Newspace parameters
| Level: | \( N \) | \(=\) | \( 820 = 2^{2} \cdot 5 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 820.w (of order \(8\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.54773296574\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
|
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| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{8}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/820\mathbb{Z}\right)^\times\).
| \(n\) | \(411\) | \(621\) | \(657\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{8}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
−1.00000 | − | 1.00000i | 0 | 2.00000i | −0.707107 | − | 2.12132i | 0 | 0 | 2.00000 | − | 2.00000i | −2.12132 | − | 2.12132i | −1.41421 | + | 2.82843i | ||||||||||||||||||||
| 219.1 | −1.00000 | + | 1.00000i | 0 | − | 2.00000i | 0.707107 | − | 2.12132i | 0 | 0 | 2.00000 | + | 2.00000i | 2.12132 | − | 2.12132i | 1.41421 | + | 2.82843i | ||||||||||||||||||||
| 519.1 | −1.00000 | + | 1.00000i | 0 | − | 2.00000i | −0.707107 | + | 2.12132i | 0 | 0 | 2.00000 | + | 2.00000i | −2.12132 | + | 2.12132i | −1.41421 | − | 2.82843i | ||||||||||||||||||||
| 659.1 | −1.00000 | − | 1.00000i | 0 | 2.00000i | 0.707107 | + | 2.12132i | 0 | 0 | 2.00000 | − | 2.00000i | 2.12132 | + | 2.12132i | 1.41421 | − | 2.82843i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 205.m | odd | 8 | 1 | inner |
| 820.w | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 820.2.w.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | CM | 820.2.w.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 820.2.w.b | yes | 4 | |
| 20.d | odd | 2 | 1 | 820.2.w.b | yes | 4 | |
| 41.e | odd | 8 | 1 | 820.2.w.b | yes | 4 | |
| 164.i | even | 8 | 1 | 820.2.w.b | yes | 4 | |
| 205.m | odd | 8 | 1 | inner | 820.2.w.a | ✓ | 4 |
| 820.w | even | 8 | 1 | inner | 820.2.w.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 820.2.w.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 820.2.w.a | ✓ | 4 | 4.b | odd | 2 | 1 | CM |
| 820.2.w.a | ✓ | 4 | 205.m | odd | 8 | 1 | inner |
| 820.2.w.a | ✓ | 4 | 820.w | even | 8 | 1 | inner |
| 820.2.w.b | yes | 4 | 5.b | even | 2 | 1 | |
| 820.2.w.b | yes | 4 | 20.d | odd | 2 | 1 | |
| 820.2.w.b | yes | 4 | 41.e | odd | 8 | 1 | |
| 820.2.w.b | yes | 4 | 164.i | even | 8 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\):
|
\( T_{3} \)
|
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\( T_{13}^{4} - 8T_{13}^{3} + 66T_{13}^{2} - 140T_{13} + 98 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 8T^{2} + 25 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 8 T^{3} + \cdots + 98 \)
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| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 98 \)
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| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} + 20 T^{3} + \cdots + 2 \)
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| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} + 50)^{2} \)
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| $41$ |
\( (T^{2} + 10 T + 41)^{2} \)
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| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
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| $53$ |
\( T^{4} + 28 T^{3} + \cdots + 578 \)
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| $59$ |
\( T^{4} \)
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| $61$ |
\( (T^{2} - 2 T + 2)^{2} \)
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| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + 1296 \)
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| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} - 32 T^{3} + \cdots + 28322 \)
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| $97$ |
\( T^{4} - 36 T^{3} + \cdots + 37538 \)
|
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