Newspace parameters
| Level: | \( N \) | \(=\) | \( 595 = 5 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 595.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.75109892027\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.10889.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{4} - x^{3} - 5x^{2} + 2x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.95197 | −0.487697 | 1.81018 | 1.00000 | 0.951969 | 1.00000 | 0.370517 | −2.76215 | −1.95197 | ||||||||||||||||||||||||||||||
| 1.2 | −0.296768 | 2.36963 | −1.91193 | 1.00000 | −0.703232 | 1.00000 | 1.16094 | 2.61516 | −0.296768 | |||||||||||||||||||||||||||||||
| 1.3 | 0.669225 | −2.49427 | −1.55214 | 1.00000 | −1.66922 | 1.00000 | −2.37718 | 3.22136 | 0.669225 | |||||||||||||||||||||||||||||||
| 1.4 | 2.57951 | −1.38767 | 4.65388 | 1.00000 | −3.57951 | 1.00000 | 6.84573 | −1.07437 | 2.57951 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 595.2.a.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | 5355.2.a.bk | 4 | ||
| 4.b | odd | 2 | 1 | 9520.2.a.bn | 4 | ||
| 5.b | even | 2 | 1 | 2975.2.a.j | 4 | ||
| 7.b | odd | 2 | 1 | 4165.2.a.bd | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 595.2.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2975.2.a.j | 4 | 5.b | even | 2 | 1 | ||
| 4165.2.a.bd | 4 | 7.b | odd | 2 | 1 | ||
| 5355.2.a.bk | 4 | 3.b | odd | 2 | 1 | ||
| 9520.2.a.bn | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(595))\):
|
\( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 2T_{2} + 1 \)
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\( T_{11}^{4} - 5T_{11}^{3} - 5T_{11}^{2} + 45T_{11} - 44 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} - 5 T^{2} + \cdots + 1 \)
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| $3$ |
\( T^{4} + 2 T^{3} + \cdots - 4 \)
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| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( T^{4} - 5 T^{3} + \cdots - 44 \)
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| $13$ |
\( T^{4} - 3 T^{3} + \cdots - 2 \)
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| $17$ |
\( (T - 1)^{4} \)
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| $19$ |
\( T^{4} + 5 T^{3} + \cdots - 44 \)
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| $23$ |
\( T^{4} - 11 T^{3} + \cdots - 8 \)
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| $29$ |
\( T^{4} - 3 T^{3} + \cdots - 22 \)
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| $31$ |
\( T^{4} + T^{3} + \cdots - 64 \)
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| $37$ |
\( T^{4} - 20 T^{3} + \cdots + 50 \)
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| $41$ |
\( T^{4} - 9 T^{3} + \cdots - 162 \)
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| $43$ |
\( T^{4} + 5 T^{3} + \cdots - 4 \)
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| $47$ |
\( T^{4} + 9 T^{3} + \cdots - 848 \)
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| $53$ |
\( T^{4} - 5 T^{3} + \cdots + 2246 \)
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| $59$ |
\( T^{4} + 5 T^{3} + \cdots + 764 \)
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| $61$ |
\( T^{4} + 11 T^{3} + \cdots - 242 \)
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| $67$ |
\( T^{4} - 7 T^{3} + \cdots + 1100 \)
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| $71$ |
\( T^{4} + 19 T^{3} + \cdots - 6824 \)
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| $73$ |
\( T^{4} - 5 T^{3} + \cdots + 2 \)
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| $79$ |
\( T^{4} + 10 T^{3} + \cdots + 16 \)
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| $83$ |
\( T^{4} - 16 T^{3} + \cdots - 508 \)
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| $89$ |
\( T^{4} + 11 T^{3} + \cdots - 4886 \)
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| $97$ |
\( T^{4} - 8 T^{3} + \cdots + 5650 \)
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