Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2475,4,Mod(1,2475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2475.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2475.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: | \(146.029727264\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1540841.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 27x^{2} - 18x + 92 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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 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
\( x^{4} - 27x^{2} - 18x + 92 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 20\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 21\beta _1 + 14 \)
|
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 |
|
−4.20196 | 0 | 9.65650 | 0 | 0 | −15.3793 | −6.96057 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.607192 | 0 | −7.63132 | 0 | 0 | −8.95080 | 9.49121 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 3.63835 | 0 | 5.23763 | 0 | 0 | −20.8444 | −10.0505 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 5.17080 | 0 | 18.7372 | 0 | 0 | 11.1745 | 55.5199 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 2475.4.a.be | 4 | |
| 3.b | odd | 2 | 1 | 825.4.a.t | 4 | ||
| 5.b | even | 2 | 1 | 495.4.a.m | 4 | ||
| 15.d | odd | 2 | 1 | 165.4.a.h | ✓ | 4 | |
| 15.e | even | 4 | 2 | 825.4.c.p | 8 | ||
| 165.d | even | 2 | 1 | 1815.4.a.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.h | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 495.4.a.m | 4 | 5.b | even | 2 | 1 | ||
| 825.4.a.t | 4 | 3.b | odd | 2 | 1 | ||
| 825.4.c.p | 8 | 15.e | even | 4 | 2 | ||
| 1815.4.a.t | 4 | 165.d | even | 2 | 1 | ||
| 2475.4.a.be | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
|
\( T_{2}^{4} - 4T_{2}^{3} - 21T_{2}^{2} + 68T_{2} + 48 \)
|
|
\( T_{7}^{4} + 34T_{7}^{3} + 140T_{7}^{2} - 4336T_{7} - 32064 \)
|
|
\( T_{29}^{4} + 52T_{29}^{3} - 57364T_{29}^{2} - 2737584T_{29} + 315474624 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots + 48 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 34 T^{3} + \cdots - 32064 \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots + 68144 \)
|
| $17$ |
\( T^{4} - 74 T^{3} + \cdots + 23770800 \)
|
| $19$ |
\( T^{4} - 136 T^{3} + \cdots + 576 \)
|
| $23$ |
\( T^{4} + 64 T^{3} + \cdots + 247529472 \)
|
| $29$ |
\( T^{4} + 52 T^{3} + \cdots + 315474624 \)
|
| $31$ |
\( T^{4} - 492 T^{3} + \cdots - 903269376 \)
|
| $37$ |
\( T^{4} + \cdots + 1260009136 \)
|
| $41$ |
\( T^{4} + \cdots - 6228069696 \)
|
| $43$ |
\( T^{4} + \cdots - 1200551616 \)
|
| $47$ |
\( T^{4} + \cdots + 5628791808 \)
|
| $53$ |
\( T^{4} + \cdots + 13030473936 \)
|
| $59$ |
\( T^{4} + \cdots - 2612441088 \)
|
| $61$ |
\( T^{4} + \cdots - 8546777488 \)
|
| $67$ |
\( T^{4} - 552 T^{3} + \cdots - 909580544 \)
|
| $71$ |
\( T^{4} + \cdots + 291456592896 \)
|
| $73$ |
\( T^{4} + \cdots - 133750796272 \)
|
| $79$ |
\( T^{4} + \cdots + 48148922944 \)
|
| $83$ |
\( T^{4} + \cdots - 17388663552 \)
|
| $89$ |
\( T^{4} + \cdots + 479041129296 \)
|
| $97$ |
\( T^{4} + \cdots + 1607443600 \)
|
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