Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(185.939049695\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{331}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 331 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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 \(\beta = 8\sqrt{331}\). 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 |
|
−32.0000 | −649.642 | 1024.00 | 8422.36 | 20788.5 | 19146.7 | −32768.0 | 244887. | −269516. | ||||||||||||||||||||||||
| 1.2 | −32.0000 | 223.642 | 1024.00 | −6132.36 | −7156.54 | 67177.3 | −32768.0 | −127131. | 196236. | |||||||||||||||||||||||||
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 | 242.12.a.a | 2 | |
| 11.b | odd | 2 | 1 | 22.12.a.a | ✓ | 2 | |
| 33.d | even | 2 | 1 | 198.12.a.c | 2 | ||
| 44.c | even | 2 | 1 | 176.12.a.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.12.a.a | ✓ | 2 | 11.b | odd | 2 | 1 | |
| 176.12.a.a | 2 | 44.c | even | 2 | 1 | ||
| 198.12.a.c | 2 | 33.d | even | 2 | 1 | ||
| 242.12.a.a | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{2} + 426T_{3} - 145287 \)
|
|
\( T_{7}^{2} - 86324T_{7} + 1286223844 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 32)^{2} \)
|
| $3$ |
\( T^{2} + 426T - 145287 \)
|
| $5$ |
\( T^{2} - 2290 T - 51648975 \)
|
| $7$ |
\( T^{2} + \cdots + 1286223844 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} + \cdots - 552368670000 \)
|
| $17$ |
\( T^{2} + \cdots - 6355251487740 \)
|
| $19$ |
\( T^{2} + \cdots + 94903126697792 \)
|
| $23$ |
\( T^{2} + \cdots - 199908316265607 \)
|
| $29$ |
\( T^{2} + \cdots - 32\!\cdots\!72 \)
|
| $31$ |
\( T^{2} + \cdots + 63\!\cdots\!25 \)
|
| $37$ |
\( T^{2} + \cdots - 71\!\cdots\!75 \)
|
| $41$ |
\( T^{2} + \cdots - 67\!\cdots\!92 \)
|
| $43$ |
\( T^{2} + \cdots - 10\!\cdots\!80 \)
|
| $47$ |
\( T^{2} + \cdots - 42\!\cdots\!00 \)
|
| $53$ |
\( T^{2} + \cdots - 12\!\cdots\!56 \)
|
| $59$ |
\( T^{2} + \cdots + 11\!\cdots\!97 \)
|
| $61$ |
\( T^{2} + \cdots - 60\!\cdots\!00 \)
|
| $67$ |
\( T^{2} + \cdots + 33\!\cdots\!97 \)
|
| $71$ |
\( T^{2} + \cdots - 17\!\cdots\!07 \)
|
| $73$ |
\( T^{2} + \cdots - 53\!\cdots\!80 \)
|
| $79$ |
\( T^{2} + \cdots + 20\!\cdots\!64 \)
|
| $83$ |
\( T^{2} + \cdots + 11\!\cdots\!00 \)
|
| $89$ |
\( T^{2} + \cdots + 17\!\cdots\!85 \)
|
| $97$ |
\( T^{2} + \cdots + 77\!\cdots\!21 \)
|
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