Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,19,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, 19, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.5");
S:= CuspForms(chi, 19);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 19 \) |
| 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: | \(12.3231682626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8797x^{4} + 1809980x^{3} + 107861490x^{2} - 7180095600x + 788142376800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{31}\cdot 3^{14} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 8797x^{4} + 1809980x^{3} + 107861490x^{2} - 7180095600x + 788142376800 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 7154 \nu^{5} - 39330 \nu^{4} - 61835324 \nu^{3} + 6710827644 \nu^{2} + 750396405600 \nu - 24732099541440 ) / 169163330175 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3792146903 \nu^{5} + 753972938703 \nu^{4} + 142676403256562 \nu^{3} + \cdots + 18\!\cdots\!40 ) / 12\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 161554484599 \nu^{5} + 49251373100433 \nu^{4} + \cdots - 14\!\cdots\!80 ) / 12\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 612110098999 \nu^{5} + 68045562827217 \nu^{4} - 331993319994418 \nu^{3} + \cdots + 52\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25222077880 \nu^{5} - 119659606848 \nu^{4} + 235802762330896 \nu^{3} + \cdots + 18\!\cdots\!40 ) / 392628089336175 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{5} - 3\beta_{4} + \beta_{3} + 190\beta_{2} - 49\beta _1 + 13824 ) / 41472 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -197\beta_{5} + 49\beta_{4} - 54\beta_{3} - 3301\beta_{2} - 362831\beta _1 + 60818688 ) / 20736 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3314\beta_{5} + 10500\beta_{4} - 27671\beta_{3} + 832333\beta_{2} - 2776374\beta _1 - 18583430400 ) / 20736 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1337893 \beta_{5} + 3415903 \beta_{4} + 1849470 \beta_{3} - 88763971 \beta_{2} + \cdots - 1005732681984 ) / 20736 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 320844943 \beta_{5} + 220908921 \beta_{4} - 230796632 \beta_{3} - 161968511 \beta_{2} + \cdots - 152243998149888 ) / 20736 \)
|
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 |
|
− | 362.039i | −19019.3 | − | 5068.08i | −131072. | − | 2.98937e6i | −1.83484e6 | + | 6.88573e6i | −4.92752e7 | 4.74531e7i | 3.36050e8 | + | 1.92783e8i | −1.08227e9 | ||||||||||||||||||||||||||||
| 5.2 | − | 362.039i | −3612.02 | − | 19348.7i | −131072. | 3.69488e6i | −7.00499e6 | + | 1.30769e6i | 3.18077e7 | 4.74531e7i | −3.61327e8 | + | 1.39776e8i | 1.33769e9 | ||||||||||||||||||||||||||||||
| 5.3 | − | 362.039i | 19502.4 | + | 2660.56i | −131072. | − | 1.30525e6i | 963226. | − | 7.06061e6i | 3.15845e7 | 4.74531e7i | 3.73263e8 | + | 1.03774e8i | −4.72550e8 | |||||||||||||||||||||||||||||
| 5.4 | 362.039i | −19019.3 | + | 5068.08i | −131072. | 2.98937e6i | −1.83484e6 | − | 6.88573e6i | −4.92752e7 | − | 4.74531e7i | 3.36050e8 | − | 1.92783e8i | −1.08227e9 | ||||||||||||||||||||||||||||||
| 5.5 | 362.039i | −3612.02 | + | 19348.7i | −131072. | − | 3.69488e6i | −7.00499e6 | − | 1.30769e6i | 3.18077e7 | − | 4.74531e7i | −3.61327e8 | − | 1.39776e8i | 1.33769e9 | |||||||||||||||||||||||||||||
| 5.6 | 362.039i | 19502.4 | − | 2660.56i | −131072. | 1.30525e6i | 963226. | + | 7.06061e6i | 3.15845e7 | − | 4.74531e7i | 3.73263e8 | − | 1.03774e8i | −4.72550e8 | ||||||||||||||||||||||||||||||
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 | 6.19.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 6.19.b.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | 48.19.e.d | 6 | ||
| 12.b | even | 2 | 1 | 48.19.e.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.19.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6.19.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 48.19.e.d | 6 | 4.b | odd | 2 | 1 | ||
| 48.19.e.d | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(6, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 131072)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 58\!\cdots\!69 \)
|
| $5$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots + 49\!\cdots\!88)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 24\!\cdots\!88 \)
|
| $13$ |
\( (T^{3} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 28\!\cdots\!52 \)
|
| $19$ |
\( (T^{3} + \cdots - 12\!\cdots\!32)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 11\!\cdots\!12 \)
|
| $29$ |
\( T^{6} + \cdots + 67\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 23\!\cdots\!88)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots - 28\!\cdots\!88)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 17\!\cdots\!48 \)
|
| $43$ |
\( (T^{3} + \cdots + 10\!\cdots\!64)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 44\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 35\!\cdots\!72 \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!68 \)
|
| $61$ |
\( (T^{3} + \cdots + 37\!\cdots\!36)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 28\!\cdots\!32)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 25\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 17\!\cdots\!88)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 45\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 52\!\cdots\!88 \)
|
| $97$ |
\( (T^{3} + \cdots - 53\!\cdots\!16)^{2} \)
|
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