Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(180,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([2, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.180");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 671.k (of order \(5\), degree \(4\), 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: | \(5.35796197563\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\left(\mathbb{Z}/671\mathbb{Z}\right)^\times\).
| \(n\) | \(123\) | \(551\) |
| \(\chi(n)\) | \(-\zeta_{10}\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 180.1 |
|
0.500000 | − | 0.363271i | 1.80902 | + | 1.31433i | −0.500000 | + | 1.53884i | 1.00000 | 1.38197 | 0.500000 | − | 0.363271i | 0.690983 | + | 2.12663i | 0.618034 | + | 1.90211i | 0.500000 | − | 0.363271i | ||||||||||||||||
| 192.1 | 0.500000 | − | 1.53884i | 0.690983 | + | 2.12663i | −0.500000 | − | 0.363271i | 1.00000 | 3.61803 | 0.500000 | − | 1.53884i | 1.80902 | − | 1.31433i | −1.61803 | + | 1.17557i | 0.500000 | − | 1.53884i | |||||||||||||||||
| 339.1 | 0.500000 | + | 1.53884i | 0.690983 | − | 2.12663i | −0.500000 | + | 0.363271i | 1.00000 | 3.61803 | 0.500000 | + | 1.53884i | 1.80902 | + | 1.31433i | −1.61803 | − | 1.17557i | 0.500000 | + | 1.53884i | |||||||||||||||||
| 630.1 | 0.500000 | + | 0.363271i | 1.80902 | − | 1.31433i | −0.500000 | − | 1.53884i | 1.00000 | 1.38197 | 0.500000 | + | 0.363271i | 0.690983 | − | 2.12663i | 0.618034 | − | 1.90211i | 0.500000 | + | 0.363271i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 671.k | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 671.2.k.a | ✓ | 4 |
| 11.c | even | 5 | 1 | 671.2.l.a | yes | 4 | |
| 61.e | even | 5 | 1 | 671.2.l.a | yes | 4 | |
| 671.k | even | 5 | 1 | inner | 671.2.k.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 671.2.k.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 671.2.k.a | ✓ | 4 | 671.k | even | 5 | 1 | inner |
| 671.2.l.a | yes | 4 | 11.c | even | 5 | 1 | |
| 671.2.l.a | yes | 4 | 61.e | even | 5 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(671, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $11$ |
\( T^{4} - 9 T^{3} + \cdots + 121 \)
|
| $13$ |
\( T^{4} + 7 T^{3} + \cdots + 121 \)
|
| $17$ |
\( (T^{2} + 2 T - 4)^{2} \)
|
| $19$ |
\( T^{4} - 9 T^{3} + \cdots + 361 \)
|
| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 121 \)
|
| $29$ |
\( T^{4} - 11 T^{3} + \cdots + 361 \)
|
| $31$ |
\( T^{4} + 6 T^{3} + \cdots + 1296 \)
|
| $37$ |
\( T^{4} - 7 T^{3} + \cdots + 7921 \)
|
| $41$ |
\( (T^{2} - 6 T - 11)^{2} \)
|
| $43$ |
\( T^{4} + 3 T^{3} + \cdots + 3481 \)
|
| $47$ |
\( T^{4} - 10 T^{3} + \cdots + 400 \)
|
| $53$ |
\( (T + 2)^{4} \)
|
| $59$ |
\( (T + 3)^{4} \)
|
| $61$ |
\( T^{4} - T^{3} + \cdots + 3721 \)
|
| $67$ |
\( T^{4} + 23 T^{3} + \cdots + 361 \)
|
| $71$ |
\( T^{4} + 12 T^{3} + \cdots + 7921 \)
|
| $73$ |
\( T^{4} - 5 T^{3} + \cdots + 3025 \)
|
| $79$ |
\( (T^{2} - 18 T + 61)^{2} \)
|
| $83$ |
\( T^{4} + 26 T^{3} + \cdots + 1936 \)
|
| $89$ |
\( T^{4} + 21 T^{3} + \cdots + 29241 \)
|
| $97$ |
\( T^{4} + 13 T^{3} + \cdots + 361 \)
|
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