Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6045,2,Mod(1,6045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, 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("6045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6045.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: | \(48.2695680219\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{9} - 3x^{8} - 7x^{7} + 22x^{6} + 14x^{5} - 52x^{4} - 5x^{3} + 41x^{2} - 4x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 7x^{7} + 22x^{6} + 14x^{5} - 52x^{4} - 5x^{3} + 41x^{2} - 4x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 7\nu^{2} + 8\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 7\nu^{3} + 9\nu^{2} - 4\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{8} - \nu^{7} - 10\nu^{6} + 4\nu^{5} + 29\nu^{4} - 2\nu^{3} - 26\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{8} - 2\nu^{7} - 8\nu^{6} + 11\nu^{5} + 20\nu^{4} - 16\nu^{3} - 17\nu^{2} + 4\nu + 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{8} - \nu^{7} - 10\nu^{6} + 4\nu^{5} + 30\nu^{4} - 4\nu^{3} - 31\nu^{2} + \nu + 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( -2\nu^{8} + 3\nu^{7} + 19\nu^{6} - 17\nu^{5} - 56\nu^{4} + 28\nu^{3} + 54\nu^{2} - 11\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + 3\beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{8} + 3\beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{4} + 11\beta_{2} + 10\beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10\beta_{8} + 12\beta_{7} + 10\beta_{6} - 2\beta_{5} - 10\beta_{4} + \beta_{3} + 33\beta_{2} + 35\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 25\beta_{8} + 35\beta_{7} + 25\beta_{6} - 10\beta_{5} - 24\beta_{4} + 2\beta_{3} + 102\beta_{2} + 90\beta _1 + 25 \)
|
| \(\nu^{7}\) | \(=\) |
\( 88\beta_{8} + 113\beta_{7} + 87\beta_{6} - 24\beta_{5} - 86\beta_{4} + 11\beta_{3} + 303\beta_{2} + 282\beta _1 + 28 \)
|
| \(\nu^{8}\) | \(=\) |
\( 242 \beta_{8} + 330 \beta_{7} + 241 \beta_{6} - 86 \beta_{5} - 230 \beta_{4} + 27 \beta_{3} + 904 \beta_{2} + \cdots + 145 \)
|
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 |
|
−2.82576 | 1.00000 | 5.98491 | 1.00000 | −2.82576 | −2.33173 | −11.2604 | 1.00000 | −2.82576 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.63414 | 1.00000 | 4.93869 | 1.00000 | −2.63414 | −0.293212 | −7.74092 | 1.00000 | −2.63414 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.12407 | 1.00000 | 2.51167 | 1.00000 | −2.12407 | 2.73139 | −1.08683 | 1.00000 | −2.12407 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.31693 | 1.00000 | −0.265687 | 1.00000 | −1.31693 | 1.83326 | 2.98376 | 1.00000 | −1.31693 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.506159 | 1.00000 | −1.74380 | 1.00000 | −0.506159 | −4.78785 | 1.89496 | 1.00000 | −0.506159 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.290796 | 1.00000 | −1.91544 | 1.00000 | 0.290796 | 1.24204 | −1.13859 | 1.00000 | 0.290796 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.505549 | 1.00000 | −1.74442 | 1.00000 | 0.505549 | −2.69242 | −1.89299 | 1.00000 | 0.505549 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.662643 | 1.00000 | −1.56090 | 1.00000 | 0.662643 | −4.22758 | −2.35961 | 1.00000 | 0.662643 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.94807 | 1.00000 | 1.79498 | 1.00000 | 1.94807 | −3.47390 | −0.399392 | 1.00000 | 1.94807 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(-1\) |
| \(31\) | \(-1\) |
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 | 6045.2.a.t | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6045.2.a.t | ✓ | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):
|
\( T_{2}^{9} + 6T_{2}^{8} + 5T_{2}^{7} - 27T_{2}^{6} - 43T_{2}^{5} + 19T_{2}^{4} + 38T_{2}^{3} - 11T_{2}^{2} - 7T_{2} + 2 \)
|
|
\( T_{7}^{9} + 12T_{7}^{8} + 33T_{7}^{7} - 103T_{7}^{6} - 545T_{7}^{5} - 36T_{7}^{4} + 2162T_{7}^{3} + 1127T_{7}^{2} - 2598T_{7} - 805 \)
|
|
\( T_{11}^{9} + 10 T_{11}^{8} - 17 T_{11}^{7} - 458 T_{11}^{6} - 1034 T_{11}^{5} + 4032 T_{11}^{4} + \cdots - 6854 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 6 T^{8} + \cdots + 2 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( (T - 1)^{9} \)
|
| $7$ |
\( T^{9} + 12 T^{8} + \cdots - 805 \)
|
| $11$ |
\( T^{9} + 10 T^{8} + \cdots - 6854 \)
|
| $13$ |
\( (T - 1)^{9} \)
|
| $17$ |
\( T^{9} + 17 T^{8} + \cdots - 662 \)
|
| $19$ |
\( T^{9} - 6 T^{8} + \cdots - 2264 \)
|
| $23$ |
\( T^{9} + 23 T^{8} + \cdots + 256246 \)
|
| $29$ |
\( T^{9} + 4 T^{8} + \cdots + 8986513 \)
|
| $31$ |
\( (T - 1)^{9} \)
|
| $37$ |
\( T^{9} - 11 T^{8} + \cdots - 615122 \)
|
| $41$ |
\( T^{9} + 8 T^{8} + \cdots - 3601211 \)
|
| $43$ |
\( T^{9} + 15 T^{8} + \cdots - 837797 \)
|
| $47$ |
\( T^{9} + 33 T^{8} + \cdots - 8468608 \)
|
| $53$ |
\( T^{9} + 38 T^{8} + \cdots + 559426 \)
|
| $59$ |
\( T^{9} + 25 T^{8} + \cdots - 5554801 \)
|
| $61$ |
\( T^{9} + 10 T^{8} + \cdots + 74390614 \)
|
| $67$ |
\( T^{9} + 19 T^{8} + \cdots - 57119801 \)
|
| $71$ |
\( T^{9} + 43 T^{8} + \cdots + 206680 \)
|
| $73$ |
\( T^{9} - T^{8} + \cdots - 20454206 \)
|
| $79$ |
\( T^{9} + 9 T^{8} + \cdots - 124186 \)
|
| $83$ |
\( T^{9} + 12 T^{8} + \cdots + 27123697 \)
|
| $89$ |
\( T^{9} + 5 T^{8} + \cdots - 2629930 \)
|
| $97$ |
\( T^{9} + 4 T^{8} + \cdots - 21047473 \)
|
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