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: | 5.5.973904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \)
|
| 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
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 4\nu^{3} + 2\nu^{2} - 12\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu + 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 3\beta_{3} + 10\beta_{2} + 20\beta _1 + 18 \)
|
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 |
|
−2.47408 | 2.38679 | 4.12105 | −1.00000 | −5.90509 | 4.78404 | −5.24765 | 2.69675 | 2.47408 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.13359 | −1.10563 | −0.714970 | −1.00000 | 1.25333 | 2.46164 | 3.07767 | −1.77758 | 1.13359 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.728950 | 2.62871 | −1.46863 | −1.00000 | 1.91620 | 2.55244 | −2.52846 | 3.91009 | −0.728950 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.15510 | 1.38311 | 2.64446 | −1.00000 | 2.98075 | −2.62521 | 1.38887 | −1.08699 | −2.15510 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.72362 | −2.29298 | 5.41809 | −1.00000 | −6.24519 | 3.82710 | 9.30957 | 2.25774 | −2.72362 | ||||||||||||||||||||||||||||||||||
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.e | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1665.2.a.p | 5 | ||
| 4.b | odd | 2 | 1 | 2960.2.a.w | 5 | ||
| 5.b | even | 2 | 1 | 925.2.a.f | 5 | ||
| 5.c | odd | 4 | 2 | 925.2.b.f | 10 | ||
| 7.b | odd | 2 | 1 | 9065.2.a.k | 5 | ||
| 15.d | odd | 2 | 1 | 8325.2.a.ch | 5 | ||
| 37.b | even | 2 | 1 | 6845.2.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 185.2.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 925.2.a.f | 5 | 5.b | even | 2 | 1 | ||
| 925.2.b.f | 10 | 5.c | odd | 4 | 2 | ||
| 1665.2.a.p | 5 | 3.b | odd | 2 | 1 | ||
| 2960.2.a.w | 5 | 4.b | odd | 2 | 1 | ||
| 6845.2.a.f | 5 | 37.b | even | 2 | 1 | ||
| 8325.2.a.ch | 5 | 15.d | odd | 2 | 1 | ||
| 9065.2.a.k | 5 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 8T_{2}^{3} + 14T_{2}^{2} + 11T_{2} - 12 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(185))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 12 \)
|
| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 22 \)
|
| $5$ |
\( (T + 1)^{5} \)
|
| $7$ |
\( T^{5} - 11 T^{4} + \cdots + 302 \)
|
| $11$ |
\( T^{5} + 5 T^{4} + \cdots + 96 \)
|
| $13$ |
\( T^{5} - 4 T^{4} + \cdots - 256 \)
|
| $17$ |
\( T^{5} - 52 T^{3} + \cdots + 192 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots - 8 \)
|
| $23$ |
\( T^{5} - 4 T^{4} + \cdots + 192 \)
|
| $29$ |
\( T^{5} + 4 T^{4} + \cdots - 192 \)
|
| $31$ |
\( T^{5} - 8 T^{4} + \cdots + 324 \)
|
| $37$ |
\( (T - 1)^{5} \)
|
| $41$ |
\( T^{5} + 5 T^{4} + \cdots + 96 \)
|
| $43$ |
\( T^{5} - 10 T^{4} + \cdots - 2528 \)
|
| $47$ |
\( T^{5} - 7 T^{4} + \cdots - 978 \)
|
| $53$ |
\( T^{5} + T^{4} + \cdots + 528 \)
|
| $59$ |
\( T^{5} + 30 T^{4} + \cdots - 576 \)
|
| $61$ |
\( T^{5} + 14 T^{4} + \cdots + 3296 \)
|
| $67$ |
\( T^{5} - 24 T^{4} + \cdots + 10952 \)
|
| $71$ |
\( T^{5} + 7 T^{4} + \cdots - 7104 \)
|
| $73$ |
\( T^{5} - 5 T^{4} + \cdots + 368 \)
|
| $79$ |
\( T^{5} - 28 T^{4} + \cdots - 19508 \)
|
| $83$ |
\( T^{5} - 27 T^{4} + \cdots + 4818 \)
|
| $89$ |
\( T^{5} - 6 T^{4} + \cdots + 22944 \)
|
| $97$ |
\( T^{5} + 26 T^{4} + \cdots - 166976 \)
|
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