Properties

Label 105.4.b.b
Level $105$
Weight $4$
Character orbit 105.b
Analytic conductor $6.195$
Analytic rank $0$
Dimension $16$
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] = [105,4,Mod(41,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.b (of order \(2\), degree \(1\), 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: \(6.19520055060\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 96 x^{14} + 3618 x^{12} + 68560 x^{10} + 697017 x^{8} + 3791184 x^{6} + 10461796 x^{4} + 13209792 x^{2} + 5760000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 4) q^{4} + 5 q^{5} + ( - \beta_{5} + 2) q^{6} - \beta_{10} q^{7} + (\beta_{4} - \beta_{3} - 4 \beta_1) q^{8} + ( - \beta_{12} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} - 4) q^{4} + 5 q^{5} + ( - \beta_{5} + 2) q^{6} - \beta_{10} q^{7} + (\beta_{4} - \beta_{3} - 4 \beta_1) q^{8} + ( - \beta_{12} + \beta_{2} + \beta_1 - 1) q^{9} + 5 \beta_1 q^{10} + ( - \beta_{13} + \beta_{3} + \beta_1) q^{11} + (\beta_{8} - \beta_{4} + 4 \beta_{3} - \beta_{2} + 3 \beta_1 + 4) q^{12} + (\beta_{12} + \beta_{9} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{13} + (\beta_{11} - \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{4} + 5 \beta_{3} - \beta_1 + 7) q^{14} - 5 \beta_{3} q^{15} + (\beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{3} + \cdots + 25) q^{16}+ \cdots + (3 \beta_{15} + 11 \beta_{14} - 16 \beta_{13} - 7 \beta_{12} + 28 \beta_{11} + \cdots + 158) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} - 64 q^{4} + 80 q^{5} + 28 q^{6} - 4 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} - 64 q^{4} + 80 q^{5} + 28 q^{6} - 4 q^{7} - 22 q^{9} + 66 q^{12} + 90 q^{14} + 10 q^{15} + 376 q^{16} + 72 q^{17} - 182 q^{18} - 320 q^{20} - 70 q^{21} - 276 q^{22} - 526 q^{24} + 400 q^{25} - 696 q^{26} + 128 q^{27} + 10 q^{28} + 140 q^{30} + 502 q^{33} - 20 q^{35} + 996 q^{36} - 812 q^{37} + 1200 q^{38} - 594 q^{39} + 936 q^{41} - 974 q^{42} - 548 q^{43} - 110 q^{45} + 1224 q^{46} - 912 q^{47} - 1850 q^{48} + 328 q^{49} + 750 q^{51} + 2950 q^{54} - 1254 q^{56} + 432 q^{57} + 576 q^{58} + 552 q^{59} + 330 q^{60} + 1860 q^{62} + 362 q^{63} - 4000 q^{64} - 1378 q^{66} + 1004 q^{67} - 3828 q^{68} - 1988 q^{69} + 450 q^{70} + 1988 q^{72} + 50 q^{75} - 1152 q^{77} + 1446 q^{78} + 1292 q^{79} + 1880 q^{80} - 2950 q^{81} + 1752 q^{83} - 420 q^{84} + 360 q^{85} - 1910 q^{87} - 912 q^{88} - 6096 q^{89} - 910 q^{90} - 552 q^{91} - 1080 q^{93} + 9546 q^{96} + 4824 q^{98} + 2530 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 96 x^{14} + 3618 x^{12} + 68560 x^{10} + 697017 x^{8} + 3791184 x^{6} + 10461796 x^{4} + 13209792 x^{2} + 5760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 547 \nu^{15} + 1777 \nu^{14} + 50246 \nu^{13} + 180054 \nu^{12} + 1775248 \nu^{11} + 7185508 \nu^{10} + 30263338 \nu^{9} + 143671962 \nu^{8} + \cdots + 12630768000 ) / 380788800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 547 \nu^{15} + 1777 \nu^{14} + 50246 \nu^{13} + 180054 \nu^{12} + 1775248 \nu^{11} + 7185508 \nu^{10} + 30263338 \nu^{9} + 143671962 \nu^{8} + \cdots + 12630768000 ) / 380788800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1777 \nu^{15} - 2266 \nu^{14} + 180054 \nu^{13} - 203798 \nu^{12} + 7185508 \nu^{11} - 7238982 \nu^{10} + 143671962 \nu^{9} - 130751978 \nu^{8} + \cdots - 2389142400 ) / 380788800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 547 \nu^{15} - 1777 \nu^{14} + 50246 \nu^{13} - 180054 \nu^{12} + 1775248 \nu^{11} - 7185508 \nu^{10} + 30263338 \nu^{9} - 143671962 \nu^{8} + \cdots - 12630768000 ) / 126929600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2521 \nu^{15} - 25944 \nu^{14} + 219100 \nu^{13} - 2400552 \nu^{12} + 6878486 \nu^{11} - 86563888 \nu^{10} + 88736116 \nu^{9} - 1552833632 \nu^{8} + \cdots - 90153456000 ) / 380788800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3751 \nu^{15} - 29009 \nu^{14} - 348908 \nu^{13} - 2736916 \nu^{12} - 12288746 \nu^{11} - 100881430 \nu^{10} - 202144740 \nu^{9} + \cdots - 125656617600 ) / 380788800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17137 \nu^{15} - 46928 \nu^{14} - 1503096 \nu^{13} - 4329248 \nu^{12} - 49505042 \nu^{11} - 153385008 \nu^{10} - 757860320 \nu^{9} + \cdots - 60084528000 ) / 761577600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27779 \nu^{15} - 35226 \nu^{14} + 2607108 \nu^{13} - 3211404 \nu^{12} + 94836522 \nu^{11} - 112663080 \nu^{10} + 1696532852 \nu^{9} + \cdots - 93289344000 ) / 761577600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 27779 \nu^{15} + 35226 \nu^{14} + 2607108 \nu^{13} + 3211404 \nu^{12} + 94836522 \nu^{11} + 112663080 \nu^{10} + 1696532852 \nu^{9} + \cdots + 93289344000 ) / 761577600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26393 \nu^{15} + 54036 \nu^{14} + 2471672 \nu^{13} + 5049464 \nu^{12} + 89784930 \nu^{11} + 182127040 \nu^{10} + 1606883040 \nu^{9} + \cdots + 120508108800 ) / 761577600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 56747 \nu^{15} + 3554 \nu^{14} - 5360852 \nu^{13} + 360108 \nu^{12} - 197441554 \nu^{11} + 14371016 \nu^{10} - 3611070764 \nu^{9} + \cdots + 25261536000 ) / 761577600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 547 \nu^{15} + 52749 \nu^{14} + 50246 \nu^{13} + 4923702 \nu^{12} + 1775248 \nu^{11} + 177708268 \nu^{10} + 30263338 \nu^{9} + 3142627082 \nu^{8} + \cdots + 162993638400 ) / 380788800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1230 \nu^{15} - 58079 \nu^{14} - 129808 \nu^{13} - 5433316 \nu^{12} - 5410260 \nu^{11} - 196551530 \nu^{10} - 113408624 \nu^{9} + \cdots - 135528019200 ) / 380788800 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{12} + 2 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{3} - 28 \beta_{2} + 249 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{13} + 4 \beta_{11} + 4 \beta_{10} + 3 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} - 36 \beta_{4} + 53 \beta_{3} + 475 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8 \beta_{15} - 34 \beta_{14} + 46 \beta_{12} - 98 \beta_{11} + 6 \beta_{10} - 46 \beta_{9} - 48 \beta_{8} - 48 \beta_{7} + 48 \beta_{6} + 144 \beta_{5} - 162 \beta_{3} + 767 \beta_{2} - 6122 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24 \beta_{15} - 192 \beta_{13} + 70 \beta_{12} - 216 \beta_{11} - 216 \beta_{10} + 22 \beta_{9} - 156 \beta_{8} + 156 \beta_{7} - 198 \beta_{6} - 180 \beta_{5} + 1135 \beta_{4} - 2025 \beta_{3} - 22 \beta_{2} - 12238 \beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 436 \beta_{15} + 923 \beta_{14} - 1645 \beta_{12} + 3734 \beta_{11} - 444 \beta_{10} + 1645 \beta_{9} + 1783 \beta_{8} + 1783 \beta_{7} - 1853 \beta_{6} - 5233 \beta_{5} + 6579 \beta_{3} - 21612 \beta_{2} + \cdots + 162433 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1484 \beta_{15} + 7132 \beta_{13} - 4370 \beta_{12} + 8656 \beta_{11} + 8656 \beta_{10} - 1402 \beta_{9} + 6101 \beta_{8} - 6101 \beta_{7} + 7218 \beta_{6} + 7585 \beta_{5} - 34938 \beta_{4} + 69083 \beta_{3} + \cdots + 1402 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17276 \beta_{15} - 24096 \beta_{14} + 54358 \beta_{12} - 129530 \beta_{11} + 20814 \beta_{10} - 54358 \beta_{9} - 60826 \beta_{8} - 60826 \beta_{7} + 65396 \beta_{6} + 173882 \beta_{5} + \cdots - 4509374 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 63684 \beta_{15} - 243304 \beta_{13} + 188220 \beta_{12} - 309708 \beta_{11} - 309708 \beta_{10} + 60852 \beta_{9} - 214150 \beta_{8} + 214150 \beta_{7} - 238720 \beta_{6} - 277834 \beta_{5} + \cdots - 60852 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 609280 \beta_{15} + 636685 \beta_{14} - 1737773 \beta_{12} + 4284758 \beta_{11} - 809212 \beta_{10} + 1737773 \beta_{9} + 1994437 \beta_{8} + 1994437 \beta_{7} - 2198457 \beta_{6} + \cdots + 129075333 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2366976 \beta_{15} + 7977748 \beta_{13} - 6992780 \beta_{12} + 10466204 \beta_{11} + 10466204 \beta_{10} - 2258828 \beta_{9} + 7123619 \beta_{8} - 7123619 \beta_{7} + \cdots + 2258828 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20324704 \beta_{15} - 17290382 \beta_{14} + 54658390 \beta_{12} - 137940986 \beta_{11} + 28624206 \beta_{10} - 54658390 \beta_{9} - 63967132 \beta_{8} - 63967132 \beta_{7} + \cdots - 3775554106 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 81854816 \beta_{15} - 255868528 \beta_{13} + 241200530 \beta_{12} - 342267664 \beta_{11} - 342267664 \beta_{10} + 77490898 \beta_{9} - 229967384 \beta_{8} + 229967384 \beta_{7} + \cdots - 77490898 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\) \(71\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
5.54840i
4.61386i
4.47752i
3.20688i
2.73921i
1.80256i
1.39379i
0.948735i
0.948735i
1.39379i
1.80256i
2.73921i
3.20688i
4.47752i
4.61386i
5.54840i
5.54840i −3.35452 + 3.96828i −22.7847 5.00000 22.0176 + 18.6122i 9.21642 + 16.0642i 82.0315i −4.49445 26.6233i 27.7420i
41.2 4.61386i 3.98691 + 3.33234i −13.2877 5.00000 15.3750 18.3951i 0.582685 18.5111i 24.3969i 4.79097 + 26.5715i 23.0693i
41.3 4.47752i −0.787490 5.13613i −12.0482 5.00000 −22.9971 + 3.52600i −18.2591 + 3.09943i 18.1258i −25.7597 + 8.08931i 22.3876i
41.4 3.20688i 2.93464 4.28811i −2.28410 5.00000 −13.7515 9.41106i 18.2867 + 2.93186i 18.3302i −9.77574 25.1681i 16.0344i
41.5 2.73921i −4.32728 + 2.87658i 0.496728 5.00000 7.87955 + 11.8533i −18.4646 1.43529i 23.2743i 10.4506 24.8955i 13.6961i
41.6 1.80256i 1.60858 + 4.94090i 4.75076 5.00000 8.90629 2.89956i 2.01165 + 18.4107i 22.9841i −21.8250 + 15.8956i 9.01282i
41.7 1.39379i 5.17944 0.416449i 6.05735 5.00000 −0.580442 7.21904i −10.8253 15.0271i 19.5930i 26.6531 4.31394i 6.96894i
41.8 0.948735i −4.24029 3.00332i 7.09990 5.00000 −2.84936 + 4.02291i 15.4514 + 10.2104i 14.3258i 8.96014 + 25.4699i 4.74368i
41.9 0.948735i −4.24029 + 3.00332i 7.09990 5.00000 −2.84936 4.02291i 15.4514 10.2104i 14.3258i 8.96014 25.4699i 4.74368i
41.10 1.39379i 5.17944 + 0.416449i 6.05735 5.00000 −0.580442 + 7.21904i −10.8253 + 15.0271i 19.5930i 26.6531 + 4.31394i 6.96894i
41.11 1.80256i 1.60858 4.94090i 4.75076 5.00000 8.90629 + 2.89956i 2.01165 18.4107i 22.9841i −21.8250 15.8956i 9.01282i
41.12 2.73921i −4.32728 2.87658i 0.496728 5.00000 7.87955 11.8533i −18.4646 + 1.43529i 23.2743i 10.4506 + 24.8955i 13.6961i
41.13 3.20688i 2.93464 + 4.28811i −2.28410 5.00000 −13.7515 + 9.41106i 18.2867 2.93186i 18.3302i −9.77574 + 25.1681i 16.0344i
41.14 4.47752i −0.787490 + 5.13613i −12.0482 5.00000 −22.9971 3.52600i −18.2591 3.09943i 18.1258i −25.7597 8.08931i 22.3876i
41.15 4.61386i 3.98691 3.33234i −13.2877 5.00000 15.3750 + 18.3951i 0.582685 + 18.5111i 24.3969i 4.79097 26.5715i 23.0693i
41.16 5.54840i −3.35452 3.96828i −22.7847 5.00000 22.0176 18.6122i 9.21642 16.0642i 82.0315i −4.49445 + 26.6233i 27.7420i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.b.b yes 16
3.b odd 2 1 105.4.b.a 16
7.b odd 2 1 105.4.b.a 16
21.c even 2 1 inner 105.4.b.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.b.a 16 3.b odd 2 1
105.4.b.a 16 7.b odd 2 1
105.4.b.b yes 16 1.a even 1 1 trivial
105.4.b.b yes 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{8} - 36 T_{17}^{7} - 17439 T_{17}^{6} + 673446 T_{17}^{5} + 80245500 T_{17}^{4} - 2696813856 T_{17}^{3} - 99699071472 T_{17}^{2} + 2613586192416 T_{17} - 14231625894720 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 96 T^{14} + 3618 T^{12} + \cdots + 5760000 \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T - 5)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + 12690 T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + 14238 T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} - 36 T^{7} + \cdots - 14231625894720)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 47508 T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + 77076 T^{14} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + 206658 T^{14} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + 232728 T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + 406 T^{7} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 468 T^{7} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 274 T^{7} + \cdots - 66\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 456 T^{7} + \cdots + 95\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 1299852 T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} - 276 T^{7} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 1827996 T^{14} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} - 502 T^{7} + \cdots - 18\!\cdots\!60)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 1941672 T^{14} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + 2857944 T^{14} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{8} - 646 T^{7} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 876 T^{7} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 3048 T^{7} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 6322590 T^{14} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
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