Newspace parameters
| Level: | \( N \) | \(=\) | \( 1664 = 2^{7} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1664.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(13.2871068963\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.592456.1 |
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 9x^{2} + 8x - 2 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 6x^{3} + 9x^{2} + 8x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} - 2\nu + 1 \)
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| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 5\nu^{2} - 8\nu - 4 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 7 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 5\beta _1 + 8 ) / 2 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{3} + 10\beta_{2} + 7\beta _1 + 35 ) / 2 \)
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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 | −3.21195 | 0 | −1.62268 | 0 | 3.82180 | 0 | 7.31660 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.738305 | 0 | −3.70891 | 0 | −1.43136 | 0 | −2.45491 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.378965 | 0 | 4.27753 | 0 | 4.43100 | 0 | −2.85639 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.40634 | 0 | 0.422133 | 0 | −4.72205 | 0 | −1.02221 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.16495 | 0 | −0.368078 | 0 | 2.90060 | 0 | 7.01690 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 1664.2.a.ba | yes | 5 |
| 4.b | odd | 2 | 1 | 1664.2.a.y | ✓ | 5 | |
| 8.b | even | 2 | 1 | 1664.2.a.z | yes | 5 | |
| 8.d | odd | 2 | 1 | 1664.2.a.bb | yes | 5 | |
| 16.e | even | 4 | 2 | 3328.2.b.bc | 10 | ||
| 16.f | odd | 4 | 2 | 3328.2.b.bd | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1664.2.a.y | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 1664.2.a.z | yes | 5 | 8.b | even | 2 | 1 | |
| 1664.2.a.ba | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 1664.2.a.bb | yes | 5 | 8.d | odd | 2 | 1 | |
| 3328.2.b.bc | 10 | 16.e | even | 4 | 2 | ||
| 3328.2.b.bd | 10 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1664))\):
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\( T_{3}^{5} - T_{3}^{4} - 11T_{3}^{3} + 11T_{3}^{2} + 8T_{3} - 4 \)
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\( T_{5}^{5} + T_{5}^{4} - 17T_{5}^{3} - 25T_{5}^{2} + 4T_{5} + 4 \)
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\( T_{7}^{5} - 5T_{7}^{4} - 21T_{7}^{3} + 127T_{7}^{2} - 26T_{7} - 332 \)
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\( T_{11}^{5} - 6T_{11}^{4} - 16T_{11}^{3} + 88T_{11}^{2} + 80T_{11} - 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - T^{4} - 11 T^{3} + \cdots - 4 \)
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| $5$ |
\( T^{5} + T^{4} - 17 T^{3} + \cdots + 4 \)
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| $7$ |
\( T^{5} - 5 T^{4} + \cdots - 332 \)
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| $11$ |
\( T^{5} - 6 T^{4} + \cdots - 256 \)
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| $13$ |
\( (T + 1)^{5} \)
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| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 604 \)
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| $19$ |
\( T^{5} - 12 T^{4} + \cdots - 32 \)
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| $23$ |
\( T^{5} + 2 T^{4} + \cdots - 1024 \)
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| $29$ |
\( T^{5} + 12 T^{4} + \cdots + 64 \)
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| $31$ |
\( T^{5} - 22 T^{4} + \cdots - 512 \)
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| $37$ |
\( T^{5} + 5 T^{4} + \cdots - 932 \)
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| $41$ |
\( T^{5} - 10 T^{4} + \cdots - 1024 \)
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| $43$ |
\( T^{5} - 7 T^{4} + \cdots + 92 \)
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| $47$ |
\( T^{5} - 5 T^{4} + \cdots + 604 \)
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| $53$ |
\( T^{5} + 10 T^{4} + \cdots + 2944 \)
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| $59$ |
\( T^{5} - 6 T^{4} + \cdots - 256 \)
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| $61$ |
\( T^{5} + 14 T^{4} + \cdots - 5248 \)
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| $67$ |
\( T^{5} - 16 T^{4} + \cdots - 7264 \)
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| $71$ |
\( T^{5} + 5 T^{4} + \cdots - 25684 \)
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| $73$ |
\( T^{5} - 4 T^{4} + \cdots - 9088 \)
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| $79$ |
\( T^{5} - 32 T^{4} + \cdots - 2048 \)
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| $83$ |
\( T^{5} - 22 T^{4} + \cdots - 16384 \)
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| $89$ |
\( T^{5} + 2 T^{4} + \cdots - 3296 \)
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| $97$ |
\( T^{5} - 160 T^{3} + \cdots - 704 \)
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