Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,4,Mod(1,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("684.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 684.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: | \(40.3573064439\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.35529.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 52x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 52x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 34 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 4\beta_{2} + 3\beta _1 + 34 \)
|
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 |
|
0 | 0 | 0 | −11.0323 | 0 | 5.70454 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −3.52382 | 0 | 6.26000 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 5.55614 | 0 | 32.0355 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 684.4.a.i | 3 | |
| 3.b | odd | 2 | 1 | 76.4.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 304.4.a.h | 3 | ||
| 15.d | odd | 2 | 1 | 1900.4.a.c | 3 | ||
| 15.e | even | 4 | 2 | 1900.4.c.c | 6 | ||
| 24.f | even | 2 | 1 | 1216.4.a.v | 3 | ||
| 24.h | odd | 2 | 1 | 1216.4.a.t | 3 | ||
| 57.d | even | 2 | 1 | 1444.4.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.4.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 304.4.a.h | 3 | 12.b | even | 2 | 1 | ||
| 684.4.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
| 1216.4.a.t | 3 | 24.h | odd | 2 | 1 | ||
| 1216.4.a.v | 3 | 24.f | even | 2 | 1 | ||
| 1444.4.a.e | 3 | 57.d | even | 2 | 1 | ||
| 1900.4.a.c | 3 | 15.d | odd | 2 | 1 | ||
| 1900.4.c.c | 6 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 9T_{5}^{2} - 42T_{5} - 216 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(684))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 9 T^{2} + \cdots - 216 \)
|
| $7$ |
\( T^{3} - 44 T^{2} + \cdots - 1144 \)
|
| $11$ |
\( T^{3} + 79 T^{2} + \cdots - 108888 \)
|
| $13$ |
\( T^{3} + 11 T^{2} + \cdots - 201816 \)
|
| $17$ |
\( T^{3} + 82 T^{2} + \cdots - 1022082 \)
|
| $19$ |
\( (T - 19)^{3} \)
|
| $23$ |
\( T^{3} + 103 T^{2} + \cdots - 235392 \)
|
| $29$ |
\( T^{3} - 93 T^{2} + \cdots + 2252268 \)
|
| $31$ |
\( T^{3} + 116 T^{2} + \cdots + 1084416 \)
|
| $37$ |
\( T^{3} + 466 T^{2} + \cdots - 15144328 \)
|
| $41$ |
\( T^{3} - 188 T^{2} + \cdots + 5041152 \)
|
| $43$ |
\( T^{3} + 11 T^{2} + \cdots + 2359248 \)
|
| $47$ |
\( T^{3} + 163 T^{2} + \cdots + 3850752 \)
|
| $53$ |
\( T^{3} + 197 T^{2} + \cdots + 11566104 \)
|
| $59$ |
\( T^{3} + 1381 T^{2} + \cdots + 52865928 \)
|
| $61$ |
\( T^{3} + 405 T^{2} + \cdots - 847124 \)
|
| $67$ |
\( T^{3} - 943 T^{2} + \cdots + 1169904 \)
|
| $71$ |
\( T^{3} + 1052 T^{2} + \cdots - 521792928 \)
|
| $73$ |
\( T^{3} + 580 T^{2} + \cdots - 726527962 \)
|
| $79$ |
\( T^{3} + 1402 T^{2} + \cdots + 18139904 \)
|
| $83$ |
\( T^{3} + 1802 T^{2} + \cdots + 91984992 \)
|
| $89$ |
\( T^{3} + 1966 T^{2} + \cdots + 47198784 \)
|
| $97$ |
\( T^{3} - 48 T^{2} + \cdots - 82010336 \)
|
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