Newspace parameters
| Level: | \( N \) | \(=\) | \( 820 = 2^{2} \cdot 5 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 820.o (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.409233310359\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(i)\) |
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|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{4}\) |
| Projective field: | Galois closure of 4.0.27568400.3 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/820\mathbb{Z}\right)^\times\).
| \(n\) | \(411\) | \(621\) | \(657\) |
| \(\chi(n)\) | \(-1\) | \(i\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 319.1 |
|
− | 1.00000i | 1.00000 | − | 1.00000i | −1.00000 | 1.00000i | −1.00000 | − | 1.00000i | 1.00000 | − | 1.00000i | 1.00000i | − | 1.00000i | 1.00000 | ||||||||||||||||
| 419.1 | 1.00000i | 1.00000 | + | 1.00000i | −1.00000 | − | 1.00000i | −1.00000 | + | 1.00000i | 1.00000 | + | 1.00000i | − | 1.00000i | 1.00000i | 1.00000 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
| 41.c | even | 4 | 1 | inner |
| 820.o | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 820.1.o.c | yes | 2 |
| 4.b | odd | 2 | 1 | 820.1.o.b | ✓ | 2 | |
| 5.b | even | 2 | 1 | 820.1.o.b | ✓ | 2 | |
| 20.d | odd | 2 | 1 | CM | 820.1.o.c | yes | 2 |
| 41.c | even | 4 | 1 | inner | 820.1.o.c | yes | 2 |
| 164.e | odd | 4 | 1 | 820.1.o.b | ✓ | 2 | |
| 205.j | even | 4 | 1 | 820.1.o.b | ✓ | 2 | |
| 820.o | odd | 4 | 1 | inner | 820.1.o.c | yes | 2 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 820.1.o.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 820.1.o.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 820.1.o.b | ✓ | 2 | 164.e | odd | 4 | 1 | |
| 820.1.o.b | ✓ | 2 | 205.j | even | 4 | 1 | |
| 820.1.o.c | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 820.1.o.c | yes | 2 | 20.d | odd | 2 | 1 | CM |
| 820.1.o.c | yes | 2 | 41.c | even | 4 | 1 | inner |
| 820.1.o.c | yes | 2 | 820.o | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(820, [\chi])\):
|
\( T_{3}^{2} - 2T_{3} + 2 \)
|
|
\( T_{13} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 1 \)
|
| $3$ |
\( T^{2} - 2T + 2 \)
|
| $5$ |
\( T^{2} + 1 \)
|
| $7$ |
\( T^{2} - 2T + 2 \)
|
| $11$ |
\( T^{2} \)
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| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( (T + 2)^{2} \)
|
| $29$ |
\( T^{2} + 2T + 2 \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( T^{2} + 1 \)
|
| $43$ |
\( T^{2} \)
|
| $47$ |
\( T^{2} - 2T + 2 \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( T^{2} + 4 \)
|
| $67$ |
\( T^{2} + 2T + 2 \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( (T - 2)^{2} \)
|
| $89$ |
\( T^{2} + 2T + 2 \)
|
| $97$ |
\( T^{2} \)
|
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