Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3724,2,Mod(1,3724)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3724.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3724.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: | \(29.7362897127\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 18x^{5} + 84x^{3} - 4x^{2} - 44x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 532) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 18x^{5} + 84x^{3} - 4x^{2} - 44x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 8\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 14\nu^{4} - 14\nu^{3} + 46\nu^{2} + 36\nu - 20 ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 20\nu^{4} - 14\nu^{3} + 94\nu^{2} + 36\nu - 8 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 20\nu^{4} - 14\nu^{3} + 106\nu^{2} + 36\nu - 68 ) / 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{6} + \nu^{5} + 34\nu^{4} - 8\nu^{3} - 146\nu^{2} + 52 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{5} - 10\beta_{4} + 2\beta_{3} + 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{6} + 2\beta_{5} + 2\beta_{4} + 4\beta_{3} + 24\beta_{2} + 72\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -4\beta_{6} + 64\beta_{5} - 96\beta_{4} + 36\beta_{3} + 4\beta_{2} + 4\beta _1 + 376 \)
|
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 | −3.11678 | 0 | −3.59376 | 0 | 0 | 0 | 6.71431 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.77904 | 0 | 3.08629 | 0 | 0 | 0 | 4.72304 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.835733 | 0 | −2.48390 | 0 | 0 | 0 | −2.30155 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.191846 | 0 | 3.88468 | 0 | 0 | 0 | −2.96320 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.678078 | 0 | −0.873900 | 0 | 0 | 0 | −2.54021 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.62398 | 0 | −1.19764 | 0 | 0 | 0 | 3.88528 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 3.23764 | 0 | 3.17823 | 0 | 0 | 0 | 7.48232 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 3724.2.a.n | 7 | |
| 7.b | odd | 2 | 1 | 3724.2.a.m | 7 | ||
| 7.d | odd | 6 | 2 | 532.2.i.c | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 532.2.i.c | ✓ | 14 | 7.d | odd | 6 | 2 | |
| 3724.2.a.m | 7 | 7.b | odd | 2 | 1 | ||
| 3724.2.a.n | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 18T_{3}^{5} + 84T_{3}^{3} - 4T_{3}^{2} - 44T_{3} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 18 T^{5} + \cdots + 8 \)
|
| $5$ |
\( T^{7} - 2 T^{6} + \cdots - 356 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} - 14 T^{6} + \cdots - 556 \)
|
| $13$ |
\( T^{7} - 72 T^{5} + \cdots - 13576 \)
|
| $17$ |
\( T^{7} - 10 T^{6} + \cdots + 444 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} - 7 T^{6} + \cdots - 419 \)
|
| $29$ |
\( T^{7} - 2 T^{6} + \cdots - 11008 \)
|
| $31$ |
\( T^{7} - 4 T^{6} + \cdots + 26848 \)
|
| $37$ |
\( T^{7} + 12 T^{6} + \cdots + 1152 \)
|
| $41$ |
\( T^{7} - 4 T^{6} + \cdots + 62464 \)
|
| $43$ |
\( T^{7} - 218 T^{5} + \cdots + 198844 \)
|
| $47$ |
\( T^{7} + 16 T^{6} + \cdots - 84652 \)
|
| $53$ |
\( T^{7} + 6 T^{6} + \cdots - 64648 \)
|
| $59$ |
\( T^{7} + 4 T^{6} + \cdots + 1223424 \)
|
| $61$ |
\( T^{7} + 28 T^{6} + \cdots - 963082 \)
|
| $67$ |
\( T^{7} - 22 T^{6} + \cdots - 6144 \)
|
| $71$ |
\( T^{7} - 34 T^{6} + \cdots - 640984 \)
|
| $73$ |
\( T^{7} - 16 T^{6} + \cdots + 295242 \)
|
| $79$ |
\( T^{7} - 14 T^{6} + \cdots + 1882288 \)
|
| $83$ |
\( T^{7} - 13 T^{6} + \cdots - 252099 \)
|
| $89$ |
\( T^{7} + 16 T^{6} + \cdots - 19200 \)
|
| $97$ |
\( T^{7} + 4 T^{6} + \cdots + 120128 \)
|
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