↑

Properties

Label 24.8.f.c
Level $24$
Weight $8$
Character orbit 24.f
Analytic conductor $7.497$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,8,Mod(11,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.11");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 24.f (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: \(7.49724061162\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 274 x^{18} + 45864 x^{16} - 5031360 x^{14} + 776389632 x^{12} - 102828146688 x^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{20} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + \beta_1 + 6) q^{3} + (\beta_{4} + \beta_{3} + 27) q^{4} + (\beta_{9} - \beta_1) q^{5} + (\beta_{6} + \beta_{3} + \beta_{2} + \cdots - 71) q^{6}+ \cdots + (\beta_{18} + \beta_{17} - \beta_{11} + \cdots + 254) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} + \beta_1 + 6) q^{3} + (\beta_{4} + \beta_{3} + 27) q^{4} + (\beta_{9} - \beta_1) q^{5} + (\beta_{6} + \beta_{3} + \beta_{2} + \cdots - 71) q^{6}+ \cdots + ( - 1214 \beta_{18} - 2582 \beta_{17} + \cdots + 515846) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 120 q^{3} + 548 q^{4} - 1416 q^{6} + 5076 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 120 q^{3} + 548 q^{4} - 1416 q^{6} + 5076 q^{9} - 3384 q^{10} - 26700 q^{12} - 33304 q^{16} - 11880 q^{18} + 133744 q^{19} - 166880 q^{22} - 74424 q^{24} + 711596 q^{25} + 85320 q^{27} - 669360 q^{28} - 41904 q^{30} + 263640 q^{33} + 24112 q^{34} - 409428 q^{36} - 437088 q^{40} + 1724040 q^{42} - 2292080 q^{43} + 1466400 q^{46} + 1846200 q^{48} + 2495228 q^{49} - 7417536 q^{51} - 2511360 q^{52} + 5680584 q^{54} - 5067120 q^{57} + 1427880 q^{58} + 2211120 q^{60} - 4070608 q^{64} + 1062024 q^{66} + 3654640 q^{67} + 11712816 q^{70} + 7004880 q^{72} + 19892680 q^{73} - 8612088 q^{75} - 11799464 q^{76} - 1226160 q^{78} - 6817932 q^{81} - 52710080 q^{82} - 6585408 q^{84} - 13470800 q^{88} + 10393704 q^{90} + 67826976 q^{91} - 24601920 q^{94} + 35857680 q^{96} + 5561800 q^{97} + 10152864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 274 x^{18} + 45864 x^{16} - 5031360 x^{14} + 776389632 x^{12} - 102828146688 x^{10} + \cdots + 11\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{19} - 274 \nu^{17} + 45864 \nu^{15} - 5031360 \nu^{13} + 776389632 \nu^{11} + \cdots - 19\!\cdots\!64 \nu ) / 92\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 67470075 \nu^{19} + 620241440 \nu^{18} + 7481718950 \nu^{17} - 87283107392 \nu^{16} + \cdots - 22\!\cdots\!72 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 176868855 \nu^{19} - 3029440960 \nu^{18} - 26451903070 \nu^{17} + 664740728704 \nu^{16} + \cdots + 36\!\cdots\!04 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 176868855 \nu^{19} + 1240482880 \nu^{18} + 26451903070 \nu^{17} - 174566214784 \nu^{16} + \cdots - 45\!\cdots\!44 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 343887579 \nu^{19} + 2480965760 \nu^{18} - 309649627546 \nu^{17} - 349132429568 \nu^{16} + \cdots - 90\!\cdots\!88 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 112575697 \nu^{19} - 5659803360 \nu^{18} + 8849711182 \nu^{17} + 846460898240 \nu^{16} + \cdots + 24\!\cdots\!20 ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 176868855 \nu^{19} + 46663936192 \nu^{18} + 26451903070 \nu^{17} - 7650400378240 \nu^{16} + \cdots - 17\!\cdots\!76 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 367713945 \nu^{19} - 25214600320 \nu^{18} + 56733294530 \nu^{17} + 4263582978304 \nu^{16} + \cdots + 13\!\cdots\!84 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8511439 \nu^{19} + 3020598158 \nu^{17} - 288761902424 \nu^{15} + \cdots + 14\!\cdots\!80 \nu ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23408920 \nu^{19} + 373120255 \nu^{18} + 6526306416 \nu^{17} - 55631403246 \nu^{16} + \cdots - 15\!\cdots\!16 ) / 20\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 183080515 \nu^{19} + 1439256000 \nu^{18} - 28153897910 \nu^{17} - 229030049664 \nu^{16} + \cdots - 80\!\cdots\!84 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 585029869 \nu^{19} + 38141204352 \nu^{18} - 50164026794 \nu^{17} - 8177313523456 \nu^{16} + \cdots - 28\!\cdots\!28 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 334078239 \nu^{19} + 620241440 \nu^{18} - 93024369454 \nu^{17} - 87283107392 \nu^{16} + \cdots - 22\!\cdots\!72 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3421024153 \nu^{19} + 81935374912 \nu^{18} + 583128146370 \nu^{17} + \cdots - 15\!\cdots\!60 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1012048949 \nu^{19} - 31317375584 \nu^{18} - 143545301178 \nu^{17} + 5528689605312 \nu^{16} + \cdots + 17\!\cdots\!64 ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 3680481851 \nu^{19} - 32083178880 \nu^{18} + 517447910182 \nu^{17} + \cdots + 10\!\cdots\!68 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1928675353 \nu^{19} + 20259606880 \nu^{18} - 271949906626 \nu^{17} - 6438475962560 \nu^{16} + \cdots - 32\!\cdots\!00 ) / 64\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 6530683329 \nu^{19} + 2480965760 \nu^{18} + 1069791510674 \nu^{17} + \cdots - 90\!\cdots\!88 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 9234345027 \nu^{19} - 43570858688 \nu^{18} + 2725068754870 \nu^{17} + \cdots - 15\!\cdots\!24 ) / 12\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} - 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{8} + 5\beta_{4} - 2\beta_{3} - 3\beta _1 + 220 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{13} + 2\beta_{11} - 3\beta_{10} - 3\beta_{6} - 7\beta_{4} + 2\beta_{3} + 6\beta_{2} + 57\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{18} + 2 \beta_{17} + 2 \beta_{16} - 4 \beta_{15} + 2 \beta_{14} - 8 \beta_{12} + 7 \beta_{11} + \cdots - 3326 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 24 \beta_{19} - 48 \beta_{18} + 28 \beta_{17} - 40 \beta_{16} - 28 \beta_{14} + 76 \beta_{13} + \cdots - 68 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 323 \beta_{18} - 902 \beta_{17} - 406 \beta_{16} - 572 \beta_{15} - 166 \beta_{14} - 256 \beta_{12} + \cdots - 200026 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3848 \beta_{19} + 656 \beta_{18} + 1324 \beta_{17} + 600 \beta_{16} - 1324 \beta_{14} + 5940 \beta_{13} + \cdots - 724 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17782 \beta_{18} - 99860 \beta_{17} - 79604 \beta_{16} + 27672 \beta_{15} - 43924 \beta_{14} + \cdots - 151943260 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 243600 \beta_{19} + 1236960 \beta_{18} - 25880 \beta_{17} - 95920 \beta_{16} + 25880 \beta_{14} + \cdots - 70040 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9852580 \beta_{18} + 15045448 \beta_{17} + 1656328 \beta_{16} + 2495376 \beta_{15} - 5780152 \beta_{14} + \cdots - 10579846824 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 67022432 \beta_{19} - 65536704 \beta_{18} - 29816336 \beta_{17} - 3694880 \beta_{16} + \cdots + 26121456 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 73104344 \beta_{18} - 637727824 \beta_{17} - 805190608 \beta_{16} - 392236192 \beta_{15} + \cdots - 469614905072 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6520396224 \beta_{19} - 9033642112 \beta_{18} - 5922604128 \beta_{17} + 9182802240 \beta_{16} + \cdots + 15105406368 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 24231938576 \beta_{18} - 31505920992 \beta_{17} + 10848194336 \beta_{16} + 149303277632 \beta_{15} + \cdots + 175411632106592 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 174302113152 \beta_{19} + 822023900416 \beta_{18} - 454890849856 \beta_{17} + 367739793280 \beta_{16} + \cdots + 822630643136 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 4918911747488 \beta_{18} + 37587778847936 \beta_{17} + 37978188674752 \beta_{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 53431420690176 \beta_{19} - 277804069933568 \beta_{18} - 24870215769472 \beta_{17} + \cdots + 76456141884032 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 31\!\cdots\!76 \beta_{18} - 589780755127168 \beta_{17} + \cdots - 24\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 17\!\cdots\!72 \beta_{19} + \cdots - 49\!\cdots\!96 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
10.9006 − 3.02933i
10.9006 + 3.02933i
10.4702 − 4.28654i
10.4702 + 4.28654i
10.0085 − 5.27546i
10.0085 + 5.27546i
6.74480 − 9.08337i
6.74480 + 9.08337i
3.79334 − 10.6588i
3.79334 + 10.6588i
−3.79334 − 10.6588i
−3.79334 + 10.6588i
−6.74480 − 9.08337i
−6.74480 + 9.08337i
−10.0085 − 5.27546i
−10.0085 + 5.27546i
−10.4702 − 4.28654i
−10.4702 + 4.28654i
−10.9006 − 3.02933i
−10.9006 + 3.02933i
−10.9006 − 3.02933i 35.1794 + 30.8125i 109.646 + 66.0430i −65.0827 −290.135 − 442.445i − 90.3177i −995.145 − 1052.06i 288.180 + 2167.93i 709.441 + 197.157i
11.2 −10.9006 + 3.02933i 35.1794 − 30.8125i 109.646 − 66.0430i −65.0827 −290.135 + 442.445i 90.3177i −995.145 + 1052.06i 288.180 − 2167.93i 709.441 − 197.157i
11.3 −10.4702 − 4.28654i −24.1914 − 40.0222i 91.2511 + 89.7621i 477.520 81.7320 + 522.739i − 632.635i −570.650 − 1330.98i −1016.56 + 1936.38i −4999.74 − 2046.91i
11.4 −10.4702 + 4.28654i −24.1914 + 40.0222i 91.2511 − 89.7621i 477.520 81.7320 − 522.739i 632.635i −570.650 + 1330.98i −1016.56 − 1936.38i −4999.74 + 2046.91i
11.5 −10.0085 − 5.27546i −43.1368 + 18.0615i 72.3391 + 105.599i −277.669 527.016 + 46.7976i 1066.27i −166.924 − 1438.50i 1534.56 − 1558.23i 2779.04 + 1464.83i
11.6 −10.0085 + 5.27546i −43.1368 − 18.0615i 72.3391 − 105.599i −277.669 527.016 − 46.7976i − 1066.27i −166.924 + 1438.50i 1534.56 + 1558.23i 2779.04 − 1464.83i
11.7 −6.74480 − 9.08337i 15.4671 − 44.1335i −37.0154 + 122.531i −320.900 −505.204 + 157.178i 37.9939i 1362.66 − 490.223i −1708.54 − 1365.24i 2164.40 + 2914.85i
11.8 −6.74480 + 9.08337i 15.4671 + 44.1335i −37.0154 − 122.531i −320.900 −505.204 − 157.178i − 37.9939i 1362.66 + 490.223i −1708.54 + 1365.24i 2164.40 − 2914.85i
11.9 −3.79334 − 10.6588i 46.6816 + 2.79715i −99.2212 + 80.8651i 395.205 −147.265 − 508.182i 1395.40i 1238.31 + 750.833i 2171.35 + 261.151i −1499.14 − 4212.42i
11.10 −3.79334 + 10.6588i 46.6816 − 2.79715i −99.2212 − 80.8651i 395.205 −147.265 + 508.182i − 1395.40i 1238.31 − 750.833i 2171.35 − 261.151i −1499.14 + 4212.42i
11.11 3.79334 − 10.6588i 46.6816 + 2.79715i −99.2212 − 80.8651i −395.205 206.894 − 486.961i − 1395.40i −1238.31 + 750.833i 2171.35 + 261.151i −1499.14 + 4212.42i
11.12 3.79334 + 10.6588i 46.6816 − 2.79715i −99.2212 + 80.8651i −395.205 206.894 + 486.961i 1395.40i −1238.31 − 750.833i 2171.35 − 261.151i −1499.14 − 4212.42i
11.13 6.74480 − 9.08337i 15.4671 − 44.1335i −37.0154 − 122.531i 320.900 −296.559 − 438.165i − 37.9939i −1362.66 − 490.223i −1708.54 − 1365.24i 2164.40 − 2914.85i
11.14 6.74480 + 9.08337i 15.4671 + 44.1335i −37.0154 + 122.531i 320.900 −296.559 + 438.165i 37.9939i −1362.66 + 490.223i −1708.54 + 1365.24i 2164.40 + 2914.85i
11.15 10.0085 − 5.27546i −43.1368 + 18.0615i 72.3391 − 105.599i 277.669 −336.450 + 408.335i − 1066.27i 166.924 − 1438.50i 1534.56 − 1558.23i 2779.04 − 1464.83i
11.16 10.0085 + 5.27546i −43.1368 − 18.0615i 72.3391 + 105.599i 277.669 −336.450 − 408.335i 1066.27i 166.924 + 1438.50i 1534.56 + 1558.23i 2779.04 + 1464.83i
11.17 10.4702 − 4.28654i −24.1914 − 40.0222i 91.2511 − 89.7621i −477.520 −424.846 − 315.344i 632.635i 570.650 − 1330.98i −1016.56 + 1936.38i −4999.74 + 2046.91i
11.18 10.4702 + 4.28654i −24.1914 + 40.0222i 91.2511 + 89.7621i −477.520 −424.846 + 315.344i − 632.635i 570.650 + 1330.98i −1016.56 − 1936.38i −4999.74 − 2046.91i
11.19 10.9006 − 3.02933i 35.1794 + 30.8125i 109.646 − 66.0430i 65.0827 476.818 + 229.305i 90.3177i 995.145 − 1052.06i 288.180 + 2167.93i 709.441 − 197.157i
11.20 10.9006 + 3.02933i 35.1794 − 30.8125i 109.646 + 66.0430i 65.0827 476.818 − 229.305i − 90.3177i 995.145 + 1052.06i 288.180 − 2167.93i 709.441 + 197.157i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.8.f.c ✓ 20
3.b odd 2 1 inner 24.8.f.c ✓ 20
4.b odd 2 1 96.8.f.c 20
8.b even 2 1 96.8.f.c 20
8.d odd 2 1 inner 24.8.f.c ✓ 20
12.b even 2 1 96.8.f.c 20
24.f even 2 1 inner 24.8.f.c ✓ 20
24.h odd 2 1 96.8.f.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.f.c ✓ 20 1.a even 1 1 trivial
24.8.f.c ✓ 20 3.b odd 2 1 inner
24.8.f.c ✓ 20 8.d odd 2 1 inner
24.8.f.c ✓ 20 24.f even 2 1 inner
96.8.f.c 20 4.b odd 2 1
96.8.f.c 20 8.b even 2 1
96.8.f.c 20 12.b even 2 1
96.8.f.c 20 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 568524 T_{5}^{8} + 115131638832 T_{5}^{6} + \cdots - 11\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots + 50\!\cdots\!07)^{2} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 45\!\cdots\!84)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 13\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 40\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 15\!\cdots\!20)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 37\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 74\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 45\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 31\!\cdots\!60)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 51\!\cdots\!40)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 70\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 23\!\cdots\!20)^{4} \) Copy content Toggle raw display
show more
show less