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Properties

Label 833.2.g.h
Level $833$
Weight $2$
Character orbit 833.g
Analytic conductor $6.652$
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] = [833,2,Mod(344,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.344");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.g (of order \(4\), 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: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 32 x^{18} + 432 x^{16} + 3200 x^{14} + 14160 x^{12} + 38162 x^{10} + 61088 x^{8} + 53872 x^{6} + \cdots + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 119)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{18} + \beta_{17} - 2) q^{4} - \beta_{13} q^{5} + (\beta_{16} - \beta_{14} + \cdots + \beta_{3}) q^{6}+ \cdots + ( - \beta_{19} + \beta_{11} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{14} q^{3} + (\beta_{18} + \beta_{17} - 2) q^{4} - \beta_{13} q^{5} + (\beta_{16} - \beta_{14} + \cdots + \beta_{3}) q^{6}+ \cdots + ( - \beta_{17} + 3 \beta_{16} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} - 24 q^{4} + 8 q^{5} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} - 24 q^{4} + 8 q^{5} - 4 q^{6} - 4 q^{10} - 4 q^{11} - 12 q^{12} + 16 q^{16} + 12 q^{17} - 8 q^{18} - 20 q^{20} - 20 q^{22} - 4 q^{24} - 8 q^{27} + 16 q^{29} - 36 q^{30} - 4 q^{31} + 16 q^{33} - 36 q^{34} - 28 q^{37} - 48 q^{38} + 20 q^{39} + 24 q^{40} + 24 q^{41} - 28 q^{44} - 36 q^{45} + 8 q^{46} - 40 q^{47} + 8 q^{48} - 28 q^{50} - 40 q^{51} + 28 q^{54} + 40 q^{55} + 36 q^{57} + 56 q^{58} + 16 q^{61} + 40 q^{62} + 32 q^{64} + 8 q^{65} + 32 q^{68} + 88 q^{69} - 8 q^{71} + 108 q^{72} - 8 q^{73} + 36 q^{74} - 8 q^{75} - 44 q^{78} - 4 q^{79} + 116 q^{80} + 4 q^{81} + 16 q^{82} + 16 q^{85} + 44 q^{86} + 72 q^{88} - 48 q^{89} - 56 q^{90} - 32 q^{92} + 44 q^{95} - 68 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 32 x^{18} + 432 x^{16} + 3200 x^{14} + 14160 x^{12} + 38162 x^{10} + 61088 x^{8} + 53872 x^{6} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8 \nu^{18} + 255 \nu^{16} + 3272 \nu^{14} + 21540 \nu^{12} + 77120 \nu^{10} + 149616 \nu^{8} + \cdots + 3136 ) / 17038 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 155 \nu^{19} + 2963 \nu^{17} + 9847 \nu^{15} - 172299 \nu^{13} - 1913400 \nu^{11} + \cdots - 528268 \nu ) / 102228 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 64 \nu^{19} - 2040 \nu^{17} - 27393 \nu^{15} - 201528 \nu^{13} - 884700 \nu^{11} + \cdots - 360980 \nu ) / 34076 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64 \nu^{19} - 2040 \nu^{17} - 27393 \nu^{15} - 201528 \nu^{13} - 884700 \nu^{11} + \cdots - 156524 \nu ) / 17038 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 620 \nu^{19} - 19154 \nu^{17} - 247495 \nu^{15} - 1731417 \nu^{13} - 7070883 \nu^{11} + \cdots + 1452241 \nu ) / 102228 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 623 \nu^{18} + 19706 \nu^{16} + 259675 \nu^{14} + 1839897 \nu^{12} + 7537923 \nu^{10} + \cdots - 19873 ) / 102228 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1025 \nu^{19} + 1025 \nu^{18} + 31607 \nu^{17} + 31607 \nu^{16} + 405838 \nu^{15} + 402187 \nu^{14} + \cdots + 214382 ) / 204456 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1025 \nu^{19} - 1025 \nu^{18} + 31607 \nu^{17} - 31607 \nu^{16} + 405838 \nu^{15} - 402187 \nu^{14} + \cdots - 214382 ) / 204456 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1351 \nu^{19} - 878 \nu^{18} - 40477 \nu^{17} - 26465 \nu^{16} - 503879 \nu^{15} - 331111 \nu^{14} + \cdots + 22141 ) / 204456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1351 \nu^{19} + 878 \nu^{18} - 40477 \nu^{17} + 26465 \nu^{16} - 503879 \nu^{15} + 331111 \nu^{14} + \cdots - 22141 ) / 204456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 571 \nu^{19} + 347 \nu^{18} + 17440 \nu^{17} + 11517 \nu^{16} + 221369 \nu^{15} + 161395 \nu^{14} + \cdots + 229733 ) / 68152 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 571 \nu^{19} - 347 \nu^{18} + 17440 \nu^{17} - 11517 \nu^{16} + 221369 \nu^{15} - 161395 \nu^{14} + \cdots - 229733 ) / 68152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2539 \nu^{19} - 310 \nu^{18} - 80170 \nu^{17} - 5926 \nu^{16} - 1062791 \nu^{15} - 23345 \nu^{14} + \cdots - 181153 ) / 204456 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2539 \nu^{19} - 310 \nu^{18} + 80170 \nu^{17} - 5926 \nu^{16} + 1062791 \nu^{15} - 23345 \nu^{14} + \cdots - 181153 ) / 204456 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 375 \nu^{18} - 10584 \nu^{16} - 120516 \nu^{14} - 699961 \nu^{12} - 2121741 \nu^{10} + \cdots + 31899 ) / 34076 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1495 \nu^{18} + 45067 \nu^{16} + 566426 \nu^{14} + 3848214 \nu^{12} + 15311163 \nu^{10} + \cdots + 287875 ) / 102228 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 1495 \nu^{18} - 45067 \nu^{16} - 566426 \nu^{14} - 3848214 \nu^{12} - 15311163 \nu^{10} + \cdots + 121037 ) / 102228 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 2108 \nu^{19} + 66584 \nu^{17} + 885295 \nu^{15} + 6438849 \nu^{13} + 27840963 \nu^{11} + \cdots + 2285519 \nu ) / 102228 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{18} + \beta_{17} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{18} - 7\beta_{17} - \beta_{16} - \beta_{13} + \beta_{12} - \beta_{7} - \beta_{2} + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 8\beta_{10} - 8\beta_{6} - 7\beta_{5} - 10\beta_{4} + 8\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36 \beta_{18} + 44 \beta_{17} + 14 \beta_{16} - 4 \beta_{15} - 4 \beta_{14} + 10 \beta_{13} - 10 \beta_{12} + \cdots - 122 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{19} - 2 \beta_{13} - 2 \beta_{12} - 56 \beta_{11} - 56 \beta_{10} + 2 \beta_{9} + \cdots - 112 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 220 \beta_{18} - 271 \beta_{17} - 135 \beta_{16} + 55 \beta_{15} + 55 \beta_{14} - 80 \beta_{13} + \cdots + 742 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{19} + \beta_{15} - \beta_{14} + 29 \beta_{13} + 29 \beta_{12} + 380 \beta_{11} + \cdots + 669 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1368 \beta_{18} + 1672 \beta_{17} + 1124 \beta_{16} - 532 \beta_{15} - 532 \beta_{14} + 592 \beta_{13} + \cdots - 4637 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 216 \beta_{19} - 12 \beta_{15} + 12 \beta_{14} - 292 \beta_{13} - 292 \beta_{12} - 2556 \beta_{11} + \cdots - 4153 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8633 \beta_{18} - 10393 \beta_{17} - 8692 \beta_{16} + 4476 \beta_{15} + 4476 \beta_{14} + \cdots + 29512 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2168 \beta_{19} + 64 \beta_{15} - 64 \beta_{14} + 2544 \beta_{13} + 2544 \beta_{12} + 17157 \beta_{11} + \cdots + 26420 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 55150 \beta_{18} + 65143 \beta_{17} + 64461 \beta_{16} - 35096 \beta_{15} - 35096 \beta_{14} + \cdots - 190345 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 19656 \beta_{19} + 144 \beta_{15} - 144 \beta_{14} - 20608 \beta_{13} - 20608 \beta_{12} + \cdots - 170892 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 355868 \beta_{18} - 411532 \beta_{17} - 466018 \beta_{16} + 264340 \beta_{15} + 264340 \beta_{14} + \cdots + 1240286 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 166921 \beta_{19} - 7928 \beta_{15} + 7928 \beta_{14} + 160158 \beta_{13} + 160158 \beta_{12} + \cdots + 1118616 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 2315316 \beta_{18} + 2618247 \beta_{17} + 3314555 \beta_{16} - 1942283 \beta_{15} - 1942283 \beta_{14} + \cdots - 8147338 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1354906 \beta_{19} + 118467 \beta_{15} - 118467 \beta_{14} - 1213037 \beta_{13} + \cdots - 7387477 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
344.1
− 2.51056i
− 2.32367i
− 1.57229i
− 1.40508i
− 0.124341i
0.582556i
0.710316i
1.88449i
2.13581i
2.62277i
− 2.62277i
− 2.13581i
− 1.88449i
− 0.710316i
− 0.582556i
0.124341i
1.40508i
1.57229i
2.32367i
2.51056i
− 2.51056i 1.85068 − 1.85068i −4.30293 2.11878 − 2.11878i −4.64626 − 4.64626i 0 5.78166i − 3.85007i −5.31933 − 5.31933i
344.2 − 2.32367i −1.48963 + 1.48963i −3.39944 1.04083 − 1.04083i 3.46141 + 3.46141i 0 3.25185i − 1.43798i −2.41854 − 2.41854i
344.3 − 1.57229i 1.43847 − 1.43847i −0.472100 −2.40051 + 2.40051i −2.26170 − 2.26170i 0 − 2.40230i − 1.13841i 3.77429 + 3.77429i
344.4 − 1.40508i −0.454236 + 0.454236i 0.0257627 −0.209507 + 0.209507i 0.638235 + 0.638235i 0 − 2.84635i 2.58734i 0.294374 + 0.294374i
344.5 − 0.124341i 1.21702 − 1.21702i 1.98454 2.27864 − 2.27864i −0.151325 − 0.151325i 0 − 0.495440i 0.0377310i −0.283327 − 0.283327i
344.6 0.582556i 0.610447 − 0.610447i 1.66063 −1.41294 + 1.41294i 0.355620 + 0.355620i 0 2.13252i 2.25471i −0.823119 − 0.823119i
344.7 0.710316i −2.11910 + 2.11910i 1.49545 2.66663 − 2.66663i −1.50523 − 1.50523i 0 2.48287i − 5.98113i 1.89415 + 1.89415i
344.8 1.88449i −0.935584 + 0.935584i −1.55131 −0.976605 + 0.976605i −1.76310 − 1.76310i 0 0.845554i 1.24937i −1.84040 − 1.84040i
344.9 2.13581i 2.18393 − 2.18393i −2.56169 −0.770779 + 0.770779i 4.66447 + 4.66447i 0 − 1.19966i − 6.53913i −1.64624 − 1.64624i
344.10 2.62277i −0.302014 + 0.302014i −4.87891 1.66547 − 1.66547i −0.792113 − 0.792113i 0 − 7.55071i 2.81758i 4.36814 + 4.36814i
540.1 − 2.62277i −0.302014 − 0.302014i −4.87891 1.66547 + 1.66547i −0.792113 + 0.792113i 0 7.55071i − 2.81758i 4.36814 − 4.36814i
540.2 − 2.13581i 2.18393 + 2.18393i −2.56169 −0.770779 − 0.770779i 4.66447 − 4.66447i 0 1.19966i 6.53913i −1.64624 + 1.64624i
540.3 − 1.88449i −0.935584 − 0.935584i −1.55131 −0.976605 − 0.976605i −1.76310 + 1.76310i 0 − 0.845554i − 1.24937i −1.84040 + 1.84040i
540.4 − 0.710316i −2.11910 − 2.11910i 1.49545 2.66663 + 2.66663i −1.50523 + 1.50523i 0 − 2.48287i 5.98113i 1.89415 − 1.89415i
540.5 − 0.582556i 0.610447 + 0.610447i 1.66063 −1.41294 − 1.41294i 0.355620 − 0.355620i 0 − 2.13252i − 2.25471i −0.823119 + 0.823119i
540.6 0.124341i 1.21702 + 1.21702i 1.98454 2.27864 + 2.27864i −0.151325 + 0.151325i 0 0.495440i − 0.0377310i −0.283327 + 0.283327i
540.7 1.40508i −0.454236 − 0.454236i 0.0257627 −0.209507 − 0.209507i 0.638235 − 0.638235i 0 2.84635i − 2.58734i 0.294374 − 0.294374i
540.8 1.57229i 1.43847 + 1.43847i −0.472100 −2.40051 − 2.40051i −2.26170 + 2.26170i 0 2.40230i 1.13841i 3.77429 − 3.77429i
540.9 2.32367i −1.48963 − 1.48963i −3.39944 1.04083 + 1.04083i 3.46141 − 3.46141i 0 − 3.25185i 1.43798i −2.41854 + 2.41854i
540.10 2.51056i 1.85068 + 1.85068i −4.30293 2.11878 + 2.11878i −4.64626 + 4.64626i 0 − 5.78166i 3.85007i −5.31933 + 5.31933i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 344.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.g.h 20
7.b odd 2 1 119.2.g.a ✓ 20
7.c even 3 2 833.2.o.f 40
7.d odd 6 2 833.2.o.g 40
17.c even 4 1 inner 833.2.g.h 20
21.c even 2 1 1071.2.n.c 20
119.f odd 4 1 119.2.g.a ✓ 20
119.l odd 8 1 2023.2.a.m 10
119.l odd 8 1 2023.2.a.n 10
119.m odd 12 2 833.2.o.g 40
119.n even 12 2 833.2.o.f 40
357.l even 4 1 1071.2.n.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.g.a ✓ 20 7.b odd 2 1
119.2.g.a ✓ 20 119.f odd 4 1
833.2.g.h 20 1.a even 1 1 trivial
833.2.g.h 20 17.c even 4 1 inner
833.2.o.f 40 7.c even 3 2
833.2.o.f 40 119.n even 12 2
833.2.o.g 40 7.d odd 6 2
833.2.o.g 40 119.m odd 12 2
1071.2.n.c 20 21.c even 2 1
1071.2.n.c 20 357.l even 4 1
2023.2.a.m 10 119.l odd 8 1
2023.2.a.n 10 119.l odd 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{20} + 32 T_{2}^{18} + 432 T_{2}^{16} + 3200 T_{2}^{14} + 14160 T_{2}^{12} + 38162 T_{2}^{10} + \cdots + 49 \) Copy content Toggle raw display
\( T_{3}^{20} - 4 T_{3}^{19} + 8 T_{3}^{18} + 102 T_{3}^{16} - 412 T_{3}^{15} + 832 T_{3}^{14} + \cdots + 3136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 32 T^{18} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{20} - 4 T^{19} + \cdots + 3136 \) Copy content Toggle raw display
$5$ \( T^{20} - 8 T^{19} + \cdots + 145924 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 4 T^{19} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( (T^{10} - 42 T^{8} + \cdots + 512)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 2015993900449 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 1073741824 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 901071765504 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 6466733056 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 58928533504 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 106390782976 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 266319987844 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 31201072267264 \) Copy content Toggle raw display
$47$ \( (T^{10} + 20 T^{9} + \cdots - 3084544)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 11516086583296 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 285993926656 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 2592621666244 \) Copy content Toggle raw display
$67$ \( (T^{10} - 410 T^{8} + \cdots - 94120384)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 346700824969216 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 900369458884 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 21870662385664 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 575545433718784 \) Copy content Toggle raw display
$89$ \( (T^{10} + 24 T^{9} + \cdots + 3584)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 35385199514116 \) Copy content Toggle raw display
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