Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,12,Mod(37,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (of order \(3\), degree \(2\), 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: | \(96.8112407505\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 6891x^{4} + 48890x^{3} + 47451100x^{2} + 144690000x + 441000000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3\cdot 7^{4} \) |
| Twist minimal: | no (minimal twist has level 42) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 6891x^{4} + 48890x^{3} + 47451100x^{2} + 144690000x + 441000000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1582633 \nu^{5} - 1583333 \nu^{4} + 10910747703 \nu^{3} + 44134810670 \nu^{2} + \cdots + 101297700000 ) / 228991154070000 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 46112581 \nu^{5} + 23344597581 \nu^{4} - 573814081671 \nu^{3} + 159263780250910 \nu^{2} + \cdots + 73\!\cdots\!00 ) / 686973462210000 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 241981569 \nu^{5} - 869130531 \nu^{4} - 1643859979879 \nu^{3} - 22038577909510 \nu^{2} + \cdots - 69\!\cdots\!00 ) / 32713022010000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 726436177 \nu^{5} - 779328177 \nu^{4} + 5370350467707 \nu^{3} + 42794941611530 \nu^{2} + \cdots + 11\!\cdots\!00 ) / 98139066030000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 242464639 \nu^{5} - 2459704839 \nu^{4} + 1683749107549 \nu^{3} - 883624967490 \nu^{2} + \cdots - 34\!\cdots\!00 ) / 32713022010000 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} - \beta_{3} - 3\beta_{2} - 27\beta_1 ) / 84 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 98\beta_{5} + 33\beta_{4} - 229\beta_{3} - 33\beta_{2} - 385857\beta _1 - 385857 ) / 84 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7021\beta_{5} + 20703\beta_{4} + 13682\beta_{3} - 2335887 ) / 84 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -1550149\beta_{5} + 930611\beta_{3} + 311073\beta_{2} + 2661457617\beta_1 ) / 84 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 91591442 \beta_{5} - 143647743 \beta_{4} - 39535141 \beta_{3} + 143647743 \beta_{2} + \cdots + 26858716047 ) / 84 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−16.0000 | + | 27.7128i | 0 | −512.000 | − | 886.810i | −2779.47 | + | 4814.19i | 0 | −30437.5 | + | 32417.4i | 32768.0 | 0 | −88943.2 | − | 154054.i | ||||||||||||||||||||||||||
| 37.2 | −16.0000 | + | 27.7128i | 0 | −512.000 | − | 886.810i | −1601.94 | + | 2774.64i | 0 | 33098.6 | + | 29695.3i | 32768.0 | 0 | −51262.1 | − | 88788.6i | |||||||||||||||||||||||||||
| 37.3 | −16.0000 | + | 27.7128i | 0 | −512.000 | − | 886.810i | 3715.92 | − | 6436.15i | 0 | −44433.6 | − | 1727.34i | 32768.0 | 0 | 118909. | + | 205957.i | |||||||||||||||||||||||||||
| 109.1 | −16.0000 | − | 27.7128i | 0 | −512.000 | + | 886.810i | −2779.47 | − | 4814.19i | 0 | −30437.5 | − | 32417.4i | 32768.0 | 0 | −88943.2 | + | 154054.i | |||||||||||||||||||||||||||
| 109.2 | −16.0000 | − | 27.7128i | 0 | −512.000 | + | 886.810i | −1601.94 | − | 2774.64i | 0 | 33098.6 | − | 29695.3i | 32768.0 | 0 | −51262.1 | + | 88788.6i | |||||||||||||||||||||||||||
| 109.3 | −16.0000 | − | 27.7128i | 0 | −512.000 | + | 886.810i | 3715.92 | + | 6436.15i | 0 | −44433.6 | + | 1727.34i | 32768.0 | 0 | 118909. | − | 205957.i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.12.g.a | 6 | |
| 3.b | odd | 2 | 1 | 42.12.e.b | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 126.12.g.a | 6 | |
| 21.h | odd | 6 | 1 | 42.12.e.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.12.e.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 42.12.e.b | ✓ | 6 | 21.h | odd | 6 | 1 | |
| 126.12.g.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 126.12.g.a | 6 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 1331 T_{5}^{5} + 49085221 T_{5}^{4} + 201750551940 T_{5}^{3} + \cdots + 17\!\cdots\!00 \)
acting on \(S_{12}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 1024)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 77\!\cdots\!07 \)
|
| $11$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 15\!\cdots\!60)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 64\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 61\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 49\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots - 27\!\cdots\!00)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 57\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 86\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} + \cdots - 18\!\cdots\!80)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 16\!\cdots\!50)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!24 \)
|
| $53$ |
\( T^{6} + \cdots + 37\!\cdots\!24 \)
|
| $59$ |
\( T^{6} + \cdots + 69\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 78\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + \cdots - 37\!\cdots\!72)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 11\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots + 83\!\cdots\!49 \)
|
| $83$ |
\( (T^{3} + \cdots + 47\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 49\!\cdots\!68)^{2} \)
|
| show more | |
| show less |