Properties

Label 380.2.f.b
Level $380$
Weight $2$
Character orbit 380.f
Analytic conductor $3.034$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [380,2,Mod(151,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.f (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} - x^{16} + 5x^{14} - 4x^{12} - 8x^{10} - 16x^{8} + 80x^{6} - 64x^{4} - 256x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{15} q^{2} + \beta_{9} q^{3} - \beta_{19} q^{4} + q^{5} - \beta_{10} q^{6} - \beta_{4} q^{7} + ( - \beta_{11} - \beta_{9}) q^{8} + ( - \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{15} q^{2} + \beta_{9} q^{3} - \beta_{19} q^{4} + q^{5} - \beta_{10} q^{6} - \beta_{4} q^{7} + ( - \beta_{11} - \beta_{9}) q^{8} + ( - \beta_{5} + 1) q^{9} - \beta_{15} q^{10} + ( - \beta_{14} + \beta_{13} + \beta_{4}) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{12} + \beta_{3} q^{13} + ( - \beta_{9} + \beta_{8} - \beta_{7}) q^{14} + \beta_{9} q^{15} + (\beta_{13} + \beta_{10} + \beta_{5}) q^{16} + (\beta_{19} + \beta_{17} + \cdots - \beta_{10}) q^{17}+ \cdots + (2 \beta_{19} - 2 \beta_{16} + \cdots + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 20 q^{5} + 2 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} + 20 q^{5} + 2 q^{6} + 16 q^{9} + 6 q^{16} - 12 q^{17} + 2 q^{20} - 38 q^{24} + 20 q^{25} + 2 q^{26} - 6 q^{28} + 2 q^{30} - 12 q^{36} - 30 q^{38} - 42 q^{42} + 8 q^{44} + 16 q^{45} - 38 q^{54} - 4 q^{57} - 2 q^{58} - 40 q^{61} + 48 q^{62} - 22 q^{64} + 48 q^{66} - 10 q^{68} - 36 q^{73} - 36 q^{74} + 36 q^{76} + 48 q^{77} + 6 q^{80} - 4 q^{81} + 32 q^{82} - 12 q^{85} + 2 q^{92} - 16 q^{93} + 46 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{18} - x^{16} + 5x^{14} - 4x^{12} - 8x^{10} - 16x^{8} + 80x^{6} - 64x^{4} - 256x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} + \nu^{5} + 2\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{19} - 11 \nu^{17} - 3 \nu^{15} + 87 \nu^{13} + 152 \nu^{11} + 168 \nu^{9} + 240 \nu^{7} + \cdots - 1280 \nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{18} + 11 \nu^{16} + 3 \nu^{14} + 41 \nu^{12} - 24 \nu^{10} + 216 \nu^{8} + 144 \nu^{6} + \cdots - 768 ) / 2048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{18} - 11 \nu^{16} - 3 \nu^{14} - 41 \nu^{12} + 24 \nu^{10} + 296 \nu^{8} - 144 \nu^{6} + \cdots + 768 ) / 2048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{18} + 11 \nu^{16} + 3 \nu^{14} - 87 \nu^{12} - 152 \nu^{10} - 168 \nu^{8} - 240 \nu^{6} + \cdots + 1280 ) / 2048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{19} + \nu^{17} + 41 \nu^{15} + 27 \nu^{13} + 152 \nu^{11} - 120 \nu^{9} - 208 \nu^{7} + \cdots + 3840 \nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{19} + \nu^{17} - 23 \nu^{15} - 37 \nu^{13} + 88 \nu^{11} - 56 \nu^{9} - 848 \nu^{7} + \cdots + 1792 \nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{19} + \nu^{17} - 23 \nu^{15} - 37 \nu^{13} + 88 \nu^{11} - 56 \nu^{9} + 176 \nu^{7} + \cdots - 256 \nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3 \nu^{18} - \nu^{16} + 23 \nu^{14} + 37 \nu^{12} - 88 \nu^{10} + 56 \nu^{8} - 176 \nu^{6} + \cdots + 256 ) / 2048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{19} + 23 \nu^{17} - 17 \nu^{15} + 45 \nu^{13} + 40 \nu^{11} - 136 \nu^{9} - 560 \nu^{7} + \cdots - 3840 \nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{18} - \nu^{16} - 9 \nu^{14} - 3 \nu^{12} - 12 \nu^{10} + 128 \nu^{7} + 32 \nu^{6} + 48 \nu^{4} + \cdots - 256 ) / 512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{18} - 3\nu^{16} + 5\nu^{14} - \nu^{12} - 16\nu^{10} - 40\nu^{8} + 176\nu^{6} - 80\nu^{4} - 768\nu^{2} + 768 ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5 \nu^{18} + 9 \nu^{16} - 15 \nu^{14} + 51 \nu^{12} - 8 \nu^{10} - 120 \nu^{8} + 432 \nu^{6} + \cdots - 256 ) / 2048 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{19} - \nu^{17} - \nu^{15} + 5\nu^{13} - 4\nu^{11} - 8\nu^{9} - 16\nu^{7} + 80\nu^{5} - 64\nu^{3} - 256\nu ) / 512 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 7 \nu^{18} - 3 \nu^{16} + 5 \nu^{14} + 15 \nu^{12} + 104 \nu^{10} - 120 \nu^{8} + 512 \nu^{7} + \cdots - 1280 ) / 2048 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 7 \nu^{18} + 3 \nu^{16} - 5 \nu^{14} - 15 \nu^{12} - 104 \nu^{10} + 120 \nu^{8} + 512 \nu^{7} + \cdots + 1280 ) / 2048 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 5 \nu^{19} + 9 \nu^{17} - 15 \nu^{15} + 51 \nu^{13} - 8 \nu^{11} - 120 \nu^{9} - 80 \nu^{7} + \cdots - 3328 \nu ) / 2048 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( \nu^{18} - \nu^{16} - \nu^{14} + 5\nu^{12} - 4\nu^{10} - 8\nu^{8} - 16\nu^{6} + 80\nu^{4} - 64\nu^{2} - 256 ) / 256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{19} - \beta_{17} + \beta_{12} + \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{17} + \beta_{16} - 2\beta_{9} + 2\beta_{8} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{17} + \beta_{16} + 2\beta_{14} + \beta_{10} + \beta_{6} + 2\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{18} + 2\beta_{15} - 2\beta_{11} + 2\beta_{9} + 2\beta_{7} - 2\beta_{3} + 2\beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{19} - 3\beta_{17} + 4\beta_{13} + 3\beta_{12} - \beta_{10} + 4\beta_{4} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{17} + 3\beta_{16} + 2\beta_{9} - 2\beta_{8} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2 \beta_{19} + 3 \beta_{17} - \beta_{16} - 2 \beta_{14} - 2 \beta_{12} - 3 \beta_{10} + \cdots + 8 \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2 \beta_{18} - 2 \beta_{17} - 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{11} + 2 \beta_{9} + \cdots - 5 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 13 \beta_{19} - 3 \beta_{17} + 6 \beta_{16} - 4 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 9 \beta_{10} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4 \beta_{18} - 3 \beta_{17} - 3 \beta_{16} - 20 \beta_{15} + 4 \beta_{11} + 26 \beta_{9} + \cdots - 5 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 12 \beta_{19} - 5 \beta_{17} + \beta_{16} + 18 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} + 29 \beta_{10} + \cdots + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 18 \beta_{18} + 4 \beta_{17} + 4 \beta_{16} + 34 \beta_{15} + 14 \beta_{11} - 38 \beta_{9} + \cdots + 31 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 39 \beta_{19} - 27 \beta_{17} + 52 \beta_{16} - 24 \beta_{14} + 36 \beta_{13} - 25 \beta_{12} + \cdots - 68 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 24 \beta_{18} - 25 \beta_{17} - 25 \beta_{16} - 24 \beta_{15} - 8 \beta_{11} - 30 \beta_{9} + \cdots - 57 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 26 \beta_{19} + 35 \beta_{17} + 35 \beta_{16} + 38 \beta_{14} - 48 \beta_{13} - 70 \beta_{12} + \cdots + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 38 \beta_{18} - 70 \beta_{17} - 70 \beta_{16} - 122 \beta_{15} + 218 \beta_{11} + 274 \beta_{9} + \cdots + 99 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 239 \beta_{19} + 13 \beta_{17} + 18 \beta_{16} - 268 \beta_{14} + 76 \beta_{13} - 31 \beta_{12} + \cdots + 244 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 268 \beta_{18} - 31 \beta_{17} - 31 \beta_{16} + 452 \beta_{15} + 332 \beta_{11} + 298 \beta_{9} + \cdots + 83 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
−1.40191 + 0.186121i
−1.40191 0.186121i
−1.27722 + 0.607219i
−1.27722 0.607219i
−1.10555 + 0.881911i
−1.10555 0.881911i
−0.641680 + 1.26026i
−0.641680 1.26026i
−0.139169 + 1.40735i
−0.139169 1.40735i
0.139169 + 1.40735i
0.139169 1.40735i
0.641680 + 1.26026i
0.641680 1.26026i
1.10555 + 0.881911i
1.10555 0.881911i
1.27722 + 0.607219i
1.27722 0.607219i
1.40191 + 0.186121i
1.40191 0.186121i
−1.40191 0.186121i −0.448438 1.93072 + 0.521850i 1.00000 0.628671 + 0.0834636i 4.24611i −2.60957 1.09093i −2.79890 −1.40191 0.186121i
151.2 −1.40191 + 0.186121i −0.448438 1.93072 0.521850i 1.00000 0.628671 0.0834636i 4.24611i −2.60957 + 1.09093i −2.79890 −1.40191 + 0.186121i
151.3 −1.27722 0.607219i 2.69112 1.26257 + 1.55110i 1.00000 −3.43715 1.63410i 0.164869i −0.670720 2.74775i 4.24214 −1.27722 0.607219i
151.4 −1.27722 + 0.607219i 2.69112 1.26257 1.55110i 1.00000 −3.43715 + 1.63410i 0.164869i −0.670720 + 2.74775i 4.24214 −1.27722 + 0.607219i
151.5 −1.10555 0.881911i −1.47694 0.444468 + 1.94999i 1.00000 1.63283 + 1.30253i 1.71604i 1.22833 2.54778i −0.818651 −1.10555 0.881911i
151.6 −1.10555 + 0.881911i −1.47694 0.444468 1.94999i 1.00000 1.63283 1.30253i 1.71604i 1.22833 + 2.54778i −0.818651 −1.10555 + 0.881911i
151.7 −0.641680 1.26026i −2.85265 −1.17649 + 1.61736i 1.00000 1.83049 + 3.59507i 2.20103i 2.79323 + 0.444852i 5.13760 −0.641680 1.26026i
151.8 −0.641680 + 1.26026i −2.85265 −1.17649 1.61736i 1.00000 1.83049 3.59507i 2.20103i 2.79323 0.444852i 5.13760 −0.641680 + 1.26026i
151.9 −0.139169 1.40735i 1.11257 −1.96126 + 0.391719i 1.00000 −0.154836 1.56578i 3.02556i 0.824234 + 2.70567i −1.76218 −0.139169 1.40735i
151.10 −0.139169 + 1.40735i 1.11257 −1.96126 0.391719i 1.00000 −0.154836 + 1.56578i 3.02556i 0.824234 2.70567i −1.76218 −0.139169 + 1.40735i
151.11 0.139169 1.40735i −1.11257 −1.96126 0.391719i 1.00000 −0.154836 + 1.56578i 3.02556i −0.824234 + 2.70567i −1.76218 0.139169 1.40735i
151.12 0.139169 + 1.40735i −1.11257 −1.96126 + 0.391719i 1.00000 −0.154836 1.56578i 3.02556i −0.824234 2.70567i −1.76218 0.139169 + 1.40735i
151.13 0.641680 1.26026i 2.85265 −1.17649 1.61736i 1.00000 1.83049 3.59507i 2.20103i −2.79323 + 0.444852i 5.13760 0.641680 1.26026i
151.14 0.641680 + 1.26026i 2.85265 −1.17649 + 1.61736i 1.00000 1.83049 + 3.59507i 2.20103i −2.79323 0.444852i 5.13760 0.641680 + 1.26026i
151.15 1.10555 0.881911i 1.47694 0.444468 1.94999i 1.00000 1.63283 1.30253i 1.71604i −1.22833 2.54778i −0.818651 1.10555 0.881911i
151.16 1.10555 + 0.881911i 1.47694 0.444468 + 1.94999i 1.00000 1.63283 + 1.30253i 1.71604i −1.22833 + 2.54778i −0.818651 1.10555 + 0.881911i
151.17 1.27722 0.607219i −2.69112 1.26257 1.55110i 1.00000 −3.43715 + 1.63410i 0.164869i 0.670720 2.74775i 4.24214 1.27722 0.607219i
151.18 1.27722 + 0.607219i −2.69112 1.26257 + 1.55110i 1.00000 −3.43715 1.63410i 0.164869i 0.670720 + 2.74775i 4.24214 1.27722 + 0.607219i
151.19 1.40191 0.186121i 0.448438 1.93072 0.521850i 1.00000 0.628671 0.0834636i 4.24611i 2.60957 1.09093i −2.79890 1.40191 0.186121i
151.20 1.40191 + 0.186121i 0.448438 1.93072 + 0.521850i 1.00000 0.628671 + 0.0834636i 4.24611i 2.60957 + 1.09093i −2.79890 1.40191 + 0.186121i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.f.b 20
4.b odd 2 1 inner 380.2.f.b 20
19.b odd 2 1 inner 380.2.f.b 20
76.d even 2 1 inner 380.2.f.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.f.b 20 1.a even 1 1 trivial
380.2.f.b 20 4.b odd 2 1 inner
380.2.f.b 20 19.b odd 2 1 inner
380.2.f.b 20 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 19T_{3}^{8} + 118T_{3}^{6} - 266T_{3}^{4} + 208T_{3}^{2} - 32 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( (T^{10} - 19 T^{8} + \cdots - 32)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + 35 T^{8} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + 68 T^{8} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 49 T^{8} + \cdots + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 3 T^{4} + \cdots + 256)^{4} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( (T^{10} + 115 T^{8} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 141 T^{8} + \cdots + 2097152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 192 T^{8} + \cdots - 524288)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + 280 T^{8} + \cdots + 6595712)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + 248 T^{8} + \cdots + 33554432)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 168 T^{8} + \cdots + 16128256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 320 T^{8} + \cdots + 17707264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 257 T^{8} + \cdots + 8096288)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 459 T^{8} + \cdots - 246597632)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 10 T^{4} + \cdots + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} - 243 T^{8} + \cdots - 110826272)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 240 T^{8} + \cdots - 524288)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 9 T^{4} + \cdots + 112)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} - 204 T^{8} + \cdots - 69337088)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + 308 T^{8} + \cdots + 123298816)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 564 T^{8} + \cdots + 553779200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 584 T^{8} + \cdots + 11254800512)^{2} \) Copy content Toggle raw display
show more
show less