↑

Properties

Label 182.2.v.a
Level $182$
Weight $2$
Character orbit 182.v
Analytic conductor $1.453$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [182,2,Mod(121,182)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(182, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("182.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 182.v (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: \(1.45327731679\)
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: \( x^{20} + 46 x^{18} + 895 x^{16} + 9634 x^{14} + 62977 x^{12} + 257850 x^{10} + 656102 x^{8} + \cdots + 32041 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{3} q^{3} + \beta_{4} q^{4} + (\beta_{16} - \beta_{8}) q^{5} - \beta_1 q^{6} + (\beta_{17} - \beta_{14} + \cdots - \beta_{9}) q^{7}+ \cdots + (\beta_{10} - \beta_{9} - \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{3} q^{3} + \beta_{4} q^{4} + (\beta_{16} - \beta_{8}) q^{5} - \beta_1 q^{6} + (\beta_{17} - \beta_{14} + \cdots - \beta_{9}) q^{7}+ \cdots + ( - \beta_{18} - \beta_{17} + \beta_{16} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 4 q^{3} + 10 q^{4} - 2 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 4 q^{3} + 10 q^{4} - 2 q^{7} + 32 q^{9} + 8 q^{10} - 2 q^{12} + 6 q^{13} - 4 q^{14} - 12 q^{15} - 10 q^{16} - 10 q^{17} - 24 q^{18} + 14 q^{21} + 2 q^{22} + 18 q^{25} - 6 q^{26} - 28 q^{27} - 4 q^{28} + 2 q^{29} + 4 q^{30} + 6 q^{31} - 36 q^{35} + 16 q^{36} - 36 q^{37} - 14 q^{38} - 26 q^{39} + 4 q^{40} + 18 q^{41} - 4 q^{42} + 8 q^{43} - 6 q^{44} - 60 q^{45} + 12 q^{46} - 18 q^{47} + 2 q^{48} - 12 q^{50} - 22 q^{51} - 4 q^{53} - 18 q^{54} + 2 q^{55} - 2 q^{56} - 12 q^{60} + 8 q^{61} - 24 q^{62} + 64 q^{63} - 20 q^{64} - 18 q^{65} + 32 q^{66} + 10 q^{68} - 34 q^{69} + 48 q^{70} - 18 q^{71} + 72 q^{73} + 12 q^{74} - 18 q^{75} + 24 q^{76} + 42 q^{77} + 20 q^{78} + 42 q^{79} + 4 q^{81} + 32 q^{82} + 16 q^{84} - 12 q^{85} - 42 q^{86} - 64 q^{87} + 4 q^{88} - 18 q^{89} + 52 q^{90} + 60 q^{91} + 60 q^{93} - 32 q^{94} - 12 q^{95} + 78 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 46 x^{18} + 895 x^{16} + 9634 x^{14} + 62977 x^{12} + 257850 x^{10} + 656102 x^{8} + \cdots + 32041 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 689 \nu^{18} - 24939 \nu^{16} - 352550 \nu^{14} - 2452752 \nu^{12} - 8592689 \nu^{10} + \cdots + 596965 ) / 2881152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 689 \nu^{18} - 24939 \nu^{16} - 352550 \nu^{14} - 2452752 \nu^{12} - 8592689 \nu^{10} + \cdots + 596965 ) / 2881152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15507 \nu^{18} - 640669 \nu^{16} - 10848930 \nu^{14} - 97358920 \nu^{12} - 499479691 \nu^{10} + \cdots + 62025827 ) / 30947008 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3335 \nu^{19} + 276741 \nu^{17} + 7448906 \nu^{15} + 95235840 \nu^{13} + 649070903 \nu^{11} + \cdots + 257863104 ) / 515726208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 127863297 \nu^{19} + 3415070116 \nu^{18} + 6905964519 \nu^{17} + 143544449700 \nu^{16} + \cdots + 63626088783556 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 127863297 \nu^{19} - 3415070116 \nu^{18} + 6905964519 \nu^{17} - 143544449700 \nu^{16} + \cdots - 63626088783556 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 231418601 \nu^{19} + 35087043 \nu^{18} + 9960821043 \nu^{17} + 4824670593 \nu^{16} + \cdots + 34344090625425 ) / 2044080825408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 231418601 \nu^{19} + 35087043 \nu^{18} - 9960821043 \nu^{17} + 4824670593 \nu^{16} + \cdots + 34344090625425 ) / 2044080825408 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 585433149 \nu^{19} + 2390817259 \nu^{18} + 31398905979 \nu^{17} + 101227621581 \nu^{16} + \cdots + 77943360809773 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 585433149 \nu^{19} - 2390817259 \nu^{18} + 31398905979 \nu^{17} - 101227621581 \nu^{16} + \cdots - 77943360809773 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 525849824 \nu^{19} + 1056701261 \nu^{18} + 21627009414 \nu^{17} + 45211308597 \nu^{16} + \cdots + 68025005329565 ) / 2044080825408 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 525849824 \nu^{19} - 1056701261 \nu^{18} + 21627009414 \nu^{17} - 45211308597 \nu^{16} + \cdots - 68025005329565 ) / 2044080825408 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1463123799 \nu^{19} - 458139760 \nu^{18} - 61994305407 \nu^{17} - 15569843514 \nu^{16} + \cdots + 269919761054 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1463123799 \nu^{19} - 458139760 \nu^{18} + 61994305407 \nu^{17} - 15569843514 \nu^{16} + \cdots + 269919761054 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1904272858 \nu^{19} + 5646642531 \nu^{18} + 84922872144 \nu^{17} + 237943453623 \nu^{16} + \cdots + 14554867430271 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1904272858 \nu^{19} + 5646642531 \nu^{18} - 84922872144 \nu^{17} + 237943453623 \nu^{16} + \cdots + 14554867430271 ) / 4088161650816 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1186693110 \nu^{19} - 817900583 \nu^{18} - 50459891028 \nu^{17} - 32335486683 \nu^{16} + \cdots - 14477324772755 ) / 2044080825408 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 1186693110 \nu^{19} + 817900583 \nu^{18} - 50459891028 \nu^{17} + 32335486683 \nu^{16} + \cdots + 14477324772755 ) / 2044080825408 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 19078604 \nu^{19} + 5722083 \nu^{18} + 801924300 \nu^{17} + 236406861 \nu^{16} + \cdots - 22887530163 ) / 22838891904 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{18} + 3 \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{14} - 3 \beta_{13} + \beta_{12} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26 \beta_{19} + 16 \beta_{18} + 16 \beta_{17} - 16 \beta_{16} + 16 \beta_{15} - 12 \beta_{14} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 43 \beta_{18} - 43 \beta_{17} + 15 \beta_{16} + 15 \beta_{15} + 45 \beta_{14} + 45 \beta_{13} + \cdots - 289 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 292 \beta_{19} - 196 \beta_{18} - 196 \beta_{17} + 193 \beta_{16} - 193 \beta_{15} + 123 \beta_{14} + \cdots + 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 483 \beta_{18} + 483 \beta_{17} - 179 \beta_{16} - 179 \beta_{15} - 529 \beta_{14} - 529 \beta_{13} + \cdots + 2621 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3144 \beta_{19} + 2199 \beta_{18} + 2199 \beta_{17} - 2121 \beta_{16} + 2121 \beta_{15} - 1217 \beta_{14} + \cdots - 805 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5060 \beta_{18} - 5060 \beta_{17} + 1970 \beta_{16} + 1970 \beta_{15} + 5766 \beta_{14} + 5766 \beta_{13} + \cdots - 25039 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 33224 \beta_{19} - 23711 \beta_{18} - 23711 \beta_{17} + 22470 \beta_{16} - 22470 \beta_{15} + \cdots + 7219 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 51833 \beta_{18} + 51833 \beta_{17} - 20819 \beta_{16} - 20819 \beta_{15} - 61057 \beta_{14} + \cdots + 246121 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 347590 \beta_{19} + 250515 \beta_{18} + 250515 \beta_{17} - 234512 \beta_{16} + 234512 \beta_{15} + \cdots - 64587 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 527632 \beta_{18} - 527632 \beta_{17} + 214918 \beta_{16} + 214918 \beta_{15} + 639102 \beta_{14} + \cdots - 2457595 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3615656 \beta_{19} - 2617830 \beta_{18} - 2617830 \beta_{17} + 2433058 \beta_{16} - 2433058 \beta_{15} + \cdots + 572952 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 5367698 \beta_{18} + 5367698 \beta_{17} - 2186816 \beta_{16} - 2186816 \beta_{15} - 6659800 \beta_{14} + \cdots + 24756783 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 37484170 \beta_{19} + 27190389 \beta_{18} + 27190389 \beta_{17} - 25189781 \beta_{16} + 25189781 \beta_{15} + \cdots - 5008137 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 54669537 \beta_{18} - 54669537 \beta_{17} + 22047095 \beta_{16} + 22047095 \beta_{15} + 69291965 \beta_{14} + \cdots - 250646175 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 387843338 \beta_{19} - 281470414 \beta_{18} - 281470414 \beta_{17} + 260652200 \beta_{16} + \cdots + 42810124 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(157\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
3.22209i
2.35757i
0.510029i
− 1.92209i
− 3.16760i
− 2.49593i
− 2.17431i
0.596224i
1.40613i
1.66789i
− 3.22209i
− 2.35757i
− 0.510029i
1.92209i
3.16760i
2.49593i
2.17431i
− 0.596224i
− 1.40613i
− 1.66789i
−0.866025 + 0.500000i −3.22209 0.500000 − 0.866025i −2.42600 − 1.40065i 2.79041 − 1.61104i 1.45795 + 2.20780i 1.00000i 7.38183 2.80130
121.2 −0.866025 + 0.500000i −2.35757 0.500000 − 0.866025i 2.89612 + 1.67208i 2.04172 − 1.17879i −2.14521 − 1.54858i 1.00000i 2.55814 −3.34415
121.3 −0.866025 + 0.500000i −0.510029 0.500000 − 0.866025i −1.23682 − 0.714077i 0.441698 − 0.255014i −0.221327 − 2.63648i 1.00000i −2.73987 1.42815
121.4 −0.866025 + 0.500000i 1.92209 0.500000 − 0.866025i 1.94934 + 1.12545i −1.66457 + 0.961043i −1.27491 + 2.31832i 1.00000i 0.694411 −2.25090
121.5 −0.866025 + 0.500000i 3.16760 0.500000 − 0.866025i −2.91470 − 1.68280i −2.74322 + 1.58380i 2.54951 − 0.707090i 1.00000i 7.03369 3.36560
121.6 0.866025 − 0.500000i −2.49593 0.500000 − 0.866025i 2.41344 + 1.39340i −2.16154 + 1.24796i 2.26257 − 1.37142i − 1.00000i 3.22966 2.78680
121.7 0.866025 − 0.500000i −2.17431 0.500000 − 0.866025i −2.17777 − 1.25733i −1.88301 + 1.08715i −2.59088 − 0.536046i − 1.00000i 1.72761 −2.51467
121.8 0.866025 − 0.500000i 0.596224 0.500000 − 0.866025i 3.25277 + 1.87799i 0.516345 − 0.298112i −2.44418 + 1.01292i − 1.00000i −2.64452 3.75598
121.9 0.866025 − 0.500000i 1.40613 0.500000 − 0.866025i −0.552947 − 0.319244i 1.21774 − 0.703063i −0.103968 − 2.64371i − 1.00000i −1.02281 −0.638488
121.10 0.866025 − 0.500000i 1.66789 0.500000 − 0.866025i −1.20345 − 0.694810i 1.44443 − 0.833944i 1.51043 + 2.17223i − 1.00000i −0.218152 −1.38962
179.1 −0.866025 − 0.500000i −3.22209 0.500000 + 0.866025i −2.42600 + 1.40065i 2.79041 + 1.61104i 1.45795 − 2.20780i − 1.00000i 7.38183 2.80130
179.2 −0.866025 − 0.500000i −2.35757 0.500000 + 0.866025i 2.89612 − 1.67208i 2.04172 + 1.17879i −2.14521 + 1.54858i − 1.00000i 2.55814 −3.34415
179.3 −0.866025 − 0.500000i −0.510029 0.500000 + 0.866025i −1.23682 + 0.714077i 0.441698 + 0.255014i −0.221327 + 2.63648i − 1.00000i −2.73987 1.42815
179.4 −0.866025 − 0.500000i 1.92209 0.500000 + 0.866025i 1.94934 − 1.12545i −1.66457 − 0.961043i −1.27491 − 2.31832i − 1.00000i 0.694411 −2.25090
179.5 −0.866025 − 0.500000i 3.16760 0.500000 + 0.866025i −2.91470 + 1.68280i −2.74322 − 1.58380i 2.54951 + 0.707090i − 1.00000i 7.03369 3.36560
179.6 0.866025 + 0.500000i −2.49593 0.500000 + 0.866025i 2.41344 − 1.39340i −2.16154 − 1.24796i 2.26257 + 1.37142i 1.00000i 3.22966 2.78680
179.7 0.866025 + 0.500000i −2.17431 0.500000 + 0.866025i −2.17777 + 1.25733i −1.88301 − 1.08715i −2.59088 + 0.536046i 1.00000i 1.72761 −2.51467
179.8 0.866025 + 0.500000i 0.596224 0.500000 + 0.866025i 3.25277 − 1.87799i 0.516345 + 0.298112i −2.44418 − 1.01292i 1.00000i −2.64452 3.75598
179.9 0.866025 + 0.500000i 1.40613 0.500000 + 0.866025i −0.552947 + 0.319244i 1.21774 + 0.703063i −0.103968 + 2.64371i 1.00000i −1.02281 −0.638488
179.10 0.866025 + 0.500000i 1.66789 0.500000 + 0.866025i −1.20345 + 0.694810i 1.44443 + 0.833944i 1.51043 − 2.17223i 1.00000i −0.218152 −1.38962
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.2.v.a yes 20
3.b odd 2 1 1638.2.cr.c 20
7.b odd 2 1 1274.2.v.h 20
7.c even 3 1 182.2.o.a ✓ 20
7.c even 3 1 1274.2.m.g 20
7.d odd 6 1 1274.2.m.f 20
7.d odd 6 1 1274.2.o.h 20
13.e even 6 1 182.2.o.a ✓ 20
21.h odd 6 1 1638.2.dt.c 20
39.h odd 6 1 1638.2.dt.c 20
91.k even 6 1 1274.2.m.g 20
91.l odd 6 1 1274.2.m.f 20
91.p odd 6 1 1274.2.v.h 20
91.t odd 6 1 1274.2.o.h 20
91.u even 6 1 inner 182.2.v.a yes 20
273.x odd 6 1 1638.2.cr.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.o.a ✓ 20 7.c even 3 1
182.2.o.a ✓ 20 13.e even 6 1
182.2.v.a yes 20 1.a even 1 1 trivial
182.2.v.a yes 20 91.u even 6 1 inner
1274.2.m.f 20 7.d odd 6 1
1274.2.m.f 20 91.l odd 6 1
1274.2.m.g 20 7.c even 3 1
1274.2.m.g 20 91.k even 6 1
1274.2.o.h 20 7.d odd 6 1
1274.2.o.h 20 91.t odd 6 1
1274.2.v.h 20 7.b odd 2 1
1274.2.v.h 20 91.p odd 6 1
1638.2.cr.c 20 3.b odd 2 1
1638.2.cr.c 20 273.x odd 6 1
1638.2.dt.c 20 21.h odd 6 1
1638.2.dt.c 20 39.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(182, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T^{10} + 2 T^{9} + \cdots - 179)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} - 34 T^{18} + \cdots + 5602689 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{20} + 84 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 542517264 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 29018781801 \) Copy content Toggle raw display
$23$ \( T^{20} + 104 T^{18} + \cdots + 15968016 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 25206655308129 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 8215428321 \) Copy content Toggle raw display
$37$ \( T^{20} + 36 T^{19} + \cdots + 15116544 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 169027054641 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 20983462954441 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 19178119594089 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 60575820604209 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 18503177547024 \) Copy content Toggle raw display
$61$ \( (T^{10} - 4 T^{9} + \cdots - 60882119)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1420790096961 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 448025929224129 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 23\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 81\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 167120049410064 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
show more
show less