Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [624,6,Mod(1,624)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(624, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("624.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 624 = 2^{4} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 624.a (trivial) |
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: | yes |
| Analytic conductor: | \(100.079503563\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 186x^{2} - 529x - 95 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 312) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 186x^{2} - 529x - 95 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 12\nu^{2} + 312\nu - 124 ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 4\nu^{2} - 328\nu - 618 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} + 8\nu^{2} + 776\nu + 1206 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 10 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + 19\beta_{2} + 45\beta _1 + 3730 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 84\beta_{3} + 213\beta_{2} + 45\beta _1 + 10730 ) / 20 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −9.00000 | 0 | −91.2603 | 0 | 155.198 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | −29.2714 | 0 | −174.732 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | 18.4040 | 0 | 228.995 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −9.00000 | 0 | 94.1278 | 0 | −33.4615 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 624.6.a.v | 4 | |
| 4.b | odd | 2 | 1 | 312.6.a.i | ✓ | 4 | |
| 12.b | even | 2 | 1 | 936.6.a.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.6.a.i | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 624.6.a.v | 4 | 1.a | even | 1 | 1 | trivial | |
| 936.6.a.m | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(624))\):
|
\( T_{5}^{4} + 8T_{5}^{3} - 9160T_{5}^{2} - 91808T_{5} + 4627584 \)
|
|
\( T_{7}^{4} - 176T_{7}^{3} - 38600T_{7}^{2} + 5152800T_{7} + 207792000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 9)^{4} \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 4627584 \)
|
| $7$ |
\( T^{4} - 176 T^{3} + \cdots + 207792000 \)
|
| $11$ |
\( T^{4} + \cdots + 1596691920 \)
|
| $13$ |
\( (T + 169)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 904643703696 \)
|
| $19$ |
\( T^{4} + \cdots - 4883857879680 \)
|
| $23$ |
\( T^{4} + \cdots + 1369714914560 \)
|
| $29$ |
\( T^{4} + \cdots + 963566008276752 \)
|
| $31$ |
\( T^{4} + \cdots - 674112129948800 \)
|
| $37$ |
\( T^{4} + \cdots + 14\!\cdots\!44 \)
|
| $41$ |
\( T^{4} + \cdots - 18\!\cdots\!40 \)
|
| $43$ |
\( T^{4} + \cdots + 896054731550464 \)
|
| $47$ |
\( T^{4} + \cdots + 76\!\cdots\!28 \)
|
| $53$ |
\( T^{4} + \cdots - 24\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 21\!\cdots\!92 \)
|
| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!08 \)
|
| $67$ |
\( T^{4} + \cdots + 18\!\cdots\!52 \)
|
| $71$ |
\( T^{4} + \cdots - 11\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 48\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 66\!\cdots\!76 \)
|
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