Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,10,Mod(49,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.49");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.c (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: | \(206.014334466\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.3241708096.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 39x^{4} + 317x^{2} + 484 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{6}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{6} + 39x^{4} + 317x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 61\nu^{3} + 757\nu ) / 451 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 168\nu^{4} + 4344\nu^{2} + 6800 ) / 41 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -196\nu^{5} - 6544\nu^{3} - 7660\nu ) / 451 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3144\nu^{4} + 115032\nu^{2} + 565840 ) / 41 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1976\nu^{5} + 68384\nu^{3} + 376040\nu ) / 41 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 106\beta_{3} - 960\beta_1 ) / 5760 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{4} - 131\beta_{2} - 74880 ) / 5760 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{5} - 1138\beta_{3} + 59520\beta_1 ) / 2880 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -181\beta_{4} + 4793\beta_{2} + 1703040 ) / 5760 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 829\beta_{5} + 58594\beta_{3} - 3936960\beta_1 ) / 5760 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 262.608i | 0 | 0 | 0 | − | 1790.86i | 0 | −49280.0 | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 192.296i | 0 | 0 | 0 | 10171.1i | 0 | −17294.6 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 13.6876i | 0 | 0 | 0 | 6441.91i | 0 | 19495.7 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 13.6876i | 0 | 0 | 0 | − | 6441.91i | 0 | 19495.7 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 192.296i | 0 | 0 | 0 | − | 10171.1i | 0 | −17294.6 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 262.608i | 0 | 0 | 0 | 1790.86i | 0 | −49280.0 | 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 | 400.10.c.r | 6 | |
| 4.b | odd | 2 | 1 | 200.10.c.f | 6 | ||
| 5.b | even | 2 | 1 | inner | 400.10.c.r | 6 | |
| 5.c | odd | 4 | 1 | 80.10.a.k | 3 | ||
| 5.c | odd | 4 | 1 | 400.10.a.x | 3 | ||
| 20.d | odd | 2 | 1 | 200.10.c.f | 6 | ||
| 20.e | even | 4 | 1 | 40.10.a.d | ✓ | 3 | |
| 20.e | even | 4 | 1 | 200.10.a.f | 3 | ||
| 40.i | odd | 4 | 1 | 320.10.a.v | 3 | ||
| 40.k | even | 4 | 1 | 320.10.a.u | 3 | ||
| 60.l | odd | 4 | 1 | 360.10.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.10.a.d | ✓ | 3 | 20.e | even | 4 | 1 | |
| 80.10.a.k | 3 | 5.c | odd | 4 | 1 | ||
| 200.10.a.f | 3 | 20.e | even | 4 | 1 | ||
| 200.10.c.f | 6 | 4.b | odd | 2 | 1 | ||
| 200.10.c.f | 6 | 20.d | odd | 2 | 1 | ||
| 320.10.a.u | 3 | 40.k | even | 4 | 1 | ||
| 320.10.a.v | 3 | 40.i | odd | 4 | 1 | ||
| 360.10.a.j | 3 | 60.l | odd | 4 | 1 | ||
| 400.10.a.x | 3 | 5.c | odd | 4 | 1 | ||
| 400.10.c.r | 6 | 1.a | even | 1 | 1 | trivial | |
| 400.10.c.r | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 106128T_{3}^{4} + 2569936896T_{3}^{2} + 477757440000 \)
acting on \(S_{10}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 477757440000 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 13\!\cdots\!36 \)
|
| $11$ |
\( (T^{3} + \cdots + 46675499449024)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $17$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + \cdots + 13\!\cdots\!96)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 35\!\cdots\!16 \)
|
| $29$ |
\( (T^{3} + \cdots - 81\!\cdots\!08)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 55\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 79\!\cdots\!16 \)
|
| $41$ |
\( (T^{3} + \cdots - 62\!\cdots\!56)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 43\!\cdots\!64 \)
|
| $47$ |
\( T^{6} + \cdots + 36\!\cdots\!76 \)
|
| $53$ |
\( T^{6} + \cdots + 58\!\cdots\!56 \)
|
| $59$ |
\( (T^{3} + \cdots + 34\!\cdots\!36)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 21\!\cdots\!84 \)
|
| $71$ |
\( (T^{3} + \cdots + 54\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 80\!\cdots\!76 \)
|
| $79$ |
\( (T^{3} + \cdots - 67\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 43\!\cdots\!96 \)
|
| $89$ |
\( (T^{3} + \cdots - 71\!\cdots\!96)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 60\!\cdots\!16 \)
|
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