Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3456,2,Mod(3455,3456)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3456, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3456.3455");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3456 = 2^{7} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3456.c (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: | \(27.5962989386\) |
| 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: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 2\nu + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} - 7\nu^{2} + 6\nu - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12\nu^{7} - 42\nu^{6} + 134\nu^{5} - 230\nu^{4} + 234\nu^{3} - 142\nu^{2} - 82\nu + 58 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -16\nu^{7} + 56\nu^{6} - 228\nu^{5} + 430\nu^{4} - 756\nu^{3} + 732\nu^{2} - 532\nu + 157 ) / 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 6\nu^{5} + 22\nu^{4} - 34\nu^{3} + 48\nu^{2} - 32\nu + 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -36\nu^{7} + 126\nu^{6} - 476\nu^{5} + 875\nu^{4} - 1442\nu^{3} + 1351\nu^{2} - 1012\nu + 307 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 - 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} + 3\beta_{5} - 2\beta_{4} + 3\beta_{2} - 5\beta _1 - 13 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{7} + 3\beta_{5} - 2\beta_{4} - 2\beta_{3} - 4\beta_{2} - 6\beta _1 + 9 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 8\beta_{7} - 15\beta_{5} + 4\beta_{4} - 10\beta_{3} - 25\beta_{2} + 11\beta _1 + 67 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 17\beta_{7} + \beta_{6} - 30\beta_{5} + 11\beta_{4} + 7\beta_{3} + 8\beta_{2} + 32\beta _1 - 10 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -8\beta_{7} + 7\beta_{6} + 14\beta_{5} + 7\beta_{4} + 84\beta_{3} + 147\beta_{2} + 6\beta _1 - 320 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).
| \(n\) | \(2053\) | \(2431\) | \(2945\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3455.1 |
|
0 | 0 | 0 | − | 3.90931i | 0 | 3.12976i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3455.2 | 0 | 0 | 0 | − | 2.23931i | 0 | − | 2.38587i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3455.3 | 0 | 0 | 0 | − | 1.90931i | 0 | 1.12976i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3455.4 | 0 | 0 | 0 | − | 0.239314i | 0 | − | 4.38587i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 3455.5 | 0 | 0 | 0 | 0.239314i | 0 | 4.38587i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 3455.6 | 0 | 0 | 0 | 1.90931i | 0 | − | 1.12976i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 3455.7 | 0 | 0 | 0 | 2.23931i | 0 | 2.38587i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 3455.8 | 0 | 0 | 0 | 3.90931i | 0 | − | 3.12976i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3456.2.c.h | yes | 8 |
| 3.b | odd | 2 | 1 | 3456.2.c.a | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 3456.2.c.a | ✓ | 8 | |
| 8.b | even | 2 | 1 | 3456.2.c.g | yes | 8 | |
| 8.d | odd | 2 | 1 | 3456.2.c.b | yes | 8 | |
| 12.b | even | 2 | 1 | inner | 3456.2.c.h | yes | 8 |
| 24.f | even | 2 | 1 | 3456.2.c.g | yes | 8 | |
| 24.h | odd | 2 | 1 | 3456.2.c.b | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3456.2.c.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 3456.2.c.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 3456.2.c.b | yes | 8 | 8.d | odd | 2 | 1 | |
| 3456.2.c.b | yes | 8 | 24.h | odd | 2 | 1 | |
| 3456.2.c.g | yes | 8 | 8.b | even | 2 | 1 | |
| 3456.2.c.g | yes | 8 | 24.f | even | 2 | 1 | |
| 3456.2.c.h | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 3456.2.c.h | yes | 8 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3456, [\chi])\):
|
\( T_{5}^{8} + 24T_{5}^{6} + 152T_{5}^{4} + 288T_{5}^{2} + 16 \)
|
|
\( T_{11}^{4} - 28T_{11}^{2} + 72T_{11} - 44 \)
|
|
\( T_{13}^{4} - 22T_{13}^{2} + 24T_{13} + 13 \)
|
|
\( T_{23}^{4} - 8T_{23}^{3} - 4T_{23}^{2} + 152T_{23} - 284 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 24 T^{6} + \cdots + 16 \)
|
| $7$ |
\( T^{8} + 36 T^{6} + \cdots + 1369 \)
|
| $11$ |
\( (T^{4} - 28 T^{2} + \cdots - 44)^{2} \)
|
| $13$ |
\( (T^{4} - 22 T^{2} + \cdots + 13)^{2} \)
|
| $17$ |
\( T^{8} + 72 T^{6} + \cdots + 1936 \)
|
| $19$ |
\( T^{8} + 92 T^{6} + \cdots + 9409 \)
|
| $23$ |
\( (T^{4} - 8 T^{3} + \cdots - 284)^{2} \)
|
| $29$ |
\( T^{8} + 192 T^{6} + \cdots + 262144 \)
|
| $31$ |
\( T^{8} + 128 T^{6} + \cdots + 16384 \)
|
| $37$ |
\( (T^{4} - 8 T^{3} + \cdots - 827)^{2} \)
|
| $41$ |
\( T^{8} + 192 T^{6} + \cdots + 262144 \)
|
| $43$ |
\( T^{8} + 192 T^{6} + \cdots + 16384 \)
|
| $47$ |
\( (T^{4} + 16 T^{3} + \cdots - 1916)^{2} \)
|
| $53$ |
\( T^{8} + 192 T^{6} + \cdots + 262144 \)
|
| $59$ |
\( (T^{4} + 16 T^{3} + \cdots + 52)^{2} \)
|
| $61$ |
\( (T^{4} + 8 T^{3} + \cdots + 229)^{2} \)
|
| $67$ |
\( (T^{2} + 3)^{4} \)
|
| $71$ |
\( (T^{2} + 8 T - 32)^{4} \)
|
| $73$ |
\( (T^{4} - 4 T^{3} + \cdots + 5689)^{2} \)
|
| $79$ |
\( T^{8} + 148 T^{6} + \cdots + 32041 \)
|
| $83$ |
\( (T^{4} + 8 T^{3} + \cdots - 6656)^{2} \)
|
| $89$ |
\( T^{8} + 328 T^{6} + \cdots + 18215824 \)
|
| $97$ |
\( (T^{4} + 4 T^{3} + \cdots - 743)^{2} \)
|
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