Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,6,Mod(1,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 17.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: | \(2.72652493682\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5416116.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 44x^{2} + 108x + 68 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 44x^{2} + 108x + 68 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} - 28\nu + 9 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} - 4\nu^{2} + 76\nu - 23 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 9\nu^{2} - 56\nu - 92 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2\beta _1 + 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + 7\beta_{2} + 23\beta _1 - 46 \)
|
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 |
|
−9.25343 | −11.4517 | 53.6260 | 29.5295 | 105.967 | 166.170 | −200.115 | −111.859 | −273.249 | ||||||||||||||||||||||||||||||
| 1.2 | −2.10712 | 18.1466 | −27.5600 | 101.720 | −38.2370 | 68.4337 | 125.500 | 86.2978 | −214.337 | |||||||||||||||||||||||||||||||
| 1.3 | 5.79803 | 23.9305 | 1.61720 | −89.8579 | 138.750 | 150.881 | −176.161 | 329.669 | −520.999 | |||||||||||||||||||||||||||||||
| 1.4 | 8.56252 | −2.62540 | 41.3168 | 38.6084 | −22.4800 | −101.485 | 79.7753 | −236.107 | 330.585 | |||||||||||||||||||||||||||||||
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 | 17.6.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 153.6.a.f | 4 | ||
| 4.b | odd | 2 | 1 | 272.6.a.j | 4 | ||
| 5.b | even | 2 | 1 | 425.6.a.c | 4 | ||
| 7.b | odd | 2 | 1 | 833.6.a.c | 4 | ||
| 8.b | even | 2 | 1 | 1088.6.a.s | 4 | ||
| 8.d | odd | 2 | 1 | 1088.6.a.bb | 4 | ||
| 17.b | even | 2 | 1 | 289.6.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.6.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 153.6.a.f | 4 | 3.b | odd | 2 | 1 | ||
| 272.6.a.j | 4 | 4.b | odd | 2 | 1 | ||
| 289.6.a.c | 4 | 17.b | even | 2 | 1 | ||
| 425.6.a.c | 4 | 5.b | even | 2 | 1 | ||
| 833.6.a.c | 4 | 7.b | odd | 2 | 1 | ||
| 1088.6.a.s | 4 | 8.b | even | 2 | 1 | ||
| 1088.6.a.bb | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 3T_{2}^{3} - 94T_{2}^{2} + 284T_{2} + 968 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(17))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 968 \)
|
| $3$ |
\( T^{4} - 28 T^{3} + \cdots + 13056 \)
|
| $5$ |
\( T^{4} - 80 T^{3} + \cdots - 10420784 \)
|
| $7$ |
\( T^{4} - 284 T^{3} + \cdots - 174124224 \)
|
| $11$ |
\( T^{4} + 412 T^{3} + \cdots + 908815104 \)
|
| $13$ |
\( T^{4} + \cdots + 175048896272 \)
|
| $17$ |
\( (T - 289)^{4} \)
|
| $19$ |
\( T^{4} + \cdots - 510891077376 \)
|
| $23$ |
\( T^{4} + \cdots - 7244195450816 \)
|
| $29$ |
\( T^{4} + \cdots - 80666284850352 \)
|
| $31$ |
\( T^{4} + \cdots - 68269807663936 \)
|
| $37$ |
\( T^{4} + \cdots + 22888518818768 \)
|
| $41$ |
\( T^{4} + \cdots - 86\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots + 256140463281408 \)
|
| $47$ |
\( T^{4} + \cdots + 63\!\cdots\!68 \)
|
| $53$ |
\( T^{4} + \cdots + 22\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots + 42\!\cdots\!04 \)
|
| $61$ |
\( T^{4} + \cdots - 25\!\cdots\!48 \)
|
| $67$ |
\( T^{4} + \cdots + 76\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 15\!\cdots\!52 \)
|
| $73$ |
\( T^{4} + \cdots + 60\!\cdots\!48 \)
|
| $79$ |
\( T^{4} + \cdots + 38\!\cdots\!52 \)
|
| $83$ |
\( T^{4} + \cdots - 39\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots + 14\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 31\!\cdots\!32 \)
|
| show more | |
| show less |