Properties

Label 7.17.d.a
Level $7$
Weight $17$
Character orbit 7.d
Analytic conductor $11.363$
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] = [7,17,Mod(3,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.3");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\), 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: \(11.3627180700\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 470221 x^{18} - 17792382 x^{17} + 149610140535 x^{16} - 5885709493452 x^{15} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{16}\cdot 7^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 9 \beta_{2} - \beta_1 + 9) q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 219 \beta_{2} + \beta_1 + 438) q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 28595 \beta_{2} + 10 \beta_1) q^{4} + ( - \beta_{10} - \beta_{9} - 3 \beta_{6} + 202 \beta_{3} - 8102 \beta_{2} + \cdots + 8103) q^{5}+ \cdots + ( - \beta_{18} - 2 \beta_{15} + 2 \beta_{14} + 3 \beta_{12} - 16 \beta_{10} + \cdots + 14605406) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 9 \beta_{2} - \beta_1 + 9) q^{2} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + 219 \beta_{2} + \beta_1 + 438) q^{3} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 28595 \beta_{2} + 10 \beta_1) q^{4} + ( - \beta_{10} - \beta_{9} - 3 \beta_{6} + 202 \beta_{3} - 8102 \beta_{2} + \cdots + 8103) q^{5}+ \cdots + (13552056 \beta_{19} - 51196010 \beta_{18} + \cdots + 71\!\cdots\!21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 92 q^{2} + 6558 q^{3} - 285924 q^{4} + 241890 q^{5} - 1847944 q^{7} + 23873872 q^{8} + 146092512 q^{9} - 567579804 q^{10} + 487037030 q^{11} - 2240722092 q^{12} + 598121216 q^{14} - 2098975212 q^{15} - 13522996616 q^{16} + 16479299850 q^{17} - 21631627512 q^{18} - 29753076894 q^{19} + 30310552398 q^{21} + 143879888720 q^{22} - 199076938822 q^{23} + 613456300512 q^{24} + 212723266388 q^{25} - 1359419612544 q^{26} + 2531834927748 q^{28} + 438309600272 q^{29} - 2296351012392 q^{30} + 1037434908306 q^{31} - 3523947158064 q^{32} + 411779151054 q^{33} - 7248579242478 q^{35} - 2493315404256 q^{36} + 5318067734218 q^{37} - 9079217417208 q^{38} + 13250117821332 q^{39} + 15010293809208 q^{40} + 16256122433712 q^{42} - 621187489400 q^{43} + 25172272315980 q^{44} - 85040191344096 q^{45} - 15614039192704 q^{46} + 106960600327866 q^{47} - 52978127838580 q^{49} + 35872835226128 q^{50} + 4235281588962 q^{51} + 126484190926632 q^{52} + 29048763888218 q^{53} - 635343594055560 q^{54} + 230352840277168 q^{56} + 366396034764636 q^{57} - 279762037805080 q^{58} - 99092116656282 q^{59} - 173605217618196 q^{60} - 904353032308434 q^{61} + 11\!\cdots\!92 q^{63}+ \cdots + 14\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 470221 x^{18} - 17792382 x^{17} + 149610140535 x^{16} - 5885709493452 x^{15} + \cdots + 26\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 81\!\cdots\!00 \nu^{19} + \cdots - 18\!\cdots\!60 ) / 12\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!30 \nu^{19} + \cdots + 55\!\cdots\!00 ) / 32\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!78 \nu^{19} + \cdots + 21\!\cdots\!75 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20\!\cdots\!61 \nu^{19} + \cdots - 17\!\cdots\!75 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 95\!\cdots\!79 \nu^{19} + \cdots + 15\!\cdots\!25 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 38\!\cdots\!71 \nu^{19} + \cdots - 35\!\cdots\!00 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 60\!\cdots\!39 \nu^{19} + \cdots + 16\!\cdots\!75 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25\!\cdots\!27 \nu^{19} + \cdots - 39\!\cdots\!90 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 91\!\cdots\!86 \nu^{19} + \cdots + 37\!\cdots\!95 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 56\!\cdots\!08 \nu^{19} + \cdots - 21\!\cdots\!35 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 57\!\cdots\!42 \nu^{19} + \cdots + 43\!\cdots\!55 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 64\!\cdots\!17 \nu^{19} + \cdots + 65\!\cdots\!95 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 59\!\cdots\!31 \nu^{19} + \cdots + 40\!\cdots\!90 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 58\!\cdots\!77 \nu^{19} + \cdots + 10\!\cdots\!30 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 39\!\cdots\!75 \nu^{19} + \cdots + 32\!\cdots\!50 ) / 18\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 16\!\cdots\!76 \nu^{19} + \cdots + 71\!\cdots\!95 ) / 54\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 32\!\cdots\!52 \nu^{19} + \cdots - 26\!\cdots\!15 ) / 90\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 10\!\cdots\!29 \nu^{19} + \cdots - 15\!\cdots\!55 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta_{4} - 28\beta_{3} - 94050\beta_{2} - 28\beta _1 - 94050 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{16} - 2 \beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + 69 \beta_{7} + 655 \beta_{6} - 69 \beta_{5} - 327 \beta_{4} + 166216 \beta_{3} - \beta_{2} - \beta _1 + 2561232 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 58 \beta_{19} + 8 \beta_{17} + 60 \beta_{16} + 322 \beta_{15} + 1018 \beta_{14} - 30 \beta_{13} - 1868 \beta_{12} + 6226 \beta_{10} + 3182 \beta_{9} - 228 \beta_{8} - 241728 \beta_{7} - 240786 \beta_{6} - 124 \beta_{5} + \cdots - 84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5330 \beta_{19} + 11462 \beta_{18} - 7614 \beta_{17} + 353662 \beta_{15} + 343722 \beta_{14} + 348250 \beta_{12} + 279724 \beta_{11} - 609942 \beta_{10} + 976548 \beta_{9} + \cdots - 1175558557942 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5822296 \beta_{19} - 9804750 \beta_{18} - 21680706 \beta_{17} - 26039904 \beta_{16} - 78063824 \beta_{15} - 764668228 \beta_{14} + 9804750 \beta_{13} + \cdots + 31\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2664712038 \beta_{19} + 5821239808 \beta_{17} + 68823969001 \beta_{16} - 42645819003 \beta_{15} + 111445124815 \beta_{14} - 4403496238 \beta_{13} + \cdots - 86287688425 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5753793941412 \beta_{19} + 2559168325212 \beta_{18} + 3525587873364 \beta_{17} - 1401784311396 \beta_{15} + 107589624644868 \beta_{14} + \cdots - 67\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18\!\cdots\!04 \beta_{19} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 75\!\cdots\!92 \beta_{19} + \cdots - 48\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 35\!\cdots\!08 \beta_{19} + \cdots - 32\!\cdots\!87 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 19\!\cdots\!68 \beta_{19} + \cdots + 36\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 33\!\cdots\!62 \beta_{19} + \cdots - 13\!\cdots\!28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 88\!\cdots\!90 \beta_{19} + \cdots - 85\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 34\!\cdots\!56 \beta_{19} + \cdots + 24\!\cdots\!26 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 76\!\cdots\!96 \beta_{19} + \cdots - 10\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 74\!\cdots\!92 \beta_{19} + \cdots - 64\!\cdots\!24 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 36\!\cdots\!84 \beta_{19} + \cdots + 50\!\cdots\!82 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 25\!\cdots\!44 \beta_{19} + \cdots - 20\!\cdots\!05 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−251.134 + 434.977i
−187.565 + 324.873i
−120.823 + 209.272i
−64.4555 + 111.640i
−58.5903 + 101.481i
45.0238 77.9836i
90.3350 156.465i
137.362 237.918i
180.712 313.002i
230.136 398.607i
−251.134 434.977i
−187.565 324.873i
−120.823 209.272i
−64.4555 111.640i
−58.5903 101.481i
45.0238 + 77.9836i
90.3350 + 156.465i
137.362 + 237.918i
180.712 + 313.002i
230.136 + 398.607i
−246.634 427.182i 6439.04 + 3717.58i −88888.5 + 153959.i 360858. 208341.i 3.66753e6i −3.56165e6 4.53294e6i 5.53649e7 6.11747e6 + 1.05958e7i −1.77999e8 1.02768e8i
3.2 −183.065 317.078i −5337.54 3081.63i −34257.8 + 59336.2i −149102. + 86084.1i 2.25656e6i 303703. + 5.75680e6i 1.09092e6 −2.53044e6 4.38285e6i 5.45908e7 + 3.15180e7i
3.3 −116.323 201.477i 2610.90 + 1507.40i 5705.88 9882.88i −301331. + 173973.i 701383.i 2.93221e6 4.96337e6i −1.79016e7 −1.69788e7 2.94082e7i 7.01035e7 + 4.04742e7i
3.4 −59.9555 103.846i −7943.79 4586.35i 25578.7 44303.6i 625327. 361033.i 1.09991e6i −1.45978e6 5.57691e6i −1.39928e7 2.05458e7 + 3.55864e7i −7.49836e7 4.32918e7i
3.5 −54.0903 93.6871i 8486.99 + 4899.97i 26916.5 46620.7i 214019. 123564.i 1.06016e6i 516705. + 5.74160e6i −1.29134e7 2.64960e7 + 4.58924e7i −2.31527e7 1.33672e7i
3.6 49.5238 + 85.7778i 289.874 + 167.359i 27862.8 48259.7i −159884. + 92309.1i 33153.0i −5.76147e6 + 195847.i 1.20107e7 −2.14673e7 3.71825e7i −1.58361e7 9.14300e6i
3.7 94.8350 + 164.259i −9421.11 5439.28i 14780.6 25600.8i −477118. + 275464.i 2.06334e6i 5.57858e6 + 1.45338e6i 1.80371e7 3.76482e7 + 6.52085e7i −9.04950e7 5.22473e7i
3.8 141.862 + 245.712i 1577.48 + 910.759i −7481.54 + 12958.4i 402298. 232267.i 516808.i 5.58849e6 + 1.41483e6i 1.43487e7 −1.98644e7 3.44061e7i 1.14141e8 + 6.58996e7i
3.9 185.212 + 320.796i 10622.1 + 6132.66i −35838.6 + 62074.3i −453136. + 261618.i 4.54336e6i 694120. 5.72286e6i −2.27485e6 5.36956e7 + 9.30036e7i −1.67852e8 9.69094e7i
3.10 234.636 + 406.401i −4044.92 2335.34i −77340.0 + 133957.i 59013.4 34071.4i 2.19181e6i −5.75487e6 + 338210.i −4.18327e7 −1.06158e7 1.83871e7i 2.76933e7 + 1.59887e7i
5.1 −246.634 + 427.182i 6439.04 3717.58i −88888.5 153959.i 360858. + 208341.i 3.66753e6i −3.56165e6 + 4.53294e6i 5.53649e7 6.11747e6 1.05958e7i −1.77999e8 + 1.02768e8i
5.2 −183.065 + 317.078i −5337.54 + 3081.63i −34257.8 59336.2i −149102. 86084.1i 2.25656e6i 303703. 5.75680e6i 1.09092e6 −2.53044e6 + 4.38285e6i 5.45908e7 3.15180e7i
5.3 −116.323 + 201.477i 2610.90 1507.40i 5705.88 + 9882.88i −301331. 173973.i 701383.i 2.93221e6 + 4.96337e6i −1.79016e7 −1.69788e7 + 2.94082e7i 7.01035e7 4.04742e7i
5.4 −59.9555 + 103.846i −7943.79 + 4586.35i 25578.7 + 44303.6i 625327. + 361033.i 1.09991e6i −1.45978e6 + 5.57691e6i −1.39928e7 2.05458e7 3.55864e7i −7.49836e7 + 4.32918e7i
5.5 −54.0903 + 93.6871i 8486.99 4899.97i 26916.5 + 46620.7i 214019. + 123564.i 1.06016e6i 516705. 5.74160e6i −1.29134e7 2.64960e7 4.58924e7i −2.31527e7 + 1.33672e7i
5.6 49.5238 85.7778i 289.874 167.359i 27862.8 + 48259.7i −159884. 92309.1i 33153.0i −5.76147e6 195847.i 1.20107e7 −2.14673e7 + 3.71825e7i −1.58361e7 + 9.14300e6i
5.7 94.8350 164.259i −9421.11 + 5439.28i 14780.6 + 25600.8i −477118. 275464.i 2.06334e6i 5.57858e6 1.45338e6i 1.80371e7 3.76482e7 6.52085e7i −9.04950e7 + 5.22473e7i
5.8 141.862 245.712i 1577.48 910.759i −7481.54 12958.4i 402298. + 232267.i 516808.i 5.58849e6 1.41483e6i 1.43487e7 −1.98644e7 + 3.44061e7i 1.14141e8 6.58996e7i
5.9 185.212 320.796i 10622.1 6132.66i −35838.6 62074.3i −453136. 261618.i 4.54336e6i 694120. + 5.72286e6i −2.27485e6 5.36956e7 9.30036e7i −1.67852e8 + 9.69094e7i
5.10 234.636 406.401i −4044.92 + 2335.34i −77340.0 133957.i 59013.4 + 34071.4i 2.19181e6i −5.75487e6 338210.i −4.18327e7 −1.06158e7 + 1.83871e7i 2.76933e7 1.59887e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.17.d.a 20
7.c even 3 1 49.17.b.a 20
7.d odd 6 1 inner 7.17.d.a 20
7.d odd 6 1 49.17.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.17.d.a 20 1.a even 1 1 trivial
7.17.d.a 20 7.d odd 6 1 inner
49.17.b.a 20 7.c even 3 1
49.17.b.a 20 7.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 92 T^{19} + \cdots + 25\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{20} - 6558 T^{19} + \cdots + 22\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{20} - 241890 T^{19} + \cdots + 56\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} + 1847944 T^{19} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{20} - 487037030 T^{19} + \cdots + 36\!\cdots\!69 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} - 16479299850 T^{19} + \cdots + 40\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{20} + 29753076894 T^{19} + \cdots + 29\!\cdots\!09 \) Copy content Toggle raw display
$23$ \( T^{20} + 199076938822 T^{19} + \cdots + 51\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{10} - 219154800136 T^{9} + \cdots + 22\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} - 1037434908306 T^{19} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{20} - 5318067734218 T^{19} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + 310593744700 T^{9} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} - 106960600327866 T^{19} + \cdots + 45\!\cdots\!61 \) Copy content Toggle raw display
$53$ \( T^{20} - 29048763888218 T^{19} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{20} + 99092116656282 T^{19} + \cdots + 10\!\cdots\!89 \) Copy content Toggle raw display
$61$ \( T^{20} + 904353032308434 T^{19} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$67$ \( T^{20} + 33326890567694 T^{19} + \cdots + 13\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 49\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{20} - 126874036566250 T^{19} + \cdots + 29\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
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