Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3800,2,Mod(3649,3800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3800.3649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3800.d (of order \(2\), degree \(1\), not 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: | \(30.3431527681\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5161984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 760) |
| 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
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} + 323 ) / 121 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -12\nu^{5} - 29\nu^{4} - 60\nu^{3} + 24\nu^{2} - 123 ) / 121 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25\nu^{5} + 10\nu^{4} + 4\nu^{3} - 50\nu^{2} + 605\nu - 258 ) / 121 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 65\nu^{5} + 26\nu^{4} - 38\nu^{3} - 372\nu^{2} + 1331\nu - 574 ) / 242 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -75\nu^{5} - 30\nu^{4} - 12\nu^{3} + 392\nu^{2} - 1815\nu + 774 ) / 242 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - 2\beta_{4} + \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 3\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -10\beta_{5} - 10\beta_{4} - 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 8 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{2} + 6\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 58\beta_{5} + 50\beta_{4} + 32\beta_{3} - 25\beta_{2} - 33\beta _1 + 64 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).
| \(n\) | \(401\) | \(951\) | \(1901\) | \(1977\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3649.1 |
|
0 | − | 3.12489i | 0 | 0 | 0 | 1.51514i | 0 | −6.76491 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.2 | 0 | − | 1.76156i | 0 | 0 | 0 | − | 4.62620i | 0 | −0.103084 | 0 | |||||||||||||||||||||||||||||||||||
| 3649.3 | 0 | − | 0.363328i | 0 | 0 | 0 | 1.14134i | 0 | 2.86799 | 0 | ||||||||||||||||||||||||||||||||||||
| 3649.4 | 0 | 0.363328i | 0 | 0 | 0 | − | 1.14134i | 0 | 2.86799 | 0 | ||||||||||||||||||||||||||||||||||||
| 3649.5 | 0 | 1.76156i | 0 | 0 | 0 | 4.62620i | 0 | −0.103084 | 0 | |||||||||||||||||||||||||||||||||||||
| 3649.6 | 0 | 3.12489i | 0 | 0 | 0 | − | 1.51514i | 0 | −6.76491 | 0 | ||||||||||||||||||||||||||||||||||||
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 | 3800.2.d.k | 6 | |
| 5.b | even | 2 | 1 | inner | 3800.2.d.k | 6 | |
| 5.c | odd | 4 | 1 | 760.2.a.h | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 3800.2.a.y | 3 | ||
| 15.e | even | 4 | 1 | 6840.2.a.bj | 3 | ||
| 20.e | even | 4 | 1 | 1520.2.a.r | 3 | ||
| 20.e | even | 4 | 1 | 7600.2.a.bo | 3 | ||
| 40.i | odd | 4 | 1 | 6080.2.a.bw | 3 | ||
| 40.k | even | 4 | 1 | 6080.2.a.bs | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 760.2.a.h | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 1520.2.a.r | 3 | 20.e | even | 4 | 1 | ||
| 3800.2.a.y | 3 | 5.c | odd | 4 | 1 | ||
| 3800.2.d.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 3800.2.d.k | 6 | 5.b | even | 2 | 1 | inner | |
| 6080.2.a.bs | 3 | 40.k | even | 4 | 1 | ||
| 6080.2.a.bw | 3 | 40.i | odd | 4 | 1 | ||
| 6840.2.a.bj | 3 | 15.e | even | 4 | 1 | ||
| 7600.2.a.bo | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3800, [\chi])\):
|
\( T_{3}^{6} + 13T_{3}^{4} + 32T_{3}^{2} + 4 \)
|
|
\( T_{7}^{6} + 25T_{7}^{4} + 80T_{7}^{2} + 64 \)
|
|
\( T_{11}^{3} + 4T_{11}^{2} - 20T_{11} - 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 13 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 25 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T^{3} + 4 T^{2} - 20 T - 64)^{2} \)
|
| $13$ |
\( T^{6} + 69 T^{4} + \cdots + 11236 \)
|
| $17$ |
\( T^{6} + 57 T^{4} + \cdots + 400 \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} + 113 T^{4} + \cdots + 25600 \)
|
| $29$ |
\( (T^{3} + 3 T^{2} + \cdots - 108)^{2} \)
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| $31$ |
\( (T^{3} + 14 T^{2} + \cdots - 64)^{2} \)
|
| $37$ |
\( T^{6} + 104 T^{4} + \cdots + 400 \)
|
| $41$ |
\( (T^{3} + 10 T^{2} + \cdots - 472)^{2} \)
|
| $43$ |
\( T^{6} + 276 T^{4} + \cdots + 719104 \)
|
| $47$ |
\( T^{6} + 80 T^{4} + \cdots + 4096 \)
|
| $53$ |
\( T^{6} + 77 T^{4} + \cdots + 4 \)
|
| $59$ |
\( (T^{3} - 7 T^{2} + 8 T + 8)^{2} \)
|
| $61$ |
\( (T^{3} - 20 T^{2} + \cdots + 640)^{2} \)
|
| $67$ |
\( T^{6} + 269 T^{4} + \cdots + 280900 \)
|
| $71$ |
\( (T^{3} + 8 T^{2} + \cdots - 512)^{2} \)
|
| $73$ |
\( T^{6} + 297 T^{4} + \cdots + 250000 \)
|
| $79$ |
\( (T^{3} + 14 T^{2} + \cdots + 16)^{2} \)
|
| $83$ |
\( T^{6} + 292 T^{4} + \cdots + 123904 \)
|
| $89$ |
\( (T^{3} + 18 T^{2} + \cdots + 40)^{2} \)
|
| $97$ |
\( T^{6} + 584 T^{4} + \cdots + 5326864 \)
|
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