Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [384,10,Mod(193,384)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("384.193");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 384 = 2^{7} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 384.d (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: | \(197.773761087\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{52}\cdot 3^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 303938737 \nu^{8} - 3215107672890 \nu^{6} + \cdots - 30\!\cdots\!72 ) / 11\!\cdots\!80 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 308496745 \nu^{8} + 5087199131946 \nu^{6} + \cdots + 16\!\cdots\!24 ) / 56\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6881235923 \nu^{8} + 103669410123246 \nu^{6} + \cdots - 71\!\cdots\!48 ) / 56\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 69206986817 \nu^{8} + \cdots - 17\!\cdots\!60 ) / 37\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2120694601 \nu^{9} - 35965121354090 \nu^{7} + \cdots - 10\!\cdots\!56 \nu ) / 71\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 105159413737129 \nu^{9} + \cdots - 88\!\cdots\!76 \nu ) / 27\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 692616151606351 \nu^{9} + \cdots + 24\!\cdots\!76 \nu ) / 55\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 125349951870869 \nu^{9} + \cdots + 82\!\cdots\!00 \nu ) / 34\!\cdots\!20 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 18\!\cdots\!27 \nu^{9} + \cdots - 60\!\cdots\!72 \nu ) / 18\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{9} + 53\beta_{7} + 148\beta_{6} + 21\beta_{5} ) / 9216 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 120\beta_{3} + 1652\beta_{2} - 1397\beta _1 - 10682541 ) / 3072 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4733\beta_{9} - 35136\beta_{8} - 467425\beta_{7} - 947588\beta_{6} - 1018767489\beta_{5} ) / 9216 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 45961\beta_{4} + 1008216\beta_{3} + 378892\beta_{2} + 15025565\beta _1 + 73666993461 ) / 3072 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 33885109 \beta_{9} + 371750688 \beta_{8} + 4421316473 \beta_{7} + 6168369604 \beta_{6} + 14034606037113 \beta_{5} ) / 9216 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 218709739 \beta_{4} - 3328147432 \beta_{3} - 23500742980 \beta_{2} - 49005318759 \beta _1 - 206497978666895 ) / 1024 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 282754258541 \beta_{9} - 3486009452544 \beta_{8} - 42756986004625 \beta_{7} + \cdots - 15\!\cdots\!01 \beta_{5} ) / 9216 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 7301243400377 \beta_{4} + 100824151254552 \beta_{3} + 963825255608204 \beta_{2} + \cdots + 56\!\cdots\!05 ) / 3072 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 25\!\cdots\!97 \beta_{9} + \cdots + 15\!\cdots\!05 \beta_{5} ) / 9216 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(133\) | \(257\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | − | 81.0000i | 0 | − | 1878.17i | 0 | −4378.40 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | − | 81.0000i | 0 | − | 332.682i | 0 | 3969.81 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | − | 81.0000i | 0 | − | 309.546i | 0 | 5272.51 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | − | 81.0000i | 0 | 553.862i | 0 | −12491.9 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 193.5 | 0 | − | 81.0000i | 0 | 2526.54i | 0 | 4839.94 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 193.6 | 0 | 81.0000i | 0 | − | 2526.54i | 0 | 4839.94 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 193.7 | 0 | 81.0000i | 0 | − | 553.862i | 0 | −12491.9 | 0 | −6561.00 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 193.8 | 0 | 81.0000i | 0 | 309.546i | 0 | 5272.51 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 193.9 | 0 | 81.0000i | 0 | 332.682i | 0 | 3969.81 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 193.10 | 0 | 81.0000i | 0 | 1878.17i | 0 | −4378.40 | 0 | −6561.00 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
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 | 384.10.d.c | ✓ | 10 |
| 4.b | odd | 2 | 1 | 384.10.d.d | yes | 10 | |
| 8.b | even | 2 | 1 | inner | 384.10.d.c | ✓ | 10 |
| 8.d | odd | 2 | 1 | 384.10.d.d | yes | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 384.10.d.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 384.10.d.c | ✓ | 10 | 8.b | even | 2 | 1 | inner |
| 384.10.d.d | yes | 10 | 4.b | odd | 2 | 1 | |
| 384.10.d.d | yes | 10 | 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 \(S_{10}^{\mathrm{new}}(384, [\chi])\):
|
\( T_{5}^{10} + 10424192 T_{5}^{8} + 27678481125376 T_{5}^{6} + \cdots + 73\!\cdots\!00 \)
|
|
\( T_{7}^{5} + 2788T_{7}^{4} - 117213856T_{7}^{3} + 236229687680T_{7}^{2} + 1882375258450176T_{7} - 5540757921577073664 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T^{2} + 6561)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 73\!\cdots\!00 \)
|
| $7$ |
\( (T^{5} + \cdots - 55\!\cdots\!64)^{2} \)
|
| $11$ |
\( T^{10} + \cdots + 11\!\cdots\!64 \)
|
| $13$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $17$ |
\( (T^{5} + \cdots - 76\!\cdots\!48)^{2} \)
|
| $19$ |
\( T^{10} + \cdots + 71\!\cdots\!16 \)
|
| $23$ |
\( (T^{5} + \cdots - 14\!\cdots\!12)^{2} \)
|
| $29$ |
\( T^{10} + \cdots + 30\!\cdots\!00 \)
|
| $31$ |
\( (T^{5} + \cdots - 10\!\cdots\!32)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 42\!\cdots\!36 \)
|
| $41$ |
\( (T^{5} + \cdots - 81\!\cdots\!88)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 45\!\cdots\!24 \)
|
| $47$ |
\( (T^{5} + \cdots - 90\!\cdots\!60)^{2} \)
|
| $53$ |
\( T^{10} + \cdots + 83\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots + 42\!\cdots\!04 \)
|
| $61$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $67$ |
\( T^{10} + \cdots + 14\!\cdots\!96 \)
|
| $71$ |
\( (T^{5} + \cdots + 48\!\cdots\!76)^{2} \)
|
| $73$ |
\( (T^{5} + \cdots - 55\!\cdots\!56)^{2} \)
|
| $79$ |
\( (T^{5} + \cdots - 13\!\cdots\!52)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 26\!\cdots\!24 \)
|
| $89$ |
\( (T^{5} + \cdots - 65\!\cdots\!48)^{2} \)
|
| $97$ |
\( (T^{5} + \cdots + 35\!\cdots\!00)^{2} \)
|
| show more | |
| show less |