Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [704,6,Mod(1,704)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(704, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("704.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 704 = 2^{6} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 704.a (trivial) |
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: | yes |
| Analytic conductor: | \(112.910209148\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{31}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 31 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 44) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 4\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −25.2711 | 0 | 55.5421 | 0 | −200.813 | 0 | 395.626 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 19.2711 | 0 | −33.5421 | 0 | −67.1868 | 0 | 128.374 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 704.6.a.m | 2 | |
| 4.b | odd | 2 | 1 | 704.6.a.n | 2 | ||
| 8.b | even | 2 | 1 | 176.6.a.g | 2 | ||
| 8.d | odd | 2 | 1 | 44.6.a.b | ✓ | 2 | |
| 24.f | even | 2 | 1 | 396.6.a.f | 2 | ||
| 40.e | odd | 2 | 1 | 1100.6.a.b | 2 | ||
| 40.k | even | 4 | 2 | 1100.6.b.c | 4 | ||
| 88.g | even | 2 | 1 | 484.6.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 44.6.a.b | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 176.6.a.g | 2 | 8.b | even | 2 | 1 | ||
| 396.6.a.f | 2 | 24.f | even | 2 | 1 | ||
| 484.6.a.d | 2 | 88.g | even | 2 | 1 | ||
| 704.6.a.m | 2 | 1.a | even | 1 | 1 | trivial | |
| 704.6.a.n | 2 | 4.b | odd | 2 | 1 | ||
| 1100.6.a.b | 2 | 40.e | odd | 2 | 1 | ||
| 1100.6.b.c | 4 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 6T_{3} - 487 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 6T - 487 \)
|
| $5$ |
\( T^{2} - 22T - 1863 \)
|
| $7$ |
\( T^{2} + 268T + 13492 \)
|
| $11$ |
\( (T + 121)^{2} \)
|
| $13$ |
\( T^{2} + 1232 T + 367056 \)
|
| $17$ |
\( T^{2} + 124 T - 1091820 \)
|
| $19$ |
\( T^{2} - 1944 T + 468128 \)
|
| $23$ |
\( T^{2} + 3346 T + 1072353 \)
|
| $29$ |
\( T^{2} - 6576 T + 7793280 \)
|
| $31$ |
\( T^{2} + 2498 T - 66709935 \)
|
| $37$ |
\( T^{2} - 14674 T + 49448913 \)
|
| $41$ |
\( T^{2} + 6504 T - 102278880 \)
|
| $43$ |
\( T^{2} - 11628 T - 171906460 \)
|
| $47$ |
\( T^{2} + 36816 T + 203702400 \)
|
| $53$ |
\( T^{2} - 3292 T - 653250684 \)
|
| $59$ |
\( T^{2} - 12126 T - 132983631 \)
|
| $61$ |
\( T^{2} + \cdots + 1180267472 \)
|
| $67$ |
\( T^{2} + \cdots - 1274510455 \)
|
| $71$ |
\( T^{2} - 46122 T + 83226825 \)
|
| $73$ |
\( T^{2} - 8240 T - 299672496 \)
|
| $79$ |
\( T^{2} - 14780 T - 305436284 \)
|
| $83$ |
\( T^{2} + \cdots + 1070918340 \)
|
| $89$ |
\( T^{2} + 33698 T + 74328801 \)
|
| $97$ |
\( T^{2} + \cdots - 1061806063 \)
|
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