Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,8,Mod(11,12)]
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, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.11");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.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: | \(3.74862030581\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 41x^{6} + 1008x^{4} + 41984x^{2} + 1048576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 41x^{6} + 1008x^{4} + 41984x^{2} + 1048576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 41\nu^{5} + 1008\nu^{3} + 41984\nu ) / 16384 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 32\nu^{6} - 87\nu^{5} - 288\nu^{4} - 144\nu^{3} - 23040\nu^{2} + 97280\nu - 827392 ) / 24576 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 9\nu^{4} + 1328\nu^{2} - 4864 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 64\nu^{6} - 87\nu^{5} + 3648\nu^{4} - 144\nu^{3} + 73728\nu^{2} + 97280\nu + 2195456 ) / 24576 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17\nu^{7} - 32\nu^{6} - 57\nu^{5} - 288\nu^{4} - 11376\nu^{3} - 23040\nu^{2} - 203776\nu - 827392 ) / 24576 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} - 64\nu^{6} + 133\nu^{5} - 576\nu^{4} - 720\nu^{3} - 46080\nu^{2} - 236544\nu - 1654784 ) / 16384 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -19\nu^{7} - 267\nu^{5} + 18224\nu^{3} - 68608\nu ) / 16384 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + \beta_{5} + 2\beta_{2} + 7\beta_1 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + \beta_{3} - 3\beta_{2} - \beta _1 - 164 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 16\beta_{7} + 13\beta_{6} - 29\beta_{5} - 10\beta_{2} + 21\beta_1 ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 73\beta_{6} + 128\beta_{4} - 9\beta_{3} + 91\beta_{2} + 73\beta _1 - 1340 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -144\beta_{7} + 475\beta_{6} + 693\beta_{5} - 2118\beta_{2} + 7955\beta_1 ) / 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1985\beta_{6} - 1152\beta_{4} - 639\beta_{3} - 4803\beta_{2} - 1985\beta _1 - 283556 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -10224\beta_{7} + 9405\beta_{6} - 41165\beta_{5} + 12950\beta_{2} - 116923\beta_1 ) / 32 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−9.03954 | − | 6.80344i | 46.2722 | + | 6.77368i | 35.4264 | + | 123.000i | − | 426.666i | −372.195 | − | 376.041i | − | 780.788i | 516.584 | − | 1352.88i | 2095.23 | + | 626.867i | −2902.79 | + | 3856.86i | ||||||||||||||||||||||||||
| 11.2 | −9.03954 | + | 6.80344i | 46.2722 | − | 6.77368i | 35.4264 | − | 123.000i | 426.666i | −372.195 | + | 376.041i | 780.788i | 516.584 | + | 1352.88i | 2095.23 | − | 626.867i | −2902.79 | − | 3856.86i | |||||||||||||||||||||||||||||
| 11.3 | −2.29930 | − | 11.0776i | −17.5181 | + | 43.3603i | −117.426 | + | 50.9415i | 151.910i | 520.608 | + | 94.3598i | 893.278i | 834.308 | + | 1183.67i | −1573.23 | − | 1519.18i | 1682.79 | − | 349.286i | |||||||||||||||||||||||||||||
| 11.4 | −2.29930 | + | 11.0776i | −17.5181 | − | 43.3603i | −117.426 | − | 50.9415i | − | 151.910i | 520.608 | − | 94.3598i | − | 893.278i | 834.308 | − | 1183.67i | −1573.23 | + | 1519.18i | 1682.79 | + | 349.286i | |||||||||||||||||||||||||||
| 11.5 | 2.29930 | − | 11.0776i | 17.5181 | − | 43.3603i | −117.426 | − | 50.9415i | 151.910i | −440.049 | − | 293.757i | − | 893.278i | −834.308 | + | 1183.67i | −1573.23 | − | 1519.18i | 1682.79 | + | 349.286i | ||||||||||||||||||||||||||||
| 11.6 | 2.29930 | + | 11.0776i | 17.5181 | + | 43.3603i | −117.426 | + | 50.9415i | − | 151.910i | −440.049 | + | 293.757i | 893.278i | −834.308 | − | 1183.67i | −1573.23 | + | 1519.18i | 1682.79 | − | 349.286i | ||||||||||||||||||||||||||||
| 11.7 | 9.03954 | − | 6.80344i | −46.2722 | − | 6.77368i | 35.4264 | − | 123.000i | − | 426.666i | −464.364 | + | 253.579i | 780.788i | −516.584 | − | 1352.88i | 2095.23 | + | 626.867i | −2902.79 | − | 3856.86i | ||||||||||||||||||||||||||||
| 11.8 | 9.03954 | + | 6.80344i | −46.2722 | + | 6.77368i | 35.4264 | + | 123.000i | 426.666i | −464.364 | − | 253.579i | − | 780.788i | −516.584 | + | 1352.88i | 2095.23 | − | 626.867i | −2902.79 | + | 3856.86i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.8.b.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 12.8.b.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 12.8.b.b | ✓ | 8 |
| 8.b | even | 2 | 1 | 192.8.c.e | 8 | ||
| 8.d | odd | 2 | 1 | 192.8.c.e | 8 | ||
| 12.b | even | 2 | 1 | inner | 12.8.b.b | ✓ | 8 |
| 24.f | even | 2 | 1 | 192.8.c.e | 8 | ||
| 24.h | odd | 2 | 1 | 192.8.c.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.8.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 12.8.b.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 12.8.b.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 12.8.b.b | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 192.8.c.e | 8 | 8.b | even | 2 | 1 | ||
| 192.8.c.e | 8 | 8.d | odd | 2 | 1 | ||
| 192.8.c.e | 8 | 24.f | even | 2 | 1 | ||
| 192.8.c.e | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 205120T_{5}^{2} + 4200928000 \)
acting on \(S_{8}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 164 T^{6} + \cdots + 268435456 \)
|
| $3$ |
\( T^{8} + \cdots + 22876792454961 \)
|
| $5$ |
\( (T^{4} + 205120 T^{2} + 4200928000)^{2} \)
|
| $7$ |
\( (T^{4} + 1407576 T^{2} + 486451938960)^{2} \)
|
| $11$ |
\( (T^{4} + \cdots + 218962899475200)^{2} \)
|
| $13$ |
\( (T^{2} - 13612 T + 44826340)^{4} \)
|
| $17$ |
\( (T^{4} + \cdots + 657720842321920)^{2} \)
|
| $19$ |
\( (T^{4} + \cdots + 27\!\cdots\!40)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 15\!\cdots\!72)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 50\!\cdots\!20)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{2} + 5444 T - 103420719740)^{4} \)
|
| $41$ |
\( (T^{4} + \cdots + 42\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{4} + \cdots + 11\!\cdots\!40)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 15\!\cdots\!92)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 34\!\cdots\!80)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots - 2927319207644)^{4} \)
|
| $67$ |
\( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 69\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{2} - 920692 T - 431561259740)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 40\!\cdots\!40)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 55\!\cdots\!32)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 54\!\cdots\!80)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 8919460638140)^{4} \)
|
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