Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [45,9,Mod(28,45)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(45, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("45.28");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.g (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(18.3320374528\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} - 30x^{3} + 1089x^{2} - 3168x + 4608 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{6} - 2x^{5} + 2x^{4} - 30x^{3} + 1089x^{2} - 3168x + 4608 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 39\nu^{5} - 22\nu^{4} - 10\nu^{3} + 790\nu^{2} + 41631\nu - 62928 ) / 66000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -273\nu^{5} + 154\nu^{4} + 70\nu^{3} - 5530\nu^{2} + 1028583\nu - 21504 ) / 66000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 761\nu^{5} - 2178\nu^{4} - 4990\nu^{3} - 45790\nu^{2} + 838569\nu - 2347872 ) / 66000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -77\nu^{5} + 146\nu^{4} - 3570\nu^{3} + 2030\nu^{2} - 83733\nu + 244704 ) / 6000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -921\nu^{5} - 1342\nu^{4} - 610\nu^{3} + 48190\nu^{2} - 823209\nu + 187392 ) / 66000 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 7\beta _1 + 7 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 10\beta_{5} + \beta_{4} - 10\beta_{3} - \beta_{2} + 446\beta_1 ) / 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -31\beta_{4} - 20\beta_{3} - 283\beta _1 + 283 ) / 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -35\beta_{5} + 5\beta_{4} - 35\beta_{3} + 5\beta_{2} - 1348 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -400\beta_{5} - 1019\beta_{2} + 17267\beta _1 + 17267 ) / 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−15.2610 | + | 15.2610i | 0 | − | 209.796i | −558.542 | − | 280.457i | 0 | −2415.21 | + | 2415.21i | −705.116 | − | 705.116i | 0 | 12804.0 | − | 4243.86i | |||||||||||||||||||||||||
| 28.2 | 4.39608 | − | 4.39608i | 0 | 217.349i | 14.1685 | + | 624.839i | 0 | 730.992 | − | 730.992i | 2080.88 | + | 2080.88i | 0 | 2809.13 | + | 2684.56i | |||||||||||||||||||||||||||
| 28.3 | 11.8649 | − | 11.8649i | 0 | − | 25.5528i | 434.373 | − | 449.383i | 0 | 508.219 | − | 508.219i | 2734.24 | + | 2734.24i | 0 | −178.084 | − | 10485.7i | ||||||||||||||||||||||||||
| 37.1 | −15.2610 | − | 15.2610i | 0 | 209.796i | −558.542 | + | 280.457i | 0 | −2415.21 | − | 2415.21i | −705.116 | + | 705.116i | 0 | 12804.0 | + | 4243.86i | |||||||||||||||||||||||||||
| 37.2 | 4.39608 | + | 4.39608i | 0 | − | 217.349i | 14.1685 | − | 624.839i | 0 | 730.992 | + | 730.992i | 2080.88 | − | 2080.88i | 0 | 2809.13 | − | 2684.56i | ||||||||||||||||||||||||||
| 37.3 | 11.8649 | + | 11.8649i | 0 | 25.5528i | 434.373 | + | 449.383i | 0 | 508.219 | + | 508.219i | 2734.24 | − | 2734.24i | 0 | −178.084 | + | 10485.7i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 45.9.g.a | 6 | |
| 3.b | odd | 2 | 1 | 5.9.c.a | ✓ | 6 | |
| 5.c | odd | 4 | 1 | inner | 45.9.g.a | 6 | |
| 12.b | even | 2 | 1 | 80.9.p.c | 6 | ||
| 15.d | odd | 2 | 1 | 25.9.c.b | 6 | ||
| 15.e | even | 4 | 1 | 5.9.c.a | ✓ | 6 | |
| 15.e | even | 4 | 1 | 25.9.c.b | 6 | ||
| 60.l | odd | 4 | 1 | 80.9.p.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.9.c.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 5.9.c.a | ✓ | 6 | 15.e | even | 4 | 1 | |
| 25.9.c.b | 6 | 15.d | odd | 2 | 1 | ||
| 25.9.c.b | 6 | 15.e | even | 4 | 1 | ||
| 45.9.g.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 45.9.g.a | 6 | 5.c | odd | 4 | 1 | inner | |
| 80.9.p.c | 6 | 12.b | even | 2 | 1 | ||
| 80.9.p.c | 6 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 2T_{2}^{5} + 2T_{2}^{4} - 2400T_{2}^{3} + 153664T_{2}^{2} - 1248128T_{2} + 5068928 \)
acting on \(S_{9}^{\mathrm{new}}(45, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots + 5068928 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 59\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 64\!\cdots\!08 \)
|
| $11$ |
\( (T^{3} + 11596 T^{2} + \cdots + 135134348768)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 22\!\cdots\!88 \)
|
| $17$ |
\( T^{6} + \cdots + 37\!\cdots\!68 \)
|
| $19$ |
\( T^{6} + \cdots + 26\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 45\!\cdots\!28 \)
|
| $29$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 53\!\cdots\!88)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 10\!\cdots\!88 \)
|
| $41$ |
\( (T^{3} + \cdots - 25\!\cdots\!52)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 92\!\cdots\!08 \)
|
| $47$ |
\( T^{6} + \cdots + 13\!\cdots\!48 \)
|
| $53$ |
\( T^{6} + \cdots + 80\!\cdots\!48 \)
|
| $59$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 48\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 18\!\cdots\!68 \)
|
| $71$ |
\( (T^{3} + \cdots + 75\!\cdots\!28)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 24\!\cdots\!28 \)
|
| $79$ |
\( T^{6} + \cdots + 38\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 87\!\cdots\!68 \)
|
| $89$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 40\!\cdots\!48 \)
|
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