Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [450,6,Mod(1,450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, 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("450.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 450.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: | \(72.1727189158\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1249}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 312 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 30) |
| 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 \(\beta = 5\sqrt{1249}\). 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 |
|
−4.00000 | 0 | 16.0000 | 0 | 0 | −119.706 | −64.0000 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | 0 | 0 | 233.706 | −64.0000 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 450.6.a.bb | 2 | |
| 3.b | odd | 2 | 1 | 150.6.a.o | 2 | ||
| 5.b | even | 2 | 1 | 450.6.a.bc | 2 | ||
| 5.c | odd | 4 | 2 | 90.6.c.c | 4 | ||
| 15.d | odd | 2 | 1 | 150.6.a.n | 2 | ||
| 15.e | even | 4 | 2 | 30.6.c.b | ✓ | 4 | |
| 20.e | even | 4 | 2 | 720.6.f.i | 4 | ||
| 60.l | odd | 4 | 2 | 240.6.f.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.6.c.b | ✓ | 4 | 15.e | even | 4 | 2 | |
| 90.6.c.c | 4 | 5.c | odd | 4 | 2 | ||
| 150.6.a.n | 2 | 15.d | odd | 2 | 1 | ||
| 150.6.a.o | 2 | 3.b | odd | 2 | 1 | ||
| 240.6.f.b | 4 | 60.l | odd | 4 | 2 | ||
| 450.6.a.bb | 2 | 1.a | even | 1 | 1 | trivial | |
| 450.6.a.bc | 2 | 5.b | even | 2 | 1 | ||
| 720.6.f.i | 4 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):
|
\( T_{7}^{2} - 114T_{7} - 27976 \)
|
|
\( T_{11}^{2} + 174T_{11} - 23656 \)
|
|
\( T_{17}^{2} + 1514T_{17} + 292024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 114T - 27976 \)
|
| $11$ |
\( T^{2} + 174T - 23656 \)
|
| $13$ |
\( T^{2} - 642T - 177984 \)
|
| $17$ |
\( T^{2} + 1514 T + 292024 \)
|
| $19$ |
\( T^{2} + 120 T - 4492800 \)
|
| $23$ |
\( T^{2} + 4352 T + 238576 \)
|
| $29$ |
\( T^{2} - 2430 T - 21286800 \)
|
| $31$ |
\( T^{2} - 11684 T + 33004864 \)
|
| $37$ |
\( T^{2} + 17586 T + 66044624 \)
|
| $41$ |
\( T^{2} + 24984 T + 119953964 \)
|
| $43$ |
\( T^{2} + 24168 T + 48101456 \)
|
| $47$ |
\( T^{2} - 13316 T + 43204864 \)
|
| $53$ |
\( T^{2} - 13698 T - 337814424 \)
|
| $59$ |
\( T^{2} - 23730 T - 927897400 \)
|
| $61$ |
\( T^{2} - 57124 T + 528018244 \)
|
| $67$ |
\( T^{2} + 38316 T + 136088864 \)
|
| $71$ |
\( T^{2} + \cdots - 2668294656 \)
|
| $73$ |
\( T^{2} + \cdots + 1660302176 \)
|
| $79$ |
\( T^{2} + \cdots - 1173592800 \)
|
| $83$ |
\( T^{2} + \cdots - 5182377584 \)
|
| $89$ |
\( T^{2} - 60420 T - 336355900 \)
|
| $97$ |
\( T^{2} + \cdots + 7320736064 \)
|
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