Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,2,Mod(79,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 0, 0, 13]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.79");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.ci (of order \(18\), degree \(6\), 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: | \(7.28235666434\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{18})\) |
| Coefficient field: | 12.0.101559956668416.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 8x^{6} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{12} - 8x^{6} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 8 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{8} ) / 16 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} ) / 16 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{10} ) / 32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{11} ) / 32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 8\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} \)
|
| \(\nu^{8}\) | \(=\) |
\( 16\beta_{8} \)
|
| \(\nu^{9}\) | \(=\) |
\( 16\beta_{9} \)
|
| \(\nu^{10}\) | \(=\) |
\( 32\beta_{10} \)
|
| \(\nu^{11}\) | \(=\) |
\( 32\beta_{11} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(-\beta_{4}\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
0 | 0.939693 | − | 0.342020i | 0 | −0.749734 | + | 4.25195i | 0 | −3.63109 | + | 2.09641i | 0 | 0.766044 | − | 0.642788i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 79.2 | 0 | 0.939693 | − | 0.342020i | 0 | 0.217645 | − | 1.23433i | 0 | −2.78039 | + | 1.60526i | 0 | 0.766044 | − | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 127.1 | 0 | 0.939693 | + | 0.342020i | 0 | −0.749734 | − | 4.25195i | 0 | −3.63109 | − | 2.09641i | 0 | 0.766044 | + | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 127.2 | 0 | 0.939693 | + | 0.342020i | 0 | 0.217645 | + | 1.23433i | 0 | −2.78039 | − | 1.60526i | 0 | 0.766044 | + | 0.642788i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 223.1 | 0 | −0.173648 | + | 0.984808i | 0 | 0.0469641 | + | 0.0394076i | 0 | 1.48976 | − | 0.860113i | 0 | −0.939693 | − | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 223.2 | 0 | −0.173648 | + | 0.984808i | 0 | 2.83242 | + | 2.37668i | 0 | −2.26308 | + | 1.30659i | 0 | −0.939693 | − | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 319.1 | 0 | −0.173648 | − | 0.984808i | 0 | 0.0469641 | − | 0.0394076i | 0 | 1.48976 | + | 0.860113i | 0 | −0.939693 | + | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −0.173648 | − | 0.984808i | 0 | 2.83242 | − | 2.37668i | 0 | −2.26308 | − | 1.30659i | 0 | −0.939693 | + | 0.342020i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 751.1 | 0 | −0.766044 | + | 0.642788i | 0 | −0.582687 | − | 0.212081i | 0 | 1.39416 | + | 0.804921i | 0 | 0.173648 | − | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 751.2 | 0 | −0.766044 | + | 0.642788i | 0 | 1.23539 | + | 0.449645i | 0 | −3.20937 | − | 1.85293i | 0 | 0.173648 | − | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 895.1 | 0 | −0.766044 | − | 0.642788i | 0 | −0.582687 | + | 0.212081i | 0 | 1.39416 | − | 0.804921i | 0 | 0.173648 | + | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 895.2 | 0 | −0.766044 | − | 0.642788i | 0 | 1.23539 | − | 0.449645i | 0 | −3.20937 | + | 1.85293i | 0 | 0.173648 | + | 0.984808i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.k | even | 18 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.2.ci.c | ✓ | 12 |
| 4.b | odd | 2 | 1 | 912.2.ci.d | yes | 12 | |
| 19.f | odd | 18 | 1 | 912.2.ci.d | yes | 12 | |
| 76.k | even | 18 | 1 | inner | 912.2.ci.c | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 912.2.ci.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 912.2.ci.c | ✓ | 12 | 76.k | even | 18 | 1 | inner |
| 912.2.ci.d | yes | 12 | 4.b | odd | 2 | 1 | |
| 912.2.ci.d | yes | 12 | 19.f | odd | 18 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):
|
\( T_{5}^{12} - 6 T_{5}^{11} + 33 T_{5}^{10} - 136 T_{5}^{9} + 450 T_{5}^{8} - 612 T_{5}^{7} + 432 T_{5}^{6} + \cdots + 1 \)
|
|
\( T_{7}^{12} + 18 T_{7}^{11} + 135 T_{7}^{10} + 486 T_{7}^{9} + 531 T_{7}^{8} - 1782 T_{7}^{7} + \cdots + 130321 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{6} - T^{3} + 1)^{2} \)
|
| $5$ |
\( T^{12} - 6 T^{11} + \cdots + 1 \)
|
| $7$ |
\( T^{12} + 18 T^{11} + \cdots + 130321 \)
|
| $11$ |
\( T^{12} - 42 T^{10} + \cdots + 5329 \)
|
| $13$ |
\( T^{12} - 18 T^{11} + \cdots + 1 \)
|
| $17$ |
\( T^{12} - 6 T^{11} + \cdots + 26569 \)
|
| $19$ |
\( T^{12} + 12 T^{11} + \cdots + 47045881 \)
|
| $23$ |
\( T^{12} - 18 T^{11} + \cdots + 3988009 \)
|
| $29$ |
\( T^{12} + \cdots + 151117849 \)
|
| $31$ |
\( T^{12} + \cdots + 254625849 \)
|
| $37$ |
\( T^{12} + \cdots + 166126321 \)
|
| $41$ |
\( T^{12} - 6 T^{11} + \cdots + 4422609 \)
|
| $43$ |
\( T^{12} - 6 T^{11} + \cdots + 71014329 \)
|
| $47$ |
\( T^{12} + 6 T^{11} + \cdots + 3052009 \)
|
| $53$ |
\( T^{12} + \cdots + 4583154601 \)
|
| $59$ |
\( T^{12} - 6 T^{11} + \cdots + 361 \)
|
| $61$ |
\( T^{12} + \cdots + 129665528281 \)
|
| $67$ |
\( T^{12} + 6 T^{11} + \cdots + 34023889 \)
|
| $71$ |
\( (T^{6} + 27 T^{5} + \cdots + 104329)^{2} \)
|
| $73$ |
\( T^{12} + \cdots + 749390625 \)
|
| $79$ |
\( T^{12} + \cdots + 4267094329 \)
|
| $83$ |
\( T^{12} + \cdots + 5079555441 \)
|
| $89$ |
\( T^{12} + \cdots + 21849865489 \)
|
| $97$ |
\( T^{12} + \cdots + 1138995001 \)
|
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