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Properties

Label 40.5.e.c
Level $40$
Weight $5$
Character orbit 40.e
Analytic conductor $4.135$
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] = [40,5,Mod(19,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 40.e (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: \(4.13479852335\)
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: \( x^{20} + 10 x^{18} - 20 x^{16} + 2640 x^{14} + 22400 x^{12} - 652288 x^{10} + 5734400 x^{8} + \cdots + 1099511627776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{33}\cdot 5^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{11} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{4} + 1) q^{6} + (\beta_{15} + 2 \beta_1) q^{7} + (\beta_{11} + \beta_{10} - \beta_1) q^{8} + ( - \beta_{6} - \beta_{3} - 2 \beta_{2} - 32) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{11} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{5} - \beta_1) q^{5} + (\beta_{4} + 1) q^{6} + (\beta_{15} + 2 \beta_1) q^{7} + (\beta_{11} + \beta_{10} - \beta_1) q^{8} + ( - \beta_{6} - \beta_{3} - 2 \beta_{2} - 32) q^{9} + ( - \beta_{16} - \beta_{11} + \cdots - 12) q^{10}+ \cdots + ( - 15 \beta_{9} - 25 \beta_{7} + \cdots - 3074) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 20 q^{4} + 12 q^{6} - 652 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 20 q^{4} + 12 q^{6} - 652 q^{9} - 240 q^{10} - 168 q^{11} + 660 q^{14} + 280 q^{16} + 728 q^{19} + 900 q^{20} - 1232 q^{24} - 1420 q^{25} + 4680 q^{26} + 1540 q^{30} - 2312 q^{34} + 4440 q^{35} - 7868 q^{36} - 4640 q^{40} - 4728 q^{41} + 1368 q^{44} - 6260 q^{46} + 540 q^{49} - 2280 q^{50} + 4352 q^{51} + 10688 q^{54} + 12960 q^{56} + 8280 q^{59} - 3480 q^{60} - 19040 q^{64} + 9480 q^{65} + 2632 q^{66} - 11020 q^{70} - 12000 q^{74} - 16000 q^{75} + 22472 q^{76} + 22440 q^{80} - 10956 q^{81} + 29000 q^{84} + 7740 q^{86} + 22248 q^{89} + 28520 q^{90} - 39760 q^{91} + 8540 q^{94} - 1328 q^{96} - 62504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 10 x^{18} - 20 x^{16} + 2640 x^{14} + 22400 x^{12} - 652288 x^{10} + 5734400 x^{8} + \cdots + 1099511627776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 147 \nu^{18} - 22274 \nu^{16} - 72444 \nu^{14} + 5950192 \nu^{12} - 28671360 \nu^{10} + \cdots + 26585847562240 ) / 3060164198400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 243 \nu^{18} - 2690 \nu^{16} + 13572 \nu^{14} + 1833456 \nu^{12} + 757376 \nu^{10} + \cdots + 15238007095296 ) / 765041049600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 513 \nu^{19} + 4288 \nu^{18} + 15882 \nu^{17} + 45952 \nu^{16} - 566292 \nu^{15} + \cdots + 905241667043328 ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 291 \nu^{18} - 3678 \nu^{16} - 96068 \nu^{14} - 2088176 \nu^{12} - 10593920 \nu^{10} + \cdots - 18191870853120 ) / 765041049600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1053 \nu^{18} - 4158 \nu^{16} + 120828 \nu^{14} + 194192 \nu^{12} - 91219072 \nu^{10} + \cdots + 49190260441088 ) / 1530082099200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1387 \nu^{19} - 4288 \nu^{18} + 15918 \nu^{17} - 45952 \nu^{16} - 1801308 \nu^{15} + \cdots - 905241667043328 ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{18} - 10 \nu^{16} + 20 \nu^{14} - 2640 \nu^{12} - 22400 \nu^{10} + 652288 \nu^{8} + \cdots - 38654705664 ) / 1073741824 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1863 \nu^{19} - 12474 \nu^{17} + 307060 \nu^{15} + 3181104 \nu^{13} + \cdots + 199853418217472 \nu ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1863 \nu^{19} + 12474 \nu^{17} - 307060 \nu^{15} - 3181104 \nu^{13} + \cdots - 101928163868672 \nu ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 513 \nu^{19} - 156384 \nu^{18} + 15882 \nu^{17} + 2230080 \nu^{16} - 566292 \nu^{15} + \cdots - 51\!\cdots\!28 ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 247 \nu^{19} - 1072 \nu^{18} + 33318 \nu^{17} - 11488 \nu^{16} + 373172 \nu^{15} + \cdots - 226310416760832 ) / 12240656793600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3851 \nu^{19} + 6464 \nu^{18} + 172946 \nu^{17} - 601984 \nu^{16} + \cdots - 27\!\cdots\!16 ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4579 \nu^{19} - 195234 \nu^{17} + 1282884 \nu^{15} + 21725936 \nu^{13} + \cdots + 315525477433344 \nu ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 6151 \nu^{19} + 10752 \nu^{18} + 33478 \nu^{17} - 556032 \nu^{16} + 6346868 \nu^{15} + \cdots - 18\!\cdots\!88 ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( \nu^{19} + 10 \nu^{17} - 20 \nu^{15} + 2640 \nu^{13} + 22400 \nu^{11} - 652288 \nu^{9} + \cdots + 42949672960 \nu ) / 8589934592 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15731 \nu^{19} - 4288 \nu^{18} - 224130 \nu^{17} - 45952 \nu^{16} - 6201596 \nu^{15} + \cdots - 905241667043328 ) / 97925254348800 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 15107 \nu^{19} + 109154 \nu^{17} - 1871556 \nu^{15} - 42514672 \nu^{13} + \cdots - 706221472481280 \nu ) / 48962627174400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} - 2 \beta_{9} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{19} + 2 \beta_{18} - 4 \beta_{17} + 4 \beta_{16} - 16 \beta_{15} - 4 \beta_{14} + \cdots + 16 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{18} + 16 \beta_{17} - 14 \beta_{16} + 16 \beta_{15} - 14 \beta_{14} - 2 \beta_{13} + \cdots - 928 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{19} + 68 \beta_{18} + 8 \beta_{17} + 48 \beta_{16} + 80 \beta_{15} - 48 \beta_{14} + \cdots - 900 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 64 \beta_{18} - 64 \beta_{17} + 80 \beta_{16} - 64 \beta_{15} + 80 \beta_{14} - 16 \beta_{13} + \cdots + 4144 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1504 \beta_{19} - 544 \beta_{18} - 1568 \beta_{17} - 976 \beta_{16} - 128 \beta_{15} + \cdots + 4376 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2880 \beta_{18} - 2880 \beta_{17} + 3960 \beta_{16} - 2880 \beta_{15} + 3960 \beta_{14} - 1080 \beta_{13} + \cdots + 264160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 944 \beta_{19} - 18448 \beta_{18} + 51488 \beta_{17} - 9056 \beta_{16} + 49728 \beta_{15} + \cdots + 225088 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9856 \beta_{18} + 9856 \beta_{17} - 43920 \beta_{16} + 9856 \beta_{15} - 43920 \beta_{14} + \cdots - 4392704 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 40288 \beta_{19} - 67104 \beta_{18} - 199232 \beta_{17} + 215168 \beta_{16} - 369024 \beta_{15} + \cdots - 4058464 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 702976 \beta_{18} + 702976 \beta_{17} + 1010944 \beta_{16} + 702976 \beta_{15} + 1010944 \beta_{14} + \cdots - 46263168 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 684032 \beta_{19} - 6868992 \beta_{18} + 2276096 \beta_{17} + 5972096 \beta_{16} + 3279872 \beta_{15} + \cdots - 52818624 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 12640768 \beta_{18} + 12640768 \beta_{17} - 21323456 \beta_{16} + 12640768 \beta_{15} + \cdots - 639727360 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 47831936 \beta_{19} - 40203648 \beta_{18} + 236149504 \beta_{17} - 1275136 \beta_{16} + \cdots - 751032320 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 207352832 \beta_{18} - 207352832 \beta_{17} + 220147840 \beta_{16} - 207352832 \beta_{15} + \cdots - 21890260992 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 1040956672 \beta_{19} - 235767552 \beta_{18} + 5270000128 \beta_{17} - 1836940288 \beta_{16} + \cdots - 25911761152 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−3.92076 − 0.792224i
−3.92076 + 0.792224i
−3.42328 − 2.06909i
−3.42328 + 2.06909i
−2.91588 − 2.73819i
−2.91588 + 2.73819i
−1.28868 − 3.78673i
−1.28868 + 3.78673i
−0.495691 − 3.96917i
−0.495691 + 3.96917i
0.495691 − 3.96917i
0.495691 + 3.96917i
1.28868 − 3.78673i
1.28868 + 3.78673i
2.91588 − 2.73819i
2.91588 + 2.73819i
3.42328 − 2.06909i
3.42328 + 2.06909i
3.92076 − 0.792224i
3.92076 + 0.792224i
−3.92076 − 0.792224i − 16.4147i 14.7448 + 6.21224i 15.0565 − 19.9575i −13.0041 + 64.3580i −3.44277 −52.8892 − 36.0379i −188.441 −74.8439 + 66.3203i
19.2 −3.92076 + 0.792224i 16.4147i 14.7448 − 6.21224i 15.0565 + 19.9575i −13.0041 − 64.3580i −3.44277 −52.8892 + 36.0379i −188.441 −74.8439 − 66.3203i
19.3 −3.42328 − 2.06909i 9.05347i 7.43775 + 14.1662i −4.74833 − 24.5449i 18.7324 − 30.9926i −65.1778 3.84950 − 63.8841i −0.965298 −34.5307 + 93.8490i
19.4 −3.42328 + 2.06909i − 9.05347i 7.43775 − 14.1662i −4.74833 + 24.5449i 18.7324 + 30.9926i −65.1778 3.84950 + 63.8841i −0.965298 −34.5307 − 93.8490i
19.5 −2.91588 − 2.73819i − 1.21614i 1.00465 + 15.9684i 13.8839 + 20.7903i −3.33001 + 3.54611i −1.36246 40.7951 − 49.3129i 79.5210 16.4443 − 98.6387i
19.6 −2.91588 + 2.73819i 1.21614i 1.00465 − 15.9684i 13.8839 − 20.7903i −3.33001 − 3.54611i −1.36246 40.7951 + 49.3129i 79.5210 16.4443 + 98.6387i
19.7 −1.28868 − 3.78673i − 10.5360i −12.6786 + 9.75979i −24.9415 − 1.70938i −39.8969 + 13.5776i 26.0178 53.2964 + 35.4330i −30.0072 25.6688 + 96.6494i
19.8 −1.28868 + 3.78673i 10.5360i −12.6786 − 9.75979i −24.9415 + 1.70938i −39.8969 − 13.5776i 26.0178 53.2964 − 35.4330i −30.0072 25.6688 − 96.6494i
19.9 −0.495691 − 3.96917i 10.2033i −15.5086 + 3.93496i 17.9141 − 17.4381i 40.4986 − 5.05768i 84.8600 23.3060 + 59.6056i −23.1073 −78.0944 − 62.4601i
19.10 −0.495691 + 3.96917i − 10.2033i −15.5086 − 3.93496i 17.9141 + 17.4381i 40.4986 + 5.05768i 84.8600 23.3060 − 59.6056i −23.1073 −78.0944 + 62.4601i
19.11 0.495691 − 3.96917i 10.2033i −15.5086 − 3.93496i −17.9141 + 17.4381i 40.4986 + 5.05768i −84.8600 −23.3060 + 59.6056i −23.1073 60.3347 + 79.7478i
19.12 0.495691 + 3.96917i − 10.2033i −15.5086 + 3.93496i −17.9141 − 17.4381i 40.4986 − 5.05768i −84.8600 −23.3060 − 59.6056i −23.1073 60.3347 − 79.7478i
19.13 1.28868 − 3.78673i − 10.5360i −12.6786 − 9.75979i 24.9415 + 1.70938i −39.8969 − 13.5776i −26.0178 −53.2964 + 35.4330i −30.0072 38.6147 − 92.2437i
19.14 1.28868 + 3.78673i 10.5360i −12.6786 + 9.75979i 24.9415 − 1.70938i −39.8969 + 13.5776i −26.0178 −53.2964 − 35.4330i −30.0072 38.6147 + 92.2437i
19.15 2.91588 − 2.73819i − 1.21614i 1.00465 − 15.9684i −13.8839 − 20.7903i −3.33001 − 3.54611i 1.36246 −40.7951 − 49.3129i 79.5210 −97.4115 − 22.6054i
19.16 2.91588 + 2.73819i 1.21614i 1.00465 + 15.9684i −13.8839 + 20.7903i −3.33001 + 3.54611i 1.36246 −40.7951 + 49.3129i 79.5210 −97.4115 + 22.6054i
19.17 3.42328 − 2.06909i 9.05347i 7.43775 − 14.1662i 4.74833 + 24.5449i 18.7324 + 30.9926i 65.1778 −3.84950 − 63.8841i −0.965298 67.0405 + 74.1995i
19.18 3.42328 + 2.06909i − 9.05347i 7.43775 + 14.1662i 4.74833 − 24.5449i 18.7324 − 30.9926i 65.1778 −3.84950 + 63.8841i −0.965298 67.0405 − 74.1995i
19.19 3.92076 − 0.792224i − 16.4147i 14.7448 − 6.21224i −15.0565 + 19.9575i −13.0041 − 64.3580i 3.44277 52.8892 − 36.0379i −188.441 −43.2224 + 90.1766i
19.20 3.92076 + 0.792224i 16.4147i 14.7448 + 6.21224i −15.0565 − 19.9575i −13.0041 + 64.3580i 3.44277 52.8892 + 36.0379i −188.441 −43.2224 − 90.1766i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.5.e.c ✓ 20
4.b odd 2 1 160.5.e.c 20
5.b even 2 1 inner 40.5.e.c ✓ 20
5.c odd 4 2 200.5.g.i 20
8.b even 2 1 160.5.e.c 20
8.d odd 2 1 inner 40.5.e.c ✓ 20
20.d odd 2 1 160.5.e.c 20
20.e even 4 2 800.5.g.i 20
40.e odd 2 1 inner 40.5.e.c ✓ 20
40.f even 2 1 160.5.e.c 20
40.i odd 4 2 800.5.g.i 20
40.k even 4 2 200.5.g.i 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.5.e.c ✓ 20 1.a even 1 1 trivial
40.5.e.c ✓ 20 5.b even 2 1 inner
40.5.e.c ✓ 20 8.d odd 2 1 inner
40.5.e.c ✓ 20 40.e odd 2 1 inner
160.5.e.c 20 4.b odd 2 1
160.5.e.c 20 8.b even 2 1
160.5.e.c 20 20.d odd 2 1
160.5.e.c 20 40.f even 2 1
200.5.g.i 20 5.c odd 4 2
200.5.g.i 20 40.k even 4 2
800.5.g.i 20 20.e even 4 2
800.5.g.i 20 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(40, [\chi])\):

\( T_{3}^{10} + 568T_{3}^{8} + 110072T_{3}^{6} + 8973408T_{3}^{4} + 268259472T_{3}^{2} + 377478144 \) Copy content Toggle raw display
\( T_{7}^{10} - 12140T_{7}^{8} + 38508440T_{7}^{6} - 21234252160T_{7}^{4} + 284733514000T_{7}^{2} - 455625000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 1099511627776 \) Copy content Toggle raw display
$3$ \( (T^{10} + 568 T^{8} + \cdots + 377478144)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 455625000000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 42 T^{4} + \cdots - 14097087072)^{4} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 182 T^{4} + \cdots - 73451402848)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 439327012557312)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 25\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 23\!\cdots\!52)^{4} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 26\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 97\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 40\!\cdots\!28)^{4} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
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