Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [139,2,Mod(1,139)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(139, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("139.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 139 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 139.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: | \(1.10992058810\) |
| 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: |
\( x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 9\nu^{4} - 6\nu^{3} + 19\nu^{2} + 4\nu - 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 11\nu^{4} + 8\nu^{3} + 31\nu^{2} - 10\nu - 16 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 9\nu^{3} - 8\nu^{2} + 17\nu + 12 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 11\nu^{4} + 8\nu^{3} + 35\nu^{2} - 14\nu - 28 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 9\nu^{4} + 8\nu^{3} + 19\nu^{2} - 12\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} - \beta_{4} + \beta_{2} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 6\beta_{5} - 8\beta_{3} + 7\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{6} - 7\beta_{5} - 7\beta_{4} + 9\beta_{2} + 20\beta _1 - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{6} + 36\beta_{5} + \beta_{4} - 53\beta_{3} + \beta_{2} + 44\beta _1 + 75 \)
|
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 |
|
−2.48318 | 1.56460 | 4.16616 | 3.76669 | −3.88518 | −3.67776 | −5.37897 | −0.552027 | −9.35335 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.63568 | −3.05533 | 0.675438 | −0.392097 | 4.99753 | −2.41294 | 2.16656 | 6.33504 | 0.641344 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.780192 | 3.15417 | −1.39130 | 1.02606 | −2.46086 | −0.216718 | 2.64586 | 6.94878 | −0.800525 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.308806 | −1.39811 | −1.90464 | 2.83261 | 0.431745 | 4.16776 | 1.20578 | −1.04529 | −0.874728 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.44228 | 1.01681 | 0.0801788 | 2.10270 | 1.46653 | −1.94441 | −2.76892 | −1.96609 | 3.03269 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.28572 | −3.03631 | 3.22451 | 3.97653 | −6.94014 | −0.589281 | 2.79888 | 6.21915 | 9.08923 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.47985 | −0.245836 | 4.14965 | −2.31250 | −0.609636 | −0.326651 | 5.33082 | −2.93956 | −5.73465 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(139\) | \(-1\) |
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 | 139.2.a.c | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1251.2.a.k | 7 | ||
| 4.b | odd | 2 | 1 | 2224.2.a.o | 7 | ||
| 5.b | even | 2 | 1 | 3475.2.a.e | 7 | ||
| 7.b | odd | 2 | 1 | 6811.2.a.p | 7 | ||
| 8.b | even | 2 | 1 | 8896.2.a.be | 7 | ||
| 8.d | odd | 2 | 1 | 8896.2.a.bd | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 139.2.a.c | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1251.2.a.k | 7 | 3.b | odd | 2 | 1 | ||
| 2224.2.a.o | 7 | 4.b | odd | 2 | 1 | ||
| 3475.2.a.e | 7 | 5.b | even | 2 | 1 | ||
| 6811.2.a.p | 7 | 7.b | odd | 2 | 1 | ||
| 8896.2.a.bd | 7 | 8.d | odd | 2 | 1 | ||
| 8896.2.a.be | 7 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - T_{2}^{6} - 11T_{2}^{5} + 8T_{2}^{4} + 35T_{2}^{3} - 10T_{2}^{2} - 32T_{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(139))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - T^{6} - 11 T^{5} + \cdots - 8 \)
|
| $3$ |
\( T^{7} + 2 T^{6} + \cdots - 16 \)
|
| $5$ |
\( T^{7} - 11 T^{6} + \cdots - 83 \)
|
| $7$ |
\( T^{7} + 5 T^{6} + \cdots - 3 \)
|
| $11$ |
\( T^{7} - 2 T^{6} + \cdots + 229 \)
|
| $13$ |
\( T^{7} - 6 T^{6} + \cdots - 1 \)
|
| $17$ |
\( T^{7} - 5 T^{6} + \cdots - 144 \)
|
| $19$ |
\( T^{7} + 10 T^{6} + \cdots - 2432 \)
|
| $23$ |
\( T^{7} + T^{6} + \cdots - 944 \)
|
| $29$ |
\( T^{7} - 30 T^{6} + \cdots + 257409 \)
|
| $31$ |
\( T^{7} + 20 T^{6} + \cdots - 2001 \)
|
| $37$ |
\( T^{7} - 6 T^{6} + \cdots - 151706 \)
|
| $41$ |
\( T^{7} - 19 T^{6} + \cdots - 191472 \)
|
| $43$ |
\( T^{7} + 12 T^{6} + \cdots - 2528 \)
|
| $47$ |
\( T^{7} + 3 T^{6} + \cdots - 1519088 \)
|
| $53$ |
\( T^{7} - 38 T^{6} + \cdots - 3168 \)
|
| $59$ |
\( T^{7} + 14 T^{6} + \cdots + 3888 \)
|
| $61$ |
\( T^{7} - 4 T^{6} + \cdots + 38176 \)
|
| $67$ |
\( T^{7} - 9 T^{6} + \cdots - 70136 \)
|
| $71$ |
\( T^{7} - 24 T^{6} + \cdots + 1068511 \)
|
| $73$ |
\( T^{7} + 5 T^{6} + \cdots - 443952 \)
|
| $79$ |
\( T^{7} - 8 T^{6} + \cdots + 1205557 \)
|
| $83$ |
\( T^{7} + 9 T^{6} + \cdots - 1088879 \)
|
| $89$ |
\( T^{7} - 10 T^{6} + \cdots + 778513 \)
|
| $97$ |
\( T^{7} + 5 T^{6} + \cdots - 260544 \)
|
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