Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1512,2,Mod(865,1512)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1512.865");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1512.s (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: | \(12.0733807856\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.9391935744.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 5x^{6} + 12x^{5} - 76x^{4} + 84x^{3} + 245x^{2} - 1372x + 2401 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} + 5x^{6} + 12x^{5} - 76x^{4} + 84x^{3} + 245x^{2} - 1372x + 2401 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} - 26\nu^{5} - 194\nu^{4} - 1282\nu^{3} + 3290\nu^{2} - 1911\nu + 343 ) / 12348 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{7} + 18\nu^{6} + 128\nu^{5} + 506\nu^{4} - 1520\nu^{3} + 574\nu^{2} + 2205\nu - 42532 ) / 37044 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\nu^{7} + 15\nu^{6} + 74\nu^{5} - 346\nu^{4} + 334\nu^{3} - 518\nu^{2} - 8967\nu + 3773 ) / 12348 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 4\nu^{6} - 10\nu^{5} + 48\nu^{4} - 146\nu^{3} - 468\nu^{2} + 1169\nu - 1960 ) / 1764 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -43\nu^{7} + 53\nu^{6} + 114\nu^{5} - 866\nu^{4} + 762\nu^{3} - 2506\nu^{2} - 5929\nu + 27783 ) / 24696 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -124\nu^{7} + 405\nu^{6} - 746\nu^{5} - 2384\nu^{4} + 5882\nu^{3} + 224\nu^{2} - 34398\nu + 154693 ) / 37044 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{6} - 2\beta_{5} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{6} - 4\beta_{5} + \beta_{4} - 6\beta_{3} - 5\beta_{2} + 2\beta _1 - 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} - 10\beta_{6} - 4\beta_{5} - 11\beta_{4} + 25\beta_{3} - 11\beta_{2} - 10\beta _1 + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( -17\beta_{7} + 32\beta_{6} - 28\beta_{5} + 14\beta_{4} + 67\beta_{3} - 17\beta_{2} + 14\beta _1 + 77 \)
|
| \(\nu^{6}\) | \(=\) |
\( 74\beta_{7} - 24\beta_{6} + 50\beta_{4} + 362\beta_{3} - 24\beta_{2} + 74\beta _1 + 119 \)
|
| \(\nu^{7}\) | \(=\) |
\( 26\beta_{7} - 272\beta_{6} + 52\beta_{5} + 338\beta_{4} + 14\beta_{3} + 169\beta _1 + 168 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).
| \(n\) | \(757\) | \(785\) | \(1081\) | \(1135\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 865.1 |
|
0 | 0 | 0 | −1.36603 | − | 2.36603i | 0 | −0.931486 | + | 2.47635i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
| 865.2 | 0 | 0 | 0 | −1.36603 | − | 2.36603i | 0 | 0.431486 | − | 2.61033i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 865.3 | 0 | 0 | 0 | 0.366025 | + | 0.633975i | 0 | −2.62920 | + | 0.295509i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 865.4 | 0 | 0 | 0 | 0.366025 | + | 0.633975i | 0 | 2.12920 | + | 1.57052i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.1 | 0 | 0 | 0 | −1.36603 | + | 2.36603i | 0 | −0.931486 | − | 2.47635i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.2 | 0 | 0 | 0 | −1.36603 | + | 2.36603i | 0 | 0.431486 | + | 2.61033i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.3 | 0 | 0 | 0 | 0.366025 | − | 0.633975i | 0 | −2.62920 | − | 0.295509i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1297.4 | 0 | 0 | 0 | 0.366025 | − | 0.633975i | 0 | 2.12920 | − | 1.57052i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
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 | 1512.2.s.m | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1512.2.s.p | yes | 8 | |
| 7.c | even | 3 | 1 | inner | 1512.2.s.m | ✓ | 8 |
| 21.h | odd | 6 | 1 | 1512.2.s.p | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1512.2.s.m | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1512.2.s.m | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 1512.2.s.p | yes | 8 | 3.b | odd | 2 | 1 | |
| 1512.2.s.p | yes | 8 | 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 \(S_{2}^{\mathrm{new}}(1512, [\chi])\):
|
\( T_{5}^{4} + 2T_{5}^{3} + 6T_{5}^{2} - 4T_{5} + 4 \)
|
|
\( T_{11}^{8} + 2T_{11}^{7} + 30T_{11}^{6} - 4T_{11}^{5} + 580T_{11}^{4} + 48T_{11}^{3} + 4320T_{11}^{2} - 3456T_{11} + 20736 \)
|
|
\( T_{13}^{4} + 4T_{13}^{3} - 35T_{13}^{2} - 204T_{13} - 264 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 2 T^{3} + 6 T^{2} + \cdots + 4)^{2} \)
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| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} + 2 T^{7} + \cdots + 20736 \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots - 264)^{2} \)
|
| $17$ |
\( (T^{4} - 2 T^{3} + \cdots + 676)^{2} \)
|
| $19$ |
\( T^{8} + 6 T^{7} + \cdots + 183184 \)
|
| $23$ |
\( T^{8} - 2 T^{7} + \cdots + 2050624 \)
|
| $29$ |
\( (T^{4} - 8 T^{3} + \cdots - 588)^{2} \)
|
| $31$ |
\( T^{8} + 6 T^{7} + \cdots + 6889 \)
|
| $37$ |
\( T^{8} + 87 T^{6} + \cdots + 5184 \)
|
| $41$ |
\( (T^{4} + 8 T^{3} + \cdots + 996)^{2} \)
|
| $43$ |
\( (T^{4} - 158 T^{2} + \cdots + 613)^{2} \)
|
| $47$ |
\( T^{8} + 20 T^{7} + \cdots + 43401744 \)
|
| $53$ |
\( T^{8} + 10 T^{7} + \cdots + 153664 \)
|
| $59$ |
\( T^{8} - 22 T^{7} + \cdots + 36864 \)
|
| $61$ |
\( T^{8} - 2 T^{7} + \cdots + 5041 \)
|
| $67$ |
\( T^{8} - 2 T^{7} + \cdots + 992016 \)
|
| $71$ |
\( (T^{4} - 22 T^{3} + \cdots - 1056)^{2} \)
|
| $73$ |
\( T^{8} + 10 T^{7} + \cdots + 65536 \)
|
| $79$ |
\( T^{8} - 8 T^{7} + \cdots + 30976 \)
|
| $83$ |
\( (T^{4} + 20 T^{3} + \cdots - 5664)^{2} \)
|
| $89$ |
\( T^{8} + 16 T^{7} + \cdots + 4717584 \)
|
| $97$ |
\( (T^{4} - 158 T^{2} + \cdots + 613)^{2} \)
|
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