Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [215,2,Mod(1,215)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(215, 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("215.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 215 = 5 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 215.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: | \(1.71678364346\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1933097.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 \)
|
| 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
\( x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + 23 \)
|
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 |
|
−2.48695 | 2.94683 | 4.18492 | 1.00000 | −7.32862 | 3.60359 | −5.43378 | 5.68382 | −2.48695 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.667116 | −3.03868 | −1.55496 | 1.00000 | 2.02715 | −4.17800 | 2.37157 | 6.23360 | −0.667116 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.434772 | 2.09168 | −1.81097 | 1.00000 | 0.909404 | 3.42802 | −1.65691 | 1.37512 | 0.434772 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.20940 | 0.261901 | 2.88146 | 1.00000 | 0.578644 | −0.988801 | 1.94750 | −2.93141 | 2.20940 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.50989 | −3.26173 | 4.29955 | 1.00000 | −8.18658 | 3.13519 | 5.77162 | 7.63887 | 2.50989 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(43\) | \(1\) |
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 | 215.2.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1935.2.a.u | 5 | ||
| 4.b | odd | 2 | 1 | 3440.2.a.w | 5 | ||
| 5.b | even | 2 | 1 | 1075.2.a.m | 5 | ||
| 5.c | odd | 4 | 2 | 1075.2.b.h | 10 | ||
| 15.d | odd | 2 | 1 | 9675.2.a.ch | 5 | ||
| 43.b | odd | 2 | 1 | 9245.2.a.l | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.2.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1075.2.a.m | 5 | 5.b | even | 2 | 1 | ||
| 1075.2.b.h | 10 | 5.c | odd | 4 | 2 | ||
| 1935.2.a.u | 5 | 3.b | odd | 2 | 1 | ||
| 3440.2.a.w | 5 | 4.b | odd | 2 | 1 | ||
| 9245.2.a.l | 5 | 43.b | odd | 2 | 1 | ||
| 9675.2.a.ch | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 7T_{2}^{3} + 13T_{2}^{2} + 5T_{2} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(215))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 4 \)
|
| $3$ |
\( T^{5} + T^{4} + \cdots - 16 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 5 T^{4} + \cdots - 160 \)
|
| $11$ |
\( T^{5} + 6 T^{4} + \cdots - 12 \)
|
| $13$ |
\( T^{5} - 5 T^{4} + \cdots - 2000 \)
|
| $17$ |
\( T^{5} + 17 T^{4} + \cdots - 16 \)
|
| $19$ |
\( T^{5} + 6 T^{4} + \cdots + 4608 \)
|
| $23$ |
\( T^{5} - T^{4} + \cdots - 384 \)
|
| $29$ |
\( T^{5} - 6 T^{4} + \cdots + 1152 \)
|
| $31$ |
\( T^{5} - 6 T^{4} + \cdots + 128 \)
|
| $37$ |
\( T^{5} - 5 T^{4} + \cdots - 400 \)
|
| $41$ |
\( T^{5} - 2 T^{4} + \cdots + 30 \)
|
| $43$ |
\( (T + 1)^{5} \)
|
| $47$ |
\( T^{5} - 124 T^{3} + \cdots - 2048 \)
|
| $53$ |
\( T^{5} + 23 T^{4} + \cdots + 400 \)
|
| $59$ |
\( T^{5} + T^{4} + \cdots - 16 \)
|
| $61$ |
\( T^{5} - 20 T^{4} + \cdots - 60672 \)
|
| $67$ |
\( T^{5} - 21 T^{4} + \cdots + 96 \)
|
| $71$ |
\( T^{5} - 4 T^{4} + \cdots - 20352 \)
|
| $73$ |
\( T^{5} - 5 T^{4} + \cdots + 1112 \)
|
| $79$ |
\( T^{5} - 41 T^{4} + \cdots - 18688 \)
|
| $83$ |
\( T^{5} + 7 T^{4} + \cdots + 2400 \)
|
| $89$ |
\( T^{5} - 20 T^{4} + \cdots - 2656 \)
|
| $97$ |
\( T^{5} - 37 T^{4} + \cdots + 1152 \)
|
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