Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,3,Mod(37,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.37");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.x (of order \(12\), degree \(4\), 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: | \(4.25069212402\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.108931682304.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 1063x^{4} - 1874x^{3} + 8090x^{2} - 7178x + 20593 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 1063x^{4} - 1874x^{3} + 8090x^{2} - 7178x + 20593 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -43\nu^{7} - 948\nu^{6} + 1512\nu^{5} - 40956\nu^{4} + 63436\nu^{3} - 502860\nu^{2} + 420108\nu - 1687080 ) / 24167 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -43\nu^{7} - 948\nu^{6} + 1512\nu^{5} - 40956\nu^{4} + 63436\nu^{3} - 527027\nu^{2} + 444275\nu - 2001251 ) / 24167 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 1095\nu^{6} - 2692\nu^{5} + 55622\nu^{4} - 91961\nu^{3} + 845226\nu^{2} - 698555\nu + 3704699 ) / 24167 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 1102\nu^{6} + 3899\nu^{5} - 58622\nu^{4} + 125542\nu^{3} - 892601\nu^{2} + 931619\nu - 3813435 ) / 24167 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 43 \nu^{7} - 1249 \nu^{6} + 5079 \nu^{5} - 49121 \nu^{4} + 105733 \nu^{3} - 582458 \nu^{2} + \cdots - 2061002 ) / 24167 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -8\nu^{7} + 28\nu^{6} - 434\nu^{5} + 1015\nu^{4} - 6898\nu^{3} + 9346\nu^{2} - 31486\nu + 13120 ) / 2197 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1208 \nu^{7} - 4228 \nu^{6} + 52352 \nu^{5} - 120310 \nu^{4} + 683487 \nu^{3} - 919118 \nu^{2} + \cdots - 1343990 ) / 24167 \)
|
| \(\nu\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 - 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} - 10\beta_{6} - 24\beta_{5} - 12\beta_{4} - 12\beta_{3} + 22\beta_{2} + 2\beta _1 - 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} - 21\beta_{6} - 48\beta_{5} - 26\beta_{4} - 24\beta_{3} + 72\beta_{2} - 22\beta _1 + 146 \)
|
| \(\nu^{5}\) | \(=\) |
\( -24\beta_{7} + 74\beta_{6} + 412\beta_{5} + 141\beta_{4} + 146\beta_{3} - 335\beta_{2} - 72\beta _1 + 460 \)
|
| \(\nu^{6}\) | \(=\) |
\( -77\beta_{7} + 275\beta_{6} + 1307\beta_{5} + 527\beta_{4} + 460\beta_{3} - 1731\beta_{2} + 335\beta _1 - 1776 \)
|
| \(\nu^{7}\) | \(=\) |
\( 424\beta_{7} - 136\beta_{6} - 5940\beta_{5} - 1524\beta_{4} - 1776\beta_{3} + 3880\beta_{2} + 1731\beta _1 - 8268 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
0 | −0.866025 | − | 1.50000i | 0 | −3.58692 | − | 3.58692i | 0 | 0.568481 | + | 2.12160i | 0 | −1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||||||||||||
| 37.2 | 0 | −0.866025 | − | 1.50000i | 0 | 3.58692 | + | 3.58692i | 0 | 1.52959 | + | 5.70853i | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 85.1 | 0 | 0.866025 | − | 1.50000i | 0 | −3.33676 | − | 3.33676i | 0 | −7.77552 | − | 2.08344i | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 85.2 | 0 | 0.866025 | − | 1.50000i | 0 | 3.33676 | + | 3.33676i | 0 | 4.67744 | + | 1.25332i | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 97.1 | 0 | −0.866025 | + | 1.50000i | 0 | −3.58692 | + | 3.58692i | 0 | 0.568481 | − | 2.12160i | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 97.2 | 0 | −0.866025 | + | 1.50000i | 0 | 3.58692 | − | 3.58692i | 0 | 1.52959 | − | 5.70853i | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 145.1 | 0 | 0.866025 | + | 1.50000i | 0 | −3.33676 | + | 3.33676i | 0 | −7.77552 | + | 2.08344i | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 145.2 | 0 | 0.866025 | + | 1.50000i | 0 | 3.33676 | − | 3.33676i | 0 | 4.67744 | − | 1.25332i | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.f | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.3.x.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 468.3.cd.c | 8 | ||
| 13.f | odd | 12 | 1 | inner | 156.3.x.a | ✓ | 8 |
| 39.k | even | 12 | 1 | 468.3.cd.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.3.x.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 156.3.x.a | ✓ | 8 | 13.f | odd | 12 | 1 | inner |
| 468.3.cd.c | 8 | 3.b | odd | 2 | 1 | ||
| 468.3.cd.c | 8 | 39.k | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 1158T_{5}^{4} + 328329 \)
acting on \(S_{3}^{\mathrm{new}}(156, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 3 T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{8} + 1158 T^{4} + 328329 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 256036 \)
|
| $11$ |
\( T^{8} + 30 T^{7} + \cdots + 36 \)
|
| $13$ |
\( T^{8} + 6 T^{7} + \cdots + 815730721 \)
|
| $17$ |
\( T^{8} - 66 T^{7} + \cdots + 530979849 \)
|
| $19$ |
\( T^{8} - 34 T^{7} + \cdots + 114244 \)
|
| $23$ |
\( T^{8} - 18 T^{7} + \cdots + 8328996 \)
|
| $29$ |
\( T^{8} + \cdots + 2058585039729 \)
|
| $31$ |
\( T^{8} + \cdots + 981494415616 \)
|
| $37$ |
\( T^{8} + \cdots + 282814430809 \)
|
| $41$ |
\( T^{8} + \cdots + 2358753489 \)
|
| $43$ |
\( T^{8} - 138 T^{7} + \cdots + 811908036 \)
|
| $47$ |
\( T^{8} + \cdots + 14112449275716 \)
|
| $53$ |
\( (T^{4} + 72 T^{3} + \cdots - 7332699)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 2283310656 \)
|
| $61$ |
\( T^{8} + \cdots + 3702534094809 \)
|
| $67$ |
\( T^{8} + \cdots + 13471058046436 \)
|
| $71$ |
\( T^{8} + \cdots + 481413712264164 \)
|
| $73$ |
\( T^{8} + \cdots + 14528164965889 \)
|
| $79$ |
\( (T^{4} + 24 T^{3} + \cdots - 8951592)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 6947726309316 \)
|
| $89$ |
\( T^{8} + \cdots + 731016023886864 \)
|
| $97$ |
\( T^{8} + \cdots + 15659495155264 \)
|
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