Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,4,Mod(1,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, 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("1216.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1216 = 2^{6} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1216.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: | \(71.7463225670\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 608) |
| 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
\( x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{3} - 15\nu^{2} + 177\nu + 24 ) / 22 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{3} - 15\nu^{2} + 89\nu + 68 ) / 22 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{4} - 13\nu^{3} - 119\nu^{2} - 17\nu + 153 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} - \nu^{3} + 111\nu^{2} + 151\nu - 270 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 5\beta_{2} + \beta _1 + 25 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{4} + 6\beta_{3} - 61\beta_{2} + 25\beta _1 + 176 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 29\beta_{4} + 36\beta_{3} - 200\beta_{2} + 58\beta _1 + 766 ) / 2 \)
|
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 | −7.68562 | 0 | 15.2627 | 0 | −2.79440 | 0 | 32.0687 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.75519 | 0 | −10.5762 | 0 | −30.4413 | 0 | −4.38817 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.55330 | 0 | −7.40075 | 0 | 10.4962 | 0 | −20.4807 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.67449 | 0 | 7.12795 | 0 | −0.511313 | 0 | −13.4981 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.31962 | 0 | −9.41363 | 0 | 16.2508 | 0 | 1.29831 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(19\) | \(-1\) |
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 | 1216.4.a.y | 5 | |
| 4.b | odd | 2 | 1 | 1216.4.a.bd | 5 | ||
| 8.b | even | 2 | 1 | 608.4.a.h | yes | 5 | |
| 8.d | odd | 2 | 1 | 608.4.a.e | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 608.4.a.e | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 608.4.a.h | yes | 5 | 8.b | even | 2 | 1 | |
| 1216.4.a.y | 5 | 1.a | even | 1 | 1 | trivial | |
| 1216.4.a.bd | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1216))\):
|
\( T_{3}^{5} + 6T_{3}^{4} - 47T_{3}^{3} - 228T_{3}^{2} + 496T_{3} + 1824 \)
|
|
\( T_{5}^{5} + 5T_{5}^{4} - 257T_{5}^{3} - 1825T_{5}^{2} + 10428T_{5} + 80160 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 6 T^{4} + \cdots + 1824 \)
|
| $5$ |
\( T^{5} + 5 T^{4} + \cdots + 80160 \)
|
| $7$ |
\( T^{5} + 7 T^{4} + \cdots + 7419 \)
|
| $11$ |
\( T^{5} + 13 T^{4} + \cdots + 122580 \)
|
| $13$ |
\( T^{5} - 72 T^{4} + \cdots - 527040 \)
|
| $17$ |
\( T^{5} + 59 T^{4} + \cdots - 297647925 \)
|
| $19$ |
\( (T - 19)^{5} \)
|
| $23$ |
\( T^{5} + 52 T^{4} + \cdots + 304895232 \)
|
| $29$ |
\( T^{5} + \cdots - 106742590368 \)
|
| $31$ |
\( T^{5} + \cdots - 53136006400 \)
|
| $37$ |
\( T^{5} + \cdots - 4385970560 \)
|
| $41$ |
\( T^{5} + \cdots - 855201369600 \)
|
| $43$ |
\( T^{5} + \cdots - 197931733952 \)
|
| $47$ |
\( T^{5} + \cdots - 2970610340352 \)
|
| $53$ |
\( T^{5} + \cdots + 173906788200 \)
|
| $59$ |
\( T^{5} + \cdots + 131722274520 \)
|
| $61$ |
\( T^{5} + \cdots + 2650589198500 \)
|
| $67$ |
\( T^{5} + \cdots + 823051244968 \)
|
| $71$ |
\( T^{5} + \cdots - 679526470240 \)
|
| $73$ |
\( T^{5} + \cdots + 2709187037875 \)
|
| $79$ |
\( T^{5} + \cdots - 17190104220000 \)
|
| $83$ |
\( T^{5} + \cdots - 297963542605824 \)
|
| $89$ |
\( T^{5} + \cdots + 414712220295168 \)
|
| $97$ |
\( T^{5} + \cdots + 212723907840 \)
|
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