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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 3364x^{3} + 79060x^{2} + 373536x - 15080832 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{23}\cdot 3\cdot 5 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 3364x^{3} + 79060x^{2} + 373536x - 15080832 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -7\nu^{4} - 211\nu^{3} + 17390\nu^{2} - 7852\nu - 3813990 ) / 2410 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -23\nu^{4} - 349\nu^{3} + 73320\nu^{2} - 754308\nu - 18528450 ) / 1205 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{4} + 1050\nu^{3} + 72666\nu^{2} - 1239784\nu - 33188583 ) / 1205 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1189\nu^{4} + 32397\nu^{3} - 3038510\nu^{2} + 8695924\nu + 604057860 ) / 2410 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + 5\beta_{3} - 22\beta_{2} - 31\beta _1 + 1020 ) / 5120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 161\beta_{4} - 5\beta_{3} + 822\beta_{2} + 21951\beta _1 + 6886500 ) / 5120 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9683\beta_{4} + 10815\beta_{3} - 67266\beta_{2} - 1215053\beta _1 - 232327020 ) / 5120 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 138593\beta_{4} - 68805\beta_{3} + 818870\beta_{2} + 17885951\beta _1 + 4264045668 ) / 1024 \)
|
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 | −241.338 | 0 | 625.000 | 0 | −4251.84 | 0 | 38561.1 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −102.050 | 0 | 625.000 | 0 | 1580.58 | 0 | −9268.88 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 26.4332 | 0 | 625.000 | 0 | 8750.46 | 0 | −18984.3 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 104.635 | 0 | 625.000 | 0 | −12330.2 | 0 | −8734.50 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 198.319 | 0 | 625.000 | 0 | 2840.98 | 0 | 19647.6 | 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.bb | 5 | |
| 4.b | odd | 2 | 1 | 320.10.a.bc | 5 | ||
| 8.b | even | 2 | 1 | 160.10.a.g | yes | 5 | |
| 8.d | odd | 2 | 1 | 160.10.a.f | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.10.a.f | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 160.10.a.g | yes | 5 | 8.b | even | 2 | 1 | |
| 320.10.a.bb | 5 | 1.a | even | 1 | 1 | trivial | |
| 320.10.a.bc | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 14T_{3}^{4} - 59720T_{3}^{3} + 1214736T_{3}^{2} + 519940368T_{3} - 13509209376 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(320))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + \cdots - 13509209376 \)
|
| $5$ |
\( (T - 625)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 20\!\cdots\!48 \)
|
| $11$ |
\( T^{5} + \cdots - 39\!\cdots\!00 \)
|
| $13$ |
\( T^{5} + \cdots + 75\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots - 17\!\cdots\!00 \)
|
| $19$ |
\( T^{5} + \cdots - 59\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 24\!\cdots\!36 \)
|
| $29$ |
\( T^{5} + \cdots + 29\!\cdots\!96 \)
|
| $31$ |
\( T^{5} + \cdots - 13\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 24\!\cdots\!00 \)
|
| $41$ |
\( T^{5} + \cdots + 17\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots - 18\!\cdots\!44 \)
|
| $47$ |
\( T^{5} + \cdots + 21\!\cdots\!08 \)
|
| $53$ |
\( T^{5} + \cdots + 20\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 71\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 91\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 84\!\cdots\!88 \)
|
| $71$ |
\( T^{5} + \cdots - 54\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 25\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 15\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 54\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots + 39\!\cdots\!44 \)
|
| $97$ |
\( T^{5} + \cdots + 14\!\cdots\!00 \)
|
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