Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,4,Mod(1,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, 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("1764.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1764.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: | \(104.079369250\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.136768.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 23x^{2} + 18x + 119 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 588) |
| 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
\( x^{4} - 2x^{3} - 23x^{2} + 18x + 119 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} + 25\nu + 2 ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{3} + 21\nu^{2} + 77\nu - 140 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 7\nu^{2} + 7\nu - 68 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 7\beta _1 + 14 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{3} - 8\beta_{2} + 7\beta _1 + 350 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} - 7\beta_{2} + 14\beta _1 + 92 ) / 4 \)
|
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 | 0 | 0 | −16.6547 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −10.6550 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 8.16940 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 19.1403 | 0 | 0 | 0 | 0 | 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 | 1764.4.a.bc | 4 | |
| 3.b | odd | 2 | 1 | 588.4.a.j | ✓ | 4 | |
| 7.b | odd | 2 | 1 | 1764.4.a.ba | 4 | ||
| 7.c | even | 3 | 2 | 1764.4.k.bb | 8 | ||
| 7.d | odd | 6 | 2 | 1764.4.k.bd | 8 | ||
| 12.b | even | 2 | 1 | 2352.4.a.cq | 4 | ||
| 21.c | even | 2 | 1 | 588.4.a.k | yes | 4 | |
| 21.g | even | 6 | 2 | 588.4.i.k | 8 | ||
| 21.h | odd | 6 | 2 | 588.4.i.l | 8 | ||
| 84.h | odd | 2 | 1 | 2352.4.a.cl | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 588.4.a.j | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 588.4.a.k | yes | 4 | 21.c | even | 2 | 1 | |
| 588.4.i.k | 8 | 21.g | even | 6 | 2 | ||
| 588.4.i.l | 8 | 21.h | odd | 6 | 2 | ||
| 1764.4.a.ba | 4 | 7.b | odd | 2 | 1 | ||
| 1764.4.a.bc | 4 | 1.a | even | 1 | 1 | trivial | |
| 1764.4.k.bb | 8 | 7.c | even | 3 | 2 | ||
| 1764.4.k.bd | 8 | 7.d | odd | 6 | 2 | ||
| 2352.4.a.cl | 4 | 84.h | odd | 2 | 1 | ||
| 2352.4.a.cq | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):
|
\( T_{5}^{4} - 412T_{5}^{2} - 576T_{5} + 27748 \)
|
|
\( T_{11}^{4} - 4136T_{11}^{2} - 82944T_{11} + 733072 \)
|
|
\( T_{13}^{4} - 7092T_{13}^{2} + 31104T_{13} + 10008036 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 412 T^{2} + \cdots + 27748 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 4136 T^{2} + \cdots + 733072 \)
|
| $13$ |
\( T^{4} - 7092 T^{2} + \cdots + 10008036 \)
|
| $17$ |
\( T^{4} - 48 T^{3} + \cdots - 4865084 \)
|
| $19$ |
\( T^{4} + 192 T^{3} + \cdots - 41971136 \)
|
| $23$ |
\( T^{4} + 192 T^{3} + \cdots - 40554608 \)
|
| $29$ |
\( T^{4} + 96 T^{3} + \cdots + 38719552 \)
|
| $31$ |
\( T^{4} + 48 T^{3} + \cdots - 189895104 \)
|
| $37$ |
\( T^{4} + \cdots - 1479272192 \)
|
| $41$ |
\( T^{4} + \cdots + 1611829828 \)
|
| $43$ |
\( T^{4} + 112 T^{3} + \cdots - 789373952 \)
|
| $47$ |
\( T^{4} + \cdots - 1168478144 \)
|
| $53$ |
\( T^{4} + \cdots - 20504773616 \)
|
| $59$ |
\( T^{4} + \cdots + 18986185792 \)
|
| $61$ |
\( T^{4} + \cdots - 106656271196 \)
|
| $67$ |
\( T^{4} + \cdots + 14336621568 \)
|
| $71$ |
\( T^{4} + \cdots - 17989567344 \)
|
| $73$ |
\( T^{4} + \cdots - 90986816444 \)
|
| $79$ |
\( T^{4} + \cdots - 1013049875456 \)
|
| $83$ |
\( T^{4} + \cdots + 74256064768 \)
|
| $89$ |
\( T^{4} + \cdots - 311467391228 \)
|
| $97$ |
\( T^{4} + \cdots - 580580611196 \)
|
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