Properties

Label 143.2.e.c
Level $143$
Weight $2$
Character orbit 143.e
Analytic conductor $1.142$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 143 = 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 143.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14186074890\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 9x^{10} - 2x^{9} + 59x^{8} - 13x^{7} + 175x^{6} - 50x^{5} + 380x^{4} - 64x^{3} + 280x^{2} + 48x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{8} + \beta_{6}) q^{3} + ( - \beta_{10} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{4} + ( - \beta_{7} - 1) q^{5} + ( - \beta_{11} - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{3} + \beta_{2} + 1) q^{8} + (\beta_{11} + \beta_{10} - \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{8} + \beta_{6}) q^{3} + ( - \beta_{10} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{4} + ( - \beta_{7} - 1) q^{5} + ( - \beta_{11} - \beta_{6} - 3 \beta_{5} - \beta_{4} + \beta_{2}) q^{6} + ( - \beta_{7} - \beta_{5} - \beta_{4} - \beta_1) q^{7} + ( - \beta_{11} - \beta_{3} + \beta_{2} + 1) q^{8} + (\beta_{11} + \beta_{10} - \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_1) q^{9} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1 + 1) q^{10} + (\beta_{5} + 1) q^{11} + ( - \beta_{10} - \beta_{8} - 2 \beta_{7} - 2 \beta_{2} - 3) q^{12} + (\beta_{11} + \beta_{8} + \beta_{7} + \beta_{5}) q^{13} + ( - \beta_{11} + \beta_{9} + \beta_{7} - \beta_{3} - \beta_{2} + 2) q^{14} + ( - \beta_{9} - 2 \beta_{8} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{15} + (\beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + 2 \beta_1 - 2) q^{16} + ( - \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{2} + \beta_1) q^{17} + (3 \beta_{10} - \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{2} + 2) q^{18} + (\beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{11} + 3 \beta_{10} + \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{20} + (2 \beta_{11} - 2 \beta_{9} + \beta_{7} + 2 \beta_{3} - 2) q^{21} + ( - \beta_{6} + \beta_{2}) q^{22} + (\beta_{6} - \beta_{3} + 2 \beta_1) q^{23} + (\beta_{8} + 3 \beta_{6} + 3 \beta_{5} - \beta_{3} - 2 \beta_1 + 3) q^{24} + (2 \beta_{7} + \beta_{2} - 1) q^{25} + ( - \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{3} + \beta_{2} + \beta_1) q^{26} + ( - \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{2} + 1) q^{27} + ( - \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_1 + 2) q^{28} + (2 \beta_{9} - 3 \beta_{8} + 2 \beta_{4} - 2) q^{29} + (2 \beta_{11} - 2 \beta_{10} - 2 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} + \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{30} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{3} - 2) q^{31} + (\beta_{11} + \beta_{10} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1) q^{32} + ( - \beta_{10} + \beta_{6} - \beta_{2}) q^{33} + ( - 2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \beta_{2}) q^{34} + ( - \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_1) q^{35} + ( - \beta_{9} - \beta_{8} - 3 \beta_{6} - 6 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_1 - 5) q^{36} + (\beta_{9} - \beta_{8} + 2 \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 3) q^{37} + ( - \beta_{9} + \beta_{7} - 2 \beta_{2} - 4) q^{38} + (2 \beta_{9} - 3 \beta_{7} + 4 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{39} + (2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{3} - 4 \beta_{2} - 2) q^{40} + ( - \beta_{9} + \beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} + 5) q^{41} + ( - 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + \cdots - 1) q^{42}+ \cdots + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} + 3 q^{7} + 6 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} - 8 q^{4} - 12 q^{5} + 12 q^{6} + 3 q^{7} + 6 q^{8} - 7 q^{9} + 3 q^{10} + 6 q^{11} - 34 q^{12} - 4 q^{13} + 24 q^{14} - 4 q^{15} - 8 q^{16} - 2 q^{17} + 12 q^{18} + 10 q^{19} + 15 q^{20} - 24 q^{21} - 3 q^{23} + 14 q^{24} - 12 q^{25} - 3 q^{26} + 20 q^{27} + 16 q^{28} - 3 q^{29} - 19 q^{30} - 10 q^{31} - q^{32} + q^{33} + 10 q^{34} + 13 q^{35} - 20 q^{36} + 25 q^{37} - 54 q^{38} - 12 q^{39} - 16 q^{40} + 24 q^{41} - 13 q^{42} + 8 q^{43} - 16 q^{44} + 27 q^{45} + 18 q^{46} - 20 q^{47} + 28 q^{48} + q^{49} - 26 q^{50} - 34 q^{51} - 39 q^{52} + 20 q^{53} + 47 q^{54} - 6 q^{55} - 15 q^{56} + 6 q^{58} - 4 q^{59} + 122 q^{60} + 21 q^{61} + 5 q^{62} + 6 q^{63} - 54 q^{64} - 32 q^{65} + 24 q^{66} + 21 q^{67} - 14 q^{68} - 5 q^{69} - 62 q^{70} - 3 q^{71} - 50 q^{72} - 26 q^{73} + 38 q^{74} + 23 q^{75} + 8 q^{76} + 6 q^{77} + 36 q^{78} - 8 q^{79} + 44 q^{80} - 34 q^{81} + 33 q^{82} - 16 q^{83} + 47 q^{84} - 13 q^{85} + 22 q^{86} + 51 q^{87} + 3 q^{88} - 9 q^{89} - 140 q^{90} - 19 q^{91} + 30 q^{92} - 21 q^{93} - 10 q^{94} - 27 q^{95} + 38 q^{96} + 15 q^{97} + 21 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 9x^{10} - 2x^{9} + 59x^{8} - 13x^{7} + 175x^{6} - 50x^{5} + 380x^{4} - 64x^{3} + 280x^{2} + 48x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 433477 \nu^{11} + 1209797 \nu^{10} - 3549502 \nu^{9} + 8605189 \nu^{8} - 25150864 \nu^{7} + 55816400 \nu^{6} - 68042765 \nu^{5} + \cdots - 736079934 ) / 321804478 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6513113 \nu^{11} + 23332617 \nu^{10} + 26020485 \nu^{9} + 178582811 \nu^{8} + 134283553 \nu^{7} + 1222140490 \nu^{6} - 896911294 \nu^{5} + \cdots + 5403214872 ) / 3861653736 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7065514 \nu^{11} + 107212017 \nu^{10} + 28665066 \nu^{9} + 964117505 \nu^{8} + 21141184 \nu^{7} + 5746960873 \nu^{6} + 1240962365 \nu^{5} + \cdots + 6598596768 ) / 3861653736 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8004451 \nu^{11} + 19111125 \nu^{10} - 66838335 \nu^{9} + 173491463 \nu^{8} - 467890835 \nu^{7} + 1128351970 \nu^{6} - 1347413182 \nu^{5} + \cdots + 792512880 ) / 3861653736 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6270543 \nu^{11} - 14271937 \nu^{10} + 52640327 \nu^{9} - 139070707 \nu^{8} + 367287379 \nu^{7} - 905086370 \nu^{6} + 1075242122 \nu^{5} + \cdots - 3736832616 ) / 1287217912 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6370375 \nu^{11} + 1733908 \nu^{10} + 52494187 \nu^{9} + 1457258 \nu^{8} + 341431369 \nu^{7} + 17788581 \nu^{6} + 891550025 \nu^{5} + \cdots + 384213648 ) / 1287217912 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26508257 \nu^{11} + 24694089 \nu^{10} + 156505095 \nu^{9} + 156404459 \nu^{8} + 773028499 \nu^{7} + 1155926710 \nu^{6} - 94460776 \nu^{5} + \cdots + 5595887088 ) / 3861653736 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15688761 \nu^{11} - 26880018 \nu^{10} - 140572891 \nu^{9} - 151808712 \nu^{8} - 808207929 \nu^{7} - 854883141 \nu^{6} + \cdots + 3212162936 ) / 1287217912 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 52748602 \nu^{11} - 7877727 \nu^{10} + 529245270 \nu^{9} - 169452311 \nu^{8} + 3481751012 \nu^{7} - 1189783075 \nu^{6} + 11213080081 \nu^{5} + \cdots + 2349022944 ) / 3861653736 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 69931387 \nu^{11} - 2525721 \nu^{10} + 603909759 \nu^{9} - 161095715 \nu^{8} + 3962892875 \nu^{7} - 1008677518 \nu^{6} + 11595511594 \nu^{5} + \cdots + 3069002640 ) / 3861653736 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 3\beta_{5} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 4\beta_{7} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - \beta_{8} - 7\beta_{6} - 13\beta_{5} - \beta_{4} - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{11} - 2\beta_{10} - 18\beta_{7} + \beta_{6} + 8\beta_{5} - \beta_{2} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{11} + 9\beta_{10} + 9\beta_{9} + 9\beta_{8} - \beta_{3} + 41\beta_{2} + 55 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{9} - 19\beta_{8} - 12\beta_{6} - 51\beta_{5} - \beta_{4} + 41\beta_{3} + 86\beta _1 - 50 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12\beta_{11} - 61\beta_{10} - 2\beta_{7} + 228\beta_{6} + 330\beta_{5} + 59\beta_{4} - 228\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -228\beta_{11} + 132\beta_{10} + 14\beta_{9} + 132\beta_{8} + 426\beta_{7} - 228\beta_{3} + 99\beta_{2} + 293 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -346\beta_{9} - 374\beta_{8} - 1242\beta_{6} - 1737\beta_{5} - 346\beta_{4} + 99\beta_{3} + 32\beta _1 - 1391 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1242 \beta_{11} - 819 \beta_{10} - 2160 \beta_{7} + 703 \beta_{6} + 1811 \beta_{5} + 127 \beta_{4} - 703 \beta_{2} - 2160 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
−1.17629 + 2.03740i
1.11519 1.93157i
−0.903935 + 1.56566i
0.790735 1.36959i
0.542744 0.940060i
−0.368446 + 0.638166i
−1.17629 2.03740i
1.11519 + 1.93157i
−0.903935 1.56566i
0.790735 + 1.36959i
0.542744 + 0.940060i
−0.368446 0.638166i
−1.26732 + 2.19506i 1.49512 2.58962i −2.21221 3.83165i −3.35258 3.78959 + 6.56377i −0.959205 1.66139i 6.14501 −2.97076 5.14551i 4.24880 7.35913i
100.2 −0.987312 + 1.71007i 0.285746 0.494927i −0.949569 1.64470i 1.23039 0.564241 + 0.977294i 1.27902 + 2.21532i −0.199164 1.33670 + 2.31523i −1.21478 + 2.10405i
100.3 −0.134198 + 0.232438i −1.17558 + 2.03617i 0.963982 + 1.66967i −2.80787 −0.315522 0.546500i −1.19233 2.06518i −1.05425 −1.26400 2.18931i 0.376811 0.652655i
100.4 0.249477 0.432106i 0.120298 0.208363i 0.875523 + 1.51645i 0.581470 −0.0600233 0.103963i −0.705086 1.22125i 1.87160 1.47106 + 2.54794i 0.145063 0.251257i
100.5 0.910859 1.57765i −1.55764 + 2.69791i −0.659327 1.14199i 0.0854874 2.83757 + 4.91482i 2.25971 + 3.91393i 1.24122 −3.35247 5.80665i 0.0778669 0.134869i
100.6 1.22850 2.12782i 0.332058 0.575141i −2.01840 3.49598i −1.73689 −0.815863 1.41312i 0.817900 + 1.41664i −5.00442 1.27948 + 2.21612i −2.13376 + 3.69579i
133.1 −1.26732 2.19506i 1.49512 + 2.58962i −2.21221 + 3.83165i −3.35258 3.78959 6.56377i −0.959205 + 1.66139i 6.14501 −2.97076 + 5.14551i 4.24880 + 7.35913i
133.2 −0.987312 1.71007i 0.285746 + 0.494927i −0.949569 + 1.64470i 1.23039 0.564241 0.977294i 1.27902 2.21532i −0.199164 1.33670 2.31523i −1.21478 2.10405i
133.3 −0.134198 0.232438i −1.17558 2.03617i 0.963982 1.66967i −2.80787 −0.315522 + 0.546500i −1.19233 + 2.06518i −1.05425 −1.26400 + 2.18931i 0.376811 + 0.652655i
133.4 0.249477 + 0.432106i 0.120298 + 0.208363i 0.875523 1.51645i 0.581470 −0.0600233 + 0.103963i −0.705086 + 1.22125i 1.87160 1.47106 2.54794i 0.145063 + 0.251257i
133.5 0.910859 + 1.57765i −1.55764 2.69791i −0.659327 + 1.14199i 0.0854874 2.83757 4.91482i 2.25971 3.91393i 1.24122 −3.35247 + 5.80665i 0.0778669 + 0.134869i
133.6 1.22850 + 2.12782i 0.332058 + 0.575141i −2.01840 + 3.49598i −1.73689 −0.815863 + 1.41312i 0.817900 1.41664i −5.00442 1.27948 2.21612i −2.13376 3.69579i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.2.e.c 12
13.c even 3 1 inner 143.2.e.c 12
13.c even 3 1 1859.2.a.k 6
13.e even 6 1 1859.2.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.c 12 1.a even 1 1 trivial
143.2.e.c 12 13.c even 3 1 inner
1859.2.a.k 6 13.c even 3 1
1859.2.a.l 6 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 10 T_{2}^{10} - 2 T_{2}^{9} + 76 T_{2}^{8} - 15 T_{2}^{7} + 235 T_{2}^{6} - 76 T_{2}^{5} + 551 T_{2}^{4} - 114 T_{2}^{3} + 97 T_{2}^{2} + 15 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(143, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 10 T^{10} - 2 T^{9} + 76 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + 13 T^{10} + 2 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{6} + 6 T^{5} + 6 T^{4} - 15 T^{3} - 14 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + 25 T^{10} + \cdots + 14884 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + 17 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + 38 T^{10} + \cdots + 19321 \) Copy content Toggle raw display
$19$ \( T^{12} - 10 T^{11} + 109 T^{10} + \cdots + 91204 \) Copy content Toggle raw display
$23$ \( T^{12} + 3 T^{11} + 65 T^{10} + \cdots + 350464 \) Copy content Toggle raw display
$29$ \( T^{12} + 3 T^{11} + 135 T^{10} + \cdots + 395651881 \) Copy content Toggle raw display
$31$ \( (T^{6} + 5 T^{5} - 31 T^{4} - 57 T^{3} + \cdots + 120)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 25 T^{11} + 449 T^{10} + \cdots + 26739241 \) Copy content Toggle raw display
$41$ \( T^{12} - 24 T^{11} + 394 T^{10} + \cdots + 41306329 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} + 107 T^{10} + \cdots + 2096704 \) Copy content Toggle raw display
$47$ \( (T^{6} + 10 T^{5} - 15 T^{4} - 183 T^{3} + \cdots - 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 10 T^{5} - 156 T^{4} + 1997 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 4 T^{11} + 165 T^{10} + \cdots + 338486404 \) Copy content Toggle raw display
$61$ \( T^{12} - 21 T^{11} + 297 T^{10} + \cdots + 429025 \) Copy content Toggle raw display
$67$ \( T^{12} - 21 T^{11} + 347 T^{10} + \cdots + 8868484 \) Copy content Toggle raw display
$71$ \( T^{12} + 3 T^{11} + \cdots + 10252777536 \) Copy content Toggle raw display
$73$ \( (T^{6} + 13 T^{5} + 49 T^{4} + 5 T^{3} + \cdots + 296)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 4 T^{5} - 408 T^{4} + \cdots - 1572016)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 8 T^{5} - 177 T^{4} - 397 T^{3} + \cdots - 12114)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 9 T^{11} + \cdots + 420375876496 \) Copy content Toggle raw display
$97$ \( T^{12} - 15 T^{11} + 492 T^{10} + \cdots + 58186384 \) Copy content Toggle raw display
show more
show less