Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [71,9,Mod(70,71)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(71, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("71.70");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 71 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 71.b (of order \(2\), degree \(1\), minimal) |
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: | \(28.9238813143\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.294755098673.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 7^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 36\nu^{2} - 7\nu - 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( -5\nu^{4} + 7\nu^{3} + 40\nu^{2} - 42\nu - 40 \)
|
| \(\beta_{5}\) | \(=\) |
\( 7\nu^{5} - 70\nu^{3} - 18\nu^{2} + 140\nu + 72 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{6} - 24\nu^{4} + 72\nu^{2} - 7\nu - 32 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - 2\beta_{3} ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{6} + \beta_{4} - 12\beta_{3} + 5\beta_{2} ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{2} + 8\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 40\beta_{6} + \beta_{5} + 10\beta_{4} - 80\beta_{3} + 50\beta_{2} + 18\beta_1 ) / 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - \beta_{3} + 12\beta_{2} + 60\beta _1 + 160 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/71\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 70.1 |
|
−31.9974 | 72.8107 | 767.835 | −1160.90 | −2329.76 | 0 | −16377.4 | −1259.60 | 37145.8 | |||||||||||||||||||||||||||||||||||||||
| 70.2 | −20.2680 | −128.390 | 154.792 | 710.182 | 2602.21 | 0 | 2051.29 | 9922.98 | −14394.0 | ||||||||||||||||||||||||||||||||||||||||
| 70.3 | −19.6321 | −2.81049 | 129.420 | −193.533 | 55.1758 | 0 | 2485.03 | −6553.10 | 3799.47 | ||||||||||||||||||||||||||||||||||||||||
| 70.4 | 6.72364 | 158.540 | −210.793 | 844.839 | 1065.97 | 0 | −3138.55 | 18573.9 | 5680.39 | ||||||||||||||||||||||||||||||||||||||||
| 70.5 | 7.51655 | −67.7464 | −199.501 | 1247.03 | −509.220 | 0 | −3423.80 | −1971.42 | 9373.37 | ||||||||||||||||||||||||||||||||||||||||
| 70.6 | 28.6522 | −157.289 | 564.951 | −1086.17 | −4506.69 | 0 | 8852.13 | 18178.9 | −31121.2 | ||||||||||||||||||||||||||||||||||||||||
| 70.7 | 29.0051 | 124.885 | 585.297 | −361.447 | 3622.31 | 0 | 9551.29 | 9035.35 | −10483.8 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 71.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-71}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 71.9.b.a | ✓ | 7 |
| 71.b | odd | 2 | 1 | CM | 71.9.b.a | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.9.b.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 71.9.b.a | ✓ | 7 | 71.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 1792T_{2}^{5} + 917504T_{2}^{3} - 117440512T_{2} + 534748063 \)
acting on \(S_{9}^{\mathrm{new}}(71, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 1792 T^{5} + \cdots + 534748063 \)
|
| $3$ |
\( T^{7} + \cdots - 5542979604322 \)
|
| $5$ |
\( T^{7} + \cdots - 65\!\cdots\!66 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} \)
|
| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} \)
|
| $19$ |
\( T^{7} + \cdots - 23\!\cdots\!82 \)
|
| $23$ |
\( T^{7} \)
|
| $29$ |
\( T^{7} + \cdots - 74\!\cdots\!22 \)
|
| $31$ |
\( T^{7} \)
|
| $37$ |
\( T^{7} + \cdots - 15\!\cdots\!42 \)
|
| $41$ |
\( T^{7} \)
|
| $43$ |
\( T^{7} + \cdots + 79\!\cdots\!98 \)
|
| $47$ |
\( T^{7} \)
|
| $53$ |
\( T^{7} \)
|
| $59$ |
\( T^{7} \)
|
| $61$ |
\( T^{7} \)
|
| $67$ |
\( T^{7} \)
|
| $71$ |
\( (T - 25411681)^{7} \)
|
| $73$ |
\( T^{7} + \cdots + 28\!\cdots\!38 \)
|
| $79$ |
\( T^{7} + \cdots - 26\!\cdots\!22 \)
|
| $83$ |
\( T^{7} + \cdots + 31\!\cdots\!18 \)
|
| $89$ |
\( T^{7} + \cdots - 71\!\cdots\!62 \)
|
| $97$ |
\( T^{7} \)
|
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