Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [320,9,Mod(191,320)]
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([1, 0, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("320.191");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 320 = 2^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 320.b (of order \(2\), degree \(1\), 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: | \(130.361155220\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5^{6} \) |
| Twist minimal: | no (minimal twist has level 80) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 25\nu^{3} - 20\nu^{2} + 80\nu + 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( -175\nu^{3} + 340\nu^{2} - 360\nu - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( 25\nu^{3} + 50 \)
|
| \(\nu\) | \(=\) |
\( ( -10\beta_{3} + \beta_{2} + 17\beta _1 + 250 ) / 1000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} + 2\beta_{2} + 9\beta _1 - 375 ) / 500 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} - 50 ) / 25 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(257\) | \(261\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
0 | − | 66.7550i | 0 | −279.508 | 0 | 4.17585i | 0 | 2104.77 | 0 | |||||||||||||||||||||||||||||
| 191.2 | 0 | − | 49.4345i | 0 | 279.508 | 0 | 3520.24i | 0 | 4117.23 | 0 | ||||||||||||||||||||||||||||||
| 191.3 | 0 | 49.4345i | 0 | 279.508 | 0 | − | 3520.24i | 0 | 4117.23 | 0 | ||||||||||||||||||||||||||||||
| 191.4 | 0 | 66.7550i | 0 | −279.508 | 0 | − | 4.17585i | 0 | 2104.77 | 0 | ||||||||||||||||||||||||||||||
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 | 320.9.b.a | 4 | |
| 4.b | odd | 2 | 1 | inner | 320.9.b.a | 4 | |
| 8.b | even | 2 | 1 | 80.9.b.a | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 80.9.b.a | ✓ | 4 | |
| 40.e | odd | 2 | 1 | 400.9.b.g | 4 | ||
| 40.f | even | 2 | 1 | 400.9.b.g | 4 | ||
| 40.i | odd | 4 | 2 | 400.9.h.c | 8 | ||
| 40.k | even | 4 | 2 | 400.9.h.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 80.9.b.a | ✓ | 4 | 8.b | even | 2 | 1 | |
| 80.9.b.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 320.9.b.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 320.9.b.a | 4 | 4.b | odd | 2 | 1 | inner | |
| 400.9.b.g | 4 | 40.e | odd | 2 | 1 | ||
| 400.9.b.g | 4 | 40.f | even | 2 | 1 | ||
| 400.9.h.c | 8 | 40.i | odd | 4 | 2 | ||
| 400.9.h.c | 8 | 40.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 6900T_{3}^{2} + 10890000 \)
acting on \(S_{9}^{\mathrm{new}}(320, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 6900 T^{2} + 10890000 \)
|
| $5$ |
\( (T^{2} - 78125)^{2} \)
|
| $7$ |
\( T^{4} + 12392100 T^{2} + 216090000 \)
|
| $11$ |
\( T^{4} + \cdots + 33\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + 12440 T + 38575900)^{2} \)
|
| $17$ |
\( (T^{2} + 24780 T - 150687900)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 27\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} - 92724 T - 200459014956)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 25\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 3355216880900)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 8532197758116)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 87\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots - 81162852740100)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 73282505965076)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots + 53\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots + 6092983336900)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 62\!\cdots\!00 \)
|
| $89$ |
\( (T^{2} + \cdots - 714605027234076)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 70\!\cdots\!00)^{2} \)
|
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