Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,10,Mod(1,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.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: | \(173.052040951\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2353}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 588 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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 \(\beta = \sqrt{2353}\). 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 |
|
0 | 81.0000 | 0 | −1112.77 | 0 | 2401.00 | 0 | 6561.00 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 81.0000 | 0 | 2282.77 | 0 | 2401.00 | 0 | 6561.00 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 336.10.a.m | 2 | |
| 4.b | odd | 2 | 1 | 21.10.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 63.10.a.c | 2 | ||
| 28.d | even | 2 | 1 | 147.10.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.10.a.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 63.10.a.c | 2 | 12.b | even | 2 | 1 | ||
| 147.10.a.d | 2 | 28.d | even | 2 | 1 | ||
| 336.10.a.m | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 1170T_{5} - 2540200 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(336))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 81)^{2} \)
|
| $5$ |
\( T^{2} - 1170 T - 2540200 \)
|
| $7$ |
\( (T - 2401)^{2} \)
|
| $11$ |
\( T^{2} + \cdots + 5132528504 \)
|
| $13$ |
\( T^{2} + \cdots + 1864912348 \)
|
| $17$ |
\( T^{2} + \cdots - 288898506792 \)
|
| $19$ |
\( T^{2} + \cdots + 12186222736 \)
|
| $23$ |
\( T^{2} + \cdots + 1016626003344 \)
|
| $29$ |
\( T^{2} + \cdots + 23456377117508 \)
|
| $31$ |
\( T^{2} + \cdots - 22088820805632 \)
|
| $37$ |
\( T^{2} + \cdots + 46813520369772 \)
|
| $41$ |
\( T^{2} + \cdots + 409048772180000 \)
|
| $43$ |
\( T^{2} + \cdots - 660121537363952 \)
|
| $47$ |
\( T^{2} + \cdots + 290934877476096 \)
|
| $53$ |
\( T^{2} + \cdots + 197034132205404 \)
|
| $59$ |
\( T^{2} + \cdots + 60\!\cdots\!56 \)
|
| $61$ |
\( T^{2} + \cdots - 86\!\cdots\!96 \)
|
| $67$ |
\( T^{2} + \cdots - 69\!\cdots\!04 \)
|
| $71$ |
\( T^{2} + \cdots + 28\!\cdots\!24 \)
|
| $73$ |
\( T^{2} + \cdots + 64\!\cdots\!76 \)
|
| $79$ |
\( T^{2} + \cdots - 517610480788736 \)
|
| $83$ |
\( T^{2} + \cdots - 11\!\cdots\!76 \)
|
| $89$ |
\( T^{2} + \cdots - 29\!\cdots\!48 \)
|
| $97$ |
\( T^{2} + \cdots + 40\!\cdots\!32 \)
|
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