 1 1728.2.r.f 12
36.f odd 6 1 576.2.r.e 12
36.f odd 6 1 5184.2.d.q 12
36.h even 6 1 1728.2.r.f 12
36.h even 6 1 5184.2.d.r 12
72.j odd 6 1 1728.2.r.e 12
72.j odd 6 1 5184.2.d.r 12
72.l even 6 1 1728.2.r.e 12
72.l even 6 1 5184.2.d.r 12
72.n even 6 1 inner 576.2.r.f yes 12
72.n even 6 1 5184.2.d.q 12
72.p odd 6 1 inner 576.2.r.f yes 12
72.p odd 6 1 5184.2.d.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 8.b even 2 1
576.2.r.e 12 8.d odd 2 1
576.2.r.e 12 9.c even 3 1
576.2.r.e 12 36.f odd 6 1
576.2.r.f yes 12 1.a even 1 1 trivial
576.2.r.f yes 12 4.b odd 2 1 inner
576.2.r.f yes 12 72.n even 6 1 inner
576.2.r.f yes 12 72.p odd 6 1 inner
1728.2.r.e 12 3.b odd 2 1
1728.2.r.e 12 12.b even 2 1
1728.2.r.e 12 72.j odd 6 1
1728.2.r.e 12 72.l even 6 1
1728.2.r.f 12 9.d odd 6 1
1728.2.r.f 12 24.f even 2 1
1728.2.r.f 12 24.h odd 2 1
1728.2.r.f 12 36.h even 6 1
5184.2.d.q 12 9.c even 3 1
5184.2.d.q 12 36.f odd 6 1
5184.2.d.q 12 72.n even 6 1
5184.2.d.q 12 72.p odd 6 1
5184.2.d.r 12 9.d odd 6 1
5184.2.d.r 12 36.h even 6 1
5184.2.d.r 12 72.j odd 6 1
5184.2.d.r 12 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{10} + 7 T^{8} + 12 T^{6} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 33 T^{10} + 810 T^{8} + \cdots + 59049 \) Copy content Toggle raw display
$11$ \( T^{12} - 39 T^{10} + 1350 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( (T^{6} - 3 T^{5} - 24 T^{4} + 81 T^{3} + \cdots + 243)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 24 T + 36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + 117 T^{10} + \cdots + 2492305929 \) Copy content Toggle raw display
$29$ \( (T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 93987)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 117 T^{10} + \cdots + 95004009 \) Copy content Toggle raw display
$37$ \( (T^{6} + 60 T^{4} + 1008 T^{2} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 9 T^{5} + 78 T^{4} - 45 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 123 T^{10} + \cdots + 47458321 \) Copy content Toggle raw display
$47$ \( T^{12} + 297 T^{10} + \cdots + 514609673769 \) Copy content Toggle raw display
$53$ \( (T^{6} + 180 T^{4} + 1728 T^{2} + \cdots + 432)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 147 T^{10} + 20898 T^{8} + \cdots + 531441 \) Copy content Toggle raw display
$61$ \( (T^{6} + 9 T^{5} - 36 T^{4} - 567 T^{3} + \cdots + 4563)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 231 T^{10} + \cdots + 352275361 \) Copy content Toggle raw display
$71$ \( (T^{6} - 288 T^{4} + 19872 T^{2} + \cdots - 124848)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 84 T - 164)^{4} \) Copy content Toggle raw display
$79$ \( T^{12} + 93 T^{10} + 7758 T^{8} + \cdots + 4782969 \) Copy content Toggle raw display
$83$ \( T^{12} - 171 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$89$ \( (T^{3} + 12 T^{2} - 36 T - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 3 T^{5} + 222 T^{4} + \cdots + 1408969)^{2} \) Copy content Toggle raw display
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