Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,6,Mod(379,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.379");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 630.g (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: | \(101.041806482\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 90x^{3} + 2304x^{2} - 4320x + 4050 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 210) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 90x^{3} + 2304x^{2} - 4320x + 4050 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 256\nu^{5} + 240\nu^{4} + 225\nu^{3} - 11520\nu^{2} + 579024\nu - 563085 ) / 542835 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2501\nu^{5} + 1425\nu^{4} + 12645\nu^{3} + 293490\nu^{2} - 6369264\nu + 10804320 ) / 542835 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2501\nu^{5} + 1425\nu^{4} + 12645\nu^{3} + 293490\nu^{2} - 5283594\nu + 10804320 ) / 542835 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2651\nu^{5} - 6255\nu^{4} + 5445\nu^{3} + 300240\nu^{2} - 5283594\nu - 490050 ) / 542835 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3131\nu^{5} + 6705\nu^{4} + 17595\nu^{3} - 321840\nu^{2} + 6369264\nu - 1583550 ) / 542835 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{5} + 3\beta_{2} + 66\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 24\beta_{5} + 24\beta_{4} - 45\beta _1 + 45 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -45\beta_{5} - 189\beta_{4} + 189\beta_{3} - 45\beta_{2} - 3168 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -135\beta_{5} + 135\beta_{4} - 2439\beta_{3} + 2439\beta_{2} + 7290\beta _1 + 7290 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 379.1 |
|
− | 4.00000i | 0 | −16.0000 | −51.0816 | + | 22.7083i | 0 | 49.0000i | 64.0000i | 0 | 90.8331 | + | 204.327i | |||||||||||||||||||||||||||||||
| 379.2 | − | 4.00000i | 0 | −16.0000 | 14.0214 | + | 54.1147i | 0 | 49.0000i | 64.0000i | 0 | 216.459 | − | 56.0854i | ||||||||||||||||||||||||||||||||
| 379.3 | − | 4.00000i | 0 | −16.0000 | 42.0603 | − | 36.8230i | 0 | 49.0000i | 64.0000i | 0 | −147.292 | − | 168.241i | ||||||||||||||||||||||||||||||||
| 379.4 | 4.00000i | 0 | −16.0000 | −51.0816 | − | 22.7083i | 0 | − | 49.0000i | − | 64.0000i | 0 | 90.8331 | − | 204.327i | |||||||||||||||||||||||||||||||
| 379.5 | 4.00000i | 0 | −16.0000 | 14.0214 | − | 54.1147i | 0 | − | 49.0000i | − | 64.0000i | 0 | 216.459 | + | 56.0854i | |||||||||||||||||||||||||||||||
| 379.6 | 4.00000i | 0 | −16.0000 | 42.0603 | + | 36.8230i | 0 | − | 49.0000i | − | 64.0000i | 0 | −147.292 | + | 168.241i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.6.g.e | 6 | |
| 3.b | odd | 2 | 1 | 210.6.g.a | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 630.6.g.e | 6 | |
| 15.d | odd | 2 | 1 | 210.6.g.a | ✓ | 6 | |
| 15.e | even | 4 | 1 | 1050.6.a.bd | 3 | ||
| 15.e | even | 4 | 1 | 1050.6.a.bn | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.6.g.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 210.6.g.a | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 630.6.g.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 630.6.g.e | 6 | 5.b | even | 2 | 1 | inner | |
| 1050.6.a.bd | 3 | 15.e | even | 4 | 1 | ||
| 1050.6.a.bn | 3 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{3} - 482T_{11}^{2} - 53268T_{11} + 7077000 \)
acting on \(S_{6}^{\mathrm{new}}(630, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 30517578125 \)
|
| $7$ |
\( (T^{2} + 2401)^{3} \)
|
| $11$ |
\( (T^{3} - 482 T^{2} + \cdots + 7077000)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 209140882583616 \)
|
| $17$ |
\( T^{6} + \cdots + 12\!\cdots\!44 \)
|
| $19$ |
\( (T^{3} - 274 T^{2} + \cdots + 923937000)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + 5006 T^{2} + \cdots - 305956186200)^{2} \)
|
| $31$ |
\( (T^{3} + 5430 T^{2} + \cdots - 133698013144)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 91\!\cdots\!24 \)
|
| $41$ |
\( (T^{3} - 33074 T^{2} + \cdots - 780820395864)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 19\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 12\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 78\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + 11688 T^{2} + \cdots + 613716110080)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 1578764358984)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + \cdots + 65512922963400)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 93\!\cdots\!76 \)
|
| $79$ |
\( (T^{3} + \cdots + 442045792923840)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 34\!\cdots\!64 \)
|
| $89$ |
\( (T^{3} + \cdots - 53904325080760)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 45\!\cdots\!76 \)
|
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