Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1859,2,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, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1859.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1859 = 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| 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: | \(14.8441897358\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 8x^{6} + 18x^{5} + 7x^{4} - 22x^{3} - 3x^{2} + 6x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 8x^{6} + 18x^{5} + 7x^{4} - 22x^{3} - 3x^{2} + 6x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + 3\nu^{2} + 5\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - \nu^{5} - 8\nu^{4} + 10\nu^{3} + 9\nu^{2} - 10\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 8\nu^{5} + 10\nu^{4} + 9\nu^{3} - 10\nu^{2} - 2\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 17\nu^{4} - \nu^{3} - 13\nu^{2} + 5\nu + 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 7\nu^{5} + 18\nu^{4} - 19\nu^{2} + 3\nu + 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 25\nu^{4} - 3\nu^{3} - 24\nu^{2} + 4\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{4} + \beta_{3} + 5\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{7} - 6\beta_{6} - 7\beta_{5} + 7\beta_{4} - \beta_{3} + 6\beta_{2} + 3\beta _1 + 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{7} + 10\beta_{6} + 3\beta_{5} - 10\beta_{4} + 7\beta_{3} - 2\beta_{2} + 27\beta _1 - 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 36\beta_{7} - 39\beta_{6} - 44\beta_{5} + 47\beta_{4} - 10\beta_{3} + 37\beta_{2} + 2\beta _1 + 62 \)
|
| \(\nu^{7}\) | \(=\) |
\( -38\beta_{7} + 82\beta_{6} + 40\beta_{5} - 83\beta_{4} + 47\beta_{3} - 29\beta_{2} + 155\beta _1 - 106 \)
|
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 |
|
−2.69931 | −1.59679 | 5.28626 | 1.39555 | 4.31023 | 4.07855 | −8.87064 | −0.450256 | −3.76701 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.920076 | 0.898842 | −1.15346 | −2.76396 | −0.827003 | 3.49246 | 2.90142 | −2.19208 | 2.54305 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.549080 | −1.45990 | −1.69851 | 0.775548 | 0.801604 | −1.41094 | 2.03078 | −0.868681 | −0.425838 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.170890 | 2.94878 | −1.97080 | 1.29847 | −0.503917 | 3.65280 | 0.678569 | 5.69533 | −0.221895 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.681121 | −0.0549966 | −1.53607 | 3.68496 | −0.0374594 | −0.234733 | −2.40849 | −2.99698 | 2.50991 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.37460 | −2.08494 | −0.110482 | −2.34290 | −2.86595 | 1.29593 | −2.90106 | 1.34697 | −3.22054 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.07742 | −1.13599 | 2.31569 | 4.26888 | −2.35994 | 3.46221 | 0.655817 | −1.70952 | 8.86827 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.20621 | 2.48500 | 2.86737 | 1.68345 | 5.48243 | −0.336274 | 1.91361 | 3.17521 | 3.71405 | |||||||||||||||||||||||||||||||||||||||||||
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 | 1859.2.a.p | 8 | |
| 13.b | even | 2 | 1 | 1859.2.a.o | 8 | ||
| 13.f | odd | 12 | 2 | 143.2.j.b | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 143.2.j.b | ✓ | 16 | 13.f | odd | 12 | 2 | |
| 1859.2.a.o | 8 | 13.b | even | 2 | 1 | ||
| 1859.2.a.p | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\):
|
\( T_{2}^{8} - 2T_{2}^{7} - 8T_{2}^{6} + 18T_{2}^{5} + 7T_{2}^{4} - 22T_{2}^{3} - 3T_{2}^{2} + 6T_{2} + 1 \)
|
|
\( T_{7}^{8} - 14T_{7}^{7} + 69T_{7}^{6} - 118T_{7}^{5} - 76T_{7}^{4} + 374T_{7}^{3} - 111T_{7}^{2} - 158T_{7} - 26 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 1 \)
|
| $3$ |
\( T^{8} - 13 T^{6} + \cdots - 2 \)
|
| $5$ |
\( T^{8} - 8 T^{7} + \cdots + 241 \)
|
| $7$ |
\( T^{8} - 14 T^{7} + \cdots - 26 \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} - 6 T^{7} + \cdots - 3119 \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots - 2018 \)
|
| $23$ |
\( T^{8} + 14 T^{7} + \cdots + 33844 \)
|
| $29$ |
\( T^{8} + 10 T^{7} + \cdots - 2063 \)
|
| $31$ |
\( T^{8} - 32 T^{7} + \cdots - 50588 \)
|
| $37$ |
\( T^{8} - 24 T^{7} + \cdots + 61 \)
|
| $41$ |
\( T^{8} - 2 T^{7} + \cdots + 1711933 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots - 3656 \)
|
| $47$ |
\( T^{8} - 18 T^{7} + \cdots + 54286 \)
|
| $53$ |
\( T^{8} + 8 T^{7} + \cdots + 1610773 \)
|
| $59$ |
\( T^{8} - 18 T^{7} + \cdots - 773474 \)
|
| $61$ |
\( T^{8} + 4 T^{7} + \cdots + 514561 \)
|
| $67$ |
\( T^{8} + 14 T^{7} + \cdots - 27643106 \)
|
| $71$ |
\( T^{8} + 12 T^{7} + \cdots - 12355388 \)
|
| $73$ |
\( T^{8} - 26 T^{7} + \cdots + 4343872 \)
|
| $79$ |
\( T^{8} - 26 T^{7} + \cdots + 2764672 \)
|
| $83$ |
\( T^{8} - 16 T^{7} + \cdots - 14622626 \)
|
| $89$ |
\( T^{8} + 8 T^{7} + \cdots - 424748 \)
|
| $97$ |
\( T^{8} - 20 T^{7} + \cdots - 3157100 \)
|
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