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Properties

Label 102.3.e.b
Level $102$
Weight $3$
Character orbit 102.e
Analytic conductor $2.779$
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] = [102,3,Mod(47,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 102.e (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: \(2.77929869648\)
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: \( x^{20} - 10 x^{18} + 149 x^{16} - 800 x^{14} - 1986 x^{12} + 2844 x^{10} - 160866 x^{8} + \cdots + 3486784401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{13} q^{3} + 2 q^{4} + (\beta_{15} - \beta_{13} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{19} + \beta_{18} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{13} q^{3} + 2 q^{4} + (\beta_{15} - \beta_{13} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - 3 \beta_{19} - \beta_{18} + \cdots + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{3} + 40 q^{4} + 4 q^{6} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{3} + 40 q^{4} + 4 q^{6} + 20 q^{7} + 44 q^{10} + 8 q^{12} - 52 q^{13} + 80 q^{16} - 16 q^{18} - 152 q^{21} + 12 q^{22} + 8 q^{24} - 68 q^{27} + 40 q^{28} - 88 q^{31} - 212 q^{33} - 172 q^{34} + 36 q^{37} - 80 q^{39} + 88 q^{40} - 232 q^{45} - 92 q^{46} + 16 q^{48} + 392 q^{51} - 104 q^{52} - 124 q^{54} + 436 q^{55} + 8 q^{57} - 288 q^{58} - 84 q^{61} + 228 q^{63} + 160 q^{64} + 768 q^{67} + 84 q^{69} - 32 q^{72} + 32 q^{73} + 628 q^{75} + 28 q^{78} + 236 q^{79} + 396 q^{81} - 148 q^{82} - 304 q^{84} - 420 q^{85} + 24 q^{88} - 92 q^{90} + 4 q^{91} + 16 q^{96} - 304 q^{97} + 568 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{18} + 149 x^{16} - 800 x^{14} - 1986 x^{12} + 2844 x^{10} - 160866 x^{8} + \cdots + 3486784401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 9931 \nu^{18} - 755375 \nu^{16} + 6505072 \nu^{14} - 61957336 \nu^{12} + \cdots - 33178913351739 ) / 2477331353376 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3317 \nu^{19} - 665617 \nu^{17} + 4571264 \nu^{15} - 28566392 \nu^{13} + \cdots - 28938287332413 \nu ) / 7773204691296 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 761 \nu^{18} - 27544 \nu^{16} + 120053 \nu^{14} - 1499549 \nu^{12} + 10089327 \nu^{10} + \cdots - 1335610612467 ) / 80009505432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 761 \nu^{18} + 27544 \nu^{16} - 120053 \nu^{14} + 1499549 \nu^{12} - 10089327 \nu^{10} + \cdots + 1335610612467 ) / 80009505432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 92035 \nu^{18} - 1009729 \nu^{16} + 6914840 \nu^{14} - 15082352 \nu^{12} + \cdots - 4843857789693 ) / 7431994060128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6284881 \nu^{19} - 148695585 \nu^{18} + 264129355 \nu^{17} + 3257310723 \nu^{16} + \cdots + 10\!\cdots\!35 ) / 36\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7295879 \nu^{18} - 18708149 \nu^{16} + 285859648 \nu^{14} - 1257003862 \nu^{12} + \cdots + 506937170099961 ) / 150497879717592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6871118 \nu^{19} + 114234381 \nu^{18} - 95239328 \nu^{17} - 247657419 \nu^{16} + \cdots + 56\!\cdots\!15 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6284881 \nu^{19} + 148695585 \nu^{18} + 264129355 \nu^{17} - 3257310723 \nu^{16} + \cdots - 10\!\cdots\!35 ) / 36\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26104 \nu^{18} - 18005 \nu^{16} - 82091 \nu^{14} - 5886328 \nu^{12} + 176827569 \nu^{10} + \cdots - 7287336351369 ) / 402400747908 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31027 \nu^{19} + 371911 \nu^{17} - 2391959 \nu^{15} + 15097307 \nu^{13} + \cdots + 8630092719522 \nu ) / 6480769939992 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6871118 \nu^{19} - 114234381 \nu^{18} - 95239328 \nu^{17} + 247657419 \nu^{16} + \cdots - 56\!\cdots\!15 ) / 27\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 828315 \nu^{19} + 6013019 \nu^{18} - 9087561 \nu^{17} - 43588289 \nu^{16} + \cdots - 99521823569229 ) / 133775893082304 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2175187 \nu^{19} + 2294866 \nu^{18} + 35905901 \nu^{17} - 144920242 \nu^{16} + \cdots - 60\!\cdots\!66 ) / 300995759435184 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2175187 \nu^{19} - 2294866 \nu^{18} + 35905901 \nu^{17} + 144920242 \nu^{16} + \cdots + 60\!\cdots\!66 ) / 300995759435184 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 828315 \nu^{19} + 6013019 \nu^{18} + 9087561 \nu^{17} - 43588289 \nu^{16} + \cdots - 99521823569229 ) / 133775893082304 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 293288 \nu^{19} + 3250012 \nu^{17} - 16784003 \nu^{15} + 210285347 \nu^{13} + \cdots + 227994073646193 \nu ) / 26558449361928 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 119235649 \nu^{19} - 1487689375 \nu^{17} + 7963603400 \nu^{15} - 43196675288 \nu^{13} + \cdots - 25\!\cdots\!15 \nu ) / 54\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 202654 \nu^{19} - 1731430 \nu^{17} + 13186310 \nu^{15} - 37506185 \nu^{13} + \cdots - 20565427468680 \nu ) / 6840812714436 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{7} + \beta_{5} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{19} - 8 \beta_{18} + 2 \beta_{17} - 4 \beta_{16} - 5 \beta_{15} - 5 \beta_{14} + \cdots - 6 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} + 6 \beta_{13} - 4 \beta_{12} + 7 \beta_{10} - 7 \beta_{9} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{19} + 24 \beta_{18} - 6 \beta_{17} - 46 \beta_{16} - 38 \beta_{15} - 38 \beta_{14} + \cdots - 196 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 78 \beta_{16} + 15 \beta_{15} - 15 \beta_{14} + 78 \beta_{13} + 12 \beta_{12} + 79 \beta_{10} + \cdots - 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24 \beta_{19} + 304 \beta_{18} + 428 \beta_{17} + 224 \beta_{16} - 245 \beta_{15} + \cdots - 30 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 186 \beta_{16} + 156 \beta_{15} - 156 \beta_{14} - 186 \beta_{13} - 352 \beta_{12} + 262 \beta_{10} + \cdots + 3983 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2496 \beta_{19} - 1824 \beta_{18} - 1596 \beta_{17} + 2444 \beta_{16} - 1628 \beta_{15} + \cdots - 6844 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2406 \beta_{16} - 3588 \beta_{15} + 3588 \beta_{14} + 2406 \beta_{13} + 1140 \beta_{12} - 1565 \beta_{10} + \cdots + 34880 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 34722 \beta_{19} - 14216 \beta_{18} - 5230 \beta_{17} + 26348 \beta_{16} - 1835 \beta_{15} + \cdots + 21258 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 15768 \beta_{16} - 27405 \beta_{15} + 27405 \beta_{14} + 15768 \beta_{13} + 1676 \beta_{12} + \cdots - 161188 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 541458 \beta_{19} - 379080 \beta_{18} - 62694 \beta_{17} - 373246 \beta_{16} - 285746 \beta_{15} + \cdots - 491608 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 431208 \beta_{16} - 435807 \beta_{15} + 435807 \beta_{14} + 431208 \beta_{13} - 32076 \beta_{12} + \cdots - 1307194 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 50376 \beta_{19} + 2257840 \beta_{18} + 1599608 \beta_{17} - 1046968 \beta_{16} + \cdots - 2137542 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 9888 \beta_{16} + 359520 \beta_{15} - 359520 \beta_{14} + 9888 \beta_{13} - 1035148 \beta_{12} + \cdots - 638479 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 12777720 \beta_{19} - 13191888 \beta_{18} + 16733064 \beta_{17} + 32478128 \beta_{16} + \cdots - 1271536 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2189064 \beta_{16} - 17343120 \beta_{15} + 17343120 \beta_{14} - 2189064 \beta_{13} + \cdots + 140332094 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 182832654 \beta_{19} + 13705576 \beta_{18} - 248025478 \beta_{17} + 338887124 \beta_{16} + \cdots - 326027478 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(-1\) \(-\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
2.40171 − 1.79772i
−0.660460 − 2.92640i
2.13636 + 2.10618i
−2.98496 − 0.299984i
−2.30685 + 1.91792i
−2.13636 + 2.10618i
−2.40171 − 1.79772i
2.30685 + 1.91792i
2.98496 − 0.299984i
0.660460 − 2.92640i
2.40171 + 1.79772i
−0.660460 + 2.92640i
2.13636 − 2.10618i
−2.98496 + 0.299984i
−2.30685 − 1.91792i
−2.13636 − 2.10618i
−2.40171 + 1.79772i
2.30685 − 1.91792i
2.98496 + 0.299984i
0.660460 + 2.92640i
−1.41421 −2.96944 − 0.427083i 2.00000 −1.26169 + 1.26169i 4.19943 + 0.603986i 2.01679 − 2.01679i −2.82843 8.63520 + 2.53640i 1.78430 − 1.78430i
47.2 −1.41421 −1.60226 + 2.53629i 2.00000 −0.525641 + 0.525641i 2.26594 − 3.58686i −3.53019 + 3.53019i −2.82843 −3.86553 − 8.12759i 0.743369 − 0.743369i
47.3 −1.41421 −0.0213387 − 2.99992i 2.00000 −2.03986 + 2.03986i 0.0301775 + 4.24253i 8.34883 − 8.34883i −2.82843 −8.99909 + 0.128029i 2.88479 − 2.88479i
47.4 −1.41421 1.89857 + 2.32281i 2.00000 3.05768 − 3.05768i −2.68498 − 3.28495i 3.50287 − 3.50287i −2.82843 −1.79088 + 8.82002i −4.32421 + 4.32421i
47.5 −1.41421 2.98737 + 0.275015i 2.00000 −7.00867 + 7.00867i −4.22478 − 0.388930i −5.33829 + 5.33829i −2.82843 8.84873 + 1.64314i 9.91175 − 9.91175i
47.6 1.41421 −2.99992 − 0.0213387i 2.00000 2.03986 − 2.03986i −4.24253 − 0.0301775i 8.34883 − 8.34883i 2.82843 8.99909 + 0.128029i 2.88479 − 2.88479i
47.7 1.41421 −0.427083 − 2.96944i 2.00000 1.26169 − 1.26169i −0.603986 − 4.19943i 2.01679 − 2.01679i 2.82843 −8.63520 + 2.53640i 1.78430 − 1.78430i
47.8 1.41421 0.275015 + 2.98737i 2.00000 7.00867 − 7.00867i 0.388930 + 4.22478i −5.33829 + 5.33829i 2.82843 −8.84873 + 1.64314i 9.91175 − 9.91175i
47.9 1.41421 2.32281 + 1.89857i 2.00000 −3.05768 + 3.05768i 3.28495 + 2.68498i 3.50287 − 3.50287i 2.82843 1.79088 + 8.82002i −4.32421 + 4.32421i
47.10 1.41421 2.53629 − 1.60226i 2.00000 0.525641 − 0.525641i 3.58686 − 2.26594i −3.53019 + 3.53019i 2.82843 3.86553 − 8.12759i 0.743369 − 0.743369i
89.1 −1.41421 −2.96944 + 0.427083i 2.00000 −1.26169 − 1.26169i 4.19943 − 0.603986i 2.01679 + 2.01679i −2.82843 8.63520 − 2.53640i 1.78430 + 1.78430i
89.2 −1.41421 −1.60226 − 2.53629i 2.00000 −0.525641 − 0.525641i 2.26594 + 3.58686i −3.53019 − 3.53019i −2.82843 −3.86553 + 8.12759i 0.743369 + 0.743369i
89.3 −1.41421 −0.0213387 + 2.99992i 2.00000 −2.03986 − 2.03986i 0.0301775 − 4.24253i 8.34883 + 8.34883i −2.82843 −8.99909 − 0.128029i 2.88479 + 2.88479i
89.4 −1.41421 1.89857 − 2.32281i 2.00000 3.05768 + 3.05768i −2.68498 + 3.28495i 3.50287 + 3.50287i −2.82843 −1.79088 − 8.82002i −4.32421 − 4.32421i
89.5 −1.41421 2.98737 − 0.275015i 2.00000 −7.00867 − 7.00867i −4.22478 + 0.388930i −5.33829 − 5.33829i −2.82843 8.84873 − 1.64314i 9.91175 + 9.91175i
89.6 1.41421 −2.99992 + 0.0213387i 2.00000 2.03986 + 2.03986i −4.24253 + 0.0301775i 8.34883 + 8.34883i 2.82843 8.99909 − 0.128029i 2.88479 + 2.88479i
89.7 1.41421 −0.427083 + 2.96944i 2.00000 1.26169 + 1.26169i −0.603986 + 4.19943i 2.01679 + 2.01679i 2.82843 −8.63520 − 2.53640i 1.78430 + 1.78430i
89.8 1.41421 0.275015 − 2.98737i 2.00000 7.00867 + 7.00867i 0.388930 − 4.22478i −5.33829 − 5.33829i 2.82843 −8.84873 − 1.64314i 9.91175 + 9.91175i
89.9 1.41421 2.32281 − 1.89857i 2.00000 −3.05768 − 3.05768i 3.28495 − 2.68498i 3.50287 + 3.50287i 2.82843 1.79088 − 8.82002i −4.32421 − 4.32421i
89.10 1.41421 2.53629 + 1.60226i 2.00000 0.525641 + 0.525641i 3.58686 + 2.26594i −3.53019 − 3.53019i 2.82843 3.86553 + 8.12759i 0.743369 + 0.743369i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.c even 4 1 inner
51.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.3.e.b ✓ 20
3.b odd 2 1 inner 102.3.e.b ✓ 20
17.c even 4 1 inner 102.3.e.b ✓ 20
51.f odd 4 1 inner 102.3.e.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.3.e.b ✓ 20 1.a even 1 1 trivial
102.3.e.b ✓ 20 3.b odd 2 1 inner
102.3.e.b ✓ 20 17.c even 4 1 inner
102.3.e.b ✓ 20 51.f odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 10081T_{5}^{16} + 4172464T_{5}^{12} + 276215392T_{5}^{8} + 2452917760T_{5}^{4} + 723394816 \) acting on \(S_{3}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 723394816 \) Copy content Toggle raw display
$7$ \( (T^{10} - 10 T^{9} + \cdots + 39533832)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{5} + 13 T^{4} + \cdots - 8028)^{4} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 40\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 601443678784)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 79\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 29829099336192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 93741278912064)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 518351995830272)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 51\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 298143585148032)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 1847949295392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 192 T^{4} + \cdots + 544398336)^{4} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 24\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 10\!\cdots\!12)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 49\!\cdots\!52)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 25\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
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