Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,5,Mod(13,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.13");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 21.d (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: | \(2.17076922476\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 37x^{4} - 84x^{3} + 1369x^{2} - 1554x + 1764 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 37x^{4} - 84x^{3} + 1369x^{2} - 1554x + 1764 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 42 ) / 37 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{3} + 222\nu - 126 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 37\nu^{3} - 42\nu^{2} + 1369\nu - 777 ) / 259 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} + 37\nu^{2} - 42\nu + 841 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -25\nu^{5} - 21\nu^{4} - 862\nu^{3} + 1827\nu^{2} - 28681\nu + 16779 ) / 777 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + 3\beta_{4} + 25\beta_{3} - 3\beta_{2} - 6\beta _1 - 75 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( 37\beta _1 + 42 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -111\beta_{5} + 111\beta_{4} - 925\beta_{3} + 153\beta_{2} - 348\beta _1 - 2775 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 126\beta_{5} + 126\beta_{4} + 2604\beta_{3} - 1495\beta_{2} - 4359\beta _1 - 7812 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−6.40664 | − | 5.19615i | 25.0451 | 40.6195i | 33.2899i | 43.3100 | + | 22.9181i | −57.9485 | −27.0000 | − | 260.235i | ||||||||||||||||||||||||||||||||
| 13.2 | −6.40664 | 5.19615i | 25.0451 | − | 40.6195i | − | 33.2899i | 43.3100 | − | 22.9181i | −57.9485 | −27.0000 | 260.235i | |||||||||||||||||||||||||||||||||
| 13.3 | −2.17948 | − | 5.19615i | −11.2499 | − | 29.5667i | 11.3249i | −41.9613 | − | 25.3032i | 59.3906 | −27.0000 | 64.4402i | |||||||||||||||||||||||||||||||||
| 13.4 | −2.17948 | 5.19615i | −11.2499 | 29.5667i | − | 11.3249i | −41.9613 | + | 25.3032i | 59.3906 | −27.0000 | − | 64.4402i | |||||||||||||||||||||||||||||||||
| 13.5 | 5.58613 | − | 5.19615i | 15.2048 | 2.80361i | − | 29.0264i | −12.3488 | + | 47.4184i | −4.44213 | −27.0000 | 15.6613i | |||||||||||||||||||||||||||||||||
| 13.6 | 5.58613 | 5.19615i | 15.2048 | − | 2.80361i | 29.0264i | −12.3488 | − | 47.4184i | −4.44213 | −27.0000 | − | 15.6613i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 21.5.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 63.5.d.d | 6 | ||
| 4.b | odd | 2 | 1 | 336.5.f.c | 6 | ||
| 5.b | even | 2 | 1 | 525.5.h.a | 6 | ||
| 5.c | odd | 4 | 2 | 525.5.e.a | 12 | ||
| 7.b | odd | 2 | 1 | inner | 21.5.d.a | ✓ | 6 |
| 7.c | even | 3 | 1 | 147.5.f.c | 6 | ||
| 7.c | even | 3 | 1 | 147.5.f.e | 6 | ||
| 7.d | odd | 6 | 1 | 147.5.f.c | 6 | ||
| 7.d | odd | 6 | 1 | 147.5.f.e | 6 | ||
| 12.b | even | 2 | 1 | 1008.5.f.k | 6 | ||
| 21.c | even | 2 | 1 | 63.5.d.d | 6 | ||
| 28.d | even | 2 | 1 | 336.5.f.c | 6 | ||
| 35.c | odd | 2 | 1 | 525.5.h.a | 6 | ||
| 35.f | even | 4 | 2 | 525.5.e.a | 12 | ||
| 84.h | odd | 2 | 1 | 1008.5.f.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.5.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 21.5.d.a | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 63.5.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 63.5.d.d | 6 | 21.c | even | 2 | 1 | ||
| 147.5.f.c | 6 | 7.c | even | 3 | 1 | ||
| 147.5.f.c | 6 | 7.d | odd | 6 | 1 | ||
| 147.5.f.e | 6 | 7.c | even | 3 | 1 | ||
| 147.5.f.e | 6 | 7.d | odd | 6 | 1 | ||
| 336.5.f.c | 6 | 4.b | odd | 2 | 1 | ||
| 336.5.f.c | 6 | 28.d | even | 2 | 1 | ||
| 525.5.e.a | 12 | 5.c | odd | 4 | 2 | ||
| 525.5.e.a | 12 | 35.f | even | 4 | 2 | ||
| 525.5.h.a | 6 | 5.b | even | 2 | 1 | ||
| 525.5.h.a | 6 | 35.c | odd | 2 | 1 | ||
| 1008.5.f.k | 6 | 12.b | even | 2 | 1 | ||
| 1008.5.f.k | 6 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + 3 T^{2} - 34 T - 78)^{2} \)
|
| $3$ |
\( (T^{2} + 27)^{3} \)
|
| $5$ |
\( T^{6} + 2532 T^{4} + \cdots + 11337408 \)
|
| $7$ |
\( T^{6} + \cdots + 13841287201 \)
|
| $11$ |
\( (T^{3} - 126 T^{2} + \cdots + 31596)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 84785692142592 \)
|
| $17$ |
\( T^{6} + \cdots + 25918947274752 \)
|
| $19$ |
\( T^{6} + \cdots + 122121162369792 \)
|
| $23$ |
\( (T^{3} - 1158 T^{2} + \cdots + 106847244)^{2} \)
|
| $29$ |
\( (T^{3} + 66 T^{2} + \cdots - 133744392)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 63207309312 \)
|
| $37$ |
\( (T^{3} - 1034 T^{2} + \cdots + 2821250336)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 20\!\cdots\!52 \)
|
| $43$ |
\( (T^{3} + 2734 T^{2} + \cdots - 540906928)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 23\!\cdots\!32 \)
|
| $53$ |
\( (T^{3} + 1482 T^{2} + \cdots - 29751816)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 69\!\cdots\!88 \)
|
| $61$ |
\( T^{6} + \cdots + 15\!\cdots\!92 \)
|
| $67$ |
\( (T^{3} - 3546 T^{2} + \cdots + 49781748256)^{2} \)
|
| $71$ |
\( (T^{3} - 17118 T^{2} + \cdots - 104928664596)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 27\!\cdots\!32 \)
|
| $79$ |
\( (T^{3} - 6930 T^{2} + \cdots - 12780457472)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 51\!\cdots\!48 \)
|
| $89$ |
\( T^{6} + \cdots + 24\!\cdots\!88 \)
|
| $97$ |
\( T^{6} + \cdots + 49\!\cdots\!08 \)
|
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