Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,4,Mod(1,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, 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("121.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.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: | \(7.13923111069\) |
| 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: | \(\Z[a_1, a_2]\) |
| 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 \(\beta = 2\sqrt{3}\). 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 |
|
−4.46410 | −7.46410 | 11.9282 | −8.46410 | 33.3205 | 20.2487 | −17.5359 | 28.7128 | 37.7846 | ||||||||||||||||||||||||
| 1.2 | 2.46410 | −0.535898 | −1.92820 | −1.53590 | −1.32051 | −28.2487 | −24.4641 | −26.7128 | −3.78461 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
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 | 121.4.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1089.4.a.x | 2 | ||
| 4.b | odd | 2 | 1 | 1936.4.a.z | 2 | ||
| 11.b | odd | 2 | 1 | 121.4.a.e | yes | 2 | |
| 11.c | even | 5 | 4 | 121.4.c.g | 8 | ||
| 11.d | odd | 10 | 4 | 121.4.c.d | 8 | ||
| 33.d | even | 2 | 1 | 1089.4.a.k | 2 | ||
| 44.c | even | 2 | 1 | 1936.4.a.y | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 121.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 121.4.a.e | yes | 2 | 11.b | odd | 2 | 1 | |
| 121.4.c.d | 8 | 11.d | odd | 10 | 4 | ||
| 121.4.c.g | 8 | 11.c | even | 5 | 4 | ||
| 1089.4.a.k | 2 | 33.d | even | 2 | 1 | ||
| 1089.4.a.x | 2 | 3.b | odd | 2 | 1 | ||
| 1936.4.a.y | 2 | 44.c | even | 2 | 1 | ||
| 1936.4.a.z | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 11 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 11 \)
|
| $3$ |
\( T^{2} + 8T + 4 \)
|
| $5$ |
\( T^{2} + 10T + 13 \)
|
| $7$ |
\( T^{2} + 8T - 572 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} - 130T + 4213 \)
|
| $17$ |
\( T^{2} + 14T - 3839 \)
|
| $19$ |
\( T^{2} + 48T - 396 \)
|
| $23$ |
\( T^{2} + 128T - 8972 \)
|
| $29$ |
\( T^{2} + 30T - 16203 \)
|
| $31$ |
\( T^{2} + 184T - 18044 \)
|
| $37$ |
\( T^{2} - 126T - 70923 \)
|
| $41$ |
\( T^{2} - 370T + 34177 \)
|
| $43$ |
\( T^{2} + 264T + 11616 \)
|
| $47$ |
\( T^{2} - 256T - 131468 \)
|
| $53$ |
\( T^{2} + 162T - 80139 \)
|
| $59$ |
\( T^{2} + 1304 T + 403936 \)
|
| $61$ |
\( T^{2} + 300T - 79068 \)
|
| $67$ |
\( T^{2} + 656T + 106132 \)
|
| $71$ |
\( T^{2} + 1176 T + 308112 \)
|
| $73$ |
\( T^{2} + 668T - 277244 \)
|
| $79$ |
\( T^{2} - 416T - 99308 \)
|
| $83$ |
\( T^{2} + 960T + 199188 \)
|
| $89$ |
\( T^{2} + 1074T - 83343 \)
|
| $97$ |
\( T^{2} + 338T - 628511 \)
|
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