Newspace parameters
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.2539180284\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 24774x^{2} - 86616x + 52534656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4}\cdot 5^{3} \) |
| 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
\( x^{4} - x^{3} - 24774x^{2} - 86616x + 52534656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6\nu - 12386 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 7\nu^{2} - 20494\nu + 8292 ) / 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6\beta _1 + 12386 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 7\beta_{2} + 20536\beta _1 + 78410 \)
|
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 |
|
−150.991 | 0 | 14606.2 | 15625.0 | 0 | −242904. | −968489. | 0 | −2.35923e6 | ||||||||||||||||||||||||||||||
| 1.2 | −54.6908 | 0 | −5200.92 | 15625.0 | 0 | 591583. | 732469. | 0 | −854543. | |||||||||||||||||||||||||||||||
| 1.3 | 42.2561 | 0 | −6406.42 | 15625.0 | 0 | −220964. | −616873. | 0 | 660252. | |||||||||||||||||||||||||||||||
| 1.4 | 148.425 | 0 | 13838.1 | 15625.0 | 0 | 215324. | 838027. | 0 | 2.31915e6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 45.14.a.g | ✓ | 4 |
| 3.b | odd | 2 | 1 | 45.14.a.h | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.14.a.g | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 45.14.a.h | yes | 4 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 15T_{2}^{3} - 24690T_{2}^{2} - 284600T_{2} + 51792000 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 15 T^{3} + \cdots + 51792000 \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( (T - 15625)^{4} \)
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| $7$ |
\( T^{4} + \cdots + 68\!\cdots\!00 \)
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| $11$ |
\( T^{4} + \cdots - 57\!\cdots\!00 \)
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| $13$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
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| $17$ |
\( T^{4} + \cdots + 57\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 16\!\cdots\!84 \)
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| $23$ |
\( T^{4} + \cdots + 86\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 38\!\cdots\!76 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 95\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 41\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
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| $53$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots - 84\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots - 77\!\cdots\!04 \)
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| $67$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
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| $71$ |
\( T^{4} + \cdots - 62\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 60\!\cdots\!44 \)
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| $83$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 35\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
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