Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [630,4,Mod(361,630)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(630, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("630.361");
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.k (of order \(3\), 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: | \(37.1712033036\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.10107317808.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 36x^{4} - 7x^{3} + 1246x^{2} - 735x + 441 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 36x^{4} - 7x^{3} + 1246x^{2} - 735x + 441 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -34\nu^{5} + 1224\nu^{4} + 57\nu^{3} + 42364\nu^{2} + 107373\nu + 1032003 ) / 132363 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 37\nu^{5} - 1332\nu^{4} + 3831\nu^{3} - 46102\nu^{2} + 159558\nu - 1111383 ) / 132363 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 60\nu^{5} - 59\nu^{4} + 2124\nu^{3} + 876\nu^{2} + 73514\nu - 43365 ) / 44121 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -1402\nu^{5} + 48\nu^{4} - 45849\nu^{3} - 24251\nu^{2} - 1692285\nu - 97083 ) / 132363 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1442\nu^{5} + 1488\nu^{4} - 53568\nu^{3} + 25589\nu^{2} - 1721685\nu + 1093680 ) / 132363 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + 2\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{5} + 5\beta_{4} + 72\beta_{3} - 4\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -71\beta_{5} + 71\beta_{4} - 40\beta_{2} + 40\beta _1 - 9 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -194\beta_{5} - 130\beta_{4} - 2529\beta_{3} - 64\beta_{2} + 130\beta _1 - 2529 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 1039\beta_{5} - 1489\beta_{4} - 1332\beta_{3} - 1039\beta_{2} - 2528\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(281\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 2.50000 | + | 4.33013i | 0 | −18.3140 | − | 2.75619i | 8.00000 | 0 | 5.00000 | − | 8.66025i | ||||||||||||||||||||||||||
| 361.2 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 2.50000 | + | 4.33013i | 0 | −1.25403 | + | 18.4778i | 8.00000 | 0 | 5.00000 | − | 8.66025i | |||||||||||||||||||||||||||
| 361.3 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 2.50000 | + | 4.33013i | 0 | 13.0681 | − | 13.1235i | 8.00000 | 0 | 5.00000 | − | 8.66025i | |||||||||||||||||||||||||||
| 541.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.50000 | − | 4.33013i | 0 | −18.3140 | + | 2.75619i | 8.00000 | 0 | 5.00000 | + | 8.66025i | |||||||||||||||||||||||||||
| 541.2 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.50000 | − | 4.33013i | 0 | −1.25403 | − | 18.4778i | 8.00000 | 0 | 5.00000 | + | 8.66025i | |||||||||||||||||||||||||||
| 541.3 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 2.50000 | − | 4.33013i | 0 | 13.0681 | + | 13.1235i | 8.00000 | 0 | 5.00000 | + | 8.66025i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 630.4.k.p | ✓ | 6 |
| 3.b | odd | 2 | 1 | 630.4.k.q | yes | 6 | |
| 7.c | even | 3 | 1 | inner | 630.4.k.p | ✓ | 6 |
| 21.h | odd | 6 | 1 | 630.4.k.q | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 630.4.k.p | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 630.4.k.p | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 630.4.k.q | yes | 6 | 3.b | odd | 2 | 1 | |
| 630.4.k.q | yes | 6 | 21.h | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\):
|
\( T_{11}^{6} + 2T_{11}^{5} + 1522T_{11}^{4} - 43104T_{11}^{3} + 2264256T_{11}^{2} - 30411612T_{11} + 401361156 \)
|
|
\( T_{13}^{3} - 55T_{13}^{2} - 107T_{13} + 11451 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} + 13 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + 2 T^{5} + \cdots + 401361156 \)
|
| $13$ |
\( (T^{3} - 55 T^{2} + \cdots + 11451)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 2760661764 \)
|
| $19$ |
\( T^{6} + \cdots + 1809396369 \)
|
| $23$ |
\( T^{6} + \cdots + 3703696164 \)
|
| $29$ |
\( (T^{3} + 216 T^{2} + \cdots - 642978)^{2} \)
|
| $31$ |
\( T^{6} - 159 T^{5} + \cdots + 484836361 \)
|
| $37$ |
\( T^{6} + \cdots + 37862054836225 \)
|
| $41$ |
\( (T^{3} - 406 T^{2} + \cdots + 41085630)^{2} \)
|
| $43$ |
\( (T^{3} + 469 T^{2} + \cdots - 61241287)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 448733724088896 \)
|
| $53$ |
\( T^{6} + \cdots + 525236190753024 \)
|
| $59$ |
\( T^{6} + \cdots + 506265390116964 \)
|
| $61$ |
\( T^{6} + \cdots + 14\!\cdots\!04 \)
|
| $67$ |
\( T^{6} + \cdots + 91\!\cdots\!21 \)
|
| $71$ |
\( (T^{3} + 8 T^{2} + \cdots + 99019710)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 31\!\cdots\!61 \)
|
| $79$ |
\( T^{6} + \cdots + 29\!\cdots\!25 \)
|
| $83$ |
\( (T^{3} + 650 T^{2} + \cdots - 517798638)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 120429597817764 \)
|
| $97$ |
\( (T^{3} - 1272 T^{2} + \cdots - 62647124)^{2} \)
|
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