Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [476,4,Mod(1,476)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(476, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("476.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 476 = 2^{2} \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 476.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: | \(28.0849091627\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 87x^{4} + 184x^{3} + 2031x^{2} - 4232x - 7516 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 2x^{5} - 87x^{4} + 184x^{3} + 2031x^{2} - 4232x - 7516 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 45\nu^{3} - 151\nu^{2} - 47\nu + 574 ) / 117 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 30 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 10\nu^{4} - 71\nu^{3} - 538\nu^{2} + 1243\nu + 4314 ) / 78 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} - 7\nu^{4} + 155\nu^{3} + 387\nu^{2} - 2811\nu - 3077 ) / 39 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} - 3\beta_{2} + 39\beta _1 - 17 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 10\beta_{4} + 53\beta_{3} + 3\beta_{2} - 14\beta _1 + 1180 \)
|
| \(\nu^{5}\) | \(=\) |
\( 51\beta_{5} + 120\beta_{4} + 8\beta_{3} - 243\beta_{2} + 1666\beta _1 - 1181 \)
|
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 |
|
0 | −6.90846 | 0 | 21.0845 | 0 | −7.00000 | 0 | 20.7268 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.11037 | 0 | −11.9145 | 0 | −7.00000 | 0 | 10.3366 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.20043 | 0 | 6.13806 | 0 | −7.00000 | 0 | −25.5590 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 4.21885 | 0 | 3.60198 | 0 | −7.00000 | 0 | −9.20134 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.08046 | 0 | 5.06198 | 0 | −7.00000 | 0 | −1.18888 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 6.91995 | 0 | −13.9720 | 0 | −7.00000 | 0 | 20.8857 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(17\) | \(1\) |
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 | 476.4.a.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 1904.4.a.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 476.4.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1904.4.a.n | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2T_{3}^{5} - 87T_{3}^{4} + 184T_{3}^{3} + 2031T_{3}^{2} - 4232T_{3} - 7516 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(476))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 7516 \)
|
| $5$ |
\( T^{6} - 10 T^{5} + \cdots + 392816 \)
|
| $7$ |
\( (T + 7)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 1238404992 \)
|
| $13$ |
\( T^{6} - 42 T^{5} + \cdots + 174989184 \)
|
| $17$ |
\( (T + 17)^{6} \)
|
| $19$ |
\( T^{6} + \cdots + 3454636032 \)
|
| $23$ |
\( T^{6} + \cdots + 3587305472 \)
|
| $29$ |
\( T^{6} + \cdots - 847432141824 \)
|
| $31$ |
\( T^{6} + \cdots - 5667213957376 \)
|
| $37$ |
\( T^{6} + \cdots + 574587071793536 \)
|
| $41$ |
\( T^{6} + \cdots + 53622726394956 \)
|
| $43$ |
\( T^{6} + \cdots - 17286410134272 \)
|
| $47$ |
\( T^{6} + \cdots + 17878379346432 \)
|
| $53$ |
\( T^{6} + \cdots - 237934889640468 \)
|
| $59$ |
\( T^{6} + \cdots + 25\!\cdots\!56 \)
|
| $61$ |
\( T^{6} + \cdots + 59\!\cdots\!36 \)
|
| $67$ |
\( T^{6} + \cdots + 61\!\cdots\!32 \)
|
| $71$ |
\( T^{6} + \cdots - 608998002329088 \)
|
| $73$ |
\( T^{6} + \cdots + 92\!\cdots\!72 \)
|
| $79$ |
\( T^{6} + \cdots + 17\!\cdots\!36 \)
|
| $83$ |
\( T^{6} + \cdots - 40\!\cdots\!16 \)
|
| $89$ |
\( T^{6} + \cdots + 19\!\cdots\!16 \)
|
| $97$ |
\( T^{6} + \cdots - 61\!\cdots\!88 \)
|
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