Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(136,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.136");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} + \zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
−1.36603 | + | 2.36603i | 0 | 0.267949 | + | 0.464102i | −2.50000 | − | 4.33013i | 0 | −3.46410 | + | 6.00000i | −23.3205 | 0 | 13.6603 | ||||||||||||||||||||||
| 136.2 | 0.366025 | − | 0.633975i | 0 | 3.73205 | + | 6.46410i | −2.50000 | − | 4.33013i | 0 | 3.46410 | − | 6.00000i | 11.3205 | 0 | −3.66025 | |||||||||||||||||||||||
| 271.1 | −1.36603 | − | 2.36603i | 0 | 0.267949 | − | 0.464102i | −2.50000 | + | 4.33013i | 0 | −3.46410 | − | 6.00000i | −23.3205 | 0 | 13.6603 | |||||||||||||||||||||||
| 271.2 | 0.366025 | + | 0.633975i | 0 | 3.73205 | − | 6.46410i | −2.50000 | + | 4.33013i | 0 | 3.46410 | + | 6.00000i | 11.3205 | 0 | −3.66025 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.e.o | 4 | |
| 3.b | odd | 2 | 1 | 405.4.e.p | 4 | ||
| 9.c | even | 3 | 1 | 405.4.a.f | yes | 2 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.o | 4 | |
| 9.d | odd | 6 | 1 | 405.4.a.c | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 405.4.e.p | 4 | ||
| 45.h | odd | 6 | 1 | 2025.4.a.m | 2 | ||
| 45.j | even | 6 | 1 | 2025.4.a.i | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 405.4.a.c | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 405.4.a.f | yes | 2 | 9.c | even | 3 | 1 | |
| 405.4.e.o | 4 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.o | 4 | 9.c | even | 3 | 1 | inner | |
| 405.4.e.p | 4 | 3.b | odd | 2 | 1 | ||
| 405.4.e.p | 4 | 9.d | odd | 6 | 1 | ||
| 2025.4.a.i | 2 | 45.j | even | 6 | 1 | ||
| 2025.4.a.m | 2 | 45.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{4} + 2T_{2}^{3} + 6T_{2}^{2} - 4T_{2} + 4 \)
|
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\( T_{7}^{4} + 48T_{7}^{2} + 2304 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{2} \)
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| $7$ |
\( T^{4} + 48T^{2} + 2304 \)
|
| $11$ |
\( T^{4} - 22 T^{3} + \cdots + 4977361 \)
|
| $13$ |
\( T^{4} - 64 T^{3} + \cdots + 952576 \)
|
| $17$ |
\( (T^{2} - 32 T - 9152)^{2} \)
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| $19$ |
\( (T^{2} + 10 T - 13847)^{2} \)
|
| $23$ |
\( T^{4} - 84 T^{3} + \cdots + 1710864 \)
|
| $29$ |
\( T^{4} - 170 T^{3} + \cdots + 30217009 \)
|
| $31$ |
\( T^{4} - 258 T^{3} + \cdots + 262731681 \)
|
| $37$ |
\( (T^{2} - 76 T - 64268)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 5868938881 \)
|
| $43$ |
\( T^{4} + \cdots + 1362200464 \)
|
| $47$ |
\( T^{4} + \cdots + 3340377616 \)
|
| $53$ |
\( (T^{2} + 544 T + 43984)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 14946774049 \)
|
| $61$ |
\( T^{4} + 668 T^{3} + \cdots + 929296 \)
|
| $67$ |
\( T^{4} + \cdots + 269323633296 \)
|
| $71$ |
\( (T^{2} + 974 T + 121921)^{2} \)
|
| $73$ |
\( (T^{2} - 1184 T + 325072)^{2} \)
|
| $79$ |
\( T^{4} + 408 T^{3} + \cdots + 84052224 \)
|
| $83$ |
\( T^{4} + \cdots + 1062375472656 \)
|
| $89$ |
\( (T + 513)^{4} \)
|
| $97$ |
\( T^{4} + \cdots + 220189931536 \)
|
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