Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [342,2,Mod(65,342)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(342, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("342.65");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 342.j (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: | \(2.73088374913\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 5 x^{17} + 13 x^{16} - 30 x^{15} + 54 x^{14} - 69 x^{13} + 66 x^{12} + 36 x^{11} - 243 x^{10} + \cdots + 19683 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{18} - 5 x^{17} + 13 x^{16} - 30 x^{15} + 54 x^{14} - 69 x^{13} + 66 x^{12} + 36 x^{11} - 243 x^{10} + \cdots + 19683 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{17} + 5 \nu^{16} - 13 \nu^{15} + 30 \nu^{14} - 54 \nu^{13} + 69 \nu^{12} - 66 \nu^{11} + \cdots + 32805 ) / 6561 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 10 \nu^{17} + 67 \nu^{16} - 140 \nu^{15} + 159 \nu^{14} - 792 \nu^{13} + 498 \nu^{12} + \cdots + 275562 ) / 32805 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 53 \nu^{17} + 74 \nu^{16} - 457 \nu^{15} + 531 \nu^{14} - 1737 \nu^{13} + 3336 \nu^{12} + \cdots + 1135053 ) / 65610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23 \nu^{17} - 28 \nu^{16} + 197 \nu^{15} - 414 \nu^{14} + 369 \nu^{13} - 1128 \nu^{12} + \cdots - 472392 ) / 32805 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 28 \nu^{17} - 119 \nu^{16} + 124 \nu^{15} - 459 \nu^{14} + 747 \nu^{13} - 150 \nu^{12} + \cdots - 111537 ) / 32805 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 47 \nu^{17} + 364 \nu^{16} - 707 \nu^{15} + 1071 \nu^{14} - 2997 \nu^{13} + 2406 \nu^{12} + \cdots + 610173 ) / 65610 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 37 \nu^{17} + 194 \nu^{16} - 553 \nu^{15} + 1065 \nu^{14} - 1863 \nu^{13} + 2580 \nu^{12} + \cdots + 1148175 ) / 32805 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 16 \nu^{17} - 71 \nu^{16} + 154 \nu^{15} - 318 \nu^{14} + 558 \nu^{13} - 591 \nu^{12} + \cdots - 223074 ) / 10935 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 145 \nu^{17} - 488 \nu^{16} + 1141 \nu^{15} - 2745 \nu^{14} + 3699 \nu^{13} - 4686 \nu^{12} + \cdots - 2066715 ) / 65610 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 85 \nu^{17} - 167 \nu^{16} + 328 \nu^{15} - 951 \nu^{14} + 765 \nu^{13} - 1005 \nu^{12} + \cdots - 249318 ) / 32805 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 76 \nu^{17} + 257 \nu^{16} - 571 \nu^{15} + 1428 \nu^{14} - 2016 \nu^{13} + 2193 \nu^{12} + \cdots + 977589 ) / 32805 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 79 \nu^{17} + 248 \nu^{16} - 535 \nu^{15} + 1377 \nu^{14} - 1773 \nu^{13} + 1914 \nu^{12} + \cdots + 929475 ) / 21870 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 79 \nu^{17} - 248 \nu^{16} + 535 \nu^{15} - 1377 \nu^{14} + 1773 \nu^{13} - 1914 \nu^{12} + \cdots - 951345 ) / 21870 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 105 \nu^{17} - 380 \nu^{16} + 877 \nu^{15} - 2009 \nu^{14} + 2925 \nu^{13} - 3546 \nu^{12} + \cdots - 1362501 ) / 21870 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 157 \nu^{17} + 665 \nu^{16} - 1378 \nu^{15} + 3051 \nu^{14} - 4896 \nu^{13} + 4866 \nu^{12} + \cdots + 1830519 ) / 32805 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 377 \nu^{17} - 1384 \nu^{16} + 2837 \nu^{15} - 6867 \nu^{14} + 10305 \nu^{13} - 9624 \nu^{12} + \cdots - 4625505 ) / 65610 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 191 \nu^{17} + 650 \nu^{16} - 1393 \nu^{15} + 3229 \nu^{14} - 4587 \nu^{13} + 4944 \nu^{12} + \cdots + 1826145 ) / 21870 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{13} + \beta_{12} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{14} + \beta_{13} - 2\beta_{12} + 3\beta_{10} - 3\beta_{9} + 3\beta_{8} - 3\beta_{3} - 3\beta _1 - 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{14} - 2\beta_{13} + \beta_{12} - 3\beta_{10} + 3\beta_{8} + 3\beta_{6} + 3\beta_{3} + 3\beta _1 + 1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 6 \beta_{17} + 6 \beta_{15} - 2 \beta_{13} + 4 \beta_{12} - 9 \beta_{10} + 9 \beta_{9} - 3 \beta_{8} + \cdots + 13 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 3 \beta_{17} + 3 \beta_{16} + 6 \beta_{15} + 6 \beta_{14} + 10 \beta_{13} - 5 \beta_{12} + 6 \beta_{10} + \cdots - 14 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3 \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 33 \beta_{14} + \beta_{13} - 8 \beta_{12} - 18 \beta_{11} + \cdots + 28 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 15 \beta_{17} - 27 \beta_{16} + 15 \beta_{15} + 34 \beta_{13} - 5 \beta_{12} - 18 \beta_{11} + \cdots - 41 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 24 \beta_{17} + 21 \beta_{16} + 42 \beta_{15} + 69 \beta_{14} - 53 \beta_{13} + 4 \beta_{12} + \cdots - 113 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 21 \beta_{17} + 42 \beta_{16} + 15 \beta_{15} + 48 \beta_{14} - 17 \beta_{13} - 62 \beta_{12} + \cdots - 71 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 24 \beta_{17} - 18 \beta_{16} + 105 \beta_{15} + 61 \beta_{13} + 58 \beta_{12} - 126 \beta_{11} + \cdots + 130 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 84 \beta_{17} - 33 \beta_{16} + 42 \beta_{15} + 69 \beta_{14} + 199 \beta_{13} - 203 \beta_{12} + \cdots + 85 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 102 \beta_{17} + 258 \beta_{16} - 255 \beta_{15} + 183 \beta_{14} - 170 \beta_{13} + 55 \beta_{12} + \cdots - 845 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 132 \beta_{17} - 18 \beta_{16} + 429 \beta_{15} + 459 \beta_{14} - 92 \beta_{13} - 284 \beta_{12} + \cdots - 968 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 462 \beta_{17} + 642 \beta_{16} + 555 \beta_{15} + 3255 \beta_{14} - 1385 \beta_{13} + 1345 \beta_{12} + \cdots + 1147 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 438 \beta_{17} + 1230 \beta_{16} - 606 \beta_{15} - 1437 \beta_{14} - 350 \beta_{13} + 10 \beta_{12} + \cdots + 6967 ) / 3 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 1515 \beta_{17} + 900 \beta_{16} + 726 \beta_{15} + 4320 \beta_{14} + 4048 \beta_{13} + 967 \beta_{12} + \cdots - 3200 ) / 3 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 1299 \beta_{17} + 3531 \beta_{16} + 1095 \beta_{15} - 1902 \beta_{14} - 4940 \beta_{13} + 3622 \beta_{12} + \cdots + 14845 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(325\) |
| \(\chi(n)\) | \(-\beta_{9}\) | \(-\beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
−0.500000 | − | 0.866025i | −1.62412 | − | 0.601846i | −0.500000 | + | 0.866025i | − | 2.95236i | 0.290848 | + | 1.70746i | 1.79740 | − | 3.11318i | 1.00000 | 2.27556 | + | 1.95495i | −2.55682 | + | 1.47618i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.2 | −0.500000 | − | 0.866025i | −1.19911 | + | 1.24985i | −0.500000 | + | 0.866025i | − | 1.87359i | 1.68196 | + | 0.413538i | −0.813726 | + | 1.40941i | 1.00000 | −0.124252 | − | 2.99743i | −1.62258 | + | 0.936797i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.3 | −0.500000 | − | 0.866025i | −1.13509 | − | 1.30827i | −0.500000 | + | 0.866025i | 2.55258i | −0.565455 | + | 1.63715i | 0.762878 | − | 1.32134i | 1.00000 | −0.423160 | + | 2.97001i | 2.21060 | − | 1.27629i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.4 | −0.500000 | − | 0.866025i | 0.266393 | − | 1.71144i | −0.500000 | + | 0.866025i | − | 3.93890i | −1.61535 | + | 0.625018i | −1.35148 | + | 2.34083i | 1.00000 | −2.85807 | − | 0.911834i | −3.41119 | + | 1.96945i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.5 | −0.500000 | − | 0.866025i | 0.696389 | + | 1.58589i | −0.500000 | + | 0.866025i | 3.97890i | 1.02522 | − | 1.39604i | 0.446005 | − | 0.772503i | 1.00000 | −2.03008 | + | 2.20879i | 3.44583 | − | 1.98945i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.6 | −0.500000 | − | 0.866025i | 0.901946 | − | 1.47868i | −0.500000 | + | 0.866025i | − | 0.947600i | −1.73155 | − | 0.0417686i | 1.09312 | − | 1.89335i | 1.00000 | −1.37299 | − | 2.66738i | −0.820646 | + | 0.473800i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.7 | −0.500000 | − | 0.866025i | 1.10774 | + | 1.33151i | −0.500000 | + | 0.866025i | − | 1.27738i | 0.599249 | − | 1.62508i | −1.74174 | + | 3.01679i | 1.00000 | −0.545825 | + | 2.94993i | −1.10624 | + | 0.638690i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.8 | −0.500000 | − | 0.866025i | 1.29076 | − | 1.15496i | −0.500000 | + | 0.866025i | 3.18866i | −1.64560 | − | 0.540356i | −1.55443 | + | 2.69235i | 1.00000 | 0.332146 | − | 2.98156i | 2.76146 | − | 1.59433i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 65.9 | −0.500000 | − | 0.866025i | 1.69509 | + | 0.355902i | −0.500000 | + | 0.866025i | − | 0.462359i | −0.539325 | − | 1.64594i | 1.36198 | − | 2.35901i | 1.00000 | 2.74667 | + | 1.20657i | −0.400415 | + | 0.231180i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.1 | −0.500000 | + | 0.866025i | −1.62412 | + | 0.601846i | −0.500000 | − | 0.866025i | 2.95236i | 0.290848 | − | 1.70746i | 1.79740 | + | 3.11318i | 1.00000 | 2.27556 | − | 1.95495i | −2.55682 | − | 1.47618i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.2 | −0.500000 | + | 0.866025i | −1.19911 | − | 1.24985i | −0.500000 | − | 0.866025i | 1.87359i | 1.68196 | − | 0.413538i | −0.813726 | − | 1.40941i | 1.00000 | −0.124252 | + | 2.99743i | −1.62258 | − | 0.936797i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.3 | −0.500000 | + | 0.866025i | −1.13509 | + | 1.30827i | −0.500000 | − | 0.866025i | − | 2.55258i | −0.565455 | − | 1.63715i | 0.762878 | + | 1.32134i | 1.00000 | −0.423160 | − | 2.97001i | 2.21060 | + | 1.27629i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.4 | −0.500000 | + | 0.866025i | 0.266393 | + | 1.71144i | −0.500000 | − | 0.866025i | 3.93890i | −1.61535 | − | 0.625018i | −1.35148 | − | 2.34083i | 1.00000 | −2.85807 | + | 0.911834i | −3.41119 | − | 1.96945i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.5 | −0.500000 | + | 0.866025i | 0.696389 | − | 1.58589i | −0.500000 | − | 0.866025i | − | 3.97890i | 1.02522 | + | 1.39604i | 0.446005 | + | 0.772503i | 1.00000 | −2.03008 | − | 2.20879i | 3.44583 | + | 1.98945i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.6 | −0.500000 | + | 0.866025i | 0.901946 | + | 1.47868i | −0.500000 | − | 0.866025i | 0.947600i | −1.73155 | + | 0.0417686i | 1.09312 | + | 1.89335i | 1.00000 | −1.37299 | + | 2.66738i | −0.820646 | − | 0.473800i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.7 | −0.500000 | + | 0.866025i | 1.10774 | − | 1.33151i | −0.500000 | − | 0.866025i | 1.27738i | 0.599249 | + | 1.62508i | −1.74174 | − | 3.01679i | 1.00000 | −0.545825 | − | 2.94993i | −1.10624 | − | 0.638690i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.8 | −0.500000 | + | 0.866025i | 1.29076 | + | 1.15496i | −0.500000 | − | 0.866025i | − | 3.18866i | −1.64560 | + | 0.540356i | −1.55443 | − | 2.69235i | 1.00000 | 0.332146 | + | 2.98156i | 2.76146 | + | 1.59433i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 221.9 | −0.500000 | + | 0.866025i | 1.69509 | − | 0.355902i | −0.500000 | − | 0.866025i | 0.462359i | −0.539325 | + | 1.64594i | 1.36198 | + | 2.35901i | 1.00000 | 2.74667 | − | 1.20657i | −0.400415 | − | 0.231180i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 171.k | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 342.2.j.f | ✓ | 18 |
| 3.b | odd | 2 | 1 | 1026.2.j.f | 18 | ||
| 9.c | even | 3 | 1 | 1026.2.n.f | 18 | ||
| 9.d | odd | 6 | 1 | 342.2.n.f | yes | 18 | |
| 19.d | odd | 6 | 1 | 342.2.n.f | yes | 18 | |
| 57.f | even | 6 | 1 | 1026.2.n.f | 18 | ||
| 171.k | even | 6 | 1 | inner | 342.2.j.f | ✓ | 18 |
| 171.s | odd | 6 | 1 | 1026.2.j.f | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 342.2.j.f | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 342.2.j.f | ✓ | 18 | 171.k | even | 6 | 1 | inner |
| 342.2.n.f | yes | 18 | 9.d | odd | 6 | 1 | |
| 342.2.n.f | yes | 18 | 19.d | odd | 6 | 1 | |
| 1026.2.j.f | 18 | 3.b | odd | 2 | 1 | ||
| 1026.2.j.f | 18 | 171.s | odd | 6 | 1 | ||
| 1026.2.n.f | 18 | 9.c | even | 3 | 1 | ||
| 1026.2.n.f | 18 | 57.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\):
|
\( T_{5}^{18} + 63 T_{5}^{16} + 1620 T_{5}^{14} + 21958 T_{5}^{12} + 169218 T_{5}^{10} + 745983 T_{5}^{8} + \cdots + 155952 \)
|
|
\( T_{7}^{18} + 30 T_{7}^{16} - 10 T_{7}^{15} + 588 T_{7}^{14} - 270 T_{7}^{13} + 6761 T_{7}^{12} + \cdots + 1926544 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{9} \)
|
| $3$ |
\( T^{18} - 4 T^{17} + \cdots + 19683 \)
|
| $5$ |
\( T^{18} + 63 T^{16} + \cdots + 155952 \)
|
| $7$ |
\( T^{18} + 30 T^{16} + \cdots + 1926544 \)
|
| $11$ |
\( T^{18} - 15 T^{17} + \cdots + 87723 \)
|
| $13$ |
\( T^{18} - 3 T^{17} + \cdots + 45349632 \)
|
| $17$ |
\( T^{18} - 15 T^{17} + \cdots + 5015547 \)
|
| $19$ |
\( T^{18} + \cdots + 322687697779 \)
|
| $23$ |
\( T^{18} - 99 T^{16} + \cdots + 5419008 \)
|
| $29$ |
\( (T^{9} - 18 T^{8} + \cdots - 13122)^{2} \)
|
| $31$ |
\( T^{18} + \cdots + 2293235712 \)
|
| $37$ |
\( T^{18} + 294 T^{16} + \cdots + 11337408 \)
|
| $41$ |
\( (T^{9} - 234 T^{7} + \cdots - 29447712)^{2} \)
|
| $43$ |
\( T^{18} + \cdots + 3396092176 \)
|
| $47$ |
\( T^{18} + \cdots + 87121505633328 \)
|
| $53$ |
\( T^{18} + \cdots + 40\!\cdots\!24 \)
|
| $59$ |
\( (T^{9} - 213 T^{7} + \cdots + 426708)^{2} \)
|
| $61$ |
\( (T^{9} - 9 T^{8} + \cdots - 281296)^{2} \)
|
| $67$ |
\( T^{18} + \cdots + 142397118885888 \)
|
| $71$ |
\( T^{18} + \cdots + 259843051643904 \)
|
| $73$ |
\( T^{18} + \cdots + 13\!\cdots\!01 \)
|
| $79$ |
\( T^{18} + \cdots + 25\!\cdots\!68 \)
|
| $83$ |
\( T^{18} + \cdots + 862152512709843 \)
|
| $89$ |
\( T^{18} + \cdots + 713791570066884 \)
|
| $97$ |
\( T^{18} + \cdots + 53197952688 \)
|
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