Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,28,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, 28, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.1");
S:= CuspForms(chi, 28);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| 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: | \(207.835008677\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 19275662x^{2} - 30468026939x + 4134032404260 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{5}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 19275662x^{2} - 30468026939x + 4134032404260 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -14\nu^{3} + 380152\nu^{2} - 82619548\nu - 3344017320395 ) / 181745885 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 99966\nu^{3} - 221933208\nu^{2} - 1396595469348\nu - 145370351920796 ) / 36349177 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -193626\nu^{3} + 584199468\nu^{2} + 1993231669968\nu - 1205864013932855 ) / 181745885 \)
|
| \(\nu\) | \(=\) |
\( ( -8\beta_{3} - 3\beta_{2} + 3537\beta _1 + 1771 ) / 25920 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6376\beta_{3} - 2013\beta_{2} + 16314399\beta _1 + 249820738901 ) / 25920 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -62960456\beta_{3} - 18478119\beta_{2} + 42816883149\beta _1 + 296170652529823 ) / 12960 \)
|
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 |
|
−19372.0 | 0 | 2.41055e8 | −1.22070e9 | 0 | −3.32780e11 | −2.06966e12 | 0 | 2.36474e13 | ||||||||||||||||||||||||||||||
| 1.2 | 1651.55 | 0 | −1.31490e8 | −1.22070e9 | 0 | 4.20350e11 | −4.38829e11 | 0 | −2.01605e12 | |||||||||||||||||||||||||||||||
| 1.3 | 7959.76 | 0 | −7.08599e7 | −1.22070e9 | 0 | −2.73788e10 | −1.63237e12 | 0 | −9.71651e12 | |||||||||||||||||||||||||||||||
| 1.4 | 21310.7 | 0 | 3.19926e8 | −1.22070e9 | 0 | −2.75207e11 | 3.95757e12 | 0 | −2.60140e13 | |||||||||||||||||||||||||||||||
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 | 45.28.a.b | 4 | |
| 3.b | odd | 2 | 1 | 5.28.a.a | ✓ | 4 | |
| 15.d | odd | 2 | 1 | 25.28.a.b | 4 | ||
| 15.e | even | 4 | 2 | 25.28.b.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.28.a.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 25.28.a.b | 4 | 15.d | odd | 2 | 1 | ||
| 25.28.b.b | 8 | 15.e | even | 4 | 2 | ||
| 45.28.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 11550T_{2}^{3} - 381050112T_{2}^{2} + 3942345932800T_{2} - 5427025739710464 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots - 54\!\cdots\!64 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T + 1220703125)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 10\!\cdots\!04 \)
|
| $11$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $13$ |
\( T^{4} + \cdots - 38\!\cdots\!44 \)
|
| $17$ |
\( T^{4} + \cdots - 19\!\cdots\!84 \)
|
| $19$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 60\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 49\!\cdots\!64 \)
|
| $37$ |
\( T^{4} + \cdots - 33\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots - 86\!\cdots\!24 \)
|
| $43$ |
\( T^{4} + \cdots + 27\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots - 82\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 94\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots - 20\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots - 13\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots - 13\!\cdots\!64 \)
|
| $79$ |
\( T^{4} + \cdots - 19\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 11\!\cdots\!84 \)
|
| $89$ |
\( T^{4} + \cdots + 90\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 56\!\cdots\!24 \)
|
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