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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2292.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 13x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 105) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 13x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 9 \)
|
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 |
|
−4.10645 | 3.00000 | 8.86293 | 0 | −12.3193 | 7.00000 | −3.54358 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | −0.0765073 | 3.00000 | −7.99415 | 0 | −0.229522 | 7.00000 | 1.22367 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 3.18296 | 3.00000 | 2.13122 | 0 | 9.54887 | 7.00000 | −18.6801 | 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.q | 3 | |
| 3.b | odd | 2 | 1 | 1575.4.a.bd | 3 | ||
| 5.b | even | 2 | 1 | 525.4.a.r | 3 | ||
| 5.c | odd | 4 | 2 | 105.4.d.a | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1575.4.a.bc | 3 | ||
| 15.e | even | 4 | 2 | 315.4.d.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.d.a | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 315.4.d.a | 6 | 15.e | even | 4 | 2 | ||
| 525.4.a.q | 3 | 1.a | even | 1 | 1 | trivial | |
| 525.4.a.r | 3 | 5.b | even | 2 | 1 | ||
| 1575.4.a.bc | 3 | 15.d | odd | 2 | 1 | ||
| 1575.4.a.bd | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):
|
\( T_{2}^{3} + T_{2}^{2} - 13T_{2} - 1 \)
|
|
\( T_{11}^{3} + 66T_{11}^{2} - 276T_{11} - 6120 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 13T - 1 \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} + 66 T^{2} + \cdots - 6120 \)
|
| $13$ |
\( T^{3} + 102 T^{2} + \cdots + 28696 \)
|
| $17$ |
\( T^{3} + 152 T^{2} + \cdots + 68272 \)
|
| $19$ |
\( T^{3} + 138 T^{2} + \cdots + 70632 \)
|
| $23$ |
\( T^{3} - 180 T^{2} + \cdots + 228816 \)
|
| $29$ |
\( T^{3} - 170 T^{2} + \cdots + 1680584 \)
|
| $31$ |
\( T^{3} + 366 T^{2} + \cdots - 3510632 \)
|
| $37$ |
\( T^{3} + 252 T^{2} + \cdots - 8221888 \)
|
| $41$ |
\( T^{3} + 206 T^{2} + \cdots - 18222200 \)
|
| $43$ |
\( T^{3} - 108 T^{2} + \cdots - 13701184 \)
|
| $47$ |
\( T^{3} - 24 T^{2} + \cdots - 20893824 \)
|
| $53$ |
\( T^{3} + 354 T^{2} + \cdots - 53538840 \)
|
| $59$ |
\( T^{3} + 880 T^{2} + \cdots - 22878976 \)
|
| $61$ |
\( T^{3} + 870 T^{2} + \cdots - 112468792 \)
|
| $67$ |
\( T^{3} - 96 T^{2} + \cdots + 35189888 \)
|
| $71$ |
\( T^{3} + 1018 T^{2} + \cdots - 133243912 \)
|
| $73$ |
\( T^{3} + 1554 T^{2} + \cdots - 199646136 \)
|
| $79$ |
\( T^{3} + 1620 T^{2} + \cdots - 258624448 \)
|
| $83$ |
\( T^{3} - 872 T^{2} + \cdots + 688370432 \)
|
| $89$ |
\( T^{3} + 1938 T^{2} + \cdots + 181473048 \)
|
| $97$ |
\( T^{3} + 1878 T^{2} + \cdots - 140508392 \)
|
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