Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [330,2,Mod(131,330)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(330, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("330.131");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 330.d (of order \(2\), degree \(1\), 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: | \(2.63506326670\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.2051727616.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 9\nu^{5} + 14\nu^{4} + 27\nu^{3} + 18\nu^{2} + 30\nu - 8 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 13\nu^{5} + 4\nu^{4} + 47\nu^{3} + 20\nu^{2} + 34\nu + 4 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} - 9\nu^{5} - 14\nu^{4} - 19\nu^{3} - 18\nu^{2} + 2\nu + 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 14\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 2\nu^{5} + 9\nu^{4} + 14\nu^{3} + 23\nu^{2} + 22\nu + 10 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 13\nu^{5} + 4\nu^{4} - 47\nu^{3} + 20\nu^{2} - 34\nu + 4 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{6} + 2\nu^{5} - 9\nu^{4} + 14\nu^{3} - 23\nu^{2} + 22\nu - 10 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{7} + 7\beta_{6} - 5\beta_{5} - 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 5\beta _1 + 28 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19\beta_{7} + 17\beta_{6} + 19\beta_{5} + 17\beta_{4} - 31\beta_{3} - 17\beta_{2} - 31\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -13\beta_{7} - 20\beta_{6} + 13\beta_{5} + 11\beta_{4} + 11\beta_{3} - 20\beta_{2} - 11\beta _1 - 67 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -93\beta_{7} - 75\beta_{6} - 93\beta_{5} - 67\beta_{4} + 155\beta_{3} + 75\beta_{2} + 155\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(211\) | \(221\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 131.1 |
|
1.00000 | −1.72809 | − | 0.117016i | 1.00000 | 1.00000i | −1.72809 | − | 0.117016i | 0.723074i | 1.00000 | 2.97261 | + | 0.404430i | 1.00000i | ||||||||||||||||||||||||||||||||||||
| 131.2 | 1.00000 | −1.72809 | + | 0.117016i | 1.00000 | − | 1.00000i | −1.72809 | + | 0.117016i | − | 0.723074i | 1.00000 | 2.97261 | − | 0.404430i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 131.3 | 1.00000 | 0.0828988 | − | 1.73007i | 1.00000 | 1.00000i | 0.0828988 | − | 1.73007i | − | 4.34658i | 1.00000 | −2.98626 | − | 0.286841i | 1.00000i | ||||||||||||||||||||||||||||||||||||
| 131.4 | 1.00000 | 0.0828988 | + | 1.73007i | 1.00000 | − | 1.00000i | 0.0828988 | + | 1.73007i | 4.34658i | 1.00000 | −2.98626 | + | 0.286841i | − | 1.00000i | |||||||||||||||||||||||||||||||||||
| 131.5 | 1.00000 | 1.25820 | − | 1.19035i | 1.00000 | 1.00000i | 1.25820 | − | 1.19035i | 3.22941i | 1.00000 | 0.166154 | − | 2.99540i | 1.00000i | |||||||||||||||||||||||||||||||||||||
| 131.6 | 1.00000 | 1.25820 | + | 1.19035i | 1.00000 | − | 1.00000i | 1.25820 | + | 1.19035i | − | 3.22941i | 1.00000 | 0.166154 | + | 2.99540i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 131.7 | 1.00000 | 1.38699 | − | 1.03743i | 1.00000 | − | 1.00000i | 1.38699 | − | 1.03743i | − | 0.394100i | 1.00000 | 0.847487 | − | 2.87781i | − | 1.00000i | ||||||||||||||||||||||||||||||||||
| 131.8 | 1.00000 | 1.38699 | + | 1.03743i | 1.00000 | 1.00000i | 1.38699 | + | 1.03743i | 0.394100i | 1.00000 | 0.847487 | + | 2.87781i | 1.00000i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 33.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 330.2.d.b | yes | 8 |
| 3.b | odd | 2 | 1 | 330.2.d.a | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 2640.2.f.a | 8 | ||
| 5.b | even | 2 | 1 | 1650.2.d.c | 8 | ||
| 5.c | odd | 4 | 1 | 1650.2.f.d | 8 | ||
| 5.c | odd | 4 | 1 | 1650.2.f.f | 8 | ||
| 11.b | odd | 2 | 1 | 330.2.d.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 2640.2.f.b | 8 | ||
| 15.d | odd | 2 | 1 | 1650.2.d.f | 8 | ||
| 15.e | even | 4 | 1 | 1650.2.f.c | 8 | ||
| 15.e | even | 4 | 1 | 1650.2.f.e | 8 | ||
| 33.d | even | 2 | 1 | inner | 330.2.d.b | yes | 8 |
| 44.c | even | 2 | 1 | 2640.2.f.b | 8 | ||
| 55.d | odd | 2 | 1 | 1650.2.d.f | 8 | ||
| 55.e | even | 4 | 1 | 1650.2.f.c | 8 | ||
| 55.e | even | 4 | 1 | 1650.2.f.e | 8 | ||
| 132.d | odd | 2 | 1 | 2640.2.f.a | 8 | ||
| 165.d | even | 2 | 1 | 1650.2.d.c | 8 | ||
| 165.l | odd | 4 | 1 | 1650.2.f.d | 8 | ||
| 165.l | odd | 4 | 1 | 1650.2.f.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 330.2.d.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 330.2.d.a | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 330.2.d.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 330.2.d.b | yes | 8 | 33.d | even | 2 | 1 | inner |
| 1650.2.d.c | 8 | 5.b | even | 2 | 1 | ||
| 1650.2.d.c | 8 | 165.d | even | 2 | 1 | ||
| 1650.2.d.f | 8 | 15.d | odd | 2 | 1 | ||
| 1650.2.d.f | 8 | 55.d | odd | 2 | 1 | ||
| 1650.2.f.c | 8 | 15.e | even | 4 | 1 | ||
| 1650.2.f.c | 8 | 55.e | even | 4 | 1 | ||
| 1650.2.f.d | 8 | 5.c | odd | 4 | 1 | ||
| 1650.2.f.d | 8 | 165.l | odd | 4 | 1 | ||
| 1650.2.f.e | 8 | 15.e | even | 4 | 1 | ||
| 1650.2.f.e | 8 | 55.e | even | 4 | 1 | ||
| 1650.2.f.f | 8 | 5.c | odd | 4 | 1 | ||
| 1650.2.f.f | 8 | 165.l | odd | 4 | 1 | ||
| 2640.2.f.a | 8 | 4.b | odd | 2 | 1 | ||
| 2640.2.f.a | 8 | 132.d | odd | 2 | 1 | ||
| 2640.2.f.b | 8 | 12.b | even | 2 | 1 | ||
| 2640.2.f.b | 8 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17}^{4} - 2T_{17}^{3} - 31T_{17}^{2} + 104T_{17} - 68 \)
acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( T^{8} + 30 T^{6} + \cdots + 16 \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots + 14641 \)
|
| $13$ |
\( T^{8} + 44 T^{6} + \cdots + 1024 \)
|
| $17$ |
\( (T^{4} - 2 T^{3} - 31 T^{2} + \cdots - 68)^{2} \)
|
| $19$ |
\( T^{8} + 98 T^{6} + \cdots + 73984 \)
|
| $23$ |
\( T^{8} + 104 T^{6} + \cdots + 25600 \)
|
| $29$ |
\( (T^{4} + 2 T^{3} + \cdots + 452)^{2} \)
|
| $31$ |
\( (T^{4} + 2 T^{3} + \cdots + 2000)^{2} \)
|
| $37$ |
\( (T^{4} + 6 T^{3} + \cdots - 1388)^{2} \)
|
| $41$ |
\( (T^{4} + 8 T^{3} - 12 T^{2} + \cdots + 64)^{2} \)
|
| $43$ |
\( T^{8} + 160 T^{6} + \cdots + 262144 \)
|
| $47$ |
\( T^{8} + 256 T^{6} + \cdots + 2715904 \)
|
| $53$ |
\( T^{8} + 58 T^{6} + \cdots + 2704 \)
|
| $59$ |
\( T^{8} + 188 T^{6} + \cdots + 640000 \)
|
| $61$ |
\( T^{8} + 310 T^{6} + \cdots + 712336 \)
|
| $67$ |
\( (T^{4} + 6 T^{3} + \cdots + 3824)^{2} \)
|
| $71$ |
\( T^{8} + 170 T^{6} + \cdots + 150544 \)
|
| $73$ |
\( T^{8} + 380 T^{6} + \cdots + 1183744 \)
|
| $79$ |
\( T^{8} + 316 T^{6} + \cdots + 173056 \)
|
| $83$ |
\( (T^{4} + 36 T^{3} + \cdots - 2944)^{2} \)
|
| $89$ |
\( T^{8} + 262 T^{6} + \cdots + 135424 \)
|
| $97$ |
\( (T^{4} + 4 T^{3} + \cdots + 11408)^{2} \)
|
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