Newspace parameters
| Level: | \( N \) | \(=\) | \( 2075 = 5^{2} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2075.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.03555990121\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 83) |
| Projective image: | \(D_{3}\) |
| Projective field: | Galois closure of 3.1.83.1 |
| Artin image: | $D_6$ |
| Artin field: | Galois closure of 6.2.861125.1 |
| Stark unit: | Root of $x^{6} - 239981x^{5} + 229464790x^{4} - 57132430745x^{3} + 229464790x^{2} - 239981x + 1$ |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2075\mathbb{Z}\right)^\times\).
| \(n\) | \(251\) | \(1827\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1576.1 |
|
0 | 1.00000 | 1.00000 | 0 | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 83.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-83}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2075.1.c.f | 1 | |
| 5.b | even | 2 | 1 | 83.1.b.a | ✓ | 1 | |
| 5.c | odd | 4 | 2 | 2075.1.d.c | 2 | ||
| 15.d | odd | 2 | 1 | 747.1.c.a | 1 | ||
| 20.d | odd | 2 | 1 | 1328.1.g.a | 1 | ||
| 83.b | odd | 2 | 1 | CM | 2075.1.c.f | 1 | |
| 415.d | odd | 2 | 1 | 83.1.b.a | ✓ | 1 | |
| 415.e | even | 4 | 2 | 2075.1.d.c | 2 | ||
| 1245.b | even | 2 | 1 | 747.1.c.a | 1 | ||
| 1660.g | even | 2 | 1 | 1328.1.g.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 83.1.b.a | ✓ | 1 | 5.b | even | 2 | 1 | |
| 83.1.b.a | ✓ | 1 | 415.d | odd | 2 | 1 | |
| 747.1.c.a | 1 | 15.d | odd | 2 | 1 | ||
| 747.1.c.a | 1 | 1245.b | even | 2 | 1 | ||
| 1328.1.g.a | 1 | 20.d | odd | 2 | 1 | ||
| 1328.1.g.a | 1 | 1660.g | even | 2 | 1 | ||
| 2075.1.c.f | 1 | 1.a | even | 1 | 1 | trivial | |
| 2075.1.c.f | 1 | 83.b | odd | 2 | 1 | CM | |
| 2075.1.d.c | 2 | 5.c | odd | 4 | 2 | ||
| 2075.1.d.c | 2 | 415.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2075, [\chi])\):
|
\( T_{2} \)
|
|
\( T_{3} - 1 \)
|
|
\( T_{7} - 1 \)
|
|
\( T_{17} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 1 \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T - 1 \)
|
| $11$ |
\( T + 1 \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T - 1 \)
|
| $19$ |
\( T \)
|
| $23$ |
\( T + 2 \)
|
| $29$ |
\( T + 1 \)
|
| $31$ |
\( T + 1 \)
|
| $37$ |
\( T - 1 \)
|
| $41$ |
\( T - 2 \)
|
| $43$ |
\( T \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T + 1 \)
|
| $61$ |
\( T + 1 \)
|
| $67$ |
\( T \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T \)
|
| $79$ |
\( T \)
|
| $83$ |
\( T + 1 \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T \)
|
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