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Properties

Label 961.2.c.j
Level $961$
Weight $2$
Character orbit 961.c
Analytic conductor $7.674$
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] = [961,2,Mod(439,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.c (of order \(3\), 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: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{3}+ \cdots + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{3}+ \cdots + ( - 2 \beta_{15} - \beta_{14} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 3 q^{3} + 16 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} - 18 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 3 q^{3} + 16 q^{4} - 3 q^{5} + 11 q^{6} + 2 q^{7} - 18 q^{8} - 5 q^{9} + 13 q^{10} + 18 q^{11} + 8 q^{13} + 9 q^{14} - 36 q^{15} + 8 q^{16} + 14 q^{17} - 23 q^{18} + 6 q^{19} + 7 q^{20} - q^{21} + 4 q^{22} - 44 q^{23} + 30 q^{24} - 13 q^{25} + 9 q^{26} - 36 q^{27} + 5 q^{28} - 24 q^{29} - 22 q^{30} - 42 q^{32} + 12 q^{33} - 7 q^{34} - 24 q^{35} + q^{36} - 8 q^{37} - 7 q^{38} + 2 q^{39} - 11 q^{40} + 22 q^{41} - 6 q^{42} - 2 q^{43} + 4 q^{44} - 36 q^{46} - 36 q^{47} - q^{48} + 2 q^{49} - 27 q^{50} - 2 q^{51} - q^{52} + 6 q^{53} + 14 q^{54} + 28 q^{55} - 30 q^{56} - 17 q^{57} + 10 q^{58} + 4 q^{59} + 80 q^{60} - 60 q^{61} - 46 q^{63} + 18 q^{64} + 3 q^{65} + 20 q^{66} + 13 q^{67} + 30 q^{68} - 22 q^{69} - 78 q^{70} + q^{71} - 3 q^{72} + 2 q^{73} + 8 q^{74} + 23 q^{75} + 33 q^{76} + 48 q^{77} + 8 q^{79} - 24 q^{80} + 28 q^{81} + 19 q^{82} + 39 q^{83} + 8 q^{84} + 52 q^{85} - 16 q^{86} + 15 q^{87} - 17 q^{88} - 54 q^{89} - 8 q^{90} - 32 q^{91} - 64 q^{92} + 44 q^{94} - 12 q^{95} - 12 q^{96} + 68 q^{97} - 10 q^{98} + 6 q^{99}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( -4\nu^{14} - 65\nu^{12} - 358\nu^{10} - 641\nu^{8} + 691\nu^{6} + 3382\nu^{4} + 2839\nu^{2} + 255 ) / 93 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + \cdots + 93 ) / 186 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23 \nu^{15} + 13 \nu^{14} - 397 \nu^{13} + 219 \nu^{12} - 2508 \nu^{11} + 1334 \nu^{10} + \cdots + 140 ) / 186 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9 \nu^{15} - 15 \nu^{14} - 185 \nu^{13} - 267 \nu^{12} - 1503 \nu^{11} - 1792 \nu^{10} - 6162 \nu^{9} + \cdots + 127 ) / 186 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{14} + 219\nu^{12} + 1334\nu^{10} + 3517\nu^{8} + 3497\nu^{6} - 33\nu^{4} - 1035\nu^{2} + 140 ) / 93 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{14} - 219\nu^{12} - 1334\nu^{10} - 3517\nu^{8} - 3497\nu^{6} + 33\nu^{4} + 1128\nu^{2} + 46 ) / 93 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{14} - 144\nu^{12} - 1374\nu^{10} - 6619\nu^{8} - 16835\nu^{6} - 21308\nu^{4} - 10699\nu^{2} - 532 ) / 93 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\nu^{14} + 267\nu^{12} + 1792\nu^{10} + 5713\nu^{8} + 8964\nu^{6} + 6398\nu^{4} + 1374\nu^{2} - 127 ) / 93 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49 \nu^{15} + 13 \nu^{14} + 897 \nu^{13} + 219 \nu^{12} + 6261 \nu^{11} + 1334 \nu^{10} + 21097 \nu^{9} + \cdots - 46 ) / 186 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 38 \nu^{15} + 25 \nu^{14} + 695 \nu^{13} + 445 \nu^{12} + 4827 \nu^{11} + 2966 \nu^{10} + 16025 \nu^{9} + \cdots - 36 ) / 186 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\nu^{14} + 445\nu^{12} + 2966\nu^{10} + 9222\nu^{8} + 13483\nu^{6} + 7987\nu^{4} + 926\nu^{2} - 129 ) / 93 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 61 \nu^{15} - 21 \nu^{14} + 1154 \nu^{13} - 380 \nu^{12} + 8451 \nu^{11} - 2608 \nu^{10} + 30553 \nu^{9} + \cdots - 126 ) / 186 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -28\nu^{14} - 517\nu^{12} - 3653\nu^{10} - 12516\nu^{8} - 21730\nu^{6} - 18145\nu^{4} - 6074\nu^{2} - 354 ) / 93 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 56 \nu^{15} - 53 \nu^{14} - 1127 \nu^{13} - 962 \nu^{12} - 8949 \nu^{11} - 6619 \nu^{10} + \cdots - 318 ) / 186 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 127 \nu^{15} - 43 \nu^{14} + 2304 \nu^{13} - 753 \nu^{12} + 15846 \nu^{11} - 4887 \nu^{10} + 52057 \nu^{9} + \cdots + 52 ) / 186 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{15} - \beta_{13} + 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 10 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} - 8 \beta_{11} + 12 \beta_{10} + 8 \beta_{9} + \cdots - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{8} - \beta_{7} - 7\beta_{6} - 5\beta_{5} - 2\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 54 \beta_{15} - 14 \beta_{14} - 20 \beta_{13} - 28 \beta_{12} + 40 \beta_{11} - 68 \beta_{10} + \cdots + 51 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{13} + 9\beta_{11} + 8\beta_{8} + 11\beta_{7} + 44\beta_{6} + 25\beta_{5} + 22\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 304 \beta_{15} + 88 \beta_{14} + 108 \beta_{13} + 170 \beta_{12} - 216 \beta_{11} + 386 \beta_{10} + \cdots - 265 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 28\beta_{13} - 61\beta_{11} - 56\beta_{8} - 92\beta_{7} - 273\beta_{6} - 128\beta_{5} - 178\beta _1 + 235 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1762 \beta_{15} - 552 \beta_{14} - 605 \beta_{13} - 1026 \beta_{12} + 1198 \beta_{11} - 2212 \beta_{10} + \cdots + 1393 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -265\beta_{13} + 375\beta_{11} + 385\beta_{8} + 688\beta_{7} + 1697\beta_{6} + 675\beta_{5} + 1289\beta _1 - 1417 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10442 \beta_{15} + 3496 \beta_{14} + 3473 \beta_{13} + 6284 \beta_{12} - 6748 \beta_{11} + \cdots - 7427 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2132 \beta_{13} - 2220 \beta_{11} - 2624 \beta_{8} - 4853 \beta_{7} - 10592 \beta_{6} - 3670 \beta_{5} + \cdots + 8757 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 62958 \beta_{15} - 22294 \beta_{14} - 20332 \beta_{13} - 39020 \beta_{12} + 38567 \beta_{11} + \cdots + 40302 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 15760 \beta_{13} + 12997 \beta_{11} + 17693 \beta_{8} + 33083 \beta_{7} + 66379 \beta_{6} + \cdots - 54813 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 384698 \beta_{15} + 142628 \beta_{14} + 121035 \beta_{13} + 244606 \beta_{12} - 223815 \beta_{11} + \cdots - 223094 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
439.1
− 0.333129i
− 0.176392i
− 1.42343i
1.03739i
2.16544i
− 2.52368i
1.14660i
1.83925i
0.333129i
0.176392i
1.42343i
− 1.03739i
− 2.16544i
2.52368i
− 1.14660i
− 1.83925i
−2.69016 −0.709582 + 1.22903i 5.23694 −0.304192 − 0.526876i 1.90889 − 3.30629i 0.863440 − 1.49552i −8.70786 0.492986 + 0.853878i 0.818323 + 1.41738i
439.2 −1.85021 −0.249677 + 0.432454i 1.42326 −0.603681 − 1.04561i 0.461954 − 0.800128i −1.86652 + 3.23291i 1.06708 1.37532 + 2.38213i 1.11693 + 1.93459i
439.3 −0.689493 0.451200 − 0.781502i −1.52460 −1.85376 − 3.21080i −0.311099 + 0.538840i −0.381697 + 0.661119i 2.43019 1.09284 + 1.89285i 1.27815 + 2.21383i
439.4 0.351432 1.44665 − 2.50566i −1.87650 −1.48661 − 2.57489i 0.508398 − 0.880572i 0.541063 − 0.937148i −1.36233 −2.68557 − 4.65154i −0.522445 − 0.904901i
439.5 1.23217 −1.03749 + 1.79698i −0.481752 0.772811 + 1.33855i −1.27836 + 2.21419i 1.90188 − 3.29415i −3.05795 −0.652760 − 1.13061i 0.952237 + 1.64932i
439.6 1.26660 −0.742157 + 1.28545i −0.395721 1.90016 + 3.29117i −0.940018 + 1.62816i −1.09449 + 1.89572i −3.03442 0.398405 + 0.690057i 2.40675 + 4.16861i
439.7 2.07212 1.06970 − 1.85277i 2.29369 −1.17396 − 2.03335i 2.21654 − 3.83916i 1.83727 − 3.18225i 0.608557 −0.788498 − 1.36572i −2.43258 − 4.21335i
439.8 2.30753 1.27136 − 2.20206i 3.32468 1.24923 + 2.16373i 2.93370 − 5.08132i −0.800939 + 1.38727i 3.05673 −1.73272 − 3.00116i 2.88263 + 4.99286i
521.1 −2.69016 −0.709582 − 1.22903i 5.23694 −0.304192 + 0.526876i 1.90889 + 3.30629i 0.863440 + 1.49552i −8.70786 0.492986 − 0.853878i 0.818323 − 1.41738i
521.2 −1.85021 −0.249677 − 0.432454i 1.42326 −0.603681 + 1.04561i 0.461954 + 0.800128i −1.86652 − 3.23291i 1.06708 1.37532 − 2.38213i 1.11693 − 1.93459i
521.3 −0.689493 0.451200 + 0.781502i −1.52460 −1.85376 + 3.21080i −0.311099 − 0.538840i −0.381697 − 0.661119i 2.43019 1.09284 − 1.89285i 1.27815 − 2.21383i
521.4 0.351432 1.44665 + 2.50566i −1.87650 −1.48661 + 2.57489i 0.508398 + 0.880572i 0.541063 + 0.937148i −1.36233 −2.68557 + 4.65154i −0.522445 + 0.904901i
521.5 1.23217 −1.03749 − 1.79698i −0.481752 0.772811 − 1.33855i −1.27836 − 2.21419i 1.90188 + 3.29415i −3.05795 −0.652760 + 1.13061i 0.952237 − 1.64932i
521.6 1.26660 −0.742157 − 1.28545i −0.395721 1.90016 − 3.29117i −0.940018 − 1.62816i −1.09449 − 1.89572i −3.03442 0.398405 − 0.690057i 2.40675 − 4.16861i
521.7 2.07212 1.06970 + 1.85277i 2.29369 −1.17396 + 2.03335i 2.21654 + 3.83916i 1.83727 + 3.18225i 0.608557 −0.788498 + 1.36572i −2.43258 + 4.21335i
521.8 2.30753 1.27136 + 2.20206i 3.32468 1.24923 − 2.16373i 2.93370 + 5.08132i −0.800939 − 1.38727i 3.05673 −1.73272 + 3.00116i 2.88263 − 4.99286i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 439.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.c.j 16
31.b odd 2 1 961.2.c.i 16
31.c even 3 1 961.2.a.i 8
31.c even 3 1 inner 961.2.c.j 16
31.d even 5 1 31.2.g.a ✓ 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.a.j 8
31.e odd 6 1 961.2.c.i 16
31.f odd 10 1 961.2.g.j 16
31.f odd 10 1 961.2.g.l 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 31.2.g.a ✓ 16
31.g even 15 2 961.2.d.o 16
31.g even 15 2 961.2.d.p 16
31.g even 15 1 961.2.g.k 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 2 961.2.d.n 16
31.h odd 30 2 961.2.d.q 16
31.h odd 30 1 961.2.g.j 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.g even 6 1 8649.2.a.be 8
93.h odd 6 1 8649.2.a.bf 8
93.l odd 10 1 279.2.y.c 16
93.o odd 30 1 279.2.y.c 16
124.l odd 10 1 496.2.bg.c 16
124.n odd 30 1 496.2.bg.c 16
155.n even 10 1 775.2.bl.a 16
155.s odd 20 2 775.2.ck.a 32
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 31.d even 5 1
31.2.g.a ✓ 16 31.g even 15 1
279.2.y.c 16 93.l odd 10 1
279.2.y.c 16 93.o odd 30 1
496.2.bg.c 16 124.l odd 10 1
496.2.bg.c 16 124.n odd 30 1
775.2.bl.a 16 155.n even 10 1
775.2.bl.a 16 155.u even 30 1
775.2.ck.a 32 155.s odd 20 2
775.2.ck.a 32 155.w odd 60 2
961.2.a.i 8 31.c even 3 1
961.2.a.j 8 31.e odd 6 1
961.2.c.i 16 31.b odd 2 1
961.2.c.i 16 31.e odd 6 1
961.2.c.j 16 1.a even 1 1 trivial
961.2.c.j 16 31.c even 3 1 inner
961.2.d.n 16 31.h odd 30 2
961.2.d.o 16 31.g even 15 2
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.f odd 10 1
961.2.g.j 16 31.h odd 30 1
961.2.g.k 16 31.d even 5 1
961.2.g.k 16 31.g even 15 1
961.2.g.l 16 31.f odd 10 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.g even 6 1
8649.2.a.bf 8 93.h odd 6 1

Hecke kernels

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

\( T_{2}^{8} - 2T_{2}^{7} - 10T_{2}^{6} + 23T_{2}^{5} + 19T_{2}^{4} - 63T_{2}^{3} + 15T_{2}^{2} + 27T_{2} - 9 \) Copy content Toggle raw display
\( T_{3}^{16} - 3 T_{3}^{15} + 19 T_{3}^{14} - 24 T_{3}^{13} + 142 T_{3}^{12} - 108 T_{3}^{11} + 763 T_{3}^{10} + \cdots + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} - 10 T^{6} + \cdots - 9)^{2} \) 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
$7$ \( T^{16} - 2 T^{15} + \cdots + 68121 \) Copy content Toggle raw display
$11$ \( T^{16} - 18 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{16} - 8 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$17$ \( T^{16} - 14 T^{15} + \cdots + 74805201 \) Copy content Toggle raw display
$19$ \( T^{16} - 6 T^{15} + \cdots + 361201 \) Copy content Toggle raw display
$23$ \( (T^{8} + 22 T^{7} + \cdots - 279)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 12 T^{7} + \cdots - 279)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 344807761 \) Copy content Toggle raw display
$41$ \( T^{16} - 22 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} + 2 T^{15} + \cdots + 7612081 \) Copy content Toggle raw display
$47$ \( (T^{8} + 18 T^{7} + \cdots + 57501)^{2} \) 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
$89$ \( (T^{8} + 27 T^{7} + \cdots - 343449)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 34 T^{7} + \cdots - 2670579)^{2} \) Copy content Toggle raw display
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