Properties

Label 1980.2.u.a
Level $1980$
Weight $2$
Character orbit 1980.u
Analytic conductor $15.810$
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] = [1980,2,Mod(1277,1980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1980, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1980.1277");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1980.u (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: \(15.8103796002\)
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: \( x^{20} - 8 x^{19} + 28 x^{18} - 84 x^{17} + 296 x^{16} - 664 x^{15} + 716 x^{14} - 360 x^{13} + 152 x^{12} - 776 x^{11} + 6408 x^{10} - 11064 x^{9} + 41992 x^{8} - 78352 x^{7} + \cdots + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + \beta_{4} q^{7} - \beta_{5} q^{11} + (\beta_{19} + \beta_{6} - \beta_{2}) q^{13} + (\beta_{17} - \beta_{15} + \beta_{14} + \beta_{10} + \beta_{9} + \beta_{5} - 1) q^{17} + ( - \beta_{17} + \beta_{16} + \beta_{13} + \beta_{11} - \beta_{6} - \beta_{5}) q^{19} + (\beta_{18} - \beta_{14} + \beta_{12} + \beta_{10} - \beta_{8} - \beta_{6} - \beta_{2}) q^{23} + ( - \beta_{18} - \beta_{12} + \beta_{6} + \beta_{2} + \beta_1) q^{25} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{4} + \beta_{2} - 1) q^{29} + (\beta_{15} - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{7} + 1) q^{31} + (\beta_{19} + \beta_{17} - \beta_{16} - \beta_{13} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 1) q^{35}+ \cdots + ( - \beta_{19} + \beta_{14} + \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{13} - 16 q^{17} - 16 q^{29} + 16 q^{31} + 16 q^{35} - 4 q^{37} + 32 q^{53} - 16 q^{59} - 16 q^{61} + 16 q^{65} - 16 q^{67} - 44 q^{73} + 64 q^{83} + 24 q^{85} - 32 q^{89} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} + 28 x^{18} - 84 x^{17} + 296 x^{16} - 664 x^{15} + 716 x^{14} - 360 x^{13} + 152 x^{12} - 776 x^{11} + 6408 x^{10} - 11064 x^{9} + 41992 x^{8} - 78352 x^{7} + \cdots + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 14\!\cdots\!19 \nu^{19} + \cdots - 12\!\cdots\!04 ) / 37\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!87 \nu^{19} + \cdots + 71\!\cdots\!92 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 68\!\cdots\!31 \nu^{19} + \cdots + 17\!\cdots\!76 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47\!\cdots\!19 \nu^{19} + \cdots + 25\!\cdots\!16 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 88\!\cdots\!99 \nu^{19} + \cdots + 91\!\cdots\!12 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!94 \nu^{19} + \cdots + 73\!\cdots\!44 ) / 67\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 80\!\cdots\!51 \nu^{19} + \cdots - 11\!\cdots\!64 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!73 \nu^{19} + \cdots + 24\!\cdots\!68 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!97 \nu^{19} + \cdots - 90\!\cdots\!72 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!49 \nu^{19} + \cdots + 15\!\cdots\!44 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32\!\cdots\!03 \nu^{19} + \cdots - 30\!\cdots\!28 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 64\!\cdots\!09 \nu^{19} + \cdots + 91\!\cdots\!52 ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 65\!\cdots\!13 \nu^{19} + \cdots - 53\!\cdots\!72 ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26\!\cdots\!51 \nu^{19} + \cdots - 81\!\cdots\!56 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27\!\cdots\!17 \nu^{19} + \cdots - 15\!\cdots\!12 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 34\!\cdots\!31 \nu^{19} + \cdots - 36\!\cdots\!76 ) / 37\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 18\!\cdots\!17 \nu^{19} + \cdots - 15\!\cdots\!92 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 96\!\cdots\!69 \nu^{19} + \cdots - 58\!\cdots\!04 ) / 74\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 96\!\cdots\!72 \nu^{19} + \cdots - 47\!\cdots\!24 ) / 74\!\cdots\!68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{11} - \beta_{10} - \beta_{5} + \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - \beta_{18} - 2 \beta_{17} + \beta_{16} + 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - 5 \beta_{6} - \beta_{5} + 2 \beta_{3} + 5 \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{19} - 8 \beta_{17} + \beta_{16} + 8 \beta_{15} + 2 \beta_{14} + 9 \beta_{13} + 7 \beta_{12} - \beta_{11} - \beta_{10} + 3 \beta_{9} + \beta_{8} + \beta_{7} - 13 \beta_{6} - 9 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{19} - 5 \beta_{18} - \beta_{17} + 7 \beta_{15} + 5 \beta_{14} + \beta_{13} + 6 \beta_{12} + \beta_{11} - 7 \beta_{10} + 3 \beta_{9} + 5 \beta_{8} - 7 \beta_{6} - 32 \beta_{5} - 2 \beta_{4} + 11 \beta_{3} + 5 \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{19} - 49 \beta_{18} - 72 \beta_{17} + 9 \beta_{16} + 29 \beta_{15} + 25 \beta_{14} - 10 \beta_{13} - 14 \beta_{12} + 86 \beta_{11} - 91 \beta_{10} + 11 \beta_{9} + 41 \beta_{8} - 11 \beta_{7} - 91 \beta_{6} - 93 \beta_{5} - 18 \beta_{4} + \cdots - 99 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 52 \beta_{19} - 23 \beta_{18} - 110 \beta_{17} + 19 \beta_{16} + 166 \beta_{14} + 67 \beta_{13} + 78 \beta_{12} + 49 \beta_{11} - 21 \beta_{10} + 52 \beta_{9} + 56 \beta_{8} + 52 \beta_{7} - 198 \beta_{6} - 77 \beta_{5} - 8 \beta_{4} - 15 \beta_{3} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 149 \beta_{19} - 113 \beta_{18} + 177 \beta_{17} - 51 \beta_{16} - 336 \beta_{15} + 1048 \beta_{14} - 200 \beta_{13} + 634 \beta_{12} + 58 \beta_{11} - 85 \beta_{10} + 101 \beta_{9} + 363 \beta_{8} + 313 \beta_{7} + 70 \beta_{6} + \cdots - 955 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 455 \beta_{19} - 455 \beta_{18} - 912 \beta_{15} + 821 \beta_{14} - 994 \beta_{13} - 223 \beta_{12} + 994 \beta_{11} - 732 \beta_{10} - 31 \beta_{9} + 641 \beta_{8} + 98 \beta_{7} + 455 \beta_{6} - 424 \beta_{4} - 232 \beta_{3} + \cdots - 2894 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 584 \beta_{19} + 828 \beta_{18} - 873 \beta_{17} + 328 \beta_{16} - 4098 \beta_{15} + 4104 \beta_{14} - 224 \beta_{13} + 1150 \beta_{12} + 687 \beta_{11} + 1515 \beta_{10} + 990 \beta_{9} + 746 \beta_{8} + 1740 \beta_{7} + \cdots - 4949 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2397 \beta_{19} + 5799 \beta_{18} + 10790 \beta_{17} - 769 \beta_{16} - 15224 \beta_{15} + 10021 \beta_{14} - 4963 \beta_{13} + 7875 \beta_{12} - 7875 \beta_{11} + 10790 \beta_{10} + 575 \beta_{9} - 769 \beta_{8} + \cdots - 8853 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 28365 \beta_{19} + 6845 \beta_{18} + 33585 \beta_{17} - 236 \beta_{16} - 44232 \beta_{15} - 14314 \beta_{14} - 42633 \beta_{13} - 16787 \beta_{12} - 1579 \beta_{11} + 11788 \beta_{10} - 9540 \beta_{9} + \cdots - 49163 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 64178 \beta_{19} + 64178 \beta_{18} + 25330 \beta_{17} + 17684 \beta_{16} - 109120 \beta_{15} - 64178 \beta_{14} - 36134 \beta_{13} - 46556 \beta_{12} - 36134 \beta_{11} + 109120 \beta_{10} + \cdots - 23124 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 87663 \beta_{19} + 295129 \beta_{18} + 340982 \beta_{17} + 10789 \beta_{16} - 215413 \beta_{15} - 243341 \beta_{14} - 40876 \beta_{13} + 65404 \beta_{12} - 433646 \beta_{11} + 553555 \beta_{10} + \cdots + 497525 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 651954 \beta_{19} + 308560 \beta_{18} + 1091028 \beta_{17} - 5660 \beta_{16} - 2045882 \beta_{14} - 884876 \beta_{13} - 888570 \beta_{12} - 545176 \beta_{11} + 691922 \beta_{10} - 651954 \beta_{9} + \cdots + 1067452 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1061325 \beta_{19} + 721625 \beta_{18} - 632083 \beta_{17} + 469845 \beta_{16} + 2328702 \beta_{15} - 6707974 \beta_{14} + 114046 \beta_{13} - 3609124 \beta_{12} - 275668 \beta_{11} + \cdots + 5950915 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3776530 \beta_{19} + 3776530 \beta_{18} + 13018928 \beta_{15} - 13788594 \beta_{14} + 7967276 \beta_{13} - 1320058 \beta_{12} - 7967276 \beta_{11} + 9300292 \beta_{10} + \cdots + 31792640 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7953552 \beta_{19} - 11403740 \beta_{18} + 5162220 \beta_{17} - 5065376 \beta_{16} + 57047838 \beta_{15} - 52826646 \beta_{14} + 3597332 \beta_{13} - 20690308 \beta_{12} + \cdots + 64465950 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 40481878 \beta_{19} - 73613798 \beta_{18} - 114088156 \beta_{17} - 5299082 \beta_{16} + 210489844 \beta_{15} - 119387238 \beta_{14} + 63447838 \beta_{13} + \cdots + 132591398 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 341576980 \beta_{19} - 153634300 \beta_{18} - 380675796 \beta_{17} - 45303902 \beta_{16} + 622961200 \beta_{15} + 126904940 \beta_{14} + 473971758 \beta_{13} + \cdots + 536273376 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(991\) \(1541\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1277.1
0.517662 0.214422i
−0.662545 1.59953i
2.61322 1.08243i
−1.19227 2.87840i
−1.71031 + 0.708433i
0.369350 + 0.891690i
−0.0600477 0.144968i
0.351799 0.145720i
0.717088 + 1.73120i
3.05606 1.26586i
0.517662 + 0.214422i
−0.662545 + 1.59953i
2.61322 + 1.08243i
−1.19227 + 2.87840i
−1.71031 0.708433i
0.369350 0.891690i
−0.0600477 + 0.144968i
0.351799 + 0.145720i
0.717088 1.73120i
3.05606 + 1.26586i
0 0 0 −2.20944 0.344027i 0 3.68720 + 3.68720i 0 0 0
1277.2 0 0 0 −1.57028 + 1.59193i 0 −3.18760 3.18760i 0 0 0
1277.3 0 0 0 −1.39901 1.74435i 0 −2.01177 2.01177i 0 0 0
1277.4 0 0 0 −1.18483 1.89636i 0 1.05334 + 1.05334i 0 0 0
1277.5 0 0 0 −0.964544 + 2.01734i 0 −1.11132 1.11132i 0 0 0
1277.6 0 0 0 0.231689 + 2.22403i 0 1.05082 + 1.05082i 0 0 0
1277.7 0 0 0 1.10041 1.94656i 0 2.97759 + 2.97759i 0 0 0
1277.8 0 0 0 1.72483 + 1.42301i 0 −0.541605 0.541605i 0 0 0
1277.9 0 0 0 2.13012 0.680150i 0 −1.89416 1.89416i 0 0 0
1277.10 0 0 0 2.14106 0.644863i 0 −0.0225108 0.0225108i 0 0 0
1673.1 0 0 0 −2.20944 + 0.344027i 0 3.68720 3.68720i 0 0 0
1673.2 0 0 0 −1.57028 1.59193i 0 −3.18760 + 3.18760i 0 0 0
1673.3 0 0 0 −1.39901 + 1.74435i 0 −2.01177 + 2.01177i 0 0 0
1673.4 0 0 0 −1.18483 + 1.89636i 0 1.05334 1.05334i 0 0 0
1673.5 0 0 0 −0.964544 2.01734i 0 −1.11132 + 1.11132i 0 0 0
1673.6 0 0 0 0.231689 2.22403i 0 1.05082 1.05082i 0 0 0
1673.7 0 0 0 1.10041 + 1.94656i 0 2.97759 2.97759i 0 0 0
1673.8 0 0 0 1.72483 1.42301i 0 −0.541605 + 0.541605i 0 0 0
1673.9 0 0 0 2.13012 + 0.680150i 0 −1.89416 + 1.89416i 0 0 0
1673.10 0 0 0 2.14106 + 0.644863i 0 −0.0225108 + 0.0225108i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1277.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1980.2.u.a 20
3.b odd 2 1 1980.2.u.b yes 20
5.c odd 4 1 1980.2.u.b yes 20
15.e even 4 1 inner 1980.2.u.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1980.2.u.a 20 1.a even 1 1 trivial
1980.2.u.a 20 15.e even 4 1 inner
1980.2.u.b yes 20 3.b odd 2 1
1980.2.u.b yes 20 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{20} + 16 T_{17}^{19} + 128 T_{17}^{18} + 344 T_{17}^{17} + 432 T_{17}^{16} + 3680 T_{17}^{15} + 62752 T_{17}^{14} + 86800 T_{17}^{13} + 59256 T_{17}^{12} + 131488 T_{17}^{11} + 4040576 T_{17}^{10} + 4798240 T_{17}^{9} + \cdots + 984064 \) acting on \(S_{2}^{\mathrm{new}}(1980, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 16 T^{17} + 21 T^{16} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 40 T^{17} + 800 T^{16} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{20} + 4 T^{19} + \cdots + 1626347584 \) Copy content Toggle raw display
$17$ \( T^{20} + 16 T^{19} + 128 T^{18} + \cdots + 984064 \) Copy content Toggle raw display
$19$ \( T^{20} + 208 T^{18} + \cdots + 125440000 \) Copy content Toggle raw display
$23$ \( T^{20} - 384 T^{17} + \cdots + 108720553984 \) Copy content Toggle raw display
$29$ \( (T^{10} + 8 T^{9} - 106 T^{8} - 896 T^{7} + \cdots - 1568)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 8 T^{9} - 128 T^{8} + \cdots - 1252352)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 4 T^{19} + \cdots + 3117569360896 \) Copy content Toggle raw display
$41$ \( T^{20} + 452 T^{18} + \cdots + 20879197029376 \) Copy content Toggle raw display
$43$ \( T^{20} - 216 T^{17} + \cdots + 29777364714496 \) Copy content Toggle raw display
$47$ \( T^{20} + 3008 T^{16} + \cdots + 67108864 \) Copy content Toggle raw display
$53$ \( T^{20} - 32 T^{19} + \cdots + 211393970176 \) Copy content Toggle raw display
$59$ \( (T^{10} + 8 T^{9} - 232 T^{8} + \cdots - 21006464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 8 T^{9} - 216 T^{8} + \cdots - 1470976)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 311673319849984 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 360346088390656 \) Copy content Toggle raw display
$73$ \( T^{20} + 44 T^{19} + \cdots + 76\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{20} + 608 T^{18} + \cdots + 1724536262656 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 697770398638336 \) Copy content Toggle raw display
$89$ \( (T^{10} + 16 T^{9} - 530 T^{8} + \cdots - 784056448)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 320733140632576 \) Copy content Toggle raw display
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