Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [230,4,Mod(1,230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(230, 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("230.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 230 = 2 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 230.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: | \(13.5704393013\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 84x^{2} - 11x + 1242 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 84x^{2} - 11x + 1242 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 50\nu - 96 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 13\nu^{2} + 50\nu - 534 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 13\beta_{2} + 50\beta _1 + 12 \)
|
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 |
|
2.00000 | −7.16920 | 4.00000 | −5.00000 | −14.3384 | −25.3945 | 8.00000 | 24.3974 | −10.0000 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.26018 | 4.00000 | −5.00000 | −6.52037 | 27.7921 | 8.00000 | −16.3712 | −10.0000 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 5.50148 | 4.00000 | −5.00000 | 11.0030 | 0.0500526 | 8.00000 | 3.26627 | −10.0000 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 8.92791 | 4.00000 | −5.00000 | 17.8558 | 23.5524 | 8.00000 | 52.7075 | −10.0000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 230.4.a.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2070.4.a.bi | 4 | ||
| 4.b | odd | 2 | 1 | 1840.4.a.l | 4 | ||
| 5.b | even | 2 | 1 | 1150.4.a.o | 4 | ||
| 5.c | odd | 4 | 2 | 1150.4.b.m | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.4.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1150.4.a.o | 4 | 5.b | even | 2 | 1 | ||
| 1150.4.b.m | 8 | 5.c | odd | 4 | 2 | ||
| 1840.4.a.l | 4 | 4.b | odd | 2 | 1 | ||
| 2070.4.a.bi | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} - 78T_{3}^{2} + 175T_{3} + 1148 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 1148 \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( T^{4} - 26 T^{3} + \cdots - 832 \)
|
| $11$ |
\( T^{4} - 93 T^{3} + \cdots - 41700 \)
|
| $13$ |
\( T^{4} - 32 T^{3} + \cdots + 682562 \)
|
| $17$ |
\( T^{4} - 108 T^{3} + \cdots + 96120 \)
|
| $19$ |
\( T^{4} - 185 T^{3} + \cdots + 1404820 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} + \cdots - 1735042500 \)
|
| $31$ |
\( T^{4} + 211 T^{3} + \cdots - 311259365 \)
|
| $37$ |
\( T^{4} - 5 T^{3} + \cdots - 14187104 \)
|
| $41$ |
\( T^{4} + 369 T^{3} + \cdots - 114199911 \)
|
| $43$ |
\( T^{4} + 100 T^{3} + \cdots + 93158400 \)
|
| $47$ |
\( T^{4} + \cdots - 9732540576 \)
|
| $53$ |
\( T^{4} + \cdots - 8104397736 \)
|
| $59$ |
\( T^{4} + 33 T^{3} + \cdots - 623752800 \)
|
| $61$ |
\( T^{4} + \cdots - 5277943032 \)
|
| $67$ |
\( T^{4} + \cdots - 49731103520 \)
|
| $71$ |
\( T^{4} + \cdots - 3703975749 \)
|
| $73$ |
\( T^{4} + \cdots + 80640087688 \)
|
| $79$ |
\( T^{4} + \cdots - 498954065920 \)
|
| $83$ |
\( T^{4} + \cdots - 424764340752 \)
|
| $89$ |
\( T^{4} + \cdots + 92383027200 \)
|
| $97$ |
\( T^{4} + \cdots + 734679597128 \)
|
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