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Properties

Label 666.2.e.e
Level $666$
Weight $2$
Character orbit 666.e
Analytic conductor $5.318$
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] = [666,2,Mod(223,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.e (of order \(3\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 28 x^{18} + 304 x^{16} + 1654 x^{14} + 4950 x^{12} + 8429 x^{10} + 8179 x^{8} + 4427 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + \beta_{7} q^{3} + (\beta_{11} - 1) q^{4} - \beta_{12} q^{5} + \beta_{14} q^{6} + (\beta_{17} + \beta_{15} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{15} + \beta_{11} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + \beta_{7} q^{3} + (\beta_{11} - 1) q^{4} - \beta_{12} q^{5} + \beta_{14} q^{6} + (\beta_{17} + \beta_{15} + \beta_{9} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{19} - 3 \beta_{18} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{2} + 2 q^{3} - 10 q^{4} - q^{5} + 2 q^{6} - q^{7} + 20 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 10 q^{2} + 2 q^{3} - 10 q^{4} - q^{5} + 2 q^{6} - q^{7} + 20 q^{8} + 2 q^{9} + 2 q^{10} - 3 q^{11} - 4 q^{12} - 12 q^{13} - q^{14} - 13 q^{15} - 10 q^{16} + 24 q^{17} + 2 q^{18} + 48 q^{19} - q^{20} - 18 q^{21} - 3 q^{22} - 3 q^{23} + 2 q^{24} - 21 q^{25} + 24 q^{26} - q^{27} + 2 q^{28} - 4 q^{29} + 2 q^{30} - 11 q^{31} - 10 q^{32} + 19 q^{33} - 12 q^{34} - 14 q^{35} - 4 q^{36} - 20 q^{37} - 24 q^{38} - 18 q^{39} - q^{40} + 3 q^{41} + 30 q^{42} - 6 q^{43} + 6 q^{44} + 18 q^{45} + 6 q^{46} - 8 q^{47} + 2 q^{48} - 35 q^{49} - 21 q^{50} + 17 q^{51} - 12 q^{52} - 20 q^{53} - 31 q^{54} + 4 q^{55} - q^{56} - 41 q^{57} - 4 q^{58} - 10 q^{59} + 11 q^{60} - 2 q^{61} + 22 q^{62} + 29 q^{63} + 20 q^{64} - 26 q^{65} - 17 q^{66} - 7 q^{67} - 12 q^{68} + 9 q^{69} + 7 q^{70} + 30 q^{71} + 2 q^{72} + 6 q^{73} + 10 q^{74} - 48 q^{75} - 24 q^{76} - 9 q^{77} + 48 q^{78} - 16 q^{79} + 2 q^{80} + 10 q^{81} - 6 q^{82} + 23 q^{83} - 12 q^{84} - 15 q^{85} - 6 q^{86} + 25 q^{87} - 3 q^{88} + 42 q^{89} - 15 q^{90} + 118 q^{91} - 3 q^{92} - 117 q^{93} - 8 q^{94} + 6 q^{95} - 4 q^{96} - 24 q^{97} + 70 q^{98} + 49 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 28 x^{18} + 304 x^{16} + 1654 x^{14} + 4950 x^{12} + 8429 x^{10} + 8179 x^{8} + 4427 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1243960 \nu^{19} - 3832932 \nu^{18} + 34732066 \nu^{17} - 106226166 \nu^{16} + 375273154 \nu^{15} + \cdots - 119249415 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1199963 \nu^{19} - 5507991 \nu^{18} - 33315647 \nu^{17} - 152699049 \nu^{16} - 357040457 \nu^{15} + \cdots - 189596025 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6397 \nu^{19} - 210889 \nu^{18} - 179116 \nu^{17} - 5844133 \nu^{16} - 1944688 \nu^{15} + \cdots - 6540174 ) / 38382 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3327062 \nu^{19} + 1024341 \nu^{18} - 92566007 \nu^{17} + 28558644 \nu^{16} - 994870559 \nu^{15} + \cdots + 49564404 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 29882 \nu^{19} + 10555605 \nu^{18} - 672746 \nu^{17} + 292799061 \nu^{16} - 4524830 \nu^{15} + \cdots + 352293300 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 150130 \nu^{18} - 4160902 \nu^{16} - 44455243 \nu^{14} - 235665379 \nu^{12} - 676107276 \nu^{10} + \cdots - 4661364 ) / 19191 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4829060 \nu^{19} - 905205 \nu^{18} - 133914299 \nu^{17} - 25078143 \nu^{16} - 1432013381 \nu^{15} + \cdots - 19834947 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 210889 \nu^{18} - 5844133 \nu^{16} - 62427025 \nu^{14} - 330838624 \nu^{12} - 948727305 \nu^{10} + \cdots - 6540174 ) / 19191 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6504119 \nu^{19} + 3282018 \nu^{18} + 180387182 \nu^{17} + 91022523 \nu^{16} + 1929414416 \nu^{15} + \cdots + 101051352 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4829060 \nu^{19} - 13731597 \nu^{18} + 133914299 \nu^{17} - 380632527 \nu^{16} + 1432013381 \nu^{15} + \cdots - 438696891 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 210889 \nu^{19} - 5844133 \nu^{17} - 62427025 \nu^{15} - 330838624 \nu^{13} - 948727305 \nu^{11} + \cdots + 19191 ) / 38382 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 217286 \nu^{19} - 150130 \nu^{18} - 6023249 \nu^{17} - 4160902 \nu^{16} - 64371713 \nu^{15} + \cdots - 4661364 ) / 38382 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2912306 \nu^{19} + 2083082 \nu^{18} + 80785565 \nu^{17} + 57739298 \nu^{16} + 864263096 \nu^{15} + \cdots + 70219644 ) / 498966 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10133216 \nu^{19} - 3131178 \nu^{18} - 280985834 \nu^{17} - 86754669 \nu^{16} + \cdots - 108379620 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15966142 \nu^{19} + 4181076 \nu^{18} - 442556590 \nu^{17} + 115790004 \nu^{16} + \cdots + 121903200 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 15973354 \nu^{19} - 9432450 \nu^{18} + 442854961 \nu^{17} - 261349731 \nu^{16} + 4733906224 \nu^{15} + \cdots - 294527052 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 16682828 \nu^{19} + 3138390 \nu^{18} + 462434138 \nu^{17} + 87053040 \nu^{16} + 4941640280 \nu^{15} + \cdots + 116322435 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 15544741 \nu^{19} - 15852549 \nu^{18} + 430578757 \nu^{17} - 439572225 \nu^{16} + 4596145657 \nu^{15} + \cdots - 512436312 ) / 1496898 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 17452682 \nu^{19} - 15038871 \nu^{18} - 483828878 \nu^{17} - 416918598 \nu^{16} + \cdots - 481276773 ) / 1496898 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{12} - 2\beta_{11} + \beta_{8} - \beta_{6} - 2\beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{18} + 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{19} + 4 \beta_{18} - \beta_{17} - \beta_{15} + \beta_{14} + 4 \beta_{13} - 13 \beta_{12} + \cdots - 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{19} + 5 \beta_{18} + \beta_{17} - 16 \beta_{16} + 8 \beta_{15} - 16 \beta_{14} - 11 \beta_{13} + \cdots + 52 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{19} - 46 \beta_{18} + 9 \beta_{17} - 2 \beta_{16} + 12 \beta_{15} - 13 \beta_{14} + \cdots + 136 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 27 \beta_{19} - 31 \beta_{18} - 24 \beta_{17} + 128 \beta_{16} - 73 \beta_{15} + 134 \beta_{14} + \cdots - 410 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 50 \beta_{19} + 455 \beta_{18} - 79 \beta_{17} + 48 \beta_{16} - 110 \beta_{15} + 158 \beta_{14} + \cdots - 1215 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 174 \beta_{19} + 244 \beta_{18} + 318 \beta_{17} - 1061 \beta_{16} + 697 \beta_{15} - 1205 \beta_{14} + \cdots + 3419 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 264 \beta_{19} - 4362 \beta_{18} + 724 \beta_{17} - 714 \beta_{16} + 900 \beta_{15} - 1785 \beta_{14} + \cdots + 10640 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 822 \beta_{19} - 2170 \beta_{18} - 3527 \beta_{17} + 8978 \beta_{16} - 6706 \beta_{15} + 11210 \beta_{14} + \cdots - 29180 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 905 \beta_{19} + 41415 \beta_{18} - 6777 \beta_{17} + 8824 \beta_{16} - 6883 \beta_{15} + 19086 \beta_{14} + \cdots - 92939 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 807 \beta_{19} + 20202 \beta_{18} + 36368 \beta_{17} - 76842 \beta_{16} + 64341 \beta_{15} + \cdots + 252117 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3593 \beta_{19} - 391307 \beta_{18} + 63848 \beta_{17} - 99385 \beta_{16} + 50007 \beta_{15} + \cdots + 813860 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 48300 \beta_{19} - 190659 \beta_{18} - 362035 \beta_{17} + 662643 \beta_{16} - 614226 \beta_{15} + \cdots - 2195742 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 127295 \beta_{19} + 3684204 \beta_{18} - 601747 \beta_{17} + 1060535 \beta_{16} - 344822 \beta_{15} + \cdots - 7155188 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 298764 \beta_{19} + 601045 \beta_{18} + 1178229 \beta_{17} - 1916161 \beta_{16} + 1944809 \beta_{15} + \cdots + 6413313 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 1921470 \beta_{19} - 34585536 \beta_{18} + 5661183 \beta_{17} - 10930767 \beta_{16} + 2215845 \beta_{15} + \cdots + 63174532 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 11888013 \beta_{19} - 17024050 \beta_{18} - 34095180 \beta_{17} + 50134172 \beta_{16} + \cdots - 169459091 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 23675603 \beta_{19} + 323858923 \beta_{18} - 53135503 \beta_{17} + 109997394 \beta_{16} + \cdots - 560117636 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(-1 + \beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
0.531449i
3.03729i
− 0.519589i
− 1.94427i
− 0.832956i
2.82646i
1.35695i
− 0.340848i
1.01469i
− 1.66508i
− 0.531449i
− 3.03729i
0.519589i
1.94427i
0.832956i
− 2.82646i
− 1.35695i
0.340848i
− 1.01469i
1.66508i
−0.500000 + 0.866025i −1.70319 + 0.314889i −0.500000 − 0.866025i 1.66931 + 2.89132i 0.578891 − 1.63245i −0.0979808 + 0.169708i 1.00000 2.80169 − 1.07263i −3.33861
223.2 −0.500000 + 0.866025i −1.51552 + 0.838574i −0.500000 − 0.866025i −1.84524 − 3.19605i 0.0315329 − 1.73176i −0.269564 + 0.466898i 1.00000 1.59359 − 2.54175i 3.69049
223.3 −0.500000 + 0.866025i −1.24741 − 1.20166i −0.500000 − 0.866025i −0.716775 − 1.24149i 1.66437 − 0.479458i −0.348992 + 0.604472i 1.00000 0.112048 + 2.99791i 1.43355
223.4 −0.500000 + 0.866025i −0.524756 + 1.65065i −0.500000 − 0.866025i 1.73836 + 3.01093i −1.16712 − 1.27977i −2.36295 + 4.09274i 1.00000 −2.44926 − 1.73237i −3.47672
223.5 −0.500000 + 0.866025i −0.120566 − 1.72785i −0.500000 − 0.866025i 0.181659 + 0.314643i 1.55664 + 0.759512i 2.51911 − 4.36323i 1.00000 −2.97093 + 0.416639i −0.363318
223.6 −0.500000 + 0.866025i 0.688178 − 1.58947i −0.500000 − 0.866025i −1.64139 − 2.84297i 1.03243 + 1.39071i 1.56646 − 2.71318i 1.00000 −2.05282 − 2.18768i 3.28277
223.7 −0.500000 + 0.866025i 0.695998 − 1.58606i −0.500000 − 0.866025i −0.0369337 − 0.0639710i 1.02557 + 1.39578i −2.33901 + 4.05129i 1.00000 −2.03117 − 2.20779i 0.0738674
223.8 −0.500000 + 0.866025i 1.41345 + 1.00108i −0.500000 − 0.866025i −1.74561 − 3.02349i −1.57368 + 0.723546i −1.43053 + 2.47776i 1.00000 0.995687 + 2.82995i 3.49123
223.9 −0.500000 + 0.866025i 1.58644 + 0.695127i −0.500000 − 0.866025i 0.474737 + 0.822268i −1.39522 + 1.02634i 2.02158 − 3.50148i 1.00000 2.03360 + 2.20556i −0.949474
223.10 −0.500000 + 0.866025i 1.72736 − 0.127329i −0.500000 − 0.866025i 1.42189 + 2.46278i −0.753412 + 1.55961i 0.241881 − 0.418950i 1.00000 2.96757 − 0.439888i −2.84378
445.1 −0.500000 − 0.866025i −1.70319 − 0.314889i −0.500000 + 0.866025i 1.66931 − 2.89132i 0.578891 + 1.63245i −0.0979808 − 0.169708i 1.00000 2.80169 + 1.07263i −3.33861
445.2 −0.500000 − 0.866025i −1.51552 − 0.838574i −0.500000 + 0.866025i −1.84524 + 3.19605i 0.0315329 + 1.73176i −0.269564 − 0.466898i 1.00000 1.59359 + 2.54175i 3.69049
445.3 −0.500000 − 0.866025i −1.24741 + 1.20166i −0.500000 + 0.866025i −0.716775 + 1.24149i 1.66437 + 0.479458i −0.348992 − 0.604472i 1.00000 0.112048 − 2.99791i 1.43355
445.4 −0.500000 − 0.866025i −0.524756 − 1.65065i −0.500000 + 0.866025i 1.73836 − 3.01093i −1.16712 + 1.27977i −2.36295 − 4.09274i 1.00000 −2.44926 + 1.73237i −3.47672
445.5 −0.500000 − 0.866025i −0.120566 + 1.72785i −0.500000 + 0.866025i 0.181659 − 0.314643i 1.55664 − 0.759512i 2.51911 + 4.36323i 1.00000 −2.97093 − 0.416639i −0.363318
445.6 −0.500000 − 0.866025i 0.688178 + 1.58947i −0.500000 + 0.866025i −1.64139 + 2.84297i 1.03243 − 1.39071i 1.56646 + 2.71318i 1.00000 −2.05282 + 2.18768i 3.28277
445.7 −0.500000 − 0.866025i 0.695998 + 1.58606i −0.500000 + 0.866025i −0.0369337 + 0.0639710i 1.02557 − 1.39578i −2.33901 − 4.05129i 1.00000 −2.03117 + 2.20779i 0.0738674
445.8 −0.500000 − 0.866025i 1.41345 − 1.00108i −0.500000 + 0.866025i −1.74561 + 3.02349i −1.57368 − 0.723546i −1.43053 − 2.47776i 1.00000 0.995687 − 2.82995i 3.49123
445.9 −0.500000 − 0.866025i 1.58644 − 0.695127i −0.500000 + 0.866025i 0.474737 − 0.822268i −1.39522 − 1.02634i 2.02158 + 3.50148i 1.00000 2.03360 − 2.20556i −0.949474
445.10 −0.500000 − 0.866025i 1.72736 + 0.127329i −0.500000 + 0.866025i 1.42189 − 2.46278i −0.753412 − 1.55961i 0.241881 + 0.418950i 1.00000 2.96757 + 0.439888i −2.84378
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.e.e ✓ 20
3.b odd 2 1 1998.2.e.e 20
9.c even 3 1 inner 666.2.e.e ✓ 20
9.c even 3 1 5994.2.a.bb 10
9.d odd 6 1 1998.2.e.e 20
9.d odd 6 1 5994.2.a.ba 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.e.e ✓ 20 1.a even 1 1 trivial
666.2.e.e ✓ 20 9.c even 3 1 inner
1998.2.e.e 20 3.b odd 2 1
1998.2.e.e 20 9.d odd 6 1
5994.2.a.ba 10 9.d odd 6 1
5994.2.a.bb 10 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + T_{5}^{19} + 36 T_{5}^{18} + 17 T_{5}^{17} + 827 T_{5}^{16} + 258 T_{5}^{15} + 11404 T_{5}^{14} + \cdots + 2601 \) acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{19} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} + T^{19} + \cdots + 2601 \) Copy content Toggle raw display
$7$ \( T^{20} + T^{19} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( T^{20} + 3 T^{19} + \cdots + 25391521 \) Copy content Toggle raw display
$13$ \( T^{20} + 12 T^{19} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{10} - 12 T^{9} + \cdots + 769872)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} - 24 T^{9} + \cdots + 7248)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 1675837969 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 64892977081 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 5236596066321 \) Copy content Toggle raw display
$37$ \( (T + 1)^{20} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 11609416009 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 115915735296 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 102012262412544 \) Copy content Toggle raw display
$53$ \( (T^{10} + 10 T^{9} + \cdots + 1011376)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 197916277065984 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 1367774691361 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 1626438551761 \) Copy content Toggle raw display
$71$ \( (T^{10} - 15 T^{9} + \cdots + 11950032)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 3 T^{9} + \cdots - 15936)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 48\!\cdots\!69 \) Copy content Toggle raw display
$89$ \( (T^{10} - 21 T^{9} + \cdots - 3312)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
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