Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,4,Mod(1,1859)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1859.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1859.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: | \(109.684550701\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - 34x^{4} - 26x^{3} + 249x^{2} + 274x - 200 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 143) |
| 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
\( x^{6} - 34x^{4} - 26x^{3} + 249x^{2} + 274x - 200 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 29\nu^{3} - 32\nu^{2} - 158\nu + 56 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} - 7\nu^{4} - 85\nu^{3} + 121\nu^{2} + 452\nu - 256 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 7\nu^{4} + 86\nu^{3} - 123\nu^{2} - 467\nu + 266 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + 4\beta_{4} + 2\beta_{2} + 17\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{5} + 4\beta_{4} - 6\beta_{3} + 29\beta_{2} + 37\beta _1 + 211 \)
|
| \(\nu^{5}\) | \(=\) |
\( 66\beta_{5} + 124\beta_{4} - 14\beta_{3} + 84\beta_{2} + 377\beta _1 + 474 \)
|
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 |
|
−4.34903 | −6.31201 | 10.9140 | 2.31865 | 27.4511 | 33.8589 | −12.6732 | 12.8415 | −10.0839 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.11199 | 4.05013 | −3.53950 | 16.1679 | −8.55384 | 9.75779 | 24.3713 | −10.5964 | −34.1465 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.487498 | 0.0966010 | −7.76235 | −13.1745 | 0.0470928 | 11.9172 | −7.68411 | −26.9907 | −6.42254 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.30499 | 0.708355 | 2.92295 | 6.96953 | 2.34111 | −22.1312 | −16.7796 | −26.4982 | 23.0342 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.36134 | −9.16081 | 3.29862 | 0.852032 | −30.7926 | −5.75009 | −15.8029 | 56.9204 | 2.86397 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.30719 | 4.61773 | 20.1662 | −5.13365 | 24.5072 | 25.3474 | 64.5685 | −5.67656 | −27.2453 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(13\) | \(1\) |
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 | 1859.4.a.c | 6 | |
| 13.b | even | 2 | 1 | 143.4.a.b | ✓ | 6 | |
| 39.d | odd | 2 | 1 | 1287.4.a.f | 6 | ||
| 52.b | odd | 2 | 1 | 2288.4.a.m | 6 | ||
| 143.d | odd | 2 | 1 | 1573.4.a.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.4.a.b | ✓ | 6 | 13.b | even | 2 | 1 | |
| 1287.4.a.f | 6 | 39.d | odd | 2 | 1 | ||
| 1573.4.a.d | 6 | 143.d | odd | 2 | 1 | ||
| 1859.4.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 2288.4.a.m | 6 | 52.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 6T_{2}^{5} - 19T_{2}^{4} + 142T_{2}^{3} - 18T_{2}^{2} - 564T_{2} + 264 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 6 T^{5} + \cdots + 264 \)
|
| $3$ |
\( T^{6} + 6 T^{5} + \cdots + 74 \)
|
| $5$ |
\( T^{6} - 8 T^{5} + \cdots + 15056 \)
|
| $7$ |
\( T^{6} - 53 T^{5} + \cdots + 12700172 \)
|
| $11$ |
\( (T + 11)^{6} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} + 117 T^{5} + \cdots + 64149376 \)
|
| $19$ |
\( T^{6} + \cdots - 49335856960 \)
|
| $23$ |
\( T^{6} + 158 T^{5} + \cdots + 158607016 \)
|
| $29$ |
\( T^{6} + \cdots - 14715687201280 \)
|
| $31$ |
\( T^{6} + \cdots - 2334074232384 \)
|
| $37$ |
\( T^{6} + \cdots - 1475360000224 \)
|
| $41$ |
\( T^{6} + \cdots + 19515694713824 \)
|
| $43$ |
\( T^{6} + \cdots - 4882557023552 \)
|
| $47$ |
\( T^{6} + \cdots - 359796596534272 \)
|
| $53$ |
\( T^{6} + \cdots - 33677222445392 \)
|
| $59$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 703896062106752 \)
|
| $67$ |
\( T^{6} + \cdots + 76\!\cdots\!24 \)
|
| $71$ |
\( T^{6} + \cdots + 73\!\cdots\!72 \)
|
| $73$ |
\( T^{6} + \cdots - 260628470656 \)
|
| $79$ |
\( T^{6} + \cdots - 30\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 24\!\cdots\!48 \)
|
| $89$ |
\( T^{6} + \cdots - 71\!\cdots\!20 \)
|
| $97$ |
\( T^{6} + \cdots + 46\!\cdots\!64 \)
|
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