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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{4729}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1182 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{4729})\). 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 |
|
−43.8839 | 0 | 1413.79 | 625.000 | 0 | −7861.50 | −39574.2 | 0 | −27427.4 | ||||||||||||||||||||||||
| 1.2 | 24.8839 | 0 | 107.207 | 625.000 | 0 | −4010.50 | −10072.8 | 0 | 15552.4 | |||||||||||||||||||||||||
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.e | 2 | |
| 3.b | odd | 2 | 1 | 15.10.a.c | ✓ | 2 | |
| 5.b | even | 2 | 1 | 225.10.a.j | 2 | ||
| 5.c | odd | 4 | 2 | 225.10.b.g | 4 | ||
| 12.b | even | 2 | 1 | 240.10.a.m | 2 | ||
| 15.d | odd | 2 | 1 | 75.10.a.g | 2 | ||
| 15.e | even | 4 | 2 | 75.10.b.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.10.a.c | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 45.10.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 75.10.a.g | 2 | 15.d | odd | 2 | 1 | ||
| 75.10.b.e | 4 | 15.e | even | 4 | 2 | ||
| 225.10.a.j | 2 | 5.b | even | 2 | 1 | ||
| 225.10.b.g | 4 | 5.c | odd | 4 | 2 | ||
| 240.10.a.m | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 19T_{2} - 1092 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 19T - 1092 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T - 625)^{2} \)
|
| $7$ |
\( T^{2} + 11872 T + 31528560 \)
|
| $11$ |
\( T^{2} + \cdots - 4189882368 \)
|
| $13$ |
\( T^{2} + \cdots + 2896150388 \)
|
| $17$ |
\( T^{2} + \cdots + 31364196084 \)
|
| $19$ |
\( T^{2} + \cdots + 39554119280 \)
|
| $23$ |
\( T^{2} + \cdots - 4977578430720 \)
|
| $29$ |
\( T^{2} + \cdots - 180861933660 \)
|
| $31$ |
\( T^{2} + \cdots - 463313088000 \)
|
| $37$ |
\( T^{2} + \cdots - 28861754638220 \)
|
| $41$ |
\( T^{2} + \cdots - 210775232832060 \)
|
| $43$ |
\( T^{2} + \cdots + 165047750825744 \)
|
| $47$ |
\( T^{2} + \cdots + 20\!\cdots\!76 \)
|
| $53$ |
\( T^{2} + \cdots + 11555844938820 \)
|
| $59$ |
\( T^{2} + \cdots + 57\!\cdots\!60 \)
|
| $61$ |
\( T^{2} + \cdots - 85608279866044 \)
|
| $67$ |
\( T^{2} + \cdots - 65\!\cdots\!56 \)
|
| $71$ |
\( T^{2} + \cdots - 45\!\cdots\!56 \)
|
| $73$ |
\( T^{2} + \cdots - 49\!\cdots\!00 \)
|
| $79$ |
\( T^{2} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots - 10\!\cdots\!68 \)
|
| $89$ |
\( T^{2} + \cdots - 89\!\cdots\!40 \)
|
| $97$ |
\( T^{2} + \cdots - 22\!\cdots\!96 \)
|
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