Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,6,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.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: | \(70.7292645375\) |
| 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: |
\( x^{4} - x^{3} - 97x^{2} + 7x + 294 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 7 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{4} - x^{3} - 97x^{2} + 7x + 294 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 49 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 89\nu + 52 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 91\beta _1 + 46 \)
|
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 |
|
−10.2239 | 0 | 72.5272 | 23.7528 | 0 | 0 | −414.344 | 0 | −242.845 | ||||||||||||||||||||||||||||||
| 1.2 | −2.74818 | 0 | −24.4475 | 58.3673 | 0 | 0 | 155.128 | 0 | −160.404 | |||||||||||||||||||||||||||||||
| 1.3 | 0.790805 | 0 | −31.3746 | −104.192 | 0 | 0 | −50.1170 | 0 | −82.3953 | |||||||||||||||||||||||||||||||
| 1.4 | 9.18123 | 0 | 52.2950 | 22.0716 | 0 | 0 | 186.333 | 0 | 202.644 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 441.6.a.v | 4 | |
| 3.b | odd | 2 | 1 | 147.6.a.l | 4 | ||
| 7.b | odd | 2 | 1 | 441.6.a.w | 4 | ||
| 7.d | odd | 6 | 2 | 63.6.e.e | 8 | ||
| 21.c | even | 2 | 1 | 147.6.a.m | 4 | ||
| 21.g | even | 6 | 2 | 21.6.e.c | ✓ | 8 | |
| 21.h | odd | 6 | 2 | 147.6.e.o | 8 | ||
| 84.j | odd | 6 | 2 | 336.6.q.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.6.e.c | ✓ | 8 | 21.g | even | 6 | 2 | |
| 63.6.e.e | 8 | 7.d | odd | 6 | 2 | ||
| 147.6.a.l | 4 | 3.b | odd | 2 | 1 | ||
| 147.6.a.m | 4 | 21.c | even | 2 | 1 | ||
| 147.6.e.o | 8 | 21.h | odd | 6 | 2 | ||
| 336.6.q.j | 8 | 84.j | odd | 6 | 2 | ||
| 441.6.a.v | 4 | 1.a | even | 1 | 1 | trivial | |
| 441.6.a.w | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{4} + 3T_{2}^{3} - 94T_{2}^{2} - 186T_{2} + 204 \)
|
|
\( T_{5}^{4} - 7657T_{5}^{2} + 302700T_{5} - 3188244 \)
|
|
\( T_{13}^{4} + 462T_{13}^{3} - 1148423T_{13}^{2} - 515112852T_{13} + 149501563456 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 3 T^{3} + \cdots + 204 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 7657 T^{2} + \cdots - 3188244 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 1682132124 \)
|
| $13$ |
\( T^{4} + \cdots + 149501563456 \)
|
| $17$ |
\( T^{4} + \cdots - 50104147968 \)
|
| $19$ |
\( T^{4} + \cdots + 7391138416576 \)
|
| $23$ |
\( T^{4} + \cdots + 3007939608576 \)
|
| $29$ |
\( T^{4} + \cdots + 408027025117872 \)
|
| $31$ |
\( T^{4} + \cdots + 86716089209547 \)
|
| $37$ |
\( T^{4} + \cdots - 50\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!52 \)
|
| $43$ |
\( T^{4} + \cdots - 991662745581932 \)
|
| $47$ |
\( T^{4} + \cdots - 270685655359056 \)
|
| $53$ |
\( T^{4} + \cdots - 85\!\cdots\!72 \)
|
| $59$ |
\( T^{4} + \cdots - 25\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots + 17\!\cdots\!24 \)
|
| $67$ |
\( T^{4} + \cdots + 55\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots + 21\!\cdots\!68 \)
|
| $73$ |
\( T^{4} + \cdots - 12\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 16\!\cdots\!59 \)
|
| $83$ |
\( T^{4} + \cdots - 41\!\cdots\!32 \)
|
| $89$ |
\( T^{4} + \cdots - 47\!\cdots\!68 \)
|
| $97$ |
\( T^{4} + \cdots - 11\!\cdots\!44 \)
|
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