Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,5,Mod(193,320)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(320, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.193");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.p (of order \(4\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(33.0783881868\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{29})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 15x^{2} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 40) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 15x^{2} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 8\nu ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 28\nu^{2} + 88\nu + 210 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - 56\nu^{2} + 44\nu - 420 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - 10\beta_1 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} + \beta_{2} - 150 ) / 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} - 4\beta_{2} + 55\beta_1 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(257\) | \(261\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | 3.00000 | + | 3.00000i | 0 | −18.5407 | − | 16.7703i | 0 | 64.8516 | − | 64.8516i | 0 | − | 63.0000i | 0 | |||||||||||||||||||||||
| 193.2 | 0 | 3.00000 | + | 3.00000i | 0 | 24.5407 | + | 4.77033i | 0 | −42.8516 | + | 42.8516i | 0 | − | 63.0000i | 0 | ||||||||||||||||||||||||
| 257.1 | 0 | 3.00000 | − | 3.00000i | 0 | −18.5407 | + | 16.7703i | 0 | 64.8516 | + | 64.8516i | 0 | 63.0000i | 0 | |||||||||||||||||||||||||
| 257.2 | 0 | 3.00000 | − | 3.00000i | 0 | 24.5407 | − | 4.77033i | 0 | −42.8516 | − | 42.8516i | 0 | 63.0000i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 320.5.p.n | 4 | |
| 4.b | odd | 2 | 1 | 320.5.p.k | 4 | ||
| 5.c | odd | 4 | 1 | inner | 320.5.p.n | 4 | |
| 8.b | even | 2 | 1 | 40.5.l.b | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 80.5.p.f | 4 | ||
| 20.e | even | 4 | 1 | 320.5.p.k | 4 | ||
| 24.h | odd | 2 | 1 | 360.5.v.b | 4 | ||
| 40.e | odd | 2 | 1 | 400.5.p.g | 4 | ||
| 40.f | even | 2 | 1 | 200.5.l.c | 4 | ||
| 40.i | odd | 4 | 1 | 40.5.l.b | ✓ | 4 | |
| 40.i | odd | 4 | 1 | 200.5.l.c | 4 | ||
| 40.k | even | 4 | 1 | 80.5.p.f | 4 | ||
| 40.k | even | 4 | 1 | 400.5.p.g | 4 | ||
| 120.w | even | 4 | 1 | 360.5.v.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.5.l.b | ✓ | 4 | 8.b | even | 2 | 1 | |
| 40.5.l.b | ✓ | 4 | 40.i | odd | 4 | 1 | |
| 80.5.p.f | 4 | 8.d | odd | 2 | 1 | ||
| 80.5.p.f | 4 | 40.k | even | 4 | 1 | ||
| 200.5.l.c | 4 | 40.f | even | 2 | 1 | ||
| 200.5.l.c | 4 | 40.i | odd | 4 | 1 | ||
| 320.5.p.k | 4 | 4.b | odd | 2 | 1 | ||
| 320.5.p.k | 4 | 20.e | even | 4 | 1 | ||
| 320.5.p.n | 4 | 1.a | even | 1 | 1 | trivial | |
| 320.5.p.n | 4 | 5.c | odd | 4 | 1 | inner | |
| 360.5.v.b | 4 | 24.h | odd | 2 | 1 | ||
| 360.5.v.b | 4 | 120.w | even | 4 | 1 | ||
| 400.5.p.g | 4 | 40.e | odd | 2 | 1 | ||
| 400.5.p.g | 4 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(320, [\chi])\):
|
\( T_{3}^{2} - 6T_{3} + 18 \)
|
|
\( T_{13}^{4} + 316T_{13}^{3} + 49928T_{13}^{2} + 2111512T_{13} + 44649124 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 6 T + 18)^{2} \)
|
| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 390625 \)
|
| $7$ |
\( T^{4} - 44 T^{3} + \cdots + 30891364 \)
|
| $11$ |
\( (T^{2} - 148 T - 6124)^{2} \)
|
| $13$ |
\( T^{4} + 316 T^{3} + \cdots + 44649124 \)
|
| $17$ |
\( T^{4} + \cdots + 7525562500 \)
|
| $19$ |
\( T^{4} + 93888 T^{2} + 563777536 \)
|
| $23$ |
\( T^{4} - 652 T^{3} + \cdots + 879844 \)
|
| $29$ |
\( T^{4} + \cdots + 167195938816 \)
|
| $31$ |
\( (T^{2} + 1380 T + 186100)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 1322182619044 \)
|
| $41$ |
\( (T^{2} + 2156 T - 2596316)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 5932527319684 \)
|
| $47$ |
\( T^{4} + \cdots + 3139055540644 \)
|
| $53$ |
\( T^{4} + \cdots + 21974737549284 \)
|
| $59$ |
\( T^{4} + \cdots + 242185305788416 \)
|
| $61$ |
\( (T^{2} - 6060 T - 4228700)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 2926898229124 \)
|
| $71$ |
\( (T^{2} - 1116 T - 15569036)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!24 \)
|
| $79$ |
\( T^{4} + \cdots + 618613200388096 \)
|
| $83$ |
\( T^{4} + \cdots + 14274961481284 \)
|
| $89$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 98478194209924 \)
|
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