Newspace parameters
| Level: | \( N \) | \(=\) | \( 3840 = 2^{8} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3840.d (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(30.6625543762\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
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| Defining polynomial: |
\( x^{2} + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1920) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).
| \(n\) | \(511\) | \(1537\) | \(2561\) | \(2821\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2689.1 |
|
0 | −1.00000 | 0 | −1.00000 | − | 2.00000i | 0 | − | 4.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||
| 2689.2 | 0 | −1.00000 | 0 | −1.00000 | + | 2.00000i | 0 | 4.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 40.f | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3840.2.d.h | 2 | |
| 4.b | odd | 2 | 1 | 3840.2.d.w | 2 | ||
| 5.b | even | 2 | 1 | 3840.2.d.z | 2 | ||
| 8.b | even | 2 | 1 | 3840.2.d.z | 2 | ||
| 8.d | odd | 2 | 1 | 3840.2.d.i | 2 | ||
| 16.e | even | 4 | 1 | 1920.2.f.c | yes | 2 | |
| 16.e | even | 4 | 1 | 1920.2.f.k | yes | 2 | |
| 16.f | odd | 4 | 1 | 1920.2.f.b | ✓ | 2 | |
| 16.f | odd | 4 | 1 | 1920.2.f.j | yes | 2 | |
| 20.d | odd | 2 | 1 | 3840.2.d.i | 2 | ||
| 40.e | odd | 2 | 1 | 3840.2.d.w | 2 | ||
| 40.f | even | 2 | 1 | inner | 3840.2.d.h | 2 | |
| 80.i | odd | 4 | 1 | 9600.2.a.bc | 1 | ||
| 80.i | odd | 4 | 1 | 9600.2.a.ca | 1 | ||
| 80.j | even | 4 | 1 | 9600.2.a.bd | 1 | ||
| 80.j | even | 4 | 1 | 9600.2.a.cb | 1 | ||
| 80.k | odd | 4 | 1 | 1920.2.f.b | ✓ | 2 | |
| 80.k | odd | 4 | 1 | 1920.2.f.j | yes | 2 | |
| 80.q | even | 4 | 1 | 1920.2.f.c | yes | 2 | |
| 80.q | even | 4 | 1 | 1920.2.f.k | yes | 2 | |
| 80.s | even | 4 | 1 | 9600.2.a.c | 1 | ||
| 80.s | even | 4 | 1 | 9600.2.a.ba | 1 | ||
| 80.t | odd | 4 | 1 | 9600.2.a.d | 1 | ||
| 80.t | odd | 4 | 1 | 9600.2.a.bb | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1920.2.f.b | ✓ | 2 | 16.f | odd | 4 | 1 | |
| 1920.2.f.b | ✓ | 2 | 80.k | odd | 4 | 1 | |
| 1920.2.f.c | yes | 2 | 16.e | even | 4 | 1 | |
| 1920.2.f.c | yes | 2 | 80.q | even | 4 | 1 | |
| 1920.2.f.j | yes | 2 | 16.f | odd | 4 | 1 | |
| 1920.2.f.j | yes | 2 | 80.k | odd | 4 | 1 | |
| 1920.2.f.k | yes | 2 | 16.e | even | 4 | 1 | |
| 1920.2.f.k | yes | 2 | 80.q | even | 4 | 1 | |
| 3840.2.d.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 3840.2.d.h | 2 | 40.f | even | 2 | 1 | inner | |
| 3840.2.d.i | 2 | 8.d | odd | 2 | 1 | ||
| 3840.2.d.i | 2 | 20.d | odd | 2 | 1 | ||
| 3840.2.d.w | 2 | 4.b | odd | 2 | 1 | ||
| 3840.2.d.w | 2 | 40.e | odd | 2 | 1 | ||
| 3840.2.d.z | 2 | 5.b | even | 2 | 1 | ||
| 3840.2.d.z | 2 | 8.b | even | 2 | 1 | ||
| 9600.2.a.c | 1 | 80.s | even | 4 | 1 | ||
| 9600.2.a.d | 1 | 80.t | odd | 4 | 1 | ||
| 9600.2.a.ba | 1 | 80.s | even | 4 | 1 | ||
| 9600.2.a.bb | 1 | 80.t | odd | 4 | 1 | ||
| 9600.2.a.bc | 1 | 80.i | odd | 4 | 1 | ||
| 9600.2.a.bd | 1 | 80.j | even | 4 | 1 | ||
| 9600.2.a.ca | 1 | 80.i | odd | 4 | 1 | ||
| 9600.2.a.cb | 1 | 80.j | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3840, [\chi])\):
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\( T_{7}^{2} + 16 \)
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\( T_{11} \)
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\( T_{13} - 6 \)
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\( T_{31} - 8 \)
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\( T_{37} - 10 \)
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\( T_{43} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( (T + 1)^{2} \)
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| $5$ |
\( T^{2} + 2T + 5 \)
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| $7$ |
\( T^{2} + 16 \)
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| $11$ |
\( T^{2} \)
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| $13$ |
\( (T - 6)^{2} \)
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| $17$ |
\( T^{2} + 4 \)
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| $19$ |
\( T^{2} + 36 \)
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| $23$ |
\( T^{2} + 36 \)
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| $29$ |
\( T^{2} + 64 \)
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| $31$ |
\( (T - 8)^{2} \)
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| $37$ |
\( (T - 10)^{2} \)
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| $41$ |
\( (T - 6)^{2} \)
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| $43$ |
\( (T - 4)^{2} \)
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| $47$ |
\( T^{2} + 4 \)
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| $53$ |
\( (T - 6)^{2} \)
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| $59$ |
\( T^{2} + 144 \)
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| $61$ |
\( T^{2} + 196 \)
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| $67$ |
\( (T - 4)^{2} \)
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| $71$ |
\( (T + 8)^{2} \)
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| $73$ |
\( T^{2} + 16 \)
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| $79$ |
\( (T + 12)^{2} \)
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| $83$ |
\( (T + 8)^{2} \)
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| $89$ |
\( (T - 6)^{2} \)
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| $97$ |
\( T^{2} \)
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