Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [672,4,Mod(193,672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(672, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("672.193");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 672 = 2^{5} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 672.q (of order \(3\), 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: | \(39.6492835239\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.2972215728.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 32x^{4} - 59x^{3} + 232x^{2} - 203x + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 32x^{4} - 59x^{3} + 232x^{2} - 203x + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{4} - 4\nu^{3} + 46\nu^{2} - 44\nu + 133 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{5} - 3\nu^{4} + 58\nu^{3} - 56\nu^{2} + 412\nu - 203 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 62\nu^{3} - 88\nu^{2} + 428\nu - 196 ) / 7 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{5} - 7\nu^{4} + 66\nu^{3} - 120\nu^{2} + 472\nu - 203 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -36\nu^{5} + 91\nu^{4} - 1104\nu^{3} + 1586\nu^{2} - 7502\nu + 3549 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + \beta_{2} + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta _1 - 18 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{5} - 13\beta_{4} + 65\beta_{3} - 16\beta_{2} + 2\beta _1 - 73 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} - 25\beta_{4} + 66\beta_{3} - 5\beta_{2} - 14\beta _1 + 230 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -52\beta_{5} + 152\beta_{4} - 1331\beta_{3} + 257\beta_{2} - 44\beta _1 + 1793 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(421\) | \(449\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 + \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −1.50000 | + | 2.59808i | 0 | −8.28471 | − | 14.3495i | 0 | −17.1867 | − | 6.90059i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||||||||||||||||||||
| 193.2 | 0 | −1.50000 | + | 2.59808i | 0 | −1.93774 | − | 3.35626i | 0 | 15.0188 | − | 10.8367i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 193.3 | 0 | −1.50000 | + | 2.59808i | 0 | 8.22245 | + | 14.2417i | 0 | 10.6679 | + | 15.1393i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 289.1 | 0 | −1.50000 | − | 2.59808i | 0 | −8.28471 | + | 14.3495i | 0 | −17.1867 | + | 6.90059i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 289.2 | 0 | −1.50000 | − | 2.59808i | 0 | −1.93774 | + | 3.35626i | 0 | 15.0188 | + | 10.8367i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 289.3 | 0 | −1.50000 | − | 2.59808i | 0 | 8.22245 | − | 14.2417i | 0 | 10.6679 | − | 15.1393i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 672.4.q.e | ✓ | 6 |
| 4.b | odd | 2 | 1 | 672.4.q.f | yes | 6 | |
| 7.c | even | 3 | 1 | inner | 672.4.q.e | ✓ | 6 |
| 28.g | odd | 6 | 1 | 672.4.q.f | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 672.4.q.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 672.4.q.e | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 672.4.q.f | yes | 6 | 4.b | odd | 2 | 1 | |
| 672.4.q.f | yes | 6 | 28.g | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(672, [\chi])\):
|
\( T_{5}^{6} + 4T_{5}^{5} + 288T_{5}^{4} + 1024T_{5}^{3} + 78208T_{5}^{2} + 287232T_{5} + 1115136 \)
|
|
\( T_{11}^{6} + 58T_{11}^{5} + 3352T_{11}^{4} + 18120T_{11}^{3} + 505440T_{11}^{2} - 104544T_{11} + 75898944 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 1115136 \)
|
| $7$ |
\( T^{6} - 17 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + 58 T^{5} + \cdots + 75898944 \)
|
| $13$ |
\( (T^{3} + 85 T^{2} + \cdots - 291753)^{2} \)
|
| $17$ |
\( T^{6} + 150 T^{5} + \cdots + 338265664 \)
|
| $19$ |
\( T^{6} + \cdots + 38738506041 \)
|
| $23$ |
\( T^{6} + \cdots + 4874273856 \)
|
| $29$ |
\( (T^{3} + 224 T^{2} + \cdots - 965888)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 638715844809 \)
|
| $37$ |
\( T^{6} + \cdots + 45092071093969 \)
|
| $41$ |
\( (T^{3} - 990 T^{2} + \cdots - 32791016)^{2} \)
|
| $43$ |
\( (T^{3} - 539 T^{2} + \cdots + 42663111)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 806372798063616 \)
|
| $53$ |
\( T^{6} + \cdots + 223497955223104 \)
|
| $59$ |
\( T^{6} + \cdots + 662267166640704 \)
|
| $61$ |
\( T^{6} + \cdots + 52638043123264 \)
|
| $67$ |
\( T^{6} + \cdots + 265443959587401 \)
|
| $71$ |
\( (T^{3} - 528 T^{2} + \cdots + 333038304)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 17\!\cdots\!09 \)
|
| $79$ |
\( T^{6} + \cdots + 2702824776729 \)
|
| $83$ |
\( (T^{3} + 668 T^{2} + \cdots - 304149984)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 59\!\cdots\!24 \)
|
| $97$ |
\( (T^{3} + 1202 T^{2} + \cdots - 1458012456)^{2} \)
|
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