Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,5,Mod(7,12)]
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([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.7");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.d (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: | \(1.24043955701\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} - 4\nu + 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + \beta _1 - 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - \beta _1 - 5 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−0.302776 | − | 3.98852i | − | 5.19615i | −15.8167 | + | 2.41526i | 34.8444 | −20.7250 | + | 1.57327i | 43.0318i | 14.4222 | + | 62.3538i | −27.0000 | −10.5500 | − | 138.978i | |||||||||||||||||||
| 7.2 | −0.302776 | + | 3.98852i | 5.19615i | −15.8167 | − | 2.41526i | 34.8444 | −20.7250 | − | 1.57327i | − | 43.0318i | 14.4222 | − | 62.3538i | −27.0000 | −10.5500 | + | 138.978i | ||||||||||||||||||||
| 7.3 | 3.30278 | − | 2.25647i | 5.19615i | 5.81665 | − | 14.9053i | −22.8444 | 11.7250 | + | 17.1617i | 56.8882i | −14.4222 | − | 62.3538i | −27.0000 | −75.4500 | + | 51.5478i | |||||||||||||||||||||
| 7.4 | 3.30278 | + | 2.25647i | − | 5.19615i | 5.81665 | + | 14.9053i | −22.8444 | 11.7250 | − | 17.1617i | − | 56.8882i | −14.4222 | + | 62.3538i | −27.0000 | −75.4500 | − | 51.5478i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.5.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 36.5.d.b | 4 | ||
| 4.b | odd | 2 | 1 | inner | 12.5.d.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 300.5.c.a | 4 | ||
| 5.c | odd | 4 | 2 | 300.5.f.a | 8 | ||
| 8.b | even | 2 | 1 | 192.5.g.d | 4 | ||
| 8.d | odd | 2 | 1 | 192.5.g.d | 4 | ||
| 12.b | even | 2 | 1 | 36.5.d.b | 4 | ||
| 16.e | even | 4 | 2 | 768.5.b.g | 8 | ||
| 16.f | odd | 4 | 2 | 768.5.b.g | 8 | ||
| 20.d | odd | 2 | 1 | 300.5.c.a | 4 | ||
| 20.e | even | 4 | 2 | 300.5.f.a | 8 | ||
| 24.f | even | 2 | 1 | 576.5.g.m | 4 | ||
| 24.h | odd | 2 | 1 | 576.5.g.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.5.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 12.5.d.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 36.5.d.b | 4 | 3.b | odd | 2 | 1 | ||
| 36.5.d.b | 4 | 12.b | even | 2 | 1 | ||
| 192.5.g.d | 4 | 8.b | even | 2 | 1 | ||
| 192.5.g.d | 4 | 8.d | odd | 2 | 1 | ||
| 300.5.c.a | 4 | 5.b | even | 2 | 1 | ||
| 300.5.c.a | 4 | 20.d | odd | 2 | 1 | ||
| 300.5.f.a | 8 | 5.c | odd | 4 | 2 | ||
| 300.5.f.a | 8 | 20.e | even | 4 | 2 | ||
| 576.5.g.m | 4 | 24.f | even | 2 | 1 | ||
| 576.5.g.m | 4 | 24.h | odd | 2 | 1 | ||
| 768.5.b.g | 8 | 16.e | even | 4 | 2 | ||
| 768.5.b.g | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{3} + \cdots + 256 \)
|
| $3$ |
\( (T^{2} + 27)^{2} \)
|
| $5$ |
\( (T^{2} - 12 T - 796)^{2} \)
|
| $7$ |
\( T^{4} + 5088 T^{2} + 5992704 \)
|
| $11$ |
\( T^{4} + 22368 T^{2} + 77158656 \)
|
| $13$ |
\( (T^{2} - 148 T - 24476)^{2} \)
|
| $17$ |
\( (T^{2} + 300 T + 19172)^{2} \)
|
| $19$ |
\( T^{4} + 325728 T^{2} + 283855104 \)
|
| $23$ |
\( T^{4} + \cdots + 44930433024 \)
|
| $29$ |
\( (T^{2} - 444 T + 8516)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 310721515776 \)
|
| $37$ |
\( (T^{2} + 2204 T + 1094596)^{2} \)
|
| $41$ |
\( (T^{2} - 276 T - 144028)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 4698628487424 \)
|
| $47$ |
\( T^{4} + \cdots + 1723191791616 \)
|
| $53$ |
\( (T^{2} - 2556 T + 1392836)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 60549065443584 \)
|
| $61$ |
\( (T^{2} - 2116 T - 10861436)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 3924392696064 \)
|
| $71$ |
\( T^{4} + \cdots + 10870367256576 \)
|
| $73$ |
\( (T^{2} - 4420 T - 53102972)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 783891587748096 \)
|
| $83$ |
\( T^{4} + \cdots + 87\!\cdots\!16 \)
|
| $89$ |
\( (T^{2} + 12540 T + 7350788)^{2} \)
|
| $97$ |
\( (T^{2} - 11524 T - 30177788)^{2} \)
|
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