Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2688,2,Mod(673,2688)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2688, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2688.673");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2688 = 2^{7} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2688.w (of order \(4\), degree \(2\), not 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: | \(21.4637880633\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 4 x^{19} + 8 x^{18} - 16 x^{17} + 35 x^{16} - 56 x^{15} + 64 x^{14} - 84 x^{13} + 125 x^{12} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| Twist minimal: | no (minimal twist has level 336) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{20} - 4 x^{19} + 8 x^{18} - 16 x^{17} + 35 x^{16} - 56 x^{15} + 64 x^{14} - 84 x^{13} + 125 x^{12} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 58 \nu^{19} - 399 \nu^{18} + 1514 \nu^{17} - 1746 \nu^{16} + 1626 \nu^{15} - 5585 \nu^{14} + \cdots + 153600 ) / 17920 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7 \nu^{19} + 38 \nu^{18} - 36 \nu^{17} - 72 \nu^{16} - 139 \nu^{15} + 318 \nu^{14} + 180 \nu^{13} + \cdots + 25088 ) / 2560 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9 \nu^{19} - 14 \nu^{18} - 22 \nu^{17} + 216 \nu^{16} - 593 \nu^{15} + 826 \nu^{14} - 1450 \nu^{13} + \cdots + 52736 ) / 2560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 61 \nu^{19} - 385 \nu^{18} + 1206 \nu^{17} - 122 \nu^{16} + 149 \nu^{15} - 3891 \nu^{14} + \cdots + 103424 ) / 17920 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3 \nu^{19} - 40 \nu^{17} + 56 \nu^{16} - 55 \nu^{15} + 188 \nu^{14} - 336 \nu^{13} + 156 \nu^{12} + \cdots - 2304 \nu ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8 \nu^{19} + 22 \nu^{18} - 69 \nu^{17} + 112 \nu^{16} - 386 \nu^{15} + 682 \nu^{14} - 755 \nu^{13} + \cdots + 26112 ) / 1280 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 59 \nu^{19} + 42 \nu^{18} - 410 \nu^{17} + 524 \nu^{16} - 1075 \nu^{15} + 2658 \nu^{14} + \cdots + 49408 ) / 8960 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 269 \nu^{19} - 224 \nu^{18} - 382 \nu^{17} - 1464 \nu^{16} + 4707 \nu^{15} - 2284 \nu^{14} + \cdots - 400384 ) / 35840 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 165 \nu^{19} + 672 \nu^{18} - 1376 \nu^{17} + 2736 \nu^{16} - 5359 \nu^{15} + 7404 \nu^{14} + \cdots + 108544 ) / 17920 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 199 \nu^{19} + 392 \nu^{18} - 542 \nu^{17} + 1828 \nu^{16} - 2873 \nu^{15} + 2340 \nu^{14} + \cdots - 15360 ) / 17920 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 101 \nu^{19} + 385 \nu^{18} - 584 \nu^{17} + 1268 \nu^{16} - 3531 \nu^{15} + 4839 \nu^{14} + \cdots + 185344 ) / 8960 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 205 \nu^{19} + 966 \nu^{18} - 1948 \nu^{17} + 4308 \nu^{16} - 8647 \nu^{15} + 12662 \nu^{14} + \cdots + 197632 ) / 17920 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 451 \nu^{19} + 1176 \nu^{18} - 822 \nu^{17} + 1576 \nu^{16} - 4973 \nu^{15} + 3516 \nu^{14} + \cdots + 70656 ) / 35840 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 114 \nu^{19} - 140 \nu^{18} + 229 \nu^{17} + 682 \nu^{16} - 324 \nu^{15} - 904 \nu^{14} + \cdots + 4096 ) / 8960 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 260 \nu^{19} + 707 \nu^{18} - 876 \nu^{17} + 2126 \nu^{16} - 4964 \nu^{15} + 5309 \nu^{14} + \cdots + 105984 ) / 17920 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 30 \nu^{19} + 71 \nu^{18} - 8 \nu^{17} - 2 \nu^{16} - 162 \nu^{15} - 63 \nu^{14} + 740 \nu^{13} + \cdots - 3328 ) / 1280 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 71 \nu^{19} - 246 \nu^{18} + 392 \nu^{17} - 616 \nu^{16} + 1293 \nu^{15} - 1846 \nu^{14} + \cdots - 11776 ) / 2560 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 1235 \nu^{19} - 3612 \nu^{18} + 6646 \nu^{17} - 10816 \nu^{16} + 18189 \nu^{15} - 25944 \nu^{14} + \cdots + 65536 ) / 35840 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 885 \nu^{19} - 3031 \nu^{18} + 5848 \nu^{17} - 12958 \nu^{16} + 25287 \nu^{15} - 34197 \nu^{14} + \cdots - 604672 ) / 17920 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{14} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - \beta_{19} + 2 \beta_{18} - 2 \beta_{17} + 2 \beta_{16} - \beta_{15} - 3 \beta_{13} - \beta_{12} + \cdots + 2 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{18} - \beta_{17} + \beta_{15} + \beta_{14} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{19} - 4 \beta_{17} - 2 \beta_{16} + \beta_{15} + 4 \beta_{14} - 3 \beta_{13} - \beta_{12} + \cdots + 2 \beta_1 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4 \beta_{19} - 4 \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - \beta_{14} + 4 \beta_{13} + \cdots - 5 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2 \beta_{19} - \beta_{18} + \beta_{17} + \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{12} + \cdots + 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 8 \beta_{19} - 8 \beta_{18} + 2 \beta_{17} - 4 \beta_{16} + 6 \beta_{15} - 3 \beta_{14} + 8 \beta_{12} + \cdots + 7 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 7 \beta_{19} - 6 \beta_{18} + 4 \beta_{16} - 13 \beta_{15} - 4 \beta_{14} - 5 \beta_{13} + 7 \beta_{12} + \cdots - 16 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 2 \beta_{19} - 2 \beta_{18} - 3 \beta_{17} + 9 \beta_{16} + 5 \beta_{15} - 4 \beta_{14} - 28 \beta_{13} + \cdots + 18 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 7 \beta_{19} + 4 \beta_{18} - 4 \beta_{17} + 8 \beta_{16} + 25 \beta_{15} - 2 \beta_{14} - 73 \beta_{13} + \cdots - 2 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 8 \beta_{19} - 10 \beta_{18} - 36 \beta_{17} + 20 \beta_{16} - 44 \beta_{15} - \beta_{14} - 80 \beta_{13} + \cdots + 7 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( \beta_{19} - 19 \beta_{18} + 10 \beta_{17} + 27 \beta_{16} - 5 \beta_{15} + 8 \beta_{14} - 23 \beta_{13} + \cdots - 4 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 72 \beta_{19} + 56 \beta_{18} + 44 \beta_{17} + 100 \beta_{16} - 56 \beta_{15} + 17 \beta_{14} + \cdots - 47 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 17 \beta_{19} - 10 \beta_{18} + 2 \beta_{17} - 34 \beta_{16} - 215 \beta_{15} - 4 \beta_{14} + \cdots - 226 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 28 \beta_{19} - 49 \beta_{18} + 13 \beta_{17} + 12 \beta_{16} - 153 \beta_{15} + 23 \beta_{14} + \cdots - 126 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 77 \beta_{19} - 16 \beta_{18} + 220 \beta_{17} + 2 \beta_{16} + 59 \beta_{15} - 220 \beta_{14} + \cdots - 152 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 36 \beta_{19} + 60 \beta_{18} + 58 \beta_{17} - 230 \beta_{16} - 290 \beta_{15} - 147 \beta_{14} + \cdots - 167 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 82 \beta_{19} - 247 \beta_{18} - 57 \beta_{17} - \beta_{16} - 246 \beta_{15} - 149 \beta_{14} + \cdots + 476 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 744 \beta_{19} + 440 \beta_{18} + 894 \beta_{17} + 780 \beta_{16} - 6 \beta_{15} - 369 \beta_{14} + \cdots + 421 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2688\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(1793\) | \(1921\) | \(2437\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(\beta_{15}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 673.1 |
|
0 | −0.707107 | − | 0.707107i | 0 | −2.01011 | + | 2.01011i | 0 | 1.00000i | 0 | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.2 | 0 | −0.707107 | − | 0.707107i | 0 | −0.616911 | + | 0.616911i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.3 | 0 | −0.707107 | − | 0.707107i | 0 | −0.134119 | + | 0.134119i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.4 | 0 | −0.707107 | − | 0.707107i | 0 | 1.17321 | − | 1.17321i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.5 | 0 | −0.707107 | − | 0.707107i | 0 | 3.00215 | − | 3.00215i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.6 | 0 | 0.707107 | + | 0.707107i | 0 | −1.67936 | + | 1.67936i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.7 | 0 | 0.707107 | + | 0.707107i | 0 | −1.18844 | + | 1.18844i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.8 | 0 | 0.707107 | + | 0.707107i | 0 | −1.13147 | + | 1.13147i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.9 | 0 | 0.707107 | + | 0.707107i | 0 | 0.894131 | − | 0.894131i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 673.10 | 0 | 0.707107 | + | 0.707107i | 0 | 1.69093 | − | 1.69093i | 0 | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.1 | 0 | −0.707107 | + | 0.707107i | 0 | −2.01011 | − | 2.01011i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.2 | 0 | −0.707107 | + | 0.707107i | 0 | −0.616911 | − | 0.616911i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.3 | 0 | −0.707107 | + | 0.707107i | 0 | −0.134119 | − | 0.134119i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.4 | 0 | −0.707107 | + | 0.707107i | 0 | 1.17321 | + | 1.17321i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.5 | 0 | −0.707107 | + | 0.707107i | 0 | 3.00215 | + | 3.00215i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.6 | 0 | 0.707107 | − | 0.707107i | 0 | −1.67936 | − | 1.67936i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.7 | 0 | 0.707107 | − | 0.707107i | 0 | −1.18844 | − | 1.18844i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.8 | 0 | 0.707107 | − | 0.707107i | 0 | −1.13147 | − | 1.13147i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.9 | 0 | 0.707107 | − | 0.707107i | 0 | 0.894131 | + | 0.894131i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 2017.10 | 0 | 0.707107 | − | 0.707107i | 0 | 1.69093 | + | 1.69093i | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 16.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2688.2.w.b | 20 | |
| 4.b | odd | 2 | 1 | 2688.2.w.a | 20 | ||
| 8.b | even | 2 | 1 | 1344.2.w.a | 20 | ||
| 8.d | odd | 2 | 1 | 336.2.w.a | ✓ | 20 | |
| 16.e | even | 4 | 1 | 1344.2.w.a | 20 | ||
| 16.e | even | 4 | 1 | inner | 2688.2.w.b | 20 | |
| 16.f | odd | 4 | 1 | 336.2.w.a | ✓ | 20 | |
| 16.f | odd | 4 | 1 | 2688.2.w.a | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 336.2.w.a | ✓ | 20 | 8.d | odd | 2 | 1 | |
| 336.2.w.a | ✓ | 20 | 16.f | odd | 4 | 1 | |
| 1344.2.w.a | 20 | 8.b | even | 2 | 1 | ||
| 1344.2.w.a | 20 | 16.e | even | 4 | 1 | ||
| 2688.2.w.a | 20 | 4.b | odd | 2 | 1 | ||
| 2688.2.w.a | 20 | 16.f | odd | 4 | 1 | ||
| 2688.2.w.b | 20 | 1.a | even | 1 | 1 | trivial | |
| 2688.2.w.b | 20 | 16.e | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2688, [\chi])\):
|
\( T_{5}^{20} + 24 T_{5}^{17} + 240 T_{5}^{16} + 336 T_{5}^{15} + 288 T_{5}^{14} + 1040 T_{5}^{13} + \cdots + 4096 \)
|
|
\( T_{11}^{20} - 12 T_{11}^{19} + 72 T_{11}^{18} - 272 T_{11}^{17} + 1620 T_{11}^{16} - 12704 T_{11}^{15} + \cdots + 22963264 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{4} + 1)^{5} \)
|
| $5$ |
\( T^{20} + 24 T^{17} + \cdots + 4096 \)
|
| $7$ |
\( (T^{2} + 1)^{10} \)
|
| $11$ |
\( T^{20} - 12 T^{19} + \cdots + 22963264 \)
|
| $13$ |
\( T^{20} + 80 T^{17} + \cdots + 16384 \)
|
| $17$ |
\( (T^{10} - 76 T^{8} + \cdots - 1264)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 102252544 \)
|
| $23$ |
\( T^{20} + 232 T^{18} + \cdots + 25080064 \)
|
| $29$ |
\( T^{20} + \cdots + 2436995743744 \)
|
| $31$ |
\( (T^{10} - 176 T^{8} + \cdots + 101888)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 1952301973504 \)
|
| $41$ |
\( T^{20} + \cdots + 9755277742336 \)
|
| $43$ |
\( T^{20} - 4 T^{19} + \cdots + 5161984 \)
|
| $47$ |
\( (T^{10} - 216 T^{8} + \cdots - 65536)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 15\!\cdots\!64 \)
|
| $59$ |
\( T^{20} + \cdots + 655360000 \)
|
| $61$ |
\( T^{20} + \cdots + 1130326196224 \)
|
| $67$ |
\( T^{20} + \cdots + 286557184 \)
|
| $71$ |
\( T^{20} + \cdots + 43\!\cdots\!36 \)
|
| $73$ |
\( T^{20} + \cdots + 29\!\cdots\!00 \)
|
| $79$ |
\( (T^{10} + 12 T^{9} + \cdots + 35885056)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 14\!\cdots\!56 \)
|
| $89$ |
\( T^{20} + \cdots + 61\!\cdots\!96 \)
|
| $97$ |
\( (T^{10} + 36 T^{9} + \cdots - 6870424576)^{2} \)
|
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