Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.379");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(37.1712033036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.43197465600.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 16x^{3} + 1521x^{2} - 624x + 128 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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
\( x^{6} - 16x^{3} + 1521x^{2} - 624x + 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 31\nu^{5} - 1513\nu^{4} + 1209\nu^{3} - 248\nu^{2} - 1500719 ) / 177765 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 86\nu^{5} + 1537\nu^{4} + 3354\nu^{3} - 688\nu^{2} - 177765\nu + 1427711 ) / 177765 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1521\nu^{5} + 312\nu^{4} + 64\nu^{3} - 12168\nu^{2} + 2310945\nu - 475064 ) / 474040 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7293\nu^{5} + 1496\nu^{4} - 11848\nu^{3} - 153152\nu^{2} + 10891069\nu - 2180632 ) / 284424 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 38337\nu^{5} + 7864\nu^{4} + 13768\nu^{3} - 780736\nu^{2} + 58721705\nu - 12071288 ) / 1422120 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - 2\beta_{2} - 2\beta_1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -12\beta_{5} - 3\beta_{4} + 125\beta_{3} - \beta_{2} - \beta_1 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 78\beta_{5} - 78\beta_{4} - 40\beta_{3} + 39\beta_{2} + 39\beta _1 + 40 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 31\beta_{5} - 31\beta_{4} + 93\beta_{2} - 492\beta _1 - 4875 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( -325\beta_{5} + 301\beta_{4} + 512\beta_{3} + 602\beta_{2} + 626\beta _1 + 512 \)
|
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 |
|
− | 2.00000i | 0 | −4.00000 | −11.1034 | + | 1.30950i | 0 | − | 7.00000i | 8.00000i | 0 | 2.61901 | + | 22.2068i | ||||||||||||||||||||||||||||||
| 379.2 | − | 2.00000i | 0 | −4.00000 | 8.27790 | − | 7.51508i | 0 | − | 7.00000i | 8.00000i | 0 | −15.0302 | − | 16.5558i | |||||||||||||||||||||||||||||||
| 379.3 | − | 2.00000i | 0 | −4.00000 | 10.8255 | − | 2.79443i | 0 | − | 7.00000i | 8.00000i | 0 | −5.58885 | − | 21.6510i | |||||||||||||||||||||||||||||||
| 379.4 | 2.00000i | 0 | −4.00000 | −11.1034 | − | 1.30950i | 0 | 7.00000i | − | 8.00000i | 0 | 2.61901 | − | 22.2068i | ||||||||||||||||||||||||||||||||
| 379.5 | 2.00000i | 0 | −4.00000 | 8.27790 | + | 7.51508i | 0 | 7.00000i | − | 8.00000i | 0 | −15.0302 | + | 16.5558i | ||||||||||||||||||||||||||||||||
| 379.6 | 2.00000i | 0 | −4.00000 | 10.8255 | + | 2.79443i | 0 | 7.00000i | − | 8.00000i | 0 | −5.58885 | + | 21.6510i | ||||||||||||||||||||||||||||||||
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 | 630.4.g.j | 6 | |
| 3.b | odd | 2 | 1 | 70.4.c.b | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 630.4.g.j | 6 | |
| 12.b | even | 2 | 1 | 560.4.g.e | 6 | ||
| 15.d | odd | 2 | 1 | 70.4.c.b | ✓ | 6 | |
| 15.e | even | 4 | 1 | 350.4.a.w | 3 | ||
| 15.e | even | 4 | 1 | 350.4.a.x | 3 | ||
| 21.c | even | 2 | 1 | 490.4.c.c | 6 | ||
| 60.h | even | 2 | 1 | 560.4.g.e | 6 | ||
| 105.g | even | 2 | 1 | 490.4.c.c | 6 | ||
| 105.k | odd | 4 | 1 | 2450.4.a.cf | 3 | ||
| 105.k | odd | 4 | 1 | 2450.4.a.cg | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.c.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 70.4.c.b | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 350.4.a.w | 3 | 15.e | even | 4 | 1 | ||
| 350.4.a.x | 3 | 15.e | even | 4 | 1 | ||
| 490.4.c.c | 6 | 21.c | even | 2 | 1 | ||
| 490.4.c.c | 6 | 105.g | even | 2 | 1 | ||
| 560.4.g.e | 6 | 12.b | even | 2 | 1 | ||
| 560.4.g.e | 6 | 60.h | even | 2 | 1 | ||
| 630.4.g.j | 6 | 1.a | even | 1 | 1 | trivial | |
| 630.4.g.j | 6 | 5.b | even | 2 | 1 | inner | |
| 2450.4.a.cf | 3 | 105.k | odd | 4 | 1 | ||
| 2450.4.a.cg | 3 | 105.k | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{3} + 19T_{11}^{2} - 1560T_{11} + 600 \)
acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 16 T^{5} + \cdots + 1953125 \)
|
| $7$ |
\( (T^{2} + 49)^{3} \)
|
| $11$ |
\( (T^{3} + 19 T^{2} + \cdots + 600)^{2} \)
|
| $13$ |
\( T^{6} + 4077 T^{4} + \cdots + 829555204 \)
|
| $17$ |
\( T^{6} + 12729 T^{4} + \cdots + 69288976 \)
|
| $19$ |
\( (T^{3} + 156 T^{2} + \cdots - 1196840)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 2480625000000 \)
|
| $29$ |
\( (T^{3} - 91 T^{2} + \cdots - 204564)^{2} \)
|
| $31$ |
\( (T^{3} - 170 T^{2} + \cdots + 2033232)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 16972878400 \)
|
| $41$ |
\( (T^{3} - 6 T^{2} + \cdots + 7243272)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 125041775755264 \)
|
| $47$ |
\( T^{6} + \cdots + 169390641480256 \)
|
| $53$ |
\( T^{6} + \cdots + 17\!\cdots\!56 \)
|
| $59$ |
\( (T^{3} - 590 T^{2} + \cdots + 88782692)^{2} \)
|
| $61$ |
\( (T^{3} - 352 T^{2} + \cdots - 3430824)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 48\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + 724 T^{2} + \cdots - 55149600)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 46\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} + 1037 T^{2} + \cdots - 276622388)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 83\!\cdots\!36 \)
|
| $89$ |
\( (T^{3} + 1048 T^{2} + \cdots - 210826480)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 48\!\cdots\!00 \)
|
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