Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,8,Mod(1,207)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.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: | \(64.6637002752\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 104x^{3} + 200x^{2} + 2037x - 3704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 23) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 104x^{3} + 200x^{2} + 2037x - 3704 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} + 13\nu^{3} - 693\nu^{2} - 791\nu + 16130 ) / 194 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{4} + 18\nu^{3} + 682\nu^{2} - 1140\nu - 6893 ) / 97 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 15\nu^{4} + 39\nu^{3} - 1303\nu^{2} - 2761\nu + 15410 ) / 194 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 3\beta_{2} + \beta _1 + 171 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{4} + 10\beta_{3} - \beta_{2} + 135\beta _1 + 55 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 55\beta_{4} - 13\beta_{3} - 129\beta_{2} + 52\beta _1 + 5485 ) / 2 \)
|
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 |
|
−13.7251 | 0 | 60.3796 | 364.747 | 0 | 165.110 | 928.099 | 0 | −5006.20 | |||||||||||||||||||||||||||||||||
| 1.2 | −6.41439 | 0 | −86.8556 | −258.464 | 0 | −1375.00 | 1378.17 | 0 | 1657.89 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.402251 | 0 | −127.838 | 384.733 | 0 | −88.4463 | −102.911 | 0 | 154.759 | ||||||||||||||||||||||||||||||||||
| 1.4 | 14.1446 | 0 | 72.0689 | −178.126 | 0 | 368.361 | −791.122 | 0 | −2519.52 | ||||||||||||||||||||||||||||||||||
| 1.5 | 21.5927 | 0 | 338.245 | −256.889 | 0 | −226.020 | 4539.77 | 0 | −5546.93 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
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 | 207.8.a.b | 5 | |
| 3.b | odd | 2 | 1 | 23.8.a.a | ✓ | 5 | |
| 12.b | even | 2 | 1 | 368.8.a.e | 5 | ||
| 15.d | odd | 2 | 1 | 575.8.a.a | 5 | ||
| 69.c | even | 2 | 1 | 529.8.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.8.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 207.8.a.b | 5 | 1.a | even | 1 | 1 | trivial | |
| 368.8.a.e | 5 | 12.b | even | 2 | 1 | ||
| 529.8.a.b | 5 | 69.c | even | 2 | 1 | ||
| 575.8.a.a | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 16T_{2}^{4} - 320T_{2}^{3} + 3136T_{2}^{2} + 25680T_{2} - 10816 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 16 T^{4} + \cdots - 10816 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots + 1659678082560 \)
|
| $7$ |
\( T^{5} + \cdots + 1671775278848 \)
|
| $11$ |
\( T^{5} + \cdots - 18\!\cdots\!80 \)
|
| $13$ |
\( T^{5} + \cdots - 17\!\cdots\!06 \)
|
| $17$ |
\( T^{5} + \cdots + 16\!\cdots\!00 \)
|
| $19$ |
\( T^{5} + \cdots + 51\!\cdots\!60 \)
|
| $23$ |
\( (T + 12167)^{5} \)
|
| $29$ |
\( T^{5} + \cdots + 20\!\cdots\!50 \)
|
| $31$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 18\!\cdots\!16 \)
|
| $41$ |
\( T^{5} + \cdots + 12\!\cdots\!30 \)
|
| $43$ |
\( T^{5} + \cdots - 97\!\cdots\!48 \)
|
| $47$ |
\( T^{5} + \cdots - 17\!\cdots\!12 \)
|
| $53$ |
\( T^{5} + \cdots - 16\!\cdots\!16 \)
|
| $59$ |
\( T^{5} + \cdots - 13\!\cdots\!48 \)
|
| $61$ |
\( T^{5} + \cdots + 54\!\cdots\!24 \)
|
| $67$ |
\( T^{5} + \cdots + 14\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots + 18\!\cdots\!20 \)
|
| $73$ |
\( T^{5} + \cdots - 38\!\cdots\!94 \)
|
| $79$ |
\( T^{5} + \cdots + 99\!\cdots\!08 \)
|
| $83$ |
\( T^{5} + \cdots - 25\!\cdots\!12 \)
|
| $89$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots + 84\!\cdots\!68 \)
|
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