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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 89x^{4} + 76x^{3} + 1552x^{2} - 2437x + 574 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 89x^{4} + 76x^{3} + 1552x^{2} - 2437x + 574 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 18\nu^{4} - 35\nu^{3} - 1309\nu^{2} - 2295\nu + 8966 ) / 432 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 6\nu^{4} - 415\nu^{3} + 319\nu^{2} + 6285\nu - 6674 ) / 432 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 2\nu^{4} - 99\nu^{3} + 3\nu^{2} + 2009\nu - 4778 ) / 144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 89\nu^{3} - 13\nu^{2} + 1539\nu - 1006 ) / 54 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( -7\beta_{5} + \beta_{4} + 10\beta_{3} + 3\beta_{2} + 56\beta _1 - 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( -54\beta_{5} + 69\beta_{4} + 42\beta_{3} + 15\beta_{2} + 45\beta _1 + 1621 \)
|
| \(\nu^{5}\) | \(=\) |
\( -582\beta_{5} + 102\beta_{4} + 903\beta_{3} + 267\beta_{2} + 3445\beta _1 + 951 \)
|
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 | −6.72305 | 4.00000 | −5.00000 | 13.4461 | 2.28447 | −8.00000 | 18.1993 | 10.0000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −4.46228 | 4.00000 | −5.00000 | 8.92456 | −21.8888 | −8.00000 | −7.08808 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 1.28936 | 4.00000 | −5.00000 | −2.57871 | −10.6852 | −8.00000 | −25.3376 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 2.34576 | 4.00000 | −5.00000 | −4.69151 | 26.5751 | −8.00000 | −21.4974 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 5.16923 | 4.00000 | −5.00000 | −10.3385 | −0.0207069 | −8.00000 | −0.279019 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 9.38098 | 4.00000 | −5.00000 | −18.7620 | −6.26483 | −8.00000 | 61.0027 | 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.c | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 670.4.a.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 7T_{3}^{5} - 69T_{3}^{4} + 402T_{3}^{3} + 815T_{3}^{2} - 4968T_{3} + 4400 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(670))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} - 7 T^{5} + \cdots + 4400 \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} + 10 T^{5} + \cdots + 1842 \)
|
| $11$ |
\( T^{6} + 39 T^{5} + \cdots + 968800 \)
|
| $13$ |
\( T^{6} - 85 T^{5} + \cdots + 189601624 \)
|
| $17$ |
\( T^{6} + \cdots - 42586357760 \)
|
| $19$ |
\( T^{6} + \cdots + 3914620832 \)
|
| $23$ |
\( T^{6} + \cdots + 63239595520 \)
|
| $29$ |
\( T^{6} + \cdots - 325374457500 \)
|
| $31$ |
\( T^{6} + \cdots + 15775394979840 \)
|
| $37$ |
\( T^{6} + \cdots + 1181221200 \)
|
| $41$ |
\( T^{6} + \cdots + 149417272503040 \)
|
| $43$ |
\( T^{6} + \cdots + 137884341428 \)
|
| $47$ |
\( T^{6} + \cdots - 41084866704000 \)
|
| $53$ |
\( T^{6} + \cdots - 95764008548352 \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!06 \)
|
| $61$ |
\( T^{6} + \cdots - 1855395681920 \)
|
| $67$ |
\( (T - 67)^{6} \)
|
| $71$ |
\( T^{6} + \cdots - 43\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 33\!\cdots\!60 \)
|
| $79$ |
\( T^{6} + \cdots + 52509336735744 \)
|
| $83$ |
\( T^{6} + \cdots + 18\!\cdots\!96 \)
|
| $89$ |
\( T^{6} + \cdots - 64\!\cdots\!25 \)
|
| $97$ |
\( T^{6} + \cdots - 11\!\cdots\!90 \)
|
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