Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,5,Mod(193,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.193");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.p (of order \(4\), degree \(2\), 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: | \(41.3479852335\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{61})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 31x^{2} + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 200) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 31x^{2} + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 16\nu ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 30\nu^{2} + 46\nu + 465 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 30\nu^{2} + 46\nu - 465 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 62 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{3} - 4\beta_{2} + 23\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | −11.8102 | − | 11.8102i | 0 | 0 | 0 | 19.8102 | − | 19.8102i | 0 | 197.964i | 0 | ||||||||||||||||||||||||||
| 193.2 | 0 | 3.81025 | + | 3.81025i | 0 | 0 | 0 | 4.18975 | − | 4.18975i | 0 | − | 51.9640i | 0 | ||||||||||||||||||||||||||
| 257.1 | 0 | −11.8102 | + | 11.8102i | 0 | 0 | 0 | 19.8102 | + | 19.8102i | 0 | − | 197.964i | 0 | ||||||||||||||||||||||||||
| 257.2 | 0 | 3.81025 | − | 3.81025i | 0 | 0 | 0 | 4.18975 | + | 4.18975i | 0 | 51.9640i | 0 | |||||||||||||||||||||||||||
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 | 400.5.p.f | 4 | |
| 4.b | odd | 2 | 1 | 200.5.l.d | yes | 4 | |
| 5.b | even | 2 | 1 | 400.5.p.m | 4 | ||
| 5.c | odd | 4 | 1 | inner | 400.5.p.f | 4 | |
| 5.c | odd | 4 | 1 | 400.5.p.m | 4 | ||
| 20.d | odd | 2 | 1 | 200.5.l.b | ✓ | 4 | |
| 20.e | even | 4 | 1 | 200.5.l.b | ✓ | 4 | |
| 20.e | even | 4 | 1 | 200.5.l.d | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.5.l.b | ✓ | 4 | 20.d | odd | 2 | 1 | |
| 200.5.l.b | ✓ | 4 | 20.e | even | 4 | 1 | |
| 200.5.l.d | yes | 4 | 4.b | odd | 2 | 1 | |
| 200.5.l.d | yes | 4 | 20.e | even | 4 | 1 | |
| 400.5.p.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.5.p.f | 4 | 5.c | odd | 4 | 1 | inner | |
| 400.5.p.m | 4 | 5.b | even | 2 | 1 | ||
| 400.5.p.m | 4 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 16T_{3}^{3} + 128T_{3}^{2} - 1440T_{3} + 8100 \)
acting on \(S_{5}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 16 T^{3} + \cdots + 8100 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 48 T^{3} + \cdots + 27556 \)
|
| $11$ |
\( (T^{2} - 120 T - 12016)^{2} \)
|
| $13$ |
\( T^{4} + 384 T^{3} + \cdots + 746496 \)
|
| $17$ |
\( T^{4} + 512 T^{3} + \cdots + 231040000 \)
|
| $19$ |
\( T^{4} + \cdots + 17041735936 \)
|
| $23$ |
\( T^{4} + \cdots + 155067413796 \)
|
| $29$ |
\( T^{4} + \cdots + 51690750736 \)
|
| $31$ |
\( (T^{2} + 1552 T + 40000)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 97863860224 \)
|
| $41$ |
\( (T^{2} + 144 T - 1884352)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 1398083948836 \)
|
| $47$ |
\( T^{4} + \cdots + 82145183306404 \)
|
| $53$ |
\( T^{4} + \cdots + 13008198329344 \)
|
| $59$ |
\( T^{4} + \cdots + 8392655352064 \)
|
| $61$ |
\( (T^{2} + 5472 T + 6923520)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 81224687600676 \)
|
| $71$ |
\( (T^{2} + 8928 T + 19677440)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 111331664896 \)
|
| $79$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 560151218940196 \)
|
| $89$ |
\( T^{4} + \cdots + 37\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots + 38\!\cdots\!44 \)
|
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