Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(102.739323672\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 289882x^{2} - 60469800x - 1406031219 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{8}\cdot 5 \) |
| 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
\( x^{4} - 289882x^{2} - 60469800x - 1406031219 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{3} - 528\nu^{2} - 2266568\nu - 286289952 ) / 735 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -136\nu^{3} + 39216\nu^{2} + 29036296\nu + 483913344 ) / 735 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 272\nu^{3} - 58272\nu^{2} - 23760272\nu - 3889837248 ) / 245 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} - 34\beta_1 ) / 165888 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 157\beta_{3} + 2644\beta_{2} + 28934\beta _1 + 12021986304 ) / 82944 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 304045\beta_{3} + 1482292\beta_{2} + 9427334\beta _1 + 7523410636800 ) / 165888 \)
|
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 | −314788. | 0 | −557948. | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 50674.5 | 0 | 1.20768e6 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 96049.6 | 0 | 2.88558e6 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 141119. | 0 | −3.12314e6 | 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 | 72.16.a.h | ✓ | 4 |
| 3.b | odd | 2 | 1 | 72.16.a.i | yes | 4 | |
| 4.b | odd | 2 | 1 | 144.16.a.w | 4 | ||
| 12.b | even | 2 | 1 | 144.16.a.x | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.16.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 72.16.a.i | yes | 4 | 3.b | odd | 2 | 1 | |
| 144.16.a.w | 4 | 4.b | odd | 2 | 1 | ||
| 144.16.a.x | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 26944T_{5}^{3} - 65036674560T_{5}^{2} + 7363159855513600T_{5} - 216216684950872064000 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
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| $7$ |
\( T^{4} + \cdots + 60\!\cdots\!28 \)
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| $11$ |
\( T^{4} + \cdots + 56\!\cdots\!32 \)
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| $13$ |
\( T^{4} + \cdots - 89\!\cdots\!76 \)
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| $17$ |
\( T^{4} + \cdots + 43\!\cdots\!12 \)
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| $19$ |
\( T^{4} + \cdots + 24\!\cdots\!04 \)
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| $23$ |
\( T^{4} + \cdots - 59\!\cdots\!72 \)
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| $29$ |
\( T^{4} + \cdots - 23\!\cdots\!60 \)
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| $31$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
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| $37$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 46\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots + 83\!\cdots\!36 \)
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| $53$ |
\( T^{4} + \cdots - 28\!\cdots\!28 \)
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| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!68 \)
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| $61$ |
\( T^{4} + \cdots + 17\!\cdots\!28 \)
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| $67$ |
\( T^{4} + \cdots - 27\!\cdots\!32 \)
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| $71$ |
\( T^{4} + \cdots + 34\!\cdots\!84 \)
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| $73$ |
\( T^{4} + \cdots - 36\!\cdots\!44 \)
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| $79$ |
\( T^{4} + \cdots - 10\!\cdots\!16 \)
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| $83$ |
\( T^{4} + \cdots + 25\!\cdots\!32 \)
|
| $89$ |
\( T^{4} + \cdots + 70\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots + 78\!\cdots\!52 \)
|
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