Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [111,4,Mod(1,111)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, 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("111.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 111.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: | \(6.54921201064\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 46x^{5} + 112x^{4} + 597x^{3} - 909x^{2} - 1968x - 744 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 46x^{5} + 112x^{4} + 597x^{3} - 909x^{2} - 1968x - 744 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{6} - 10\nu^{5} - 270\nu^{4} + 450\nu^{3} + 2313\nu^{2} - 5166\nu - 1828 ) / 316 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 17\nu^{6} - 92\nu^{5} - 746\nu^{4} + 3666\nu^{3} + 9003\nu^{2} - 33560\nu - 24844 ) / 316 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\nu^{6} - 51\nu^{5} - 508\nu^{4} + 1900\nu^{3} + 5658\nu^{2} - 15887\nu - 12388 ) / 158 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -39\nu^{6} + 146\nu^{5} + 1730\nu^{4} - 5622\nu^{3} - 20561\nu^{2} + 49638\nu + 47292 ) / 316 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 22\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 3\beta_{5} + \beta_{4} + 4\beta_{3} + 30\beta_{2} + 32\beta _1 + 308 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{6} - 42\beta_{5} + 32\beta_{4} + 44\beta_{3} + 10\beta_{2} + 533\beta _1 + 216 \)
|
| \(\nu^{6}\) | \(=\) |
\( 110\beta_{6} + 120\beta_{5} + 20\beta_{4} + 198\beta_{3} + 841\beta_{2} + 989\beta _1 + 7438 \)
|
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 |
|
−5.02671 | 3.00000 | 17.2679 | 14.9094 | −15.0801 | 20.4469 | −46.5869 | 9.00000 | −74.9454 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.77216 | 3.00000 | 6.22916 | −18.6623 | −11.3165 | 10.4856 | 6.67988 | 9.00000 | 70.3972 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.808825 | 3.00000 | −7.34580 | 20.7438 | −2.42648 | −10.2667 | 12.4121 | 9.00000 | −16.7781 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.604189 | 3.00000 | −7.63496 | −5.63218 | −1.81257 | 15.1092 | 9.44647 | 9.00000 | 3.40290 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.30662 | 3.00000 | 2.93376 | 4.72314 | 9.91987 | 26.2302 | −16.7522 | 9.00000 | 15.6176 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.45621 | 3.00000 | 11.8578 | 10.4248 | 13.3686 | −21.1358 | 17.1911 | 9.00000 | 46.4552 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.44905 | 3.00000 | 21.6922 | −12.5067 | 16.3472 | 2.13079 | 74.6095 | 9.00000 | −68.1494 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(37\) | \(-1\) |
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 | 111.4.a.f | ✓ | 7 |
| 3.b | odd | 2 | 1 | 333.4.a.h | 7 | ||
| 4.b | odd | 2 | 1 | 1776.4.a.x | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.4.a.f | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 333.4.a.h | 7 | 3.b | odd | 2 | 1 | ||
| 1776.4.a.x | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 3T_{2}^{6} - 46T_{2}^{5} + 112T_{2}^{4} + 597T_{2}^{3} - 909T_{2}^{2} - 1968T_{2} - 744 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(111))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots - 744 \)
|
| $3$ |
\( (T - 3)^{7} \)
|
| $5$ |
\( T^{7} - 14 T^{6} + \cdots + 20018496 \)
|
| $7$ |
\( T^{7} - 43 T^{6} + \cdots - 39287296 \)
|
| $11$ |
\( T^{7} + 31 T^{6} + \cdots + 177168384 \)
|
| $13$ |
\( T^{7} + \cdots + 27022844224 \)
|
| $17$ |
\( T^{7} + \cdots - 23935450606512 \)
|
| $19$ |
\( T^{7} + \cdots - 347830264576 \)
|
| $23$ |
\( T^{7} + \cdots + 43845065856 \)
|
| $29$ |
\( T^{7} + \cdots + 10\!\cdots\!32 \)
|
| $31$ |
\( T^{7} + \cdots - 2982773550080 \)
|
| $37$ |
\( (T - 37)^{7} \)
|
| $41$ |
\( T^{7} + \cdots + 10\!\cdots\!40 \)
|
| $43$ |
\( T^{7} + \cdots + 12\!\cdots\!52 \)
|
| $47$ |
\( T^{7} + \cdots - 21\!\cdots\!48 \)
|
| $53$ |
\( T^{7} + \cdots - 11\!\cdots\!32 \)
|
| $59$ |
\( T^{7} + \cdots - 23\!\cdots\!52 \)
|
| $61$ |
\( T^{7} + \cdots + 11\!\cdots\!16 \)
|
| $67$ |
\( T^{7} + \cdots - 15\!\cdots\!44 \)
|
| $71$ |
\( T^{7} + \cdots + 22\!\cdots\!92 \)
|
| $73$ |
\( T^{7} + \cdots - 34\!\cdots\!24 \)
|
| $79$ |
\( T^{7} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 85\!\cdots\!28 \)
|
| $89$ |
\( T^{7} + \cdots - 23\!\cdots\!68 \)
|
| $97$ |
\( T^{7} + \cdots - 18\!\cdots\!04 \)
|
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