Properties

Label 363.3.h.l
Level $363$
Weight $3$
Character orbit 363.h
Analytic conductor $9.891$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,3,Mod(245,363)]
 
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([5, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.245");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.h (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: 16.0.343361479062744140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + 168 x^{7} + 54 x^{6} + 189 x^{5} + 648 x^{4} - 1944 x^{3} + 2187 x^{2} - 2187 x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_1) q^{2} + ( - \beta_{12} + \beta_{11} + \beta_{2}) q^{3} + ( - \beta_{9} - \beta_{7} + \beta_{3}) q^{4} + ( - \beta_{10} - \beta_{6}) q^{5} + ( - \beta_{10} - 3 \beta_{6} + 3 \beta_1) q^{6} + ( - 2 \beta_{9} - 2 \beta_{7}) q^{7} + ( - 3 \beta_{14} - 3 \beta_{12} - \beta_{11}) q^{8} + ( - 3 \beta_{15} + 3 \beta_{11} - 3 \beta_{8} + 3 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_1) q^{2} + ( - \beta_{12} + \beta_{11} + \beta_{2}) q^{3} + ( - \beta_{9} - \beta_{7} + \beta_{3}) q^{4} + ( - \beta_{10} - \beta_{6}) q^{5} + ( - \beta_{10} - 3 \beta_{6} + 3 \beta_1) q^{6} + ( - 2 \beta_{9} - 2 \beta_{7}) q^{7} + ( - 3 \beta_{14} - 3 \beta_{12} - \beta_{11}) q^{8} + ( - 3 \beta_{15} + 3 \beta_{11} - 3 \beta_{8} + 3 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + \cdots + 3) q^{9}+ \cdots + ( - 9 \beta_{14} - 9 \beta_{13} - 9 \beta_{12} + 9 \beta_{10} + 9 \beta_{9} - 9 \beta_{7} + \cdots + 9 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 5 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 5 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 7 q^{9} + 32 q^{10} + 56 q^{12} - 8 q^{13} - 13 q^{15} + 22 q^{16} + 38 q^{18} - 36 q^{19} + 152 q^{21} + 24 q^{24} - 86 q^{25} + 20 q^{27} - 64 q^{28} + 10 q^{30} - 46 q^{31} + 448 q^{34} + 86 q^{36} + 90 q^{37} - 56 q^{39} - 36 q^{40} + 68 q^{42} - 384 q^{43} + 68 q^{45} - 88 q^{46} - 110 q^{48} + 60 q^{49} + 214 q^{51} - 136 q^{52} + 704 q^{54} + 144 q^{57} - 216 q^{58} - 56 q^{60} - 24 q^{61} + 158 q^{63} - 34 q^{64} - 232 q^{67} + 253 q^{69} + 8 q^{70} + 72 q^{72} - 284 q^{73} + 124 q^{75} - 720 q^{76} - 512 q^{78} - 76 q^{79} - 113 q^{81} - 40 q^{82} + 80 q^{84} - 68 q^{85} + 1008 q^{87} + 14 q^{90} - 256 q^{91} - 25 q^{93} + 260 q^{94} + 272 q^{96} - 218 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 3 x^{14} - 8 x^{13} + 8 x^{12} + 7 x^{11} + 6 x^{10} + 56 x^{9} - 137 x^{8} + 168 x^{7} + 54 x^{6} + 189 x^{5} + 648 x^{4} - 1944 x^{3} + 2187 x^{2} - 2187 x + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5 \nu^{15} + 15 \nu^{14} - 40 \nu^{13} + 40 \nu^{12} - 5709 \nu^{11} + 30 \nu^{10} + 280 \nu^{9} - 685 \nu^{8} + 840 \nu^{7} - 90557 \nu^{6} + 945 \nu^{5} + 3240 \nu^{4} - 9720 \nu^{3} + \cdots + 32805 ) / 261711 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 160 \nu^{15} + 400 \nu^{14} + 3467 \nu^{12} - 1040 \nu^{11} - 4960 \nu^{10} + 6480 \nu^{9} - 2480 \nu^{8} + 39680 \nu^{7} - 19440 \nu^{6} - 66960 \nu^{5} + 103680 \nu^{4} + \cdots - 524880 ) / 785133 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 3 \nu^{12} - 48 \nu^{11} + 93 \nu^{10} + 81 \nu^{9} - 496 \nu^{8} + 93 \nu^{7} - 729 \nu^{6} + 1674 \nu^{5} + 1053 \nu^{4} - 8920 \nu^{3} - 10935 \nu + 13122 ) / 9693 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{15} - 40 \nu^{14} + 27 \nu^{13} - 72 \nu^{12} + 72 \nu^{11} + 63 \nu^{10} - 664 \nu^{9} + 504 \nu^{8} - 1233 \nu^{7} + 1512 \nu^{6} + 486 \nu^{5} - 20557 \nu^{4} + 5832 \nu^{3} + \cdots - 19683 ) / 58158 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\nu^{15} + 718\nu^{10} + 22258\nu^{5} + 146286 ) / 87237 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31 \nu^{15} + 93 \nu^{14} - 248 \nu^{13} + 248 \nu^{12} - 501 \nu^{11} + 186 \nu^{10} + 1736 \nu^{9} - 4247 \nu^{8} + 5208 \nu^{7} - 20584 \nu^{6} + 5859 \nu^{5} + 20088 \nu^{4} + \cdots + 203391 ) / 174474 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 403 \nu^{15} - 806 \nu^{14} + 1053 \nu^{13} - 403 \nu^{12} + 6448 \nu^{11} - 12493 \nu^{10} - 10881 \nu^{9} + 28336 \nu^{8} - 12493 \nu^{7} + 97929 \nu^{6} - 224874 \nu^{5} + \cdots - 1762722 ) / 523422 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 842 \nu^{15} + 1684 \nu^{14} - 324 \nu^{13} + 842 \nu^{12} - 13472 \nu^{11} + 26102 \nu^{10} + 22734 \nu^{9} + 4150 \nu^{8} + 26102 \nu^{7} - 204606 \nu^{6} + 469836 \nu^{5} + \cdots + 3682908 ) / 785133 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1079 \nu^{15} - 2158 \nu^{14} - 6561 \nu^{13} - 1079 \nu^{12} + 17264 \nu^{11} - 33449 \nu^{10} - 29133 \nu^{9} - 79846 \nu^{8} - 33449 \nu^{7} + 262197 \nu^{6} + \cdots - 4719546 ) / 1570266 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 3 \nu^{14} + 8 \nu^{13} - 8 \nu^{12} - 7 \nu^{11} - 6 \nu^{10} - 56 \nu^{9} + 137 \nu^{8} - 168 \nu^{7} - 54 \nu^{6} - 189 \nu^{5} - 648 \nu^{4} + 1944 \nu^{3} - 2187 \nu^{2} + 386 \nu - 6561 ) / 2154 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 646 \nu^{15} - 1615 \nu^{14} + 178 \nu^{12} + 4199 \nu^{11} + 20026 \nu^{10} - 26163 \nu^{9} + 10013 \nu^{8} + 14266 \nu^{7} + 78489 \nu^{6} + 270351 \nu^{5} - 418608 \nu^{4} + \cdots + 2119203 ) / 785133 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2 \nu^{15} + 5 \nu^{14} - 15 \nu^{12} - 13 \nu^{11} - 62 \nu^{10} + 81 \nu^{9} - 31 \nu^{8} - 222 \nu^{7} - 243 \nu^{6} - 837 \nu^{5} + 1296 \nu^{4} - 243 \nu^{3} - 3955 \nu^{2} - 4374 \nu - 6561 ) / 2154 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 599 \nu^{15} - 130 \nu^{14} - 1797 \nu^{13} + 4792 \nu^{12} - 4792 \nu^{11} - 4193 \nu^{10} - 3594 \nu^{9} - 33544 \nu^{8} + 82063 \nu^{7} - 100632 \nu^{6} - 32346 \nu^{5} + \cdots + 1310013 ) / 523422 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2170 \nu^{15} - 5425 \nu^{14} - 8855 \nu^{12} + 14105 \nu^{11} + 67270 \nu^{10} - 87885 \nu^{9} + 33635 \nu^{8} - 189212 \nu^{7} + 263655 \nu^{6} + 908145 \nu^{5} + \cdots + 7118685 ) / 1570266 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 248 \nu^{15} + 744 \nu^{14} - 1984 \nu^{13} + 1984 \nu^{12} - 4008 \nu^{11} + 1488 \nu^{10} + 13888 \nu^{9} - 33976 \nu^{8} + 41664 \nu^{7} - 77435 \nu^{6} + 46872 \nu^{5} + \cdots + 1627128 ) / 261711 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} + 2\beta_{10} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{12} - 3\beta_{11} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{8} - 2\beta_{7} - 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{15} + \beta_{11} - \beta_{8} + \beta_{5} - 10\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{13} + 2\beta_{12} + 2\beta_{7} - 2\beta_{6} + 15\beta_{5} - 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{15} - 16\beta_{10} - 16\beta_{6} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -26\beta_{14} + 35\beta_{11} - 35\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 39\beta_{8} + 70\beta_{7} + 39\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68 \beta_{15} - 74 \beta_{13} + 37 \beta_{11} - 37 \beta_{8} + 37 \beta_{5} + 74 \beta_{4} - 68 \beta_{3} + 68 \beta_{2} + 37 \beta _1 - 68 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 62\beta_{14} - 62\beta_{10} - 62\beta_{9} - 253\beta_{5} - 62\beta_{4} - 253 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -93\beta_{15} + 506\beta_{6} - 93\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 160\beta_{14} + 160\beta_{12} + 80\beta_{11} + 679\beta_{2} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -1198\beta_{9} - 1079\beta_{8} + 1079\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1797 \beta_{15} + 2158 \beta_{13} - 1797 \beta_{11} + 1797 \beta_{8} - 1797 \beta_{5} + 1797 \beta_{3} - 1797 \beta_{2} - 1797 \beta _1 + 1797 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 718 \beta_{14} - 718 \beta_{13} - 718 \beta_{12} + 718 \beta_{10} + 718 \beta_{9} - 718 \beta_{7} + 718 \beta_{6} + 359 \beta_{5} + 718 \beta_{4} + 3596 \) Copy content Toggle raw display

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
245.1
−0.217724 1.71831i
1.59696 + 0.670602i
1.13127 + 1.31158i
−1.70149 + 0.323920i
1.56693 + 0.738055i
−0.144291 1.72603i
−0.897801 1.48120i
−0.833856 + 1.51812i
1.56693 0.738055i
−0.144291 + 1.72603i
−0.897801 + 1.48120i
−0.833856 1.51812i
−0.217724 + 1.71831i
1.59696 0.670602i
1.13127 1.31158i
−1.70149 0.323920i
−1.48377 2.04223i 0.440460 2.96749i −0.733075 + 2.25617i −0.465695 + 0.640974i −6.71384 + 3.50355i −2.08418 + 6.41446i −3.90781 + 1.26972i −8.61199 2.61412i 2.00000
245.2 −0.465695 0.640974i −2.79015 + 1.10230i 1.04209 3.20723i −1.48377 + 2.04223i 2.00590 + 1.27508i 1.46615 4.51235i −5.55509 + 1.80496i 6.56989 6.15114i 2.00000
245.3 0.465695 + 0.640974i 2.90519 0.748234i 1.04209 3.20723i 1.48377 2.04223i 1.83253 + 1.51370i 1.46615 4.51235i 5.55509 1.80496i 7.88029 4.34753i 2.00000
245.4 1.48377 + 2.04223i −2.10059 2.14185i −0.733075 + 2.25617i 0.465695 0.640974i 1.25738 7.46790i −2.08418 + 6.41446i 3.90781 1.26972i −0.175073 + 8.99830i 2.00000
251.1 −2.40079 0.780063i 1.38791 2.65965i 1.91922 + 1.39439i −0.753510 + 0.244830i −5.40676 + 5.30259i 5.45647 + 3.96435i 2.41516 + 3.32418i −5.14743 7.38268i 2.00000
251.2 −0.753510 0.244830i 1.60937 + 2.53179i −2.72823 1.98218i −2.40079 + 0.780063i −0.592816 2.30175i −3.83843 2.78878i 3.43323 + 4.72544i −3.81987 + 8.14914i 2.00000
251.3 0.753510 + 0.244830i −1.91055 2.31296i −2.72823 1.98218i 2.40079 0.780063i −0.873334 2.21060i −3.83843 2.78878i −3.43323 4.72544i −1.69960 + 8.83806i 2.00000
251.4 2.40079 + 0.780063i 2.95836 0.498102i 1.91922 + 1.39439i 0.753510 0.244830i 7.49095 + 1.11187i 5.45647 + 3.96435i −2.41516 3.32418i 8.50379 2.94713i 2.00000
269.1 −2.40079 + 0.780063i 1.38791 + 2.65965i 1.91922 1.39439i −0.753510 0.244830i −5.40676 5.30259i 5.45647 3.96435i 2.41516 3.32418i −5.14743 + 7.38268i 2.00000
269.2 −0.753510 + 0.244830i 1.60937 2.53179i −2.72823 + 1.98218i −2.40079 0.780063i −0.592816 + 2.30175i −3.83843 + 2.78878i 3.43323 4.72544i −3.81987 8.14914i 2.00000
269.3 0.753510 0.244830i −1.91055 + 2.31296i −2.72823 + 1.98218i 2.40079 + 0.780063i −0.873334 + 2.21060i −3.83843 + 2.78878i −3.43323 + 4.72544i −1.69960 8.83806i 2.00000
269.4 2.40079 0.780063i 2.95836 + 0.498102i 1.91922 1.39439i 0.753510 + 0.244830i 7.49095 1.11187i 5.45647 3.96435i −2.41516 + 3.32418i 8.50379 + 2.94713i 2.00000
323.1 −1.48377 + 2.04223i 0.440460 + 2.96749i −0.733075 2.25617i −0.465695 0.640974i −6.71384 3.50355i −2.08418 6.41446i −3.90781 1.26972i −8.61199 + 2.61412i 2.00000
323.2 −0.465695 + 0.640974i −2.79015 1.10230i 1.04209 + 3.20723i −1.48377 2.04223i 2.00590 1.27508i 1.46615 + 4.51235i −5.55509 1.80496i 6.56989 + 6.15114i 2.00000
323.3 0.465695 0.640974i 2.90519 + 0.748234i 1.04209 + 3.20723i 1.48377 + 2.04223i 1.83253 1.51370i 1.46615 + 4.51235i 5.55509 + 1.80496i 7.88029 + 4.34753i 2.00000
323.4 1.48377 2.04223i −2.10059 + 2.14185i −0.733075 2.25617i 0.465695 + 0.640974i 1.25738 + 7.46790i −2.08418 6.41446i 3.90781 + 1.26972i −0.175073 8.99830i 2.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.c even 5 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.h.l 16
3.b odd 2 1 inner 363.3.h.l 16
11.b odd 2 1 363.3.h.m 16
11.c even 5 1 363.3.b.h 4
11.c even 5 3 inner 363.3.h.l 16
11.d odd 10 1 33.3.b.b 4
11.d odd 10 3 363.3.h.m 16
33.d even 2 1 363.3.h.m 16
33.f even 10 1 33.3.b.b 4
33.f even 10 3 363.3.h.m 16
33.h odd 10 1 363.3.b.h 4
33.h odd 10 3 inner 363.3.h.l 16
44.g even 10 1 528.3.i.d 4
132.n odd 10 1 528.3.i.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.b.b 4 11.d odd 10 1
33.3.b.b 4 33.f even 10 1
363.3.b.h 4 11.c even 5 1
363.3.b.h 4 33.h odd 10 1
363.3.h.l 16 1.a even 1 1 trivial
363.3.h.l 16 3.b odd 2 1 inner
363.3.h.l 16 11.c even 5 3 inner
363.3.h.l 16 33.h odd 10 3 inner
363.3.h.m 16 11.b odd 2 1
363.3.h.m 16 11.d odd 10 3
363.3.h.m 16 33.d even 2 1
363.3.h.m 16 33.f even 10 3
528.3.i.d 4 44.g even 10 1
528.3.i.d 4 132.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{16} - 7T_{2}^{14} + 45T_{2}^{12} - 287T_{2}^{10} + 1829T_{2}^{8} - 1148T_{2}^{6} + 720T_{2}^{4} - 448T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{5}^{16} - 7T_{5}^{14} + 45T_{5}^{12} - 287T_{5}^{10} + 1829T_{5}^{8} - 1148T_{5}^{6} + 720T_{5}^{4} - 448T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{7} + 36T_{7}^{6} - 136T_{7}^{5} + 1424T_{7}^{4} + 4352T_{7}^{3} + 36864T_{7}^{2} + 65536T_{7} + 1048576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 7 T^{14} + 45 T^{12} - 287 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} - 5 T^{15} + 9 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} - 7 T^{14} + 45 T^{12} - 287 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + 36 T^{6} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + 144 T^{6} + \cdots + 268435456)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 1372 T^{14} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + 18 T^{7} + 540 T^{6} + \cdots + 2176782336)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1639 T^{2} + 662596)^{4} \) Copy content Toggle raw display
$29$ \( T^{16} - 3552 T^{14} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{8} + 23 T^{7} + 603 T^{6} + \cdots + 29986576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 45 T^{7} + 1923 T^{6} + \cdots + 108243216)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 1264 T^{14} + \cdots + 76\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 48 T + 444)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} - 2716 T^{14} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} - 8124 T^{14} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} - 4003 T^{14} + \cdots + 46\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 113501238460416)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 29 T - 2174)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 360040606269696 \) Copy content Toggle raw display
$73$ \( (T^{8} + 142 T^{7} + \cdots + 506499150647296)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 38 T^{7} + \cdots + 740202130702336)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 5436 T^{14} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + 20787 T^{2} + 4112784)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 109 T^{7} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
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