Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1232,4,Mod(1,1232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, 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("1232.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1232 = 2^{4} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1232.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: | \(72.6903531271\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - 40x^{2} - 41x + 194 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 616) |
| 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
\( x^{4} - 40x^{2} - 41x + 194 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 25\nu - 9 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} + 39\nu + 9 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 5\nu^{2} + 23\nu - 69 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 8\beta _1 + 78 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 51\beta_{2} + 86\beta _1 + 114 ) / 4 \)
|
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 | −7.15990 | 0 | −9.75471 | 0 | 7.00000 | 0 | 24.2641 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.43230 | 0 | 5.36131 | 0 | 7.00000 | 0 | −21.0839 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.22993 | 0 | 1.36923 | 0 | 7.00000 | 0 | −9.10770 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 8.36227 | 0 | −15.9758 | 0 | 7.00000 | 0 | 42.9275 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 1232.4.a.t | 4 | |
| 4.b | odd | 2 | 1 | 616.4.a.e | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.4.a.e | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 1232.4.a.t | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):
|
\( T_{3}^{4} - 3T_{3}^{3} - 68T_{3}^{2} + 120T_{3} + 616 \)
|
|
\( T_{5}^{4} + 19T_{5}^{3} - 10T_{5}^{2} - 860T_{5} + 1144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 616 \)
|
| $5$ |
\( T^{4} + 19 T^{3} + \cdots + 1144 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( (T + 11)^{4} \)
|
| $13$ |
\( T^{4} - 2 T^{3} + \cdots + 63616 \)
|
| $17$ |
\( T^{4} - 8 T^{3} + \cdots + 37838944 \)
|
| $19$ |
\( T^{4} - 6 T^{3} + \cdots + 8981504 \)
|
| $23$ |
\( T^{4} + 159 T^{3} + \cdots - 27228096 \)
|
| $29$ |
\( T^{4} - 144 T^{3} + \cdots + 32288592 \)
|
| $31$ |
\( T^{4} - 183 T^{3} + \cdots - 967624 \)
|
| $37$ |
\( T^{4} + 475 T^{3} + \cdots - 672375512 \)
|
| $41$ |
\( T^{4} + 768 T^{3} + \cdots + 40685664 \)
|
| $43$ |
\( T^{4} - 152 T^{3} + \cdots - 162100224 \)
|
| $47$ |
\( T^{4} + \cdots + 10623846656 \)
|
| $53$ |
\( T^{4} + \cdots - 2515760528 \)
|
| $59$ |
\( T^{4} + \cdots + 28943918424 \)
|
| $61$ |
\( T^{4} + 1012 T^{3} + \cdots + 772785216 \)
|
| $67$ |
\( T^{4} + \cdots + 327391058816 \)
|
| $71$ |
\( T^{4} + \cdots - 16928024256 \)
|
| $73$ |
\( T^{4} + \cdots - 54340975104 \)
|
| $79$ |
\( T^{4} + 1374 T^{3} + \cdots - 954747648 \)
|
| $83$ |
\( T^{4} + \cdots + 31070871552 \)
|
| $89$ |
\( T^{4} + \cdots - 94585872472 \)
|
| $97$ |
\( T^{4} + \cdots - 13654212712 \)
|
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