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Properties

Label 39.4.f.b
Level $39$
Weight $4$
Character orbit 39.f
Analytic conductor $2.301$
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] = [39,4,Mod(5,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.f (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: \(2.30107449022\)
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: \( x^{20} + 1316x^{16} + 520390x^{12} + 64668772x^{8} + 2536036097x^{4} + 8509693504 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{5} q^{3} + ( - \beta_{14} - 6 \beta_{4}) q^{4} + (\beta_{18} - \beta_{8}) q^{5} + ( - \beta_{15} - \beta_{13} - \beta_{9} + \cdots + 3) q^{6}+ \cdots + (\beta_{17} + \beta_{16} + \beta_{15} + \cdots - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{5} q^{3} + ( - \beta_{14} - 6 \beta_{4}) q^{4} + (\beta_{18} - \beta_{8}) q^{5} + ( - \beta_{15} - \beta_{13} - \beta_{9} + \cdots + 3) q^{6}+ \cdots + ( - \beta_{19} + \beta_{17} + \cdots + 102) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} + 44 q^{6} + 44 q^{7} - 112 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} + 44 q^{6} + 44 q^{7} - 112 q^{9} - 76 q^{13} - 76 q^{15} - 16 q^{16} + 296 q^{18} + 260 q^{19} - 532 q^{21} - 224 q^{22} + 36 q^{24} - 592 q^{27} + 584 q^{28} - 700 q^{31} + 872 q^{33} + 816 q^{34} - 1660 q^{37} + 1016 q^{39} + 3288 q^{40} + 124 q^{42} + 260 q^{45} - 1560 q^{46} - 1084 q^{48} - 3456 q^{52} - 232 q^{54} - 872 q^{55} + 2648 q^{57} - 1352 q^{58} - 1064 q^{60} + 1960 q^{61} + 428 q^{63} - 7664 q^{66} - 916 q^{67} + 1192 q^{70} + 6984 q^{72} + 1964 q^{73} + 1816 q^{76} + 728 q^{78} + 6544 q^{79} + 200 q^{81} + 2612 q^{84} - 8304 q^{85} + 3136 q^{87} + 4580 q^{91} - 2536 q^{93} - 6056 q^{94} - 5956 q^{96} - 2572 q^{97} + 1700 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 1316x^{16} + 520390x^{12} + 64668772x^{8} + 2536036097x^{4} + 8509693504 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23767 \nu^{16} - 32543063 \nu^{12} - 14325458897 \nu^{8} - 2966035095265 \nu^{4} - 268310682520096 ) / 21323479843024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 570230007 \nu^{17} + 720341716997 \nu^{13} + 260128598557125 \nu^{9} + \cdots + 43\!\cdots\!04 \nu ) / 29\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40946735 \nu^{18} + 54159960537 \nu^{14} + 21683525486103 \nu^{10} + \cdots + 13\!\cdots\!10 \nu^{2} ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 604446590075 \nu^{19} + 11331613224061 \nu^{17} - 8498976062174 \nu^{16} + \cdots + 10\!\cdots\!12 ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 604446590075 \nu^{19} + 11331613224061 \nu^{17} - 36827476834070 \nu^{16} + \cdots + 16\!\cdots\!40 ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 604446590075 \nu^{19} - 11331613224061 \nu^{17} - 103677115612486 \nu^{16} + \cdots + 13\!\cdots\!04 ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40946735 \nu^{19} - 54159960537 \nu^{15} - 21683525486103 \nu^{11} + \cdots - 13\!\cdots\!10 \nu^{3} ) / 24\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 604446590075 \nu^{19} + 285517120886 \nu^{18} + 11331613224061 \nu^{17} + \cdots - 87\!\cdots\!52 \nu ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 604446590075 \nu^{19} + 1351078504546 \nu^{18} - 11331613224061 \nu^{17} + \cdots + 87\!\cdots\!52 \nu ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3998307626995 \nu^{18} - 31162202510296 \nu^{17} + 48520929609451 \nu^{16} + \cdots - 43\!\cdots\!64 ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 604446590075 \nu^{19} + 5983373564614 \nu^{18} - 11331613224061 \nu^{17} + \cdots + 87\!\cdots\!52 \nu ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2559971684696 \nu^{19} - 3998307626995 \nu^{18} - 48520929609451 \nu^{16} + \cdots + 43\!\cdots\!64 ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 286627145 \nu^{18} - 379119723759 \nu^{14} - 151784678402721 \nu^{10} + \cdots - 84\!\cdots\!98 \nu^{2} ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66173415485 \nu^{18} + 569354828965 \nu^{16} + 81995100649031 \nu^{14} + \cdots + 75\!\cdots\!20 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 66173415485 \nu^{18} + 569354828965 \nu^{16} - 81995100649031 \nu^{14} + \cdots + 75\!\cdots\!20 ) / 15\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 15492219928435 \nu^{19} - 53046163367275 \nu^{17} - 110066069038900 \nu^{16} + \cdots - 36\!\cdots\!00 ) / 40\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 1555923605885 \nu^{19} + \cdots + 23\!\cdots\!16 \nu^{3} ) / 33\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 15492219928435 \nu^{19} - 16787224074244 \nu^{18} + 53046163367275 \nu^{17} + \cdots + 58\!\cdots\!32 \nu ) / 40\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + 14\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{17} + \beta_{15} - \beta_{9} - 20\beta_{8} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{16} - 3\beta_{15} - 5\beta_{7} - 7\beta_{6} + 2\beta_{5} - 27\beta_{2} - 271 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32 \beta_{19} - 32 \beta_{17} - 34 \beta_{16} - 2 \beta_{14} - 4 \beta_{12} + 4 \beta_{11} - 10 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 110\beta_{16} - 110\beta_{15} - 685\beta_{14} + 178\beta_{12} + 140\beta_{10} + 318\beta_{9} - 6024\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 863 \beta_{19} - 1083 \beta_{18} - 863 \beta_{17} - 1003 \beta_{15} + 68 \beta_{14} - 280 \beta_{13} + \cdots + 140 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3185\beta_{16} + 3185\beta_{15} + 4887\beta_{7} + 10901\beta_{6} - 6014\beta_{5} + 17391\beta_{2} + 142849 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 22278 \beta_{19} + 22278 \beta_{17} + 28292 \beta_{16} + 1702 \beta_{14} + 7716 \beta_{12} + \cdots - 6014 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 86852 \beta_{16} + 86852 \beta_{15} + 443697 \beta_{14} - 124924 \beta_{12} + \cdots + 3497442 \beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 568621 \beta_{19} + 742325 \beta_{18} + 568621 \beta_{17} + 780821 \beta_{15} - 38072 \beta_{14} + \cdots - 212200 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2324959 \beta_{16} - 2324959 \beta_{15} - 3130601 \beta_{7} - 9897987 \beta_{6} + 6767386 \beta_{5} + \cdots - 87142739 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 14504428 \beta_{19} - 14504428 \beta_{17} - 21271814 \beta_{16} - 805642 \beta_{14} - 7573028 \beta_{12} + \cdots + 6767386 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 61813498 \beta_{16} - 61813498 \beta_{15} - 292779893 \beta_{14} + 78277158 \beta_{12} + \cdots - 2195406972 \beta_{4} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 371057051 \beta_{19} - 494684047 \beta_{18} - 371057051 \beta_{17} - 574499391 \beta_{15} + \cdots + 203442340 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1638103501 \beta_{16} + 1638103501 \beta_{15} + 1964951547 \beta_{7} + 7855902417 \beta_{6} + \cdots + 55749526469 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 9528572562 \beta_{19} + 9528572562 \beta_{17} + 15419523432 \beta_{16} + 326848046 \beta_{14} + \cdots - 5890950870 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 43324745424 \beta_{16} + 43324745424 \beta_{15} + 195986430201 \beta_{14} + \cdots + 1424436851734 \beta_{4} \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 245598148329 \beta_{19} + 332247639177 \beta_{18} + 245598148329 \beta_{17} + 411936872745 \beta_{15} + \cdots - 166338724416 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−3.62388 + 3.62388i
−3.24982 + 3.24982i
−2.26335 + 2.26335i
−2.05488 + 2.05488i
−0.980238 + 0.980238i
0.980238 − 0.980238i
2.05488 − 2.05488i
2.26335 − 2.26335i
3.24982 − 3.24982i
3.62388 − 3.62388i
−3.62388 − 3.62388i
−3.24982 − 3.24982i
−2.26335 − 2.26335i
−2.05488 − 2.05488i
−0.980238 − 0.980238i
0.980238 + 0.980238i
2.05488 + 2.05488i
2.26335 + 2.26335i
3.24982 + 3.24982i
3.62388 + 3.62388i
−3.62388 − 3.62388i 4.19035 + 3.07262i 18.2650i 10.8970 + 10.8970i −4.05051 − 26.3201i −5.52580 − 5.52580i 37.1990 − 37.1990i 8.11802 + 25.7507i − 78.9791i
5.2 −3.24982 − 3.24982i −5.00236 − 1.40585i 13.1227i −2.07291 − 2.07291i 11.6880 + 20.8255i 7.93427 + 7.93427i 16.6478 − 16.6478i 23.0472 + 14.0651i 13.4732i
5.3 −2.26335 − 2.26335i 2.15347 − 4.72891i 2.24555i 0.350510 + 0.350510i −15.5773 + 5.82912i −7.24142 − 7.24142i −13.0244 + 13.0244i −17.7251 − 20.3671i − 1.58666i
5.4 −2.05488 − 2.05488i 0.159445 + 5.19371i 0.445059i −14.0419 − 14.0419i 10.3448 − 11.0001i −5.62483 − 5.62483i −15.5245 + 15.5245i −26.9492 + 1.65622i 57.7088i
5.5 −0.980238 − 0.980238i −2.50090 + 4.55472i − 6.07827i 12.5563 + 12.5563i 6.91619 − 2.01323i 21.4578 + 21.4578i −13.8001 + 13.8001i −14.4909 − 22.7818i − 24.6163i
5.6 0.980238 + 0.980238i −2.50090 − 4.55472i − 6.07827i −12.5563 − 12.5563i 2.01323 − 6.91619i 21.4578 + 21.4578i 13.8001 − 13.8001i −14.4909 + 22.7818i − 24.6163i
5.7 2.05488 + 2.05488i 0.159445 − 5.19371i 0.445059i 14.0419 + 14.0419i 11.0001 − 10.3448i −5.62483 − 5.62483i 15.5245 − 15.5245i −26.9492 − 1.65622i 57.7088i
5.8 2.26335 + 2.26335i 2.15347 + 4.72891i 2.24555i −0.350510 − 0.350510i −5.82912 + 15.5773i −7.24142 − 7.24142i 13.0244 − 13.0244i −17.7251 + 20.3671i − 1.58666i
5.9 3.24982 + 3.24982i −5.00236 + 1.40585i 13.1227i 2.07291 + 2.07291i −20.8255 − 11.6880i 7.93427 + 7.93427i −16.6478 + 16.6478i 23.0472 − 14.0651i 13.4732i
5.10 3.62388 + 3.62388i 4.19035 − 3.07262i 18.2650i −10.8970 − 10.8970i 26.3201 + 4.05051i −5.52580 − 5.52580i −37.1990 + 37.1990i 8.11802 − 25.7507i − 78.9791i
8.1 −3.62388 + 3.62388i 4.19035 − 3.07262i − 18.2650i 10.8970 − 10.8970i −4.05051 + 26.3201i −5.52580 + 5.52580i 37.1990 + 37.1990i 8.11802 − 25.7507i 78.9791i
8.2 −3.24982 + 3.24982i −5.00236 + 1.40585i − 13.1227i −2.07291 + 2.07291i 11.6880 − 20.8255i 7.93427 − 7.93427i 16.6478 + 16.6478i 23.0472 − 14.0651i − 13.4732i
8.3 −2.26335 + 2.26335i 2.15347 + 4.72891i − 2.24555i 0.350510 − 0.350510i −15.5773 − 5.82912i −7.24142 + 7.24142i −13.0244 − 13.0244i −17.7251 + 20.3671i 1.58666i
8.4 −2.05488 + 2.05488i 0.159445 − 5.19371i − 0.445059i −14.0419 + 14.0419i 10.3448 + 11.0001i −5.62483 + 5.62483i −15.5245 − 15.5245i −26.9492 − 1.65622i − 57.7088i
8.5 −0.980238 + 0.980238i −2.50090 − 4.55472i 6.07827i 12.5563 − 12.5563i 6.91619 + 2.01323i 21.4578 − 21.4578i −13.8001 − 13.8001i −14.4909 + 22.7818i 24.6163i
8.6 0.980238 − 0.980238i −2.50090 + 4.55472i 6.07827i −12.5563 + 12.5563i 2.01323 + 6.91619i 21.4578 − 21.4578i 13.8001 + 13.8001i −14.4909 − 22.7818i 24.6163i
8.7 2.05488 − 2.05488i 0.159445 + 5.19371i − 0.445059i 14.0419 − 14.0419i 11.0001 + 10.3448i −5.62483 + 5.62483i 15.5245 + 15.5245i −26.9492 + 1.65622i − 57.7088i
8.8 2.26335 − 2.26335i 2.15347 − 4.72891i − 2.24555i −0.350510 + 0.350510i −5.82912 − 15.5773i −7.24142 + 7.24142i 13.0244 + 13.0244i −17.7251 − 20.3671i 1.58666i
8.9 3.24982 − 3.24982i −5.00236 − 1.40585i − 13.1227i 2.07291 − 2.07291i −20.8255 + 11.6880i 7.93427 − 7.93427i −16.6478 − 16.6478i 23.0472 + 14.0651i − 13.4732i
8.10 3.62388 − 3.62388i 4.19035 + 3.07262i − 18.2650i −10.8970 + 10.8970i 26.3201 − 4.05051i −5.52580 + 5.52580i −37.1990 − 37.1990i 8.11802 + 25.7507i 78.9791i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.f.b ✓ 20
3.b odd 2 1 inner 39.4.f.b ✓ 20
13.d odd 4 1 inner 39.4.f.b ✓ 20
39.f even 4 1 inner 39.4.f.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.f.b ✓ 20 1.a even 1 1 trivial
39.4.f.b ✓ 20 3.b odd 2 1 inner
39.4.f.b ✓ 20 13.d odd 4 1 inner
39.4.f.b ✓ 20 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 1316T_{2}^{16} + 520390T_{2}^{12} + 64668772T_{2}^{8} + 2536036097T_{2}^{4} + 8509693504 \) acting on \(S_{4}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 8509693504 \) Copy content Toggle raw display
$3$ \( (T^{10} + 2 T^{9} + \cdots + 14348907)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( (T^{10} - 22 T^{9} + \cdots + 46988290568)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 51\!\cdots\!57)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 73\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 22\!\cdots\!28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 983394687135744)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 58\!\cdots\!52)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{5} - 490 T^{4} + \cdots + 139800572416)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 31\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 24\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 29382707320672)^{4} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 25\!\cdots\!08)^{2} \) Copy content Toggle raw display
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