Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [212,1,Mod(15,212)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(212, base_ring=CyclotomicField(26))
chi = DirichletCharacter(H, H._module([13, 6]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("212.15");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 212 = 2^{2} \cdot 53 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 212.i (of order \(26\), degree \(12\), 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: | \(0.105801782678\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\Q(\zeta_{26})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{13}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/212\mathbb{Z}\right)^\times\).
| \(n\) | \(107\) | \(161\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{26}^{12}\) |
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 |
|
−0.748511 | − | 0.663123i | 0 | 0.120537 | + | 0.992709i | 0.213460 | − | 0.112032i | 0 | 0 | 0.568065 | − | 0.822984i | 0.885456 | − | 0.464723i | −0.234068 | − | 0.0576926i | ||||||||||||||||||||||||||||||||||||||||||
| 47.1 | 0.885456 | − | 0.464723i | 0 | 0.568065 | − | 0.822984i | −0.850405 | + | 0.753393i | 0 | 0 | 0.120537 | − | 0.992709i | −0.748511 | + | 0.663123i | −0.402877 | + | 1.06230i | |||||||||||||||||||||||||||||||||||||||||||
| 63.1 | −0.970942 | + | 0.239316i | 0 | 0.885456 | − | 0.464723i | −0.627974 | − | 1.65583i | 0 | 0 | −0.748511 | + | 0.663123i | −0.354605 | − | 0.935016i | 1.00599 | + | 1.45743i | |||||||||||||||||||||||||||||||||||||||||||
| 95.1 | −0.354605 | + | 0.935016i | 0 | −0.748511 | − | 0.663123i | 1.45352 | − | 0.358261i | 0 | 0 | 0.885456 | − | 0.464723i | −0.970942 | + | 0.239316i | −0.180446 | + | 1.48611i | |||||||||||||||||||||||||||||||||||||||||||
| 99.1 | −0.748511 | + | 0.663123i | 0 | 0.120537 | − | 0.992709i | 0.213460 | + | 0.112032i | 0 | 0 | 0.568065 | + | 0.822984i | 0.885456 | + | 0.464723i | −0.234068 | + | 0.0576926i | |||||||||||||||||||||||||||||||||||||||||||
| 119.1 | 0.120537 | + | 0.992709i | 0 | −0.970942 | + | 0.239316i | −1.10312 | + | 1.59814i | 0 | 0 | −0.354605 | − | 0.935016i | 0.568065 | − | 0.822984i | −1.71945 | − | 0.902438i | |||||||||||||||||||||||||||||||||||||||||||
| 155.1 | 0.120537 | − | 0.992709i | 0 | −0.970942 | − | 0.239316i | −1.10312 | − | 1.59814i | 0 | 0 | −0.354605 | + | 0.935016i | 0.568065 | + | 0.822984i | −1.71945 | + | 0.902438i | |||||||||||||||||||||||||||||||||||||||||||
| 175.1 | −0.970942 | − | 0.239316i | 0 | 0.885456 | + | 0.464723i | −0.627974 | + | 1.65583i | 0 | 0 | −0.748511 | − | 0.663123i | −0.354605 | + | 0.935016i | 1.00599 | − | 1.45743i | |||||||||||||||||||||||||||||||||||||||||||
| 183.1 | −0.354605 | − | 0.935016i | 0 | −0.748511 | + | 0.663123i | 1.45352 | + | 0.358261i | 0 | 0 | 0.885456 | + | 0.464723i | −0.970942 | − | 0.239316i | −0.180446 | − | 1.48611i | |||||||||||||||||||||||||||||||||||||||||||
| 187.1 | 0.568065 | + | 0.822984i | 0 | −0.354605 | + | 0.935016i | −0.0854858 | − | 0.704039i | 0 | 0 | −0.970942 | + | 0.239316i | 0.120537 | + | 0.992709i | 0.530851 | − | 0.470293i | |||||||||||||||||||||||||||||||||||||||||||
| 195.1 | 0.568065 | − | 0.822984i | 0 | −0.354605 | − | 0.935016i | −0.0854858 | + | 0.704039i | 0 | 0 | −0.970942 | − | 0.239316i | 0.120537 | − | 0.992709i | 0.530851 | + | 0.470293i | |||||||||||||||||||||||||||||||||||||||||||
| 203.1 | 0.885456 | + | 0.464723i | 0 | 0.568065 | + | 0.822984i | −0.850405 | − | 0.753393i | 0 | 0 | 0.120537 | + | 0.992709i | −0.748511 | − | 0.663123i | −0.402877 | − | 1.06230i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
| 53.d | even | 13 | 1 | inner |
| 212.i | odd | 26 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 212.1.i.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 1908.1.bb.a | 12 | ||
| 4.b | odd | 2 | 1 | CM | 212.1.i.a | ✓ | 12 |
| 8.b | even | 2 | 1 | 3392.1.bp.a | 12 | ||
| 8.d | odd | 2 | 1 | 3392.1.bp.a | 12 | ||
| 12.b | even | 2 | 1 | 1908.1.bb.a | 12 | ||
| 53.d | even | 13 | 1 | inner | 212.1.i.a | ✓ | 12 |
| 159.j | odd | 26 | 1 | 1908.1.bb.a | 12 | ||
| 212.i | odd | 26 | 1 | inner | 212.1.i.a | ✓ | 12 |
| 424.o | odd | 26 | 1 | 3392.1.bp.a | 12 | ||
| 424.t | even | 26 | 1 | 3392.1.bp.a | 12 | ||
| 636.s | even | 26 | 1 | 1908.1.bb.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 212.1.i.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 212.1.i.a | ✓ | 12 | 4.b | odd | 2 | 1 | CM |
| 212.1.i.a | ✓ | 12 | 53.d | even | 13 | 1 | inner |
| 212.1.i.a | ✓ | 12 | 212.i | odd | 26 | 1 | inner |
| 1908.1.bb.a | 12 | 3.b | odd | 2 | 1 | ||
| 1908.1.bb.a | 12 | 12.b | even | 2 | 1 | ||
| 1908.1.bb.a | 12 | 159.j | odd | 26 | 1 | ||
| 1908.1.bb.a | 12 | 636.s | even | 26 | 1 | ||
| 3392.1.bp.a | 12 | 8.b | even | 2 | 1 | ||
| 3392.1.bp.a | 12 | 8.d | odd | 2 | 1 | ||
| 3392.1.bp.a | 12 | 424.o | odd | 26 | 1 | ||
| 3392.1.bp.a | 12 | 424.t | even | 26 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(212, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} \)
|
| $13$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $17$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} \)
|
| $29$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $31$ |
\( T^{12} \)
|
| $37$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $41$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $43$ |
\( T^{12} \)
|
| $47$ |
\( T^{12} \)
|
| $53$ |
\( T^{12} + T^{11} + \cdots + 1 \)
|
| $59$ |
\( T^{12} \)
|
| $61$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $67$ |
\( T^{12} \)
|
| $71$ |
\( T^{12} \)
|
| $73$ |
\( T^{12} + 2 T^{11} + \cdots + 1 \)
|
| $79$ |
\( T^{12} \)
|
| $83$ |
\( T^{12} \)
|
| $89$ |
\( T^{12} - 11 T^{11} + \cdots + 1 \)
|
| $97$ |
\( T^{12} - 11 T^{11} + \cdots + 1 \)
|
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