Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,15,Mod(5,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.5");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.c (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(14.9194761782\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 440x^{2} + 48015 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{9}\cdot 5^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 440x^{2} + 48015 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu^{3} + 60\nu^{2} + 1350\nu + 13200 \)
|
| \(\beta_{2}\) | \(=\) |
\( -42\nu^{3} - 60\nu^{2} - 3690\nu - 13200 \)
|
| \(\beta_{3}\) | \(=\) |
\( 300\nu^{3} - 6720\nu^{2} + 67500\nu - 1478400 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 27\beta_{2} + 139\beta_1 ) / 155520 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 50\beta _1 - 2138400 ) / 9720 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{3} - 1215\beta_{2} - 2671\beta_1 ) / 31104 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | −640.285 | − | 2091.17i | 0 | 68988.7i | 0 | 1.38092e6 | 0 | −3.96304e6 | + | 2.67789e6i | 0 | ||||||||||||||||||||||||||
| 5.2 | 0 | −640.285 | + | 2091.17i | 0 | − | 68988.7i | 0 | 1.38092e6 | 0 | −3.96304e6 | − | 2.67789e6i | 0 | ||||||||||||||||||||||||||
| 5.3 | 0 | 1714.29 | − | 1358.01i | 0 | − | 97311.2i | 0 | −526279. | 0 | 1.09458e6 | − | 4.65604e6i | 0 | ||||||||||||||||||||||||||
| 5.4 | 0 | 1714.29 | + | 1358.01i | 0 | 97311.2i | 0 | −526279. | 0 | 1.09458e6 | + | 4.65604e6i | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.15.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 12.15.c.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 48.15.e.c | 4 | ||
| 12.b | even | 2 | 1 | 48.15.e.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.15.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 12.15.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 48.15.e.c | 4 | 4.b | odd | 2 | 1 | ||
| 48.15.e.c | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 14228913600T_{5}^{2} + 45069410819119104000 \)
acting on \(S_{15}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 22876792454961 \)
|
| $5$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} - 854644 T - 726750508316)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots + 448093831556644)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $19$ |
\( (T^{2} + \cdots - 11\!\cdots\!24)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 36\!\cdots\!44)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots + 55\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 37\!\cdots\!56)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 43\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 51\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 81\!\cdots\!24)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 91\!\cdots\!44)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots - 11\!\cdots\!64)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 37\!\cdots\!76)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 94\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 20\!\cdots\!84)^{2} \)
|
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