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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 41x^{3} + 3x^{2} + 294x - 200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 825) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 41x^{3} + 3x^{2} + 294x - 200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} - 23\nu + 48 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 27\nu^{2} + 31\nu + 44 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 27\beta _1 + 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 6\beta_{3} + 39\beta_{2} + 77\beta _1 + 436 \)
|
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 |
|
−5.35463 | 0 | 20.6721 | 0 | 0 | 19.1169 | −67.8543 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.40874 | 0 | −6.01545 | 0 | 0 | −13.7148 | 19.7441 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.270826 | 0 | −7.92665 | 0 | 0 | 33.4133 | −4.31336 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.91216 | 0 | 16.1293 | 0 | 0 | −35.2335 | 39.9323 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.58039 | 0 | 23.1407 | 0 | 0 | 34.4181 | 84.4912 | 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.bm | 5 | |
| 3.b | odd | 2 | 1 | 825.4.a.w | ✓ | 5 | |
| 5.b | even | 2 | 1 | 2475.4.a.bf | 5 | ||
| 15.d | odd | 2 | 1 | 825.4.a.z | yes | 5 | |
| 15.e | even | 4 | 2 | 825.4.c.r | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.4.a.w | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 825.4.a.z | yes | 5 | 15.d | odd | 2 | 1 | |
| 825.4.c.r | 10 | 15.e | even | 4 | 2 | ||
| 2475.4.a.bf | 5 | 5.b | even | 2 | 1 | ||
| 2475.4.a.bm | 5 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):
|
\( T_{2}^{5} - 4T_{2}^{4} - 35T_{2}^{3} + 116T_{2}^{2} + 178T_{2} - 56 \)
|
|
\( T_{7}^{5} - 38T_{7}^{4} - 1326T_{7}^{3} + 55764T_{7}^{2} + 106197T_{7} - 10623494 \)
|
|
\( T_{29}^{5} - 153T_{29}^{4} - 33404T_{29}^{3} + 1022464T_{29}^{2} + 127607008T_{29} - 1390240880 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots - 56 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 38 T^{4} + \cdots - 10623494 \)
|
| $11$ |
\( (T + 11)^{5} \)
|
| $13$ |
\( T^{5} + 43 T^{4} + \cdots - 167445904 \)
|
| $17$ |
\( T^{5} - 150 T^{4} + \cdots + 243358192 \)
|
| $19$ |
\( T^{5} + \cdots - 7593907635 \)
|
| $23$ |
\( T^{5} + 201 T^{4} + \cdots + 188998876 \)
|
| $29$ |
\( T^{5} + \cdots - 1390240880 \)
|
| $31$ |
\( T^{5} + \cdots - 6663042216 \)
|
| $37$ |
\( T^{5} + \cdots + 6140614744 \)
|
| $41$ |
\( T^{5} + \cdots + 314650458072 \)
|
| $43$ |
\( T^{5} + \cdots - 11880158400 \)
|
| $47$ |
\( T^{5} + \cdots - 1024222396544 \)
|
| $53$ |
\( T^{5} + \cdots + 1753265144576 \)
|
| $59$ |
\( T^{5} + \cdots + 950084217280 \)
|
| $61$ |
\( T^{5} + \cdots + 67775166628112 \)
|
| $67$ |
\( T^{5} + \cdots - 5123600366224 \)
|
| $71$ |
\( T^{5} + \cdots + 246364319624144 \)
|
| $73$ |
\( T^{5} + \cdots - 715542976576 \)
|
| $79$ |
\( T^{5} + \cdots + 694623214400 \)
|
| $83$ |
\( T^{5} + \cdots + 60161708395008 \)
|
| $89$ |
\( T^{5} + \cdots - 40747061333440 \)
|
| $97$ |
\( T^{5} + \cdots + 10878747957581 \)
|
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