Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [289,4,Mod(1,289)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("289.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 289.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: | \(17.0515519917\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2555057.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 21x^{2} - 4x + 36 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 21x^{2} - 4x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 15\nu + 18 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 19\nu - 26 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + \beta_{2} + 18\beta _1 + 15 \)
|
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 |
|
−3.68488 | 9.05894 | 5.57832 | −7.08909 | −33.3811 | −28.1854 | 8.92359 | 55.0643 | 26.1224 | ||||||||||||||||||||||||||||||
| 1.2 | −1.58184 | −4.98387 | −5.49778 | −14.4471 | 7.88368 | 29.4104 | 21.3513 | −2.16104 | 22.8530 | |||||||||||||||||||||||||||||||
| 1.3 | 1.22501 | 0.534684 | −6.49935 | 20.9528 | 0.654993 | −15.0235 | −17.7618 | −26.7141 | 25.6674 | |||||||||||||||||||||||||||||||
| 1.4 | 5.04171 | −2.60975 | 17.4188 | −13.4166 | −13.1576 | −22.2015 | 47.4869 | −20.1892 | −67.6428 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(-1\) |
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 | 289.4.a.d | yes | 4 |
| 17.b | even | 2 | 1 | 289.4.a.c | ✓ | 4 | |
| 17.c | even | 4 | 2 | 289.4.b.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 289.4.a.c | ✓ | 4 | 17.b | even | 2 | 1 | |
| 289.4.a.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 289.4.b.d | 8 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(289))\):
|
\( T_{2}^{4} - T_{2}^{3} - 21T_{2}^{2} - 4T_{2} + 36 \)
|
|
\( T_{3}^{4} - 2T_{3}^{3} - 55T_{3}^{2} - 88T_{3} + 63 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 36 \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 63 \)
|
| $5$ |
\( T^{4} + 14 T^{3} + \cdots - 28791 \)
|
| $7$ |
\( T^{4} + 36 T^{3} + \cdots - 276489 \)
|
| $11$ |
\( T^{4} + 10 T^{3} + \cdots + 316467 \)
|
| $13$ |
\( T^{4} + 22 T^{3} + \cdots - 26579 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} - 22 T^{3} + \cdots + 4371507 \)
|
| $23$ |
\( T^{4} + 380 T^{3} + \cdots - 176752917 \)
|
| $29$ |
\( T^{4} - 78 T^{3} + \cdots - 107811 \)
|
| $31$ |
\( T^{4} + 362 T^{3} + \cdots - 38411429 \)
|
| $37$ |
\( T^{4} + \cdots - 1758989168 \)
|
| $41$ |
\( T^{4} + \cdots + 1180838736 \)
|
| $43$ |
\( T^{4} + \cdots + 4351627359 \)
|
| $47$ |
\( T^{4} - 10 T^{3} + \cdots - 153320769 \)
|
| $53$ |
\( T^{4} + \cdots + 3826783197 \)
|
| $59$ |
\( T^{4} + \cdots - 7997430672 \)
|
| $61$ |
\( T^{4} + \cdots + 26730644633 \)
|
| $67$ |
\( T^{4} + \cdots + 30297331184 \)
|
| $71$ |
\( T^{4} + 1116 T^{3} + \cdots + 589600944 \)
|
| $73$ |
\( T^{4} + \cdots - 46400632803 \)
|
| $79$ |
\( T^{4} + \cdots + 229066578288 \)
|
| $83$ |
\( T^{4} + \cdots + 1637812071 \)
|
| $89$ |
\( T^{4} + \cdots + 24704450064 \)
|
| $97$ |
\( T^{4} + \cdots + 2200880766749 \)
|
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