Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,4,Mod(37,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.37");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 99.f (of order \(5\), degree \(4\), 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: | \(5.84118909057\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 33) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) |
| \(\chi(n)\) | \(-\zeta_{10}^{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−3.61803 | − | 2.62866i | 0 | 3.70820 | + | 11.4127i | 4.69098 | − | 3.40820i | 0 | −4.72542 | − | 14.5434i | 5.52786 | − | 17.0130i | 0 | −25.9311 | ||||||||||||||||||||
| 64.1 | −1.38197 | + | 4.25325i | 0 | −9.70820 | − | 7.05342i | 5.80902 | + | 17.8783i | 0 | 23.2254 | + | 16.8743i | 14.4721 | − | 10.5146i | 0 | −84.0689 | |||||||||||||||||||||
| 82.1 | −1.38197 | − | 4.25325i | 0 | −9.70820 | + | 7.05342i | 5.80902 | − | 17.8783i | 0 | 23.2254 | − | 16.8743i | 14.4721 | + | 10.5146i | 0 | −84.0689 | |||||||||||||||||||||
| 91.1 | −3.61803 | + | 2.62866i | 0 | 3.70820 | − | 11.4127i | 4.69098 | + | 3.40820i | 0 | −4.72542 | + | 14.5434i | 5.52786 | + | 17.0130i | 0 | −25.9311 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 99.4.f.a | 4 | |
| 3.b | odd | 2 | 1 | 33.4.e.a | ✓ | 4 | |
| 11.c | even | 5 | 1 | inner | 99.4.f.a | 4 | |
| 11.c | even | 5 | 1 | 1089.4.a.p | 2 | ||
| 11.d | odd | 10 | 1 | 1089.4.a.q | 2 | ||
| 33.f | even | 10 | 1 | 363.4.a.o | 2 | ||
| 33.h | odd | 10 | 1 | 33.4.e.a | ✓ | 4 | |
| 33.h | odd | 10 | 1 | 363.4.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.4.e.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 33.4.e.a | ✓ | 4 | 33.h | odd | 10 | 1 | |
| 99.4.f.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 99.4.f.a | 4 | 11.c | even | 5 | 1 | inner | |
| 363.4.a.n | 2 | 33.h | odd | 10 | 1 | ||
| 363.4.a.o | 2 | 33.f | even | 10 | 1 | ||
| 1089.4.a.p | 2 | 11.c | even | 5 | 1 | ||
| 1089.4.a.q | 2 | 11.d | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 10T_{2}^{3} + 60T_{2}^{2} + 200T_{2} + 400 \)
acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 10 T^{3} + \cdots + 400 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 21 T^{3} + \cdots + 11881 \)
|
| $7$ |
\( T^{4} - 37 T^{3} + \cdots + 192721 \)
|
| $11$ |
\( T^{4} + 41 T^{3} + \cdots + 1771561 \)
|
| $13$ |
\( T^{4} + 77 T^{3} + \cdots + 14641 \)
|
| $17$ |
\( T^{4} + 192 T^{3} + \cdots + 2595321 \)
|
| $19$ |
\( T^{4} + 52 T^{3} + \cdots + 1274641 \)
|
| $23$ |
\( (T^{2} - 74 T - 1051)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 1740892176 \)
|
| $31$ |
\( T^{4} + 198 T^{3} + \cdots + 54479161 \)
|
| $37$ |
\( T^{4} + \cdots + 4216034761 \)
|
| $41$ |
\( T^{4} - 129 T^{3} + \cdots + 241081 \)
|
| $43$ |
\( (T^{2} + 66 T - 7731)^{2} \)
|
| $47$ |
\( T^{4} - 35 T^{3} + \cdots + 674700625 \)
|
| $53$ |
\( T^{4} + 188 T^{3} + \cdots + 10042561 \)
|
| $59$ |
\( T^{4} + \cdots + 47173668025 \)
|
| $61$ |
\( T^{4} + \cdots + 55792802025 \)
|
| $67$ |
\( (T^{2} - 75 T - 391995)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 53332821721 \)
|
| $73$ |
\( T^{4} + \cdots + 8360542096 \)
|
| $79$ |
\( T^{4} + \cdots + 285379255681 \)
|
| $83$ |
\( T^{4} + \cdots + 53935882081 \)
|
| $89$ |
\( (T^{2} - 2712 T + 1809091)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 932420053161 \)
|
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