Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,7,Mod(7,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.7");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.d (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: | \(2.76064900344\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.50898483.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{5} \) |
| 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
\( x^{6} + 8x^{4} - 10x^{3} + 64x^{2} - 40x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 5\nu^{4} - 13\nu^{3} - 35\nu^{2} - 79\nu - 175 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -9\nu^{5} - 72\nu^{3} + 45\nu^{2} - 576\nu + 180 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 25\nu^{4} + 31\nu^{3} - 210\nu^{2} + 693\nu - 1175 ) / 20 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -33\nu^{5} - 15\nu^{4} - 279\nu^{3} + 285\nu^{2} - 1917\nu + 735 ) / 20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 63\nu^{5} + 10\nu^{4} + 314\nu^{3} - 715\nu^{2} + 3262\nu - 1170 ) / 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} + 5\beta_{3} + 4\beta_{2} - 29\beta _1 - 11 ) / 144 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 10\beta_{4} - 5\beta_{3} - 44\beta_{2} - 7\beta _1 - 385 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} - 4\beta_{4} - \beta_{3} - \beta_{2} + 13\beta _1 + 94 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 21\beta_{5} - 54\beta_{4} + 33\beta_{3} + 412\beta_{2} - 537\beta _1 - 3255 ) / 144 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 59\beta_{5} + 178\beta_{4} - 281\beta_{3} - 732\beta_{2} + 989\beta _1 - 4357 ) / 144 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−7.12493 | − | 3.63805i | 15.5885i | 37.5291 | + | 51.8417i | 172.232 | 56.7117 | − | 111.067i | 545.623i | −78.7890 | − | 505.901i | −243.000 | −1227.14 | − | 626.588i | ||||||||||||||||||||||||||
| 7.2 | −7.12493 | + | 3.63805i | − | 15.5885i | 37.5291 | − | 51.8417i | 172.232 | 56.7117 | + | 111.067i | − | 545.623i | −78.7890 | + | 505.901i | −243.000 | −1227.14 | + | 626.588i | |||||||||||||||||||||||||
| 7.3 | −5.33979 | − | 5.95706i | − | 15.5885i | −6.97319 | + | 63.6190i | −212.349 | −92.8614 | + | 83.2392i | − | 87.6116i | 416.218 | − | 298.173i | −243.000 | 1133.90 | + | 1264.98i | |||||||||||||||||||||||||
| 7.4 | −5.33979 | + | 5.95706i | 15.5885i | −6.97319 | − | 63.6190i | −212.349 | −92.8614 | − | 83.2392i | 87.6116i | 416.218 | + | 298.173i | −243.000 | 1133.90 | − | 1264.98i | |||||||||||||||||||||||||||
| 7.5 | 7.46472 | − | 2.87714i | − | 15.5885i | 47.4441 | − | 42.9542i | 18.1171 | −44.8502 | − | 116.363i | 321.465i | 230.571 | − | 457.144i | −243.000 | 135.239 | − | 52.1255i | ||||||||||||||||||||||||||
| 7.6 | 7.46472 | + | 2.87714i | 15.5885i | 47.4441 | + | 42.9542i | 18.1171 | −44.8502 | + | 116.363i | − | 321.465i | 230.571 | + | 457.144i | −243.000 | 135.239 | + | 52.1255i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.7.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 36.7.d.e | 6 | ||
| 4.b | odd | 2 | 1 | inner | 12.7.d.a | ✓ | 6 |
| 8.b | even | 2 | 1 | 192.7.g.e | 6 | ||
| 8.d | odd | 2 | 1 | 192.7.g.e | 6 | ||
| 12.b | even | 2 | 1 | 36.7.d.e | 6 | ||
| 16.e | even | 4 | 2 | 768.7.b.h | 12 | ||
| 16.f | odd | 4 | 2 | 768.7.b.h | 12 | ||
| 24.f | even | 2 | 1 | 576.7.g.p | 6 | ||
| 24.h | odd | 2 | 1 | 576.7.g.p | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.7.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 12.7.d.a | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
| 36.7.d.e | 6 | 3.b | odd | 2 | 1 | ||
| 36.7.d.e | 6 | 12.b | even | 2 | 1 | ||
| 192.7.g.e | 6 | 8.b | even | 2 | 1 | ||
| 192.7.g.e | 6 | 8.d | odd | 2 | 1 | ||
| 576.7.g.p | 6 | 24.f | even | 2 | 1 | ||
| 576.7.g.p | 6 | 24.h | odd | 2 | 1 | ||
| 768.7.b.h | 12 | 16.e | even | 4 | 2 | ||
| 768.7.b.h | 12 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 10 T^{5} + \cdots + 262144 \)
|
| $3$ |
\( (T^{2} + 243)^{3} \)
|
| $5$ |
\( (T^{3} + 22 T^{2} + \cdots + 662600)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 236143965745152 \)
|
| $11$ |
\( T^{6} + \cdots + 36\!\cdots\!68 \)
|
| $13$ |
\( (T^{3} + 1674 T^{2} + \cdots - 94471112)^{2} \)
|
| $17$ |
\( (T^{3} - 6110 T^{2} + \cdots + 2355174680)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 19\!\cdots\!48 \)
|
| $23$ |
\( T^{6} + \cdots + 26\!\cdots\!92 \)
|
| $29$ |
\( (T^{3} + 42430 T^{2} + \cdots + 391390375016)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 15\!\cdots\!28 \)
|
| $37$ |
\( (T^{3} + \cdots - 154889066690024)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots - 1472139052456)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 13\!\cdots\!68 \)
|
| $47$ |
\( T^{6} + \cdots + 54\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots - 201035875990744)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!28 \)
|
| $61$ |
\( (T^{3} + \cdots - 59\!\cdots\!84)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!12 \)
|
| $71$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 54\!\cdots\!28)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 12\!\cdots\!72 \)
|
| $83$ |
\( T^{6} + \cdots + 55\!\cdots\!52 \)
|
| $89$ |
\( (T^{3} + \cdots + 49\!\cdots\!24)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 65\!\cdots\!16)^{2} \)
|
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