Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(148,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("847.148");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 847.f (of order \(5\), degree \(4\), 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: | \(6.76332905120\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 77) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(365\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 148.1 |
|
0.690983 | + | 2.12663i | −2.61803 | − | 1.90211i | −2.42705 | + | 1.76336i | −0.618034 | + | 1.90211i | 2.23607 | − | 6.88191i | 0.809017 | − | 0.587785i | −1.80902 | − | 1.31433i | 2.30902 | + | 7.10642i | −4.47214 | ||||||||||||||
| 323.1 | 1.80902 | + | 1.31433i | −0.381966 | + | 1.17557i | 0.927051 | + | 2.85317i | 1.61803 | − | 1.17557i | −2.23607 | + | 1.62460i | −0.309017 | − | 0.951057i | −0.690983 | + | 2.12663i | 1.19098 | + | 0.865300i | 4.47214 | |||||||||||||||
| 372.1 | 0.690983 | − | 2.12663i | −2.61803 | + | 1.90211i | −2.42705 | − | 1.76336i | −0.618034 | − | 1.90211i | 2.23607 | + | 6.88191i | 0.809017 | + | 0.587785i | −1.80902 | + | 1.31433i | 2.30902 | − | 7.10642i | −4.47214 | |||||||||||||||
| 729.1 | 1.80902 | − | 1.31433i | −0.381966 | − | 1.17557i | 0.927051 | − | 2.85317i | 1.61803 | + | 1.17557i | −2.23607 | − | 1.62460i | −0.309017 | + | 0.951057i | −0.690983 | − | 2.12663i | 1.19098 | − | 0.865300i | 4.47214 | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 847.2.f.m | 4 | |
| 11.b | odd | 2 | 1 | 847.2.f.a | 4 | ||
| 11.c | even | 5 | 1 | 847.2.a.f | 2 | ||
| 11.c | even | 5 | 2 | 847.2.f.b | 4 | ||
| 11.c | even | 5 | 1 | inner | 847.2.f.m | 4 | |
| 11.d | odd | 10 | 1 | 77.2.a.d | ✓ | 2 | |
| 11.d | odd | 10 | 1 | 847.2.f.a | 4 | ||
| 11.d | odd | 10 | 2 | 847.2.f.n | 4 | ||
| 33.f | even | 10 | 1 | 693.2.a.h | 2 | ||
| 33.h | odd | 10 | 1 | 7623.2.a.bl | 2 | ||
| 44.g | even | 10 | 1 | 1232.2.a.m | 2 | ||
| 55.h | odd | 10 | 1 | 1925.2.a.r | 2 | ||
| 55.l | even | 20 | 2 | 1925.2.b.h | 4 | ||
| 77.j | odd | 10 | 1 | 5929.2.a.m | 2 | ||
| 77.l | even | 10 | 1 | 539.2.a.f | 2 | ||
| 77.n | even | 30 | 2 | 539.2.e.j | 4 | ||
| 77.o | odd | 30 | 2 | 539.2.e.i | 4 | ||
| 88.k | even | 10 | 1 | 4928.2.a.bv | 2 | ||
| 88.p | odd | 10 | 1 | 4928.2.a.bm | 2 | ||
| 231.r | odd | 10 | 1 | 4851.2.a.y | 2 | ||
| 308.s | odd | 10 | 1 | 8624.2.a.ce | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.2.a.d | ✓ | 2 | 11.d | odd | 10 | 1 | |
| 539.2.a.f | 2 | 77.l | even | 10 | 1 | ||
| 539.2.e.i | 4 | 77.o | odd | 30 | 2 | ||
| 539.2.e.j | 4 | 77.n | even | 30 | 2 | ||
| 693.2.a.h | 2 | 33.f | even | 10 | 1 | ||
| 847.2.a.f | 2 | 11.c | even | 5 | 1 | ||
| 847.2.f.a | 4 | 11.b | odd | 2 | 1 | ||
| 847.2.f.a | 4 | 11.d | odd | 10 | 1 | ||
| 847.2.f.b | 4 | 11.c | even | 5 | 2 | ||
| 847.2.f.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 847.2.f.m | 4 | 11.c | even | 5 | 1 | inner | |
| 847.2.f.n | 4 | 11.d | odd | 10 | 2 | ||
| 1232.2.a.m | 2 | 44.g | even | 10 | 1 | ||
| 1925.2.a.r | 2 | 55.h | odd | 10 | 1 | ||
| 1925.2.b.h | 4 | 55.l | even | 20 | 2 | ||
| 4851.2.a.y | 2 | 231.r | odd | 10 | 1 | ||
| 4928.2.a.bm | 2 | 88.p | odd | 10 | 1 | ||
| 4928.2.a.bv | 2 | 88.k | even | 10 | 1 | ||
| 5929.2.a.m | 2 | 77.j | odd | 10 | 1 | ||
| 7623.2.a.bl | 2 | 33.h | odd | 10 | 1 | ||
| 8624.2.a.ce | 2 | 308.s | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\):
|
\( T_{2}^{4} - 5T_{2}^{3} + 15T_{2}^{2} - 25T_{2} + 25 \)
|
|
\( T_{3}^{4} + 6T_{3}^{3} + 16T_{3}^{2} + 16T_{3} + 16 \)
|
|
\( T_{13}^{4} - 6T_{13}^{3} + 16T_{13}^{2} - 16T_{13} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $5$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $7$ |
\( T^{4} - T^{3} + T^{2} + \cdots + 1 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 6 T^{3} + \cdots + 16 \)
|
| $17$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $19$ |
\( T^{4} + 8 T^{3} + \cdots + 256 \)
|
| $23$ |
\( (T^{2} + 4 T - 16)^{2} \)
|
| $29$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $31$ |
\( T^{4} + 40 T^{2} + \cdots + 400 \)
|
| $37$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 5776 \)
|
| $43$ |
\( (T + 8)^{4} \)
|
| $47$ |
\( T^{4} + 10 T^{3} + \cdots + 400 \)
|
| $53$ |
\( T^{4} - 6 T^{3} + \cdots + 16 \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $61$ |
\( T^{4} + 10 T^{3} + \cdots + 400 \)
|
| $67$ |
\( (T^{2} - 20 T + 80)^{2} \)
|
| $71$ |
\( T^{4} - 16 T^{3} + \cdots + 256 \)
|
| $73$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
|
| $79$ |
\( T^{4} + 20 T^{3} + \cdots + 6400 \)
|
| $83$ |
\( T^{4} + 28 T^{3} + \cdots + 30976 \)
|
| $89$ |
\( (T - 2)^{4} \)
|
| $97$ |
\( T^{4} + 34 T^{3} + \cdots + 26896 \)
|
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