Newspace parameters
| Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1600.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(94.4030560092\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{19}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - 19 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 20) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −8.71780 | 0 | 0 | 0 | 8.71780 | 0 | 49.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 8.71780 | 0 | 0 | 0 | −8.71780 | 0 | 49.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
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 | 1600.4.a.ck | 2 | |
| 4.b | odd | 2 | 1 | 1600.4.a.cj | 2 | ||
| 5.b | even | 2 | 1 | inner | 1600.4.a.ck | 2 | |
| 5.c | odd | 4 | 2 | 320.4.c.b | 2 | ||
| 8.b | even | 2 | 1 | 400.4.a.w | 2 | ||
| 8.d | odd | 2 | 1 | 100.4.a.d | 2 | ||
| 20.d | odd | 2 | 1 | 1600.4.a.cj | 2 | ||
| 20.e | even | 4 | 2 | 320.4.c.a | 2 | ||
| 24.f | even | 2 | 1 | 900.4.a.s | 2 | ||
| 40.e | odd | 2 | 1 | 100.4.a.d | 2 | ||
| 40.f | even | 2 | 1 | 400.4.a.w | 2 | ||
| 40.i | odd | 4 | 2 | 80.4.c.b | 2 | ||
| 40.k | even | 4 | 2 | 20.4.c.a | ✓ | 2 | |
| 120.m | even | 2 | 1 | 900.4.a.s | 2 | ||
| 120.q | odd | 4 | 2 | 180.4.d.a | 2 | ||
| 120.w | even | 4 | 2 | 720.4.f.a | 2 | ||
| 280.y | odd | 4 | 2 | 980.4.e.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.4.c.a | ✓ | 2 | 40.k | even | 4 | 2 | |
| 80.4.c.b | 2 | 40.i | odd | 4 | 2 | ||
| 100.4.a.d | 2 | 8.d | odd | 2 | 1 | ||
| 100.4.a.d | 2 | 40.e | odd | 2 | 1 | ||
| 180.4.d.a | 2 | 120.q | odd | 4 | 2 | ||
| 320.4.c.a | 2 | 20.e | even | 4 | 2 | ||
| 320.4.c.b | 2 | 5.c | odd | 4 | 2 | ||
| 400.4.a.w | 2 | 8.b | even | 2 | 1 | ||
| 400.4.a.w | 2 | 40.f | even | 2 | 1 | ||
| 720.4.f.a | 2 | 120.w | even | 4 | 2 | ||
| 900.4.a.s | 2 | 24.f | even | 2 | 1 | ||
| 900.4.a.s | 2 | 120.m | even | 2 | 1 | ||
| 980.4.e.a | 2 | 280.y | odd | 4 | 2 | ||
| 1600.4.a.cj | 2 | 4.b | odd | 2 | 1 | ||
| 1600.4.a.cj | 2 | 20.d | odd | 2 | 1 | ||
| 1600.4.a.ck | 2 | 1.a | even | 1 | 1 | trivial | |
| 1600.4.a.ck | 2 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):
|
\( T_{3}^{2} - 76 \)
|
|
\( T_{7}^{2} - 76 \)
|
|
\( T_{11} - 20 \)
|
|
\( T_{13}^{2} - 2736 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 76 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 76 \)
|
| $11$ |
\( (T - 20)^{2} \)
|
| $13$ |
\( T^{2} - 2736 \)
|
| $17$ |
\( T^{2} - 4864 \)
|
| $19$ |
\( (T - 84)^{2} \)
|
| $23$ |
\( T^{2} - 3724 \)
|
| $29$ |
\( (T - 6)^{2} \)
|
| $31$ |
\( (T - 224)^{2} \)
|
| $37$ |
\( T^{2} - 14896 \)
|
| $41$ |
\( (T - 266)^{2} \)
|
| $43$ |
\( T^{2} - 93100 \)
|
| $47$ |
\( T^{2} - 140524 \)
|
| $53$ |
\( T^{2} - 134064 \)
|
| $59$ |
\( (T - 28)^{2} \)
|
| $61$ |
\( (T + 182)^{2} \)
|
| $67$ |
\( T^{2} - 182476 \)
|
| $71$ |
\( (T + 408)^{2} \)
|
| $73$ |
\( T^{2} - 1168576 \)
|
| $79$ |
\( (T - 48)^{2} \)
|
| $83$ |
\( T^{2} - 40204 \)
|
| $89$ |
\( (T - 1526)^{2} \)
|
| $97$ |
\( T^{2} - 311296 \)
|
| show more | |
| show less |