Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,6,Mod(1,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 819.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: | \(131.354348427\) |
| 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: |
\( x^{6} - 3x^{5} - 111x^{4} + 75x^{3} + 2750x^{2} + 1800x - 1600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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
\( x^{6} - 3x^{5} - 111x^{4} + 75x^{3} + 2750x^{2} + 1800x - 1600 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 6\nu^{4} - 77\nu^{3} + 178\nu^{2} + 1240\nu + 2080 ) / 160 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 6\nu^{4} - 77\nu^{3} + 338\nu^{2} + 760\nu - 3840 ) / 160 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 2\nu^{4} + 141\nu^{3} + 230\nu^{2} - 3720\nu - 2720 ) / 160 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 14\nu^{4} + 303\nu^{3} - 610\nu^{2} - 6880\nu - 320 ) / 160 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 3\beta _1 + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} - 2\beta_{4} + 7\beta_{3} + 3\beta_{2} + 69\beta _1 + 103 \)
|
| \(\nu^{4}\) | \(=\) |
\( 32\beta_{5} - 36\beta_{4} + 107\beta_{3} - 47\beta_{2} + 395\beta _1 + 2631 \)
|
| \(\nu^{5}\) | \(=\) |
\( 500\beta_{5} - 370\beta_{4} + 1003\beta_{3} + 287\beta_{2} + 5909\beta _1 + 15051 \)
|
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 |
|
−7.39489 | 0 | 22.6844 | 8.21614 | 0 | −49.0000 | 68.8876 | 0 | −60.7574 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.14462 | 0 | −22.1114 | −19.5650 | 0 | −49.0000 | 170.160 | 0 | 61.5245 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.49777 | 0 | −25.7611 | 68.1279 | 0 | −49.0000 | −144.274 | 0 | 170.168 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 4.23912 | 0 | −14.0298 | −51.7406 | 0 | −49.0000 | −195.126 | 0 | −219.335 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 8.55022 | 0 | 41.1063 | 6.55528 | 0 | −49.0000 | 77.8610 | 0 | 56.0491 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.2524 | 0 | 73.1116 | −26.5938 | 0 | −49.0000 | 421.492 | 0 | −272.650 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
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 | 819.6.a.f | 6 | |
| 3.b | odd | 2 | 1 | 273.6.a.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.6.a.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 819.6.a.f | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 15T_{2}^{5} - 21T_{2}^{4} + 987T_{2}^{3} - 2164T_{2}^{2} - 8580T_{2} + 21584 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(819))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 15 T^{5} + \cdots + 21584 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 15 T^{5} + \cdots - 98781400 \)
|
| $7$ |
\( (T + 49)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 489735592441216 \)
|
| $13$ |
\( (T - 169)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 11\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots - 29\!\cdots\!88 \)
|
| $23$ |
\( T^{6} + \cdots + 19\!\cdots\!92 \)
|
| $29$ |
\( T^{6} + \cdots - 43\!\cdots\!88 \)
|
| $31$ |
\( T^{6} + \cdots - 41\!\cdots\!24 \)
|
| $37$ |
\( T^{6} + \cdots + 19\!\cdots\!92 \)
|
| $41$ |
\( T^{6} + \cdots - 53\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 25\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots - 89\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 34\!\cdots\!32 \)
|
| $59$ |
\( T^{6} + \cdots - 85\!\cdots\!28 \)
|
| $61$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 44\!\cdots\!64 \)
|
| $71$ |
\( T^{6} + \cdots + 60\!\cdots\!20 \)
|
| $73$ |
\( T^{6} + \cdots + 63\!\cdots\!60 \)
|
| $79$ |
\( T^{6} + \cdots - 11\!\cdots\!12 \)
|
| $83$ |
\( T^{6} + \cdots - 25\!\cdots\!04 \)
|
| $89$ |
\( T^{6} + \cdots + 15\!\cdots\!60 \)
|
| $97$ |
\( T^{6} + \cdots + 18\!\cdots\!80 \)
|
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