Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [117,10,Mod(1,117)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(117, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("117.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.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: | \(60.2591928312\) |
| 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: |
\( x^{8} - 2613x^{6} + 2010020x^{4} - 399948096x^{2} + 6898597888 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{4}\cdot 7 \) |
| 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
\( x^{8} - 2613x^{6} + 2010020x^{4} - 399948096x^{2} + 6898597888 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{7} + 36753\nu^{5} - 20387252\nu^{3} - 4606636608\nu ) / 9832192 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -13\nu^{7} + 36753\nu^{5} - 30219444\nu^{3} + 5953137600\nu ) / 9832192 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} - 653 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -13\nu^{6} + 36753\nu^{4} - 26532372\nu^{2} + 2208301472 ) / 1229024 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -93\nu^{6} + 168385\nu^{4} - 68324212\nu^{2} + 2797233120 ) / 1229024 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -197\nu^{7} + 462409\nu^{5} - 303934644\nu^{3} + 43614769984\nu ) / 9832192 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 653 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 1074\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -13\beta_{6} + 93\beta_{5} + 1285\beta_{4} + 701591 \)
|
| \(\nu^{5}\) | \(=\) |
\( -104\beta_{7} - 53\beta_{3} + 1629\beta_{2} + 1256654\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -36753\beta_{6} + 168385\beta_{5} + 1591941\beta_{4} + 820633583 \)
|
| \(\nu^{7}\) | \(=\) |
\( -294024\beta_{7} + 1418411\beta_{3} + 2280861\beta_{2} + 1514096862\beta_1 \)
|
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 |
|
−36.0840 | 0 | 790.056 | −548.189 | 0 | −6372.32 | −10033.4 | 0 | 19780.9 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −31.9131 | 0 | 506.444 | 1635.60 | 0 | 9377.46 | 177.316 | 0 | −52197.1 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −16.5372 | 0 | −238.522 | −172.959 | 0 | 1545.12 | 12411.5 | 0 | 2860.24 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.36150 | 0 | −492.977 | −2361.10 | 0 | 327.747 | 4383.21 | 0 | 10298.0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.36150 | 0 | −492.977 | 2361.10 | 0 | 327.747 | −4383.21 | 0 | 10298.0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 16.5372 | 0 | −238.522 | 172.959 | 0 | 1545.12 | −12411.5 | 0 | 2860.24 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 31.9131 | 0 | 506.444 | −1635.60 | 0 | 9377.46 | −177.316 | 0 | −52197.1 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 36.0840 | 0 | 790.056 | 548.189 | 0 | −6372.32 | 10033.4 | 0 | 19780.9 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 117.10.a.g | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 117.10.a.g | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.10.a.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 117.10.a.g | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2613T_{2}^{6} + 2010020T_{2}^{4} - 399948096T_{2}^{2} + 6898597888 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(117))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 6898597888 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 30260923880096)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 39\!\cdots\!88 \)
|
| $13$ |
\( (T + 28561)^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 42\!\cdots\!28 \)
|
| $19$ |
\( (T^{4} + \cdots + 24\!\cdots\!24)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots - 22\!\cdots\!60)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 75\!\cdots\!48)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 43\!\cdots\!52 \)
|
| $43$ |
\( (T^{4} + \cdots + 65\!\cdots\!84)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 21\!\cdots\!92 \)
|
| $53$ |
\( T^{8} + \cdots + 22\!\cdots\!92 \)
|
| $59$ |
\( T^{8} + \cdots + 13\!\cdots\!32 \)
|
| $61$ |
\( (T^{4} + \cdots - 60\!\cdots\!64)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 36\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 26\!\cdots\!00 \)
|
| $73$ |
\( (T^{4} + \cdots - 57\!\cdots\!28)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 32\!\cdots\!20)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 27\!\cdots\!48 \)
|
| $89$ |
\( T^{8} + \cdots + 58\!\cdots\!28 \)
|
| $97$ |
\( (T^{4} + \cdots - 45\!\cdots\!20)^{2} \)
|
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