Newspace parameters
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.4328556826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 14) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | −8.00000 | 0 | 14.0000 | 0 | −7.00000 | 0 | 37.0000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 448.4.a.b | 1 | |
| 4.b | odd | 2 | 1 | 448.4.a.o | 1 | ||
| 8.b | even | 2 | 1 | 14.4.a.a | ✓ | 1 | |
| 8.d | odd | 2 | 1 | 112.4.a.a | 1 | ||
| 24.f | even | 2 | 1 | 1008.4.a.s | 1 | ||
| 24.h | odd | 2 | 1 | 126.4.a.h | 1 | ||
| 40.f | even | 2 | 1 | 350.4.a.l | 1 | ||
| 40.i | odd | 4 | 2 | 350.4.c.b | 2 | ||
| 56.e | even | 2 | 1 | 784.4.a.s | 1 | ||
| 56.h | odd | 2 | 1 | 98.4.a.a | 1 | ||
| 56.j | odd | 6 | 2 | 98.4.c.f | 2 | ||
| 56.p | even | 6 | 2 | 98.4.c.d | 2 | ||
| 88.b | odd | 2 | 1 | 1694.4.a.g | 1 | ||
| 104.e | even | 2 | 1 | 2366.4.a.h | 1 | ||
| 168.i | even | 2 | 1 | 882.4.a.i | 1 | ||
| 168.s | odd | 6 | 2 | 882.4.g.b | 2 | ||
| 168.ba | even | 6 | 2 | 882.4.g.k | 2 | ||
| 280.c | odd | 2 | 1 | 2450.4.a.bo | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.4.a.a | ✓ | 1 | 8.b | even | 2 | 1 | |
| 98.4.a.a | 1 | 56.h | odd | 2 | 1 | ||
| 98.4.c.d | 2 | 56.p | even | 6 | 2 | ||
| 98.4.c.f | 2 | 56.j | odd | 6 | 2 | ||
| 112.4.a.a | 1 | 8.d | odd | 2 | 1 | ||
| 126.4.a.h | 1 | 24.h | odd | 2 | 1 | ||
| 350.4.a.l | 1 | 40.f | even | 2 | 1 | ||
| 350.4.c.b | 2 | 40.i | odd | 4 | 2 | ||
| 448.4.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
| 448.4.a.o | 1 | 4.b | odd | 2 | 1 | ||
| 784.4.a.s | 1 | 56.e | even | 2 | 1 | ||
| 882.4.a.i | 1 | 168.i | even | 2 | 1 | ||
| 882.4.g.b | 2 | 168.s | odd | 6 | 2 | ||
| 882.4.g.k | 2 | 168.ba | even | 6 | 2 | ||
| 1008.4.a.s | 1 | 24.f | even | 2 | 1 | ||
| 1694.4.a.g | 1 | 88.b | odd | 2 | 1 | ||
| 2366.4.a.h | 1 | 104.e | even | 2 | 1 | ||
| 2450.4.a.bo | 1 | 280.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\):
|
\( T_{3} + 8 \)
|
|
\( T_{5} - 14 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T + 8 \)
|
| $5$ |
\( T - 14 \)
|
| $7$ |
\( T + 7 \)
|
| $11$ |
\( T - 28 \)
|
| $13$ |
\( T + 18 \)
|
| $17$ |
\( T - 74 \)
|
| $19$ |
\( T + 80 \)
|
| $23$ |
\( T + 112 \)
|
| $29$ |
\( T + 190 \)
|
| $31$ |
\( T - 72 \)
|
| $37$ |
\( T - 346 \)
|
| $41$ |
\( T - 162 \)
|
| $43$ |
\( T - 412 \)
|
| $47$ |
\( T - 24 \)
|
| $53$ |
\( T + 318 \)
|
| $59$ |
\( T - 200 \)
|
| $61$ |
\( T - 198 \)
|
| $67$ |
\( T - 716 \)
|
| $71$ |
\( T - 392 \)
|
| $73$ |
\( T - 538 \)
|
| $79$ |
\( T - 240 \)
|
| $83$ |
\( T - 1072 \)
|
| $89$ |
\( T - 810 \)
|
| $97$ |
\( T - 1354 \)
|
| show more | |
| show less |