Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.4");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(10.8157525594\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} + 1600 x^{8} - 7420 x^{7} + 2144441 x^{6} - 9353044 x^{5} + 682842856 x^{4} + \cdots + 6442540462656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3}\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} + 1600 x^{8} - 7420 x^{7} + 2144441 x^{6} - 9353044 x^{5} + 682842856 x^{4} + \cdots + 6442540462656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 33\!\cdots\!15 \nu^{9} + \cdots + 32\!\cdots\!48 ) / 34\!\cdots\!56 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 35\!\cdots\!89 \nu^{9} + \cdots - 22\!\cdots\!80 ) / 36\!\cdots\!07 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 67\!\cdots\!40 \nu^{9} + \cdots - 20\!\cdots\!64 ) / 33\!\cdots\!37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 21\!\cdots\!71 \nu^{9} + \cdots + 14\!\cdots\!40 ) / 13\!\cdots\!28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 24\!\cdots\!03 \nu^{9} + \cdots - 51\!\cdots\!80 ) / 10\!\cdots\!96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 25\!\cdots\!38 \nu^{9} + \cdots - 16\!\cdots\!24 ) / 42\!\cdots\!57 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 49\!\cdots\!27 \nu^{9} + \cdots + 56\!\cdots\!72 ) / 47\!\cdots\!84 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 13\!\cdots\!04 \nu^{9} + \cdots + 49\!\cdots\!64 ) / 11\!\cdots\!96 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{4} - 2\beta_{3} - 640\beta_{2} - 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 10\beta_{6} + 8\beta_{5} - 2\beta_{4} + 1067\beta_{3} + 1696 \)
|
| \(\nu^{4}\) | \(=\) |
\( -34\beta_{9} - 184\beta_{8} - 1367\beta_{7} - 34\beta_{6} + 685752\beta_{2} + 2444\beta _1 - 685752 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15892 \beta_{9} + 12400 \beta_{8} - 620 \beta_{7} - 12400 \beta_{5} + 620 \beta_{4} + \cdots - 1295269 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 75540\beta_{6} + 311312\beta_{5} + 1758973\beta_{4} + 3606598\beta_{3} + 834505616 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 21259086 \beta_{9} - 16836984 \beta_{8} - 1034394 \beta_{7} - 21259086 \beta_{6} + 3143952144 \beta_{2} + \cdots - 3143952144 \)
|
| \(\nu^{8}\) | \(=\) |
\( 122929302 \beta_{9} + 432605448 \beta_{8} + 2241938803 \beta_{7} - 432605448 \beta_{5} + \cdots - 5656546160 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 27428612344\beta_{6} + 22065406496\beta_{5} + 3286684624\beta_{4} + 2062615185737\beta_{3} + 4699650485848 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).
| \(n\) | \(8\) | \(10\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 |
|
−14.1118 | − | 24.4423i | 40.5000 | − | 70.1481i | −142.286 | + | 246.446i | −101.336 | − | 175.518i | −2286.11 | −4617.22 | − | 4362.90i | −6418.86 | −3280.50 | − | 5681.99i | −2860.06 | + | 4953.76i | ||||||||||||||||||||||||||||||||||
| 4.2 | −4.21783 | − | 7.30550i | 40.5000 | − | 70.1481i | 220.420 | − | 381.778i | 1069.58 | + | 1852.56i | −683.289 | 6190.35 | + | 1425.89i | −8037.83 | −3280.50 | − | 5681.99i | 9022.59 | − | 15627.6i | |||||||||||||||||||||||||||||||||||
| 4.3 | 0.148406 | + | 0.257047i | 40.5000 | − | 70.1481i | 255.956 | − | 443.329i | −734.607 | − | 1272.38i | 24.0418 | −2964.10 | + | 5618.52i | 303.910 | −3280.50 | − | 5681.99i | 218.040 | − | 377.657i | |||||||||||||||||||||||||||||||||||
| 4.4 | 13.1632 | + | 22.7993i | 40.5000 | − | 70.1481i | −90.5399 | + | 156.820i | −337.364 | − | 584.331i | 2132.44 | 5108.86 | − | 3775.33i | 8711.94 | −3280.50 | − | 5681.99i | 8881.57 | − | 15383.3i | |||||||||||||||||||||||||||||||||||
| 4.5 | 21.5180 | + | 37.2703i | 40.5000 | − | 70.1481i | −670.050 | + | 1160.56i | 21.2298 | + | 36.7711i | 3485.92 | −4708.39 | + | 4264.34i | −35638.2 | −3280.50 | − | 5681.99i | −913.647 | + | 1582.48i | |||||||||||||||||||||||||||||||||||
| 16.1 | −14.1118 | + | 24.4423i | 40.5000 | + | 70.1481i | −142.286 | − | 246.446i | −101.336 | + | 175.518i | −2286.11 | −4617.22 | + | 4362.90i | −6418.86 | −3280.50 | + | 5681.99i | −2860.06 | − | 4953.76i | |||||||||||||||||||||||||||||||||||
| 16.2 | −4.21783 | + | 7.30550i | 40.5000 | + | 70.1481i | 220.420 | + | 381.778i | 1069.58 | − | 1852.56i | −683.289 | 6190.35 | − | 1425.89i | −8037.83 | −3280.50 | + | 5681.99i | 9022.59 | + | 15627.6i | |||||||||||||||||||||||||||||||||||
| 16.3 | 0.148406 | − | 0.257047i | 40.5000 | + | 70.1481i | 255.956 | + | 443.329i | −734.607 | + | 1272.38i | 24.0418 | −2964.10 | − | 5618.52i | 303.910 | −3280.50 | + | 5681.99i | 218.040 | + | 377.657i | |||||||||||||||||||||||||||||||||||
| 16.4 | 13.1632 | − | 22.7993i | 40.5000 | + | 70.1481i | −90.5399 | − | 156.820i | −337.364 | + | 584.331i | 2132.44 | 5108.86 | + | 3775.33i | 8711.94 | −3280.50 | + | 5681.99i | 8881.57 | + | 15383.3i | |||||||||||||||||||||||||||||||||||
| 16.5 | 21.5180 | − | 37.2703i | 40.5000 | + | 70.1481i | −670.050 | − | 1160.56i | 21.2298 | − | 36.7711i | 3485.92 | −4708.39 | − | 4264.34i | −35638.2 | −3280.50 | + | 5681.99i | −913.647 | − | 1582.48i | |||||||||||||||||||||||||||||||||||
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 | 21.10.e.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 63.10.e.a | 10 | ||
| 7.c | even | 3 | 1 | inner | 21.10.e.a | ✓ | 10 |
| 7.c | even | 3 | 1 | 147.10.a.i | 5 | ||
| 7.d | odd | 6 | 1 | 147.10.a.j | 5 | ||
| 21.h | odd | 6 | 1 | 63.10.e.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.10.e.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 21.10.e.a | ✓ | 10 | 7.c | even | 3 | 1 | inner |
| 63.10.e.a | 10 | 3.b | odd | 2 | 1 | ||
| 63.10.e.a | 10 | 21.h | odd | 6 | 1 | ||
| 147.10.a.i | 5 | 7.c | even | 3 | 1 | ||
| 147.10.a.j | 5 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 33 T_{2}^{9} + 2251 T_{2}^{8} - 12390 T_{2}^{7} + 1925068 T_{2}^{6} - 12244560 T_{2}^{5} + \cdots + 6410244096 \)
acting on \(S_{10}^{\mathrm{new}}(21, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 6410244096 \)
|
| $3$ |
\( (T^{2} - 81 T + 6561)^{5} \)
|
| $5$ |
\( T^{10} + \cdots + 33\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 10\!\cdots\!07 \)
|
| $11$ |
\( T^{10} + \cdots + 36\!\cdots\!00 \)
|
| $13$ |
\( (T^{5} + \cdots + 34\!\cdots\!32)^{2} \)
|
| $17$ |
\( T^{10} + \cdots + 26\!\cdots\!76 \)
|
| $19$ |
\( T^{10} + \cdots + 21\!\cdots\!36 \)
|
| $23$ |
\( T^{10} + \cdots + 10\!\cdots\!04 \)
|
| $29$ |
\( (T^{5} + \cdots + 13\!\cdots\!16)^{2} \)
|
| $31$ |
\( T^{10} + \cdots + 12\!\cdots\!41 \)
|
| $37$ |
\( T^{10} + \cdots + 76\!\cdots\!44 \)
|
| $41$ |
\( (T^{5} + \cdots - 31\!\cdots\!00)^{2} \)
|
| $43$ |
\( (T^{5} + \cdots + 94\!\cdots\!76)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 12\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots + 61\!\cdots\!64 \)
|
| $59$ |
\( T^{10} + \cdots + 25\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 16\!\cdots\!84 \)
|
| $67$ |
\( T^{10} + \cdots + 43\!\cdots\!00 \)
|
| $71$ |
\( (T^{5} + \cdots - 12\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 62\!\cdots\!00 \)
|
| $79$ |
\( T^{10} + \cdots + 70\!\cdots\!89 \)
|
| $83$ |
\( (T^{5} + \cdots - 25\!\cdots\!08)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 26\!\cdots\!04 \)
|
| $97$ |
\( (T^{5} + \cdots - 37\!\cdots\!36)^{2} \)
|
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