Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6720,2,Mod(3361,6720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6720, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6720.3361");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6720.g (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: | \(53.6594701583\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.56070144.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 2\nu + 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 6\nu^{7} - 21\nu^{6} + 67\nu^{5} - 115\nu^{4} + 117\nu^{3} - 71\nu^{2} - 41\nu + 29 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{6} + 3\nu^{5} - 10\nu^{4} + 15\nu^{3} - 18\nu^{2} + 11\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\nu^{7} - 35\nu^{6} + 161\nu^{5} - 315\nu^{4} + 639\nu^{3} - 661\nu^{2} + 647\nu - 223 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} + 12\nu^{4} - 19\nu^{3} + 32\nu^{2} - 23\nu + 15 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -26\nu^{7} + 91\nu^{6} - 315\nu^{5} + 560\nu^{4} - 803\nu^{3} + 690\nu^{2} - 439\nu + 121 ) / 37 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -58\nu^{7} + 203\nu^{6} - 771\nu^{5} + 1420\nu^{4} - 2315\nu^{3} + 2154\nu^{2} - 1503\nu + 435 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + 2\beta_{4} + 2\beta_{2} + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + 2\beta_{4} + 2\beta_{2} + 2\beta _1 - 8 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} - 5\beta_{4} - 11\beta_{2} + 3\beta _1 - 13 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -5\beta_{7} + \beta_{6} + 2\beta_{5} - 12\beta_{4} + 2\beta_{3} - 24\beta_{2} - 8\beta _1 + 18 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{7} + 6\beta_{6} + 5\beta_{5} + 11\beta_{4} + 5\beta_{3} + 27\beta_{2} - 25\beta _1 + 67 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 19\beta_{7} + 15\beta_{6} - 5\beta_{5} + 64\beta_{4} - 9\beta_{3} + 142\beta_{2} + 14\beta _1 - 20 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 6\beta_{7} - 14\beta_{6} - 35\beta_{5} + 6\beta_{4} - 49\beta_{3} + 12\beta_{2} + 140\beta _1 - 320 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6720\mathbb{Z}\right)^\times\).
| \(n\) | \(1471\) | \(1921\) | \(3781\) | \(4481\) | \(5377\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3361.1 |
|
0 | − | 1.00000i | 0 | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3361.2 | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3361.3 | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3361.4 | 0 | − | 1.00000i | 0 | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3361.5 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3361.6 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3361.7 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3361.8 | 0 | 1.00000i | 0 | − | 1.00000i | 0 | −1.00000 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6720.2.g.p | ✓ | 8 |
| 4.b | odd | 2 | 1 | 6720.2.g.r | yes | 8 | |
| 8.b | even | 2 | 1 | inner | 6720.2.g.p | ✓ | 8 |
| 8.d | odd | 2 | 1 | 6720.2.g.r | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6720.2.g.p | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 6720.2.g.p | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 6720.2.g.r | yes | 8 | 4.b | odd | 2 | 1 | |
| 6720.2.g.r | yes | 8 | 8.d | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6720, [\chi])\):
|
\( T_{11}^{8} + 48T_{11}^{6} + 560T_{11}^{4} + 1344T_{11}^{2} + 64 \)
|
|
\( T_{13}^{8} + 64T_{13}^{6} + 1200T_{13}^{4} + 7360T_{13}^{2} + 7744 \)
|
|
\( T_{23}^{4} + 4T_{23}^{3} - 16T_{23}^{2} - 64T_{23} - 32 \)
|
|
\( T_{29}^{8} + 48T_{29}^{6} + 416T_{29}^{4} + 768T_{29}^{2} + 256 \)
|
|
\( T_{47}^{4} + 4T_{47}^{3} - 64T_{47}^{2} - 208T_{47} + 208 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( (T + 1)^{8} \)
|
| $11$ |
\( T^{8} + 48 T^{6} + \cdots + 64 \)
|
| $13$ |
\( T^{8} + 64 T^{6} + \cdots + 7744 \)
|
| $17$ |
\( (T^{4} - 4 T^{3} - 16 T^{2} + \cdots - 32)^{2} \)
|
| $19$ |
\( T^{8} + 48 T^{6} + \cdots + 10816 \)
|
| $23$ |
\( (T^{4} + 4 T^{3} - 16 T^{2} + \cdots - 32)^{2} \)
|
| $29$ |
\( T^{8} + 48 T^{6} + \cdots + 256 \)
|
| $31$ |
\( (T^{4} - 96 T^{2} + \cdots + 792)^{2} \)
|
| $37$ |
\( T^{8} + 80 T^{6} + \cdots + 1024 \)
|
| $41$ |
\( (T^{2} + 4 T - 8)^{4} \)
|
| $43$ |
\( T^{8} + 144 T^{6} + \cdots + 256 \)
|
| $47$ |
\( (T^{4} + 4 T^{3} + \cdots + 208)^{2} \)
|
| $53$ |
\( T^{8} + 224 T^{6} + \cdots + 2050624 \)
|
| $59$ |
\( T^{8} + 400 T^{6} + \cdots + 45373696 \)
|
| $61$ |
\( T^{8} + 48 T^{6} + \cdots + 256 \)
|
| $67$ |
\( T^{8} + 80 T^{6} + \cdots + 30976 \)
|
| $71$ |
\( (T^{4} + 4 T^{3} + \cdots - 296)^{2} \)
|
| $73$ |
\( (T^{4} - 16 T^{2} - 24 T - 8)^{2} \)
|
| $79$ |
\( (T^{4} - 216 T^{2} + \cdots + 5328)^{2} \)
|
| $83$ |
\( T^{8} + 224 T^{6} + \cdots + 4096 \)
|
| $89$ |
\( (T^{4} + 16 T^{3} + \cdots - 12416)^{2} \)
|
| $97$ |
\( (T^{4} - 8 T^{3} - 112 T^{2} + \cdots - 8)^{2} \)
|
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