Properties

Label 63.2.h.b
Level $63$
Weight $2$
Character orbit 63.h
Analytic conductor $0.503$
Analytic rank $0$
Dimension $10$
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] = [63,2,Mod(25,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 63.h (of order \(3\), degree \(2\), 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: \(0.503057532734\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_1) q^{2} - \beta_{8} q^{3} + (\beta_{7} - \beta_{4} + 1) q^{4} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{6} + (\beta_{9} - \beta_{5} + \beta_1 - 1) q^{7} + (\beta_{9} - \beta_{3} - 1) q^{8} + ( - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_1) q^{2} - \beta_{8} q^{3} + (\beta_{7} - \beta_{4} + 1) q^{4} + (\beta_{4} + \beta_{2} + \beta_1) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{6} + (\beta_{9} - \beta_{5} + \beta_1 - 1) q^{7} + (\beta_{9} - \beta_{3} - 1) q^{8} + ( - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{9} + ( - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{10} + ( - \beta_{9} + \beta_{8} - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{11} + (\beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{12} + (\beta_{8} - \beta_{7} + 2 \beta_{6} - 2) q^{13} + ( - \beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta_1 - 2) q^{14} + ( - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_1) q^{15} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{16} + (\beta_{8} + \beta_{7} + 3 \beta_{6} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 + 1) q^{18} + (\beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{2}) q^{19} + (\beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{20} + (\beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 1) q^{21} + (\beta_{7} + \beta_{4} - \beta_{2}) q^{22} + (2 \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - \beta_1) q^{23} + (\beta_{7} - \beta_{5} + \beta_{2} + 2 \beta_1) q^{24} + ( - 3 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{25} + (\beta_{9} - 2 \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{26} + ( - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{27} + ( - 2 \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{28} + ( - 2 \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{29} + (\beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{30} + ( - \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{31} + ( - \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{32} + ( - \beta_{9} - 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} + 3) q^{33} + (\beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{3} - \beta_{2} - 4 \beta_1) q^{34} + (\beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} + 3 \beta_{4} + \beta_{2} - 1) q^{35} + (\beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 3) q^{36} + ( - 2 \beta_{9} - 2 \beta_{5}) q^{37} + (\beta_{9} - 3 \beta_{8} - 5 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{2} + 5) q^{38} + (\beta_{8} - \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 2) q^{39} + ( - 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{40} + (\beta_{9} - 3 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} + 2) q^{41} + (\beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{42}+ \cdots + ( - 3 \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} - q^{3} + 8 q^{4} + 4 q^{5} - 2 q^{6} - 4 q^{7} - 6 q^{8} + 11 q^{9} - 7 q^{10} + 4 q^{11} - 20 q^{12} - 8 q^{13} - 20 q^{14} - 19 q^{15} - 4 q^{16} + 12 q^{17} + 4 q^{18} + q^{19} + 5 q^{20} + 13 q^{21} - q^{22} + 3 q^{23} + 6 q^{24} - q^{25} + 11 q^{26} - 7 q^{27} - 2 q^{28} + 7 q^{29} + 16 q^{30} + 6 q^{31} + 4 q^{32} + 14 q^{33} + 3 q^{34} + 5 q^{35} + 34 q^{36} + 20 q^{38} + 2 q^{39} - 3 q^{40} + 5 q^{41} - 4 q^{42} - 7 q^{43} - 10 q^{44} - 16 q^{45} + 3 q^{46} - 54 q^{47} - 5 q^{48} - 8 q^{49} + 19 q^{50} - 9 q^{51} - 10 q^{52} - 21 q^{53} + q^{54} + 4 q^{55} + 18 q^{56} - 4 q^{57} - 10 q^{58} - 60 q^{59} + 10 q^{60} + 28 q^{61} - 12 q^{62} - 59 q^{63} - 50 q^{64} + 22 q^{65} + 19 q^{66} + 4 q^{67} + 27 q^{68} + 15 q^{69} + 40 q^{70} - 6 q^{71} - 36 q^{72} + 15 q^{73} - 36 q^{74} - 14 q^{75} + 5 q^{76} + 11 q^{77} - 20 q^{78} + 8 q^{79} + 20 q^{80} + 23 q^{81} - 5 q^{82} + 9 q^{83} + 35 q^{84} - 6 q^{85} - 8 q^{86} + 2 q^{87} - 18 q^{88} + 28 q^{89} + 28 q^{90} - 4 q^{91} + 27 q^{92} - 6 q^{93} + 6 q^{94} + 28 q^{95} + 59 q^{96} - 12 q^{97} + 59 q^{98} + 35 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{9} + 9\nu^{8} - 3\nu^{7} + 95\nu^{6} + 18\nu^{5} + 402\nu^{4} - 87\nu^{3} + 936\nu^{2} + 342\nu + 72 ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{9} + \nu^{8} - 12\nu^{7} - 8\nu^{6} - 68\nu^{5} - 30\nu^{4} - 123\nu^{3} - 204\nu^{2} - 270\nu - 63 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17 \nu^{9} - 24 \nu^{8} + 159 \nu^{7} - 106 \nu^{6} + 786 \nu^{5} - 417 \nu^{4} + 1893 \nu^{3} - 27 \nu^{2} + 1395 \nu + 639 ) / 567 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 ) / 567 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu + 504 ) / 567 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{9} - 12\nu^{8} + 69\nu^{7} - 43\nu^{6} + 330\nu^{5} - 219\nu^{4} + 732\nu^{3} - 45\nu^{2} + 477\nu - 306 ) / 189 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71 \nu^{9} + 123 \nu^{8} - 591 \nu^{7} + 403 \nu^{6} - 2604 \nu^{5} + 1794 \nu^{4} - 5214 \nu^{3} - 1458 \nu^{2} - 1476 \nu - 234 ) / 567 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 ) / 567 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{6} + \beta_{4} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 4\beta_{5} - \beta_{3} - 4\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{8} - 5\beta_{7} - 13\beta_{6} - \beta_{4} - \beta_{3} + 4\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{9} + \beta_{8} - 2\beta_{7} - 7\beta_{6} - 19\beta_{5} - \beta_{4} + \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{9} + 9\beta_{8} + 15\beta_{7} - 10\beta_{5} - 15\beta_{4} + 10\beta_{3} + 9\beta_{2} + 10\beta _1 + 61 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{8} + 11\beta_{7} + 46\beta_{6} + 19\beta_{4} + 43\beta_{3} + 8\beta_{2} + 94\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 73\beta_{9} + 56\beta_{8} + 62\beta_{7} + 298\beta_{6} + 76\beta_{5} + 118\beta_{4} - 118\beta_{2} - 298 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 253 \beta_{9} - 135 \beta_{8} + 48 \beta_{7} + 478 \beta_{5} - 48 \beta_{4} - 253 \beta_{3} - 135 \beta_{2} - 478 \beta _1 - 295 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(-1 + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
1.19343 2.06709i
0.920620 1.59456i
0.247934 0.429435i
−0.335166 + 0.580525i
−1.02682 + 1.77851i
1.19343 + 2.06709i
0.920620 + 1.59456i
0.247934 + 0.429435i
−0.335166 0.580525i
−1.02682 1.77851i
−2.38687 −1.61557 + 0.624446i 3.69714 1.46043 2.52954i 3.85615 1.49047i −0.138560 2.64212i −4.05086 2.22013 2.01767i −3.48586 + 6.03769i
25.2 −1.84124 1.39291 + 1.02946i 1.39017 −0.667377 + 1.15593i −2.56469 1.89549i 1.90267 + 1.83844i 1.12285 0.880416 + 2.86790i 1.22880 2.12835i
25.3 −0.495868 −0.221298 1.71786i −1.75411 1.84629 3.19787i 0.109735 + 0.851830i 0.926641 + 2.47817i 1.86155 −2.90205 + 0.760316i −0.915516 + 1.58572i
25.4 0.670333 1.65263 0.518475i −1.55065 −0.712469 + 1.23403i 1.10781 0.347551i −2.36039 1.19522i −2.38012 2.46237 1.71369i −0.477591 + 0.827212i
25.5 2.05365 −1.70867 0.283604i 2.21746 0.0731228 0.126652i −3.50901 0.582422i −2.33035 + 1.25278i 0.446582 2.83914 + 0.969173i 0.150168 0.260099i
58.1 −2.38687 −1.61557 0.624446i 3.69714 1.46043 + 2.52954i 3.85615 + 1.49047i −0.138560 + 2.64212i −4.05086 2.22013 + 2.01767i −3.48586 6.03769i
58.2 −1.84124 1.39291 1.02946i 1.39017 −0.667377 1.15593i −2.56469 + 1.89549i 1.90267 1.83844i 1.12285 0.880416 2.86790i 1.22880 + 2.12835i
58.3 −0.495868 −0.221298 + 1.71786i −1.75411 1.84629 + 3.19787i 0.109735 0.851830i 0.926641 2.47817i 1.86155 −2.90205 0.760316i −0.915516 1.58572i
58.4 0.670333 1.65263 + 0.518475i −1.55065 −0.712469 1.23403i 1.10781 + 0.347551i −2.36039 + 1.19522i −2.38012 2.46237 + 1.71369i −0.477591 0.827212i
58.5 2.05365 −1.70867 + 0.283604i 2.21746 0.0731228 + 0.126652i −3.50901 + 0.582422i −2.33035 1.25278i 0.446582 2.83914 0.969173i 0.150168 + 0.260099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.2.h.b yes 10
3.b odd 2 1 189.2.h.b 10
4.b odd 2 1 1008.2.q.i 10
7.b odd 2 1 441.2.h.f 10
7.c even 3 1 63.2.g.b 10
7.c even 3 1 441.2.f.e 10
7.d odd 6 1 441.2.f.f 10
7.d odd 6 1 441.2.g.f 10
9.c even 3 1 63.2.g.b 10
9.c even 3 1 567.2.e.f 10
9.d odd 6 1 189.2.g.b 10
9.d odd 6 1 567.2.e.e 10
12.b even 2 1 3024.2.q.i 10
21.c even 2 1 1323.2.h.f 10
21.g even 6 1 1323.2.f.f 10
21.g even 6 1 1323.2.g.f 10
21.h odd 6 1 189.2.g.b 10
21.h odd 6 1 1323.2.f.e 10
28.g odd 6 1 1008.2.t.i 10
36.f odd 6 1 1008.2.t.i 10
36.h even 6 1 3024.2.t.i 10
63.g even 3 1 441.2.f.e 10
63.g even 3 1 567.2.e.f 10
63.h even 3 1 inner 63.2.h.b yes 10
63.h even 3 1 3969.2.a.z 5
63.i even 6 1 1323.2.h.f 10
63.i even 6 1 3969.2.a.bb 5
63.j odd 6 1 189.2.h.b 10
63.j odd 6 1 3969.2.a.bc 5
63.k odd 6 1 441.2.f.f 10
63.l odd 6 1 441.2.g.f 10
63.n odd 6 1 567.2.e.e 10
63.n odd 6 1 1323.2.f.e 10
63.o even 6 1 1323.2.g.f 10
63.s even 6 1 1323.2.f.f 10
63.t odd 6 1 441.2.h.f 10
63.t odd 6 1 3969.2.a.ba 5
84.n even 6 1 3024.2.t.i 10
252.u odd 6 1 1008.2.q.i 10
252.bb even 6 1 3024.2.q.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 7.c even 3 1
63.2.g.b 10 9.c even 3 1
63.2.h.b yes 10 1.a even 1 1 trivial
63.2.h.b yes 10 63.h even 3 1 inner
189.2.g.b 10 9.d odd 6 1
189.2.g.b 10 21.h odd 6 1
189.2.h.b 10 3.b odd 2 1
189.2.h.b 10 63.j odd 6 1
441.2.f.e 10 7.c even 3 1
441.2.f.e 10 63.g even 3 1
441.2.f.f 10 7.d odd 6 1
441.2.f.f 10 63.k odd 6 1
441.2.g.f 10 7.d odd 6 1
441.2.g.f 10 63.l odd 6 1
441.2.h.f 10 7.b odd 2 1
441.2.h.f 10 63.t odd 6 1
567.2.e.e 10 9.d odd 6 1
567.2.e.e 10 63.n odd 6 1
567.2.e.f 10 9.c even 3 1
567.2.e.f 10 63.g even 3 1
1008.2.q.i 10 4.b odd 2 1
1008.2.q.i 10 252.u odd 6 1
1008.2.t.i 10 28.g odd 6 1
1008.2.t.i 10 36.f odd 6 1
1323.2.f.e 10 21.h odd 6 1
1323.2.f.e 10 63.n odd 6 1
1323.2.f.f 10 21.g even 6 1
1323.2.f.f 10 63.s even 6 1
1323.2.g.f 10 21.g even 6 1
1323.2.g.f 10 63.o even 6 1
1323.2.h.f 10 21.c even 2 1
1323.2.h.f 10 63.i even 6 1
3024.2.q.i 10 12.b even 2 1
3024.2.q.i 10 252.bb even 6 1
3024.2.t.i 10 36.h even 6 1
3024.2.t.i 10 84.n even 6 1
3969.2.a.z 5 63.h even 3 1
3969.2.a.ba 5 63.t odd 6 1
3969.2.a.bb 5 63.i even 6 1
3969.2.a.bc 5 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 9T_{2}^{2} + 3T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + 2 T^{4} - 5 T^{3} - 9 T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} - 5 T^{8} - 3 T^{7} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} - 4 T^{9} + 21 T^{8} - 16 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{10} + 4 T^{9} + 12 T^{8} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} - 4 T^{9} + 24 T^{8} + 2 T^{7} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( T^{10} + 8 T^{9} + 51 T^{8} + 130 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{10} - 12 T^{9} + 99 T^{8} - 420 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{10} - T^{9} + 42 T^{8} + \cdots + 185761 \) Copy content Toggle raw display
$23$ \( T^{10} - 3 T^{9} + 72 T^{8} + \cdots + 2595321 \) Copy content Toggle raw display
$29$ \( T^{10} - 7 T^{9} + 69 T^{8} - 190 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( (T^{5} - 3 T^{4} - 21 T^{3} + 64 T^{2} + \cdots - 285)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 96 T^{8} + 560 T^{7} + \cdots + 82944 \) Copy content Toggle raw display
$41$ \( T^{10} - 5 T^{9} + 69 T^{8} + \cdots + 2025 \) Copy content Toggle raw display
$43$ \( T^{10} + 7 T^{9} + 138 T^{8} + \cdots + 687241 \) Copy content Toggle raw display
$47$ \( (T^{5} + 27 T^{4} + 213 T^{3} + 93 T^{2} + \cdots - 6615)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 21 T^{9} + 306 T^{8} + \cdots + 178929 \) Copy content Toggle raw display
$59$ \( (T^{5} + 30 T^{4} + 306 T^{3} + 1113 T^{2} + \cdots - 5625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 14 T^{4} + 34 T^{3} + 7 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 2 T^{4} - 203 T^{3} + 340 T^{2} + \cdots - 7121)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + 3 T^{4} - 168 T^{3} - 567 T^{2} + \cdots - 81)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 15 T^{9} + 231 T^{8} + \cdots + 772641 \) Copy content Toggle raw display
$79$ \( (T^{5} - 4 T^{4} - 95 T^{3} - 224 T^{2} + \cdots + 193)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} - 9 T^{9} + 267 T^{8} + \cdots + 218123361 \) Copy content Toggle raw display
$89$ \( T^{10} - 28 T^{9} + 549 T^{8} + \cdots + 7080921 \) Copy content Toggle raw display
$97$ \( T^{10} + 12 T^{9} + \cdots + 2307745521 \) Copy content Toggle raw display
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