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Properties

Label 931.2.e.d
Level $931$
Weight $2$
Character orbit 931.e
Analytic conductor $7.434$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(197,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.e (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: \(7.43407242818\)
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: \( x^{20} + 23 x^{18} + 358 x^{16} + 3025 x^{14} + 18501 x^{12} + 63983 x^{10} + 153847 x^{8} + 115798 x^{6} + \cdots + 3249 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} + \beta_{5}) q^{2} + \beta_{3} q^{3} + ( - \beta_{4} - \beta_{2} - 1) q^{4} - \beta_{15} q^{5} + (\beta_{19} + \beta_{17} + \cdots - \beta_{12}) q^{6}+ \cdots + (\beta_{6} - \beta_{5} - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} + \beta_{5}) q^{2} + \beta_{3} q^{3} + ( - \beta_{4} - \beta_{2} - 1) q^{4} - \beta_{15} q^{5} + (\beta_{19} + \beta_{17} + \cdots - \beta_{12}) q^{6}+ \cdots + (3 \beta_{9} + 5 \beta_{5} + 6 \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} - 14 q^{4} + 12 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} - 14 q^{4} + 12 q^{8} - 16 q^{9} + 4 q^{11} - 8 q^{15} - 14 q^{16} + 60 q^{18} + 14 q^{22} + 12 q^{23} - 20 q^{29} + 32 q^{30} + 16 q^{32} - 56 q^{36} + 4 q^{37} - 52 q^{39} + 4 q^{43} + 48 q^{44} + 48 q^{46} + 28 q^{50} - 46 q^{51} - 10 q^{53} + 28 q^{57} + 36 q^{58} - 62 q^{60} - 76 q^{64} - 32 q^{65} - 30 q^{67} - 40 q^{71} - 62 q^{72} + 28 q^{74} - 76 q^{78} - 6 q^{79} - 50 q^{81} - 8 q^{85} - 40 q^{86} - 152 q^{88} + 72 q^{92} - 36 q^{93} - 20 q^{95} + 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 23 x^{18} + 358 x^{16} + 3025 x^{14} + 18501 x^{12} + 63983 x^{10} + 153847 x^{8} + 115798 x^{6} + \cdots + 3249 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 203921130386591 \nu^{18} + \cdots + 10\!\cdots\!25 ) / 21\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 203921130386591 \nu^{19} + \cdots + 31\!\cdots\!42 \nu ) / 21\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\!\cdots\!10 \nu^{18} + \cdots + 52\!\cdots\!61 ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!86 \nu^{18} + \cdots + 63\!\cdots\!95 ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!74 \nu^{18} + \cdots + 73\!\cdots\!67 ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 112361838833825 \nu^{18} + \cdots - 45\!\cdots\!82 ) / 42\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 914481020686953 \nu^{18} + \cdots + 40\!\cdots\!22 ) / 29\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 62\!\cdots\!90 \nu^{18} + \cdots - 13\!\cdots\!31 ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 88\!\cdots\!19 \nu^{18} + \cdots - 43\!\cdots\!41 ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!69 \nu^{18} + \cdots + 84\!\cdots\!06 ) / 29\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 40\!\cdots\!58 \nu^{19} + \cdots - 80\!\cdots\!75 \nu ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!74 \nu^{19} + \cdots - 24\!\cdots\!45 \nu ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\!\cdots\!75 \nu^{19} + \cdots + 62\!\cdots\!99 \nu ) / 25\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 45\!\cdots\!38 \nu^{19} + \cdots + 10\!\cdots\!63 \nu ) / 85\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 13\!\cdots\!89 \nu^{19} + \cdots + 23\!\cdots\!00 \nu ) / 25\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 93\!\cdots\!03 \nu^{19} + \cdots - 46\!\cdots\!41 \nu ) / 17\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 14\!\cdots\!61 \nu^{19} + \cdots + 62\!\cdots\!14 \nu ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 29\!\cdots\!85 \nu^{19} + \cdots + 14\!\cdots\!47 \nu ) / 28\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{5} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 2\beta_{17} + 7\beta_{3} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} + 9\beta_{6} - 14\beta_{5} - 2\beta_{4} - 29\beta_{2} - 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{16} + 3\beta_{15} - 2\beta_{14} + 19\beta_{13} - 28\beta_{12} - 56\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -52\beta_{11} - 82\beta_{10} + 161\beta_{8} - 43\beta_{7} - 82\beta_{6} + 243 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 170 \beta_{19} - 52 \beta_{18} - 338 \beta_{17} - 86 \beta_{16} - 43 \beta_{14} - 213 \beta_{13} + \cdots + 575 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 689 \beta_{11} + 784 \beta_{10} + 689 \beta_{9} - 1761 \beta_{8} + 648 \beta_{7} + 1761 \beta_{5} + \cdots + 2212 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1802 \beta_{19} + 689 \beta_{18} + 3882 \beta_{17} + 648 \beta_{16} - 689 \beta_{15} + 1296 \beta_{14} + \cdots - 5860 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -8317\beta_{9} + 7798\beta_{6} - 19030\beta_{5} - 8459\beta_{4} - 21354\beta_{2} - 21354 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8459\beta_{16} + 8317\beta_{15} - 8459\beta_{14} + 35806\beta_{13} - 43604\beta_{12} - 44748\beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -96186\beta_{11} - 79893\beta_{10} + 205538\beta_{8} - 102675\beta_{7} - 79893\beta_{6} + 214798 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 199049 \beta_{19} - 96186 \beta_{18} - 484292 \beta_{17} - 205350 \beta_{16} - 102675 \beta_{14} + \cdots + 659827 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1087552 \beta_{11} + 836094 \beta_{10} + 1087552 \beta_{9} - 2225558 \beta_{8} + \cdots + 2222307 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2116582 \beta_{19} + 1087552 \beta_{18} + 5345732 \beta_{17} + 1196528 \beta_{16} - 1087552 \beta_{15} + \cdots - 7123645 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( -12139450\beta_{9} + 8879793\beta_{6} - 24164855\beta_{5} - 13612452\beta_{4} - 23431994\beta_{2} - 23431994 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 13612452 \beta_{16} + 12139450 \beta_{15} - 13612452 \beta_{14} + 49916757 \beta_{13} + \cdots - 50071373 \beta_{3} \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 134465209 \beta_{11} - 95255471 \beta_{10} + 262964936 \beta_{8} - 152610712 \beta_{7} + \cdots + 250202249 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 244819433 \beta_{19} - 134465209 \beta_{18} - 645296328 \beta_{17} - 305221424 \beta_{16} + \cdots + 841190086 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
1.65341 − 2.86380i
−1.65341 + 2.86380i
0.319539 − 0.553458i
−0.319539 + 0.553458i
1.32681 − 2.29810i
−1.32681 + 2.29810i
0.329190 − 0.570173i
−0.329190 + 0.570173i
1.02242 − 1.77088i
−1.02242 + 1.77088i
1.65341 + 2.86380i
−1.65341 − 2.86380i
0.319539 + 0.553458i
−0.319539 − 0.553458i
1.32681 + 2.29810i
−1.32681 − 2.29810i
0.329190 + 0.570173i
−0.329190 − 0.570173i
1.02242 + 1.77088i
−1.02242 − 1.77088i
−1.24809 − 2.16176i −1.65341 − 2.86380i −2.11547 + 3.66410i −0.953114 − 1.65084i −4.12723 + 7.14856i 0 5.56882 −3.96756 + 6.87201i −2.37915 + 4.12080i
197.2 −1.24809 − 2.16176i 1.65341 + 2.86380i −2.11547 + 3.66410i 0.953114 + 1.65084i 4.12723 − 7.14856i 0 5.56882 −3.96756 + 6.87201i 2.37915 − 4.12080i
197.3 −0.961368 − 1.66514i −0.319539 − 0.553458i −0.848456 + 1.46957i 1.41499 + 2.45084i −0.614389 + 1.06415i 0 −0.582759 1.29579 − 2.24437i 2.72066 − 4.71232i
197.4 −0.961368 − 1.66514i 0.319539 + 0.553458i −0.848456 + 1.46957i −1.41499 − 2.45084i 0.614389 − 1.06415i 0 −0.582759 1.29579 − 2.24437i −2.72066 + 4.71232i
197.5 −0.0402239 − 0.0696698i −1.32681 − 2.29810i 0.996764 − 1.72645i 0.639740 + 1.10806i −0.106739 + 0.184877i 0 −0.321270 −2.02083 + 3.50018i 0.0514657 − 0.0891411i
197.6 −0.0402239 − 0.0696698i 1.32681 + 2.29810i 0.996764 − 1.72645i −0.639740 − 1.10806i 0.106739 − 0.184877i 0 −0.321270 −2.02083 + 3.50018i −0.0514657 + 0.0891411i
197.7 0.531531 + 0.920638i −0.329190 − 0.570173i 0.434950 − 0.753355i −1.70422 − 2.95179i 0.349949 − 0.606129i 0 3.05088 1.28327 − 2.22269i 1.81169 − 3.13793i
197.8 0.531531 + 0.920638i 0.329190 + 0.570173i 0.434950 − 0.753355i 1.70422 + 2.95179i −0.349949 + 0.606129i 0 3.05088 1.28327 − 2.22269i −1.81169 + 3.13793i
197.9 1.21815 + 2.10990i −1.02242 − 1.77088i −1.96779 + 3.40831i −0.160457 − 0.277920i 2.49092 − 4.31440i 0 −4.71567 −0.590671 + 1.02307i 0.390923 − 0.677099i
197.10 1.21815 + 2.10990i 1.02242 + 1.77088i −1.96779 + 3.40831i 0.160457 + 0.277920i −2.49092 + 4.31440i 0 −4.71567 −0.590671 + 1.02307i −0.390923 + 0.677099i
638.1 −1.24809 + 2.16176i −1.65341 + 2.86380i −2.11547 − 3.66410i −0.953114 + 1.65084i −4.12723 − 7.14856i 0 5.56882 −3.96756 − 6.87201i −2.37915 − 4.12080i
638.2 −1.24809 + 2.16176i 1.65341 − 2.86380i −2.11547 − 3.66410i 0.953114 − 1.65084i 4.12723 + 7.14856i 0 5.56882 −3.96756 − 6.87201i 2.37915 + 4.12080i
638.3 −0.961368 + 1.66514i −0.319539 + 0.553458i −0.848456 − 1.46957i 1.41499 − 2.45084i −0.614389 − 1.06415i 0 −0.582759 1.29579 + 2.24437i 2.72066 + 4.71232i
638.4 −0.961368 + 1.66514i 0.319539 − 0.553458i −0.848456 − 1.46957i −1.41499 + 2.45084i 0.614389 + 1.06415i 0 −0.582759 1.29579 + 2.24437i −2.72066 − 4.71232i
638.5 −0.0402239 + 0.0696698i −1.32681 + 2.29810i 0.996764 + 1.72645i 0.639740 − 1.10806i −0.106739 − 0.184877i 0 −0.321270 −2.02083 − 3.50018i 0.0514657 + 0.0891411i
638.6 −0.0402239 + 0.0696698i 1.32681 − 2.29810i 0.996764 + 1.72645i −0.639740 + 1.10806i 0.106739 + 0.184877i 0 −0.321270 −2.02083 − 3.50018i −0.0514657 − 0.0891411i
638.7 0.531531 − 0.920638i −0.329190 + 0.570173i 0.434950 + 0.753355i −1.70422 + 2.95179i 0.349949 + 0.606129i 0 3.05088 1.28327 + 2.22269i 1.81169 + 3.13793i
638.8 0.531531 − 0.920638i 0.329190 − 0.570173i 0.434950 + 0.753355i 1.70422 − 2.95179i −0.349949 − 0.606129i 0 3.05088 1.28327 + 2.22269i −1.81169 − 3.13793i
638.9 1.21815 − 2.10990i −1.02242 + 1.77088i −1.96779 − 3.40831i −0.160457 + 0.277920i 2.49092 + 4.31440i 0 −4.71567 −0.590671 − 1.02307i 0.390923 + 0.677099i
638.10 1.21815 − 2.10990i 1.02242 − 1.77088i −1.96779 − 3.40831i 0.160457 − 0.277920i −2.49092 − 4.31440i 0 −4.71567 −0.590671 − 1.02307i −0.390923 − 0.677099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.c even 3 1 inner
133.m odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.e.d ✓ 20
7.b odd 2 1 inner 931.2.e.d ✓ 20
7.c even 3 1 931.2.g.g 20
7.c even 3 1 931.2.h.g 20
7.d odd 6 1 931.2.g.g 20
7.d odd 6 1 931.2.h.g 20
19.c even 3 1 inner 931.2.e.d ✓ 20
133.g even 3 1 931.2.h.g 20
133.h even 3 1 931.2.g.g 20
133.k odd 6 1 931.2.h.g 20
133.m odd 6 1 inner 931.2.e.d ✓ 20
133.t odd 6 1 931.2.g.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.e.d ✓ 20 1.a even 1 1 trivial
931.2.e.d ✓ 20 7.b odd 2 1 inner
931.2.e.d ✓ 20 19.c even 3 1 inner
931.2.e.d ✓ 20 133.m odd 6 1 inner
931.2.g.g 20 7.c even 3 1
931.2.g.g 20 7.d odd 6 1
931.2.g.g 20 133.h even 3 1
931.2.g.g 20 133.t odd 6 1
931.2.h.g 20 7.c even 3 1
931.2.h.g 20 7.d odd 6 1
931.2.h.g 20 133.g even 3 1
931.2.h.g 20 133.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{10} + T_{2}^{9} + 9T_{2}^{8} + 4T_{2}^{7} + 58T_{2}^{6} + 25T_{2}^{5} + 131T_{2}^{4} - 56T_{2}^{3} + 150T_{2}^{2} + 12T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{20} + 23 T_{3}^{18} + 358 T_{3}^{16} + 3025 T_{3}^{14} + 18501 T_{3}^{12} + 63983 T_{3}^{10} + \cdots + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + T^{9} + 9 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} + 23 T^{18} + \cdots + 3249 \) Copy content Toggle raw display
$5$ \( T^{20} + 25 T^{18} + \cdots + 3249 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{5} - T^{4} - 28 T^{3} + \cdots - 28)^{4} \) Copy content Toggle raw display
$13$ \( T^{20} + 86 T^{18} + \cdots + 2030625 \) Copy content Toggle raw display
$17$ \( T^{20} + 23 T^{18} + \cdots + 3249 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} - 6 T^{9} + 42 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 10 T^{9} + \cdots + 24649)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 183 T^{8} + \cdots - 19707408)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - T^{4} - 82 T^{3} + \cdots + 252)^{4} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( (T^{10} - 2 T^{9} + \cdots + 5546025)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{10} + 5 T^{9} + \cdots + 257827249)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 7516369809 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 72\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( (T^{10} + 15 T^{9} + \cdots + 40233649)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 20 T^{9} + \cdots + 3613801)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 49\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( (T^{10} + 3 T^{9} + \cdots + 1849)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 666 T^{8} + \cdots - 54742800)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 598018902080625 \) Copy content Toggle raw display
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