Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [120,6,Mod(49,120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("120.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.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: | \(19.2460583776\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.25787221056.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 61x^{4} + 852x^{2} + 576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 61x^{4} + 852x^{2} + 576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 85\nu^{3} + 1596\nu ) / 1296 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} - 28\nu^{4} - 379\nu^{3} - 1516\nu^{2} - 2748\nu - 11856 ) / 432 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} + 28\nu^{4} - 379\nu^{3} + 1516\nu^{2} - 2748\nu + 11856 ) / 432 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 65\nu^{5} - 108\nu^{4} + 3581\nu^{3} - 3996\nu^{2} + 40884\nu - 9072 ) / 1296 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -65\nu^{5} - 108\nu^{4} - 3581\nu^{3} - 3996\nu^{2} - 40884\nu - 9072 ) / 1296 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} - 4\beta_1 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{5} + 7\beta_{4} + 9\beta_{3} - 9\beta_{2} - 396 ) / 20 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 39\beta_{5} - 39\beta_{4} - 97\beta_{3} - 97\beta_{2} + 996\beta_1 ) / 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -379\beta_{5} - 379\beta_{4} - 333\beta_{3} + 333\beta_{2} + 12972 ) / 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1719\beta_{5} + 1719\beta_{4} + 3457\beta_{3} + 3457\beta_{2} - 52356\beta_1 ) / 20 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(41\) | \(61\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 9.00000i | 0 | −51.5439 | + | 21.6386i | 0 | − | 21.3742i | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 9.00000i | 0 | 33.8706 | + | 44.4723i | 0 | 85.8595i | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 9.00000i | 0 | 42.6733 | − | 36.1108i | 0 | − | 168.485i | 0 | −81.0000 | 0 | |||||||||||||||||||||||||||||||||
| 49.4 | 0 | 9.00000i | 0 | −51.5439 | − | 21.6386i | 0 | 21.3742i | 0 | −81.0000 | 0 | |||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 9.00000i | 0 | 33.8706 | − | 44.4723i | 0 | − | 85.8595i | 0 | −81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 9.00000i | 0 | 42.6733 | + | 36.1108i | 0 | 168.485i | 0 | −81.0000 | 0 | |||||||||||||||||||||||||||||||||||
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 | 120.6.f.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 360.6.f.a | 6 | ||
| 4.b | odd | 2 | 1 | 240.6.f.e | 6 | ||
| 5.b | even | 2 | 1 | inner | 120.6.f.a | ✓ | 6 |
| 5.c | odd | 4 | 1 | 600.6.a.r | 3 | ||
| 5.c | odd | 4 | 1 | 600.6.a.s | 3 | ||
| 12.b | even | 2 | 1 | 720.6.f.l | 6 | ||
| 15.d | odd | 2 | 1 | 360.6.f.a | 6 | ||
| 20.d | odd | 2 | 1 | 240.6.f.e | 6 | ||
| 60.h | even | 2 | 1 | 720.6.f.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.6.f.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 120.6.f.a | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 240.6.f.e | 6 | 4.b | odd | 2 | 1 | ||
| 240.6.f.e | 6 | 20.d | odd | 2 | 1 | ||
| 360.6.f.a | 6 | 3.b | odd | 2 | 1 | ||
| 360.6.f.a | 6 | 15.d | odd | 2 | 1 | ||
| 600.6.a.r | 3 | 5.c | odd | 4 | 1 | ||
| 600.6.a.s | 3 | 5.c | odd | 4 | 1 | ||
| 720.6.f.l | 6 | 12.b | even | 2 | 1 | ||
| 720.6.f.l | 6 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 36216T_{7}^{4} + 225603600T_{7}^{2} + 95604640000 \)
acting on \(S_{6}^{\mathrm{new}}(120, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 81)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 30517578125 \)
|
| $7$ |
\( T^{6} + \cdots + 95604640000 \)
|
| $11$ |
\( (T^{3} + 332 T^{2} + \cdots + 751600)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 743071584256 \)
|
| $17$ |
\( T^{6} + \cdots + 31\!\cdots\!04 \)
|
| $19$ |
\( (T^{3} - 144 T^{2} + \cdots + 2584131584)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 21\!\cdots\!24 \)
|
| $29$ |
\( (T^{3} + 946 T^{2} + \cdots + 37303134464)^{2} \)
|
| $31$ |
\( (T^{3} + 5124 T^{2} + \cdots - 71486323200)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 54\!\cdots\!36 \)
|
| $41$ |
\( (T^{3} + 662 T^{2} + \cdots + 289547160040)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 40\!\cdots\!96 \)
|
| $47$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + \cdots - 6990217203600)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 30927698302664)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 34\!\cdots\!56 \)
|
| $71$ |
\( (T^{3} + \cdots - 37186025753600)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots + 4303737800704)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 33\!\cdots\!04 \)
|
| $89$ |
\( (T^{3} + \cdots + 11306161037592)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 34\!\cdots\!24 \)
|
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