Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,13,Mod(3,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.3");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.c (of order \(4\), 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: | \(9.13993817276\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 43009x^{4} + 461169144x^{2} + 392422062096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 5^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{6} + 43009x^{4} + 461169144x^{2} + 392422062096 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 669445\nu^{3} - 13932675324\nu ) / 405892941840 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 1566090 \nu^{4} + 669445 \nu^{3} - 33678765450 \nu^{2} + 1028665029924 \nu + 1386948097080 ) / 202946470920 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 1566090 \nu^{4} + 669445 \nu^{3} + 33678765450 \nu^{2} + 1028665029924 \nu - 1386948097080 ) / 202946470920 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 266983 \nu^{5} + 22447290 \nu^{4} - 9608375335 \nu^{3} + 820973089650 \nu^{2} + \cdots + 48\!\cdots\!40 ) / 608839412760 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1067929 \nu^{5} + 22447290 \nu^{4} + 38431493005 \nu^{3} + 820973089650 \nu^{2} + \cdots + 48\!\cdots\!00 ) / 1217678825520 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 4\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 36\beta_{5} + 72\beta_{4} - 215\beta_{3} + 215\beta_{2} - 36\beta _1 - 716948 ) / 50 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -72\beta_{5} + 36\beta_{4} - 107735\beta_{3} - 107735\beta_{2} - 32468828\beta _1 + 36 ) / 50 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -154836\beta_{5} - 309672\beta_{4} + 1572655\beta_{3} - 1572655\beta_{2} + 154836\beta _1 + 3092449468 ) / 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 9640008 \beta_{5} - 4820004 \beta_{4} + 491856091 \beta_{3} + 491856091 \beta_{2} + 232558792396 \beta _1 - 4820004 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
32.0000 | − | 32.0000i | −748.698 | − | 748.698i | − | 2048.00i | 11268.2 | − | 10824.4i | −47916.6 | −121018. | + | 121018.i | −65536.0 | − | 65536.0i | 589655.i | 14201.0 | − | 706964.i | |||||||||||||||||||||||
| 3.2 | 32.0000 | − | 32.0000i | 203.464 | + | 203.464i | − | 2048.00i | −11784.0 | − | 10260.5i | 13021.7 | 68548.2 | − | 68548.2i | −65536.0 | − | 65536.0i | − | 448646.i | −705424. | + | 48750.4i | |||||||||||||||||||||||
| 3.3 | 32.0000 | − | 32.0000i | 693.233 | + | 693.233i | − | 2048.00i | 7745.77 | + | 13570.0i | 44366.9 | −108582. | + | 108582.i | −65536.0 | − | 65536.0i | 429704.i | 682103. | + | 186374.i | ||||||||||||||||||||||||
| 7.1 | 32.0000 | + | 32.0000i | −748.698 | + | 748.698i | 2048.00i | 11268.2 | + | 10824.4i | −47916.6 | −121018. | − | 121018.i | −65536.0 | + | 65536.0i | − | 589655.i | 14201.0 | + | 706964.i | ||||||||||||||||||||||||
| 7.2 | 32.0000 | + | 32.0000i | 203.464 | − | 203.464i | 2048.00i | −11784.0 | + | 10260.5i | 13021.7 | 68548.2 | + | 68548.2i | −65536.0 | + | 65536.0i | 448646.i | −705424. | − | 48750.4i | |||||||||||||||||||||||||
| 7.3 | 32.0000 | + | 32.0000i | 693.233 | − | 693.233i | 2048.00i | 7745.77 | − | 13570.0i | 44366.9 | −108582. | − | 108582.i | −65536.0 | + | 65536.0i | − | 429704.i | 682103. | − | 186374.i | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 10.13.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.13.g.a | 6 | ||
| 4.b | odd | 2 | 1 | 80.13.p.a | 6 | ||
| 5.b | even | 2 | 1 | 50.13.c.c | 6 | ||
| 5.c | odd | 4 | 1 | inner | 10.13.c.b | ✓ | 6 |
| 5.c | odd | 4 | 1 | 50.13.c.c | 6 | ||
| 15.e | even | 4 | 1 | 90.13.g.a | 6 | ||
| 20.e | even | 4 | 1 | 80.13.p.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.13.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 10.13.c.b | ✓ | 6 | 5.c | odd | 4 | 1 | inner |
| 50.13.c.c | 6 | 5.b | even | 2 | 1 | ||
| 50.13.c.c | 6 | 5.c | odd | 4 | 1 | ||
| 80.13.p.a | 6 | 4.b | odd | 2 | 1 | ||
| 80.13.p.a | 6 | 20.e | even | 4 | 1 | ||
| 90.13.g.a | 6 | 3.b | odd | 2 | 1 | ||
| 90.13.g.a | 6 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 296 T_{3}^{5} + 43808 T_{3}^{4} - 108468072 T_{3}^{3} + 1124902056996 T_{3}^{2} + \cdots + 89\!\cdots\!28 \)
acting on \(S_{13}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 64 T + 2048)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 89\!\cdots\!28 \)
|
| $5$ |
\( T^{6} + \cdots + 14\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 64\!\cdots\!28 \)
|
| $11$ |
\( (T^{3} + \cdots - 16\!\cdots\!08)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 20\!\cdots\!48 \)
|
| $17$ |
\( T^{6} + \cdots + 20\!\cdots\!68 \)
|
| $19$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 50\!\cdots\!28 \)
|
| $29$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 55\!\cdots\!48)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 38\!\cdots\!28 \)
|
| $41$ |
\( (T^{3} + \cdots - 18\!\cdots\!48)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 78\!\cdots\!68 \)
|
| $47$ |
\( T^{6} + \cdots + 57\!\cdots\!48 \)
|
| $53$ |
\( T^{6} + \cdots + 11\!\cdots\!48 \)
|
| $59$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 10\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 11\!\cdots\!48 \)
|
| $71$ |
\( (T^{3} + \cdots - 28\!\cdots\!28)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 81\!\cdots\!48 \)
|
| $79$ |
\( T^{6} + \cdots + 32\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 27\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 44\!\cdots\!28 \)
|
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