Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(218,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.218");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 525.j (of order \(4\), degree \(2\), 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: | \(4.19214610612\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
218.1 | −1.90099 | − | 1.90099i | 0.394708 | − | 1.68648i | 5.22756i | 0 | −3.95632 | + | 2.45565i | −0.707107 | + | 0.707107i | 6.13557 | − | 6.13557i | −2.68841 | − | 1.33133i | 0 | ||||||
218.2 | −1.90099 | − | 1.90099i | 1.68648 | − | 0.394708i | 5.22756i | 0 | −3.95632 | − | 2.45565i | 0.707107 | − | 0.707107i | 6.13557 | − | 6.13557i | 2.68841 | − | 1.33133i | 0 | ||||||
218.3 | −1.27211 | − | 1.27211i | −1.54904 | + | 0.774907i | 1.23654i | 0 | 2.95632 | + | 0.984783i | 0.707107 | − | 0.707107i | −0.971203 | + | 0.971203i | 1.79904 | − | 2.40072i | 0 | ||||||
218.4 | −1.27211 | − | 1.27211i | −0.774907 | + | 1.54904i | 1.23654i | 0 | 2.95632 | − | 0.984783i | −0.707107 | + | 0.707107i | −0.971203 | + | 0.971203i | −1.79904 | − | 2.40072i | 0 | ||||||
218.5 | −1.11527 | − | 1.11527i | −1.51282 | − | 0.843438i | 0.487636i | 0 | 0.746534 | + | 2.62785i | 0.707107 | − | 0.707107i | −1.68669 | + | 1.68669i | 1.57722 | + | 2.55193i | 0 | ||||||
218.6 | −1.11527 | − | 1.11527i | 0.843438 | + | 1.51282i | 0.487636i | 0 | 0.746534 | − | 2.62785i | −0.707107 | + | 0.707107i | −1.68669 | + | 1.68669i | −1.57722 | + | 2.55193i | 0 | ||||||
218.7 | −0.723969 | − | 0.723969i | 0.994020 | − | 1.41842i | − | 0.951738i | 0 | −1.74653 | + | 0.307254i | −0.707107 | + | 0.707107i | −2.13697 | + | 2.13697i | −1.02385 | − | 2.81988i | 0 | |||||
218.8 | −0.723969 | − | 0.723969i | 1.41842 | − | 0.994020i | − | 0.951738i | 0 | −1.74653 | − | 0.307254i | 0.707107 | − | 0.707107i | −2.13697 | + | 2.13697i | 1.02385 | − | 2.81988i | 0 | |||||
218.9 | 0.723969 | + | 0.723969i | −1.41842 | + | 0.994020i | − | 0.951738i | 0 | −1.74653 | − | 0.307254i | −0.707107 | + | 0.707107i | 2.13697 | − | 2.13697i | 1.02385 | − | 2.81988i | 0 | |||||
218.10 | 0.723969 | + | 0.723969i | −0.994020 | + | 1.41842i | − | 0.951738i | 0 | −1.74653 | + | 0.307254i | 0.707107 | − | 0.707107i | 2.13697 | − | 2.13697i | −1.02385 | − | 2.81988i | 0 | |||||
218.11 | 1.11527 | + | 1.11527i | −0.843438 | − | 1.51282i | 0.487636i | 0 | 0.746534 | − | 2.62785i | 0.707107 | − | 0.707107i | 1.68669 | − | 1.68669i | −1.57722 | + | 2.55193i | 0 | ||||||
218.12 | 1.11527 | + | 1.11527i | 1.51282 | + | 0.843438i | 0.487636i | 0 | 0.746534 | + | 2.62785i | −0.707107 | + | 0.707107i | 1.68669 | − | 1.68669i | 1.57722 | + | 2.55193i | 0 | ||||||
218.13 | 1.27211 | + | 1.27211i | 0.774907 | − | 1.54904i | 1.23654i | 0 | 2.95632 | − | 0.984783i | 0.707107 | − | 0.707107i | 0.971203 | − | 0.971203i | −1.79904 | − | 2.40072i | 0 | ||||||
218.14 | 1.27211 | + | 1.27211i | 1.54904 | − | 0.774907i | 1.23654i | 0 | 2.95632 | + | 0.984783i | −0.707107 | + | 0.707107i | 0.971203 | − | 0.971203i | 1.79904 | − | 2.40072i | 0 | ||||||
218.15 | 1.90099 | + | 1.90099i | −1.68648 | + | 0.394708i | 5.22756i | 0 | −3.95632 | − | 2.45565i | −0.707107 | + | 0.707107i | −6.13557 | + | 6.13557i | 2.68841 | − | 1.33133i | 0 | ||||||
218.16 | 1.90099 | + | 1.90099i | −0.394708 | + | 1.68648i | 5.22756i | 0 | −3.95632 | + | 2.45565i | 0.707107 | − | 0.707107i | −6.13557 | + | 6.13557i | −2.68841 | − | 1.33133i | 0 | ||||||
407.1 | −1.90099 | + | 1.90099i | 0.394708 | + | 1.68648i | − | 5.22756i | 0 | −3.95632 | − | 2.45565i | −0.707107 | − | 0.707107i | 6.13557 | + | 6.13557i | −2.68841 | + | 1.33133i | 0 | |||||
407.2 | −1.90099 | + | 1.90099i | 1.68648 | + | 0.394708i | − | 5.22756i | 0 | −3.95632 | + | 2.45565i | 0.707107 | + | 0.707107i | 6.13557 | + | 6.13557i | 2.68841 | + | 1.33133i | 0 | |||||
407.3 | −1.27211 | + | 1.27211i | −1.54904 | − | 0.774907i | − | 1.23654i | 0 | 2.95632 | − | 0.984783i | 0.707107 | + | 0.707107i | −0.971203 | − | 0.971203i | 1.79904 | + | 2.40072i | 0 | |||||
407.4 | −1.27211 | + | 1.27211i | −0.774907 | − | 1.54904i | − | 1.23654i | 0 | 2.95632 | + | 0.984783i | −0.707107 | − | 0.707107i | −0.971203 | − | 0.971203i | −1.79904 | + | 2.40072i | 0 | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
15.d | odd | 2 | 1 | inner |
15.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 525.2.j.c | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 525.2.j.c | ✓ | 32 |
5.b | even | 2 | 1 | inner | 525.2.j.c | ✓ | 32 |
5.c | odd | 4 | 2 | inner | 525.2.j.c | ✓ | 32 |
15.d | odd | 2 | 1 | inner | 525.2.j.c | ✓ | 32 |
15.e | even | 4 | 2 | inner | 525.2.j.c | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
525.2.j.c | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
525.2.j.c | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
525.2.j.c | ✓ | 32 | 5.b | even | 2 | 1 | inner |
525.2.j.c | ✓ | 32 | 5.c | odd | 4 | 2 | inner |
525.2.j.c | ✓ | 32 | 15.d | odd | 2 | 1 | inner |
525.2.j.c | ✓ | 32 | 15.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 70T_{2}^{12} + 1011T_{2}^{8} + 4414T_{2}^{4} + 3721 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\).