Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,4,Mod(31,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.31");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.bl (of order \(6\), 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: | \(19.8246417619\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.633537072.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 13x^{4} - 16x^{3} + 158x^{2} - 168x + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{6} - x^{5} + 13x^{4} - 16x^{3} + 158x^{2} - 168x + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -6\nu^{5} + 1021\nu^{4} - 71\nu^{3} + 13207\nu^{2} - 29298\nu + 105518 ) / 13202 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -78\nu^{5} + 71\nu^{4} - 923\nu^{3} + 65\nu^{2} - 11218\nu - 1274 ) / 13202 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 148\nu^{5} - 981\nu^{4} - 449\nu^{3} - 11125\nu^{2} - 3426\nu - 85596 ) / 13202 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -604\nu^{5} - 635\nu^{4} - 4947\nu^{3} - 1697\nu^{2} - 58094\nu - 27468 ) / 13202 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -702\nu^{5} + 639\nu^{4} - 8307\nu^{3} + 13787\nu^{2} - 74558\nu + 94150 ) / 13202 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} - \beta_{3} - 3\beta_{2} - 2\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + \beta_{4} + \beta_{3} - 24\beta_{2} + 2\beta _1 - 24 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{5} + 11\beta_{4} - 7\beta_{3} + 7\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -4\beta_{5} - 9\beta_{4} - 9\beta_{3} + 89\beta_{2} - 5\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 49\beta_{5} - 82\beta_{4} + 131\beta_{3} - 69\beta_{2} + 49\beta _1 - 69 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | 1.50000 | − | 2.59808i | 0 | −8.37970 | + | 4.83802i | 0 | 15.8619 | + | 9.56041i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||||||||||||||||||||
| 31.2 | 0 | 1.50000 | − | 2.59808i | 0 | 6.21705 | − | 3.58942i | 0 | −1.45104 | − | 18.4633i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 31.3 | 0 | 1.50000 | − | 2.59808i | 0 | 8.16265 | − | 4.71271i | 0 | −10.9108 | + | 14.9651i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 271.1 | 0 | 1.50000 | + | 2.59808i | 0 | −8.37970 | − | 4.83802i | 0 | 15.8619 | − | 9.56041i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 271.2 | 0 | 1.50000 | + | 2.59808i | 0 | 6.21705 | + | 3.58942i | 0 | −1.45104 | + | 18.4633i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
| 271.3 | 0 | 1.50000 | + | 2.59808i | 0 | 8.16265 | + | 4.71271i | 0 | −10.9108 | − | 14.9651i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 28.f | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.4.bl.h | yes | 6 |
| 4.b | odd | 2 | 1 | 336.4.bl.f | ✓ | 6 | |
| 7.d | odd | 6 | 1 | 336.4.bl.f | ✓ | 6 | |
| 28.f | even | 6 | 1 | inner | 336.4.bl.h | yes | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 336.4.bl.f | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 336.4.bl.f | ✓ | 6 | 7.d | odd | 6 | 1 | |
| 336.4.bl.h | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 336.4.bl.h | yes | 6 | 28.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):
|
\( T_{5}^{6} - 12T_{5}^{5} - 45T_{5}^{4} + 1116T_{5}^{3} + 4113T_{5}^{2} - 105462T_{5} + 428652 \)
|
|
\( T_{11}^{6} - 48T_{11}^{5} - 657T_{11}^{4} + 68400T_{11}^{3} + 2342529T_{11}^{2} + 27778950T_{11} + 126672012 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} - 12 T^{5} + \cdots + 428652 \)
|
| $7$ |
\( T^{6} - 7 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} - 48 T^{5} + \cdots + 126672012 \)
|
| $13$ |
\( T^{6} + 579 T^{4} + \cdots + 3048192 \)
|
| $17$ |
\( T^{6} + 84 T^{5} + \cdots + 248832 \)
|
| $19$ |
\( T^{6} + \cdots + 71197181584 \)
|
| $23$ |
\( T^{6} + \cdots + 429632335872 \)
|
| $29$ |
\( (T^{3} - 186 T^{2} + \cdots - 5184)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 6131274251881 \)
|
| $37$ |
\( T^{6} + \cdots + 6785587367056 \)
|
| $41$ |
\( T^{6} + \cdots + 702930633408 \)
|
| $43$ |
\( T^{6} + \cdots + 4094157830832 \)
|
| $47$ |
\( T^{6} + \cdots + 16\!\cdots\!84 \)
|
| $53$ |
\( T^{6} + \cdots + 525419367632784 \)
|
| $59$ |
\( T^{6} + \cdots + 35939689280784 \)
|
| $61$ |
\( T^{6} + \cdots + 12\!\cdots\!12 \)
|
| $67$ |
\( T^{6} + \cdots + 145376921152512 \)
|
| $71$ |
\( T^{6} + \cdots + 831854523313152 \)
|
| $73$ |
\( T^{6} + \cdots + 29\!\cdots\!48 \)
|
| $79$ |
\( T^{6} + \cdots + 64\!\cdots\!43 \)
|
| $83$ |
\( (T^{3} - 732 T^{2} + \cdots + 5856786)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 14\!\cdots\!72 \)
|
| $97$ |
\( T^{6} + \cdots + 19\!\cdots\!32 \)
|
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