Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(23.1766126274\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{241}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 60 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{241})\). 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 |
|
−38.7863 | 0 | 992.374 | −625.000 | 0 | 12272.1 | −18631.9 | 0 | 24241.4 | ||||||||||||||||||||||||
| 1.2 | 7.78626 | 0 | −451.374 | −625.000 | 0 | 1839.88 | −7501.08 | 0 | −4866.41 | |||||||||||||||||||||||||
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 | 45.10.a.d | 2 | |
| 3.b | odd | 2 | 1 | 15.10.a.d | ✓ | 2 | |
| 5.b | even | 2 | 1 | 225.10.a.k | 2 | ||
| 5.c | odd | 4 | 2 | 225.10.b.i | 4 | ||
| 12.b | even | 2 | 1 | 240.10.a.r | 2 | ||
| 15.d | odd | 2 | 1 | 75.10.a.f | 2 | ||
| 15.e | even | 4 | 2 | 75.10.b.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.10.a.d | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 45.10.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 75.10.a.f | 2 | 15.d | odd | 2 | 1 | ||
| 75.10.b.f | 4 | 15.e | even | 4 | 2 | ||
| 225.10.a.k | 2 | 5.b | even | 2 | 1 | ||
| 225.10.b.i | 4 | 5.c | odd | 4 | 2 | ||
| 240.10.a.r | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 31T_{2} - 302 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 31T - 302 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T + 625)^{2} \)
|
| $7$ |
\( T^{2} - 14112 T + 22579200 \)
|
| $11$ |
\( T^{2} + \cdots - 2924934128 \)
|
| $13$ |
\( T^{2} + \cdots - 15338329532 \)
|
| $17$ |
\( T^{2} + \cdots - 24505657916 \)
|
| $19$ |
\( T^{2} + \cdots - 15492203120 \)
|
| $23$ |
\( T^{2} + \cdots - 239117414400 \)
|
| $29$ |
\( T^{2} + \cdots - 13616383922300 \)
|
| $31$ |
\( T^{2} + \cdots + 16415447040000 \)
|
| $37$ |
\( T^{2} + \cdots + 99286893737380 \)
|
| $41$ |
\( T^{2} + \cdots + 38475315093220 \)
|
| $43$ |
\( T^{2} + \cdots - 254550637865456 \)
|
| $47$ |
\( T^{2} + \cdots - 18\!\cdots\!24 \)
|
| $53$ |
\( T^{2} + \cdots - 14677210114460 \)
|
| $59$ |
\( T^{2} + \cdots - 99\!\cdots\!20 \)
|
| $61$ |
\( T^{2} + \cdots + 28\!\cdots\!96 \)
|
| $67$ |
\( T^{2} + \cdots + 20\!\cdots\!64 \)
|
| $71$ |
\( T^{2} + \cdots - 147594805309376 \)
|
| $73$ |
\( T^{2} + \cdots + 12\!\cdots\!60 \)
|
| $79$ |
\( T^{2} + \cdots - 19\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots + 60\!\cdots\!92 \)
|
| $89$ |
\( T^{2} + \cdots + 13\!\cdots\!20 \)
|
| $97$ |
\( T^{2} + \cdots + 68\!\cdots\!64 \)
|
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