Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [115,4,Mod(1,115)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("115.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 115 = 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 115.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: | \(6.78521965066\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{109})\). 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 |
|
−3.00000 | −6.72015 | 1.00000 | 5.00000 | 20.1605 | 26.6008 | 21.0000 | 18.1605 | −15.0000 | ||||||||||||||||||||||||
| 1.2 | −3.00000 | 3.72015 | 1.00000 | 5.00000 | −11.1605 | −25.6008 | 21.0000 | −13.1605 | −15.0000 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 115.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 1035.4.a.g | 2 | ||
| 4.b | odd | 2 | 1 | 1840.4.a.h | 2 | ||
| 5.b | even | 2 | 1 | 575.4.a.h | 2 | ||
| 5.c | odd | 4 | 2 | 575.4.b.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 575.4.a.h | 2 | 5.b | even | 2 | 1 | ||
| 575.4.b.f | 4 | 5.c | odd | 4 | 2 | ||
| 1035.4.a.g | 2 | 3.b | odd | 2 | 1 | ||
| 1840.4.a.h | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 3 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 3)^{2} \)
|
| $3$ |
\( T^{2} + 3T - 25 \)
|
| $5$ |
\( (T - 5)^{2} \)
|
| $7$ |
\( T^{2} - T - 681 \)
|
| $11$ |
\( T^{2} + 27T - 499 \)
|
| $13$ |
\( T^{2} + 15T - 189 \)
|
| $17$ |
\( T^{2} + 79T + 225 \)
|
| $19$ |
\( T^{2} + 71T - 3345 \)
|
| $23$ |
\( (T + 23)^{2} \)
|
| $29$ |
\( T^{2} + 430T + 43500 \)
|
| $31$ |
\( T^{2} + 305T + 15381 \)
|
| $37$ |
\( T^{2} + 68T - 20208 \)
|
| $41$ |
\( T^{2} + 593T + 84615 \)
|
| $43$ |
\( T^{2} - 648T + 103232 \)
|
| $47$ |
\( T^{2} - 382T - 112740 \)
|
| $53$ |
\( T^{2} + 464T + 1068 \)
|
| $59$ |
\( T^{2} + 18T - 79380 \)
|
| $61$ |
\( T^{2} + 7T - 88523 \)
|
| $67$ |
\( T^{2} - 60T - 156496 \)
|
| $71$ |
\( T^{2} + 1029T + 8315 \)
|
| $73$ |
\( T^{2} - 74T - 103380 \)
|
| $79$ |
\( T^{2} - 692T - 686448 \)
|
| $83$ |
\( T^{2} + 1460T - 32156 \)
|
| $89$ |
\( T^{2} + 220T - 870800 \)
|
| $97$ |
\( T^{2} - 1339 T - 674715 \)
|
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