Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(1,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("405.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.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: | \(23.8957735523\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{33}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). 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 |
|
−2.37228 | 0 | −2.37228 | −5.00000 | 0 | 7.37228 | 24.6060 | 0 | 11.8614 | ||||||||||||||||||||||||
| 1.2 | 3.37228 | 0 | 3.37228 | −5.00000 | 0 | 1.62772 | −15.6060 | 0 | −16.8614 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 405.4.a.e | 2 | |
| 3.b | odd | 2 | 1 | 405.4.a.d | 2 | ||
| 5.b | even | 2 | 1 | 2025.4.a.j | 2 | ||
| 9.c | even | 3 | 2 | 135.4.e.a | 4 | ||
| 9.d | odd | 6 | 2 | 45.4.e.a | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 2025.4.a.l | 2 | ||
| 45.h | odd | 6 | 2 | 225.4.e.a | 4 | ||
| 45.l | even | 12 | 4 | 225.4.k.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.e.a | ✓ | 4 | 9.d | odd | 6 | 2 | |
| 135.4.e.a | 4 | 9.c | even | 3 | 2 | ||
| 225.4.e.a | 4 | 45.h | odd | 6 | 2 | ||
| 225.4.k.a | 8 | 45.l | even | 12 | 4 | ||
| 405.4.a.d | 2 | 3.b | odd | 2 | 1 | ||
| 405.4.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 2025.4.a.j | 2 | 5.b | even | 2 | 1 | ||
| 2025.4.a.l | 2 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 8 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T - 8 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} - 9T + 12 \)
|
| $11$ |
\( T^{2} - 37T + 136 \)
|
| $13$ |
\( T^{2} + 112T + 2608 \)
|
| $17$ |
\( T^{2} + 77T - 3674 \)
|
| $19$ |
\( T^{2} - 35T - 4850 \)
|
| $23$ |
\( T^{2} - 267T + 13458 \)
|
| $29$ |
\( T^{2} + 325T + 13858 \)
|
| $31$ |
\( (T + 6)^{2} \)
|
| $37$ |
\( T^{2} + 638T + 100936 \)
|
| $41$ |
\( T^{2} + 238T - 15539 \)
|
| $43$ |
\( T^{2} + 97T - 60092 \)
|
| $47$ |
\( T^{2} - 901T + 201952 \)
|
| $53$ |
\( T^{2} + 224T - 69956 \)
|
| $59$ |
\( T^{2} - 85T - 18002 \)
|
| $61$ |
\( T^{2} + 247T - 298454 \)
|
| $67$ |
\( T^{2} + 606T - 187503 \)
|
| $71$ |
\( T^{2} + 394T - 61016 \)
|
| $73$ |
\( T^{2} + 811T - 182276 \)
|
| $79$ |
\( T^{2} + 840T + 128748 \)
|
| $83$ |
\( T^{2} - 387 T - 1655532 \)
|
| $89$ |
\( T^{2} - 1065 T - 535050 \)
|
| $97$ |
\( T^{2} + 1031 T + 124162 \)
|
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