Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,92,Mod(1,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 92, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 92);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 92 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.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: | \(471.979402479\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 2 x^{6} + \cdots + 37\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{83}\cdot 3^{43}\cdot 5^{8}\cdot 7^{6}\cdot 11\cdot 13^{3}\cdot 23 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 2 x^{6} + \cdots + 37\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 24\nu - 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 84\!\cdots\!37 \nu^{6} + \cdots - 39\!\cdots\!96 ) / 10\!\cdots\!04 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 84\!\cdots\!37 \nu^{6} + \cdots - 64\!\cdots\!60 ) / 10\!\cdots\!04 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13\!\cdots\!39 \nu^{6} + \cdots + 11\!\cdots\!00 ) / 62\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 75\!\cdots\!21 \nu^{6} + \cdots + 14\!\cdots\!00 ) / 78\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 72\!\cdots\!49 \nu^{6} + \cdots - 30\!\cdots\!00 ) / 62\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 7 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 1270960310652\beta _1 + 3285816439692620103304959270 ) / 576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 39 \beta_{6} + 1432 \beta_{5} - 12682235 \beta_{4} + 8849674811434 \beta_{3} + \cdots - 52\!\cdots\!74 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 141481024033993 \beta_{6} + \cdots + 27\!\cdots\!72 ) / 5184 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 24\!\cdots\!87 \beta_{6} + \cdots - 93\!\cdots\!02 ) / 7776 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 76\!\cdots\!74 \beta_{6} + \cdots + 84\!\cdots\!10 ) / 15552 \)
|
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 |
|
−7.65028e13 | 0 | 3.37680e27 | 4.45639e31 | 0 | 2.84237e38 | −6.89228e40 | 0 | −3.40927e45 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.15766e13 | 0 | 2.64733e27 | −1.09575e32 | 0 | −1.62560e38 | −1.22718e40 | 0 | 7.84298e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.16242e13 | 0 | −2.00827e27 | −4.97855e31 | 0 | −2.82571e38 | 9.69662e40 | 0 | 1.07657e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −9.51137e12 | 0 | −2.38541e27 | 1.05575e32 | 0 | −1.59557e38 | 4.62376e40 | 0 | −1.00417e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.49886e13 | 0 | −1.25168e27 | −4.98385e31 | 0 | 4.56727e38 | −1.30422e41 | 0 | −1.74378e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 5.62150e13 | 0 | 6.84242e26 | 3.78165e31 | 0 | 1.92974e38 | −1.00717e41 | 0 | 2.12585e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.41697e13 | 0 | 4.60866e27 | −2.26211e30 | 0 | −5.00335e38 | 1.79515e41 | 0 | −1.90401e44 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 9.92.a.b | 7 | |
| 3.b | odd | 2 | 1 | 1.92.a.a | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.92.a.a | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 9.92.a.b | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 3841716838056 T_{2}^{6} + \cdots - 18\!\cdots\!16 \)
acting on \(S_{92}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots - 18\!\cdots\!16 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + \cdots - 10\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 91\!\cdots\!64 \)
|
| $11$ |
\( T^{7} + \cdots - 17\!\cdots\!92 \)
|
| $13$ |
\( T^{7} + \cdots + 32\!\cdots\!28 \)
|
| $17$ |
\( T^{7} + \cdots - 95\!\cdots\!76 \)
|
| $19$ |
\( T^{7} + \cdots - 27\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots + 11\!\cdots\!32 \)
|
| $29$ |
\( T^{7} + \cdots - 35\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 10\!\cdots\!68 \)
|
| $37$ |
\( T^{7} + \cdots - 83\!\cdots\!44 \)
|
| $41$ |
\( T^{7} + \cdots + 10\!\cdots\!48 \)
|
| $43$ |
\( T^{7} + \cdots - 47\!\cdots\!52 \)
|
| $47$ |
\( T^{7} + \cdots + 22\!\cdots\!04 \)
|
| $53$ |
\( T^{7} + \cdots + 51\!\cdots\!12 \)
|
| $59$ |
\( T^{7} + \cdots - 36\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 38\!\cdots\!92 \)
|
| $67$ |
\( T^{7} + \cdots + 54\!\cdots\!76 \)
|
| $71$ |
\( T^{7} + \cdots + 75\!\cdots\!88 \)
|
| $73$ |
\( T^{7} + \cdots - 57\!\cdots\!32 \)
|
| $79$ |
\( T^{7} + \cdots - 67\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots - 12\!\cdots\!08 \)
|
| $89$ |
\( T^{7} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots + 53\!\cdots\!96 \)
|
| show more | |
| show less |