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Properties

Label 961.2.g.j
Level $961$
Weight $2$
Character orbit 961.g
Analytic conductor $7.674$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [961,2,Mod(235,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([26])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.235"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.g (of order \(15\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-6,-3,-14,-3,-11,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 1) q^{2} + ( - \beta_{14} - \beta_{11} - \beta_{9} + \cdots + 1) q^{3} + (\beta_{12} - \beta_{11} + \cdots - \beta_{3}) q^{4} + (2 \beta_{13} - \beta_{9} - \beta_{4} - 1) q^{5}+ \cdots + ( - 2 \beta_{15} + 3 \beta_{14} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} - 3 q^{3} - 14 q^{4} - 3 q^{5} - 11 q^{6} - 13 q^{7} + 17 q^{8} + 5 q^{9} - 17 q^{10} + 7 q^{11} + 10 q^{12} - 8 q^{13} - 21 q^{14} - 14 q^{15} - 2 q^{16} - 9 q^{17} + 12 q^{18} - 29 q^{19}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{15} - 4 \nu^{14} + 48 \nu^{13} - 65 \nu^{12} + 458 \nu^{11} - 358 \nu^{10} + 2196 \nu^{9} + \cdots + 255 ) / 186 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{15} + 4 \nu^{14} + 48 \nu^{13} + 65 \nu^{12} + 458 \nu^{11} + 358 \nu^{10} + 2196 \nu^{9} + \cdots - 255 ) / 186 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6 \nu^{15} - 10 \nu^{14} + 144 \nu^{13} - 209 \nu^{12} + 1374 \nu^{11} - 1732 \nu^{10} + 6619 \nu^{9} + \cdots - 463 ) / 186 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + \cdots + 93 ) / 186 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6 \nu^{15} + 10 \nu^{14} + 144 \nu^{13} + 209 \nu^{12} + 1374 \nu^{11} + 1732 \nu^{10} + 6619 \nu^{9} + \cdots + 463 ) / 186 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{15} + 28 \nu^{14} - 144 \nu^{13} + 517 \nu^{12} - 1374 \nu^{11} + 3653 \nu^{10} - 6619 \nu^{9} + \cdots + 354 ) / 186 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8 \nu^{15} + 33 \nu^{14} + 130 \nu^{13} + 575 \nu^{12} + 716 \nu^{11} + 3713 \nu^{10} + 1282 \nu^{9} + \cdots - 143 ) / 186 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6 \nu^{15} + 28 \nu^{14} + 144 \nu^{13} + 517 \nu^{12} + 1374 \nu^{11} + 3653 \nu^{10} + 6619 \nu^{9} + \cdots + 354 ) / 186 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36 \nu^{15} + 38 \nu^{14} - 709 \nu^{13} + 695 \nu^{12} - 5485 \nu^{11} + 4827 \nu^{10} + \cdots + 538 ) / 186 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53 \nu^{15} + 5 \nu^{14} - 962 \nu^{13} + 89 \nu^{12} - 6619 \nu^{11} + 587 \nu^{10} - 21738 \nu^{9} + \cdots + 61 ) / 186 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} + \cdots + 36 ) / 186 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 57 \nu^{15} + 21 \nu^{14} + 1058 \nu^{13} + 380 \nu^{12} + 7535 \nu^{11} + 2608 \nu^{10} + 26161 \nu^{9} + \cdots + 126 ) / 186 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27 \nu^{15} + 53 \nu^{14} + 524 \nu^{13} + 962 \nu^{12} + 3982 \nu^{11} + 6619 \nu^{10} + 15200 \nu^{9} + \cdots + 597 ) / 186 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36 \nu^{15} + 68 \nu^{14} + 709 \nu^{13} + 1229 \nu^{12} + 5485 \nu^{11} + 8411 \nu^{10} + 21362 \nu^{9} + \cdots + 470 ) / 186 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 135 \nu^{15} + 34 \nu^{14} + 2558 \nu^{13} + 599 \nu^{12} + 18763 \nu^{11} + 3942 \nu^{10} + \cdots + 359 ) / 186 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{4} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{8} + 6 \beta_{6} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 8 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 7 \beta_{9} + 12 \beta_{8} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{15} - 6 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} - 14 \beta_{11} - 7 \beta_{10} + 2 \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 19 \beta_{15} + 55 \beta_{14} - 8 \beta_{12} + 38 \beta_{11} - 19 \beta_{10} + 47 \beta_{9} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 42 \beta_{15} + 35 \beta_{14} + 44 \beta_{13} + 29 \beta_{12} + 86 \beta_{11} + 42 \beta_{10} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 145 \beta_{15} - 365 \beta_{14} + 14 \beta_{12} - 248 \beta_{11} + 145 \beta_{10} - 309 \beta_{9} + \cdots + 455 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 245 \beta_{15} - 212 \beta_{14} - 356 \beta_{13} - 128 \beta_{12} - 518 \beta_{11} - 245 \beta_{10} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1022 \beta_{15} + 2385 \beta_{14} + 34 \beta_{12} + 1625 \beta_{11} - 1022 \beta_{10} + 2000 \beta_{9} + \cdots - 2780 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1432 \beta_{15} + 1322 \beta_{14} + 2578 \beta_{13} + 518 \beta_{12} + 3129 \beta_{11} + 1432 \beta_{10} + \cdots - 1831 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6922 \beta_{15} - 15445 \beta_{14} - 619 \beta_{12} - 10601 \beta_{11} + 6922 \beta_{10} - 12821 \beta_{9} + \cdots + 17280 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8460 \beta_{15} - 8372 \beta_{14} - 17726 \beta_{13} - 1817 \beta_{12} - 19052 \beta_{11} - 8460 \beta_{10} + \cdots + 13340 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 45841 \beta_{15} + 99462 \beta_{14} + 5217 \beta_{12} + 68772 \beta_{11} - 45841 \beta_{10} + \cdots - 108434 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 50619 \beta_{15} + 53382 \beta_{14} + 118538 \beta_{13} + 4347 \beta_{12} + 116998 \beta_{11} + \cdots - 93153 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{11}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
− 1.03739i
2.52368i
− 1.42343i
1.83925i
2.16544i
− 0.176392i
− 2.16544i
0.176392i
1.03739i
− 2.52368i
1.42343i
− 1.83925i
1.14660i
− 0.333129i
− 1.14660i
0.333129i
−1.02470 + 0.744490i −1.35599 + 0.603725i −0.122284 + 0.376353i 1.90016 + 3.29117i 0.940018 − 1.62816i 1.46472 + 1.62674i −0.937688 − 2.88591i −0.533169 + 0.592145i −4.39734 − 1.95782i
235.2 −0.284315 + 0.206567i 2.64315 − 1.17681i −0.579869 + 1.78465i −1.48661 − 2.57489i −0.508398 + 0.880572i −0.724084 − 0.804176i −0.420982 − 1.29565i 3.59399 − 3.99153i 0.954554 + 0.424995i
338.1 −1.86683 + 1.35633i −0.265787 + 2.52879i 1.02738 − 3.16196i 1.24923 − 2.16373i −2.93370 − 5.08132i −1.56687 + 0.333049i 0.944583 + 2.90713i −3.38972 − 0.720507i 0.602634 + 5.73368i
338.2 0.557811 − 0.405274i −0.0943266 + 0.897457i −0.471127 + 1.44998i −1.85376 + 3.21080i 0.311099 + 0.538840i −0.746712 + 0.158719i 0.750969 + 2.31124i 2.13791 + 0.454427i 0.267207 + 2.54230i
448.1 −0.571745 + 1.75965i −0.334133 + 0.371093i −1.15144 − 0.836573i −0.603681 + 1.04561i −0.461954 − 0.800128i −0.390209 + 3.71259i −0.863288 + 0.627215i 0.287521 + 2.73558i −1.49475 − 1.66009i
448.2 0.380762 − 1.17187i −1.38843 + 1.54201i 0.389745 + 0.283166i 0.772811 − 1.33855i 1.27836 + 2.21419i 0.397601 − 3.78292i 2.47393 − 1.79742i −0.136464 − 1.29837i −1.27434 − 1.41530i
547.1 −0.571745 − 1.75965i −0.334133 − 0.371093i −1.15144 + 0.836573i −0.603681 − 1.04561i −0.461954 + 0.800128i −0.390209 − 3.71259i −0.863288 − 0.627215i 0.287521 − 2.73558i −1.49475 + 1.66009i
547.2 0.380762 + 1.17187i −1.38843 − 1.54201i 0.389745 − 0.283166i 0.772811 + 1.33855i 1.27836 − 2.21419i 0.397601 + 3.78292i 2.47393 + 1.79742i −0.136464 + 1.29837i −1.27434 + 1.41530i
732.1 −1.02470 − 0.744490i −1.35599 − 0.603725i −0.122284 − 0.376353i 1.90016 − 3.29117i 0.940018 + 1.62816i 1.46472 − 1.62674i −0.937688 + 2.88591i −0.533169 − 0.592145i −4.39734 + 1.95782i
732.2 −0.284315 − 0.206567i 2.64315 + 1.17681i −0.579869 − 1.78465i −1.48661 + 2.57489i −0.508398 − 0.880572i −0.724084 + 0.804176i −0.420982 + 1.29565i 3.59399 + 3.99153i 0.954554 − 0.424995i
816.1 −1.86683 − 1.35633i −0.265787 − 2.52879i 1.02738 + 3.16196i 1.24923 + 2.16373i −2.93370 + 5.08132i −1.56687 − 0.333049i 0.944583 − 2.90713i −3.38972 + 0.720507i 0.602634 − 5.73368i
816.2 0.557811 + 0.405274i −0.0943266 − 0.897457i −0.471127 − 1.44998i −1.85376 − 3.21080i 0.311099 − 0.538840i −0.746712 − 0.158719i 0.750969 − 2.31124i 2.13791 − 0.454427i 0.267207 − 2.54230i
844.1 −0.831304 − 2.55849i 1.38815 − 0.295061i −4.23677 + 3.07819i −0.304192 + 0.526876i −1.90889 − 3.30629i −1.57758 − 0.702385i 7.04481 + 5.11835i −0.900731 + 0.401031i 1.60088 + 0.340278i
844.2 0.640321 + 1.97070i −2.09264 + 0.444804i −1.85563 + 1.34820i −1.17396 + 2.03335i −2.21654 − 3.83916i −3.35686 − 1.49457i −0.492333 − 0.357701i 1.44066 − 0.641422i −4.75884 − 1.01152i
846.1 −0.831304 + 2.55849i 1.38815 + 0.295061i −4.23677 − 3.07819i −0.304192 − 0.526876i −1.90889 + 3.30629i −1.57758 + 0.702385i 7.04481 − 5.11835i −0.900731 − 0.401031i 1.60088 − 0.340278i
846.2 0.640321 − 1.97070i −2.09264 − 0.444804i −1.85563 − 1.34820i −1.17396 − 2.03335i −2.21654 + 3.83916i −3.35686 + 1.49457i −0.492333 + 0.357701i 1.44066 + 0.641422i −4.75884 + 1.01152i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.g.j 16
31.b odd 2 1 961.2.g.k 16
31.c even 3 1 961.2.d.n 16
31.c even 3 1 961.2.g.l 16
31.d even 5 1 961.2.c.i 16
31.d even 5 1 961.2.g.l 16
31.d even 5 1 961.2.g.m 16
31.d even 5 1 961.2.g.n 16
31.e odd 6 1 31.2.g.a ✓ 16
31.e odd 6 1 961.2.d.o 16
31.f odd 10 1 31.2.g.a ✓ 16
31.f odd 10 1 961.2.c.j 16
31.f odd 10 1 961.2.g.s 16
31.f odd 10 1 961.2.g.t 16
31.g even 15 1 961.2.a.j 8
31.g even 15 1 961.2.c.i 16
31.g even 15 1 961.2.d.n 16
31.g even 15 2 961.2.d.q 16
31.g even 15 1 inner 961.2.g.j 16
31.g even 15 1 961.2.g.m 16
31.g even 15 1 961.2.g.n 16
31.h odd 30 1 961.2.a.i 8
31.h odd 30 1 961.2.c.j 16
31.h odd 30 1 961.2.d.o 16
31.h odd 30 2 961.2.d.p 16
31.h odd 30 1 961.2.g.k 16
31.h odd 30 1 961.2.g.s 16
31.h odd 30 1 961.2.g.t 16
93.g even 6 1 279.2.y.c 16
93.k even 10 1 279.2.y.c 16
93.o odd 30 1 8649.2.a.be 8
93.p even 30 1 8649.2.a.bf 8
124.g even 6 1 496.2.bg.c 16
124.j even 10 1 496.2.bg.c 16
155.i odd 6 1 775.2.bl.a 16
155.m odd 10 1 775.2.bl.a 16
155.p even 12 2 775.2.ck.a 32
155.r even 20 2 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 31.e odd 6 1
31.2.g.a ✓ 16 31.f odd 10 1
279.2.y.c 16 93.g even 6 1
279.2.y.c 16 93.k even 10 1
496.2.bg.c 16 124.g even 6 1
496.2.bg.c 16 124.j even 10 1
775.2.bl.a 16 155.i odd 6 1
775.2.bl.a 16 155.m odd 10 1
775.2.ck.a 32 155.p even 12 2
775.2.ck.a 32 155.r even 20 2
961.2.a.i 8 31.h odd 30 1
961.2.a.j 8 31.g even 15 1
961.2.c.i 16 31.d even 5 1
961.2.c.i 16 31.g even 15 1
961.2.c.j 16 31.f odd 10 1
961.2.c.j 16 31.h odd 30 1
961.2.d.n 16 31.c even 3 1
961.2.d.n 16 31.g even 15 1
961.2.d.o 16 31.e odd 6 1
961.2.d.o 16 31.h odd 30 1
961.2.d.p 16 31.h odd 30 2
961.2.d.q 16 31.g even 15 2
961.2.g.j 16 1.a even 1 1 trivial
961.2.g.j 16 31.g even 15 1 inner
961.2.g.k 16 31.b odd 2 1
961.2.g.k 16 31.h odd 30 1
961.2.g.l 16 31.c even 3 1
961.2.g.l 16 31.d even 5 1
961.2.g.m 16 31.d even 5 1
961.2.g.m 16 31.g even 15 1
961.2.g.n 16 31.d even 5 1
961.2.g.n 16 31.g even 15 1
961.2.g.s 16 31.f odd 10 1
961.2.g.s 16 31.h odd 30 1
961.2.g.t 16 31.f odd 10 1
961.2.g.t 16 31.h odd 30 1
8649.2.a.be 8 93.o odd 30 1
8649.2.a.bf 8 93.p even 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{16} + 6 T_{2}^{15} + 29 T_{2}^{14} + 91 T_{2}^{13} + 246 T_{2}^{12} + 523 T_{2}^{11} + 1011 T_{2}^{10} + \cdots + 81 \) Copy content Toggle raw display
\( T_{3}^{16} + 3 T_{3}^{15} - T_{3}^{14} - 21 T_{3}^{13} - 48 T_{3}^{12} + 48 T_{3}^{11} + 583 T_{3}^{10} + \cdots + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 961 \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$7$ \( T^{16} + 13 T^{15} + \cdots + 68121 \) Copy content Toggle raw display
$11$ \( T^{16} - 7 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$17$ \( T^{16} + 9 T^{15} + \cdots + 74805201 \) Copy content Toggle raw display
$19$ \( T^{16} + 29 T^{15} + \cdots + 361201 \) Copy content Toggle raw display
$23$ \( T^{16} + T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$29$ \( T^{16} - 14 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 344807761 \) Copy content Toggle raw display
$41$ \( T^{16} + 8 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} - 37 T^{15} + \cdots + 7612081 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3306365001 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 366207732801 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 167728401 \) Copy content Toggle raw display
$61$ \( (T^{8} - 30 T^{7} + \cdots + 38161)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 7485883441 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 214944921 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17441907675201 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 84609661119201 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1446653267361 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 117957215601 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 7131992195241 \) Copy content Toggle raw display
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