Newspace parameters
| Level: | \( N \) | \(=\) | \( 1840 = 2^{4} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1840.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(108.563514411\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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| Defining polynomial: |
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 115) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 7\nu^{3} + 9\nu^{2} - 77\nu - 2 ) / 16 \)
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| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 9\nu^{2} - 43\nu - 78 ) / 16 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - \nu^{3} + 25\nu^{2} + 43\nu - 98 ) / 16 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} + 5\nu^{3} + 59\nu^{2} - 55\nu - 182 ) / 16 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{4} + 2\beta_{3} + \beta _1 + 1 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 11 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -15\beta_{4} + 30\beta_{3} + 8\beta_{2} + 23\beta _1 + 55 ) / 4 \)
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| \(\nu^{4}\) | \(=\) |
\( -7\beta_{4} + 23\beta_{3} + 23\beta_{2} + 5\beta _1 + 174 \)
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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 | −9.02447 | 0 | −5.00000 | 0 | −4.33445 | 0 | 54.4411 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.84147 | 0 | −5.00000 | 0 | 8.97260 | 0 | 34.4886 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.89520 | 0 | −5.00000 | 0 | −11.4426 | 0 | −23.4082 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.85751 | 0 | −5.00000 | 0 | 23.5932 | 0 | −12.1196 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 7.11323 | 0 | −5.00000 | 0 | −13.7888 | 0 | 23.5981 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 1840.4.a.n | 5 | |
| 4.b | odd | 2 | 1 | 115.4.a.e | ✓ | 5 | |
| 12.b | even | 2 | 1 | 1035.4.a.k | 5 | ||
| 20.d | odd | 2 | 1 | 575.4.a.j | 5 | ||
| 20.e | even | 4 | 2 | 575.4.b.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.e | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 575.4.a.j | 5 | 20.d | odd | 2 | 1 | ||
| 575.4.b.i | 10 | 20.e | even | 4 | 2 | ||
| 1035.4.a.k | 5 | 12.b | even | 2 | 1 | ||
| 1840.4.a.n | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):
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\( T_{3}^{5} + 4T_{3}^{4} - 98T_{3}^{3} - 149T_{3}^{2} + 2536T_{3} - 3680 \)
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\( T_{7}^{5} - 3T_{7}^{4} - 484T_{7}^{3} - 1757T_{7}^{2} + 34281T_{7} + 144774 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} + 4 T^{4} + \cdots - 3680 \)
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| $5$ |
\( (T + 5)^{5} \)
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| $7$ |
\( T^{5} - 3 T^{4} + \cdots + 144774 \)
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| $11$ |
\( T^{5} + 23 T^{4} + \cdots + 74136848 \)
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| $13$ |
\( T^{5} - 132 T^{4} + \cdots + 1550116 \)
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| $17$ |
\( T^{5} + \cdots - 1039045340 \)
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| $19$ |
\( T^{5} - 161 T^{4} + \cdots + 801280 \)
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| $23$ |
\( (T - 23)^{5} \)
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| $29$ |
\( T^{5} + \cdots - 6149898500 \)
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| $31$ |
\( T^{5} + 32 T^{4} + \cdots + 438072447 \)
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| $37$ |
\( T^{5} + \cdots + 1590700778176 \)
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| $41$ |
\( T^{5} + \cdots + 114116030755 \)
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| $43$ |
\( T^{5} + \cdots - 504784881664 \)
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| $47$ |
\( T^{5} + \cdots - 117787714816 \)
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| $53$ |
\( T^{5} + \cdots + 5720332226904 \)
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| $59$ |
\( T^{5} + \cdots - 24279649927232 \)
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| $61$ |
\( T^{5} + \cdots - 34095834816896 \)
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| $67$ |
\( T^{5} + \cdots - 5644442112 \)
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| $71$ |
\( T^{5} + \cdots - 15638892903635 \)
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| $73$ |
\( T^{5} + \cdots - 100895881632176 \)
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| $79$ |
\( T^{5} + \cdots + 90481602379776 \)
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| $83$ |
\( T^{5} + \cdots - 18307318870176 \)
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| $89$ |
\( T^{5} + \cdots - 115104799418880 \)
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| $97$ |
\( T^{5} + \cdots - 480989167569272 \)
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