Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4005,2,Mod(1,4005)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4005.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4005 = 3^{2} \cdot 5 \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4005.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: | \(31.9800860095\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 445) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 4\nu^{2} - 2\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} + 8\nu - 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 7\nu^{4} + 5\nu^{3} + 13\nu^{2} - 4\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{4} + 9\beta_{3} - 2\beta_{2} + 6\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 8\beta_{3} + \beta_{2} + 23\beta _1 + 29 \)
|
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 |
|
−2.11962 | 0 | 2.49279 | −1.00000 | 0 | −4.83304 | −1.04452 | 0 | 2.11962 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.856822 | 0 | −1.26586 | −1.00000 | 0 | −0.580377 | 2.79826 | 0 | 0.856822 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.584976 | 0 | −1.65780 | −1.00000 | 0 | −2.74591 | 2.13973 | 0 | 0.584976 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.755898 | 0 | −1.42862 | −1.00000 | 0 | 0.0498231 | −2.59169 | 0 | −0.755898 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.83770 | 0 | 1.37716 | −1.00000 | 0 | −4.72699 | −1.14460 | 0 | −1.83770 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.21675 | 0 | 2.91399 | −1.00000 | 0 | −3.75132 | 2.02609 | 0 | −2.21675 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.75106 | 0 | 5.56834 | −1.00000 | 0 | 0.587818 | 9.81674 | 0 | −2.75106 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(89\) | \(1\) |
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 | 4005.2.a.o | 7 | |
| 3.b | odd | 2 | 1 | 445.2.a.f | ✓ | 7 | |
| 12.b | even | 2 | 1 | 7120.2.a.bj | 7 | ||
| 15.d | odd | 2 | 1 | 2225.2.a.k | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 445.2.a.f | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 2225.2.a.k | 7 | 15.d | odd | 2 | 1 | ||
| 4005.2.a.o | 7 | 1.a | even | 1 | 1 | trivial | |
| 7120.2.a.bj | 7 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):
|
\( T_{2}^{7} - 4T_{2}^{6} - 3T_{2}^{5} + 24T_{2}^{4} - 8T_{2}^{3} - 29T_{2}^{2} + 6T_{2} + 9 \)
|
|
\( T_{7}^{7} + 16T_{7}^{6} + 94T_{7}^{5} + 236T_{7}^{4} + 189T_{7}^{3} - 96T_{7}^{2} - 76T_{7} + 4 \)
|
|
\( T_{11}^{7} - 10T_{11}^{6} + 14T_{11}^{5} + 149T_{11}^{4} - 639T_{11}^{3} + 968T_{11}^{2} - 608T_{11} + 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots + 9 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} + 16 T^{6} + \cdots + 4 \)
|
| $11$ |
\( T^{7} - 10 T^{6} + \cdots + 128 \)
|
| $13$ |
\( T^{7} + 7 T^{6} + \cdots + 5816 \)
|
| $17$ |
\( T^{7} - 13 T^{6} + \cdots - 53832 \)
|
| $19$ |
\( T^{7} + 7 T^{6} + \cdots - 7164 \)
|
| $23$ |
\( T^{7} - 13 T^{6} + \cdots - 10772 \)
|
| $29$ |
\( T^{7} - 4 T^{6} + \cdots - 22744 \)
|
| $31$ |
\( T^{7} - T^{6} + \cdots + 2084 \)
|
| $37$ |
\( T^{7} + 5 T^{6} + \cdots + 1544 \)
|
| $41$ |
\( T^{7} + 5 T^{6} + \cdots + 57688 \)
|
| $43$ |
\( T^{7} + 31 T^{6} + \cdots + 724 \)
|
| $47$ |
\( T^{7} - 14 T^{6} + \cdots + 128944 \)
|
| $53$ |
\( T^{7} - 13 T^{6} + \cdots + 371944 \)
|
| $59$ |
\( T^{7} - 14 T^{6} + \cdots + 1813884 \)
|
| $61$ |
\( T^{7} - 3 T^{6} + \cdots - 608264 \)
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| $67$ |
\( T^{7} - T^{6} + \cdots + 95952 \)
|
| $71$ |
\( T^{7} - 8 T^{6} + \cdots + 3171744 \)
|
| $73$ |
\( T^{7} - 9 T^{6} + \cdots - 275592 \)
|
| $79$ |
\( T^{7} - 9 T^{6} + \cdots + 6368 \)
|
| $83$ |
\( T^{7} - 42 T^{6} + \cdots - 3622012 \)
|
| $89$ |
\( (T + 1)^{7} \)
|
| $97$ |
\( T^{7} + 7 T^{6} + \cdots - 2264 \)
|
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