Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [936,4,Mod(1,936)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(936, 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("936.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 936.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: | \(55.2257877654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.18257.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 26x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 104) |
| 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
\( x^{3} - x^{2} - 26x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 18 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 3\nu + 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 3\beta _1 + 35 ) / 2 \)
|
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 | 0 | 0 | −7.51634 | 0 | 12.0519 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −6.19042 | 0 | 3.19918 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 21.7068 | 0 | 20.7489 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 936.4.a.m | 3 | |
| 3.b | odd | 2 | 1 | 104.4.a.e | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 1872.4.a.bm | 3 | ||
| 12.b | even | 2 | 1 | 208.4.a.l | 3 | ||
| 24.f | even | 2 | 1 | 832.4.a.bb | 3 | ||
| 24.h | odd | 2 | 1 | 832.4.a.bc | 3 | ||
| 39.d | odd | 2 | 1 | 1352.4.a.h | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.4.a.e | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 208.4.a.l | 3 | 12.b | even | 2 | 1 | ||
| 832.4.a.bb | 3 | 24.f | even | 2 | 1 | ||
| 832.4.a.bc | 3 | 24.h | odd | 2 | 1 | ||
| 936.4.a.m | 3 | 1.a | even | 1 | 1 | trivial | |
| 1352.4.a.h | 3 | 39.d | odd | 2 | 1 | ||
| 1872.4.a.bm | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 8T_{5}^{2} - 251T_{5} - 1010 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(936))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots - 1010 \)
|
| $7$ |
\( T^{3} - 36 T^{2} + \cdots - 800 \)
|
| $11$ |
\( T^{3} + 52 T^{2} + \cdots - 59184 \)
|
| $13$ |
\( (T - 13)^{3} \)
|
| $17$ |
\( T^{3} - 116 T^{2} + \cdots + 1048898 \)
|
| $19$ |
\( T^{3} - 124 T^{2} + \cdots - 34864 \)
|
| $23$ |
\( T^{3} + 232 T^{2} + \cdots - 65536 \)
|
| $29$ |
\( T^{3} - 30 T^{2} + \cdots - 1387112 \)
|
| $31$ |
\( T^{3} - 240 T^{2} + \cdots + 311936 \)
|
| $37$ |
\( T^{3} + 264 T^{2} + \cdots + 99730 \)
|
| $41$ |
\( T^{3} - 374 T^{2} + \cdots + 29246464 \)
|
| $43$ |
\( T^{3} - 248 T^{2} + \cdots + 52832740 \)
|
| $47$ |
\( T^{3} - 412 T^{2} + \cdots + 2411208 \)
|
| $53$ |
\( T^{3} - 386 T^{2} + \cdots + 4448256 \)
|
| $59$ |
\( T^{3} - 940 T^{2} + \cdots + 37508496 \)
|
| $61$ |
\( T^{3} - 1206 T^{2} + \cdots - 46097792 \)
|
| $67$ |
\( T^{3} + 564 T^{2} + \cdots + 2603408 \)
|
| $71$ |
\( T^{3} + 1260 T^{2} + \cdots - 198899280 \)
|
| $73$ |
\( T^{3} - 142 T^{2} + \cdots - 146317608 \)
|
| $79$ |
\( T^{3} - 1040 T^{2} + \cdots + 373329920 \)
|
| $83$ |
\( T^{3} + 756 T^{2} + \cdots - 64918848 \)
|
| $89$ |
\( T^{3} - 18 T^{2} + \cdots - 121968344 \)
|
| $97$ |
\( T^{3} - 2174 T^{2} + \cdots + 7072888 \)
|
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