Newspace parameters
| Level: | \( N \) | \(=\) | \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1452.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(39.5641343851\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 132) |
| 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
\( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} + 38\nu - 18 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 2\nu + 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} - 12\nu^{2} + 88\nu - 42 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - 2\beta _1 - 18 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -8\beta_{3} + 3\beta_{2} + 19\beta _1 - 28 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(1333\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 485.1 |
|
0 | −1.00000 | − | 2.82843i | 0 | − | 3.80482i | 0 | 9.38083 | 0 | −7.00000 | + | 5.65685i | 0 | |||||||||||||||||||||||||
| 485.2 | 0 | −1.00000 | − | 2.82843i | 0 | 9.46168i | 0 | −9.38083 | 0 | −7.00000 | + | 5.65685i | 0 | |||||||||||||||||||||||||||
| 485.3 | 0 | −1.00000 | + | 2.82843i | 0 | − | 9.46168i | 0 | −9.38083 | 0 | −7.00000 | − | 5.65685i | 0 | ||||||||||||||||||||||||||
| 485.4 | 0 | −1.00000 | + | 2.82843i | 0 | 3.80482i | 0 | 9.38083 | 0 | −7.00000 | − | 5.65685i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1452.3.e.h | 4 | |
| 3.b | odd | 2 | 1 | inner | 1452.3.e.h | 4 | |
| 11.b | odd | 2 | 1 | 132.3.e.b | ✓ | 4 | |
| 33.d | even | 2 | 1 | 132.3.e.b | ✓ | 4 | |
| 44.c | even | 2 | 1 | 528.3.i.c | 4 | ||
| 132.d | odd | 2 | 1 | 528.3.i.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 132.3.e.b | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 132.3.e.b | ✓ | 4 | 33.d | even | 2 | 1 | |
| 528.3.i.c | 4 | 44.c | even | 2 | 1 | ||
| 528.3.i.c | 4 | 132.d | odd | 2 | 1 | ||
| 1452.3.e.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 1452.3.e.h | 4 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1452, [\chi])\):
|
\( T_{5}^{4} + 104T_{5}^{2} + 1296 \)
|
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\( T_{7}^{2} - 88 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 2 T + 9)^{2} \)
|
| $5$ |
\( T^{4} + 104T^{2} + 1296 \)
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| $7$ |
\( (T^{2} - 88)^{2} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( (T^{2} + 24 T + 56)^{2} \)
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| $17$ |
\( T^{4} + 608T^{2} + 2304 \)
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| $19$ |
\( (T^{2} - 28 T - 156)^{2} \)
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| $23$ |
\( T^{4} + 936 T^{2} + 104976 \)
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| $29$ |
\( T^{4} + 416 T^{2} + 20736 \)
|
| $31$ |
\( (T^{2} - 12 T - 316)^{2} \)
|
| $37$ |
\( (T^{2} - 60 T + 548)^{2} \)
|
| $41$ |
\( T^{4} + 3488 T^{2} + 1937664 \)
|
| $43$ |
\( (T^{2} - 60 T + 548)^{2} \)
|
| $47$ |
\( T^{4} + 872 T^{2} + 121104 \)
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| $53$ |
\( T^{4} + 4712 T^{2} + 3825936 \)
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| $59$ |
\( T^{4} + 12384 T^{2} + 9144576 \)
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| $61$ |
\( (T^{2} - 24 T + 56)^{2} \)
|
| $67$ |
\( (T^{2} - 68 T - 4476)^{2} \)
|
| $71$ |
\( T^{4} + 11432 T^{2} + 24324624 \)
|
| $73$ |
\( (T^{2} - 52 T - 8124)^{2} \)
|
| $79$ |
\( (T^{2} - 96 T + 2216)^{2} \)
|
| $83$ |
\( T^{4} + 34208 T^{2} + 280629504 \)
|
| $89$ |
\( T^{4} + 1952 T^{2} + 389376 \)
|
| $97$ |
\( (T^{2} + 180 T + 6692)^{2} \)
|
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