Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,10,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.c (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(25.2367559720\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{193})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 49\nu^{2} + 4753\nu - 2304 ) / 2352 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 145 ) / 49 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 97\beta _1 - 97 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 49\beta_{3} - 145 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−5.44622 | − | 9.43313i | 97.9084 | − | 169.582i | 196.677 | − | 340.655i | −100.391 | − | 173.882i | −2132.92 | 0 | −9861.52 | −9330.63 | − | 16161.1i | −1093.50 | + | 1894.01i | ||||||||||||||||||
| 18.2 | 8.44622 | + | 14.6293i | −54.9084 | + | 95.1042i | 113.323 | − | 196.281i | 1219.39 | + | 2112.05i | −1855.08 | 0 | 12477.5 | 3811.63 | + | 6601.93i | −20598.5 | + | 35677.6i | |||||||||||||||||||
| 30.1 | −5.44622 | + | 9.43313i | 97.9084 | + | 169.582i | 196.677 | + | 340.655i | −100.391 | + | 173.882i | −2132.92 | 0 | −9861.52 | −9330.63 | + | 16161.1i | −1093.50 | − | 1894.01i | |||||||||||||||||||
| 30.2 | 8.44622 | − | 14.6293i | −54.9084 | − | 95.1042i | 113.323 | + | 196.281i | 1219.39 | − | 2112.05i | −1855.08 | 0 | 12477.5 | 3811.63 | − | 6601.93i | −20598.5 | − | 35677.6i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.10.c.c | 4 | |
| 7.b | odd | 2 | 1 | 49.10.c.b | 4 | ||
| 7.c | even | 3 | 1 | 7.10.a.a | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 49.10.c.c | 4 | |
| 7.d | odd | 6 | 1 | 49.10.a.b | 2 | ||
| 7.d | odd | 6 | 1 | 49.10.c.b | 4 | ||
| 21.h | odd | 6 | 1 | 63.10.a.d | 2 | ||
| 28.g | odd | 6 | 1 | 112.10.a.e | 2 | ||
| 35.j | even | 6 | 1 | 175.10.a.b | 2 | ||
| 35.l | odd | 12 | 2 | 175.10.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.a.a | ✓ | 2 | 7.c | even | 3 | 1 | |
| 49.10.a.b | 2 | 7.d | odd | 6 | 1 | ||
| 49.10.c.b | 4 | 7.b | odd | 2 | 1 | ||
| 49.10.c.b | 4 | 7.d | odd | 6 | 1 | ||
| 49.10.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.10.c.c | 4 | 7.c | even | 3 | 1 | inner | |
| 63.10.a.d | 2 | 21.h | odd | 6 | 1 | ||
| 112.10.a.e | 2 | 28.g | odd | 6 | 1 | ||
| 175.10.a.b | 2 | 35.j | even | 6 | 1 | ||
| 175.10.b.b | 4 | 35.l | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{4} - 6T_{2}^{3} + 220T_{2}^{2} + 1104T_{2} + 33856 \)
|
|
\( T_{3}^{4} - 86T_{3}^{3} + 28900T_{3}^{2} + 1849344T_{3} + 462422016 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6 T^{3} + \cdots + 33856 \)
|
| $3$ |
\( T^{4} - 86 T^{3} + \cdots + 462422016 \)
|
| $5$ |
\( T^{4} + \cdots + 239770832896 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 33\!\cdots\!16 \)
|
| $13$ |
\( (T^{2} + 26530 T - 22750162568)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 13\!\cdots\!24 \)
|
| $19$ |
\( T^{4} + \cdots + 45\!\cdots\!04 \)
|
| $23$ |
\( T^{4} + \cdots + 16\!\cdots\!16 \)
|
| $29$ |
\( (T^{2} + \cdots + 23287739754332)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 50\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + \cdots - 51779041048756)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots - 207953886197312)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 21\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $59$ |
\( T^{4} + \cdots + 21\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots + 30\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots + 26\!\cdots\!56 \)
|
| $71$ |
\( (T^{2} + \cdots - 33\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 36\!\cdots\!84 \)
|
| $79$ |
\( T^{4} + \cdots + 49\!\cdots\!16 \)
|
| $83$ |
\( (T^{2} + \cdots - 40\!\cdots\!84)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 10\!\cdots\!84 \)
|
| $97$ |
\( (T^{2} + \cdots + 66\!\cdots\!24)^{2} \)
|
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