Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,8,Mod(1,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.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: | \(124.954010194\) |
| 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: | \( 2\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 10) |
| 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 \(\beta = 10\sqrt{31}\). 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 | −83.6776 | 0 | 0 | 0 | 185.744 | 0 | 4814.95 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 27.6776 | 0 | 0 | 0 | −593.744 | 0 | −1420.95 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 400.8.a.t | 2 | |
| 4.b | odd | 2 | 1 | 50.8.a.j | 2 | ||
| 5.b | even | 2 | 1 | 400.8.a.bf | 2 | ||
| 5.c | odd | 4 | 2 | 80.8.c.d | 4 | ||
| 12.b | even | 2 | 1 | 450.8.a.bd | 2 | ||
| 20.d | odd | 2 | 1 | 50.8.a.i | 2 | ||
| 20.e | even | 4 | 2 | 10.8.b.a | ✓ | 4 | |
| 40.i | odd | 4 | 2 | 320.8.c.g | 4 | ||
| 40.k | even | 4 | 2 | 320.8.c.f | 4 | ||
| 60.h | even | 2 | 1 | 450.8.a.bi | 2 | ||
| 60.l | odd | 4 | 2 | 90.8.c.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.8.b.a | ✓ | 4 | 20.e | even | 4 | 2 | |
| 50.8.a.i | 2 | 20.d | odd | 2 | 1 | ||
| 50.8.a.j | 2 | 4.b | odd | 2 | 1 | ||
| 80.8.c.d | 4 | 5.c | odd | 4 | 2 | ||
| 90.8.c.c | 4 | 60.l | odd | 4 | 2 | ||
| 320.8.c.f | 4 | 40.k | even | 4 | 2 | ||
| 320.8.c.g | 4 | 40.i | odd | 4 | 2 | ||
| 400.8.a.t | 2 | 1.a | even | 1 | 1 | trivial | |
| 400.8.a.bf | 2 | 5.b | even | 2 | 1 | ||
| 450.8.a.bd | 2 | 12.b | even | 2 | 1 | ||
| 450.8.a.bi | 2 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 56T_{3} - 2316 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 56T - 2316 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 408T - 110284 \)
|
| $11$ |
\( T^{2} + 8904 T + 19026704 \)
|
| $13$ |
\( T^{2} + 3504 T - 15790896 \)
|
| $17$ |
\( T^{2} - 43648 T + 388792576 \)
|
| $19$ |
\( T^{2} - 31800 T - 954255600 \)
|
| $23$ |
\( T^{2} + 71656 T - 950837516 \)
|
| $29$ |
\( T^{2} + \cdots - 19030326300 \)
|
| $31$ |
\( T^{2} + \cdots + 4879504384 \)
|
| $37$ |
\( T^{2} + \cdots - 51511044784 \)
|
| $41$ |
\( T^{2} + \cdots - 183681487516 \)
|
| $43$ |
\( T^{2} + \cdots + 17685717524 \)
|
| $47$ |
\( T^{2} + \cdots - 599256039404 \)
|
| $53$ |
\( T^{2} + \cdots + 504522481104 \)
|
| $59$ |
\( T^{2} + \cdots - 159475001200 \)
|
| $61$ |
\( T^{2} + \cdots + 1716951910564 \)
|
| $67$ |
\( T^{2} + \cdots - 2422233726604 \)
|
| $71$ |
\( T^{2} + \cdots - 4934358737856 \)
|
| $73$ |
\( T^{2} + \cdots + 296239095104 \)
|
| $79$ |
\( T^{2} + \cdots - 3963839328000 \)
|
| $83$ |
\( T^{2} + \cdots - 36935334540236 \)
|
| $89$ |
\( T^{2} + \cdots + 66278240560100 \)
|
| $97$ |
\( T^{2} + \cdots - 1192948931584 \)
|
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