Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [455,2,Mod(1,455)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(455, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("455.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 455 = 5 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 455.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: | \(3.63319329197\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 15x^{5} - 2x^{4} + 66x^{3} + 17x^{2} - 72x - 19 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 15x^{5} - 2x^{4} + 66x^{3} + 17x^{2} - 72x - 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 6\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} - \nu^{5} - 26\nu^{4} + 12\nu^{3} + 26\nu^{2} - 37\nu + 39 ) / 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{5} + 11\nu^{4} + 17\nu^{3} - 25\nu^{2} - 25\nu + 1 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} - 11\nu^{4} - 17\nu^{3} + 32\nu^{2} + 18\nu - 29 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{5} + 18\nu^{4} - 46\nu^{3} - 88\nu^{2} + 73\nu + 71 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 8\beta_{5} + 9\beta_{4} + \beta_{3} + 9\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 9\beta_{2} + 40\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{6} + 61\beta_{5} + 69\beta_{4} + 13\beta_{3} - \beta_{2} + 71\beta _1 + 160 \)
|
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 |
|
−2.74966 | 2.33807 | 5.56065 | −1.00000 | −6.42890 | 1.00000 | −9.79059 | 2.46656 | 2.74966 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.57143 | −3.23325 | 4.61227 | −1.00000 | 8.31409 | 1.00000 | −6.71728 | 7.45393 | 2.57143 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.24464 | −0.624550 | −0.450880 | −1.00000 | 0.777337 | 1.00000 | 3.05045 | −2.60994 | 1.24464 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.264180 | 3.32159 | −1.93021 | −1.00000 | 0.877498 | 1.00000 | −1.03828 | 8.03297 | −0.264180 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.33646 | −3.10904 | −0.213862 | −1.00000 | −4.15513 | 1.00000 | −2.95875 | 6.66616 | −1.33646 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.26348 | 2.09164 | 3.12334 | −1.00000 | 4.73438 | 1.00000 | 2.54265 | 1.37496 | −2.26348 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.70161 | −0.784452 | 5.29870 | −1.00000 | −2.11928 | 1.00000 | 8.91179 | −2.38464 | −2.70161 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 455.2.a.f | ✓ | 7 |
| 3.b | odd | 2 | 1 | 4095.2.a.bo | 7 | ||
| 4.b | odd | 2 | 1 | 7280.2.a.cg | 7 | ||
| 5.b | even | 2 | 1 | 2275.2.a.x | 7 | ||
| 7.b | odd | 2 | 1 | 3185.2.a.u | 7 | ||
| 13.b | even | 2 | 1 | 5915.2.a.ba | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.a.f | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2275.2.a.x | 7 | 5.b | even | 2 | 1 | ||
| 3185.2.a.u | 7 | 7.b | odd | 2 | 1 | ||
| 4095.2.a.bo | 7 | 3.b | odd | 2 | 1 | ||
| 5915.2.a.ba | 7 | 13.b | even | 2 | 1 | ||
| 7280.2.a.cg | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 15T_{2}^{5} + 2T_{2}^{4} + 66T_{2}^{3} - 17T_{2}^{2} - 72T_{2} + 19 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(455))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 15 T^{5} + \cdots + 19 \)
|
| $3$ |
\( T^{7} - 21 T^{5} + \cdots - 80 \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} - 6 T^{6} + \cdots + 256 \)
|
| $13$ |
\( (T + 1)^{7} \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots - 50056 \)
|
| $19$ |
\( T^{7} - 6 T^{6} + \cdots - 5808 \)
|
| $23$ |
\( T^{7} - 4 T^{6} + \cdots - 19200 \)
|
| $29$ |
\( T^{7} - 16 T^{6} + \cdots - 4952 \)
|
| $31$ |
\( T^{7} + 6 T^{6} + \cdots - 120784 \)
|
| $37$ |
\( T^{7} + 6 T^{6} + \cdots + 1000 \)
|
| $41$ |
\( T^{7} - 14 T^{6} + \cdots + 6024 \)
|
| $43$ |
\( T^{7} - 22 T^{6} + \cdots - 108800 \)
|
| $47$ |
\( T^{7} + 10 T^{6} + \cdots + 8192 \)
|
| $53$ |
\( T^{7} + 20 T^{6} + \cdots + 70016 \)
|
| $59$ |
\( T^{7} + 22 T^{6} + \cdots - 912 \)
|
| $61$ |
\( T^{7} - 10 T^{6} + \cdots + 776576 \)
|
| $67$ |
\( T^{7} - 12 T^{6} + \cdots - 153152 \)
|
| $71$ |
\( T^{7} - 12 T^{6} + \cdots + 43776 \)
|
| $73$ |
\( T^{7} - 28 T^{6} + \cdots - 117888 \)
|
| $79$ |
\( T^{7} - 28 T^{6} + \cdots - 75648 \)
|
| $83$ |
\( T^{7} - 6 T^{6} + \cdots - 74752 \)
|
| $89$ |
\( T^{7} + 14 T^{6} + \cdots + 858568 \)
|
| $97$ |
\( T^{7} - 8 T^{6} + \cdots - 47744 \)
|
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