Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1224,2,Mod(613,1224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1224.613");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1224.f (of order \(2\), degree \(1\), 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: | \(9.77368920740\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1649659456.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 136) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 2\nu^{3} ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 4\nu^{2} - 8\nu - 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 4\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 2\nu^{4} - 4\nu^{3} + 4\nu^{2} + 8 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 4\nu^{4} + 4\nu^{3} + 4\nu^{2} - 16 ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 2\nu^{5} + 4\nu^{3} - 4\nu^{2} - 8 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} - 2\nu^{5} + 2\nu^{4} - 4\nu^{3} + 4\nu^{2} + 8\nu + 16 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2\beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{7} + 3\beta_{6} - \beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} + 2\beta _1 - 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} + 5\beta_{6} + \beta_{5} + 5\beta_{4} - \beta_{3} - \beta_{2} - 2\beta _1 + 2 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -3\beta_{7} - \beta_{6} + 3\beta_{5} + 7\beta_{4} + 5\beta_{3} - 3\beta_{2} + 2\beta _1 + 6 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 7\beta_{7} + 5\beta_{6} + \beta_{5} - 3\beta_{4} + 7\beta_{3} + 7\beta_{2} - 2\beta _1 + 10 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).
| \(n\) | \(137\) | \(613\) | \(649\) | \(919\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 613.1 |
|
−1.13622 | − | 0.842022i | 0 | 0.581998 | + | 1.91345i | 0.436910i | 0 | −1.90455 | 0.949886 | − | 2.66415i | 0 | 0.367888 | − | 0.496427i | ||||||||||||||||||||||||||||||||||
| 613.2 | −1.13622 | + | 0.842022i | 0 | 0.581998 | − | 1.91345i | − | 0.436910i | 0 | −1.90455 | 0.949886 | + | 2.66415i | 0 | 0.367888 | + | 0.496427i | ||||||||||||||||||||||||||||||||||
| 613.3 | −0.578647 | − | 1.29041i | 0 | −1.33034 | + | 1.49339i | 3.12087i | 0 | 2.86993 | 2.69688 | + | 0.852541i | 0 | 4.02722 | − | 1.80588i | |||||||||||||||||||||||||||||||||||
| 613.4 | −0.578647 | + | 1.29041i | 0 | −1.33034 | − | 1.49339i | − | 3.12087i | 0 | 2.86993 | 2.69688 | − | 0.852541i | 0 | 4.02722 | + | 1.80588i | ||||||||||||||||||||||||||||||||||
| 613.5 | 0.814732 | − | 1.15595i | 0 | −0.672424 | − | 1.88357i | − | 1.77580i | 0 | −0.423267 | −2.72515 | − | 0.757320i | 0 | −2.05273 | − | 1.44680i | ||||||||||||||||||||||||||||||||||
| 613.6 | 0.814732 | + | 1.15595i | 0 | −0.672424 | + | 1.88357i | 1.77580i | 0 | −0.423267 | −2.72515 | + | 0.757320i | 0 | −2.05273 | + | 1.44680i | |||||||||||||||||||||||||||||||||||
| 613.7 | 1.40014 | − | 0.199044i | 0 | 1.92076 | − | 0.557378i | 3.30391i | 0 | 3.45790 | 2.57839 | − | 1.16272i | 0 | 0.657624 | + | 4.62592i | |||||||||||||||||||||||||||||||||||
| 613.8 | 1.40014 | + | 0.199044i | 0 | 1.92076 | + | 0.557378i | − | 3.30391i | 0 | 3.45790 | 2.57839 | + | 1.16272i | 0 | 0.657624 | − | 4.62592i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1224.2.f.d | 8 | |
| 3.b | odd | 2 | 1 | 136.2.c.a | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 4896.2.f.c | 8 | ||
| 8.b | even | 2 | 1 | inner | 1224.2.f.d | 8 | |
| 8.d | odd | 2 | 1 | 4896.2.f.c | 8 | ||
| 12.b | even | 2 | 1 | 544.2.c.a | 8 | ||
| 24.f | even | 2 | 1 | 544.2.c.a | 8 | ||
| 24.h | odd | 2 | 1 | 136.2.c.a | ✓ | 8 | |
| 48.i | odd | 4 | 2 | 4352.2.a.bc | 8 | ||
| 48.k | even | 4 | 2 | 4352.2.a.be | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.2.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 136.2.c.a | ✓ | 8 | 24.h | odd | 2 | 1 | |
| 544.2.c.a | 8 | 12.b | even | 2 | 1 | ||
| 544.2.c.a | 8 | 24.f | even | 2 | 1 | ||
| 1224.2.f.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 1224.2.f.d | 8 | 8.b | even | 2 | 1 | inner | |
| 4352.2.a.bc | 8 | 48.i | odd | 4 | 2 | ||
| 4352.2.a.be | 8 | 48.k | even | 4 | 2 | ||
| 4896.2.f.c | 8 | 4.b | odd | 2 | 1 | ||
| 4896.2.f.c | 8 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\):
|
\( T_{5}^{8} + 24T_{5}^{6} + 176T_{5}^{4} + 368T_{5}^{2} + 64 \)
|
|
\( T_{23}^{4} + 6T_{23}^{3} - 12T_{23}^{2} - 70T_{23} + 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} + \cdots + 16 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 24 T^{6} + \cdots + 64 \)
|
| $7$ |
\( (T^{4} - 4 T^{3} - 4 T^{2} + \cdots + 8)^{2} \)
|
| $11$ |
\( T^{8} + 44 T^{6} + \cdots + 4096 \)
|
| $13$ |
\( T^{8} + 60 T^{6} + \cdots + 9216 \)
|
| $17$ |
\( (T - 1)^{8} \)
|
| $19$ |
\( T^{8} + 112 T^{6} + \cdots + 20736 \)
|
| $23$ |
\( (T^{4} + 6 T^{3} - 12 T^{2} + \cdots + 48)^{2} \)
|
| $29$ |
\( T^{8} + 152 T^{6} + \cdots + 1032256 \)
|
| $31$ |
\( (T^{4} + 6 T^{3} - 40 T^{2} + \cdots + 64)^{2} \)
|
| $37$ |
\( T^{8} + 64 T^{6} + \cdots + 5184 \)
|
| $41$ |
\( (T^{4} - 64 T^{2} + \cdots - 48)^{2} \)
|
| $43$ |
\( T^{8} + 156 T^{6} + \cdots + 589824 \)
|
| $47$ |
\( (T^{4} - 14 T^{3} + \cdots - 192)^{2} \)
|
| $53$ |
\( T^{8} + 176 T^{6} + \cdots + 1048576 \)
|
| $59$ |
\( T^{8} + 140 T^{6} + \cdots + 4096 \)
|
| $61$ |
\( T^{8} + 504 T^{6} + \cdots + 25644096 \)
|
| $67$ |
\( T^{8} + 240 T^{6} + \cdots + 2304 \)
|
| $71$ |
\( (T^{4} - 208 T^{2} + \cdots + 4608)^{2} \)
|
| $73$ |
\( (T^{4} - 4 T^{3} + \cdots + 30704)^{2} \)
|
| $79$ |
\( (T^{4} + 22 T^{3} + \cdots + 216)^{2} \)
|
| $83$ |
\( T^{8} + 444 T^{6} + \cdots + 34668544 \)
|
| $89$ |
\( (T^{4} + 4 T^{3} - 88 T^{2} + \cdots - 72)^{2} \)
|
| $97$ |
\( (T^{4} - 4 T^{3} + \cdots - 1168)^{2} \)
|
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