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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 34x^{3} - 9x^{2} + 260x + 60 \)
|
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 34x^{3} - 9x^{2} + 260x + 60 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 14\nu^{3} - 30\nu^{2} - 237\nu + 78 ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 14\nu^{2} - 19\nu + 50 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 30\nu^{2} - 35\nu - 158 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 17\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{4} - 30\beta_{3} + 8\beta_{2} + 29\beta _1 + 242 \)
|
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 |
|
−5.34884 | 8.28008 | 20.6101 | −5.00000 | −44.2888 | −30.9947 | −67.4492 | 41.5597 | 26.7442 | |||||||||||||||||||||||||||||||||
| 1.2 | −4.67392 | −9.09294 | 13.8456 | −5.00000 | 42.4997 | 6.08885 | −27.3217 | 55.6816 | 23.3696 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.23053 | 2.78435 | −6.48580 | −5.00000 | −3.42623 | 24.1991 | 17.8252 | −19.2474 | 6.15265 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.26689 | −0.577397 | −2.86119 | −5.00000 | −1.30890 | −10.7178 | −24.6212 | −26.6666 | −11.3345 | ||||||||||||||||||||||||||||||||||
| 1.5 | 3.98640 | −7.39409 | 7.89136 | −5.00000 | −29.4758 | −3.57544 | −0.433099 | 27.6726 | −19.9320 | ||||||||||||||||||||||||||||||||||
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.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1035.4.a.m | 5 | ||
| 4.b | odd | 2 | 1 | 1840.4.a.p | 5 | ||
| 5.b | even | 2 | 1 | 575.4.a.k | 5 | ||
| 5.c | odd | 4 | 2 | 575.4.b.h | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.4.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 575.4.a.k | 5 | 5.b | even | 2 | 1 | ||
| 575.4.b.h | 10 | 5.c | odd | 4 | 2 | ||
| 1035.4.a.m | 5 | 3.b | odd | 2 | 1 | ||
| 1840.4.a.p | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 5T_{2}^{4} - 24T_{2}^{3} - 101T_{2}^{2} + 145T_{2} + 278 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 5 T^{4} + \cdots + 278 \)
|
| $3$ |
\( T^{5} + 6 T^{4} + \cdots + 895 \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( T^{5} + 15 T^{4} + \cdots + 175008 \)
|
| $11$ |
\( T^{5} + 153 T^{4} + \cdots - 58105432 \)
|
| $13$ |
\( T^{5} - 28 T^{4} + \cdots - 42528287 \)
|
| $17$ |
\( T^{5} + \cdots - 1269566848 \)
|
| $19$ |
\( T^{5} - 3 T^{4} + \cdots - 5566328 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 63213004636 \)
|
| $31$ |
\( T^{5} + \cdots + 341100199935 \)
|
| $37$ |
\( T^{5} + \cdots + 337199293312 \)
|
| $41$ |
\( T^{5} + \cdots - 554461833173 \)
|
| $43$ |
\( T^{5} + \cdots + 278531891200 \)
|
| $47$ |
\( T^{5} + \cdots + 6082562728660 \)
|
| $53$ |
\( T^{5} + \cdots - 1069522603168 \)
|
| $59$ |
\( T^{5} + \cdots - 446741827072 \)
|
| $61$ |
\( T^{5} + \cdots + 53636443112 \)
|
| $67$ |
\( T^{5} + \cdots - 19766839800960 \)
|
| $71$ |
\( T^{5} + \cdots + 256345940645 \)
|
| $73$ |
\( T^{5} + \cdots + 3947399121116 \)
|
| $79$ |
\( T^{5} + \cdots - 22913376438144 \)
|
| $83$ |
\( T^{5} + \cdots + 2501024408832 \)
|
| $89$ |
\( T^{5} + \cdots + 279901843479552 \)
|
| $97$ |
\( T^{5} + \cdots - 113500838416 \)
|
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