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Properties

Label 1140.2.b.f
Level $1140$
Weight $2$
Character orbit 1140.b
Analytic conductor $9.103$
Analytic rank $0$
Dimension $20$
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] = [1140,2,Mod(151,1140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("1140.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1140.b (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: \(9.10294583043\)
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: \( x^{20} - x^{19} + x^{18} - 3 x^{17} + 5 x^{16} - 3 x^{15} + 5 x^{14} - 7 x^{13} - 4 x^{12} + 2 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + x^{18} - 3 x^{17} + 5 x^{16} - 3 x^{15} + 5 x^{14} - 7 x^{13} - 4 x^{12} + 2 x^{11} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{19} + 3 \nu^{18} + 9 \nu^{17} + 49 \nu^{16} + 25 \nu^{15} + 17 \nu^{14} - 103 \nu^{13} + \cdots + 11776 ) / 2048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{19} - \nu^{18} - 3 \nu^{17} + 5 \nu^{16} - 3 \nu^{15} + 5 \nu^{14} - 19 \nu^{13} + 9 \nu^{12} + \cdots + 1536 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{19} - 7 \nu^{18} - 5 \nu^{17} - 45 \nu^{16} + 43 \nu^{15} - 45 \nu^{14} + 107 \nu^{13} + \cdots - 11776 ) / 2048 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{19} - 15 \nu^{18} + 3 \nu^{17} - 29 \nu^{16} + 43 \nu^{15} - 13 \nu^{14} + 11 \nu^{13} + \cdots - 512 ) / 2048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{19} + 2 \nu^{18} - 2 \nu^{17} - 4 \nu^{16} + 8 \nu^{15} + 32 \nu^{13} - 12 \nu^{12} + \cdots - 2816 ) / 512 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{19} + 9 \nu^{18} - 21 \nu^{17} - 13 \nu^{16} - 21 \nu^{15} + 19 \nu^{14} + 43 \nu^{13} + \cdots - 7680 ) / 2048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - \nu^{19} + \nu^{18} - \nu^{17} + 3 \nu^{16} - 5 \nu^{15} + 3 \nu^{14} - 5 \nu^{13} + 7 \nu^{12} + \cdots + 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{19} + \nu^{18} - \nu^{17} - \nu^{16} - \nu^{15} + 7 \nu^{14} - \nu^{13} + 3 \nu^{12} + \cdots - 512 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9 \nu^{19} + 17 \nu^{18} + 35 \nu^{17} - 13 \nu^{16} - 5 \nu^{15} - 61 \nu^{14} - 37 \nu^{13} + \cdots + 1536 ) / 2048 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11 \nu^{19} - 5 \nu^{18} + 17 \nu^{17} - 47 \nu^{16} + 9 \nu^{15} - 31 \nu^{14} - 23 \nu^{13} + \cdots - 3584 ) / 2048 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11 \nu^{19} + 11 \nu^{18} + 17 \nu^{17} - 47 \nu^{16} + 9 \nu^{15} - 31 \nu^{14} - 23 \nu^{13} + \cdots - 5632 ) / 2048 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 7 \nu^{19} - 7 \nu^{18} - 5 \nu^{17} - 33 \nu^{16} - \nu^{15} - 9 \nu^{14} + 47 \nu^{13} + \cdots - 7168 ) / 1024 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 13 \nu^{19} - 21 \nu^{18} - 15 \nu^{17} + 33 \nu^{16} - 7 \nu^{15} + 81 \nu^{14} - 39 \nu^{13} + \cdots + 6656 ) / 2048 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 7 \nu^{19} + 13 \nu^{18} - \nu^{17} + 23 \nu^{16} - 17 \nu^{15} - 9 \nu^{14} - 17 \nu^{13} + \cdots + 2560 ) / 1024 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 27 \nu^{19} - 5 \nu^{18} + \nu^{17} - 95 \nu^{16} - 7 \nu^{15} - 47 \nu^{14} + 89 \nu^{13} + \cdots - 17920 ) / 2048 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{19} - \beta_{18} - 2 \beta_{16} - \beta_{15} + \beta_{13} + \beta_{9} + 2 \beta_{6} - \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{19} - \beta_{17} + \beta_{14} - \beta_{13} - \beta_{11} - \beta_{7} - \beta_{5} - \beta_{4} + 2\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{19} - 2 \beta_{18} - \beta_{17} - 2 \beta_{16} - 2 \beta_{15} + \beta_{14} + \beta_{13} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{19} - 2 \beta_{18} + \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - 3 \beta_{14} + \beta_{13} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{19} + \beta_{18} - \beta_{17} + 5 \beta_{15} - 3 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2 \beta_{18} + 6 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} - \beta_{12} + 2 \beta_{11} - 4 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2 \beta_{18} + 6 \beta_{17} + 2 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} + 8 \beta_{10} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 16 \beta_{18} - 4 \beta_{17} - 8 \beta_{16} + 4 \beta_{15} - 12 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9 \beta_{19} - 13 \beta_{18} - 4 \beta_{17} - 10 \beta_{16} - 13 \beta_{15} - 12 \beta_{14} + 9 \beta_{13} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5 \beta_{19} + 4 \beta_{18} + 15 \beta_{17} + 4 \beta_{15} + 17 \beta_{14} - 5 \beta_{13} + 20 \beta_{12} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3 \beta_{19} - 18 \beta_{18} - 13 \beta_{17} - 34 \beta_{16} - 2 \beta_{15} + 5 \beta_{14} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 33 \beta_{19} - 54 \beta_{18} - 11 \beta_{17} - 50 \beta_{16} - 18 \beta_{15} - 63 \beta_{14} + \cdots - 125 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 10 \beta_{19} - 7 \beta_{18} + 15 \beta_{17} + 8 \beta_{16} + 61 \beta_{15} - 19 \beta_{14} - 30 \beta_{13} + \cdots - 46 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 60 \beta_{19} - 58 \beta_{18} - 40 \beta_{17} - 130 \beta_{16} - 26 \beta_{15} - 2 \beta_{14} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 4 \beta_{19} - 106 \beta_{18} - 46 \beta_{17} + 58 \beta_{15} - 98 \beta_{14} - 62 \beta_{12} + \cdots + 160 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times\).

\(n\) \(457\) \(571\) \(761\) \(781\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
−1.37956 − 0.311154i
−1.37956 + 0.311154i
−1.05532 − 0.941431i
−1.05532 + 0.941431i
−0.947529 − 1.04985i
−0.947529 + 1.04985i
−0.615692 − 1.27315i
−0.615692 + 1.27315i
−0.242808 − 1.39321i
−0.242808 + 1.39321i
0.198597 − 1.40020i
0.198597 + 1.40020i
0.787786 − 1.17448i
0.787786 + 1.17448i
1.15223 − 0.819979i
1.15223 + 0.819979i
1.19251 − 0.760212i
1.19251 + 0.760212i
1.40979 − 0.111768i
1.40979 + 0.111768i
−1.37956 − 0.311154i 1.00000 1.80637 + 0.858511i 1.00000 −1.37956 − 0.311154i − 0.407387i −2.22486 − 1.74642i 1.00000 −1.37956 − 0.311154i
151.2 −1.37956 + 0.311154i 1.00000 1.80637 − 0.858511i 1.00000 −1.37956 + 0.311154i 0.407387i −2.22486 + 1.74642i 1.00000 −1.37956 + 0.311154i
151.3 −1.05532 − 0.941431i 1.00000 0.227414 + 1.98703i 1.00000 −1.05532 − 0.941431i 4.12606i 1.63066 − 2.31105i 1.00000 −1.05532 − 0.941431i
151.4 −1.05532 + 0.941431i 1.00000 0.227414 − 1.98703i 1.00000 −1.05532 + 0.941431i − 4.12606i 1.63066 + 2.31105i 1.00000 −1.05532 + 0.941431i
151.5 −0.947529 − 1.04985i 1.00000 −0.204378 + 1.98953i 1.00000 −0.947529 − 1.04985i − 3.89458i 2.28237 − 1.67057i 1.00000 −0.947529 − 1.04985i
151.6 −0.947529 + 1.04985i 1.00000 −0.204378 − 1.98953i 1.00000 −0.947529 + 1.04985i 3.89458i 2.28237 + 1.67057i 1.00000 −0.947529 + 1.04985i
151.7 −0.615692 − 1.27315i 1.00000 −1.24185 + 1.56774i 1.00000 −0.615692 − 1.27315i − 3.45218i 2.76057 + 0.615816i 1.00000 −0.615692 − 1.27315i
151.8 −0.615692 + 1.27315i 1.00000 −1.24185 − 1.56774i 1.00000 −0.615692 + 1.27315i 3.45218i 2.76057 − 0.615816i 1.00000 −0.615692 + 1.27315i
151.9 −0.242808 − 1.39321i 1.00000 −1.88209 + 0.676567i 1.00000 −0.242808 − 1.39321i 0.221493i 1.39959 + 2.45788i 1.00000 −0.242808 − 1.39321i
151.10 −0.242808 + 1.39321i 1.00000 −1.88209 − 0.676567i 1.00000 −0.242808 + 1.39321i − 0.221493i 1.39959 − 2.45788i 1.00000 −0.242808 + 1.39321i
151.11 0.198597 − 1.40020i 1.00000 −1.92112 − 0.556150i 1.00000 0.198597 − 1.40020i 1.99302i −1.16025 + 2.57950i 1.00000 0.198597 − 1.40020i
151.12 0.198597 + 1.40020i 1.00000 −1.92112 + 0.556150i 1.00000 0.198597 + 1.40020i − 1.99302i −1.16025 − 2.57950i 1.00000 0.198597 + 1.40020i
151.13 0.787786 − 1.17448i 1.00000 −0.758786 − 1.85047i 1.00000 0.787786 − 1.17448i − 2.15810i −2.77109 − 0.566600i 1.00000 0.787786 − 1.17448i
151.14 0.787786 + 1.17448i 1.00000 −0.758786 + 1.85047i 1.00000 0.787786 + 1.17448i 2.15810i −2.77109 + 0.566600i 1.00000 0.787786 + 1.17448i
151.15 1.15223 − 0.819979i 1.00000 0.655267 − 1.88961i 1.00000 1.15223 − 0.819979i 4.94469i −0.794422 − 2.71457i 1.00000 1.15223 − 0.819979i
151.16 1.15223 + 0.819979i 1.00000 0.655267 + 1.88961i 1.00000 1.15223 + 0.819979i − 4.94469i −0.794422 + 2.71457i 1.00000 1.15223 + 0.819979i
151.17 1.19251 − 0.760212i 1.00000 0.844154 − 1.81312i 1.00000 1.19251 − 0.760212i − 0.235069i −0.371695 − 2.80390i 1.00000 1.19251 − 0.760212i
151.18 1.19251 + 0.760212i 1.00000 0.844154 + 1.81312i 1.00000 1.19251 + 0.760212i 0.235069i −0.371695 + 2.80390i 1.00000 1.19251 + 0.760212i
151.19 1.40979 − 0.111768i 1.00000 1.97502 − 0.315138i 1.00000 1.40979 − 0.111768i 2.55745i 2.74914 − 0.665022i 1.00000 1.40979 − 0.111768i
151.20 1.40979 + 0.111768i 1.00000 1.97502 + 0.315138i 1.00000 1.40979 + 0.111768i − 2.55745i 2.74914 + 0.665022i 1.00000 1.40979 + 0.111768i
\(n\): 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
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 1140.2.b.f yes 20
4.b odd 2 1 1140.2.b.e ✓ 20
19.b odd 2 1 1140.2.b.e ✓ 20
76.d even 2 1 inner 1140.2.b.f yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.b.e ✓ 20 4.b odd 2 1
1140.2.b.e ✓ 20 19.b odd 2 1
1140.2.b.f yes 20 1.a even 1 1 trivial
1140.2.b.f yes 20 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1140, [\chi])\):

\( T_{7}^{20} + 84 T_{7}^{18} + 2858 T_{7}^{16} + 50896 T_{7}^{14} + 510828 T_{7}^{12} + 2891712 T_{7}^{10} + \cdots + 4096 \) Copy content Toggle raw display
\( T_{31}^{10} - 10 T_{31}^{9} - 94 T_{31}^{8} + 1084 T_{31}^{7} + 1404 T_{31}^{6} - 34184 T_{31}^{5} + \cdots - 278528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( (T - 1)^{20} \) Copy content Toggle raw display
$5$ \( (T - 1)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 84 T^{18} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( T^{20} + 126 T^{18} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{20} + 152 T^{18} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( (T^{10} - 4 T^{9} + \cdots + 16768)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 9764601856 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 89865650176 \) Copy content Toggle raw display
$31$ \( (T^{10} - 10 T^{9} + \cdots - 278528)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 352 T^{18} + \cdots + 67108864 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 27673323175936 \) Copy content Toggle raw display
$43$ \( T^{20} + 316 T^{18} + \cdots + 67108864 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 23908581376 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 664394530816 \) Copy content Toggle raw display
$59$ \( (T^{10} + 4 T^{9} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 14 T^{9} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} - 14 T^{9} + \cdots - 10747904)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + 4 T^{9} + \cdots - 223608832)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 6 T^{9} + \cdots - 20107264)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + 16 T^{9} + \cdots + 67108864)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 46429941858304 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 2363229798400 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 43191604609024 \) Copy content Toggle raw display
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