Newspace parameters
| Level: | \( N \) | \(=\) | \( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4788.i (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(38.2323724878\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
|
|
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| Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 532) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 4\nu + 5 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( -4\beta_{3} + 2\beta_{2} + 4\beta _1 + 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times\).
| \(n\) | \(533\) | \(1009\) | \(2395\) | \(4105\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3457.1 |
|
0 | 0 | 0 | − | 3.04547i | 0 | 2.63746 | + | 0.209313i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 3457.2 | 0 | 0 | 0 | − | 1.31342i | 0 | −1.13746 | − | 2.38876i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 3457.3 | 0 | 0 | 0 | 1.31342i | 0 | −1.13746 | + | 2.38876i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 3457.4 | 0 | 0 | 0 | 3.04547i | 0 | 2.63746 | − | 0.209313i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
| 7.b | odd | 2 | 1 | inner |
| 133.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4788.2.i.c | 4 | |
| 3.b | odd | 2 | 1 | 532.2.g.a | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 4788.2.i.c | 4 | |
| 12.b | even | 2 | 1 | 2128.2.m.c | 4 | ||
| 19.b | odd | 2 | 1 | CM | 4788.2.i.c | 4 | |
| 21.c | even | 2 | 1 | 532.2.g.a | ✓ | 4 | |
| 57.d | even | 2 | 1 | 532.2.g.a | ✓ | 4 | |
| 84.h | odd | 2 | 1 | 2128.2.m.c | 4 | ||
| 133.c | even | 2 | 1 | inner | 4788.2.i.c | 4 | |
| 228.b | odd | 2 | 1 | 2128.2.m.c | 4 | ||
| 399.h | odd | 2 | 1 | 532.2.g.a | ✓ | 4 | |
| 1596.p | even | 2 | 1 | 2128.2.m.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 532.2.g.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 532.2.g.a | ✓ | 4 | 21.c | even | 2 | 1 | |
| 532.2.g.a | ✓ | 4 | 57.d | even | 2 | 1 | |
| 532.2.g.a | ✓ | 4 | 399.h | odd | 2 | 1 | |
| 2128.2.m.c | 4 | 12.b | even | 2 | 1 | ||
| 2128.2.m.c | 4 | 84.h | odd | 2 | 1 | ||
| 2128.2.m.c | 4 | 228.b | odd | 2 | 1 | ||
| 2128.2.m.c | 4 | 1596.p | even | 2 | 1 | ||
| 4788.2.i.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 4788.2.i.c | 4 | 7.b | odd | 2 | 1 | inner | |
| 4788.2.i.c | 4 | 19.b | odd | 2 | 1 | CM | |
| 4788.2.i.c | 4 | 133.c | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4788, [\chi])\):
|
\( T_{5}^{4} + 11T_{5}^{2} + 16 \)
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\( T_{13} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 11T^{2} + 16 \)
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| $7$ |
\( T^{4} - 3 T^{3} + \cdots + 49 \)
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| $11$ |
\( (T^{2} - 5 T - 8)^{2} \)
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| $13$ |
\( T^{4} \)
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| $17$ |
\( T^{4} + 83T^{2} + 1024 \)
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| $19$ |
\( (T^{2} + 19)^{2} \)
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| $23$ |
\( (T - 4)^{4} \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
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| $43$ |
\( (T^{2} - T - 128)^{2} \)
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| $47$ |
\( T^{4} + 263 T^{2} + 14884 \)
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| $53$ |
\( T^{4} \)
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| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} + 347 T^{2} + 26896 \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + 267T^{2} + 2304 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( (T^{2} + 76)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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