Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [154,4,Mod(15,154)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(154, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("154.15");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 154 = 2 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 154.f (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: | \(9.08629414088\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(\left(\mathbb{Z}/154\mathbb{Z}\right)^\times\).
| \(n\) | \(45\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
1.61803 | + | 1.17557i | 1.61803 | − | 4.97980i | 1.23607 | + | 3.80423i | 8.23607 | − | 5.98385i | 8.47214 | − | 6.15537i | −2.16312 | − | 6.65740i | −2.47214 | + | 7.60845i | −0.336881 | − | 0.244758i | 20.3607 | ||||||||||||||
| 71.1 | −0.618034 | − | 1.90211i | −0.618034 | − | 0.449028i | −3.23607 | + | 2.35114i | 3.76393 | − | 11.5842i | −0.472136 | + | 1.45309i | 5.66312 | − | 4.11450i | 6.47214 | + | 4.70228i | −8.16312 | − | 25.1235i | −24.3607 | |||||||||||||||
| 113.1 | 1.61803 | − | 1.17557i | 1.61803 | + | 4.97980i | 1.23607 | − | 3.80423i | 8.23607 | + | 5.98385i | 8.47214 | + | 6.15537i | −2.16312 | + | 6.65740i | −2.47214 | − | 7.60845i | −0.336881 | + | 0.244758i | 20.3607 | |||||||||||||||
| 141.1 | −0.618034 | + | 1.90211i | −0.618034 | + | 0.449028i | −3.23607 | − | 2.35114i | 3.76393 | + | 11.5842i | −0.472136 | − | 1.45309i | 5.66312 | + | 4.11450i | 6.47214 | − | 4.70228i | −8.16312 | + | 25.1235i | −24.3607 | |||||||||||||||
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 | 154.4.f.b | ✓ | 4 |
| 11.c | even | 5 | 1 | inner | 154.4.f.b | ✓ | 4 |
| 11.c | even | 5 | 1 | 1694.4.a.k | 2 | ||
| 11.d | odd | 10 | 1 | 1694.4.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 154.4.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 154.4.f.b | ✓ | 4 | 11.c | even | 5 | 1 | inner |
| 1694.4.a.k | 2 | 11.c | even | 5 | 1 | ||
| 1694.4.a.o | 2 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} + 24T_{3}^{2} + 32T_{3} + 16 \)
acting on \(S_{4}^{\mathrm{new}}(154, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $5$ |
\( T^{4} - 24 T^{3} + \cdots + 15376 \)
|
| $7$ |
\( T^{4} - 7 T^{3} + \cdots + 2401 \)
|
| $11$ |
\( T^{4} - 89 T^{3} + \cdots + 1771561 \)
|
| $13$ |
\( T^{4} + 148 T^{3} + \cdots + 952576 \)
|
| $17$ |
\( T^{4} - 206 T^{3} + \cdots + 26666896 \)
|
| $19$ |
\( T^{4} + 260 T^{3} + \cdots + 11424400 \)
|
| $23$ |
\( (T^{2} + 209 T - 1831)^{2} \)
|
| $29$ |
\( T^{4} - 70 T^{3} + \cdots + 4644025 \)
|
| $31$ |
\( T^{4} + 212 T^{3} + \cdots + 175933696 \)
|
| $37$ |
\( T^{4} - 321 T^{3} + \cdots + 617472801 \)
|
| $41$ |
\( T^{4} + 302 T^{3} + \cdots + 5779216 \)
|
| $43$ |
\( (T^{2} - 201 T + 5139)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 11379768976 \)
|
| $53$ |
\( T^{4} + \cdots + 28614767281 \)
|
| $59$ |
\( T^{4} + \cdots + 6577210000 \)
|
| $61$ |
\( T^{4} + \cdots + 249604156816 \)
|
| $67$ |
\( (T^{2} - 53 T - 555409)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 123256464241 \)
|
| $73$ |
\( T^{4} + \cdots + 11993754256 \)
|
| $79$ |
\( T^{4} + \cdots + 407114183025 \)
|
| $83$ |
\( T^{4} + \cdots + 169213758736 \)
|
| $89$ |
\( (T^{2} - 2530 T + 1597100)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 1614841143696 \)
|
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