Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,6,Mod(1,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, 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("208.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 208.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: | \(33.3598345211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{849}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 212 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 26) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{849})\). 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 | −19.0688 | 0 | −7.20641 | 0 | 53.6192 | 0 | 120.619 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 10.0688 | 0 | 80.2064 | 0 | −208.619 | 0 | −141.619 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 208.6.a.g | 2 | |
| 4.b | odd | 2 | 1 | 26.6.a.c | ✓ | 2 | |
| 8.b | even | 2 | 1 | 832.6.a.m | 2 | ||
| 8.d | odd | 2 | 1 | 832.6.a.k | 2 | ||
| 12.b | even | 2 | 1 | 234.6.a.h | 2 | ||
| 20.d | odd | 2 | 1 | 650.6.a.b | 2 | ||
| 20.e | even | 4 | 2 | 650.6.b.h | 4 | ||
| 52.b | odd | 2 | 1 | 338.6.a.f | 2 | ||
| 52.f | even | 4 | 2 | 338.6.b.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.6.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 208.6.a.g | 2 | 1.a | even | 1 | 1 | trivial | |
| 234.6.a.h | 2 | 12.b | even | 2 | 1 | ||
| 338.6.a.f | 2 | 52.b | odd | 2 | 1 | ||
| 338.6.b.b | 4 | 52.f | even | 4 | 2 | ||
| 650.6.a.b | 2 | 20.d | odd | 2 | 1 | ||
| 650.6.b.h | 4 | 20.e | even | 4 | 2 | ||
| 832.6.a.k | 2 | 8.d | odd | 2 | 1 | ||
| 832.6.a.m | 2 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 9T_{3} - 192 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(208))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 9T - 192 \)
|
| $5$ |
\( T^{2} - 73T - 578 \)
|
| $7$ |
\( T^{2} + 155T - 11186 \)
|
| $11$ |
\( T^{2} - 220T - 110156 \)
|
| $13$ |
\( (T + 169)^{2} \)
|
| $17$ |
\( T^{2} + 189 T - 3523122 \)
|
| $19$ |
\( T^{2} - 2496 T + 793404 \)
|
| $23$ |
\( T^{2} - 3044 T - 159200 \)
|
| $29$ |
\( T^{2} - 1900 T - 13890476 \)
|
| $31$ |
\( T^{2} + 2798 T - 28369928 \)
|
| $37$ |
\( T^{2} - 17805 T + 72604926 \)
|
| $41$ |
\( T^{2} - 11634 T - 11466000 \)
|
| $43$ |
\( T^{2} - 4069 T - 6040532 \)
|
| $47$ |
\( T^{2} - 25489 T + 127607974 \)
|
| $53$ |
\( T^{2} + 4614 T - 129839400 \)
|
| $59$ |
\( T^{2} - 23420 T + 92989684 \)
|
| $61$ |
\( T^{2} + \cdots + 2309711776 \)
|
| $67$ |
\( T^{2} + \cdots + 1295720044 \)
|
| $71$ |
\( T^{2} + \cdots - 1862988962 \)
|
| $73$ |
\( T^{2} + \cdots - 1462893860 \)
|
| $79$ |
\( T^{2} + 52024 T + 439936528 \)
|
| $83$ |
\( T^{2} - 37758 T + 83472480 \)
|
| $89$ |
\( T^{2} + \cdots + 1072134396 \)
|
| $97$ |
\( T^{2} + \cdots - 4229773940 \)
|
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