Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,9,Mod(351,400)]
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([1, 0, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.351");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.b (of order \(2\), degree \(1\), 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: | \(162.951444024\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 16) |
| 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 \(\beta = 8\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 351.1 |
|
0 | − | 13.8564i | 0 | 0 | 0 | − | 3297.82i | 0 | 6369.00 | 0 | ||||||||||||||||||||||
| 351.2 | 0 | 13.8564i | 0 | 0 | 0 | 3297.82i | 0 | 6369.00 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 400.9.b.e | 2 | |
| 4.b | odd | 2 | 1 | inner | 400.9.b.e | 2 | |
| 5.b | even | 2 | 1 | 16.9.c.b | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 400.9.h.a | 4 | ||
| 15.d | odd | 2 | 1 | 144.9.g.d | 2 | ||
| 20.d | odd | 2 | 1 | 16.9.c.b | ✓ | 2 | |
| 20.e | even | 4 | 2 | 400.9.h.a | 4 | ||
| 40.e | odd | 2 | 1 | 64.9.c.c | 2 | ||
| 40.f | even | 2 | 1 | 64.9.c.c | 2 | ||
| 60.h | even | 2 | 1 | 144.9.g.d | 2 | ||
| 80.k | odd | 4 | 2 | 256.9.d.d | 4 | ||
| 80.q | even | 4 | 2 | 256.9.d.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 16.9.c.b | ✓ | 2 | 5.b | even | 2 | 1 | |
| 16.9.c.b | ✓ | 2 | 20.d | odd | 2 | 1 | |
| 64.9.c.c | 2 | 40.e | odd | 2 | 1 | ||
| 64.9.c.c | 2 | 40.f | even | 2 | 1 | ||
| 144.9.g.d | 2 | 15.d | odd | 2 | 1 | ||
| 144.9.g.d | 2 | 60.h | even | 2 | 1 | ||
| 256.9.d.d | 4 | 80.k | odd | 4 | 2 | ||
| 256.9.d.d | 4 | 80.q | even | 4 | 2 | ||
| 400.9.b.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 400.9.b.e | 2 | 4.b | odd | 2 | 1 | inner | |
| 400.9.h.a | 4 | 5.c | odd | 4 | 2 | ||
| 400.9.h.a | 4 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(400, [\chi])\):
|
\( T_{3}^{2} + 192 \)
|
|
\( T_{13} + 19138 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 192 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 10875648 \)
|
| $11$ |
\( T^{2} + 536110272 \)
|
| $13$ |
\( (T + 19138)^{2} \)
|
| $17$ |
\( (T - 58686)^{2} \)
|
| $19$ |
\( T^{2} + 23278487232 \)
|
| $23$ |
\( T^{2} + 72963078912 \)
|
| $29$ |
\( (T - 842178)^{2} \)
|
| $31$ |
\( T^{2} + 1105026527232 \)
|
| $37$ |
\( (T + 2548610)^{2} \)
|
| $41$ |
\( (T + 4324158)^{2} \)
|
| $43$ |
\( T^{2} + 4149436047552 \)
|
| $47$ |
\( T^{2} + 52319333403648 \)
|
| $53$ |
\( (T + 1192194)^{2} \)
|
| $59$ |
\( T^{2} + 114412208832 \)
|
| $61$ |
\( (T - 8414786)^{2} \)
|
| $67$ |
\( T^{2} + 303620940564672 \)
|
| $71$ |
\( T^{2} + 953360435651328 \)
|
| $73$ |
\( (T + 12735874)^{2} \)
|
| $79$ |
\( T^{2} + 39521988480000 \)
|
| $83$ |
\( T^{2} + 69\!\cdots\!28 \)
|
| $89$ |
\( (T + 16802814)^{2} \)
|
| $97$ |
\( (T + 120994882)^{2} \)
|
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