Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(1,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(26.4328556826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{37}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 224) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{37}\). 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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| 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 | −3.08276 | 0 | 9.08276 | 0 | 7.00000 | 0 | −17.4966 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.08276 | 0 | −3.08276 | 0 | 7.00000 | 0 | 55.4966 | 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.t | 2 | |
| 4.b | odd | 2 | 1 | 448.4.a.q | 2 | ||
| 8.b | even | 2 | 1 | 224.4.a.c | ✓ | 2 | |
| 8.d | odd | 2 | 1 | 224.4.a.d | yes | 2 | |
| 24.f | even | 2 | 1 | 2016.4.a.o | 2 | ||
| 24.h | odd | 2 | 1 | 2016.4.a.p | 2 | ||
| 56.e | even | 2 | 1 | 1568.4.a.p | 2 | ||
| 56.h | odd | 2 | 1 | 1568.4.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 224.4.a.c | ✓ | 2 | 8.b | even | 2 | 1 | |
| 224.4.a.d | yes | 2 | 8.d | odd | 2 | 1 | |
| 448.4.a.q | 2 | 4.b | odd | 2 | 1 | ||
| 448.4.a.t | 2 | 1.a | even | 1 | 1 | trivial | |
| 1568.4.a.p | 2 | 56.e | even | 2 | 1 | ||
| 1568.4.a.u | 2 | 56.h | odd | 2 | 1 | ||
| 2016.4.a.o | 2 | 24.f | even | 2 | 1 | ||
| 2016.4.a.p | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(448))\):
|
\( T_{3}^{2} - 6T_{3} - 28 \)
|
|
\( T_{5}^{2} - 6T_{5} - 28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 6T - 28 \)
|
| $5$ |
\( T^{2} - 6T - 28 \)
|
| $7$ |
\( (T - 7)^{2} \)
|
| $11$ |
\( T^{2} - 44T + 336 \)
|
| $13$ |
\( T^{2} + 14T + 12 \)
|
| $17$ |
\( T^{2} - 96T - 1396 \)
|
| $19$ |
\( T^{2} - 170T + 4228 \)
|
| $23$ |
\( T^{2} + 152T - 3696 \)
|
| $29$ |
\( T^{2} - 128T - 38676 \)
|
| $31$ |
\( T^{2} + 68T - 23856 \)
|
| $37$ |
\( T^{2} - 256T + 16236 \)
|
| $41$ |
\( T^{2} - 88T - 122532 \)
|
| $43$ |
\( T^{2} - 724T + 119056 \)
|
| $47$ |
\( T^{2} - 244T - 3024 \)
|
| $53$ |
\( T^{2} + 188T - 142716 \)
|
| $59$ |
\( T^{2} - 138T - 35532 \)
|
| $61$ |
\( T^{2} - 358T + 29044 \)
|
| $67$ |
\( T^{2} - 200T - 757232 \)
|
| $71$ |
\( T^{2} - 1400 T + 460992 \)
|
| $73$ |
\( T^{2} + 628T - 399276 \)
|
| $79$ |
\( T^{2} + 200T - 985152 \)
|
| $83$ |
\( T^{2} - 618T - 839916 \)
|
| $89$ |
\( T^{2} + 908T - 30684 \)
|
| $97$ |
\( T^{2} - 1520 T + 524172 \)
|
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