Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [832,4,Mod(1,832)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(832, 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("832.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 832 = 2^{6} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 832.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: | \(49.0895891248\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{73}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). 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 | −5.77200 | 0 | −11.3160 | 0 | 8.22800 | 0 | 6.31601 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 2.77200 | 0 | 14.3160 | 0 | 16.7720 | 0 | −19.3160 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 832.4.a.u | 2 | |
| 4.b | odd | 2 | 1 | 832.4.a.y | 2 | ||
| 8.b | even | 2 | 1 | 208.4.a.j | 2 | ||
| 8.d | odd | 2 | 1 | 104.4.a.c | ✓ | 2 | |
| 24.f | even | 2 | 1 | 936.4.a.e | 2 | ||
| 24.h | odd | 2 | 1 | 1872.4.a.bc | 2 | ||
| 104.h | odd | 2 | 1 | 1352.4.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.4.a.c | ✓ | 2 | 8.d | odd | 2 | 1 | |
| 208.4.a.j | 2 | 8.b | even | 2 | 1 | ||
| 832.4.a.u | 2 | 1.a | even | 1 | 1 | trivial | |
| 832.4.a.y | 2 | 4.b | odd | 2 | 1 | ||
| 936.4.a.e | 2 | 24.f | even | 2 | 1 | ||
| 1352.4.a.f | 2 | 104.h | odd | 2 | 1 | ||
| 1872.4.a.bc | 2 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(832))\):
|
\( T_{3}^{2} + 3T_{3} - 16 \)
|
|
\( T_{5}^{2} - 3T_{5} - 162 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 3T - 16 \)
|
| $5$ |
\( T^{2} - 3T - 162 \)
|
| $7$ |
\( T^{2} - 25T + 138 \)
|
| $11$ |
\( T^{2} + 56T + 492 \)
|
| $13$ |
\( (T - 13)^{2} \)
|
| $17$ |
\( T^{2} + 13T - 6546 \)
|
| $19$ |
\( T^{2} + 124T + 2676 \)
|
| $23$ |
\( T^{2} - 172T + 4768 \)
|
| $29$ |
\( T^{2} + 196T + 4932 \)
|
| $31$ |
\( T^{2} - 78T - 19576 \)
|
| $37$ |
\( T^{2} + 161T - 65954 \)
|
| $41$ |
\( T^{2} - 234T - 97344 \)
|
| $43$ |
\( T^{2} - 135T - 146572 \)
|
| $47$ |
\( T^{2} - 237T + 13586 \)
|
| $53$ |
\( T^{2} + 666T + 109064 \)
|
| $59$ |
\( T^{2} - 136T - 353076 \)
|
| $61$ |
\( T^{2} - 146T + 3504 \)
|
| $67$ |
\( T^{2} + 4T - 74748 \)
|
| $71$ |
\( T^{2} - 563T + 71194 \)
|
| $73$ |
\( T^{2} + 1480 T + 533292 \)
|
| $79$ |
\( T^{2} - 896T + 190192 \)
|
| $83$ |
\( T^{2} - 1902 T + 903744 \)
|
| $89$ |
\( T^{2} + 272 T - 1068036 \)
|
| $97$ |
\( T^{2} + 2160 T + 521372 \)
|
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