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Properties

Label 31.2.g.a
Level $31$
Weight $2$
Character orbit 31.g
Analytic conductor $0.248$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,2,Mod(7,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 31.g (of order \(15\), degree \(8\), 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: \(0.247536246266\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 1) q^{2}+ \cdots + (\beta_{13} - \beta_{8} + \beta_{7} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{14} + \beta_{13} + \cdots - 1) q^{2}+ \cdots + (2 \beta_{15} - \beta_{14} - \beta_{13} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} + 17 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} - 12 q^{3} - 14 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} + 17 q^{8} - 10 q^{9} - 2 q^{10} - 7 q^{11} + 5 q^{12} - 7 q^{13} - 6 q^{14} + 14 q^{15} - 2 q^{16} - 6 q^{17} - 3 q^{18} + 16 q^{19} + 37 q^{20} + 9 q^{21} + 9 q^{22} + q^{23} - 20 q^{24} - 13 q^{25} + 9 q^{26} + 9 q^{27} - 30 q^{28} - 14 q^{29} - 22 q^{30} + 15 q^{31} - 42 q^{32} - 13 q^{33} - 32 q^{34} - 9 q^{35} + q^{36} - 8 q^{37} + 8 q^{38} - 3 q^{39} - q^{40} - 8 q^{41} + 69 q^{42} + 23 q^{43} + 39 q^{44} + 65 q^{45} + 34 q^{46} + 14 q^{47} + 34 q^{48} + 2 q^{49} + 3 q^{50} - 42 q^{51} + 29 q^{52} + 6 q^{53} - 46 q^{54} - 7 q^{55} - 30 q^{56} - 17 q^{57} - 15 q^{58} + 4 q^{59} - 75 q^{60} - 60 q^{61} - 25 q^{62} - 46 q^{63} + 23 q^{64} - 12 q^{65} - 30 q^{66} + 13 q^{67} + 30 q^{68} + 38 q^{69} + 12 q^{70} - 14 q^{71} + 37 q^{72} + 2 q^{73} + 13 q^{74} + 13 q^{75} - 12 q^{76} + 18 q^{77} - 15 q^{78} + 18 q^{79} + 36 q^{80} + 23 q^{81} + 14 q^{82} - 16 q^{83} + 8 q^{84} + 37 q^{85} - 26 q^{86} + 15 q^{87} - 17 q^{88} + q^{89} - 23 q^{90} + 8 q^{91} - 64 q^{92} + 17 q^{93} + 44 q^{94} - 22 q^{95} + 8 q^{96} + 3 q^{97} - 10 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) : 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 \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
− 2.16544i
0.176392i
2.16544i
− 0.176392i
1.03739i
− 2.52368i
− 1.14660i
0.333129i
− 1.42343i
1.83925i
1.42343i
− 1.83925i
1.14660i
− 0.333129i
− 1.03739i
2.52368i
−0.571745 − 1.75965i −0.488442 + 0.103822i −1.15144 + 0.836573i −0.603681 + 1.04561i 0.461954 + 0.800128i 3.41030 + 1.51837i −0.863288 − 0.627215i −2.51284 + 1.11879i 2.18505 + 0.464447i
7.2 0.380762 + 1.17187i −2.02963 + 0.431412i 0.389745 − 0.283166i 0.772811 − 1.33855i −1.27836 − 2.21419i −3.47491 − 1.54713i 2.47393 + 1.79742i 1.19265 − 0.531003i 1.86286 + 0.395962i
9.1 −0.571745 + 1.75965i −0.488442 − 0.103822i −1.15144 − 0.836573i −0.603681 − 1.04561i 0.461954 − 0.800128i 3.41030 − 1.51837i −0.863288 + 0.627215i −2.51284 − 1.11879i 2.18505 − 0.464447i
9.2 0.380762 − 1.17187i −2.02963 − 0.431412i 0.389745 + 0.283166i 0.772811 + 1.33855i −1.27836 + 2.21419i −3.47491 + 1.54713i 2.47393 − 1.79742i 1.19265 + 0.531003i 1.86286 − 0.395962i
10.1 −1.02470 − 0.744490i −0.155153 − 1.47618i −0.122284 − 0.376353i 1.90016 + 3.29117i −0.940018 + 1.62816i −2.14115 − 0.455117i −0.937688 + 2.88591i 0.779397 − 0.165666i 0.503147 − 4.78712i
10.2 −0.284315 − 0.206567i 0.302431 + 2.87744i −0.579869 − 1.78465i −1.48661 − 2.57489i 0.508398 − 0.880572i 1.05848 + 0.224987i −0.420982 + 1.29565i −5.25377 + 1.11672i −0.109221 + 1.03917i
14.1 −0.831304 + 2.55849i 0.949606 − 1.05464i −4.23677 − 3.07819i −0.304192 + 0.526876i 1.90889 + 3.30629i 0.180508 − 1.71742i 7.04481 − 5.11835i 0.103062 + 0.980572i −1.09513 − 1.21627i
14.2 0.640321 − 1.97070i −1.43153 + 1.58988i −1.85563 − 1.34820i −1.17396 + 2.03335i 2.21654 + 3.83916i 0.384094 − 3.65441i −0.492333 + 0.357701i −0.164841 − 1.56836i 3.25543 + 3.61552i
18.1 −1.86683 + 1.35633i −2.32289 + 1.03422i 1.02738 − 3.16196i 1.24923 + 2.16373i 2.93370 − 5.08132i 1.07187 + 1.19043i 0.944583 + 2.90713i 2.31884 − 2.57533i −5.26683 − 2.34494i
18.2 0.557811 − 0.405274i −0.824384 + 0.367040i −0.471127 + 1.44998i −1.85376 − 3.21080i −0.311099 + 0.538840i 0.510810 + 0.567312i 0.750969 + 2.31124i −1.46250 + 1.62427i −2.33530 − 1.03974i
19.1 −1.86683 − 1.35633i −2.32289 − 1.03422i 1.02738 + 3.16196i 1.24923 − 2.16373i 2.93370 + 5.08132i 1.07187 − 1.19043i 0.944583 − 2.90713i 2.31884 + 2.57533i −5.26683 + 2.34494i
19.2 0.557811 + 0.405274i −0.824384 − 0.367040i −0.471127 − 1.44998i −1.85376 + 3.21080i −0.311099 − 0.538840i 0.510810 − 0.567312i 0.750969 − 2.31124i −1.46250 − 1.62427i −2.33530 + 1.03974i
20.1 −0.831304 − 2.55849i 0.949606 + 1.05464i −4.23677 + 3.07819i −0.304192 − 0.526876i 1.90889 − 3.30629i 0.180508 + 1.71742i 7.04481 + 5.11835i 0.103062 − 0.980572i −1.09513 + 1.21627i
20.2 0.640321 + 1.97070i −1.43153 − 1.58988i −1.85563 + 1.34820i −1.17396 − 2.03335i 2.21654 − 3.83916i 0.384094 + 3.65441i −0.492333 − 0.357701i −0.164841 + 1.56836i 3.25543 − 3.61552i
28.1 −1.02470 + 0.744490i −0.155153 + 1.47618i −0.122284 + 0.376353i 1.90016 − 3.29117i −0.940018 − 1.62816i −2.14115 + 0.455117i −0.937688 − 2.88591i 0.779397 + 0.165666i 0.503147 + 4.78712i
28.2 −0.284315 + 0.206567i 0.302431 − 2.87744i −0.579869 + 1.78465i −1.48661 + 2.57489i 0.508398 + 0.880572i 1.05848 − 0.224987i −0.420982 − 1.29565i −5.25377 − 1.11672i −0.109221 − 1.03917i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
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 31.2.g.a ✓ 16
3.b odd 2 1 279.2.y.c 16
4.b odd 2 1 496.2.bg.c 16
5.b even 2 1 775.2.bl.a 16
5.c odd 4 2 775.2.ck.a 32
31.b odd 2 1 961.2.g.l 16
31.c even 3 1 961.2.d.o 16
31.c even 3 1 961.2.g.k 16
31.d even 5 1 961.2.c.j 16
31.d even 5 1 961.2.g.k 16
31.d even 5 1 961.2.g.s 16
31.d even 5 1 961.2.g.t 16
31.e odd 6 1 961.2.d.n 16
31.e odd 6 1 961.2.g.j 16
31.f odd 10 1 961.2.c.i 16
31.f odd 10 1 961.2.g.j 16
31.f odd 10 1 961.2.g.m 16
31.f odd 10 1 961.2.g.n 16
31.g even 15 1 inner 31.2.g.a ✓ 16
31.g even 15 1 961.2.a.i 8
31.g even 15 1 961.2.c.j 16
31.g even 15 1 961.2.d.o 16
31.g even 15 2 961.2.d.p 16
31.g even 15 1 961.2.g.s 16
31.g even 15 1 961.2.g.t 16
31.h odd 30 1 961.2.a.j 8
31.h odd 30 1 961.2.c.i 16
31.h odd 30 1 961.2.d.n 16
31.h odd 30 2 961.2.d.q 16
31.h odd 30 1 961.2.g.l 16
31.h odd 30 1 961.2.g.m 16
31.h odd 30 1 961.2.g.n 16
93.o odd 30 1 279.2.y.c 16
93.o odd 30 1 8649.2.a.bf 8
93.p even 30 1 8649.2.a.be 8
124.n odd 30 1 496.2.bg.c 16
155.u even 30 1 775.2.bl.a 16
155.w odd 60 2 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 1.a even 1 1 trivial
31.2.g.a ✓ 16 31.g even 15 1 inner
279.2.y.c 16 3.b odd 2 1
279.2.y.c 16 93.o odd 30 1
496.2.bg.c 16 4.b odd 2 1
496.2.bg.c 16 124.n odd 30 1
775.2.bl.a 16 5.b even 2 1
775.2.bl.a 16 155.u even 30 1
775.2.ck.a 32 5.c odd 4 2
775.2.ck.a 32 155.w odd 60 2
961.2.a.i 8 31.g even 15 1
961.2.a.j 8 31.h odd 30 1
961.2.c.i 16 31.f odd 10 1
961.2.c.i 16 31.h odd 30 1
961.2.c.j 16 31.d even 5 1
961.2.c.j 16 31.g even 15 1
961.2.d.n 16 31.e odd 6 1
961.2.d.n 16 31.h odd 30 1
961.2.d.o 16 31.c even 3 1
961.2.d.o 16 31.g even 15 1
961.2.d.p 16 31.g even 15 2
961.2.d.q 16 31.h odd 30 2
961.2.g.j 16 31.e odd 6 1
961.2.g.j 16 31.f odd 10 1
961.2.g.k 16 31.c even 3 1
961.2.g.k 16 31.d even 5 1
961.2.g.l 16 31.b odd 2 1
961.2.g.l 16 31.h odd 30 1
961.2.g.m 16 31.f odd 10 1
961.2.g.m 16 31.h odd 30 1
961.2.g.n 16 31.f odd 10 1
961.2.g.n 16 31.h odd 30 1
961.2.g.s 16 31.d even 5 1
961.2.g.s 16 31.g even 15 1
961.2.g.t 16 31.d even 5 1
961.2.g.t 16 31.g even 15 1
8649.2.a.be 8 93.p even 30 1
8649.2.a.bf 8 93.o odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(31, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 12 T^{15} + \cdots + 961 \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 68121 \) Copy content Toggle raw display
$11$ \( T^{16} + 7 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{16} + 7 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$17$ \( T^{16} + 6 T^{15} + \cdots + 74805201 \) Copy content Toggle raw display
$19$ \( T^{16} - 16 T^{15} + \cdots + 361201 \) 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
$31$ \( T^{16} + \cdots + 852891037441 \) 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
$43$ \( T^{16} - 23 T^{15} + \cdots + 7612081 \) 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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