Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1110,2,Mod(253,1110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1110.253");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1110.o (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: | \(8.86339462436\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
253.1 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −1.40994 | − | 1.73553i | −0.707107 | + | 0.707107i | −3.05055 | + | 3.05055i | 1.00000 | − | 1.00000i | −1.40994 | − | 1.73553i | |||||||
253.2 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −1.17802 | + | 1.90060i | −0.707107 | + | 0.707107i | −2.67041 | + | 2.67041i | 1.00000 | − | 1.00000i | −1.17802 | + | 1.90060i | |||||||
253.3 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | 1.26431 | − | 1.84432i | −0.707107 | + | 0.707107i | 1.64144 | − | 1.64144i | 1.00000 | − | 1.00000i | 1.26431 | − | 1.84432i | |||||||
253.4 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −0.825074 | + | 2.07828i | −0.707107 | + | 0.707107i | 1.45633 | − | 1.45633i | 1.00000 | − | 1.00000i | −0.825074 | + | 2.07828i | |||||||
253.5 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −2.23211 | + | 0.132937i | −0.707107 | + | 0.707107i | 1.47899 | − | 1.47899i | 1.00000 | − | 1.00000i | −2.23211 | + | 0.132937i | |||||||
253.6 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | −1.19936 | − | 1.88721i | −0.707107 | + | 0.707107i | −1.50593 | + | 1.50593i | 1.00000 | − | 1.00000i | −1.19936 | − | 1.88721i | |||||||
253.7 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | 2.22571 | − | 0.215025i | −0.707107 | + | 0.707107i | −1.35684 | + | 1.35684i | 1.00000 | − | 1.00000i | 2.22571 | − | 0.215025i | |||||||
253.8 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | 0.906760 | − | 2.04396i | −0.707107 | + | 0.707107i | 1.03373 | − | 1.03373i | 1.00000 | − | 1.00000i | 0.906760 | − | 2.04396i | |||||||
253.9 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | 1.62651 | + | 1.53443i | −0.707107 | + | 0.707107i | −2.68468 | + | 2.68468i | 1.00000 | − | 1.00000i | 1.62651 | + | 1.53443i | |||||||
253.10 | 1.00000 | −0.707107 | + | 0.707107i | 1.00000 | 0.821222 | + | 2.07981i | −0.707107 | + | 0.707107i | 1.82950 | − | 1.82950i | 1.00000 | − | 1.00000i | 0.821222 | + | 2.07981i | |||||||
253.11 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | 0.837144 | + | 2.07345i | 0.707107 | − | 0.707107i | −2.93723 | + | 2.93723i | 1.00000 | − | 1.00000i | 0.837144 | + | 2.07345i | |||||||
253.12 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −0.161376 | + | 2.23024i | 0.707107 | − | 0.707107i | 1.80704 | − | 1.80704i | 1.00000 | − | 1.00000i | −0.161376 | + | 2.23024i | |||||||
253.13 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −2.22514 | + | 0.220830i | 0.707107 | − | 0.707107i | −1.18593 | + | 1.18593i | 1.00000 | − | 1.00000i | −2.22514 | + | 0.220830i | |||||||
253.14 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | 2.21223 | + | 0.325656i | 0.707107 | − | 0.707107i | 0.597471 | − | 0.597471i | 1.00000 | − | 1.00000i | 2.21223 | + | 0.325656i | |||||||
253.15 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −0.314477 | − | 2.21384i | 0.707107 | − | 0.707107i | −0.376687 | + | 0.376687i | 1.00000 | − | 1.00000i | −0.314477 | − | 2.21384i | |||||||
253.16 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | 0.620781 | − | 2.14817i | 0.707107 | − | 0.707107i | −0.434986 | + | 0.434986i | 1.00000 | − | 1.00000i | 0.620781 | − | 2.14817i | |||||||
253.17 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | 2.00267 | + | 0.994632i | 0.707107 | − | 0.707107i | 0.593998 | − | 0.593998i | 1.00000 | − | 1.00000i | 2.00267 | + | 0.994632i | |||||||
253.18 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −2.22216 | − | 0.248996i | 0.707107 | − | 0.707107i | −2.88943 | + | 2.88943i | 1.00000 | − | 1.00000i | −2.22216 | − | 0.248996i | |||||||
253.19 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | −2.18349 | + | 0.482069i | 0.707107 | − | 0.707107i | 3.34184 | − | 3.34184i | 1.00000 | − | 1.00000i | −2.18349 | + | 0.482069i | |||||||
253.20 | 1.00000 | 0.707107 | − | 0.707107i | 1.00000 | 1.43381 | − | 1.71586i | 0.707107 | − | 0.707107i | 3.31235 | − | 3.31235i | 1.00000 | − | 1.00000i | 1.43381 | − | 1.71586i | |||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
185.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1110.2.o.b | yes | 40 |
5.c | odd | 4 | 1 | 1110.2.l.b | ✓ | 40 | |
37.d | odd | 4 | 1 | 1110.2.l.b | ✓ | 40 | |
185.k | even | 4 | 1 | inner | 1110.2.o.b | yes | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1110.2.l.b | ✓ | 40 | 5.c | odd | 4 | 1 | |
1110.2.l.b | ✓ | 40 | 37.d | odd | 4 | 1 | |
1110.2.o.b | yes | 40 | 1.a | even | 1 | 1 | trivial |
1110.2.o.b | yes | 40 | 185.k | even | 4 | 1 | inner |