Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,14,Mod(9,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.9");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.b (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: | \(10.7230928952\) |
| 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: |
\( x^{6} - 2x^{5} + 2x^{4} + 160950x^{3} + 43599609x^{2} + 975553632x + 10914144768 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 5^{5} \) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 160950x^{3} + 43599609x^{2} + 975553632x + 10914144768 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2902156 \nu^{5} - 38332616 \nu^{4} + 467184200 \nu^{3} + 252252560280 \nu^{2} + \cdots + 14\!\cdots\!56 ) / 21586271220765 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8857615 \nu^{5} - 124928126 \nu^{4} + 53829158030 \nu^{3} + 717491956170 \nu^{2} + \cdots + 82\!\cdots\!16 ) / 34538033953224 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 8853546833 \nu^{5} + 730507083298 \nu^{4} - 24595132422850 \nu^{3} + \cdots + 11\!\cdots\!52 ) / 949795933713660 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 117783373 \nu^{5} - 586906082 \nu^{4} + 15870764330 \nu^{3} + 23032569783630 \nu^{2} + \cdots + 86\!\cdots\!72 ) / 5276644076187 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 14657213123 \nu^{5} + 279534354118 \nu^{4} + 5659609021850 \nu^{3} + \cdots - 41\!\cdots\!88 ) / 474897966856830 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{5} - 4\beta_{4} + 2\beta_{3} + 64\beta_{2} - 8\beta _1 + 426 ) / 1280 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -66\beta_{5} + 66\beta_{4} + 760\beta_{2} - 27559\beta_1 ) / 400 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 98691\beta_{5} + 134404\beta_{4} - 65090\beta_{3} + 2125120\beta_{2} - 8225496\beta _1 - 515022570 ) / 6400 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 88137\beta_{5} + 146578\beta_{4} + 57490\beta_{3} - 116882\beta _1 - 2342499750 ) / 80 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 161899005 \beta_{5} + 179082748 \beta_{4} - 71150078 \beta_{3} - 3012454336 \beta_{2} + \cdots - 1133922685014 ) / 1280 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
− | 64.0000i | − | 1311.29i | −4096.00 | −34287.6 | + | 6712.97i | −83922.3 | − | 77461.9i | 262144.i | −125146. | 429630. | + | 2.19441e6i | |||||||||||||||||||||||||||||
| 9.2 | − | 64.0000i | 180.328i | −4096.00 | 33325.2 | + | 10494.5i | 11541.0 | − | 386506.i | 262144.i | 1.56180e6 | 671650. | − | 2.13281e6i | |||||||||||||||||||||||||||||||
| 9.3 | − | 64.0000i | 948.958i | −4096.00 | −272.592 | − | 34937.5i | 60733.3 | 454502.i | 262144.i | 693802. | −2.23600e6 | + | 17445.9i | ||||||||||||||||||||||||||||||||
| 9.4 | 64.0000i | − | 948.958i | −4096.00 | −272.592 | + | 34937.5i | 60733.3 | − | 454502.i | − | 262144.i | 693802. | −2.23600e6 | − | 17445.9i | ||||||||||||||||||||||||||||||
| 9.5 | 64.0000i | − | 180.328i | −4096.00 | 33325.2 | − | 10494.5i | 11541.0 | 386506.i | − | 262144.i | 1.56180e6 | 671650. | + | 2.13281e6i | |||||||||||||||||||||||||||||||
| 9.6 | 64.0000i | 1311.29i | −4096.00 | −34287.6 | − | 6712.97i | −83922.3 | 77461.9i | − | 262144.i | −125146. | 429630. | − | 2.19441e6i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.14.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.14.c.b | 6 | ||
| 4.b | odd | 2 | 1 | 80.14.c.b | 6 | ||
| 5.b | even | 2 | 1 | inner | 10.14.b.a | ✓ | 6 |
| 5.c | odd | 4 | 1 | 50.14.a.i | 3 | ||
| 5.c | odd | 4 | 1 | 50.14.a.j | 3 | ||
| 15.d | odd | 2 | 1 | 90.14.c.b | 6 | ||
| 20.d | odd | 2 | 1 | 80.14.c.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.14.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 10.14.b.a | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 50.14.a.i | 3 | 5.c | odd | 4 | 1 | ||
| 50.14.a.j | 3 | 5.c | odd | 4 | 1 | ||
| 80.14.c.b | 6 | 4.b | odd | 2 | 1 | ||
| 80.14.c.b | 6 | 20.d | odd | 2 | 1 | ||
| 90.14.c.b | 6 | 3.b | odd | 2 | 1 | ||
| 90.14.c.b | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4096)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 50\!\cdots\!56 \)
|
| $5$ |
\( T^{6} + \cdots + 18\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 18\!\cdots\!04 \)
|
| $11$ |
\( (T^{3} + \cdots - 17\!\cdots\!28)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 47\!\cdots\!56 \)
|
| $17$ |
\( T^{6} + \cdots + 14\!\cdots\!84 \)
|
| $19$ |
\( (T^{3} + \cdots + 79\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!76 \)
|
| $29$ |
\( (T^{3} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots - 13\!\cdots\!08)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 24\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots + 40\!\cdots\!72)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 92\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots + 32\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 52\!\cdots\!96 \)
|
| $59$ |
\( (T^{3} + \cdots - 72\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 22\!\cdots\!08)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 58\!\cdots\!24 \)
|
| $71$ |
\( (T^{3} + \cdots + 10\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 25\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 19\!\cdots\!96 \)
|
| $89$ |
\( (T^{3} + \cdots - 32\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
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