Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2075,2,Mod(1,2075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2075, 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("2075.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2075 = 5^{2} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2075.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: | \(16.5689584194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.179711353.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 8x^{5} + 6x^{4} + 16x^{3} - 9x^{2} - 8x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 415) |
| 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
\( x^{7} - x^{6} - 8x^{5} + 6x^{4} + 16x^{3} - 9x^{2} - 8x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 6\nu^{4} + 18\nu^{3} + 8\nu^{2} - 17\nu - 2 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 10\nu^{4} - 6\nu^{3} + 24\nu^{2} + 7\nu - 10 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 10\nu^{4} - 10\nu^{3} + 28\nu^{2} + 19\nu - 18 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{6} - \nu^{5} - 22\nu^{4} + 2\nu^{3} + 32\nu^{2} + \nu - 6 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 16\nu^{2} - 9\nu - 6 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{6} - \nu^{5} - 26\nu^{4} + 2\nu^{3} + 56\nu^{2} + 5\nu - 22 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{6} + \beta_{5} + 2\beta_{3} + \beta_{2} + \beta _1 + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} - \beta_{5} + 3\beta_{2} + 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -14\beta_{6} + 5\beta_{5} + 2\beta_{4} + 12\beta_{3} + 7\beta_{2} + 7\beta _1 + 16 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -18\beta_{6} - 5\beta_{5} + 2\beta_{4} + 4\beta_{3} + 35\beta_{2} + 19\beta _1 + 4 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -43\beta_{6} + 13\beta_{5} + 9\beta_{4} + 34\beta_{3} + 24\beta_{2} + 22\beta _1 + 40 \)
|
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 |
|
−2.04504 | 2.67664 | 2.18221 | 0 | −5.47385 | 1.97797 | −0.372621 | 4.16440 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.24504 | 0.166440 | −0.449886 | 0 | −0.207224 | 2.60638 | 3.05020 | −2.97230 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.655504 | −0.817162 | −1.57032 | 0 | 0.535653 | 4.05109 | 2.34035 | −2.33225 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.561310 | −0.294778 | −1.68493 | 0 | −0.165462 | −2.56309 | −2.06839 | −2.91311 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.35377 | 2.92758 | −0.167302 | 0 | 3.96328 | −0.477354 | −2.93403 | 5.57073 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.37878 | −1.62161 | 3.65857 | 0 | −3.85743 | 0.159231 | 3.94537 | −0.370396 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.65173 | 1.96289 | 5.03166 | 0 | 5.20504 | 0.245775 | 8.03912 | 0.852920 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(83\) | \( -1 \) |
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 | 2075.2.a.h | 7 | |
| 5.b | even | 2 | 1 | 415.2.a.d | ✓ | 7 | |
| 15.d | odd | 2 | 1 | 3735.2.a.o | 7 | ||
| 20.d | odd | 2 | 1 | 6640.2.a.be | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 415.2.a.d | ✓ | 7 | 5.b | even | 2 | 1 | |
| 2075.2.a.h | 7 | 1.a | even | 1 | 1 | trivial | |
| 3735.2.a.o | 7 | 15.d | odd | 2 | 1 | ||
| 6640.2.a.be | 7 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2075))\):
|
\( T_{2}^{7} - 3T_{2}^{6} - 6T_{2}^{5} + 19T_{2}^{4} + 9T_{2}^{3} - 28T_{2}^{2} - 4T_{2} + 8 \)
|
|
\( T_{3}^{7} - 5T_{3}^{6} + T_{3}^{5} + 21T_{3}^{4} - 10T_{3}^{3} - 23T_{3}^{2} - 2T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 8 \)
|
| $3$ |
\( T^{7} - 5 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 6 T^{6} + \cdots - 1 \)
|
| $11$ |
\( T^{7} + 2 T^{6} + \cdots - 508 \)
|
| $13$ |
\( T^{7} - 5 T^{6} + \cdots + 256 \)
|
| $17$ |
\( T^{7} - 25 T^{6} + \cdots + 4729 \)
|
| $19$ |
\( T^{7} - 6 T^{6} + \cdots + 47648 \)
|
| $23$ |
\( T^{7} - 9 T^{6} + \cdots + 16 \)
|
| $29$ |
\( T^{7} + 2 T^{6} + \cdots + 205301 \)
|
| $31$ |
\( T^{7} - T^{6} + \cdots - 706421 \)
|
| $37$ |
\( T^{7} - 13 T^{6} + \cdots + 433 \)
|
| $41$ |
\( T^{7} + 15 T^{6} + \cdots - 51904 \)
|
| $43$ |
\( T^{7} - 5 T^{6} + \cdots + 16768 \)
|
| $47$ |
\( T^{7} - 12 T^{6} + \cdots - 512 \)
|
| $53$ |
\( T^{7} - 31 T^{6} + \cdots + 42592 \)
|
| $59$ |
\( T^{7} + 14 T^{6} + \cdots + 401671 \)
|
| $61$ |
\( T^{7} - 7 T^{6} + \cdots - 1496441 \)
|
| $67$ |
\( T^{7} + 10 T^{6} + \cdots - 3275872 \)
|
| $71$ |
\( T^{7} - 9 T^{6} + \cdots + 337312 \)
|
| $73$ |
\( T^{7} - 15 T^{6} + \cdots - 2865536 \)
|
| $79$ |
\( T^{7} - 6 T^{6} + \cdots - 49408 \)
|
| $83$ |
\( (T - 1)^{7} \)
|
| $89$ |
\( T^{7} + 37 T^{6} + \cdots - 2454656 \)
|
| $97$ |
\( T^{7} - 20 T^{6} + \cdots + 303104 \)
|
| show more | |
| show less |