Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,8,Mod(1,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.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: | \(4.68577538226\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{601}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 150 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{601})\). 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 |
|
−8.75765 | 27.0000 | −51.3036 | 125.000 | −236.457 | 1338.43 | 1570.28 | 729.000 | −1094.71 | ||||||||||||||||||||||||
| 1.2 | 15.7577 | 27.0000 | 120.304 | 125.000 | 425.457 | −34.4284 | −121.278 | 729.000 | 1969.71 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 15.8.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 45.8.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 240.8.a.p | 2 | ||
| 5.b | even | 2 | 1 | 75.8.a.e | 2 | ||
| 5.c | odd | 4 | 2 | 75.8.b.d | 4 | ||
| 15.d | odd | 2 | 1 | 225.8.a.t | 2 | ||
| 15.e | even | 4 | 2 | 225.8.b.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.8.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 45.8.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 75.8.a.e | 2 | 5.b | even | 2 | 1 | ||
| 75.8.b.d | 4 | 5.c | odd | 4 | 2 | ||
| 225.8.a.t | 2 | 15.d | odd | 2 | 1 | ||
| 225.8.b.n | 4 | 15.e | even | 4 | 2 | ||
| 240.8.a.p | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 7T_{2} - 138 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 7T - 138 \)
|
| $3$ |
\( (T - 27)^{2} \)
|
| $5$ |
\( (T - 125)^{2} \)
|
| $7$ |
\( T^{2} - 1304T - 46080 \)
|
| $11$ |
\( T^{2} - 3448 T - 29376048 \)
|
| $13$ |
\( T^{2} + 8988 T - 81820108 \)
|
| $17$ |
\( T^{2} + 5492 T - 286949484 \)
|
| $19$ |
\( T^{2} + 49584 T - 20483920 \)
|
| $23$ |
\( T^{2} + \cdots - 1966256640 \)
|
| $29$ |
\( T^{2} + \cdots - 2060549340 \)
|
| $31$ |
\( T^{2} + \cdots + 22433068800 \)
|
| $37$ |
\( T^{2} + \cdots + 31820561620 \)
|
| $41$ |
\( T^{2} + \cdots + 30837469380 \)
|
| $43$ |
\( T^{2} + \cdots + 23614207376 \)
|
| $47$ |
\( T^{2} + \cdots + 49382888064 \)
|
| $53$ |
\( T^{2} + \cdots + 54364123620 \)
|
| $59$ |
\( T^{2} + \cdots - 3349911332400 \)
|
| $61$ |
\( T^{2} + \cdots + 1579956747716 \)
|
| $67$ |
\( T^{2} + \cdots + 3899842029456 \)
|
| $71$ |
\( T^{2} + \cdots + 2634564492864 \)
|
| $73$ |
\( T^{2} + \cdots - 22097955229180 \)
|
| $79$ |
\( T^{2} + \cdots + 4381741411200 \)
|
| $83$ |
\( T^{2} + \cdots + 7951958141328 \)
|
| $89$ |
\( T^{2} + \cdots + 68134385955780 \)
|
| $97$ |
\( T^{2} + \cdots - 25751505484604 \)
|
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