Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [644,4,Mod(1,644)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, 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("644.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 644 = 2^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 644.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: | \(37.9972300437\) |
| 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: |
\( x^{6} - 2x^{5} - 85x^{4} + 112x^{3} + 1778x^{2} - 1532x - 704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{6} - 2x^{5} - 85x^{4} + 112x^{3} + 1778x^{2} - 1532x - 704 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} - 44\nu^{3} + 266\nu^{2} + 196\nu + 56 ) / 36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} - 32\nu^{3} + 254\nu^{2} - 356\nu + 212 ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} - 44\nu^{3} + 302\nu^{2} + 160\nu - 988 ) / 36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{3} - 40\nu^{2} + 268\nu - 18 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{2} + \beta _1 + 29 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} - 4\beta_{2} + 47\beta _1 + 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{5} + 47\beta_{4} + 21\beta_{3} - 68\beta_{2} + 101\beta _1 + 1290 \)
|
| \(\nu^{5}\) | \(=\) |
\( 42\beta_{5} + 107\beta_{4} + 279\beta_{3} - 350\beta_{2} + 2313\beta _1 + 1964 \)
|
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 | −6.31651 | 0 | −18.8482 | 0 | −7.00000 | 0 | 12.8984 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.16713 | 0 | 6.60663 | 0 | −7.00000 | 0 | 11.0334 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.333695 | 0 | 0.604431 | 0 | −7.00000 | 0 | −26.8886 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.18726 | 0 | 16.0686 | 0 | −7.00000 | 0 | −25.5904 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 5.90459 | 0 | 2.71955 | 0 | −7.00000 | 0 | 7.86421 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 7.72548 | 0 | −7.15099 | 0 | −7.00000 | 0 | 32.6830 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(1\) |
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 | 644.4.a.b | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 644.4.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2T_{3}^{5} - 85T_{3}^{4} + 112T_{3}^{3} + 1778T_{3}^{2} - 1532T_{3} - 704 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(644))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 704 \)
|
| $5$ |
\( T^{6} - 358 T^{4} + \cdots + 23520 \)
|
| $7$ |
\( (T + 7)^{6} \)
|
| $11$ |
\( T^{6} - 42 T^{5} + \cdots + 855680 \)
|
| $13$ |
\( T^{6} + \cdots + 1174055040 \)
|
| $17$ |
\( T^{6} + \cdots + 3830120448 \)
|
| $19$ |
\( T^{6} + \cdots - 31983735168 \)
|
| $23$ |
\( (T + 23)^{6} \)
|
| $29$ |
\( T^{6} + \cdots + 291282885120 \)
|
| $31$ |
\( T^{6} + \cdots - 1094914089504 \)
|
| $37$ |
\( T^{6} + \cdots - 259606066176 \)
|
| $41$ |
\( T^{6} + \cdots + 1541616609456 \)
|
| $43$ |
\( T^{6} + \cdots - 6416661381120 \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
|
| $53$ |
\( T^{6} + \cdots + 3270111926400 \)
|
| $59$ |
\( T^{6} + \cdots - 289897121243136 \)
|
| $61$ |
\( T^{6} + \cdots - 676205230169056 \)
|
| $67$ |
\( T^{6} + \cdots + 389170521389056 \)
|
| $71$ |
\( T^{6} + \cdots + 364503618902016 \)
|
| $73$ |
\( T^{6} + \cdots - 10\!\cdots\!32 \)
|
| $79$ |
\( T^{6} + \cdots - 49\!\cdots\!24 \)
|
| $83$ |
\( T^{6} + \cdots - 701557173577216 \)
|
| $89$ |
\( T^{6} + \cdots + 87\!\cdots\!28 \)
|
| $97$ |
\( T^{6} + \cdots + 388748311443008 \)
|
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