Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,6,Mod(1,864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, 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("864.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 864.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: | \(138.571620318\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - x^{5} - 286x^{4} + 1126x^{3} + 1345x^{2} - 3193x - 384 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 286x^{4} + 1126x^{3} + 1345x^{2} - 3193x - 384 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} - 13\nu^{4} - 887\nu^{3} + 6323\nu^{2} + 4558\nu - 43404 ) / 588 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 4\nu^{4} + 1462\nu^{3} - 2920\nu^{2} - 23349\nu + 2984 ) / 49 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -33\nu^{5} + 59\nu^{4} + 9337\nu^{3} - 44437\nu^{2} - 158\nu + 62148 ) / 588 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 171\nu^{5} + 71\nu^{4} - 48515\nu^{3} + 125687\nu^{2} + 335098\nu - 169068 ) / 196 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} + 32\nu^{4} + 6574\nu^{3} - 28300\nu^{2} - 18651\nu + 54584 ) / 49 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 6\beta_{3} - \beta_{2} + 6\beta _1 + 24 ) / 144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{4} + 25\beta_{3} + 9\beta_{2} + 113\beta _1 + 6876 ) / 72 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 277\beta_{5} + 2\beta_{4} - 1868\beta_{3} - 205\beta_{2} + 494\beta _1 - 60456 ) / 144 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -395\beta_{5} + 593\beta_{4} + 9856\beta_{3} + 2906\beta_{2} + 28793\beta _1 + 1858236 ) / 72 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 76957\beta_{5} - 2700\beta_{4} - 563154\beta_{3} - 71845\beta_{2} - 61626\beta _1 - 28707816 ) / 144 \)
|
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 | 0 | 0 | −74.5644 | 0 | −104.697 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −44.5328 | 0 | 5.49567 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −43.4968 | 0 | 238.200 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 43.4968 | 0 | −238.200 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 44.5328 | 0 | −5.49567 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0 | 0 | 74.5644 | 0 | 104.697 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 864.6.a.o | ✓ | 6 |
| 3.b | odd | 2 | 1 | 864.6.a.p | yes | 6 | |
| 4.b | odd | 2 | 1 | 864.6.a.p | yes | 6 | |
| 12.b | even | 2 | 1 | inner | 864.6.a.o | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 864.6.a.o | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 864.6.a.o | ✓ | 6 | 12.b | even | 2 | 1 | inner |
| 864.6.a.p | yes | 6 | 3.b | odd | 2 | 1 | |
| 864.6.a.p | yes | 6 | 4.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(864))\):
|
\( T_{5}^{6} - 9435T_{5}^{4} + 25297347T_{5}^{2} - 20861166313 \)
|
|
\( T_{7}^{6} - 67731T_{7}^{4} + 623992563T_{7}^{2} - 18784337377 \)
|
|
\( T_{11}^{3} + 231T_{11}^{2} - 363573T_{11} - 71232539 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 20861166313 \)
|
| $7$ |
\( T^{6} + \cdots - 18784337377 \)
|
| $11$ |
\( (T^{3} + 231 T^{2} + \cdots - 71232539)^{2} \)
|
| $13$ |
\( (T^{3} + 192 T^{2} + \cdots - 326879104)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 56\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots - 47\!\cdots\!28 \)
|
| $23$ |
\( (T^{3} + 690 T^{2} + \cdots - 7585480008)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 34\!\cdots\!52 \)
|
| $31$ |
\( T^{6} + \cdots - 14\!\cdots\!13 \)
|
| $37$ |
\( (T^{3} + 198 T^{2} + \cdots - 98644735960)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 65\!\cdots\!88 \)
|
| $43$ |
\( T^{6} + \cdots - 18\!\cdots\!28 \)
|
| $47$ |
\( (T^{3} - 7518 T^{2} + \cdots + 772525842200)^{2} \)
|
| $53$ |
\( T^{6} + \cdots - 19\!\cdots\!97 \)
|
| $59$ |
\( (T^{3} + \cdots + 7172534924864)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 25826641551360)^{2} \)
|
| $67$ |
\( T^{6} + \cdots - 24\!\cdots\!28 \)
|
| $71$ |
\( (T^{3} + \cdots - 19781756300032)^{2} \)
|
| $73$ |
\( (T^{3} - 30867 T^{2} + \cdots - 63501060321)^{2} \)
|
| $79$ |
\( T^{6} + \cdots - 39\!\cdots\!52 \)
|
| $83$ |
\( (T^{3} + \cdots + 125072922751749)^{2} \)
|
| $89$ |
\( T^{6} + \cdots - 10\!\cdots\!32 \)
|
| $97$ |
\( (T^{3} + \cdots + 11\!\cdots\!61)^{2} \)
|
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