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Properties

Label 9.12.c.a
Level $9$
Weight $12$
Character orbit 9.c
Analytic conductor $6.915$
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] = [9,12,Mod(4,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.4");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), 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: \(6.91508862504\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 9863 x^{18} + 40416552 x^{16} + 89424581388 x^{14} + 116167273852206 x^{12} + \cdots + 59\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{45} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 3 \beta_{3} + \beta_1) q^{2} + (\beta_{6} + \beta_{4} - 51 \beta_{3} + 24) q^{3} + (\beta_{10} - \beta_{6} + \beta_{5} + \cdots - 921) q^{4}+ \cdots + (\beta_{19} - \beta_{18} + \cdots + 12264) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta_{3} + \beta_1) q^{2} + (\beta_{6} + \beta_{4} - 51 \beta_{3} + 24) q^{3} + (\beta_{10} - \beta_{6} + \beta_{5} + \cdots - 921) q^{4}+ \cdots + ( - 67977 \beta_{19} + \cdots + 22458395685) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 33 q^{2} - 12 q^{3} - 9217 q^{4} - 7230 q^{5} + 20583 q^{6} + 8512 q^{7} - 29118 q^{8} + 135504 q^{9} + 4092 q^{10} - 112776 q^{11} + 1027860 q^{12} + 279706 q^{13} - 3901584 q^{14} - 6358608 q^{15} - 7342081 q^{16} + 27765792 q^{17} + 8682876 q^{18} + 7029400 q^{19} - 34163508 q^{20} + 55012206 q^{21} + 2274591 q^{22} - 69371616 q^{23} - 211100355 q^{24} - 45286204 q^{25} + 481929144 q^{26} - 83699352 q^{27} - 61345796 q^{28} - 25437246 q^{29} - 23582592 q^{30} + 114575368 q^{31} + 80396559 q^{32} + 31338342 q^{33} - 243855063 q^{34} - 178147464 q^{35} - 19984653 q^{36} - 134218328 q^{37} + 489799995 q^{38} - 1999064976 q^{39} + 107425416 q^{40} + 331873026 q^{41} + 4171968882 q^{42} - 1118847584 q^{43} + 278477274 q^{44} + 1749349170 q^{45} + 2882537592 q^{46} - 1469650704 q^{47} - 9335236125 q^{48} - 3553434720 q^{49} - 6643771701 q^{50} + 1736777052 q^{51} + 3632448874 q^{52} + 14914261944 q^{53} + 18127857753 q^{54} + 4981449984 q^{55} - 27669139026 q^{56} - 7855424196 q^{57} - 1387480560 q^{58} - 26505032592 q^{59} - 10283356116 q^{60} + 990409066 q^{61} + 91044996180 q^{62} + 51565206888 q^{63} - 7516709566 q^{64} - 39045315390 q^{65} - 93201828246 q^{66} + 6557215720 q^{67} - 77299152993 q^{68} + 6907292550 q^{69} - 785437278 q^{70} + 122053719744 q^{71} + 161899013547 q^{72} - 12612893936 q^{73} - 109519086216 q^{74} - 218383044348 q^{75} + 14574055597 q^{76} - 88616208018 q^{77} + 150319870614 q^{78} - 7621233248 q^{79} + 399166683072 q^{80} + 222307104312 q^{81} - 59168477334 q^{82} - 99007044180 q^{83} - 711968015814 q^{84} + 12911595156 q^{85} - 214357830519 q^{86} + 99715491216 q^{87} - 54423523605 q^{88} + 476597704824 q^{89} + 620021743884 q^{90} + 138211652216 q^{91} - 461776423998 q^{92} - 572981484354 q^{93} + 13393667064 q^{94} - 418952909328 q^{95} + 118587589272 q^{96} + 123483551938 q^{97} + 1310123604078 q^{98} + 621154334268 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 9863 x^{18} + 40416552 x^{16} + 89424581388 x^{14} + 116167273852206 x^{12} + \cdots + 59\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31\!\cdots\!77 \nu^{18} + \cdots + 84\!\cdots\!28 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\!\cdots\!77 \nu^{18} + \cdots + 84\!\cdots\!28 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 55\!\cdots\!73 \nu^{19} + \cdots + 59\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!59 \nu^{19} + \cdots - 13\!\cdots\!96 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 62\!\cdots\!23 \nu^{19} + \cdots + 28\!\cdots\!60 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 62\!\cdots\!23 \nu^{19} + \cdots + 38\!\cdots\!40 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 76\!\cdots\!52 \nu^{19} + \cdots - 47\!\cdots\!64 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!76 \nu^{19} + \cdots + 35\!\cdots\!96 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 70\!\cdots\!90 \nu^{19} + \cdots + 32\!\cdots\!44 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!51 \nu^{19} + \cdots - 17\!\cdots\!60 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!73 \nu^{19} + \cdots + 51\!\cdots\!80 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 29\!\cdots\!63 \nu^{19} + \cdots - 56\!\cdots\!12 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 34\!\cdots\!53 \nu^{19} + \cdots - 46\!\cdots\!28 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 40\!\cdots\!27 \nu^{19} + \cdots - 11\!\cdots\!12 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{19} + \cdots - 72\!\cdots\!12 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 37\!\cdots\!29 \nu^{19} + \cdots - 42\!\cdots\!12 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 55\!\cdots\!53 \nu^{19} + \cdots - 29\!\cdots\!56 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 41\!\cdots\!33 \nu^{19} + \cdots + 90\!\cdots\!16 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 15\!\cdots\!35 \nu^{19} + \cdots - 42\!\cdots\!92 ) / 10\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} - 5\beta_{2} - 5\beta _1 - 2960 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{19} - 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{13} - 3 \beta_{12} - \beta_{11} + \cdots - 16054 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 20 \beta_{18} + 5 \beta_{16} - 6 \beta_{15} + 56 \beta_{14} + 13 \beta_{13} + 51 \beta_{12} + \cdots + 14814645 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15832 \beta_{19} - 552 \beta_{18} + 18904 \beta_{17} - 1428 \beta_{16} - 7916 \beta_{15} + \cdots + 180221954 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 77888 \beta_{18} - 13892 \beta_{16} + 36330 \beta_{15} - 242210 \beta_{14} - 105166 \beta_{13} + \cdots - 28707727592 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 12108962 \beta_{19} + 662708 \beta_{18} - 16789658 \beta_{17} + 2157862 \beta_{16} + \cdots - 191653437692 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 227714548 \beta_{18} + 29592577 \beta_{16} - 138269400 \beta_{15} + 762227890 \beta_{14} + \cdots + 60229425527857 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 27427343600 \beta_{19} - 1927847452 \beta_{18} + 42161993096 \beta_{17} - 7076732714 \beta_{16} + \cdots + 573524051995116 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 604759247088 \beta_{18} - 56014019556 \beta_{16} + 437997118332 \beta_{15} - 2140246820484 \beta_{14} + \cdots - 13\!\cdots\!32 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 188491657070390 \beta_{19} + 16429507804632 \beta_{18} - 310436022233990 \beta_{17} + \cdots - 49\!\cdots\!50 ) / 27 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 15\!\cdots\!92 \beta_{18} + 95655799280645 \beta_{16} + \cdots + 30\!\cdots\!85 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 14\!\cdots\!16 \beta_{19} + \cdots + 45\!\cdots\!42 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 38\!\cdots\!20 \beta_{18} + \cdots - 70\!\cdots\!32 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 34\!\cdots\!62 \beta_{19} + \cdots - 12\!\cdots\!76 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 95\!\cdots\!48 \beta_{18} + \cdots + 16\!\cdots\!33 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 24\!\cdots\!56 \beta_{19} + \cdots + 96\!\cdots\!48 ) / 27 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 78\!\cdots\!00 \beta_{18} + \cdots - 13\!\cdots\!36 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 66\!\cdots\!70 \beta_{19} + \cdots - 27\!\cdots\!50 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
44.9985i
39.8650i
26.4646i
20.7390i
5.09438i
− 1.73070i
− 15.2468i
− 31.4671i
− 37.0247i
− 49.9602i
− 44.9985i
− 39.8650i
− 26.4646i
− 20.7390i
− 5.09438i
1.73070i
15.2468i
31.4671i
37.0247i
49.9602i
−40.4699 − 70.0959i −411.436 − 88.6985i −2251.62 + 3899.92i −156.497 + 271.061i 10433.4 + 32429.6i −40821.5 − 70704.9i 198726. 161412. + 72987.5i 25333.7
4.2 −36.0241 − 62.3956i 417.750 + 51.3055i −1571.47 + 2721.87i −5310.88 + 9198.71i −11847.8 − 27914.0i 26579.3 + 46036.7i 78888.8 171882. + 42865.7i 765278.
4.3 −24.4190 − 42.2949i 129.541 − 400.457i −168.574 + 291.979i 5966.72 − 10334.7i −20100.6 + 4299.85i 19533.0 + 33832.1i −83554.5 −143585. − 103751.i −582805.
4.4 −19.4605 − 33.7066i 5.51306 + 420.852i 266.578 − 461.726i 1691.47 − 2929.71i 14078.2 − 8375.82i −16014.5 − 27738.0i −100461. −177086. + 4640.36i −131667.
4.5 −5.91186 − 10.2396i −417.157 + 55.9222i 954.100 − 1652.55i −2014.61 + 3489.40i 3038.79 + 3940.93i 34855.3 + 60371.1i −46777.0 170892. − 46656.6i 47640.3
4.6 −0.00116679 − 0.00202094i 35.8201 − 419.361i 1024.00 − 1773.62i −5478.49 + 9489.02i −0.889299 + 0.416917i −31655.6 − 54829.1i −9.55835 −174581. − 30043.1i 25.5690
4.7 11.7041 + 20.2721i 409.205 + 98.4789i 750.028 − 1299.09i 1606.12 − 2781.88i 2793.01 + 9448.06i −185.973 − 322.114i 83053.6 157751. + 80596.1i 75192.8
4.8 25.7513 + 44.6025i −329.417 − 261.976i −302.258 + 523.527i 4464.02 − 7731.91i 3201.87 − 21439.1i −5876.40 − 10178.2i 74343.1 39884.4 + 172599.i 459817.
4.9 30.5643 + 52.9389i −141.830 + 396.272i −844.353 + 1462.46i −1684.05 + 2916.87i −25313.1 + 4603.46i −5398.44 − 9350.38i 21963.1 −136916. − 112406.i −205888.
4.10 41.7668 + 72.3422i 296.011 − 299.207i −2464.93 + 4269.38i −2698.81 + 4674.47i 34008.7 + 8917.18i 23240.9 + 40254.5i −240732. −1902.15 − 177137.i −450882.
7.1 −40.4699 + 70.0959i −411.436 + 88.6985i −2251.62 − 3899.92i −156.497 − 271.061i 10433.4 − 32429.6i −40821.5 + 70704.9i 198726. 161412. − 72987.5i 25333.7
7.2 −36.0241 + 62.3956i 417.750 − 51.3055i −1571.47 − 2721.87i −5310.88 − 9198.71i −11847.8 + 27914.0i 26579.3 − 46036.7i 78888.8 171882. − 42865.7i 765278.
7.3 −24.4190 + 42.2949i 129.541 + 400.457i −168.574 − 291.979i 5966.72 + 10334.7i −20100.6 − 4299.85i 19533.0 − 33832.1i −83554.5 −143585. + 103751.i −582805.
7.4 −19.4605 + 33.7066i 5.51306 − 420.852i 266.578 + 461.726i 1691.47 + 2929.71i 14078.2 + 8375.82i −16014.5 + 27738.0i −100461. −177086. − 4640.36i −131667.
7.5 −5.91186 + 10.2396i −417.157 − 55.9222i 954.100 + 1652.55i −2014.61 − 3489.40i 3038.79 − 3940.93i 34855.3 − 60371.1i −46777.0 170892. + 46656.6i 47640.3
7.6 −0.00116679 + 0.00202094i 35.8201 + 419.361i 1024.00 + 1773.62i −5478.49 − 9489.02i −0.889299 − 0.416917i −31655.6 + 54829.1i −9.55835 −174581. + 30043.1i 25.5690
7.7 11.7041 − 20.2721i 409.205 − 98.4789i 750.028 + 1299.09i 1606.12 + 2781.88i 2793.01 − 9448.06i −185.973 + 322.114i 83053.6 157751. − 80596.1i 75192.8
7.8 25.7513 − 44.6025i −329.417 + 261.976i −302.258 − 523.527i 4464.02 + 7731.91i 3201.87 + 21439.1i −5876.40 + 10178.2i 74343.1 39884.4 − 172599.i 459817.
7.9 30.5643 − 52.9389i −141.830 − 396.272i −844.353 − 1462.46i −1684.05 − 2916.87i −25313.1 − 4603.46i −5398.44 + 9350.38i 21963.1 −136916. + 112406.i −205888.
7.10 41.7668 − 72.3422i 296.011 + 299.207i −2464.93 − 4269.38i −2698.81 − 4674.47i 34008.7 − 8917.18i 23240.9 − 40254.5i −240732. −1902.15 + 177137.i −450882.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.12.c.a ✓ 20
3.b odd 2 1 27.12.c.a 20
9.c even 3 1 inner 9.12.c.a ✓ 20
9.c even 3 1 81.12.a.e 10
9.d odd 6 1 27.12.c.a 20
9.d odd 6 1 81.12.a.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.12.c.a ✓ 20 1.a even 1 1 trivial
9.12.c.a ✓ 20 9.c even 3 1 inner
27.12.c.a 20 3.b odd 2 1
27.12.c.a 20 9.d odd 6 1
81.12.a.c 10 9.d odd 6 1
81.12.a.e 10 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(9, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 30\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 46\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 36\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 94\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 29\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 53\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 68\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
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