Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [670,4,Mod(1,670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(670, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("670.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 670 = 2 \cdot 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 670.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: | \(39.5312797038\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{7} - 51x^{5} - 39x^{4} + 693x^{3} + 1016x^{2} - 1428x - 1872 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 51x^{5} - 39x^{4} + 693x^{3} + 1016x^{2} - 1428x - 1872 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 91\nu^{6} - 258\nu^{5} - 7485\nu^{4} + 12309\nu^{3} + 163437\nu^{2} - 132550\nu - 695688 ) / 54228 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 383\nu^{6} - 192\nu^{5} - 16605\nu^{4} + 4431\nu^{3} + 167343\nu^{2} - 24236\nu - 254148 ) / 54228 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 297\nu^{6} - 1140\nu^{5} - 11319\nu^{4} + 37889\nu^{3} + 92341\nu^{2} - 230892\nu - 70832 ) / 18076 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 178\nu^{6} - 455\nu^{5} - 7788\nu^{4} + 12407\nu^{3} + 88923\nu^{2} - 30791\nu - 169866 ) / 9038 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -178\nu^{6} + 455\nu^{5} + 7788\nu^{4} - 12407\nu^{3} - 84404\nu^{2} + 21753\nu + 106600 ) / 4519 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 2\beta_{5} + 2\beta _1 + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{4} + 3\beta_{3} + 3\beta_{2} + 26\beta _1 + 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( 35\beta_{6} + 67\beta_{5} + 3\beta_{4} + 4\beta_{3} - 11\beta_{2} + 73\beta _1 + 323 \)
|
| \(\nu^{5}\) | \(=\) |
\( 52\beta_{6} - 5\beta_{5} + 56\beta_{4} + 153\beta_{3} + 87\beta_{2} + 729\beta _1 + 732 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1095\beta_{6} + 2040\beta_{5} + 135\beta_{4} + 357\beta_{3} - 468\beta_{2} + 2419\beta _1 + 8709 \)
|
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 |
|
2.00000 | −8.46543 | 4.00000 | 5.00000 | −16.9309 | −28.1459 | 8.00000 | 44.6635 | 10.0000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −7.00427 | 4.00000 | 5.00000 | −14.0085 | 28.4000 | 8.00000 | 22.0598 | 10.0000 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.87699 | 4.00000 | 5.00000 | −7.75398 | 8.58814 | 8.00000 | −11.9689 | 10.0000 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −1.84564 | 4.00000 | 5.00000 | −3.69128 | 9.27641 | 8.00000 | −23.5936 | 10.0000 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | −0.388796 | 4.00000 | 5.00000 | −0.777592 | −20.3416 | 8.00000 | −26.8488 | 10.0000 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 3.09264 | 4.00000 | 5.00000 | 6.18527 | −15.8184 | 8.00000 | −17.4356 | 10.0000 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 5.48850 | 4.00000 | 5.00000 | 10.9770 | −30.9587 | 8.00000 | 3.12360 | 10.0000 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(67\) | \(-1\) |
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 | 670.4.a.d | ✓ | 7 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 670.4.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} + 13T_{3}^{6} - 5T_{3}^{5} - 524T_{3}^{4} - 1009T_{3}^{3} + 3684T_{3}^{2} + 8756T_{3} + 2800 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(670))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{7} \)
|
| $3$ |
\( T^{7} + 13 T^{6} + \cdots + 2800 \)
|
| $5$ |
\( (T - 5)^{7} \)
|
| $7$ |
\( T^{7} + 49 T^{6} + \cdots - 634369110 \)
|
| $11$ |
\( T^{7} + \cdots - 89637351936 \)
|
| $13$ |
\( T^{7} + \cdots + 15670846432 \)
|
| $17$ |
\( T^{7} + \cdots - 836088611328 \)
|
| $19$ |
\( T^{7} + \cdots - 81379500395056 \)
|
| $23$ |
\( T^{7} + \cdots - 29926447799808 \)
|
| $29$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots + 61\!\cdots\!40 \)
|
| $37$ |
\( T^{7} + \cdots + 89\!\cdots\!28 \)
|
| $41$ |
\( T^{7} + \cdots - 11\!\cdots\!04 \)
|
| $43$ |
\( T^{7} + \cdots - 53\!\cdots\!64 \)
|
| $47$ |
\( T^{7} + \cdots - 42\!\cdots\!12 \)
|
| $53$ |
\( T^{7} + \cdots + 19\!\cdots\!48 \)
|
| $59$ |
\( T^{7} + \cdots + 13\!\cdots\!80 \)
|
| $61$ |
\( T^{7} + \cdots - 13\!\cdots\!68 \)
|
| $67$ |
\( (T - 67)^{7} \)
|
| $71$ |
\( T^{7} + \cdots + 22\!\cdots\!40 \)
|
| $73$ |
\( T^{7} + \cdots - 12\!\cdots\!28 \)
|
| $79$ |
\( T^{7} + \cdots - 12\!\cdots\!68 \)
|
| $83$ |
\( T^{7} + \cdots - 20\!\cdots\!80 \)
|
| $89$ |
\( T^{7} + \cdots + 67\!\cdots\!65 \)
|
| $97$ |
\( T^{7} + \cdots - 21\!\cdots\!84 \)
|
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