Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.n (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.574922894553\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{15} - \nu^{14} - 3 \nu^{13} - 8 \nu^{12} + 2 \nu^{11} + 8 \nu^{10} - 16 \nu^{9} - 48 \nu^{8} + \cdots - 128 ) / 384 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{15} - \nu^{14} - 3 \nu^{13} - 6 \nu^{10} - 4 \nu^{9} - 4 \nu^{8} + 4 \nu^{7} + 8 \nu^{6} + \cdots + 64 ) / 192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{15} + \nu^{14} + 3 \nu^{13} + 4 \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} + 4 \nu^{8} + \cdots - 256 ) / 192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{15} + \nu^{14} - \nu^{13} - 2 \nu^{11} + 4 \nu^{10} + 8 \nu^{8} - 4 \nu^{7} + 16 \nu^{6} + \cdots + 128 ) / 128 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} + \cdots + 512 ) / 384 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{15} - \nu^{14} - 2 \nu^{13} - 3 \nu^{12} - 5 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 8 \nu^{8} + \cdots + 128 ) / 96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{15} + \nu^{14} + 7 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + 20 \nu^{10} + 24 \nu^{9} + 16 \nu^{8} + \cdots - 640 ) / 384 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{15} - 7 \nu^{14} - 9 \nu^{13} - 2 \nu^{12} - 10 \nu^{11} - 4 \nu^{10} - 16 \nu^{9} + 36 \nu^{7} + \cdots - 128 ) / 384 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - \nu^{15} - \nu^{14} - 7 \nu^{13} - 14 \nu^{12} - 22 \nu^{11} - 20 \nu^{10} - 24 \nu^{9} + \cdots + 640 ) / 384 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{15} - \nu^{14} + 4 \nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} + 8 \nu^{7} + 20 \nu^{5} + \cdots - 288 ) / 96 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - \nu^{15} + \nu^{14} - 4 \nu^{11} - \nu^{10} - 4 \nu^{9} + 2 \nu^{8} - 8 \nu^{7} - 20 \nu^{5} + \cdots + 192 ) / 96 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3 \nu^{15} - \nu^{14} + 9 \nu^{13} + 6 \nu^{12} + 26 \nu^{11} + 12 \nu^{10} + 16 \nu^{9} - 16 \nu^{8} + \cdots - 896 ) / 384 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 2 \nu^{15} + \nu^{14} - 5 \nu^{13} - 5 \nu^{12} - 16 \nu^{11} - 10 \nu^{10} - 28 \nu^{9} + \cdots + 704 ) / 192 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 9 \nu^{15} + \nu^{14} - 9 \nu^{13} - 8 \nu^{12} - 42 \nu^{11} - 12 \nu^{10} - 24 \nu^{9} + \cdots + 1664 ) / 384 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{8} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{12} - \beta_{11} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} + \beta_{14} + 2\beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} - \beta_{2} \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{15} + \beta_{13} - 2\beta_{10} + \beta_{9} - \beta_{7} + 5\beta_{6} + 5\beta_{5} - \beta_{3} - \beta_{2} + \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \cdots + \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots + 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( 5 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 10 \beta_{10} - \beta_{9} + \cdots + 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( 6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 3 \beta_{12} - 7 \beta_{11} - 3 \beta_{10} + 7 \beta_{9} + \cdots + 5 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 7 \beta_{15} - \beta_{14} + 12 \beta_{13} + \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 2 \beta_{9} + \cdots + 26 \)
|
| \(\nu^{12}\) | \(=\) |
\( 9 \beta_{15} + 12 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 6 \beta_{10} + 19 \beta_{9} + \cdots - 10 \)
|
| \(\nu^{13}\) | \(=\) |
\( 14 \beta_{15} - 3 \beta_{14} + 45 \beta_{13} + 13 \beta_{12} - 7 \beta_{11} - 11 \beta_{10} + \cdots + 5 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 47 \beta_{15} + 31 \beta_{14} - 28 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} + 41 \beta_{10} + \cdots - 78 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 55 \beta_{15} + 12 \beta_{14} - 65 \beta_{13} - 22 \beta_{12} - 22 \beta_{11} + 10 \beta_{10} + \cdots + 46 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1 - \beta_{4}\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−1.34532 | + | 0.436011i | 1.52768 | + | 0.816201i | 1.61979 | − | 1.17315i | −0.602794 | − | 0.348023i | −2.41110 | − | 0.431967i | 0.795065 | + | 1.37709i | −1.66763 | + | 2.28452i | 1.66763 | + | 2.49379i | 0.962695 | + | 0.205379i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 13.2 | −1.12494 | − | 0.857038i | −0.986088 | + | 1.42395i | 0.530970 | + | 1.92823i | 1.19115 | + | 0.687709i | 2.32967 | − | 0.756738i | 1.80469 | + | 3.12581i | 1.05526 | − | 2.62420i | −1.05526 | − | 2.80828i | −0.750573 | − | 1.79449i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 13.3 | −0.722180 | + | 1.21592i | 0.294546 | − | 1.70682i | −0.956913 | − | 1.75622i | 3.17262 | + | 1.83171i | 1.86264 | + | 1.59078i | −0.191926 | − | 0.332426i | 2.82649 | + | 0.104780i | −2.82649 | − | 1.00547i | −4.51841 | + | 2.53482i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 13.4 | −0.179748 | − | 1.40274i | 0.986088 | − | 1.42395i | −1.93538 | + | 0.504281i | −1.19115 | − | 0.687709i | −2.17468 | − | 1.12728i | 1.80469 | + | 3.12581i | 1.05526 | + | 2.62420i | −1.05526 | − | 2.80828i | −0.750573 | + | 1.79449i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 13.5 | 0.587625 | + | 1.28635i | 1.69028 | − | 0.378078i | −1.30939 | + | 1.51178i | −1.97542 | − | 1.14051i | 1.47959 | + | 1.95213i | −0.907824 | − | 1.57240i | −2.71411 | − | 0.795980i | 2.71411 | − | 1.27812i | 0.306290 | − | 3.21128i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 13.6 | 0.820200 | + | 1.15207i | −1.69028 | + | 0.378078i | −0.654545 | + | 1.88986i | 1.97542 | + | 1.14051i | −1.82194 | − | 1.63723i | −0.907824 | − | 1.57240i | −2.71411 | + | 0.795980i | 2.71411 | − | 1.27812i | 0.306290 | + | 3.21128i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 13.7 | 1.05026 | − | 0.947078i | −1.52768 | − | 0.816201i | 0.206086 | − | 1.98935i | 0.602794 | + | 0.348023i | −2.37747 | + | 0.589613i | 0.795065 | + | 1.37709i | −1.66763 | − | 2.28452i | 1.66763 | + | 2.49379i | 0.962695 | − | 0.205379i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 13.8 | 1.41411 | − | 0.0174668i | −0.294546 | + | 1.70682i | 1.99939 | − | 0.0493999i | −3.17262 | − | 1.83171i | −0.386706 | + | 2.41877i | −0.191926 | − | 0.332426i | 2.82649 | − | 0.104780i | −2.82649 | − | 1.00547i | −4.51841 | − | 2.53482i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.1 | −1.34532 | − | 0.436011i | 1.52768 | − | 0.816201i | 1.61979 | + | 1.17315i | −0.602794 | + | 0.348023i | −2.41110 | + | 0.431967i | 0.795065 | − | 1.37709i | −1.66763 | − | 2.28452i | 1.66763 | − | 2.49379i | 0.962695 | − | 0.205379i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.2 | −1.12494 | + | 0.857038i | −0.986088 | − | 1.42395i | 0.530970 | − | 1.92823i | 1.19115 | − | 0.687709i | 2.32967 | + | 0.756738i | 1.80469 | − | 3.12581i | 1.05526 | + | 2.62420i | −1.05526 | + | 2.80828i | −0.750573 | + | 1.79449i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.3 | −0.722180 | − | 1.21592i | 0.294546 | + | 1.70682i | −0.956913 | + | 1.75622i | 3.17262 | − | 1.83171i | 1.86264 | − | 1.59078i | −0.191926 | + | 0.332426i | 2.82649 | − | 0.104780i | −2.82649 | + | 1.00547i | −4.51841 | − | 2.53482i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.4 | −0.179748 | + | 1.40274i | 0.986088 | + | 1.42395i | −1.93538 | − | 0.504281i | −1.19115 | + | 0.687709i | −2.17468 | + | 1.12728i | 1.80469 | − | 3.12581i | 1.05526 | − | 2.62420i | −1.05526 | + | 2.80828i | −0.750573 | − | 1.79449i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.5 | 0.587625 | − | 1.28635i | 1.69028 | + | 0.378078i | −1.30939 | − | 1.51178i | −1.97542 | + | 1.14051i | 1.47959 | − | 1.95213i | −0.907824 | + | 1.57240i | −2.71411 | + | 0.795980i | 2.71411 | + | 1.27812i | 0.306290 | + | 3.21128i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.6 | 0.820200 | − | 1.15207i | −1.69028 | − | 0.378078i | −0.654545 | − | 1.88986i | 1.97542 | − | 1.14051i | −1.82194 | + | 1.63723i | −0.907824 | + | 1.57240i | −2.71411 | − | 0.795980i | 2.71411 | + | 1.27812i | 0.306290 | − | 3.21128i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.7 | 1.05026 | + | 0.947078i | −1.52768 | + | 0.816201i | 0.206086 | + | 1.98935i | 0.602794 | − | 0.348023i | −2.37747 | − | 0.589613i | 0.795065 | − | 1.37709i | −1.66763 | + | 2.28452i | 1.66763 | − | 2.49379i | 0.962695 | + | 0.205379i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 61.8 | 1.41411 | + | 0.0174668i | −0.294546 | − | 1.70682i | 1.99939 | + | 0.0493999i | −3.17262 | + | 1.83171i | −0.386706 | − | 2.41877i | −0.191926 | + | 0.332426i | 2.82649 | + | 0.104780i | −2.82649 | + | 1.00547i | −4.51841 | + | 2.53482i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 72.n | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.2.n.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | 216.2.n.b | 16 | ||
| 4.b | odd | 2 | 1 | 288.2.r.b | 16 | ||
| 8.b | even | 2 | 1 | inner | 72.2.n.b | ✓ | 16 |
| 8.d | odd | 2 | 1 | 288.2.r.b | 16 | ||
| 9.c | even | 3 | 1 | inner | 72.2.n.b | ✓ | 16 |
| 9.c | even | 3 | 1 | 648.2.d.j | 8 | ||
| 9.d | odd | 6 | 1 | 216.2.n.b | 16 | ||
| 9.d | odd | 6 | 1 | 648.2.d.k | 8 | ||
| 12.b | even | 2 | 1 | 864.2.r.b | 16 | ||
| 24.f | even | 2 | 1 | 864.2.r.b | 16 | ||
| 24.h | odd | 2 | 1 | 216.2.n.b | 16 | ||
| 36.f | odd | 6 | 1 | 288.2.r.b | 16 | ||
| 36.f | odd | 6 | 1 | 2592.2.d.j | 8 | ||
| 36.h | even | 6 | 1 | 864.2.r.b | 16 | ||
| 36.h | even | 6 | 1 | 2592.2.d.k | 8 | ||
| 72.j | odd | 6 | 1 | 216.2.n.b | 16 | ||
| 72.j | odd | 6 | 1 | 648.2.d.k | 8 | ||
| 72.l | even | 6 | 1 | 864.2.r.b | 16 | ||
| 72.l | even | 6 | 1 | 2592.2.d.k | 8 | ||
| 72.n | even | 6 | 1 | inner | 72.2.n.b | ✓ | 16 |
| 72.n | even | 6 | 1 | 648.2.d.j | 8 | ||
| 72.p | odd | 6 | 1 | 288.2.r.b | 16 | ||
| 72.p | odd | 6 | 1 | 2592.2.d.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.2.n.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 72.2.n.b | ✓ | 16 | 8.b | even | 2 | 1 | inner |
| 72.2.n.b | ✓ | 16 | 9.c | even | 3 | 1 | inner |
| 72.2.n.b | ✓ | 16 | 72.n | even | 6 | 1 | inner |
| 216.2.n.b | 16 | 3.b | odd | 2 | 1 | ||
| 216.2.n.b | 16 | 9.d | odd | 6 | 1 | ||
| 216.2.n.b | 16 | 24.h | odd | 2 | 1 | ||
| 216.2.n.b | 16 | 72.j | odd | 6 | 1 | ||
| 288.2.r.b | 16 | 4.b | odd | 2 | 1 | ||
| 288.2.r.b | 16 | 8.d | odd | 2 | 1 | ||
| 288.2.r.b | 16 | 36.f | odd | 6 | 1 | ||
| 288.2.r.b | 16 | 72.p | odd | 6 | 1 | ||
| 648.2.d.j | 8 | 9.c | even | 3 | 1 | ||
| 648.2.d.j | 8 | 72.n | even | 6 | 1 | ||
| 648.2.d.k | 8 | 9.d | odd | 6 | 1 | ||
| 648.2.d.k | 8 | 72.j | odd | 6 | 1 | ||
| 864.2.r.b | 16 | 12.b | even | 2 | 1 | ||
| 864.2.r.b | 16 | 24.f | even | 2 | 1 | ||
| 864.2.r.b | 16 | 36.h | even | 6 | 1 | ||
| 864.2.r.b | 16 | 72.l | even | 6 | 1 | ||
| 2592.2.d.j | 8 | 36.f | odd | 6 | 1 | ||
| 2592.2.d.j | 8 | 72.p | odd | 6 | 1 | ||
| 2592.2.d.k | 8 | 36.h | even | 6 | 1 | ||
| 2592.2.d.k | 8 | 72.l | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{16} - 21T_{5}^{14} + 326T_{5}^{12} - 2049T_{5}^{10} + 9318T_{5}^{8} - 18357T_{5}^{6} + 26129T_{5}^{4} - 11712T_{5}^{2} + 4096 \)
acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - T^{15} + \cdots + 256 \)
|
| $3$ |
\( T^{16} - T^{14} + \cdots + 6561 \)
|
| $5$ |
\( T^{16} - 21 T^{14} + \cdots + 4096 \)
|
| $7$ |
\( (T^{8} - 3 T^{7} + 14 T^{6} + \cdots + 16)^{2} \)
|
| $11$ |
\( T^{16} - 40 T^{14} + \cdots + 10556001 \)
|
| $13$ |
\( T^{16} - 53 T^{14} + \cdots + 20736 \)
|
| $17$ |
\( (T^{4} + 7 T^{3} - 2 T^{2} + \cdots + 36)^{4} \)
|
| $19$ |
\( (T^{8} + 83 T^{6} + \cdots + 5184)^{2} \)
|
| $23$ |
\( (T^{8} + 5 T^{7} + \cdots + 90000)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 4032758016 \)
|
| $31$ |
\( (T^{8} + 5 T^{7} + \cdots + 92416)^{2} \)
|
| $37$ |
\( (T^{8} + 152 T^{6} + \cdots + 1327104)^{2} \)
|
| $41$ |
\( (T^{8} + 4 T^{7} + \cdots + 335241)^{2} \)
|
| $43$ |
\( T^{16} - 20 T^{14} + \cdots + 81 \)
|
| $47$ |
\( (T^{8} - 3 T^{7} + \cdots + 2178576)^{2} \)
|
| $53$ |
\( (T^{8} + 160 T^{6} + \cdots + 451584)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 131079601 \)
|
| $61$ |
\( T^{16} + \cdots + 176319369216 \)
|
| $67$ |
\( T^{16} + \cdots + 9881774573841 \)
|
| $71$ |
\( (T^{4} - 18 T^{3} + \cdots - 864)^{4} \)
|
| $73$ |
\( (T^{4} + 11 T^{3} + \cdots + 36)^{4} \)
|
| $79$ |
\( (T^{8} + 15 T^{7} + \cdots + 3136)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 31713911056 \)
|
| $89$ |
\( (T^{4} - 16 T^{3} + \cdots - 504)^{4} \)
|
| $97$ |
\( (T^{8} + 134 T^{6} + \cdots + 1324801)^{2} \)
|
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