Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(7.21727189158\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{145}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 36 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{145})\). 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 |
|
−8.52080 | 0 | 40.6040 | −25.0000 | 0 | 160.416 | −73.3128 | 0 | 213.020 | ||||||||||||||||||||||||
| 1.2 | 3.52080 | 0 | −19.6040 | −25.0000 | 0 | −80.4159 | −181.687 | 0 | −88.0199 | |||||||||||||||||||||||||
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 | 45.6.a.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 45.6.a.f | yes | 2 | |
| 4.b | odd | 2 | 1 | 720.6.a.y | 2 | ||
| 5.b | even | 2 | 1 | 225.6.a.r | 2 | ||
| 5.c | odd | 4 | 2 | 225.6.b.j | 4 | ||
| 12.b | even | 2 | 1 | 720.6.a.be | 2 | ||
| 15.d | odd | 2 | 1 | 225.6.a.k | 2 | ||
| 15.e | even | 4 | 2 | 225.6.b.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.6.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 45.6.a.f | yes | 2 | 3.b | odd | 2 | 1 | |
| 225.6.a.k | 2 | 15.d | odd | 2 | 1 | ||
| 225.6.a.r | 2 | 5.b | even | 2 | 1 | ||
| 225.6.b.j | 4 | 5.c | odd | 4 | 2 | ||
| 225.6.b.k | 4 | 15.e | even | 4 | 2 | ||
| 720.6.a.y | 2 | 4.b | odd | 2 | 1 | ||
| 720.6.a.be | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 5T_{2} - 30 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 5T - 30 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( T^{2} - 80T - 12900 \)
|
| $11$ |
\( T^{2} + 800T + 145500 \)
|
| $13$ |
\( T^{2} + 120T - 228400 \)
|
| $17$ |
\( T^{2} + 1940 T + 903780 \)
|
| $19$ |
\( T^{2} + 1512 T - 3140464 \)
|
| $23$ |
\( T^{2} - 1320 T - 8772480 \)
|
| $29$ |
\( T^{2} + 1300 T - 58969500 \)
|
| $31$ |
\( T^{2} + 5824 T + 1461744 \)
|
| $37$ |
\( T^{2} + 12560 T - 35729600 \)
|
| $41$ |
\( T^{2} + 400 T - 67008000 \)
|
| $43$ |
\( T^{2} - 25680 T + 105473600 \)
|
| $47$ |
\( T^{2} + 18920 T + 88264320 \)
|
| $53$ |
\( T^{2} + 49460 T + 598172580 \)
|
| $59$ |
\( T^{2} + 63200 T + 963745500 \)
|
| $61$ |
\( T^{2} + 49116 T - 469672636 \)
|
| $67$ |
\( T^{2} + \cdots - 2380880400 \)
|
| $71$ |
\( T^{2} + 65200 T + 983358000 \)
|
| $73$ |
\( T^{2} + \cdots + 1607828900 \)
|
| $79$ |
\( T^{2} + 46288 T + 275282736 \)
|
| $83$ |
\( T^{2} + \cdots - 3349469520 \)
|
| $89$ |
\( T^{2} - 87000 T - 665550000 \)
|
| $97$ |
\( T^{2} - 10180 T - 274763900 \)
|
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