Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [51,3,Mod(50,51)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(51, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("51.50");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.c (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: | \(1.38964934824\) |
| 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: |
\( x^{8} + 14x^{6} + 142x^{4} + 294x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 14x^{6} + 142x^{4} + 294x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{4} - \nu^{2} + 537 ) / 117 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\nu^{6} + 139\nu^{4} + 1415\nu^{2} + 1581 ) / 234 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -11\nu^{6} - 139\nu^{4} - 1181\nu^{2} - 645 ) / 234 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{7} + 139\nu^{5} + 1532\nu^{3} + 1464\nu ) / 1755 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23\nu^{7} + 397\nu^{5} + 4001\nu^{3} + 12687\nu ) / 1170 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -121\nu^{7} - 1529\nu^{5} - 15097\nu^{3} - 14349\nu ) / 3510 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 11\beta_{5} - \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -12\beta_{4} - 10\beta_{3} + 11\beta_{2} - 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -15\beta_{7} + 11\beta_{6} - 117\beta_{5} - 83\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 23\beta_{4} + 19\beta_{3} - 139\beta_{2} + 573 \)
|
| \(\nu^{7}\) | \(=\) |
\( -89\beta_{7} - 139\beta_{6} + 106\beta_{5} + 1055\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).
| \(n\) | \(35\) | \(37\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 50.1 |
|
− | 3.54719i | −2.77253 | + | 1.14589i | −8.58258 | −2.44949 | 4.06470 | + | 9.83470i | − | 10.9806i | 16.2553i | 6.37386 | − | 6.35404i | 8.68881i | ||||||||||||||||||||||||||||||||||
| 50.2 | − | 3.54719i | 2.77253 | − | 1.14589i | −8.58258 | 2.44949 | −4.06470 | − | 9.83470i | 10.9806i | 16.2553i | 6.37386 | − | 6.35404i | − | 8.68881i | |||||||||||||||||||||||||||||||||||
| 50.3 | − | 1.84863i | −0.901703 | − | 2.86128i | 0.582576 | −2.44949 | −5.28944 | + | 1.66691i | 1.19437i | − | 8.47148i | −7.37386 | + | 5.16005i | 4.52819i | |||||||||||||||||||||||||||||||||||
| 50.4 | − | 1.84863i | 0.901703 | + | 2.86128i | 0.582576 | 2.44949 | 5.28944 | − | 1.66691i | − | 1.19437i | − | 8.47148i | −7.37386 | + | 5.16005i | − | 4.52819i | |||||||||||||||||||||||||||||||||
| 50.5 | 1.84863i | −0.901703 | + | 2.86128i | 0.582576 | −2.44949 | −5.28944 | − | 1.66691i | − | 1.19437i | 8.47148i | −7.37386 | − | 5.16005i | − | 4.52819i | |||||||||||||||||||||||||||||||||||
| 50.6 | 1.84863i | 0.901703 | − | 2.86128i | 0.582576 | 2.44949 | 5.28944 | + | 1.66691i | 1.19437i | 8.47148i | −7.37386 | − | 5.16005i | 4.52819i | |||||||||||||||||||||||||||||||||||||
| 50.7 | 3.54719i | −2.77253 | − | 1.14589i | −8.58258 | −2.44949 | 4.06470 | − | 9.83470i | 10.9806i | − | 16.2553i | 6.37386 | + | 6.35404i | − | 8.68881i | |||||||||||||||||||||||||||||||||||
| 50.8 | 3.54719i | 2.77253 | + | 1.14589i | −8.58258 | 2.44949 | −4.06470 | + | 9.83470i | − | 10.9806i | − | 16.2553i | 6.37386 | + | 6.35404i | 8.68881i | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 51.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 51.3.c.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 51.3.c.c | ✓ | 8 |
| 4.b | odd | 2 | 1 | 816.3.m.f | 8 | ||
| 12.b | even | 2 | 1 | 816.3.m.f | 8 | ||
| 17.b | even | 2 | 1 | inner | 51.3.c.c | ✓ | 8 |
| 51.c | odd | 2 | 1 | inner | 51.3.c.c | ✓ | 8 |
| 68.d | odd | 2 | 1 | 816.3.m.f | 8 | ||
| 204.h | even | 2 | 1 | 816.3.m.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.3.c.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 51.3.c.c | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 51.3.c.c | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 51.3.c.c | ✓ | 8 | 51.c | odd | 2 | 1 | inner |
| 816.3.m.f | 8 | 4.b | odd | 2 | 1 | ||
| 816.3.m.f | 8 | 12.b | even | 2 | 1 | ||
| 816.3.m.f | 8 | 68.d | odd | 2 | 1 | ||
| 816.3.m.f | 8 | 204.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(51, [\chi])\):
|
\( T_{2}^{4} + 16T_{2}^{2} + 43 \)
|
|
\( T_{5}^{2} - 6 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 16 T^{2} + 43)^{2} \)
|
| $3$ |
\( T^{8} + 2 T^{6} + \cdots + 6561 \)
|
| $5$ |
\( (T^{2} - 6)^{4} \)
|
| $7$ |
\( (T^{4} + 122 T^{2} + 172)^{2} \)
|
| $11$ |
\( (T^{4} - 250 T^{2} + 13924)^{2} \)
|
| $13$ |
\( (T^{2} - 4 T - 80)^{4} \)
|
| $17$ |
\( T^{8} + \cdots + 6975757441 \)
|
| $19$ |
\( (T^{2} - 22 T - 68)^{4} \)
|
| $23$ |
\( (T^{4} - 514 T^{2} + 62500)^{2} \)
|
| $29$ |
\( (T^{4} - 2316 T^{2} + 1232100)^{2} \)
|
| $31$ |
\( (T^{4} + 2042 T^{2} + 107500)^{2} \)
|
| $37$ |
\( (T^{4} + 2672 T^{2} + 1754572)^{2} \)
|
| $41$ |
\( (T^{4} - 3400 T^{2} + 107584)^{2} \)
|
| $43$ |
\( (T^{2} - 16 T - 20)^{4} \)
|
| $47$ |
\( (T^{4} + 3364 T^{2} + 430000)^{2} \)
|
| $53$ |
\( (T^{4} + 5196 T^{2} + 1789488)^{2} \)
|
| $59$ |
\( (T^{4} + 1632 T^{2} + 154800)^{2} \)
|
| $61$ |
\( (T^{4} + 11072 T^{2} + 26836300)^{2} \)
|
| $67$ |
\( (T^{2} - 102 T + 2580)^{4} \)
|
| $71$ |
\( (T^{4} - 714 T^{2} + 44100)^{2} \)
|
| $73$ |
\( (T^{4} + 22272 T^{2} + 114527232)^{2} \)
|
| $79$ |
\( (T^{4} + 23450 T^{2} + 131687500)^{2} \)
|
| $83$ |
\( (T^{4} + 5184 T^{2} + 4513968)^{2} \)
|
| $89$ |
\( (T^{4} + 29668 T^{2} + 31802800)^{2} \)
|
| $97$ |
\( (T^{4} + 1784 T^{2} + 795328)^{2} \)
|
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