Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,12,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, 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("45.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.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: | \(34.5754431252\) |
| 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: |
\( x^{4} - 795x^{2} - 6374x + 16128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4}\cdot 5^{3} \) |
| 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 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
\( x^{4} - 795x^{2} - 6374x + 16128 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + 779\nu + 3204 ) / 54 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 76\nu^{3} - 979\nu^{2} - 35579\nu + 25848 ) / 54 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -10\nu^{3} + 310\nu^{2} + 5630\nu - 75420 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( 5\beta_{3} + 4\beta_{2} + 204\beta _1 - 52 ) / 1350 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 175\beta_{3} + 32\beta_{2} - 1068\beta _1 + 536884 ) / 1350 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4595\beta_{3} + 3244\beta_{2} + 81744\beta _1 + 6432428 ) / 1350 \)
|
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 |
|
−69.6616 | 0 | 2804.74 | −3125.00 | 0 | −79546.9 | −52716.0 | 0 | 217693. | ||||||||||||||||||||||||||||||
| 1.2 | 4.71838 | 0 | −2025.74 | −3125.00 | 0 | −6243.26 | −19221.4 | 0 | −14744.9 | |||||||||||||||||||||||||||||||
| 1.3 | 50.7631 | 0 | 528.891 | −3125.00 | 0 | −34993.8 | −77114.7 | 0 | −158635. | |||||||||||||||||||||||||||||||
| 1.4 | 89.1802 | 0 | 5905.10 | −3125.00 | 0 | 58704.0 | 343977. | 0 | −278688. | |||||||||||||||||||||||||||||||
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 | 45.12.a.h | yes | 4 |
| 3.b | odd | 2 | 1 | 45.12.a.g | ✓ | 4 | |
| 5.b | even | 2 | 1 | 225.12.a.o | 4 | ||
| 5.c | odd | 4 | 2 | 225.12.b.m | 8 | ||
| 15.d | odd | 2 | 1 | 225.12.a.u | 4 | ||
| 15.e | even | 4 | 2 | 225.12.b.l | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.12.a.g | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 45.12.a.h | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 225.12.a.o | 4 | 5.b | even | 2 | 1 | ||
| 225.12.a.u | 4 | 15.d | odd | 2 | 1 | ||
| 225.12.b.l | 8 | 15.e | even | 4 | 2 | ||
| 225.12.b.m | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 75T_{2}^{3} - 4890T_{2}^{2} + 340000T_{2} - 1488000 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 75 T^{3} + \cdots - 1488000 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 3125)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots - 71\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots - 70\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 50\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 59\!\cdots\!56 \)
|
| $23$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 62\!\cdots\!96 \)
|
| $37$ |
\( T^{4} + \cdots - 90\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 24\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 86\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 88\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 22\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!36 \)
|
| $83$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
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