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: | \(6\) |
| Coefficient field: | 6.6.32503921.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 5x^{4} + 13x^{3} + 9x^{2} - 11x - 7 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 5x^{4} + 13x^{3} + 9x^{2} - 11x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 4\nu^{3} + 11\nu^{2} + 3\nu - 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} - \nu^{3} + 14\nu^{2} - 5\nu - 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} - 2\nu^{3} + 17\nu^{2} - 3\nu - 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 3\beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{5} + 2\beta_{4} + \beta_{3} + 12\beta_{2} + 10\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( -13\beta_{5} + 10\beta_{4} + 4\beta_{3} + 37\beta_{2} + 36\beta _1 + 42 \)
|
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 |
|
−2.10292 | 0.742586 | 2.42225 | −1.00000 | −1.56160 | 2.60406 | −0.887964 | −2.44857 | 2.10292 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.961086 | 3.34672 | −1.07631 | −1.00000 | −3.21649 | −1.04792 | 2.95660 | 8.20055 | 0.961086 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.176734 | −2.74829 | −1.96877 | −1.00000 | 0.485715 | 4.38443 | 0.701414 | 4.55310 | 0.176734 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.67360 | 2.56351 | 0.800923 | −1.00000 | 4.29028 | −0.173417 | −2.00677 | 3.57157 | −1.67360 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.84056 | 0.291442 | 1.38764 | −1.00000 | 0.536415 | 5.13977 | −1.12707 | −2.91506 | −1.84056 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.72658 | −0.195967 | 5.43426 | −1.00000 | −0.534322 | −2.90692 | 9.36379 | −2.96160 | −2.72658 | |||||||||||||||||||||||||||||||||||||
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.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1935.2.a.z | 6 | ||
| 4.b | odd | 2 | 1 | 3440.2.a.x | 6 | ||
| 5.b | even | 2 | 1 | 1075.2.a.p | 6 | ||
| 5.c | odd | 4 | 2 | 1075.2.b.k | 12 | ||
| 15.d | odd | 2 | 1 | 9675.2.a.cl | 6 | ||
| 43.b | odd | 2 | 1 | 9245.2.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 215.2.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1075.2.a.p | 6 | 5.b | even | 2 | 1 | ||
| 1075.2.b.k | 12 | 5.c | odd | 4 | 2 | ||
| 1935.2.a.z | 6 | 3.b | odd | 2 | 1 | ||
| 3440.2.a.x | 6 | 4.b | odd | 2 | 1 | ||
| 9245.2.a.n | 6 | 43.b | odd | 2 | 1 | ||
| 9675.2.a.cl | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} - 5T_{2}^{4} + 17T_{2}^{3} + 3T_{2}^{2} - 17T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(215))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots - 3 \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 8 T^{5} + \cdots - 31 \)
|
| $11$ |
\( T^{6} - 41 T^{4} + \cdots - 93 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots - 448 \)
|
| $17$ |
\( T^{6} - 6 T^{5} + \cdots + 1344 \)
|
| $19$ |
\( T^{6} - 6 T^{5} + \cdots - 512 \)
|
| $23$ |
\( T^{6} - 96 T^{4} + \cdots - 5952 \)
|
| $29$ |
\( T^{6} + 10 T^{5} + \cdots + 5952 \)
|
| $31$ |
\( T^{6} - 97 T^{4} + \cdots - 10133 \)
|
| $37$ |
\( T^{6} - 28 T^{5} + \cdots - 29813 \)
|
| $41$ |
\( T^{6} + 6 T^{5} + \cdots - 10911 \)
|
| $43$ |
\( (T - 1)^{6} \)
|
| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 4416 \)
|
| $53$ |
\( T^{6} + 4 T^{5} + \cdots + 17088 \)
|
| $59$ |
\( T^{6} + 20 T^{5} + \cdots + 6987 \)
|
| $61$ |
\( T^{6} + 8 T^{5} + \cdots + 6848 \)
|
| $67$ |
\( T^{6} - 22 T^{5} + \cdots + 32192 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots + 192 \)
|
| $73$ |
\( T^{6} - 34 T^{5} + \cdots - 10133 \)
|
| $79$ |
\( T^{6} + 16 T^{5} + \cdots + 194267 \)
|
| $83$ |
\( T^{6} + 14 T^{5} + \cdots - 101952 \)
|
| $89$ |
\( T^{6} - 264 T^{4} + \cdots + 265152 \)
|
| $97$ |
\( T^{6} - 34 T^{5} + \cdots - 11776 \)
|
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