Newspace parameters
| Level: | \( N \) | \(=\) | \( 1072 = 2^{4} \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1072.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.55996309668\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{13})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 7x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 7x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 7\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} - 4\nu \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 4 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{2} - 12\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1072\mathbb{Z}\right)^\times\).
| \(n\) | \(337\) | \(671\) | \(805\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1071.1 |
|
0 | −2.30278 | 0 | − | 3.00000i | 0 | −0.697224 | 0 | 2.30278 | 0 | |||||||||||||||||||||||||||||
| 1071.2 | 0 | −2.30278 | 0 | 3.00000i | 0 | −0.697224 | 0 | 2.30278 | 0 | |||||||||||||||||||||||||||||||
| 1071.3 | 0 | 1.30278 | 0 | − | 3.00000i | 0 | −4.30278 | 0 | −1.30278 | 0 | ||||||||||||||||||||||||||||||
| 1071.4 | 0 | 1.30278 | 0 | 3.00000i | 0 | −4.30278 | 0 | −1.30278 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 268.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1072.2.g.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1072.2.g.d | yes | 4 | |
| 67.b | odd | 2 | 1 | 1072.2.g.d | yes | 4 | |
| 268.d | even | 2 | 1 | inner | 1072.2.g.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1072.2.g.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1072.2.g.b | ✓ | 4 | 268.d | even | 2 | 1 | inner |
| 1072.2.g.d | yes | 4 | 4.b | odd | 2 | 1 | |
| 1072.2.g.d | yes | 4 | 67.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 \(S_{2}^{\mathrm{new}}(1072, [\chi])\):
|
\( T_{3}^{2} + T_{3} - 3 \)
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\( T_{17}^{2} - 2T_{17} - 12 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + T - 3)^{2} \)
|
| $5$ |
\( (T^{2} + 9)^{2} \)
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| $7$ |
\( (T^{2} + 5 T + 3)^{2} \)
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| $11$ |
\( (T + 3)^{4} \)
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| $13$ |
\( T^{4} + 63T^{2} + 729 \)
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| $17$ |
\( (T^{2} - 2 T - 12)^{2} \)
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| $19$ |
\( T^{4} + 31T^{2} + 81 \)
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| $23$ |
\( T^{4} + 58T^{2} + 9 \)
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| $29$ |
\( (T^{2} + 8 T + 3)^{2} \)
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| $31$ |
\( (T^{2} + 4 T - 9)^{2} \)
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| $37$ |
\( (T^{2} - 7 T + 9)^{2} \)
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| $41$ |
\( T^{4} + 63T^{2} + 729 \)
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| $43$ |
\( (T^{2} + 13 T + 39)^{2} \)
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| $47$ |
\( T^{4} + 67T^{2} + 729 \)
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| $53$ |
\( (T^{2} + 9)^{2} \)
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| $59$ |
\( T^{4} + 112T^{2} + 2304 \)
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| $61$ |
\( T^{4} + 63T^{2} + 729 \)
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| $67$ |
\( T^{4} + 22 T^{3} + \cdots + 4489 \)
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| $71$ |
\( T^{4} + 34T^{2} + 81 \)
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| $73$ |
\( (T^{2} + 10 T + 12)^{2} \)
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| $79$ |
\( (T^{2} + 13 T + 39)^{2} \)
|
| $83$ |
\( T^{4} + 151T^{2} + 4761 \)
|
| $89$ |
\( (T^{2} - 16 T + 51)^{2} \)
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| $97$ |
\( T^{4} + 306T^{2} + 6561 \)
|
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