Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [119,2,Mod(1,119)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(119, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("119.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 119 = 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 119.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: | \(0.950219784053\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.453749.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \)
|
| 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 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
\( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 13 \)
|
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 |
|
−2.32942 | −3.21594 | 3.42621 | −3.16902 | 7.49128 | −1.00000 | −3.32226 | 7.34225 | 7.38200 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.40868 | 2.55982 | −0.0156267 | 1.76660 | −3.60597 | −1.00000 | 2.83937 | 3.55270 | −2.48857 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.877834 | 0.907578 | −1.22941 | 3.03818 | 0.796703 | −1.00000 | −2.83488 | −2.17630 | 2.66702 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.36800 | 0.569378 | 3.60742 | −4.15465 | 1.34829 | −1.00000 | 3.80636 | −2.67581 | −9.83819 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.49227 | −2.82084 | 4.21140 | 2.51889 | −7.03030 | −1.00000 | 5.51141 | 4.95716 | 6.27775 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(17\) | \(-1\) |
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 | 119.2.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1071.2.a.m | 5 | ||
| 4.b | odd | 2 | 1 | 1904.2.a.t | 5 | ||
| 5.b | even | 2 | 1 | 2975.2.a.m | 5 | ||
| 7.b | odd | 2 | 1 | 833.2.a.g | 5 | ||
| 7.c | even | 3 | 2 | 833.2.e.i | 10 | ||
| 7.d | odd | 6 | 2 | 833.2.e.h | 10 | ||
| 8.b | even | 2 | 1 | 7616.2.a.bt | 5 | ||
| 8.d | odd | 2 | 1 | 7616.2.a.bq | 5 | ||
| 17.b | even | 2 | 1 | 2023.2.a.j | 5 | ||
| 21.c | even | 2 | 1 | 7497.2.a.br | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.2.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 833.2.a.g | 5 | 7.b | odd | 2 | 1 | ||
| 833.2.e.h | 10 | 7.d | odd | 6 | 2 | ||
| 833.2.e.i | 10 | 7.c | even | 3 | 2 | ||
| 1071.2.a.m | 5 | 3.b | odd | 2 | 1 | ||
| 1904.2.a.t | 5 | 4.b | odd | 2 | 1 | ||
| 2023.2.a.j | 5 | 17.b | even | 2 | 1 | ||
| 2975.2.a.m | 5 | 5.b | even | 2 | 1 | ||
| 7497.2.a.br | 5 | 21.c | even | 2 | 1 | ||
| 7616.2.a.bq | 5 | 8.d | odd | 2 | 1 | ||
| 7616.2.a.bt | 5 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 8T_{2}^{3} + 14T_{2}^{2} + 14T_{2} - 17 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(119))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 17 \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots - 12 \)
|
| $5$ |
\( T^{5} - 23 T^{3} + \cdots - 178 \)
|
| $7$ |
\( (T + 1)^{5} \)
|
| $11$ |
\( T^{5} + 2 T^{4} + \cdots - 192 \)
|
| $13$ |
\( T^{5} - 2 T^{4} + \cdots - 544 \)
|
| $17$ |
\( (T - 1)^{5} \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots - 64 \)
|
| $23$ |
\( T^{5} + 10 T^{4} + \cdots - 128 \)
|
| $29$ |
\( T^{5} + 8 T^{4} + \cdots + 2592 \)
|
| $31$ |
\( T^{5} - 33 T^{3} + \cdots - 16 \)
|
| $37$ |
\( T^{5} - 8 T^{4} + \cdots + 4384 \)
|
| $41$ |
\( T^{5} - 18 T^{4} + \cdots + 162 \)
|
| $43$ |
\( T^{5} - 8 T^{4} + \cdots - 1052 \)
|
| $47$ |
\( T^{5} + 10 T^{4} + \cdots - 2304 \)
|
| $53$ |
\( T^{5} - 4 T^{4} + \cdots + 138 \)
|
| $59$ |
\( T^{5} - 8 T^{4} + \cdots - 3072 \)
|
| $61$ |
\( T^{5} - 22 T^{4} + \cdots + 5542 \)
|
| $67$ |
\( T^{5} - 16 T^{4} + \cdots + 1868 \)
|
| $71$ |
\( T^{5} + 2 T^{4} + \cdots + 13696 \)
|
| $73$ |
\( T^{5} - 10 T^{4} + \cdots - 11118 \)
|
| $79$ |
\( T^{5} - 18 T^{4} + \cdots + 3072 \)
|
| $83$ |
\( T^{5} + 12 T^{4} + \cdots + 1984 \)
|
| $89$ |
\( T^{5} - 20 T^{4} + \cdots + 7456 \)
|
| $97$ |
\( T^{5} - 12 T^{4} + \cdots + 218 \)
|
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