Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,6,Mod(199,225)]
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([0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.199");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.b (of order \(2\), degree \(1\), 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: | \(36.0863594579\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{145})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 73x^{2} + 1296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 5^{2} \) |
| 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
\( x^{4} + 73x^{2} + 1296 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 19\nu ) / 18 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} + 83\nu ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( 5\nu^{2} + 183 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 4\beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 183 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -19\beta_{2} + 166\beta_1 ) / 5 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 199.1 |
|
− | 8.52080i | 0 | −40.6040 | 0 | 0 | − | 160.416i | 73.3128i | 0 | 0 | ||||||||||||||||||||||||||||
| 199.2 | − | 3.52080i | 0 | 19.6040 | 0 | 0 | − | 80.4159i | − | 181.687i | 0 | 0 | ||||||||||||||||||||||||||||
| 199.3 | 3.52080i | 0 | 19.6040 | 0 | 0 | 80.4159i | 181.687i | 0 | 0 | |||||||||||||||||||||||||||||||
| 199.4 | 8.52080i | 0 | −40.6040 | 0 | 0 | 160.416i | − | 73.3128i | 0 | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.6.b.k | 4 | |
| 3.b | odd | 2 | 1 | 225.6.b.j | 4 | ||
| 5.b | even | 2 | 1 | inner | 225.6.b.k | 4 | |
| 5.c | odd | 4 | 1 | 45.6.a.f | yes | 2 | |
| 5.c | odd | 4 | 1 | 225.6.a.k | 2 | ||
| 15.d | odd | 2 | 1 | 225.6.b.j | 4 | ||
| 15.e | even | 4 | 1 | 45.6.a.d | ✓ | 2 | |
| 15.e | even | 4 | 1 | 225.6.a.r | 2 | ||
| 20.e | even | 4 | 1 | 720.6.a.be | 2 | ||
| 60.l | odd | 4 | 1 | 720.6.a.y | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.6.a.d | ✓ | 2 | 15.e | even | 4 | 1 | |
| 45.6.a.f | yes | 2 | 5.c | odd | 4 | 1 | |
| 225.6.a.k | 2 | 5.c | odd | 4 | 1 | ||
| 225.6.a.r | 2 | 15.e | even | 4 | 1 | ||
| 225.6.b.j | 4 | 3.b | odd | 2 | 1 | ||
| 225.6.b.j | 4 | 15.d | odd | 2 | 1 | ||
| 225.6.b.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 225.6.b.k | 4 | 5.b | even | 2 | 1 | inner | |
| 720.6.a.y | 2 | 60.l | odd | 4 | 1 | ||
| 720.6.a.be | 2 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(225, [\chi])\):
|
\( T_{2}^{4} + 85T_{2}^{2} + 900 \)
|
|
\( T_{7}^{4} + 32200T_{7}^{2} + 166410000 \)
|
|
\( T_{11}^{2} - 800T_{11} + 145500 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 85T^{2} + 900 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 32200 T^{2} + 166410000 \)
|
| $11$ |
\( (T^{2} - 800 T + 145500)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 52166560000 \)
|
| $17$ |
\( T^{4} + \cdots + 816818288400 \)
|
| $19$ |
\( (T^{2} - 1512 T - 3140464)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 76956405350400 \)
|
| $29$ |
\( (T^{2} + 1300 T - 58969500)^{2} \)
|
| $31$ |
\( (T^{2} + 5824 T + 1461744)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} - 400 T - 67008000)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 77\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 35\!\cdots\!00 \)
|
| $59$ |
\( (T^{2} + 63200 T + 963745500)^{2} \)
|
| $61$ |
\( (T^{2} + 49116 T - 469672636)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 56\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} - 65200 T + 983358000)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $79$ |
\( (T^{2} - 46288 T + 275282736)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} - 87000 T - 665550000)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 75\!\cdots\!00 \)
|
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