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Properties

Label 330.2.j.a
Level $330$
Weight $2$
Character orbit 330.j
Analytic conductor $2.635$
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] = [330,2,Mod(23,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.j (of order \(4\), 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: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 18 x^{18} - 52 x^{17} + 146 x^{16} - 348 x^{15} + 794 x^{14} - 1652 x^{13} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{3} q^{3} - \beta_{14} q^{4} + ( - \beta_{18} - \beta_{14} + \cdots + \beta_{6}) q^{5}+ \cdots + (\beta_{18} + \beta_{14} - \beta_{13} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{3} q^{3} - \beta_{14} q^{4} + ( - \beta_{18} - \beta_{14} + \cdots + \beta_{6}) q^{5}+ \cdots + (\beta_{16} - \beta_{15} - \beta_{14} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{6} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{6} + 20 q^{9} - 4 q^{12} + 8 q^{13} - 8 q^{15} - 20 q^{16} + 8 q^{17} - 8 q^{18} - 12 q^{20} + 36 q^{23} + 12 q^{25} - 16 q^{29} + 32 q^{30} + 16 q^{31} - 4 q^{33} - 8 q^{35} - 4 q^{36} - 4 q^{37} + 32 q^{39} - 8 q^{40} - 16 q^{42} - 20 q^{44} + 20 q^{45} - 24 q^{46} + 12 q^{47} - 16 q^{50} - 16 q^{51} - 8 q^{52} + 20 q^{53} - 12 q^{54} - 12 q^{55} - 28 q^{57} - 8 q^{58} - 112 q^{59} + 24 q^{60} + 8 q^{61} + 16 q^{62} - 8 q^{63} - 56 q^{65} + 20 q^{67} - 8 q^{68} + 28 q^{69} - 8 q^{70} + 8 q^{72} - 8 q^{73} + 8 q^{74} + 8 q^{75} - 16 q^{76} + 36 q^{78} + 24 q^{82} + 16 q^{83} - 12 q^{84} - 72 q^{87} - 32 q^{89} + 40 q^{90} + 36 q^{92} - 76 q^{93} - 4 q^{96} - 44 q^{97} - 56 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 18 x^{18} - 52 x^{17} + 146 x^{16} - 348 x^{15} + 794 x^{14} - 1652 x^{13} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 173095 \nu^{19} + 3693562 \nu^{18} - 14861805 \nu^{17} + 50267200 \nu^{16} + \cdots + 54968950296 ) / 12880555200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 983 \nu^{19} - 42193 \nu^{18} + 106597 \nu^{17} - 480353 \nu^{16} + 1166177 \nu^{15} + \cdots - 773749665 ) / 57246912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 260987 \nu^{19} + 1143137 \nu^{18} - 2032065 \nu^{17} + 2278079 \nu^{16} + \cdots - 30981888369 ) / 6440277600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{19} - 4 \nu^{18} + 18 \nu^{17} - 52 \nu^{16} + 146 \nu^{15} - 348 \nu^{14} + 794 \nu^{13} + \cdots - 78732 ) / 19683 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 731789 \nu^{19} - 4063298 \nu^{18} + 12444615 \nu^{17} - 41110508 \nu^{16} + \cdots - 33748235604 ) / 12880555200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 126238 \nu^{19} - 80843 \nu^{18} - 339720 \nu^{17} - 907109 \nu^{16} + 2655562 \nu^{15} + \cdots - 4801267629 ) / 1431172800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1714588 \nu^{19} + 4662985 \nu^{18} - 18672690 \nu^{17} + 51824731 \nu^{16} + \cdots + 31312995795 ) / 12880555200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2254477 \nu^{19} - 6349861 \nu^{18} + 28141725 \nu^{17} - 83894329 \nu^{16} + \cdots - 86938452873 ) / 12880555200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 930904 \nu^{19} - 3550521 \nu^{18} + 13062710 \nu^{17} - 33545203 \nu^{16} + 85644784 \nu^{15} + \cdots + 5120972037 ) / 4293518400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1000394 \nu^{19} - 3915365 \nu^{18} + 13755420 \nu^{17} - 36857603 \nu^{16} + 93282082 \nu^{15} + \cdots + 3407028885 ) / 4293518400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1523201 \nu^{19} - 11337446 \nu^{18} + 37908045 \nu^{17} - 124245242 \nu^{16} + \cdots - 102633184998 ) / 6440277600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1125907 \nu^{19} - 8583711 \nu^{18} + 30452675 \nu^{17} - 104149339 \nu^{16} + \cdots - 115551301923 ) / 4293518400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 46125 \nu^{19} + 124291 \nu^{18} - 531469 \nu^{17} + 1309695 \nu^{16} - 3440725 \nu^{15} + \cdots + 58045167 ) / 171740736 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 4301663 \nu^{19} + 23224163 \nu^{18} - 80765535 \nu^{17} + 246340871 \nu^{16} + \cdots + 162965260719 ) / 12880555200 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 60209 \nu^{19} + 298781 \nu^{18} - 1088805 \nu^{17} + 3293525 \nu^{16} - 8429545 \nu^{15} + \cdots + 2723635125 ) / 171740736 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 772207 \nu^{19} + 2663811 \nu^{18} - 8992325 \nu^{17} + 24390289 \nu^{16} - 58660963 \nu^{15} + \cdots + 953923473 ) / 2146759200 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 5038949 \nu^{19} - 17291429 \nu^{18} + 75798405 \nu^{17} - 201300233 \nu^{16} + \cdots - 38893981977 ) / 12880555200 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 3463327 \nu^{19} - 13222186 \nu^{18} + 50632275 \nu^{17} - 144403954 \nu^{16} + \cdots - 65361456198 ) / 6440277600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{19} + \beta_{18} - \beta_{15} + \beta_{14} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} + 2\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} - 2 \beta_{14} - 2 \beta_{12} + \beta_{10} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{19} - 2 \beta_{18} - \beta_{17} + \beta_{16} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{19} + \beta_{18} + 2 \beta_{17} + \beta_{16} + 2 \beta_{15} - 2 \beta_{14} - \beta_{13} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{19} + 5 \beta_{17} - 2 \beta_{16} + 7 \beta_{15} - 5 \beta_{14} - 4 \beta_{13} - 5 \beta_{12} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{19} + 5 \beta_{18} + 4 \beta_{17} - 18 \beta_{15} + 22 \beta_{14} - 3 \beta_{13} - 4 \beta_{12} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{19} - 3 \beta_{18} + 22 \beta_{16} - 12 \beta_{15} - 17 \beta_{14} - 5 \beta_{13} + 7 \beta_{12} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25 \beta_{19} - 15 \beta_{17} - 2 \beta_{16} + 67 \beta_{15} + 6 \beta_{14} - 14 \beta_{13} + 32 \beta_{12} + \cdots - 104 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 65 \beta_{19} + 22 \beta_{18} - 11 \beta_{17} - 27 \beta_{16} + 35 \beta_{15} + 105 \beta_{14} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 17 \beta_{19} - 3 \beta_{18} - 67 \beta_{17} - 35 \beta_{16} + 49 \beta_{15} - 92 \beta_{14} + 23 \beta_{13} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 47 \beta_{19} + 31 \beta_{18} + 143 \beta_{17} - 57 \beta_{16} - 67 \beta_{15} - 104 \beta_{14} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 125 \beta_{19} + 11 \beta_{18} + 347 \beta_{17} - 85 \beta_{16} - 109 \beta_{15} - 12 \beta_{14} + \cdots + 504 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 144 \beta_{19} + 180 \beta_{18} - 811 \beta_{17} - 507 \beta_{16} + 380 \beta_{15} + 1199 \beta_{14} + \cdots - 959 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1136 \beta_{19} - 66 \beta_{18} - 683 \beta_{17} + 588 \beta_{16} - 878 \beta_{15} + 206 \beta_{14} + \cdots + 136 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1321 \beta_{19} - 719 \beta_{18} + 112 \beta_{17} - 402 \beta_{16} + 77 \beta_{15} - 5071 \beta_{14} + \cdots + 313 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1300 \beta_{19} - 534 \beta_{18} + 1961 \beta_{17} - 5046 \beta_{16} + 785 \beta_{15} + 1198 \beta_{14} + \cdots + 1870 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 4275 \beta_{19} - 3179 \beta_{18} + 6536 \beta_{17} + 5095 \beta_{16} - 4740 \beta_{15} - 1139 \beta_{14} + \cdots + 8156 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1676 \beta_{19} + 672 \beta_{18} - 10727 \beta_{17} + 6077 \beta_{16} + 1715 \beta_{15} - 6266 \beta_{14} + \cdots - 10682 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(211\) \(221\)
\(\chi(n)\) \(\beta_{14}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.136880 − 1.72663i
−0.580307 − 1.63194i
1.61547 − 0.624716i
0.756111 + 1.55830i
−0.221043 + 1.71789i
0.862185 − 1.50221i
−1.12724 − 1.31504i
1.56031 + 0.751945i
−1.35653 + 1.07696i
0.354163 + 1.69546i
0.136880 + 1.72663i
−0.580307 + 1.63194i
1.61547 + 0.624716i
0.756111 − 1.55830i
−0.221043 − 1.71789i
0.862185 + 1.50221i
−1.12724 + 1.31504i
1.56031 − 0.751945i
−1.35653 − 1.07696i
0.354163 − 1.69546i
−0.707107 − 0.707107i −1.72663 + 0.136880i 1.00000i −1.61645 + 1.54502i 1.31770 + 1.12413i 0.907887 − 0.907887i 0.707107 − 0.707107i 2.96253 − 0.472684i 2.23550 + 0.0505092i
23.2 −0.707107 − 0.707107i −1.63194 − 0.580307i 1.00000i 2.01596 − 0.967426i 0.743620 + 1.56430i −3.05246 + 3.05246i 0.707107 − 0.707107i 2.32649 + 1.89406i −2.10957 − 0.741424i
23.3 −0.707107 − 0.707107i −0.624716 + 1.61547i 1.00000i 1.74863 + 1.39366i 1.58405 − 0.700566i 2.25290 − 2.25290i 0.707107 − 0.707107i −2.21946 − 2.01841i −0.251002 − 2.22194i
23.4 −0.707107 − 0.707107i 1.55830 + 0.756111i 1.00000i −1.63724 − 1.52298i −0.567233 − 1.63654i 3.08242 − 3.08242i 0.707107 − 0.707107i 1.85659 + 2.35649i 0.0807924 + 2.23461i
23.5 −0.707107 − 0.707107i 1.71789 − 0.221043i 1.00000i −1.92512 + 1.13751i −1.37103 − 1.05843i −3.19074 + 3.19074i 0.707107 − 0.707107i 2.90228 − 0.759454i 2.16560 + 0.556922i
23.6 0.707107 + 0.707107i −1.50221 + 0.862185i 1.00000i 0.940148 + 2.02882i −1.67188 − 0.452567i −1.62899 + 1.62899i −0.707107 + 0.707107i 1.51327 − 2.59037i −0.769809 + 2.09938i
23.7 0.707107 + 0.707107i −1.31504 − 1.12724i 1.00000i −0.674951 + 2.13177i −0.132792 − 1.72695i 3.33351 − 3.33351i −0.707107 + 0.707107i 0.458652 + 2.96473i −1.98465 + 1.03013i
23.8 0.707107 + 0.707107i 0.751945 + 1.56031i 1.00000i −2.21146 + 0.330832i −0.571603 + 1.63501i −0.310025 + 0.310025i −0.707107 + 0.707107i −1.86916 + 2.34654i −1.79767 − 1.32980i
23.9 0.707107 + 0.707107i 1.07696 − 1.35653i 1.00000i 1.71412 + 1.43590i 1.72073 − 0.197686i −0.332447 + 0.332447i −0.707107 + 0.707107i −0.680331 − 2.92184i 0.196729 + 2.22740i
23.10 0.707107 + 0.707107i 1.69546 + 0.354163i 1.00000i 1.64636 − 1.51311i 0.948437 + 1.44930i −1.06205 + 1.06205i −0.707107 + 0.707107i 2.74914 + 1.20094i 2.23408 + 0.0942213i
287.1 −0.707107 + 0.707107i −1.72663 − 0.136880i − 1.00000i −1.61645 − 1.54502i 1.31770 − 1.12413i 0.907887 + 0.907887i 0.707107 + 0.707107i 2.96253 + 0.472684i 2.23550 − 0.0505092i
287.2 −0.707107 + 0.707107i −1.63194 + 0.580307i − 1.00000i 2.01596 + 0.967426i 0.743620 − 1.56430i −3.05246 − 3.05246i 0.707107 + 0.707107i 2.32649 − 1.89406i −2.10957 + 0.741424i
287.3 −0.707107 + 0.707107i −0.624716 − 1.61547i − 1.00000i 1.74863 − 1.39366i 1.58405 + 0.700566i 2.25290 + 2.25290i 0.707107 + 0.707107i −2.21946 + 2.01841i −0.251002 + 2.22194i
287.4 −0.707107 + 0.707107i 1.55830 − 0.756111i − 1.00000i −1.63724 + 1.52298i −0.567233 + 1.63654i 3.08242 + 3.08242i 0.707107 + 0.707107i 1.85659 − 2.35649i 0.0807924 − 2.23461i
287.5 −0.707107 + 0.707107i 1.71789 + 0.221043i − 1.00000i −1.92512 − 1.13751i −1.37103 + 1.05843i −3.19074 − 3.19074i 0.707107 + 0.707107i 2.90228 + 0.759454i 2.16560 − 0.556922i
287.6 0.707107 − 0.707107i −1.50221 − 0.862185i − 1.00000i 0.940148 − 2.02882i −1.67188 + 0.452567i −1.62899 − 1.62899i −0.707107 − 0.707107i 1.51327 + 2.59037i −0.769809 − 2.09938i
287.7 0.707107 − 0.707107i −1.31504 + 1.12724i − 1.00000i −0.674951 − 2.13177i −0.132792 + 1.72695i 3.33351 + 3.33351i −0.707107 − 0.707107i 0.458652 − 2.96473i −1.98465 − 1.03013i
287.8 0.707107 − 0.707107i 0.751945 − 1.56031i − 1.00000i −2.21146 − 0.330832i −0.571603 − 1.63501i −0.310025 − 0.310025i −0.707107 − 0.707107i −1.86916 − 2.34654i −1.79767 + 1.32980i
287.9 0.707107 − 0.707107i 1.07696 + 1.35653i − 1.00000i 1.71412 − 1.43590i 1.72073 + 0.197686i −0.332447 − 0.332447i −0.707107 − 0.707107i −0.680331 + 2.92184i 0.196729 − 2.22740i
287.10 0.707107 − 0.707107i 1.69546 − 0.354163i − 1.00000i 1.64636 + 1.51311i 0.948437 − 1.44930i −1.06205 − 1.06205i −0.707107 − 0.707107i 2.74914 − 1.20094i 2.23408 − 0.0942213i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.j.a ✓ 20
3.b odd 2 1 330.2.j.b yes 20
5.c odd 4 1 330.2.j.b yes 20
15.e even 4 1 inner 330.2.j.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.j.a ✓ 20 1.a even 1 1 trivial
330.2.j.a ✓ 20 15.e even 4 1 inner
330.2.j.b yes 20 3.b odd 2 1
330.2.j.b yes 20 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{20} - 8 T_{17}^{19} + 32 T_{17}^{18} - 328 T_{17}^{17} + 6186 T_{17}^{16} - 52024 T_{17}^{15} + \cdots + 198697216 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} - 10 T^{18} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} - 6 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 16 T^{17} + \cdots + 1364224 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 1491813376 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 198697216 \) Copy content Toggle raw display
$19$ \( T^{20} + 148 T^{18} + \cdots + 39337984 \) Copy content Toggle raw display
$23$ \( T^{20} - 36 T^{19} + \cdots + 14868736 \) Copy content Toggle raw display
$29$ \( (T^{10} + 8 T^{9} + \cdots + 3275200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 8 T^{9} + \cdots - 11950976)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 1733372363776 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 16046625918976 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 5116716384256 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 13690872064 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{10} + 56 T^{9} + \cdots - 12732416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 4 T^{9} + \cdots + 7214600)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1218250817536 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 380702846745664 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 14848884736 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 577020355477504 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{10} + 16 T^{9} + \cdots + 44429312)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
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