Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5929,2,Mod(1,5929)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5929, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5929.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5929 = 7^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5929.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: | \(47.3433033584\) |
| 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: |
\( x^{7} - 11x^{5} - x^{4} + 33x^{3} - x^{2} - 27x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 847) |
| 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
\( x^{7} - 11x^{5} - x^{4} + 33x^{3} - x^{2} - 27x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} - \nu^{3} + 30\nu^{2} + 2\nu - 15 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 11\nu^{4} + 4\nu^{3} - 33\nu^{2} - 17\nu + 21 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} - 19\nu^{4} - 5\nu^{3} + 42\nu^{2} + 13\nu - 18 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 12\nu - 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} - \beta_{3} + 7\beta_{2} + 2\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 9\beta_{4} + 7\beta_{3} + 9\beta_{2} + 30\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{5} + 12\beta_{4} - 7\beta_{3} + 48\beta_{2} + 25\beta _1 + 91 \)
|
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 |
|
−2.35905 | 2.65247 | 3.56512 | 3.32612 | −6.25730 | 0 | −3.69218 | 4.03558 | −7.84648 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.88846 | −0.133852 | 1.56628 | −2.34222 | 0.252774 | 0 | 0.819057 | −2.98208 | 4.42318 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.36049 | −2.99364 | −0.149055 | 1.74433 | 4.07283 | 0 | 2.92378 | 5.96189 | −2.37315 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.401136 | 3.23854 | −1.83909 | −1.27779 | 1.29909 | 0 | −1.54000 | 7.48816 | −0.512568 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.812047 | −0.458260 | −1.34058 | 4.29549 | −0.372128 | 0 | −2.71271 | −2.79000 | 3.48814 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.67775 | −1.07480 | 0.814844 | −1.50548 | −1.80324 | 0 | −1.98840 | −1.84481 | −2.52582 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.71707 | 1.76954 | 5.38248 | −0.240449 | 4.80797 | 0 | 9.19045 | 0.131270 | −0.653318 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 5929.2.a.bq | 7 | |
| 7.b | odd | 2 | 1 | 5929.2.a.bn | 7 | ||
| 7.c | even | 3 | 2 | 847.2.e.g | yes | 14 | |
| 11.b | odd | 2 | 1 | 5929.2.a.bp | 7 | ||
| 77.b | even | 2 | 1 | 5929.2.a.bo | 7 | ||
| 77.h | odd | 6 | 2 | 847.2.e.f | ✓ | 14 | |
| 77.m | even | 15 | 8 | 847.2.n.l | 56 | ||
| 77.o | odd | 30 | 8 | 847.2.n.m | 56 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 847.2.e.f | ✓ | 14 | 77.h | odd | 6 | 2 | |
| 847.2.e.g | yes | 14 | 7.c | even | 3 | 2 | |
| 847.2.n.l | 56 | 77.m | even | 15 | 8 | ||
| 847.2.n.m | 56 | 77.o | odd | 30 | 8 | ||
| 5929.2.a.bn | 7 | 7.b | odd | 2 | 1 | ||
| 5929.2.a.bo | 7 | 77.b | even | 2 | 1 | ||
| 5929.2.a.bp | 7 | 11.b | odd | 2 | 1 | ||
| 5929.2.a.bq | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5929))\):
|
\( T_{2}^{7} - 11T_{2}^{5} - T_{2}^{4} + 33T_{2}^{3} - T_{2}^{2} - 27T_{2} + 9 \)
|
|
\( T_{3}^{7} - 3T_{3}^{6} - 11T_{3}^{5} + 32T_{3}^{4} + 21T_{3}^{3} - 47T_{3}^{2} - 29T_{3} - 3 \)
|
|
\( T_{5}^{7} - 4T_{5}^{6} - 13T_{5}^{5} + 39T_{5}^{4} + 73T_{5}^{3} - 71T_{5}^{2} - 133T_{5} - 27 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 11 T^{5} + \cdots + 9 \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots - 3 \)
|
| $5$ |
\( T^{7} - 4 T^{6} + \cdots - 27 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} \)
|
| $13$ |
\( T^{7} + 6 T^{6} + \cdots + 169 \)
|
| $17$ |
\( T^{7} - 3 T^{6} + \cdots + 633 \)
|
| $19$ |
\( T^{7} - 9 T^{6} + \cdots + 5913 \)
|
| $23$ |
\( T^{7} + 6 T^{6} + \cdots - 27 \)
|
| $29$ |
\( T^{7} + 6 T^{6} + \cdots - 45 \)
|
| $31$ |
\( T^{7} - 14 T^{6} + \cdots + 70505 \)
|
| $37$ |
\( T^{7} + 8 T^{6} + \cdots - 106065 \)
|
| $41$ |
\( T^{7} - 10 T^{6} + \cdots + 49431 \)
|
| $43$ |
\( T^{7} + 17 T^{6} + \cdots - 1089 \)
|
| $47$ |
\( T^{7} - 30 T^{6} + \cdots - 52869 \)
|
| $53$ |
\( T^{7} + 10 T^{6} + \cdots - 177147 \)
|
| $59$ |
\( T^{7} - 20 T^{6} + \cdots - 1647 \)
|
| $61$ |
\( T^{7} + 7 T^{6} + \cdots - 49695 \)
|
| $67$ |
\( T^{7} - 7 T^{6} + \cdots - 4425 \)
|
| $71$ |
\( T^{7} + 11 T^{6} + \cdots - 19845 \)
|
| $73$ |
\( T^{7} - 6 T^{6} + \cdots + 12589 \)
|
| $79$ |
\( T^{7} - 14 T^{6} + \cdots + 288375 \)
|
| $83$ |
\( T^{7} + 17 T^{6} + \cdots + 3915 \)
|
| $89$ |
\( T^{7} - 40 T^{6} + \cdots + 5265 \)
|
| $97$ |
\( T^{7} + 22 T^{6} + \cdots - 134047 \)
|
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