Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [304,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("304.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(94.9650477472\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2737}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 684 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{2737})\). We also show the integral \(q\)-expansion of the trace form.
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 | 4.34183 | 0 | 479.873 | 0 | 1139.05 | 0 | −2168.15 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 56.6582 | 0 | −304.873 | 0 | 1452.95 | 0 | 1023.15 | 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 | 304.8.a.c | 2 | |
| 4.b | odd | 2 | 1 | 38.8.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 342.8.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.a.b | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 304.8.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 342.8.a.h | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 61T_{3} + 246 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 61T + 246 \)
|
| $5$ |
\( T^{2} - 175T - 146300 \)
|
| $7$ |
\( T^{2} - 2592 T + 1654983 \)
|
| $11$ |
\( T^{2} + 1045 T - 50723462 \)
|
| $13$ |
\( T^{2} - 14647 T + 48454564 \)
|
| $17$ |
\( T^{2} - 29616 T + 213734439 \)
|
| $19$ |
\( (T - 6859)^{2} \)
|
| $23$ |
\( T^{2} + 69985 T - 84282392 \)
|
| $29$ |
\( T^{2} + \cdots - 18767350094 \)
|
| $31$ |
\( T^{2} + \cdots - 6382036064 \)
|
| $37$ |
\( T^{2} + \cdots - 142395679268 \)
|
| $41$ |
\( T^{2} + 539350 T - 64948688 \)
|
| $43$ |
\( T^{2} + \cdots + 90498085252 \)
|
| $47$ |
\( T^{2} + \cdots - 263524205000 \)
|
| $53$ |
\( T^{2} + \cdots + 147117109708 \)
|
| $59$ |
\( T^{2} + \cdots + 3864174091866 \)
|
| $61$ |
\( T^{2} + \cdots - 4064856437738 \)
|
| $67$ |
\( T^{2} + \cdots - 2495267011548 \)
|
| $71$ |
\( T^{2} + \cdots - 8985205484828 \)
|
| $73$ |
\( T^{2} + \cdots + 20561142356433 \)
|
| $79$ |
\( T^{2} + \cdots - 50903350780448 \)
|
| $83$ |
\( T^{2} + \cdots + 14334319474056 \)
|
| $89$ |
\( T^{2} + \cdots - 513986601600 \)
|
| $97$ |
\( T^{2} + \cdots - 48822541081424 \)
|
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