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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 11x^{7} + 35x^{6} + 32x^{5} - 117x^{4} - 17x^{3} + 109x^{2} - 17x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 11x^{7} + 35x^{6} + 32x^{5} - 117x^{4} - 17x^{3} + 109x^{2} - 17x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 8\nu^{4} + 40\nu^{3} - 11\nu^{2} - 31\nu + 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - 13\nu^{6} - 2\nu^{5} + 48\nu^{4} + 13\nu^{3} - 42\nu^{2} - 9\nu - 2 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{8} + \nu^{7} + 25\nu^{6} - 6\nu^{5} - 88\nu^{4} - 2\nu^{3} + 73\nu^{2} + 7\nu + 2 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{8} - 3\nu^{7} - 64\nu^{6} + 24\nu^{5} + 238\nu^{4} - 39\nu^{3} - 237\nu^{2} + 30\nu + 12 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{8} + 4\nu^{7} + 63\nu^{6} - 36\nu^{5} - 228\nu^{4} + 79\nu^{3} + 212\nu^{2} - 63\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{4} + 7\beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} - \beta_{4} + 8\beta_{3} + 30\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11\beta_{8} + 10\beta_{7} - \beta_{6} + 3\beta_{5} - 13\beta_{4} + 47\beta_{2} + 12\beta _1 + 98 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3\beta_{8} + 2\beta_{7} + 11\beta_{6} + 27\beta_{5} - 15\beta_{4} + 56\beta_{3} + 2\beta_{2} + 195\beta _1 + 81 \)
|
| \(\nu^{8}\) | \(=\) |
\( 95\beta_{8} + 82\beta_{7} - 11\beta_{6} + 45\beta_{5} - 123\beta_{4} + 3\beta_{3} + 317\beta_{2} + 112\beta _1 + 641 \)
|
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 |
|
−2.56568 | 1.00000 | 4.58269 | 1.00000 | −2.56568 | −1.37486 | −6.62635 | 1.00000 | −2.56568 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.85591 | 1.00000 | 1.44439 | 1.00000 | −1.85591 | 4.18592 | 1.03116 | 1.00000 | −1.85591 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.23871 | 1.00000 | −0.465602 | 1.00000 | −1.23871 | −3.38053 | 3.05416 | 1.00000 | −1.23871 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.128778 | 1.00000 | −1.98342 | 1.00000 | −0.128778 | 2.86344 | 0.512975 | 1.00000 | −0.128778 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.317716 | 1.00000 | −1.89906 | 1.00000 | 0.317716 | −3.81738 | −1.23879 | 1.00000 | 0.317716 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.15888 | 1.00000 | −0.657004 | 1.00000 | 1.15888 | 1.64408 | −3.07914 | 1.00000 | 1.15888 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.26604 | 1.00000 | 3.13492 | 1.00000 | 2.26604 | 3.03256 | 2.57176 | 1.00000 | 2.26604 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.28892 | 1.00000 | 3.23914 | 1.00000 | 2.28892 | 2.05243 | 2.83628 | 1.00000 | 2.28892 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.75752 | 1.00000 | 5.60394 | 1.00000 | 2.75752 | −3.20566 | 9.93796 | 1.00000 | 2.75752 | ||||||||||||||||||||||||||||||||||||||||||||||
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.v | ✓ | 9 |
| 3.b | odd | 2 | 1 | 8415.2.a.bu | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2805.2.a.v | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 8415.2.a.bu | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2805))\):
|
\( T_{2}^{9} - 3T_{2}^{8} - 11T_{2}^{7} + 35T_{2}^{6} + 32T_{2}^{5} - 117T_{2}^{4} - 17T_{2}^{3} + 109T_{2}^{2} - 17T_{2} - 4 \)
|
|
\( T_{7}^{9} - 2T_{7}^{8} - 38T_{7}^{7} + 76T_{7}^{6} + 492T_{7}^{5} - 1015T_{7}^{4} - 2413T_{7}^{3} + 5247T_{7}^{2} + 2904T_{7} - 6976 \)
|
|
\( T_{19}^{9} - 5 T_{19}^{8} - 67 T_{19}^{7} + 193 T_{19}^{6} + 1261 T_{19}^{5} - 628 T_{19}^{4} + \cdots - 4448 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - 3 T^{8} + \cdots - 4 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( (T - 1)^{9} \)
|
| $7$ |
\( T^{9} - 2 T^{8} + \cdots - 6976 \)
|
| $11$ |
\( (T + 1)^{9} \)
|
| $13$ |
\( T^{9} + 6 T^{8} + \cdots - 2344 \)
|
| $17$ |
\( (T - 1)^{9} \)
|
| $19$ |
\( T^{9} - 5 T^{8} + \cdots - 4448 \)
|
| $23$ |
\( T^{9} - 14 T^{8} + \cdots + 640 \)
|
| $29$ |
\( T^{9} - 5 T^{8} + \cdots + 7664 \)
|
| $31$ |
\( T^{9} - 3 T^{8} + \cdots + 164096 \)
|
| $37$ |
\( T^{9} - T^{8} + \cdots - 1744 \)
|
| $41$ |
\( T^{9} - 11 T^{8} + \cdots + 800 \)
|
| $43$ |
\( T^{9} - 14 T^{8} + \cdots - 10240 \)
|
| $47$ |
\( T^{9} - 34 T^{8} + \cdots - 1571840 \)
|
| $53$ |
\( T^{9} - 23 T^{8} + \cdots - 1583008 \)
|
| $59$ |
\( T^{9} - 13 T^{8} + \cdots - 112576 \)
|
| $61$ |
\( T^{9} + 3 T^{8} + \cdots - 11475592 \)
|
| $67$ |
\( T^{9} - 17 T^{8} + \cdots - 5017264 \)
|
| $71$ |
\( T^{9} - 5 T^{8} + \cdots - 4096 \)
|
| $73$ |
\( T^{9} + 17 T^{8} + \cdots - 10394288 \)
|
| $79$ |
\( T^{9} - 2 T^{8} + \cdots + 19259392 \)
|
| $83$ |
\( T^{9} - 29 T^{8} + \cdots + 155601280 \)
|
| $89$ |
\( T^{9} - 25 T^{8} + \cdots + 366823568 \)
|
| $97$ |
\( T^{9} + 11 T^{8} + \cdots - 591472 \)
|
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