Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,4,Mod(1,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, 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("368.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 368.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: | \(21.7127028821\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{41}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 10 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 46) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). 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 | −9.10469 | 0 | −1.40312 | 0 | 3.40312 | 0 | 55.8953 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 10.1047 | 0 | 11.4031 | 0 | −9.40312 | 0 | 75.1047 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 368.4.a.g | 2 | |
| 4.b | odd | 2 | 1 | 46.4.a.c | ✓ | 2 | |
| 8.b | even | 2 | 1 | 1472.4.a.l | 2 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.m | 2 | ||
| 12.b | even | 2 | 1 | 414.4.a.j | 2 | ||
| 20.d | odd | 2 | 1 | 1150.4.a.k | 2 | ||
| 20.e | even | 4 | 2 | 1150.4.b.i | 4 | ||
| 28.d | even | 2 | 1 | 2254.4.a.d | 2 | ||
| 92.b | even | 2 | 1 | 1058.4.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.4.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 368.4.a.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 414.4.a.j | 2 | 12.b | even | 2 | 1 | ||
| 1058.4.a.f | 2 | 92.b | even | 2 | 1 | ||
| 1150.4.a.k | 2 | 20.d | odd | 2 | 1 | ||
| 1150.4.b.i | 4 | 20.e | even | 4 | 2 | ||
| 1472.4.a.l | 2 | 8.b | even | 2 | 1 | ||
| 1472.4.a.m | 2 | 8.d | odd | 2 | 1 | ||
| 2254.4.a.d | 2 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - T_{3} - 92 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(368))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - T - 92 \)
|
| $5$ |
\( T^{2} - 10T - 16 \)
|
| $7$ |
\( T^{2} + 6T - 32 \)
|
| $11$ |
\( T^{2} + 72T + 640 \)
|
| $13$ |
\( T^{2} + 111T + 2578 \)
|
| $17$ |
\( T^{2} - 124T - 2060 \)
|
| $19$ |
\( T^{2} + 22T - 3200 \)
|
| $23$ |
\( (T - 23)^{2} \)
|
| $29$ |
\( T^{2} - 15T - 43250 \)
|
| $31$ |
\( T^{2} + 67T - 1840 \)
|
| $37$ |
\( T^{2} - 18T - 50144 \)
|
| $41$ |
\( T^{2} - 485T + 54286 \)
|
| $43$ |
\( T^{2} - 440T + 16256 \)
|
| $47$ |
\( T^{2} + 215T - 77096 \)
|
| $53$ |
\( T^{2} - 240T - 143204 \)
|
| $59$ |
\( T^{2} + 792T + 146320 \)
|
| $61$ |
\( T^{2} - 456T - 408692 \)
|
| $67$ |
\( T^{2} + 240T - 9216 \)
|
| $71$ |
\( T^{2} - 705T + 39376 \)
|
| $73$ |
\( T^{2} - 27T - 8438 \)
|
| $79$ |
\( T^{2} + 594T + 32080 \)
|
| $83$ |
\( T^{2} + 394T - 285952 \)
|
| $89$ |
\( T^{2} + 486T - 31520 \)
|
| $97$ |
\( T^{2} - 2152 T + 882092 \)
|
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