Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,2,Mod(81,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.81");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.g (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: | \(4.83094932229\) |
| 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 \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(486\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 81.1 |
|
0.809017 | + | 0.587785i | −0.927051 | + | 2.85317i | −0.309017 | − | 0.951057i | −0.809017 | + | 0.587785i | −2.42705 | + | 1.76336i | −0.927051 | − | 2.85317i | 0.927051 | − | 2.85317i | −4.85410 | − | 3.52671i | −1.00000 | ||||||||||||||
| 251.1 | −0.309017 | + | 0.951057i | 2.42705 | − | 1.76336i | 0.809017 | + | 0.587785i | 0.309017 | + | 0.951057i | 0.927051 | + | 2.85317i | 2.42705 | + | 1.76336i | −2.42705 | + | 1.76336i | 1.85410 | − | 5.70634i | −1.00000 | |||||||||||||||
| 366.1 | 0.809017 | − | 0.587785i | −0.927051 | − | 2.85317i | −0.309017 | + | 0.951057i | −0.809017 | − | 0.587785i | −2.42705 | − | 1.76336i | −0.927051 | + | 2.85317i | 0.927051 | + | 2.85317i | −4.85410 | + | 3.52671i | −1.00000 | |||||||||||||||
| 511.1 | −0.309017 | − | 0.951057i | 2.42705 | + | 1.76336i | 0.809017 | − | 0.587785i | 0.309017 | − | 0.951057i | 0.927051 | − | 2.85317i | 2.42705 | − | 1.76336i | −2.42705 | − | 1.76336i | 1.85410 | + | 5.70634i | −1.00000 | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 605.2.g.d | 4 | |
| 11.b | odd | 2 | 1 | 605.2.g.b | 4 | ||
| 11.c | even | 5 | 1 | 605.2.a.a | ✓ | 1 | |
| 11.c | even | 5 | 3 | inner | 605.2.g.d | 4 | |
| 11.d | odd | 10 | 1 | 605.2.a.c | yes | 1 | |
| 11.d | odd | 10 | 3 | 605.2.g.b | 4 | ||
| 33.f | even | 10 | 1 | 5445.2.a.d | 1 | ||
| 33.h | odd | 10 | 1 | 5445.2.a.h | 1 | ||
| 44.g | even | 10 | 1 | 9680.2.a.be | 1 | ||
| 44.h | odd | 10 | 1 | 9680.2.a.bf | 1 | ||
| 55.h | odd | 10 | 1 | 3025.2.a.c | 1 | ||
| 55.j | even | 10 | 1 | 3025.2.a.g | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.2.a.a | ✓ | 1 | 11.c | even | 5 | 1 | |
| 605.2.a.c | yes | 1 | 11.d | odd | 10 | 1 | |
| 605.2.g.b | 4 | 11.b | odd | 2 | 1 | ||
| 605.2.g.b | 4 | 11.d | odd | 10 | 3 | ||
| 605.2.g.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 605.2.g.d | 4 | 11.c | even | 5 | 3 | inner | |
| 3025.2.a.c | 1 | 55.h | odd | 10 | 1 | ||
| 3025.2.a.g | 1 | 55.j | even | 10 | 1 | ||
| 5445.2.a.d | 1 | 33.f | even | 10 | 1 | ||
| 5445.2.a.h | 1 | 33.h | odd | 10 | 1 | ||
| 9680.2.a.be | 1 | 44.g | even | 10 | 1 | ||
| 9680.2.a.bf | 1 | 44.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):
|
\( T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1 \)
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\( T_{3}^{4} - 3T_{3}^{3} + 9T_{3}^{2} - 27T_{3} + 81 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + T^{2} + \cdots + 1 \)
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| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} + T^{3} + T^{2} + \cdots + 1 \)
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| $7$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
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| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
|
| $23$ |
\( (T + 8)^{4} \)
|
| $29$ |
\( T^{4} + 6 T^{3} + \cdots + 1296 \)
|
| $31$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 4096 \)
|
| $41$ |
\( T^{4} - 5 T^{3} + \cdots + 625 \)
|
| $43$ |
\( (T - 5)^{4} \)
|
| $47$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
|
| $53$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
|
| $59$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $61$ |
\( T^{4} - 11 T^{3} + \cdots + 14641 \)
|
| $67$ |
\( (T + 13)^{4} \)
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| $71$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
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| $73$ |
\( T^{4} - 8 T^{3} + \cdots + 4096 \)
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| $79$ |
\( T^{4} + 10 T^{3} + \cdots + 10000 \)
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| $83$ |
\( T^{4} + 4 T^{3} + \cdots + 256 \)
|
| $89$ |
\( (T - 1)^{4} \)
|
| $97$ |
\( T^{4} - 8 T^{3} + \cdots + 4096 \)
|
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