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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 105406182x^{3} - 8285617904x^{2} + 1593173725628800x - 1939393055148057600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{16}\cdot 3^{9}\cdot 5^{4}\cdot 7 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 105406182x^{3} - 8285617904x^{2} + 1593173725628800x - 1939393055148057600 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 482\nu - 168649702 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 387\nu^{4} + 128122\nu^{3} - 37592785714\nu^{2} - 31082496429040\nu + 357705740702875488 ) / 608394416 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -119\nu^{4} - 488562\nu^{3} + 12666282090\nu^{2} + 43191416556112\nu - 154378913370408864 ) / 260740464 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 241\beta _1 + 168649943 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -1161\beta_{4} - 833\beta_{3} + 1232\beta_{2} + 75177471\beta _1 + 10210752443 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 384366\beta_{4} + 3419934\beta_{3} + 48161619\beta_{2} + 67133176145\beta _1 + 6339333237738587 ) / 2 \)
|
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 |
|
−22161.7 | 0 | 3.56923e8 | 1.22070e9 | 0 | −8.77786e10 | −4.93553e12 | 0 | −2.70528e13 | |||||||||||||||||||||||||||||||||
| 1.2 | −14024.4 | 0 | 6.24671e7 | 1.22070e9 | 0 | 3.73002e11 | 1.00626e12 | 0 | −1.71197e13 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1165.24 | 0 | −1.32860e8 | 1.22070e9 | 0 | −4.04373e11 | 3.11211e11 | 0 | −1.42242e12 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2338.06 | 0 | −1.28751e8 | 1.22070e9 | 0 | −1.25721e10 | −6.14836e11 | 0 | 2.85407e12 | ||||||||||||||||||||||||||||||||||
| 1.5 | 15097.3 | 0 | 9.37113e7 | 1.22070e9 | 0 | 2.87369e11 | −6.11538e11 | 0 | 1.84293e13 | ||||||||||||||||||||||||||||||||||
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.d | 5 | |
| 3.b | odd | 2 | 1 | 5.28.a.b | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 25.28.a.c | 5 | ||
| 15.e | even | 4 | 2 | 25.28.b.c | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.28.a.b | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 25.28.a.c | 5 | 15.d | odd | 2 | 1 | ||
| 25.28.b.c | 10 | 15.e | even | 4 | 2 | ||
| 45.28.a.d | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 19916T_{2}^{4} - 262965912T_{2}^{3} - 4473573947392T_{2}^{2} + 6144845117710336T_{2} + 12783803780642635776 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots + 12\!\cdots\!76 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 1220703125)^{5} \)
|
| $7$ |
\( T^{5} + \cdots + 47\!\cdots\!24 \)
|
| $11$ |
\( T^{5} + \cdots - 40\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots - 71\!\cdots\!32 \)
|
| $17$ |
\( T^{5} + \cdots + 63\!\cdots\!76 \)
|
| $19$ |
\( T^{5} + \cdots - 26\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 33\!\cdots\!68 \)
|
| $29$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 56\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots + 10\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots - 10\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 14\!\cdots\!32 \)
|
| $47$ |
\( T^{5} + \cdots + 49\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots - 18\!\cdots\!68 \)
|
| $59$ |
\( T^{5} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 27\!\cdots\!68 \)
|
| $67$ |
\( T^{5} + \cdots - 71\!\cdots\!76 \)
|
| $71$ |
\( T^{5} + \cdots + 66\!\cdots\!32 \)
|
| $73$ |
\( T^{5} + \cdots + 20\!\cdots\!68 \)
|
| $79$ |
\( T^{5} + \cdots - 36\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 20\!\cdots\!32 \)
|
| $89$ |
\( T^{5} + \cdots - 35\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 29\!\cdots\!76 \)
|
| show more | |
| show less |