Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(131.919902888\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
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|
|
| Defining polynomial: |
\( x^{4} - 43098290x^{2} - 18986612040x + 22664628899989 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{8} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 43098290x^{2} - 18986612040x + 22664628899989 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -568\nu^{3} + 3651712\nu^{2} + 15383263192\nu - 70602974657200 ) / 79518607 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -7168\nu^{3} + 4384\nu^{2} + 291728178496\nu + 101977554875360 ) / 11359801 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -806216\nu^{3} + 990012896\nu^{2} + 43907311246184\nu - 9853448638443440 ) / 79518607 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 13\beta_{2} - 271\beta_1 ) / 165888 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 353\beta_{3} - 11405\beta_{2} + 506449\beta _1 + 595790760960 ) / 27648 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 40699981\beta_{3} - 792023017\beta_{2} - 11027485315\beta _1 + 2362238323568640 ) / 165888 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −1.65331e6 | 0 | 6.66132e6 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −484700. | 0 | −8.65967e6 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 601950. | 0 | 2.51263e7 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 858240. | 0 | −1.34951e7 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.18.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 72.18.a.h | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.18.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 72.18.a.h | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 677824T_{5}^{3} - 1803927221760T_{5}^{2} - 65604870625280000T_{5} + 413997294267080089600000 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
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| $7$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
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| $11$ |
\( T^{4} + \cdots + 14\!\cdots\!64 \)
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| $13$ |
\( T^{4} + \cdots + 24\!\cdots\!36 \)
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| $17$ |
\( T^{4} + \cdots + 62\!\cdots\!64 \)
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| $19$ |
\( T^{4} + \cdots - 17\!\cdots\!24 \)
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| $23$ |
\( T^{4} + \cdots + 32\!\cdots\!64 \)
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| $29$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
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| $31$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
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| $37$ |
\( T^{4} + \cdots - 89\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 98\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots + 43\!\cdots\!76 \)
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| $53$ |
\( T^{4} + \cdots - 42\!\cdots\!76 \)
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| $59$ |
\( T^{4} + \cdots + 60\!\cdots\!04 \)
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| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!64 \)
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| $67$ |
\( T^{4} + \cdots + 77\!\cdots\!04 \)
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| $71$ |
\( T^{4} + \cdots - 75\!\cdots\!84 \)
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| $73$ |
\( T^{4} + \cdots + 93\!\cdots\!96 \)
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| $79$ |
\( T^{4} + \cdots - 69\!\cdots\!04 \)
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| $83$ |
\( T^{4} + \cdots - 31\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots + 95\!\cdots\!24 \)
|
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