Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,4,Mod(4,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.e (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: | \(1.23904011012\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.9924270768.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{6} - x^{5} + 25x^{4} + 12x^{3} + 582x^{2} - 144x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 25\nu^{4} + 625\nu^{3} - 582\nu^{2} + 144\nu - 3600 ) / 14406 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -25\nu^{5} + 625\nu^{4} - 1219\nu^{3} + 14550\nu^{2} - 3600\nu + 234060 ) / 14406 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 100\nu^{5} - 99\nu^{4} + 2475\nu^{3} + 1825\nu^{2} + 57618\nu + 150 ) / 14406 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -1601\nu^{5} + 1609\nu^{4} - 40225\nu^{3} - 14212\nu^{2} - 936438\nu + 231696 ) / 14406 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 16\beta_{4} + \beta_{2} + \beta _1 - 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 25\beta_{2} - 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( -25\beta_{5} - 394\beta_{4} + 25\beta_{3} - 43\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -43\beta_{5} - 538\beta_{4} - 637\beta_{2} - 637\beta _1 + 538 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−2.65415 | − | 4.59712i | 1.50000 | − | 2.59808i | −10.0890 | + | 17.4746i | −2.78070 | − | 4.81631i | −15.9249 | 9.67799 | − | 15.7904i | 64.6443 | −4.50000 | − | 7.79423i | −14.7608 | + | 25.5664i | ||||||||||||||||||||||
| 4.2 | −0.124036 | − | 0.214837i | 1.50000 | − | 2.59808i | 3.96923 | − | 6.87491i | 6.21730 | + | 10.7687i | −0.744216 | −18.4385 | + | 1.73873i | −3.95388 | −4.50000 | − | 7.79423i | 1.54234 | − | 2.67141i | |||||||||||||||||||||||
| 4.3 | 2.27818 | + | 3.94593i | 1.50000 | − | 2.59808i | −6.38024 | + | 11.0509i | −8.93660 | − | 15.4786i | 13.6691 | 2.26047 | + | 18.3818i | −21.6905 | −4.50000 | − | 7.79423i | 40.7184 | − | 70.5264i | |||||||||||||||||||||||
| 16.1 | −2.65415 | + | 4.59712i | 1.50000 | + | 2.59808i | −10.0890 | − | 17.4746i | −2.78070 | + | 4.81631i | −15.9249 | 9.67799 | + | 15.7904i | 64.6443 | −4.50000 | + | 7.79423i | −14.7608 | − | 25.5664i | |||||||||||||||||||||||
| 16.2 | −0.124036 | + | 0.214837i | 1.50000 | + | 2.59808i | 3.96923 | + | 6.87491i | 6.21730 | − | 10.7687i | −0.744216 | −18.4385 | − | 1.73873i | −3.95388 | −4.50000 | + | 7.79423i | 1.54234 | + | 2.67141i | |||||||||||||||||||||||
| 16.3 | 2.27818 | − | 3.94593i | 1.50000 | + | 2.59808i | −6.38024 | − | 11.0509i | −8.93660 | + | 15.4786i | 13.6691 | 2.26047 | − | 18.3818i | −21.6905 | −4.50000 | + | 7.79423i | 40.7184 | + | 70.5264i | |||||||||||||||||||||||
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 | 21.4.e.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 63.4.e.c | 6 | ||
| 4.b | odd | 2 | 1 | 336.4.q.k | 6 | ||
| 7.b | odd | 2 | 1 | 147.4.e.n | 6 | ||
| 7.c | even | 3 | 1 | inner | 21.4.e.b | ✓ | 6 |
| 7.c | even | 3 | 1 | 147.4.a.l | 3 | ||
| 7.d | odd | 6 | 1 | 147.4.a.m | 3 | ||
| 7.d | odd | 6 | 1 | 147.4.e.n | 6 | ||
| 21.c | even | 2 | 1 | 441.4.e.w | 6 | ||
| 21.g | even | 6 | 1 | 441.4.a.t | 3 | ||
| 21.g | even | 6 | 1 | 441.4.e.w | 6 | ||
| 21.h | odd | 6 | 1 | 63.4.e.c | 6 | ||
| 21.h | odd | 6 | 1 | 441.4.a.s | 3 | ||
| 28.f | even | 6 | 1 | 2352.4.a.cg | 3 | ||
| 28.g | odd | 6 | 1 | 336.4.q.k | 6 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.ci | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 21.4.e.b | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 63.4.e.c | 6 | 3.b | odd | 2 | 1 | ||
| 63.4.e.c | 6 | 21.h | odd | 6 | 1 | ||
| 147.4.a.l | 3 | 7.c | even | 3 | 1 | ||
| 147.4.a.m | 3 | 7.d | odd | 6 | 1 | ||
| 147.4.e.n | 6 | 7.b | odd | 2 | 1 | ||
| 147.4.e.n | 6 | 7.d | odd | 6 | 1 | ||
| 336.4.q.k | 6 | 4.b | odd | 2 | 1 | ||
| 336.4.q.k | 6 | 28.g | odd | 6 | 1 | ||
| 441.4.a.s | 3 | 21.h | odd | 6 | 1 | ||
| 441.4.a.t | 3 | 21.g | even | 6 | 1 | ||
| 441.4.e.w | 6 | 21.c | even | 2 | 1 | ||
| 441.4.e.w | 6 | 21.g | even | 6 | 1 | ||
| 2352.4.a.cg | 3 | 28.f | even | 6 | 1 | ||
| 2352.4.a.ci | 3 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} + 25T_{2}^{4} - 12T_{2}^{3} + 582T_{2}^{2} + 144T_{2} + 36 \)
acting on \(S_{4}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 36 \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $5$ |
\( T^{6} + 11 T^{5} + \cdots + 1527696 \)
|
| $7$ |
\( T^{6} + 13 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + 35 T^{5} + \cdots + 91470096 \)
|
| $13$ |
\( (T^{3} - 62 T^{2} + \cdots + 18452)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 12745506816 \)
|
| $19$ |
\( T^{6} + \cdots + 54664310416 \)
|
| $23$ |
\( T^{6} + \cdots + 2498119335936 \)
|
| $29$ |
\( (T^{3} - 53 T^{2} + \cdots - 824976)^{2} \)
|
| $31$ |
\( T^{6} - 95 T^{5} + \cdots + 139783329 \)
|
| $37$ |
\( T^{6} + \cdots + 2415919104 \)
|
| $41$ |
\( (T^{3} - 244 T^{2} + \cdots - 300384)^{2} \)
|
| $43$ |
\( (T^{3} - 360 T^{2} + \cdots + 18269746)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 26205471480384 \)
|
| $53$ |
\( T^{6} + \cdots + 11\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
| $61$ |
\( T^{6} + \cdots + 71\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 783608160972004 \)
|
| $71$ |
\( (T^{3} - 318 T^{2} + \cdots - 28535976)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 20\!\cdots\!24 \)
|
| $79$ |
\( T^{6} + \cdots + 37\!\cdots\!69 \)
|
| $83$ |
\( (T^{3} - 519 T^{2} + \cdots + 47916036)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 169118164647936 \)
|
| $97$ |
\( (T^{3} - 19 T^{2} + \cdots + 44776452)^{2} \)
|
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