Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1224,2,Mod(1189,1224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1224.1189");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1224.l (of order \(2\), degree \(1\), 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: | \(9.77368920740\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 408) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{16} - \nu^{14} - 2 \nu^{12} + 2 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + 8 \nu^{6} - 16 \nu^{5} + \cdots - 128 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{16} - \nu^{14} + 4 \nu^{13} - 2 \nu^{12} + 4 \nu^{11} + 2 \nu^{10} - 4 \nu^{9} + 4 \nu^{8} + \cdots - 256 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{17} - 3 \nu^{15} + 6 \nu^{13} + 6 \nu^{11} + 4 \nu^{10} - 12 \nu^{9} + 16 \nu^{8} + \cdots + 384 \nu ) / 256 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{17} + \nu^{15} + 2 \nu^{13} - 2 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 8 \nu^{7} + 16 \nu^{6} + \cdots + 128 \nu ) / 256 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{15} + 2 \nu^{14} - 3 \nu^{13} + 2 \nu^{12} - 2 \nu^{11} - 2 \nu^{9} - 8 \nu^{8} + 12 \nu^{7} + \cdots + 128 ) / 64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{17} + 2 \nu^{16} - \nu^{15} + 4 \nu^{14} - 4 \nu^{13} + 2 \nu^{12} + 2 \nu^{11} - 8 \nu^{10} + \cdots - 256 ) / 128 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3 \nu^{17} - 6 \nu^{16} + 9 \nu^{15} - 10 \nu^{14} + 6 \nu^{13} - 4 \nu^{12} + 6 \nu^{11} + 8 \nu^{10} + \cdots + 256 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 3 \nu^{17} - 6 \nu^{16} + 5 \nu^{15} - 6 \nu^{14} + 2 \nu^{13} - 10 \nu^{11} + 8 \nu^{10} + 4 \nu^{9} + \cdots + 768 ) / 256 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3 \nu^{17} - 8 \nu^{16} + 5 \nu^{15} - 12 \nu^{14} + 2 \nu^{13} + 12 \nu^{12} - 10 \nu^{11} + \cdots + 1536 ) / 256 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 2 \nu^{17} + \nu^{16} - 2 \nu^{15} + \nu^{14} + 4 \nu^{13} - 8 \nu^{12} + 8 \nu^{11} + 2 \nu^{10} + \cdots - 384 ) / 128 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 3 \nu^{17} + 5 \nu^{16} - 5 \nu^{15} + 7 \nu^{14} - 2 \nu^{13} - 6 \nu^{12} + 10 \nu^{11} + \cdots - 768 ) / 128 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 2 \nu^{17} - 3 \nu^{16} - 2 \nu^{15} + 3 \nu^{14} - 12 \nu^{13} + 14 \nu^{12} - 20 \nu^{11} + \cdots + 1280 ) / 128 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 3 \nu^{17} - 2 \nu^{16} + 7 \nu^{15} - 18 \nu^{14} + 26 \nu^{13} - 32 \nu^{12} + 26 \nu^{11} + \cdots - 1280 ) / 256 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 5 \nu^{17} - 4 \nu^{16} - \nu^{15} + 8 \nu^{14} - 22 \nu^{13} + 28 \nu^{12} - 22 \nu^{11} + \cdots + 1792 ) / 256 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 9 \nu^{17} - 10 \nu^{16} + 3 \nu^{15} - 2 \nu^{14} - 22 \nu^{13} + 40 \nu^{12} - 46 \nu^{11} + \cdots + 3328 ) / 256 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 11 \nu^{17} + 14 \nu^{16} - 5 \nu^{15} - 2 \nu^{14} + 30 \nu^{13} - 48 \nu^{12} + 50 \nu^{11} + \cdots - 3840 ) / 256 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{17} + \beta_{15} - \beta_{11} - \beta_{7} - \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{17} + \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{17} + 2 \beta_{16} - \beta_{15} + 2 \beta_{14} + \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots - \beta_{2} ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - \beta_{17} + 2 \beta_{14} + \beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{8} - \beta_{6} + \cdots + 2 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - \beta_{17} + 2 \beta_{16} - 3 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{9} + \cdots - 2 \beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( \beta_{17} + 4 \beta_{15} + 2 \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} - 3 \beta_{8} + \cdots - \beta_{2} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3 \beta_{17} - 2 \beta_{16} + 5 \beta_{15} + 2 \beta_{13} - 2 \beta_{12} - \beta_{11} + 6 \beta_{9} + \cdots - 2 \beta_1 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( \beta_{17} + 8 \beta_{16} - 8 \beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{12} + 5 \beta_{11} + \cdots - 9 \beta_{2} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 5 \beta_{17} + 14 \beta_{16} - 3 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} + 10 \beta_{12} + 11 \beta_{11} + \cdots - 16 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 3 \beta_{17} + 12 \beta_{15} + 6 \beta_{14} - 9 \beta_{13} + 10 \beta_{12} + \beta_{11} + 8 \beta_{10} + \cdots + 32 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 5 \beta_{17} - 2 \beta_{16} - 7 \beta_{15} - 12 \beta_{14} - 10 \beta_{13} - 6 \beta_{12} - 5 \beta_{11} + \cdots - 48 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 5 \beta_{17} + 24 \beta_{16} - 8 \beta_{15} - 10 \beta_{14} - 29 \beta_{13} - 14 \beta_{12} + \beta_{11} + \cdots + 32 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 5 \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 12 \beta_{14} - 6 \beta_{13} + 2 \beta_{12} + 19 \beta_{11} + \cdots + 48 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 19 \beta_{17} + 32 \beta_{16} + 20 \beta_{15} + 14 \beta_{14} - 57 \beta_{13} - 14 \beta_{12} + \cdots + 96 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 59 \beta_{17} - 34 \beta_{16} + 41 \beta_{15} - 52 \beta_{14} + 6 \beta_{13} + 2 \beta_{12} - 77 \beta_{11} + \cdots + 240 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 29 \beta_{17} + 136 \beta_{16} - 80 \beta_{15} - 42 \beta_{14} - 93 \beta_{13} - 14 \beta_{12} + \cdots + 32 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).
| \(n\) | \(137\) | \(613\) | \(649\) | \(919\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1189.1 |
|
−1.36011 | − | 0.387425i | 0 | 1.69980 | + | 1.05388i | −0.665591 | 0 | − | 0.998700i | −1.90362 | − | 2.09194i | 0 | 0.905277 | + | 0.257866i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.2 | −1.36011 | + | 0.387425i | 0 | 1.69980 | − | 1.05388i | −0.665591 | 0 | 0.998700i | −1.90362 | + | 2.09194i | 0 | 0.905277 | − | 0.257866i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.3 | −1.35042 | − | 0.419960i | 0 | 1.64727 | + | 1.13424i | 3.71339 | 0 | 5.19876i | −1.74817 | − | 2.22349i | 0 | −5.01464 | − | 1.55948i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.4 | −1.35042 | + | 0.419960i | 0 | 1.64727 | − | 1.13424i | 3.71339 | 0 | − | 5.19876i | −1.74817 | + | 2.22349i | 0 | −5.01464 | + | 1.55948i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.5 | −0.943929 | − | 1.05309i | 0 | −0.217998 | + | 1.98808i | −1.57273 | 0 | − | 1.68207i | 2.29941 | − | 1.64704i | 0 | 1.48454 | + | 1.65622i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.6 | −0.943929 | + | 1.05309i | 0 | −0.217998 | − | 1.98808i | −1.57273 | 0 | 1.68207i | 2.29941 | + | 1.64704i | 0 | 1.48454 | − | 1.65622i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.7 | −0.410541 | − | 1.35331i | 0 | −1.66291 | + | 1.11118i | −1.09133 | 0 | 4.50716i | 2.18647 | + | 1.79426i | 0 | 0.448036 | + | 1.47691i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.8 | −0.410541 | + | 1.35331i | 0 | −1.66291 | − | 1.11118i | −1.09133 | 0 | − | 4.50716i | 2.18647 | − | 1.79426i | 0 | 0.448036 | − | 1.47691i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.9 | −0.0535843 | − | 1.41320i | 0 | −1.99426 | + | 0.151450i | 3.17378 | 0 | − | 2.51539i | 0.320890 | + | 2.81017i | 0 | −0.170065 | − | 4.48518i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.10 | −0.0535843 | + | 1.41320i | 0 | −1.99426 | − | 0.151450i | 3.17378 | 0 | 2.51539i | 0.320890 | − | 2.81017i | 0 | −0.170065 | + | 4.48518i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.11 | 0.480188 | − | 1.33020i | 0 | −1.53884 | − | 1.27749i | 1.12134 | 0 | − | 0.430012i | −2.43824 | + | 1.43352i | 0 | 0.538452 | − | 1.49159i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.12 | 0.480188 | + | 1.33020i | 0 | −1.53884 | + | 1.27749i | 1.12134 | 0 | 0.430012i | −2.43824 | − | 1.43352i | 0 | 0.538452 | + | 1.49159i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.13 | 0.937200 | − | 1.05908i | 0 | −0.243310 | − | 1.98514i | −4.06326 | 0 | 1.33474i | −2.33046 | − | 1.60279i | 0 | −3.80809 | + | 4.30332i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.14 | 0.937200 | + | 1.05908i | 0 | −0.243310 | + | 1.98514i | −4.06326 | 0 | − | 1.33474i | −2.33046 | + | 1.60279i | 0 | −3.80809 | − | 4.30332i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.15 | 1.29188 | − | 0.575356i | 0 | 1.33793 | − | 1.48659i | 2.84912 | 0 | − | 1.38176i | 0.873137 | − | 2.69028i | 0 | 3.68073 | − | 1.63925i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.16 | 1.29188 | + | 0.575356i | 0 | 1.33793 | + | 1.48659i | 2.84912 | 0 | 1.38176i | 0.873137 | + | 2.69028i | 0 | 3.68073 | + | 1.63925i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.17 | 1.40931 | − | 0.117654i | 0 | 1.97232 | − | 0.331622i | −1.46472 | 0 | 3.26018i | 2.74059 | − | 0.699408i | 0 | −2.06424 | + | 0.172330i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1189.18 | 1.40931 | + | 0.117654i | 0 | 1.97232 | + | 0.331622i | −1.46472 | 0 | − | 3.26018i | 2.74059 | + | 0.699408i | 0 | −2.06424 | − | 0.172330i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 136.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1224.2.l.d | 18 | |
| 3.b | odd | 2 | 1 | 408.2.l.b | yes | 18 | |
| 4.b | odd | 2 | 1 | 4896.2.l.d | 18 | ||
| 8.b | even | 2 | 1 | 1224.2.l.c | 18 | ||
| 8.d | odd | 2 | 1 | 4896.2.l.c | 18 | ||
| 12.b | even | 2 | 1 | 1632.2.l.a | 18 | ||
| 17.b | even | 2 | 1 | 1224.2.l.c | 18 | ||
| 24.f | even | 2 | 1 | 1632.2.l.b | 18 | ||
| 24.h | odd | 2 | 1 | 408.2.l.a | ✓ | 18 | |
| 51.c | odd | 2 | 1 | 408.2.l.a | ✓ | 18 | |
| 68.d | odd | 2 | 1 | 4896.2.l.c | 18 | ||
| 136.e | odd | 2 | 1 | 4896.2.l.d | 18 | ||
| 136.h | even | 2 | 1 | inner | 1224.2.l.d | 18 | |
| 204.h | even | 2 | 1 | 1632.2.l.b | 18 | ||
| 408.b | odd | 2 | 1 | 408.2.l.b | yes | 18 | |
| 408.h | even | 2 | 1 | 1632.2.l.a | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 408.2.l.a | ✓ | 18 | 24.h | odd | 2 | 1 | |
| 408.2.l.a | ✓ | 18 | 51.c | odd | 2 | 1 | |
| 408.2.l.b | yes | 18 | 3.b | odd | 2 | 1 | |
| 408.2.l.b | yes | 18 | 408.b | odd | 2 | 1 | |
| 1224.2.l.c | 18 | 8.b | even | 2 | 1 | ||
| 1224.2.l.c | 18 | 17.b | even | 2 | 1 | ||
| 1224.2.l.d | 18 | 1.a | even | 1 | 1 | trivial | |
| 1224.2.l.d | 18 | 136.h | even | 2 | 1 | inner | |
| 1632.2.l.a | 18 | 12.b | even | 2 | 1 | ||
| 1632.2.l.a | 18 | 408.h | even | 2 | 1 | ||
| 1632.2.l.b | 18 | 24.f | even | 2 | 1 | ||
| 1632.2.l.b | 18 | 204.h | even | 2 | 1 | ||
| 4896.2.l.c | 18 | 8.d | odd | 2 | 1 | ||
| 4896.2.l.c | 18 | 68.d | odd | 2 | 1 | ||
| 4896.2.l.d | 18 | 4.b | odd | 2 | 1 | ||
| 4896.2.l.d | 18 | 136.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{9} - 2T_{5}^{8} - 26T_{5}^{7} + 44T_{5}^{6} + 197T_{5}^{5} - 166T_{5}^{4} - 648T_{5}^{3} - 60T_{5}^{2} + 552T_{5} + 256 \)
acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - T^{16} + \cdots + 512 \)
|
| $3$ |
\( T^{18} \)
|
| $5$ |
\( (T^{9} - 2 T^{8} + \cdots + 256)^{2} \)
|
| $7$ |
\( T^{18} + 72 T^{16} + \cdots + 65536 \)
|
| $11$ |
\( (T^{9} - 54 T^{7} + \cdots - 512)^{2} \)
|
| $13$ |
\( T^{18} + 124 T^{16} + \cdots + 1048576 \)
|
| $17$ |
\( T^{18} + \cdots + 118587876497 \)
|
| $19$ |
\( T^{18} + \cdots + 5110534144 \)
|
| $23$ |
\( T^{18} + \cdots + 57431163904 \)
|
| $29$ |
\( (T^{9} + 6 T^{8} + \cdots - 8192)^{2} \)
|
| $31$ |
\( T^{18} + 192 T^{16} + \cdots + 75759616 \)
|
| $37$ |
\( (T^{9} + 8 T^{8} + \cdots + 182272)^{2} \)
|
| $41$ |
\( T^{18} + \cdots + 1474922807296 \)
|
| $43$ |
\( T^{18} + \cdots + 22641152425984 \)
|
| $47$ |
\( (T^{9} - 220 T^{7} + \cdots - 8781824)^{2} \)
|
| $53$ |
\( T^{18} + \cdots + 3561644621824 \)
|
| $59$ |
\( T^{18} + \cdots + 10417711611904 \)
|
| $61$ |
\( (T^{9} + 8 T^{8} + \cdots - 22528)^{2} \)
|
| $67$ |
\( T^{18} + \cdots + 8875147264 \)
|
| $71$ |
\( T^{18} + \cdots + 34530008498176 \)
|
| $73$ |
\( T^{18} + \cdots + 3427652337664 \)
|
| $79$ |
\( T^{18} + \cdots + 223514460160000 \)
|
| $83$ |
\( T^{18} + \cdots + 945638269517824 \)
|
| $89$ |
\( (T^{9} + 10 T^{8} + \cdots + 192794368)^{2} \)
|
| $97$ |
\( T^{18} + \cdots + 9088150798336 \)
|
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