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Properties

Label 42.5.h.a
Level $42$
Weight $5$
Character orbit 42.h
Analytic conductor $4.342$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [42,5,Mod(11,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 42.h (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: \(4.34153844952\)
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: \( x^{20} - 20 x^{18} - 684 x^{17} - 3593 x^{16} - 62280 x^{15} + 1340206 x^{14} + 6921828 x^{13} + \cdots + 81\!\cdots\!09 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_1) q^{2} - \beta_{10} q^{3} + (8 \beta_{2} + 8) q^{4} + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{18} - \beta_{17} - \beta_{13} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_1) q^{2} - \beta_{10} q^{3} + (8 \beta_{2} + 8) q^{4} + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + (138 \beta_{19} + 98 \beta_{18} + \cdots + 3312) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 80 q^{4} - 64 q^{6} + 130 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 80 q^{4} - 64 q^{6} + 130 q^{7} + 52 q^{9} + 32 q^{10} - 324 q^{13} + 964 q^{15} - 640 q^{16} + 576 q^{18} + 1030 q^{19} - 1442 q^{21} - 1280 q^{22} - 256 q^{24} + 22 q^{25} - 3780 q^{27} + 544 q^{28} - 128 q^{30} - 318 q^{31} + 842 q^{33} + 1024 q^{34} + 832 q^{36} + 2806 q^{37} + 5542 q^{39} - 256 q^{40} + 1632 q^{42} - 6644 q^{43} + 1526 q^{45} - 3008 q^{46} - 8350 q^{49} - 5266 q^{51} - 1296 q^{52} + 7040 q^{54} + 49696 q^{55} - 3840 q^{57} + 7904 q^{58} + 3856 q^{60} - 15960 q^{61} - 144 q^{63} - 10240 q^{64} + 896 q^{66} - 12490 q^{67} - 56932 q^{69} - 19904 q^{70} - 4608 q^{72} - 23746 q^{73} + 31738 q^{75} + 16480 q^{76} + 5888 q^{78} + 15474 q^{79} + 29788 q^{81} - 9984 q^{82} + 7168 q^{84} - 39368 q^{85} + 16084 q^{87} - 5120 q^{88} - 30208 q^{90} + 16074 q^{91} - 15176 q^{93} + 15360 q^{94} + 2048 q^{96} + 102512 q^{97} + 73276 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 20 x^{18} - 684 x^{17} - 3593 x^{16} - 62280 x^{15} + 1340206 x^{14} + 6921828 x^{13} + \cdots + 81\!\cdots\!09 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30\!\cdots\!36 \nu^{19} + \cdots + 67\!\cdots\!33 ) / 61\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!78 \nu^{19} + \cdots + 14\!\cdots\!96 ) / 38\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!70 \nu^{19} + \cdots - 16\!\cdots\!14 ) / 23\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79\!\cdots\!36 \nu^{19} + \cdots + 13\!\cdots\!14 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 89\!\cdots\!38 \nu^{19} + \cdots - 77\!\cdots\!03 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38\!\cdots\!95 \nu^{19} + \cdots - 15\!\cdots\!04 ) / 61\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!39 \nu^{19} + \cdots + 70\!\cdots\!36 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43\!\cdots\!62 \nu^{19} + \cdots - 38\!\cdots\!82 ) / 46\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36\!\cdots\!31 \nu^{19} + \cdots - 17\!\cdots\!13 ) / 30\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!98 \nu^{19} + \cdots + 50\!\cdots\!72 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 38\!\cdots\!31 \nu^{19} + \cdots + 22\!\cdots\!53 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 40\!\cdots\!31 \nu^{19} + \cdots - 20\!\cdots\!52 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!09 \nu^{19} + \cdots + 14\!\cdots\!17 ) / 46\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 85\!\cdots\!53 \nu^{19} + \cdots - 14\!\cdots\!59 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 87\!\cdots\!66 \nu^{19} + \cdots - 79\!\cdots\!28 ) / 14\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 33\!\cdots\!85 \nu^{19} + \cdots - 89\!\cdots\!13 ) / 49\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 17\!\cdots\!29 \nu^{19} + \cdots + 31\!\cdots\!63 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 12\!\cdots\!81 \nu^{19} + \cdots - 13\!\cdots\!90 ) / 13\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 46\!\cdots\!07 \nu^{19} + \cdots + 74\!\cdots\!23 ) / 49\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - 2\beta_{6} - \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{18} - 2 \beta_{14} - \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 20 \beta_{9} - 2 \beta_{7} + \cdots + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{19} - 30 \beta_{18} + 34 \beta_{17} - 9 \beta_{16} + 81 \beta_{14} - 108 \beta_{13} + \cdots + 684 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 67 \beta_{18} + 175 \beta_{17} - 162 \beta_{16} - 162 \beta_{15} + 1769 \beta_{13} - 27 \beta_{12} + \cdots - 350 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 171 \beta_{19} - 915 \beta_{18} + 968 \beta_{17} - 5661 \beta_{15} + 18 \beta_{14} + 897 \beta_{12} + \cdots + 229113 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3726 \beta_{19} - 14204 \beta_{18} + 46376 \beta_{17} - 3726 \beta_{16} + 14714 \beta_{14} + \cdots - 922514 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 107730 \beta_{18} + 282186 \beta_{17} - 365742 \beta_{16} - 14364 \beta_{15} + 593532 \beta_{13} + \cdots - 564372 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 762048 \beta_{19} - 2566221 \beta_{18} + 5546556 \beta_{17} - 2217132 \beta_{15} + 3267943 \beta_{14} + \cdots - 11101253 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 31415319 \beta_{19} + 12145398 \beta_{18} - 12089360 \beta_{17} - 31415319 \beta_{16} + \cdots - 646515720 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 166336618 \beta_{18} + 12098947 \beta_{17} + 31361661 \beta_{16} - 37650339 \beta_{15} + \cdots - 24197894 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 145401093 \beta_{19} - 193733565 \beta_{18} + 5615750282 \beta_{17} - 581073363 \beta_{15} + \cdots - 231911540541 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 9231877476 \beta_{19} + 1609927576 \beta_{18} - 43886334196 \beta_{17} - 9231877476 \beta_{16} + \cdots - 548456356895 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 226325774196 \beta_{18} - 374326276932 \beta_{17} + 65429013600 \beta_{16} - 272614474152 \beta_{15} + \cdots + 748652553864 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1861710160212 \beta_{19} + 1139650858875 \beta_{18} - 6404077494828 \beta_{17} + \cdots - 63571242122375 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 20767690360071 \beta_{19} - 2557491078594 \beta_{18} - 88091402034098 \beta_{17} + \cdots - 678867951503232 ) / 6 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 115629477871595 \beta_{18} - 242946401154881 \beta_{17} + 53846711818704 \beta_{16} + \cdots + 485892802309762 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 11\!\cdots\!87 \beta_{19} + \cdots - 38\!\cdots\!07 ) / 6 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 93\!\cdots\!90 \beta_{19} + \cdots + 12\!\cdots\!88 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 16\!\cdots\!26 \beta_{18} + \cdots + 44\!\cdots\!44 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
9.84502 − 3.29377i
1.03362 − 9.70508i
0.495401 + 8.26329i
−2.76523 + 7.36012i
−7.38407 − 3.33167i
7.75666 + 1.28530i
6.90852 + 4.56068i
0.806726 − 8.06063i
−7.77500 + 6.87915i
−8.92165 − 3.95740i
9.84502 + 3.29377i
1.03362 + 9.70508i
0.495401 − 8.26329i
−2.76523 − 7.36012i
−7.38407 + 3.33167i
7.75666 − 1.28530i
6.90852 − 4.56068i
0.806726 + 8.06063i
−7.77500 − 6.87915i
−8.92165 + 3.95740i
−2.44949 + 1.41421i −8.62028 − 2.58667i 4.00000 − 6.92820i −2.28929 + 1.32172i 24.7734 − 5.85490i 47.8951 + 10.3472i 22.6274i 67.6183 + 44.5955i 3.73839 − 6.47509i
11.2 −2.44949 + 1.41421i 0.191126 − 8.99797i 4.00000 − 6.92820i −11.9899 + 6.92235i 12.2569 + 22.3107i −39.6197 + 28.8319i 22.6274i −80.9269 − 3.43949i 19.5794 − 33.9125i
11.3 −2.44949 + 1.41421i 0.729344 + 8.97040i 4.00000 − 6.92820i −30.1838 + 17.4266i −14.4726 − 20.9415i 31.6264 − 37.4269i 22.6274i −79.9361 + 13.0850i 49.2899 − 85.3727i
11.4 −2.44949 + 1.41421i 3.98997 + 8.06723i 4.00000 − 6.92820i 35.7395 − 20.6342i −21.1822 − 14.1179i −9.50910 + 48.0685i 22.6274i −49.1603 + 64.3760i −58.3624 + 101.087i
11.5 −2.44949 + 1.41421i 8.60881 − 2.62456i 4.00000 − 6.92820i 3.82445 − 2.20805i −17.3755 + 18.6035i 2.10732 − 48.9547i 22.6274i 67.2234 − 45.1887i −6.24530 + 10.8172i
11.6 2.44949 − 1.41421i −8.98141 + 0.578196i 4.00000 − 6.92820i −35.7395 + 20.6342i −21.1822 + 14.1179i −9.50910 + 48.0685i − 22.6274i 80.3314 − 10.3860i −58.3624 + 101.087i
11.7 2.44949 − 1.41421i −8.13327 + 3.85357i 4.00000 − 6.92820i 30.1838 − 17.4266i −14.4726 + 20.9415i 31.6264 − 37.4269i − 22.6274i 51.3000 − 62.6842i 49.2899 − 85.3727i
11.8 2.44949 − 1.41421i −2.03147 − 8.76773i 4.00000 − 6.92820i −3.82445 + 2.20805i −17.3755 − 18.6035i 2.10732 − 48.9547i − 22.6274i −72.7463 + 35.6228i −6.24530 + 10.8172i
11.9 2.44949 − 1.41421i 6.55026 + 6.17205i 4.00000 − 6.92820i 2.28929 − 1.32172i 24.7734 + 5.85490i 47.8951 + 10.3472i − 22.6274i 4.81172 + 80.8570i 3.73839 − 6.47509i
11.10 2.44949 − 1.41421i 7.69691 − 4.66451i 4.00000 − 6.92820i 11.9899 − 6.92235i 12.2569 − 22.3107i −39.6197 + 28.8319i − 22.6274i 37.4848 − 71.8045i 19.5794 − 33.9125i
23.1 −2.44949 − 1.41421i −8.62028 + 2.58667i 4.00000 + 6.92820i −2.28929 − 1.32172i 24.7734 + 5.85490i 47.8951 − 10.3472i − 22.6274i 67.6183 − 44.5955i 3.73839 + 6.47509i
23.2 −2.44949 − 1.41421i 0.191126 + 8.99797i 4.00000 + 6.92820i −11.9899 − 6.92235i 12.2569 − 22.3107i −39.6197 − 28.8319i − 22.6274i −80.9269 + 3.43949i 19.5794 + 33.9125i
23.3 −2.44949 − 1.41421i 0.729344 − 8.97040i 4.00000 + 6.92820i −30.1838 − 17.4266i −14.4726 + 20.9415i 31.6264 + 37.4269i − 22.6274i −79.9361 − 13.0850i 49.2899 + 85.3727i
23.4 −2.44949 − 1.41421i 3.98997 − 8.06723i 4.00000 + 6.92820i 35.7395 + 20.6342i −21.1822 + 14.1179i −9.50910 − 48.0685i − 22.6274i −49.1603 − 64.3760i −58.3624 − 101.087i
23.5 −2.44949 − 1.41421i 8.60881 + 2.62456i 4.00000 + 6.92820i 3.82445 + 2.20805i −17.3755 − 18.6035i 2.10732 + 48.9547i − 22.6274i 67.2234 + 45.1887i −6.24530 − 10.8172i
23.6 2.44949 + 1.41421i −8.98141 − 0.578196i 4.00000 + 6.92820i −35.7395 − 20.6342i −21.1822 − 14.1179i −9.50910 − 48.0685i 22.6274i 80.3314 + 10.3860i −58.3624 − 101.087i
23.7 2.44949 + 1.41421i −8.13327 − 3.85357i 4.00000 + 6.92820i 30.1838 + 17.4266i −14.4726 − 20.9415i 31.6264 + 37.4269i 22.6274i 51.3000 + 62.6842i 49.2899 + 85.3727i
23.8 2.44949 + 1.41421i −2.03147 + 8.76773i 4.00000 + 6.92820i −3.82445 − 2.20805i −17.3755 + 18.6035i 2.10732 + 48.9547i 22.6274i −72.7463 − 35.6228i −6.24530 − 10.8172i
23.9 2.44949 + 1.41421i 6.55026 − 6.17205i 4.00000 + 6.92820i 2.28929 + 1.32172i 24.7734 − 5.85490i 47.8951 − 10.3472i 22.6274i 4.81172 − 80.8570i 3.73839 + 6.47509i
23.10 2.44949 + 1.41421i 7.69691 + 4.66451i 4.00000 + 6.92820i 11.9899 + 6.92235i 12.2569 + 22.3107i −39.6197 − 28.8319i 22.6274i 37.4848 + 71.8045i 19.5794 + 33.9125i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.5.h.a ✓ 20
3.b odd 2 1 inner 42.5.h.a ✓ 20
7.c even 3 1 inner 42.5.h.a ✓ 20
7.c even 3 1 294.5.b.e 10
7.d odd 6 1 294.5.b.f 10
21.g even 6 1 294.5.b.f 10
21.h odd 6 1 inner 42.5.h.a ✓ 20
21.h odd 6 1 294.5.b.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.h.a ✓ 20 1.a even 1 1 trivial
42.5.h.a ✓ 20 3.b odd 2 1 inner
42.5.h.a ✓ 20 7.c even 3 1 inner
42.5.h.a ✓ 20 21.h odd 6 1 inner
294.5.b.e 10 7.c even 3 1
294.5.b.e 10 21.h odd 6 1
294.5.b.f 10 7.d odd 6 1
294.5.b.f 10 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 64)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 79\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{5} + 81 T^{4} + \cdots + 79012524816)^{4} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 42\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 12\!\cdots\!69)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 75\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 15\!\cdots\!24)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 47\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 50\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 96\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 97\!\cdots\!09)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 35\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 44\!\cdots\!68)^{4} \) Copy content Toggle raw display
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