Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,6,Mod(1,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.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: | \(115.797117905\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 382x^{2} - 1336x + 14544 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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
\( x^{4} - 2x^{3} - 382x^{2} - 1336x + 14544 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 8\nu - 188 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 14\nu^{2} - 214\nu + 1218 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 8\beta _1 + 188 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 28\beta_{2} + 326\beta _1 + 1414 \)
|
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 |
|
−4.00000 | −24.3388 | 16.0000 | 57.3176 | 97.3553 | 33.7394 | −64.0000 | 349.379 | −229.271 | ||||||||||||||||||||||||||||||
| 1.2 | −4.00000 | −7.72295 | 16.0000 | −92.7387 | 30.8918 | 13.5472 | −64.0000 | −183.356 | 370.955 | |||||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 8.37406 | 16.0000 | 25.1809 | −33.4962 | −187.086 | −64.0000 | −172.875 | −100.724 | |||||||||||||||||||||||||||||||
| 1.4 | −4.00000 | 9.68773 | 16.0000 | 46.2401 | −38.7509 | 177.800 | −64.0000 | −149.148 | −184.961 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
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 | 722.6.a.g | 4 | |
| 19.b | odd | 2 | 1 | 722.6.a.j | 4 | ||
| 19.d | odd | 6 | 2 | 38.6.c.b | ✓ | 8 | |
| 57.f | even | 6 | 2 | 342.6.g.d | 8 | ||
| 76.f | even | 6 | 2 | 304.6.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.c.b | ✓ | 8 | 19.d | odd | 6 | 2 | |
| 304.6.i.b | 8 | 76.f | even | 6 | 2 | ||
| 342.6.g.d | 8 | 57.f | even | 6 | 2 | ||
| 722.6.a.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 722.6.a.j | 4 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 14T_{3}^{3} - 310T_{3}^{2} - 794T_{3} + 15249 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{4} \)
|
| $3$ |
\( T^{4} + 14 T^{3} + \cdots + 15249 \)
|
| $5$ |
\( T^{4} - 36 T^{3} + \cdots - 6189270 \)
|
| $7$ |
\( T^{4} - 38 T^{3} + \cdots - 15204096 \)
|
| $11$ |
\( T^{4} - 72 T^{3} + \cdots - 423612144 \)
|
| $13$ |
\( T^{4} + \cdots - 104482298012 \)
|
| $17$ |
\( T^{4} + \cdots + 488685884832 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 65332086048162 \)
|
| $29$ |
\( T^{4} + \cdots - 644908895564790 \)
|
| $31$ |
\( T^{4} + \cdots - 9213535294816 \)
|
| $37$ |
\( T^{4} + \cdots + 27\!\cdots\!80 \)
|
| $41$ |
\( T^{4} + \cdots + 22\!\cdots\!45 \)
|
| $43$ |
\( T^{4} + \cdots + 88\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 93\!\cdots\!50 \)
|
| $53$ |
\( T^{4} + \cdots + 45\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 77\!\cdots\!85 \)
|
| $61$ |
\( T^{4} + \cdots - 40\!\cdots\!54 \)
|
| $67$ |
\( T^{4} + \cdots - 22\!\cdots\!55 \)
|
| $71$ |
\( T^{4} + \cdots - 55\!\cdots\!64 \)
|
| $73$ |
\( T^{4} + \cdots - 53\!\cdots\!47 \)
|
| $79$ |
\( T^{4} + \cdots - 80\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots - 61\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots - 78\!\cdots\!04 \)
|
| $97$ |
\( T^{4} + \cdots + 23\!\cdots\!85 \)
|
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