Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1840,4,Mod(1,1840)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1840.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1840 = 2^{4} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1840.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: | \(108.563514411\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 920) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} + 86\nu^{4} - 1888\nu^{3} - 5583\nu^{2} + 44176\nu + 40584 ) / 4824 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{5} + 32\nu^{4} - 796\nu^{3} - 2643\nu^{2} + 8248\nu + 24132 ) / 1206 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} - 22\nu^{4} - 307\nu^{3} + 1503\nu^{2} + 1264\nu - 15234 ) / 603 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\nu^{5} - 30\nu^{4} - 1448\nu^{3} + 539\nu^{2} + 19704\nu + 16320 ) / 1608 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 27 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + \beta_{4} + 4\beta_{3} - 6\beta_{2} + 50\beta _1 + 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( -77\beta_{5} + 59\beta_{4} + 21\beta_{3} + 65\beta_{2} + 81\beta _1 + 1305 \)
|
| \(\nu^{5}\) | \(=\) |
\( -161\beta_{5} + 141\beta_{4} + 338\beta_{3} - 383\beta_{2} + 2873\beta _1 + 1655 \)
|
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 | −7.08353 | 0 | 5.00000 | 0 | 9.74988 | 0 | 23.1764 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.61348 | 0 | 5.00000 | 0 | −3.48289 | 0 | −13.9428 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.24528 | 0 | 5.00000 | 0 | 19.3962 | 0 | −25.4493 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.67963 | 0 | 5.00000 | 0 | −24.9473 | 0 | −19.8196 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 4.88687 | 0 | 5.00000 | 0 | 22.2944 | 0 | −3.11847 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 8.37578 | 0 | 5.00000 | 0 | −9.01032 | 0 | 43.1537 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(23\) | \(1\) |
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 | 1840.4.a.u | 6 | |
| 4.b | odd | 2 | 1 | 920.4.a.a | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 920.4.a.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 1840.4.a.u | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):
|
\( T_{3}^{6} - 4T_{3}^{5} - 75T_{3}^{4} + 215T_{3}^{3} + 1158T_{3}^{2} - 1831T_{3} - 3496 \)
|
|
\( T_{7}^{6} - 14T_{7}^{5} - 744T_{7}^{4} + 10329T_{7}^{3} + 89664T_{7}^{2} - 789580T_{7} - 3300776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots - 3496 \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 14 T^{5} + \cdots - 3300776 \)
|
| $11$ |
\( T^{6} - 23 T^{5} + \cdots + 398480 \)
|
| $13$ |
\( T^{6} + \cdots + 1972631470 \)
|
| $17$ |
\( T^{6} + \cdots - 1487271232 \)
|
| $19$ |
\( T^{6} + \cdots - 22042764464 \)
|
| $23$ |
\( (T + 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 3217328048828 \)
|
| $31$ |
\( T^{6} + \cdots - 2038672875799 \)
|
| $37$ |
\( T^{6} + \cdots - 120683526562304 \)
|
| $41$ |
\( T^{6} + \cdots - 4268456808415 \)
|
| $43$ |
\( T^{6} + \cdots + 1641239221248 \)
|
| $47$ |
\( T^{6} + \cdots + 99563112748880 \)
|
| $53$ |
\( T^{6} + \cdots - 39\!\cdots\!08 \)
|
| $59$ |
\( T^{6} + \cdots - 882676639673600 \)
|
| $61$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 607060879004288 \)
|
| $71$ |
\( T^{6} + \cdots - 42369419745775 \)
|
| $73$ |
\( T^{6} + \cdots + 802499042949320 \)
|
| $79$ |
\( T^{6} + \cdots + 29\!\cdots\!12 \)
|
| $83$ |
\( T^{6} + \cdots + 85\!\cdots\!88 \)
|
| $89$ |
\( T^{6} + \cdots + 25\!\cdots\!68 \)
|
| $97$ |
\( T^{6} + \cdots - 11\!\cdots\!00 \)
|
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