Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [608,4,Mod(1,608)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(608, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("608.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 608 = 2^{5} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 608.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: | \(35.8731612835\) |
| 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: |
\( x^{7} - 2x^{6} - 130x^{5} + 212x^{4} + 4589x^{3} - 4178x^{2} - 46788x + 7848 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{7} - 2x^{6} - 130x^{5} + 212x^{4} + 4589x^{3} - 4178x^{2} - 46788x + 7848 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 92\nu^{5} - 146\nu^{4} - 10320\nu^{3} + 8069\nu^{2} + 233100\nu - 70812 ) / 10944 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -11\nu^{6} + 14\nu^{5} + 1492\nu^{4} - 1050\nu^{3} - 48517\nu^{2} + 3180\nu + 219420 ) / 10944 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} - 4\nu^{5} + 274\nu^{4} + 72\nu^{3} + 18479\nu^{2} + 3684\nu - 515988 ) / 10944 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{6} + 5\nu^{5} + 883\nu^{4} - 489\nu^{3} - 20491\nu^{2} + 3432\nu + 59652 ) / 5472 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - 22\nu^{5} - 32\nu^{4} + 2106\nu^{3} - 4813\nu^{2} - 31380\nu + 80124 ) / 1824 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{6} - 4\beta_{5} - 2\beta_{4} + 4\beta_{3} - 6\beta_{2} + 61\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{6} - 95\beta_{5} + 77\beta_{4} + 107\beta_{3} + 6\beta_{2} - 22\beta _1 + 2296 \)
|
| \(\nu^{5}\) | \(=\) |
\( -446\beta_{6} - 418\beta_{5} - 254\beta_{4} + 430\beta_{3} - 552\beta_{2} + 4307\beta _1 - 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( 628\beta_{6} - 8625\beta_{5} + 5901\beta_{4} + 9273\beta_{3} + 684\beta_{2} - 3036\beta _1 + 163166 \)
|
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 |
|
0 | −9.91449 | 0 | −8.60335 | 0 | −34.9181 | 0 | 71.2970 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −7.44524 | 0 | 18.2823 | 0 | 6.49029 | 0 | 28.4317 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −5.53243 | 0 | −15.3317 | 0 | 23.3717 | 0 | 3.60773 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.16573 | 0 | 9.90212 | 0 | −23.9754 | 0 | −25.6411 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.90450 | 0 | −12.8220 | 0 | 11.9961 | 0 | −18.5639 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 4.20371 | 0 | 3.63602 | 0 | 8.86224 | 0 | −9.32886 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 7.94968 | 0 | −0.0633525 | 0 | −19.8268 | 0 | 36.1974 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(1\) |
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 | 608.4.a.i | ✓ | 7 |
| 4.b | odd | 2 | 1 | 608.4.a.l | yes | 7 | |
| 8.b | even | 2 | 1 | 1216.4.a.bh | 7 | ||
| 8.d | odd | 2 | 1 | 1216.4.a.be | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 608.4.a.i | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 608.4.a.l | yes | 7 | 4.b | odd | 2 | 1 | |
| 1216.4.a.be | 7 | 8.d | odd | 2 | 1 | ||
| 1216.4.a.bh | 7 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(608))\):
|
\( T_{3}^{7} + 9T_{3}^{6} - 97T_{3}^{5} - 797T_{3}^{4} + 2516T_{3}^{3} + 15424T_{3}^{2} - 26144T_{3} - 46208 \)
|
|
\( T_{5}^{7} + 5T_{5}^{6} - 447T_{5}^{5} - 2537T_{5}^{4} + 46234T_{5}^{3} + 193600T_{5}^{2} - 1101184T_{5} - 70528 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} + 9 T^{6} + \cdots - 46208 \)
|
| $5$ |
\( T^{7} + 5 T^{6} + \cdots - 70528 \)
|
| $7$ |
\( T^{7} + 28 T^{6} + \cdots + 267674508 \)
|
| $11$ |
\( T^{7} + \cdots + 8913577312 \)
|
| $13$ |
\( T^{7} + \cdots + 3972040064 \)
|
| $17$ |
\( T^{7} + \cdots + 1272145656338 \)
|
| $19$ |
\( (T + 19)^{7} \)
|
| $23$ |
\( T^{7} + \cdots + 43343115459584 \)
|
| $29$ |
\( T^{7} + \cdots + 53253690048 \)
|
| $31$ |
\( T^{7} + \cdots - 312246972080128 \)
|
| $37$ |
\( T^{7} + \cdots + 1907389311488 \)
|
| $41$ |
\( T^{7} + \cdots - 460009274261504 \)
|
| $43$ |
\( T^{7} + \cdots + 372474759986432 \)
|
| $47$ |
\( T^{7} + \cdots + 829054396295168 \)
|
| $53$ |
\( T^{7} + \cdots - 27\!\cdots\!48 \)
|
| $59$ |
\( T^{7} + \cdots - 56\!\cdots\!76 \)
|
| $61$ |
\( T^{7} + \cdots + 24\!\cdots\!64 \)
|
| $67$ |
\( T^{7} + \cdots + 42\!\cdots\!76 \)
|
| $71$ |
\( T^{7} + \cdots - 68\!\cdots\!76 \)
|
| $73$ |
\( T^{7} + \cdots - 10\!\cdots\!54 \)
|
| $79$ |
\( T^{7} + \cdots + 34\!\cdots\!96 \)
|
| $83$ |
\( T^{7} + \cdots - 16\!\cdots\!92 \)
|
| $89$ |
\( T^{7} + \cdots - 25\!\cdots\!36 \)
|
| $97$ |
\( T^{7} + \cdots + 26\!\cdots\!56 \)
|
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