Newspace parameters
| Level: | \( N \) | \(=\) | \( 740 = 2^{2} \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 740.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.90892974957\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.286164.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 11x^{2} - 3x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} - 11x^{2} - 3x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 7\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 11\beta _1 + 8 \)
|
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 | −2.84956 | 0 | 1.00000 | 0 | −4.84956 | 0 | 5.11999 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0.0858680 | 0 | 1.00000 | 0 | −1.91413 | 0 | −2.99263 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.51951 | 0 | 1.00000 | 0 | 0.519509 | 0 | 3.34793 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.24418 | 0 | 1.00000 | 0 | 1.24418 | 0 | 7.52471 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 740.2.a.f | ✓ | 4 |
| 3.b | odd | 2 | 1 | 6660.2.a.r | 4 | ||
| 4.b | odd | 2 | 1 | 2960.2.a.v | 4 | ||
| 5.b | even | 2 | 1 | 3700.2.a.j | 4 | ||
| 5.c | odd | 4 | 2 | 3700.2.d.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 740.2.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2960.2.a.v | 4 | 4.b | odd | 2 | 1 | ||
| 3700.2.a.j | 4 | 5.b | even | 2 | 1 | ||
| 3700.2.d.i | 8 | 5.c | odd | 4 | 2 | ||
| 6660.2.a.r | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 3T_{3}^{3} - 8T_{3}^{2} + 24T_{3} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(740))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots - 2 \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( T^{4} + 5 T^{3} + \cdots + 6 \)
|
| $11$ |
\( T^{4} - 5 T^{3} + \cdots - 240 \)
|
| $13$ |
\( T^{4} + 6 T^{3} + \cdots + 160 \)
|
| $17$ |
\( T^{4} - 10 T^{3} + \cdots + 48 \)
|
| $19$ |
\( T^{4} - 38 T^{2} + \cdots + 64 \)
|
| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 192 \)
|
| $29$ |
\( T^{4} - 2 T^{3} + \cdots + 192 \)
|
| $31$ |
\( T^{4} + 4 T^{3} + \cdots + 148 \)
|
| $37$ |
\( (T - 1)^{4} \)
|
| $41$ |
\( T^{4} - 29 T^{3} + \cdots - 168 \)
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| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 1296 \)
|
| $47$ |
\( T^{4} + 9 T^{3} + \cdots + 54 \)
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| $53$ |
\( T^{4} + 3 T^{3} + \cdots + 72 \)
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| $59$ |
\( T^{4} + 10 T^{3} + \cdots - 1968 \)
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| $61$ |
\( (T - 2)^{4} \)
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| $67$ |
\( T^{4} - 14 T^{3} + \cdots - 8 \)
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| $71$ |
\( T^{4} + 17 T^{3} + \cdots - 288 \)
|
| $73$ |
\( T^{4} - 7 T^{3} + \cdots + 360 \)
|
| $79$ |
\( T^{4} + 24 T^{3} + \cdots - 332 \)
|
| $83$ |
\( T^{4} - 31 T^{3} + \cdots - 5238 \)
|
| $89$ |
\( T^{4} + 4 T^{3} + \cdots + 3504 \)
|
| $97$ |
\( T^{4} - 12 T^{3} + \cdots + 400 \)
|
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