Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2112,2,Mod(65,2112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2112.65");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2112 = 2^{6} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2112.b (of order \(2\), degree \(1\), not 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: | \(16.8644049069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.7388168.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - x^{4} + 2x^{3} - 3x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 264) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - x^{4} + 2x^{3} - 3x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} + 5\nu^{3} - \nu^{2} - 6\nu - 9 ) / 18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + \nu^{3} - 2\nu^{2} + 3\nu + 9 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 2\nu^{4} + 4\nu^{3} + \nu^{2} - 3\nu + 9 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{5} + 3\beta_{4} + 2\beta_{3} + \beta_{2} - \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} - 6\beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 + 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).
| \(n\) | \(133\) | \(1409\) | \(1729\) | \(2047\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −1.54194 | − | 0.788944i | 0 | − | 0.957146i | 0 | 2.53503i | 0 | 1.75513 | + | 2.43300i | 0 | |||||||||||||||||||||||||||||||
| 65.2 | 0 | −1.54194 | + | 0.788944i | 0 | 0.957146i | 0 | − | 2.53503i | 0 | 1.75513 | − | 2.43300i | 0 | ||||||||||||||||||||||||||||||||
| 65.3 | 0 | 0.393401 | − | 1.68678i | 0 | 2.18788i | 0 | 1.18569i | 0 | −2.69047 | − | 1.32716i | 0 | |||||||||||||||||||||||||||||||||
| 65.4 | 0 | 0.393401 | + | 1.68678i | 0 | − | 2.18788i | 0 | − | 1.18569i | 0 | −2.69047 | + | 1.32716i | 0 | |||||||||||||||||||||||||||||||
| 65.5 | 0 | 1.64854 | − | 0.531349i | 0 | − | 2.70131i | 0 | 3.76401i | 0 | 2.43534 | − | 1.75189i | 0 | ||||||||||||||||||||||||||||||||
| 65.6 | 0 | 1.64854 | + | 0.531349i | 0 | 2.70131i | 0 | − | 3.76401i | 0 | 2.43534 | + | 1.75189i | 0 | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2112.2.b.q | 6 | |
| 3.b | odd | 2 | 1 | 2112.2.b.r | 6 | ||
| 4.b | odd | 2 | 1 | 2112.2.b.p | 6 | ||
| 8.b | even | 2 | 1 | 528.2.b.f | 6 | ||
| 8.d | odd | 2 | 1 | 264.2.b.a | ✓ | 6 | |
| 11.b | odd | 2 | 1 | 2112.2.b.r | 6 | ||
| 12.b | even | 2 | 1 | 2112.2.b.o | 6 | ||
| 24.f | even | 2 | 1 | 264.2.b.b | yes | 6 | |
| 24.h | odd | 2 | 1 | 528.2.b.e | 6 | ||
| 33.d | even | 2 | 1 | inner | 2112.2.b.q | 6 | |
| 44.c | even | 2 | 1 | 2112.2.b.o | 6 | ||
| 88.b | odd | 2 | 1 | 528.2.b.e | 6 | ||
| 88.g | even | 2 | 1 | 264.2.b.b | yes | 6 | |
| 132.d | odd | 2 | 1 | 2112.2.b.p | 6 | ||
| 264.m | even | 2 | 1 | 528.2.b.f | 6 | ||
| 264.p | odd | 2 | 1 | 264.2.b.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 264.2.b.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 264.2.b.a | ✓ | 6 | 264.p | odd | 2 | 1 | |
| 264.2.b.b | yes | 6 | 24.f | even | 2 | 1 | |
| 264.2.b.b | yes | 6 | 88.g | even | 2 | 1 | |
| 528.2.b.e | 6 | 24.h | odd | 2 | 1 | ||
| 528.2.b.e | 6 | 88.b | odd | 2 | 1 | ||
| 528.2.b.f | 6 | 8.b | even | 2 | 1 | ||
| 528.2.b.f | 6 | 264.m | even | 2 | 1 | ||
| 2112.2.b.o | 6 | 12.b | even | 2 | 1 | ||
| 2112.2.b.o | 6 | 44.c | even | 2 | 1 | ||
| 2112.2.b.p | 6 | 4.b | odd | 2 | 1 | ||
| 2112.2.b.p | 6 | 132.d | odd | 2 | 1 | ||
| 2112.2.b.q | 6 | 1.a | even | 1 | 1 | trivial | |
| 2112.2.b.q | 6 | 33.d | even | 2 | 1 | inner | |
| 2112.2.b.r | 6 | 3.b | odd | 2 | 1 | ||
| 2112.2.b.r | 6 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2112, [\chi])\):
|
\( T_{5}^{6} + 13T_{5}^{4} + 46T_{5}^{2} + 32 \)
|
|
\( T_{7}^{6} + 22T_{7}^{4} + 120T_{7}^{2} + 128 \)
|
|
\( T_{17}^{3} + 2T_{17}^{2} - 40T_{17} - 64 \)
|
|
\( T_{29}^{3} - 8T_{29}^{2} - 20T_{29} + 128 \)
|
|
\( T_{31}^{3} + 3T_{31}^{2} - 28T_{31} + 32 \)
|
|
\( T_{83}^{3} + 8T_{83}^{2} - 144T_{83} - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + \cdots + 27 \)
|
| $5$ |
\( T^{6} + 13 T^{4} + \cdots + 32 \)
|
| $7$ |
\( T^{6} + 22 T^{4} + \cdots + 128 \)
|
| $11$ |
\( T^{6} + 2 T^{5} + \cdots + 1331 \)
|
| $13$ |
\( T^{6} + 58 T^{4} + \cdots + 512 \)
|
| $17$ |
\( (T^{3} + 2 T^{2} - 40 T - 64)^{2} \)
|
| $19$ |
\( T^{6} + 108 T^{4} + \cdots + 32768 \)
|
| $23$ |
\( T^{6} + 73 T^{4} + \cdots + 4232 \)
|
| $29$ |
\( (T^{3} - 8 T^{2} + \cdots + 128)^{2} \)
|
| $31$ |
\( (T^{3} + 3 T^{2} - 28 T + 32)^{2} \)
|
| $37$ |
\( (T^{3} - 3 T^{2} - 28 T + 92)^{2} \)
|
| $41$ |
\( (T^{3} - 4 T^{2} - 36 T + 16)^{2} \)
|
| $43$ |
\( T^{6} + 68 T^{4} + \cdots + 8192 \)
|
| $47$ |
\( T^{6} + 130 T^{4} + \cdots + 70688 \)
|
| $53$ |
\( T^{6} + 58 T^{4} + \cdots + 512 \)
|
| $59$ |
\( T^{6} + 55 T^{4} + \cdots + 512 \)
|
| $61$ |
\( T^{6} + 234 T^{4} + \cdots + 131072 \)
|
| $67$ |
\( (T^{3} + 11 T^{2} + \cdots + 16)^{2} \)
|
| $71$ |
\( T^{6} + 145 T^{4} + \cdots + 17672 \)
|
| $73$ |
\( T^{6} + 240 T^{4} + \cdots + 131072 \)
|
| $79$ |
\( T^{6} + 166 T^{4} + \cdots + 512 \)
|
| $83$ |
\( (T^{3} + 8 T^{2} + \cdots - 128)^{2} \)
|
| $89$ |
\( T^{6} + 387 T^{4} + \cdots + 1492992 \)
|
| $97$ |
\( (T^{3} + 13 T^{2} + \cdots + 32)^{2} \)
|
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