Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [105,4,Mod(4,105)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(105, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("105.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 105 = 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 105.q (of order \(6\), degree \(2\), 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: | \(6.19520055060\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/105\mathbb{Z}\right)^\times\).
| \(n\) | \(22\) | \(31\) | \(71\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \zeta_{12}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−4.33013 | + | 2.50000i | 2.59808 | + | 1.50000i | 8.50000 | − | 14.7224i | −10.5263 | + | 3.76795i | −15.0000 | 15.5885 | − | 10.0000i | 45.0000i | 4.50000 | + | 7.79423i | 36.1603 | − | 42.6314i | ||||||||||||||||
| 4.2 | 4.33013 | − | 2.50000i | −2.59808 | − | 1.50000i | 8.50000 | − | 14.7224i | 8.52628 | − | 7.23205i | −15.0000 | −15.5885 | + | 10.0000i | − | 45.0000i | 4.50000 | + | 7.79423i | 18.8397 | − | 52.6314i | ||||||||||||||||
| 79.1 | −4.33013 | − | 2.50000i | 2.59808 | − | 1.50000i | 8.50000 | + | 14.7224i | −10.5263 | − | 3.76795i | −15.0000 | 15.5885 | + | 10.0000i | − | 45.0000i | 4.50000 | − | 7.79423i | 36.1603 | + | 42.6314i | ||||||||||||||||
| 79.2 | 4.33013 | + | 2.50000i | −2.59808 | + | 1.50000i | 8.50000 | + | 14.7224i | 8.52628 | + | 7.23205i | −15.0000 | −15.5885 | − | 10.0000i | 45.0000i | 4.50000 | − | 7.79423i | 18.8397 | + | 52.6314i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 35.j | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 105.4.q.a | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 105.4.q.a | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 105.4.q.a | ✓ | 4 |
| 35.j | even | 6 | 1 | inner | 105.4.q.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 105.4.q.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 105.4.q.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 105.4.q.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 105.4.q.a | ✓ | 4 | 35.j | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 25T_{2}^{2} + 625 \)
acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 25T^{2} + 625 \)
|
| $3$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 15625 \)
|
| $7$ |
\( T^{4} - 286 T^{2} + 117649 \)
|
| $11$ |
\( (T^{2} - 45 T + 2025)^{2} \)
|
| $13$ |
\( (T^{2} + 1369)^{2} \)
|
| $17$ |
\( T^{4} - 3364 T^{2} + 11316496 \)
|
| $19$ |
\( (T^{2} - 135 T + 18225)^{2} \)
|
| $23$ |
\( T^{4} - 3969 T^{2} + 15752961 \)
|
| $29$ |
\( (T - 184)^{4} \)
|
| $31$ |
\( (T^{2} + 170 T + 28900)^{2} \)
|
| $37$ |
\( T^{4} - 5041 T^{2} + 25411681 \)
|
| $41$ |
\( (T - 247)^{4} \)
|
| $43$ |
\( (T^{2} + 58564)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 103355177121 \)
|
| $53$ |
\( T^{4} - 2025 T^{2} + 4100625 \)
|
| $59$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $61$ |
\( (T^{2} + 262 T + 68644)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 583506543376 \)
|
| $71$ |
\( (T + 1138)^{4} \)
|
| $73$ |
\( T^{4} - 11664 T^{2} + 136048896 \)
|
| $79$ |
\( (T^{2} + 62 T + 3844)^{2} \)
|
| $83$ |
\( (T^{2} + 206116)^{2} \)
|
| $89$ |
\( (T^{2} + 1206 T + 1454436)^{2} \)
|
| $97$ |
\( (T^{2} + 1849600)^{2} \)
|
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