Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,2,Mod(49,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([0, 0, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("368.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 368.m (of order \(11\), degree \(10\), 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: | \(2.93849479438\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\Q(\zeta_{22})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 46) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).
| \(n\) | \(47\) | \(97\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{22}^{4}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | −0.580699 | + | 1.27155i | 0 | −1.66741 | + | 1.92429i | 0 | −1.75667 | + | 1.12894i | 0 | 0.684944 | + | 0.790468i | 0 | ||||||||||||||||||||||||||||||||||||||||
| 81.1 | 0 | 0.0530529 | − | 0.368991i | 0 | −3.26024 | + | 0.957293i | 0 | 0.297176 | + | 0.342959i | 0 | 2.74514 | + | 0.806046i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 177.1 | 0 | 2.71616 | − | 1.74557i | 0 | 0.985691 | + | 2.15836i | 0 | −0.381761 | + | 0.112095i | 0 | 3.08427 | − | 6.75361i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 193.1 | 0 | −0.712591 | − | 0.822373i | 0 | 0.174863 | + | 1.21620i | 0 | −0.260554 | + | 0.570534i | 0 | 0.258432 | − | 1.79743i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 209.1 | 0 | 0.0530529 | + | 0.368991i | 0 | −3.26024 | − | 0.957293i | 0 | 0.297176 | − | 0.342959i | 0 | 2.74514 | − | 0.806046i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 225.1 | 0 | −0.712591 | + | 0.822373i | 0 | 0.174863 | − | 1.21620i | 0 | −0.260554 | − | 0.570534i | 0 | 0.258432 | + | 1.79743i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 257.1 | 0 | 0.524075 | − | 0.153882i | 0 | 0.767092 | + | 0.492980i | 0 | 0.601808 | + | 4.18567i | 0 | −2.27279 | + | 1.46063i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 289.1 | 0 | 2.71616 | + | 1.74557i | 0 | 0.985691 | − | 2.15836i | 0 | −0.381761 | − | 0.112095i | 0 | 3.08427 | + | 6.75361i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 305.1 | 0 | 0.524075 | + | 0.153882i | 0 | 0.767092 | − | 0.492980i | 0 | 0.601808 | − | 4.18567i | 0 | −2.27279 | − | 1.46063i | 0 | |||||||||||||||||||||||||||||||||||||||||
| 353.1 | 0 | −0.580699 | − | 1.27155i | 0 | −1.66741 | − | 1.92429i | 0 | −1.75667 | − | 1.12894i | 0 | 0.684944 | − | 0.790468i | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.c | even | 11 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 368.2.m.b | 10 | |
| 4.b | odd | 2 | 1 | 46.2.c.a | ✓ | 10 | |
| 12.b | even | 2 | 1 | 414.2.i.f | 10 | ||
| 23.c | even | 11 | 1 | inner | 368.2.m.b | 10 | |
| 23.c | even | 11 | 1 | 8464.2.a.bx | 5 | ||
| 23.d | odd | 22 | 1 | 8464.2.a.bw | 5 | ||
| 92.g | odd | 22 | 1 | 46.2.c.a | ✓ | 10 | |
| 92.g | odd | 22 | 1 | 1058.2.a.m | 5 | ||
| 92.h | even | 22 | 1 | 1058.2.a.l | 5 | ||
| 276.j | odd | 22 | 1 | 9522.2.a.bu | 5 | ||
| 276.o | even | 22 | 1 | 414.2.i.f | 10 | ||
| 276.o | even | 22 | 1 | 9522.2.a.bp | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.2.c.a | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 46.2.c.a | ✓ | 10 | 92.g | odd | 22 | 1 | |
| 368.2.m.b | 10 | 1.a | even | 1 | 1 | trivial | |
| 368.2.m.b | 10 | 23.c | even | 11 | 1 | inner | |
| 414.2.i.f | 10 | 12.b | even | 2 | 1 | ||
| 414.2.i.f | 10 | 276.o | even | 22 | 1 | ||
| 1058.2.a.l | 5 | 92.h | even | 22 | 1 | ||
| 1058.2.a.m | 5 | 92.g | odd | 22 | 1 | ||
| 8464.2.a.bw | 5 | 23.d | odd | 22 | 1 | ||
| 8464.2.a.bx | 5 | 23.c | even | 11 | 1 | ||
| 9522.2.a.bp | 5 | 276.o | even | 22 | 1 | ||
| 9522.2.a.bu | 5 | 276.j | odd | 22 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 4T_{3}^{9} + 5T_{3}^{8} + 2T_{3}^{7} + 25T_{3}^{6} - T_{3}^{5} + 4T_{3}^{4} - 16T_{3}^{3} + 9T_{3}^{2} - 3T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( T^{10} - 4 T^{9} + \cdots + 1 \)
|
| $5$ |
\( T^{10} + 6 T^{9} + \cdots + 529 \)
|
| $7$ |
\( T^{10} + 3 T^{9} + \cdots + 1 \)
|
| $11$ |
\( T^{10} - 12 T^{9} + \cdots + 109561 \)
|
| $13$ |
\( T^{10} + 14 T^{9} + \cdots + 1 \)
|
| $17$ |
\( T^{10} - 15 T^{9} + \cdots + 214369 \)
|
| $19$ |
\( T^{10} + 2 T^{9} + \cdots + 4489 \)
|
| $23$ |
\( T^{10} - T^{9} + \cdots + 6436343 \)
|
| $29$ |
\( T^{10} + 8 T^{9} + \cdots + 4489 \)
|
| $31$ |
\( T^{10} - 21 T^{9} + \cdots + 20529961 \)
|
| $37$ |
\( T^{10} - 28 T^{9} + \cdots + 49857721 \)
|
| $41$ |
\( T^{10} + \cdots + 172475689 \)
|
| $43$ |
\( T^{10} + 11 T^{9} + \cdots + 7027801 \)
|
| $47$ |
\( (T^{5} + 9 T^{4} + \cdots + 529)^{2} \)
|
| $53$ |
\( T^{10} + 21 T^{9} + \cdots + 31236921 \)
|
| $59$ |
\( T^{10} - 5 T^{9} + \cdots + 4489 \)
|
| $61$ |
\( T^{10} + \cdots + 349727401 \)
|
| $67$ |
\( T^{10} - 13 T^{9} + \cdots + 94109401 \)
|
| $71$ |
\( T^{10} + 49 T^{9} + \cdots + 8300161 \)
|
| $73$ |
\( T^{10} + 8 T^{9} + \cdots + 7921 \)
|
| $79$ |
\( T^{10} + 8 T^{9} + \cdots + 17161 \)
|
| $83$ |
\( T^{10} + \cdots + 667137241 \)
|
| $89$ |
\( T^{10} + 13 T^{9} + \cdots + 26739241 \)
|
| $97$ |
\( T^{10} + 32 T^{9} + \cdots + 5031049 \)
|
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