Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [675,4,Mod(1,675)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("675.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 675 = 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 675.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: | \(39.8262892539\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.183945.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} + 3x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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
\( x^{4} - x^{3} - 12x^{2} + 3x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 6\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 7\nu^{2} + 6\nu - 39 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} + 2\nu^{2} + 24\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 6\beta _1 + 4 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 13\beta _1 + 17 ) / 2 \)
|
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 |
|
−4.72651 | 0 | 14.3399 | 0 | 0 | 17.6256 | −29.9655 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.33012 | 0 | −6.23078 | 0 | 0 | −10.6979 | 18.9286 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 2.53008 | 0 | −1.59867 | 0 | 0 | 27.8841 | −24.2855 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 4.52654 | 0 | 12.4896 | 0 | 0 | −30.8119 | 20.3223 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 675.4.a.z | yes | 4 |
| 3.b | odd | 2 | 1 | 675.4.a.v | yes | 4 | |
| 5.b | even | 2 | 1 | 675.4.a.u | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 675.4.b.p | 8 | ||
| 15.d | odd | 2 | 1 | 675.4.a.y | yes | 4 | |
| 15.e | even | 4 | 2 | 675.4.b.q | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 675.4.a.u | ✓ | 4 | 5.b | even | 2 | 1 | |
| 675.4.a.v | yes | 4 | 3.b | odd | 2 | 1 | |
| 675.4.a.y | yes | 4 | 15.d | odd | 2 | 1 | |
| 675.4.a.z | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 675.4.b.p | 8 | 5.c | odd | 4 | 2 | ||
| 675.4.b.q | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(675))\):
|
\( T_{2}^{4} - T_{2}^{3} - 25T_{2}^{2} + 25T_{2} + 72 \)
|
|
\( T_{7}^{4} - 4T_{7}^{3} - 1068T_{7}^{2} + 5400T_{7} + 162000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 72 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 162000 \)
|
| $11$ |
\( T^{4} + 52 T^{3} + \cdots + 93825 \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots + 126275 \)
|
| $17$ |
\( T^{4} - 64 T^{3} + \cdots + 23263056 \)
|
| $19$ |
\( T^{4} + 46 T^{3} + \cdots - 23014672 \)
|
| $23$ |
\( T^{4} + 90 T^{3} + \cdots + 261711 \)
|
| $29$ |
\( T^{4} + 470 T^{3} + \cdots - 776494800 \)
|
| $31$ |
\( T^{4} + 262 T^{3} + \cdots + 254274336 \)
|
| $37$ |
\( T^{4} + \cdots - 4695491525 \)
|
| $41$ |
\( T^{4} + \cdots + 1081900800 \)
|
| $43$ |
\( T^{4} + \cdots + 4052243600 \)
|
| $47$ |
\( T^{4} + \cdots - 3790497105 \)
|
| $53$ |
\( T^{4} - 910 T^{3} + \cdots + 339474384 \)
|
| $59$ |
\( T^{4} + \cdots + 39525145425 \)
|
| $61$ |
\( T^{4} + \cdots + 13956747407 \)
|
| $67$ |
\( T^{4} + \cdots + 28371870000 \)
|
| $71$ |
\( T^{4} + \cdots - 83536909425 \)
|
| $73$ |
\( T^{4} + \cdots + 16967166800 \)
|
| $79$ |
\( T^{4} + \cdots + 3926930544 \)
|
| $83$ |
\( T^{4} + \cdots - 975383057664 \)
|
| $89$ |
\( T^{4} + \cdots + 671702058000 \)
|
| $97$ |
\( T^{4} + \cdots - 14510701775 \)
|
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