Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [368,4,Mod(1,368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(368, 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("368.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 368 = 2^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 368.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: | \(21.7127028821\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.28669.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 34x - 69 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 92) |
| 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
\( x^{3} - 34x - 69 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + 3\beta _1 + 46 ) / 2 \)
|
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 | −10.0649 | 0 | −8.36924 | 0 | −22.3692 | 0 | 74.3025 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −4.37890 | 0 | 13.3206 | 0 | −0.679360 | 0 | −7.82521 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 6.44381 | 0 | −4.95140 | 0 | −18.9514 | 0 | 14.5228 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 368.4.a.h | 3 | |
| 4.b | odd | 2 | 1 | 92.4.a.b | ✓ | 3 | |
| 8.b | even | 2 | 1 | 1472.4.a.x | 3 | ||
| 8.d | odd | 2 | 1 | 1472.4.a.o | 3 | ||
| 12.b | even | 2 | 1 | 828.4.a.e | 3 | ||
| 20.d | odd | 2 | 1 | 2300.4.a.a | 3 | ||
| 20.e | even | 4 | 2 | 2300.4.c.a | 6 | ||
| 92.b | even | 2 | 1 | 2116.4.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 92.4.a.b | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 368.4.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 828.4.a.e | 3 | 12.b | even | 2 | 1 | ||
| 1472.4.a.o | 3 | 8.d | odd | 2 | 1 | ||
| 1472.4.a.x | 3 | 8.b | even | 2 | 1 | ||
| 2116.4.a.b | 3 | 92.b | even | 2 | 1 | ||
| 2300.4.a.a | 3 | 20.d | odd | 2 | 1 | ||
| 2300.4.c.a | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 8T_{3}^{2} - 49T_{3} - 284 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(368))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 8 T^{2} + \cdots - 284 \)
|
| $5$ |
\( T^{3} - 136T - 552 \)
|
| $7$ |
\( T^{3} + 42 T^{2} + \cdots + 288 \)
|
| $11$ |
\( T^{3} + 6 T^{2} + \cdots - 89424 \)
|
| $13$ |
\( T^{3} - 100 T^{2} + \cdots + 24394 \)
|
| $17$ |
\( T^{3} + 28 T^{2} + \cdots - 56376 \)
|
| $19$ |
\( T^{3} + 120 T^{2} + \cdots + 3984 \)
|
| $23$ |
\( (T + 23)^{3} \)
|
| $29$ |
\( T^{3} + 128 T^{2} + \cdots - 231174 \)
|
| $31$ |
\( T^{3} - 76 T^{2} + \cdots - 382208 \)
|
| $37$ |
\( T^{3} + 212 T^{2} + \cdots + 64536 \)
|
| $41$ |
\( T^{3} + 580 T^{2} + \cdots - 509682 \)
|
| $43$ |
\( T^{3} - 20 T^{2} + \cdots + 25961472 \)
|
| $47$ |
\( T^{3} - 396 T^{2} + \cdots + 5642304 \)
|
| $53$ |
\( T^{3} + 122 T^{2} + \cdots + 28384056 \)
|
| $59$ |
\( T^{3} + 632 T^{2} + \cdots - 9077184 \)
|
| $61$ |
\( T^{3} - 338 T^{2} + \cdots + 2020904 \)
|
| $67$ |
\( T^{3} + 442 T^{2} + \cdots - 105984432 \)
|
| $71$ |
\( T^{3} + 888 T^{2} + \cdots - 150510312 \)
|
| $73$ |
\( T^{3} - 376 T^{2} + \cdots + 431656494 \)
|
| $79$ |
\( T^{3} + 1540 T^{2} + \cdots + 66491136 \)
|
| $83$ |
\( T^{3} + 454 T^{2} + \cdots - 241050384 \)
|
| $89$ |
\( T^{3} + 810 T^{2} + \cdots - 178848 \)
|
| $97$ |
\( T^{3} - 1284 T^{2} + \cdots - 22204008 \)
|
| show more | |
| show less |