Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,4,Mod(1,1014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, 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("1014.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.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: | \(59.8279367458\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.30907152.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 347x^{2} - 444x + 13156 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 78) |
| 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
\( x^{4} - 347x^{2} - 444x + 13156 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 74\nu^{2} + 175\nu - 12506 ) / 5304 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 37\nu^{3} - 86\nu^{2} - 9127\nu + 1274 ) / 2652 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 74\beta_{2} + \beta _1 + 174 \)
|
| \(\nu^{3}\) | \(=\) |
\( 74\beta_{3} + 172\beta_{2} + 249\beta _1 + 370 \)
|
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 | −3.00000 | 4.00000 | −19.6772 | −6.00000 | 4.31596 | 8.00000 | 9.00000 | −39.3544 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | −0.346567 | −6.00000 | −22.8819 | 8.00000 | 9.00000 | −0.693134 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | 13.0068 | −6.00000 | 26.9537 | 8.00000 | 9.00000 | 26.0136 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −3.00000 | 4.00000 | 15.0170 | −6.00000 | 13.6122 | 8.00000 | 9.00000 | 30.0339 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1014.4.a.ba | 4 | |
| 13.b | even | 2 | 1 | 1014.4.a.z | 4 | ||
| 13.d | odd | 4 | 2 | 1014.4.b.o | 8 | ||
| 13.f | odd | 12 | 2 | 78.4.i.b | ✓ | 8 | |
| 39.k | even | 12 | 2 | 234.4.l.c | 8 | ||
| 52.l | even | 12 | 2 | 624.4.bv.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.4.i.b | ✓ | 8 | 13.f | odd | 12 | 2 | |
| 234.4.l.c | 8 | 39.k | even | 12 | 2 | ||
| 624.4.bv.g | 8 | 52.l | even | 12 | 2 | ||
| 1014.4.a.z | 4 | 13.b | even | 2 | 1 | ||
| 1014.4.a.ba | 4 | 1.a | even | 1 | 1 | trivial | |
| 1014.4.b.o | 8 | 13.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1014))\):
|
\( T_{5}^{4} - 8T_{5}^{3} - 359T_{5}^{2} + 3720T_{5} + 1332 \)
|
|
\( T_{7}^{4} - 22T_{7}^{3} - 485T_{7}^{2} + 10818T_{7} - 36234 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} - 8 T^{3} + \cdots + 1332 \)
|
| $7$ |
\( T^{4} - 22 T^{3} + \cdots - 36234 \)
|
| $11$ |
\( T^{4} + 18 T^{3} + \cdots + 38232 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 76 T^{3} + \cdots + 14773968 \)
|
| $19$ |
\( T^{4} - 102 T^{3} + \cdots - 7098696 \)
|
| $23$ |
\( T^{4} - 182 T^{3} + \cdots - 70163496 \)
|
| $29$ |
\( T^{4} - 38 T^{3} + \cdots - 209109708 \)
|
| $31$ |
\( T^{4} - 188 T^{3} + \cdots - 385430376 \)
|
| $37$ |
\( T^{4} + \cdots - 1504814016 \)
|
| $41$ |
\( T^{4} + \cdots + 1031797008 \)
|
| $43$ |
\( T^{4} + \cdots - 3095063114 \)
|
| $47$ |
\( T^{4} + 942 T^{3} + \cdots + 211537656 \)
|
| $53$ |
\( T^{4} + \cdots - 6144065136 \)
|
| $59$ |
\( T^{4} + \cdots - 28774232064 \)
|
| $61$ |
\( T^{4} + \cdots + 18296406997 \)
|
| $67$ |
\( T^{4} + \cdots + 21861095334 \)
|
| $71$ |
\( T^{4} + \cdots - 194905126152 \)
|
| $73$ |
\( T^{4} + \cdots + 103967382129 \)
|
| $79$ |
\( T^{4} + \cdots + 368623918104 \)
|
| $83$ |
\( T^{4} + \cdots + 11436677856 \)
|
| $89$ |
\( T^{4} + \cdots + 13032734976 \)
|
| $97$ |
\( T^{4} - 972 T^{3} + \cdots + 707697972 \)
|
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