Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1323,4,Mod(1,1323)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, 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("1323.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1323 = 3^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1323.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: | \(78.0595269376\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 54x^{6} + 887x^{4} - 4176x^{2} + 3136 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 38\nu^{4} - 367\nu^{2} + 680 ) / 264 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 38\nu^{5} - 367\nu^{3} + 680\nu ) / 264 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 38\nu^{4} - 103\nu^{2} - 3016 ) / 264 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} + 354\nu^{5} - 4945\nu^{3} + 13032\nu ) / 1056 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 6\nu^{4} - 821\nu^{2} + 2796 ) / 132 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 278\nu^{5} - 4739\nu^{3} + 22760\nu ) / 1056 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + 2\beta_{5} - \beta_{3} + 21\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 27\beta_{4} - 21\beta_{2} + 299 \)
|
| \(\nu^{5}\) | \(=\) |
\( -54\beta_{7} + 66\beta_{5} - 48\beta_{3} + 473\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 114\beta_{6} + 659\beta_{4} - 695\beta_{2} + 6904 \)
|
| \(\nu^{7}\) | \(=\) |
\( -1318\beta_{7} + 1774\beta_{5} - 1721\beta_{3} + 10947\beta_1 \)
|
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 |
|
−5.00098 | 0 | 17.0098 | 3.98256 | 0 | 0 | −45.0578 | 0 | −19.9167 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.67291 | 0 | 13.8361 | 15.5408 | 0 | 0 | −27.2713 | 0 | −72.6208 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.49657 | 0 | −1.76715 | −14.5857 | 0 | 0 | 24.3844 | 0 | 36.4142 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.959848 | 0 | −7.07869 | 10.2898 | 0 | 0 | 14.4733 | 0 | −9.87663 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.959848 | 0 | −7.07869 | −10.2898 | 0 | 0 | −14.4733 | 0 | −9.87663 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.49657 | 0 | −1.76715 | 14.5857 | 0 | 0 | −24.3844 | 0 | 36.4142 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.67291 | 0 | 13.8361 | −15.5408 | 0 | 0 | 27.2713 | 0 | −72.6208 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.00098 | 0 | 17.0098 | −3.98256 | 0 | 0 | 45.0578 | 0 | −19.9167 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1323.4.a.bl | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 1323.4.a.bl | ✓ | 8 |
| 7.b | odd | 2 | 1 | 1323.4.a.bm | yes | 8 | |
| 21.c | even | 2 | 1 | 1323.4.a.bm | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1323.4.a.bl | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1323.4.a.bl | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 1323.4.a.bm | yes | 8 | 7.b | odd | 2 | 1 | |
| 1323.4.a.bm | yes | 8 | 21.c | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):
|
\( T_{2}^{8} - 54T_{2}^{6} + 887T_{2}^{4} - 4176T_{2}^{2} + 3136 \)
|
|
\( T_{5}^{8} - 576T_{5}^{6} + 108362T_{5}^{4} - 7017984T_{5}^{2} + 86285521 \)
|
|
\( T_{13}^{4} + 168T_{13}^{3} + 8328T_{13}^{2} + 100608T_{13} - 932864 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 54 T^{6} + \cdots + 3136 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 576 T^{6} + \cdots + 86285521 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} - 4864 T^{6} + \cdots + 625681 \)
|
| $13$ |
\( (T^{4} + 168 T^{3} + \cdots - 932864)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 47\!\cdots\!96 \)
|
| $19$ |
\( (T^{4} - 144 T^{3} + \cdots - 53761631)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 12\!\cdots\!41 \)
|
| $29$ |
\( T^{8} + \cdots + 50\!\cdots\!36 \)
|
| $31$ |
\( (T^{4} + 60 T^{3} + \cdots - 21279671)^{2} \)
|
| $37$ |
\( (T^{4} - 296 T^{3} + \cdots - 2024956199)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 20\!\cdots\!61 \)
|
| $43$ |
\( (T^{4} + 936 T^{3} + \cdots - 1639683644)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 69\!\cdots\!36 \)
|
| $53$ |
\( T^{8} + \cdots + 19\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 16\!\cdots\!56 \)
|
| $61$ |
\( (T^{4} + 1200 T^{3} + \cdots + 451950064)^{2} \)
|
| $67$ |
\( (T^{4} - 179656 T^{2} + \cdots - 798350336)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 67\!\cdots\!61 \)
|
| $73$ |
\( (T^{4} + 912 T^{3} + \cdots - 33168592604)^{2} \)
|
| $79$ |
\( (T^{4} - 1184 T^{3} + \cdots + 898048804)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 45\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 61\!\cdots\!21 \)
|
| $97$ |
\( (T^{4} + 2856 T^{3} + \cdots - 356770409984)^{2} \)
|
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