Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,4,Mod(1,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, 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("525.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.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: | \(30.9760027530\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.26729725.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 18x^{2} + 19x + 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 2x^{3} - 18x^{2} + 19x + 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 14\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 15\beta _1 + 8 \)
|
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 |
|
−2.56826 | 3.00000 | −1.40404 | 0 | −7.70478 | −7.00000 | 24.1520 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.217342 | 3.00000 | −7.95276 | 0 | −0.652027 | −7.00000 | 3.46721 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 3.21734 | 3.00000 | 2.35129 | 0 | 9.65203 | −7.00000 | −18.1738 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 5.56826 | 3.00000 | 23.0055 | 0 | 16.7048 | −7.00000 | 83.5546 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 525.4.a.v | yes | 4 |
| 3.b | odd | 2 | 1 | 1575.4.a.bf | 4 | ||
| 5.b | even | 2 | 1 | 525.4.a.s | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 525.4.d.o | 8 | ||
| 15.d | odd | 2 | 1 | 1575.4.a.bm | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.4.a.s | ✓ | 4 | 5.b | even | 2 | 1 | |
| 525.4.a.v | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 525.4.d.o | 8 | 5.c | odd | 4 | 2 | ||
| 1575.4.a.bf | 4 | 3.b | odd | 2 | 1 | ||
| 1575.4.a.bm | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):
|
\( T_{2}^{4} - 6T_{2}^{3} - 6T_{2}^{2} + 45T_{2} + 10 \)
|
|
\( T_{11}^{4} - 57T_{11}^{3} - 1323T_{11}^{2} + 44829T_{11} + 99334 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{3} + \cdots + 10 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 57 T^{3} + \cdots + 99334 \)
|
| $13$ |
\( T^{4} - 43 T^{3} + \cdots + 3240760 \)
|
| $17$ |
\( T^{4} - 99 T^{3} + \cdots + 3462544 \)
|
| $19$ |
\( T^{4} + 12 T^{3} + \cdots + 329386624 \)
|
| $23$ |
\( T^{4} - 156 T^{3} + \cdots + 92278225 \)
|
| $29$ |
\( T^{4} - 378 T^{3} + \cdots + 29419291 \)
|
| $31$ |
\( T^{4} + \cdots + 1187132760 \)
|
| $37$ |
\( T^{4} - 81 T^{3} + \cdots + 210866814 \)
|
| $41$ |
\( T^{4} + \cdots - 7122182144 \)
|
| $43$ |
\( T^{4} + \cdots + 7838190085 \)
|
| $47$ |
\( T^{4} - 744 T^{3} + \cdots + 425590240 \)
|
| $53$ |
\( T^{4} + \cdots + 8629646344 \)
|
| $59$ |
\( T^{4} - 231 T^{3} + \cdots + 323902840 \)
|
| $61$ |
\( T^{4} + \cdots - 9759896064 \)
|
| $67$ |
\( T^{4} + \cdots - 109414277684 \)
|
| $71$ |
\( T^{4} + \cdots - 423829861100 \)
|
| $73$ |
\( T^{4} + \cdots + 67091874784 \)
|
| $79$ |
\( T^{4} + \cdots + 28849899226 \)
|
| $83$ |
\( T^{4} + \cdots - 22647834344 \)
|
| $89$ |
\( T^{4} + \cdots - 108748692656 \)
|
| $97$ |
\( T^{4} + \cdots - 5353888496 \)
|
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