Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,9,Mod(37,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.37");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.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: | \(36.6640749055\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{601})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 301x^{2} + 22500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 10) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 301x^{2} + 22500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 151\nu ) / 150 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 1125\nu^{2} + 1729\nu + 169350 ) / 75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{3} - 2250\nu^{2} + 3307\nu - 338700 ) / 150 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 15\beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - \beta _1 - 4516 ) / 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -151\beta_{3} - 151\beta_{2} + 6765\beta_1 ) / 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−8.00000 | − | 8.00000i | 0 | 128.000i | 33.6352 | + | 624.094i | 0 | 2718.55 | + | 2718.55i | 1024.00 | − | 1024.00i | 0 | 4723.67 | − | 5261.84i | ||||||||||||||||||||
| 37.2 | −8.00000 | − | 8.00000i | 0 | 128.000i | 401.365 | − | 479.094i | 0 | 144.447 | + | 144.447i | 1024.00 | − | 1024.00i | 0 | −7043.67 | + | 621.836i | |||||||||||||||||||||
| 73.1 | −8.00000 | + | 8.00000i | 0 | − | 128.000i | 33.6352 | − | 624.094i | 0 | 2718.55 | − | 2718.55i | 1024.00 | + | 1024.00i | 0 | 4723.67 | + | 5261.84i | ||||||||||||||||||||
| 73.2 | −8.00000 | + | 8.00000i | 0 | − | 128.000i | 401.365 | + | 479.094i | 0 | 144.447 | − | 144.447i | 1024.00 | + | 1024.00i | 0 | −7043.67 | − | 621.836i | ||||||||||||||||||||
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 | 90.9.g.a | 4 | |
| 3.b | odd | 2 | 1 | 10.9.c.b | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 90.9.g.a | 4 | |
| 12.b | even | 2 | 1 | 80.9.p.a | 4 | ||
| 15.d | odd | 2 | 1 | 50.9.c.b | 4 | ||
| 15.e | even | 4 | 1 | 10.9.c.b | ✓ | 4 | |
| 15.e | even | 4 | 1 | 50.9.c.b | 4 | ||
| 60.l | odd | 4 | 1 | 80.9.p.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.9.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 10.9.c.b | ✓ | 4 | 15.e | even | 4 | 1 | |
| 50.9.c.b | 4 | 15.d | odd | 2 | 1 | ||
| 50.9.c.b | 4 | 15.e | even | 4 | 1 | ||
| 80.9.p.a | 4 | 12.b | even | 2 | 1 | ||
| 80.9.p.a | 4 | 60.l | odd | 4 | 1 | ||
| 90.9.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 90.9.g.a | 4 | 5.c | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(90, [\chi])\):
|
\( T_{7}^{4} - 5726T_{7}^{3} + 16393538T_{7}^{2} - 4497040072T_{7} + 616809178384 \)
|
|
\( T_{11}^{2} - 7366T_{11} - 285147536 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 128)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 152587890625 \)
|
| $7$ |
\( T^{4} + \cdots + 616809178384 \)
|
| $11$ |
\( (T^{2} - 7366 T - 285147536)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 25\!\cdots\!84 \)
|
| $17$ |
\( T^{4} + \cdots + 966589219240324 \)
|
| $19$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 24\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots + 30\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots + 1247722520344)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 14\!\cdots\!24 \)
|
| $41$ |
\( (T^{2} + \cdots + 6327500249104)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!44 \)
|
| $47$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!24 \)
|
| $59$ |
\( T^{4} + \cdots + 60\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots + 48503902803984)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 87\!\cdots\!44 \)
|
| $71$ |
\( (T^{2} + \cdots - 151499857447496)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 22\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 50\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 21\!\cdots\!04 \)
|
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