Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [385,2,Mod(36,385)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(385, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("385.36");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 385 = 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 385.n (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: | \(3.07424047782\) |
| 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 \(\left(\mathbb{Z}/385\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(232\) | \(276\) |
| \(\chi(n)\) | \(-\zeta_{10}^{3}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 36.1 |
|
0.809017 | − | 0.587785i | −0.190983 | − | 0.587785i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −0.500000 | − | 0.363271i | 0.309017 | − | 0.951057i | 0.927051 | + | 2.85317i | 2.11803 | − | 1.53884i | −1.00000 | ||||||||||||||
| 71.1 | −0.309017 | − | 0.951057i | −1.30902 | − | 0.951057i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | −0.500000 | + | 1.53884i | −0.809017 | + | 0.587785i | −2.42705 | − | 1.76336i | −0.118034 | − | 0.363271i | −1.00000 | |||||||||||||||
| 141.1 | −0.309017 | + | 0.951057i | −1.30902 | + | 0.951057i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | −0.500000 | − | 1.53884i | −0.809017 | − | 0.587785i | −2.42705 | + | 1.76336i | −0.118034 | + | 0.363271i | −1.00000 | |||||||||||||||
| 246.1 | 0.809017 | + | 0.587785i | −0.190983 | + | 0.587785i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −0.500000 | + | 0.363271i | 0.309017 | + | 0.951057i | 0.927051 | − | 2.85317i | 2.11803 | + | 1.53884i | −1.00000 | |||||||||||||||
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 | 385.2.n.a | ✓ | 4 |
| 11.c | even | 5 | 1 | inner | 385.2.n.a | ✓ | 4 |
| 11.c | even | 5 | 1 | 4235.2.a.i | 2 | ||
| 11.d | odd | 10 | 1 | 4235.2.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 385.2.n.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 385.2.n.a | ✓ | 4 | 11.c | even | 5 | 1 | inner |
| 4235.2.a.i | 2 | 11.c | even | 5 | 1 | ||
| 4235.2.a.n | 2 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(385, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + T^{2} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{4} + T^{3} + T^{2} + \cdots + 1 \)
|
| $7$ |
\( T^{4} + T^{3} + T^{2} + \cdots + 1 \)
|
| $11$ |
\( T^{4} - T^{3} + \cdots + 121 \)
|
| $13$ |
\( T^{4} - 6 T^{3} + \cdots + 121 \)
|
| $17$ |
\( T^{4} + T^{3} + \cdots + 121 \)
|
| $19$ |
\( T^{4} - 8 T^{3} + \cdots + 16 \)
|
| $23$ |
\( (T^{2} - 20)^{2} \)
|
| $29$ |
\( T^{4} + 14 T^{3} + \cdots + 841 \)
|
| $31$ |
\( T^{4} + 14 T^{3} + \cdots + 1936 \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
|
| $41$ |
\( T^{4} - 20 T^{3} + \cdots + 6400 \)
|
| $43$ |
\( (T^{2} - 2 T - 4)^{2} \)
|
| $47$ |
\( T^{4} + 4 T^{3} + \cdots + 1 \)
|
| $53$ |
\( T^{4} + 160 T^{2} + \cdots + 6400 \)
|
| $59$ |
\( T^{4} - 14 T^{3} + \cdots + 5776 \)
|
| $61$ |
\( T^{4} + 2 T^{3} + \cdots + 1936 \)
|
| $67$ |
\( (T^{2} + 6 T - 36)^{2} \)
|
| $71$ |
\( T^{4} - 22 T^{3} + \cdots + 10201 \)
|
| $73$ |
\( T^{4} - 18 T^{3} + \cdots + 6561 \)
|
| $79$ |
\( T^{4} + 18 T^{3} + \cdots + 9801 \)
|
| $83$ |
\( T^{4} - 6 T^{3} + \cdots + 121 \)
|
| $89$ |
\( (T^{2} + 18 T + 36)^{2} \)
|
| $97$ |
\( T^{4} - 7 T^{3} + \cdots + 361 \)
|
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