Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,3,Mod(181,210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.181");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.f (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: | \(5.72208555157\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.3317760000.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -10\nu^{7} + 49\nu^{5} - 133\nu^{3} + 801\nu ) / 189 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{7} + \nu^{5} + 5\nu^{3} - 63\nu ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{4} + 2\nu^{2} + 18 ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{6} + 14\nu^{4} - 56\nu^{2} + 225 ) / 63 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 22 ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} - 7\nu^{5} + 19\nu^{3} - 81\nu ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - \beta_{3} + \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - 2\beta_{5} + \beta_{4} + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 7\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{6} + \beta_{5} + 4\beta_{4} + 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -5\beta_{7} + 19\beta_{3} + 5\beta_{2} + 19\beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{6} + 22 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 29\beta_{7} + 13\beta_{3} + 29\beta_{2} - 13\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 181.1 |
|
−1.41421 | − | 1.73205i | 2.00000 | − | 2.23607i | 2.44949i | −6.70141 | − | 2.02265i | −2.82843 | −3.00000 | 3.16228i | ||||||||||||||||||||||||||||||||||||||
| 181.2 | −1.41421 | − | 1.73205i | 2.00000 | 2.23607i | 2.44949i | 1.04456 | + | 6.92163i | −2.82843 | −3.00000 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||
| 181.3 | −1.41421 | 1.73205i | 2.00000 | − | 2.23607i | − | 2.44949i | 1.04456 | − | 6.92163i | −2.82843 | −3.00000 | 3.16228i | |||||||||||||||||||||||||||||||||||||||
| 181.4 | −1.41421 | 1.73205i | 2.00000 | 2.23607i | − | 2.44949i | −6.70141 | + | 2.02265i | −2.82843 | −3.00000 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||
| 181.5 | 1.41421 | − | 1.73205i | 2.00000 | − | 2.23607i | − | 2.44949i | −1.04456 | − | 6.92163i | 2.82843 | −3.00000 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||
| 181.6 | 1.41421 | − | 1.73205i | 2.00000 | 2.23607i | − | 2.44949i | 6.70141 | + | 2.02265i | 2.82843 | −3.00000 | 3.16228i | |||||||||||||||||||||||||||||||||||||||
| 181.7 | 1.41421 | 1.73205i | 2.00000 | − | 2.23607i | 2.44949i | 6.70141 | − | 2.02265i | 2.82843 | −3.00000 | − | 3.16228i | |||||||||||||||||||||||||||||||||||||||
| 181.8 | 1.41421 | 1.73205i | 2.00000 | 2.23607i | 2.44949i | −1.04456 | + | 6.92163i | 2.82843 | −3.00000 | 3.16228i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 210.3.f.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 630.3.f.c | 8 | ||
| 4.b | odd | 2 | 1 | 1680.3.s.a | 8 | ||
| 5.b | even | 2 | 1 | 1050.3.f.b | 8 | ||
| 5.c | odd | 4 | 2 | 1050.3.h.b | 16 | ||
| 7.b | odd | 2 | 1 | inner | 210.3.f.a | ✓ | 8 |
| 21.c | even | 2 | 1 | 630.3.f.c | 8 | ||
| 28.d | even | 2 | 1 | 1680.3.s.a | 8 | ||
| 35.c | odd | 2 | 1 | 1050.3.f.b | 8 | ||
| 35.f | even | 4 | 2 | 1050.3.h.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.3.f.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 210.3.f.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 630.3.f.c | 8 | 3.b | odd | 2 | 1 | ||
| 630.3.f.c | 8 | 21.c | even | 2 | 1 | ||
| 1050.3.f.b | 8 | 5.b | even | 2 | 1 | ||
| 1050.3.f.b | 8 | 35.c | odd | 2 | 1 | ||
| 1050.3.h.b | 16 | 5.c | odd | 4 | 2 | ||
| 1050.3.h.b | 16 | 35.f | even | 4 | 2 | ||
| 1680.3.s.a | 8 | 4.b | odd | 2 | 1 | ||
| 1680.3.s.a | 8 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{4} \)
|
| $3$ |
\( (T^{2} + 3)^{4} \)
|
| $5$ |
\( (T^{2} + 5)^{4} \)
|
| $7$ |
\( T^{8} + 12 T^{6} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{4} + 8 T^{3} + \cdots + 964)^{2} \)
|
| $13$ |
\( T^{8} + 264 T^{6} + \cdots + 14714896 \)
|
| $17$ |
\( (T^{4} + 440 T^{2} + 19600)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 2869102096 \)
|
| $23$ |
\( (T^{4} - 1336 T^{2} + \cdots + 54544)^{2} \)
|
| $29$ |
\( (T^{4} + 72 T^{3} + \cdots - 281456)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 79120688656 \)
|
| $37$ |
\( (T^{4} + 24 T^{3} + \cdots - 883184)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 7126061056 \)
|
| $43$ |
\( (T^{4} + 32 T^{3} + \cdots + 2654464)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 1081300500736 \)
|
| $53$ |
\( (T^{4} - 64 T^{3} + \cdots + 3722116)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 9016903929856 \)
|
| $61$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
| $67$ |
\( (T^{4} + 96 T^{3} + \cdots - 7371504)^{2} \)
|
| $71$ |
\( (T^{4} - 88 T^{3} + \cdots + 25706884)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 483086161936 \)
|
| $79$ |
\( (T^{4} + 144 T^{3} + \cdots - 2641664)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 177346111242496 \)
|
| $89$ |
\( T^{8} + \cdots + 86\!\cdots\!96 \)
|
| $97$ |
\( T^{8} + \cdots + 76019496215056 \)
|
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