Newspace parameters
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.72015280046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-19}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 20) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 8.71780i | 0 | 7.00000 | − | 8.71780i | 0 | 8.71780i | 0 | −49.0000 | 0 | |||||||||||||||||||||
| 49.2 | 0 | 8.71780i | 0 | 7.00000 | + | 8.71780i | 0 | − | 8.71780i | 0 | −49.0000 | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 80.4.c.b | 2 | |
| 3.b | odd | 2 | 1 | 720.4.f.a | 2 | ||
| 4.b | odd | 2 | 1 | 20.4.c.a | ✓ | 2 | |
| 5.b | even | 2 | 1 | inner | 80.4.c.b | 2 | |
| 5.c | odd | 4 | 2 | 400.4.a.w | 2 | ||
| 8.b | even | 2 | 1 | 320.4.c.b | 2 | ||
| 8.d | odd | 2 | 1 | 320.4.c.a | 2 | ||
| 12.b | even | 2 | 1 | 180.4.d.a | 2 | ||
| 15.d | odd | 2 | 1 | 720.4.f.a | 2 | ||
| 20.d | odd | 2 | 1 | 20.4.c.a | ✓ | 2 | |
| 20.e | even | 4 | 2 | 100.4.a.d | 2 | ||
| 28.d | even | 2 | 1 | 980.4.e.a | 2 | ||
| 40.e | odd | 2 | 1 | 320.4.c.a | 2 | ||
| 40.f | even | 2 | 1 | 320.4.c.b | 2 | ||
| 40.i | odd | 4 | 2 | 1600.4.a.ck | 2 | ||
| 40.k | even | 4 | 2 | 1600.4.a.cj | 2 | ||
| 60.h | even | 2 | 1 | 180.4.d.a | 2 | ||
| 60.l | odd | 4 | 2 | 900.4.a.s | 2 | ||
| 140.c | even | 2 | 1 | 980.4.e.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.4.c.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 20.4.c.a | ✓ | 2 | 20.d | odd | 2 | 1 | |
| 80.4.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 80.4.c.b | 2 | 5.b | even | 2 | 1 | inner | |
| 100.4.a.d | 2 | 20.e | even | 4 | 2 | ||
| 180.4.d.a | 2 | 12.b | even | 2 | 1 | ||
| 180.4.d.a | 2 | 60.h | even | 2 | 1 | ||
| 320.4.c.a | 2 | 8.d | odd | 2 | 1 | ||
| 320.4.c.a | 2 | 40.e | odd | 2 | 1 | ||
| 320.4.c.b | 2 | 8.b | even | 2 | 1 | ||
| 320.4.c.b | 2 | 40.f | even | 2 | 1 | ||
| 400.4.a.w | 2 | 5.c | odd | 4 | 2 | ||
| 720.4.f.a | 2 | 3.b | odd | 2 | 1 | ||
| 720.4.f.a | 2 | 15.d | odd | 2 | 1 | ||
| 900.4.a.s | 2 | 60.l | odd | 4 | 2 | ||
| 980.4.e.a | 2 | 28.d | even | 2 | 1 | ||
| 980.4.e.a | 2 | 140.c | even | 2 | 1 | ||
| 1600.4.a.cj | 2 | 40.k | even | 4 | 2 | ||
| 1600.4.a.ck | 2 | 40.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 76 \)
acting on \(S_{4}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 76 \)
|
| $5$ |
\( T^{2} - 14T + 125 \)
|
| $7$ |
\( T^{2} + 76 \)
|
| $11$ |
\( (T + 20)^{2} \)
|
| $13$ |
\( T^{2} + 2736 \)
|
| $17$ |
\( T^{2} + 4864 \)
|
| $19$ |
\( (T - 84)^{2} \)
|
| $23$ |
\( T^{2} + 3724 \)
|
| $29$ |
\( (T - 6)^{2} \)
|
| $31$ |
\( (T - 224)^{2} \)
|
| $37$ |
\( T^{2} + 14896 \)
|
| $41$ |
\( (T - 266)^{2} \)
|
| $43$ |
\( T^{2} + 93100 \)
|
| $47$ |
\( T^{2} + 140524 \)
|
| $53$ |
\( T^{2} + 134064 \)
|
| $59$ |
\( (T - 28)^{2} \)
|
| $61$ |
\( (T - 182)^{2} \)
|
| $67$ |
\( T^{2} + 182476 \)
|
| $71$ |
\( (T + 408)^{2} \)
|
| $73$ |
\( T^{2} + 1168576 \)
|
| $79$ |
\( (T + 48)^{2} \)
|
| $83$ |
\( T^{2} + 40204 \)
|
| $89$ |
\( (T + 1526)^{2} \)
|
| $97$ |
\( T^{2} + 311296 \)
|
| show more | |
| show less |