Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1617,4,Mod(1,1617)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, 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("1617.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1617 = 3 \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1617.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: | \(95.4060884793\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{5} - 28x^{3} - 11x^{2} + 108x - 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 231) |
| 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
\( x^{5} - 28x^{3} - 11x^{2} + 108x - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 26\nu^{2} - 17\nu + 64 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 28\nu^{2} - 57\nu + 74 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 21\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + 26\beta_{2} + 43\beta _1 + 222 \)
|
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 |
|
−4.15106 | −3.00000 | 9.23129 | −3.26204 | 12.4532 | 0 | −5.11115 | 9.00000 | 13.5409 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.300143 | −3.00000 | −7.90991 | 20.3930 | 0.900430 | 0 | 4.77526 | 9.00000 | −6.12084 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.232912 | −3.00000 | −7.94575 | −7.75746 | −0.698736 | 0 | −3.71396 | 9.00000 | −1.80680 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.85551 | −3.00000 | 6.86494 | −18.0422 | −11.5665 | 0 | −4.37623 | 9.00000 | −69.5618 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.36278 | −3.00000 | 20.7594 | 1.66863 | −16.0883 | 0 | 68.4261 | 9.00000 | 8.94849 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1617.4.a.p | 5 | |
| 7.b | odd | 2 | 1 | 231.4.a.l | ✓ | 5 | |
| 21.c | even | 2 | 1 | 693.4.a.n | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.4.a.l | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 693.4.a.n | 5 | 21.c | even | 2 | 1 | ||
| 1617.4.a.p | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1617))\):
|
\( T_{2}^{5} - 5T_{2}^{4} - 18T_{2}^{3} + 85T_{2}^{2} + 7T_{2} - 6 \)
|
|
\( T_{5}^{5} + 7T_{5}^{4} - 383T_{5}^{3} - 3499T_{5}^{2} - 2446T_{5} + 15536 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 5 T^{4} + \cdots - 6 \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( T^{5} + 7 T^{4} + \cdots + 15536 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} + 111 T^{4} + \cdots + 276332 \)
|
| $17$ |
\( T^{5} + \cdots + 3184406528 \)
|
| $19$ |
\( T^{5} + 111 T^{4} + \cdots + 2596968 \)
|
| $23$ |
\( T^{5} + \cdots - 9848447488 \)
|
| $29$ |
\( T^{5} + \cdots - 1678137108 \)
|
| $31$ |
\( T^{5} + \cdots + 97533466624 \)
|
| $37$ |
\( T^{5} + \cdots - 82159135548 \)
|
| $41$ |
\( T^{5} + \cdots + 2830976555008 \)
|
| $43$ |
\( T^{5} + \cdots - 1905935989632 \)
|
| $47$ |
\( T^{5} + \cdots - 417603397024 \)
|
| $53$ |
\( T^{5} + \cdots + 3927875616864 \)
|
| $59$ |
\( T^{5} + \cdots + 211529924045472 \)
|
| $61$ |
\( T^{5} + \cdots - 14480622579168 \)
|
| $67$ |
\( T^{5} + \cdots - 8392173156048 \)
|
| $71$ |
\( T^{5} + \cdots + 2460667275264 \)
|
| $73$ |
\( T^{5} + \cdots + 1051447598248 \)
|
| $79$ |
\( T^{5} + \cdots + 93230064402432 \)
|
| $83$ |
\( T^{5} + \cdots - 15123167361792 \)
|
| $89$ |
\( T^{5} + \cdots + 1811669774112 \)
|
| $97$ |
\( T^{5} + \cdots - 346762905716192 \)
|
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