Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,94,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, 94, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 94);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 94 \) |
| 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: | \(492.952887545\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - x^{6} + \cdots - 13\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{88}\cdot 3^{47}\cdot 5^{10}\cdot 7^{6}\cdot 13^{2}\cdot 19\cdot 23\cdot 31^{2} \) |
| 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
\( x^{7} - x^{6} + \cdots - 13\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 576\nu - 82 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 42\!\cdots\!39 \nu^{6} + \cdots - 25\!\cdots\!36 ) / 15\!\cdots\!56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 42\!\cdots\!39 \nu^{6} + \cdots + 22\!\cdots\!20 ) / 15\!\cdots\!56 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\!\cdots\!91 \nu^{6} + \cdots - 43\!\cdots\!40 ) / 76\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 12\!\cdots\!81 \nu^{6} + \cdots - 44\!\cdots\!20 ) / 38\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 58\!\cdots\!23 \nu^{6} + \cdots + 85\!\cdots\!36 ) / 21\!\cdots\!08 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 82 ) / 576 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 4721797413976\beta _1 + 15212166611438802125090782379 ) / 331776 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 27 \beta_{6} + 20 \beta_{5} + 37744530 \beta_{4} + 8524894457564 \beta_{3} + \cdots - 44\!\cdots\!87 ) / 11943936 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4463654783831 \beta_{6} + 798238219324828 \beta_{5} + \cdots + 34\!\cdots\!95 ) / 107495424 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 10\!\cdots\!11 \beta_{6} + \cdots - 41\!\cdots\!29 ) / 483729408 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 36\!\cdots\!43 \beta_{6} + \cdots + 70\!\cdots\!83 ) / 2902376448 \)
|
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 |
|
−1.64459e14 | 0 | 1.71431e28 | −3.18897e32 | 0 | −2.05341e39 | −1.19061e42 | 0 | 5.24454e46 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.55678e14 | 0 | 1.43322e28 | 3.85879e32 | 0 | −1.93687e39 | −6.89447e41 | 0 | −6.00730e46 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.17029e13 | 0 | −3.22816e27 | −2.56302e32 | 0 | 2.42990e39 | 1.07290e42 | 0 | 2.09406e46 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 5.20495e12 | 0 | −9.87643e27 | 3.34454e32 | 0 | −2.46426e38 | −1.02954e41 | 0 | 1.74082e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.05764e13 | 0 | −7.34554e27 | −1.20105e32 | 0 | −1.85159e39 | −8.72396e41 | 0 | −6.07451e45 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.35097e14 | 0 | 8.34770e27 | −3.52234e32 | 0 | −7.46408e38 | −2.10186e41 | 0 | −4.75858e46 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.67226e14 | 0 | 1.80609e28 | 5.69707e32 | 0 | 3.48367e39 | 1.36413e42 | 0 | 9.52696e46 | ||||||||||||||||||||||||||||||||||||||||
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 | 9.94.a.b | 7 | |
| 3.b | odd | 2 | 1 | 1.94.a.a | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.94.a.a | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 9.94.a.b | 7 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 43735426713792 T_{2}^{6} + \cdots + 12\!\cdots\!12 \)
acting on \(S_{94}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 12\!\cdots\!12 \)
|
| $3$ |
\( T^{7} \)
|
| $5$ |
\( T^{7} + \cdots - 25\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots + 11\!\cdots\!48 \)
|
| $11$ |
\( T^{7} + \cdots + 97\!\cdots\!68 \)
|
| $13$ |
\( T^{7} + \cdots - 68\!\cdots\!44 \)
|
| $17$ |
\( T^{7} + \cdots - 99\!\cdots\!68 \)
|
| $19$ |
\( T^{7} + \cdots + 60\!\cdots\!00 \)
|
| $23$ |
\( T^{7} + \cdots + 14\!\cdots\!84 \)
|
| $29$ |
\( T^{7} + \cdots - 26\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots + 60\!\cdots\!52 \)
|
| $37$ |
\( T^{7} + \cdots - 17\!\cdots\!92 \)
|
| $41$ |
\( T^{7} + \cdots - 53\!\cdots\!12 \)
|
| $43$ |
\( T^{7} + \cdots - 21\!\cdots\!64 \)
|
| $47$ |
\( T^{7} + \cdots + 59\!\cdots\!72 \)
|
| $53$ |
\( T^{7} + \cdots - 67\!\cdots\!96 \)
|
| $59$ |
\( T^{7} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( T^{7} + \cdots + 11\!\cdots\!32 \)
|
| $67$ |
\( T^{7} + \cdots - 40\!\cdots\!32 \)
|
| $71$ |
\( T^{7} + \cdots + 47\!\cdots\!08 \)
|
| $73$ |
\( T^{7} + \cdots - 86\!\cdots\!84 \)
|
| $79$ |
\( T^{7} + \cdots + 23\!\cdots\!00 \)
|
| $83$ |
\( T^{7} + \cdots + 27\!\cdots\!24 \)
|
| $89$ |
\( T^{7} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 50\!\cdots\!72 \)
|
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