Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,2,Mod(13,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.m (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: | \(1.67685844245\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.1698758656.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 18\nu^{5} + 28\nu^{4} + 89\nu^{3} + 74\nu^{2} + 104\nu - 16 ) / 64 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} + 18\nu^{5} - 28\nu^{4} + 89\nu^{3} - 74\nu^{2} + 104\nu + 16 ) / 64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 18\nu^{5} - 8\nu^{4} + 105\nu^{3} - 72\nu^{2} + 248\nu - 64 ) / 64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 6\nu^{6} + 10\nu^{5} - 92\nu^{4} - 15\nu^{3} - 358\nu^{2} - 120\nu - 336 ) / 64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} + \beta_{4} + 3\beta_{3} - \beta_{2} - 3\beta _1 - 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} - 5\beta_{4} - 13\beta_{3} - 5\beta_{2} - 13\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{7} + 9\beta_{6} - 17\beta_{4} - 27\beta_{3} + 17\beta_{2} + 27\beta _1 + 74 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} + 37\beta_{4} + 149\beta_{3} + 37\beta_{2} + 149\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 89\beta_{7} - 89\beta_{6} + 201\beta_{4} + 235\beta_{3} - 201\beta_{2} - 235\beta _1 - 650 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} - 325\beta_{4} - 1565\beta_{3} - 325\beta_{2} - 1565\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | − | 1.00000i | −1.52773 | + | 1.63280i | − | 1.00000i | 2.33991 | + | 1.23483i | 0.707107 | + | 0.707107i | − | 1.00000i | −0.0743018 | − | 2.23483i | |||||||||||||||||||||||||||
| 13.2 | −0.707107 | + | 0.707107i | −0.707107 | + | 0.707107i | − | 1.00000i | 2.23483 | + | 0.0743018i | − | 1.00000i | 0.781409 | − | 2.52773i | 0.707107 | + | 0.707107i | − | 1.00000i | −1.63280 | + | 1.52773i | ||||||||||||||||||||||||||||
| 13.3 | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | − | 1.00000i | −1.19663 | − | 1.88893i | − | 1.00000i | −2.59604 | − | 0.510472i | −0.707107 | − | 0.707107i | − | 1.00000i | −2.18183 | − | 0.489528i | ||||||||||||||||||||||||||||
| 13.4 | 0.707107 | − | 0.707107i | 0.707107 | − | 0.707107i | − | 1.00000i | 0.489528 | + | 2.18183i | − | 1.00000i | 1.47472 | − | 2.19663i | −0.707107 | − | 0.707107i | − | 1.00000i | 1.88893 | + | 1.19663i | ||||||||||||||||||||||||||||
| 97.1 | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | 1.00000i | −1.52773 | − | 1.63280i | 1.00000i | 2.33991 | − | 1.23483i | 0.707107 | − | 0.707107i | 1.00000i | −0.0743018 | + | 2.23483i | |||||||||||||||||||||||||||||||
| 97.2 | −0.707107 | − | 0.707107i | −0.707107 | − | 0.707107i | 1.00000i | 2.23483 | − | 0.0743018i | 1.00000i | 0.781409 | + | 2.52773i | 0.707107 | − | 0.707107i | 1.00000i | −1.63280 | − | 1.52773i | |||||||||||||||||||||||||||||||
| 97.3 | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | 1.00000i | −1.19663 | + | 1.88893i | 1.00000i | −2.59604 | + | 0.510472i | −0.707107 | + | 0.707107i | 1.00000i | −2.18183 | + | 0.489528i | |||||||||||||||||||||||||||||||
| 97.4 | 0.707107 | + | 0.707107i | 0.707107 | + | 0.707107i | 1.00000i | 0.489528 | − | 2.18183i | 1.00000i | 1.47472 | + | 2.19663i | −0.707107 | + | 0.707107i | 1.00000i | 1.88893 | − | 1.19663i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 210.2.m.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 630.2.p.b | 8 | ||
| 4.b | odd | 2 | 1 | 1680.2.cz.b | 8 | ||
| 5.b | even | 2 | 1 | 1050.2.m.b | 8 | ||
| 5.c | odd | 4 | 1 | 210.2.m.b | yes | 8 | |
| 5.c | odd | 4 | 1 | 1050.2.m.a | 8 | ||
| 7.b | odd | 2 | 1 | 210.2.m.b | yes | 8 | |
| 15.e | even | 4 | 1 | 630.2.p.c | 8 | ||
| 20.e | even | 4 | 1 | 1680.2.cz.a | 8 | ||
| 21.c | even | 2 | 1 | 630.2.p.c | 8 | ||
| 28.d | even | 2 | 1 | 1680.2.cz.a | 8 | ||
| 35.c | odd | 2 | 1 | 1050.2.m.a | 8 | ||
| 35.f | even | 4 | 1 | inner | 210.2.m.a | ✓ | 8 |
| 35.f | even | 4 | 1 | 1050.2.m.b | 8 | ||
| 105.k | odd | 4 | 1 | 630.2.p.b | 8 | ||
| 140.j | odd | 4 | 1 | 1680.2.cz.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.2.m.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 210.2.m.a | ✓ | 8 | 35.f | even | 4 | 1 | inner |
| 210.2.m.b | yes | 8 | 5.c | odd | 4 | 1 | |
| 210.2.m.b | yes | 8 | 7.b | odd | 2 | 1 | |
| 630.2.p.b | 8 | 3.b | odd | 2 | 1 | ||
| 630.2.p.b | 8 | 105.k | odd | 4 | 1 | ||
| 630.2.p.c | 8 | 15.e | even | 4 | 1 | ||
| 630.2.p.c | 8 | 21.c | even | 2 | 1 | ||
| 1050.2.m.a | 8 | 5.c | odd | 4 | 1 | ||
| 1050.2.m.a | 8 | 35.c | odd | 2 | 1 | ||
| 1050.2.m.b | 8 | 5.b | even | 2 | 1 | ||
| 1050.2.m.b | 8 | 35.f | even | 4 | 1 | ||
| 1680.2.cz.a | 8 | 20.e | even | 4 | 1 | ||
| 1680.2.cz.a | 8 | 28.d | even | 2 | 1 | ||
| 1680.2.cz.b | 8 | 4.b | odd | 2 | 1 | ||
| 1680.2.cz.b | 8 | 140.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{4} + 8T_{13}^{3} + 32T_{13}^{2} + 32T_{13} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 1)^{2} \)
|
| $3$ |
\( (T^{4} + 1)^{2} \)
|
| $5$ |
\( T^{8} + 2 T^{6} + \cdots + 625 \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{4} - 4 T^{3} - 32 T^{2} + \cdots - 4)^{2} \)
|
| $13$ |
\( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \)
|
| $17$ |
\( T^{8} - 12 T^{7} + \cdots + 40000 \)
|
| $19$ |
\( (T^{4} + 4 T^{3} - 14 T^{2} + \cdots + 32)^{2} \)
|
| $23$ |
\( T^{8} - 16 T^{7} + \cdots + 1024 \)
|
| $29$ |
\( T^{8} + 228 T^{6} + \cdots + 2560000 \)
|
| $31$ |
\( T^{8} + 100 T^{6} + \cdots + 4096 \)
|
| $37$ |
\( T^{8} + 28 T^{7} + \cdots + 61504 \)
|
| $41$ |
\( T^{8} + 152 T^{6} + \cdots + 1024 \)
|
| $43$ |
\( T^{8} - 32 T^{5} + \cdots + 59474944 \)
|
| $47$ |
\( T^{8} - 24 T^{7} + \cdots + 1024 \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots + 16384 \)
|
| $59$ |
\( (T^{4} - 4 T^{3} + \cdots - 496)^{2} \)
|
| $61$ |
\( T^{8} + 224 T^{6} + \cdots + 1364224 \)
|
| $67$ |
\( T^{8} - 64 T^{5} + \cdots + 16384 \)
|
| $71$ |
\( (T^{4} - 4 T^{3} + \cdots - 2336)^{2} \)
|
| $73$ |
\( T^{8} + 28 T^{7} + \cdots + 24364096 \)
|
| $79$ |
\( T^{8} + 360 T^{6} + \cdots + 9634816 \)
|
| $83$ |
\( T^{8} + 16 T^{7} + \cdots + 1024 \)
|
| $89$ |
\( (T^{4} + 32 T^{3} + \cdots - 7312)^{2} \)
|
| $97$ |
\( T^{8} + 28 T^{7} + \cdots + 64 \)
|
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