Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1085,2,Mod(1,1085)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1085, 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("1085.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1085 = 5 \cdot 7 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1085.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: | \(8.66376861931\) |
| 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: |
\( x^{7} - 3x^{6} - 9x^{5} + 29x^{4} + 18x^{3} - 70x^{2} - 10x + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{7} - 3x^{6} - 9x^{5} + 29x^{4} + 18x^{3} - 70x^{2} - 10x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} + 14\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 16\nu^{2} + 4\nu - 20 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 9\nu^{3} + 16\nu^{2} + 14\nu - 20 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 16\nu^{3} + 18\nu^{2} - 20\nu - 12 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + 8\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{5} + 9\beta_{4} + 2\beta_{3} + 31\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{6} - 11\beta_{5} + 2\beta_{4} + 13\beta_{3} + 54\beta_{2} + 2\beta _1 + 138 \)
|
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 |
|
−2.53671 | 2.16264 | 4.43491 | −1.00000 | −5.48599 | 1.00000 | −6.17668 | 1.67701 | 2.53671 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.35041 | −0.875910 | −0.176382 | −1.00000 | 1.18284 | 1.00000 | 2.93902 | −2.23278 | 1.35041 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.10106 | 3.23919 | −0.787672 | −1.00000 | −3.56653 | 1.00000 | 3.06939 | 7.49233 | 1.10106 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.13413 | −2.48623 | −0.713738 | −1.00000 | −2.81973 | 1.00000 | −3.07775 | 3.18136 | −1.13413 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.68842 | 1.88203 | 0.850758 | −1.00000 | 3.17765 | 1.00000 | −1.94040 | 0.542036 | −1.68842 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.42431 | −2.45978 | 3.87727 | −1.00000 | −5.96327 | 1.00000 | 4.55109 | 3.05052 | −2.42431 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.74132 | 0.538070 | 5.51485 | −1.00000 | 1.47502 | 1.00000 | 9.63533 | −2.71048 | −2.74132 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(31\) | \( +1 \) |
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 | 1085.2.a.p | ✓ | 7 |
| 3.b | odd | 2 | 1 | 9765.2.a.bc | 7 | ||
| 5.b | even | 2 | 1 | 5425.2.a.ba | 7 | ||
| 7.b | odd | 2 | 1 | 7595.2.a.u | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1085.2.a.p | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 5425.2.a.ba | 7 | 5.b | even | 2 | 1 | ||
| 7595.2.a.u | 7 | 7.b | odd | 2 | 1 | ||
| 9765.2.a.bc | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1085))\):
|
\( T_{2}^{7} - 3T_{2}^{6} - 9T_{2}^{5} + 29T_{2}^{4} + 18T_{2}^{3} - 70T_{2}^{2} - 10T_{2} + 48 \)
|
|
\( T_{3}^{7} - 2T_{3}^{6} - 14T_{3}^{5} + 24T_{3}^{4} + 55T_{3}^{3} - 80T_{3}^{2} - 46T_{3} + 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 48 \)
|
| $3$ |
\( T^{7} - 2 T^{6} + \cdots + 38 \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( (T - 1)^{7} \)
|
| $11$ |
\( T^{7} - 5 T^{6} + \cdots + 2216 \)
|
| $13$ |
\( T^{7} - 9 T^{6} + \cdots + 32 \)
|
| $17$ |
\( T^{7} + 2 T^{6} + \cdots - 912 \)
|
| $19$ |
\( T^{7} - 13 T^{6} + \cdots - 608 \)
|
| $23$ |
\( T^{7} - 8 T^{6} + \cdots - 512 \)
|
| $29$ |
\( T^{7} + T^{6} + \cdots + 4208 \)
|
| $31$ |
\( (T + 1)^{7} \)
|
| $37$ |
\( T^{7} - 37 T^{6} + \cdots + 64 \)
|
| $41$ |
\( T^{7} - 9 T^{6} + \cdots + 144192 \)
|
| $43$ |
\( T^{7} - 19 T^{6} + \cdots - 245248 \)
|
| $47$ |
\( T^{7} + T^{6} + \cdots - 90592 \)
|
| $53$ |
\( T^{7} - 25 T^{6} + \cdots + 7716 \)
|
| $59$ |
\( T^{7} - 7 T^{6} + \cdots - 1336 \)
|
| $61$ |
\( T^{7} + 34 T^{6} + \cdots - 243632 \)
|
| $67$ |
\( T^{7} - 148 T^{5} + \cdots + 4672 \)
|
| $71$ |
\( T^{7} + 23 T^{6} + \cdots + 53632 \)
|
| $73$ |
\( T^{7} + 3 T^{6} + \cdots - 420864 \)
|
| $79$ |
\( T^{7} - 13 T^{6} + \cdots - 336828 \)
|
| $83$ |
\( T^{7} + 13 T^{6} + \cdots - 335976 \)
|
| $89$ |
\( T^{7} + 10 T^{6} + \cdots + 185856 \)
|
| $97$ |
\( T^{7} - 3 T^{6} + \cdots + 59832 \)
|
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