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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
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| Defining polynomial: |
\( x^{2} - x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).
| \(n\) | \(363\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 | 1.50000 | − | 2.59808i | −0.500000 | − | 0.866025i | −1.00000 | + | 1.73205i | −1.50000 | − | 2.59808i | −3.00000 | −1.00000 | −3.00000 | − | 5.19615i | 1.00000 | + | 1.73205i | ||||||||||
| 653.1 | 0.500000 | + | 0.866025i | 1.50000 | + | 2.59808i | −0.500000 | + | 0.866025i | −1.00000 | − | 1.73205i | −1.50000 | + | 2.59808i | −3.00000 | −1.00000 | −3.00000 | + | 5.19615i | 1.00000 | − | 1.73205i | |||||||||||
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.g | 2 | |
| 19.b | odd | 2 | 1 | 722.2.c.a | 2 | ||
| 19.c | even | 3 | 1 | 722.2.a.a | ✓ | 1 | |
| 19.c | even | 3 | 1 | inner | 722.2.c.g | 2 | |
| 19.d | odd | 6 | 1 | 722.2.a.f | yes | 1 | |
| 19.d | odd | 6 | 1 | 722.2.c.a | 2 | ||
| 19.e | even | 9 | 6 | 722.2.e.h | 6 | ||
| 19.f | odd | 18 | 6 | 722.2.e.g | 6 | ||
| 57.f | even | 6 | 1 | 6498.2.a.a | 1 | ||
| 57.h | odd | 6 | 1 | 6498.2.a.m | 1 | ||
| 76.f | even | 6 | 1 | 5776.2.a.a | 1 | ||
| 76.g | odd | 6 | 1 | 5776.2.a.q | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 722.2.a.a | ✓ | 1 | 19.c | even | 3 | 1 | |
| 722.2.a.f | yes | 1 | 19.d | odd | 6 | 1 | |
| 722.2.c.a | 2 | 19.b | odd | 2 | 1 | ||
| 722.2.c.a | 2 | 19.d | odd | 6 | 1 | ||
| 722.2.c.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 722.2.c.g | 2 | 19.c | even | 3 | 1 | inner | |
| 722.2.e.g | 6 | 19.f | odd | 18 | 6 | ||
| 722.2.e.h | 6 | 19.e | even | 9 | 6 | ||
| 5776.2.a.a | 1 | 76.f | even | 6 | 1 | ||
| 5776.2.a.q | 1 | 76.g | odd | 6 | 1 | ||
| 6498.2.a.a | 1 | 57.f | even | 6 | 1 | ||
| 6498.2.a.m | 1 | 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 \(S_{2}^{\mathrm{new}}(722, [\chi])\):
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\( T_{3}^{2} - 3T_{3} + 9 \)
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\( T_{5}^{2} + 2T_{5} + 4 \)
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\( T_{7} + 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T + 1 \)
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| $3$ |
\( T^{2} - 3T + 9 \)
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| $5$ |
\( T^{2} + 2T + 4 \)
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| $7$ |
\( (T + 3)^{2} \)
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| $11$ |
\( (T + 2)^{2} \)
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| $13$ |
\( T^{2} + 3T + 9 \)
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| $17$ |
\( T^{2} - T + 1 \)
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| $19$ |
\( T^{2} \)
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| $23$ |
\( T^{2} + 5T + 25 \)
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| $29$ |
\( T^{2} + 3T + 9 \)
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| $31$ |
\( (T - 6)^{2} \)
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| $37$ |
\( (T + 6)^{2} \)
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| $41$ |
\( T^{2} - 12T + 144 \)
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| $43$ |
\( T^{2} - 10T + 100 \)
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| $47$ |
\( T^{2} - 8T + 64 \)
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| $53$ |
\( T^{2} + 3T + 9 \)
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| $59$ |
\( T^{2} - 3T + 9 \)
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| $61$ |
\( T^{2} \)
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| $67$ |
\( T^{2} - 15T + 225 \)
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| $71$ |
\( T^{2} \)
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| $73$ |
\( T^{2} - 11T + 121 \)
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| $79$ |
\( T^{2} + 12T + 144 \)
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| $83$ |
\( (T - 2)^{2} \)
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| $89$ |
\( T^{2} - 6T + 36 \)
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| $97$ |
\( T^{2} - 12T + 144 \)
|
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