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Properties

Label 111.4.j.b
Level $111$
Weight $4$
Character orbit 111.j
Analytic conductor $6.549$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [111,4,Mod(64,111)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(111, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("111.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 111.j (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.54921201064\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 125 x^{18} + 6503 x^{16} + 182846 x^{14} + 3019535 x^{12} + 29805869 x^{10} + 171637273 x^{8} + \cdots + 147456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 3 \beta_{6} q^{3} + (\beta_{13} + 5 \beta_{6} - \beta_{2}) q^{4} + (\beta_{10} + \beta_{9} - 1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{15} - 3 \beta_{6}) q^{7} + (\beta_{6} - \beta_{5} + 5 \beta_1) q^{8} + (9 \beta_{6} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 3 \beta_{6} q^{3} + (\beta_{13} + 5 \beta_{6} - \beta_{2}) q^{4} + (\beta_{10} + \beta_{9} - 1) q^{5} + 3 \beta_1 q^{6} + ( - \beta_{15} - 3 \beta_{6}) q^{7} + (\beta_{6} - \beta_{5} + 5 \beta_1) q^{8} + (9 \beta_{6} - 9) q^{9} + ( - \beta_{19} + \beta_{18} - \beta_{17} + \cdots + 1) q^{10}+ \cdots + ( - 9 \beta_{18} - 9 \beta_{6} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 3 q^{2} + 30 q^{3} + 45 q^{4} - 21 q^{5} - 29 q^{7} - 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 3 q^{2} + 30 q^{3} + 45 q^{4} - 21 q^{5} - 29 q^{7} - 90 q^{9} - 12 q^{11} - 135 q^{12} + 111 q^{13} - 63 q^{15} - 259 q^{16} + 321 q^{17} - 27 q^{18} + 402 q^{19} - 345 q^{20} + 87 q^{21} + 489 q^{22} - 72 q^{24} + 333 q^{25} + 78 q^{26} - 540 q^{27} + 373 q^{28} - 639 q^{32} - 18 q^{33} - 463 q^{34} + 246 q^{35} - 810 q^{36} - 83 q^{37} + 1194 q^{38} + 333 q^{39} - 25 q^{40} - 369 q^{41} + 279 q^{42} - 78 q^{44} - 996 q^{46} - 456 q^{47} - 1554 q^{48} - 701 q^{49} - 186 q^{50} + 1398 q^{52} - 354 q^{53} - 81 q^{54} - 582 q^{55} + 3606 q^{56} + 1206 q^{57} - 91 q^{58} + 2658 q^{59} - 1185 q^{61} + 1512 q^{62} + 522 q^{63} - 2316 q^{64} - 684 q^{65} + 73 q^{67} + 342 q^{69} - 3112 q^{70} - 2028 q^{71} - 216 q^{72} - 2426 q^{73} - 1053 q^{74} + 1998 q^{75} + 3750 q^{76} - 540 q^{77} + 117 q^{78} - 525 q^{79} - 810 q^{81} - 2112 q^{83} + 2238 q^{84} - 4734 q^{85} + 5385 q^{86} - 441 q^{87} + 1725 q^{89} - 3585 q^{91} + 4281 q^{92} - 1593 q^{93} + 4653 q^{94} + 900 q^{95} - 1917 q^{96} - 8010 q^{98} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 125 x^{18} + 6503 x^{16} + 182846 x^{14} + 3019535 x^{12} + 29805869 x^{10} + 171637273 x^{8} + \cdots + 147456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 46962371 \nu^{18} + 5533538282 \nu^{16} + 268055244023 \nu^{14} + 6903033956507 \nu^{12} + \cdots + 22042961600256 ) / 13\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1303256691038 \nu^{18} - 163138804656677 \nu^{16} + \cdots + 17\!\cdots\!16 ) / 56\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9567257639 \nu^{19} + 1192901613131 \nu^{17} + 61861729976369 \nu^{15} + \cdots + 42\!\cdots\!44 ) / 85\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9567257639 \nu^{19} + 1192901613131 \nu^{17} + 61861729976369 \nu^{15} + \cdots + 42\!\cdots\!44 ) / 85\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 324725849 \nu^{18} - 39431624666 \nu^{16} - 1972656196562 \nu^{14} + \cdots - 227988655113600 ) / 670092568593696 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23048666432557 \nu^{18} + \cdots - 18\!\cdots\!72 ) / 45\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 118108931827013 \nu^{19} + \cdots - 18\!\cdots\!76 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19821108047793 \nu^{19} + 47412036301987 \nu^{18} + \cdots + 10\!\cdots\!76 ) / 90\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51669340139577 \nu^{19} + 20563127606648 \nu^{18} + \cdots + 22\!\cdots\!76 ) / 60\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51669340139577 \nu^{19} + 20563127606648 \nu^{18} + \cdots + 22\!\cdots\!76 ) / 60\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4495139169 \nu^{19} - 561243500765 \nu^{17} - 29149887187975 \nu^{15} + \cdots + 20\!\cdots\!36 ) / 31\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!65 \nu^{19} - 358164019705472 \nu^{18} + \cdots - 23\!\cdots\!16 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!21 \nu^{19} + 41704214113216 \nu^{18} + \cdots - 57\!\cdots\!12 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!65 \nu^{19} + 358164019705472 \nu^{18} + \cdots + 23\!\cdots\!16 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 16\!\cdots\!49 \nu^{19} - 48246760608852 \nu^{18} + \cdots + 19\!\cdots\!52 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 18\!\cdots\!19 \nu^{19} + 98549989472916 \nu^{18} + \cdots + 75\!\cdots\!56 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{19} + 224077917109080 \nu^{18} + \cdots + 19\!\cdots\!60 ) / 36\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{8} + \cdots + 275 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{19} + \beta_{18} + 3 \beta_{17} + 4 \beta_{16} + 12 \beta_{15} + \beta_{14} + 3 \beta_{13} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{19} + 49 \beta_{18} - 49 \beta_{17} + 8 \beta_{16} + 41 \beta_{14} - 49 \beta_{13} + \cdots - 6635 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 150 \beta_{19} - 18 \beta_{18} - 150 \beta_{17} - 192 \beta_{16} - 608 \beta_{15} - 42 \beta_{14} + \cdots + 1136 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1824 \beta_{19} - 1824 \beta_{18} + 1824 \beta_{17} - 436 \beta_{16} - 1388 \beta_{14} + 1824 \beta_{13} + \cdots + 170925 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5490 \beta_{19} - 42 \beta_{18} + 5490 \beta_{17} + 6958 \beta_{16} + 22644 \beta_{15} + 1468 \beta_{14} + \cdots - 51792 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 60787 \beta_{19} + 60787 \beta_{18} - 60787 \beta_{17} + 16700 \beta_{16} + 44087 \beta_{14} + \cdots - 4586527 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 179499 \beta_{19} + 15951 \beta_{18} - 179499 \beta_{17} - 229806 \beta_{16} - 754040 \beta_{15} + \cdots + 2108372 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1916019 \beta_{19} - 1916019 \beta_{18} + 1916019 \beta_{17} - 555304 \beta_{16} - 1360715 \beta_{14} + \cdots + 126440943 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5573628 \beta_{19} - 799548 \beta_{18} + 5573628 \beta_{17} + 7296886 \beta_{16} + 23850276 \beta_{15} + \cdots - 79447968 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 58597960 \beta_{19} + 58597960 \beta_{18} - 58597960 \beta_{17} + 17182904 \beta_{16} + \cdots - 3549653053 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 168599544 \beta_{19} + 30719280 \beta_{18} - 168599544 \beta_{17} - 227309820 \beta_{16} + \cdots + 2836432520 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1762331313 \beta_{19} - 1762331313 \beta_{18} + 1762331313 \beta_{17} - 510460852 \beta_{16} + \cdots + 100896721083 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 5029972227 \beta_{19} - 1062459111 \beta_{18} + 5029972227 \beta_{17} + 7009288480 \beta_{16} + \cdots - 97472478828 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 52512974497 \beta_{19} + 52512974497 \beta_{18} - 52512974497 \beta_{17} + 14799502820 \beta_{16} + \cdots - 2892929169955 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 148967488062 \beta_{19} + 34869695310 \beta_{18} - 148967488062 \beta_{17} - 214863477204 \beta_{16} + \cdots + 3258725263064 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).

\(n\) \(38\) \(76\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
− 5.20628i
− 4.37156i
− 2.92495i
− 2.02391i
0.0180470i
1.08038i
1.65892i
3.92033i
4.10298i
5.47810i
5.20628i
4.37156i
2.92495i
2.02391i
− 0.0180470i
− 1.08038i
− 1.65892i
− 3.92033i
− 4.10298i
− 5.47810i
−4.50877 − 2.60314i 1.50000 + 2.59808i 9.55268 + 16.5457i 4.40999 − 2.54611i − 15.6188i 6.81642 + 11.8064i − 57.8176i −4.50000 + 7.79423i −26.5115
64.2 −3.78588 − 2.18578i 1.50000 + 2.59808i 5.55528 + 9.62203i −17.9356 + 10.3551i − 13.1147i −11.7220 − 20.3031i − 13.5980i −4.50000 + 7.79423i 90.5362
64.3 −2.53308 − 1.46247i 1.50000 + 2.59808i 0.277661 + 0.480922i 11.4761 − 6.62570i − 8.77484i −0.218775 − 0.378930i 21.7753i −4.50000 + 7.79423i −38.7597
64.4 −1.75276 − 1.01196i 1.50000 + 2.59808i −1.95189 − 3.38078i −1.45670 + 0.841025i − 6.07173i −3.66569 − 6.34915i 24.0922i −4.50000 + 7.79423i 3.40432
64.5 0.0156292 + 0.00902350i 1.50000 + 2.59808i −3.99984 − 6.92792i −0.371052 + 0.214227i 0.0541410i 14.3611 + 24.8742i − 0.288746i −4.50000 + 7.79423i −0.00773231
64.6 0.935634 + 0.540188i 1.50000 + 2.59808i −3.41639 − 5.91737i −12.0699 + 6.96855i 3.24113i −4.54002 − 7.86354i − 16.0250i −4.50000 + 7.79423i −15.0573
64.7 1.43667 + 0.829460i 1.50000 + 2.59808i −2.62399 − 4.54489i 13.7692 − 7.94965i 4.97676i −16.2296 − 28.1105i − 21.9773i −4.50000 + 7.79423i 26.3757
64.8 3.39511 + 1.96016i 1.50000 + 2.59808i 3.68449 + 6.38173i −16.2087 + 9.35812i 11.7610i 7.79609 + 13.5032i − 2.47379i −4.50000 + 7.79423i −73.3738
64.9 3.55329 + 2.05149i 1.50000 + 2.59808i 4.41723 + 7.65087i 10.3878 − 5.99737i 12.3089i 7.83490 + 13.5705i 3.42380i −4.50000 + 7.79423i 49.2142
64.10 4.74417 + 2.73905i 1.50000 + 2.59808i 11.0048 + 19.0608i −2.50102 + 1.44397i 16.4343i −14.9325 − 25.8638i 76.7456i −4.50000 + 7.79423i −15.8204
85.1 −4.50877 + 2.60314i 1.50000 − 2.59808i 9.55268 − 16.5457i 4.40999 + 2.54611i 15.6188i 6.81642 − 11.8064i 57.8176i −4.50000 − 7.79423i −26.5115
85.2 −3.78588 + 2.18578i 1.50000 − 2.59808i 5.55528 − 9.62203i −17.9356 − 10.3551i 13.1147i −11.7220 + 20.3031i 13.5980i −4.50000 − 7.79423i 90.5362
85.3 −2.53308 + 1.46247i 1.50000 − 2.59808i 0.277661 − 0.480922i 11.4761 + 6.62570i 8.77484i −0.218775 + 0.378930i − 21.7753i −4.50000 − 7.79423i −38.7597
85.4 −1.75276 + 1.01196i 1.50000 − 2.59808i −1.95189 + 3.38078i −1.45670 − 0.841025i 6.07173i −3.66569 + 6.34915i − 24.0922i −4.50000 − 7.79423i 3.40432
85.5 0.0156292 − 0.00902350i 1.50000 − 2.59808i −3.99984 + 6.92792i −0.371052 − 0.214227i − 0.0541410i 14.3611 − 24.8742i 0.288746i −4.50000 − 7.79423i −0.00773231
85.6 0.935634 − 0.540188i 1.50000 − 2.59808i −3.41639 + 5.91737i −12.0699 − 6.96855i − 3.24113i −4.54002 + 7.86354i 16.0250i −4.50000 − 7.79423i −15.0573
85.7 1.43667 − 0.829460i 1.50000 − 2.59808i −2.62399 + 4.54489i 13.7692 + 7.94965i − 4.97676i −16.2296 + 28.1105i 21.9773i −4.50000 − 7.79423i 26.3757
85.8 3.39511 − 1.96016i 1.50000 − 2.59808i 3.68449 − 6.38173i −16.2087 − 9.35812i − 11.7610i 7.79609 − 13.5032i 2.47379i −4.50000 − 7.79423i −73.3738
85.9 3.55329 − 2.05149i 1.50000 − 2.59808i 4.41723 − 7.65087i 10.3878 + 5.99737i − 12.3089i 7.83490 − 13.5705i − 3.42380i −4.50000 − 7.79423i 49.2142
85.10 4.74417 − 2.73905i 1.50000 − 2.59808i 11.0048 − 19.0608i −2.50102 − 1.44397i − 16.4343i −14.9325 + 25.8638i − 76.7456i −4.50000 − 7.79423i −15.8204
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 111.4.j.b ✓ 20
3.b odd 2 1 333.4.s.d 20
37.e even 6 1 inner 111.4.j.b ✓ 20
111.h odd 6 1 333.4.s.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.4.j.b ✓ 20 1.a even 1 1 trivial
111.4.j.b ✓ 20 37.e even 6 1 inner
333.4.s.d 20 3.b odd 2 1
333.4.s.d 20 111.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 3 T_{2}^{19} - 58 T_{2}^{18} + 183 T_{2}^{17} + 2330 T_{2}^{16} - 7071 T_{2}^{15} + \cdots + 147456 \) acting on \(S_{4}^{\mathrm{new}}(111, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 3 T^{19} + \cdots + 147456 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 25072104296448)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 65\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 92\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 11\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 45\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 40\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 28\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 77\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 33\!\cdots\!21 \) Copy content Toggle raw display
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