Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,11,Mod(5,6)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.5");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6.b (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: | \(3.81214351604\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{85})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 37x^{2} + 38x + 531 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 37x^{2} + 38x + 531 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -32\nu^{3} + 48\nu^{2} + 464\nu - 240 ) / 93 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -36\nu^{3} - 132\nu^{2} + 2196\nu + 2520 ) / 31 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -64\nu^{3} + 3072\nu^{2} + 928\nu - 58512 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 16\beta_{2} - 60\beta _1 + 432 ) / 864 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 6\beta _1 + 1872 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14\beta_{3} + 116\beta_{2} - 1731\beta _1 + 12528 ) / 432 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
− | 22.6274i | −200.269 | + | 137.627i | −512.000 | 3630.47i | 3114.15 | + | 4531.57i | −23226.5 | 11585.2i | 21166.4 | − | 55125.0i | 82148.2 | |||||||||||||||||||||||
| 5.2 | − | 22.6274i | 242.269 | − | 18.8335i | −512.000 | − | 4818.41i | −426.153 | − | 5481.92i | 670.530 | 11585.2i | 58339.6 | − | 9125.53i | −109028. | |||||||||||||||||||||||
| 5.3 | 22.6274i | −200.269 | − | 137.627i | −512.000 | − | 3630.47i | 3114.15 | − | 4531.57i | −23226.5 | − | 11585.2i | 21166.4 | + | 55125.0i | 82148.2 | |||||||||||||||||||||||
| 5.4 | 22.6274i | 242.269 | + | 18.8335i | −512.000 | 4818.41i | −426.153 | + | 5481.92i | 670.530 | − | 11585.2i | 58339.6 | + | 9125.53i | −109028. | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6.11.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 6.11.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 48.11.e.d | 4 | ||
| 5.b | even | 2 | 1 | 150.11.d.a | 4 | ||
| 5.c | odd | 4 | 2 | 150.11.b.a | 8 | ||
| 8.b | even | 2 | 1 | 192.11.e.g | 4 | ||
| 8.d | odd | 2 | 1 | 192.11.e.h | 4 | ||
| 9.c | even | 3 | 2 | 162.11.d.d | 8 | ||
| 9.d | odd | 6 | 2 | 162.11.d.d | 8 | ||
| 12.b | even | 2 | 1 | 48.11.e.d | 4 | ||
| 15.d | odd | 2 | 1 | 150.11.d.a | 4 | ||
| 15.e | even | 4 | 2 | 150.11.b.a | 8 | ||
| 24.f | even | 2 | 1 | 192.11.e.h | 4 | ||
| 24.h | odd | 2 | 1 | 192.11.e.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.11.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 6.11.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 48.11.e.d | 4 | 4.b | odd | 2 | 1 | ||
| 48.11.e.d | 4 | 12.b | even | 2 | 1 | ||
| 150.11.b.a | 8 | 5.c | odd | 4 | 2 | ||
| 150.11.b.a | 8 | 15.e | even | 4 | 2 | ||
| 150.11.d.a | 4 | 5.b | even | 2 | 1 | ||
| 150.11.d.a | 4 | 15.d | odd | 2 | 1 | ||
| 162.11.d.d | 8 | 9.c | even | 3 | 2 | ||
| 162.11.d.d | 8 | 9.d | odd | 6 | 2 | ||
| 192.11.e.g | 4 | 8.b | even | 2 | 1 | ||
| 192.11.e.g | 4 | 24.h | odd | 2 | 1 | ||
| 192.11.e.h | 4 | 8.d | odd | 2 | 1 | ||
| 192.11.e.h | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(6, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 512)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 3486784401 \)
|
| $5$ |
\( T^{4} + \cdots + 306009247334400 \)
|
| $7$ |
\( (T^{2} + 22556 T - 15574076)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 21\!\cdots\!24 \)
|
| $13$ |
\( (T^{2} - 137620 T - 52372127900)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 32\!\cdots\!84 \)
|
| $19$ |
\( (T^{2} + \cdots - 1189491369116)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 61\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots + 20\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 12\!\cdots\!56)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 42\!\cdots\!96)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 28\!\cdots\!04 \)
|
| $43$ |
\( (T^{2} + \cdots + 72\!\cdots\!16)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 26\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 37\!\cdots\!04 \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!04 \)
|
| $61$ |
\( (T^{2} + \cdots + 32\!\cdots\!36)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots - 35\!\cdots\!96)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots + 23\!\cdots\!40)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 36\!\cdots\!76)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 57\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 15\!\cdots\!84 \)
|
| $97$ |
\( (T^{2} + \cdots + 95\!\cdots\!96)^{2} \)
|
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