Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [165,6,Mod(1,165)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(165, 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("165.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.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: | \(26.4633302691\) |
| 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: |
\( x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 5 \) |
| 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
\( x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 60 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 88\nu - 78 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -15\nu^{6} + 71\nu^{5} + 2323\nu^{4} - 11587\nu^{3} - 70328\nu^{2} + 346212\nu - 34776 ) / 2344 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7\nu^{6} + 45\nu^{5} - 1631\nu^{4} - 7641\nu^{3} + 104468\nu^{2} + 291764\nu - 1252344 ) / 2344 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -9\nu^{6} + 277\nu^{5} + 2097\nu^{4} - 41409\nu^{3} - 113220\nu^{2} + 1195020\nu + 889544 ) / 2344 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 60 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 88\beta _1 + 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - 3\beta_{5} - 2\beta_{4} - 2\beta_{3} + 124\beta_{2} - 17\beta _1 + 5272 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{6} + 9\beta_{5} + 153\beta_{3} - 216\beta_{2} + 8775\beta _1 + 1126 \)
|
| \(\nu^{6}\) | \(=\) |
\( 188\beta_{6} - 422\beta_{5} - 466\beta_{4} - 358\beta_{3} + 14265\beta_{2} - 5994\beta _1 + 524252 \)
|
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 |
|
−10.8333 | 9.00000 | 85.3606 | 25.0000 | −97.4998 | −188.476 | −578.072 | 81.0000 | −270.833 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −6.96291 | 9.00000 | 16.4822 | 25.0000 | −62.6662 | 244.038 | 108.049 | 81.0000 | −174.073 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −5.88025 | 9.00000 | 2.57733 | 25.0000 | −52.9222 | −219.194 | 173.013 | 81.0000 | −147.006 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0402667 | 9.00000 | −31.9984 | 25.0000 | −0.362400 | 37.9248 | 2.57700 | 81.0000 | −1.00667 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.17374 | 9.00000 | −5.23241 | 25.0000 | 46.5637 | 24.5405 | −192.631 | 81.0000 | 129.343 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 9.12102 | 9.00000 | 51.1931 | 25.0000 | 82.0892 | 202.409 | 175.061 | 81.0000 | 228.026 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 10.4220 | 9.00000 | 76.6176 | 25.0000 | 93.7978 | −101.242 | 465.003 | 81.0000 | 260.549 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 165.6.a.h | ✓ | 7 |
| 3.b | odd | 2 | 1 | 495.6.a.n | 7 | ||
| 5.b | even | 2 | 1 | 825.6.a.n | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.h | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 495.6.a.n | 7 | 3.b | odd | 2 | 1 | ||
| 825.6.a.n | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - T_{2}^{6} - 209T_{2}^{5} + 137T_{2}^{4} + 12724T_{2}^{3} - 1040T_{2}^{2} - 218208T_{2} - 8784 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} + \cdots - 8784 \)
|
| $3$ |
\( (T - 9)^{7} \)
|
| $5$ |
\( (T - 25)^{7} \)
|
| $7$ |
\( T^{7} + \cdots + 192283411073024 \)
|
| $11$ |
\( (T - 121)^{7} \)
|
| $13$ |
\( T^{7} + \cdots + 62\!\cdots\!76 \)
|
| $17$ |
\( T^{7} + \cdots - 73\!\cdots\!36 \)
|
| $19$ |
\( T^{7} + \cdots - 20\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots - 52\!\cdots\!96 \)
|
| $29$ |
\( T^{7} + \cdots - 26\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 15\!\cdots\!24 \)
|
| $37$ |
\( T^{7} + \cdots - 38\!\cdots\!96 \)
|
| $41$ |
\( T^{7} + \cdots + 71\!\cdots\!68 \)
|
| $43$ |
\( T^{7} + \cdots + 11\!\cdots\!00 \)
|
| $47$ |
\( T^{7} + \cdots + 24\!\cdots\!32 \)
|
| $53$ |
\( T^{7} + \cdots + 47\!\cdots\!08 \)
|
| $59$ |
\( T^{7} + \cdots - 29\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 55\!\cdots\!84 \)
|
| $67$ |
\( T^{7} + \cdots - 12\!\cdots\!44 \)
|
| $71$ |
\( T^{7} + \cdots + 83\!\cdots\!40 \)
|
| $73$ |
\( T^{7} + \cdots + 45\!\cdots\!28 \)
|
| $79$ |
\( T^{7} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 39\!\cdots\!92 \)
|
| $89$ |
\( T^{7} + \cdots + 59\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 29\!\cdots\!68 \)
|
| show more | |
| show less |