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: | \(3\) |
| Coefficient field: | 3.3.2292.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 13x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 13x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 4\nu - 11 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 2\nu - 9 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + 2\beta _1 + 29 ) / 3 \)
|
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 |
|
−4.57358 | 0 | 12.9176 | 5.00000 | 0 | −20.1145 | −22.4912 | 0 | −22.8679 | |||||||||||||||||||||||||||
| 1.2 | −0.174985 | 0 | −7.96938 | 5.00000 | 0 | 8.46371 | 2.79440 | 0 | −0.874923 | ||||||||||||||||||||||||||||
| 1.3 | 3.74857 | 0 | 6.05174 | 5.00000 | 0 | −31.3492 | −7.30318 | 0 | 18.7428 | ||||||||||||||||||||||||||||
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.h | 3 | |
| 3.b | odd | 2 | 1 | 405.4.a.j | 3 | ||
| 5.b | even | 2 | 1 | 2025.4.a.s | 3 | ||
| 9.c | even | 3 | 2 | 45.4.e.b | ✓ | 6 | |
| 9.d | odd | 6 | 2 | 135.4.e.b | 6 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.q | 3 | ||
| 45.j | even | 6 | 2 | 225.4.e.c | 6 | ||
| 45.k | odd | 12 | 4 | 225.4.k.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.e.b | ✓ | 6 | 9.c | even | 3 | 2 | |
| 135.4.e.b | 6 | 9.d | odd | 6 | 2 | ||
| 225.4.e.c | 6 | 45.j | even | 6 | 2 | ||
| 225.4.k.c | 12 | 45.k | odd | 12 | 4 | ||
| 405.4.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 405.4.a.j | 3 | 3.b | odd | 2 | 1 | ||
| 2025.4.a.q | 3 | 15.d | odd | 2 | 1 | ||
| 2025.4.a.s | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + T_{2}^{2} - 17T_{2} - 3 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 17T - 3 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} + 43 T^{2} + \cdots - 5337 \)
|
| $11$ |
\( T^{3} - 14 T^{2} + \cdots - 43548 \)
|
| $13$ |
\( T^{3} - 40 T^{2} + \cdots + 75364 \)
|
| $17$ |
\( T^{3} + 166 T^{2} + \cdots + 156324 \)
|
| $19$ |
\( T^{3} + 164 T^{2} + \cdots + 57316 \)
|
| $23$ |
\( T^{3} - 171 T^{2} + \cdots + 61209 \)
|
| $29$ |
\( T^{3} + 335 T^{2} + \cdots - 107067 \)
|
| $31$ |
\( T^{3} + 352 T^{2} + \cdots - 9860940 \)
|
| $37$ |
\( T^{3} - 402 T^{2} + \cdots + 3335284 \)
|
| $41$ |
\( T^{3} - 187 T^{2} + \cdots + 7228275 \)
|
| $43$ |
\( T^{3} + 602 T^{2} + \cdots - 15444524 \)
|
| $47$ |
\( T^{3} - 665 T^{2} + \cdots - 2487483 \)
|
| $53$ |
\( T^{3} + 730 T^{2} + \cdots + 3250536 \)
|
| $59$ |
\( T^{3} + 298 T^{2} + \cdots - 127375896 \)
|
| $61$ |
\( T^{3} + 1439 T^{2} + \cdots + 55613497 \)
|
| $67$ |
\( T^{3} + 1849 T^{2} + \cdots + 208776159 \)
|
| $71$ |
\( T^{3} - 70 T^{2} + \cdots + 223775052 \)
|
| $73$ |
\( T^{3} + 368 T^{2} + \cdots - 134927744 \)
|
| $79$ |
\( T^{3} + 382 T^{2} + \cdots - 138322584 \)
|
| $83$ |
\( T^{3} + 831 T^{2} + \cdots - 955843821 \)
|
| $89$ |
\( T^{3} - 1719 T^{2} + \cdots + 125506395 \)
|
| $97$ |
\( T^{3} + 282 T^{2} + \cdots - 16898264 \)
|
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