Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(45.7457221925\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 25027x^{2} + 25028x + 156700326 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 2x^{3} - 25027x^{2} + 25028x + 156700326 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 25019\nu + 12510 ) / 50073 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -32\nu^{3} + 48\nu^{2} + 1201472\nu - 600744 ) / 16691 \)
|
| \(\beta_{3}\) | \(=\) |
\( 12\nu^{2} - 12\nu - 150168 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 48\beta _1 + 24 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 48\beta _1 + 600696 ) / 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 12511\beta_{2} + 1802280\beta _1 + 901032 ) / 48 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | 0 | 0 | − | 5628.61i | 0 | 7950.05 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 17.2 | 0 | 0 | 0 | − | 1965.79i | 0 | −2790.05 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 17.3 | 0 | 0 | 0 | 1965.79i | 0 | −2790.05 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 17.4 | 0 | 0 | 0 | 5628.61i | 0 | 7950.05 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.11.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 72.11.e.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 144.11.e.c | 4 | ||
| 12.b | even | 2 | 1 | 144.11.e.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.11.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 72.11.e.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 144.11.e.c | 4 | 4.b | odd | 2 | 1 | ||
| 144.11.e.c | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 35545540T_{5}^{2} + 122426922208900 \)
acting on \(S_{11}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 122426922208900 \)
|
| $7$ |
\( (T^{2} - 5160 T - 22181040)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 37\!\cdots\!44 \)
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| $13$ |
\( (T^{2} - 205184 T - 37950618176)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $19$ |
\( (T^{2} + \cdots - 2095718172416)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 35\!\cdots\!84 \)
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| $29$ |
\( T^{4} + \cdots + 10\!\cdots\!24 \)
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| $31$ |
\( (T^{2} + \cdots - 274102731369776)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 325550457908604)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 57\!\cdots\!84 \)
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| $43$ |
\( (T^{2} + \cdots + 88\!\cdots\!56)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 64\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots + 32\!\cdots\!44 \)
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| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!44 \)
|
| $61$ |
\( (T^{2} + \cdots + 47\!\cdots\!84)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 13\!\cdots\!24)^{2} \)
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| $71$ |
\( T^{4} + \cdots + 18\!\cdots\!44 \)
|
| $73$ |
\( (T^{2} + \cdots + 76\!\cdots\!84)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots + 12\!\cdots\!36)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 50\!\cdots\!24 \)
|
| $89$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots - 13\!\cdots\!60)^{2} \)
|
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