Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9200,2,Mod(1,9200)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9200.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9200.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: | \(73.4623698596\) |
| 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: |
\( x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 920) |
| 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
\( x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 11\nu^{4} - 3\nu^{3} + 26\nu^{2} + 7\nu - 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 15\nu^{3} + 16\nu^{2} - 21\nu - 2 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 10\nu^{3} + 6\nu^{2} + 20\nu - 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{6} - 8\nu^{5} - 23\nu^{4} + 63\nu^{3} + 20\nu^{2} - 91\nu + 22 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\nu^{6} + 4\nu^{5} + 17\nu^{4} - 30\nu^{3} - 22\nu^{2} + 42\nu - 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2\beta_{2} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{6} + 10\beta_{5} - 4\beta_{3} + 10\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{6} + 14\beta_{5} - 9\beta_{4} - 14\beta_{3} + 24\beta_{2} + 40\beta _1 - 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 76\beta_{6} + 87\beta_{5} - 3\beta_{4} - 47\beta_{3} + 92\beta_{2} + 11\beta _1 + 90 \)
|
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 | −2.49931 | 0 | 0 | 0 | 2.92777 | 0 | 3.24657 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.40334 | 0 | 0 | 0 | 1.57117 | 0 | −1.03063 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.189375 | 0 | 0 | 0 | −1.65661 | 0 | −2.96414 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.356372 | 0 | 0 | 0 | −2.46376 | 0 | −2.87300 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.58319 | 0 | 0 | 0 | 2.84620 | 0 | −0.493499 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.78665 | 0 | 0 | 0 | 1.75578 | 0 | 0.192116 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.98707 | 0 | 0 | 0 | −0.980560 | 0 | 5.92257 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(23\) | \( +1 \) |
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 | 9200.2.a.dc | 7 | |
| 4.b | odd | 2 | 1 | 4600.2.a.bh | 7 | ||
| 5.b | even | 2 | 1 | 9200.2.a.cz | 7 | ||
| 5.c | odd | 4 | 2 | 1840.2.e.g | 14 | ||
| 20.d | odd | 2 | 1 | 4600.2.a.bi | 7 | ||
| 20.e | even | 4 | 2 | 920.2.e.b | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 920.2.e.b | ✓ | 14 | 20.e | even | 4 | 2 | |
| 1840.2.e.g | 14 | 5.c | odd | 4 | 2 | ||
| 4600.2.a.bh | 7 | 4.b | odd | 2 | 1 | ||
| 4600.2.a.bi | 7 | 20.d | odd | 2 | 1 | ||
| 9200.2.a.cz | 7 | 5.b | even | 2 | 1 | ||
| 9200.2.a.dc | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):
|
\( T_{3}^{7} - 3T_{3}^{6} - 7T_{3}^{5} + 24T_{3}^{4} + T_{3}^{3} - 35T_{3}^{2} + 17T_{3} - 2 \)
|
|
\( T_{7}^{7} - 4T_{7}^{6} - 8T_{7}^{5} + 41T_{7}^{4} + 10T_{7}^{3} - 116T_{7}^{2} + 12T_{7} + 92 \)
|
|
\( T_{11}^{7} - 7T_{11}^{6} - 27T_{11}^{5} + 198T_{11}^{4} + 248T_{11}^{3} - 1224T_{11}^{2} - 1780T_{11} - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 3 T^{6} + \cdots - 2 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 4 T^{6} + \cdots + 92 \)
|
| $11$ |
\( T^{7} - 7 T^{6} + \cdots - 128 \)
|
| $13$ |
\( T^{7} - 7 T^{6} + \cdots + 2 \)
|
| $17$ |
\( T^{7} - 58 T^{5} + \cdots - 52 \)
|
| $19$ |
\( T^{7} - 7 T^{6} + \cdots - 1936 \)
|
| $23$ |
\( (T + 1)^{7} \)
|
| $29$ |
\( T^{7} + 11 T^{6} + \cdots + 9244 \)
|
| $31$ |
\( T^{7} - 10 T^{6} + \cdots + 225251 \)
|
| $37$ |
\( T^{7} - 19 T^{6} + \cdots - 42832 \)
|
| $41$ |
\( T^{7} + 16 T^{6} + \cdots + 110153 \)
|
| $43$ |
\( T^{7} - 6 T^{6} + \cdots + 64 \)
|
| $47$ |
\( T^{7} - 6 T^{6} + \cdots + 5752 \)
|
| $53$ |
\( T^{7} - 15 T^{6} + \cdots - 156352 \)
|
| $59$ |
\( T^{7} - 11 T^{6} + \cdots - 486592 \)
|
| $61$ |
\( T^{7} - 5 T^{6} + \cdots + 2669336 \)
|
| $67$ |
\( T^{7} - 9 T^{6} + \cdots + 462832 \)
|
| $71$ |
\( T^{7} - 14 T^{6} + \cdots - 632317 \)
|
| $73$ |
\( T^{7} - 10 T^{6} + \cdots - 16 \)
|
| $79$ |
\( T^{7} - 32 T^{6} + \cdots - 473312 \)
|
| $83$ |
\( T^{7} - T^{6} + \cdots - 11984 \)
|
| $89$ |
\( T^{7} + 24 T^{6} + \cdots + 311456 \)
|
| $97$ |
\( T^{7} + 7 T^{6} + \cdots + 4038856 \)
|
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