Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,18,Mod(1,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.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: | \(82.4499393051\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 572070x^{4} - 41331730x^{3} + 64380256405x^{2} + 3992267176659x - 902152704489264 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{12}\cdot 5^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 572070x^{4} - 41331730x^{3} + 64380256405x^{2} + 3992267176659x - 902152704489264 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 110\nu - 190672 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 162\nu^{4} - 490692\nu^{3} + 26665778\nu^{2} + 37944944163\nu - 682661955888 ) / 184320 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 162\nu^{4} + 675012\nu^{3} - 62423858\nu^{2} - 98003035683\nu + 3649531603248 ) / 184320 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} + 3798\nu^{4} - 3278292\nu^{3} - 1870823942\nu^{2} + 295134983343\nu + 90992941916688 ) / 184320 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 110\beta _1 + 190672 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 194\beta_{2} + 347176\beta _1 + 20894070 \)
|
| \(\nu^{4}\) | \(=\) |
\( 40\beta_{5} + 179\beta_{4} - 21\beta_{3} + 469655\beta_{2} + 87111203\beta _1 + 66181353002 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6480\beta_{5} + 519690\beta_{4} + 671610\beta_{3} + 144612580\beta_{2} + 143590320935\beta _1 + 16572176915836 \)
|
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 |
|
−591.349 | 0 | 218621. | 390625. | 0 | −2.96765e6 | −5.17722e7 | 0 | −2.30996e8 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −234.567 | 0 | −76050.5 | 390625. | 0 | 1.02957e7 | 4.85840e7 | 0 | −9.16276e7 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 14.2492 | 0 | −130869. | 390625. | 0 | −1.46281e7 | −3.73246e6 | 0 | 5.56611e6 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 278.360 | 0 | −53587.8 | 390625. | 0 | 2.63677e7 | −5.14019e7 | 0 | 1.08734e8 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 505.682 | 0 | 124642. | 390625. | 0 | −2.08413e7 | −3.25143e6 | 0 | 1.97532e8 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 692.625 | 0 | 348657. | 390625. | 0 | 2.35624e7 | 1.50705e8 | 0 | 2.70556e8 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
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 | 45.18.a.h | yes | 6 |
| 3.b | odd | 2 | 1 | 45.18.a.g | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.18.a.g | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 45.18.a.h | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 665 T_{2}^{5} - 387810 T_{2}^{4} + 268101400 T_{2}^{3} + 10589372800 T_{2}^{2} + \cdots + 192700846080000 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 192700846080000 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 390625)^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 57\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots - 36\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots - 41\!\cdots\!04 \)
|
| $23$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 30\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 20\!\cdots\!64 \)
|
| $37$ |
\( T^{6} + \cdots - 53\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 27\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 61\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 16\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots + 58\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 83\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 10\!\cdots\!64 \)
|
| $83$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
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