Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,6,Mod(67,147)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.67");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.e (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: | \(23.5764215125\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 49\nu^{2} - 49\nu + 2304 ) / 2352 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 97 ) / 49 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 48\beta_{2} + \beta _1 - 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( 49\beta_{3} - 97 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−2.72311 | − | 4.71657i | −4.50000 | + | 7.79423i | 1.16933 | − | 2.02534i | 18.0000 | + | 31.1769i | 49.0160 | 0 | −187.016 | −40.5000 | − | 70.1481i | 98.0320 | − | 169.796i | ||||||||||||||||||
| 67.2 | 4.22311 | + | 7.31464i | −4.50000 | + | 7.79423i | −19.6693 | + | 34.0683i | 18.0000 | + | 31.1769i | −76.0160 | 0 | −61.9840 | −40.5000 | − | 70.1481i | −152.032 | + | 263.327i | |||||||||||||||||||
| 79.1 | −2.72311 | + | 4.71657i | −4.50000 | − | 7.79423i | 1.16933 | + | 2.02534i | 18.0000 | − | 31.1769i | 49.0160 | 0 | −187.016 | −40.5000 | + | 70.1481i | 98.0320 | + | 169.796i | |||||||||||||||||||
| 79.2 | 4.22311 | − | 7.31464i | −4.50000 | − | 7.79423i | −19.6693 | − | 34.0683i | 18.0000 | − | 31.1769i | −76.0160 | 0 | −61.9840 | −40.5000 | + | 70.1481i | −152.032 | − | 263.327i | |||||||||||||||||||
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 | 147.6.e.m | 4 | |
| 7.b | odd | 2 | 1 | 147.6.e.n | 4 | ||
| 7.c | even | 3 | 1 | 147.6.a.j | yes | 2 | |
| 7.c | even | 3 | 1 | inner | 147.6.e.m | 4 | |
| 7.d | odd | 6 | 1 | 147.6.a.h | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 147.6.e.n | 4 | ||
| 21.g | even | 6 | 1 | 441.6.a.q | 2 | ||
| 21.h | odd | 6 | 1 | 441.6.a.r | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.6.a.h | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 147.6.a.j | yes | 2 | 7.c | even | 3 | 1 | |
| 147.6.e.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 147.6.e.m | 4 | 7.c | even | 3 | 1 | inner | |
| 147.6.e.n | 4 | 7.b | odd | 2 | 1 | ||
| 147.6.e.n | 4 | 7.d | odd | 6 | 1 | ||
| 441.6.a.q | 2 | 21.g | even | 6 | 1 | ||
| 441.6.a.r | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{4} - 3T_{2}^{3} + 55T_{2}^{2} + 138T_{2} + 2116 \)
|
|
\( T_{5}^{2} - 36T_{5} + 1296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + 55 T^{2} + \cdots + 2116 \)
|
| $3$ |
\( (T^{2} + 9 T + 81)^{2} \)
|
| $5$ |
\( (T^{2} - 36 T + 1296)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 480 T^{3} + \cdots + 2971558144 \)
|
| $13$ |
\( (T^{2} + 1296 T + 169776)^{2} \)
|
| $17$ |
\( T^{4} - 936 T^{3} + \cdots + 4129544208384 \)
|
| $19$ |
\( T^{4} + 216 T^{3} + \cdots + 15923164467456 \)
|
| $23$ |
\( T^{4} - 504 T^{3} + \cdots + 7710244864 \)
|
| $29$ |
\( (T^{2} - 6372 T - 9612604)^{2} \)
|
| $31$ |
\( T^{4} - 9936 T^{3} + \cdots + 75218014900224 \)
|
| $37$ |
\( T^{4} + \cdots + 523012566603024 \)
|
| $41$ |
\( (T^{2} + 20952 T + 107495424)^{2} \)
|
| $43$ |
\( (T^{2} + 6264 T - 268110576)^{2} \)
|
| $47$ |
\( T^{4} - 7920 T^{3} + \cdots + 31\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + 2220 T^{3} + \cdots + 44\!\cdots\!04 \)
|
| $59$ |
\( T^{4} - 29736 T^{3} + \cdots + 34\!\cdots\!64 \)
|
| $61$ |
\( T^{4} + 17280 T^{3} + \cdots + 96\!\cdots\!00 \)
|
| $67$ |
\( T^{4} - 20680 T^{3} + \cdots + 36\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} + 92280 T + 1814661632)^{2} \)
|
| $73$ |
\( T^{4} - 56592 T^{3} + \cdots + 39\!\cdots\!44 \)
|
| $79$ |
\( T^{4} - 56096 T^{3} + \cdots + 15\!\cdots\!16 \)
|
| $83$ |
\( (T^{2} - 71352 T - 3085453296)^{2} \)
|
| $89$ |
\( T^{4} - 123192 T^{3} + \cdots + 98\!\cdots\!44 \)
|
| $97$ |
\( (T^{2} + 35856 T - 18184056768)^{2} \)
|
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