Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,4,Mod(1,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, 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("693.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 693.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: | \(40.8883236340\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 231) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 30\nu^{2} - \nu + 82 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 28\nu^{2} - 45\nu - 58 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + \beta_{2} + 23\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{3} + 30\beta_{2} + \beta _1 + 278 \)
|
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 |
|
−5.02173 | 0 | 17.2178 | −20.4371 | 0 | −7.00000 | −46.2894 | 0 | 102.630 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.50528 | 0 | −5.73413 | 0.155265 | 0 | −7.00000 | 20.6737 | 0 | −0.233718 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.183399 | 0 | −7.96636 | 17.9271 | 0 | −7.00000 | 2.92821 | 0 | −3.28780 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.48454 | 0 | −1.82705 | −13.9229 | 0 | −7.00000 | −24.4157 | 0 | −34.5919 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.22587 | 0 | 19.3097 | 9.27763 | 0 | −7.00000 | 59.1032 | 0 | 48.4837 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 693.4.a.q | 5 | |
| 3.b | odd | 2 | 1 | 231.4.a.j | ✓ | 5 | |
| 21.c | even | 2 | 1 | 1617.4.a.o | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.4.a.j | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 693.4.a.q | 5 | 1.a | even | 1 | 1 | trivial | |
| 1617.4.a.o | 5 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\):
|
\( T_{2}^{5} - T_{2}^{4} - 30T_{2}^{3} + 21T_{2}^{2} + 103T_{2} + 18 \)
|
|
\( T_{5}^{5} + 7T_{5}^{4} - 485T_{5}^{3} - 1951T_{5}^{2} + 47640T_{5} - 7348 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} + \cdots + 18 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 7 T^{4} + \cdots - 7348 \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} - 23 T^{4} + \cdots + 122236 \)
|
| $17$ |
\( T^{5} - 102 T^{4} + \cdots + 159061888 \)
|
| $19$ |
\( T^{5} + \cdots + 1688336424 \)
|
| $23$ |
\( T^{5} + \cdots + 2862756736 \)
|
| $29$ |
\( T^{5} + \cdots - 144277546716 \)
|
| $31$ |
\( T^{5} + \cdots - 131921372032 \)
|
| $37$ |
\( T^{5} + \cdots - 244887003636 \)
|
| $41$ |
\( T^{5} + \cdots + 1412698624 \)
|
| $43$ |
\( T^{5} + \cdots - 7075944192 \)
|
| $47$ |
\( T^{5} + \cdots - 384442814216 \)
|
| $53$ |
\( T^{5} + \cdots - 299252301984 \)
|
| $59$ |
\( T^{5} + \cdots + 896870843568 \)
|
| $61$ |
\( T^{5} + \cdots - 10265429183328 \)
|
| $67$ |
\( T^{5} + \cdots + 1110831992688 \)
|
| $71$ |
\( T^{5} + \cdots - 27412560709632 \)
|
| $73$ |
\( T^{5} + \cdots - 25877686218916 \)
|
| $79$ |
\( T^{5} + \cdots - 264536335362048 \)
|
| $83$ |
\( T^{5} + \cdots - 11601305339904 \)
|
| $89$ |
\( T^{5} + \cdots - 7305960217248 \)
|
| $97$ |
\( T^{5} + \cdots + 2636714723104 \)
|
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