Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,6,Mod(1,225)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.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: | \(36.0863594579\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{241}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 60 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 25) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{241})\). 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 |
|
−10.2621 | 0 | 73.3104 | 0 | 0 | 68.9517 | −423.931 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 5.26209 | 0 | −4.31044 | 0 | 0 | 131.048 | −191.069 | 0 | 0 | |||||||||||||||||||||||||
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 | 225.6.a.l | 2 | |
| 3.b | odd | 2 | 1 | 25.6.a.d | yes | 2 | |
| 5.b | even | 2 | 1 | 225.6.a.s | 2 | ||
| 5.c | odd | 4 | 2 | 225.6.b.i | 4 | ||
| 12.b | even | 2 | 1 | 400.6.a.o | 2 | ||
| 15.d | odd | 2 | 1 | 25.6.a.b | ✓ | 2 | |
| 15.e | even | 4 | 2 | 25.6.b.b | 4 | ||
| 60.h | even | 2 | 1 | 400.6.a.w | 2 | ||
| 60.l | odd | 4 | 2 | 400.6.c.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.6.a.b | ✓ | 2 | 15.d | odd | 2 | 1 | |
| 25.6.a.d | yes | 2 | 3.b | odd | 2 | 1 | |
| 25.6.b.b | 4 | 15.e | even | 4 | 2 | ||
| 225.6.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
| 225.6.a.s | 2 | 5.b | even | 2 | 1 | ||
| 225.6.b.i | 4 | 5.c | odd | 4 | 2 | ||
| 400.6.a.o | 2 | 12.b | even | 2 | 1 | ||
| 400.6.a.w | 2 | 60.h | even | 2 | 1 | ||
| 400.6.c.n | 4 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):
|
\( T_{2}^{2} + 5T_{2} - 54 \)
|
|
\( T_{7}^{2} - 200T_{7} + 9036 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 5T - 54 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 200T + 9036 \)
|
| $11$ |
\( T^{2} - 196T - 141021 \)
|
| $13$ |
\( T^{2} + 360T - 29296 \)
|
| $17$ |
\( T^{2} + 1490 T - 559359 \)
|
| $19$ |
\( T^{2} + 3180 T + 2232875 \)
|
| $23$ |
\( T^{2} + 1560 T + 599724 \)
|
| $29$ |
\( T^{2} - 3920 T + 3456000 \)
|
| $31$ |
\( T^{2} + 1096 T - 72602196 \)
|
| $37$ |
\( T^{2} - 2020 T - 7864124 \)
|
| $41$ |
\( T^{2} + 27754 T + 182931129 \)
|
| $43$ |
\( T^{2} + 3000 T - 270585136 \)
|
| $47$ |
\( T^{2} + 25760 T + 22262256 \)
|
| $53$ |
\( T^{2} - 26980 T + 147908484 \)
|
| $59$ |
\( T^{2} + 11960 T - 195696000 \)
|
| $61$ |
\( T^{2} + 24396 T - 92208796 \)
|
| $67$ |
\( T^{2} + 40060 T + 249648291 \)
|
| $71$ |
\( T^{2} + \cdots + 1844897904 \)
|
| $73$ |
\( T^{2} + \cdots + 1160669249 \)
|
| $79$ |
\( T^{2} - 65480 T - 446416500 \)
|
| $83$ |
\( T^{2} + \cdots + 2098410219 \)
|
| $89$ |
\( T^{2} + \cdots - 5241540375 \)
|
| $97$ |
\( T^{2} + \cdots - 3238386044 \)
|
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