Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [11,7,Mod(10,11)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(11, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("11.10");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 11 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 11.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: | \(2.53059491982\) |
| 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: |
\( x^{4} + 270x^{2} + 16680 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{4} + 270x^{2} + 16680 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 135 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 146\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 135 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 146\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
− | 13.2025i | 45.3065 | −110.306 | −65.0000 | − | 598.160i | 452.932i | 611.362i | 1323.68 | 858.164i | ||||||||||||||||||||||||||||
| 10.2 | − | 9.78231i | −33.3065 | −31.6935 | −65.0000 | 325.814i | − | 433.420i | − | 316.032i | 380.322 | 635.850i | ||||||||||||||||||||||||||||
| 10.3 | 9.78231i | −33.3065 | −31.6935 | −65.0000 | − | 325.814i | 433.420i | 316.032i | 380.322 | − | 635.850i | |||||||||||||||||||||||||||||
| 10.4 | 13.2025i | 45.3065 | −110.306 | −65.0000 | 598.160i | − | 452.932i | − | 611.362i | 1323.68 | − | 858.164i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 11.7.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 99.7.c.b | 4 | ||
| 4.b | odd | 2 | 1 | 176.7.h.c | 4 | ||
| 11.b | odd | 2 | 1 | inner | 11.7.b.b | ✓ | 4 |
| 33.d | even | 2 | 1 | 99.7.c.b | 4 | ||
| 44.c | even | 2 | 1 | 176.7.h.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.7.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 11.7.b.b | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 99.7.c.b | 4 | 3.b | odd | 2 | 1 | ||
| 99.7.c.b | 4 | 33.d | even | 2 | 1 | ||
| 176.7.h.c | 4 | 4.b | odd | 2 | 1 | ||
| 176.7.h.c | 4 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 270T_{2}^{2} + 16680 \)
acting on \(S_{7}^{\mathrm{new}}(11, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 270 T^{2} + 16680 \)
|
| $3$ |
\( (T^{2} - 12 T - 1509)^{2} \)
|
| $5$ |
\( (T + 65)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 38537472000 \)
|
| $11$ |
\( T^{4} + \cdots + 3138428376721 \)
|
| $13$ |
\( T^{4} + \cdots + 7116861204480 \)
|
| $17$ |
\( T^{4} + \cdots + 580640870983680 \)
|
| $19$ |
\( T^{4} + \cdots + 996840956897280 \)
|
| $23$ |
\( (T^{2} - 5732 T - 18299789)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 37\!\cdots\!80 \)
|
| $31$ |
\( (T^{2} + 2756 T - 677235221)^{2} \)
|
| $37$ |
\( (T^{2} - 38038 T - 1146420119)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 10\!\cdots\!80 \)
|
| $43$ |
\( T^{4} + \cdots + 21\!\cdots\!00 \)
|
| $47$ |
\( (T^{2} + 74152 T + 613247596)^{2} \)
|
| $53$ |
\( (T^{2} - 107132 T - 6059572364)^{2} \)
|
| $59$ |
\( (T^{2} - 383396 T + 33869440579)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 43\!\cdots\!80 \)
|
| $67$ |
\( (T^{2} - 440908 T - 70365999509)^{2} \)
|
| $71$ |
\( (T^{2} - 640484 T + 18019322659)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!80 \)
|
| $79$ |
\( T^{4} + \cdots + 26\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!80 \)
|
| $89$ |
\( (T^{2} + 1245634 T + 301272643009)^{2} \)
|
| $97$ |
\( (T^{2} - 740038 T - 309801084359)^{2} \)
|
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