Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,3,Mod(26,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.26");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.c (of order \(2\), degree \(1\), 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: | \(6.13080594811\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 45) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 4x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 7\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 3\nu^{2} + \nu + 6 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 2\nu + 6 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} - 2\beta_{2} + 6 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} - 7\beta_{2} + 3\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
− | 3.65028i | 0 | −9.32456 | 0 | 0 | −7.16228 | 19.4361i | 0 | 0 | |||||||||||||||||||||||||||||
| 26.2 | − | 0.821854i | 0 | 3.32456 | 0 | 0 | −0.837722 | − | 6.01972i | 0 | 0 | |||||||||||||||||||||||||||||
| 26.3 | 0.821854i | 0 | 3.32456 | 0 | 0 | −0.837722 | 6.01972i | 0 | 0 | |||||||||||||||||||||||||||||||
| 26.4 | 3.65028i | 0 | −9.32456 | 0 | 0 | −7.16228 | − | 19.4361i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.3.c.c | 4 | |
| 3.b | odd | 2 | 1 | inner | 225.3.c.c | 4 | |
| 4.b | odd | 2 | 1 | 3600.3.l.v | 4 | ||
| 5.b | even | 2 | 1 | 45.3.c.a | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 225.3.d.b | 8 | ||
| 12.b | even | 2 | 1 | 3600.3.l.v | 4 | ||
| 15.d | odd | 2 | 1 | 45.3.c.a | ✓ | 4 | |
| 15.e | even | 4 | 2 | 225.3.d.b | 8 | ||
| 20.d | odd | 2 | 1 | 720.3.l.a | 4 | ||
| 20.e | even | 4 | 2 | 3600.3.c.i | 8 | ||
| 40.e | odd | 2 | 1 | 2880.3.l.c | 4 | ||
| 40.f | even | 2 | 1 | 2880.3.l.g | 4 | ||
| 45.h | odd | 6 | 2 | 405.3.i.d | 8 | ||
| 45.j | even | 6 | 2 | 405.3.i.d | 8 | ||
| 60.h | even | 2 | 1 | 720.3.l.a | 4 | ||
| 60.l | odd | 4 | 2 | 3600.3.c.i | 8 | ||
| 120.i | odd | 2 | 1 | 2880.3.l.g | 4 | ||
| 120.m | even | 2 | 1 | 2880.3.l.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.3.c.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 45.3.c.a | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 225.3.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 225.3.c.c | 4 | 3.b | odd | 2 | 1 | inner | |
| 225.3.d.b | 8 | 5.c | odd | 4 | 2 | ||
| 225.3.d.b | 8 | 15.e | even | 4 | 2 | ||
| 405.3.i.d | 8 | 45.h | odd | 6 | 2 | ||
| 405.3.i.d | 8 | 45.j | even | 6 | 2 | ||
| 720.3.l.a | 4 | 20.d | odd | 2 | 1 | ||
| 720.3.l.a | 4 | 60.h | even | 2 | 1 | ||
| 2880.3.l.c | 4 | 40.e | odd | 2 | 1 | ||
| 2880.3.l.c | 4 | 120.m | even | 2 | 1 | ||
| 2880.3.l.g | 4 | 40.f | even | 2 | 1 | ||
| 2880.3.l.g | 4 | 120.i | odd | 2 | 1 | ||
| 3600.3.c.i | 8 | 20.e | even | 4 | 2 | ||
| 3600.3.c.i | 8 | 60.l | odd | 4 | 2 | ||
| 3600.3.l.v | 4 | 4.b | odd | 2 | 1 | ||
| 3600.3.l.v | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\):
|
\( T_{2}^{4} + 14T_{2}^{2} + 9 \)
|
|
\( T_{7}^{2} + 8T_{7} + 6 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 14T^{2} + 9 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 8 T + 6)^{2} \)
|
| $11$ |
\( T^{4} + 236T^{2} + 6084 \)
|
| $13$ |
\( (T^{2} - 12 T - 214)^{2} \)
|
| $17$ |
\( T^{4} + 704 T^{2} + 82944 \)
|
| $19$ |
\( (T^{2} - 40)^{2} \)
|
| $23$ |
\( T^{4} + 1656 T^{2} + 219024 \)
|
| $29$ |
\( T^{4} + 2024T^{2} + 144 \)
|
| $31$ |
\( (T^{2} + 28 T - 1764)^{2} \)
|
| $37$ |
\( (T^{2} + 76 T + 634)^{2} \)
|
| $41$ |
\( T^{4} + 644 T^{2} + 101124 \)
|
| $43$ |
\( (T^{2} - 24 T - 856)^{2} \)
|
| $47$ |
\( T^{4} + 2096 T^{2} + 788544 \)
|
| $53$ |
\( T^{4} + 2576 T^{2} + 419904 \)
|
| $59$ |
\( T^{4} + 5996 T^{2} + 3392964 \)
|
| $61$ |
\( (T^{2} + 84 T + 1724)^{2} \)
|
| $67$ |
\( (T^{2} + 104 T + 2064)^{2} \)
|
| $71$ |
\( T^{4} + 19424 T^{2} + 93083904 \)
|
| $73$ |
\( (T^{2} + 108 T + 2876)^{2} \)
|
| $79$ |
\( (T^{2} + 28 T - 6564)^{2} \)
|
| $83$ |
\( T^{4} + 7056 T^{2} + 4981824 \)
|
| $89$ |
\( T^{4} + 14436 T^{2} + 33385284 \)
|
| $97$ |
\( (T^{2} - 116 T + 124)^{2} \)
|
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