Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2736,3,Mod(305,2736)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2736.305");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2736.h (of order \(2\), degree \(1\), not 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: | \(74.5506003290\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.691798081536.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 14x^{6} - 12x^{5} + 127x^{4} - 144x^{3} - 282x^{2} + 900x + 1350 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2\cdot 3^{3}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 342) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 14x^{6} - 12x^{5} + 127x^{4} - 144x^{3} - 282x^{2} + 900x + 1350 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 48316 \nu^{7} + 85773 \nu^{6} - 1779500 \nu^{5} + 522333 \nu^{4} + 14944036 \nu^{3} + \cdots + 278918685 ) / 41176665 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 67412 \nu^{7} + 428751 \nu^{6} - 587440 \nu^{5} - 3792624 \nu^{4} - 7184788 \nu^{3} + \cdots + 179989965 ) / 41176665 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7454 \nu^{7} + 2968 \nu^{6} - 115450 \nu^{5} - 137627 \nu^{4} + 1137374 \nu^{3} - 447413 \nu^{2} + \cdots + 5419443 ) / 2745111 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 64139 \nu^{7} + 352811 \nu^{6} - 220768 \nu^{5} - 3612013 \nu^{4} + 505466 \nu^{3} + \cdots - 63649725 ) / 13725555 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 200\nu^{7} - 233\nu^{6} - 1969\nu^{5} + 2560\nu^{4} + 21101\nu^{3} - 71453\nu^{2} + 13578\nu + 147000 ) / 32915 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 146294 \nu^{7} - 249626 \nu^{6} - 1991854 \nu^{5} + 711322 \nu^{4} + 17330060 \nu^{3} + \cdots + 120165450 ) / 13725555 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 510358 \nu^{7} + 579492 \nu^{6} + 6288608 \nu^{5} - 740859 \nu^{4} - 62007385 \nu^{3} + \cdots - 385086375 ) / 41176665 \)
|
| \(\nu\) | \(=\) |
\( ( -6\beta_{7} - \beta_{6} - 6\beta_{5} + 2\beta_{4} - 14\beta_{3} - 2\beta_{2} + 4\beta _1 - 2 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 12\beta_{7} + 12\beta_{6} + 2\beta_{5} + 6\beta_{4} - 3\beta_{3} - 9\beta_{2} + 3\beta _1 + 51 ) / 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -108\beta_{7} - 93\beta_{6} - 38\beta_{5} + 6\beta_{4} - 30\beta_{3} - 60\beta_{2} + 30\beta _1 + 120 ) / 30 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 132\beta_{7} + 72\beta_{6} + 152\beta_{5} + 36\beta_{4} - 102\beta_{3} - 51\beta_{2} + 117\beta _1 - 276 ) / 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -456\beta_{7} - 651\beta_{6} + 304\beta_{5} + 102\beta_{4} + 144\beta_{3} - 468\beta_{2} - 564\beta _1 + 6132 ) / 30 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -684\beta_{7} - 924\beta_{6} + 356\beta_{5} + 288\beta_{4} - 1530\beta_{3} + 135\beta_{2} + 1530\beta _1 - 6495 ) / 15 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 10914 \beta_{7} + 8049 \beta_{6} + 10714 \beta_{5} + 4272 \beta_{4} + 2436 \beta_{3} - 2952 \beta_{2} + \cdots + 54168 ) / 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).
| \(n\) | \(1009\) | \(1217\) | \(1711\) | \(2053\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 305.1 |
|
0 | 0 | 0 | − | 8.85053i | 0 | −9.86326 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 305.2 | 0 | 0 | 0 | − | 5.06122i | 0 | 3.50436 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 305.3 | 0 | 0 | 0 | − | 2.80489i | 0 | −7.23641 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 305.4 | 0 | 0 | 0 | − | 0.429787i | 0 | 9.59531 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 305.5 | 0 | 0 | 0 | 0.429787i | 0 | 9.59531 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 305.6 | 0 | 0 | 0 | 2.80489i | 0 | −7.23641 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 305.7 | 0 | 0 | 0 | 5.06122i | 0 | 3.50436 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 305.8 | 0 | 0 | 0 | 8.85053i | 0 | −9.86326 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2736.3.h.b | 8 | |
| 3.b | odd | 2 | 1 | inner | 2736.3.h.b | 8 | |
| 4.b | odd | 2 | 1 | 342.3.c.b | ✓ | 8 | |
| 12.b | even | 2 | 1 | 342.3.c.b | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 342.3.c.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 342.3.c.b | ✓ | 8 | 12.b | even | 2 | 1 | |
| 2736.3.h.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 2736.3.h.b | 8 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 112T_{5}^{6} + 2845T_{5}^{4} + 16308T_{5}^{2} + 2916 \)
acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 112 T^{6} + \cdots + 2916 \)
|
| $7$ |
\( (T^{4} + 4 T^{3} + \cdots + 2400)^{2} \)
|
| $11$ |
\( T^{8} + 580 T^{6} + \cdots + 26132544 \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 17728)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 3458145636 \)
|
| $19$ |
\( (T^{2} - 19)^{4} \)
|
| $23$ |
\( T^{8} + \cdots + 56933777664 \)
|
| $29$ |
\( T^{8} + \cdots + 1494904896 \)
|
| $31$ |
\( (T^{4} - 12 T^{3} + \cdots + 840064)^{2} \)
|
| $37$ |
\( (T^{4} - 36 T^{3} + \cdots + 1291648)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 7364840702976 \)
|
| $43$ |
\( (T^{4} - 90 T^{3} + \cdots - 9680000)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 122595218496 \)
|
| $53$ |
\( T^{8} + \cdots + 3771925056 \)
|
| $59$ |
\( T^{8} + \cdots + 247669456896 \)
|
| $61$ |
\( (T^{4} - 20 T^{3} + \cdots - 226412)^{2} \)
|
| $67$ |
\( (T^{4} + 64 T^{3} + \cdots + 1132800)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 2721391312896 \)
|
| $73$ |
\( (T^{4} + 158 T^{3} + \cdots - 16900608)^{2} \)
|
| $79$ |
\( (T^{4} + 200 T^{3} + \cdots + 1561600)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 79\!\cdots\!04 \)
|
| $89$ |
\( T^{8} + \cdots + 84\!\cdots\!00 \)
|
| $97$ |
\( (T^{4} + 144 T^{3} + \cdots - 21980096)^{2} \)
|
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