Properties

Label 15.11.f.a
Level $15$
Weight $11$
Character orbit 15.f
Analytic conductor $9.530$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 15.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.53035879011\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \( x^{20} - 4 x^{19} + 8 x^{18} - 49316 x^{17} + 18276332 x^{16} - 230627572 x^{15} + 1992333560 x^{14} - 394917674060 x^{13} + 80687382741790 x^{12} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{34}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 3 \beta_{2} - 3) q^{2} + \beta_{6} q^{3} + (\beta_{9} + 2 \beta_{7} + 2 \beta_{6} + 451 \beta_{2}) q^{4} + (\beta_{10} - \beta_{9} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 15 \beta_{3} - 466 \beta_{2} + \cdots + 538) q^{5}+ \cdots - 19683 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 3 \beta_{2} - 3) q^{2} + \beta_{6} q^{3} + (\beta_{9} + 2 \beta_{7} + 2 \beta_{6} + 451 \beta_{2}) q^{4} + (\beta_{10} - \beta_{9} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 15 \beta_{3} - 466 \beta_{2} + \cdots + 538) q^{5}+ \cdots + (275562 \beta_{19} - 118098 \beta_{17} - 39366 \beta_{16} + \cdots + 39366) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 64 q^{2} + 10676 q^{5} - 4860 q^{6} + 10604 q^{7} - 39948 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 64 q^{2} + 10676 q^{5} - 4860 q^{6} + 10604 q^{7} - 39948 q^{8} + 153704 q^{10} + 32080 q^{11} - 620136 q^{12} - 69352 q^{13} + 662904 q^{15} - 3111700 q^{16} + 347360 q^{17} - 1259712 q^{18} + 24132564 q^{20} + 1448280 q^{21} - 27255980 q^{22} - 30995704 q^{23} + 36824936 q^{25} + 80977840 q^{26} - 104897636 q^{28} - 6436584 q^{30} - 8846480 q^{31} - 81249676 q^{32} + 86048244 q^{33} + 91578920 q^{35} + 177540660 q^{36} - 27635896 q^{37} + 44187744 q^{38} - 887009352 q^{40} + 149264920 q^{41} + 442105452 q^{42} + 675552392 q^{43} - 183996684 q^{45} - 1916100680 q^{46} - 257112832 q^{47} + 1182772368 q^{48} + 909704384 q^{50} - 711183240 q^{51} + 1397512520 q^{52} - 152646064 q^{53} - 1181518004 q^{55} + 1735516800 q^{56} + 342507528 q^{57} - 1947576252 q^{58} - 1084331124 q^{60} + 2582791000 q^{61} - 969372632 q^{62} + 208718532 q^{63} + 7250334488 q^{65} - 1968290280 q^{66} - 6731030200 q^{67} - 12869460704 q^{68} + 7421027700 q^{70} + 7511442640 q^{71} + 786296484 q^{72} + 1660222316 q^{73} + 72454824 q^{75} + 9998646360 q^{76} - 13264676792 q^{77} - 5574993480 q^{78} - 15692039116 q^{80} - 7748409780 q^{81} + 27089146528 q^{82} + 30753878864 q^{83} - 2653017808 q^{85} - 46532117120 q^{86} + 3048661476 q^{87} + 5813201532 q^{88} + 8161910244 q^{90} + 14175275920 q^{91} + 30045377384 q^{92} - 6062778072 q^{93} - 58265269776 q^{95} - 18513718020 q^{96} - 32149992820 q^{97} + 54432471592 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 8 x^{18} - 49316 x^{17} + 18276332 x^{16} - 230627572 x^{15} + 1992333560 x^{14} - 394917674060 x^{13} + 80687382741790 x^{12} + \cdots + 14\!\cdots\!04 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 47\!\cdots\!94 \nu^{19} + \cdots + 75\!\cdots\!44 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53\!\cdots\!77 \nu^{19} + \cdots + 25\!\cdots\!48 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53\!\cdots\!28 \nu^{19} + \cdots - 66\!\cdots\!72 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!59 \nu^{19} + \cdots + 34\!\cdots\!84 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16\!\cdots\!02 \nu^{19} + \cdots + 16\!\cdots\!52 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!82 \nu^{19} + \cdots + 22\!\cdots\!32 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23\!\cdots\!88 \nu^{19} + \cdots - 20\!\cdots\!12 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 77\!\cdots\!07 \nu^{19} + \cdots + 37\!\cdots\!68 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!99 \nu^{19} + \cdots - 12\!\cdots\!76 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 51\!\cdots\!57 \nu^{19} + \cdots - 55\!\cdots\!64 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 36\!\cdots\!16 \nu^{19} + \cdots - 74\!\cdots\!84 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 86\!\cdots\!09 \nu^{19} + \cdots + 67\!\cdots\!16 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!59 \nu^{19} + \cdots + 40\!\cdots\!16 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 40\!\cdots\!78 \nu^{19} + \cdots - 75\!\cdots\!72 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59\!\cdots\!69 \nu^{19} + \cdots + 62\!\cdots\!56 ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 27\!\cdots\!37 \nu^{19} + \cdots - 12\!\cdots\!12 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14\!\cdots\!98 \nu^{19} + \cdots - 46\!\cdots\!52 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 29\!\cdots\!51 \nu^{19} + \cdots + 36\!\cdots\!24 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 10\!\cdots\!59 \nu^{19} + \cdots + 54\!\cdots\!16 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + 81\beta_1 ) / 81 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -81\beta_{9} - 6\beta_{8} - 164\beta_{7} - 164\beta_{6} + 438\beta_{3} - 117756\beta_{2} - 438\beta_1 ) / 81 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 11 \beta_{19} + 9 \beta_{18} + 2 \beta_{17} + 9 \beta_{16} + 9 \beta_{15} - 3 \beta_{14} + 32 \beta_{13} - 2 \beta_{12} - 27 \beta_{11} + 41 \beta_{10} + 22 \beta_{9} - 72 \beta_{8} + 134 \beta_{7} + 8 \beta_{6} - 38 \beta_{5} - 79 \beta_{4} + \cdots + 71030 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1917 \beta_{19} - 3834 \beta_{18} - 507 \beta_{17} + 2169 \beta_{15} - 3033 \beta_{14} - 17169 \beta_{13} - 4926 \beta_{12} + 4380 \beta_{11} - 3672 \beta_{10} + 5163 \beta_{9} - 3246 \beta_{8} + \cdots - 293795235 ) / 81 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13422 \beta_{19} + 416664 \beta_{18} + 921942 \beta_{17} - 416664 \beta_{16} - 25632 \beta_{15} + 437724 \beta_{14} + 1806186 \beta_{13} - 430086 \beta_{12} - 7638 \beta_{11} + \cdots + 3321567852 ) / 81 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2532484 \beta_{19} - 2615180 \beta_{17} + 2339136 \beta_{16} + 2314992 \beta_{15} + 1329096 \beta_{14} - 2764548 \beta_{13} + 1169568 \beta_{12} - 266368 \beta_{11} + 11329616 \beta_{10} + \cdots - 2581360 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1558330731 \beta_{19} - 1766753289 \beta_{18} + 280052478 \beta_{17} - 1766753289 \beta_{16} - 1838383209 \beta_{15} - 524178297 \beta_{14} - 5511615192 \beta_{13} + \cdots - 14663812861566 ) / 81 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 47156659461 \beta_{19} + 94313318922 \beta_{18} + 11335635891 \beta_{17} - 49893993657 \beta_{15} + 102251544129 \beta_{14} + 468432325737 \beta_{13} + \cdots + 30\!\cdots\!59 ) / 81 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 61221448742 \beta_{19} - 260185352976 \beta_{18} - 395057340258 \beta_{17} + 260185352976 \beta_{16} + 120498638508 \beta_{15} - 265958646396 \beta_{14} + \cdots - 22\!\cdots\!08 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 313839976508808 \beta_{19} + 427914435625320 \beta_{17} - 401941184329872 \beta_{16} - 447345855019944 \beta_{15} - 186996420151992 \beta_{14} + \cdots + 502041471433440 ) / 81 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18\!\cdots\!39 \beta_{19} + \cdots + 25\!\cdots\!38 ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 93\!\cdots\!49 \beta_{19} + \cdots - 42\!\cdots\!43 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 42\!\cdots\!14 \beta_{19} + \cdots + 10\!\cdots\!96 ) / 81 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 39\!\cdots\!08 \beta_{19} + \cdots - 87\!\cdots\!60 ) / 81 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 23\!\cdots\!99 \beta_{19} + \cdots - 47\!\cdots\!02 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 13\!\cdots\!77 \beta_{19} + \cdots + 52\!\cdots\!27 ) / 81 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 79\!\cdots\!78 \beta_{19} + \cdots - 17\!\cdots\!12 ) / 81 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 18\!\cdots\!48 \beta_{19} + \cdots + 53\!\cdots\!80 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 26\!\cdots\!79 \beta_{19} + \cdots + 70\!\cdots\!66 ) / 81 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
39.4498 39.4498i
28.5746 28.5746i
23.8698 23.8698i
17.5986 17.5986i
10.7802 10.7802i
−1.37019 + 1.37019i
−15.9113 + 15.9113i
−22.5896 + 22.5896i
−34.1628 + 34.1628i
−44.2389 + 44.2389i
39.4498 + 39.4498i
28.5746 + 28.5746i
23.8698 + 23.8698i
17.5986 + 17.5986i
10.7802 + 10.7802i
−1.37019 1.37019i
−15.9113 15.9113i
−22.5896 22.5896i
−34.1628 34.1628i
−44.2389 44.2389i
−41.2250 41.2250i −99.2043 + 99.2043i 2375.00i 953.956 2975.83i 8179.40 16792.7 + 16792.7i 55695.2 55695.2i 19683.0i −162006. + 83352.0i
7.2 −32.7993 32.7993i 99.2043 99.2043i 1127.59i −1119.22 2917.70i −6507.67 −16059.1 16059.1i 3397.64 3397.64i 19683.0i −58989.1 + 132408.i
7.3 −28.0945 28.0945i 99.2043 99.2043i 554.607i 2550.39 + 1805.86i −5574.20 20119.5 + 20119.5i −13187.4 + 13187.4i 19683.0i −20917.3 122387.i
7.4 −19.3738 19.3738i −99.2043 + 99.2043i 273.309i −2998.53 + 880.025i 3843.94 3127.99 + 3127.99i −25133.9 + 25133.9i 19683.0i 75142.5 + 41043.6i
7.5 −12.5555 12.5555i −99.2043 + 99.2043i 708.721i 3105.25 + 350.813i 2491.11 −8580.16 8580.16i −21755.1 + 21755.1i 19683.0i −34583.2 43392.4i
7.6 −2.85456 2.85456i 99.2043 99.2043i 1007.70i −3074.01 + 562.191i −566.369 5862.00 + 5862.00i −5799.62 + 5799.62i 19683.0i 10379.8 + 7170.15i
7.7 11.6866 + 11.6866i 99.2043 99.2043i 750.847i 2257.04 + 2161.34i 2318.72 −21416.5 21416.5i 20741.9 20741.9i 19683.0i 1118.39 + 51635.8i
7.8 20.8144 + 20.8144i −99.2043 + 99.2043i 157.526i −1515.02 2733.19i −4129.75 −6932.43 6932.43i 24592.7 24592.7i 19683.0i 25355.6 88423.8i
7.9 29.9381 + 29.9381i 99.2043 99.2043i 768.579i 2095.22 2318.55i 5939.98 15970.0 + 15970.0i 7646.81 7646.81i 19683.0i 132140. 6686.35i
7.10 42.4637 + 42.4637i −99.2043 + 99.2043i 2582.33i 3082.93 + 511.056i −8425.16 −3582.02 3582.02i −66172.3 + 66172.3i 19683.0i 109211. + 152614.i
13.1 −41.2250 + 41.2250i −99.2043 99.2043i 2375.00i 953.956 + 2975.83i 8179.40 16792.7 16792.7i 55695.2 + 55695.2i 19683.0i −162006. 83352.0i
13.2 −32.7993 + 32.7993i 99.2043 + 99.2043i 1127.59i −1119.22 + 2917.70i −6507.67 −16059.1 + 16059.1i 3397.64 + 3397.64i 19683.0i −58989.1 132408.i
13.3 −28.0945 + 28.0945i 99.2043 + 99.2043i 554.607i 2550.39 1805.86i −5574.20 20119.5 20119.5i −13187.4 13187.4i 19683.0i −20917.3 + 122387.i
13.4 −19.3738 + 19.3738i −99.2043 99.2043i 273.309i −2998.53 880.025i 3843.94 3127.99 3127.99i −25133.9 25133.9i 19683.0i 75142.5 41043.6i
13.5 −12.5555 + 12.5555i −99.2043 99.2043i 708.721i 3105.25 350.813i 2491.11 −8580.16 + 8580.16i −21755.1 21755.1i 19683.0i −34583.2 + 43392.4i
13.6 −2.85456 + 2.85456i 99.2043 + 99.2043i 1007.70i −3074.01 562.191i −566.369 5862.00 5862.00i −5799.62 5799.62i 19683.0i 10379.8 7170.15i
13.7 11.6866 11.6866i 99.2043 + 99.2043i 750.847i 2257.04 2161.34i 2318.72 −21416.5 + 21416.5i 20741.9 + 20741.9i 19683.0i 1118.39 51635.8i
13.8 20.8144 20.8144i −99.2043 99.2043i 157.526i −1515.02 + 2733.19i −4129.75 −6932.43 + 6932.43i 24592.7 + 24592.7i 19683.0i 25355.6 + 88423.8i
13.9 29.9381 29.9381i 99.2043 + 99.2043i 768.579i 2095.22 + 2318.55i 5939.98 15970.0 15970.0i 7646.81 + 7646.81i 19683.0i 132140. + 6686.35i
13.10 42.4637 42.4637i −99.2043 99.2043i 2582.33i 3082.93 511.056i −8425.16 −3582.02 + 3582.02i −66172.3 66172.3i 19683.0i 109211. 152614.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.11.f.a 20
3.b odd 2 1 45.11.g.c 20
5.b even 2 1 75.11.f.d 20
5.c odd 4 1 inner 15.11.f.a 20
5.c odd 4 1 75.11.f.d 20
15.e even 4 1 45.11.g.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.11.f.a 20 1.a even 1 1 trivial
15.11.f.a 20 5.c odd 4 1 inner
45.11.g.c 20 3.b odd 2 1
45.11.g.c 20 15.e even 4 1
75.11.f.d 20 5.b even 2 1
75.11.f.d 20 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 64 T^{19} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( (T^{4} + 387420489)^{5} \) Copy content Toggle raw display
$5$ \( T^{20} - 10676 T^{19} + \cdots + 78\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} - 10604 T^{19} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{10} - 16040 T^{9} + \cdots + 18\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + 69352 T^{19} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{20} - 347360 T^{19} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + 54779671100520 T^{18} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + 30995704 T^{19} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + 4423240 T^{9} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 27635896 T^{19} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} - 74632460 T^{9} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} - 675552392 T^{19} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} + 257112832 T^{19} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{20} + 152646064 T^{19} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} - 1291395500 T^{9} + \cdots - 79\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 6731030200 T^{19} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{10} - 3755721320 T^{9} + \cdots + 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} - 1660222316 T^{19} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} - 30753878864 T^{19} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + 32149992820 T^{19} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
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