Properties

Label 363.3.g.g
Level $363$
Weight $3$
Character orbit 363.g
Analytic conductor $9.891$
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] = [363,3,Mod(40,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.g (of order \(10\), degree \(4\), 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: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + 1868 x^{7} - 1619 x^{6} - 16804 x^{5} + 32427 x^{4} + 43316 x^{3} + \cdots + 83521 \) 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 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{2} - \beta_{15} q^{3} + ( - \beta_{15} - \beta_{14} + 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{9} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{7} - 3 \beta_{5} + \cdots - 1) q^{7}+ \cdots - 3 \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + \beta_1 + 1) q^{2} - \beta_{15} q^{3} + ( - \beta_{15} - \beta_{14} + 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{5} + (\beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{9} + \beta_{6} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{6} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{7} - 3 \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - 47 \beta_{15} - 20 \beta_{14} - 5 \beta_{13} + 5 \beta_{12} + 5 \beta_{11} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{2} - 10 q^{4} + 6 q^{5} - 20 q^{7} - 50 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{2} - 10 q^{4} + 6 q^{5} - 20 q^{7} - 50 q^{8} - 12 q^{9} - 24 q^{12} + 10 q^{13} + 28 q^{14} + 6 q^{15} + 6 q^{16} - 50 q^{17} + 70 q^{19} + 12 q^{20} + 132 q^{23} - 42 q^{25} - 44 q^{26} + 90 q^{28} - 80 q^{29} + 120 q^{30} - 30 q^{31} - 368 q^{34} - 170 q^{35} - 30 q^{36} + 134 q^{37} - 10 q^{38} + 120 q^{39} + 370 q^{40} - 150 q^{41} + 186 q^{42} - 12 q^{45} + 80 q^{46} + 110 q^{47} + 24 q^{48} - 140 q^{49} - 350 q^{50} + 90 q^{51} + 40 q^{52} - 278 q^{53} + 524 q^{56} + 240 q^{57} - 220 q^{58} + 156 q^{60} + 260 q^{61} - 770 q^{62} + 60 q^{63} + 172 q^{64} + 36 q^{67} - 290 q^{68} - 120 q^{69} - 290 q^{70} - 86 q^{71} + 120 q^{72} + 140 q^{73} - 700 q^{74} + 252 q^{75} - 312 q^{78} + 380 q^{79} + 674 q^{80} - 36 q^{81} + 124 q^{82} - 620 q^{83} + 540 q^{84} + 450 q^{85} - 774 q^{86} + 76 q^{89} + 120 q^{90} + 6 q^{91} + 90 q^{92} + 24 q^{93} + 330 q^{94} - 550 q^{95} + 360 q^{96} + 246 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + 1868 x^{7} - 1619 x^{6} - 16804 x^{5} + 32427 x^{4} + 43316 x^{3} + \cdots + 83521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 138203247537734 \nu^{15} - 291050501651580 \nu^{14} - 300665341738367 \nu^{13} + \cdots - 10\!\cdots\!14 ) / 10\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1196620054805 \nu^{15} + 7612440593330 \nu^{14} - 13694873409440 \nu^{13} + 37431650235165 \nu^{12} + \cdots + 87\!\cdots\!85 ) / 78\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79\!\cdots\!69 \nu^{15} + \cdots + 55\!\cdots\!82 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!15 \nu^{15} + \cdots - 57\!\cdots\!30 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!90 \nu^{15} + \cdots - 14\!\cdots\!47 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!48 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!52 \nu^{15} + \cdots - 12\!\cdots\!26 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!14 \nu^{15} + \cdots + 29\!\cdots\!54 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!12 \nu^{15} + \cdots + 19\!\cdots\!73 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31\!\cdots\!70 \nu^{15} + \cdots - 62\!\cdots\!81 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!33 \nu^{15} + \cdots - 91\!\cdots\!68 ) / 24\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35\!\cdots\!76 \nu^{15} + \cdots - 19\!\cdots\!69 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 47\!\cdots\!48 \nu^{15} + \cdots + 91\!\cdots\!78 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 57\!\cdots\!67 \nu^{15} + \cdots + 27\!\cdots\!46 ) / 41\!\cdots\!94 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} + \beta_{10} + 3\beta_{8} - 3\beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} + 6 \beta_{6} - 11 \beta_{5} - 9 \beta_{4} - 2 \beta_{2} - 3 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} - 9 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} - 27 \beta_{8} - 27 \beta_{7} + 13 \beta_{6} + 29 \beta_{5} - 19 \beta_{4} - 9 \beta_{3} - 13 \beta_{2} - 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{15} - 36 \beta_{14} + 13 \beta_{13} - 13 \beta_{12} - 9 \beta_{11} + 23 \beta_{10} + 9 \beta_{9} + 34 \beta_{8} - 10 \beta_{7} - 21 \beta_{6} + 61 \beta_{5} + 42 \beta_{4} - 44 \beta_{3} + 42 \beta_{2} - 30 \beta _1 - 73 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + 7 \beta_{14} + 44 \beta_{13} - 76 \beta_{12} - 44 \beta_{11} - 142 \beta_{10} - 65 \beta_{8} + 209 \beta_{7} + 98 \beta_{3} + 105 \beta_{2} - 191 \beta _1 + 209 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 449 \beta_{15} + 59 \beta_{14} - 59 \beta_{13} + 157 \beta_{12} + 98 \beta_{11} - 504 \beta_{10} + 59 \beta_{9} - 551 \beta_{8} + 489 \beta_{7} - 143 \beta_{6} + 710 \beta_{5} + 257 \beta_{4} + 390 \beta_{3} - 59 \beta_{2} + 316 \beta _1 - 59 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11 \beta_{15} + 378 \beta_{14} + 602 \beta_{12} + 390 \beta_{11} + 591 \beta_{10} + 602 \beta_{9} + 2340 \beta_{8} + 1312 \beta_{7} - 1589 \beta_{6} - 434 \beta_{5} + 2901 \beta_{4} + 710 \beta_{2} + 1312 \beta _1 + 1028 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2504 \beta_{15} + 4164 \beta_{14} - 201 \beta_{13} - 367 \beta_{9} + 3660 \beta_{8} + 3660 \beta_{7} + 635 \beta_{6} - 12024 \beta_{5} - 1320 \beta_{4} + 4164 \beta_{3} - 635 \beta_{2} + 11389 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2504 \beta_{15} + 4696 \beta_{14} - 6668 \beta_{13} + 6668 \beta_{12} + 4164 \beta_{11} + 1972 \beta_{10} - 4164 \beta_{9} - 15494 \beta_{8} - 18735 \beta_{7} + 9866 \beta_{6} - 4121 \beta_{5} - 18692 \beta_{4} + \cdots - 11612 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22394 \beta_{15} - 22394 \beta_{14} - 3312 \beta_{13} + 5504 \beta_{12} + 3312 \beta_{11} + 67722 \beta_{10} + 42552 \beta_{8} - 76812 \beta_{7} - 64410 \beta_{3} - 7433 \beta_{2} + 12584 \beta _1 - 76812 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 102120 \beta_{15} - 39824 \beta_{14} + 39824 \beta_{13} - 104234 \beta_{12} - 64410 \beta_{11} + 35457 \beta_{10} - 39824 \beta_{9} - 2325 \beta_{8} - 69559 \beta_{7} + 93965 \beta_{6} + \cdots + 39824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 300533 \beta_{15} - 247371 \beta_{14} - 99867 \beta_{12} - 62296 \beta_{11} - 400400 \beta_{10} - 99867 \beta_{9} - 905140 \beta_{8} - 209826 \beta_{7} + 241814 \beta_{6} + 945646 \beta_{5} + \cdots - 695314 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 594635 \beta_{15} - 966540 \beta_{14} + 338104 \beta_{13} + 547904 \beta_{9} + 786321 \beta_{8} + 786321 \beta_{7} - 1283750 \beta_{6} + 2729161 \beta_{5} + 2417009 \beta_{4} + \cdots - 1445411 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 594635 \beta_{15} + 1832695 \beta_{14} + 1561175 \beta_{13} - 1561175 \beta_{12} - 966540 \beta_{11} - 3393870 \beta_{10} + 966540 \beta_{9} + 3573048 \beta_{8} + 9449472 \beta_{7} + \cdots + 11063785 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−1.95510 + 0.109518i
−1.29715 0.104262i
0.988132 0.846795i
1.64608 1.06057i
−1.43448 2.82504i
−0.797732 1.94863i
1.60675 + 1.36085i
2.24350 + 2.23726i
−1.43448 + 2.82504i
−0.797732 + 1.94863i
1.60675 1.36085i
2.24350 2.23726i
−1.95510 0.109518i
−1.29715 + 0.104262i
0.988132 + 0.846795i
1.64608 + 1.06057i
−1.45510 + 2.00277i 0.535233 1.64728i −0.657709 2.02422i 2.06583 1.50091i 2.52030 + 3.46890i −0.162060 + 0.0526565i −4.40651 1.43176i −2.42705 1.76336i 6.32135i
40.2 −0.797149 + 1.09718i −0.535233 + 1.64728i 0.667707 + 2.05499i −1.85613 + 1.34856i −1.38070 1.90037i −9.29312 + 3.01952i −7.94622 2.58188i −2.42705 1.76336i 3.11151i
40.3 1.48813 2.04824i −0.535233 + 1.64728i −0.744674 2.29187i 6.83369 4.96497i 2.57752 + 3.54765i 2.32099 0.754134i 3.82892 + 1.24409i −2.42705 1.76336i 21.3855i
40.4 2.14608 2.95383i 0.535233 1.64728i −2.88336 8.87407i −4.42535 + 3.21521i −3.71712 5.11618i −6.81008 + 2.21273i −18.5106 6.01447i −2.42705 1.76336i 19.9718i
94.1 −0.934478 + 0.303630i 1.40126 + 1.01807i −2.45501 + 1.78367i 0.122858 0.378117i −1.61856 0.525903i −4.24781 5.84661i 4.06274 5.59188i 0.927051 + 2.85317i 0.390645i
94.2 −0.297732 + 0.0967389i −1.40126 1.01807i −3.15678 + 2.29354i −2.29149 + 7.05247i 0.515686 + 0.167557i 5.89378 + 8.11209i 1.45403 2.00130i 0.927051 + 2.85317i 2.32142i
94.3 2.10675 0.684524i −1.40126 1.01807i 0.733748 0.533099i 2.68691 8.26946i −3.64900 1.18563i −3.92164 5.39767i −4.02726 + 5.54305i 0.927051 + 2.85317i 19.2609i
94.4 2.74350 0.891416i 1.40126 + 1.01807i 3.49608 2.54005i −0.136315 + 0.419536i 4.75187 + 1.54398i 6.21995 + 8.56103i 0.544939 0.750044i 0.927051 + 2.85317i 1.27251i
112.1 −0.934478 0.303630i 1.40126 1.01807i −2.45501 1.78367i 0.122858 + 0.378117i −1.61856 + 0.525903i −4.24781 + 5.84661i 4.06274 + 5.59188i 0.927051 2.85317i 0.390645i
112.2 −0.297732 0.0967389i −1.40126 + 1.01807i −3.15678 2.29354i −2.29149 7.05247i 0.515686 0.167557i 5.89378 8.11209i 1.45403 + 2.00130i 0.927051 2.85317i 2.32142i
112.3 2.10675 + 0.684524i −1.40126 + 1.01807i 0.733748 + 0.533099i 2.68691 + 8.26946i −3.64900 + 1.18563i −3.92164 + 5.39767i −4.02726 5.54305i 0.927051 2.85317i 19.2609i
112.4 2.74350 + 0.891416i 1.40126 1.01807i 3.49608 + 2.54005i −0.136315 0.419536i 4.75187 1.54398i 6.21995 8.56103i 0.544939 + 0.750044i 0.927051 2.85317i 1.27251i
118.1 −1.45510 2.00277i 0.535233 + 1.64728i −0.657709 + 2.02422i 2.06583 + 1.50091i 2.52030 3.46890i −0.162060 0.0526565i −4.40651 + 1.43176i −2.42705 + 1.76336i 6.32135i
118.2 −0.797149 1.09718i −0.535233 1.64728i 0.667707 2.05499i −1.85613 1.34856i −1.38070 + 1.90037i −9.29312 3.01952i −7.94622 + 2.58188i −2.42705 + 1.76336i 3.11151i
118.3 1.48813 + 2.04824i −0.535233 1.64728i −0.744674 + 2.29187i 6.83369 + 4.96497i 2.57752 3.54765i 2.32099 + 0.754134i 3.82892 1.24409i −2.42705 + 1.76336i 21.3855i
118.4 2.14608 + 2.95383i 0.535233 + 1.64728i −2.88336 + 8.87407i −4.42535 3.21521i −3.71712 + 5.11618i −6.81008 2.21273i −18.5106 + 6.01447i −2.42705 + 1.76336i 19.9718i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.g.g 16
11.b odd 2 1 363.3.g.a 16
11.c even 5 1 33.3.g.a 16
11.c even 5 1 363.3.c.e 16
11.c even 5 1 363.3.g.a 16
11.c even 5 1 363.3.g.f 16
11.d odd 10 1 33.3.g.a 16
11.d odd 10 1 363.3.c.e 16
11.d odd 10 1 363.3.g.f 16
11.d odd 10 1 inner 363.3.g.g 16
33.f even 10 1 99.3.k.c 16
33.f even 10 1 1089.3.c.m 16
33.h odd 10 1 99.3.k.c 16
33.h odd 10 1 1089.3.c.m 16
44.g even 10 1 528.3.bf.b 16
44.h odd 10 1 528.3.bf.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.g.a 16 11.c even 5 1
33.3.g.a 16 11.d odd 10 1
99.3.k.c 16 33.f even 10 1
99.3.k.c 16 33.h odd 10 1
363.3.c.e 16 11.c even 5 1
363.3.c.e 16 11.d odd 10 1
363.3.g.a 16 11.b odd 2 1
363.3.g.a 16 11.c even 5 1
363.3.g.f 16 11.c even 5 1
363.3.g.f 16 11.d odd 10 1
363.3.g.g 16 1.a even 1 1 trivial
363.3.g.g 16 11.d odd 10 1 inner
528.3.bf.b 16 44.g even 10 1
528.3.bf.b 16 44.h odd 10 1
1089.3.c.m 16 33.f even 10 1
1089.3.c.m 16 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 10 T_{2}^{15} + 47 T_{2}^{14} - 120 T_{2}^{13} + 195 T_{2}^{12} - 500 T_{2}^{11} + 1772 T_{2}^{10} - 2950 T_{2}^{9} - 331 T_{2}^{8} + 5750 T_{2}^{7} + 5588 T_{2}^{6} - 17740 T_{2}^{5} - 24260 T_{2}^{4} + 27410 T_{2}^{3} + \cdots + 3721 \) acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 10 T^{15} + 47 T^{14} + \cdots + 3721 \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{6} + 9 T^{4} + 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 6 T^{15} + 89 T^{14} + \cdots + 9369721 \) Copy content Toggle raw display
$7$ \( T^{16} + 20 T^{15} + \cdots + 22159001881 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{15} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + 50 T^{15} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} - 70 T^{15} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{8} - 66 T^{7} + 862 T^{6} + \cdots + 255717136)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 80 T^{15} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{16} + 30 T^{15} + \cdots + 59\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{16} - 134 T^{15} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + 150 T^{15} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{16} + 16708 T^{14} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} - 110 T^{15} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + 278 T^{15} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{16} + 5384 T^{14} + \cdots + 34\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{16} - 260 T^{15} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( (T^{8} - 18 T^{7} - 13937 T^{6} + \cdots + 47006885776)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 86 T^{15} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} - 140 T^{15} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{16} - 380 T^{15} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + 620 T^{15} + \cdots + 16\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{8} - 38 T^{7} + \cdots - 15121642690304)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 246 T^{15} + \cdots + 29\!\cdots\!01 \) Copy content Toggle raw display
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