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: |
\( x^{4} - x^{3} - 924x^{2} + 3360x + 110592 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 924x^{2} + 3360x + 110592 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 732\nu - 2496 ) / 96 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 23\nu^{2} - 564\nu - 8640 ) / 48 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} - 7\beta _1 + 464 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} - 92\beta_{2} + 725\beta _1 - 2032 \)
|
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 | −25.4173 | 16.0000 | 17.9284 | −101.669 | −245.183 | 64.0000 | 403.039 | 71.7138 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −15.0297 | 16.0000 | −44.5593 | −60.1187 | 199.099 | 64.0000 | −17.1088 | −178.237 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 9.74758 | 16.0000 | 75.9404 | 38.9903 | 14.5353 | 64.0000 | −147.985 | 303.762 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 29.6994 | 16.0000 | −63.3095 | 118.798 | 128.549 | 64.0000 | 639.054 | −253.238 | |||||||||||||||||||||||||||||||
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.i | yes | 4 |
| 19.b | odd | 2 | 1 | 722.6.a.h | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 722.6.a.h | ✓ | 4 | 19.b | odd | 2 | 1 | |
| 722.6.a.i | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + T_{3}^{3} - 924T_{3}^{2} - 3360T_{3} + 110592 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( T^{4} + T^{3} + \cdots + 110592 \)
|
| $5$ |
\( T^{4} + 14 T^{3} + \cdots + 3840812 \)
|
| $7$ |
\( T^{4} - 97 T^{3} + \cdots - 91211508 \)
|
| $11$ |
\( T^{4} + \cdots - 6028722256 \)
|
| $13$ |
\( T^{4} - 1465 T^{3} + \cdots - 55296000 \)
|
| $17$ |
\( T^{4} + \cdots - 1007551906110 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + \cdots - 2602166692160 \)
|
| $29$ |
\( T^{4} + \cdots - 63244006861824 \)
|
| $31$ |
\( T^{4} + \cdots + 47054539063296 \)
|
| $37$ |
\( T^{4} + \cdots - 187250082600960 \)
|
| $41$ |
\( T^{4} + \cdots - 33\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 994614017440000 \)
|
| $47$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 12\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots + 58\!\cdots\!92 \)
|
| $61$ |
\( T^{4} + \cdots + 35\!\cdots\!40 \)
|
| $67$ |
\( T^{4} + \cdots - 54\!\cdots\!20 \)
|
| $71$ |
\( T^{4} + \cdots - 66\!\cdots\!20 \)
|
| $73$ |
\( T^{4} + \cdots - 18\!\cdots\!38 \)
|
| $79$ |
\( T^{4} + \cdots - 11\!\cdots\!76 \)
|
| $83$ |
\( T^{4} + \cdots - 52\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 53\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots - 10\!\cdots\!92 \)
|
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