Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,2,Mod(305,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 0, 6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.305");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.cn (of order \(12\), degree \(4\), 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: | \(4.98266508613\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.56070144.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(209\) | \(469\) |
| \(\chi(n)\) | \(1\) | \(\beta_{5}\) | \(-1\) | \(1\) |
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 | −0.239203 | − | 1.71545i | 0 | 1.06488 | + | 1.06488i | 0 | −0.366025 | + | 1.36603i | 0 | −2.88556 | + | 0.820682i | 0 | ||||||||||||||||||||||||||||||||||
| 305.2 | 0 | 1.60523 | − | 0.650571i | 0 | −1.06488 | − | 1.06488i | 0 | −0.366025 | + | 1.36603i | 0 | 2.15351 | − | 2.08863i | 0 | |||||||||||||||||||||||||||||||||||
| 353.1 | 0 | −1.64914 | + | 0.529480i | 0 | −1.69293 | − | 1.69293i | 0 | 1.36603 | − | 0.366025i | 0 | 2.43930 | − | 1.74637i | 0 | |||||||||||||||||||||||||||||||||||
| 353.2 | 0 | 1.28311 | − | 1.16345i | 0 | 1.69293 | + | 1.69293i | 0 | 1.36603 | − | 0.366025i | 0 | 0.292748 | − | 2.98568i | 0 | |||||||||||||||||||||||||||||||||||
| 401.1 | 0 | −0.239203 | + | 1.71545i | 0 | 1.06488 | − | 1.06488i | 0 | −0.366025 | − | 1.36603i | 0 | −2.88556 | − | 0.820682i | 0 | |||||||||||||||||||||||||||||||||||
| 401.2 | 0 | 1.60523 | + | 0.650571i | 0 | −1.06488 | + | 1.06488i | 0 | −0.366025 | − | 1.36603i | 0 | 2.15351 | + | 2.08863i | 0 | |||||||||||||||||||||||||||||||||||
| 449.1 | 0 | −1.64914 | − | 0.529480i | 0 | −1.69293 | + | 1.69293i | 0 | 1.36603 | + | 0.366025i | 0 | 2.43930 | + | 1.74637i | 0 | |||||||||||||||||||||||||||||||||||
| 449.2 | 0 | 1.28311 | + | 1.16345i | 0 | 1.69293 | − | 1.69293i | 0 | 1.36603 | + | 0.366025i | 0 | 0.292748 | + | 2.98568i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.f | odd | 12 | 1 | inner |
| 39.k | even | 12 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):
|
\( T_{5}^{8} + 38T_{5}^{4} + 169 \)
|
|
\( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} - 4T_{7} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 38T^{4} + 169 \)
|
| $7$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $11$ |
\( T^{8} + 24 T^{6} + \cdots + 2704 \)
|
| $13$ |
\( (T^{4} - 4 T^{3} + \cdots + 169)^{2} \)
|
| $17$ |
\( T^{8} + 30 T^{6} + \cdots + 13689 \)
|
| $19$ |
\( (T^{4} - 8 T^{3} + 20 T^{2} + \cdots + 16)^{2} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} - 82 T^{6} + \cdots + 2474329 \)
|
| $31$ |
\( (T^{4} + 4 T^{3} + \cdots + 484)^{2} \)
|
| $37$ |
\( (T^{4} + 14 T^{3} + \cdots + 1369)^{2} \)
|
| $41$ |
\( T^{8} - 54 T^{6} + \cdots + 169 \)
|
| $43$ |
\( (T^{4} + 18 T^{3} + \cdots + 324)^{2} \)
|
| $47$ |
\( T^{8} + 9728 T^{4} + 11075584 \)
|
| $53$ |
\( (T^{4} + 22 T^{2} + 13)^{2} \)
|
| $59$ |
\( T^{8} + 24 T^{6} + \cdots + 43264 \)
|
| $61$ |
\( (T^{2} - 7 T + 49)^{4} \)
|
| $67$ |
\( (T^{4} - 20 T^{3} + \cdots + 2704)^{2} \)
|
| $71$ |
\( T^{8} - 24 T^{6} + \cdots + 43264 \)
|
| $73$ |
\( (T^{4} + 14 T^{3} + \cdots + 121)^{2} \)
|
| $79$ |
\( (T + 2)^{8} \)
|
| $83$ |
\( T^{8} + 296T^{4} + 2704 \)
|
| $89$ |
\( T^{8} - 24 T^{6} + \cdots + 77228944 \)
|
| $97$ |
\( (T^{4} - 10 T^{3} + \cdots + 484)^{2} \)
|
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