Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,6,Mod(1,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 136.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: | \(21.8121994946\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 223x^{2} - 604x + 320 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 223x^{2} - 604x + 320 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 5\nu^{2} - 375\nu - 329 ) / 39 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 5\nu^{2} + 453\nu + 329 ) / 39 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 233\nu - 899 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + 7\beta_{2} + \beta _1 + 226 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{3} + 205\beta_{2} + 229\beta _1 + 894 ) / 2 \)
|
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 |
|
0 | −17.0883 | 0 | −72.1807 | 0 | −209.255 | 0 | 49.0099 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −14.5928 | 0 | 54.2414 | 0 | 125.992 | 0 | −30.0514 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 15.8213 | 0 | 82.0080 | 0 | −86.6300 | 0 | 7.31211 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 25.8598 | 0 | 39.9313 | 0 | 255.893 | 0 | 425.729 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
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 | 136.6.a.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | 272.6.a.m | 4 | ||
| 8.b | even | 2 | 1 | 1088.6.a.u | 4 | ||
| 8.d | odd | 2 | 1 | 1088.6.a.y | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.6.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 272.6.a.m | 4 | 4.b | odd | 2 | 1 | ||
| 1088.6.a.u | 4 | 8.b | even | 2 | 1 | ||
| 1088.6.a.y | 4 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 10T_{3}^{3} - 662T_{3}^{2} + 2568T_{3} + 102024 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(136))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 10 T^{3} + \cdots + 102024 \)
|
| $5$ |
\( T^{4} - 104 T^{3} + \cdots - 12820992 \)
|
| $7$ |
\( T^{4} - 86 T^{3} + \cdots + 584450208 \)
|
| $11$ |
\( T^{4} - 338 T^{3} + \cdots - 37269784 \)
|
| $13$ |
\( T^{4} + \cdots + 5178083040 \)
|
| $17$ |
\( (T + 289)^{4} \)
|
| $19$ |
\( T^{4} + \cdots + 122102872448 \)
|
| $23$ |
\( T^{4} + \cdots - 9804410831872 \)
|
| $29$ |
\( T^{4} + \cdots + 34154783989248 \)
|
| $31$ |
\( T^{4} + \cdots + 71284364541728 \)
|
| $37$ |
\( T^{4} + \cdots - 803999544522304 \)
|
| $41$ |
\( T^{4} + \cdots + 332840657425296 \)
|
| $43$ |
\( T^{4} + \cdots - 21\!\cdots\!68 \)
|
| $47$ |
\( T^{4} + \cdots + 31\!\cdots\!56 \)
|
| $53$ |
\( T^{4} + \cdots + 36\!\cdots\!68 \)
|
| $59$ |
\( T^{4} + \cdots - 70\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots - 26\!\cdots\!04 \)
|
| $67$ |
\( T^{4} + \cdots + 33\!\cdots\!36 \)
|
| $71$ |
\( T^{4} + \cdots + 31\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 97\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 24\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots - 26\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 59\!\cdots\!52 \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!84 \)
|
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