Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(11.5251477084\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1609}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 402 \)
|
| 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 \(\beta = \sqrt{1609}\). 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 |
|
−51.1123 | 243.000 | 564.472 | −3125.00 | −12420.3 | 45751.3 | 75826.6 | 59049.0 | 159726. | ||||||||||||||||||||||||
| 1.2 | 29.1123 | 243.000 | −1200.47 | −3125.00 | 7074.30 | −56615.3 | −94570.6 | 59049.0 | −90976.1 | |||||||||||||||||||||||||
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 | 15.12.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 45.12.a.e | 2 | ||
| 4.b | odd | 2 | 1 | 240.12.a.j | 2 | ||
| 5.b | even | 2 | 1 | 75.12.a.d | 2 | ||
| 5.c | odd | 4 | 2 | 75.12.b.c | 4 | ||
| 15.d | odd | 2 | 1 | 225.12.a.g | 2 | ||
| 15.e | even | 4 | 2 | 225.12.b.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.12.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 45.12.a.e | 2 | 3.b | odd | 2 | 1 | ||
| 75.12.a.d | 2 | 5.b | even | 2 | 1 | ||
| 75.12.b.c | 4 | 5.c | odd | 4 | 2 | ||
| 225.12.a.g | 2 | 15.d | odd | 2 | 1 | ||
| 225.12.b.h | 4 | 15.e | even | 4 | 2 | ||
| 240.12.a.j | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 22T_{2} - 1488 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 22T - 1488 \)
|
| $3$ |
\( (T - 243)^{2} \)
|
| $5$ |
\( (T + 3125)^{2} \)
|
| $7$ |
\( T^{2} + \cdots - 2590228560 \)
|
| $11$ |
\( T^{2} + \cdots - 140771013888 \)
|
| $13$ |
\( T^{2} + \cdots + 1132296332852 \)
|
| $17$ |
\( T^{2} + \cdots + 7888201038036 \)
|
| $19$ |
\( T^{2} + \cdots - 9309926445520 \)
|
| $23$ |
\( T^{2} + \cdots - 419924260414080 \)
|
| $29$ |
\( T^{2} + \cdots - 14\!\cdots\!20 \)
|
| $31$ |
\( T^{2} + \cdots + 20\!\cdots\!00 \)
|
| $37$ |
\( T^{2} + \cdots - 11\!\cdots\!80 \)
|
| $41$ |
\( T^{2} + \cdots - 74\!\cdots\!60 \)
|
| $43$ |
\( T^{2} + \cdots - 26\!\cdots\!24 \)
|
| $47$ |
\( T^{2} + \cdots + 41\!\cdots\!84 \)
|
| $53$ |
\( T^{2} + \cdots - 21\!\cdots\!20 \)
|
| $59$ |
\( T^{2} + \cdots - 26\!\cdots\!60 \)
|
| $61$ |
\( T^{2} + \cdots + 52\!\cdots\!36 \)
|
| $67$ |
\( T^{2} + \cdots + 59\!\cdots\!36 \)
|
| $71$ |
\( T^{2} + \cdots - 14\!\cdots\!56 \)
|
| $73$ |
\( T^{2} + \cdots + 14\!\cdots\!00 \)
|
| $79$ |
\( T^{2} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots - 30\!\cdots\!32 \)
|
| $89$ |
\( T^{2} + \cdots - 34\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots + 64\!\cdots\!56 \)
|
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