Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,13,Mod(55,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.55");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.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: | \(57.5816104884\) |
| 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: |
\( x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3^{2}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{6} - x^{5} + 238188x^{4} - 14589496x^{3} + 11212054600x^{2} - 101757597480x + 81251686776288 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 316322973 \nu^{5} - 1880954567990 \nu^{4} - 229888629934646 \nu^{3} + \cdots + 67\!\cdots\!71 ) / 78\!\cdots\!45 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 829712739 \nu^{5} - 436403341435 \nu^{4} - 197574368898172 \nu^{3} + \cdots - 23\!\cdots\!88 ) / 15\!\cdots\!90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 214467963392 \nu^{5} - 2208641196555 \nu^{4} + \cdots - 34\!\cdots\!96 ) / 36\!\cdots\!10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13584463019209 \nu^{5} + 603445770410790 \nu^{4} + \cdots + 26\!\cdots\!32 ) / 49\!\cdots\!35 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 36542487068476 \nu^{5} + \cdots + 31\!\cdots\!08 ) / 98\!\cdots\!70 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{5} + 24\beta_{4} + 97\beta_{3} + 396\beta_{2} + 36\beta _1 + 324 ) / 2016 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -129\beta_{5} - 480\beta_{4} - 5125\beta_{3} - 210828\beta_{2} + 22620\beta _1 - 53361540 ) / 672 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12957\beta_{5} - 1307040\beta_{4} - 6212689\beta_{3} - 3458460\beta_{2} - 4760052\beta _1 + 4823626284 ) / 672 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6442253 \beta_{5} + 79955072 \beta_{4} + 498689921 \beta_{3} + 8005050140 \beta_{2} + \cdots + 2564427752724 ) / 224 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2655070297 \beta_{5} + 71240832448 \beta_{4} + 374055289213 \beta_{3} - 1339548073876 \beta_{2} + \cdots - 621325934265084 ) / 224 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−91.2237 | 0 | 4225.75 | − | 20289.4i | 0 | 57579.4 | − | 102596.i | −11836.7 | 0 | 1.85087e6i | |||||||||||||||||||||||||||||||||
| 55.2 | −91.2237 | 0 | 4225.75 | 20289.4i | 0 | 57579.4 | + | 102596.i | −11836.7 | 0 | − | 1.85087e6i | ||||||||||||||||||||||||||||||||||
| 55.3 | −17.4333 | 0 | −3792.08 | − | 18388.3i | 0 | −116360. | − | 17366.5i | 137515. | 0 | 320568.i | ||||||||||||||||||||||||||||||||||
| 55.4 | −17.4333 | 0 | −3792.08 | 18388.3i | 0 | −116360. | + | 17366.5i | 137515. | 0 | − | 320568.i | ||||||||||||||||||||||||||||||||||
| 55.5 | 108.657 | 0 | 7710.33 | − | 23251.4i | 0 | −98544.2 | + | 64267.7i | 392722. | 0 | − | 2.52642e6i | |||||||||||||||||||||||||||||||||
| 55.6 | 108.657 | 0 | 7710.33 | 23251.4i | 0 | −98544.2 | − | 64267.7i | 392722. | 0 | 2.52642e6i | |||||||||||||||||||||||||||||||||||
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 | 63.13.d.d | 6 | |
| 3.b | odd | 2 | 1 | 7.13.b.b | ✓ | 6 | |
| 7.b | odd | 2 | 1 | inner | 63.13.d.d | 6 | |
| 12.b | even | 2 | 1 | 112.13.c.b | 6 | ||
| 21.c | even | 2 | 1 | 7.13.b.b | ✓ | 6 | |
| 21.g | even | 6 | 2 | 49.13.d.b | 12 | ||
| 21.h | odd | 6 | 2 | 49.13.d.b | 12 | ||
| 84.h | odd | 2 | 1 | 112.13.c.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.13.b.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 7.13.b.b | ✓ | 6 | 21.c | even | 2 | 1 | |
| 49.13.d.b | 12 | 21.g | even | 6 | 2 | ||
| 49.13.d.b | 12 | 21.h | odd | 6 | 2 | ||
| 63.13.d.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 63.13.d.d | 6 | 7.b | odd | 2 | 1 | inner | |
| 112.13.c.b | 6 | 12.b | even | 2 | 1 | ||
| 112.13.c.b | 6 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 10216T_{2} - 172800 \)
acting on \(S_{13}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 10216 T - 172800)^{2} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 75\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 26\!\cdots\!01 \)
|
| $11$ |
\( (T^{3} + \cdots - 22\!\cdots\!48)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots + 62\!\cdots\!00 \)
|
| $23$ |
\( (T^{3} + \cdots + 23\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{3} + \cdots - 41\!\cdots\!92)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 37\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 49\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots + 36\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots - 49\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots - 76\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{3} + \cdots + 18\!\cdots\!12)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots - 66\!\cdots\!68)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
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