Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [210,3,Mod(43,210)]
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, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("210.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 210.l (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: | \(5.72208555157\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.12745506816.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 23x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 23x^{4} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 19\nu ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 24\nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 24\nu^{3} + 5\nu ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} - 24\nu^{3} + 24\nu ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{6} - \nu^{4} - 67\nu^{2} - 9 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + \nu^{4} - 67\nu^{2} + 9 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -4\nu^{7} - 91\nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + 6\beta_{2} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 2\beta_{4} + 2\beta_{3} + 2\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{6} - 5\beta_{5} - 18 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -19\beta_{4} - 19\beta_{3} + 29\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{6} - 12\beta_{5} - 67\beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -48\beta_{7} + 91\beta_{4} - 91\beta_{3} - 91\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(71\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
1.00000 | − | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | −4.32032 | + | 2.51691i | −2.44949 | −1.87083 | + | 1.87083i | −2.00000 | − | 2.00000i | 3.00000i | −1.80341 | + | 6.83723i | |||||||||||||||||||||||||||||
| 43.2 | 1.00000 | − | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | −0.578661 | − | 4.96640i | −2.44949 | 1.87083 | − | 1.87083i | −2.00000 | − | 2.00000i | 3.00000i | −5.54506 | − | 4.38774i | ||||||||||||||||||||||||||||||
| 43.3 | 1.00000 | − | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0.578661 | + | 4.96640i | 2.44949 | −1.87083 | + | 1.87083i | −2.00000 | − | 2.00000i | 3.00000i | 5.54506 | + | 4.38774i | ||||||||||||||||||||||||||||||
| 43.4 | 1.00000 | − | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 4.32032 | − | 2.51691i | 2.44949 | 1.87083 | − | 1.87083i | −2.00000 | − | 2.00000i | 3.00000i | 1.80341 | − | 6.83723i | ||||||||||||||||||||||||||||||
| 127.1 | 1.00000 | + | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | −4.32032 | − | 2.51691i | −2.44949 | −1.87083 | − | 1.87083i | −2.00000 | + | 2.00000i | − | 3.00000i | −1.80341 | − | 6.83723i | ||||||||||||||||||||||||||||||
| 127.2 | 1.00000 | + | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | −0.578661 | + | 4.96640i | −2.44949 | 1.87083 | + | 1.87083i | −2.00000 | + | 2.00000i | − | 3.00000i | −5.54506 | + | 4.38774i | ||||||||||||||||||||||||||||||
| 127.3 | 1.00000 | + | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0.578661 | − | 4.96640i | 2.44949 | −1.87083 | − | 1.87083i | −2.00000 | + | 2.00000i | − | 3.00000i | 5.54506 | − | 4.38774i | ||||||||||||||||||||||||||||||
| 127.4 | 1.00000 | + | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 4.32032 | + | 2.51691i | 2.44949 | 1.87083 | + | 1.87083i | −2.00000 | + | 2.00000i | − | 3.00000i | 1.80341 | + | 6.83723i | ||||||||||||||||||||||||||||||
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 | 210.3.l.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 630.3.o.b | 8 | ||
| 5.b | even | 2 | 1 | 1050.3.l.b | 8 | ||
| 5.c | odd | 4 | 1 | inner | 210.3.l.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 1050.3.l.b | 8 | ||
| 15.e | even | 4 | 1 | 630.3.o.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.3.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 210.3.l.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 630.3.o.b | 8 | 3.b | odd | 2 | 1 | ||
| 630.3.o.b | 8 | 15.e | even | 4 | 1 | ||
| 1050.3.l.b | 8 | 5.b | even | 2 | 1 | ||
| 1050.3.l.b | 8 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{4} + 4T_{11}^{3} - 364T_{11}^{2} - 2416T_{11} + 10000 \)
acting on \(S_{3}^{\mathrm{new}}(210, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $3$ |
\( (T^{4} + 9)^{2} \)
|
| $5$ |
\( T^{8} + 24 T^{6} + \cdots + 390625 \)
|
| $7$ |
\( (T^{4} + 49)^{2} \)
|
| $11$ |
\( (T^{4} + 4 T^{3} + \cdots + 10000)^{2} \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 473344 \)
|
| $17$ |
\( T^{8} + 32 T^{7} + \cdots + 565504 \)
|
| $19$ |
\( T^{8} + 1256 T^{6} + \cdots + 254083600 \)
|
| $23$ |
\( T^{8} + 40 T^{7} + \cdots + 6250000 \)
|
| $29$ |
\( T^{8} + \cdots + 1914937600 \)
|
| $31$ |
\( (T^{4} - 72 T^{3} + \cdots - 332604)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 2032050250000 \)
|
| $41$ |
\( (T^{4} + 160 T^{3} + \cdots - 9703676)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 1562460000256 \)
|
| $47$ |
\( T^{8} + \cdots + 94568550400 \)
|
| $53$ |
\( T^{8} + \cdots + 53935417600 \)
|
| $59$ |
\( T^{8} + \cdots + 107464902774784 \)
|
| $61$ |
\( (T^{4} - 144 T^{3} + \cdots + 4954000)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 653738987622400 \)
|
| $71$ |
\( (T^{4} + 140 T^{3} + \cdots + 478864)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 10\!\cdots\!44 \)
|
| $83$ |
\( T^{8} + \cdots + 471997524054016 \)
|
| $89$ |
\( T^{8} + \cdots + 3929513290000 \)
|
| $97$ |
\( T^{8} + \cdots + 17\!\cdots\!00 \)
|
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