Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,18,Mod(1,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.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: | \(128.255461141\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 147628703x^{2} - 106835879976x + 1396373230200024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{3}\cdot 5^{2}\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 147628703x^{2} - 106835879976x + 1396373230200024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -767\nu^{3} + 329891\nu^{2} + 112439865840\nu + 37224431644872 ) / 16621431360 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 62127\nu^{3} - 26721171\nu^{2} - 4453628352240\nu - 3017167994520712 ) / 2770238560 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 167773\nu^{3} - 4623008329\nu^{2} - 13122322862160\nu + 327770647489702632 ) / 16621431360 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 486\beta _1 + 718 ) / 1680 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -6136\beta_{3} + 2521\beta_{2} - 116978\beta _1 + 124008141718 ) / 1680 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2639128\beta_{3} + 147681253\beta_{2} + 34789022326\beta _1 + 134976462668254 ) / 1680 \)
|
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 |
|
−256.000 | −21152.0 | 65536.0 | 390625. | 5.41491e6 | 5.76480e6 | −1.67772e7 | 3.18267e8 | −1.00000e8 | ||||||||||||||||||||||||||||||
| 1.2 | −256.000 | −5770.92 | 65536.0 | 390625. | 1.47736e6 | 5.76480e6 | −1.67772e7 | −9.58366e7 | −1.00000e8 | |||||||||||||||||||||||||||||||
| 1.3 | −256.000 | 4536.13 | 65536.0 | 390625. | −1.16125e6 | 5.76480e6 | −1.67772e7 | −1.08564e8 | −1.00000e8 | |||||||||||||||||||||||||||||||
| 1.4 | −256.000 | 19105.8 | 65536.0 | 390625. | −4.89108e6 | 5.76480e6 | −1.67772e7 | 2.35891e8 | −1.00000e8 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 70.18.a.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.18.a.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3281T_{3}^{3} - 427776633T_{3}^{2} - 552575360997T_{3} + 10579075365787020 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(70))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 256)^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 10\!\cdots\!20 \)
|
| $5$ |
\( (T - 390625)^{4} \)
|
| $7$ |
\( (T - 5764801)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 39\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots + 89\!\cdots\!26 \)
|
| $17$ |
\( T^{4} + \cdots - 20\!\cdots\!02 \)
|
| $19$ |
\( T^{4} + \cdots + 36\!\cdots\!40 \)
|
| $23$ |
\( T^{4} + \cdots - 82\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 76\!\cdots\!26 \)
|
| $31$ |
\( T^{4} + \cdots - 37\!\cdots\!32 \)
|
| $37$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 26\!\cdots\!28 \)
|
| $43$ |
\( T^{4} + \cdots - 27\!\cdots\!28 \)
|
| $47$ |
\( T^{4} + \cdots + 65\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots - 64\!\cdots\!08 \)
|
| $59$ |
\( T^{4} + \cdots + 15\!\cdots\!68 \)
|
| $61$ |
\( T^{4} + \cdots - 15\!\cdots\!28 \)
|
| $67$ |
\( T^{4} + \cdots + 52\!\cdots\!12 \)
|
| $71$ |
\( T^{4} + \cdots + 63\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots - 38\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots - 16\!\cdots\!76 \)
|
| $83$ |
\( T^{4} + \cdots - 15\!\cdots\!32 \)
|
| $89$ |
\( T^{4} + \cdots + 14\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 17\!\cdots\!50 \)
|
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