Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [386,2,Mod(1,386)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(386, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("386.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 386 = 2 \cdot 193 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 386.a (trivial) |
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: | yes |
| Analytic conductor: | \(3.08222551802\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 10x^{5} + 33x^{4} + 14x^{3} - 91x^{2} + 45x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 10x^{5} + 33x^{4} + 14x^{3} - 91x^{2} + 45x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{6} - 3\nu^{5} - 24\nu^{4} + 29\nu^{3} + 68\nu^{2} - 75\nu - 18 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{6} + 4\nu^{5} + 37\nu^{4} - 38\nu^{3} - 109\nu^{2} + 98\nu + 36 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{6} - 8\nu^{5} - 61\nu^{4} + 78\nu^{3} + 181\nu^{2} - 194\nu - 62 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -6\nu^{6} + 9\nu^{5} + 74\nu^{4} - 89\nu^{3} - 220\nu^{2} + 229\nu + 70 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -13\nu^{6} + 20\nu^{5} + 159\nu^{4} - 196\nu^{3} - 467\nu^{2} + 498\nu + 142 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 2\beta_{5} + \beta_{3} + 2\beta_{2} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{6} - 9\beta_{5} + 8\beta_{4} + \beta_{3} + 13\beta_{2} + 4\beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{6} - 26\beta_{5} + 2\beta_{4} + 9\beta_{3} + 28\beta_{2} + 45\beta _1 - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 82\beta_{6} - 84\beta_{5} + 65\beta_{4} + 11\beta_{3} + 136\beta_{2} + 66\beta _1 + 159 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.60783 | 1.00000 | 3.42620 | −2.60783 | −1.91434 | 1.00000 | 3.80080 | 3.42620 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.08661 | 1.00000 | −1.77777 | −2.08661 | 2.36210 | 1.00000 | 1.35393 | −1.77777 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.238671 | 1.00000 | 3.12720 | −0.238671 | 3.46803 | 1.00000 | −2.94304 | 3.12720 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.39942 | 1.00000 | 2.37946 | 1.39942 | −1.05546 | 1.00000 | −1.04164 | 2.37946 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.47587 | 1.00000 | −1.61512 | 1.47587 | 3.66066 | 1.00000 | −0.821817 | −1.61512 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.87281 | 1.00000 | 1.71811 | 1.87281 | −3.99175 | 1.00000 | 0.507432 | 1.71811 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.18502 | 1.00000 | −2.25808 | 3.18502 | −0.529232 | 1.00000 | 7.14433 | −2.25808 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(193\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 386.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 3474.2.a.v | 7 | ||
| 4.b | odd | 2 | 1 | 3088.2.a.k | 7 | ||
| 5.b | even | 2 | 1 | 9650.2.a.bd | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 386.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 3088.2.a.k | 7 | 4.b | odd | 2 | 1 | ||
| 3474.2.a.v | 7 | 3.b | odd | 2 | 1 | ||
| 9650.2.a.bd | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 3T_{3}^{6} - 10T_{3}^{5} + 33T_{3}^{4} + 14T_{3}^{3} - 91T_{3}^{2} + 45T_{3} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(386))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots + 16 \)
|
| $5$ |
\( T^{7} - 5 T^{6} + \cdots + 284 \)
|
| $7$ |
\( T^{7} - 2 T^{6} + \cdots - 128 \)
|
| $11$ |
\( T^{7} - 2 T^{6} + \cdots + 352 \)
|
| $13$ |
\( T^{7} - 5 T^{6} + \cdots + 3076 \)
|
| $17$ |
\( T^{7} - 8 T^{6} + \cdots + 160 \)
|
| $19$ |
\( T^{7} - 4 T^{6} + \cdots + 864 \)
|
| $23$ |
\( T^{7} + 8 T^{6} + \cdots + 640 \)
|
| $29$ |
\( T^{7} - 76 T^{5} + \cdots - 640 \)
|
| $31$ |
\( T^{7} + 6 T^{6} + \cdots + 2048 \)
|
| $37$ |
\( T^{7} + T^{6} + \cdots + 165884 \)
|
| $41$ |
\( T^{7} + 4 T^{6} + \cdots - 881248 \)
|
| $43$ |
\( T^{7} + 13 T^{6} + \cdots - 143068 \)
|
| $47$ |
\( T^{7} + 17 T^{6} + \cdots - 152 \)
|
| $53$ |
\( T^{7} + 5 T^{6} + \cdots + 375140 \)
|
| $59$ |
\( T^{7} + 3 T^{6} + \cdots + 35800 \)
|
| $61$ |
\( T^{7} - 12 T^{6} + \cdots + 1077568 \)
|
| $67$ |
\( T^{7} - T^{6} + \cdots - 662912 \)
|
| $71$ |
\( T^{7} + 25 T^{6} + \cdots + 205200 \)
|
| $73$ |
\( T^{7} - 28 T^{6} + \cdots - 3271840 \)
|
| $79$ |
\( T^{7} + 5 T^{6} + \cdots + 169732 \)
|
| $83$ |
\( T^{7} + 7 T^{6} + \cdots + 94496 \)
|
| $89$ |
\( T^{7} - 4 T^{6} + \cdots + 109600 \)
|
| $97$ |
\( T^{7} - 13 T^{6} + \cdots - 101770 \)
|
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