Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7600,2,Mod(1,7600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, 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("7600.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7600.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: | \(60.6863055362\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.253565184.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 760) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 11\nu^{3} - 16\nu^{2} - 26\nu + 12 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 11\nu^{3} + 20\nu^{2} + 22\nu - 28 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 11\nu^{3} + 9\nu^{2} + 24\nu - 10 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 4\nu^{4} + 37\nu^{3} - 38\nu^{2} - 98\nu + 48 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 7\beta_{3} + 11\beta_{2} + 9\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{5} + 15\beta_{4} + 9\beta_{3} + 2\beta_{2} + 53\beta_1 \)
|
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 | −2.84742 | 0 | 0 | 0 | −0.145034 | 0 | 5.10782 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.59277 | 0 | 0 | 0 | 1.66290 | 0 | −0.463073 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.366738 | 0 | 0 | 0 | −3.08675 | 0 | −2.86550 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.664406 | 0 | 0 | 0 | −0.345799 | 0 | −2.55856 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.43031 | 0 | 0 | 0 | 3.60737 | 0 | 2.90640 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.97875 | 0 | 0 | 0 | 4.30732 | 0 | 5.87292 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 7600.2.a.cn | 6 | |
| 4.b | odd | 2 | 1 | 3800.2.a.z | 6 | ||
| 5.b | even | 2 | 1 | 7600.2.a.cg | 6 | ||
| 5.c | odd | 4 | 2 | 1520.2.d.k | 12 | ||
| 20.d | odd | 2 | 1 | 3800.2.a.be | 6 | ||
| 20.e | even | 4 | 2 | 760.2.d.e | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 760.2.d.e | ✓ | 12 | 20.e | even | 4 | 2 | |
| 1520.2.d.k | 12 | 5.c | odd | 4 | 2 | ||
| 3800.2.a.z | 6 | 4.b | odd | 2 | 1 | ||
| 3800.2.a.be | 6 | 20.d | odd | 2 | 1 | ||
| 7600.2.a.cg | 6 | 5.b | even | 2 | 1 | ||
| 7600.2.a.cn | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7600))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 11T_{3}^{4} + 20T_{3}^{3} + 22T_{3}^{2} - 32T_{3} + 8 \)
|
|
\( T_{7}^{6} - 6T_{7}^{5} - 4T_{7}^{4} + 62T_{7}^{3} - 49T_{7}^{2} - 36T_{7} - 4 \)
|
|
\( T_{11}^{6} - 2T_{11}^{5} - 53T_{11}^{4} + 76T_{11}^{3} + 762T_{11}^{2} - 456T_{11} - 2584 \)
|
|
\( T_{13}^{6} + 14T_{13}^{5} + 27T_{13}^{4} - 352T_{13}^{3} - 1270T_{13}^{2} + 2080T_{13} + 9376 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 8 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 6 T^{5} + \cdots - 4 \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots - 2584 \)
|
| $13$ |
\( T^{6} + 14 T^{5} + \cdots + 9376 \)
|
| $17$ |
\( T^{6} + 10 T^{5} + \cdots + 15864 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} - 2 T^{5} + \cdots + 272 \)
|
| $29$ |
\( T^{6} + 2 T^{5} + \cdots + 10064 \)
|
| $31$ |
\( T^{6} + 8 T^{5} + \cdots - 2304 \)
|
| $37$ |
\( T^{6} + 4 T^{5} + \cdots - 12352 \)
|
| $41$ |
\( T^{6} - 4 T^{5} + \cdots - 4832 \)
|
| $43$ |
\( T^{6} - 4 T^{5} + \cdots - 4632 \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots - 1152 \)
|
| $53$ |
\( T^{6} - 14 T^{5} + \cdots + 2752 \)
|
| $59$ |
\( T^{6} - 2 T^{5} + \cdots - 374496 \)
|
| $61$ |
\( T^{6} - 10 T^{5} + \cdots + 40256 \)
|
| $67$ |
\( T^{6} - 42 T^{5} + \cdots + 2752 \)
|
| $71$ |
\( T^{6} - 8 T^{5} + \cdots - 64 \)
|
| $73$ |
\( T^{6} - 2 T^{5} + \cdots + 3208 \)
|
| $79$ |
\( T^{6} - 20 T^{5} + \cdots + 1024 \)
|
| $83$ |
\( T^{6} - 16 T^{5} + \cdots + 6208 \)
|
| $89$ |
\( T^{6} - 32 T^{5} + \cdots + 69504 \)
|
| $97$ |
\( T^{6} + 40 T^{5} + \cdots - 196992 \)
|
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