Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,10,Mod(1,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.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: | \(164.811467572\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{22}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 22 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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 \(\beta = 40\sqrt{22}\). 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 | −245.617 | 0 | 625.000 | 0 | −12208.6 | 0 | 40644.5 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 129.617 | 0 | 625.000 | 0 | 924.582 | 0 | −2882.53 | 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 | 320.10.a.m | 2 | |
| 4.b | odd | 2 | 1 | 320.10.a.r | 2 | ||
| 8.b | even | 2 | 1 | 80.10.a.i | 2 | ||
| 8.d | odd | 2 | 1 | 40.10.a.a | ✓ | 2 | |
| 24.f | even | 2 | 1 | 360.10.a.i | 2 | ||
| 40.e | odd | 2 | 1 | 200.10.a.e | 2 | ||
| 40.f | even | 2 | 1 | 400.10.a.n | 2 | ||
| 40.i | odd | 4 | 2 | 400.10.c.k | 4 | ||
| 40.k | even | 4 | 2 | 200.10.c.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.10.a.a | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 80.10.a.i | 2 | 8.b | even | 2 | 1 | ||
| 200.10.a.e | 2 | 40.e | odd | 2 | 1 | ||
| 200.10.c.c | 4 | 40.k | even | 4 | 2 | ||
| 320.10.a.m | 2 | 1.a | even | 1 | 1 | trivial | |
| 320.10.a.r | 2 | 4.b | odd | 2 | 1 | ||
| 360.10.a.i | 2 | 24.f | even | 2 | 1 | ||
| 400.10.a.n | 2 | 40.f | even | 2 | 1 | ||
| 400.10.c.k | 4 | 40.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 116T_{3} - 31836 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 116T - 31836 \)
|
| $5$ |
\( (T - 625)^{2} \)
|
| $7$ |
\( T^{2} + 11284 T - 11287836 \)
|
| $11$ |
\( T^{2} + \cdots + 2175388416 \)
|
| $13$ |
\( T^{2} - 21372 T - 12529404 \)
|
| $17$ |
\( T^{2} + \cdots - 92535851100 \)
|
| $19$ |
\( T^{2} + \cdots - 33027868144 \)
|
| $23$ |
\( T^{2} + \cdots - 171919939836 \)
|
| $29$ |
\( T^{2} + \cdots + 21864370578084 \)
|
| $31$ |
\( T^{2} + \cdots - 16130186713200 \)
|
| $37$ |
\( T^{2} + \cdots - 19449536203644 \)
|
| $41$ |
\( T^{2} + \cdots + 429561344293284 \)
|
| $43$ |
\( T^{2} + \cdots - 79279162592124 \)
|
| $47$ |
\( T^{2} + \cdots - 29\!\cdots\!24 \)
|
| $53$ |
\( T^{2} + \cdots + 216519484773156 \)
|
| $59$ |
\( T^{2} + \cdots + 15\!\cdots\!44 \)
|
| $61$ |
\( T^{2} + \cdots - 63\!\cdots\!00 \)
|
| $67$ |
\( T^{2} + \cdots - 12\!\cdots\!04 \)
|
| $71$ |
\( T^{2} + \cdots - 94\!\cdots\!56 \)
|
| $73$ |
\( T^{2} + \cdots - 77\!\cdots\!56 \)
|
| $79$ |
\( T^{2} + \cdots - 27\!\cdots\!04 \)
|
| $83$ |
\( T^{2} + \cdots - 13\!\cdots\!76 \)
|
| $89$ |
\( T^{2} + \cdots - 37\!\cdots\!24 \)
|
| $97$ |
\( T^{2} + \cdots - 57\!\cdots\!44 \)
|
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