Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,4,Mod(1,304)]
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, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.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: | \(17.9365806417\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3221.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 152) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 9x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 3\nu + 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} - \beta _1 + 26 ) / 4 \)
|
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 | −4.73435 | 0 | −18.0754 | 0 | −0.213413 | 0 | −4.58596 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.573746 | 0 | 5.92862 | 0 | 31.1522 | 0 | −26.6708 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 10.3081 | 0 | 14.1468 | 0 | 4.06119 | 0 | 79.2568 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(19\) | \( +1 \) |
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 | 304.4.a.j | 3 | |
| 4.b | odd | 2 | 1 | 152.4.a.b | ✓ | 3 | |
| 8.b | even | 2 | 1 | 1216.4.a.q | 3 | ||
| 8.d | odd | 2 | 1 | 1216.4.a.x | 3 | ||
| 12.b | even | 2 | 1 | 1368.4.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.4.a.b | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 304.4.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
| 1216.4.a.q | 3 | 8.b | even | 2 | 1 | ||
| 1216.4.a.x | 3 | 8.d | odd | 2 | 1 | ||
| 1368.4.a.e | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 5T_{3}^{2} - 52T_{3} - 28 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 5 T^{2} + \cdots - 28 \)
|
| $5$ |
\( T^{3} - 2 T^{2} + \cdots + 1516 \)
|
| $7$ |
\( T^{3} - 35 T^{2} + \cdots + 27 \)
|
| $11$ |
\( T^{3} - 28 T^{2} + \cdots - 358 \)
|
| $13$ |
\( T^{3} + 109 T^{2} + \cdots - 113456 \)
|
| $17$ |
\( T^{3} + 123 T^{2} + \cdots + 6637 \)
|
| $19$ |
\( (T + 19)^{3} \)
|
| $23$ |
\( T^{3} - 193 T^{2} + \cdots + 246848 \)
|
| $29$ |
\( T^{3} + 297 T^{2} + \cdots - 1167636 \)
|
| $31$ |
\( T^{3} - 140 T^{2} + \cdots + 3668096 \)
|
| $37$ |
\( T^{3} - 38 T^{2} + \cdots - 3429384 \)
|
| $41$ |
\( T^{3} - 736 T^{2} + \cdots - 7266624 \)
|
| $43$ |
\( T^{3} - 514 T^{2} + \cdots + 25097948 \)
|
| $47$ |
\( T^{3} + 134 T^{2} + \cdots - 58888776 \)
|
| $53$ |
\( T^{3} - 311 T^{2} + \cdots - 13619792 \)
|
| $59$ |
\( T^{3} + 199 T^{2} + \cdots - 117609444 \)
|
| $61$ |
\( T^{3} - 56 T^{2} + \cdots + 120500582 \)
|
| $67$ |
\( T^{3} - 509 T^{2} + \cdots + 177822064 \)
|
| $71$ |
\( T^{3} - 874 T^{2} + \cdots + 112230216 \)
|
| $73$ |
\( T^{3} - 203 T^{2} + \cdots + 474103 \)
|
| $79$ |
\( T^{3} + 242 T^{2} + \cdots - 201599456 \)
|
| $83$ |
\( T^{3} - 62 T^{2} + \cdots - 83648992 \)
|
| $89$ |
\( T^{3} - 1764 T^{2} + \cdots - 127322496 \)
|
| $97$ |
\( T^{3} + 2178 T^{2} + \cdots + 99903104 \)
|
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