Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,4,Mod(31,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.e (of order \(3\), 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: | \(5.31017190052\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.41783472.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 28x^{4} - 51x^{3} + 196x^{2} - 171x + 183 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - 3x^{5} + 28x^{4} - 51x^{3} + 196x^{2} - 171x + 183 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 26\nu^{3} + 34\nu^{2} + 10\nu - 38 ) / 55 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} - 10\nu^{4} - 91\nu^{3} - 266\nu^{2} + 35\nu - 958 ) / 110 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{5} + 5\nu^{4} + 117\nu^{3} + 67\nu^{2} + 285\nu - 489 ) / 110 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 13\nu^{3} - 18\nu^{2} - 25\nu - 74 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 15\nu^{4} + 43\nu^{3} - 207\nu^{2} + 235\nu - 271 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{4} - 2\beta_{3} + \beta _1 - 23 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + 7\beta_{4} - \beta_{3} - 13\beta_{2} + 4\beta _1 - 36 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 28\beta_{4} + 28\beta_{3} - 12\beta_{2} + 7\beta _1 + 232 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -13\beta_{5} - 60\beta_{4} + 49\beta_{3} + 144\beta_{2} - 95\beta _1 + 610 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
1.00000 | + | 1.73205i | −5.03546 | + | 1.28223i | −2.00000 | + | 3.46410i | −2.50000 | + | 4.33013i | −7.25635 | − | 7.43945i | −14.5482 | − | 25.1981i | −8.00000 | 23.7118 | − | 12.9132i | −10.0000 | ||||||||||||||||||||||
| 31.2 | 1.00000 | + | 1.73205i | −2.91431 | − | 4.30195i | −2.00000 | + | 3.46410i | −2.50000 | + | 4.33013i | 4.53690 | − | 9.34968i | 11.1595 | + | 19.3288i | −8.00000 | −10.0136 | + | 25.0744i | −10.0000 | |||||||||||||||||||||||
| 31.3 | 1.00000 | + | 1.73205i | 3.44977 | + | 3.88575i | −2.00000 | + | 3.46410i | −2.50000 | + | 4.33013i | −3.28055 | + | 9.86093i | 1.88867 | + | 3.27127i | −8.00000 | −3.19816 | + | 26.8099i | −10.0000 | |||||||||||||||||||||||
| 61.1 | 1.00000 | − | 1.73205i | −5.03546 | − | 1.28223i | −2.00000 | − | 3.46410i | −2.50000 | − | 4.33013i | −7.25635 | + | 7.43945i | −14.5482 | + | 25.1981i | −8.00000 | 23.7118 | + | 12.9132i | −10.0000 | |||||||||||||||||||||||
| 61.2 | 1.00000 | − | 1.73205i | −2.91431 | + | 4.30195i | −2.00000 | − | 3.46410i | −2.50000 | − | 4.33013i | 4.53690 | + | 9.34968i | 11.1595 | − | 19.3288i | −8.00000 | −10.0136 | − | 25.0744i | −10.0000 | |||||||||||||||||||||||
| 61.3 | 1.00000 | − | 1.73205i | 3.44977 | − | 3.88575i | −2.00000 | − | 3.46410i | −2.50000 | − | 4.33013i | −3.28055 | − | 9.86093i | 1.88867 | − | 3.27127i | −8.00000 | −3.19816 | − | 26.8099i | −10.0000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 90.4.e.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 270.4.e.d | 6 | ||
| 9.c | even | 3 | 1 | inner | 90.4.e.d | ✓ | 6 |
| 9.c | even | 3 | 1 | 810.4.a.q | 3 | ||
| 9.d | odd | 6 | 1 | 270.4.e.d | 6 | ||
| 9.d | odd | 6 | 1 | 810.4.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 90.4.e.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 90.4.e.d | ✓ | 6 | 9.c | even | 3 | 1 | inner |
| 270.4.e.d | 6 | 3.b | odd | 2 | 1 | ||
| 270.4.e.d | 6 | 9.d | odd | 6 | 1 | ||
| 810.4.a.q | 3 | 9.c | even | 3 | 1 | ||
| 810.4.a.r | 3 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 3T_{7}^{5} + 684T_{7}^{4} - 6931T_{7}^{3} + 448266T_{7}^{2} - 1655775T_{7} + 6017209 \)
acting on \(S_{4}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} + 9 T^{5} + \cdots + 19683 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots + 6017209 \)
|
| $11$ |
\( T^{6} - 18 T^{5} + \cdots + 291600 \)
|
| $13$ |
\( T^{6} + 36 T^{5} + \cdots + 128232976 \)
|
| $17$ |
\( (T^{3} + 66 T^{2} + \cdots - 67716)^{2} \)
|
| $19$ |
\( (T^{3} + 84 T^{2} + \cdots - 473444)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 4146716025 \)
|
| $29$ |
\( T^{6} + \cdots + 140760641619081 \)
|
| $31$ |
\( T^{6} + \cdots + 51604970995600 \)
|
| $37$ |
\( (T^{3} - 30 T^{2} + \cdots + 16733596)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 118453162537161 \)
|
| $43$ |
\( T^{6} + \cdots + 19203922243984 \)
|
| $47$ |
\( T^{6} + \cdots + 695562504443361 \)
|
| $53$ |
\( (T^{3} + 354 T^{2} + \cdots + 903528)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 18\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 290032105848241 \)
|
| $67$ |
\( T^{6} + \cdots + 11\!\cdots\!81 \)
|
| $71$ |
\( (T^{3} + 1206 T^{2} + \cdots - 103343796)^{2} \)
|
| $73$ |
\( (T^{3} - 840 T^{2} + \cdots + 215989888)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 55\!\cdots\!24 \)
|
| $83$ |
\( T^{6} + \cdots + 40\!\cdots\!61 \)
|
| $89$ |
\( (T^{3} + 465 T^{2} + \cdots - 373653405)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 16\!\cdots\!04 \)
|
| show more | |
| show less |