Newspace parameters
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.76519902594\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).
| \(n\) | \(363\) |
| \(\chi(n)\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 429.1 |
|
0.500000 | − | 0.866025i | −1.61803 | + | 2.80252i | −0.500000 | − | 0.866025i | 0.690983 | − | 1.19682i | 1.61803 | + | 2.80252i | 1.23607 | −1.00000 | −3.73607 | − | 6.47106i | −0.690983 | − | 1.19682i | ||||||||||||||||
| 429.2 | 0.500000 | − | 0.866025i | 0.618034 | − | 1.07047i | −0.500000 | − | 0.866025i | 1.80902 | − | 3.13331i | −0.618034 | − | 1.07047i | −3.23607 | −1.00000 | 0.736068 | + | 1.27491i | −1.80902 | − | 3.13331i | |||||||||||||||||
| 653.1 | 0.500000 | + | 0.866025i | −1.61803 | − | 2.80252i | −0.500000 | + | 0.866025i | 0.690983 | + | 1.19682i | 1.61803 | − | 2.80252i | 1.23607 | −1.00000 | −3.73607 | + | 6.47106i | −0.690983 | + | 1.19682i | |||||||||||||||||
| 653.2 | 0.500000 | + | 0.866025i | 0.618034 | + | 1.07047i | −0.500000 | + | 0.866025i | 1.80902 | + | 3.13331i | −0.618034 | + | 1.07047i | −3.23607 | −1.00000 | 0.736068 | − | 1.27491i | −1.80902 | + | 3.13331i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 722.2.c.i | 4 | |
| 19.b | odd | 2 | 1 | 722.2.c.h | 4 | ||
| 19.c | even | 3 | 1 | 722.2.a.h | ✓ | 2 | |
| 19.c | even | 3 | 1 | inner | 722.2.c.i | 4 | |
| 19.d | odd | 6 | 1 | 722.2.a.i | yes | 2 | |
| 19.d | odd | 6 | 1 | 722.2.c.h | 4 | ||
| 19.e | even | 9 | 6 | 722.2.e.q | 12 | ||
| 19.f | odd | 18 | 6 | 722.2.e.p | 12 | ||
| 57.f | even | 6 | 1 | 6498.2.a.be | 2 | ||
| 57.h | odd | 6 | 1 | 6498.2.a.bk | 2 | ||
| 76.f | even | 6 | 1 | 5776.2.a.be | 2 | ||
| 76.g | odd | 6 | 1 | 5776.2.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 722.2.a.h | ✓ | 2 | 19.c | even | 3 | 1 | |
| 722.2.a.i | yes | 2 | 19.d | odd | 6 | 1 | |
| 722.2.c.h | 4 | 19.b | odd | 2 | 1 | ||
| 722.2.c.h | 4 | 19.d | odd | 6 | 1 | ||
| 722.2.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 722.2.c.i | 4 | 19.c | even | 3 | 1 | inner | |
| 722.2.e.p | 12 | 19.f | odd | 18 | 6 | ||
| 722.2.e.q | 12 | 19.e | even | 9 | 6 | ||
| 5776.2.a.t | 2 | 76.g | odd | 6 | 1 | ||
| 5776.2.a.be | 2 | 76.f | even | 6 | 1 | ||
| 6498.2.a.be | 2 | 57.f | even | 6 | 1 | ||
| 6498.2.a.bk | 2 | 57.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):
|
\( T_{3}^{4} + 2T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 16 \)
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\( T_{5}^{4} - 5T_{5}^{3} + 20T_{5}^{2} - 25T_{5} + 25 \)
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\( T_{7}^{2} + 2T_{7} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $5$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
|
| $7$ |
\( (T^{2} + 2 T - 4)^{2} \)
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| $11$ |
\( (T^{2} + 2 T - 4)^{2} \)
|
| $13$ |
\( T^{4} + 5 T^{3} + \cdots + 25 \)
|
| $17$ |
\( T^{4} - 9 T^{3} + \cdots + 361 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
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| $29$ |
\( T^{4} - 7 T^{3} + \cdots + 361 \)
|
| $31$ |
\( (T^{2} - 2 T - 4)^{2} \)
|
| $37$ |
\( (T^{2} - 19 T + 89)^{2} \)
|
| $41$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
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| $43$ |
\( T^{4} - 14 T^{3} + \cdots + 1936 \)
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| $47$ |
\( T^{4} + 20T^{2} + 400 \)
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| $53$ |
\( T^{4} - T^{3} + \cdots + 961 \)
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| $59$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $61$ |
\( T^{4} + 5 T^{3} + \cdots + 25 \)
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| $67$ |
\( T^{4} + 10 T^{3} + \cdots + 400 \)
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| $71$ |
\( T^{4} - 12 T^{3} + \cdots + 1936 \)
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| $73$ |
\( T^{4} - 9 T^{3} + \cdots + 361 \)
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| $79$ |
\( T^{4} - 10 T^{3} + \cdots + 400 \)
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| $83$ |
\( (T^{2} - 8 T - 4)^{2} \)
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| $89$ |
\( T^{4} + 11 T^{3} + \cdots + 361 \)
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| $97$ |
\( T^{4} + 15 T^{3} + \cdots + 3025 \)
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