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Properties

Label 228.2.k.a
Level $228$
Weight $2$
Character orbit 228.k
Analytic conductor $1.821$
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] = [228,2,Mod(31,228)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(228, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("228.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 228.k (of order \(6\), 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: \(1.82058916609\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 3 x^{18} - 2 x^{17} + x^{16} + 3 x^{14} - 12 x^{13} + 28 x^{12} - 24 x^{11} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 3 x^{18} - 2 x^{17} + x^{16} + 3 x^{14} - 12 x^{13} + 28 x^{12} - 24 x^{11} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{18} - 6 \nu^{14} - 6 \nu^{13} - 10 \nu^{12} - 16 \nu^{11} - 7 \nu^{10} + 10 \nu^{9} + \cdots - 448 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{19} + 6 \nu^{18} - 6 \nu^{17} - 2 \nu^{16} + 4 \nu^{13} + 66 \nu^{12} - 19 \nu^{11} + \cdots + 5120 ) / 1792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{19} - 60 \nu^{18} - 3 \nu^{17} - 120 \nu^{16} - 161 \nu^{15} - 182 \nu^{14} - 215 \nu^{13} + \cdots - 18944 ) / 7168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{19} - 9 \nu^{17} - 4 \nu^{16} + 13 \nu^{15} + 10 \nu^{14} + 3 \nu^{13} + 34 \nu^{12} + \cdots - 512 ) / 1024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3 \nu^{19} + 10 \nu^{18} - 10 \nu^{17} + 20 \nu^{16} - 42 \nu^{14} - 12 \nu^{13} + 12 \nu^{12} + \cdots + 6144 ) / 1792 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4 \nu^{19} + 4 \nu^{18} - 4 \nu^{17} + 15 \nu^{16} - 21 \nu^{14} + 12 \nu^{13} + 37 \nu^{12} + \cdots + 4608 ) / 1792 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4 \nu^{19} - 3 \nu^{18} - 4 \nu^{17} + \nu^{16} + 7 \nu^{14} + 54 \nu^{13} + 9 \nu^{12} + 118 \nu^{11} + \cdots + 1024 ) / 1792 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2 \nu^{19} + 5 \nu^{18} - 5 \nu^{17} + 10 \nu^{16} + 21 \nu^{15} + 29 \nu^{13} + 62 \nu^{12} + \cdots + 3968 ) / 896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{19} - 13 \nu^{17} - 4 \nu^{16} + \nu^{15} - 14 \nu^{14} - 17 \nu^{13} + 26 \nu^{12} + \cdots + 512 ) / 1024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{19} + 9 \nu^{18} - 9 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 15 \nu^{13} + 64 \nu^{12} + \cdots + 5888 ) / 1792 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 5 \nu^{19} + 9 \nu^{18} - 9 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 15 \nu^{13} + 64 \nu^{12} + \cdots + 5888 ) / 1792 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 6 \nu^{19} + 13 \nu^{18} - 6 \nu^{17} + 12 \nu^{16} + 14 \nu^{15} - 14 \nu^{14} - 52 \nu^{13} + \cdots + 1536 ) / 1792 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11 \nu^{19} + 38 \nu^{18} + 11 \nu^{17} - 22 \nu^{16} + 49 \nu^{15} - 28 \nu^{14} - 61 \nu^{13} + \cdots + 6144 ) / 3584 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13 \nu^{19} - 34 \nu^{18} - 15 \nu^{17} - 26 \nu^{16} - 21 \nu^{15} - 28 \nu^{14} - 39 \nu^{13} + \cdots - 8704 ) / 3584 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( \nu^{19} + \nu^{18} - \nu^{17} + 3 \nu^{16} + \nu^{15} - \nu^{14} + 5 \nu^{13} + 13 \nu^{12} + \cdots + 768 ) / 256 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( \nu^{19} - 2 \nu^{18} + 3 \nu^{17} - 2 \nu^{16} + \nu^{15} + 3 \nu^{13} - 12 \nu^{12} + 28 \nu^{11} + \cdots - 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 8 \nu^{19} - \nu^{18} + \nu^{17} - 9 \nu^{16} + 7 \nu^{15} + 21 \nu^{14} - 17 \nu^{13} + \cdots - 2048 ) / 896 \) 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
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{18} - \beta_{17} + \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + \beta_{16} + \beta_{15} - \beta_{14} + \beta_{13} - \beta_{10} - \beta_{9} + \cdots - \beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - \beta_{15} - 3 \beta_{12} + \beta_{11} + \beta_{8} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4 \beta_{19} + 2 \beta_{17} + 2 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{11} + 6 \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{18} - \beta_{17} - 4 \beta_{16} - 2 \beta_{15} - 3 \beta_{14} - 4 \beta_{13} + 4 \beta_{12} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{19} + 6 \beta_{18} - 2 \beta_{17} - \beta_{16} - 5 \beta_{15} - \beta_{14} - \beta_{13} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{19} - \beta_{18} - 9 \beta_{17} + \beta_{16} - \beta_{15} + 4 \beta_{14} - 5 \beta_{12} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6 \beta_{19} + 2 \beta_{18} - 4 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2 \beta_{19} + 5 \beta_{18} + 21 \beta_{17} + 14 \beta_{16} + 10 \beta_{15} + 9 \beta_{14} + 4 \beta_{13} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - \beta_{19} + 20 \beta_{17} - 11 \beta_{16} + \beta_{15} - 33 \beta_{14} - 23 \beta_{13} + 12 \beta_{12} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13 \beta_{19} + 15 \beta_{18} + 9 \beta_{17} - 13 \beta_{16} - 21 \beta_{15} - 20 \beta_{14} - 4 \beta_{13} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 4 \beta_{19} - 8 \beta_{18} - 50 \beta_{17} - 8 \beta_{16} - 4 \beta_{15} + 38 \beta_{14} - 39 \beta_{13} + \cdots + 60 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 56 \beta_{19} + 37 \beta_{18} - 21 \beta_{17} + 44 \beta_{16} + 6 \beta_{15} + 9 \beta_{14} + \cdots - 52 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7 \beta_{19} + 14 \beta_{18} + 42 \beta_{17} + 15 \beta_{16} + 23 \beta_{15} + 11 \beta_{14} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 5 \beta_{19} - 69 \beta_{18} + 75 \beta_{17} + 17 \beta_{16} + 103 \beta_{15} + 72 \beta_{14} + \cdots - 66 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 182 \beta_{19} - 150 \beta_{18} + 24 \beta_{17} + 70 \beta_{16} + 150 \beta_{15} - 54 \beta_{14} + \cdots + 176 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/228\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\) \(115\)
\(\chi(n)\) \(1\) \(1 - \beta_{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.818463 − 1.15331i
0.478868 − 1.33067i
1.32127 − 0.504239i
−0.874835 − 1.11115i
1.34699 + 0.430844i
−1.25575 − 0.650459i
0.894531 + 1.09536i
−1.34963 + 0.422503i
−0.141961 + 1.40707i
−0.237943 + 1.39405i
0.818463 + 1.15331i
0.478868 + 1.33067i
1.32127 + 0.504239i
−0.874835 + 1.11115i
1.34699 − 0.430844i
−1.25575 + 0.650459i
0.894531 − 1.09536i
−1.34963 − 0.422503i
−0.141961 − 1.40707i
−0.237943 − 1.39405i
−1.40803 − 0.132156i −0.500000 − 0.866025i 1.96507 + 0.372157i −2.18826 − 3.79018i 0.589563 + 1.28546i 0.239504i −2.71769 − 0.783701i −0.500000 + 0.866025i 2.58024 + 5.62586i
31.2 −1.39183 + 0.250623i −0.500000 − 0.866025i 1.87438 − 0.697649i 1.38652 + 2.40152i 0.912961 + 1.08005i − 0.149639i −2.43396 + 1.44077i −0.500000 + 0.866025i −2.53167 − 2.99501i
31.3 −1.09732 − 0.892130i −0.500000 − 0.866025i 0.408207 + 1.95790i 0.127074 + 0.220099i −0.223949 + 1.39637i − 1.43470i 1.29877 − 2.51261i −0.500000 + 0.866025i 0.0569164 − 0.354885i
31.4 −0.524870 + 1.31321i −0.500000 − 0.866025i −1.44902 − 1.37853i −0.490176 − 0.849009i 1.39971 − 0.202053i 4.28530i 2.57084 − 1.17932i −0.500000 + 0.866025i 1.37220 − 0.198083i
31.5 −0.300371 − 1.38195i −0.500000 − 0.866025i −1.81955 + 0.830194i 1.45389 + 2.51821i −1.04662 + 0.951102i 5.22116i 1.69383 + 2.26516i −0.500000 + 0.866025i 3.04332 − 2.76559i
31.6 0.0645600 + 1.41274i −0.500000 − 0.866025i −1.99166 + 0.182413i −0.542680 − 0.939950i 1.19119 − 0.762280i − 4.07547i −0.386284 − 2.80193i −0.500000 + 0.866025i 1.29287 − 0.827349i
31.7 0.501344 − 1.32237i −0.500000 − 0.866025i −1.49731 − 1.32592i −0.720511 − 1.24796i −1.39588 + 0.227006i − 1.30023i −2.50402 + 1.31525i −0.500000 + 0.866025i −2.01149 + 0.327121i
31.8 1.04071 + 0.957560i −0.500000 − 0.866025i 0.166160 + 1.99309i 0.984694 + 1.70554i 0.308915 − 1.38006i 0.355075i −1.73557 + 2.23333i −0.500000 + 0.866025i −0.608374 + 2.71788i
31.9 1.28954 − 0.580593i −0.500000 − 0.866025i 1.32582 − 1.49740i −1.44325 − 2.49978i −1.14758 − 0.826477i 2.96861i 0.840324 − 2.70071i −0.500000 + 0.866025i −3.31248 − 2.38562i
31.10 1.32626 − 0.490961i −0.500000 − 0.866025i 1.51791 − 1.30228i 1.43271 + 2.48152i −1.08831 − 0.903091i − 2.64551i 1.37377 − 2.47240i −0.500000 + 0.866025i 3.11846 + 2.58773i
103.1 −1.40803 + 0.132156i −0.500000 + 0.866025i 1.96507 − 0.372157i −2.18826 + 3.79018i 0.589563 − 1.28546i − 0.239504i −2.71769 + 0.783701i −0.500000 − 0.866025i 2.58024 − 5.62586i
103.2 −1.39183 − 0.250623i −0.500000 + 0.866025i 1.87438 + 0.697649i 1.38652 − 2.40152i 0.912961 − 1.08005i 0.149639i −2.43396 − 1.44077i −0.500000 − 0.866025i −2.53167 + 2.99501i
103.3 −1.09732 + 0.892130i −0.500000 + 0.866025i 0.408207 − 1.95790i 0.127074 − 0.220099i −0.223949 − 1.39637i 1.43470i 1.29877 + 2.51261i −0.500000 − 0.866025i 0.0569164 + 0.354885i
103.4 −0.524870 − 1.31321i −0.500000 + 0.866025i −1.44902 + 1.37853i −0.490176 + 0.849009i 1.39971 + 0.202053i − 4.28530i 2.57084 + 1.17932i −0.500000 − 0.866025i 1.37220 + 0.198083i
103.5 −0.300371 + 1.38195i −0.500000 + 0.866025i −1.81955 − 0.830194i 1.45389 − 2.51821i −1.04662 − 0.951102i − 5.22116i 1.69383 − 2.26516i −0.500000 − 0.866025i 3.04332 + 2.76559i
103.6 0.0645600 − 1.41274i −0.500000 + 0.866025i −1.99166 − 0.182413i −0.542680 + 0.939950i 1.19119 + 0.762280i 4.07547i −0.386284 + 2.80193i −0.500000 − 0.866025i 1.29287 + 0.827349i
103.7 0.501344 + 1.32237i −0.500000 + 0.866025i −1.49731 + 1.32592i −0.720511 + 1.24796i −1.39588 − 0.227006i 1.30023i −2.50402 − 1.31525i −0.500000 − 0.866025i −2.01149 − 0.327121i
103.8 1.04071 − 0.957560i −0.500000 + 0.866025i 0.166160 − 1.99309i 0.984694 − 1.70554i 0.308915 + 1.38006i − 0.355075i −1.73557 − 2.23333i −0.500000 − 0.866025i −0.608374 − 2.71788i
103.9 1.28954 + 0.580593i −0.500000 + 0.866025i 1.32582 + 1.49740i −1.44325 + 2.49978i −1.14758 + 0.826477i − 2.96861i 0.840324 + 2.70071i −0.500000 − 0.866025i −3.31248 + 2.38562i
103.10 1.32626 + 0.490961i −0.500000 + 0.866025i 1.51791 + 1.30228i 1.43271 − 2.48152i −1.08831 + 0.903091i 2.64551i 1.37377 + 2.47240i −0.500000 − 0.866025i 3.11846 − 2.58773i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 228.2.k.a ✓ 20
3.b odd 2 1 684.2.r.c 20
4.b odd 2 1 228.2.k.b yes 20
12.b even 2 1 684.2.r.b 20
19.d odd 6 1 228.2.k.b yes 20
57.f even 6 1 684.2.r.b 20
76.f even 6 1 inner 228.2.k.a ✓ 20
228.n odd 6 1 684.2.r.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.k.a ✓ 20 1.a even 1 1 trivial
228.2.k.a ✓ 20 76.f even 6 1 inner
228.2.k.b yes 20 4.b odd 2 1
228.2.k.b yes 20 19.d odd 6 1
684.2.r.b 20 12.b even 2 1
684.2.r.b 20 57.f even 6 1
684.2.r.c 20 3.b odd 2 1
684.2.r.c 20 228.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23}^{20} - 78 T_{23}^{18} + 4714 T_{23}^{16} - 16284 T_{23}^{15} - 74860 T_{23}^{14} + 467976 T_{23}^{13} + \cdots + 295936 \) acting on \(S_{2}^{\mathrm{new}}(228, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + T^{19} + \cdots + 1024 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + 30 T^{18} + \cdots + 50176 \) Copy content Toggle raw display
$7$ \( T^{20} + 82 T^{18} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{20} + 116 T^{18} + \cdots + 10445824 \) Copy content Toggle raw display
$13$ \( T^{20} - 6 T^{19} + \cdots + 117649 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 1107558400 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} - 78 T^{18} + \cdots + 295936 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 708837376 \) Copy content Toggle raw display
$31$ \( (T^{10} + 6 T^{9} + \cdots + 133909)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 22559739601 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 124084289536 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 11511112196809 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 94686674944 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 876665877799936 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 437545206784 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 55822580045401 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 17\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 78\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 70\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 80\!\cdots\!36 \) Copy content Toggle raw display
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