Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(39.3605132110\) |
| 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: |
\( x^{6} - x^{5} + 1111x^{4} + 2838x^{3} + 1231236x^{2} + 959040x + 746496 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - x^{5} + 1111x^{4} + 2838x^{3} + 1231236x^{2} + 959040x + 746496 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 1111\nu^{4} - 1234321\nu^{3} + 1231236\nu^{2} + 343174599\nu - 1407708999 ) / 342215559 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -277\nu^{5} + 307747\nu^{4} + 308642\nu^{3} + 341052372\nu^{2} + 1292300757\nu + 254114289315 ) / 1026646677 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 205535 \nu^{5} - 205679 \nu^{4} + 228509369 \nu^{3} + 405566106 \nu^{2} + 253239389244 \nu + 138226176 ) / 197116161984 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 205607 \nu^{5} - 285671 \nu^{4} + 317380481 \nu^{3} + 316917114 \nu^{2} + 745960743324 \nu + 76853753856 ) / 98558080992 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 50767097 \nu^{5} + 50749385 \nu^{4} - 56382566735 \nu^{3} - 165939314838 \nu^{2} + \cdots - 48670690190400 ) / 197116161984 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{4} - 8\beta_{3} + \beta _1 + 1 ) / 21 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -9\beta_{5} - \beta_{4} - 2221\beta_{3} - \beta _1 - 2224 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1096\beta_{4} - 2192\beta_{3} + 63\beta_{2} - 5543\beta _1 - 39248 ) / 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9999\beta_{5} - 2239\beta_{4} + 2476823\beta_{3} + 9999\beta_{2} + 1940\beta _1 + 1940 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 194355\beta_{5} - 6148775\beta_{4} + 80461219\beta_{3} + 4957891\beta _1 + 73121560 ) / 21 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{3}\) |
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 |
|
−4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −6.21726 | + | 10.7686i | 0 | −879.593 | + | 223.290i | 512.000 | 0 | −49.7381 | − | 86.1488i | ||||||||||||||||||||||||||
| 37.2 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 95.0129 | − | 164.567i | 0 | −515.817 | − | 746.643i | 512.000 | 0 | 760.103 | + | 1316.54i | |||||||||||||||||||||||||||
| 37.3 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | 270.204 | − | 468.008i | 0 | 659.910 | + | 622.946i | 512.000 | 0 | 2161.63 | + | 3744.06i | |||||||||||||||||||||||||||
| 109.1 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −6.21726 | − | 10.7686i | 0 | −879.593 | − | 223.290i | 512.000 | 0 | −49.7381 | + | 86.1488i | |||||||||||||||||||||||||||
| 109.2 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 95.0129 | + | 164.567i | 0 | −515.817 | + | 746.643i | 512.000 | 0 | 760.103 | − | 1316.54i | |||||||||||||||||||||||||||
| 109.3 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | 270.204 | + | 468.008i | 0 | 659.910 | − | 622.946i | 512.000 | 0 | 2161.63 | − | 3744.06i | |||||||||||||||||||||||||||
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 | 126.8.g.i | ✓ | 6 |
| 3.b | odd | 2 | 1 | 126.8.g.j | yes | 6 | |
| 7.c | even | 3 | 1 | inner | 126.8.g.i | ✓ | 6 |
| 21.h | odd | 6 | 1 | 126.8.g.j | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.8.g.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 126.8.g.i | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 126.8.g.j | yes | 6 | 3.b | odd | 2 | 1 | |
| 126.8.g.j | yes | 6 | 21.h | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 718T_{5}^{5} + 421915T_{5}^{4} - 69765102T_{5}^{3} + 9679473441T_{5}^{2} + 119531204280T_{5} + 1630524686400 \)
acting on \(S_{8}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 1630524686400 \)
|
| $7$ |
\( T^{6} + \cdots + 55\!\cdots\!07 \)
|
| $11$ |
\( T^{6} + \cdots + 80\!\cdots\!36 \)
|
| $13$ |
\( (T^{3} - 9697 T^{2} + \cdots + 201095083596)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 96\!\cdots\!04 \)
|
| $19$ |
\( T^{6} + \cdots + 41\!\cdots\!44 \)
|
| $23$ |
\( T^{6} + \cdots + 48\!\cdots\!84 \)
|
| $29$ |
\( (T^{3} + \cdots - 115409062150992)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 14\!\cdots\!21 \)
|
| $37$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} + \cdots - 43\!\cdots\!20)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 38\!\cdots\!48)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 25\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots + 61\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots + 35\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots + 19\!\cdots\!44 \)
|
| $67$ |
\( T^{6} + \cdots + 37\!\cdots\!76 \)
|
| $71$ |
\( (T^{3} + \cdots - 10\!\cdots\!40)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 41\!\cdots\!96 \)
|
| $79$ |
\( T^{6} + \cdots + 17\!\cdots\!25 \)
|
| $83$ |
\( (T^{3} + \cdots - 22\!\cdots\!72)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 13\!\cdots\!64 \)
|
| $97$ |
\( (T^{3} + \cdots - 95\!\cdots\!74)^{2} \)
|
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