Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,10,Mod(1,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.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: | \(3.60525085315\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 426x + 2016 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 426x + 2016 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 25\nu + 276 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 11\nu - 288 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 25\beta_{2} - 11\beta _1 + 1706 ) / 6 \)
|
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 |
|
−34.1627 | −79.6469 | 655.088 | 1423.70 | 2720.95 | 2401.00 | −4888.28 | −13339.4 | −48637.4 | |||||||||||||||||||||||||||
| 1.2 | 13.3607 | 163.415 | −333.491 | 1922.19 | 2183.34 | 2401.00 | −11296.4 | 7021.32 | 25681.8 | ||||||||||||||||||||||||||||
| 1.3 | 41.8019 | 0.232339 | 1235.40 | −1791.89 | 9.71222 | 2401.00 | 30239.6 | −19682.9 | −74904.4 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( -1 \) |
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 | 7.10.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 63.10.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 112.10.a.h | 3 | ||
| 5.b | even | 2 | 1 | 175.10.a.d | 3 | ||
| 5.c | odd | 4 | 2 | 175.10.b.d | 6 | ||
| 7.b | odd | 2 | 1 | 49.10.a.c | 3 | ||
| 7.c | even | 3 | 2 | 49.10.c.d | 6 | ||
| 7.d | odd | 6 | 2 | 49.10.c.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 49.10.a.c | 3 | 7.b | odd | 2 | 1 | ||
| 49.10.c.d | 6 | 7.c | even | 3 | 2 | ||
| 49.10.c.e | 6 | 7.d | odd | 6 | 2 | ||
| 63.10.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 112.10.a.h | 3 | 4.b | odd | 2 | 1 | ||
| 175.10.a.d | 3 | 5.b | even | 2 | 1 | ||
| 175.10.b.d | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 21T_{2}^{2} - 1326T_{2} + 19080 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 21 T^{2} + \cdots + 19080 \)
|
| $3$ |
\( T^{3} - 84 T^{2} + \cdots + 3024 \)
|
| $5$ |
\( T^{3} + \cdots + 4903718400 \)
|
| $7$ |
\( (T - 2401)^{3} \)
|
| $11$ |
\( T^{3} + \cdots + 108859759460352 \)
|
| $13$ |
\( T^{3} + \cdots - 41548412541440 \)
|
| $17$ |
\( T^{3} + \cdots - 21\!\cdots\!32 \)
|
| $19$ |
\( T^{3} + \cdots - 43\!\cdots\!60 \)
|
| $23$ |
\( T^{3} + \cdots + 97\!\cdots\!36 \)
|
| $29$ |
\( T^{3} + \cdots + 44\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 74\!\cdots\!84 \)
|
| $37$ |
\( T^{3} + \cdots - 34\!\cdots\!28 \)
|
| $41$ |
\( T^{3} + \cdots - 19\!\cdots\!12 \)
|
| $43$ |
\( T^{3} + \cdots + 68\!\cdots\!80 \)
|
| $47$ |
\( T^{3} + \cdots - 43\!\cdots\!16 \)
|
| $53$ |
\( T^{3} + \cdots + 23\!\cdots\!28 \)
|
| $59$ |
\( T^{3} + \cdots + 42\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 51\!\cdots\!08 \)
|
| $67$ |
\( T^{3} + \cdots - 20\!\cdots\!64 \)
|
| $71$ |
\( T^{3} + \cdots + 16\!\cdots\!80 \)
|
| $73$ |
\( T^{3} + \cdots + 19\!\cdots\!48 \)
|
| $79$ |
\( T^{3} + \cdots - 11\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 18\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots + 19\!\cdots\!40 \)
|
| $97$ |
\( T^{3} + \cdots - 49\!\cdots\!16 \)
|
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