Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,12,Mod(1,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("144.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.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: | \(110.641418001\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{109}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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 \(\beta = 192\sqrt{109}\). 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 | 0 | 0 | −11952.2 | 0 | −17464.5 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 4084.16 | 0 | −73591.5 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
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 | 144.12.a.p | 2 | |
| 3.b | odd | 2 | 1 | 16.12.a.d | 2 | ||
| 4.b | odd | 2 | 1 | 72.12.a.e | 2 | ||
| 12.b | even | 2 | 1 | 8.12.a.b | ✓ | 2 | |
| 24.f | even | 2 | 1 | 64.12.a.h | 2 | ||
| 24.h | odd | 2 | 1 | 64.12.a.k | 2 | ||
| 48.i | odd | 4 | 2 | 256.12.b.k | 4 | ||
| 48.k | even | 4 | 2 | 256.12.b.h | 4 | ||
| 60.h | even | 2 | 1 | 200.12.a.d | 2 | ||
| 60.l | odd | 4 | 2 | 200.12.c.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.12.a.b | ✓ | 2 | 12.b | even | 2 | 1 | |
| 16.12.a.d | 2 | 3.b | odd | 2 | 1 | ||
| 64.12.a.h | 2 | 24.f | even | 2 | 1 | ||
| 64.12.a.k | 2 | 24.h | odd | 2 | 1 | ||
| 72.12.a.e | 2 | 4.b | odd | 2 | 1 | ||
| 144.12.a.p | 2 | 1.a | even | 1 | 1 | trivial | |
| 200.12.a.d | 2 | 60.h | even | 2 | 1 | ||
| 200.12.c.c | 4 | 60.l | odd | 4 | 2 | ||
| 256.12.b.h | 4 | 48.k | even | 4 | 2 | ||
| 256.12.b.k | 4 | 48.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 7868T_{5} - 48814460 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(144))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 7868 T - 48814460 \)
|
| $7$ |
\( T^{2} + \cdots + 1285236288 \)
|
| $11$ |
\( T^{2} + \cdots - 208087277936 \)
|
| $13$ |
\( T^{2} + \cdots - 125364026012 \)
|
| $17$ |
\( T^{2} + \cdots - 51795407805500 \)
|
| $19$ |
\( T^{2} + \cdots + 110143192291984 \)
|
| $23$ |
\( T^{2} + \cdots + 47308617068608 \)
|
| $29$ |
\( T^{2} + \cdots + 11\!\cdots\!96 \)
|
| $31$ |
\( T^{2} + \cdots + 681230587110400 \)
|
| $37$ |
\( T^{2} + \cdots - 15\!\cdots\!44 \)
|
| $41$ |
\( T^{2} + \cdots + 17\!\cdots\!88 \)
|
| $43$ |
\( T^{2} + \cdots - 46\!\cdots\!08 \)
|
| $47$ |
\( T^{2} + \cdots + 80\!\cdots\!24 \)
|
| $53$ |
\( T^{2} + \cdots - 18\!\cdots\!56 \)
|
| $59$ |
\( T^{2} + \cdots + 75\!\cdots\!48 \)
|
| $61$ |
\( T^{2} + \cdots + 14\!\cdots\!20 \)
|
| $67$ |
\( T^{2} + \cdots - 14\!\cdots\!52 \)
|
| $71$ |
\( T^{2} + \cdots - 17\!\cdots\!60 \)
|
| $73$ |
\( T^{2} + \cdots + 25\!\cdots\!08 \)
|
| $79$ |
\( T^{2} + \cdots - 89\!\cdots\!80 \)
|
| $83$ |
\( T^{2} + \cdots + 28\!\cdots\!72 \)
|
| $89$ |
\( T^{2} + \cdots - 24\!\cdots\!96 \)
|
| $97$ |
\( T^{2} + \cdots - 18\!\cdots\!56 \)
|
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