Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,6,Mod(1,55)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(55, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("55.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.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: | \(8.82111008971\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 129x^{3} + 45x^{2} + 2924x - 5216 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 129x^{3} + 45x^{2} + 2924x - 5216 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 111\nu^{2} - 150\nu + 1400 ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 6\nu^{3} - 147\nu^{2} - 624\nu + 3056 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 52 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{4} - 4\beta_{3} + 6\beta_{2} + 85\beta _1 + 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( 24\beta_{3} + 111\beta_{2} + 261\beta _1 + 4372 \)
|
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 |
|
−8.68310 | 4.24113 | 43.3963 | 25.0000 | −36.8261 | −61.3633 | −98.9550 | −225.013 | −217.078 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.83289 | 28.4084 | −28.6405 | 25.0000 | −52.0695 | 92.2140 | 111.147 | 564.037 | −45.8222 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.134973 | −22.6392 | −31.9818 | 25.0000 | 3.05567 | −118.127 | 8.63578 | 269.534 | −3.37431 | ||||||||||||||||||||||||||||||||||
| 1.4 | 8.57133 | 16.0464 | 41.4676 | 25.0000 | 137.539 | 3.71474 | 81.1502 | 14.4878 | 214.283 | ||||||||||||||||||||||||||||||||||
| 1.5 | 11.0796 | −26.0567 | 90.7584 | 25.0000 | −288.699 | 153.562 | 651.022 | 435.954 | 276.991 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
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 | 55.6.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 495.6.a.h | 5 | ||
| 4.b | odd | 2 | 1 | 880.6.a.r | 5 | ||
| 5.b | even | 2 | 1 | 275.6.a.e | 5 | ||
| 5.c | odd | 4 | 2 | 275.6.b.e | 10 | ||
| 11.b | odd | 2 | 1 | 605.6.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 275.6.a.e | 5 | 5.b | even | 2 | 1 | ||
| 275.6.b.e | 10 | 5.c | odd | 4 | 2 | ||
| 495.6.a.h | 5 | 3.b | odd | 2 | 1 | ||
| 605.6.a.d | 5 | 11.b | odd | 2 | 1 | ||
| 880.6.a.r | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 9T_{2}^{4} - 97T_{2}^{3} + 673T_{2}^{2} + 1604T_{2} + 204 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(55))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 9 T^{4} + \cdots + 204 \)
|
| $3$ |
\( T^{5} - 1137 T^{3} + \cdots - 1140480 \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} - 70 T^{4} + \cdots - 381300480 \)
|
| $11$ |
\( (T + 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots - 436629940000 \)
|
| $17$ |
\( T^{5} + \cdots + 333141603962904 \)
|
| $19$ |
\( T^{5} + \cdots + 35\!\cdots\!64 \)
|
| $23$ |
\( T^{5} + \cdots + 38\!\cdots\!76 \)
|
| $29$ |
\( T^{5} + \cdots + 12\!\cdots\!24 \)
|
| $31$ |
\( T^{5} + \cdots + 15\!\cdots\!12 \)
|
| $37$ |
\( T^{5} + \cdots + 26\!\cdots\!12 \)
|
| $41$ |
\( T^{5} + \cdots - 11\!\cdots\!32 \)
|
| $43$ |
\( T^{5} + \cdots + 40\!\cdots\!76 \)
|
| $47$ |
\( T^{5} + \cdots + 86\!\cdots\!88 \)
|
| $53$ |
\( T^{5} + \cdots - 29\!\cdots\!48 \)
|
| $59$ |
\( T^{5} + \cdots - 69\!\cdots\!56 \)
|
| $61$ |
\( T^{5} + \cdots - 60\!\cdots\!20 \)
|
| $67$ |
\( T^{5} + \cdots - 30\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots - 46\!\cdots\!12 \)
|
| $73$ |
\( T^{5} + \cdots + 61\!\cdots\!36 \)
|
| $79$ |
\( T^{5} + \cdots + 49\!\cdots\!92 \)
|
| $83$ |
\( T^{5} + \cdots - 11\!\cdots\!24 \)
|
| $89$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 76\!\cdots\!80 \)
|
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