Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("693.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(111.145987130\) |
| 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: |
\( x^{5} - x^{4} - 71x^{3} - 101x^{2} + 269x + 427 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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
\( x^{5} - x^{4} - 71x^{3} - 101x^{2} + 269x + 427 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{4} - 11\nu^{3} - 264\nu^{2} + 64\nu + 919 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{4} + 11\nu^{3} + 264\nu^{2} - 52\nu - 922 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{4} + 20\nu^{3} + 459\nu^{2} - 142\nu - 1594 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8\nu^{4} + 23\nu^{3} + 526\nu^{2} - 178\nu - 1875 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{4} - 8\beta_{3} + 5\beta_{2} + 6\beta _1 + 117 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{4} - 4\beta_{3} + 15\beta_{2} + 40\beta _1 + 108 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 308\beta_{4} - 572\beta_{3} + 479\beta_{2} + 810\beta _1 + 7975 ) / 4 \)
|
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 |
|
−10.2849 | 0 | 73.7791 | −84.1631 | 0 | −49.0000 | −429.694 | 0 | 865.608 | |||||||||||||||||||||||||||||||||
| 1.2 | −9.08702 | 0 | 50.5739 | −6.85516 | 0 | −49.0000 | −168.781 | 0 | 62.2930 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.85420 | 0 | −23.8535 | −2.49184 | 0 | −49.0000 | 159.417 | 0 | 7.11222 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.93798 | 0 | −7.61637 | 56.6939 | 0 | −49.0000 | −195.625 | 0 | 279.953 | ||||||||||||||||||||||||||||||||||
| 1.5 | 7.28814 | 0 | 21.1169 | −82.1838 | 0 | −49.0000 | −79.3173 | 0 | −598.967 | ||||||||||||||||||||||||||||||||||
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 | 693.6.a.e | 5 | |
| 3.b | odd | 2 | 1 | 77.6.a.c | ✓ | 5 | |
| 21.c | even | 2 | 1 | 539.6.a.g | 5 | ||
| 33.d | even | 2 | 1 | 847.6.a.e | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.6.a.c | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 539.6.a.g | 5 | 21.c | even | 2 | 1 | ||
| 693.6.a.e | 5 | 1.a | even | 1 | 1 | trivial | |
| 847.6.a.e | 5 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(693))\):
|
\( T_{2}^{5} + 10T_{2}^{4} - 87T_{2}^{3} - 752T_{2}^{2} + 2092T_{2} + 9600 \)
|
|
\( T_{5}^{5} + 119T_{5}^{4} - 1472T_{5}^{3} - 413768T_{5}^{2} - 3708304T_{5} - 6698576 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 10 T^{4} + \cdots + 9600 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 119 T^{4} + \cdots - 6698576 \)
|
| $7$ |
\( (T + 49)^{5} \)
|
| $11$ |
\( (T + 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 15026837904640 \)
|
| $17$ |
\( T^{5} + \cdots + 78795399796096 \)
|
| $19$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots + 495447588607232 \)
|
| $29$ |
\( T^{5} + \cdots - 25\!\cdots\!32 \)
|
| $31$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 88\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 64\!\cdots\!84 \)
|
| $43$ |
\( T^{5} + \cdots + 28\!\cdots\!60 \)
|
| $47$ |
\( T^{5} + \cdots + 16\!\cdots\!04 \)
|
| $53$ |
\( T^{5} + \cdots + 67\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots + 25\!\cdots\!60 \)
|
| $61$ |
\( T^{5} + \cdots - 11\!\cdots\!68 \)
|
| $67$ |
\( T^{5} + \cdots - 44\!\cdots\!28 \)
|
| $71$ |
\( T^{5} + \cdots - 86\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots - 11\!\cdots\!92 \)
|
| $79$ |
\( T^{5} + \cdots - 41\!\cdots\!84 \)
|
| $83$ |
\( T^{5} + \cdots - 71\!\cdots\!56 \)
|
| $89$ |
\( T^{5} + \cdots - 15\!\cdots\!48 \)
|
| $97$ |
\( T^{5} + \cdots - 29\!\cdots\!92 \)
|
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