Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,7,Mod(65,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.65");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 192.e (of order \(2\), degree \(1\), 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: | \(44.1703840550\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.1173604352.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 14x^{4} - 12x^{3} + 112x^{2} + 192x + 324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 14x^{4} - 12x^{3} + 112x^{2} + 192x + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 17\nu^{4} - 40\nu^{3} - 152\nu^{2} + 138\nu + 1143 ) / 45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -53\nu^{5} + 19\nu^{4} + 1160\nu^{3} + 896\nu^{2} - 8994\nu - 12069 ) / 135 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -128\nu^{5} + 64\nu^{4} + 1280\nu^{3} + 896\nu^{2} - 15744\nu - 13824 ) / 135 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{5} - 13\nu^{4} - 200\nu^{3} + 448\nu^{2} + 1758\nu - 792 ) / 15 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} + 49\nu^{4} + 200\nu^{3} - 304\nu^{2} - 54\nu + 216 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( 6\beta_{5} + 2\beta_{4} - 9\beta_{3} - 36\beta _1 + 12 ) / 1152 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 16\beta_{4} + 3\beta_{3} + 24\beta_{2} + 24\beta _1 + 2688 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} - 2\beta_{4} - 18\beta_{3} + 27\beta_{2} - 18\beta _1 + 879 ) / 144 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9\beta_{5} + 59\beta_{4} - 30\beta_{3} + 192\beta_{2} + 786\beta _1 - 2886 ) / 288 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -240\beta_{5} - 32\beta_{4} - 783\beta_{3} + 1440\beta_{2} + 2448\beta _1 - 11856 ) / 576 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).
| \(n\) | \(65\) | \(127\) | \(133\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −22.0014 | − | 15.6505i | 0 | − | 161.417i | 0 | −562.144 | 0 | 239.124 | + | 688.666i | 0 | |||||||||||||||||||||||||||||||
| 65.2 | 0 | −22.0014 | + | 15.6505i | 0 | 161.417i | 0 | −562.144 | 0 | 239.124 | − | 688.666i | 0 | |||||||||||||||||||||||||||||||||
| 65.3 | 0 | −6.43940 | − | 26.2209i | 0 | − | 10.3581i | 0 | 540.917 | 0 | −646.068 | + | 337.693i | 0 | ||||||||||||||||||||||||||||||||
| 65.4 | 0 | −6.43940 | + | 26.2209i | 0 | 10.3581i | 0 | 540.917 | 0 | −646.068 | − | 337.693i | 0 | |||||||||||||||||||||||||||||||||
| 65.5 | 0 | 23.4408 | − | 13.3988i | 0 | 100.147i | 0 | −56.7723 | 0 | 369.944 | − | 628.158i | 0 | |||||||||||||||||||||||||||||||||
| 65.6 | 0 | 23.4408 | + | 13.3988i | 0 | − | 100.147i | 0 | −56.7723 | 0 | 369.944 | + | 628.158i | 0 | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 192.7.e.g | 6 | |
| 3.b | odd | 2 | 1 | inner | 192.7.e.g | 6 | |
| 4.b | odd | 2 | 1 | 192.7.e.h | 6 | ||
| 8.b | even | 2 | 1 | 48.7.e.d | 6 | ||
| 8.d | odd | 2 | 1 | 24.7.e.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 192.7.e.h | 6 | ||
| 24.f | even | 2 | 1 | 24.7.e.a | ✓ | 6 | |
| 24.h | odd | 2 | 1 | 48.7.e.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.7.e.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 24.7.e.a | ✓ | 6 | 24.f | even | 2 | 1 | |
| 48.7.e.d | 6 | 8.b | even | 2 | 1 | ||
| 48.7.e.d | 6 | 24.h | odd | 2 | 1 | ||
| 192.7.e.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 192.7.e.g | 6 | 3.b | odd | 2 | 1 | inner | |
| 192.7.e.h | 6 | 4.b | odd | 2 | 1 | ||
| 192.7.e.h | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(192, [\chi])\):
|
\( T_{5}^{6} + 36192T_{5}^{4} + 265190400T_{5}^{2} + 28037120000 \)
|
|
\( T_{7}^{3} + 78T_{7}^{2} - 302868T_{7} - 17262936 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 10 T^{5} + \cdots + 387420489 \)
|
| $5$ |
\( T^{6} + \cdots + 28037120000 \)
|
| $7$ |
\( (T^{3} + 78 T^{2} + \cdots - 17262936)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 14\!\cdots\!32 \)
|
| $13$ |
\( (T^{3} + 78 T^{2} + \cdots + 4988511400)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 49\!\cdots\!48 \)
|
| $19$ |
\( (T^{3} + 2250 T^{2} + \cdots + 111704435512)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 20\!\cdots\!92 \)
|
| $29$ |
\( T^{6} + \cdots + 49\!\cdots\!12 \)
|
| $31$ |
\( (T^{3} - 37122 T^{2} + \cdots - 242319962776)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 58243421953944)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 45\!\cdots\!68 \)
|
| $43$ |
\( (T^{3} + \cdots - 162635792076808)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 86\!\cdots\!72 \)
|
| $53$ |
\( T^{6} + \cdots + 21\!\cdots\!72 \)
|
| $59$ |
\( T^{6} + \cdots + 67\!\cdots\!68 \)
|
| $61$ |
\( (T^{3} + \cdots - 26\!\cdots\!84)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 14\!\cdots\!92)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 74\!\cdots\!92 \)
|
| $73$ |
\( (T^{3} + \cdots + 22\!\cdots\!12)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 26\!\cdots\!20)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!92 \)
|
| $89$ |
\( T^{6} + \cdots + 45\!\cdots\!28 \)
|
| $97$ |
\( (T^{3} + \cdots + 36\!\cdots\!16)^{2} \)
|
| show more | |
| show less |