Properties

Label 819.2.g.d
Level $819$
Weight $2$
Character orbit 819.g
Analytic conductor $6.540$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(818,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.818");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.g (of order \(2\), degree \(1\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 14 x^{10} - 36 x^{9} + 78 x^{8} - 140 x^{7} + 208 x^{6} - 196 x^{5} + 40 x^{4} - 64 x^{3} + 236 x^{2} - 264 x + 194 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 1) q^{2} + ( - \beta_{5} - \beta_1 + 2) q^{4} + \beta_{3} q^{5} + (\beta_{9} - \beta_{7}) q^{7} + ( - 3 \beta_1 + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + 1) q^{2} + ( - \beta_{5} - \beta_1 + 2) q^{4} + \beta_{3} q^{5} + (\beta_{9} - \beta_{7}) q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{11} + 2 \beta_{3}) q^{10} + (\beta_{5} - \beta_1 - 3) q^{11} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{4}) q^{13} + (\beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{6}) q^{14} + ( - 2 \beta_{5} - 4 \beta_1 + 3) q^{16} + ( - \beta_{9} + \beta_{8} + 2 \beta_{2}) q^{17} - 2 \beta_{2} q^{19} + (2 \beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{4} + 3 \beta_{3}) q^{20} + (3 \beta_{5} - \beta_1 - 5) q^{22} + (2 \beta_{7} - \beta_{6}) q^{23} + (2 \beta_{5} - 1) q^{25} + ( - \beta_{11} + \beta_{9} + 2 \beta_{8} - 3 \beta_{4} - \beta_{3} + \beta_{2}) q^{26} + (2 \beta_{10} + \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - \beta_{2}) q^{28} + ( - 2 \beta_{7} + 3 \beta_{6}) q^{29} + (\beta_{9} - \beta_{8} + 2 \beta_{2}) q^{31} + ( - 3 \beta_{5} - 4 \beta_1 + 5) q^{32} + ( - 6 \beta_{10} + \beta_{9} + 5 \beta_{8} + 2 \beta_{2}) q^{34} + (2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{7} - 4 \beta_{6} - \beta_{2}) q^{35} + ( - \beta_{9} - \beta_{8} - 4 \beta_{6}) q^{37} + (4 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{2}) q^{38} + (3 \beta_{11} - 3 \beta_{9} - 3 \beta_{8} + 6 \beta_{4} + 4 \beta_{3}) q^{40} + ( - 2 \beta_{11} + \beta_{3}) q^{41} + ( - 4 \beta_{5} + 4 \beta_1 - 2) q^{43} + (3 \beta_{5} + 3 \beta_1 - 7) q^{44} + ( - \beta_{9} - \beta_{8} + 5 \beta_{7} - 4 \beta_{6}) q^{46} + ( - 2 \beta_{11} + \beta_{9} + \beta_{8} - 2 \beta_{4} - \beta_{3}) q^{47} + ( - 2 \beta_{11} + \beta_{9} + \beta_{8} - 2 \beta_{5} - 2 \beta_{4} - 1) q^{49} + (\beta_{5} + 2 \beta_1 - 7) q^{50} + ( - 2 \beta_{11} - \beta_{10} + \beta_{9} + 3 \beta_{8} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2}) q^{52} + ( - 2 \beta_{9} - 2 \beta_{8} + 4 \beta_{7} - \beta_{6}) q^{53} + ( - \beta_{9} - \beta_{8} + 2 \beta_{4} - 4 \beta_{3}) q^{55} + (3 \beta_{10} + 4 \beta_{9} - 4 \beta_{7} + 6 \beta_{6} - 3 \beta_{2}) q^{56} + (3 \beta_{9} + 3 \beta_{8} - 7 \beta_{7} + 8 \beta_{6}) q^{58} + ( - 2 \beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{4} - \beta_{3}) q^{59} + ( - 2 \beta_{11} - \beta_{9} - \beta_{8} + 2 \beta_{4} + 2 \beta_{3}) q^{61} + ( - 2 \beta_{10} + 3 \beta_{9} - \beta_{8} + 2 \beta_{2}) q^{62} + ( - \beta_{5} - 3 \beta_1 + 12) q^{64} + (\beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} - \beta_{5} - 5 \beta_1 + 1) q^{65} + ( - 8 \beta_{10} - \beta_{9} + 9 \beta_{8} + 4 \beta_{2}) q^{68} + (2 \beta_{10} - 3 \beta_{9} - 5 \beta_{8} + 4 \beta_{7} - 9 \beta_{6} - 3 \beta_{2}) q^{70} + ( - 3 \beta_{5} - \beta_1 + 1) q^{71} + (2 \beta_{10} - 2 \beta_{8} + 2 \beta_{2}) q^{73} + ( - 3 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} - 8 \beta_{6}) q^{74} + (4 \beta_{10} - 4 \beta_{8} - 2 \beta_{2}) q^{76} + ( - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + \beta_{6} - \beta_{2}) q^{77} + (4 \beta_{5} + 2 \beta_1 - 4) q^{79} + (6 \beta_{11} - 4 \beta_{9} - 4 \beta_{8} + 8 \beta_{4} + 5 \beta_{3}) q^{80} + (\beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 4 \beta_{4}) q^{82} + ( - \beta_{9} - \beta_{8} + 2 \beta_{4} + 3 \beta_{3}) q^{83} + ( - 6 \beta_{7} + 8 \beta_{6}) q^{85} + (2 \beta_{5} + 4 \beta_1 + 6) q^{86} + (\beta_{5} + 11 \beta_1 - 9) q^{88} + (2 \beta_{9} + 2 \beta_{8} - 4 \beta_{4} + \beta_{3}) q^{89} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{8} - 3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{91}+ \cdots + ( - 2 \beta_{11} + 3 \beta_{9} + 3 \beta_{8} + \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 20 q^{4} + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 20 q^{4} + 36 q^{8} - 40 q^{11} + 20 q^{16} - 64 q^{22} - 12 q^{25} + 44 q^{32} - 8 q^{43} - 72 q^{44} - 12 q^{49} - 76 q^{50} + 132 q^{64} - 8 q^{65} + 8 q^{71} - 40 q^{79} + 88 q^{86} - 64 q^{88} + 4 q^{91} + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 14 x^{10} - 36 x^{9} + 78 x^{8} - 140 x^{7} + 208 x^{6} - 196 x^{5} + 40 x^{4} - 64 x^{3} + 236 x^{2} - 264 x + 194 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6258 \nu^{11} + 7900 \nu^{10} - 22162 \nu^{9} + 4215 \nu^{8} + 65136 \nu^{7} - 256186 \nu^{6} + 625792 \nu^{5} - 1479489 \nu^{4} + 1755524 \nu^{3} + \cdots - 771987 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8512 \nu^{11} + 36285 \nu^{10} - 149568 \nu^{9} + 383995 \nu^{8} - 926996 \nu^{7} + 1772196 \nu^{6} - 2795742 \nu^{5} + 3302064 \nu^{4} - 1446244 \nu^{3} + \cdots + 3423782 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9171 \nu^{11} + 11225 \nu^{10} + 15964 \nu^{9} + 144795 \nu^{8} - 221562 \nu^{7} + 859957 \nu^{6} - 1283104 \nu^{5} + 2991258 \nu^{4} - 1657868 \nu^{3} + \cdots + 1525554 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2790 \nu^{11} + 8056 \nu^{10} - 29771 \nu^{9} + 65716 \nu^{8} - 132036 \nu^{7} + 213650 \nu^{6} - 271768 \nu^{5} + 111659 \nu^{4} + 201716 \nu^{3} + \cdots + 330332 ) / 428025 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22994 \nu^{11} + 65800 \nu^{10} - 230946 \nu^{9} + 524620 \nu^{8} - 1044232 \nu^{7} + 1702802 \nu^{6} - 2220744 \nu^{5} + 938863 \nu^{4} + 1466652 \nu^{3} + \cdots + 618194 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25907 \nu^{11} - 46675 \nu^{10} + 224748 \nu^{9} - 375610 \nu^{8} + 887806 \nu^{7} - 1099031 \nu^{6} + 1563432 \nu^{5} + 572906 \nu^{4} - 1368996 \nu^{3} + \cdots - 2004752 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28463 \nu^{11} - 98400 \nu^{10} + 341782 \nu^{9} - 800190 \nu^{8} + 1675204 \nu^{7} - 2685429 \nu^{6} + 3539138 \nu^{5} - 1591946 \nu^{4} - 2973214 \nu^{3} + \cdots - 4358568 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7773 \nu^{11} + 17382 \nu^{10} - 78572 \nu^{9} + 148172 \nu^{8} - 351969 \nu^{7} + 511224 \nu^{6} - 785374 \nu^{5} + 328828 \nu^{4} - 56412 \nu^{3} + \cdots + 1846603 ) / 428025 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 56953 \nu^{11} + 137950 \nu^{10} - 557042 \nu^{9} + 1132940 \nu^{8} - 2527949 \nu^{7} + 3825224 \nu^{6} - 5650678 \nu^{5} + 2405276 \nu^{4} + \cdots + 2885833 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 69229 \nu^{11} + 165670 \nu^{10} - 672406 \nu^{9} + 1341490 \nu^{8} - 2962457 \nu^{7} + 4417157 \nu^{6} - 6197264 \nu^{5} + 2333163 \nu^{4} + \cdots + 7244069 ) / 2140125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 113909 \nu^{11} + 315700 \nu^{10} - 1227336 \nu^{9} + 2602945 \nu^{8} - 5797882 \nu^{7} + 8826197 \nu^{6} - 12964584 \nu^{5} + 6118108 \nu^{4} + \cdots + 16352144 ) / 2140125 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} + 2\beta_{5} - 2\beta_{4} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} - 3\beta_{9} - 3\beta_{8} + 3\beta_{7} - 5\beta_{6} - \beta_{5} + 2\beta_{3} + 3\beta_{2} + 5\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} + 4\beta_{10} - 3\beta_{9} - 5\beta_{8} + 4\beta_{7} - 3\beta_{5} + 4\beta_{4} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9 \beta_{11} + 20 \beta_{10} - \beta_{9} - \beta_{8} - 15 \beta_{7} + 9 \beta_{6} - 6 \beta_{5} - 8 \beta_{4} - 3 \beta_{3} - 10 \beta_{2} - 24 \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{11} - 7 \beta_{10} + 18 \beta_{9} + 19 \beta_{8} - 25 \beta_{7} + 12 \beta_{6} - 2 \beta_{5} - 14 \beta_{4} - 4 \beta_{3} + 12 \beta_{2} + 21 \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{11} - 35 \beta_{10} - \beta_{9} + 27 \beta_{8} + 21 \beta_{7} + 10 \beta_{6} + 26 \beta_{5} + 44 \beta_{4} - 5 \beta_{3} + 21 \beta_{2} + 45 \beta _1 - 38 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 36 \beta_{11} + 12 \beta_{10} - 84 \beta_{9} - 48 \beta_{8} + 88 \beta_{7} - 144 \beta_{6} + 69 \beta_{5} + 56 \beta_{4} + 68 \beta_{3} - 64 \beta_{2} - 113 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20 \beta_{11} + 72 \beta_{10} + 92 \beta_{9} - 184 \beta_{8} - 48 \beta_{7} - 157 \beta_{6} - 96 \beta_{5} - 145 \beta_{4} + 81 \beta_{3} - 60 \beta_{2} - 119 \beta _1 + 469 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 91 \beta_{11} - 56 \beta_{10} + 428 \beta_{9} + 130 \beta_{8} - 295 \beta_{7} + 1001 \beta_{6} - 137 \beta_{5} - 78 \beta_{4} - 493 \beta_{3} + 101 \beta_{2} + 193 \beta _1 + 443 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4 \beta_{11} - 341 \beta_{10} - 474 \beta_{9} + 1121 \beta_{8} - 176 \beta_{7} + 941 \beta_{6} + 938 \beta_{5} - 108 \beta_{4} - 478 \beta_{3} - 55 \beta_{2} - 27 \beta _1 - 2319 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
818.1
1.45161 0.0803372i
1.45161 + 1.33388i
1.45161 + 0.0803372i
1.45161 1.33388i
−0.854638 + 0.623154i
−0.854638 + 2.03737i
−0.854638 0.623154i
−0.854638 2.03737i
0.403032 + 0.821803i
0.403032 + 2.23602i
0.403032 0.821803i
0.403032 2.23602i
−1.21432 0 −0.525428 1.25354i 0 −0.886386 + 2.49285i 3.06668 0 1.52220i
818.2 −1.21432 0 −0.525428 1.25354i 0 0.886386 2.49285i 3.06668 0 1.52220i
818.3 −1.21432 0 −0.525428 1.25354i 0 −0.886386 2.49285i 3.06668 0 1.52220i
818.4 −1.21432 0 −0.525428 1.25354i 0 0.886386 + 2.49285i 3.06668 0 1.52220i
818.5 1.53919 0 0.369102 2.66052i 0 −1.88127 + 1.86033i −2.51026 0 4.09505i
818.6 1.53919 0 0.369102 2.66052i 0 1.88127 1.86033i −2.51026 0 4.09505i
818.7 1.53919 0 0.369102 2.66052i 0 −1.88127 1.86033i −2.51026 0 4.09505i
818.8 1.53919 0 0.369102 2.66052i 0 1.88127 + 1.86033i −2.51026 0 4.09505i
818.9 2.67513 0 5.15633 3.05782i 0 −2.16221 1.52475i 8.44358 0 8.18007i
818.10 2.67513 0 5.15633 3.05782i 0 2.16221 + 1.52475i 8.44358 0 8.18007i
818.11 2.67513 0 5.15633 3.05782i 0 −2.16221 + 1.52475i 8.44358 0 8.18007i
818.12 2.67513 0 5.15633 3.05782i 0 2.16221 1.52475i 8.44358 0 8.18007i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 818.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
39.d odd 2 1 inner
273.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.g.d yes 12
3.b odd 2 1 819.2.g.c 12
7.b odd 2 1 inner 819.2.g.d yes 12
13.b even 2 1 819.2.g.c 12
21.c even 2 1 819.2.g.c 12
39.d odd 2 1 inner 819.2.g.d yes 12
91.b odd 2 1 819.2.g.c 12
273.g even 2 1 inner 819.2.g.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.g.c 12 3.b odd 2 1
819.2.g.c 12 13.b even 2 1
819.2.g.c 12 21.c even 2 1
819.2.g.c 12 91.b odd 2 1
819.2.g.d yes 12 1.a even 1 1 trivial
819.2.g.d yes 12 7.b odd 2 1 inner
819.2.g.d yes 12 39.d odd 2 1 inner
819.2.g.d yes 12 273.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{3} - 3 T^{2} - T + 5)^{4} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 18 T^{4} + 92 T^{2} + 104)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{10} + 95 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{3} + 10 T^{2} + 28 T + 20)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} + 34 T^{10} + 647 T^{8} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 116 T^{4} + 3696 T^{2} + \cdots - 20800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 80 T^{4} + 1920 T^{2} + \cdots - 13312)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 54 T^{4} + 716 T^{2} + 200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 86 T^{4} + 1100 T^{2} + \cdots + 2888)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 116 T^{4} + 1264 T^{2} + \cdots - 832)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 88 T^{4} + 1536 T^{2} + \cdots + 3200)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 114 T^{4} + 2396 T^{2} + \cdots + 2600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 2 T^{2} - 84 T - 104)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + 122 T^{4} + 2876 T^{2} + \cdots + 17576)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 166 T^{4} + 8844 T^{2} + \cdots + 150152)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 154 T^{4} + 5404 T^{2} + \cdots + 55016)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 232 T^{4} + 6304 T^{2} + \cdots + 41600)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( (T^{3} - 2 T^{2} - 44 T + 20)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 168 T^{4} + 8064 T^{2} + \cdots - 83200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 10 T^{2} - 60 T - 136)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 202 T^{4} + 9660 T^{2} + \cdots + 87464)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 178 T^{4} + 5628 T^{2} + \cdots + 17576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 172 T^{4} + 5264 T^{2} + \cdots - 20800)^{2} \) Copy content Toggle raw display
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