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Properties

Label 931.2.f.q
Level $931$
Weight $2$
Character orbit 931.f
Analytic conductor $7.434$
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] = [931,2,Mod(324,931)]
 
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([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.f (of order \(3\), degree \(2\), not 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: \( x^{20} - 2 x^{19} + 17 x^{18} - 22 x^{17} + 160 x^{16} - 182 x^{15} + 935 x^{14} - 842 x^{13} + \cdots + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{18} - \beta_{11}) q^{3} + (\beta_{19} - \beta_{14}) q^{4} + (\beta_{15} + 2 \beta_{14} + \beta_{2} - 2) q^{5} + (\beta_{8} + \beta_{4} + \beta_{2}) q^{6} + (\beta_{6} - \beta_{5} + \beta_{4} - 1) q^{8} + (\beta_{19} - \beta_{18} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{18} - \beta_{11}) q^{3} + (\beta_{19} - \beta_{14}) q^{4} + (\beta_{15} + 2 \beta_{14} + \beta_{2} - 2) q^{5} + (\beta_{8} + \beta_{4} + \beta_{2}) q^{6} + (\beta_{6} - \beta_{5} + \beta_{4} - 1) q^{8} + (\beta_{19} - \beta_{18} + \cdots - \beta_1) q^{9}+ \cdots + (\beta_{11} - \beta_{9} + 6 \beta_{8} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 4 q^{3} - 10 q^{4} - 16 q^{5} + 16 q^{6} - 12 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} - 4 q^{3} - 10 q^{4} - 16 q^{5} + 16 q^{6} - 12 q^{8} - 10 q^{9} + 12 q^{10} - 12 q^{12} + 24 q^{13} - 2 q^{16} - 16 q^{17} - 2 q^{18} - 10 q^{19} + 64 q^{20} + 8 q^{22} - 12 q^{23} + 8 q^{24} - 14 q^{25} - 24 q^{26} + 32 q^{27} - 24 q^{29} + 12 q^{30} - 8 q^{31} + 34 q^{32} + 4 q^{33} + 32 q^{34} - 12 q^{36} - 4 q^{37} + 2 q^{38} + 4 q^{39} + 20 q^{40} + 80 q^{41} + 8 q^{43} + 20 q^{44} - 24 q^{45} + 32 q^{46} - 16 q^{47} + 24 q^{48} - 68 q^{50} + 28 q^{51} + 40 q^{52} - 8 q^{54} + 32 q^{55} + 8 q^{57} + 8 q^{58} - 36 q^{59} - 32 q^{60} - 16 q^{61} - 32 q^{62} + 36 q^{64} - 8 q^{65} - 8 q^{66} + 28 q^{67} - 40 q^{68} + 96 q^{69} - 24 q^{71} - 34 q^{72} - 24 q^{74} + 32 q^{75} + 20 q^{76} + 56 q^{78} + 8 q^{79} - 8 q^{80} - 14 q^{81} + 8 q^{82} + 80 q^{85} + 52 q^{86} + 8 q^{87} + 4 q^{88} - 48 q^{89} - 128 q^{90} + 56 q^{92} - 40 q^{93} + 36 q^{94} - 16 q^{95} + 8 q^{96} - 32 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 17 x^{18} - 22 x^{17} + 160 x^{16} - 182 x^{15} + 935 x^{14} - 842 x^{13} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!31 \nu^{19} + \cdots + 21\!\cdots\!62 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 62\!\cdots\!93 \nu^{19} + \cdots + 73\!\cdots\!80 ) / 62\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!60 \nu^{19} + \cdots - 53\!\cdots\!23 ) / 20\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70\!\cdots\!00 \nu^{19} + \cdots - 18\!\cdots\!95 ) / 62\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26\!\cdots\!68 \nu^{19} + \cdots + 16\!\cdots\!60 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 42\!\cdots\!33 \nu^{19} + \cdots + 31\!\cdots\!98 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 260963440717660 \nu^{19} + \cdots + 11\!\cdots\!87 ) / 81\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 72\!\cdots\!99 \nu^{19} + \cdots - 19\!\cdots\!08 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!62 \nu^{19} + \cdots + 18\!\cdots\!00 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!48 \nu^{19} + \cdots + 44\!\cdots\!73 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18\!\cdots\!05 \nu^{19} + \cdots - 15\!\cdots\!54 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33\!\cdots\!40 \nu^{19} + \cdots + 19\!\cdots\!43 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!20 \nu^{19} + \cdots + 58\!\cdots\!87 ) / 62\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 51\!\cdots\!56 \nu^{19} + \cdots - 20\!\cdots\!99 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 52\!\cdots\!60 \nu^{19} + \cdots + 23\!\cdots\!46 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 79\!\cdots\!75 \nu^{19} + \cdots + 16\!\cdots\!35 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 10\!\cdots\!84 \nu^{19} + \cdots + 32\!\cdots\!79 ) / 14\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 44\!\cdots\!60 \nu^{19} + \cdots + 17\!\cdots\!61 ) / 62\!\cdots\!46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{19} - 3\beta_{14} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} + 4\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{19} - \beta_{18} - \beta_{17} + 14 \beta_{14} + 2 \beta_{13} - \beta_{12} - \beta_{10} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{19} - \beta_{18} - 8 \beta_{17} + 11 \beta_{14} + \beta_{13} - 2 \beta_{12} + \cdots - 20 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{11} + 12\beta_{9} + 21\beta_{8} - 13\beta_{6} + 48\beta_{5} - 11\beta_{4} - 13\beta_{3} + \beta_{2} + 78 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 85 \beta_{19} + 11 \beta_{18} + 56 \beta_{17} + \beta_{16} + 4 \beta_{15} - 97 \beta_{14} - 20 \beta_{13} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 337 \beta_{19} + 56 \beta_{18} + 95 \beta_{17} + 4 \beta_{16} + 17 \beta_{15} - 480 \beta_{14} + \cdots + 127 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 95 \beta_{11} - 253 \beta_{9} - 239 \beta_{8} - 17 \beta_{7} + 619 \beta_{6} - 689 \beta_{5} + \cdots - 788 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2423 \beta_{19} - 389 \beta_{18} - 767 \beta_{17} - 62 \beta_{16} - 192 \beta_{15} + 3160 \beta_{14} + \cdots - 3160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5459 \beta_{19} - 767 \beta_{18} - 2750 \beta_{17} - 192 \beta_{16} - 652 \beta_{15} + \cdots - 4878 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2750 \beta_{11} + 7322 \beta_{9} + 10436 \beta_{8} + 652 \beta_{7} - 9814 \beta_{6} + 17738 \beta_{5} + \cdots + 21807 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 42676 \beta_{19} + 6034 \beta_{18} + 19836 \beta_{17} + 1828 \beta_{16} + 5876 \beta_{15} - 47910 \beta_{14} + \cdots + 47910 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 131519 \beta_{19} + 19836 \beta_{18} + 46882 \beta_{17} + 5876 \beta_{16} + 15934 \beta_{15} + \cdots + 72578 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 46882 \beta_{11} - 147080 \beta_{9} - 172736 \beta_{8} - 15934 \beta_{7} + 261509 \beta_{6} + \cdots - 368449 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 983767 \beta_{19} - 145479 \beta_{18} - 361745 \beta_{17} - 49120 \beta_{16} - 132066 \beta_{15} + \cdots - 1133234 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2549892 \beta_{19} - 361745 \beta_{18} - 1080126 \beta_{17} - 132066 \beta_{16} + \cdots - 1804376 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 1080126 \beta_{11} + 3390458 \beta_{9} + 4463893 \beta_{8} + 394536 \beta_{7} - 4974039 \beta_{6} + \cdots + 8384478 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 19582733 \beta_{19} + 2779571 \beta_{18} + 8088572 \beta_{17} + 1061787 \beta_{16} + 3099404 \beta_{15} + \cdots + 21610117 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-1 + \beta_{14}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
−1.13645 + 1.96838i
−0.970129 + 1.68031i
−0.874206 + 1.51417i
−0.343238 + 0.594506i
−0.0579927 + 0.100446i
0.448842 − 0.777418i
0.657416 − 1.13868i
0.794922 − 1.37685i
1.10057 − 1.90624i
1.38026 − 2.39068i
−1.13645 − 1.96838i
−0.970129 − 1.68031i
−0.874206 − 1.51417i
−0.343238 − 0.594506i
−0.0579927 − 0.100446i
0.448842 + 0.777418i
0.657416 + 1.13868i
0.794922 + 1.37685i
1.10057 + 1.90624i
1.38026 + 2.39068i
−1.13645 + 1.96838i −0.944630 − 1.63615i −1.58302 − 2.74187i −0.861046 + 1.49137i 4.29408 0 2.65028 −0.284651 + 0.493030i −1.95706 − 3.38973i
324.2 −0.970129 + 1.68031i −1.37123 − 2.37504i −0.882299 − 1.52819i 0.341748 − 0.591925i 5.32108 0 −0.456739 −2.26055 + 3.91538i 0.663080 + 1.14849i
324.3 −0.874206 + 1.51417i 0.465650 + 0.806530i −0.528471 − 0.915339i −2.04740 + 3.54620i −1.62830 0 −1.64885 1.06634 − 1.84695i −3.57970 − 6.20022i
324.4 −0.343238 + 0.594506i 0.561165 + 0.971966i 0.764375 + 1.32394i 0.368655 − 0.638530i −0.770452 0 −2.42240 0.870188 − 1.50721i 0.253073 + 0.438335i
324.5 −0.0579927 + 0.100446i −1.49086 − 2.58224i 0.993274 + 1.72040i −0.227711 + 0.394407i 0.345835 0 −0.462382 −2.94530 + 5.10141i −0.0264111 − 0.0457454i
324.6 0.448842 − 0.777418i −0.157632 − 0.273027i 0.597081 + 1.03417i −0.636233 + 1.10199i −0.283008 0 2.86735 1.45030 − 2.51200i 0.571136 + 0.989237i
324.7 0.657416 − 1.13868i 1.50722 + 2.61059i 0.135608 + 0.234880i −2.01261 + 3.48595i 3.96349 0 2.98627 −3.04344 + 5.27139i 2.64625 + 4.58344i
324.8 0.794922 − 1.37685i 0.668922 + 1.15861i −0.263803 − 0.456921i 0.460537 − 0.797673i 2.12696 0 2.34088 0.605087 − 1.04804i −0.732182 − 1.26818i
324.9 1.10057 − 1.90624i −1.31282 − 2.27386i −1.42251 − 2.46386i −2.11865 + 3.66961i −5.77938 0 −1.86000 −1.94697 + 3.37225i 4.66345 + 8.07733i
324.10 1.38026 − 2.39068i 0.0742045 + 0.128526i −2.81024 − 4.86748i −1.26728 + 2.19500i 0.409686 0 −9.99440 1.48899 − 2.57900i 3.49836 + 6.05934i
704.1 −1.13645 − 1.96838i −0.944630 + 1.63615i −1.58302 + 2.74187i −0.861046 − 1.49137i 4.29408 0 2.65028 −0.284651 − 0.493030i −1.95706 + 3.38973i
704.2 −0.970129 − 1.68031i −1.37123 + 2.37504i −0.882299 + 1.52819i 0.341748 + 0.591925i 5.32108 0 −0.456739 −2.26055 − 3.91538i 0.663080 − 1.14849i
704.3 −0.874206 − 1.51417i 0.465650 − 0.806530i −0.528471 + 0.915339i −2.04740 − 3.54620i −1.62830 0 −1.64885 1.06634 + 1.84695i −3.57970 + 6.20022i
704.4 −0.343238 − 0.594506i 0.561165 − 0.971966i 0.764375 − 1.32394i 0.368655 + 0.638530i −0.770452 0 −2.42240 0.870188 + 1.50721i 0.253073 − 0.438335i
704.5 −0.0579927 − 0.100446i −1.49086 + 2.58224i 0.993274 − 1.72040i −0.227711 − 0.394407i 0.345835 0 −0.462382 −2.94530 − 5.10141i −0.0264111 + 0.0457454i
704.6 0.448842 + 0.777418i −0.157632 + 0.273027i 0.597081 − 1.03417i −0.636233 − 1.10199i −0.283008 0 2.86735 1.45030 + 2.51200i 0.571136 − 0.989237i
704.7 0.657416 + 1.13868i 1.50722 − 2.61059i 0.135608 − 0.234880i −2.01261 − 3.48595i 3.96349 0 2.98627 −3.04344 − 5.27139i 2.64625 − 4.58344i
704.8 0.794922 + 1.37685i 0.668922 − 1.15861i −0.263803 + 0.456921i 0.460537 + 0.797673i 2.12696 0 2.34088 0.605087 + 1.04804i −0.732182 + 1.26818i
704.9 1.10057 + 1.90624i −1.31282 + 2.27386i −1.42251 + 2.46386i −2.11865 − 3.66961i −5.77938 0 −1.86000 −1.94697 − 3.37225i 4.66345 − 8.07733i
704.10 1.38026 + 2.39068i 0.0742045 − 0.128526i −2.81024 + 4.86748i −1.26728 − 2.19500i 0.409686 0 −9.99440 1.48899 + 2.57900i 3.49836 − 6.05934i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.f.q 20
7.b odd 2 1 931.2.f.r 20
7.c even 3 1 931.2.a.q yes 10
7.c even 3 1 inner 931.2.f.q 20
7.d odd 6 1 931.2.a.p ✓ 10
7.d odd 6 1 931.2.f.r 20
21.g even 6 1 8379.2.a.ct 10
21.h odd 6 1 8379.2.a.cs 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.a.p ✓ 10 7.d odd 6 1
931.2.a.q yes 10 7.c even 3 1
931.2.f.q 20 1.a even 1 1 trivial
931.2.f.q 20 7.c even 3 1 inner
931.2.f.r 20 7.b odd 2 1
931.2.f.r 20 7.d odd 6 1
8379.2.a.cs 10 21.h odd 6 1
8379.2.a.ct 10 21.g even 6 1

Hecke kernels

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

\( T_{2}^{20} - 2 T_{2}^{19} + 17 T_{2}^{18} - 22 T_{2}^{17} + 160 T_{2}^{16} - 182 T_{2}^{15} + 935 T_{2}^{14} + \cdots + 49 \) Copy content Toggle raw display
\( T_{3}^{20} + 4 T_{3}^{19} + 28 T_{3}^{18} + 64 T_{3}^{17} + 335 T_{3}^{16} + 616 T_{3}^{15} + 2552 T_{3}^{14} + \cdots + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 2 T^{19} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{20} + 4 T^{19} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{20} + 16 T^{19} + \cdots + 6724 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 62 T^{18} + \cdots + 40551424 \) Copy content Toggle raw display
$13$ \( (T^{10} - 12 T^{9} + \cdots + 1022)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1417221316 \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 841608081664 \) Copy content Toggle raw display
$29$ \( (T^{10} + 12 T^{9} + \cdots + 168952)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 79438295104 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 627479442496 \) Copy content Toggle raw display
$41$ \( (T^{10} - 40 T^{9} + \cdots - 10740226)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} - 4 T^{9} + \cdots - 1282048)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 18402795863104 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 83\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 3812365590784 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1516552842256 \) Copy content Toggle raw display
$71$ \( (T^{10} + 12 T^{9} + \cdots + 7611772)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 915657795786244 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{10} - 254 T^{8} + \cdots + 328888)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{10} + 16 T^{9} + \cdots - 223920578)^{2} \) Copy content Toggle raw display
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