Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,21,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, 21, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.5");
S:= CuspForms(chi, 21);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 21 \) |
| 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: | \(15.2108259062\) |
| 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: |
\( x^{6} + 3396x^{4} + 2813589x^{2} + 548136050 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{31}\cdot 3^{17} \) |
| 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
\( x^{6} + 3396x^{4} + 2813589x^{2} + 548136050 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 128\nu^{5} + 2553728\nu^{3} + 3960388352\nu ) / 75540465 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 238684 \nu^{5} - 8939700 \nu^{4} - 682799374 \nu^{3} - 23346920520 \nu^{2} + \cdots - 8830503352440 ) / 75540465 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 47804 \nu^{5} + 5363820 \nu^{4} - 137900582 \nu^{3} + 14008152312 \nu^{2} + \cdots + 5298302011464 ) / 15108093 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 1192876 \nu^{5} + 19667340 \nu^{4} + 3403143526 \nu^{3} + 155790363960 \nu^{2} + \cdots + 137638628515080 ) / 75540465 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 334052 \nu^{5} - 2979900 \nu^{4} + 953812298 \nu^{3} - 7782306840 \nu^{2} + 431213871014 \nu - 2943501117480 ) / 5036031 \)
|
| \(\nu\) | \(=\) |
\( ( -16\beta_{5} - 104\beta_{3} - 232\beta_{2} - 471\beta_1 ) / 746496 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 5\beta_{4} - 3\beta_{3} + 7\beta_{2} + \beta _1 - 7824384 ) / 6912 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 27152\beta_{5} + 179944\beta_{3} + 404072\beta_{2} + 26586231\beta_1 ) / 746496 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 765\beta_{5} - 6529\beta_{4} + 7703\beta_{3} - 24283\beta_{2} + 1939\beta _1 + 6803294976 ) / 3456 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -46661008\beta_{5} - 372247208\beta_{3} - 883436584\beta_{2} - 75296911287\beta_1 ) / 746496 \)
|
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 |
|
− | 724.077i | −57693.2 | + | 12580.8i | −524288. | 8.94833e6i | 9.10944e6 | + | 4.17744e7i | −3.40159e7 | 3.79625e8i | 3.17023e9 | − | 1.45165e9i | 6.47928e9 | |||||||||||||||||||||||||||||
| 5.2 | − | 724.077i | 25323.8 | + | 53343.1i | −524288. | − | 6.05191e6i | 3.86246e7 | − | 1.83364e7i | 1.14251e8 | 3.79625e8i | −2.20420e9 | + | 2.70170e9i | −4.38205e9 | |||||||||||||||||||||||||||||
| 5.3 | − | 724.077i | 41792.5 | − | 41715.4i | −524288. | 805527.i | −3.02052e7 | − | 3.02610e7i | −3.63571e8 | 3.79625e8i | 6.43531e6 | − | 3.48678e9i | 5.83264e8 | ||||||||||||||||||||||||||||||
| 5.4 | 724.077i | −57693.2 | − | 12580.8i | −524288. | − | 8.94833e6i | 9.10944e6 | − | 4.17744e7i | −3.40159e7 | − | 3.79625e8i | 3.17023e9 | + | 1.45165e9i | 6.47928e9 | |||||||||||||||||||||||||||||
| 5.5 | 724.077i | 25323.8 | − | 53343.1i | −524288. | 6.05191e6i | 3.86246e7 | + | 1.83364e7i | 1.14251e8 | − | 3.79625e8i | −2.20420e9 | − | 2.70170e9i | −4.38205e9 | ||||||||||||||||||||||||||||||
| 5.6 | 724.077i | 41792.5 | + | 41715.4i | −524288. | − | 805527.i | −3.02052e7 | + | 3.02610e7i | −3.63571e8 | − | 3.79625e8i | 6.43531e6 | + | 3.48678e9i | 5.83264e8 | |||||||||||||||||||||||||||||
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 | 6.21.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 6.21.b.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | 48.21.e.b | 6 | ||
| 12.b | even | 2 | 1 | 48.21.e.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.21.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6.21.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 48.21.e.b | 6 | 4.b | odd | 2 | 1 | ||
| 48.21.e.b | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{21}^{\mathrm{new}}(6, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 524288)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 42\!\cdots\!01 \)
|
| $5$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots - 14\!\cdots\!76)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 18\!\cdots\!68 \)
|
| $13$ |
\( (T^{3} + \cdots - 24\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 38\!\cdots\!48 \)
|
| $19$ |
\( (T^{3} + \cdots + 67\!\cdots\!52)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 46\!\cdots\!68 \)
|
| $29$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 68\!\cdots\!28)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 52\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 35\!\cdots\!08 \)
|
| $43$ |
\( (T^{3} + \cdots - 96\!\cdots\!52)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!08 \)
|
| $59$ |
\( T^{6} + \cdots + 58\!\cdots\!08 \)
|
| $61$ |
\( (T^{3} + \cdots + 94\!\cdots\!16)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 92\!\cdots\!56)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 19\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 79\!\cdots\!68)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 92\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots + 37\!\cdots\!48 \)
|
| $97$ |
\( (T^{3} + \cdots + 17\!\cdots\!92)^{2} \)
|
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