Newspace parameters
| Level: | \( N \) | \(=\) | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1078.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.60787333789\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
|
|
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| Defining polynomial: |
\( x^{4} + 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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
\( x^{4} + 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(981\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−0.500000 | − | 0.866025i | −1.41421 | + | 2.44949i | −0.500000 | + | 0.866025i | 0 | 2.82843 | 0 | 1.00000 | −2.50000 | − | 4.33013i | 0 | ||||||||||||||||||||||
| 67.2 | −0.500000 | − | 0.866025i | 1.41421 | − | 2.44949i | −0.500000 | + | 0.866025i | 0 | −2.82843 | 0 | 1.00000 | −2.50000 | − | 4.33013i | 0 | |||||||||||||||||||||||
| 177.1 | −0.500000 | + | 0.866025i | −1.41421 | − | 2.44949i | −0.500000 | − | 0.866025i | 0 | 2.82843 | 0 | 1.00000 | −2.50000 | + | 4.33013i | 0 | |||||||||||||||||||||||
| 177.2 | −0.500000 | + | 0.866025i | 1.41421 | + | 2.44949i | −0.500000 | − | 0.866025i | 0 | −2.82843 | 0 | 1.00000 | −2.50000 | + | 4.33013i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1078.2.e.o | 4 | |
| 7.b | odd | 2 | 1 | inner | 1078.2.e.o | 4 | |
| 7.c | even | 3 | 1 | 1078.2.a.v | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 1078.2.e.o | 4 | |
| 7.d | odd | 6 | 1 | 1078.2.a.v | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 1078.2.e.o | 4 | |
| 21.g | even | 6 | 1 | 9702.2.a.co | 2 | ||
| 21.h | odd | 6 | 1 | 9702.2.a.co | 2 | ||
| 28.f | even | 6 | 1 | 8624.2.a.bz | 2 | ||
| 28.g | odd | 6 | 1 | 8624.2.a.bz | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1078.2.a.v | ✓ | 2 | 7.c | even | 3 | 1 | |
| 1078.2.a.v | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 1078.2.e.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 1078.2.e.o | 4 | 7.b | odd | 2 | 1 | inner | |
| 1078.2.e.o | 4 | 7.c | even | 3 | 1 | inner | |
| 1078.2.e.o | 4 | 7.d | odd | 6 | 1 | inner | |
| 8624.2.a.bz | 2 | 28.f | even | 6 | 1 | ||
| 8624.2.a.bz | 2 | 28.g | odd | 6 | 1 | ||
| 9702.2.a.co | 2 | 21.g | even | 6 | 1 | ||
| 9702.2.a.co | 2 | 21.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 \(S_{2}^{\mathrm{new}}(1078, [\chi])\):
|
\( T_{3}^{4} + 8T_{3}^{2} + 64 \)
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\( T_{5} \)
|
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\( T_{13}^{2} - 18 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{2} \)
|
| $3$ |
\( T^{4} + 8T^{2} + 64 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - T + 1)^{2} \)
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| $13$ |
\( (T^{2} - 18)^{2} \)
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| $17$ |
\( T^{4} + 8T^{2} + 64 \)
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| $19$ |
\( T^{4} + 18T^{2} + 324 \)
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| $23$ |
\( (T^{2} + 6 T + 36)^{2} \)
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| $29$ |
\( (T + 4)^{4} \)
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| $31$ |
\( T^{4} + 50T^{2} + 2500 \)
|
| $37$ |
\( (T^{2} + 2 T + 4)^{2} \)
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| $41$ |
\( (T^{2} - 8)^{2} \)
|
| $43$ |
\( (T - 10)^{4} \)
|
| $47$ |
\( T^{4} + 162 T^{2} + 26244 \)
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| $53$ |
\( (T^{2} + 2 T + 4)^{2} \)
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| $59$ |
\( T^{4} + 128 T^{2} + 16384 \)
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| $61$ |
\( T^{4} + 98T^{2} + 9604 \)
|
| $67$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $71$ |
\( (T - 16)^{4} \)
|
| $73$ |
\( T^{4} + 72T^{2} + 5184 \)
|
| $79$ |
\( (T^{2} - 8 T + 64)^{2} \)
|
| $83$ |
\( (T^{2} - 162)^{2} \)
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| $89$ |
\( T^{4} + 50T^{2} + 2500 \)
|
| $97$ |
\( (T^{2} - 50)^{2} \)
|
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