Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2352,4,Mod(1,2352)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2352.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2352.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: | \(138.772492334\) |
| 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: |
\( x^{4} - 2x^{3} - 152x^{2} - 177x + 2922 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 168) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 152x^{2} - 177x + 2922 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} + 7\nu^{2} - 213\nu - 1158 ) / 105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 98\nu^{2} + 942\nu + 6297 ) / 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{3} + 77\nu^{2} + 677\nu - 3348 ) / 35 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + 9\beta _1 + 15 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{3} + 5\beta_{2} + 127\beta _1 + 2153 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 34\beta_{3} + 151\beta_{2} + 992\beta _1 + 5137 ) / 14 \)
|
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 | 3.00000 | 0 | −18.9580 | 0 | 0 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | −0.726342 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | −0.128591 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 15.8130 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 2352.4.a.cp | 4 | |
| 4.b | odd | 2 | 1 | 1176.4.a.ba | 4 | ||
| 7.b | odd | 2 | 1 | 2352.4.a.cm | 4 | ||
| 7.d | odd | 6 | 2 | 336.4.q.m | 8 | ||
| 28.d | even | 2 | 1 | 1176.4.a.bd | 4 | ||
| 28.f | even | 6 | 2 | 168.4.q.f | ✓ | 8 | |
| 84.j | odd | 6 | 2 | 504.4.s.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 168.4.q.f | ✓ | 8 | 28.f | even | 6 | 2 | |
| 336.4.q.m | 8 | 7.d | odd | 6 | 2 | ||
| 504.4.s.j | 8 | 84.j | odd | 6 | 2 | ||
| 1176.4.a.ba | 4 | 4.b | odd | 2 | 1 | ||
| 1176.4.a.bd | 4 | 28.d | even | 2 | 1 | ||
| 2352.4.a.cm | 4 | 7.b | odd | 2 | 1 | ||
| 2352.4.a.cp | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2352))\):
|
\( T_{5}^{4} + 4T_{5}^{3} - 297T_{5}^{2} - 256T_{5} - 28 \)
|
|
\( T_{11}^{4} + 14T_{11}^{3} - 5395T_{11}^{2} - 67916T_{11} + 5765844 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 4 T^{3} + \cdots - 28 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 14 T^{3} + \cdots + 5765844 \)
|
| $13$ |
\( T^{4} + 22 T^{3} + \cdots + 13795008 \)
|
| $17$ |
\( T^{4} + 96 T^{3} + \cdots - 44203008 \)
|
| $19$ |
\( T^{4} + 26 T^{3} + \cdots + 5487296 \)
|
| $23$ |
\( T^{4} + 96 T^{3} + \cdots - 98304 \)
|
| $29$ |
\( T^{4} + 76 T^{3} + \cdots - 802800 \)
|
| $31$ |
\( T^{4} - 238 T^{3} + \cdots - 165027985 \)
|
| $37$ |
\( T^{4} + \cdots - 4624450848 \)
|
| $41$ |
\( T^{4} + \cdots - 3866949120 \)
|
| $43$ |
\( T^{4} + \cdots - 2654719484 \)
|
| $47$ |
\( T^{4} + \cdots + 23866588368 \)
|
| $53$ |
\( T^{4} - 135309 T^{2} + \cdots - 5074800 \)
|
| $59$ |
\( T^{4} + \cdots - 1642011120 \)
|
| $61$ |
\( T^{4} + \cdots + 9863636400 \)
|
| $67$ |
\( T^{4} + \cdots + 26529903468 \)
|
| $71$ |
\( T^{4} + \cdots + 1940742864 \)
|
| $73$ |
\( T^{4} - 218 T^{3} + \cdots + 280393876 \)
|
| $79$ |
\( T^{4} + \cdots - 93048620201 \)
|
| $83$ |
\( T^{4} + \cdots + 352673538780 \)
|
| $89$ |
\( T^{4} + \cdots - 45796564224 \)
|
| $97$ |
\( T^{4} + \cdots + 79407506004 \)
|
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