Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,102,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, 102, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.1");
S:= CuspForms(chi, 102);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 102 \) |
| 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: | \(581.406281043\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - x^{7} + \cdots + 14\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{119}\cdot 3^{56}\cdot 5^{14}\cdot 7^{7}\cdot 11^{2}\cdot 13^{2}\cdot 17^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + \cdots + 14\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 96\nu - 12 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 19\!\cdots\!41 \nu^{7} + \cdots - 45\!\cdots\!28 ) / 68\!\cdots\!48 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19\!\cdots\!41 \nu^{7} + \cdots - 20\!\cdots\!36 ) / 68\!\cdots\!48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25\!\cdots\!97 \nu^{7} + \cdots - 20\!\cdots\!60 ) / 86\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14\!\cdots\!23 \nu^{7} + \cdots + 38\!\cdots\!40 ) / 10\!\cdots\!20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 33\!\cdots\!91 \nu^{7} + \cdots + 23\!\cdots\!56 ) / 98\!\cdots\!64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!67 \nu^{7} + \cdots - 10\!\cdots\!60 ) / 49\!\cdots\!20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 12 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 120674951473495\beta _1 + 3658465598508525014815957089408 ) / 9216 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + 76 \beta_{6} + 183 \beta_{5} - 1912855272 \beta_{4} - 287410639150763 \beta_{3} + \cdots + 44\!\cdots\!32 ) / 884736 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 6908623838013 \beta_{7} + \cdots + 62\!\cdots\!04 ) / 2654208 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 82\!\cdots\!57 \beta_{7} + \cdots + 24\!\cdots\!88 ) / 7962624 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 17\!\cdots\!11 \beta_{7} + \cdots + 12\!\cdots\!12 ) / 7962624 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 19\!\cdots\!67 \beta_{7} + \cdots + 39\!\cdots\!64 ) / 23887872 \)
|
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 |
|
−2.56375e15 | 0 | 4.03750e30 | −2.88877e35 | 0 | −6.25978e42 | −3.85125e45 | 0 | 7.40607e50 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.17839e15 | 0 | 2.21006e30 | 2.56275e35 | 0 | 8.01618e42 | 7.08493e44 | 0 | −5.58265e50 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.29597e15 | 0 | −8.55769e29 | 1.52987e35 | 0 | −2.30272e42 | 4.39472e45 | 0 | −1.98266e50 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −5.93582e14 | 0 | −2.18296e30 | −2.07725e34 | 0 | −6.15512e42 | 2.80068e45 | 0 | 1.23302e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.56863e14 | 0 | −2.46932e30 | −1.16158e35 | 0 | 2.79663e42 | −1.28550e45 | 0 | −2.98366e49 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.76666e15 | 0 | 5.85773e29 | 3.44135e35 | 0 | −5.35995e42 | −3.44415e45 | 0 | 6.07967e50 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.38447e15 | 0 | 3.15040e30 | −7.99157e34 | 0 | 3.14769e42 | 1.46670e45 | 0 | −1.90557e50 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.65868e15 | 0 | 4.53328e30 | −2.85912e35 | 0 | 3.30273e41 | 5.31200e45 | 0 | −7.60148e50 | |||||||||||||||||||||||||||||||||||||||||||
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 | 9.102.a.b | 8 | |
| 3.b | odd | 2 | 1 | 1.102.a.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.102.a.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 9.102.a.b | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 434989091795040 T_{2}^{7} + \cdots + 12\!\cdots\!16 \)
acting on \(S_{102}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 12\!\cdots\!16 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots - 21\!\cdots\!00 \)
|
| $7$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
| $11$ |
\( T^{8} + \cdots + 34\!\cdots\!96 \)
|
| $13$ |
\( T^{8} + \cdots - 42\!\cdots\!64 \)
|
| $17$ |
\( T^{8} + \cdots + 65\!\cdots\!16 \)
|
| $19$ |
\( T^{8} + \cdots - 58\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots - 47\!\cdots\!04 \)
|
| $29$ |
\( T^{8} + \cdots - 37\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots - 17\!\cdots\!24 \)
|
| $37$ |
\( T^{8} + \cdots - 65\!\cdots\!84 \)
|
| $41$ |
\( T^{8} + \cdots - 40\!\cdots\!84 \)
|
| $43$ |
\( T^{8} + \cdots - 46\!\cdots\!84 \)
|
| $47$ |
\( T^{8} + \cdots - 63\!\cdots\!84 \)
|
| $53$ |
\( T^{8} + \cdots - 10\!\cdots\!24 \)
|
| $59$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 12\!\cdots\!04 \)
|
| $67$ |
\( T^{8} + \cdots - 16\!\cdots\!84 \)
|
| $71$ |
\( T^{8} + \cdots + 16\!\cdots\!36 \)
|
| $73$ |
\( T^{8} + \cdots - 25\!\cdots\!04 \)
|
| $79$ |
\( T^{8} + \cdots - 26\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 11\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 28\!\cdots\!16 \)
|
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