Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,5,Mod(113,304)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(304, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.113");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.e (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: | \(31.4244687775\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 269x^{2} + 17592 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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
\( x^{4} + 269x^{2} + 17592 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 135 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 138\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 135 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 138\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(191\) | \(229\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 113.1 |
|
0 | − | 12.5228i | 0 | 27.8215 | 0 | −50.6430 | 0 | −75.8215 | 0 | |||||||||||||||||||||||||||||
| 113.2 | 0 | − | 10.5914i | 0 | −16.8215 | 0 | 38.6430 | 0 | −31.1785 | 0 | ||||||||||||||||||||||||||||||
| 113.3 | 0 | 10.5914i | 0 | −16.8215 | 0 | 38.6430 | 0 | −31.1785 | 0 | |||||||||||||||||||||||||||||||
| 113.4 | 0 | 12.5228i | 0 | 27.8215 | 0 | −50.6430 | 0 | −75.8215 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 304.5.e.d | 4 | |
| 4.b | odd | 2 | 1 | 76.5.c.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 684.5.h.c | 4 | ||
| 19.b | odd | 2 | 1 | inner | 304.5.e.d | 4 | |
| 76.d | even | 2 | 1 | 76.5.c.b | ✓ | 4 | |
| 228.b | odd | 2 | 1 | 684.5.h.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.5.c.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 76.5.c.b | ✓ | 4 | 76.d | even | 2 | 1 | |
| 304.5.e.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 304.5.e.d | 4 | 19.b | odd | 2 | 1 | inner | |
| 684.5.h.c | 4 | 12.b | even | 2 | 1 | ||
| 684.5.h.c | 4 | 228.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(304, [\chi])\):
|
\( T_{3}^{4} + 269T_{3}^{2} + 17592 \)
|
|
\( T_{5}^{2} - 11T_{5} - 468 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 269 T^{2} + 17592 \)
|
| $5$ |
\( (T^{2} - 11 T - 468)^{2} \)
|
| $7$ |
\( (T^{2} + 12 T - 1957)^{2} \)
|
| $11$ |
\( (T^{2} - 97 T + 1854)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 4362886368 \)
|
| $17$ |
\( (T^{2} + 538 T + 40473)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 16983563041 \)
|
| $23$ |
\( (T^{2} + 545 T - 237150)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 9349110072 \)
|
| $31$ |
\( T^{4} + \cdots + 325578732768 \)
|
| $37$ |
\( T^{4} + \cdots + 1170993888 \)
|
| $41$ |
\( T^{4} + \cdots + 4155160032 \)
|
| $43$ |
\( (T^{2} + 2053 T + 941596)^{2} \)
|
| $47$ |
\( (T^{2} + 2627 T - 9126)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 20392614892728 \)
|
| $59$ |
\( T^{4} + \cdots + 219595642144992 \)
|
| $61$ |
\( (T^{2} - 3039 T - 17027704)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 221664248921592 \)
|
| $71$ |
\( T^{4} + \cdots + 581898049763328 \)
|
| $73$ |
\( (T^{2} + 4928 T - 3417377)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 48366114070752 \)
|
| $83$ |
\( (T^{2} + 1934 T - 19395504)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 27\!\cdots\!48 \)
|
| $97$ |
\( T^{4} + \cdots + 30\!\cdots\!88 \)
|
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