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Properties

Label 245.4.j.f
Level $245$
Weight $4$
Character orbit 245.j
Analytic conductor $14.455$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(79,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.79");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.j (of order \(6\), degree \(2\), not 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: \(14.4554679514\)
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: \( x^{20} - 55 x^{18} + 2042 x^{16} - 41247 x^{14} + 600234 x^{12} - 4812047 x^{10} + 27547801 x^{8} + \cdots + 12960000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2}\cdot 7^{8} \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + \beta_{11} q^{3} + (\beta_{8} - 4 \beta_1 + 4) q^{4} + ( - \beta_{13} - \beta_{10} + \beta_1) q^{5} + (\beta_{4} - 1) q^{6} + (\beta_{19} - \beta_{16} + \cdots + 3 \beta_{7}) q^{8}+ \cdots + (44 \beta_{14} + 44 \beta_{13} + \cdots - 536) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 36 q^{4} + 6 q^{5} - 24 q^{6} + 46 q^{9} - 16 q^{10} - 84 q^{11} + 16 q^{15} - 148 q^{16} + 72 q^{19} + 136 q^{20} + 72 q^{24} + 362 q^{25} - 620 q^{26} + 176 q^{29} - 52 q^{30} + 120 q^{31} - 1928 q^{34}+ \cdots - 10608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 55 x^{18} + 2042 x^{16} - 41247 x^{14} + 600234 x^{12} - 4812047 x^{10} + 27547801 x^{8} + \cdots + 12960000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23\!\cdots\!19 \nu^{18} + \cdots - 14\!\cdots\!00 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23\!\cdots\!57 \nu^{18} + \cdots - 87\!\cdots\!00 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 82\!\cdots\!27 \nu^{18} + \cdots - 37\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!13 \nu^{18} + \cdots + 99\!\cdots\!16 ) / 11\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!01 \nu^{18} + \cdots - 15\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!59 \nu^{18} + \cdots - 11\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 51\!\cdots\!89 \nu^{19} + \cdots - 30\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!31 \nu^{18} + \cdots - 81\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 66\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!43 \nu^{19} + \cdots + 39\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 76\!\cdots\!49 \nu^{19} + \cdots + 35\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73\!\cdots\!71 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 73\!\cdots\!71 \nu^{19} + \cdots + 10\!\cdots\!00 ) / 88\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 74\!\cdots\!53 \nu^{19} + \cdots + 19\!\cdots\!00 \nu ) / 63\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 84\!\cdots\!53 \nu^{19} + \cdots - 22\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 88\!\cdots\!43 \nu^{19} + \cdots + 46\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 33\!\cdots\!11 \nu^{19} + \cdots + 16\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 53\!\cdots\!33 \nu^{19} + \cdots - 26\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{11} - 13\beta_{7} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + 2 \beta_{5} + \cdots + 81 \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{19} + 25 \beta_{18} - 61 \beta_{17} + 8 \beta_{16} - 25 \beta_{15} - 4 \beta_{14} + \cdots - 241 \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -48\beta_{10} - 48\beta_{9} - 115\beta_{8} + 10\beta_{6} + 70\beta_{5} + 10\beta_{3} + 1581\beta _1 - 1581 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 204 \beta_{19} + 609 \beta_{18} - 1793 \beta_{17} - 228 \beta_{14} + 228 \beta_{13} + \cdots - 228 \beta_{9} ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1330\beta_{14} + 1330\beta_{13} + 400\beta_{6} - 1986\beta_{4} - 2425\beta_{2} - 35201 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -4536\beta_{16} + 15737\beta_{15} - 48077\beta_{11} + 8036\beta_{10} - 8036\beta_{9} + 120609\beta_{7} ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 36864 \beta_{14} + 36864 \beta_{13} + 36864 \beta_{10} + 36864 \beta_{9} + 50875 \beta_{8} + \cdots - 823061 \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 99020 \beta_{19} - 411393 \beta_{18} + 1260785 \beta_{17} - 99020 \beta_{16} + 411393 \beta_{15} + \cdots + 2841353 \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1000642 \beta_{10} + 1000642 \beta_{9} + 1083585 \beta_{8} - 346840 \beta_{6} - 1417162 \beta_{5} + \cdots + 19725241 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2186168 \beta_{19} - 10724825 \beta_{18} + 32751261 \beta_{17} + 6912884 \beta_{14} + \cdots + 6912884 \beta_{9} ) / 14 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 26678288 \beta_{14} - 26678288 \beta_{13} - 9423130 \beta_{6} + 37043350 \beta_{4} + 23587475 \beta_{2} + 480275741 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 49236364 \beta_{16} - 278156449 \beta_{15} + 845999713 \beta_{11} - 189045508 \beta_{10} + \cdots - 1659785417 \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 702074130 \beta_{14} - 702074130 \beta_{13} - 702074130 \beta_{10} - 702074130 \beta_{9} + \cdots + 11828617841 \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1132842296 \beta_{19} + 7181518137 \beta_{18} - 21770396397 \beta_{17} + \cdots - 40854349729 \beta_{7} ) / 14 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 18307346784 \beta_{10} - 18307346784 \beta_{9} - 11992550315 \beta_{8} + 6583635570 \beta_{6} + \cdots - 293848013861 ) / 7 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 26591749580 \beta_{19} + 184769275713 \beta_{18} - 558733748305 \beta_{17} + \cdots - 133280657652 \beta_{9} ) / 14 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 474310356002 \beta_{14} + 474310356002 \beta_{13} + 171388869800 \beta_{6} - 638259667162 \beta_{4} + \cdots - 7347685400361 ) / 7 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 635230140728 \beta_{16} + 4741634904985 \beta_{15} - 14312500385981 \beta_{11} + \cdots + 25351742981601 \beta_{7} ) / 14 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−3.73574 + 2.15683i
−4.37214 + 2.52426i
−1.60625 + 0.927371i
−2.31676 + 1.33758i
−0.480883 + 0.277638i
0.480883 − 0.277638i
2.31676 − 1.33758i
1.60625 − 0.927371i
4.37214 − 2.52426i
3.73574 − 2.15683i
−3.73574 − 2.15683i
−4.37214 − 2.52426i
−1.60625 − 0.927371i
−2.31676 − 1.33758i
−0.480883 − 0.277638i
0.480883 + 0.277638i
2.31676 + 1.33758i
1.60625 + 0.927371i
4.37214 + 2.52426i
3.73574 + 2.15683i
−4.60176 − 2.65683i 1.67956 − 0.969693i 10.1175 + 17.5240i 8.30643 + 7.48353i −10.3052 0 − 65.0123i −11.6194 + 20.1254i −18.3418 − 56.5062i
79.2 −3.50611 − 2.02426i −5.65297 + 3.26374i 4.19523 + 7.26634i −5.00915 − 9.99542i 26.4266 0 − 1.58074i 7.80403 − 13.5170i −2.67065 + 45.1849i
79.3 −2.47228 − 1.42737i 7.78434 − 4.49429i 0.0747741 + 0.129513i −7.11340 − 8.62551i −25.6601 0 22.4110i 26.8973 − 46.5874i 5.27451 + 31.4781i
79.4 −1.45073 − 0.837581i 2.15983 − 1.24698i −2.59692 − 4.49799i 11.1437 − 0.904354i −4.17779 0 22.1018i −10.3901 + 17.9961i −16.9240 − 8.02178i
79.5 −1.34691 − 0.777638i −4.29677 + 2.48074i −2.79056 − 4.83339i −4.33318 + 10.3065i 7.71648 0 21.1224i −1.19182 + 2.06430i 13.8511 − 10.5122i
79.6 1.34691 + 0.777638i 4.29677 − 2.48074i −2.79056 − 4.83339i −6.75908 + 8.90588i 7.71648 0 − 21.1224i −1.19182 + 2.06430i −16.0294 + 6.73929i
79.7 1.45073 + 0.837581i −2.15983 + 1.24698i −2.59692 − 4.49799i −4.78866 − 10.1029i −4.17779 0 − 22.1018i −10.3901 + 17.9961i 1.51494 − 18.6675i
79.8 2.47228 + 1.42737i −7.78434 + 4.49429i 0.0747741 + 0.129513i 11.0266 + 1.84763i −25.6601 0 − 22.4110i 26.8973 − 46.5874i 24.6236 + 20.3069i
79.9 3.50611 + 2.02426i 5.65297 − 3.26374i 4.19523 + 7.26634i 11.1609 − 0.659663i 26.4266 0 1.58074i 7.80403 − 13.5170i 40.4666 + 20.2796i
79.10 4.60176 + 2.65683i −1.67956 + 0.969693i 10.1175 + 17.5240i −10.6341 − 3.45182i −10.3052 0 65.0123i −11.6194 + 20.1254i −39.7649 − 44.1375i
214.1 −4.60176 + 2.65683i 1.67956 + 0.969693i 10.1175 − 17.5240i 8.30643 − 7.48353i −10.3052 0 65.0123i −11.6194 − 20.1254i −18.3418 + 56.5062i
214.2 −3.50611 + 2.02426i −5.65297 − 3.26374i 4.19523 − 7.26634i −5.00915 + 9.99542i 26.4266 0 1.58074i 7.80403 + 13.5170i −2.67065 − 45.1849i
214.3 −2.47228 + 1.42737i 7.78434 + 4.49429i 0.0747741 − 0.129513i −7.11340 + 8.62551i −25.6601 0 − 22.4110i 26.8973 + 46.5874i 5.27451 − 31.4781i
214.4 −1.45073 + 0.837581i 2.15983 + 1.24698i −2.59692 + 4.49799i 11.1437 + 0.904354i −4.17779 0 − 22.1018i −10.3901 − 17.9961i −16.9240 + 8.02178i
214.5 −1.34691 + 0.777638i −4.29677 − 2.48074i −2.79056 + 4.83339i −4.33318 − 10.3065i 7.71648 0 − 21.1224i −1.19182 − 2.06430i 13.8511 + 10.5122i
214.6 1.34691 − 0.777638i 4.29677 + 2.48074i −2.79056 + 4.83339i −6.75908 − 8.90588i 7.71648 0 21.1224i −1.19182 − 2.06430i −16.0294 − 6.73929i
214.7 1.45073 − 0.837581i −2.15983 − 1.24698i −2.59692 + 4.49799i −4.78866 + 10.1029i −4.17779 0 22.1018i −10.3901 − 17.9961i 1.51494 + 18.6675i
214.8 2.47228 − 1.42737i −7.78434 − 4.49429i 0.0747741 − 0.129513i 11.0266 − 1.84763i −25.6601 0 22.4110i 26.8973 + 46.5874i 24.6236 − 20.3069i
214.9 3.50611 − 2.02426i 5.65297 + 3.26374i 4.19523 − 7.26634i 11.1609 + 0.659663i 26.4266 0 − 1.58074i 7.80403 + 13.5170i 40.4666 − 20.2796i
214.10 4.60176 − 2.65683i −1.67956 − 0.969693i 10.1175 − 17.5240i −10.6341 + 3.45182i −10.3052 0 − 65.0123i −11.6194 − 20.1254i −39.7649 + 44.1375i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.j.f 20
5.b even 2 1 inner 245.4.j.f 20
7.b odd 2 1 245.4.j.e 20
7.c even 3 1 245.4.b.d 10
7.c even 3 1 inner 245.4.j.f 20
7.d odd 6 1 35.4.b.a ✓ 10
7.d odd 6 1 245.4.j.e 20
21.g even 6 1 315.4.d.c 10
28.f even 6 1 560.4.g.f 10
35.c odd 2 1 245.4.j.e 20
35.i odd 6 1 35.4.b.a ✓ 10
35.i odd 6 1 245.4.j.e 20
35.j even 6 1 245.4.b.d 10
35.j even 6 1 inner 245.4.j.f 20
35.k even 12 1 175.4.a.i 5
35.k even 12 1 175.4.a.j 5
35.l odd 12 1 1225.4.a.be 5
35.l odd 12 1 1225.4.a.bh 5
105.p even 6 1 315.4.d.c 10
105.w odd 12 1 1575.4.a.bn 5
105.w odd 12 1 1575.4.a.bq 5
140.s even 6 1 560.4.g.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.b.a ✓ 10 7.d odd 6 1
35.4.b.a ✓ 10 35.i odd 6 1
175.4.a.i 5 35.k even 12 1
175.4.a.j 5 35.k even 12 1
245.4.b.d 10 7.c even 3 1
245.4.b.d 10 35.j even 6 1
245.4.j.e 20 7.b odd 2 1
245.4.j.e 20 7.d odd 6 1
245.4.j.e 20 35.c odd 2 1
245.4.j.e 20 35.i odd 6 1
245.4.j.f 20 1.a even 1 1 trivial
245.4.j.f 20 5.b even 2 1 inner
245.4.j.f 20 7.c even 3 1 inner
245.4.j.f 20 35.j even 6 1 inner
315.4.d.c 10 21.g even 6 1
315.4.d.c 10 105.p even 6 1
560.4.g.f 10 28.f even 6 1
560.4.g.f 10 140.s even 6 1
1225.4.a.be 5 35.l odd 12 1
1225.4.a.bh 5 35.l odd 12 1
1575.4.a.bn 5 105.w odd 12 1
1575.4.a.bq 5 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{20} - 58 T_{2}^{18} + 2255 T_{2}^{16} - 47426 T_{2}^{14} + 714581 T_{2}^{12} - 6457776 T_{2}^{10} + \cdots + 655360000 \) Copy content Toggle raw display
\( T_{19}^{10} - 36 T_{19}^{9} + 18646 T_{19}^{8} - 493432 T_{19}^{7} + 295286596 T_{19}^{6} + \cdots + 405342520934400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 655360000 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 3930163511296 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 124078925021184)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 405342520934400)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{5} - 44 T^{4} + \cdots - 126081243400)^{4} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{5} - 426 T^{4} + \cdots - 109849343072)^{4} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 63\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 7767441797120)^{4} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 78\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 78\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
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