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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 982x^{4} + 197305x^{2} - 11233800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{33}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 160) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 982x^{4} + 197305x^{2} - 11233800 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{5} + 6671\nu^{3} - 598445\nu ) / 5925 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -368\nu^{5} + 318716\nu^{3} - 34105220\nu ) / 5925 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6104\nu^{5} - 5244023\nu^{3} + 572447285\nu ) / 5925 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 32\nu^{4} - 27104\nu^{2} + 2795095 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -96\nu^{4} + 83872\nu^{2} - 9223255 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 13\beta_{2} + 165\beta_1 ) / 5120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 3\beta_{4} + 167594 ) / 512 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 111\beta_{3} + 1955\beta_{2} - 5237\beta_1 ) / 1024 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 847\beta_{5} + 2621\beta_{4} + 97230598 ) / 512 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 77599\beta_{3} + 1435731\beta_{2} - 7593989\beta_1 ) / 1024 \)
|
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 | −276.838 | 0 | −625.000 | 0 | −2184.14 | 0 | 56956.1 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −122.500 | 0 | −625.000 | 0 | 4187.52 | 0 | −4676.77 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −56.2373 | 0 | −625.000 | 0 | 7284.72 | 0 | −16520.4 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 56.2373 | 0 | −625.000 | 0 | −7284.72 | 0 | −16520.4 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 122.500 | 0 | −625.000 | 0 | −4187.52 | 0 | −4676.77 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 276.838 | 0 | −625.000 | 0 | 2184.14 | 0 | 56956.1 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.10.a.bd | 6 | |
| 4.b | odd | 2 | 1 | inner | 320.10.a.bd | 6 | |
| 8.b | even | 2 | 1 | 160.10.a.h | ✓ | 6 | |
| 8.d | odd | 2 | 1 | 160.10.a.h | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.10.a.h | ✓ | 6 | 8.b | even | 2 | 1 | |
| 160.10.a.h | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 320.10.a.bd | 6 | 1.a | even | 1 | 1 | trivial | |
| 320.10.a.bd | 6 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 94808T_{3}^{4} + 1439904960T_{3}^{2} - 3637228147200 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 3637228147200 \)
|
| $5$ |
\( (T + 625)^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 44\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 26\!\cdots\!72)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots + 57\!\cdots\!92)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 74\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots + 69\!\cdots\!60)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots + 40\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 77\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots - 22\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots - 26\!\cdots\!64)^{2} \)
|
| $59$ |
\( T^{6} + \cdots - 69\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 23\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots - 28\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 24\!\cdots\!16)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 18\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 57\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots + 80\!\cdots\!52)^{2} \)
|
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