Newspace parameters
| Level: | \( N \) | = | \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | = | \( 1 \) |
| Character orbit: | \([\chi]\) | = | 1800.l (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.898317022739\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-2}) \) |
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image | \(S_{4}\) |
| Projective field | Galois closure of 4.2.10800.2 |
| Artin image | $\GL(2,3)$ |
| Artin field | Galois closure of 8.2.1399680000.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(901\) | \(1001\) | \(1351\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1601.1 |
|
0 | 0 | 0 | 0 | 0 | −1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1601.2 | 0 | 0 | 0 | 0 | 0 | −1.00000 | 0 | 0 | 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 | 1800.1.l.a | ✓ | 2 |
| 3.b | odd | 2 | 1 | inner | 1800.1.l.a | ✓ | 2 |
| 4.b | odd | 2 | 1 | 3600.1.l.b | 2 | ||
| 5.b | even | 2 | 1 | 1800.1.l.b | yes | 2 | |
| 5.c | odd | 4 | 2 | 1800.1.c.a | 4 | ||
| 12.b | even | 2 | 1 | 3600.1.l.b | 2 | ||
| 15.d | odd | 2 | 1 | 1800.1.l.b | yes | 2 | |
| 15.e | even | 4 | 2 | 1800.1.c.a | 4 | ||
| 20.d | odd | 2 | 1 | 3600.1.l.a | 2 | ||
| 20.e | even | 4 | 2 | 3600.1.c.a | 4 | ||
| 60.h | even | 2 | 1 | 3600.1.l.a | 2 | ||
| 60.l | odd | 4 | 2 | 3600.1.c.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1800.1.c.a | 4 | 5.c | odd | 4 | 2 | ||
| 1800.1.c.a | 4 | 15.e | even | 4 | 2 | ||
| 1800.1.l.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1800.1.l.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
| 1800.1.l.b | yes | 2 | 5.b | even | 2 | 1 | |
| 1800.1.l.b | yes | 2 | 15.d | odd | 2 | 1 | |
| 3600.1.c.a | 4 | 20.e | even | 4 | 2 | ||
| 3600.1.c.a | 4 | 60.l | odd | 4 | 2 | ||
| 3600.1.l.a | 2 | 20.d | odd | 2 | 1 | ||
| 3600.1.l.a | 2 | 60.h | even | 2 | 1 | ||
| 3600.1.l.b | 2 | 4.b | odd | 2 | 1 | ||
| 3600.1.l.b | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).