Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3525,2,Mod(1,3525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3525, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3525.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3525 = 3 \cdot 5^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3525.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: | \(28.1472667125\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 9x^{5} + 6x^{4} + 20x^{3} - 9x^{2} - 12x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 9x^{5} + 6x^{4} + 20x^{3} - 9x^{2} - 12x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - \nu^{5} + 7\nu^{4} + 13\nu^{3} + \nu^{2} - 24\nu - 16 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 12\nu^{4} - 8\nu^{3} + 34\nu^{2} + 9\nu - 19 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{6} - 3\nu^{5} - 14\nu^{4} + 19\nu^{3} + 13\nu^{2} - 27\nu + 2 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} + 2\nu^{5} - 19\nu^{4} - 21\nu^{3} + 38\nu^{2} + 28\nu - 18 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + 2\nu^{5} + 26\nu^{4} - 6\nu^{3} - 52\nu^{2} - 2\nu + 22 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 8\beta_{5} + \beta_{4} - 8\beta_{3} + 7\beta_{2} + 9\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{6} + 12\beta_{5} + 8\beta_{4} - 3\beta_{3} + 10\beta_{2} + 33\beta _1 + 21 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{6} + 58\beta_{5} + 12\beta_{4} - 54\beta_{3} + 48\beta_{2} + 72\beta _1 + 104 \)
|
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 |
|
−2.74386 | 1.00000 | 5.52877 | 0 | −2.74386 | 1.02926 | −9.68245 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.49858 | 1.00000 | 0.245756 | 0 | −1.49858 | −3.73400 | 2.62888 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.10030 | 1.00000 | −0.789350 | 0 | −1.10030 | −1.25231 | 3.06911 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.231416 | 1.00000 | −1.94645 | 0 | −0.231416 | −4.71752 | 0.913273 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.913267 | 1.00000 | −1.16594 | 0 | 0.913267 | 2.03699 | −2.89135 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.36884 | 1.00000 | −0.126278 | 0 | 1.36884 | 0.111162 | −2.91053 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.29205 | 1.00000 | 3.25349 | 0 | 2.29205 | −4.47358 | 2.87307 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(47\) | \(-1\) |
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 | 3525.2.a.y | ✓ | 7 |
| 5.b | even | 2 | 1 | 3525.2.a.bb | yes | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3525.2.a.y | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 3525.2.a.bb | yes | 7 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3525))\):
|
\( T_{2}^{7} + T_{2}^{6} - 9T_{2}^{5} - 6T_{2}^{4} + 20T_{2}^{3} + 9T_{2}^{2} - 12T_{2} - 3 \)
|
|
\( T_{7}^{7} + 11T_{7}^{6} + 29T_{7}^{5} - 45T_{7}^{4} - 201T_{7}^{3} + 31T_{7}^{2} + 206T_{7} - 23 \)
|
|
\( T_{11}^{7} + 8T_{11}^{6} - 3T_{11}^{5} - 152T_{11}^{4} - 365T_{11}^{3} - 216T_{11}^{2} + 21T_{11} + 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} - 9 T^{5} + \cdots - 3 \)
|
| $3$ |
\( (T - 1)^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} + 11 T^{6} + \cdots - 23 \)
|
| $11$ |
\( T^{7} + 8 T^{6} + \cdots + 5 \)
|
| $13$ |
\( T^{7} + 5 T^{6} + \cdots - 2783 \)
|
| $17$ |
\( T^{7} + 10 T^{6} + \cdots + 17567 \)
|
| $19$ |
\( T^{7} - 7 T^{6} + \cdots - 15 \)
|
| $23$ |
\( T^{7} + 4 T^{6} + \cdots - 121 \)
|
| $29$ |
\( T^{7} + 11 T^{6} + \cdots + 825 \)
|
| $31$ |
\( T^{7} - 3 T^{6} + \cdots + 43059 \)
|
| $37$ |
\( T^{7} + 11 T^{6} + \cdots - 30047 \)
|
| $41$ |
\( T^{7} + 20 T^{6} + \cdots + 135321 \)
|
| $43$ |
\( T^{7} + 18 T^{6} + \cdots + 2321 \)
|
| $47$ |
\( (T - 1)^{7} \)
|
| $53$ |
\( T^{7} + 12 T^{6} + \cdots - 290305 \)
|
| $59$ |
\( T^{7} - 18 T^{6} + \cdots + 782875 \)
|
| $61$ |
\( T^{7} + 4 T^{6} + \cdots - 483289 \)
|
| $67$ |
\( T^{7} + 22 T^{6} + \cdots + 272539 \)
|
| $71$ |
\( T^{7} + 14 T^{6} + \cdots + 972757 \)
|
| $73$ |
\( T^{7} + 30 T^{6} + \cdots + 104159 \)
|
| $79$ |
\( T^{7} + T^{6} + \cdots + 57725 \)
|
| $83$ |
\( T^{7} + 54 T^{6} + \cdots - 896587 \)
|
| $89$ |
\( T^{7} + 14 T^{6} + \cdots - 121325 \)
|
| $97$ |
\( T^{7} + 24 T^{6} + \cdots - 564059 \)
|
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