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Properties

Label 63.4.f.c
Level $63$
Weight $4$
Character orbit 63.f
Analytic conductor $3.717$
Analytic rank $0$
Dimension $18$
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] = [63,4,Mod(22,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.22");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.f (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: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} - 6 x^{14} + 255 x^{13} - 56 x^{12} - 81 x^{11} + \cdots + 387420489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{2} + (\beta_{8} + 1) q^{3} + ( - \beta_{15} + \beta_{7} + \cdots - \beta_{3}) q^{4}+ \cdots + (2 \beta_{15} - \beta_{14} - \beta_{12} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{2} + (\beta_{8} + 1) q^{3} + ( - \beta_{15} + \beta_{7} + \cdots - \beta_{3}) q^{4}+ \cdots + ( - \beta_{17} + 42 \beta_{16} + \cdots + 106) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 6 q^{2} + 9 q^{3} - 36 q^{4} + 24 q^{5} - 63 q^{7} - 150 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 6 q^{2} + 9 q^{3} - 36 q^{4} + 24 q^{5} - 63 q^{7} - 150 q^{8} + 63 q^{9} + 111 q^{11} - 18 q^{13} + 42 q^{14} - 36 q^{15} - 144 q^{16} - 546 q^{17} - 45 q^{18} + 90 q^{19} + 402 q^{20} - 63 q^{21} + 162 q^{22} + 312 q^{23} - 36 q^{24} - 279 q^{25} + 102 q^{26} + 432 q^{27} + 504 q^{28} + 378 q^{29} - 864 q^{30} - 18 q^{31} + 891 q^{32} + 513 q^{33} + 324 q^{34} - 336 q^{35} + 414 q^{36} - 72 q^{37} + 147 q^{38} - 810 q^{39} - 405 q^{40} + 477 q^{41} + 315 q^{42} + 171 q^{43} - 1896 q^{44} - 720 q^{45} - 756 q^{46} + 654 q^{47} - 2709 q^{48} - 441 q^{49} + 429 q^{50} + 1341 q^{51} - 747 q^{52} - 1896 q^{53} - 108 q^{54} - 432 q^{55} + 525 q^{56} - 1143 q^{57} - 297 q^{58} + 957 q^{59} + 5400 q^{60} + 198 q^{61} - 600 q^{62} - 504 q^{63} + 4770 q^{64} + 2478 q^{65} - 2646 q^{66} + 333 q^{67} + 1443 q^{68} + 3366 q^{69} - 5652 q^{71} - 3681 q^{72} + 306 q^{73} + 2100 q^{74} - 4113 q^{75} + 144 q^{76} + 777 q^{77} + 6336 q^{78} - 1152 q^{79} - 8418 q^{80} - 1917 q^{81} - 6048 q^{82} + 1890 q^{83} + 1008 q^{84} + 648 q^{85} + 3837 q^{86} + 4212 q^{87} + 2268 q^{88} - 2604 q^{89} - 135 q^{90} + 252 q^{91} + 987 q^{92} + 378 q^{93} - 324 q^{94} + 3144 q^{95} + 5643 q^{96} + 1737 q^{97} - 588 q^{98} + 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 6 x^{16} - 23 x^{15} - 6 x^{14} + 255 x^{13} - 56 x^{12} - 81 x^{11} + \cdots + 387420489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 35137211695 \nu^{17} - 1622378379288 \nu^{16} - 1349491982943 \nu^{15} + \cdots + 51\!\cdots\!52 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 33291362 \nu^{17} + 49543827 \nu^{16} - 1047096165 \nu^{15} + 321711301 \nu^{14} + \cdots + 11\!\cdots\!96 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 100884080039 \nu^{17} + 189431315142 \nu^{16} + 536203636539 \nu^{15} + \cdots + 34\!\cdots\!28 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 934928795 \nu^{17} + 1529236356 \nu^{16} - 4688064201 \nu^{15} + 16147747558 \nu^{14} + \cdots + 10\!\cdots\!28 ) / 36\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 934928795 \nu^{17} + 1529236356 \nu^{16} - 4688064201 \nu^{15} + 16147747558 \nu^{14} + \cdots + 13\!\cdots\!90 ) / 36\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43671897251 \nu^{17} + 431314232919 \nu^{16} - 1386353129901 \nu^{15} + \cdots + 24\!\cdots\!68 ) / 15\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 125740550 \nu^{17} - 239756565 \nu^{16} - 3019695132 \nu^{15} + 6332440258 \nu^{14} + \cdots + 39\!\cdots\!90 ) / 32\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4601274938 \nu^{17} + 11439252651 \nu^{16} - 13681731984 \nu^{15} + 20748409853 \nu^{14} + \cdots - 42\!\cdots\!47 ) / 10\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2891721292 \nu^{17} - 1235289351 \nu^{16} + 11378779980 \nu^{15} - 163481001263 \nu^{14} + \cdots - 94\!\cdots\!75 ) / 36\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 9425808251 \nu^{17} - 6860997375 \nu^{16} + 18029993601 \nu^{15} - 106282700527 \nu^{14} + \cdots - 98\!\cdots\!51 ) / 10\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64909219150 \nu^{17} + 559430453634 \nu^{16} + 1212366559218 \nu^{15} + \cdots + 10\!\cdots\!40 ) / 67\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 471729716684 \nu^{17} - 1671338975199 \nu^{16} + 124452258888 \nu^{15} + \cdots - 93\!\cdots\!97 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 573834086020 \nu^{17} + 1839711567324 \nu^{16} - 4474830173874 \nu^{15} + \cdots - 24\!\cdots\!29 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 690830509814 \nu^{17} - 685272601038 \nu^{16} - 6766420033119 \nu^{15} + \cdots + 15\!\cdots\!34 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 705247659434 \nu^{17} - 1068128624931 \nu^{16} + 2710590961395 \nu^{15} + \cdots - 96\!\cdots\!63 ) / 47\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 193556779 \nu^{17} + 276829698 \nu^{16} + 2133154329 \nu^{15} + 3107835478 \nu^{14} + \cdots - 29\!\cdots\!04 ) / 10\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 806254553 \nu^{17} - 655552110 \nu^{16} - 2598929994 \nu^{15} - 8913639005 \nu^{14} + \cdots + 13\!\cdots\!46 ) / 32\!\cdots\!46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} + \beta_{16} - 2\beta_{15} + \beta_{12} + \beta_{10} + 2\beta_{8} + \beta_{7} - 5\beta_{5} - \beta_{2} + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{17} + \beta_{16} - 2 \beta_{15} + 2 \beta_{13} - \beta_{11} - 3 \beta_{10} - 2 \beta_{9} + \cdots + 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} + 9 \beta_{13} + 15 \beta_{12} - 3 \beta_{11} + \beta_{9} + \cdots + 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17 \beta_{17} - 11 \beta_{16} - 4 \beta_{15} - 4 \beta_{14} - 6 \beta_{13} + 23 \beta_{12} + 27 \beta_{11} + \cdots - 202 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 38 \beta_{17} + 37 \beta_{16} - 120 \beta_{15} - 56 \beta_{14} - 22 \beta_{13} + 92 \beta_{12} + \cdots - 546 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 60 \beta_{17} + 102 \beta_{15} + 99 \beta_{14} - 6 \beta_{13} + 60 \beta_{12} - 57 \beta_{11} + \cdots + 231 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 420 \beta_{17} - 683 \beta_{16} + 1233 \beta_{15} - 143 \beta_{14} - 471 \beta_{13} - 210 \beta_{12} + \cdots - 9930 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 81 \beta_{17} - 389 \beta_{16} - 475 \beta_{15} + 32 \beta_{14} - 1753 \beta_{13} + 1165 \beta_{12} + \cdots - 19318 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5367 \beta_{17} - 4704 \beta_{16} + 216 \beta_{15} - 2235 \beta_{14} + 3786 \beta_{13} - 4575 \beta_{12} + \cdots + 28210 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3529 \beta_{17} + 12574 \beta_{16} - 5303 \beta_{15} + 18801 \beta_{14} + 5103 \beta_{13} - 11825 \beta_{12} + \cdots - 2174 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1628 \beta_{17} - 86318 \beta_{16} - 46361 \beta_{15} + 4536 \beta_{14} - 34837 \beta_{13} + \cdots - 238958 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 86211 \beta_{17} + 58911 \beta_{16} - 11847 \beta_{15} - 77784 \beta_{14} + 5229 \beta_{13} + \cdots - 993532 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 224765 \beta_{17} + 413332 \beta_{16} + 244841 \beta_{15} + 367079 \beta_{14} + 614037 \beta_{13} + \cdots - 2927218 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1219543 \beta_{17} + 340426 \beta_{16} + 1357989 \beta_{15} + 184588 \beta_{14} + 248207 \beta_{13} + \cdots + 3716139 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 1012533 \beta_{17} - 2860344 \beta_{16} - 1287222 \beta_{15} - 1252890 \beta_{14} - 3570288 \beta_{13} + \cdots + 3248052 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 4016220 \beta_{17} - 10996082 \beta_{16} + 21339252 \beta_{15} - 12737096 \beta_{14} + \cdots + 150484434 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.960810 + 2.84198i
−2.90795 − 0.737429i
−0.0718325 − 2.99914i
−0.831471 + 2.88247i
2.85089 − 0.934028i
2.81021 + 1.05012i
−2.61694 + 1.46684i
−1.07135 − 2.80218i
2.37763 + 1.82944i
0.960810 − 2.84198i
−2.90795 + 0.737429i
−0.0718325 + 2.99914i
−0.831471 − 2.88247i
2.85089 + 0.934028i
2.81021 − 1.05012i
−2.61694 − 1.46684i
−1.07135 + 2.80218i
2.37763 − 1.82944i
−2.32254 − 4.02275i 3.90244 − 3.43088i −6.78835 + 11.7578i 8.24268 − 14.2767i −22.8651 − 7.73020i −3.50000 − 6.06218i 25.9042 3.45809 − 26.7776i −76.5757
22.2 −1.61753 − 2.80164i −5.00056 − 1.41222i −1.23279 + 2.13526i −4.95111 + 8.57557i 4.13202 + 16.2941i −3.50000 − 6.06218i −17.9041 23.0113 + 14.1238i 32.0342
22.3 −1.21836 − 2.11026i −2.70508 + 4.43650i 1.03120 − 1.78609i 2.61762 − 4.53384i 12.6579 + 0.303171i −3.50000 − 6.06218i −24.5192 −12.3651 − 24.0022i −12.7568
22.4 −0.231183 − 0.400421i 1.24909 − 5.04379i 3.89311 − 6.74306i −2.26496 + 3.92303i −2.30841 + 0.665877i −3.50000 − 6.06218i −7.29902 −23.8796 − 12.6003i 2.09449
22.5 0.377604 + 0.654029i 3.46745 + 3.86999i 3.71483 − 6.43428i 6.62462 − 11.4742i −1.22176 + 3.72913i −3.50000 − 6.06218i 11.6526 −2.95361 + 26.8380i 10.0059
22.6 1.32666 + 2.29785i 5.12474 + 0.858536i 0.479932 − 0.831266i −10.0300 + 17.3725i 4.82601 + 12.9148i −3.50000 − 6.06218i 23.7734 25.5258 + 8.79954i −53.2258
22.7 1.66614 + 2.88585i −2.65509 − 4.46660i −1.55207 + 2.68826i 8.37356 − 14.5034i 8.46616 − 15.1042i −3.50000 − 6.06218i 16.3144 −12.9010 + 23.7184i 55.8062
22.8 2.21232 + 3.83185i −4.03378 + 3.27546i −5.78871 + 10.0263i −1.92707 + 3.33779i −21.4751 − 8.21047i −3.50000 − 6.06218i −15.8288 5.54273 − 26.4250i −17.0532
22.9 2.80688 + 4.86166i 5.15079 − 0.685073i −11.7571 + 20.3640i 5.31469 − 9.20530i 17.7882 + 23.1185i −3.50000 − 6.06218i −87.0935 26.0614 − 7.05734i 59.6707
43.1 −2.32254 + 4.02275i 3.90244 + 3.43088i −6.78835 − 11.7578i 8.24268 + 14.2767i −22.8651 + 7.73020i −3.50000 + 6.06218i 25.9042 3.45809 + 26.7776i −76.5757
43.2 −1.61753 + 2.80164i −5.00056 + 1.41222i −1.23279 − 2.13526i −4.95111 − 8.57557i 4.13202 − 16.2941i −3.50000 + 6.06218i −17.9041 23.0113 − 14.1238i 32.0342
43.3 −1.21836 + 2.11026i −2.70508 − 4.43650i 1.03120 + 1.78609i 2.61762 + 4.53384i 12.6579 − 0.303171i −3.50000 + 6.06218i −24.5192 −12.3651 + 24.0022i −12.7568
43.4 −0.231183 + 0.400421i 1.24909 + 5.04379i 3.89311 + 6.74306i −2.26496 − 3.92303i −2.30841 − 0.665877i −3.50000 + 6.06218i −7.29902 −23.8796 + 12.6003i 2.09449
43.5 0.377604 − 0.654029i 3.46745 − 3.86999i 3.71483 + 6.43428i 6.62462 + 11.4742i −1.22176 − 3.72913i −3.50000 + 6.06218i 11.6526 −2.95361 − 26.8380i 10.0059
43.6 1.32666 − 2.29785i 5.12474 − 0.858536i 0.479932 + 0.831266i −10.0300 − 17.3725i 4.82601 − 12.9148i −3.50000 + 6.06218i 23.7734 25.5258 − 8.79954i −53.2258
43.7 1.66614 − 2.88585i −2.65509 + 4.46660i −1.55207 − 2.68826i 8.37356 + 14.5034i 8.46616 + 15.1042i −3.50000 + 6.06218i 16.3144 −12.9010 − 23.7184i 55.8062
43.8 2.21232 − 3.83185i −4.03378 − 3.27546i −5.78871 − 10.0263i −1.92707 − 3.33779i −21.4751 + 8.21047i −3.50000 + 6.06218i −15.8288 5.54273 + 26.4250i −17.0532
43.9 2.80688 − 4.86166i 5.15079 + 0.685073i −11.7571 − 20.3640i 5.31469 + 9.20530i 17.7882 − 23.1185i −3.50000 + 6.06218i −87.0935 26.0614 + 7.05734i 59.6707
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.9
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 63.4.f.c ✓ 18
3.b odd 2 1 189.4.f.c 18
9.c even 3 1 inner 63.4.f.c ✓ 18
9.c even 3 1 567.4.a.j 9
9.d odd 6 1 189.4.f.c 18
9.d odd 6 1 567.4.a.k 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.c ✓ 18 1.a even 1 1 trivial
63.4.f.c ✓ 18 9.c even 3 1 inner
189.4.f.c 18 3.b odd 2 1
189.4.f.c 18 9.d odd 6 1
567.4.a.j 9 9.c even 3 1
567.4.a.k 9 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 6 T_{2}^{17} + 72 T_{2}^{16} - 246 T_{2}^{15} + 2340 T_{2}^{14} - 6831 T_{2}^{13} + \cdots + 7884864 \) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 6 T^{17} + \cdots + 7884864 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 7625597484987 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots + 50\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} - 45 T^{8} + \cdots - 278256660857)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 93\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 84\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 32\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 15\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
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