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: |
\( x^{4} - 68800128x^{2} - 210523015768x - 161308399173705 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{3}\cdot 5^{2}\cdot 7 \) |
| 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
\( x^{4} - 68800128x^{2} - 210523015768x - 161308399173705 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 265\nu^{3} - 481997\nu^{2} - 17110895423\nu - 25260796713621 ) / 299910156 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 129\nu^{3} + 8819259\nu^{2} - 24726052263\nu - 323751150815617 ) / 99970052 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -943\nu^{3} + 829475\nu^{2} + 70707905273\nu + 120358413075411 ) / 13039572 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{3} + 3\beta_{2} + 323\beta _1 + 79 ) / 2520 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3622\beta_{3} + 16629\beta_{2} + 272159\beta _1 + 43344143617 ) / 1260 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 90484488\beta_{3} + 84733261\beta_{2} + 8232644601\beta _1 + 132631514290553 ) / 840 \)
|
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 | −18411.5 | 65536.0 | −390625. | −4.71333e6 | 5.76480e6 | 1.67772e7 | 2.09842e8 | −1.00000e8 | ||||||||||||||||||||||||||||||
| 1.2 | 256.000 | −12589.4 | 65536.0 | −390625. | −3.22289e6 | 5.76480e6 | 1.67772e7 | 2.93533e7 | −1.00000e8 | |||||||||||||||||||||||||||||||
| 1.3 | 256.000 | −1002.61 | 65536.0 | −390625. | −256667. | 5.76480e6 | 1.67772e7 | −1.28135e8 | −1.00000e8 | |||||||||||||||||||||||||||||||
| 1.4 | 256.000 | 12030.5 | 65536.0 | −390625. | 3.07980e6 | 5.76480e6 | 1.67772e7 | 1.55924e7 | −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.e | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.18.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 19973T_{3}^{3} - 122146245T_{3}^{2} - 2930075563833T_{3} - 2795810218789896 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(70))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 256)^{4} \)
|
| $3$ |
\( T^{4} + \cdots - 27\!\cdots\!96 \)
|
| $5$ |
\( (T + 390625)^{4} \)
|
| $7$ |
\( (T - 5764801)^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 11\!\cdots\!04 \)
|
| $13$ |
\( T^{4} + \cdots + 51\!\cdots\!82 \)
|
| $17$ |
\( T^{4} + \cdots + 38\!\cdots\!66 \)
|
| $19$ |
\( T^{4} + \cdots + 10\!\cdots\!40 \)
|
| $23$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 26\!\cdots\!90 \)
|
| $31$ |
\( T^{4} + \cdots + 23\!\cdots\!92 \)
|
| $37$ |
\( T^{4} + \cdots - 40\!\cdots\!24 \)
|
| $41$ |
\( T^{4} + \cdots - 43\!\cdots\!88 \)
|
| $43$ |
\( T^{4} + \cdots + 17\!\cdots\!28 \)
|
| $47$ |
\( T^{4} + \cdots - 60\!\cdots\!88 \)
|
| $53$ |
\( T^{4} + \cdots + 12\!\cdots\!68 \)
|
| $59$ |
\( T^{4} + \cdots - 12\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots - 11\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 38\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots + 47\!\cdots\!72 \)
|
| $73$ |
\( T^{4} + \cdots + 49\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 65\!\cdots\!40 \)
|
| $83$ |
\( T^{4} + \cdots + 20\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 64\!\cdots\!50 \)
|
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