Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1694,4,Mod(1,1694)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, 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("1694.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1694 = 2 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1694.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: | \(99.9492355497\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.7636.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 16x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 154) |
| 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
\( x^{3} - 16x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 4\nu - 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 21 ) / 2 \)
|
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 |
|
−2.00000 | −6.94933 | 4.00000 | 7.85338 | 13.8987 | 7.00000 | −8.00000 | 21.2932 | −15.7068 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 4.49172 | 4.00000 | 21.9122 | −8.98345 | 7.00000 | −8.00000 | −6.82442 | −43.8244 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 8.45761 | 4.00000 | −3.76558 | −16.9152 | 7.00000 | −8.00000 | 44.5312 | 7.53117 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1694.4.a.s | 3 | |
| 11.b | odd | 2 | 1 | 154.4.a.i | ✓ | 3 | |
| 33.d | even | 2 | 1 | 1386.4.a.bc | 3 | ||
| 44.c | even | 2 | 1 | 1232.4.a.q | 3 | ||
| 77.b | even | 2 | 1 | 1078.4.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.a.i | ✓ | 3 | 11.b | odd | 2 | 1 | |
| 1078.4.a.p | 3 | 77.b | even | 2 | 1 | ||
| 1232.4.a.q | 3 | 44.c | even | 2 | 1 | ||
| 1386.4.a.bc | 3 | 33.d | even | 2 | 1 | ||
| 1694.4.a.s | 3 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1694))\):
|
\( T_{3}^{3} - 6T_{3}^{2} - 52T_{3} + 264 \)
|
|
\( T_{5}^{3} - 26T_{5}^{2} + 60T_{5} + 648 \)
|
|
\( T_{13}^{3} + 72T_{13}^{2} - 4624T_{13} - 331632 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} - 6 T^{2} + \cdots + 264 \)
|
| $5$ |
\( T^{3} - 26 T^{2} + \cdots + 648 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + 72 T^{2} + \cdots - 331632 \)
|
| $17$ |
\( T^{3} + 56 T^{2} + \cdots - 88976 \)
|
| $19$ |
\( T^{3} - 48 T^{2} + \cdots - 28512 \)
|
| $23$ |
\( T^{3} + 244 T^{2} + \cdots + 219584 \)
|
| $29$ |
\( T^{3} + 198 T^{2} + \cdots - 3523896 \)
|
| $31$ |
\( T^{3} - 374 T^{2} + \cdots + 15721176 \)
|
| $37$ |
\( T^{3} - 110 T^{2} + \cdots + 5981784 \)
|
| $41$ |
\( T^{3} - 648 T^{2} + \cdots + 28837872 \)
|
| $43$ |
\( T^{3} - 500 T^{2} + \cdots + 2503232 \)
|
| $47$ |
\( T^{3} + 326 T^{2} + \cdots - 136297368 \)
|
| $53$ |
\( T^{3} - 206 T^{2} + \cdots + 863064 \)
|
| $59$ |
\( T^{3} + 266 T^{2} + \cdots - 4809816 \)
|
| $61$ |
\( T^{3} + 308 T^{2} + \cdots - 81057168 \)
|
| $67$ |
\( T^{3} - 256 T^{2} + \cdots - 1711488 \)
|
| $71$ |
\( T^{3} + 484 T^{2} + \cdots - 19953216 \)
|
| $73$ |
\( T^{3} - 1028 T^{2} + \cdots + 550878192 \)
|
| $79$ |
\( T^{3} - 1072 T^{2} + \cdots + 148401792 \)
|
| $83$ |
\( T^{3} + 912 T^{2} + \cdots - 33774048 \)
|
| $89$ |
\( T^{3} + \cdots - 1452062376 \)
|
| $97$ |
\( T^{3} - 2042 T^{2} + \cdots - 259709688 \)
|
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