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Properties

Label 114.2.l.a
Level $114$
Weight $2$
Character orbit 114.l
Analytic conductor $0.910$
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] = [114,2,Mod(29,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 114.l (of order \(18\), degree \(6\), 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: \(0.910294583043\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{6}) q^{2} + ( - \beta_{17} + \beta_1) q^{3} + (\beta_{8} + \beta_{4}) q^{4} + ( - \beta_{15} + \beta_{14} + \beta_{2}) q^{5} + \beta_{5} q^{6} + (\beta_{16} - \beta_{15} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{16} - 2 \beta_{9} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{6}) q^{2} + ( - \beta_{17} + \beta_1) q^{3} + (\beta_{8} + \beta_{4}) q^{4} + ( - \beta_{15} + \beta_{14} + \beta_{2}) q^{5} + \beta_{5} q^{6} + (\beta_{16} - \beta_{15} + \cdots - \beta_{3}) q^{7}+ \cdots + (\beta_{17} + 2 \beta_{16} - \beta_{14} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{6} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{6} + 9 q^{8} - 12 q^{13} + 12 q^{14} - 24 q^{15} - 6 q^{17} - 6 q^{19} - 24 q^{22} - 18 q^{25} - 18 q^{26} - 3 q^{27} + 6 q^{28} + 6 q^{29} - 27 q^{33} - 6 q^{34} + 24 q^{35} - 3 q^{36} - 3 q^{38} + 6 q^{39} - 3 q^{41} - 6 q^{43} - 24 q^{44} + 54 q^{45} + 18 q^{46} - 30 q^{47} + 6 q^{48} + 21 q^{49} - 3 q^{50} - 33 q^{51} - 6 q^{52} + 60 q^{53} + 30 q^{55} + 12 q^{57} + 12 q^{58} - 3 q^{59} + 18 q^{60} + 54 q^{61} - 6 q^{62} + 84 q^{63} - 9 q^{64} - 24 q^{65} + 30 q^{66} - 15 q^{67} + 27 q^{68} + 24 q^{69} + 24 q^{70} - 36 q^{71} - 42 q^{73} - 6 q^{74} - 18 q^{78} - 6 q^{79} + 3 q^{82} - 36 q^{83} - 24 q^{84} + 6 q^{86} - 54 q^{87} + 60 q^{89} - 12 q^{90} - 18 q^{91} - 84 q^{93} - 6 q^{95} + 9 q^{97} - 12 q^{98} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{15} - 18 x^{14} + 36 x^{13} + 10 x^{12} + 18 x^{11} + 90 x^{10} - 567 x^{9} + 270 x^{8} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{17} + \nu^{14} + 18 \nu^{13} - 36 \nu^{12} - 10 \nu^{11} - 18 \nu^{10} - 90 \nu^{9} + \cdots + 729 \nu^{2} ) / 6561 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{17} + 21 \nu^{16} - 81 \nu^{15} + 97 \nu^{14} - 57 \nu^{13} - 198 \nu^{12} + 1568 \nu^{11} + \cdots - 65610 ) / 4374 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{17} + 99 \nu^{16} - 153 \nu^{15} - 22 \nu^{14} + 207 \nu^{13} - 1485 \nu^{12} + \cdots + 334611 ) / 13122 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} - 27 \nu^{15} + 50 \nu^{14} - 47 \nu^{13} + 9 \nu^{12} + 409 \nu^{11} + \cdots - 72171 ) / 1458 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16 \nu^{17} + 81 \nu^{16} + 36 \nu^{15} - 335 \nu^{14} + 477 \nu^{13} - 1422 \nu^{12} + \cdots + 603612 ) / 13122 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{17} - 3 \nu^{16} - 6 \nu^{15} + 92 \nu^{14} + 48 \nu^{13} - 255 \nu^{12} - 137 \nu^{11} + \cdots - 225261 ) / 4374 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{17} - 105 \nu^{16} + 90 \nu^{15} + 275 \nu^{14} - 318 \nu^{13} + 1296 \nu^{12} + \cdots - 787320 ) / 13122 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{17} + 147 \nu^{16} - 216 \nu^{15} - 43 \nu^{14} + 375 \nu^{13} - 2259 \nu^{12} + \cdots + 452709 ) / 13122 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17 \nu^{17} - 5 \nu^{16} + 99 \nu^{15} - 136 \nu^{14} + 284 \nu^{13} - 405 \nu^{12} - 1655 \nu^{11} + \cdots + 63423 ) / 4374 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4 \nu^{17} + 17 \nu^{16} - 10 \nu^{15} - 23 \nu^{14} + 82 \nu^{13} - 332 \nu^{12} + 536 \nu^{11} + \cdots + 56862 ) / 1458 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 23 \nu^{17} + 10 \nu^{16} + 6 \nu^{15} + 86 \nu^{14} + 161 \nu^{13} - 717 \nu^{12} - 41 \nu^{11} + \cdots - 247131 ) / 4374 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9 \nu^{17} - 4 \nu^{16} + 39 \nu^{15} - 21 \nu^{14} + 94 \nu^{13} - 174 \nu^{12} - 717 \nu^{11} + \cdots - 34992 ) / 1458 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 76 \nu^{17} - 108 \nu^{16} + 459 \nu^{15} - 194 \nu^{14} + 747 \nu^{13} - 522 \nu^{12} + \cdots - 551124 ) / 13122 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11 \nu^{17} - 17 \nu^{16} - 3 \nu^{15} + 13 \nu^{14} - 145 \nu^{13} + 525 \nu^{12} - 283 \nu^{11} + \cdots + 10935 ) / 1458 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 103 \nu^{17} - 99 \nu^{16} - 27 \nu^{15} - 157 \nu^{14} - 1026 \nu^{13} + 4140 \nu^{12} + \cdots + 669222 ) / 13122 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17 \nu^{17} + 10 \nu^{16} + 27 \nu^{15} - 10 \nu^{14} + 188 \nu^{13} - 576 \nu^{12} - 269 \nu^{11} + \cdots - 78732 ) / 1458 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{16} + \beta_{15} + \beta_{9} - \beta_{8} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - 4\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} - 3 \beta_{7} - 3 \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} - 2 \beta_{16} + \beta_{15} + 3 \beta_{13} + 3 \beta_{10} - \beta_{9} - 2 \beta_{8} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{16} + 10 \beta_{15} + 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{17} + 9 \beta_{16} - 9 \beta_{15} + 15 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18 \beta_{17} - \beta_{16} + 15 \beta_{15} - 9 \beta_{14} + 21 \beta_{12} + 24 \beta_{11} + \cdots + 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 27 \beta_{17} - 27 \beta_{16} + 11 \beta_{15} + \beta_{14} - 4 \beta_{13} + 3 \beta_{12} - 30 \beta_{11} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 33 \beta_{17} - 54 \beta_{16} + 63 \beta_{15} + 24 \beta_{14} - 89 \beta_{13} + \beta_{12} + 2 \beta_{11} + \cdots + 140 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15 \beta_{17} + 100 \beta_{16} - 103 \beta_{15} + 126 \beta_{14} - 129 \beta_{13} + 102 \beta_{12} + \cdots + 120 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 90 \beta_{17} - 81 \beta_{16} + 136 \beta_{15} - 154 \beta_{14} + 33 \beta_{13} + 249 \beta_{12} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 62 \beta_{17} - 171 \beta_{16} - 189 \beta_{15} + 153 \beta_{14} - 19 \beta_{13} - 100 \beta_{12} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 372 \beta_{17} - 106 \beta_{16} + 206 \beta_{15} + 243 \beta_{14} - 426 \beta_{13} + 9 \beta_{12} + \cdots + 354 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 351 \beta_{17} + 765 \beta_{16} - 802 \beta_{15} + 916 \beta_{14} + 46 \beta_{13} + 327 \beta_{12} + \cdots - 2208 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 878 \beta_{17} - 630 \beta_{16} + 927 \beta_{15} - 1554 \beta_{14} + 1726 \beta_{13} + 430 \beta_{12} + \cdots - 4210 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 918 \beta_{17} + 406 \beta_{16} - 3786 \beta_{15} + 1656 \beta_{14} - 378 \beta_{13} - 3126 \beta_{12} + \cdots + 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(-1\) \(-\beta_{4} - \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−1.72388 + 0.168030i
1.47158 − 0.913487i
0.0786547 + 1.73026i
−0.442647 + 1.67453i
−1.73189 − 0.0237018i
1.40849 − 1.00804i
−0.363139 − 1.69356i
1.69944 − 0.334495i
−0.396613 + 1.68603i
−1.72388 − 0.168030i
1.47158 + 0.913487i
0.0786547 − 1.73026i
−0.363139 + 1.69356i
1.69944 + 0.334495i
−0.396613 − 1.68603i
−0.442647 − 1.67453i
−1.73189 + 0.0237018i
1.40849 + 1.00804i
−0.173648 − 0.984808i −0.716422 + 1.57694i −0.939693 + 0.342020i −1.14133 + 3.13578i 1.67739 + 0.431705i −1.07356 + 1.85947i 0.500000 + 0.866025i −1.97348 − 2.25951i 3.28633 + 0.579469i
29.2 −0.173648 − 0.984808i −0.0553136 − 1.73117i −0.939693 + 0.342020i 0.882820 − 2.42553i −1.69526 + 0.355087i −1.58376 + 2.74316i 0.500000 + 0.866025i −2.99388 + 0.191514i −2.54198 − 0.448219i
29.3 −0.173648 − 0.984808i 1.53778 + 0.797015i −0.939693 + 0.342020i 0.258510 − 0.710252i 0.517874 − 1.65282i 0.777943 − 1.34744i 0.500000 + 0.866025i 1.72953 + 2.45127i −0.744351 − 0.131249i
41.1 −0.766044 − 0.642788i −1.67151 + 0.453924i 0.173648 + 0.984808i 1.96615 + 0.346685i 1.57223 + 0.726702i 0.910931 + 1.57778i 0.500000 − 0.866025i 2.58791 − 1.51748i −1.28331 − 1.52939i
41.2 −0.766044 − 0.642788i −0.845418 − 1.51171i 0.173648 + 0.984808i −2.22841 − 0.392929i −0.324081 + 1.70146i −1.16829 − 2.02354i 0.500000 − 0.866025i −1.57054 + 2.55605i 1.45449 + 1.73339i
41.3 −0.766044 − 0.642788i 1.57724 + 0.715766i 0.173648 + 0.984808i 0.262261 + 0.0462437i −0.748148 − 1.56214i 0.604656 + 1.04730i 0.500000 − 0.866025i 1.97536 + 2.25787i −0.171179 − 0.204003i
53.1 0.939693 + 0.342020i −1.64823 − 0.532290i 0.766044 + 0.642788i 2.20556 + 2.62849i −1.36678 − 1.06392i 1.68651 − 2.92113i 0.500000 + 0.866025i 2.43333 + 1.75467i 1.17355 + 3.22432i
53.2 0.939693 + 0.342020i 0.560041 − 1.63901i 0.766044 + 0.642788i −0.343148 − 0.408948i 1.08684 − 1.34862i −0.716507 + 1.24103i 0.500000 + 0.866025i −2.37271 − 1.83583i −0.182585 − 0.501649i
53.3 0.939693 + 0.342020i 1.26184 + 1.18649i 0.766044 + 0.642788i −1.86241 − 2.21954i 0.779936 + 1.54651i 0.562083 − 0.973556i 0.500000 + 0.866025i 0.184473 + 2.99432i −0.990970 − 2.72267i
59.1 −0.173648 + 0.984808i −0.716422 − 1.57694i −0.939693 − 0.342020i −1.14133 − 3.13578i 1.67739 − 0.431705i −1.07356 − 1.85947i 0.500000 − 0.866025i −1.97348 + 2.25951i 3.28633 − 0.579469i
59.2 −0.173648 + 0.984808i −0.0553136 + 1.73117i −0.939693 − 0.342020i 0.882820 + 2.42553i −1.69526 − 0.355087i −1.58376 − 2.74316i 0.500000 − 0.866025i −2.99388 − 0.191514i −2.54198 + 0.448219i
59.3 −0.173648 + 0.984808i 1.53778 − 0.797015i −0.939693 − 0.342020i 0.258510 + 0.710252i 0.517874 + 1.65282i 0.777943 + 1.34744i 0.500000 − 0.866025i 1.72953 − 2.45127i −0.744351 + 0.131249i
71.1 0.939693 − 0.342020i −1.64823 + 0.532290i 0.766044 − 0.642788i 2.20556 − 2.62849i −1.36678 + 1.06392i 1.68651 + 2.92113i 0.500000 − 0.866025i 2.43333 − 1.75467i 1.17355 − 3.22432i
71.2 0.939693 − 0.342020i 0.560041 + 1.63901i 0.766044 − 0.642788i −0.343148 + 0.408948i 1.08684 + 1.34862i −0.716507 − 1.24103i 0.500000 − 0.866025i −2.37271 + 1.83583i −0.182585 + 0.501649i
71.3 0.939693 − 0.342020i 1.26184 − 1.18649i 0.766044 − 0.642788i −1.86241 + 2.21954i 0.779936 − 1.54651i 0.562083 + 0.973556i 0.500000 − 0.866025i 0.184473 − 2.99432i −0.990970 + 2.72267i
89.1 −0.766044 + 0.642788i −1.67151 − 0.453924i 0.173648 − 0.984808i 1.96615 − 0.346685i 1.57223 − 0.726702i 0.910931 − 1.57778i 0.500000 + 0.866025i 2.58791 + 1.51748i −1.28331 + 1.52939i
89.2 −0.766044 + 0.642788i −0.845418 + 1.51171i 0.173648 − 0.984808i −2.22841 + 0.392929i −0.324081 − 1.70146i −1.16829 + 2.02354i 0.500000 + 0.866025i −1.57054 − 2.55605i 1.45449 − 1.73339i
89.3 −0.766044 + 0.642788i 1.57724 − 0.715766i 0.173648 − 0.984808i 0.262261 − 0.0462437i −0.748148 + 1.56214i 0.604656 − 1.04730i 0.500000 + 0.866025i 1.97536 − 2.25787i −0.171179 + 0.204003i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.l.a ✓ 18
3.b odd 2 1 114.2.l.b yes 18
4.b odd 2 1 912.2.cc.d 18
12.b even 2 1 912.2.cc.c 18
19.f odd 18 1 114.2.l.b yes 18
57.j even 18 1 inner 114.2.l.a ✓ 18
76.k even 18 1 912.2.cc.c 18
228.u odd 18 1 912.2.cc.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.l.a ✓ 18 1.a even 1 1 trivial
114.2.l.a ✓ 18 57.j even 18 1 inner
114.2.l.b yes 18 3.b odd 2 1
114.2.l.b yes 18 19.f odd 18 1
912.2.cc.c 18 12.b even 2 1
912.2.cc.c 18 76.k even 18 1
912.2.cc.d 18 4.b odd 2 1
912.2.cc.d 18 228.u odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 9 T_{5}^{16} + 108 T_{5}^{14} + 162 T_{5}^{13} + 46 T_{5}^{12} - 324 T_{5}^{11} + \cdots + 1728 \) acting on \(S_{2}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{3} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} + T^{15} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} + 9 T^{16} + \cdots + 1728 \) Copy content Toggle raw display
$7$ \( T^{18} + 21 T^{16} + \cdots + 87616 \) Copy content Toggle raw display
$11$ \( T^{18} - 51 T^{16} + \cdots + 35769627 \) Copy content Toggle raw display
$13$ \( T^{18} + 12 T^{17} + \cdots + 2365632 \) Copy content Toggle raw display
$17$ \( T^{18} + 6 T^{17} + \cdots + 3878307 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 123187392 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 52719833664 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 6231379854528 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 7868768303808 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 1846709769969 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 390621250009 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 3499077312 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 3426463296 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 38983402581561 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 65033160256 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 56\!\cdots\!23 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 404099233344 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 192753487369 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 21259626441408 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 176145902499843 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 381874169643201 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 17447631785307 \) Copy content Toggle raw display
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