Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2805,2,Mod(1,2805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, 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("2805.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2805 = 3 \cdot 5 \cdot 11 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2805.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: | \(22.3980377670\) |
| 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: |
\( x^{7} - 3x^{6} - 10x^{5} + 34x^{4} + 14x^{3} - 85x^{2} + 21x + 32 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{7} - 3x^{6} - 10x^{5} + 34x^{4} + 14x^{3} - 85x^{2} + 21x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 11\nu^{4} + \nu^{3} + 28\nu^{2} - 3\nu - 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 11\nu^{4} + 2\nu^{3} + 28\nu^{2} - 9\nu - 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 11\nu^{4} + 11\nu^{3} + 27\nu^{2} - 22\nu - 11 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 12\nu^{4} + 11\nu^{3} + 36\nu^{2} - 22\nu - 23 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 6\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 9\beta_{2} + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} + 10\beta_{4} - 9\beta_{3} - \beta_{2} + 41\beta _1 - 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( -11\beta_{6} + 11\beta_{5} - \beta_{4} + 2\beta_{3} + 71\beta_{2} - 3\beta _1 + 166 \)
|
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 |
|
−2.81208 | −1.00000 | 5.90780 | −1.00000 | 2.81208 | −4.11303 | −10.9891 | 1.00000 | 2.81208 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.72384 | −1.00000 | 0.971632 | −1.00000 | 1.72384 | 4.57975 | 1.77274 | 1.00000 | 1.72384 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.509058 | −1.00000 | −1.74086 | −1.00000 | 0.509058 | −2.14767 | 1.90432 | 1.00000 | 0.509058 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.15534 | −1.00000 | −0.665196 | −1.00000 | −1.15534 | −0.159722 | −3.07920 | 1.00000 | −1.15534 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.61209 | −1.00000 | 0.598835 | −1.00000 | −1.61209 | −3.10331 | −2.25880 | 1.00000 | −1.61209 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.61143 | −1.00000 | 4.81958 | −1.00000 | −2.61143 | −4.35442 | 7.36313 | 1.00000 | −2.61143 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.66612 | −1.00000 | 5.10821 | −1.00000 | −2.66612 | 3.29840 | 8.28687 | 1.00000 | −2.66612 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(11\) | \(1\) |
| \(17\) | \(-1\) |
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 | 2805.2.a.r | ✓ | 7 |
| 3.b | odd | 2 | 1 | 8415.2.a.bl | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2805.2.a.r | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 8415.2.a.bl | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2805))\):
|
\( T_{2}^{7} - 3T_{2}^{6} - 10T_{2}^{5} + 34T_{2}^{4} + 14T_{2}^{3} - 85T_{2}^{2} + 21T_{2} + 32 \)
|
|
\( T_{7}^{7} + 6T_{7}^{6} - 23T_{7}^{5} - 190T_{7}^{4} - 53T_{7}^{3} + 1329T_{7}^{2} + 2016T_{7} + 288 \)
|
|
\( T_{19}^{7} - 25T_{19}^{6} + 194T_{19}^{5} + 64T_{19}^{4} - 8897T_{19}^{3} + 50058T_{19}^{2} - 112320T_{19} + 88512 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 32 \)
|
| $3$ |
\( (T + 1)^{7} \)
|
| $5$ |
\( (T + 1)^{7} \)
|
| $7$ |
\( T^{7} + 6 T^{6} + \cdots + 288 \)
|
| $11$ |
\( (T + 1)^{7} \)
|
| $13$ |
\( T^{7} + 10 T^{6} + \cdots + 2184 \)
|
| $17$ |
\( (T - 1)^{7} \)
|
| $19$ |
\( T^{7} - 25 T^{6} + \cdots + 88512 \)
|
| $23$ |
\( T^{7} - 2 T^{6} + \cdots + 49152 \)
|
| $29$ |
\( T^{7} + 7 T^{6} + \cdots - 238996 \)
|
| $31$ |
\( T^{7} + 3 T^{6} + \cdots - 15232 \)
|
| $37$ |
\( T^{7} + 5 T^{6} + \cdots + 136256 \)
|
| $41$ |
\( T^{7} + 19 T^{6} + \cdots - 3276 \)
|
| $43$ |
\( T^{7} - 24 T^{6} + \cdots + 27392 \)
|
| $47$ |
\( T^{7} + 14 T^{6} + \cdots + 48 \)
|
| $53$ |
\( T^{7} - 17 T^{6} + \cdots - 18916 \)
|
| $59$ |
\( T^{7} - T^{6} + \cdots + 2497632 \)
|
| $61$ |
\( T^{7} - 25 T^{6} + \cdots + 26152 \)
|
| $67$ |
\( T^{7} - 17 T^{6} + \cdots - 1737264 \)
|
| $71$ |
\( T^{7} + T^{6} + \cdots + 5825536 \)
|
| $73$ |
\( T^{7} + 17 T^{6} + \cdots - 10668 \)
|
| $79$ |
\( T^{7} - 14 T^{6} + \cdots - 1550592 \)
|
| $83$ |
\( T^{7} + 15 T^{6} + \cdots - 28608 \)
|
| $89$ |
\( T^{7} + 5 T^{6} + \cdots + 48916 \)
|
| $97$ |
\( T^{7} + 21 T^{6} + \cdots + 128352 \)
|
| show more | |
| show less |