Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1672,2,Mod(1,1672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1672, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1672.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1672 = 2^{3} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1672.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: | \(13.3509872180\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.57500224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 6\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 3\nu^{2} + 10\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 9\beta_{3} + 10\beta_{2} + 20\beta _1 + 14 \)
|
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 | −2.94641 | 0 | −3.26823 | 0 | −4.50474 | 0 | 5.68132 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.67361 | 0 | 3.02460 | 0 | −0.969288 | 0 | 4.14820 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.878814 | 0 | −1.66503 | 0 | 1.34977 | 0 | −2.22769 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.129383 | 0 | −1.42012 | 0 | 5.10098 | 0 | −2.98326 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0.648873 | 0 | 1.55597 | 0 | −3.00900 | 0 | −2.57896 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.72058 | 0 | −2.22719 | 0 | −1.96772 | 0 | −0.0396114 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(19\) | \(1\) |
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 | 1672.2.a.h | ✓ | 6 |
| 4.b | odd | 2 | 1 | 3344.2.a.y | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1672.2.a.h | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3344.2.a.y | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1672))\):
|
\( T_{3}^{6} + 4T_{3}^{5} - 2T_{3}^{4} - 16T_{3}^{3} + 8T_{3} - 1 \)
|
|
\( T_{5}^{6} + 4T_{5}^{5} - 8T_{5}^{4} - 46T_{5}^{3} - 20T_{5}^{2} + 88T_{5} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 4 T^{5} - 2 T^{4} - 16 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{6} + 4 T^{5} - 8 T^{4} - 46 T^{3} + \cdots + 81 \)
|
| $7$ |
\( T^{6} + 4 T^{5} - 23 T^{4} - 116 T^{3} + \cdots + 178 \)
|
| $11$ |
\( (T - 1)^{6} \)
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| $13$ |
\( T^{6} - 2 T^{5} - 67 T^{4} + 90 T^{3} + \cdots - 962 \)
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| $17$ |
\( T^{6} + 10 T^{5} - 6 T^{4} + \cdots + 1184 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} + 14 T^{5} + 5 T^{4} + \cdots - 4304 \)
|
| $29$ |
\( T^{6} - 6 T^{5} - 65 T^{4} + \cdots + 11038 \)
|
| $31$ |
\( T^{6} - 4 T^{5} - 156 T^{4} + \cdots + 82899 \)
|
| $37$ |
\( T^{6} + 4 T^{5} - 195 T^{4} + \cdots - 48848 \)
|
| $41$ |
\( T^{6} + 8 T^{5} - 97 T^{4} + \cdots - 4160 \)
|
| $43$ |
\( T^{6} + 26 T^{5} + 163 T^{4} + \cdots + 61972 \)
|
| $47$ |
\( T^{6} + 24 T^{5} + 64 T^{4} + \cdots - 32448 \)
|
| $53$ |
\( T^{6} + 8 T^{5} - 72 T^{4} + \cdots - 18176 \)
|
| $59$ |
\( T^{6} + 4 T^{5} - 111 T^{4} + \cdots - 5200 \)
|
| $61$ |
\( T^{6} + 6 T^{5} - 262 T^{4} + \cdots - 408608 \)
|
| $67$ |
\( T^{6} + 44 T^{5} + 660 T^{4} + \cdots - 96461 \)
|
| $71$ |
\( T^{6} + 16 T^{5} + 42 T^{4} + \cdots + 2319 \)
|
| $73$ |
\( T^{6} - 12 T^{5} - 276 T^{4} + \cdots - 470656 \)
|
| $79$ |
\( T^{6} - 6 T^{5} - 186 T^{4} + \cdots - 13152 \)
|
| $83$ |
\( T^{6} + 28 T^{5} + 117 T^{4} + \cdots + 516266 \)
|
| $89$ |
\( T^{6} - 239 T^{4} - 704 T^{3} + \cdots + 69376 \)
|
| $97$ |
\( T^{6} - 4 T^{5} - 259 T^{4} + \cdots + 144880 \)
|
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