Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,4,Mod(1,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, 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("242.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(14.2784622214\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \sqrt{3}\). 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 | −2.73205 | 4.00000 | 6.12436 | −5.46410 | −22.0526 | 8.00000 | −19.5359 | 12.2487 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 0.732051 | 4.00000 | −18.1244 | 1.46410 | 16.0526 | 8.00000 | −26.4641 | −36.2487 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
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 | 242.4.a.l | yes | 2 |
| 3.b | odd | 2 | 1 | 2178.4.a.bd | 2 | ||
| 4.b | odd | 2 | 1 | 1936.4.a.s | 2 | ||
| 11.b | odd | 2 | 1 | 242.4.a.i | ✓ | 2 | |
| 11.c | even | 5 | 4 | 242.4.c.p | 8 | ||
| 11.d | odd | 10 | 4 | 242.4.c.t | 8 | ||
| 33.d | even | 2 | 1 | 2178.4.a.bn | 2 | ||
| 44.c | even | 2 | 1 | 1936.4.a.r | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 242.4.a.i | ✓ | 2 | 11.b | odd | 2 | 1 | |
| 242.4.a.l | yes | 2 | 1.a | even | 1 | 1 | trivial |
| 242.4.c.p | 8 | 11.c | even | 5 | 4 | ||
| 242.4.c.t | 8 | 11.d | odd | 10 | 4 | ||
| 1936.4.a.r | 2 | 44.c | even | 2 | 1 | ||
| 1936.4.a.s | 2 | 4.b | odd | 2 | 1 | ||
| 2178.4.a.bd | 2 | 3.b | odd | 2 | 1 | ||
| 2178.4.a.bn | 2 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{2} + 2T_{3} - 2 \)
|
|
\( T_{5}^{2} + 12T_{5} - 111 \)
|
|
\( T_{7}^{2} + 6T_{7} - 354 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} + 2T - 2 \)
|
| $5$ |
\( T^{2} + 12T - 111 \)
|
| $7$ |
\( T^{2} + 6T - 354 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} + 114T + 3141 \)
|
| $17$ |
\( T^{2} - 72T + 1053 \)
|
| $19$ |
\( T^{2} + 150T - 1002 \)
|
| $23$ |
\( T^{2} + 222T + 10446 \)
|
| $29$ |
\( T^{2} + 462T + 48069 \)
|
| $31$ |
\( T^{2} - 14T - 81626 \)
|
| $37$ |
\( T^{2} - 464T + 41917 \)
|
| $41$ |
\( T^{2} + 60T - 93087 \)
|
| $43$ |
\( T^{2} + 432T + 19008 \)
|
| $47$ |
\( T^{2} - 450T - 85482 \)
|
| $53$ |
\( T^{2} - 288T + 19413 \)
|
| $59$ |
\( T^{2} - 204T - 16104 \)
|
| $61$ |
\( T^{2} + 372T - 276456 \)
|
| $67$ |
\( T^{2} - 374T - 309794 \)
|
| $71$ |
\( T^{2} - 1092 T + 296088 \)
|
| $73$ |
\( T^{2} + 444T + 46932 \)
|
| $79$ |
\( T^{2} - 726T + 97422 \)
|
| $83$ |
\( T^{2} - 282T + 19854 \)
|
| $89$ |
\( T^{2} + 1578 T + 616173 \)
|
| $97$ |
\( T^{2} + 1274 T - 807719 \)
|
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