Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.18");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(15.3068662487\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{865})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 217x^{2} + 216x + 46656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{4} - x^{3} + 217x^{2} + 216x + 46656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 217\nu^{2} - 217\nu + 46656 ) / 46872 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 433 ) / 217 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 216\beta_{2} + \beta _1 - 217 \)
|
| \(\nu^{3}\) | \(=\) |
\( 217\beta_{3} - 433 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 |
|
−6.60272 | − | 11.4362i | 8.79456 | − | 15.2326i | −23.1918 | + | 40.1694i | 8.97279 | + | 15.5413i | −232.272 | 0 | −1077.78 | 938.811 | + | 1626.07i | 118.490 | − | 205.230i | ||||||||||||||||||
| 18.2 | 8.10272 | + | 14.0343i | 38.2054 | − | 66.1738i | −67.3082 | + | 116.581i | 156.027 | + | 270.247i | 1238.27 | 0 | −107.220 | −1825.81 | − | 3162.40i | −2528.49 | + | 4379.47i | |||||||||||||||||||
| 30.1 | −6.60272 | + | 11.4362i | 8.79456 | + | 15.2326i | −23.1918 | − | 40.1694i | 8.97279 | − | 15.5413i | −232.272 | 0 | −1077.78 | 938.811 | − | 1626.07i | 118.490 | + | 205.230i | |||||||||||||||||||
| 30.2 | 8.10272 | − | 14.0343i | 38.2054 | + | 66.1738i | −67.3082 | − | 116.581i | 156.027 | − | 270.247i | 1238.27 | 0 | −107.220 | −1825.81 | + | 3162.40i | −2528.49 | − | 4379.47i | |||||||||||||||||||
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 | 49.8.c.f | 4 | |
| 7.b | odd | 2 | 1 | 49.8.c.e | 4 | ||
| 7.c | even | 3 | 1 | 49.8.a.c | 2 | ||
| 7.c | even | 3 | 1 | inner | 49.8.c.f | 4 | |
| 7.d | odd | 6 | 1 | 7.8.a.b | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 49.8.c.e | 4 | ||
| 21.g | even | 6 | 1 | 63.8.a.e | 2 | ||
| 21.h | odd | 6 | 1 | 441.8.a.l | 2 | ||
| 28.f | even | 6 | 1 | 112.8.a.f | 2 | ||
| 35.i | odd | 6 | 1 | 175.8.a.c | 2 | ||
| 35.k | even | 12 | 2 | 175.8.b.b | 4 | ||
| 56.j | odd | 6 | 1 | 448.8.a.k | 2 | ||
| 56.m | even | 6 | 1 | 448.8.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.8.a.b | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 49.8.a.c | 2 | 7.c | even | 3 | 1 | ||
| 49.8.c.e | 4 | 7.b | odd | 2 | 1 | ||
| 49.8.c.e | 4 | 7.d | odd | 6 | 1 | ||
| 49.8.c.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.8.c.f | 4 | 7.c | even | 3 | 1 | inner | |
| 63.8.a.e | 2 | 21.g | even | 6 | 1 | ||
| 112.8.a.f | 2 | 28.f | even | 6 | 1 | ||
| 175.8.a.c | 2 | 35.i | odd | 6 | 1 | ||
| 175.8.b.b | 4 | 35.k | even | 12 | 2 | ||
| 441.8.a.l | 2 | 21.h | odd | 6 | 1 | ||
| 448.8.a.k | 2 | 56.j | odd | 6 | 1 | ||
| 448.8.a.t | 2 | 56.m | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{4} - 3T_{2}^{3} + 223T_{2}^{2} + 642T_{2} + 45796 \)
|
|
\( T_{3}^{4} - 94T_{3}^{3} + 7492T_{3}^{2} - 126336T_{3} + 1806336 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots + 45796 \)
|
| $3$ |
\( T^{4} - 94 T^{3} + \cdots + 1806336 \)
|
| $5$ |
\( T^{4} - 330 T^{3} + \cdots + 31360000 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 788146226176 \)
|
| $13$ |
\( (T^{2} + 2534 T - 166620776)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 490512819330576 \)
|
| $19$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 106351863791616 \)
|
| $29$ |
\( (T^{2} - 20640 T - 18920124100)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 13\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!56 \)
|
| $41$ |
\( (T^{2} + 734664 T + 13303276364)^{2} \)
|
| $43$ |
\( (T^{2} + 480476 T + 50864711104)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 43\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 41\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 37\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $67$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + \cdots + 10359492378624)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $83$ |
\( (T^{2} + \cdots + 30181573873584)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 27021168617436)^{2} \)
|
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