Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(42.5993790241\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{177}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{177})\). We also show the integral \(q\)-expansion of the trace form.
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 | −7.15207 | 4.00000 | −8.30413 | −14.3041 | 35.1521 | 8.00000 | 24.1521 | −16.6083 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 6.15207 | 4.00000 | 18.3041 | 12.3041 | 21.8479 | 8.00000 | 10.8479 | 36.6083 | |||||||||||||||||||||||||
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 | 722.4.a.i | 2 | |
| 19.b | odd | 2 | 1 | 38.4.a.b | ✓ | 2 | |
| 57.d | even | 2 | 1 | 342.4.a.k | 2 | ||
| 76.d | even | 2 | 1 | 304.4.a.d | 2 | ||
| 95.d | odd | 2 | 1 | 950.4.a.h | 2 | ||
| 95.g | even | 4 | 2 | 950.4.b.g | 4 | ||
| 133.c | even | 2 | 1 | 1862.4.a.b | 2 | ||
| 152.b | even | 2 | 1 | 1216.4.a.l | 2 | ||
| 152.g | odd | 2 | 1 | 1216.4.a.j | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.4.a.b | ✓ | 2 | 19.b | odd | 2 | 1 | |
| 304.4.a.d | 2 | 76.d | even | 2 | 1 | ||
| 342.4.a.k | 2 | 57.d | even | 2 | 1 | ||
| 722.4.a.i | 2 | 1.a | even | 1 | 1 | trivial | |
| 950.4.a.h | 2 | 95.d | odd | 2 | 1 | ||
| 950.4.b.g | 4 | 95.g | even | 4 | 2 | ||
| 1216.4.a.j | 2 | 152.g | odd | 2 | 1 | ||
| 1216.4.a.l | 2 | 152.b | even | 2 | 1 | ||
| 1862.4.a.b | 2 | 133.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):
|
\( T_{3}^{2} + T_{3} - 44 \)
|
|
\( T_{5}^{2} - 10T_{5} - 152 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} + T - 44 \)
|
| $5$ |
\( T^{2} - 10T - 152 \)
|
| $7$ |
\( T^{2} - 57T + 768 \)
|
| $11$ |
\( T^{2} - 10T - 152 \)
|
| $13$ |
\( T^{2} + 13T - 2126 \)
|
| $17$ |
\( T^{2} + 51T - 9306 \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} + 155T - 1472 \)
|
| $29$ |
\( T^{2} - 79T - 35654 \)
|
| $31$ |
\( T^{2} - 16T - 11264 \)
|
| $37$ |
\( T^{2} + 380T + 24772 \)
|
| $41$ |
\( T^{2} - 790T + 154432 \)
|
| $43$ |
\( T^{2} - 296T - 80048 \)
|
| $47$ |
\( T^{2} + 200T - 60800 \)
|
| $53$ |
\( T^{2} + 397T + 35818 \)
|
| $59$ |
\( T^{2} + 201T - 212964 \)
|
| $61$ |
\( T^{2} + 680T + 29932 \)
|
| $67$ |
\( T^{2} - 939T + 138612 \)
|
| $71$ |
\( T^{2} + 406T + 19792 \)
|
| $73$ |
\( T^{2} - 123T + 3738 \)
|
| $79$ |
\( T^{2} + 106T - 146048 \)
|
| $83$ |
\( T^{2} - 2226 T + 1237176 \)
|
| $89$ |
\( T^{2} - 870T + 184800 \)
|
| $97$ |
\( T^{2} - 1864 T + 613036 \)
|
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