Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3675,2,Mod(1,3675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3675, 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("3675.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3675.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: | \(29.3450227428\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.88404.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 7x^{2} + 5x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 525) |
| 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
\( x^{4} - x^{3} - 7x^{2} + 5x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
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 |
|
−2.43736 | 1.00000 | 3.94072 | 0 | −2.43736 | 0 | −4.73024 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −0.494173 | 1.00000 | −1.75579 | 0 | −0.494173 | 0 | 1.85601 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.22868 | 1.00000 | −0.490347 | 0 | 1.22868 | 0 | −3.05984 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.70285 | 1.00000 | 5.30542 | 0 | 2.70285 | 0 | 8.93406 | 1.00000 | 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 | 3675.2.a.bx | 4 | |
| 5.b | even | 2 | 1 | 3675.2.a.bq | 4 | ||
| 7.b | odd | 2 | 1 | 3675.2.a.bw | 4 | ||
| 7.d | odd | 6 | 2 | 525.2.i.i | ✓ | 8 | |
| 35.c | odd | 2 | 1 | 3675.2.a.br | 4 | ||
| 35.i | odd | 6 | 2 | 525.2.i.j | yes | 8 | |
| 35.k | even | 12 | 4 | 525.2.r.h | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.2.i.i | ✓ | 8 | 7.d | odd | 6 | 2 | |
| 525.2.i.j | yes | 8 | 35.i | odd | 6 | 2 | |
| 525.2.r.h | 16 | 35.k | even | 12 | 4 | ||
| 3675.2.a.bq | 4 | 5.b | even | 2 | 1 | ||
| 3675.2.a.br | 4 | 35.c | odd | 2 | 1 | ||
| 3675.2.a.bw | 4 | 7.b | odd | 2 | 1 | ||
| 3675.2.a.bx | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):
|
\( T_{2}^{4} - T_{2}^{3} - 7T_{2}^{2} + 5T_{2} + 4 \)
|
|
\( T_{11}^{4} - 8T_{11}^{3} - 8T_{11}^{2} + 180T_{11} - 328 \)
|
|
\( T_{13}^{4} - 7T_{13}^{3} - 19T_{13}^{2} + 179T_{13} - 194 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} - 7 T^{2} + \cdots + 4 \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 8 T^{3} + \cdots - 328 \)
|
| $13$ |
\( T^{4} - 7 T^{3} + \cdots - 194 \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 40 \)
|
| $19$ |
\( T^{4} + 3 T^{3} + \cdots - 196 \)
|
| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 960 \)
|
| $29$ |
\( (T - 4)^{4} \)
|
| $31$ |
\( T^{4} + 9 T^{3} + \cdots - 768 \)
|
| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 773 \)
|
| $41$ |
\( T^{4} + 4 T^{3} + \cdots - 200 \)
|
| $43$ |
\( T^{4} - 5 T^{3} + \cdots + 160 \)
|
| $47$ |
\( T^{4} + 6 T^{3} + \cdots - 856 \)
|
| $53$ |
\( T^{4} + 6 T^{3} + \cdots + 1112 \)
|
| $59$ |
\( T^{4} + 10 T^{3} + \cdots - 8 \)
|
| $61$ |
\( (T - 5)^{4} \)
|
| $67$ |
\( T^{4} - T^{3} + \cdots + 60 \)
|
| $71$ |
\( T^{4} - 22 T^{3} + \cdots - 12736 \)
|
| $73$ |
\( T^{4} + 4 T^{3} + \cdots - 1567 \)
|
| $79$ |
\( T^{4} - 8 T^{3} + \cdots + 153 \)
|
| $83$ |
\( T^{4} + 2 T^{3} + \cdots - 312 \)
|
| $89$ |
\( T^{4} - 10 T^{3} + \cdots + 24 \)
|
| $97$ |
\( T^{4} - 12 T^{3} + \cdots - 79 \)
|
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