Newspace parameters
| Level: | \( N \) | \(=\) | \( 575 = 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 575.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(33.9260982533\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{109})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 55x^{2} + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 115) |
| 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
\( x^{4} + 55x^{2} + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 28\nu ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 28 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( 27\beta_{2} - 28\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 24.1 |
|
− | 3.00000i | − | 3.72015i | −1.00000 | 0 | −11.1605 | − | 25.6008i | − | 21.0000i | 13.1605 | 0 | ||||||||||||||||||||||||||
| 24.2 | − | 3.00000i | 6.72015i | −1.00000 | 0 | 20.1605 | 26.6008i | − | 21.0000i | −18.1605 | 0 | |||||||||||||||||||||||||||||
| 24.3 | 3.00000i | − | 6.72015i | −1.00000 | 0 | 20.1605 | − | 26.6008i | 21.0000i | −18.1605 | 0 | |||||||||||||||||||||||||||||
| 24.4 | 3.00000i | 3.72015i | −1.00000 | 0 | −11.1605 | 25.6008i | 21.0000i | 13.1605 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 575.4.b.f | 4 | |
| 5.b | even | 2 | 1 | inner | 575.4.b.f | 4 | |
| 5.c | odd | 4 | 1 | 115.4.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 575.4.a.h | 2 | ||
| 15.e | even | 4 | 1 | 1035.4.a.g | 2 | ||
| 20.e | even | 4 | 1 | 1840.4.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.c | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 575.4.a.h | 2 | 5.c | odd | 4 | 1 | ||
| 575.4.b.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 575.4.b.f | 4 | 5.b | even | 2 | 1 | inner | |
| 1035.4.a.g | 2 | 15.e | even | 4 | 1 | ||
| 1840.4.a.h | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(575, [\chi])\):
|
\( T_{2}^{2} + 9 \)
|
|
\( T_{3}^{4} + 59T_{3}^{2} + 625 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 9)^{2} \)
|
| $3$ |
\( T^{4} + 59T^{2} + 625 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 1363 T^{2} + 463761 \)
|
| $11$ |
\( (T^{2} + 27 T - 499)^{2} \)
|
| $13$ |
\( T^{4} + 603 T^{2} + 35721 \)
|
| $17$ |
\( T^{4} + 5791 T^{2} + 50625 \)
|
| $19$ |
\( (T^{2} - 71 T - 3345)^{2} \)
|
| $23$ |
\( (T^{2} + 529)^{2} \)
|
| $29$ |
\( (T^{2} - 430 T + 43500)^{2} \)
|
| $31$ |
\( (T^{2} + 305 T + 15381)^{2} \)
|
| $37$ |
\( T^{4} + 45040 T^{2} + 408363264 \)
|
| $41$ |
\( (T^{2} + 593 T + 84615)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 10656845824 \)
|
| $47$ |
\( T^{4} + \cdots + 12710307600 \)
|
| $53$ |
\( T^{4} + 213160 T^{2} + 1140624 \)
|
| $59$ |
\( (T^{2} - 18 T - 79380)^{2} \)
|
| $61$ |
\( (T^{2} + 7 T - 88523)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 24490998016 \)
|
| $71$ |
\( (T^{2} + 1029 T + 8315)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 10687424400 \)
|
| $79$ |
\( (T^{2} + 692 T - 686448)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 1034008336 \)
|
| $89$ |
\( (T^{2} - 220 T - 870800)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 455240331225 \)
|
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