Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [185,2,Mod(1,185)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, 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("185.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 185 = 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 185.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: | \(1.47723243739\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.368464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \)
|
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 2\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 4\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 10\nu^{2} - 4\nu + 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} - 10\beta_{3} + 10\beta_{2} + 14\beta _1 + 18 \)
|
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 |
|
−2.66356 | −0.671838 | 5.09455 | 1.00000 | 1.78948 | −0.305070 | −8.24252 | −2.54863 | −2.66356 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.25268 | 2.51369 | −0.430798 | 1.00000 | −3.14884 | 0.281955 | 3.04501 | 3.31863 | −1.25268 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.180152 | −3.06709 | −1.96755 | 1.00000 | −0.552543 | 4.41500 | −0.714762 | 6.40702 | 0.180152 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.46516 | 0.744131 | 0.146703 | 1.00000 | 1.09027 | 3.94357 | −2.71538 | −2.44627 | 1.46516 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.27092 | −0.518894 | 3.15709 | 1.00000 | −1.17837 | −1.33546 | 2.62766 | −2.73075 | 2.27092 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(37\) | \(1\) |
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 | 185.2.a.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1665.2.a.q | 5 | ||
| 4.b | odd | 2 | 1 | 2960.2.a.ba | 5 | ||
| 5.b | even | 2 | 1 | 925.2.a.h | 5 | ||
| 5.c | odd | 4 | 2 | 925.2.b.g | 10 | ||
| 7.b | odd | 2 | 1 | 9065.2.a.j | 5 | ||
| 15.d | odd | 2 | 1 | 8325.2.a.cc | 5 | ||
| 37.b | even | 2 | 1 | 6845.2.a.g | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 925.2.a.h | 5 | 5.b | even | 2 | 1 | ||
| 925.2.b.g | 10 | 5.c | odd | 4 | 2 | ||
| 1665.2.a.q | 5 | 3.b | odd | 2 | 1 | ||
| 2960.2.a.ba | 5 | 4.b | odd | 2 | 1 | ||
| 6845.2.a.g | 5 | 37.b | even | 2 | 1 | ||
| 8325.2.a.cc | 5 | 15.d | odd | 2 | 1 | ||
| 9065.2.a.j | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 8T_{2}^{3} + 2T_{2}^{2} + 11T_{2} - 2 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(185))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 8 T^{3} + \cdots - 2 \)
|
| $3$ |
\( T^{5} + T^{4} - 8 T^{3} + \cdots + 2 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 7 T^{4} + \cdots - 2 \)
|
| $11$ |
\( T^{5} - 7 T^{4} + \cdots - 32 \)
|
| $13$ |
\( T^{5} - 2 T^{4} + \cdots - 88 \)
|
| $17$ |
\( T^{5} + 8 T^{4} + \cdots - 32 \)
|
| $19$ |
\( T^{5} - 14 T^{4} + \cdots + 2224 \)
|
| $23$ |
\( T^{5} - 2 T^{4} + \cdots + 1504 \)
|
| $29$ |
\( T^{5} - 2 T^{4} + \cdots + 32 \)
|
| $31$ |
\( T^{5} - 8 T^{4} + \cdots - 3016 \)
|
| $37$ |
\( (T + 1)^{5} \)
|
| $41$ |
\( T^{5} + 9 T^{4} + \cdots - 928 \)
|
| $43$ |
\( T^{5} - 14 T^{4} + \cdots - 1312 \)
|
| $47$ |
\( T^{5} + 5 T^{4} + \cdots - 3994 \)
|
| $53$ |
\( T^{5} + 15 T^{4} + \cdots - 16 \)
|
| $59$ |
\( T^{5} - 12 T^{4} + \cdots + 5456 \)
|
| $61$ |
\( T^{5} - 12 T^{4} + \cdots - 512 \)
|
| $67$ |
\( T^{5} + 2 T^{4} + \cdots - 7568 \)
|
| $71$ |
\( T^{5} - 13 T^{4} + \cdots - 291136 \)
|
| $73$ |
\( T^{5} + 5 T^{4} + \cdots + 176 \)
|
| $79$ |
\( T^{5} - 36 T^{4} + \cdots + 5912 \)
|
| $83$ |
\( T^{5} + 9 T^{4} + \cdots + 314 \)
|
| $89$ |
\( T^{5} + 16 T^{4} + \cdots + 1856 \)
|
| $97$ |
\( T^{5} - 464 T^{3} + \cdots - 193408 \)
|
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