Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2312,4,Mod(1,2312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2312, 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("2312.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2312 = 2^{3} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2312.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: | \(136.412415933\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 81x^{4} + 222x^{2} - 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 136) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 81x^{4} + 222x^{2} - 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 85\nu^{3} + 562\nu ) / 68 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 85\nu^{3} - 426\nu ) / 34 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} + 391\nu^{2} - 362 ) / 17 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\nu^{5} - 1207\nu^{3} + 2650\nu ) / 68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{4} + 731\nu^{2} - 1386 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{5} - 9\beta_{3} + 216 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{4} + 85\beta_{2} + 110\beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 391\beta_{5} - 731\beta_{3} + 16312 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 340\beta_{4} + 6663\beta_{2} + 8498\beta_1 ) / 4 \)
|
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 | −8.27144 | 0 | 6.42821 | 0 | 24.9151 | 0 | 41.4167 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −4.49442 | 0 | −18.9829 | 0 | −7.78768 | 0 | −6.80022 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.65834 | 0 | 5.50702 | 0 | −18.8836 | 0 | −13.6165 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.65834 | 0 | −5.50702 | 0 | 18.8836 | 0 | −13.6165 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.49442 | 0 | 18.9829 | 0 | 7.78768 | 0 | −6.80022 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.27144 | 0 | −6.42821 | 0 | −24.9151 | 0 | 41.4167 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2312.4.a.h | 6 | |
| 17.b | even | 2 | 1 | inner | 2312.4.a.h | 6 | |
| 17.c | even | 4 | 2 | 136.4.b.a | ✓ | 6 | |
| 51.f | odd | 4 | 2 | 1224.4.c.c | 6 | ||
| 68.f | odd | 4 | 2 | 272.4.b.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.4.b.a | ✓ | 6 | 17.c | even | 4 | 2 | |
| 272.4.b.e | 6 | 68.f | odd | 4 | 2 | ||
| 1224.4.c.c | 6 | 51.f | odd | 4 | 2 | ||
| 2312.4.a.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 2312.4.a.h | 6 | 17.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 102T_{3}^{4} + 2568T_{3}^{2} - 18496 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 102 T^{4} + \cdots - 18496 \)
|
| $5$ |
\( T^{6} - 432 T^{4} + \cdots - 451584 \)
|
| $7$ |
\( T^{6} - 1038 T^{4} + \cdots - 13424896 \)
|
| $11$ |
\( T^{6} - 1062 T^{4} + \cdots - 25482304 \)
|
| $13$ |
\( (T^{3} + 36 T^{2} + \cdots + 28016)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{3} - 12 T^{2} + \cdots - 178496)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 3953894400 \)
|
| $29$ |
\( T^{6} + \cdots - 1770986054656 \)
|
| $31$ |
\( T^{6} + \cdots - 10987474749696 \)
|
| $37$ |
\( T^{6} + \cdots - 7252378264576 \)
|
| $41$ |
\( T^{6} + \cdots - 8454416891904 \)
|
| $43$ |
\( (T^{3} + 96 T^{2} + \cdots + 20641152)^{2} \)
|
| $47$ |
\( (T^{3} + 36 T^{2} + \cdots - 11699200)^{2} \)
|
| $53$ |
\( (T^{3} + 462 T^{2} + \cdots - 1630776)^{2} \)
|
| $59$ |
\( (T^{3} + 840 T^{2} + \cdots - 116758016)^{2} \)
|
| $61$ |
\( T^{6} + \cdots - 11\!\cdots\!76 \)
|
| $67$ |
\( (T^{3} + 156 T^{2} + \cdots - 49226176)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 47\!\cdots\!76 \)
|
| $73$ |
\( T^{6} + \cdots - 25\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots - 10\!\cdots\!44 \)
|
| $83$ |
\( (T^{3} + 648 T^{2} + \cdots + 10110464)^{2} \)
|
| $89$ |
\( (T^{3} - 12 T^{2} + \cdots + 2334896)^{2} \)
|
| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!16 \)
|
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