Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,29,Mod(5,6)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 29, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.5");
S:= CuspForms(chi, 29);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 29 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(29.8010845489\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + \cdots + 25\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{85}\cdot 3^{50}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + \cdots + 25\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 35\!\cdots\!63 \nu^{9} + \cdots - 18\!\cdots\!00 \nu ) / 11\!\cdots\!50 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 25\!\cdots\!93 \nu^{9} + \cdots - 15\!\cdots\!50 ) / 14\!\cdots\!25 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 16\!\cdots\!94 \nu^{9} + \cdots - 45\!\cdots\!50 ) / 14\!\cdots\!25 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 16\!\cdots\!53 \nu^{9} + \cdots + 43\!\cdots\!50 ) / 28\!\cdots\!50 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!04 \nu^{9} + \cdots - 18\!\cdots\!75 ) / 14\!\cdots\!25 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!71 \nu^{9} + \cdots + 48\!\cdots\!50 ) / 28\!\cdots\!50 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 81\!\cdots\!64 \nu^{9} + \cdots + 42\!\cdots\!50 ) / 14\!\cdots\!25 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 77\!\cdots\!79 \nu^{9} + \cdots + 10\!\cdots\!25 ) / 14\!\cdots\!25 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 81\!\cdots\!88 \nu^{9} + \cdots + 19\!\cdots\!75 ) / 14\!\cdots\!25 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 299\beta_{2} - 20954\beta _1 + 120 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 3524703 \beta_{9} + 7503320 \beta_{8} + 27620246 \beta_{7} + 907828 \beta_{6} + 25845866 \beta_{5} + 159947376 \beta_{4} + 691296382 \beta_{3} + \cdots - 49\!\cdots\!85 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 36\!\cdots\!75 \beta_{9} + 551355785305500 \beta_{8} + \cdots - 32\!\cdots\!95 ) / 96 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 32\!\cdots\!56 \beta_{9} + \cdots + 50\!\cdots\!95 ) / 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 11\!\cdots\!00 \beta_{9} + \cdots + 80\!\cdots\!45 ) / 288 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 78\!\cdots\!13 \beta_{9} + \cdots - 15\!\cdots\!85 ) / 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 64\!\cdots\!75 \beta_{9} + \cdots - 47\!\cdots\!10 ) / 192 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 40\!\cdots\!67 \beta_{9} + \cdots + 92\!\cdots\!65 ) / 72 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 46\!\cdots\!75 \beta_{9} + \cdots + 35\!\cdots\!60 ) / 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
− | 11585.2i | −4.66950e6 | − | 1.03566e6i | −1.34218e8 | − | 8.17387e9i | −1.19983e10 | + | 5.40972e10i | 9.16442e11 | 1.55494e12i | 2.07316e13 | + | 9.67200e12i | −9.46962e13 | ||||||||||||||||||||||||||||||||||||||||
| 5.2 | − | 11585.2i | −4.35859e6 | − | 1.96965e6i | −1.34218e8 | 1.12356e10i | −2.28189e10 | + | 5.04953e10i | −1.18961e12 | 1.55494e12i | 1.51177e13 | + | 1.71698e13i | 1.30167e14 | ||||||||||||||||||||||||||||||||||||||||||
| 5.3 | − | 11585.2i | −1.12657e6 | + | 4.64840e6i | −1.34218e8 | − | 2.00507e9i | 5.38528e10 | + | 1.30516e10i | −2.77395e11 | 1.55494e12i | −2.03385e13 | − | 1.04735e13i | −2.32292e13 | |||||||||||||||||||||||||||||||||||||||||
| 5.4 | − | 11585.2i | 1.23098e6 | − | 4.62185e6i | −1.34218e8 | 3.66400e9i | −5.35452e10 | − | 1.42612e10i | 1.09838e12 | 1.55494e12i | −1.98462e13 | − | 1.13788e13i | 4.24483e13 | ||||||||||||||||||||||||||||||||||||||||||
| 5.5 | − | 11585.2i | 4.52069e6 | − | 1.56210e6i | −1.34218e8 | − | 5.92767e9i | −1.80973e10 | − | 5.23733e10i | −8.68104e11 | 1.55494e12i | 1.79965e13 | − | 1.41236e13i | −6.86735e13 | |||||||||||||||||||||||||||||||||||||||||
| 5.6 | 11585.2i | −4.66950e6 | + | 1.03566e6i | −1.34218e8 | 8.17387e9i | −1.19983e10 | − | 5.40972e10i | 9.16442e11 | − | 1.55494e12i | 2.07316e13 | − | 9.67200e12i | −9.46962e13 | ||||||||||||||||||||||||||||||||||||||||||
| 5.7 | 11585.2i | −4.35859e6 | + | 1.96965e6i | −1.34218e8 | − | 1.12356e10i | −2.28189e10 | − | 5.04953e10i | −1.18961e12 | − | 1.55494e12i | 1.51177e13 | − | 1.71698e13i | 1.30167e14 | |||||||||||||||||||||||||||||||||||||||||
| 5.8 | 11585.2i | −1.12657e6 | − | 4.64840e6i | −1.34218e8 | 2.00507e9i | 5.38528e10 | − | 1.30516e10i | −2.77395e11 | − | 1.55494e12i | −2.03385e13 | + | 1.04735e13i | −2.32292e13 | ||||||||||||||||||||||||||||||||||||||||||
| 5.9 | 11585.2i | 1.23098e6 | + | 4.62185e6i | −1.34218e8 | − | 3.66400e9i | −5.35452e10 | + | 1.42612e10i | 1.09838e12 | − | 1.55494e12i | −1.98462e13 | + | 1.13788e13i | 4.24483e13 | |||||||||||||||||||||||||||||||||||||||||
| 5.10 | 11585.2i | 4.52069e6 | + | 1.56210e6i | −1.34218e8 | 5.92767e9i | −1.80973e10 | + | 5.23733e10i | −8.68104e11 | − | 1.55494e12i | 1.79965e13 | + | 1.41236e13i | −6.86735e13 | ||||||||||||||||||||||||||||||||||||||||||
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 | 6.29.b.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | inner | 6.29.b.a | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.29.b.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 6.29.b.a | ✓ | 10 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{29}^{\mathrm{new}}(6, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 134217728)^{5} \)
|
| $3$ |
\( T^{10} + 8805966 T^{9} + \cdots + 62\!\cdots\!01 \)
|
| $5$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $7$ |
\( (T^{5} + 320290748654 T^{4} + \cdots + 28\!\cdots\!64)^{2} \)
|
| $11$ |
\( T^{10} + \cdots + 16\!\cdots\!32 \)
|
| $13$ |
\( (T^{5} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{10} + \cdots + 25\!\cdots\!52 \)
|
| $19$ |
\( (T^{5} + \cdots + 34\!\cdots\!28)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 14\!\cdots\!72 \)
|
| $29$ |
\( T^{10} + \cdots + 25\!\cdots\!00 \)
|
| $31$ |
\( (T^{5} + \cdots - 88\!\cdots\!08)^{2} \)
|
| $37$ |
\( (T^{5} + \cdots - 41\!\cdots\!24)^{2} \)
|
| $41$ |
\( T^{10} + \cdots + 30\!\cdots\!72 \)
|
| $43$ |
\( (T^{5} + \cdots + 10\!\cdots\!52)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 93\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots + 16\!\cdots\!32 \)
|
| $59$ |
\( T^{10} + \cdots + 22\!\cdots\!32 \)
|
| $61$ |
\( (T^{5} + \cdots - 49\!\cdots\!44)^{2} \)
|
| $67$ |
\( (T^{5} + \cdots + 20\!\cdots\!36)^{2} \)
|
| $71$ |
\( T^{10} + \cdots + 38\!\cdots\!00 \)
|
| $73$ |
\( (T^{5} + \cdots - 56\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{5} + \cdots - 77\!\cdots\!08)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 38\!\cdots\!08 \)
|
| $89$ |
\( T^{10} + \cdots + 43\!\cdots\!12 \)
|
| $97$ |
\( (T^{5} + \cdots - 10\!\cdots\!12)^{2} \)
|
| show more | |
| show less |