Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6171,2,Mod(1,6171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6171, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6171.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6171 = 3 \cdot 11^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6171.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: | \(49.2756830873\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.46051664.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 16x^{2} - 5x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 8x^{4} + 5x^{3} + 16x^{2} - 5x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} - 3\nu^{2} + 8\nu + 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 4\nu^{2} + 9\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 7\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 8\beta_{4} - 7\beta_{3} + 10\beta_{2} + 20\beta _1 + 11 \)
|
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 |
|
−2.15303 | −1.00000 | 2.63554 | 1.59222 | 2.15303 | 4.72881 | −1.36833 | 1.00000 | −3.42809 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.23201 | −1.00000 | −0.482154 | 3.37910 | 1.23201 | −4.47126 | 3.05804 | 1.00000 | −4.16308 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.800810 | −1.00000 | −1.35870 | −1.23251 | 0.800810 | −0.260573 | 2.68968 | 1.00000 | 0.987003 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.10458 | −1.00000 | −0.779902 | −4.01858 | −1.10458 | −1.25270 | −3.07063 | 1.00000 | −4.43884 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.46755 | −1.00000 | 0.153708 | −0.272030 | −1.46755 | 2.99647 | −2.70953 | 1.00000 | −0.399218 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.61372 | −1.00000 | 4.83151 | 0.551794 | −2.61372 | −1.74074 | 7.40077 | 1.00000 | 1.44223 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 6171.2.a.ba | yes | 6 |
| 11.b | odd | 2 | 1 | 6171.2.a.w | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6171.2.a.w | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 6171.2.a.ba | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6171))\):
|
\( T_{2}^{6} - T_{2}^{5} - 8T_{2}^{4} + 5T_{2}^{3} + 16T_{2}^{2} - 5T_{2} - 9 \)
|
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\( T_{5}^{6} - 16T_{5}^{4} + 8T_{5}^{3} + 28T_{5}^{2} - 8T_{5} - 4 \)
|
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\( T_{7}^{6} - 28T_{7}^{4} - 12T_{7}^{3} + 144T_{7}^{2} + 176T_{7} + 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} - 8 T^{4} + \cdots - 9 \)
|
| $3$ |
\( (T + 1)^{6} \)
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| $5$ |
\( T^{6} - 16 T^{4} + \cdots - 4 \)
|
| $7$ |
\( T^{6} - 28 T^{4} + \cdots + 36 \)
|
| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} + 14 T^{5} + \cdots - 1519 \)
|
| $17$ |
\( (T - 1)^{6} \)
|
| $19$ |
\( T^{6} + 8 T^{5} + \cdots - 1117 \)
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| $23$ |
\( T^{6} + 4 T^{5} + \cdots - 7936 \)
|
| $29$ |
\( T^{6} + 12 T^{5} + \cdots - 132 \)
|
| $31$ |
\( T^{6} - 20 T^{5} + \cdots - 2628 \)
|
| $37$ |
\( T^{6} - 8 T^{5} + \cdots + 1344 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots + 4032 \)
|
| $43$ |
\( T^{6} - 151 T^{4} + \cdots - 60453 \)
|
| $47$ |
\( T^{6} - 127 T^{4} + \cdots - 9029 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 48784 \)
|
| $59$ |
\( T^{6} - 120 T^{4} + \cdots - 1792 \)
|
| $61$ |
\( T^{6} + 16 T^{5} + \cdots + 3836 \)
|
| $67$ |
\( T^{6} + 32 T^{5} + \cdots + 3603 \)
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| $71$ |
\( T^{6} - 180 T^{4} + \cdots - 3652 \)
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| $73$ |
\( T^{6} + 8 T^{5} + \cdots - 704 \)
|
| $79$ |
\( T^{6} - 272 T^{4} + \cdots + 87296 \)
|
| $83$ |
\( T^{6} - 12 T^{5} + \cdots + 187099 \)
|
| $89$ |
\( T^{6} + 14 T^{5} + \cdots + 225477 \)
|
| $97$ |
\( T^{6} - 28 T^{5} + \cdots - 1868 \)
|
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